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UBC Theses and Dissertations

Lithium intercalation in crystalline Li MoS₂ Mulhern, Peter John 1986

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LITHIUM INTERCALATION IN CRYSTALLINE L i x M o S 2 by PETER JOHN MULHERN B.Sc. Simon F r a s e r U n i v e r s i t y , 1980 M.Sc. The U n i v e r s i t y of B r i t i s h C o lumbia, 1982 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department of P h y s i c s ) We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA J u l y , 1986 © P e t e r John M u l h e r n , 1986 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of P h y s i c s  The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date J u l y 15, 1986 D E - 6 n/R-n ABSTRACT i i T h i s t h e s i s r e p o r t s the f i r s t s u c c e s s f u l s y n t h e s i s of h i g h l y c r y s t a l l i n e L i x M o S 2 , and d e s c r i b e s the e l e c t r o c h e m i c a l and s t r u c t u r a l p r o p e r t i e s of t h i s m a t e r i a l . F i v e c r y s t a l phases were found when the l i t h i u m c o n t e n t was v a r i e d between x^O and x-1 by e l e c t r o c h e m i c a l i n t e r c a l a t i o n . Each phase e x i s t e d over a s m a l l range of x, and the c r y s t a l l a t t i c e s were found t o have s m a l l t r i c l i n i c or m o n o c l i n i c d i s t o r t i o n s from 1T-hexagonal symmetry. These d i s c o v e r i e s have r e s o l v e d a c o n t r o v e r s y i n the l i t e r a t u r e . Many r e f e r e n c e s have c l a i m e d t h a t c r y s t a l l i n e MoS 2 d i s p r o p o r t i o n a t e s when the l i t h i u m c o n t e n t exceeds x^0.2, w h i l e o t h e r s have c l a i m e d an e l e c t r o c h e m i c a l l y produced MoS 2 p o l y t y p e i s an i n t e r c a l a t i o n host w i t h Ax—1. A s t r o n g c o r r e s p o n d e n c e has been found between the p r o p e r t i e s of the e l e c t r o c h e m i c a l l y produced h o s t and the c r y s t a l l i n e 1 T - L i x M o S 2 . A g r a p h i c a l t e c h n i q u e t o index the x- r a y p a t t e r n s of an a r b i t r a r y l a t t i c e i s a l s o p r e s e n t e d . i i i T A B L E OF CONTENTS P a g e A B S T R A C T . . . . i i T A B L E OF CONTENTS . i i i L I S T OF F I G U R E S . . . . v L I S T OF T A B L E S . . v i i L I S T OF SYMBOLS v i i i ACKNOWLEDGEMENTS x i i CHAPTER 1 I n t r o d u c t i o n 1 . 1 M o t i v a t i o n f o r R e s e a r c h 1 1 . 2 I n t e r c a l a t i o n M a t e r i a l s 3 1 . 3 I n t e r c a l a t i o n T e c h n i q u e s 9 CHAPTER 2 O v e r v i e w o f t h e L i x M o S 2 S y s t e m 2 . 1 S u m m a r y 14 2 . 2 L i t e r a t u r e R e v i e w . 16 2 . 3 B e t a P h a s e M o S 2 2 6 2 . 4 B a s i c M o S 2 E l e c t r o c h e m i s t r y 31 CHAPTER 3 G r o w t h o f L i x M o S 2 3 . 1 I n i t i a l D i s c o v e r y o f C r y s t a l l i n e L i x M o S 2 36 3 . 2 G e n e r a l P r o c e d u r e f o r S a m p l e P r e p a r a t i o n 37 3 . 3 V a r i a t i o n s i n P r o c e d u r e 4 0 3 . 4 P r o b l e m s i n S a m p l e P r e p a r a t i o n 4 2 3 . 5 C h e m i c a l T e c h n i q u e s 4 6 3 . 6 A n o t h e r P o s s i b l e P r e p a r a t i o n T e c h n i q u e 48 CHAPTER 4 E x p e r i m e n t a l T e c h n i q u e s 4 . 1 C a t h o d e P r e p a r a t i o n 5 0 4 . 2 C e l l C o n s t r u c t i o n 56 4 . 3 C o n s t a n t C u r r e n t C y c l i n g 58 4 . 4 C o n s t a n t C u r r e n t V o l t a m m e t r y 61 4 . 5 X - R a y D i f f r a c t i o n 66 4 . 6 O t h e r T e c h n i q u e s a n d D e v i c e s . . . 67 i v CHAPTER 5 ELECTROCHEMICAL INVESTIGATIONS 5.1 Overview 68 5.2 C y c l i n g B e h a v i o u r . . . . . 70 5.3 Voltammetry I n v e s t i g a t i o n s . . 78 5.4 C o n v e r s i o n t o 2H-MoS 2 96 CHAPTER 6 X-Ray I n v e s t i g a t i o n s 6. 1 Overview 1-01 6.2 X-Ray S t u d i e s of 0-Phase 102 6.3 B a s i c X-Ray S t u d i e s of L i x M o S 2 105 6.4 D e t a i l e d S t r u c t u r a l A n a l y s i s — O v e r v i e w 109 6.5 A p p l i c a t i o n of the G r a p h i c a l I n d e x i n g Technique . . .115 6.6 A n a l y s i s of L i M o S 2 119 6.7 A n a l y s i s of the T r i c l i n i c Phases 124 6.8 L i x M o S 2 L a t t i c e P a r a m e t e r s . . . 127 6.9 C o n f i r m a t i o n of Near 1T Symmetry. . . 130 CHAPTER 7 Summary and C o n c l u s i o n s 7.1 I n t e r p r e t a t i o n of R e s u l t s .142 7.2 S u g g e s t i o n s f o r F u t u r e Work 148 7.3 Summary 151 BIBLIOGRAPHY .152 APPENDIX I X-Ray D i f f r a c t i o n : Theory and G r a p h i c a l A n a l y s i s Technique A1 . 1 Overview 1 58 A1.2 B a s i c s of C r y s t a l S t r u c t u r e and X-Ray D i f f r a c t i o n 1 59 A1.3 I n t e n s i t y C a l c u l a t i o n s 167 A1.4 I n d e x i n g Hexagonal L a t t i c e s 171 A1.5 I n d e x i n g M o n o c l i n i c L a t t i c e s 181 A1.6 I n d e x i n g T r i c l i n i c L a t t i c e s 188 APPENDIX I I Computer Programs 200 APPENDIX I I I L i x M o S 2 X-Ray Data 207 APPENDIX IV Phenomena C a u s i n g N o n - R e v e r s i b i l i t y . . . . . .214 y L I S T OF FIGURES F i g u r e Page 1 S t r u c t u r e of t h e T r a n s i t i o n M e t a l D i c h a l c o g e n i d e s . . . . .5 2 N o t a t i o n f o r S t a c k i n g A r r a n g e m e n t s . .6 3 S c h e m a t i c o f t h e E l e c t r o c h e m i c a l C e l l 10 4 D e n s i t y o f S t a t e s f o r 2H and 1T-MoS 2 30 5 C y c l i n g o f D i f f e r e n t MoS 2 P h a s e s 32 6 a t o j3 Phase C o n v e r s i o n 33 7 F l a n g e C e l l . . . . . . 57 8 X-Ray C e l l 59 9 C o n s t a n t C u r r e n t V o l t a m m e t r y 64 10 /3-Phase C y c l e 5 71 11 /3-Phase C y c l e 10. 73 12 C y c l i n g o f 4p138 L i x M o S 2 74 13 C o m p a r i s o n of /3-Phase and L i x M o S 2 C y c l i n g 76 14 dQ/dV o f /3-Phase from 1.4V t o 2.8V 80 15 dQ/dV o f /3-Phase C y c l e 3 82 16 dQ/dV o f /3-Phase C y c l e 101 83 17 dQ/dV o f /3-Phase C y c l e 202 84 18 dQ/dV o f /3-Phase C y c l e 303 85 19 dQ/dV o f C r y s t a l l i n e L i x M o S 2 . 88 20 H y s t e r e s i s i n L i x M o S 2 91 21 C r y s t a l l i n e L i x M o S 2 A f t e r E x p o s u r e t o A i r 94 22 C o n v e r s i o n o f 5p53 t o a-Phase 98 23 /3-Phase L a t t i c e P a r a m e t e r s 104 24 /3-Phase — I n d i c a t i o n o f C o - E x i s t i n g P h a s e s 106 v i 25 L i x M o S 2 S i n g l e Phase C a p a c i t y . . . . .108 26 O u t - o f - P l a n e Geometry. 113 27 D i f f r a c t i o n P a t t e r n o f L i M o S 2 f o r 3 O ° < 2 0 < 3 3 ° . . . . . .121 28 G r a p h i c a l A n a l y s i s — L i M o S 2 122 29 P r e f e r r e d O r i e n t a t i o n . 132 30 I n t e r f e r e n c e f r o m D i f f e r e n t A t o m i c P l a n e s . . . . . . . .134 31 C a l c u l a t e d D i f f r a c t i o n P a t t e r n s . . . . .139 32 T h r e e C r y s t a l L a t t i c e s . .161 33 C r y s t a l l o g r a p h i c P l a n e s and T h e i r M i l l e r I n d i c e s . . . .163 34 A t o m i c P l a n e s i n a H e x a g o n a l L a t t i c e 165 35 B r a g g Law o f D i f f r a c t i o n 166 36 Von Laue D i f f r a c t i o n C o n d i t i o n 169 37 G r a p h i c a l A n a l y s i s : H e x a g o n a l L a t t i c e — I n i t i a l P l o t . . .176 38 G r a p h i c a l A n a l y s i s : H e x a g o n a l L a t t i c e — C a n d i d a t e L i n e s .178 39 G r a p h i c a l A n a l y s i s : H e x a g o n a l L a t t i c e — F i n a l R e s u l t s . .179 40 Peak S p l i t t i n g Due t o M o n o c l i n i c D i s t o r t i o n s 183 41 G r a p h i c a l A n a l y s i s : M o n o c l i n i c L a t t i c e — I n i t i a l P l o t . .187 42 G r a p h i c a l A n a l y s i s : M o n o c l i n i c L a t t i c e - - F i n a l R e s u l t s . .189 43 T r i c l i n i c D i s t o r t i o n — T o p View 190 44 G r a p h i c a l A n a l y s i s : T r i c l i n i c L a t t i c e - - S 3 3 / V 2 C o n t r i b u t i o n Removed 197 45 G r a p h i c a l A n a l y s i s : T r i c l i n i c L a t t i c e — F i n a l R e s u l t s . .198 v i i LIST OF TABLES Table Page I Preparation of Li xMoS 2 Samples. . . . . . . . . . . . .41 II x-Values Associated with the Features of Li xMoS 2. . . . 92 III Li xMoS 2 L a t t i c e Parameters. . . . .128 IV . Models of Atomic Stacking Tested. 137 V Hexagonal L a t t i c e Data. 175 VI Monoclinic L a t t i c e Data 185 VII T r i c l i n i c L a t t i c e Data 194 VIII L a t t i c e A (1.7V) X-Ray Data 209 IX L a t t i c e B (1.93V) X-Ray Data 210 X La t t i c e C (2.23V) X-Ray Data 211 XI La t t i c e D (2.56V) X-Ray Data. . 212 XII L a t t i c e E (2.63V) X-Ray Data. .213 LIST OF SYMBOLS A — c h a l c o g e n s i t e l a b e l - d e f i n e d i n s e c t i o n 1.2 B — c h a l c o g e n s i t e l a b e l - d e f i n e d i n s e c t i o n 1.2 C — c h a l c o g e n s i t e l a b e l - d e f i n e d i n s e c t i o n 1.2 a — me t a l s i t e l a b e l - d e f i n e d i n s e c t i o n 1.2 b -- me t a l s i t e l a b e l - d e f i n e d i n s e c t i o n 1.2 c — me t a l s i t e l a b e l - d e f i n e d i n s e c t i o n 1.2 A L a t t i c e -- l a b e l f o r 1T-LiMoS 2 B L a t t i c e — l a b e l f o r 1 T - L i 0 . 7 5 M o S 2 C L a t t i c e — l a b e l f o r 1 T - L i 0 . 5 M o S 2 D L a t t i c e — l a b e l f o r 1 T - L i 0 . 2 ? M o S 2 E L a t t i c e — l a b e l f o r 1 T - L i 0 . , 2 M o S 2 F L a t t i c e — l a b e l f o r 1T-MoS2 A* -- r e c i p r o c a l l a t t i c e v e c t o r 1 — r e c i p r o c a l l a t t i c e v e c t o r C — r e c i p r o c a l l a t t i c e v e c t o r a — l a t t i c e v e c t o r B — l a t t i c e v e c t o r c -- l a t t i c e v e c t o r a — magnitude of l a t t i c e v e c t o r a b -- magnitude of l a t t i c e v e c t o r B c — magnitude of l a t t i c e v e c t o r c a — a n g l e between l a t t i c e v e c t o r s B and c 0 — a n g l e between l a t t i c e v e c t o r s a and c 7 — a n g l e between l a t t i c e v e c t o r s a and B a-phase — 2H-MoS2 p o l y t y p e j3-phase -- e l e c t r o c h e m i c a l l y produced 1T-MoS2 p o l y t y p e 7-phase — e l e c t r o c h e m i c a l l y produced MoS 2 p o l y t y p e I X D 0 — d i s p l a c e m e n t of atom from i d e a l i z e d s i t e p o s i t i o n d -- i n t e r - p l a n a r s p a c i n g 1/d 2 — r e c i p r o c a l of the square of t h e i n t e r - p l a n a r s p a c i n g d -- l a b e l on e l e c t r o n band dz2 — l a b e l on s p e c i f i c d band o r b i t a l e -- e l e c t r i c charge F -- Faraday c o n s t a n t . 96,500 coulombs/mole F -- f u n c t i o n (Appendix I I o n l y ) / -- number ( d e f i n e d and used i n Appendix I I o n l y ) f j -- atomic form f a c t o r f o r atom j G -- Gibbs f r e e energy 5 -- g e n e r a l r e c i p r o c a l l a t t i c e v e c t o r h -- M i l l e r index I — c u r r e n t I 0 — i n t e n s i t y K T -•- i s o t h e r m a l c o m p r e s s i b i l i t y of a gas K*i -- wave v e c t o r k — M i l l e r index / — M i l l e r index M — m o l e c u l a r weight m — mass of powder n -- number of p a r t i c l e s n^ -- i n t e g e r f»i -- normal t o wave v e c t o r p -- p r e s s u r e p — m u l t i p l i c i t y of x - r a y l i n e s P" — s e t of parameters Q — charge X R — sample t o d e t e c t o r d i s t a n c e R* — g e n e r a l l a t t i c e v e c t o r r -- r e g r e s s i o n c o e f f i c i e n t r c — c r i t i c a l r a d i u s 5 .-- e n t r o p y -- s t r u c t u r e f a c t o r S j j - - p a r a m e t e r s f o r t r i c l i n i c l a t t i c e — d e f i n e d i n e q u a t i o n A1.18 T -- t e m p e r a t u r e t — time V -- v o l t a g e V — u n i t c e l l volume — always appears as V 2 except where noted v -- volume X -- c h a l c o g e n atom x - - u s e d i n L i x M o S 2 — mole f r a c t i o n of l i t h i u m t o MoS 2 x — s e t of independent v a r i a b l e s (Appendix I I o n l y ) y -- dependent v a r i a b l e (Appendix I I o n l y ) 8 — o u t - o f - p l a n e d i s p l a c e m e n t X — w a v e l e n g t h 4> — a n g l e between c a x i s and p r e f e r r e d d i r e c t i o n 6 — a n g l e 6-Q — Bragg a n g l e 0 m — measured a n g l e u — c h e m i c a l p o t e n t i a l XI 1T -- c r y s t a l p o l y t y p e - one l a y e r u n i t c e l l w i t h t r i g o n a l symmetry 2 H -- c r y s t a l p o l y t y p e - two l a y e r u n i t c e l l w i t h h e x a g o n a l symmetry 3R -- c r y s t a l p o l y t y p e - t h r e e l a y e r u n i t c e l l w i t h r h o m b o h e d r a l symmetry I A -- e l e m e n t s from t h e f i r s t c o lumn of t h e p e r i o d i c t a b l e I V B •-- T i , Z r , o r Hf V B — V, Nb, o r Ta V I B -- C r , Mo, or W x i i ACKNOWLEDGEMENTS I would l i k e t o thank my s u p e r v i s o r , Rudi H a e r i n g , f o r h i s s u g g e s t i o n s and a d v i c e r e g a r d i n g t h i s t h e s i s . I have b e n e f i t t e d from the p r i o r work done by the many p e o p l e i n the l a b and by d i s c u s s i o n s w i t h them. Space l i m i t a t i o n s p r e v e n t me from l i s t i n g them a l l , but t h e r e a r e t h r e e t h a t s h o u l d be mentioned by name. Dr. J e f f Dahn p r o v i d e d both o p i n i o n s and guidance a t s e v e r a l p o i n t s i n the c o u r s e of work on t h i s t h e s i s , A l e c Rivers-Bowerman d e s i g n e d and b u i l t many of the w i d g e t s used i n the e x p e r i m e n t s , and some of the d a t a p r e s e n t e d i n Chapter 6 was c o l l e c t e d by Dr. M a r c e l Py. I g r e a t l y a p p r e c i a t e the work done by the s m a l l army of p r o o f r e a d e r s who went over my t h e s i s and c o r r e c t e d my s p e l i n g e r r o r s , found t p y o s , and i n c o n s i s t a n c i e s i n s t y l e . I am a l s o i n d e b t e d t o the many f r i e n d s who p u l l e d me thr o u g h the rough t i m e s . F i n a l l y , I would l i k e t o thank t h e N a t u r a l S c i e n c e s and E n g i n e e r i n g R e s e a r c h C o u n c i l and t h e T r u s t e e s of the E s t a t e of D.J. K i l l a m f o r f i n a n c i a l s u p p o r t . x i i i Philosophy have I digested, The whole of Law and Medicine, From each its secrets I have wrested, Theology, alas, thrown in. Poor fool, with all this sweated lore, I stand no wiser than I was before. Master and Doctor are my t i t l e s ; For ten years now, without repose, I've led my pupils by the nose. And round we go, on crooked ways or straight, And well I know that ignorance is our fate, And t hi s I hat e. opening lines to "Faust" --Goet he 1 CHAPTER 1 INTRODUCTION The gods hav e not revealed all t hi ngs from t he beginning, but men seek and so find out better in time. --Xenophanes 1.1 M o t i v a t i o n f o r Research I n t e r c a l a t i o n i s d e f i n e d here as the i n s e r t i o n of an atom or m o l e c u l e i n t o a host l a t t i c e . 1 F u r t h e r m o r e , the i n s e r t i o n must o n l y cause minor s t r u c t u r a l changes i n the h o s t , such as an ex p a n s i o n of t h e l a t t i c e , and i t must be p o s s i b l e t o remove the i n t e r c a l a n t from the h o s t . I n t e r c a l a t i o n systems have been s t u d i e d t o i n v e s t i g a t e t h e i r u n d e r l y i n g p r i n c i p l e s , and a r e known t o have a wide v a r i e t y of p r a c t i c a l a p p l i c a t i o n s . There a r e two c o n f l i c t i n g v iews on the lithium/molybdenum d i s u l f i d e i n t e r c a l a t i o n system. The g e n e r a l c o n c l u s i o n of s e v e r a l s u r v e y s t u d i e s was t h a t c r y s t a l l i n e MoS 2 i s an u n s u i t a b l e h o s t f o r l i t h i u m i n t e r c a l a t i o n , an i d e a which now pervades the l i t e r a t u r e . Other s t u d i e s c o n c l u d e d t h a t a p a r t i c u l a r form of MoS 2 c r y s t a l was not o n l y a s u i t a b l e h o s t , but a l s o had a p p l i c a t i o n s as a commercial b a t t e r y system. 1 I n t e r c a l a t i o n has o t h e r d e f i n i t i o n s i n o t h e r f i e l d s . Recombinant DNA r e s e a r c h i s a l s o c a l l e d i n t e r c a l a t i o n . The o r i g i n a l meaning of the word r e f e r s t o a d d i n g an e x t r a day t o the c a l e n d a r . 2 T h i s c h a p t e r r e v i e w s i n t e r c a l a t i o n i n g e n e r a l , and p r e s e n t s some n o t a t i o n c o n v e n t i o n s . Chapter 2 t r a c e s t h e h i s t o r y of the l i t h i u m / M o S 2 i n t e r c a l a t i o n system, e x p l a i n s the o r i g i n s of the b e l i e f t h a t c r y s t a l l i n e MoS 2 i s a poor i n t e r c a l a t i o n h o s t , and a l s o d e a l s w i t h the b e h a v i o r of a p o o r l y c r y s t a l l i n e form of MoS 2 r e f e r r e d t o as the /3-phase. The /3-phase i s a good i n t e r c a l a t i o n h o s t , but i t s b e h a v i o r was not u n d e r s t o o d , and i t s s t r u c t u r e was i l l - d e f i n e d . My c o n t r i b u t i o n was the growth and stud y of h i g h l y c r y s t a l l i n e l i t h i a t e d MoS 2 powders. Chapter 3 d e s c r i b e s a method f o r the p r e p a r a t i o n of such m a t e r i a l s . Chapter 4 d e a l s w i t h e x p e r i m e n t a l t e c h n i q u e s . C h a p t e r s 5 and 6 c h r o n i c l e my work on the e l e c t r o c h e m i c a l l i t h i u m i n t e r c a l a t i o n and the s t r u c t u r a l a n a l y s i s of /3-phase and h i g h l y c r y s t a l l i n e powders. A g r a p h i c a l t e c h n i q u e t o index powder x - r a y d i f f r a c t i o n p a t t e r n s was de v e l o p e d t o a i d i n t h i s work. The c r y s t a l l i n e L i x M o S 2 has s e v e r a l d i s t i n c t phases t h a t a r e almost 1T-hexagonal, but have s m a l l m o n o c l i n i c and t r i c l i n i c d i s t o r t i o n s . The L i / L i x M o S 2 e l e c t r o c h e m i c a l system was found t o be an i n t e r c a l a t i o n system. A s t r o n g c o r r e s p o n d e n c e was found between the h i g h l y c r y s t a l l i n e powders and the /3-phase. T h i s t h e s i s r e s o l v e s the c o n t r o v e r s y i n the l i t e r a t u r e r e g a r d i n g l i t h i u m i n MoS 2 by unambiguously p r o v i n g t h e e x i s t e n c e of a c r y s t a l l i n e L i x M o S 2 i n t e r c a l a t i o n h o s t t h a t i s s t a b l e f o r v a l u e s of x between ~0 and ~ 1 . 3 1.2 I n t e r c a l a t i o n M a t e r i a l s The d e f i n i t i o n of i n t e r c a l a t i o n r e s t r i c t s t he s t u d i e s t o host and guest m a t e r i a l s which have c o m p a t i b l e p r o p e r t i e s . The prope r h o s t - g u e s t r e l a t i o n s h i p r e q u i r e s t h a t the guest has easy a c c e s s t o s i t e s i n the h o s t ' s s t r u c t u r e w i t h o u t the d i s r u p t i o n of the h o s t and w i t h o u t any ad v e r s e r e a c t i o n between the two. There a r e many c o m b i n a t i o n s of host m a t e r i a l and i n t e r c a l a n t , and many t e c h n i q u e s f o r i n s e r t i o n . N o n - e x h a u s t i v e l i s t s can be found i n c o l l e c t i o n s e d i t e d by Levy (1979), P i e t r o n e r o and T o s a t i (1981), or Whittingham and Jacobson (1982) . The h o s t s d e s c r i b e d here a r e the t r a n s i t i o n m e t a l d i c h a l c o g e n i d e s , and the guest s p e c i e s i s l i t h i u m . The h o s t s have a l a y e r e d s t r u c t u r e t h a t i s open enough t o p e r m i t c e r t a i n g u e s t s t o t r a v e l between the l a y e r s . L i t h i u m i s a s u i t a b l e guest because i t i s l i g h t , m o b i l e i n s i d e the c r y s t a l a t room t e m p e r a t u r e , and produces a p r a c t i c a l v o l t a g e (~2V) when used i n an e l e c t r o c h e m i c a l c e l l . S i n c e i n t e r c a l a t i o n r e a c t i o n s don't cause major s t r u c t u r a l changes, an e l e c t r o c h e m i c a l c e l l can be r e p e a t e d l y d i s c h a r g e d and charged w i t h o u t d o i n g major damage t o the h o s t . For these r e a s o n s l i t h i u m / t r a n s i t i o n m e t a l d i c h a l c o g e n i d e s were c o n s i d e r e d t o be prime m a t e r i a l s f o r use i n r e c h a r g e a b l e h i g h energy d e n s i t y b a t t e r i e s . The t r a n s i t i o n m e t a l d i c h a l c o g e n i d e s a r e m a t e r i a l s of the form MX 2, where X r e p r e s e n t s s u l f u r , s e l e n i u m , or t e l l u r i u m , and 4 M r e p r e s e n t s one of the f o l l o w i n g t r a n s i t i o n m e t a l s : T i , Z r , Hf, V, Nb, Ta, Mo, W, Tc, Re, P t , Ge, Sn or Pb. Most of the work r e p o r t e d i n the l i t e r a t u r e has s t u d i e d t r a n s i t i o n m e t a l s from Groups IVB, VB, or VIB. . The t r a n s i t i o n m e t a l d i c h a l c o g e n i d e s t r u c t u r e c o n s i s t s of l a y e r s h e l d t o g e t h e r by van der Waals f o r c e s . Each l a y e r i s a c h a l c o g e n - m e t a l - c h a l c o g e n sandwich. ( F i g u r e 1a). L i t h i u m can be i n t e r c a l a t e d i n t o s i t e s i n the van der Waals gap. I d e a l l y , the metal atoms i n each l a y e r w i l l be a r r a n g e d i n one of two ways. One i s the t r i g o n a l p r i s m a t i c c o o r d i n a t i o n of the metal as shown i n F i g u r e 1b, and the o t h e r has the m e t a l o c t a h e d r a l l y c o o r d i n a t e d as shown i n F i g u r e 1c. In both c a s e s the metal atom i s surrounded by s i x c h a l c o g e n atoms, but the symmetry of the environment i s d i f f e r e n t . The d i f f e r e n t environment w i l l l e a d t o a d i f f e r e n t band s t r u c t u r e and so t o d i f f e r e n t p h y s i c a l p r o p e r t i e s of the c r y s t a l . In p r a c t i c e , the s t r u c t u r e may have minor d e v i a t i o n s from the p e r f e c t symmetry d e s c r i b e d h e r e , but the g e n e r a l d i s t i n c t i o n between t r i g o n a l p r i s m a t i c and o c t a h e d r a l w i l l p e r s i s t . I t i s c o n v e n i e n t t o e x p r e s s the c r y s t a l s t r u c t u r e i n terms of the s t a c k i n g arrangement of the atoms. T h i s may be most e a s i l y v i s u a l i z e d i n terms of the s t a c k i n g of h a r d s p h e r e s . F i g u r e 2 d e p i c t s a monolayer of c l o s e packed s p h e r e s . Denote the c e n t r e s of t h e s e spheres as type A s i t e s . The next such l a y e r can s e t t l e i n one of two s e t s of s i t e s — t h o s e denoted by B or t h o s e denoted by C. The l a y e r above t h a t w i l l have t o s e t t l e i n one of the p o s i t i o n s A, B, or C, and so on f o r the subsequent 5 von der Wools gap AbA A b C trigonal prism octahedron d) 1 T - T I S 2 2 H -MoS 2 A b C A b C AbA BaB F i g u r e 1 S t r u c t u r e of the T r a n s i t i o n Metal D i c h a l c o g e n i d e s a) Two chalcogen-metal-chalcogen l a y e r s w i t h the van der Waals gap i n d i c a t e d . b) T r i g o n a l P r i s m a t i c c o o r d i n a t i o n of a metal (•) by s i x chalcogen atoms ( o ) . c) Octahedral c o o r d i n a t i o n of a metal (•) by s i x chalcogen atoms ( o ) . d) The s t r u c t u r e s of the two most commonly named p o l y t y p e s i n t h i s t h e s i s . The 1T-polytype shown f o r T i S 2 i s s i m i l a r t o the 0-phase. I t only has one l a y e r per u n i t c e l l . The 2Hb-polytype of MoS 2 i s c a l l e d the o-phase. I t has two l a y e r s per u n i t c e l l . T h i s f i g u r e i s based on one i n McKinnon (1980). F i g u r e 2 N o t a t i o n f o r S t a c k i n g Arrangements The c e n t r e s of the spheres, marked A, d e f i n e a two-dimensional hexagonal l a t t i c e . The next monolayer of spheres can s e t t l e , e i t h e r i n t o i n t e r s t i t i a l s i t e s marked by a B or by a C. A l l subsequent l a y e r s of spheres w i l l be over one of these three s i t e s . The same l a b e l l i n g of s i t e s can e q u a l l y w e l l serve to d e s c r i b e the s t a c k i n g arrangement of the t r a n s i t i o n metal d i c h a l c o g e n i d e s . Each sheet of atoms w i l l be i n an A, B, or C p o s i t i o n . A chalcogen sheet-metal sheet-chalcogen sheet makes up one l a y e r of the c r y s t a l . 7 l a y e r s . Each l a y e r can then be l a b e l l e d by which of the t h r e e t y p e s of s i t e i t o c c u p i e s . The same system can be used f o r the t r a n s i t i o n m e tal d i c h a l c o g e n i d e s . Each l a y e r of the c r y s t a l can be thought of as t h r e e t w o - d i m e n s i o n a l s h e e t s . F i g u r e 2 can a l s o be used t o d e p i c t one such s h e e t . The atoms of the next sheet may be a t or near a B or C s i t e , and so on. I t i s e x p e r i m e n t a l l y observed t h a t t h e s e m a t e r i a l s u s u a l l y form h e x a g o n a l c r y s t a l s w i t h atomic p o s i t i o n s v e r y c l o s e t o t h i s i d e a l i z e d s t a c k i n g . Note t h a t t h i s system of i d e n t i f y i n g atomic p o s i t i o n s w i t h one of t h r e e i d e a l i z e d s i t e s i s r e l a t i v e l y i n s e n s i t i v e t o minor v a r i a t i o n s i n t h e e x a c t p o s i t i o n of the atom. By c o n v e n t i o n a t h r e e l e t t e r system has been adopted t o d e s c r i b e the s t a c k i n g arrangement i n the t r a n s i t i o n m e t a l d i c h a l c o g e n i d e s . Chalcogen atoms a r e denoted by upper case l e t t e r s , and the t r a n s i t i o n m e t a l by a lower case l e t t e r . L i t h i u m s i t e s i n the van der Waals gap a r e s y m b o l i z e d by a f o r A s i t e s , /3 f o r B s i t e s and 7 f o r C s i t e s . The t r i g o n a l p r i s m a t i c c o o r d i n a t i o n d e s c r i b e d above can t h e r e f o r e be e x p r e s s e d as AbA, and the o c t a h e d r a l form i s AbC. A c r y s t a l may have d i f f e r e n t s t a c k i n g i n each l a y e r of the u n i t c e l l . There a r e two commonly mentioned p o l y t y p e s i n t h i s t h e s i s . The 1T c r y s t a l i s t y p i c a l of the Group IVB d i c h a l c o g e n i d e s , such as T i S 2 . The 1 r e f e r s t o the number of l a y e r s i n the u n i t c e l l , and the T r e p r e s e n t s the t r i g o n a l symmetry of the c r y s t a l . A 1T c r y s t a l has AbC AbC s t a c k i n g . The o t h e r c r y s t a l mentioned i s the 8 2Hb form of MoS 2. I t has a two l a y e r u n i t c e l l and hexagonal symmetry. 2 The 2Hb s t a c k i n g i s AbA BaB ( F i g u r e 1d). There i s an o t h e r n a t u r a l p o l y t y p e of MoS 2, the 3R, a t h r e e l a y e r rhombohedral c r y s t a l . The p e r i o d i c i t i e s and symmetries of a c r y s t a l a r e o f t e n d e s c r i b e d i n terms of t h r e e l a t t i c e v e c t o r s a, E, and c . 3 A c o n v e n t i o n a l c h o i c e of v e c t o r s f o r a hexagonal l a t t i c e i s shown i n F i g u r e 2: v e c t o r s a and B d e f i n e a p l a n e p a r a l l e l t o the sh e e t s of atoms, and c i s p e r p e n d i c u l a r ' t o the l a y e r s . Any s i t e s c o n n e c t e d by an i n t e g e r number of l a t t i c e v e c t o r s a r e i d e n t i c a l . The i l l u s t r a t i o n i n F i g u r e 2 shows i d e n t i c a l a d j a c e n t s i t e s . Many c r y s t a l s have s u p e r l a t t i c e s i n which the a d j a c e n t s i t e s may not be i d e n t i c a l . A n o t a t i o n of the form " n a by n^ by n c " w i l l be used t o d e s c r i b e the s u p e r l a t t i c e . n a and n^ a r e the number of n o n - i d e n t i c a l s i t e s a l o n g the a and B l a t t i c e v e c t o r s r e s p e c t i v e l y , and n c i s the number of l a y e r s i n the u n i t c e l l . For example, i f the s p a c i n g between a d j a c e n t type A s i t e s was 3A, but the c r y s t a l p e r i o d i c i t y was 6A a l o n g the a a x i s , 9A on the B a x i s and i d e n t i c a l i n e v e r y l a y e r , then the s u p e r l a t t i c e would be d e s c r i b e d as 2 by 3 by 1. An i d e a l 1T c r y s t a l i s 1 by 1 by 1, and an i d e a l 2Hb c r y s t a l i s 1 by 1 by 2. 2 The m e t a l i n the 2H c r y s t a l has t r i g o n a l p r i s m a t i c c o o r d i n a t i o n and the m e t a l i n the 1T c r y s t a l has o c t a h e d r a l c o o r d i n a t i o n . The T and H d e s i g n a t i o n r e f e r s t o the symmetry of the c r y s t a l , and not t o t h e c o o r d i n a t i o n of the m e t a l . 3 These v e c t o r s a r e i l l u s t r a t e d i n F i g u r e 32 of Appendix I . 9 1.3 I n t e r c a l a t i o n Techniques I n t e r c a l a t i o n t e c h n i q u e s a l l i n v o l v e a way of b r i n g i n g the i n t e r c a l a n t atom or m o l e c u l e t o the h o s t . An e l e c t r o c h e m i c a l t e c h n i q u e i s t h e one most commonly used i n t h i s t h e s i s . I t i s p o s s i b l e t o c o n s t r u c t a c e l l u s i n g l i t h i u m m e t a l as one e l e c t r o d e and the host m a t e r i a l as the o t h e r . Such a c e l l i s shown s c h e m a t i c a l l y i n F i g u r e 3. The two e l e c t r o d e s a r e con n e c t e d i o n i c a l l y t h r o u g h a l i t h i u m s a l t e l e c t r o l y t e , and connected e l e c t r o n i c a l l y t h r o u g h an e x t e r n a l c i r c u i t . A l i t h i u m atom i s t r a n s f e r r e d between one e l e c t r o d e and the o t h e r i n two p a r t s : an e l e c t r o n and a L i + i o n . The e l e c t r o n can not pass th r o u g h the e l e c t r o l y t e , and the i o n can not pass t h r o u g h the e x t e r n a l c i r c u i t . As a c e l l d i s c h a r g e s , the p o t e n t i a l d i f f e r e n c e between the anode and the hos t cathode w i l l draw an i o n from the s o l u t i o n and p u l l an e l e c t r o n t h r o u g h the e x t e r n a l c i r c u i t . When one i o n l e a v e s the e l e c t r o l y t e t o e n t e r the c a t h o d e , a n o t h e r i o n from the anode must e n t e r the s o l u t i o n t o m a i n t a i n charge n e u t r a l i t y i n the e l e c t r o l y t e . The net r e s u l t of t h i s i s the t r a n s f e r of one atom from the anode t o the c a t h o d e . T h i s p r o c e s s can be r e v e r s e d by d r i v i n g the c u r r e n t i n the o p p o s i t e d i r e c t i o n w i t h an a p p l i e d v o l t a g e . T h i s would r e s u l t i n the t r a n s f e r of a l i t h i u m atom from the hos t s t r u c t u r e t o the b u l k l i t h i u m m e t a l . The l i t h i u m e l e c t r o d e s h o u l d be c a l l e d the anode d u r i n g the c e l l d i s c h a r g e and c a l l e d the cathode w h i l e the c e l l i s c h a r g i n g , and t h e o p p o s i t e c o n v e n t i o n s h o u l d a p p l y t o the h o s t 10 External Load F i g u r e 3 Schematic o f the E l e c t r o c h e m i c a l C e l l The e l e c t r o l y t e i n these experiments was 1 molar L i A s F e i n propylene carbonate. 11 e l e c t r o d e . However, the host m a t e r i a l i s o f t e n i n c o r r e c t l y r e f e r r e d t o as the cathode r e g a r d l e s s of the d i r e c t i o n of c u r r e n t f l o w . T h i s usage i s c o n v e n i e n t f o r the work done here because i t a l l o w s a s i m p l e unambiguous i d e n t i f i c a t i o n of the e l e c t r o d e s even a t z e r o c u r r e n t f l o w . A c h e m i c a l p o t e n t i a l d i f f e r e n c e i s the d r i v i n g f o r c e f o r i n t e r c a l a t i o n r e a c t i o n s . In the case of l i t h i u m c e l l s , the A c h e m i c a l p o t e n t i a l f o r the b u l k l i t h i u m , M . i s d i f f e r e n t than L i the c h e m i c a l p o t e n t i a l f o r the l i t h i u m i n t e r c a l a t e d i n the host C m a t e r i a l , n . ( x ) . T h i s d i f f e r e n c e can be r e l a t e d t o the v o l t a g e Li 1 of the c e l l V = 4~ { A " A (x) } 1.1 e L i L i The c h e m i c a l p o t e n t i a l f o r b u l k l i t h i u m i s a c o n s t a n t , but l i t h i u m i n d u c e d v a r i a t i o n s i n the environment of the h o s t may cause ^ L j ^ ^ t o ^e dependent on the l i t h i u m c o n t e n t . For t h i s r e a s o n , the v o l t a g e of an i n t e r c a l a t i o n c e l l o f t e n does not remain c o n s t a n t as the c e l l i s c h a r g e d or d i s c h a r g e d . A u s e f u l c o r o l l a r y of t h i s i s t h a t the l i t h i u m c o n t e n t of the h o s t can be a l t e r e d by e x t e r n a l l y c o n t r o l l i n g the c e l l v o l t a g e . The v a r i a b l e x i n t h e above e q u a t i o n r e f e r s t o the mole f r a c t i o n of l i t h i u m i n t h e h o s t . I n g e n e r a l , the l i t h i u m c o n t e n t of a h o s t can o f t e n be v a r i e d c o n t i n o u s l y . The parameter x, as i n L i x M o S 2 , g i v e s the r a t i o of the number of moles of the guest s p e c i e s t o the number of moles of the f o r m u l a u n i t of the h o s t . The v a l u e of x can be d e t e r m i n e d e x p e r i m e n t a l l y i f t h e host i n i t i a l l y has no l i t h i u m i n i t and i f t h e weight of the cathode 12 powder, m, and the m o l e c u l a r weight of the h o s t , M, a r e known. One e l e c t r o n must be passed t h r o u g h the e x t e r n a l c i r c u i t f o r each l i t h i u m t h a t i s t r a n s f e r r e d t o the h o s t , so a measurement of the net number of coulombs t h a t have f l o w e d , Q, can be s i m p l y r e l a t e d t o x: Q M x = ~ 1 .2 F m where F i s the Faraday c o n s t a n t . The d i f f e r e n t i a l change i n the Gibbs f r e e energy of the system i s dG = v dn - S dT + v dp 1.3 where S i s the e n t r o p y , T i s the t e m p e r a t u r e , v i s t h e volume and p i s the p r e s s u r e . U s u a l l y t h e c e l l s a r e run a t c o n s t a n t t e m p e r a t u r e and c o n s t a n t p r e s s u r e , so the l a s t two terms of e q u a t i o n 1.3 a r e u s u a l l y z e r o . The v o l t a g e , V, can be r e l a t e d t o the c h e m i c a l p o t e n t i a l , and the t o t a l number of l i t h i u m atoms i n the h o s t i s n. Both the v o l t a g e and the amount of charge t r a n s f e r r e d a r e r e a d i l y measured. The e q u a t i o n of s t a t e can thus be r e l a t e d ' t o V(x) of the system. A way of e x p l o i t i n g the a n a l o g y between V ( x ) and the e q u a t i o n of s t a t e w i l l be d i s c u s s e d i n Chapter 4. Other i n t e r c a l a t i o n t e c h n i q u e s a r e a l s o mentioned i n t h i s t h e s i s . One i n s e r t i o n t e c h n i q u e i s the l i q u i d ammonia method f o r a l k a l i m e t a l s ( R u d o r f f , 1959). The m e t a l i s d i s s o l v e d i n the 13 ammonia and can then d i f f u s e t o a h o s t m a t e r i a l immersed i n the s o l u t i o n . A problem w i t h t h i s method i s t h a t the ammonia s o l v a t i o n c l o u d may a l s o be i n t e r c a l a t e d a l o n g w i t h the a l k a l i atom. There a r e a l s o many o r g a n o m e t a l l i c c h e m i c a l s t h a t can be used as l i t h i a t o r s . The work here d i s c u s s e s n - b u t y l l i t h i u m Cj,H 9Li ( D i n e s , 1975). T h i s c h e m i c a l i s d i s o l v e d i n hexane and the h o s t i s immersed i n the l i q u i d . The net r e a c t i o n has two l i t h i u m atoms go i n t o t h e h o s t a l o n g w i t h one octan e m o l e c u l e b e i n g formed i n the s o l u t i o n . Guest atoms can a l s o be put i n t o d i r e c t c o n t a c t w i t h the h o s t by h e a t i n g the m a t e r i a l s . The vapor p r e s s u r e r i s e s a t e l e v a t e d t e m p e r a t u r e s and the hos t i s bathed i n a gas of guest atoms. T h i s i s c l o s e l y r e l a t e d t o the p r e s s u r e i n t e r c a l a t i o n t e c h n i q u e s known f o r m a t e r i a l s l i k e g r a p h i t e ( H o o l e y , 1977). A problem w i t h t h i s t e c h n i q u e i s the h o s t may be a l t e r e d a t the h i g h e r t e m p e r a t u r e . 14 CHAPTER 2 Overview of the L i x M o S 2 System Exper ience does not ever err, it . is on Iy your judgement that errs in promising itself results which are not caused by your exper iment s . --Leonardo DaVinci 2.1 Summary There a r e many r e f e r e n c e s t o l i t h i u m i n t e r c a l a t i o n i n c r y s t a l l i n e MoS 2 i n the l i t e r a t u r e . Most of them c i t e a s e r i e s of t h r e e e x p e r i m e n t s (Somoano, 1973; Whittingham, 1975; and Besenhard, 1976). These e x p e r i m e n t s d e t a i l t h r e e i n t e r c a l a t i o n t e c h n i q u e s , and a l l come t o the same c o n c l u s i o n : MoS 2 i s an u n s u i t a b l e i n t e r c a l a t i o n h o s t because i t decomposes when l i t h i u m i s i n s e r t e d . Other work (on which t h i s t h e s i s i s based) r e p o r t s t h a t l i t h i u m i n s e r t i o n i n MoS 2 causes a phase t r a n s i t i o n t o a c r y s t a l p o l y t y p e r e f e r r e d t o as /3-phase. T h i s form of MoS 2 has the p r o p e r t i e s a p p r o p r i a t e t o a good i n t e r c a l a t i o n h o s t . Most of the i n f o r m a t i o n p u b l i s h e d r e f e r s back t o the e a r l i e s t work on the t o p i c , which as a r e s u l t , dominates the l i t e r a t u r e . T h i s work l e d t o the f i r s t of the two c o n c l u s i o n s mentioned above. Most i f not a l l of the b a s i c i n f o r m a t i o n i s c o r r e c t , b u t , as w i l l be shown l a t e r , i t was i n c o m p l e t e . L a t e r r e v i e w s and papers o f t e n summarized the data by o m i t t i n g the c o n d i t i o n s of the e x p e r i m e n t s and l e a v i n g j u s t the f i n a l c o n c l u s i o n s of the work. These s i m p l i f i e d c o n c l u s i o n s u s u a l l y 15 d i d not c o n t r a d i c t the d a t a , but r e s u l t e d i n s t a t e m e n t s t h a t were more g e n e r a l than was v a l i d . The l o s s of i n f o r m a t i o n can be seen i n the f o l l o w i n g example. I t was much more d i f f i c u l t t o i n t e r c a l a t e the Group VTB d i c h a l c o g e n i d e s , i n c l u d i n g MoS 2, than i t was t o i n t e r c a l a t e the Group IVB and VB compounds. MoS 2 was o n l y o b s e r v e d t o tak e up a s m a l l amount of l i t h i u m b e f o r e i t s b e h a v i o u r began t o change r a d i c a l l y , but many o t h e r m a t e r i a l s r e a c t e d t o the l i m i t of a s t o i c h i o m e t r i c r a t i o . G e n e r a l t h e o r i e s were put f o r w a r d t o e x p l a i n why an " i d e a l " Group VIB m a t e r i a l would not be e x p e c t e d t o i n t e r c a l a t e , and why Group IVB and VB complexes had such c o n s i s t e n t p r o p e r t i e s . E v e n t u a l l y the concept of MoS 2 b e i n g an u n s u i t a b l e l i t h i u m h o s t became so common t h a t s p e c i f i c r e f e r e n c e s t o the sour c e of the i n f o r m a t i o n were o f t e n o m i t t e d . One purpose of t h i s t h e s i s i s t o r e s o l v e the c l a i m s i n t h e l i t e r a t u r e r e g a r d i n g the s u i t a b i l i t y of MoS 2 as a host f o r l i t h i u m i n t e r c a l a t i o n . A d e t a i l e d l i t e r a t u r e r e v i e w w i l l be p r e s e n t e d i n s e c t i o n 2.2 t o i l l u s t r a t e the c o n t r o v e r s y . A d i s c u s s i o n of the work on which t h i s t h e s i s i s based w i l l f o l l o w i n the s e c t i o n s a f t e r t h a t . 16 2.2 L i t e r a t u r e Review I n t e r c a l a t i o n has been r e p o r t e d as e a r l y as 1842 when s u l f u r i c and n i t r i c a c i d s were i n s e r t e d i n t o the l a y e r s of g r a p h i t e ( S c h a f h a u l t , 1842), and the l a y e r e d s t r u c t u r e of MoS 2 has been known s i n c e 1923 ( D i c k i n s o n and P a u l i n g , 1923). Rather than t r a c e e i t h e r of t h e s e i d e a s f u l l y , I w i l l i m m e d i a t e l y s p e c i a l i z e t o r e p o r t s on l i t h i u m i n s e r t i o n i n MoS 2 and v e r y c l o s e l y r e l a t e d m a t t e r s . The e a r l i e s t s t u d i e s of l i t h i u m i n t e r c a l a t i o n i n MoS 2 were done by R i i d o r f f i n 1959 by u s i n g a l i q u i d ammonia t e c h n i q u e . He d i s c o v e r e d t h a t ammonia c o - i n t e r c a l a t e d w i t h the l i t h i u m . P r o d u c t s of the form L i , m , ( N H 3 ) 0 . 6 M o S 2 were found, but the e x a c t c o m p o s i t i o n of the p r o d u c t s c o u l d be v a r i e d . A d i f f e r e n t t e c h n i q u e was used i n 1965 when Sergent and P r i g e n t s t u d i e d a l k a l i s u l f i d e s of molybdenum, t u n g s t e n , and chromium. L i M o S 2 was formed by r e a c t i n g L i 2 M o S 3 . 5 under a hydrogen atmosphere a t 600°C. The way the Li/Mo r a t i o became b a l a n c e d was not mentioned. The r e s u l t was a b l a c k powder, the x - r a y p a t t e r n of which was p u b l i s h e d . The p a t t e r n was c o n s i s t e n t w i t h the d a t a I o b t a i n e d when the l i t h i a t e d MoS 2 powder r e a c t e d w i t h a i r . S i n c e t h e r e was no i n d i c a t i o n i n t h e i r paper of any a t t e mpt made t o p r o t e c t t h e i r samples from w a t e r , i t i s r e a s o n a b l e t o assume t h e i r L i M o S 2 d a t a had been d i s t o r t e d by a r e a c t i o n w i t h the atmosphere. Somoano e t . a l ; (1973) i n v e s t i g a t e d the i n t e r c a l a t i o n of the a l k a l i m e t a l s i n t o 2H and 3R p o l y t y p e s of MoS 2 v i a a l i q u i d 17 ammonia t e c h n i q u e . T h i s paper was o f t e n c i t e d over the next decade. The r e s u l t s f o r l i t h i u m were not as w e l l d e f i n e d as they were f o r most of the o t h e r m e t a l s . The s t o i c h i o m e t r y was de t e r m i n e d o n l y t o the l i m i t s 0.4<x<1.0 i n L i x M o S 2 . The x - r a y a n a l y s i s was f a r from d e f i n i t i v e : the hexagonal c a x i s was measured a t 19.039A per u n i t c e l l which was a 3.372A i n c r e a s e i n each of the two l a y e r s ; and no a a x i s d a t a was p u b l i s h e d because the p a t t e r n was not f u l l y i n d e x e d . NH 3 or l i t h i u m amide i n t e r c a l a t i o n , or N H 3 - L i c o - i n t e r c a l a t i o n were put f o r w a r d t o e x p l a i n t h e s e r e s u l t s . The compounds were a l s o found t o be a i r s e n s i t i v e . S e v e r a l p o s s i b l e i n t e r p r e t i o n s of the o v e r a l l r e s u l t s were s u g g e s t e d . The l i t h i a t e d m a t e r i a l was s u s p e c t e d t o be a d i s o r d e r e d i n t e r c a l a t i o n compound where perhaps e i t h e r the hexagonal u n i t c e l l was d i s t o r t e d , or the b a s i c s t r u c t u r e was a l t e r e d . MoS 2, n o r m a l l y a s e m i c o n d u c t o r , when i n t e r c a l a t e d was found t o be a s u p e r c o n d u c t o r . I t was thought t h a t the MoS 2 was c o n v e r t e d i n t o a metal by the d o n a t i o n of an e l e c t r o n from the i n t e r c a l a n t i n t o the d bands of the h o s t . Somoano made r e f e r e n c e t o t h i s work l a t e r (1975) when he suggested t h a t L i x ( N H 3 ) y M o S 2 was not a hexagonal c r y s t a l , but he was unable t o de t e r m i n e what i t was. He a l s o n o t e d a t r e n d i n d e c r e a s i n g c r y s t a l symmetry w i t h d e c r e a s i n g i n t e r c a l a t e s i z e . He r e p e a t e d t h i s a g a i n i n another paper (Woollam e t . a l . , 1976) Work by Subba Rao (1974) s u p p o r t e d the s u s p i c i o n t h a t t h e r e was some form of ammonia c o - i n t e r c a l a t i o n w i t h the l i t h i u m i n the MoS 2. S c h o l l h o r n and Weiss (1974) a l s o used the l i q u i d ammonia t e c h n i q u e t o i n t e r c a l a t e l i t h i u m , but the t h r u s t of t h a t paper 18 was t o i n v e s t i g a t e the h y d r a t i o n complexes A x ( H 2 0 ) y M o S 2 . L a t e r S c h o l l h o r n e t . a l . (1978) examined a l k a l i molybdenum s u l f i d e s produced from h i g h temperature m e l t s . However, the t h r u s t was a g a i n towards the h y d r a t i o n compounds. No i n v e s t i g a t i o n of pure L i x M o S 2 was made i n e i t h e r of the s e p a p e r s . They a l s o s u s p e c t e d t h a t the 2H-MoS2 became d i s o r d e r e d a t h i g h t e m p e r a t u r e . 1975 saw s e v e r a l papers on i n t e r c a l a t i o n systems p r e s e n t e d by p e o p l e from the Exxon Research and E n g i n e e r i n g Company. The t h r e e mentioned here c i t e each o t h e r and seem t o have been w r i t t e n as a s e t . A l l were r e c e i v e d f o r p u b l i c a t i o n w i t h i n a one month p e r i o d . D i nes (1975) r e p o r t e d on u s i n g n - b u t y l l i t h i u m as a c h e m i c a l t e c h n i q u e f o r the i n s e r t i o n of l i t h i u m i n t o a h o s t . In p a r t i c u l a r he s t u d i e d the t r a n s i t i o n m e t a l d i c h a l c o g e n i d e s . The Group IVB and VB compounds a l l r e a c t e d t o x=1. He d e s c r i b e d the work w i t h the Group VIB compounds as anomalous because they e i t h e r d i d not seem t o have a f i x e d e n d p o i n t f o r the r e a c t i o n , or they d i d not r e a c t even as f a r as x=1. The MoS 2 r e s u l t s were p a r t i c u l a r l y odd. The r e a c t i o n showed both a sample dependence and a time dependence. The MoS 2 consumed the e n t i r e 1.5 l i t h i u m s per molybdenum t h a t were a v a i l a b l e whenever the r e a c t i o n was a l l o w e d t o p r o c e e d t o c o m p l e t i o n . He r e f e r r e d t o the o t h e r papers i n the s e r i e s and mentioned x-ray d a t a which seemed t o i n d i c a t e MoS 2 i n t e r c a l a t e d b e f o r e the host degraded i n t o amorphous p r o d u c t s . The next paper i n the s e r i e s , Whittingham and Gamble (1975), proved t o be the one most o f t e n c i t e d i n the Li / M o S 2 l i t e r a t u r e . I t used the t e c h n i q u e s and da t a from the above paper 19 and examined the s t r u c t u r e s of the r e s u l t i n g m a t e r i a l s . The MoS 2 r e s u l t s showed a d e g r a d a t i o n of the h o s t . Only a s i n g l e x - r a y l i n e was found, and t h a t one i n d i c a t e d a c s p a c i n g of some m u l t i p l e of 6.40A. No a a x i s i n f o r m a t i o n was a v a i l a b l e . By c o n t r a s t , the Group IVB and VB m a t e r i a l s y i e l d e d s e v e r a l c l e a r d i f f r a c t i o n l i n e s t h a t a l l o w e d the u n i t c e l l d i m e n s i o n s t o be d e t e r m i n e d . MoS 2 a l s o r e a c t e d f a s t e r w i t h the n - b u t y l l i t h i u m than d i d the o t h e r Group VIB compounds. The MoS 2 was proposed t o be r e a c t i n g v i a a d i f f e r e n t mechanism where a low x i n t e r c a l a t i o n compound, L i x M o S 2 , f i r s t formed, and then the r e a c t i o n went t o c o m p l e t i o n . The u l t i m a t e r e a c t i o n f o r a l l t h e s e compounds was assumed t o i n v o l v e L i 2 S or L i 2 S e f o r m a t i o n . The l i t h i u m c h a l c o g e n s were not found i n the x-ray p a t t e r n s , but they were p r e d i c t e d on the b a s i s of the L i 7 NMR d a t a d e s c r i b e d below. F u r t h e r m o r e , Group VIB d i c h a l c o g e n i d e s were e x p l a i n e d t o be t h e r m o d y n a m i c a l l y u n s t a b l e i n the presence of l i t h i u m . MoS 2 may have i n t e r c a l a t e d a s m a l l amount of the m e t a l because i t was near the l i m i t between s t a b i l i t y and d i s p r o p o r t i o n a t i o n . I t was f e l t t h a t the l a r g e x v a l u e seen by Somoano (1973) was p o s s i b l e because the r e a c t i v i t y of t h e l i t h i u m had been reduced by the ammonia t h a t had c o - i n t e r c a l a t e d . The d e t a i l s of the thermodynamic argument were not p r e s e n t e d i n t h i s paper. A l s o mentioned was t h a t the l i t h i a t e d m a t e r i a l s were a i r s e n s i t i v e . The h y d r a t e d L i x M o S 2 had a d i f f r a c t i o n l i n e a t 12A. T h i s was c o n s i s t e n t w i t h the work by Sergent (1965). The t h i r d Exxon paper was NMR work r e p o r t e d by S i l b e r n a g e l ( 1 9 7 5). A l l the i n v e s t i g a t e d d i c h a l c o g e n i d e compounds of 20 Groups IVB and VB showed K n i g h t s h i f t s , but the Group VIB m a t e r i a l s d i d n o t . The L i 7 NMR s i g n a l s measured f o r the l i t h i a t e d Group VIB d i c h a l c o g e n i d e s were the same as the s i g n a l s d e t e c t e d from samples of pure L i 2 S or L i 2 S e . The x - r a y work was unable t o d e t e r m i n e i f the Group VIB compounds had decomposed or i f they had j u s t l o s t t h e i r c r y s t a l l i n i t y . The NMR work c o n c l u d e d d e c o m p o s i t i o n was the e x p l a n a t i o n . A t h i r d o f t e n c i t e d s e t of e x p e r i m e n t s were performed by Besenhard e t . a l . (1976). They s t u d i e d the e l e c t r o c h e m i c a l i n t e r c a l a t i o n of a l k a l i m e t a l s i n t o MoS 2. The d a t a r e p o r t e d shows V v s . x c u r v e s f o r x up t o 0.25 f o r l i t h i u m , sodium, p o t a s s i u m , r u b i d i u m and cesium. C e l l s made w i t h a l k a l i m e t a l s o t h e r than l i t h i u m a l l had a f e a t u r e a t x=0.125, but t h i s was not seen w i t h l i t h i u m samples u n l e s s ammonia was added t o the l i t h i u m s a l t . The d i f f e r e n c e was a t t r i b u t e d t o a p o s s i b l e k i n e t i c e f f e c t . The t e x t mentions the d e c o m p o s i t i o n of m a t e r i a l s f o r x>0.125 except i n the case where ammonia was added. Whittingham (1975) was r e f e r e n c e d t o g i v e L i 2 S f o r m a t i o n as an example of the p o s s i b l e d e c o m p o s i t i o n t h a t o c c u r r e d a t h i g h x v a l u e s . I t was a l s o p o s s i b l e t h a t they c o n v e r t e d some of the MoS 2 i n t o /3-phase by p o t a s s i u m i n t e r c a l a t i o n , but i n s u f f i c i e n t i n f o r m a t i o n was g i v e n t o c o n f i r m t h i s . Most papers w r i t t e n a f t e r 1976 t h a t mentioned l i t h i u m i n t e r c a l a t i o n i n c r y s t a l l i n e MoS 2 merely r e s t a t e d t h e r e s u l t s of e a r l i e r e x p e r i m e n t s . I t was d u r i n g t h i s r e p e t i t i o n of o t h e r r e s u l t s t h a t the concept of the u n s u i t a b i l i t y of MoS 2 as an i n t e r c a l a t i o n h o s t seemed t o become p a r t of the a c c e p t e d wisdom. The remainder of t h i s s e c t i o n c o n t a i n s the c h r o n o l o g y of the 21 p r o p a g a t i o n of t h i s i d e a as w e l l as the new i d e a s t h a t were put f o r w a r d . Woollam (1977) r e v i e w e d the p h y s i c s and c h e m i s t r y of MoS 2 i n t e r c a l a t i o n compounds, but r e p e a t e d no p a r t i c u l a r i n f o r m a t i o n about l i t h i u m except t o say t h a t i t c o - i n t e r c a l a t e d w i t h ammonia. He mentioned the MoS 2 band s t r u c t u r e c a l c u l a t i o n of M a t t h e i s s (1973) and p o i n t e d out the d i f f i c u l t y i n making a q u a n t i t a t i v e e x t e n s i o n of t h i s t o an i n t e r c a l a t e d m a t e r i a l . When Somoano and Woollam (1979) l a t e r r e v i e w e d MoS 2 i n t e r c a l a t i o n the c r y s t a l s t r u c t u r e of L i x ( N H 3 ) y M o S 2 was s t i l l unknown, and a t e t r a g o n a l symmetry was r e p o r t e d f o r MoS 2 c r y s t a l s t h a t had been sodium i n t e r c a l a t e d . P u l l h a m (1977) r e v i e w e d r e c e n t c h e m i c a l work on the Group IA e lements. He r e p o r t e d t h a t by u s i n g n - b u t y l l i t h i u m i t was p o s s i b l e t o make p u r e , c r y s t a l l i n e l i t h i a t e d d i s u l f i d e s and d i s e l e n i d e s of Groups IVB and VB, but not Group VIB. The L i x M o S 2 compound was d e s c r i b e d as m e t a s t a b l e . The source was S i l b e r n a g e l ' s (1975) work. Whittingham (1977) reviewed m a t e r i a l s f o r b a t t e r y c a t hodes w i t h p a r t i c u l a r emphasis on t h e t r a n s i t i o n m e t a l d i c h a l c o g e n i d e s . He c i t e d h i s 1975 work and mentioned t h a t MoS 2 seemed t o decompose when l i t h i u m was i n t e r c a l a t e d . Most of the remainder of the paper (and many o t h e r s a t t h a t time) s t r e s s e d the L i / L i x T i S 2 system and i t s s u i t a b i l i t y f o r commercial development. Gamble's 1978 r e v i e w of the c h e m i s t r y and p h y s i c s of the l a y e r e d d i c h a l c o g e n i d e s i n c l u d e d s e v e r a l r e f e r e n c e s t o i n t e r c a l a t i o n . I t was r e p o r t e d t h a t the Group VIB compounds d i d 22 not r e a d i l y i n t e r c a l a t e , but l i t h i u m i n MoS 2 was not mentioned e x p l i c i t l y . He p o i n t e d out a l l t h a t a l l i n t e r c a l a t i o n complexes but one, T a S 2 , had a p a r t i a l y f i l l e d l o w - l y i n g band. He proposed an e m p i r i c a l r u l e t h a t such a s t a t e was r e q u i r e d f o r the i n t e r c a l a t i o n p r o c e s s . A major paper by Whittingham (1978) r e v i e w e d much of the known i n t e r c a l a t i o n c h e m i s t r y of m e t a l c h a l c o g e n i d e s and added a few o t h e r i m p o r t a n t p o i n t s . He began by d i s c u s s i n g i n t e r c a l a t i o n t e c h n i q u e s . He r e p e a t e d the problem of c o - i n t e r c a l a t i o n when u s i n g l i q u i d ammonia, and he r e s t a t e d h i s own 1975 work w i t h n - b u t y l l i t h i u m . F u r t h e r m o r e , he s a i d a l i t h i u m n a p t h a l i n e s o l u t i o n would reduce MoS 2 t o molybdenum and l i t h i u m s u l f i d e , but added t h a t c r y s t a l l i n e samples w i t h low s u r f a c e a r e a s r e a c t e d more s l o w l y and formed the i n t e r c a l a t i o n compound b e f o r e t h e y d i s p r o p o r t i o n a t e d . I t was assumed t h a t t h e r e d u c t i o n t o L i 2 S was k i n e t i c a l l y slow i n t h a t r e a c t i o n . E l e c t r o c h e m i c a l c e l l s w i t h MoS 2 cathodes were d i s c h a r g e d t o 1.3V. An i n t e r c a l a t i o n compound w i t h x=0.1 was so produced. I t was thought t h a t the l a t t i c e would decompose i n t o L i 2 S and molybdenum i f the x v a l u e was l a r g e enough, but t h e r e was no i n d i c a t i o n of d i s p r o p o r t i o n a t i o n above 1.3V. The s t r u c t u r e of the L i x M o S 2 m a t e r i a l s s t i l l was not known, but i t was assumed t o be hexagonal because o t h e r a l k a l i / M o S 2 compounds had hexagonal s t r u c t u r e s . P o t a s s i u m i n t e r c a l a t i o n i n 3R-MoS 2 was a l s o r e p o r t e d . That s t u d y showed the {1,0,1) x - r a y l i n e s c h a n g i n g i n a way c o n s i s t e n t w i t h the s u l f u r l a y e r s s h i f t i n g , but u n f o r t u n a t e l y no f u r t h e r e l a b o r a t i o n or r e f e r e n c e was g i v e n . 23 The paper a l s o d i s c u s s e d the thermodynamics of i n t e r c a l a t i o n systems. A t a b l e of AG of f o r m a t i o n s and AG f o r d e c o m p o s i t i o n f o r a v a r i e t y of t r a n s i t i o n m e t a l d i c h a l c o g e n i d e s was p r o v i d e d . The Group VIB m a t e r i a l s were grouped a t the u n s t a b l e end of the t a b l e w i t h MoS 2 b e i n g the most r o b u s t of the group. The argument was then put f o r w a r d t h a t i f a m a t e r i a l was o n l y b o r d e r l i n e s t a b l e , then the energy change from i n t e r c a l a t i o n would be enough t o cause the d i s p r o p o r t i o n a t i o n of the m a t e r i a l . MoS 2, which c o u l d i n t e r c a l a t e a s m a l l amount of l i t h i u m and r e t a i n i t s s t r u c t u r e , was c o n s i d e r e d t o be the d i v i d i n g p o i n t between s t a b l e and u n s t a b l e m a t e r i a l s . T h i s t h e o r y q u a l i t a t i v e l y e x p l a i n e d the o b s e r v a t i o n s , however i t d i d not p r e d i c t the v a l u e of x a t which L i x M o S 2 would become u n s t a b l e . A l s o i n 1978 Wainwright performed the f i r s t s t r u c t u r a l a n a l y s i s of the /3-phase. H a e r i n g e t . a l . f i l e d a p a t e n t i n 1979 which d e s c r i b e d the p r o d u c t i o n and use of j3-phase as an e l e c t r o c h e m i c a l c e l l . The p a t e n t was g r a n t e d i n 1980. Both t h e s e works w i l l be d i s c u s s e d i n the next s e c t i o n . C h i a n e l l i e t . a l . (1978) d e s c r i b e d the p r e p a r a t i o n of amorphous t r a n s i t i o n m e t a l d i c h a l c o g e n i d e s by a p r e c i p i t a t i o n t e c h n i q u e . One mole of the MoS 2 so produced would r e a c t w i t h 1.51 moles of n - b u t y l l i t h i u m . The amorphous m a t e r i a l c o u l d be made " p o o r l y c r y s t a l l i n e " by a n n e a l i n g a t 400°C. F u r t h e r e l a b o r a t i o n , C h i a n e l l i e t . a l . (1979), of t h i s " r a g " s t r u c t u r e d e s c r i b e d i t as s i n g l e , f o l d e d , h i g h l y d i s o r d e r e d MoS 2 l a y e r s . C e l l s made w i t h t h i s m a t e r i a l (Jacobson e t . a l . , 1979a) showed a d i s t i n c t l y d i f f e r e n t b e h a v i o u r t h a n c e l l s made w i t h c r y s t a l l i n e 24 MoS 2. Whereas the c r y s t a l l i n e m a t e r i a l was e x p e c t e d t o decompose f o r x>0.1, the amorphous m a t e r i a l c o u l d a t t a i n v a l u e s of. x=0.8. The r e s u l t was c o n s i d e r e d s u r p r i s i n g , and the e x p l a n a t i o n put fo r w a r d was t h a t the f o r m a t i o n of L i 2 S and Mo was k i n e t i c a l l y i n h i b i t e d i n the amorphous m a t e r i a l . Cathodes c o u l d a l s o c y c l e hundreds of tim e s and r e t a i n up t o h a l f t h e i r c a p a c i t y . I f the amorphous m a t e r i a l was a n n e a l e d then the c a p a c i t y dropped. The h i g h e r the a n n e a l t e m p e r a t u r e , the s m a l l e r was the c a p a c i t y . The c a p a c i t y l i m i t of c r y s t a l l i n e MoS 2 was reached a t about 400°C. In a n o t h e r paper, Jacobson (1979b) a g a i n s t a t e d t h a t amorphous MoS 2 was s u p e r i o r t o c r y s t a l l i n e MoS 2 i n e l e c t r o c h e m i c a l c e l l s . T h i s i n f o r m a t i o n was l a t e r r e p e a t e d i n a re v i e w by Johnson and W o r r e l l (1982b). R o u x e l ' s (1979a) d i s c u s s i o n on the p r e p a r a t i o n and s t r u c t u r e of a l k a l i m e t a l i n t e r c a l a t i o n compounds d i s m i s s e d MoS 2 i n t e r c a l a t i o n i n one l i n e because the f i l l e d dz2 band l e d t o u n s t a b l e i n t e r c a l a t i o n compounds. A re v i e w ( R o u x e l , 1979b) c i t e d the e a r l i e r work by S e r g e n t ( 1 9 6 5 ) , W h i t t i n g h a m ( 1 9 7 5 ) , and Besenhard(1976) and r e p e a t e d t h a t t h e r e was something odd about the l i t h i u m i n t e r c a l a t e s of MoS 2, but t h e r e was no e v i d e n c e f o r l a r g e x L i x M o S 2 i n t e r c a l a t i o n compounds. Some work done by S u i b e t . a l . (1979) on copper i n MoS 2 was i n t e r e s t i n g i n p a r t because they showed a c y c l i c voltammogram of Cu/MoS 2 which was q u i t e s i m i l a r t o what one would expect from /3-phase. I t was p o s s i b l e they had a p a r t i a l c o n v e r s i o n t o the /3 s t r u c t u r e . However, any c o n v e r s i o n which o c c u r r e d was o n l y near the s u r f a c e of the c r y s t a l s , and t h e i r f i n a l c o n c l u s i o n s s u p p o r t e d the e a r l i e r c l a i m s the MoS 2 c o u l d not i n t e r c a l a t e w i t h 25 c o pper. R e f e r e n c e s t o l i t h i u m i n t e r c a l a t e d MoS 2 reached t h e i r peak i n 1979. L i t t l e new work was be i n g p u b l i s h e d , and most r e f e r e n c e s were c i t a t i o n s t o o l d e r work. At about t h e same time the number of r e f e r e n c e s t o l i t h i u m i n t e r c a l a t i o n i n o t h e r m a t e r i a l s , p a r t i c u l a r l y T i S 2 , a l s o reached a peak. T i S 2 was b e i n g e x p l o r e d as a cathode m a t e r i a l f o r a c o m m e r c i a l l y v i a b l e c e l l . V i r t u a l l y no mention was made of the p o s s i b i l i t y of u s i n g L i / M o S 2 c e l l s f o r p r a c t i c a l p u rposes u n l e s s amorphous MoS 2 was used. By 1980 i t was g e n e r a l l y a c c e p t e d t h a t MoS 2 was not a s u i t a b l e i n t e r c a l a t i o n h o s t . R o u x e l (1980) d e s c r i b e d the MoS 2 i n t e r c a l a t i o n compounds as u n s t a b l e . Armand (1980) e x p l a i n e d how the d o n a t i o n of an e l e c t r o n t o the MoS 2 bands d u r i n g i n t e r c a l a t i o n made the Mo-S bond u n s t a b l e . W h i ttingham e t . a l . (1980) r e p e a t e d t h e i r e a r l i e r r e s u l t t h a t e l e c t r o c h e m i c a l c e l l s made w i t h c r y s t a l l i n e MoS 2 d i d not i n t e r c a l a t e beyond 0.1 l i t h i u m s per molybdenum, but major improvements c o u l d be made w i t h amorphous c a t h o d e s . The f i g u r e f o r c r y s t a l l i n e MoS 2 was l a t e r upgraded t o 0.2 Li/Mo (Whittingham, 1981), and r e p e a t e d a g a i n l a t e r (Whittingham, 1982) a l o n g w i t h a warning t h a t the samples d i s p r o p o r t i o n a t e t o L i 2 S . A re v i e w by Thompson and D i S a l v o (1982) a g a i n c i t e d t he work on d i s p r o p o r t i o n a t i o n . Abraham and Brummer (1983) r e f e r r e d t o Ja c o b s o n ' s work on amorphous MoS 2 and i t s s u p e r i o r i t y t o c r y s t a l l i n e MoS 2. Somoano's 1973 r e s e a r c h was c i t e d by K a n z a k i e t . a l . (1982) i n t h e i r ESR work which used a d i f f e r e n t c h e m i c a l l i t h i a t i o n 26 t e c h n i q u e . They c l a i m e d t o see no e v i d e n c e f o r c o - i n t e r c a l a t i o n and measured a 12.6A1 c a x i s s p a c i n g and ambiguous a a x i s d a t a . They c l a i m e d the powder t o be L i 0 . 6 7MoS 2. T h i s x-ray d a t a i s c o n s i s t e n t w i t h the /3-phase work p r e s e n t e d h e r e . Py and H a e r i n g (1983) d i s c u s s e d the f o r m a t i o n of /3-phase, and t h i s i s d e t a i l e d i n the next s e c t i o n . Laman (1985) mentioned the L i / L i x M o S 2 as a p r a c t i c a l c e l l i n a paper on the modeling of e l e c t r o d e s . I l i c e t . a l . (1985) s t u d i e d a l k a l i i n t e r c a l a t i o n i n 2H and 3R MoS 2. I t would seem as i f the /3-phase was formed i n t h e i r work, but what may have been the a-/3 phase t r a n s i t i o n was e x p l a i n e d as e i t h e r a phase t r a n s i t i o n of the form 3R t o 2H, or the f o r m a t i o n of Mo 6S 8. These i n t e r p r e t a t i o n s were i n c o n s i s t e n t w i t h the d a t a t h a t w i l l be p r e s e n t e d i n t h i s t h e s i s . 2.3 Beta Phase MoS 2 The phases of L i x M o S 2 were d e s c r i b e d by H a e r i n g e t . a l . (1980) and Wainwright (1978). The f i r s t phase, or a-phase, was the normal 2H p o l y t y p e . The second, or /3-phase, was produced by d i s c h a r g i n g an a-phase c e l l t h r o u g h a phase t r a n s i t i o n between 1.2 and 1.0 v o l t s . Enough charge was t r a n s f e r r e d i n t h i s t r a n s i t i o n t o change the c o m p o s i t i o n from x=0.2 t o at l e a s t x=1.0. E x t r a l i t h i u m t r a n s f e r , up t o a t o t a l e q u i v a l e n t of x=1.5, was o f t e n o b s e r v e d . T h i s e x c e s s may have r e s u l t e d from i m p u r i t y o r e l e c t r o l y t e r e a c t i o n s , and c o u l d be reduced by p e r f o r m i n g the t r a n s i t i o n a t lower t e m p e r a t u r e s . The t h i r d phase, 7-phase, was produced by the d i s c h a r g e of /3-phase t h r o u g h a phase t r a n s i t i o n a t about 0.6V where the x v a l u e rose t o 3. 27 The / 3 and 7 phases s e r v e d as a r e c y c l a b l e h o s t m a t e r i a l s , b o t h of which have s i g n i f i c a n t l y h i g h e r c a p a c i t y than the a-phase. Examples of the c y c l i n g b e h a v i o u r w i l l f o l l o w . W ainwright made s t r u c t u r a l measurements u s i n g D e b y e - S c h e r r e r x - r a y t e c h n i q u e s . The samples were t a k e n from the c a t h o d e s of L i / M o S 2 c e l l s which had been brought t o v a r i o u s s t a t e s of charge and d i s c h a r g e . The a and /3 s t r u c t u r e s were i n d e x a b l e on hexagonal l a t t i c e s t o w i t h i n the l i m i t s of the e x p e r i m e n t a l e r r o r . The 7-phase was found t o be too p o o r l y c r y s t a l l i n e t o i n d e x , and t h e r e was some i n d i c a t i o n of L i 2 S f o r m a t i o n . The i n t e n s i t y of the d i f f r a c t i o n l i n e s of /3-phase m a t e r i a l s were most c o n s i s t e n t w i t h a s t a c k i n g arrangement t h a t had the molybdenum atoms a l i g n e d a l o n g the c a x i s . The 1T (AbC) s t r u c t u r e or the 2Ha-NbS 2 (AbA CbC) s t r u c t u r e were prime c a n d i d a t e s . S t a c k i n g arrangements t h a t had the molybdenum atoms a l i g n e d a l o n g the c a x i s had the {1,0,1) l i n e s w i t h even / M i l l e r i n d i c e s more i n t e n s e than l i n e s w i t h odd / i n d i c e s . The r e v e r s e i n t e n s i t y p a t t e r n would be seen f o r arrangements where the molybdenum atoms were s t a g g e r e d a l o n g the c a x i s . I n s p i t e of the two above s t r u c t u r e s b e i n g good f i t s t o the d a t a , the x - r a y l i n e s were t o o broad t o p r e c l u d e o t h e r p o s s i b i l i t i e s . E x posures made from m a t e r i a l s p r e p a r e d a t 2.3V and h i g h e r v o l t a g e s showed e x t r a l i n e s t h a t c o u l d be indexed as (1,0,3) or (1,0,5) on a two l a y e r u n i t c e l l . These e x t r a l i n e s were not c o n s i s t e n t w i t h the s i n g l e l a y e r 1T s t r u c t u r e , but t h e r e were d i s t o r t i o n s , such as s t a g i n g , which i n c r e a s e d t h e s i z e of the u n i t c e l l and r e t a i n e d the 1T arrangement i n each l a y e r . 28 The l a t t i c e parameters f o r the /3-phase v a r i e d as a f u n c t i o n of l i t h i u m c o n t e n t . The a parameter ranged from ~3.23A t o ~ 3.37A as x ranged from z e r o t o one. The c a x i s parameter was not monotonic. At low x i t was ~ 6 . 1 - 8 A and i t reached a maximum of " ~ 6 i 4 2 A a t a p p r o x i m a t e l y x=0.5. The f u l l y d i s c h a r g e d powder had a c s p a c i n g of ~ 6 . 3 0 A . The c e l l volume i n c r e a s e d m o n o t o n i c a l l y w i t h i n c r e a s i n g x. The f i r s t d e t a i l e d i n v e s t i g a t i o n s of the a t o /3 t r a n s i t i o n were done by Py and H a e r i n g ( 1 9 8 3 ) by u s i n g in situ x - r a y d i f f r a c t i o n . The l i t h i u m c o n t e n t of the L i x M o S 2 was c o n t i n u o u s l y a l t e r e d from x=0 t o x= 1 w h i l e the d i f f r a c t i o n p a t t e r n was m o n i t o r e d . The i n t e n s i t i e s of the 2H-MoS 2 peaks dropped and the i n t e n s i t i e s of a new s e t of l i n e s r o s e as the phase t r a n s i t i o n proceeded a l o n g the 1 . 1 V p l a t e a u . The f i n a l m a t e r i a l d i d not have the s t r u c t u r e s of e i t h e r L i 2 S or molybdenum as would have been e x p e c t e d i f the e a r l i e r work i n d i c a t i n g d i s p r o p o r t i o n a t i o n a p p l i e d i n t h i s c a s e . The r e s u l t i n g x - ray l i n e s were l e s s i n t e n s e and broader than t h o s e of the the i n i t i a l h i g h l y c r y s t a l l i n e m a t e r i a l . T h i s i n d i c a t e d a somewhat d i s o r d e r e d m a t e r i a l , but the presence of x - r a y l i n e s i n d i c a t e d an u n d e r l y i n g c r y s t a l s t r u c t u r e . The L i M o S 2 s t r u c t u r e was indexed on a hexagonal l a t t i c e w i t h a = 3 . 3 6 A and c = 6 . 2 9 4 A . F u r t h e r m o r e , an attempt was made t o d e t e r m i n e the atomic s t a c k i n g arrangement i n the c r y s t a l . Four models were t e s t e d u s i n g t h e c o n s t r a i n t of a l i g n m e n t of molybdenum atoms a l o n g the c a x i s . The b e s t f i t of the i n t e n s i t y d a t a was f o r a 1T s t a c k i n g (AbC). The o t h e r models e i t h e r d i d not f i t the observed i n t e n s i t y r a t i o s q u i t e as w e l l (AbA model) or p r e d i c t e d l i n e s 29 t h a t were not obser v e d (AbC CbA and AbA CbC m o d e l s ) . . The d a t a was e x p l a i n e d i n terms of e l e c t r o n d o n a t i o n t o the MoS 2 d bands. F i g u r e 4 shows the band s t r u c t u r e s f o r 2H and 1T TaS 2 c r y s t a l s as based on the work by M a t t h e i s s (1973). T h i s compound can e x i s t as e i t h e r p o l y t y p e and was used as a model f o r the MoS 2. The c r o s s h a t c h e d r e g i o n of the diagram r e p r e s e n t s the e l e c t r o n s i n the d bands. The maximum e l e c t r o n i c energy f o r the 2H p o l y t y p e i s lower than t h a t f o r the 1T form i f no guest s p e c i e s i s p r e s e n t . I f something i s i n t e r c a l a t e d i n t o the h o s t , then i t i s p o s s i b l e the 1T form becomes f a v o u r e d . Charge t r a n s f e r from i n t e r c a l a n t t o hos t i s known t o o c c u r . These e x t r a e l e c t r o n s have t o be accommodated i n the bands of the c r y s t a l . The b l a c k e n e d a r e a on the F i g u r e r e p r e s e n t s the t r a n s f e r of one e x t r a e l e c t r o n per molybdenum i n t o the bands. The 1T s t r u c t u r e has the lower e l e c t r o n i c energy a f t e r the i n t e r c a l a t i o n . The p o s s i b i l i t y of c o n v e r s i o n of 2H-MoS2 t o 1T-MoS 2 was not c o n s i d e r e d by Gamble (1978) or Whittingham (1978) when they t r i e d t o e x p l a i n l i t h i u m i n t e r c a l a t i o n i n Group V I B d i c h a l c o g e n i d e s . The w i d e l y a c c e p t e d b e l i e f t h a t c r y s t a l l i n e MoS 2 cannot i n t e r c a l a t e i s based on work done on the 2H-MoS2 p o l y t y p e . Gamble's band s t r u c t u r e argument s t a t e d t h a t the 2H c r y s t a l d i d not have a p a r t i a l l y f i l l e d band, and t h i s would i n h i b i t i n t e r c a l a t i o n . F u r t h e r m o r e , the energy a s s o c i a t e d w i t h the d o n a t i o n of the i n t e r c a l a n t e l e c t r o n t o the next a v a i l a b l e d band was assumed t o put the complex i n a s t a t e where the d e c o m p o s i t i o n t o L i 2 S and Mo was e n e r g e t i c a l l y f a v o u r a b l e . Whittingham's thermodynamic argument came t o the same c o n c l u s i o n by comparing the heat of f o r m a t i o n of L i 2 S and Mo w i t h the heat 30 T 1 1 1 1 1 1 I 1 1 1 1 1 r 6 STATES/eV- CATION STATES/eV- CATION F i g u r e 4 D e n s i t y of S t a t e s f o r 2H and 1T-MoS 2 Both t h e s e f i g u r e s a r e based on t h e band s t r u c t u r e c a l c u l a t i o n s of M a t t h e i s s (1973) and were f i r s t p r e s e n t e d by Py (1983). The c r o s s h a t c h e d r e g i o n s r e p r e s e n t the s t a t e s o c c u p i e d by e l e c t o n s from the MoS 2 c r y s t a l s . The darkened r e g i o n s r e p r e s e n t the e x t r a s t a t e s o c c u p i e d by e l e c t r o n s donated from an i n t e r c a l a n t . The 2H-MoS 2 ( F i g u r e 4a) has the lower e l e c t r o n i c energy i f no e l e c t r o n s have been donated t o the bands. The 1T-MoS 2 ( F i g u r e 4b) has the lower e l e c t r o n i c energy i f t h e m a t e r i a l has been i n t e r c a l a t e d . These diagrams a r e o n l y d e s i g n e d t o " i l l u s t r a t e some of the d i f f e r e n c e s between the two p o l y t y p e s . The a c t u a l d e n s i t y of s t a t e s w i l l p r o b a b l y d i f f e r from t h o s e shown. 31 of f o r m a t i o n of 2H-LiMoS 2. The 1T~LiMoS 2 p o l y t y p e i s e n e r g e t i c a l l y f a v o u r a b l e compared t o both 2H-LiMoS 2 and a l s o L i 2S and Mo. The d a t a support the t h e o r y t h a t as l i t h i u m i s added t o 2H-MoS 2 the h o s t i s d r i v e n by the e l e c t r o n i c energy i n t o a phase t r a n s i t i o n t o the 1T s t a c k i n g . 2.4 B a s i c MoS 2 E l e c t r o c h e m i s t r y The c y c l i n g of the t h r e e phases of L i x M o S 2 i s shown i n F i g u r e 5. The c u r v e s were produced by c o n s t a n t c u r r e n t c y c l i n g , the f u l l d e t a i l s of which w i l l be d i s c u s s e d i n Chapter 4. The c e l l v o l t a g e was r e c o r d e d as the l i t h i u m c o n t e n t was v a r i e d . Each h a l f c y c l e shown r e f l e c t e d a c o n s t a n t v a r i a t i o n i n x between s e t v o l t a g e l i m i t s . The b a r s on the graph r e f l e c t the time r e q u i r e d t o change the x c o n t e n t by the i n d i c a t e d amount. Note the s c a l e i s d i f f e r e n t on a l l t h r e e g r a p h s . F i g u r e 5a i s the a-phase c y c l i n g between 1.4V and 2.7V. The c e l l has o n l y m i n i m a l c a p a c i t y i n t h i s s t r u c t u r e , w i t h a Ax of 0.02 shown. A l a r g e r Ax can be o b t a i n e d by c y c l i n g down t o 1.3V. The c a p a c i t y i n t h i s phase i s r e v e r s i b l e over many c y c l e s . T h i s i s the l i m i t e d x i n t e r c a l a t i o n phase d i s c u s s e d i n the l i t e r a t u r e . The c a p a c i t y i n the a-phase w i l l not be f u r t h e r d i s c u s s e d i n t h i s t h e s i s . F i g u r e 6 shows the a t o j3 phase t r a n s i t i o n p l u s the f i r s t c y c l e of /3-phase. The d r a m a t i c i n c r e a s e i n c a p a c i t y by c o n v e r s i o n t o 0-phase i s c l e a r l y e v i d e n t . The t r a n s i t i o n t y p i c a l l y o c u r r e d a t about 1.1V, but t h i s p o t e n t i a l may v a r y f o r a number of r e a s o n s . The t r a n s i t i o n may n u c l e a t e a t some 32 J I I I 1 1 1 1 I I • 1 « b) 1? 2 3 4 Time (hours) : - * I — — I \ - -1 t i l l i i D 5 10 Time (hours) 15 ± ± 0 10 20 Time (hours) Figure 5 C y c l i n g of D i f f e r e n t MoS 2 Phases These f i g u r e s show c e l l PM-94 c y c l i n g at currents of 100MA. The i n i t i a l cathode m a t e r i a l was 2H-MoS2. Figure 5a represents the a-phase. Figure 5b represents the 0-phase. Figure 5c represents the 7-phase. Each f i g u r e i n c l u d e s a s c a l e that i n d i c a t e s the time i t would take to change the x content of Li x M o S 2 by the i n d i c a t e d amount. T h i s f i g u r e shows s e v e r a l a-phase c y c l e s , t h e 1 .1V a t o 0 t r a n s i t i o n , and the f i r s t c y c l e of /3-phase. The c e l l was c y c l e d a t SOuk and i t i n i t i a l l y had a 2H-MoS2 c a t h o d e . 34 u n d e r p o t e n t i a l r a t h e r than a t the e q u i l i b r i u m between the two phases. The magnitude of t h e u n d e r p o t e n t i a l was p r o b a b l y r e l a t e d t o the energy r e q u i r e d to form the i n t e r f a c e between the a and |3 m a t e r i a l s . D i f f e r e n t p a r t i c l e s i z e s or c o n d i t i o n s can a l t e r the t r a n s i t i o n v o l t a g e . E l e c t r o l y t e d e c o m p o s i t i o n o f t e n o c c u r r e d d u r i n g the a t o /3 t r a n s i t i o n and t h i s c o u l d be reduced i n two ways. F i r s t , the c e l l c o u l d be d i s c h a r g e d at 0°C or some o t h e r low temperature t h a t w i l l slow the k i n e t i c s of the d e c o m p o s i t i o n more than the k i n e t i c s of the c o n v e r s i o n . U n p u b l i s h e d low t emperature c o n v e r s i o n work by M a r c e l Py i n d i c a t e d t h a t x=1 a t the end of the a t o /3 p l a t e a u . Second, and e a s i e s t , was t o c y c l e the c e l l a t a h i g h c u r r e n t so the c e l l spent l e s s time a t the low v o l t a g e s which caused d e c o m p o s i t i o n . T y p i c a l l y a s i n g l e c y c l e such as the one shown d i d not l e a d t o 100% c o n v e r s i o n , but the r e s i d u a l a-phase c o u l d be c o n v e r t e d on subsequent c y c l e s . I t was p r o b a b l y because of known e l e c t r o l y t e d e c o m p o s i t i o n problems t h a t e l e c t r o c h e m i c a l c e l l s were not d i s c h a r g e d below 1.3V by o t h e r w o r k e r s . The /3-phase was not s t a b l e a t h i g h v o l t a g e s and low x v a l u e s . I t c o n v e r t e d back t o a-phase i f i t was c h a r g e d t o above about 2.6V. T h i s was c o n s i s t e n t w i t h the argument t h a t e l e c t r o n i c energy d r i v e s t h e phase t r a n s i t i o n . R e f e r r i n g back t o F i g u r e 4, a t x=1 MoS 2 was i n t h e 1T p o l y t y p e because th e 1T form had a lower energy than the 2H p o l y t y p e . I f the sample was d e i n t e r c a l a t e d then a p o i n t was reached when enough e l e c t r o n s were removed from the 1T bands t o make i t u n s t a b l e w i t h r e s p e c t t o the 2H form. The a-phase produced by such a c o n v e r s i o n was s l i g h t l y d i s o r d e r e d compared t o the s t a r t i n g m a t e r i a l . T h i s was 35 l i k e l y due t o the d i s t o r t i o n s i n v o l v e d i n the t r a n s i t i o n s l e a d i n g t o s t r a i n s and d e f e c t s i n the m a t e r i a l . A c e l l can a l s o be r e c o n v e r t e d t o 0-phase by d i s c h a r g i n g t h r o u g h the a t o 0 t r a n s i t i o n , but the t r a n s i t i o n v o l t a g e f o r r e c o n v e r s i o n i s t y p i c a l l y about 1.3V. T h i s may be e x p l a i n e d e i t h e r by the a d d i t i o n a l d e f e c t s i n the a-phase e i t h e r g i v i n g more b o u n d a r i e s f o r n u c l e a t i o n , or by the energy r e q u i r e d t o s u p p o r t the a-/3 i n t e r f a c e b e i n g reduced. The 0-phase c y c l i n g b e h a v i o u r i s shown i n F i g u r e 5b. There were two major v o l t a g e f e a t u r e s , one a t ~1.9V and the o t h e r near 2.6V. The f e a t u r e s i n i t i a l l y were rounded and l a c k e d d e f i n i t i o n , but w i t h i n c r e a s i n g c y c l e number they tended t o sharpen and become more d i s t i n c t . A d e t a i l e d a n a l y s i s of the e l e c t r o c h e m i s t r y of /3-phase i s i n Chapter 5. I t i s the /3-phase t h a t has a p o t e n t i a l f o r c ommercial development. The c e l l can be made t o c y c l e hundreds of t i m e s over a l a r g e range i n x and has many o t h e r f e a t u r e s s u i t a b l e f o r the market. The t h i r d phase, 7-phase, was produced i n a 0.6V phase t r a n s i t i o n . F i g u r e 5c shows the c y c l i n g of 7-phase. I t had a l a r g e r c a p a c i t y than the o t h e r phases but c y c l e d p o o r l y . T y p i c a l l y most of the c a p a c i t y was l o s t i n l e s s than a dozen c y c l e s . The 7-phase may be the d i s p r o p o r t i o n a t e d L i x M o S 2 r e p o r t e d . The x - r a y p a t t e r n s showed broad l i n e s t h a t were c o n s i s t e n t w i t h L i 2 S f o r m a t i o n . T h i s m a t e r i a l d i d not r e a d i l y c o n v e r t back t o the o t h e r phases. D e t a i l s of the 7-phase w i l l not be d i s c u s s e d h e r e . 36 CHAPTER 3 GROWTH OF L i x M o S 2 It must not be supposed that atoms of every sort can be linked in every variety of combination. If that wer e so, you woul d see monsters coming i nt o being everywhe re. --Luc r e tius 3.1 I n i t i a l D i s c o v e r y of H i g h l y C r y s t a l l i n e L i x M o S 2 I was a t t e m p t i n g t o p r e p a r e a l i t h i a t e d C h e v r e l phase compound, L i f l M o 6 S B , i n 1982 by r e a c t i n g m a t e r i a l s a t h i g h t e m p e r a t u r e s . 1 One b a t c h produced unexpected r e s u l t s . The sample showed a c o m b i n a t i o n of bo t h f r e e molybdenum and an o t h e r m a t e r i a l t h a t had the s h a r p x - r a y p a t t e r n a s s o c i a t e d w i t h c r y s t a l l i n e powders. The p o s i t i o n s of the d i f f r a c t i o n peaks of t h i s m a t e r i a l c o r r e s p o n d e d q u i t e w e l l t o the c e n t r e s of the broad d i f f r a c t i o n peaks of /3-phase. The e l e c t r o c h e m i s t r y of c e l l s made w i t h t h i s m a t e r i a l l o o k e d v e r y s i m i l a r t o t h a t of /3-phase c e l l s e x cept t h a t the f e a t u r e s were s h a r p e r and b e t t e r d e f i n e d . Two y e a r s l a t e r I t r i e d t o r e p e a t the above experiment i n o r d e r t o produce a p u r e , c r y s t a l l i n e m a t e r i a l t h a t may a l l o w some i n s i g h t s t o the /3-phase b e h a v i o u r . T h i s c h a p t e r w i l l d e s c r i b e the p r o c e s s d e v e l o p e d t o produce c r y s t a l l i n e MoS 2. A 1 C h e v r e l phase m a t e r i a l s a r e t e r n a r y compounds of the form A x M o 6 X 8 , where A i s a m e t a l and X i s a c h a l c o g e n . 37 d i s c u s s i o n of some of the problems a s s o c i a t e d w i t h the p r o d u c t i o n of such powders i s a l s o p r e s e n t e d . Mention w i l l a l s o be made of how some v a r i a t i o n s i n the p r o c e s s a f f e c t e d the r e s u l t s . C h e m i c a l t e c h n i q u e s used t o d r i v e the system i n t o a pure phase w i l l a l s o be d i s c u s s e d . 3.2 G e n e r a l P r o c e d u r e f o r Sample P r e p a r a t i o n Most samples f o r t h i s t h e s i s were p r e p a r e d by r e a c t i n g l i t h i u m s u l f i d e , molybdenum, and s u l f u r i n s e a l e d f u s e d s i l i c a t u b e s a t e l e v a t e d t e m p e r a t u r e s . T h i s s e c t i o n d e s c r i b e s a s u c c e s s f u l t e c h n i q u e f o r the p r e p a r a t i o n of c r y s t a l l i n e L i x M o S 2 . High p u r i t y r e a c t a n t s were used t o p r e p a r e the samples. The molybdenum used was f o u r n i n e s pure powder, the s u l f u r powder was s i x n i n e s p u r e . Both were from Spex I n d u s t r i e s I n c . The l i t h i u m s u l f i d e from A l d r i c h C h emical Company I n c . was 98% pure. These f i g u r e s a r e the m a n u f a c t u r e r ' s quoted p u r i t i e s , and no t e s t i n g was done t o c o n f i r m them. The s o u r c e of 2H-MoS 2 was 3-10/um powder from Atomergic C h e m i c a l s Corp. T h i s powder was used both as a r e a c t a n t i n some ex p e r i m e n t s and d i r e c t l y as a cathode powder i n o t h e r s . A few ex p e r i m e n t s a l s o used l i t h i u m f o i l from Foote M i n e r a l Company as a r e a c t a n t . The most s u c c e s s f u l samples were made by m i x i n g l i t h i u m s u l f i d e , molybdenum, and s u l f u r i n a s t o i c h i o m e t r i c r a t i o of L i M o S 2 . The weighed a c c u r a c y of the s t o i c h i o m e t r y was p r o b a b l y good t o a t l e a s t one p e r c e n t , but no b e t t e r than 0.1%. The u l t i m a t e a c c u r a c y was l i m i t e d by p u r i t y , t h e s c a l e s used, and a 38 l o s s of m a t e r i a l i n t h e l a s t s t e p of the p r o c e d u r e . P r o d u c i n g o t h e r t r a n s i t i o n m e tal d i c h a l c o g e n i d e s , such as T i S 2 , o f t e n r e q u i r e d the a d d i t i o n of e x c e s s s u l f u r t o compensate f o r the vapor p r e s s u r e of s u l f u r over the sample, but t h i s d i d not seem t o be n e c e s s a r y h e r e . The powders were mixed t o g e t h e r and then were ground w i t h a mortar and p e s t l e . T h i s s t e p was d e s i g n e d t o o b t a i n as i n t i m a t e a m i x t u r e as was c o n v e n i e n t l y p o s s i b l e . T h i s would seem p o i n t l e s s because the r e a c t i o n took p l a c e a t a temperature where the s u l f u r would be a vapour and the l i t h i u m a l i q u i d . However, p r i o r t o t h e i n t r o d u c t i o n of t h i s s t e p the f i n a l samples o f t e n were no n - u n i f o r m . T h i s problem d i s a p p e a r e d when the i n i t i a l m a t e r i a l s were ground t o g e t h e r . The powder was then t r a n s f e r r e d i n t o a c a r r i u s t ube. These tubes were 15mm d i a m e t e r f u s e d s i l i c a c y l i n d e r s s e a l e d a t one end and necked down t o a 3mm opening near the m i d d l e of the tube. P r i o r t o use, the tubes were c l e a n e d w i t h a t r i c h l o r o e t h y l e n e r i n s e and an HF e t c h . The tubes had a volume of about 15cc when s e a l e d , and t y p i c a l l y two t o t h r e e grams of powder were used i n each b a t c h . A s m a l l amount of s u l f u r powder, and perhaps L i 2 S , would o f t e n s t i c k t o the tube above the c o n s t r i c t i o n , and t h i s imposed a l i m i t on the a c c u r a c y of the mass of m a t e r i a l used. A l l t h i s sample p r e p a r a t i o n took p l a c e under a p u r i f i e d argon atmosphere because L i 2 S i s water s e n s i t i v e . The gas i n the tube had t o be removed b e f o r e the powders were r e a c t e d . I f the gas was not removed then the p r e s s u r e would b u i l d up and p o s s i b l y e x p l o d e the t u b e . Removing th e gas a l s o reduced the 39 chance of c o n t a m i n a t i o n from r e s i d u a l r e a c t i v e gas. The tube was eva c u a t e d t o about 10" 5 t o r r . A t o r c h was used t o s e a l the tube a t the c o n s t r i c t i o n when the d e s i r e d p r e s s u r e was reached. The te m p e r a t u r e s used here were not h i g h enough t o make the f u s e d s i l i c a become p l a s t i c and c o l l a p s e under a i r p r e s s u r e . The s e a l e d tubes were then baked a t h i g h t e m p e r a t u r e . P r e s s u r e b u i l d u p from the v a p o r i z e d s u l f u r was o c c a s i o n a l l y enough t o e x p l o d e the t u b e s . T h i s was p r e v e n t e d by g i v i n g the m a t e r i a l s time t o r e a c t p a r t i a l l y b e f o r e the tube was brought up t o the f u l l t e m p e r a t u r e . T h i s c o u l d be done by e i t h e r h o l d i n g the tube a t 400°C o v e r n i g h t (44°C below the s u l f u r b o i l i n g p o i n t ) , or by r a i s i n g the temperature s l o w l y . Most samples were p r e p a r e d by r a i s i n g the temp e r a t u r e from 200°C a t a r a t e of about 100°C per hour u n t i l t he tube reached the maximum t e m p e r a t u r e . Good r e s u l t s were o b t a i n e d f o r maximum te m p e r a t u r e s between 800°C and 1000°C. The tube would remain a t t h i s t e m p e r a t u r e f o r about s i x t e e n h o u r s . The sample would then be c o o l e d a t about 100°C per hour. The tube was f i n a l l y wrapped i n tape and t r a n s f e r r e d back i n t o the g l o v e box. The tub e ' s end was then s c o r e d and broken o f f . The f i n a l r e s u l t was a ~5 t o ~15AOII b l a c k powder. Some ba t c h e s were f r e e f l o w i n g powders and o t h e r s were weakly bonded a g g r e g a t e s . The f i n a l powders were a l s o ground as a matter of p r o c e d u r e . The l a s t s t e p was i n t r o d u c e d because s e v e r a l e a r l y e x p e r i m e n t s i n d i c a t e d the q u a l i t y of the e l e c t r o c h e m i s t r y was improved by g r i n d i n g . 40 3.3 V a r i a t i o n s i n P r o c e d u r e T h e r e was a p o s s i b i l i t y t h a t e a c h and e v e r y a t t e m p t a t p r o d u c i n g L i x M o S 2 would r e s u l t i n a powder d i s t i n c t l y d i f f e r e n t f r o m a l l o t h e r s . I f t h i s were t r u e t h e n i t would be v i r t u a l l y i m p o s s i b l e t o r e p r o d u c e d a t a from one sample t o t h e n e x t . T h i s was t e s t e d by a l t e r i n g t h e p r o c e d u r e and c o m p a r i n g t h e f i n a l p r o d u c t s . I t was f o u n d t h a t t h e x - r a y and e l e c t r o c h e m i c a l b e h a v i o u r o f many o f t h e s e s a m p l e s were so s i m i l a r a s t o be a l m o s t i n d i s t i n g u i s h a b l e . The i n v e s t i g a t i o n s were c e n t r e d on t h e s e n s i t i v i t y t o s t o i c h i o m e t r y v a r i a t i o n and t h e s e n s i t i v i t y t o r e a c t i o n t e m p e r a t u r e . T h e s e two t h i n g s were t h e h a r d e s t t o c o n t r o l a c c u r a t e l y i n t h e p r o c e d u r e d e s c r i b e d i n s e c t i o n 3.2. V a r i a t i o n s i n p r o c e d u r e t h a t l e d t o s e v e r e l y d e g r a d e d s a m p l e s a r e l i s t e d i n th e n e x t s e c t i o n . The h i g h t e m p e r a t u r e grown m a t e r i a l s t h a t were s t u d i e d a r e l i s t e d i n T a b l e I. The sample name r e f e r s t o t h e l a b book and page number where t h e f u l l d a t a i s l o c a t e d . The n o m i n a l s t o i c h i o m e t r y i s l i s t e d a s w e l l a s t h e maximum t e m p e r a t u r e s e t t i n g o f t h e f u r n a c e . ' R i s e Time' was t h e i n t e r v a l between t h e t u b e b e i n g a t room t e m p e r a t u r e and t h e t i m e t h e f u r n a c e was s e t t o t h e maximum t e m p e r a t u r e . The ' C o o l Time' was t h e i n t e r v a l between t h e t e m p e r a t u r e f i r s t b e i n g l o w e r e d f r o m t h e maximum and th e t i m e t h e t u b e was removed from t h e f u r n a c e . The f u r n a c e s e t t i n g was u s u a l l y c h a n g e d i n i n c r e m e n t s o f 100°C. 41 TABLE I P r e p a r a t i o n of L i x M o S 2 Samples Sample S t o i c h i o - B a k i n g Bake R i s e C o o l metry Temp. (°C) Time Time Time ( h r s ) ( h r s ) ( h r s ) 2p1 1 2 Li«Mo 6S 8 750 15 7 6 4p1 1 7 L i M o S 2 1000 1 6 7 7 4p1 24 L i M o S 2 800 16 6 7 4p125 L i M o S 2 1000 1 6 7 7 4p134 L i M o 0 . g S 2 900 18 5 5 4p135 L i M o S , . 9 900 16 7 7 4p138 L i M o S 2 800 18 6 8 4pl39 L i M o s 2 1000 18 6 8 4p1 40 L i 0 . 8 M o S 2 1 000 17 7 8 4p1 41 L i , . 2 M o S 2 800 1 7 7 8 5p53 L i M o S 2 800 17 8 7 T h i s t a b l e i s a l i s t of some of the d i f f e r e n t s t a r t i n g c o m p o s i t i o n s and d i f f e r e n t r e a c t i o n t e m p e r a t u r e s t h a t produced c r y s t a l l i n e L i x M o S 2 . "Samples" i s a r e f e r e n c e t o the l a b book and page t h a t f u l l y d e s c r i b e the sample. The " s t o i c h i o m e t r y " l i s t e d i s the i n i t i a l s t o i c h i o m e t r y of the m a t e r i a l s i n the r e a c t i o n t u b e , not the s t o i c h i o m e t r y of the f i n a l p r o d u c t s . The samples were r a i s e d from room temperature t o the "Baking Temp." d u r i n g the " R i s e Time", l e f t t o r e a c t f o r t h e "Bake Time" and then c o o l e d back t o room te m p e r a t u r e d u r i n g the "Cool Time". The major c o n s t i t u e n t i n the f i n a l powders of a l l t h e 11 batc h e s l i s t e d above was c r y s t a l l i n e L i x M o S 2 . The f i r s t e n t r y i n the t a b l e r e f e r s t o the i n i t i a l a c c i d e n t a l d i s c o v e r y of the c r y s t a l l i n e m a t e r i a l , and t h a t b a t c h a l s o had f r e e molybdenum i n the f i n a l powder. 42 The s t o i c h i o m e t r i c s were v a r i e d by about 10% and the b a k i n g t e m p e r a t u r e s ranged from 800°C t o 1000°C. No s y s t e m a t i c v a r i a t i o n i n r e s u l t s was o b s e r v e d . Some samples had a s m a l l amount of a-phase c o n t a m i n a t i o n and some showed the presence of C h e v r e l compound, L i x M o 6 S 8 . Even b a t c h e s t h a t seemed t o be p r e p a r e d under i d e n t i c a l c o n d i t i o n s had d i f f e r e n t amounts 'of t h e s e i m p u r i t i e s i n the f i n a l m a t e r i a l s . 3.4 Problems i n Sample P r e p a r a t i o n The p r o c e d u r e o u t l i n e d i n s e c t i o n 3.2 d e s c r i b e d a s u c c e s s f u l p r o c e s s f o r making L i x M o S 2 . The p o t e n t i a l d i f f i c u l t i e s would a l s o be of i n t e r e s t i f any a t t e m p t s were made to r eproduce t h i s work. There a r e more problems i n d e t e r m i n i n g the s t o i c h i o m e t r i e s t han j u s t s u l f u r s t i c k i n g t o the w a l l s of the r e a c t i o n tube. Sample p u r i t y must a l s o be c o n s i d e r e d . The molybdenum powder may have had an Mo0 2 or Mo0 3 c o a t . I f an o x i d e were p r e s e n t , then a n o n - s t o i c h i o m e t r i c amount of molybdenum would be weighed out and t h e r e would be p o s s i b l e oxygen c o n t a m i n a t i o n d u r i n g the b a k i n g p r o c e s s . X-ray scans d i d not d e t e c t the o x i d e s , but s m a l l amounts or p o o r l y c r y s t a l l i n e m a t e r i a l s would not be d e t e c t e d i n t h i s manner. A few e a r l y e x p e r i m e n t s used molybdenum t h a t had been reduced under hydrogen at h i g h t e m p e r a t u r e . No o b v i o u s d i f f e r e n c e s were found between samples made w i t h t h e reduced Mo and t h o s e made w i t h the non-reduced m e t a l . The samples d i s c u s s e d here were a l l made w i t h non-reduced molybdenum. 43 S i m i l a r l y the l i t h i u m s u l f i d e may have been impure. I t r e a d i l y r e a c t e d w i t h water and had t o be s t o r e d under an i n e r t gas. The atmosphere was not s u f f i c i e n t l y pure t o guarantee t h e r e would be no c o n v e r s i o n t o LiOH. The presence of the h y d r o x i d e would l e a d t o i n a c c u r a t e s t o i c h i o m e t r i e s and t o i m p u r i t i e s i n the h i g h temperature r e a c t i o n . The c h o i c e of r e a g e n t s a l s o p r o v e d t o be i m p o r t a n t . Another p o s s i b l e s o u r c e of molybdenum and s u l f u r was MoS 2 powder. The a t t e m p t s t o grow c r y s t a l l i n e L i x M o S 2 u s i n g 2H-MoS 2 were u n i f o r m l y d i s a s t r o u s . A l l e x p e r i m e n t s t h a t had the a-phase as a s t a r t i n g m a t e r i a l had a-phase i n the f i n a l r e s u l t s . No t e c h n i q u e was d e v i s e d t h a t c o u l d y i e l d a pure l i t h i a t e d MoS 2 m a t e r i a l when 2H-MoS 2 was i n the tube. L i t h i u m r e a c t e d w i t h the f u s e d s i l i c a tube a t t e m p e r a t u r e s above about 400°C. T h i s put a c o n s t r a i n t on the b a k i n g p r o c e d u r e s and a f f e c t e d the u l t i m a t e c o m p o s i t i o n of t h e m a t e r i a l . The l i t h i u m - s i l i c a r e a c t i o n eroded away the tube w a l l s . Some b a t c h e s e x p l o d e d a t h i g h t e m p e r a t u r e s . T h i s may have been caused by p r e s s u r e b u i l d - u p i n tubes w i t h weakened w a l l s . The a t t a c k o c c u r r e d w i t h b oth l i t h i u m m e tal and L i 2 S r e a c t a n t s . The damage seemed t o s e t i n a t a lower temperature i f m e t a l l i c l i t h i u m was used, but t h i s was not i n v e s t i g a t e d i n d e t a i l . The tube a t t a c k a l s o consumed l i t h i u m from the L i x M o S 2 m a t e r i a l . As a r e s u l t , the f i n a l l i t h i u m c o n t e n t was always lower than i n the o r i g i n a l s t o i c h i o m e t r y . I f t h e b a k i n g were t o p e r s i s t over an extended p e r i o d , such as a week, then so much 44 l i t h i u m would be l o s t t h a t t h e r e would be more a-phase than the l i t h i a t e d m a t e r i a l . The s o l u t i o n t o the l i t h i u m - t u b e problem was t o l i m i t the l e n g t h of time the tube spent a t e l e v a t e d t e m p e r a t u r e s . F o r t u n a t e l y the d e s i r e d r e a c t i o n c o u l d s t i l l t a k e p l a c e under t h i s r e s t r i c t i o n . Another p o s s i b l e s o l u t i o n would have been t o r e p l a c e the f u s e d s i l i c a tube w i t h something e l s e , such as a g r a p h i t e c r u c i b l e . The s i n g l e b i g g e s t problem e n c o u n t e r e d i n the p r e p a r a t i o n of these samples was t h e i r a i r s e n s i t i v i t y . Even a v e r y b r i e f exposure t o m o i s t a i r was s u f f i c i e n t t o render the m a t e r i a l s u s e l e s s f o r t h e s e e x p e r i m e n t s . A l t h o u g h i t was not s p e c i f i c a l l y t e s t e d h e r e , water and not oxygen was b e l i e v e d t o cause the r e a c t i o n . In e a r l y e x p e r i m e n t s , the tubes were c r a c k e d open i n a g l o v e bag r a t h e r than a g l o v e box because i t was f e a r e d any s u l f u r v a p o r s i n the box may damage the gas p u r i f i c a t i o n system. The sample tube and t o o l s were put i n t o a g l o v e bag. The gas was p r e s s e d out of the bag and i t was then i n f l a t e d w i t h h i g h p u r i t y a r g o n . T h i s p r o c e d u r e was r e p e a t e d t h r e e t i m e s . The tubes were then opened and the powder was put i n t o sample b o t t l e s t h a t were i m m e d i a t e l y t r a n s f e r r e d i n t o the g l o v e box t o be pumped down t o <60mtorr. The t o t a l time between th e opening of the tubes t o the pumping on the sample b o t t l e s was l e s s than f i v e m i n u t e s . No sample was a b l e t o s u r v i v e the exposure t o the r e s i d u a l a i r f o r t h i s l e n g t h of t i m e . The x - r a y p a t t e r n showed both broad l i n e s when s h a r p ones were e x p e c t e d , and a l s o a s e t of low a n g l e d i f f r a c t i o n peaks assumed t o be from the h y d r a t i o n compound. The 45 gas f r o m t h e g l o v e bag s m e l l e d o f H 2 S when t h e bag was o p e n e d . I t was n o t c l e a r i f t h i s was a r e s u l t o f a r e a c t i o n o f t h e sample w i t h w ater o r i f S 0 2 o r some o t h e r gas was l i b e r a t e d when t h e t u b e was o pened. A s i m p l e t e s t of t h e a i r s e n s i t i v i t y was p e r f o r m e d on a sample t h a t had been opened i n a g l o v e box. T h i s sample showed no s i g n s o f w ater c o n t a m i n a t i o n . An x - r a y sample was p r e p a r e d w i t h a powder under a m y l a r window. The t e s t c o n s i s t e d o f r e m o v i n g t h e window and i m m e d i a t e l y making a s e r i e s of f i v e m i n u t e s c a n s and w a t c h i n g how t h e d i f f r a c t i o n p a t t e r n v a r i e d i n a r e g i o n of i n t e r e s t . The r e a c t i o n was w e l l a d v a n c e d i n f o u r m i n u t e s , and t h e d i f f r a c t i o n l i n e s o f L i x M o S 2 were gone i n l e s s t h a n e i g h t m i n u t e s . The p r o c e s s seemed t o go i n two s t a g e s w i t h t h e new l i n e s b e i n g r e p l a c e d w i t h y e t a n o t h e r new s e t a f t e r a b o u t h a l f an h o u r . No a t t e m p t was made t o make t h i s e x p e r i m e n t q u a n t i t a t i v e by m e a s u r i n g t h e h u m i d i t y and m o n i t o r i n g a s i n g l e l i n e a t a t i m e . The q u a l i t a t i v e c o n c l u s i o n s were t h a t g r e a t c a r e had t o be t a k e n t o keep t h e sample d r y , and t h a t t h e r e a c t i o n w i t h a i r had a u n i q u e x - r a y s i g n a t u r e . 46 3.5 C h e m i c a l Techniques L i x M o S 2 c o n s i s t s of s e v e r a l d i s t i n c t c r y s t a l phases. U s u a l l y the m a t e r i a l s produced by the h i g h - t e m p e r a t u r e b a k i n g p r o c e s s were a m i x t u r e of two of t h e s e phases. I t was d i f f i c u l t t o do x - r a y a n a l y s i s on samples w i t h two o v e r l a p p i n g d i f f r a c t i o n p a t t e r n s , and i t was o f t e n d i f f i c u l t t o e l i m i n a t e one phase e l e c t r o c h e m i c a l l y . The s o l u t i o n was t o use c h e m i c a l t e c h n i q u e s t o d r i v e t h e sample i n t o a pure phase. One of the c h e m i c a l s used was n - b u t y l l i t h i u m manufactured by the Foote M i n e r a l Company. E a r l i e r work i n t h i s l a b , such as t h a t by Wainwright (1978), had shown t h a t r e a c t i n g 2H-MoS 2 w i t h n - b u t y l l i t h i u m produced the 0-phase. T h i s was d e f i n i t e l y not the r e s u l t r e p o r t e d by D i n e s (1975), Whittingham (1975), and S i l b e r n a g e l (1975) who thought t h i s t o be a d i s p r o p o r t i o n a t i o n r e a c t i o n . The reason f o r t h i s d i s c r e p a n c y i s not known. A p p r o x i m a t e l y 0.1g of the 5p53 b a t c h of L i x M o S 2 was put i n t o a septum b o t t l e . About t h r e e mis of d r y c y c l o h e x a n e and a s p i n bar were added."The b o t t l e was s e a l e d and removed from t h e g l o v e box. About 3ml of c o l d n - b u t y l l i t h i u m was i n j e c t e d i n t o the b o t t l e . The sample then spent t h r e e days a t room temperature w h i l e the s p i n bar and magnetic s t i r r e r kept t h e s o l u t i o n a g i t a t e d . The m a t e r i a l was then d e c a n t e d , r i n s e d i n dry c y c l o h e x a n e , and x - r a y e d . The t r e a t m e n t had made e s s e n t i a l l y no change i n the d i f f r a c t i o n p a t t e r n . The n - b u t y l l i t h i u m was o r i g i n a l l y 1.0 m o l a r , but t h i s had p r o b a b l y changed over t i m e . To t e s t the s o l u t i o n , a second 47 experiment had been conducted s i m u l t a n e o u s l y w i t h the one d e s c r i b e d above. A s i m i l a r mass of 2H-MoS 2 was i n j e c t e d w i t h a s i m i l a r amount of the r e a g e n t . The m a t e r i a l f u l l y c o n v e r t e d t o /3-phase. Thus, t h e r e was more than enough l i t h i u m a v a i l a b l e t o change the x c o n t e n t of the c r y s t a l l i n e m a t e r i a l by a s m a l l amount. One p o s s i b l e e x p l a n a t i o n f o r the n o n - r e a c t i o n of the c r y s t a l l i n e m a t e r i a l was t h a t perhaps the sample had a l r e a d y been i n t e r c a l a t e d t o the maximum l i t h i u m c o n t e n t . T h i s seemed i n c o n s i s t e n t w i t h o t h e r d a t a . A second p o s s i b i l i t y was t h a t the r e a c t i o n was t o o slow t o be o b s e r v e d over t h r e e days. Other samples were p r e p a r e d i n a s i m i l a r way, but were l e f t f o r over a month a t 50°C. They r e a c t e d . The r e s u l t was a s i n g l e phase powder t h a t had a d i f f r a c t i o n p a t t e r n s u f f i c i e n t l y s i m p l e f o r a n a l y s i s . I t was a l s o p o s s i b l e t o produce a pure phase by a d e - l i t h i a t i o n p r o c e s s . A d r y p r o p a n o l reagent was p r o v i d e d by Dr. J e f f Dahn of M o l i Energy L i m i t e d . The r e a c t i o n i n the p resence of l i t h i u m i s 2 C 3H 7OH + 2 L i -» C 6 H , 4 + 2 LiOH (3.1) The c h e m i c a l p o t e n t i a l f o r t h i s r e a c t i o n was e q u i v a l e n t t o 2.23V i n a l i t h i u m c e l l . A 5p53 sample was p l a c e d i n a septum b o t t l e w i t h e x c e s s p r o p a n o l f o r a month. The r e s u l t was a s i n g l e phase powder w i t h c l e a n x - r a y l i n e s . 48 3.6 Another P o s s i b l e P r e p a r a t i o n Technique The s o r t of h i g h t emperature r e a c t i o n a l r e a d y d e s c r i b e d was not the o n l y way t o produce c r y s t a l l i n e MoS 2. The /3-phase can be made more c r y s t a l l i n e by a n n e a l i n g the powder. U n f o r t u n a t e l y t h i s t e c h n i q u e had most of the problems a s s o c i a t e d w i t h the h i g h t e mperature growth p l u s a few of i t s own. Large b a t c h e s of 2-3 grams of /3-phase were produced e l e c t r o c h e m i c a l l y . O f t e n not a l l the cathode powder was e l e c t r i c a l l y c o n n e c t e d , and so some MoS 2 would remain i n the a-phase. The f i r s t problem was o b t a i n i n g a h i g h y i e l d of /3-phase. The c e l l s were d i s m a n t l e d a f t e r the c o n v e r s i o n and the cathode powder was c o l l e c t e d . The m a t e r i a l was decanted and washed w i t h a c e t o n i t r i l e t o remove the e l e c t r o l y t e . R e s i d u a l s o l v e n t was removed by pumping on the sample. T h i s procedure had t h r e e d i f f i c u l t i e s a s s o c i a t e d w i t h i t : the a c e t o n i t r i l e must be dr y enough so t h e r e would be no water r e a c t i o n w i t h the l i t h i a t e d powder; e x t r a c a r e must be t a k e n when u s i n g the a c e t o n i t r i l e ( a l s o known as meth y l c y a n i d e ) ; and a s m a l l amount of r e s i d u a l e l e c t r o l y t e seemed t o remain on the powder. The washed powder was t r a n s f e r r e d and s e a l e d i n t o f u s e d s i l i c a c a r r i u s t u b e s . The next s t e p was t o determine the temp e r a t u r e and time r e q u i r e d t o a n n e a l the powder. A sample heated t o 100°C f o r one month showed a b s o l u t e l y no change i n i t s p r o p e r t i e s . Other samples were a n n e a l e d a t 200°C and 400°C e i t h e r i n c a r r i u s tubes f o r extended p e r i o d s , or under 49 argon f l o w f o r up t o e i g h t h o u r s . No improvement was o b s e r v e d i n any of th e s e i n v e s t i g a t i o n s , but not a l l of t h e s e t e s t s were performed under the d r y c o n d i t i o n s t h a t were l a t e r found t o be a b s o l u t e l y n e c e s s a r y . A n n e a l i n g was not a d e q u a t e l y t e s t e d t o i n v e s t i g a t e i t s s u i t a b i l i t y as a t e c h n i q u e f o r p r o d u c i n g c r y s t a l l i n e MoS 2 samples. 50 CHAPTER 4 EXPERIMENTAL TECHNIQUES You instruments, you mock me to my face, With wheel and gimbal, cylinder and cog; You were my key to unlock the secret place: The wards are cunning, but the levers clog. --Goet he 4.1 Cathode P r e p a r a t i o n -Most of the e x p e r i m e n t s d i s c u s s e d i n t h i s t h e s i s i n v o l v e d s t u d i e s of e l e c t r o c h e m i c a l c e l l s . T h i s c h a p t e r w i l l d e s c r i b e the p r e p a r a t i o n of the c e l l s and some of the e x p e r i m e n t a l t e c h n i q u e s . The f i r s t s e c t i o n w i l l o u t l i n e the s t e p s i n v o l v e d i n p r e p a r i n g the h o s t e l e c t r o d e . The cathode powders had t o be d e p o s i t e d onto a m e t a l s u b s t r a t e t h a t would g i v e them both p h y s i c a l s u pport and e l e c t r i c a l c o n t a c t . A v a r i e t y of t e c h n i q u e s were t e s t e d t o o p t i m i z e t h i s p r o c e d u r e . The g e n e r a l p r o c e d u r e was the same i n a l l c a s e s . A s u b s t r a t e was p r e p a r e d , powder was put on the s u b s t r a t e , a few drops of s o l v e n t were added, a s l u r r y was made, and the l i q u i d was removed t o l e a v e the powder s t u c k t o the m e t a l s u r f a c e . I d e a l l y , t h i s would have been a l l t h a t was r e q u i r e d , but many samples needed f u r t h e r work b e f o r e they were ready f o r use. An i n e r t gas atmosphere had t o be m a i n t a i n e d over most of the 51 samples a t a l l t i m e s . Two t y p e s of cathode s u b s t r a t e were commonly used. N i c k e l f o i l r o u g h l y 0.0125cm X 1.2cm X 1.2cm was used f o r most t e s t s . The n i c k e l was etched i n n i t r i c a c i d t o c l e a n and p i t the s u r f a c e . These s u b s t r a t e s were c o n v e n i e n t because they were ro b u s t and easy t o work w i t h . The o t h e r common s u b s t r a t e s were one i n c h square and 0.25mm t h i c k b e r y l l i u m s h e e t s . They were used as x - r a y c e l l windows, as d e s c r i b e d , i n the next s e c t i o n . These s u b s t r a t e s were b r i t t l e , and c o u l d o n l y be c l e a n e d w i t h a l c o h o l s because e t c h i n g might have l i b e r a t e d t o x i c m a t e r i a l . A d d i t i o n a l s u b s t r a t e s were used f o r s p e c i a l p u r p o s e s , the most common of which were q u i c k t e s t s of a powder's p r o p e r t i e s . These c a t h o d e s , which d i d n ' t r e q u i r e h i g h u t i l i z a t i o n , were pr e p a r e d by s p r i n k l i n g the powder onto e i t h e r a p i e c e of aluminum f o i l or the e l e c t r o d e of the c e l l c a s e . The c e l l was then c o n s t r u c t e d w i t h j u s t the l o o s e powder as the ca t h o d e . Aluminum f o i l s u b s t r a t e s were used when p r e p a r i n g the l a r g e 2-3 gram b a t c h e s of powder mentioned i n s e c t i o n 3.6. The a-phase MoS 2 cathodes were the e a s i e s t t o make. F i r s t the s u b s t r a t e was weighed, about 1Omg of cathode powder was p l a c e d on i t , and a c o u p l e of drops of l i q u i d were added. The bes t r e s u l t s were o b t a i n e d when the l i q u i d was p r o p y l e n e g l y c o l but i t was a l s o p o s s i b l e t o use methanol or c y c l o h e x a n e . The m i x t u r e was made i n t o a s l u r r y , and a s p a t u l a was used t o sp r e a d i t as e v e n l y as p o s s i b l e over the s u b s t r a t e . The powder then adhered t o t h e s u b s t r a t e when the l i q u i d e v a p o r a t e d . The p r o p y l e n e g l y c o l was e v a p o r a t e d by h e a t i n g i t t o 200°C under a f l o w of a r g o n , and the o t h e r s o l v e n t s e v a p o r a t e d a t room 52 temperature i n a i r . The p r o p y l e n e g l y c o l c a t h o d e s u s u a l l y r e q u i r e d no f u r t h e r p r e p a r a t i o n , but the o t h e r s worked b e t t e r i f they were r o l l e d , as e x p l a i n e d below. The f i n i s h e d cathode was weighed, and the mass of the cathode powder was d e t e r m i n e d by s u b t r a c t i n g the s u b s t r a t e w e i g h t . Cathodes made w i t h a i r - s e n s i t i v e compounds c o u l d o n l y be produced i n an i n e r t atmosphere, and t h i s l e d t o a d d i t i o n a l d i f f i c u l t i e s . The s c a l e s a v a i l a b l e f o r use i n the g l o v e box were o n l y a c c u r a t e t o 0.1mg, and the r e s t r i c t i v e w o r k i n g c o n d i t i o n s o f t e n reduced t h e a c c u r a c y t o no b e t t e r than ±0.5mg. T h i s made i t d i f f i c u l t t o d e t e r m i n e the x v a l u e of the samples. The l i q u i d i n the s l u r r y was removed from the a i r - s e n s i t i v e compounds i n one of two ways. In one, t h e samples were put i n t o a s m a l l d r y box t h a t would f i t i n t o a c o n v e n t i o n a l s t o v e . The cathodes were then baked a t 200°C as d e s c r i b e d above. T h i s p r o c e s s always y i e l d e d poor r e s u l t s , p r o b a b l y because the samples were c o n t a m i n a t e d by a i r somewhere i n the l o n g p r o c e d u r e . The o t h e r t e c h n i q u e p r o v e d t o be more p r a c t i c a l : a c y c l o h e x a n e s l u r r y was made and the s o l v e n t was removed by pumping on the c a t h o d e . The b i g g e s t s i n g l e problem i n cathode p r o d u c t i o n c o n c e r n e d cathode u t i l i z a t i o n . Each i n d i v i d u a l powder g r a i n i n the f i n i s h e d c e l l had t o be i n e l e c t r i c a l c o n t a c t w i t h the c u r r e n t - c a r r y i n g s u b s t r a t e and i n p h y s i c a l c o n t a c t w i t h the i o n - c a r r y i n g e l e c t r o l y t e . I f e i t h e r of t h e s e c o n d i t i o n s were not s a t i s f i e d then t h e g r a i n was s a i d t o be unconnected and i t d i d not p a r t i c i p a t e i n the i n t e r c a l a t i o n r e a c t i o n . The presence of 53 such unconnected powder hampered both the e l e c t r o c h e m i c a l and x - r a y a n a l y s e s . The x v a l u e c o u l d not be d e t e r m i n e d i f the weight of the a c t i v e powder was unknown. Unconnected powder a l s o l e f t i t s x - r a y p a t t e r n superimposed on the one of i n t e r e s t . Three p r o c e d u r e s t o improve t h e cathode u t i l i z a t i o n were t e s t e d . They i n v o l v e d the use of b i n d e r s t o i n c r e a s e the a d h e s i o n of the g r a i n s , the use of r o l l i n g and p r e s s i n g t o compact the powders, and the use of c o n d u c t i n g powders t o i n c r e a s e the e l e c t r i c a l c o n d u c t i v i t y of the c a t h o d e . Compaction, b r i e f l y mentioned b e f o r e , was e f f e c t i v e i n i m p r o v i n g the c e l l s . The c e l l k i n e t i c s were a f f e c t e d by the arrangement of the cathode powder. The b e s t cathode s t r u c t u r e was a t r a d e - o f f between optimum p a r t i c l e - p a r t i c l e c o n t a c t f o r e l e c t r o n t r a n s f e r , and optimum pore s i z e between the p a r t i c l e s t o a l l o w i o n d i f f u s i o n . O u t s i d e the g l o v e box the powder was compacted w i t h a p a i r of l a r g e s t e e l r o l l e r s . Two t e c h n i q u e s were used i n s i d e t h e box. The f i r s t was t o use a m o d i f i e d C-clamp arrangement t o p r e s s the powder, and the second was t o use a s m a l l s e t of b r a s s r o l l e r s . The cathode was p l a c e d on a p i e c e of s t a i n l e s s s t e e l shim s t o c k , and c o v e r e d w i t h a p i e c e of w e i g h i n g paper and a n o t h e r p i e c e of shim s t o c k . The m e t a l p r o v i d e d s u p p o r t f o r the c a t h o d e . The compressed cathode powder would u s u a l l y s t i c k t o the s u b s t r a t e and not t o t h e w e i g h i n g paper. There was some s u c c e s s w i t h t h e s e t e c h n i q u e s . C a l c u l a t e d u t i l i z a t i o n s based on the approximate cathode weight improved w i t h c o m p a c t i o n , but the equipment i n the g l o v e box d i d not a c h i e v e as h i g h a p r e s s u r e as was d e s i r e d . C o n t i n u e d work a l o n g 54 t h e s e l i n e s might have l e d t o the d e t e r m i n a t i o n of the optimum c o n d i t i o n s , but the t e c h n i q u e would have been u n s u i t a b l e f o r x- r a y s t u d i e s . The b e r y l l i u m s u b s t r a t e used i n x - r a y c e l l s would s p r e a d and perhaps c r a c k under severe c o m p r e s s i o n , and, more i m p o r t a n t l y , MoS 2 c r y s t a l s t e n d t o be s m a l l p l a t e l e t s , and when p r e s s e d t h e p l a t e l e t s t e n d t o l i n e up w i t h the c a x i s p e r p e n d i c u l a r t o the s u b s t r a t e . T h i s p r e f e r r e d o r i e n t a t i o n l e a d s t o x - r a y p a t t e r n s i n which i t i s ha r d t o d e t e c t c e r t a i n l i n e s . B i n d e r s can be used t o i n c r e a s e the a d h e s i o n of the p a r t i c l e s . The same pr o c e d u r e as d e s c r i b e d above was used i n ano t h e r s e t of e x p e r i m e n t s , except t h a t some of U n i r o y a l C h e m i c a l ' s R-512 rubber was d i s s o l v e d i n the c y c l o h e x a n e . I f l a r g e amounts of the b i n d e r were used, then the cathodes adhered q u i t e w e l l and were v e r y r o b u s t , but they a l s o had no c a p a c i t y , p r o b a b l y because the rubber c o a t e d the g r a i n s . E x p e r i m e n t s performed by R. Thompson i n t h i s l a b i n d i c a t e d an e f f e c t i v e c o n c e n t r a t i o n of b i n d e r i n the cathode was about 0.3% by w e i g h t . Only a rough e s t i m a t e of t h e rubber c o n t e n t was made f o r the e x p e r i m e n t s r e p o r t e d here because of the d i f f i c u l t y i n making a c c u r a t e measurements on the s m a l l amounts of l i q u i d s and powders used. The f i n a l rubber c o n t e n t s were <0.5% by w e i g h t . No e f f o r t s were made t o r e f i n e the p r o c e s s because a l l the r e s u l t s were poor. E x p e r i m e n t s on samples p r e p a r e d w i t h a b i n d e r , whether r o l l e d or n o t , always i n d i c a t e d t h a t the s m a l l e r the amount of b i n d e r the b e t t e r . E v e n t u a l l y the use of the b i n d e r was dropped and none of the work d e s c r i b e d i n t h i s t h e s i s made use of t h i s t e c h n i q u e . 55 The t h i r d t e c h n i q u e was t o add a c o n d u c t i n g m a t e r i a l t o the cathode powder t o improve the e l e c t r o n i c c o n t a c t between g r a i n s . 1T-MoS 2 was a l r e a d y a m e t a l , and i t was assumed t h a t t h i s would not be n e c e s s a r y , but s u r p r i s i n g l y i t proved t o be the most e f f e c t i v e way of i n c r e a s i n g the u t i l i z a t i o n . X-ray scans i n d i c a t e d t h a t the u t i l i z a t i o n of c e l l s made w i t h a c e t y l e n e b l a c k i n the cathode was near 100%. U t i l i z a t i o n s of 50 t o 75% were t y p i c a l i n c e l l s w i t h o u t t h i s c o n d u c t o r . The a c e t y l e n e b l a c k was a f i n e l y d i v i d e d , c o n d u c t i n g , amorphous form of ca r b o n . I t had no x- r a y p a t t e r n of i t s own t o obscure the p a t t e r n of i n t e r e s t , and had no c a p a c i t y of i t s own i n the v o l t a g e range i n v e s t i g a t e d . The f i r s t s t e p s i n the u l t i m a t e p r o c e d u r e were t o mix the cathode powder w i t h an a p p r o x i m a t e l y e q u a l volume of a c e t y l e n e b l a c k and then t o g r i n d the two powders t o g e t h e r . About 1Omg of the m i x t u r e was put on a s u b s t r a t e and enough d r y c y c l o h e x a n e was added t o make a s l u r r y . A s p a t u l a was used t o p a i n t the s l u r r y i n t o a f a i r l y u n i f o r m c o a t w i t h no major p r e f e r r e d o r i e n t a t i o n . The cathode was pumped t o remove the c y c l o h e x a n e . Some of the cathodes t h a t were used o n l y f o r e l e c t r o c h e m i c a l i n v e s t i g a t i o n s were r o l l e d t o i n c r e a s e the c o m p a c t i o n . 56 4.2 C e l l C o n s t r u c t i o n I t was p o s s i b l e t o p e r f o r m e x p e r i m e n t s o u t s i d e the glovebox by u s i n g a c e l l t h a t m a i n t a i n e d a s u i t a b l e environment f o r the a c t i v e c e l l components. There were two g e n e r a l t y p e s of c e l l - - a f l a n g e c e l l f o r g e n e r a l use, and a s p e c i a l i z e d x - r a y c e l l . The a c t i v e c e l l components were assembled i n the same way i n a l l c a s e s : a sandwich was made of the anode and cathode w i t h a p l a s t i c s e p a r a t o r between them. F i g u r e 7 shows a s t a i n l e s s s t e e l f l a n g e c e l l d e s i g n e d by Dr. Ross McKinnon. D i r e c t e l e c t r i c a l c o n t a c t between the two f l a n g e s was p r e v e n t e d by u s i n g i n s u l a t e d b o l t s and an O - r i n g . The O - r i n g a l s o s e a l e d i n the i n e r t atmosphere. The f l a n g e s s e r v e d b oth as e l e c t r i c a l c o n t a c t s f o r the e l e c t r o d e s and a l s o as p h y s i c a l s u p p o r t f o r the c e l l . The c e l l was assembled by p l a c i n g the cathode on one f l a n g e and a d d i n g a drop of e l e c t r o l y t e t o the powder. The cathode was c o v e r e d w i t h a s e p a r a t o r and the l i t h i u m anode was p l a c e d on t o p . The o t h e r f l a n g e was then put on and the c e l l was b o l t e d t o g e t h e r . A second v e r s i o n of t h i s c e l l had p o l y p r o p y l e n e f l a n g e s . T h i s c e l l c a s e was q u i t e s i m i l a r t o the s t a i n l e s s s t e e l case except t h a t each f l a n g e had a b r a s s b o l t t h r o u g h i t which p r o v i d e d e l e c t r i c a l c o n t a c t t o the a c t i v e c e l l components. The s e p a r a t o r was a porous p l a s t i c sheet soaked i n e l e c t r o l y t e . The s h e e t , a C e l g a r d M i c r o p o r o u s F i l m P r o d u c t s type #3501, a l l o w e d the i o n i c c o n t a c t needed between the anode and c a t h o d e and i n s u l a t e d the two e l e c t r o d e s from d i r e c t e l e c t r o n i c 57 The f l a n g e c e l l i s designed to p r o t e c t the a c t i v e c e l l components'from the e x t e r n a l environment. I o n i c c o n t a c t between the anode and cathode i s p r o v i d e d by a non-aqueous l i t h i u m s a l t e l e c t r o l y t e , but d i r e c t e l e c t r i c a l c o n t a c t i s prevented by a p l a s t i c s e p a r a t o r . The s t a i n l e s s s t e e l f l a n g e c e l l (shown) has two 3 i n c h diameter metal p l a t e s t h a t serve as e l e c t r i c a l c o n t a c t s f o r the anode and cathode. An O-ring prevents the two p l a t e s from s h o r t i n g , and s e a l s i n an i n e r t atmosphere. The b o l t s are i n s u l a t e d with h e a t - s h r i n k t u b i n g and nylon spacers. T h i s diagram i s from Von Sacken (1980). 58 c o n t a c t . The e l e c t r o l y t e was one molar L i A s F 6 d i s s o l v e d i n p r o p y l e n e c a r b o n a t e . The s a l t was from USS A g r i - C h e m i c a l s and the p r o p y l e n e c a r b o n a t e was from the J.T.Baker C h e m i c a l Company. The PC had been f u r t h e r d i s t i l l e d t o l e s s than 20ppm of water and p r o p y l e n e g l y c o l . The l i t h i u m anodes were made of 0.010 or 0.005 i n c h t h i c k l i t h i u m f o i l from the Fbote M i n e r a l Company. The m e t a l was c u t t o c o v e r c o m p l e t e l y the powder on the c a t h o d e . The l i t h i u m s u r f a c e s were s c r a p e d c l e a n of any c o a t i n g i m m e d i a t e l y b e f o r e the c e l l was assembled. The somewhat d i f f e r e n t type of c e l l needed f o r in situ x - r a y d i f f r a c t i o n i s shown i n F i g u r e 8 (Dahn e t . a l . , 1982a). I t had a b e r y l l i u m window which a l l o w e d copper K a x - r a y s t o probe the m a t e r i a l i n s i d e the c e l l . The c e l l was d e s i g n e d t o f i t on a m o d i f i e d x-ray machine goniometer ( s e c t i o n 4.5). The c e l l was assembled i n e s s e n t i a l l y the same way as the f l a n g e c e l l . One o t h e r type of f l a n g e c e l l was used when p r e p a r i n g m a t e r i a l s f o r a n n e a l i n g . I t was o n l y unique i n i t s s i z e — 1 0 c m X 25cm. 4.3 Constant C u r r e n t C y c l i n g The b a s i c i n f o r m a t i o n on v o l t a g e s , c a p a c i t y , and r e v e r s i b i l i t y of c e l l s was o b t a i n e d by c o n t i n u o u s l y c h a r g i n g and d i s c h a r g i n g the c e l l s between s e t v o l t a g e l i m i t s . The t e c h n i q u e i n v o l v e d c y c l i n g the c e l l a t a c o n s t a n t c u r r e n t and r e c o r d i n g the c e l l v o l t a g e as a f u n c t i o n of t i m e . ANGLE NYLON SCREW CELL TOP CELL VOLTAGE CONTACT CELL CATHODE ON .010" THICK BERYLLIUM. WINDOW POLYPROPYLENE GASKET EPARATQR LITHIUM ANODE, FLUID INLET  THERMISTOR  CELL VOLTAGE CONTACT FLUID OUTLET CELL BASE f i g u r e 8 X-Ray Cel1 T h i s type of c e l l c a s e was used when c o l l e c t i n g x-ray d a t a from a cathode powder. The c e l l had a b e r y l l i u m window through which the x- r a y s c o u l d pass. Most of the c e l l c a ses d i d not have the f l u i d p o r t s o r t h e r m i s t o r shown i n the diagram. The diagram i s from Dahn e t . a l . (1982). 6 0 The v o l t a g e of a c e l l i s r e l a t e d t o i t s c h e m i c a l p o t e n t i a l . Time i s d i r e c t l y p r o p o r t i o n a l t o the charge t r a n s f e r r e d i f a c e l l i s run a t c o n s t a n t c u r r e n t , so the V v s . t c y c l i n g c u r v e s a r e r e l a t e d by s c a l i n g f a c t o r s and c o n s t a n t s t o u v s . n f o r the system. The V v s . t c u r v e s a r e t h u s r e l a t e d t o the e q u a t i o n of s t a t e . The c o r r e s p o n d e n c e between a c y c l i n g c u r v e and the e q u a t i o n of s t a t e assumed t h a t the v o l t a g e measured was the p o t e n t i a l f o r the system, and the number of l i t h i u m atoms i n the host was e q u a l t o the number of e l e c t r o n s t h a t f l o w e d i n the e x t e r n a l c i r c u i t . In p r a c t i c e , the v o l t a g e and c u r r e n t f l o w c o u l d be o b s c u r e d by s e v e r a l f a c t o r s . P a r a s i t i c s i d e r e a c t i o n s c o u l d change the l i t h i u m c o n t e n t of the h o s t w i t h o u t c u r r e n t f l o w i n g e x t e r n a l l y ; f o r example, t h e r e might be a d e - l i t h i a t i o n of the cathode caused by a c h e m i c a l i m p u r i t y i n the e l e c t r o l y t e . Another p o s s i b l e s i d e r e a c t i o n was a secondary r e a c t i o n t h a t used e x t e r n a l c u r r e n t . An example would be the e l e c t r o l y t e d e c o m p o s i t i o n r e a c t i o n - - l i t h i u m b e i n g t r a n s f e r r e d t o the c a t h o d e , w i t h o u t p a r t i c i p a t i n g i n an i n t e r c a l a t i o n r e a c t i o n . The thermodynamic r e l a t i o n s assumed the system t o be a t e q u i l i b r i u m . T h i s was o b v i o u s l y not the case when c u r r e n t was f l o w i n g . The k i n e t i c e f f e c t s c o u l d r a d i c a l l y d i s t o r t the appearance of a v o l t a g e c u r v e . The r a t e a t which l i t h i u m c o u l d be i n t e r c a l a t e d was a f f e c t e d by the L i + i o n d i f f u s i o n i n the e l e c t r o l y t e and the s o l i d s t a t e d i f f u s i o n i n the h o s t . For example, i f i nadequate time was g i v e n f o r s o l i d s t a t e d i f f u s i o n , then the s u r f a c e of a p a r t i c l e i n the cathode might be a t a d i f f e r e n t l i t h i u m c o n t e n t than the c e n t r e of the p a r t i c l e . T h i s 61 would cause a p o t e n t i a l g r a d i e n t through the g r a i n , but the s u r f a c e v o l t a g e was the measured v o l t a g e . I f t h e r e was such a g r a d i e n t then the c o r r e l a t i o n between measured v o l t a g e and charge t r a n s f e r r e d would not be an a c c u r a t e r e p r e s e n t a t i o n of the e q u i l i b r i u m c o n d i t i o n s . The d i f f u s i o n problems were reduced by c y c l i n g the c e l l s a t low c u r r e n t s t o t r y and keep the cathodes i n a q u a s i - e q u i l i b r i u m s t a t e . The s i d e r e a c t i o n s were reduced by u s i n g a h i g h p u r i t y e l e c t r o l y t e . The c e l l s were c y c l e d on Model D66 C y c l e r s made by the UBC P h y s i c s department e l e c t r o n i c s shop. These machines c o u l d produce a c o n s t a n t c u r r e n t (±<0.1MA) from as low as a few microamperes up t o 100 m i l l i a m p e r e s . T y p i c a l c u r r e n t s f o r the p r e c i s i o n e l e c t r o c h e m i c a l e x p e r i m e n t s d e s c r i b e d here were about 10MA. The c e l l would c y c l e between two v o l t a g e l i m i t s . When the c e l l had d i s c h a r g e d t o the low t r i p v o l t a g e i t would a u t o m a t i c a l l y b e g i n t o charge and v i c e v e r s a . The c e l l v o l t a g e as a f u n c t i o n of time was d i s p l a y e d on a s t r i p c h a r t r e c o r d e r . 4.4 C o n s t a n t C u r r e n t Voltammetry The drawback t o c o n s t a n t c u r r e n t c y c l i n g was t h a t i t was d i f f i c u l t t o see s u b t l e e f f e c t s on the V v s . t c u r v e s . C o n stant c u r r e n t voltammetry p r o v i d e d a more s e n s i t i v e t e c h n i q u e f o r the a n a l y s i s of the e l e c t r o c h e m i s t r y . The t e c h n i q u e , a l s o c a l l e d c o n s t a n t c u r r e n t c h r o n o p o t e n t i o m e t r y , e x p l o i t e d the r e l a t i o n s h i p between a V v s . t c u r v e and the thermodynamics i t r e p r e s e n t e d . A g r e a t 62 d e a l of i n f o r m a t i o n c o u l d be o b t a i n e d by l o o k i n g a t the response of the system t o a s m a l l change. The e q u a t i o n of s t a t e of the e l e c t r o c h e m i c a l system u(n, T ) , i s analogous t o the e q u a t i o n of s t a t e f o r a gas, p ( v , T ) , where p i s the p r e s s u r e and v i s the volume. J u s t as the i s o t h e r m a l c o m p r e s s i b i l i t y of a gas can be d e f i n e d as _9y_ 9p K T = - - v - ' - | 5 - ) T (4.1) a s i m i l a r q u a n t i t y e x i s t s f o r a c e l l 1 9n \ 1 9x ) T = - T - - S * " ) T (4.2) n 9M ' t " x bu The i s o t h e r m a l c o m p r e s s i b i l i t y of a gas i s r e l a t e d t o the volume change as the p r e s s u r e i s a l t e r e d . The e q u i v a l e n t parameter f o r a c e l l i s r e l a t e d t o the change i n number of p a r t i c l e s i n a system as the c h e m i c a l p o t e n t i a l i s a l t e r e d . P r o p e r t i e s of the i s o t h e r m a l c o m p r e s s i b i l i t y of a gas, such as i t s d i v e r g e n c e a t a phase t r a n s i t i o n , a l s o h o l d f o r the e x t e n s i o n t o a c e l l . The c l o s e a n a l o g t o Kr j i t h a t w i l l be used here i s dQ/dV. T h i s r a t i o i s p r o p o r t i o n a l t o dx/dV where n 9M T ex 9V ;T V ^ . J ; and i s e s t i m a t e d from the d a t a by dQ ~ T At ( 4 4 ) dV 1 AV v*.«; where At i s the time i t w i l l t a k e t o change the c e l l v o l t a g e by AV when a c u r r e n t I i s f l o w i n g . 63 J u s t as t h e c a p a c i t y of a c o n t a i n e r i s i t s volume, v, the c a p a c i t y of a c e l l i s the amount of charge i t can h o l d , Q. The q u a n t i t y of i n t e r e s t i n t h i s t h e s i s w i l l u s u a l l y be the amount of c a p a c i t y i n some l i m i t e d v o l t a g e range r a t h e r than the t o t a l c a p a c i t y of the c e l l . The r e l a t i v e c a p a c i t y i n d i f f e r e n t v o l t a g e r e g i o n s i s u s e f u l i n the comparison of two samples. The measure of c a p a c i t y a t a v o l t a g e i s dQ/dV. A more i n t u i t i v e way of s e e i n g the use of these r e l a t i o n s h i p s i s t o c o n s i d e r how the V v s . t c u r v e s (the e q u a t i o n of s t a t e ) were r e l a t e d t o the " i n v e r s e d e r i v a t i v e " dQ/dV. F i g u r e 9a shows a h y p o t h e t i c a l example of a V v s . t c u r v e f o r a m a t e r i a l d i s c h a r g e d a t c o n s t a n t c u r r e n t . The r e g i o n marked "A" has e s s e n t i a l l y no c a p a c i t y , "B" i n d i c a t e s a r e g i o n where the p o t e n t i a l v a r i e s c o n t i n u o u s l y w i t h l i t h i u m c o n t e n t , and "C" r e p r e s e n t s a f i r s t o r d e r phase t r a n s i t i o n where two phases w i t h the same c h e m i c a l p o t e n t i a l c o - e x i s t . S i n c e the c u r r e n t was c o n s t a n t , an i n t e r v a l i n the time a x i s of F i g u r e 9a c o u l d be r e a d i l y c o n v e r t e d t o charge t r a n s f e r r e d i n the same time i n t e r v a l . I n r e g i o n "A" o n l y a s m a l l change i n l i t h i u m c o n t e n t was r e q u i r e d t o change the c e l l ' s v o l t a g e by a l a r g e amount * The q u a n t i t y AQ/AV was then s m a l l . T h i s s m a l l v a l u e i s b a r e l y v i s i b l e i n r e g i o n "A" of F i g u r e 9b. In r e g i o n "B" the c a p a c i t y s t a r t e d s m a l l , as o n l y a s m a l l AQ was needed t o change the v o l t a g e . The c a p a c i t y c o n t i n u o u s l y r o s e t o a maximum, where a l a r g e change i n l i t h i u m c o n t e n t was r e q u i r e d t o change the v o l t a g e , and then dropped back t o low c a p a c i t y . F e a t u r e "B" i n F i g u r e 9b shows the dQ/dV b e h a v i o u r i n t h i s v o l t a g e r e g i o n . R e gion "C" was a phase t r a n s i t i o n where the h o s t c o n v e r t e d from 64 Charge (coulombs) a) 0.0 0.1 0.2 OJ 0.4 0.5 0.6 0.7 2.4 2.2 2.0 18 -% 16 14 12 10 I~I—i 1 I 7 7 1 1 ; A --B — c I i i i i i } 1 1 1 1 u e 12 16 Time (hours) 20 4 8 12 16 - d Q / d V (coul/volt) 20 C) 10 12 14 16 18 2.0 2.2 2.4 Voltage (V) F i g u r e 9 C o n s t a n t C u r r e n t Voltammetry C o n s t a n t c u r r e n t voltammetry h i g h l i g h t s s u b t l e f e a t u r e s i n c y c l i n g c u r v e s . F i g u r e 9a r e p r e s e n t s a c y c l i n g c u r v e of a c e l l d i s c h a r g i n g a t 10/iA. The c u r v e shows a r e g i o n of v e r y l i t t l e c a p a c i t y ( A ) , s i n g l e phase c a p a c i t y (B), and a f i r s t o r d e r phase t r a n s i t i o n ( C ) . F i g u r e 9b i s the " i n v e r s e d e r i v a t i v e " of th e F i g u r e 9a c y c l i n g c u r v e . The dQ/dV v s . V f e a t u r e s of s i n g l e phase c a p a c i t y a r e broad. F i r s t o r d e r phase t r a n s i t i o n s appear as sh a r p s p i k e s . I t i s much e a s i e r t o d i s t i n g u i s h between t h e s e two t y p e s of c a p a c i t y by exa m i n i n g t h e voltammograms ( F i g . 9b) than by examining the c y c l i n g c u r v e s ( F i g . 9 a ) . F i g u r e 9c shows the d a t a of F i g u r e 9b o r i e n t e d i n t h e way the voltammograms w i l l be d i s p l a y e d i n t h e r e s t of the t h e s i s . The q u a n t i t y dQ/dV i s a l w a y s n e g a t i v e , but f o r i l l u s t r a t i o n p u r p o s e s t h e da t a measured on charge w i l l be shown w i t h a p o s i t i v e dQ/dV. 65 a m a t e r i a l w i t h t o t a l l i t h i u m c o n t e n t n, t o a m a t e r i a l w i t h c o n t e n t n 2. In t h e o r y , w i t h AQ = n 2 - n, over a AV of z e r o , AQ/AV s h o u l d be i n f i n i t e . In p r a c t i c e i t was l i m i t e d by two f a c t o r s . F i r s t , t he i n s t r u m e n t a t i o n was l i m i t e d t o measuring some f i x e d AV, so the r a t i o was always f i n i t e . Second, the l i t h i u m d i d not i n s t a n t a n e o u s l y d i f f u s e t h r o u g h t h e c r y s t a l , so t h e r e was a c o n c e n t r a t i o n g r a d i e n t (and a p o t e n t i a l g r a d i e n t ) t h r o u g h each p a r t i c l e i n the cathode. F e a t u r e "C" on F i g u r e 9b shows the s h a r p but f i n i t e s p i k e t h a t was ob s e r v e d a t the onset of a phase t r a n s i t i o n . I t a l s o shows the d i f f u s i o n t a i l formed as l i t h i u m c o n t i n u e d t o i n t e r c a l a t e i n t o the cathode p a r t i c l e s and c o n v e r t m a t e r i a l i n s i d e . F i g u r e 9c shows F i g u r e 9b r e p l o t t e d i n the c o n f i g u r a t i o n t h a t w i l l be used i n t h i s t h e s i s . These c u r v e s a r e a l s o c a l l e d voltammograms. F i r s t o r d e r phase t r a n s i t i o n s were e v i d e n t as sharp s p i k e s w i t h d i f f u s i o n l i m i t e d t a i l s , and s i n g l e phase c a p a c i t y had broad f e a t u r e s w i t h no d i s c o n t i n u i t i e s . The be s t dQ/dV c u r v e s were o b t a i n e d when s m a l l c u r r e n t s were used t o keep the system c l o s e t o e q u i l i b r i u m . In the l i m i t of n e g l i g i b l e c u r r e n t the f i r s t o r d e r t r a n s i t i o n s sharpened up and were d i s t i n c t l y d i f f e r e n t from s i n g l e phase c a p a c i t y f e a t u r e s . S m a l l c u r r e n t s a l s o kept the measured v o l t a g e c l o s e r t o the t r u e v o l t a g e because the IR l o s s e s i n the system were m i n i m i z e d . The d i f f u s i o n e f f e c t s a t h i g h c u r r e n t s c o u l d broaden f i r s t o r d e r t r a n s i t i o n s so much t h a t they l o o k e d l i k e s i n g l e phase c a p a c i t y . V o l t a g e drops c o u l d a l s o s h i f t t h e p o s i t i o n s of the f e a t u r e s . The c o n s t a n t c u r r e n t voltammetry t e c h n i q u e was d e s c r i b e d i n d e t a i l by Johnson (1982a). 66 The Model F06 dQ/dV A n a l y z e r d e s i g n e d by the UBC P h y s i c s department e l e c t r o n i c s shop d i d the d a t a p r o c e s s i n g t o dete r m i n e At/AV. The machine m o n i t o r e d the v o l t a g e o u t p u t of a c y c l e r . I t measured the time i t t o o k , A t , f o r the c e l l v o l t a g e t o change by some AV of a t l e a s t 500MV. The r e s o l u t i o n of the machine was 45MV. M u l t i p l e r e a d i n g s were av e r a g e d t o reduce the e f f e c t of n o i s e and minor f l u c t u a t i o n s i n the system. The i n t e r n a l s o f t w a r e then computed the r a t i o At/AV and c o n v e r t e d i t i n t o an a n a l o g o u t p u t . The dQ/dV v s . V graphs p r e s e n t e d here have been d i g i t i z e d from the a n a l o g r e c o r d i n g s . 4.5 X-Ray D i f f r a c t i o n X-ray d i f f r a c t i o n was used t o study the corres p o n d e n c e between the e l e c t r o c h e m i s t r y and t h e s t r u c t u r a l changes i n the h o s t . These e x p e r i m e n t s were performed on a P h i l i p s PW 1730 x - r a y powder d i f f T a c t o m e t e r w i t h a PW 1386/50 a u t o m a t i c d i v e r g e n c e s l i t . The s l i t was used t o keep the same a r e a of the sample i r r a d i a t e d a t a l l d i f f r a c t i o n a n g l e s . T h i s reduced the background a t low a n g l e s and i n c r e a s e d the i n t e n s i t y a t h i g h a n g l e s . The PW 1050/80 goniometer s h a f t had been m o d i f i e d t o a l l o w the x- r a y c e l l s d e s c r i b e d i n s e c t i o n 4.2 t o be mounted i n the beam. I t was p o s s i b l e t o c y c l e the c e l l or c o n t r o l i t s v o l t a g e w h i l e s t r u c t u r a l measurements were b e i n g made. The r a d i a t i o n , copper K a x - r a y s , had a main w a v e l e n g t h , K t t i, of 1.54056A, but a t low a n g l e s (<30°) the K f l i and the K a 2 l i n e s c o u l d not be r e s o l v e d and a we i g h t e d mean wa v e l e n g t h , 1.54184A, was used. 67 The d e v i c e had two forms of o u t p u t . There was a s t r i p c h a r t a n a l o g o u t p u t s u i t a b l e f o r f a s t s u r v e y s of broad r e g i o n s , and a l s o a d i g i t a l o u tput s u i t a b l e f o r the p r e c i s e measurements d e s c r i b e d i n Chapter 6. Appendix I c o n t a i n s both a more complete d e s c r i p t i o n of x - r a y d i f f r a c t i o n , and a d i s c u s s i o n of the a n a l y s i s t e c h n i q u e s d e v i s e d f o r u n d e r s t a n d i n g L i x M o S 2 . 4.6 Other T e c h n i q u e s and D e v i c e s At t i m e s i t was n e c e s s a r y t o h o l d a c e l l a t some p a r t i c u l a r v o l t a g e . T h i s was done w i t h a PAR 173 p o t e n t i o s t a t / g a l v a n o s t a t . The d e v i c e had a PAR 179 d i g i t a l c o u l o m e t e r module so i t was p o s s i b l e t o r e c o r d the amount of charge t r a n s f e r r e d as the c e l l v o l t a g e was stepped t o a new v a l u e . A c e l l was c o n s i d e r e d t o be at e q u i l i b r i u m i f the c u r r e n t f l o w dropped t o the p a r a s i t i c s i d e r e a c t i o n l e v e l of a few t e n t h s of a microampere. The i n e r t atmosphere c o n s i s t e d of h i g h p u r i t y argon i n a Vacuum Atmospheres g l o v e box. A p u r i f i c a t i o n system of m o l e c u l a r s i e v e s t o a b s o r b water and copper monoxide t o r e a c t w i t h oxygen was used t o m a i n t a i n the q u a l i t y of the atmosphere. Haake F3 or N e s t l a b RTE 9ED temperature b a t h s were used t o m a i n t a i n s t a b l e t e m p e r a t u r e s i n many dQ/dV e x p e r i m e n t s and i n a few c h e m i c a l p r o c e s s e s . 68 CHAPTER 5 ELECTROCHEMICAL INVESTIGATIONS Concerning the objects we propose to study, we should investigate not what others have thought nor what we ourselves conjecture, but what we can intuit clearly and evidently or deduce with certainty, for there is no other way to acquire knowledge. --Rene Desartes 5.1 Overview The o r i g i n a l m o t i v a t i o n f o r s t u d y i n g the h i g h temperature grown L i x M o S 2 was t o use the r e s u l t s t o u n d e r s t a n d /3-phase. The /3-phase e l e c t r o c h e m i s t r y had not been s u c c e s s f u l l y modeled, and both the e l e c t r o c h e m i c a l b e h a v i o u r and the s t a b i l i t y a g a i n s t c o n v e r s i o n t o a-phase changed as the c e l l s were c y c l e d . E x p l a n a t i o n s of t h e s e p r o p e r t i e s were of i n t e r e s t t o both pure and a p p l i e d r e s e a r c h , and the h i g h temperature grown m a t e r i a l 1 seemed t o p r o v i d e a way t o u n d e r s t a n d them. I t i s i m p o r t a n t t o note t h a t the h i g h t e m p e r a t u r e grown m a t e r i a l i s not the same as /3-phase. The former i s the c r y s t a l l i n e m a t e r i a l produced by the p r o c e d u r e d e s c r i b e d i n 1 I t w i l l be n e c e s s a r y t o d i f f e r e n t i a t e between samples of /3-phase and samples grown a t h i g h t e m p e r a t u r e . The new m a t e r i a l w i l l commonly be r e f e r r e d t o as c r y s t a l l i n e L i x M o S 2 , or as the h i g h t e m p e r a t u r e m a t e r i a l . 69 Chapter 3 , and the l a t t e r , 0 -phase, i s by d e f i n i t i o n the m a t e r i a l produced by the e l e c t r o c h e m i c a l c o n v e r s i o n of 2H-MoS 2. The two m a t e r i a l s each have t h e i r own d i s t i n c t , though s i m i l a r , p r o p e r t i e s . In many ways i t was easy t o t h i n k of the h i g h t e m p e r a t u r e m a t e r i a l as a h i g h l y c r y s t a l l i n e form of 0-phase. E v e n t u a l l y I w i l l show t h a t the h i g h t e m p e r a t u r e grown L i x M o S 2 m a t e r i a l i s an i n t e r c a l a t i o n compound w i t h r o u g h l y a 1T c r y s t a l symmetry. T h i s d e s c r i p t i o n a l s o h o l d s f o r the 0-phase. Even the c e n t r e s of the v o l t a g e f e a t u r e s f o r the two m a t e r i a l s a r e almost i d e n t i c a l . The major d i f f e r e n c e s between the two m a t e r i a l s were i n the c r y s t a l l i n i t y of the samples and t h e s h arpness of the e l e c t r o c h e m i c a l f e a t u r e s . There was t h e r e f o r e r e ason t o i n f e r t h a t the two m a t e r i a l s were d i f f e r e n t forms of the same t h i n g , but t h i s was not proven. T h i s c h a p t e r w i l l d e s c r i b e the c y c l i n g , v oltammetry and s t a b i l i t y f o r b oth the 0-phase and the c r y s t a l l i n e L i x M o S 2 . The s i m i l a r i t i e s and d i f f e r e n c e s between t h e s e m a t e r i a l s w i l l be d i s c u s s e d , and i n d o i n g so, the e l e c t r o c h e m i s t r y f o r the c r y s t a l l i n e L i x M o S 2 w i l l be d e s c r i b e d f o r the f i r s t t i m e . 70 5.2 C y c l i n g B e h a v i o u r The e a s i e s t way t o i n v e s t i g a t e the e l e c t r o c h e m i c a l p r o p e r t i e s of a c e l l i s t o c y c l e i t a t c o n s t a n t c u r r e n t . The major f e a t u r e s a r e apparent and comparisons between d i f f e r e n t samples are easy t o make. T h i s s e c t i o n w i l l d e s c r i b e the c y c l i n g b e h a v i o u r of both the /3-phase and the h i g h t e m p e r a t u r e m a t e r i a l . The next s e c t i o n w i l l d e s c r i b e t h e p r e c i s i o n e l e c t r o c h e m i c a l measurements. The /3-phase c y c l i n g was mentioned b r i e f l y a t the end of Chapter 2. F i g u r e 6 showed the a t o /3 phase t r a n s i t i o n and a /3 c y c l e , and F i g u r e 5b showed t y p i c a l /3-phase c y c l e s . Each f i g u r e showed two major c a p a c i t y f e a t u r e s , one a t ""1.9V and the o t h e r at ~2.6V, The shapes of the two /3-phase c y c l i n g c u r v e s were d i f f e r e n t , as w i l l be seen more c l e a r l y i n the f i g u r e s of t h i s c h a p t e r . F i g u r e 10 shows the f i f t h c y c l e of a c e l l i n /3-phase. The 2.6V and 1.9V f e a t u r e s a r e rounded and la.ck d e f i n i t i o n . The c e l l was c y c l e d a t 100MA, and each h a l f c y c l e took about ten h o u r s . T h i s was a f a i r l y f a s t c y c l e t i m e , but i f a lower c u r r e n t were used, the m a t e r i a l c o u l d have c o n v e r t e d t o a-phase a t h i g h v o l t a g e . The a x i s of the graph was s c a l e d t o r e a d x=0 a t the h i g h v o l t a g e t r i p and x=1 a t the low v o l t a g e t r i p . In f a c t the Ax v a l u e would have been l e s s than one because not a l l the l i t h i u m had been removed from t h e cathode a t h i g h v o l t a g e s . I f the c e l l had been f u l l y c h a r g e d i t would have c o n v e r t e d t o 71 F i g u r e 10 0-Phase C y c l e 5 C y c l i n g c u r v e measured on t h e f i f t h c y c l e of c e l l PM-41. c e l l was c y c l e d a t 100>A. The i n i t i a l c athode m a t e r i a l w The was A t o m e r g i c 2H-MoS 2. 72 a-phase. In a d d i t i o n , the c y c l e was not slow enough t o guarantee a q u a s i - e q u i l i b r i u m s t a t e . The e l e c t r o c h e m i c a l b e h a v i o u r of /3-phase c e l l s a l t e r e d as they were c y c l e d . F i g u r e 11 shows the I80 t* 1 c y c l e of the same c e l l t h a t was shown i n F i g u r e 10. The c y c l i n g c u r v e s became much f l a t t e r and s h a r p e r . S u b t l e f e a t u r e s were a l s o v i s i b l e which were not e a s i l y seen i n e a r l y c y c l e s . The m a t e r i a l a l s o became more s t a b l e a g a i n s t c o n v e r s i o n t o a-phase. The d a t a of F i g u r e 11 was measured at 20MA and no a p p r e c i a b l e c o n v e r s i o n was o b s e r v e d . T h i s s t a b i l i t y w i l l be d i s c u s s e d i n more d e t a i l l a t e r on. A s i m p l e t h e o r y t o e x p l a i n how c y c l i n g the /3-phase can a l t e r i t s p r o p e r t i e s w i l l be d i s c u s s e d a t t h e end of the s e c t i o n . The h i g h temperature grown m a t e r i a l had a s i m i l a r but d i s t i n c t l y d i f f e r e n t c y c l i n g b e h a v i o u r . F i g u r e 12 shows an e a r l y c y c l e of t h i s m a t e r i a l . The f i r s t , and most i m p o r t a n t , o b s e r v a t i o n i s t h a t the m a t e r i a l d i d a c t as a c e l l c athode. I t was e n t i r e l y p o s s i b l e t h a t the m a t e r i a l might have had no c a p a c i t y a t a l l . The v o l t a g e c u r v e s were v e r y f l a t and w e l l d e f i n e d , and many f e a t u r e s l o o k e d l i k e the s i g n a t u r e s of f i r s t o r d e r phase t r a n s i t i o n s . The m a t e r i a l appeared t o be q u i t e s t a b l e a g a i n s t c o n v e r s i o n a t h i g h v o l t a g e s . I t was a l s o o b v i o u s t h a t the charge and d i s c h a r g e f e a t u r e s were not symmetric, i n d i c a t i n g some s o r t of i r r e v e r s i b i l i t y i n the system. There was a l s o an i m p u r i t y i n the sample. The s m a l l f e a t u r e near 2.1V was a s s o c i a t e d w i t h i n t e r c a l a t i o n i n M o 6 S 8 . 3.0 0.5 0.0 0.0 0.5 1.0 .5 .0 F i g u r e 11 0-Phase C y c l e 180 C y c l i n g c u r v e measured on t h e 1 8 0 t h c y c l e of PM-41 The c e l l c y c l e d a t 20MA. 74 F i g u r e 12 C y c l i n g of 4 P138 L i x M o S 2 SK 1cSIiSS rI? T o " " r e d o n t h e s e c o n d c y c l e o f P M " 2 8 0 - T h e c e l 1 75 The s t r o n g s i m i l a r i t y between the /3-phase and the h i g h t e m p e r a t u r e m a t e r i a l i s e v i d e n t i n F i g u r e 13, which shows the t h r e e p r e v i o u s f i g u r e s o v e r l a i d . Note t h a t the d i f f e r e n t samples were run w i t h d i f f e r e n t c u r r e n t s and t r i p v o l t a g e s . T h i s a l l o w e d some d i s t i n c t i o n between the t h r e e c u r v e s f o r d i s p l a y p u r p o s e s . The s i m i l a r i t i e s i n the t h r e e samples a r e s t r i k i n g . The v o l t a g e f e a t u r e s were c e n t e r e d a t about 1.9V and 2.6V. The c a p a c i t y near each of t h e s e v o l t a g e s was about the same i n each sample. In g e n e r a l , t h e r e i s a s t r o n g c o r r e s p o n d e n c e between the /3-phase and the h i g h t emperature m a t e r i a l . The major d i f f e r e n c e between the two m a t e r i a l s was seen t o be i n the sharpness of the f e a t u r e s . The h i g h t e m p e r a t u r e m a t e r i a l had s h a r p , d i s t i n c t f e a t u r e s , but the /3-phase f e a t u r e s were broad and i n d i s t i n c t . However, the sharpness of the /3-phase f e a t u r e s was improved by c y c l i n g the c e l l . The /3 m a t e r i a l which had been c y c l e d many tim e s seemed t o be i n t e r m e d i a t e between the e a r l y c y c l e s of /3-phase and the h i g h temperature m a t e r i a l . The s i m i l a r i t i e s between t h e s e m a t e r i a l s may be r e l a t e d t o the c r y s t a l l i n i t y of the samples. The e l e c t r o c h e m i c a l p r o d u c t i o n of l i t h i a t e d MoS 2 can l e a d t o i m p e r f e c t c r y s t a l s . The a t o /3 phase t r a n s i t i o n r e q u i r e d a major r e s t r u c t u r i n g of the host l a t t i c e . The e x a c t mechanism of the change i s not known, but i t must i n v o l v e the r e l a t i v e s h i f t i n g of most of the atoms i n the c r y s t a l . The e l e c t r o c h e m i s t r y i s s e n s i t i v e t o the l o c a l environment of the l i t h i u m . The h i g h l y d i s o r d e r e d /3-phase, i n i t i a l l y s t r a i n e d and f u l l of d e f e c t s , would have a 76 3.0 2.5 2.0 u> 1.5 o o > 1.0 0.5 0.0 I 1 1 1 1 i i - w f~-- N — \ V \ < \ — 1 1 1 1 i 1 1 0.0 0.5 1.0 .5 .0 F i g u r e 13 Comparison of 0-Phase and L i x M o S 2 C y c l i n g T h i s i s an o v e r l a y of the pre c e e d i n g three f i g u r e s . 4p138 L i x M o S 2 i s the s o l i d l i n e . 0-phase (5 c y c l e s ) i s the ( — ) l i n e . 0-phase (180 c y c l e s ) i s the ( ) l i n e . The 0-phase c y c l i n g curves f l a t t e n as the c e l l s are c y c l e d , but the 4p138 sample i s much f l a t t e r than the 0-phase. 77 d i s t r i b u t i o n of s l i g h t l y d i f f e r e n t s i t e s f o r the l i t h i u m > and the e l e c t r o c h e m i s t r y would show broad f e a t u r e s c o r r e s p o n d i n g t o the v a r i e t y of s i t e s . Many of the d e f e c t s i n the /3-phase can be removed by c y c l i n g . A c r y s t a l w i t h d e f e c t s may be i n a m e t a s t a b l e s t a t e , but t h e s e d i s t o r t i o n s can be removed i f t h e r e i s a mechanism f o r the c r y s t a l t o r e l a x i n t o a lower energy c o n f i g u r a t i o n . Changing the l i t h i u m c o n t e n t of a h o s t makes i t s l a y e r s expand or c o n t r a c t , and t h i s movement of atoms may a l l o w the c r y s t a l t o s e t t l e i n t o a more f a v o u r a b l e s t a t e . The c y c l i n g of l i t h i u m may t h u s put the h o s t i n t o a more c r y s t a l l i n e form. The f e a t u r e s i n the e l e c t r o c h e m i s t r y of /3-phase c e l l s may have sharpened w i t h c y c l e number because the m a t e r i a l became more c r y s t a l l i n e . However, c y c l i n g d i d not remove a l l the d e f e c t s , so t h e r e was a l i m i t t o how s h a r p the f e a t u r e s c o u l d become. The i n c r e a s e d c r y s t a l l i n i t y a l s o e x p l a i n s why the s t a b i l i t y of /3-phase changed. A phase t r a n s i t i o n w i l l u s u a l l y b e g i n by heterogeneous n u c l e a t i o n a t a d e f e c t r a t h e r than by homogeneous n u c l e a t i o n i n the b u l k . The g r e a t e r t h e number of d e f e c t s , the e a s i e r i t i s f o r the t r a n s i t i o n t o s t a r t . The /3-phase may have become more s t a b l e because the c y c l i n g removed many of t h e d e f e c t s a t which the a-phase c o u l d i n i t i a l l y form. 2 The argument above can be f u r t h e r e x t r a p o l a t e d t o . t h e l i m i t of a v e r y c r y s t a l l i n e m a t e r i a l . Such a m a t e r i a l would have no 2 Appendix IV has a b r i e f d e s c r i p t i o n of n u c l e a t i o n . 78 d e f e c t i n d u c e d b r o a d e n i n g of the e l e c t r o c h e m i c a l f e a t u r e s and i t would be q u i t e s t a b l e a t h i g h v o l t a g e s . These a r e the p r o p e r t i e s of the h i g h t e m p e r a t u r e grown m a t e r i a l . I t must be s t r e s s e d t h a t t h e r e was o n l y a cor r e s p o n d e n c e between the b e h a v i o u r of /3-phase and the b e h a v i o u r of the h i g h t e mperature m a t e r i a l . There was no p r o o f t h a t the c r y s t a l l i n e L i x M o S 2 s t u d i e d was a c r y s t a l l i n e form of /3-phase. I t i s p o s s i b l e t h a t the most s t a b l e L i x M o S 2 s t r u c t u r e a t e l e v a t e d t e m p e r a t u r e s may not be the most s t a b l e s t r u c t u r e a t room t e m p e r a t u r e . The h i g h t e m p e r a t u r e grown m a t e r i a l may be i n a l o n g - l i v e d m e t a s t a b l e s t a t e a t room t e m p e r a t u r e . I t i s p o s s i b l e t h a t the d e f e c t f r e e form of /3-phase i s d i f f e r e n t than the h i g h l y c r y s t a l l i n e form of L i x M o S 2 d e s c r i b e d h e r e . 5.3 Voltammetry I n v e s t i g a t i o n s C y c l i n g was u s e f u l t o d e t e r m i n e the most g e n e r a l a s p e c t s of the e l e c t r o c h e m i c a l b e h a v i o u r of the m a t e r i a l s , but d e t a i l e d i n v e s t i g a t i o n s r e q u i r e d a more p r e c i s e method of e x a m i n a t i o n . T h i s s e c t i o n w i l l d e s c r i b e the use of c o n s t a n t c u r r e n t voltammetry t o i n v e s t i g a t e the e l e c t r o c h e m i s t r y of /3-phase and the h i g h t e m p e r a t u r e m a t e r i a l . The s h a r p e n i n g of /3-phase f e a t u r e s w i t h c y c l e number was c l e a r l y e v i d e n t , and the h i g h t e mperature m a t e r i a l proved t o have v e r y d i s t i n c t i v e voltammetry f e a t u r e s . P r e d i c t i o n s of the s o u r c e of the c a p a c i t y were made based on t h e s e o b s e r v a t i o n s . The corr e s p o n d e n c e between p a r t i c u l a r f e a t u r e s was s t u d i e d and an i r r e v e r s i b i l i t y was 79 n o t e d . Some p a r t i a l c o n f i r m a t i o n of the i d e a s put f o r w a r d a t the end of the p r e v i o u s s e c t i o n i s a l s o p r e s e n t e d . The voltammetry f e a t u r e s of /3-phase were u s u a l l y q u i t e b r o a d . F i g u r e 14 shows a f a i r l y t y p i c a l /3-phase voltammogram measured on the 1 1 6 t h and 1 1 7 t h c y c l e s of c e l l PM-289. The 1.9V f e a t u r e can be r e s o l v e d i n t o two s e p a r a t e p a r t s . The charge h a l f - c y c l e ( t o p of the graph) shows two d i s t i n c t peaks, one a t 1.89V and the o t h e r a t 1.94V. There appear t o be two c o r r e s p o n d i n g f e a t u r e s on the d i s c h a r g e , one a t 1.82V and the o t h e r a t about 1.87V, but thes e were not c l e a r l y r e s o l v e d . I f a c e l l was p e r f e c t l y r e v e r s i b l e , then the s e t of peaks on charge would l o o k a l m o s t i d e n t i c a l t o the f e a t u r e s on d i s c h a r g e . These f e a t u r e s were not symmetric on charge and d i s c h a r g e even i f week lo n g c y c l e s were used t o m i n i m i z e the d i f f u s i o n g r a d i e n t s . T h i s can be be s t e x p l a i n e d by the presence of some h y s t e r e s i s as the m a t e r i a l was c y c l e d . The major c a p a c i t y near 2.6V was a l s o r e s o l v e d i n t o two broad f e a t u r e s . There were v e r y broad f e a t u r e s a t ~2.53V on charge and ~2.49V on d i s c h a r g e , and s h a r p e r f e a t u r e s a t 2.67V on charge and 2.63V on d i s c h a r g e . Again t h e r e was a l a c k of symmetry between the s e f e a t u r e s . A l s o note the s m a l l r i s e i n c a p a c i t y near 1.45V on d i s c h a r g e . T h i s c e l l had spent about 20 hours above 2.6V w h i l e the voltammogram was b e i n g r e c o r d e d . Some of the /3-phase had c o n v e r t e d t o a-phase d u r i n g t h i s t i m e , and the low v o l t a g e f e a t u r e a t 1.4V was the p a r t i a l r e c o n v e r s i o n t o /3-phase. Some of the c a p a c i t y measured above ~2.6V i s l i n k e d t o the /3 t o a c o n v e r s i o n . I n g e n e r a l t h i s 8 0 I I I I I I I I I I I I I I I I I I I i ' i ' ' i ' ' I 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 Voltage (V) F i g u r e 14 dQ/dV of 0-Phase from 1.4V t o 2.8V C o n s t a n t c u r r e n t voltammogram measured on t h e 11 6 t h c y c l e of PM-289. The c u r r e n t was 5MA. The o r i g i n a l cathode m a t e r i a l was A t o m e r g i c 2H-MoS 3. The c e l l was c o n v e r t e d t o 0-phase on t h e f i r s t c y c l e . 81 voltammogram shows a m a t e r i a l w i t h : broad f e a t u r e s t h a t l o o k e d l i k e s i n g l e phase c a p a c i t y ; some h y s t e r e s i s ; and some phase c o n v e r s i o n . No i n t e r v a l of v o l t a g e had z e r o c a p a c i t y . I t o r i g i n a l l y p r o v e d t o be q u i t e d i f f i c u l t t o u n d e r s t a n d t h i s d a t a . The /3-phase b e h a v i o u r a l s o v a r i e d as a f u n c t i o n of c y c l e number. F i g u r e s 1 5 t o 18 show a s e r i e s of voltammograms of a c e l l c y c l i n g i n /3-phase. The c e l l , which had u s u a l l y been c y c l e d between 2 . 7 V and 0.8V at a t y p i c a l c u r r e n t of 2 0 0 M A , was c y c l e d a t 10MA i n a r e s t r i c t e d v o l t a g e range ( ~ 1 . 3 V t o ~ 2 . 3 V ) d u r i n g the p r e c i s i o n measurements. T h i s reduced the d i f f u s i o n g r a d i e n t s i n the h o s t which c o u l d broaden the f e a t u r e s , and a v o i d e d /3 t o a c o n v e r s i o n d u r i n g the slow c y c l e s . F i g u r e 1 5 shows the t h i r d c y c l e of a /3-phase c e l l . The v o l t a g e f e a t u r e s were q u i t e b r o ad, and they a l l l o o k s i m i l a r t o what c o u l d be e x p e c t e d from s i n g l e phase c a p a c i t y . F i g u r e 1 6 shows the 1 0 1 S T c y c l e of the same c e l l . A l l the f e a t u r e s have become much s h a r p e r . The f e a t u r e which o c c u r r e d a t 1 . 8 2 V on the 3 R D d i s c h a r g e appears a t 1 . 8 6 V on the 1 0 1 S T and b e g i n s t o l o o k l i k e a f i r s t o r d e r phase t r a n s i t i o n . A l s o , the c a p a c i t y between 1.3V and 1.7V and the c a p a c i t y between 2 . 1 V and 2 . 3 V on the 1 0 1 S T c y c l e was g r e a t l y reduced compared t o the 3 R D c y c l e . F i g u r e 17 shows the 2 0 2 N D c y c l e of the c e l l and F i g u r e 18 shows the 3 0 3 R D c y c l e . Both show improved s h a r p n e s s of the f e a t u r e s as compared t o the 1 0 1 S T c y c l e . However the 2 0 2 N D and 3 0 3 R D c y c l e s a r e not s i g n i f i c a n t l y d i f f e r e n t . There appeared t o i— i—i—i—i—i—i—i—i— i—r • i i i i i i i 1 i J i i 1.3 1.5 1.7 1.9 2.1 2.3 2.5 Voltage (V) F i g u r e 15 dQ/dV of 0-Phase C y c l e 3 Constant c u r r e n t voltammogram of PM-212 measured at 10MA. The i n i t i a l cathode m a t e r i a l was Atomergic 2H-MoS2. 83 i — i — i — i — i — i — i — i — r Voltage (V) F i g u r e 16 <3Q/dV o f 0-Phase C y c l e 101 C o n s t a n t c u r r e n t voltammogram o f PM-212 measured a t 10MA. 84 F i g u r e 17 dQ/dV of 0-Phase C y c l e 202 C o n s t a n t c u r r e n t voltammogram of PM-212 measured a t 10*iA. 85 i — i — i — i — i — i — i — i — r Voltage (V) F i g u r e 18 dQ/dV of 0-Phase C y c l e 303 Constant c u r r e n t voltammogram of PM-212 measured a t 10*/A. 86 be a l i m i t t o how much the e l e c t r o c h e m i s t r y c o u l d be changed by c y c l i n g . These f i g u r e s each show t h r e e s h a r p f e a t u r e s t h a t a re p r o b a b l y due t o f i r s t o r d e r phase t r a n s i t i o n s . The f o u r t h f e a t u r e , 1.89V on d i s c h a r g e , never sharpened up. There was a l s o v e r y l i t t l e c a p a c i t y below 1.7V or above 2.1V. Some c a p a c i t y was l o s t as the c e l l was c y c l e d , so a q u a n t i t a t i v e comparison of F i g u r e s 15 t h r o u g h 18 would have t o i n c o r p o r a t e s c a l i n g f a c t o r s . The c e l l had 75% of i t s c a p a c i t y r e m a i n i n g on the I 0 5 t n c y c l e , 55% on the 2 0 5 t n , and about 50% on the 3 0 1 s t . The q u a l i t a t i v e comparison d e s c r i b e d above i s v a l i d , however. The major o b s e r v a t i o n s were r e l a t e d t o the s h a r p e n i n g of f e a t u r e s . The l o s s of c a p a c i t y would t e n d t o broaden the f e a t u r e s because t h e r e was l e s s time f o r d i f f u s i o n i n each c y c l e . Another t e s t had t o be made t o det e r m i n e whether the s h a r p e n i n g of the f e a t u r e s was r e l a t e d t o c y c l i n g or t o the time i n t e r v a l between /^-conversion and measurement. I t u s u a l l y took two or t h r e e months t o c y c l e a c e l l a few hundred t i m e s . I f the /3-phase were m e t a s t a b l e , i t s h o u l d r e l a x w i t h some time c o n s t a n t i n t o a lower energy form. S e t s of e x p e r i m e n t s were performed t o f i n d out i f /3-phase e l e c t r o c h e m i s t r y sharpened w i t h o u t c y c l i n g . Voltammograms were made of one of the f i r s t c y c l e s a f t e r c e l l s were c o n v e r t e d t o /3-phase. The c e l l s were l e f t 'open c i r c u i t ' f o r f i v e months and then a n o t h e r voltammogram was made. There was no s h a r p e n i n g of the /3-phase f e a t u r e s . 87 These e x a m i n a t i o n s of the voltammetry of /3-phase have shown s e v e r a l t h i n g s . T h e - e l e c t r o c h e m i s t r y of the c e l l s changed w i t h c y c l e number. The f e a t u r e s a c t u a l l y t e n d t o sharpen t o f i r s t o r d e r phase t r a n s i t i o n s , so the term "/3-phase" may be i n a p p r o p r i a t e because t h e r e a r e a c t u a l l y s e v e r a l /3-phases. There seem t o be l i m i t s t o the amount of f e a t u r e s h a r p e n i n g t h a t can be a c h i e v e d by c y c l i n g . These o b s e r v a t i o n s a r e i n k e e p i n g w i t h the e f f e c t s of i n c r e a s e d c r y s t a l l i n i t y as d e s c r i b e d a t the end of s e c t i o n 5.2. The v oltammetry of the h i g h temperature grown m a t e r i a l was much sharper than the s h a r p e s t of the 0-phase d a t a . F i g u r e 19 shows a t y p i c a l voltammogram of a h i g h temperature grown m a t e r i a l . E s s e n t i a l l y a l l t h e c a p a c i t y was c o n t a i n e d i n a s e t of f i r s t o r d e r phase t r a n s i t i o n s and a r e g i o n of s i n g l e phase c a p a c i t y . The t r a n s i t i o n s on the charge h a l f - c y c l e o c c u r r e d a t 1.89V, 1.94V, 2.58V, and 2.68V. The d i s c h a r g e peaks o c c u r r e d a t 2.64V, 2.58V, 1.89V, and 1.85V. There was a l s o a broad f e a t u r e c e n t r e d at ~2.53V t h a t seemed t o be s i n g l e phase c a p a c i t y . These p o t e n t i a l s c o r r e s p o n d e d t o t h e c e n t r e s of t h e major c a p a c i t y f e a t u r e s of /3-phase. A p a r t from t h e s e f e a t u r e s t h e r e was no o t h e r L i x M o S 2 c a p a c i t y between 1.4V and 2.8V. The f i r s t o r d e r phase t r a n s i t i o n s i m p l i e d t h a t the h i g h temperature grown L i x M o S 2 c o n s i s t e d of a v a r i e t y of s e p a r a t e phases. The l e t t e r s on F i g u r e 19 c o r r e s p o n d t o a l a b e l l i n g 88 o > ° 1 o "D -1 T i i i—i—i—i—i—i—i—i—r B D E F T J I I I I I I I l i l i 1.5 1.7 1.9 2.1 2.3 2.5 2.7 Voltage (V) F i g u r e 19 dQ/dV of C r y s t a l l i n e L i x M o S 2 C o n s t a n t c u r r e n t voltammogram o f t h e t e n t h c y c l e o f PM-176. The cathode m a t e r i a l was the o r i g i n a l L i x M o S 3 p r o d u c e d , 2p112. The c e l l was c y c l e d a t 5^A. The s i x l e t t e r s on the graph r e f e r t o the c r y s t a l phases t h a t e x i s t on e i t h e r s i d e of t h e f i r s t o r d e r phase t r a n s i t i o n s . C and D l a b e l t h e o p p o s i t e ends of the r e g i o n of s i n g l e phase c a p a c i t y . 89 scheme f o r t h e s e phases. L a t t i c e A e x i s t e d below 1.8V, l a t t i c e B e x i s t e d between the two f i r s t o r d e r phase t r a n s i t i o n s near 1.9V, and so on. L a t t i c e C and l a t t i c e D a r e l a b e l s on the o p p o s i t e ends of the s i n g l e phase c a p a c i t y r e g i o n near 2.53V. The presence of s e v e r a l d i s t i n c t phases and a s i n g l e phase r e g i o n w i l l be c o n f i r m e d i n the next c h a p t e r . The l a b e l l i n g p r e s e n t e d here w i l l s e r v e as a c o n v e n i e n c e l a t e r . D e t a i l e d e x a m i n a t i o n s of v a r i o u s voltammograms showed t h a t t h e r e were some d i s s i m i l a r i t i e s between the charge and d i s c h a r g e h a l f - c y c l e s of v a r i o u s c e l l s . I t was c o n s i d e r e d p o s s i b l e t h a t t h e r e were two d i f f e r e n t t y p e s of c r y s t a l i n the m a t e r i a l . D i f f e r e n t b a t c h e s may have had d i f f e r e n t amounts of t h e s e c r y s t a l s . I f t h i s were t r u e , i t might e x p l a i n a p e c u l i a r i t y i n the d a t a . The f e a t u r e a t ~1.89V on d i s c h a r g e was always a t a lower p o t e n t i a l than the ~1.89V f e a t u r e on c h a r g e . I t might have been p o s s i b l e t h a t one of the two m a t e r i a l s was a s s o c i a t e d w i t h the ~1.89V f e a t u r e s , and the o t h e r m a t e r i a l w i t h the 1.94V charge and 1.85V d i s c h a r g e f e a t u r e s . C a r e f u l c y c l i n g over v e r y l i m i t e d v o l t a g e ranges showed t h a t t h i s was not the c a s e . The l o w e s t v o l t a g e f e a t u r e on d i s c h a r g e c o r r e s p o n d e d t o the l o w e s t v o l t a g e f e a t u r e on c h a r g e , the next l o w e s t v o l t a g e d i s c h a r g e f e a t u r e on d i s c h a r g e c o r r e s p o n d e d t o the next l o w e s t f e a t u r e on c h a r g e , and so on. Another e x p l a n a t i o n was needed f o r the d i s c r e p a n c y i n a few c u r v e s . C l o s e r e x a m i n a t i o n of the d a t a r e v e a l e d the shape of t h e voltammetry c u r v e s t o be dependent on the upper and lower t r i p 90 v o l t a g e s . T h i s i n t u r n i n d i c a t e d a h y s t e r e s i s i n the c y c l i n g of the sample. 3 The e x t e n t of the h y s t e r e s i s i s i l l u s t r a t e d i n F i g u r e 20. T h i s c e l l was c y c l e d over s e v e r a l d i f f e r e n t v o l t a g e r a nges. The amount of c a p a c i t y measured depended on the p r i o r c y c l e of the c e l l . F i g u r e 20 i s a c t u a l l y a compos i t e of t h r e e s e p a r a t e c y c l e s : 1.6V t o 2.8V; 1.6V t o 2.3V; and 2.3V t o 2.8V. The 1.6V t o 2.8V c y c l e had the maximum amount of c a p a c i t y . The c a p a c i t y measured between 1.7V and 1.8V on d i s c h a r g e was reduced i f t he c e l l had not been a l l o w e d t o f u l l y c h a r g e . S i m i l a r l y , t he c a p a c i t y measured between 2.4V and 2.6V on charge was reduced i f the c e l l was not p e r m i t t e d t o f u l l y d i s c h a r g e . The d i f f e r e n c e between the maximum c a p a c i t y and the c a p a c i t y measured on r e s t r i c t e d c y c l i n g ranges i s shown i n the two shaded a r e a s i n d i c a t e d by the arrows on F i g u r e 20. A f u l l i n v e s t i g a t i o n of the h y s t e r e s i s was not performed f o r t h i s t h e s i s . The approximate Ax v a l u e a s s o c i a t e d w i t h each f e a t u r e i s l i s t e d i n T a b l e I I . The i r r e v e r s i b i l i t y i s c l e a r l y e v i d e n t . The t r a n s i t i o n s between phase A and phase B always had a Ax of ~0.25, and the t r a n s i t i o n s between phase E and phase F always had a Ax of ""0.12, but the c a p a c i t y of the o t h e r f e a t u r e s were d i f f e r e n t on charge and d i s c h a r g e . The Ax v a l u e s were e s t i m a t e d by: c y c l i n g a c e l l , measuring the AQ a s s o c i a t e d w i t h each f e a t u r e , and t a k i n g the r a t i o of AQ and the t o t a l c a p a c i t y of the c e l l . No c o r r e c t i o n was made f o r the r e s i d u a l l i t h i u m c o n t e n t i n the 2.7V phase. There were some s m a l l v a r i a t i o n s i n the Ax v a l u e s computed f o r d i f f e r e n t samples, but i t was not 3 A b r i e f d i s c u s s i o n of h y s t e r e s i s i s p r e s e n t e d i n Appendix IV. 91 o > 3 O o o 1.6 1.8 2.0 2.2 2.4 2.6 2.8 Voltage (V) F i g u r e 20 H y s t e r e s i s i n L i x M o S 2 T h i s f i g u r e shows th r e e o v e r l a i d voltammograms of 4p138 measured at 5MA. These voltammograms were made on c y c l e s between: 1.6V and 2.8V, 1.6V and 2.3V, and 2.3V and 2.8V. The c a p a c i t y measured on a d i s c h a r g e from 2.3V to 1.6V was reduced i f the c e l l had not been f u l l y charged, and the c a p a c i t y measured on a charge from 2.3V t o 2.8V was reduced i f the c e l l had not been f u l l y d i s c h a r g e d . These d i f f e r e n c e s have been drawn s o l i d black and are i n d i c a t e d by the arrows. The dQ/dV measured on charge from 1.6V to 2.3V was the same r e g a r d l e s s of the upper t r i p v o l t a g e , and the c a p a c i t y on d i s c h a r g e from 2.8V to 2.3V was the same r e g a r d l e s s of the lower t r i p v o l t a g e . 92 TABLE I I x-V a l u e s A s s o c i a t e d w i t h the F e a t u r e s of L i x M o S 2 L a t t i c e x x Charge D i s c h a r g e A 1 . 0 0 1 . 0 0 B 0 . 7 5 0 . 7 5 C 0 . 5 0 0 . 4 5 D 0 . 2 7 0 . 2 5 E 0 . 1 2 0 . 1 2 F 0 . 0 0 0 . 0 0 Charge D i s c h a r g e A t o B Ax 0 . 2 5 F t o E Ax 0 . 1 2 B t o c 0 . 2 5 E t o D 0 . 1 3 C t o D 0 . 2 3 D t o C 0 . 2 D t o E 0 . 1 5 C t o B 0 . 3 E t o F 0 . 1 2 B t o A 0 . 2 5 The t o p h a l f of t h i s t a b l e i s a summary of t h e x v a l u e s a s s o c i a t e d w i t h each phase of L i x M o S 2 . The x v a l u e of l a t t i c e s C and D a r e s l i g h t l y d i f f e r e n t on charge and d i s c h a r g e , i n d i c a t i n g a h y s t e r e s i s i n the sample. The bottom h a l f of the t a b l e i s a summary of t h e t y p i c a l Ax v a l u e s a s s o c i a t e d w i t h the t r a n s i t i o n s between the phases of L i x M o S 2 . The l e t t e r s used t o l a b e l t h e phases a r e d e f i n e d i n F i g u r e 19. The x and Ax v a l u e s l i s t e d a r e the avera g e s from f i v e c e l l s . 93 c l e a r i f the v a r i a t i o n s were due t o d i f f e r i n g amounts of h y s t e r e s i s or i f the r e s i d u a l l i t h i u m c o n t e n t was d i f f e r e n t i n each sample. P r e c i s e Ax v a l u e s were not o b t a i n e d here because the amount of a c t i v e m a t e r i a l was not a c c u r a t e l y known. D i f f u s i o n g r a d i e n t s c o u l d broaden the voltammetry f e a t u r e s . The c e l l i n F i g u r e 20 was b e i n g c y c l e d s u f f i c i e n t l y f a s t so t h a t the f i r s t o r d e r phase t r a n s i t i o n s near 1.9V were b l u r r e d t o g e t h e r . U s i n g h i g h e r c u r r e n t s o f t e n made the voltammograms of c r y s t a l l i n e MoS 2 l o o k q u a l i t a t i v e l y l i k e /3-phase voltammograms. However, c l o s e i n s p e c t i o n showed an i m p o r t a n t d i f f e r e n c e . I f , f o r example, the c r y s t a l l i n e m a t e r i a l were b e i n g charged from a low v o l t a g e , no a p p r e c i a b l e c a p a c i t y would be measured below the p o t e n t i a l of the f i r s t phase t r a n s i t i o n , 1.89V. Under s i m i l a r c o n d i t i o n s the /3-phase has a s i g n i f i c a n t c a p a c i t y a t 1.8V. The onset of c a p a c i t y i s lower i n the /3-phase c e l l , p r o b a b l y because of the d i s t o r t e d l i t h i u m s i t e environment. I t may be p o s s i b l e t o use the h i g h t emperature m a t e r i a l as a c r y s t a l l i n e a n a l o g of /3-phase, but the d e f e c t s and s t r a i n s i n /3-phase g i v e i t i t s own unique c h a r a c t e r . The f e a t u r e s i n the voltammetry of the h i g h t emperature grown m a t e r i a l can a l s o be broadened by e x p o s i n g the m a t e r i a l t o w a t e r . F i g u r e 21 i s the voltammogram of a c e l l made w i t h a sample which had been l e f t i n room a i r f o r s e v e r a l h o u r s . The c e l l had not been c y c l e d below 1.4V b e f o r e t h i s d a t a was r e c o r d e d . There was no a t o /3 c o n v e r s i o n of any r e s i d u a l 2H phase m a t e r i a l . Many of the f e a t u r e s of t h i s powder were 94 8 o > o o o - 4 - 8 1 I I — 1 — I — I — I — I — I — I — I — r J I I I I L J 1 I L 1.4 1.6 1.8 2.0 2.2 2.4 Voltage (V) 2.6 F i g u r e 21 C r y s t a l l i n e L i x M o S 2 A f t e r Exposure to A i r af?er ait h^TL™1^^9!am ? f a " ^ t a l l i n e L i x M o S 2 sample Volt. exposed to a i r . The curve was measured a t 95 q u a l i t a t i v e l y s i m i l a r t o t h a t of 0-phase. The main d i f f e r e n c e was t h a t the 1.94V f e a t u r e on charge was not v i s i b l e even though i t i s n o r m a l l y v e r y d i s t i n c t i v e . I t would seem t h a t exposure t o a i r d i s o r d e r e d the h i g h t e m p e r a t u r e grown m a t e r i a l , y e t some of the 1T s t r u c t u r e was r e t a i n e d . I t was not c l e a r i f L i / L i x M o S 2 was an i n t e r c a l a t i o n system. The e l e c t r o c h e m i s t r y p r e d i c t e d t h e h i g h t e m p e r a t u r e grown m a t e r i a l had f o u r f i r s t o r d e r phase t r a n s i t i o n s and a r e g i o n of s i n g l e phase c a p a c i t y . I n t e r c a l a t i o n , by d e f i n i t i o n , must not cause major d i s t o r t i o n s i n the h o s t . S i n g l e phase c a p a c i t y i s o f t e n a s s o c i a t e d w i t h a c o n t i n u o u s s m a l l v a r i a t i o n i n l a t t i c e p a r a m e t e r s . T h i s type of c a p a c i t y may be an i n t e r c a l a t i o n r e a c t i o n . However, a phase t r a n s i t i o n i m p l i e s a c o n v e r s i o n between two d i f f e r e n t m a t e r i a l s . For example, the a t o )3 t r a n s i t i o n i s not an i n t e r c a l a t i o n r e a c t i o n . I t causes a major r e s t r u c t u r i n g of the MoS 2 l a t t i c e and produces many s t r a i n s and d e f e c t s . I n t e r c a l a t i o n c e l l s a r e h i g h l y r e v e r s i b l e , and c y c l i n g does not s e v e r e l y degrade the h o s t , but c e l l s w i t h f i r s t o r d e r phase t r a n s i t i o n s as the major s o u r c e of c a p a c i t y o f t e n do not have t h e s e f a v o u r a b l e p r o p e r t i e s . I t had t o be shown t h a t the d i f f e r e n t phases of L i x M o S 2 were s u f f i c i e n t l y s i m i l a r t o m e r i t c a l l i n g t he t r a n s i t i o n s an i n t e r c a l a t i o n p r o c e s s . Much of Chapter 6 w i l l be devoted t o p r o d u c i n g e v i d e n c e f o r t h i s p r o p o s i t i o n . 96 5.4 C o n v e r s i o n t o 2H-MoS 2 /3-phase i s not s t a b l e a g a i n s t c o n v e r s i o n i n t o a-phase a t h i g h v o l t a g e s , or 7-phase a t low v o l t a g e s . T h i s s e c t i o n w i l l compare the /3-phase and h i g h t e m p e r a t u r e grown m a t e r i a l s i n the s e extremes. C o n v e r s i o n from /3 t o a phase was d i s c u s s e d i n s e c t i o n 2.3. I f enough l i t h i u m was removed from L i x M o S 2 then the 2H form would become e n e r g e t i c a l l y f a v o u r a b l e . A 1T t o 2H c o n v e r s i o n c o u l d be a f f e c t e d by many f a c t o r s . I f n u c l e a t i o n problems were a major f a c t o r i n the /3 t o a c o n v e r s i o n then one would expect a sample would be more s t a b l e i f i t had fewer d e f e c t s . I t was ob s e r v e d t h a t the more c r y s t a l l i n e the 1T sample the l e s s l i k e l y i t was t o c o n v e r t . The c o n v e r s i o n t o a-phase was a more s e r i o u s problem than was the c o n v e r s i o n t o 7-phase. A 7 c o n v e r s i o n o n l y o c c u r r e d i f the c e l l was d i s c h a r g e d t o about 0.6V and a l a r g e amount of c u r r e n t was passed t h r o u g h i t . C y c l i n g a IT c e l l over the main v o l t a g e f e a t u r e s (1.8V t o 2.7V) d i d not c o n v e r t any of the m a t e r i a l t o the 7-phase. However, c o n v e r s i o n t o a-phase o c c u r r e d a t p o t e n t i a l s above ~2.6V. A c e l l c y c l e d over the f u l l IT c a p a c i t y c o u l d c o n v e r t t o a-phase. F u r t h e r m o r e , the t r a n s i t i o n from a low x 1T c r y s t a l t o a low x 2H c r y s t a l d i d not n e c e s s a r i l y r e q u i r e any e x t e r n a l charge t o f l o w . As a r e s u l t , f u l l y c h a r g e d /3-phase c e l l s l e f t open c i r c u i t c o u l d c o n v e r t t o a - p h a s e . T h i s has s e r i o u s i m p l i c a t i o n s f o r a commercial c e l l . 97 The time s c a l e f o r c o n v e r s i o n to a-phase can be estimated i n s e v e r a l d i f f e r e n t ways. One p o s s i b l e method i s to examine the l o s s of 1T c a p a c i t y . F i r s t the c a p a c i t y of the c e l l i s measured i n some v o l t a g e i n t e r v a l , such as 1.6V to 2.3V. The c e l l i s then charged and h e l d at a high v o l t a g e f o r a p e r i o d of time, a f t e r which i t i s c y c l e d between the lower v o l t a g e l i m i t s to measure i t s c a p a c i t y . Any l o s s of c a p a c i t y c o u l d be r e l a t e d to the c o n v e r s i o n t o a-phase. There i s another technique that i s o f t e n f a s t e r and s i m p l e r . An x-ray c e l l can be prepared and c y c l e d to some v o l t a g e , and the d i f f r a c t i o n p a t t e r n can be monitored to watch growth of the a-phase. T h i s technique w i l l be d i s c u s s e d i n more d e t a i l i n the next c h a p t e r . Both these techniques were used to determine the time constant f o r c o n v e r s i o n . /3-phase c e l l s which had been f i x e d at 2.7V and had not been p r e v i o u s l y c y c l e d , o f t e n were very u n s t a b l e . The " h a l f - l i f e " of /3-phase at t h i s p o t e n t i a l seemed to be about 5 to 10 minutes. The m a t e r i a l f u l l y converted to a-phase i n l e s s than one hour. The h i g h temperature grown m a t e r i a l was c o n s i d e r a b l y more s t a b l e . F i g u r e 22 shows the c o n v e r s i o n of a 5p53 sample to the a-phase. The graph shows the i n t e n s i t y of three d i f f r a c t i o n l i n e s as a f u n c t i o n of the time a f t e r the c e l l p o t e n t i a l was stepped to 2.7V. The i n t e n s i t y of the peak was r e l a t e d to the amount of m a t e r i a l p r e s e n t . The l i n e t h a t corresponded to m a t e r i a l E (2.63V) decreased m o n o t o n i c a l l y . The one that corresponded to phase F (2.7V) rose to a maximum a f t e r about 10-15 hours and then decayed away. A s t a b l e 1T s t r u c t u r e would 98 6 -I I I I I I I I I I I I I I I I I I I I I I I I I 1 I 20 40 60 80 100 120 140 Time (hours) F i g u r e 22 Conversion of 5p53 L i x M o S 2 to a-Phase. The p o t e n t i a l of PMX-90 was f i x e d a t 2.7V a t t=0 and the i n t e n s i t i e s of three (0,0,1) x-ray d i f f r a c t i o n l i n e s between 14° and 15° of 26 were monitored. The amount of L i x M o S 2 phase E (•) decreased. The amount of L i x M o S 2 phase F (X) o r i g i n a l l y i n c r e a s e d and then decreased. The amount of a-phase ( A ) i n c r e a s e d m o n o t o n i c a l l y . 99 not show any l o s s of phase F. The amount of a-phase m a t e r i a l grew m o n o t o n i c a l l y . The time r e q u i r e d t o c o n v e r t h a l f the m a t e r i a l t o a-phase was about t h r e e days. F i g u r e 22 a l s o h i g h l i g h t s a problem encountered when t r y i n g t o a n a l y z e phase F. A l l measurements had t o be performed b e f o r e the c r y s t a l decayed t o a-phase. The s t a b i l i t y of /3-phase i n c r e a s e d w i t h c y c l i n g . A c e l l t h a t had been c y c l e d about one hundred t i m e s would l o s e a p p r o x i m a t e l y h a l f of i t s c a p a c i t y i f i t were kept above 2.6V f o r a day. These t e s t s showed t h a t the r a t e f o r the 1T t o 2H t r a n s i t i o n was reduced i f the 1T m a t e r i a l was made more c r y s t a l l i n e . The h i g h t e m p e r a t u r e m a t e r i a l was q u i t e s t a b l e a g a i n s t c o n v e r s i o n t o 7-phase. C y c l i n g a c e l l down t o 0.5V a t h i g h c u r r e n t d i d not s i g n i f i c a n t l y a l t e r the e l e c t r o c h e m i s t r y . I t was p o s s i b l e t o a f f e c t a c e l l by k e e p i n g i t a t low v o l t a g e s f o r p e r i o d s of a day or more, but the cause of the change was not c l e a r . There was p r o b a b l y some 7 - c o n v e r s i o n , but t h e r e was a l s o a g r e a t d e a l of what appeared t o be e l e c t r o l y t e d e c o m p o s i t i o n . The l o s s of s h a r p voltammetry f e a t u r e s may t h e r e f o r e have been caused i n p a r t be the degraded c e l l performance due t o poor k i n e t i c s r a t h e r than t o a c o n v e r s i o n of the h o s t . T h i s p o s s i b i l i t y was not examined i n d e t a i l . 100 F i n a l l y , t he h i g h temperature grown m a t e r i a l which had been c o n v e r t e d t o a-phase c o u l d be d r i v e n t h r o u g h the a t o 0 t r a n s i t i o n . The e l e c t r o c h e m i s t r y of these c e l l s was almost i n d i s t i n g u i s h a b l e from t y p i c a l /3-phase b e h a v i o u r . The voltammetry of the c o n v e r t e d h i g h temperature m a t e r i a l was s l i g h t l y s h a r p e r than t h a t of the f i r s t c y c l e s of /3-phase, but t h i s may have been due t o i n c o m p l e t e c o n v e r s i o n t o a-phase. I t was not p o s s i b l e t o c y c l e the /3-phase enough t o make the voltammetry l o o k l i k e t h a t of the h i g h t e m p e r a t u r e grown m a t e r i a l , but the c r y s t a l l i n e m a t e r i a l c o u l d be d i s o r d e r e d t o make i t behave l i k e /3-phase. 101 CHAPTER 6 X-RAY INVESTIGATIONS Try to improve, and soon I'll make it good; You'll learn with sage obedience, by and by, To systemize and then to classify. -Me phi st ophel es to Student --Goethe 6.1 Overview T h i s c h a p t e r d i s c u s s e s the x - r a y i n v e s t i g a t i o n of L i x M o S 2 . A r e v i e w of the /3-phase i s p r e s e n t e d i n s e c t i o n 6.2. The key o b s e r v a t i o n s were t h a t the /?-phase i s a somewhat d i s o r d e r e d c r y s t a l , p r o b a b l y h a v i n g a 1T hexagonal s t r u c t u r e . The h i g h temperature grown m a t e r i a l had the sh a r p d i f f r a c t i o n l i n e s c h a r a c t e r i s t i c of h i g h l y c r y s t a l l i n e m a t e r i a l s . I t was p o s s i b l e t o c o n f i r m t h a t the c e l l ' s c a p a c i t y was a s s o c i a t e d w i t h f i r s t o r d e r phase t r a n s i t i o n s and a r e g i o n of s i n g l e phase c a p a c i t y . I t was a l s o d e t e r m i n e d t h a t the c r y s t a l s were not q u i t e h e x a g o n a l . The b u l k of t h i s c h a p t e r i s devoted t o d e t a i l e d s t r u c t u r a l a n a l y s i s of the v a r i o u s L i x M o S 2 phases. S e c t i o n 6.4 e x p l a i n s how the g r a p h i c a l t e c h n i q u e d e s c r i b e d i n Appendix I was used t o index t h e s e phases. I t i s an ove r v i e w of the ass u m p t i o n s , p r o c e s s e s , and problems a s s o c i a t e d w i t h the i n d e x i n g . 102 S e c t i o n 6.5 c o v e r s the s t e p s i n v o l v e d i n t h e i n d e x i n g of a d i f f r a c t i o n p a t t e r n . S e c t i o n s 6.6 and 6.7 d i s c u s s d e t a i l s of the i n d e x i n g of the phases of L i x M o S 2 . The l a t t e r t h r e e s e c t i o n s a r e u s e f u l f o r u n d e r s t a n d i n g the d e t a i l s of the i n d e x i n g p r o c e s s and would be v e r y i m p o r t a n t i f the work i s r e p e a t e d . They a r e not e s s e n t i a l t o u n d e r s t a n d i n g the f i n a l r e s u l t s , which a r e d e s c r i b e d i n S e c t i o n 6.8. S e c t i o n 6.9 a d d r e s s e s the q u e s t i o n of a t o m i c p o s i t i o n s i n L i x M o S 2 . The 1 T - c r y s t a l form was found t o be the b e s t model t o e x p l a i n the d a t a . 6.2 X-Ray S t u d i e s of /3-Phase Some of the x - r a y work done by Wainwright(1978) and P y ( l 9 8 3 ) has been summarized i n Chapter 2. T h i s s e c t i o n r e v i e w s t h e i r work i n more d e t a i l and adds i n f o r m a t i o n t h a t w i l l be u s e f u l f o r l a t e r r e f e r e n c e . The most i m p o r t a n t s i n g l e o b s e r v a t i o n was t h a t /3.-phase had x - r a y d i f f r a c t i o n l i n e s , i m p l y i n g i t was c r y s t a l l i n e r a t h e r than amorphous. T h i s c o n t r a d i c t s the e a r l i e r work which had c l a i m e d MoS 2 d i s p r o p o r t i o n a t e d f o r x>0.2. D i f f r a c t i o n l i n e s of /3-phase were much br o a d e r than l i n e s f o r h i g h l y c r y s t a l l i n e m a t e r i a l s , such as a-phase. Broad l i n e s a r e u s u a l l y a s s o c i a t e d w i t h a l a c k of l o n g - r a n g e c r y s t a l o r d e r . S m a l l c r y s t a l l i t e s have broad d i f f r a c t i o n l i n e s because they do not have enough atomic p l a n e s t o sharpen the p r i n c i p a l maxima. The i n d i v i d u a l c r y s t a l domains i n /3-phase would have t o be ~20A t o a c c o u n t f o r the b r e a d t h of the l i n e s . T h i s i s o n l y about t h r e e l a t t i c e s p a c i n g s . I n t e r n a l s t r a i n can a l s o broaden x - r a y 103 l i n e s , as u n i t c e l l s would be s l i g h t l y d i f f e r e n t s i z e s i n d i f f e r e n t p a r t s of the c r y s t a l . The peak w i d t h s c o r r e s p o n d e d t o a v a r i a t i o n i n l a y e r s p a c i n g of ~0.04A and an a a x i s v a r i a t i o n of <0.02A. These d i s t o r t i o n s r e p r e s e n t l e s s than a 1% s h i f t i n l a t t i c e p a r a m e t e r s . The a c t u a l s o u r c e of the broad d i f f r a c t i o n l i n e s was p r o b a b l y a c o m b i n a t i o n of f a c t o r s , but t h e /3-phase c y c l i n g d a t a and i n f o r m a t i o n p r e s e n t e d below i m p l y t h a t s t r a i n was the dominant f a c t o r . Any f i n e d e t a i l t h a t may have been p r e s e n t was o b s c u r e d by the w i d t h s of the /3-phase d i f f r a c t i o n l i n e s . The d i f f r a c t i o n p a t t e r n s i n d i c a t e d t h a t /3-phase p r o b a b l y had a hexagonal symmetry. P y ( l 9 8 3 ) measured L i M o S 2 t o have a=3.36±0.0lA and c=6.294±0.001 A. For c o m p a r i s o n , the 2H-MoS 2 had a=3.l625A and C=2X6.1551A. U n p u b l i s h e d d a t a from Py i s p r e s e n t e d i n F i g u r e 23. T h i s d a t a was measured on the 2 9 t n charge of a /3-phase c e l l . There appeared t o be a c o n t i n u o u s v a r i a t i o n i n the l a t t i c e parameters as the l i t h i u m c o n t e n t was v a r i e d . The c a x i s d a t a was p a r t i c u l a r l y s t r i k i n g . The a d d i t i o n of l i t h i u m t o a s i n g l e phase c r y s t a l was e x p e c t e d t o cause a monotonic i n c r e a s e i n the c a x i s (Dahn e t . a l . , 1982b), but the MoS 2 c a x i s was s h o r t e r a t x=1 than i t was a t x=0.75. The /3-phase MoS 2 d a t a was not u n d e r s t o o d . The x - r a y d a t a changed w i t h c y c l e number j u s t as the e l e c t r o c h e m i c a l d a t a d i d . S e c t i o n 5.2 d i s c u s s e d how t h e c y c l i n g of /3-phase l e d t o the s h a r p e n i n g of the voltammetry f e a t u r e s . F i g u r e s 16, 17 , and 18 showed one f e a t u r e t h a t sharpened so much t h a t i t began t o l o o k l i k e a f i r s t o r d e r phase t r a n s i t i o n . 1 04 12.8 12.7 12.6 12.5 12.4 12.3 -12.2 -12.1 T — i — i — i — i — i — i — i — r 3.8 - 3.7 xxxx x ***** — x » * x M U X ' J I I I I I I I L 0.0 0.2 0.4 0.6 0.8 1.0 x 3.6 3.5 3.4 3.3 3.2 3.1 ti F i g u r e 23 0-Phase L a t t i c e Parameters T h i s graph shows the a and c l a t t i c e parameters of 0-phase as measured by M a r c e l Py. The s c a l e f o r the c a x i s d ata (o) i s on the l e f t of the graph, and the s c a l e f o r the a a x i s d ata (x) i s on the r i g h t of the graph. The m a t e r i a l was indexed assuming a two l a y e r u n i t c e l l . Note the non-monotonic v a r i a t i o n i n c a x i s l e n g t h as the x co n t e n t of L i x M o S 3 i s v a r i e d . 105 F i g u r e 24 shows a s u c c e s s i o n of x- r a y scans of /3-phase made on the 8 6 t n c y c l e as the c e l l was c y c l e d a t 100MA. There was a peak s p l i t t i n g i n d i c a t i v e of the d i s c o n t i n u o u s change i n l a t t i c e p a rameters t h a t i s a s s o c i a t e d w i t h a f i r s t o r d e r phase t r a n s i t i o n . 6.3 B a s i c X-Ray S t u d i e s of L i x M o S 2 T h i s s e c t i o n w i l l o u t l i n e the most g e n e r a l s t r u c t u r a l o b s e r v a t i o n s made of the h i g h t e m p e r a t u r e grown m a t e r i a l . A comparison w i t h /3-phase i s made, and a c o n f i r m a t i o n of s e v e r a l f i r s t o r d e r phase t r a n s i t i o n s and a r e g i o n of s i n g l e phase c a p a c i t y i s a l s o p r e s e n t e d . The most s t r i k i n g d i f f e r e n c e between the x - r a y p a t t e r n s of the /3-phase and the h i g h t e m p e r a t u r e grown m a t e r i a l i s i n the sharpness of the d i f f r a c t i o n l i n e s . The l i n e w i d t h s of the h i g h t e m p e r a t u r e m a t e r i a l s a r e u s u a l l y l i m i t e d by the r e s o l u t i o n of the x - r a y machine, and the sh a r p l i n e s made i t p o s s i b l e t o r e s o l v e d i f f r a c t i o n peaks of the c r y s t a l l i n e L i x M o S 2 t h a t were o n l y s e p a r a t e d by ~0.1°. The g r e a t e s t s i m i l a r i t y between the two m a t e r i a l s was i n the p o s i t i o n s of the peaks. The broad /3-phase l i n e s c o v e r a l a r g e a n g u l a r r e g i o n , but i t was p o s s i b l e t o e s t i m a t e the c e n t r e of the peak. The c r y s t a l l i n e m a t e r i a l u s u a l l y has s e v e r a l d i f f r a c t i o n l i n e s i n the same a n g u l a r r e g i o n , but t h e i r mean p o s i t i o n i s the same as the p o s i t i o n of the /3-phase l i n e . T h i s c o r r e s p o n d e n c e i n d i c a t e s t h a t the u n d e r l y i n g c r y s t a l s t r u c t u r e s of b o t h m a t e r i a l s a r e n e a r l y i d e n t i c a l . 106 57 6 0 57 6 0 57 60 57 6 0 F i g u r e 24 20 (°) 0-Phase — I n d i c a t i o n of C o - E x i s t i n g Phases These graphs show a s e r i e s of s i x x - r a y scans of the (0,0,8) 0-phase d i f f r a c t i o n peak (assuming a 2 l a y e r u n i t c e l l ) measured on the 8 6 t h c y c l e . The c e l l was d i s c h a r g e d a t 100MA. The v o l t a g e s i n the upper l e f t - h a n d c o r n e r s of the p l o t s a r e t h e p o t e n t i a l s a t which the scans were made, but these a r e not the e q u i l i b r i u m v o l t a g e s . The t o p row of graphs show a peak s h i f t t o lower a n g l e as the c a x i s expands, and the lower row of graphs show a l o s s of i n t e n s i t y a t 58.3° as a peak a t 58.8° grows. The d a t a was c o l l e c t e d by M a r c e l Py. 107 Note a g a i n t h a t t h i s i s not a p r o o f t h a t the two m a t e r i a l s a r e the same, but r a t h e r e v i d e n c e t h a t they a r e s i m i l a r enough f o r knowledge about one t o have i m p l i c a t i o n s f o r u n d e r s t a n d i n g the o t h e r . One of the e a s i e s t t e s t s made w i t h in situ x - r a y d i f f r a c t i o n was a c o n f i r m a t i o n of the e x i s t e n c e of e i t h e r a f i r s t o r d e r phase t r a n s i t i o n or s i n g l e phase c a p a c i t y . A t r a n s i t i o n i n v o l v e s the c o - e x i s t e n c e of two d i f f e r e n t l a t t i c e s , each w i t h i t s own x- r a y p a t t e r n . As the c o n v e r s i o n p r o c e e d s , the i n t e n s i t y of one p a t t e r n drops as the o t h e r ' s i n c r e a s e s . The s i g n a t u r e of s i n g l e phase c a p a c i t y i s a c o n t i n u o u s change i n the l a t t i c e parameters as the l i t h i u m c o n t e n t i s a l t e r e d . The experiment was performed by c h a r g i n g or d i s c h a r g i n g a c e l l s l o w l y enough t h a t the m a t e r i a l remained i n a s t a t e of q u a s i - e q u i l i b r i u m w h i l e a s u c c e s s i o n of x- r a y scans i s made. F i g u r e 22 from the p r e v i o u s c h a p t e r shows the x - r a y peak p o s i t i o n s between 14° and 15° a f t e r a c e l l was f i x e d a t 2.7V. The l o s s of one d i f f r a c t i o n peak w h i l e a n o t h e r appeared i s t h e x- r a y s i g n a t u r e of a f i r s t o r d e r phase t r a n s i t i o n . A l l t h e f e a t u r e s t h a t the e l e c t r o c h e m i s t r y suggested t o be phase t r a n s i t i o n s were c o n f i r m e d as such. The b e h a v i o u r of an x- r a y peak as a c e l l was c y c l e d over the 2.5V s i n g l e phase r e g i o n i s p l o t t e d i n F i g u r e 25. The l i n e was t r a c k e d over s e v e r a l p o s i t i o n s , and a t no time were two l i n e s o b s e r v e d . T h i s i n d i c a t e d s i n g l e phase c a p a c i t y . 108 40.0 CM 39.6 -39.5 2.5 2.6 Voltage (V) 2.7 F i g u r e 25 L i x M o S 2 S i n g l e Phase C a p a c i t y T h i s g r a p h shows t h e a n g u l a r p o s i t i o n of t h e (/,-/,-5) peak as a c e l l was c y c l e d between 2.4V and 2.7V. The peak showed a c o n t i n u o u s s h i f t i n p o s i t i o n . At no time were two peaks o b s e r v e d . The graph a l s o shows a h y s t e r e s i s . The c e l l v o l t a g e s and peak p o s i t i o n s measured w h i l e the c e l l was c h a r g i n g (x) were not t h e same as t h e d a t a c o l l e c t e d on the d i s c h a r g e ( A ) . The d i f f e r e n c e i n p o t e n t i a l i s t o o l a r g e t o a t t r i b u t e t o an IR s h i f t . 109 6.4 D e t a i l e d S t r u c t u r a l A n a l y s i s — O v e r v i e w The p r e c e d i n g s e c t i o n s e s t a b l i s h e d t h a t L i / L i x M o S 2 i s a m u l t i - p h a s e system. They d i d not show L i / L i x M o S 2 t o be an i n t e r c a l a t i o n system. I n t e r c a l a t i o n i s , by d e f i n i t i o n , r e s t r i c t e d t o systems i n which no major s t r u c t u r a l changes o c c u r i n the h o s t . The e x t e n t of rearrangement of t h e host had t o be deter m i n e d by d e t a i l e d s t r u c t u r a l a n a l y s i s . F u r t h e r m o r e , d e t a i l e d m o d e l i n g of the system would a l s o r e q u i r e s t r u c t u r a l i n f o r m a t i o n . I t i s t e d i o u s t o f o l l o w a l l the minor d e t a i l s i n v o l v e d i n i n d e x i n g d i f f r a c t i o n l i n e s and r e f i n i n g p a r a m e t e r s , so the p r o c e s s has been d i v i d e d i n t o s e v e r a l p a r t s i n an attempt t o a v o i d c o n f u s i o n and l o s s of i n t e r e s t . T h i s s e c t i o n p r e s e n t s a summary of the p r o c e d u r e . I t d e s c r i b e s a s e r i e s of a p p r o x i m a t i o n s made; why many were r e j e c t e d ; and how they were m o d i f i e d . C o m p l i c a t i o n s u s u a l l y p r e s e n t i n a l l e x p e r i m e n t s a r e a l s o p r e s e n t e d h e r e . I t s h o u l d be p o s s i b l e t o proceed t o s e c t i o n 6.8, the r e s u l t s , i m m e d i a t e l y a f t e r r e a d i n g t h i s s e c t i o n . I n f o r m a t i o n t h a t d e a l s w i t h the p r o c e s s of i n d e x i n g L i x M o S 2 i s i n c l u d e d i n s e c t i o n s 6.5, 6.6, and 6.7. The geometry, a l g e b r a , and t h e o r y of x- r a y d i f f r a c t i o n have been r e l e g a t e d t o Appendix I , which i s c o m p l e t e l y g e n e r a l and s t a n d s a p a r t from the b u l k of the t h e s i s . Appendix I a l s o i n c l u d e s s e v e r a l examples of my g r a p h i c a l a n a l y s i s t e c h n i q u e a p p l i e d t o ' h y p o t h e t i c a l ' c r y s t a l s h a v i n g i d e n t i c a l parameters t o t h o s e of the c r y s t a l s d e s c r i b e d h e r e . Appendix I I summarizes the computer programs used t o p r o c e s s the d a t a , and Appendix I I I c o n t a i n s the 1 10 a c t u a l d a t a from the L i x M o S 2 c r y s t a l s . In p a r t , t h i s t h e s i s d e s c r i b e s the i n d e x i n g of t r i c l i n i c l a t t i c e s from powder p a t t e r n i n f o r m a t i o n . N o r m a l l y , a complex c r y s t a l i s examined by growing and s t u d y i n g a s i n g l e c r y s t a l . T h i s proved t o be i m p r a c t i c a l f o r t h r e e r e a s o n s : no s i n g l e c r y s t a l s had ever been grown; even i f such c r y s t a l s were a v a i l a b l e , i n t e r c a l a t i o n i n l a r g e s i n g l e c r y s t a l s i s u s u a l l y v e r y slow and l a t t i c e parameter mismatch may cause them t o c r a c k ; and no a p p a r a t u s f o r such work was c o n v e n i e n t l y a v a i l a b l e . A g r a p h i c a l t e c h n i q u e was d e v i s e d t o a i d i n the i n d e x i n g of an a r b i t r a r y l a t t i c e u s i n g o n l y powder d i f f r a c t i o n d a t a . The s t r u c t u r e s of the t r a n s i t i o n m e t a l d i c h a l c o g e n i d e s a r e e i t h e r hexagonal or s l i g h t l y d i s t o r t e d h e x a g o n a l . The f i r s t a s sumption used i n u n d e r s t a n d i n g c r y s t a l l i n e L i x M o S 2 phases was t h a t they had hexagonal symmetry and l a t t i c e p arameters v e r y s i m i l a r t o t h o s e of /3-phase. Each phase of t h e h i g h t e m p e r a t u r e grown m a t e r i a l had d i f f r a c t i o n l i n e s a t r o u g h l y the p o s i t i o n s e x p e c t e d , but t h e r e were m u l t i p l e l i n e s where o n l y one was e x p e c t e d , or e x t r a l i n e s where none were supposed t o o c c u r . A l l the c r y s t a l s had l e s s than hexagonal symmetry. The next assumption was t h a t t h e L i x M o S 2 s t r u c t u r e was a s l i g h t l y d i s t o r t e d h e x a g o n a l . Much, but not a l l , of the d a t a c o u l d be e x p l a i n e d by p e r t u r b i n g the h i g h l y symmetric b a s a l p l a n e . The next a p p r o x i m a t i o n was t o t i p the c a x i s away from the normal t o t h e b a s a l p l a n e . The f i r s t of t h e s e a p p r o x i m a t i o n s 111 was a m o n o c l i n i c d i s t o r t i o n , and the second was t r i c l i n i c . E v e n t u a l l y a l l the d a t a were e x p l a i n e d by c o n s i d e r i n g these r e l a t i v e l y minor v a r i a t i o n s from hexagonal symmetry. The 1.7V m a t e r i a l , the A l a t t i c e , was the f i r s t t o be i n d e x e d s u c c e s s f u l l y . The m a t e r i a l had been c h e m i c a l l y t r e a t e d t o make i t p u r e , and i t s d i f f r a c t i o n p a t t e r n was p a r t i c u l a r l y easy t o u n d e r s t a n d . The l a y e r s p a c i n g was r e a d i l y c a l c u l a t e d and t h e r e was a d i s t i n c t i v e l i n e s p l i t t i n g t h a t i n d i c a t e d a d i s t o r t i o n of the b a s a l p l a n e . A m o n o c l i n i c d i s t o r t i o n e x p l a i n e d a l l but t h r e e l i n e s i n the d a t a : the l i n e s a t 32.67°, 46.39°, and 61.44°. These t h r e e l i n e s were be s t a c c o u n t e d f o r by d o u b l i n g the s i z e of the b a s a l p l a n e i n the a and B d i r e c t i o n s , so t h a t the u n i t c e l l c o n t a i n e d f o u r f o r m u l a u n i t s r a t h e r than one. The o t h e r phases of L i x M o S 2 c o u l d not be e x p l a i n e d i n terms of a m o n o c l i n i c l a t t i c e , but t h e r e were fewer d i f f r a c t i o n l i n e s than n o r m a l l y e x p e c t e d f o r a t r i c l i n i c l a t t i c e . I t was assumed t h a t the c r y s t a l was t r i c l i n i c , but t h a t s e v e r a l d i f f r a c t i o n l i n e s were e x t i n c t . I n t e n s i t y p a t t e r n s were c a l c u l a t e d based on 1T models t o a i d i n the i n d e x i n g , and I w i l l show i n s e c t i o n 6.9 t h a t a 1T model does e x p l a i n t h e d a t a . Three of the L i x M o S 2 c r y s t a l s c o u l d not be indexed on a 2 by 2 by 1 u n i t c e l l , but they had v e r y s i m i l a r d i f f r a c t i o n p a t t e r n s . These l a t t i c e s were ind e x e d by imposing the r e s t r i c t i o n t h a t c o r r e s p o n d i n g d i f f r a c t i o n l i n e s i n the t h r e e p a t t e r n s had t o have the same M i l l e r i n d i c e s . E v e n t u a l l y , v e r y s i m i l a r l a t t i c e p arameters were de t e r m i n e d f o r each c r y s t a l phase. 112 The i n d e x i n g of a m o n o c l i n i c or t r i c l i n i c l a t t i c e i s not un i q u e . The c o n v e n t i o n adopted here was d e s i g n e d t o r e f l e c t the near hexagonal symmetry. The b a s a l p l a n e was d e f i n e d by r e s t r i c t i n g the a n g l e between a and B t o ~120°. The c a x i s was r o u g h l y i n the d i r e c t i o n p e r p e n d i c u l a r t o the l a y e r s . Other c o n v e n t i o n s were used t o make the d a t a p r e s e n t a t i o n s y s t e m a t i c : a>b; 7^120°; a>/3.1 There were s e v e r a l s y s t e m a t i c c o m p l i c a t i o n s t o i n d e x i n g a c r y s t a l s t r u c t u r e . A c c u r a t e parameters can o n l y be de t e r m i n e d i f the d i f f r a c t i o n a n g l e has been measured c o r r e c t l y . The e r r o r i n the x- r a y machine i s m i n i m i z e d by c a r e f u l a l i g n m e n t and p e r i o d i c checks were made t o ensure no m i s a l i g n m e n t had o c c u r r e d . The d i f f T a c t o m e t e r c o u l d g i v e an a c c u r a t e measurement of the d i f f r a c t i o n a n g l e o n l y i f the sample was mounted e x a c t l y on the a x i s of the goniometer. However, a v a r i e t y of g a s k e t s and s p a c e r s on the x - r a y c e l l s t y p i c a l l y d i s p l a c e d the a c t i v e powder away from the i d e a l p o s i t i o n . The d i s p l a c e m e n t was s l i g h t l y d i f f e r e n t each time the c e l l was mounted i n t h e machine. F i g u r e 26 shows the geometry of the o u t - o f - p l a n e e r r o r . The sample has been d i s p l a c e d from the i d e a l p o s i t i o n by a d i s t a n c e 6. The x - r a y s , which a r e d i f f r a c t e d a t the Bragg a n g l e , 0 B, a r e then e r r o n e o u s l y r e c o r d e d as i f they had been d i f f r a c t e d a t 6m. The c o r r e c t d i f f r a c t i o n a n g l e can be computed from t h e d a t a by the r e l a t i o n 1 These parameters a r e i l l u s t r a t e d i n F i g u r e 32 of Appendix I . 1 1 3 F i g u r e 26 Out-of-Plane Geometry An e r r o r i s i n t r o d u c e d i n t o the measurement of the Bragg angle i f the sample i s d i s p l a c e d a d i s t a n c e 6 away from the a x i s of the goniometer. The x-rays are d i f f r a c t e d at 0Q, but the machine r e c o r d s the angular p o s i t i o n as 28m. 1 14 tan 8 n = t a n 8m - — — 6.1 B m R cos 8m where R i s the a x i s - d e t e c t o r d i s t a n c e of 173mm. The v a l u e of 5 i n t h e s e e x p e r i m e n t s was t y p i c a l l y about 0.25mm. D i f f e r e n c e s between measured and t r u e d i f f r a c t i o n a n g l e s of up t o t e n t h s of a degree were common, but a c c u r a c y on the o r d e r of 0.01° was r e q u i r e d . A c c u r a t e e s t i m a t e s of 5 were p o s s i b l e i n p a r t because of another s y s t e m a t i c problem. The cathode powders were t i n y p l a t e l e t s t h a t tended t o a l i g n w i t h t h e i r c axes p e r p e n d i c u l a r t o the s u b s t r a t e . T h i s p r e f e r r e d o r i e n t a t i o n produced s t r o n g (0,0,1) l i n e s , and i t was p o s s i b l e t o f i t the d a t a t o these l i n e s i n such a way as t o de t e r m i n e the o u t - o f - p l a n e c o r r e c t i o n , as mentioned i n Appendix I I . 2 Another consequence of the p r e f e r r e d o r i e n t a t i o n was t h a t o b s e r v e d i n t e n s i t i e s were not the same as would be e x p e c t e d from a random powder. P l a n e s p a r a l l e l t o t he c a x i s produced s t r o n g d i f f r a c t i o n s p o t s , but p l a n e s p e r p e n d i c u l a r t o i t d i f f r a c t e d so few x - r a y s t h a t the l i n e s were o c c a s i o n a l l y i n v i s i b l e . A more d e t a i l e d d i s c u s s i o n of p r e f e r r e d o r i e n t a t i o n i s i n s e c t i o n 6.9. D i f f r a c t i o n l i n e s were o f t e n h idden or o b s c u r e d . There were s e v e r a l a n g u l a r r e g i o n s i n which the i n t e n s i t y s c a t t e r e d from the c e l l c a s e was much s t r o n g e r than any s c a t t e r i n g from the cathode powder. The background o f t e n made s m a l l d i f f r a c t i o n peaks e i t h e r h a r d t o f i n d or i l l - d e f i n e d . O c c a s i o n a l l y two or 2 The (0,0,1) t r i p l e t r e f e r s t o the M i l l e r i n d i c e s . These a r e d e f i n e d and e x p l a i n e d i n the second s e c t i o n of Appendix I . 115 more d i f f r a c t i o n l i n e s would occur a t almost the same a n g u l a r p o s i t i o n . In a l l t h e s e c a s e s the d a t a of i n t e r e s t was s t i l l p r e s e n t , but was " l o s t " i n the random f l u c t u a t i o n s of the c o u n t i n g s t a t i s t i c s . S e p a r a t i n g t h e s i g n a l from the n o i s e o f t e n r e q u i r e d a p r o h i b i t i v e amount of t i m e . The s u p e r p o s i t i o n of d i f f r a c t i o n l i n e s from d i f f e r e n t c r y s t a l phases was best a v o i d e d by making c e l l s w i t h good cathode u t i l i z a t i o n . F o r t u n a t e l y , most of the d i f f r a c t i o n l i n e s of i n t e r e s t were not a f f e c t e d by these problems, or were o n l y a f f e c t e d by one of them. The u l t i m a t e t e s t of the q u a l i t y of the f i n a l r e s u l t s was i t s c a p a b i l i t y t o e x p l a i n a l l the d a t a o b s e r v e d . I t was p o s s i b l e t o p r e d i c t the p o s i t i o n s of t h e d i f f r a c t i o n peaks w i t h the parameters c a l c u l a t e d h e r e , but u n f o r t u n a t e l y the atomic p o s i t i o n s were not d e t e r m i n e d , making i t i m p o s s i b l e t o p r e d i c t c o r r e c t l y the i n t e n s i t i e s of the s c a t t e r e d x - r a y s . A t e c h n i q u e t o p a r t i a l l y c i r c u m v e n t t h i s problem w i l l be d i s c u s s e d l a t e r . 6.5 A p p l i c a t i o n of the G r a p h i c a l I n d e x i n g Technique T h i s s e c t i o n d e s c r i b e s t h e p r o c e s s of i n d e x i n g a c r y s t a l . The d a t a i s i n i t i a l l y c o n v e r t e d i n t o a form which l e n d s i t s e l f t o easy a n a l y s i s , and then an i t e r a t i v e p r o c e d u r e makes a s e r i e s of r e f i n e m e n t s t o a model of the c r y s t a l . Both g r a p h i c a l and n u m e r i c a l t e c h n i q u e s were used t o r e j e c t u n s u i t a b l e i n d e x i n g schemes. I t was u s u a l l y p o s s i b l e t o i d e n t i f y i m m e d i a t e l y the (0,0,1) d i f f r a c t i o n l i n e s because p r e f e r r e d o r i e n t a t i o n made thes e peaks 1 16 v e r y i n t e n s e . O u t - o f - p l a n e d i s p l a c e m e n t and l a y e r s p a c i n g were d e t e r m i n e d from t h i s d a t a . The former was used t o compute the Bragg a n g l e s from t h e measured p o s i t i o n s of t h e d i f f r a c t i o n peaks, and t h e Bragg a n g l e s were used i n t u r n t o c a l c u l a t e the 1/d 2 v a l u e f o r each peak. Appendix I d e s c r i b e s how the 1/d 2 v a l u e s , r e c i p r o c a l s of the i n t e r - p l a n a r s p a c i n g s , a r e d i r e c t l y r e l a t e d t o the M i l l e r i n d i c e s and how they can be graphed i n a way t o a i d i n t h e i n d e x i n g . P l o t s of 1/d 2 v s . I2 were produced by the t e c h n i q u e d e s c r i b e d i n Appendix I . A l l the c r y s t a l s of i n t e r e s t t o t h i s t h e s i s had a p p r o x i m a t e l y h e x a g o n a l symmetry, which made a n a l y s i s of the d a t a much s i m p l e r . I f a c r y s t a l i s h e x a g o n a l , i t i s p o s s i b l e t o draw s t r a i g h t l i n e s t h r o u g h d a t a p o i n t s t h a t had the same h and k i n d i c e s but d i f f e r e n t / i n d i c e s , and a l l of the l i n e s would be p a r a l l e l t o the {0,0,1) d a t a . The t r i c l i n i c d i s t o r t i o n s i n L i x M o S 2 a r e so s m a l l t h a t l i n e s drawn t h r o u g h the da t a o f t e n appeared s t r a i g h t and p a r a l l e l t o the naked eye, and the peaks c o u l d be in d e x e d u s i n g a scheme v e r y s i m i l a r t o t h a t used f o r a hexagonal l a t t i c e . I n s p i t e of the n e a r - h e x a g o n a l symmetry, i t was not im m e d i a t e l y o b v i o u s which index s h o u l d be a f f i x e d t o a peak. I f a c r y s t a l i s m o n o c l i n i c , a l l l i n e s on t h e graph s h o u l d be p a r a l l e l t o the (0,0,1) d a t a . However i t was v e r y common t o be a b l e t o draw s e v e r a l l i n e s t h a t were a l l p a r a l l e l t o w i t h i n the u n c e r t a i n t y of the p l o t t i n g , but p a s s i n g t h r o u g h d i f f e r e n t , c l o s e l y spaced, d i f f r a c t i o n peaks. The b u l k of the i n d e x i n g p r o c e d u r e i n v o l v e d e x a m i n a t i o n and p o s s i b l e r e j e c t i o n of c a n d i d a t e l i n e s . I f a l l c a n d i d a t e l i n e s from the 1/d 2 v s . I2 1 17 graph were r e j e c t e d , the c r y s t a l was assumed t o be t r i c l i n i c and a graph of 1/d 2 - I 2 S 3 3 / V 2 v s . / was p l o t t e d . 3 L i n e s on t h i s graph were drawn, examined, and r e j e c t e d i n the same way. E v e n t u a l l y , when a l l the c a n d i d a t e l i n e s were r e j e c t e d except f o r those c o n s i s t e n t w i t h the d a t a , the c r y s t a l was f u l l y i n d e x e d . The b e s t p r o c e d u r e was t o index the most i n t e n s e l i n e s f i r s t . Low a n g l e peaks were u s u a l l y b oth w e l l s e p a r a t e d and i n t e n s e , and t h e i r / index c o u l d o f t e n be unambiguously r e a d o f f the graph. Each time another d i f f r a c t i o n spot was i n d e x e d , the h and k i n d i c e s chosen had t o be c o n s i s t e n t w i t h the p r e v i o u s l y i n d exed peaks. A l l l i n e s on the graph p a s s i n g t h r o u g h a newly indexed peak were used t o p r e d i c t the i n d i c e s of o t h e r peaks and then a f i t was made t o the d a t a . The a c c u r a c y of t h e u n i t c e l l p arameters improves each time a peak i s s u c c e s s f u l l y i n d e x e d , and u s u a l l y a l l the weak d i f f r a c t i o n l i n e s can be ac c o u n t e d f o r once the major peaks are i n d e x e d . The d e c i s i o n t o ac c e p t or r e j e c t a f i t was based on d i f f e r e n c e s between p r e d i c t e d and obser v e d d i f f r a c t i o n peak p o s i t i o n s . A d e v i a t i o n of 0.1° was u n a c c e p t a b l e , and agreement of 0.01° or b e t t e r was common. I f the d e v i a t i o n was t o o g r e a t then the i n d e x i n g of a t l e a s t one l i n e was wrong, and the most r e c e n t a s s u m p t i o n s were r e j e c t e d . I f a l l the c a n d i d a t e l i n e s were r e j e c t e d , an e a r l i e r i n d e x i n g was i n e r r o r and the p r o c e s s would s t a r t a g a i n . 3 The S 3 3 and V terms are d e f i n e d i n e q u a t i o n A1.18 of Appendix I . 1 18 I f a f i t was not r e j e c t e d i m m e d i a t e l y , i t would be used t o g e n e r a t e a s e t of l a t t i c e parameters which were used i n t u r n t o produce a l i s t of a l l p o s s i b l e d i f f r a c t i o n l i n e p o s i t i o n s . The l i s t was compared w i t h the o r i g i n a l d a t a . I f a p r e d i c t e d l i n e c o r r e s p o n d e d w i t h one t h a t was not y e t i n d e x e d , the p o s s i b l e index was r e c o r d e d . I f t h e r e were u n p r e d i c t e d l i n e s i n the d a t a , the i n d e x i n g scheme, was wrong or a t l e a s t i n a d e q u a t e , and the p r o c e s s would s t a r t a g a i n . E a r l y i t e r a t i o n s would u s u a l l y index l i n e s w i t h low 20 v a l u e s because th e s e peaks were u s u a l l y w e l l s e p a r a t e d and i n t e n s e . There was a l a r g e u n c e r t a i n t y i n the l a t t i c e p arameters d e t e r m i n e d from t h i s r e s t r i c t e d d a t a s e t , so i t was n e c e s s a r y t o index the h i g h a n g l e peaks t o o b t a i n good e s t i m a t e s of the p a r a m e t e r s . The computer programs used i n the p r o c e s s d e s c r i b e d above a r e l i s t e d i n Appendix I I . Much of the g r a p h i c a l work r e q u i r e d f i n d i n g s t r a i g h t l i n e s on the graphs. The f a s t e s t way of c h e c k i n g the q u a l i t y of a s t r a i g h t l i n e was t o use a pocket c a l c u l a t o r t o compute a l i n e a r r e g r e s s i o n c o e f f i c i e n t , r . For example, f o u r p o i n t s i n the (1,0,1) f a m i l y of the 1.7V m a t e r i a l had r=0.9999. The (1,0,1) f a m i l y of p o i n t s i n the t r i c l i n i c l a t t i c e were almost l i n e a r , t y p i c a l l y h a v i n g r=*0.98. The d e v i a t i o n of the t r i c l i n i c d a t a from l i n e a r i t y was d i f f i c u l t t o d e t e c t w i t h the naked eye, but t h e r e were s i g n i f i c a n t d i f f e r e n c e s between p r e d i c t e d and o b s e r v e d l i n e p o s i t i o n s . As a r u l e - o f - t h u m b , a l i n e was o n l y c o n s i d e r e d t o be s t r a i g h t i f the r v a l u e was g r e a t e r than 0.99. T h i s c u t - o f f based s o l e l y on the need t o reduce the number of 119 c a n d i d a t e l i n e s and a s s o c i a t e d i n d e x i n g schemes t o a manageable number. The d e t a i l e d a n a l y s i s used a s o p h i s t i c a t e d c u r v e f i t t i n g r o u t i n e , and made no use of r v a l u e s . 6.6 A n a l y s i s of L i M o S 2 The s i m p l e s t c r y s t a l examined was the 1.7V m a t e r i a l , l a t t i c e A. The m a t e r i a l had been d r i v e n i n t o a pure phase by l e t t i n g the powder r e a c t w i t h hot n - b u t y l l i t h i u m f o r one month. I t was b e l i e v e d t h a t x=1 f o r t h i s m a t e r i a l . The parameters d e t e r m i n e d f o r l a t t i c e A were v e r y s i m i l a r t o those d e t e r m i n e d f o r the x=1 /3-phase. T h i s s e c t i o n d e s c r i b e s i n d e t a i l t he v a r i o u s s t e p s and a p p r o x i m a t i o n s made t o do the s t r u c t u r a l d e t e r m i n a t i o n . The f i r s t a p p r o x i m a t i o n assumed a hexagonal l a t t i c e w i t h the parameters of /3-phase: a=3.36A and c=6.294A. The {0,0,1) peaks were c o n s i s t e n t w i t h t h i s e s t i m a t e . The complete d a t a s e t was graphed a c c o r d i n g t o the proce d u r e o u t l i n e d i n s e c t i o n s A1.4 and A1.5, and s e v e r a l l i n e s were drawn p a r a l l e l t o the {0,0,1) d a t a . Some of t h e s e l i n e s had s l o p e s and i n t e r c e p t s c o n s i s t e n t w i t h the hexagonal a p p r o x i m a t i o n , but where the a p p r o x i m a t i o n p r e d i c t e d two peaks between 30° and 35° of 26, seven were o b s e r v e d . T h i s i m m e d i a t e l y suggested e i t h e r lower symmetry, c o - e x i s t i n g l a t t i c e s , or i n c o r r e c t a s s u m p t i o n s . B r i e f e x a m i n a t i o n s were made of a l l t h r e e p o s s i b i l i t i e s , and a low symmetry c r y s t a l seemed t o be the be s t e x p l a n a t i o n . A l o s s of symmetry i s u s u a l l y accompanied by an i n c r e a s e i n the number of d i f f r a c t i o n peaks. As d e s c r i b e d i n s e c t i o n A1.5, 120 the hexagonal (1,0,0) peak may be s p l i t i n t o e i t h e r two or t h r e e peaks i f the b a s a l p l a n e i s d i s t o r t e d . F i g u r e 27 shows d a t a from 30° t o 33°. A s i n g l e (1,0,0) l i n e was e x p e c t e d t h e r e , but f o u r l i n e s appeared i n the d a t a . At l e a s t one would have t o be i g n o r e d a t t h i s l e v e l of a p p r o x i m a t i o n . The t h r e e low a n g l e peaks i n F i g u r e 27 c o r r e s p o n d e d t o the i n t e r c e p t s of t h r e e of the l i n e s drawn on the graph. The 30.58°, 31.09°, and ~31.25° peaks were i n i t i a l l y i n d e xed as (1,-1,0), (1,0,0), and (0,1,0) r e s p e c t i v e l y . D i f f r a c t i o n p a t t e r n s were g e n e r a t e d based on t h i s i n d e x i n g , and the (1,0,1), (0,1,1), and (1,-1,1) peaks were q u i c k l y i d e n t i f i e d and i n d e x e d . I t was p o s s i b l e t o account f o r a l l but t h r e e peaks i n the d a t a by assuming t h a t L i x M o S 2 has a s m a l l m o n o c l i n i c p e r t u r b a t i o n from the hexagonal l a t t i c e . However, s i n c e a s u c c e s s f u l i n d e x i n g scheme must account f o r a l l the d a t a , a t l e a s t one of the assumptions mentioned was wrong. F i g u r e 28 i s a p l o t of 1/d 2 v s . I2 f o r the d a t a measured f o r the 1.7V m a t e r i a l . I t was p o s s i b l e t o draw a l i n e p a r a l l e l t o the (0,0,1) d a t a t h r o u g h the t h r e e e x t r a p o i n t s . The i n t e r c e p t was a t a 1/d 2 v a l u e one q u a r t e r of t h a t a s s o c i a t e d w i t h the (1,-1,0) peak. I f t h e s e p o i n t s were c o r r e c t l y i n d e x e d , one would e x p e c t d i f f r a c t i o n l i n e s c o r r e s p o n d i n g t o 1=0 and 1=1. F u r t h e r measurements were made t o s e a r c h f o r t h e s e e x t r a l i n e s , and weak d i f f r a c t i o n peaks were found a t e x a c t l y the p o s i t i o n s p r e d i c t e d . T h i s i m p l i e d t h e r e was an e r r o r i n the assumption of the s i z e of t h e u n i t c e l l . The d a t a c o u l d be u n d e r s t o o d by 121 CO o o 12000 11500 11000 -10500 -10000 9500 9000 I I I I I I I I I I I I I* I — • . '. t * — * • . • * — • • • * *5 % / + -• • & • • • ***** I I I I I I I I I I I I I I 30.0 30.6 31.2 31.8 Two Theta 32.4 33.0 F i g u r e 27 D i f f r a c t i o n P a t t e r n of LiMoS 2 f o r 3O°£20£33° If LiMoS 2 was hexagonal, then one d i f f r a c t i o n l i n e would be expected i n t h i s angular r e g i o n , but four were observed. The broad peak near 31.2° i s a c t u a l l y a doublet that can be r e s o l v e d by expanding the 20 s c a l e . The lowest peak was indexed as (2,-2,0), the doublet as (2,0,0) and (0,2,0), and the 32.65° peak as (1,-1,2), 122 0 2 4 6 8 10 13 16 19 22 25 F i g u r e 28 G r a p h i c a l A n a l y s i s — L i M o S j The L i M o S 2 d a t a was p l o t t e d by the t e c h n i g u e d e s c r i b e d i n Appendix I . The t r i p l e t of l i n e s i n t e r s e c t i n g 1/d 2 a t ~0.12 (l/A2) can not be r e s o l v e d on th e graph above. The t h r e e e x t r a peaks d e s c r i b e d i n t h e t e x t a r e a l s o shown (•), but t h e two peaks t h a t c o r r e s p o n d e d t o (1,-1,0) and (1.-1,1) were not p l o t t e d because t h e y were not o b s e r v e d d u r i n g t h e p r e l i m i n a r y i n v e s t i g a t i o n s of the m a t e r i a l . L i M o S 2 was c o n f i r m e d t o have a m o n o c l i n i c symmetry, and the u n i t c e l l had t o have more t h a n one f o r m u l a u n i t . 123 d o u b l i n g the s i z e of the u n i t c e l l ' s b a s a l p l a n e i n both the a and B d i r e c t i o n s . " Expanding the s i z e of the u n i t c e l l r e q u i r e s some j u s t i f i c a t i o n because a s u f f i c i e n t l y l a r g e u n i t c e l l would have so many p o s s i b l e d s p a c i n g s t h a t i t would be p o s s i b l e t o f i n d an index f o r a l m o s t any peak. The d i f f r a c t i o n p a t t e r n s of a l l the L i x M o S 2 phases were almost h e x a g o n a l , and the i n t e n s e x-ray peaks were a c c o u n t e d f o r by a s m a l l u n i t c e l l t h a t c o n t a i n e d one f o r m u l a u n i t . However, a l l the phases a l s o had weak d i f f r a c t i o n l i n e s t h a t c o u l d be u n d e r s t o o d by assuming t h e r e were two or more n o n - i d e n t i c a l f o r m u l a u n i t s i n the u n i t c e l l . Other t r a n s i t i o n m e t a l d i c h a l c o g e n i d e s , R eS 2, ReSe 2, and T c T e 2 , were a l s o found t o have f o u r f o r m u l a u n i t s i n a 2 by 2 by 1 n e a r - h e x a g o n a l t r i c l i n i c u n i t c e l l . ( W i l d e r v a n c k and J e l l i n e k , 1971; A l c o c k and K j e k s h u s , 1965). S t a g i n g can a l s o i n c r e a s e the number of f o r m u l a u n i t s i n the u n i t c e l l . A t t e m p t s were a l s o made t o index the d i f f r a c t i o n p a t t e r n s on the a s s u m p t i o n of c o - e x i s t i n g hexagonal l a t t i c e s . I t was i m p o s s i b l e t o e x p l a i n a l l of the d a t a t h i s way. I f t h e r e were hexagonal l a t t i c e s then the 1/d 2 v a l u e of the (1,0,0) peak would have one t h i r d the 1/d 2 v a l u e of the (1,1,0) peak. None of the L i x M o S 2 phases had d a t a t h a t s a t i s f i e d t h e s e r e l a t i o n s . 4 I t was p o s s i b l e t o index t h i s m a t e r i a l on a 2 by 1 by 1 u n i t c e l l , but the r e s u l t s p r e s e n t e d here use a 2 by 2 by 1 u n i t c e l l t o be c o n s i s t e n t w i t h the o t h e r phases. 124 6.7 A n a l y s i s of the T r i c l i n i c L a t t i c e s A l l the phases of c r y s t a l l i n e L i x M o S 2 were examined u s i n g the g r a p h i c a l p r o c e d u r e s of s e c t i o n A1.5. Each phase was found t o be a p p r o x i m a t e l y h e x a g o n a l . The l a y e r s p a c i n g and o u t - o f - p l a n e d i s p l a c e m e n t were d e t e r m i n e d f o r each sample, but o n l y one l a t t i c e was f u l l y i n d exed assuming a m o n o c l i n i c s t r u c t u r e . The o t h e r phases had t o have the lower symmetry t r i c l i n i c l a t t i c e . T h i s s e c t i o n w i l l f i r s t d e s c r i b e the problems e n c o u n t e r e d when t r y i n g t o index L i x M o S 2 , and a d e s c r i p t i o n of the t e c h n i q u e s used t o overcome t h e s e d i f f i c u l t i e s w i l l be g i v e n . The i n d e x i n g p r o c e d u r e w i l l then be d e s c r i b e d . Most of the i n d e x i n g was done w i t h the a i d of the g r a p h i c a l t e c h n i q u e d e s c r i b e d i n s e c t i o n A1.6. I d e a l l y , a l i n e would o n l y be drawn on a graph i f i t were d e f i n e d by f i v e or more p o i n t s . However, most L i x M o S 2 phases had fewer d i f f r a c t i o n peaks than e x p e c t e d , and the c a n d i d a t e l i n e s used i n t h e i n d e x i n g schemes were o c c a s i o n a l l y based o n l y on two p o i n t s . The number of c a n d i d a t e s was q u i t e l a r g e because t h e r e were so many ways of d r a w i n g l i n e s t h r o u g h two p o i n t s of the d a t a . There were o f t e n t o o few p o i n t s d e f i n i n g a c a n d i d a t e l i n e t o use, v a l i d l y , a r e g r e s s i o n c o e f f i c i e n t c r i t e r i o n t o r e j e c t the l i n e s . S e c t i o n A1.6 d e s c r i b e s how the s l o p e s of each of two of the c a n d i d a t e l i n e s had t o e q u a l the s l o p e of the t h i r d , so o n l y s e t s of t h r e e c a n d i d a t e l i n e s t h a t s a t i s f i e d t h i s c o n d i t i o n need be c o n s i d e r e d . T h i s c o n d i t i o n was not u s e f u l here because the s l o p e s were so s m a l l t h a t the u n c e r t a i n t y i n the peak p o s i t i o n s caused s i g n i f i c a n t u n c e r t a i n t i e s i n the s l o p e s . E v e n t u a l l y , a l l 125 the p o s s i b i l i t i e s c o u l d have been checked, but i t was more u s e f u l t o f i n d o t h e r ways t o r e s t r i c t the number of c a n d i d a t e s t o be examined. S i n c e the l a t t i c e s were almost hexagonal i t was p o s s i b l e t o i d e n t i f y the 1=0 peaks g r a p h i c a l l y and use the n u m e r i c a l r e l a t i o n s h i p s t o c o n f i r m t h e i r i n d e x i n g . T h i s d e t e r m i n e d the shape of the b a s a l p l a n e . A l l l i n e s drawn on the 1/d 2 - I2 S 3 3 / V 2 v s . / graph were c o n s t r a i n e d t o pass through one of the 1=0 p o i n t s . Hexagonal l a t t i c e s have many s i x - f o l d degenerate i n t e r - p l a n a r s p a c i n g s , d, so the d i f f r a c t i o n p a t t e r n of a s m a l l t r i c l i n i c d i s t o r t i o n of a hexagonal l a t t i c e would be e x p e c t e d t o have s e v e r a l a n g u l a r r e g i o n s i n which t h e r e were s i x c l o s e l y - s p a c e d peaks. The t r i c l i n i c phases of L i x M o S 2 u s u a l l y o n l y had t h r e e peaks i n t h e s e r e g i o n s . C l o s e i n s p e c t i o n of the d a t a r e v e a l e d t h a t t h e r e were no d e g e n e r a c i e s c a u s i n g peaks t o o v e r l a p , so the m i s s i n g peaks had t o be e x t i n c t . S e v e r a l d i f f e r e n t c r y s t a l s t r u c t u r e s were modeled, and the 1T s t r u c t u r e was found t o have a d i s t i n c t i v e e x t i n c t i o n p a t t e r n . 5 The / =-5, -1, 3, and 4 peaks were a l l v e r y weak or e x t i n c t i n the (-2,0,-1), (0,2,1), and (2,-2,1) s e r i e s . The number of c a n d i d a t e l i n e s t h a t f u l f i l l e d t h e s e c o n d i t i o n s was s u f f i c i e n t l y s m a l l t o a l l o w r e l a t i v e l y easy i n d e x i n g of the d i f f r a c t i o n p a t t e r n s . L a t t i c e B (1.93V) was r e l a t i v e l y easy t o index because t h e r e was o n l y one peak i n the r e g i o n where the (2,0,1) d a t a was e x p e c t e d . T h i s l i n e was assumed t o be t h r e e - f o l d d e g e n e r a t e , and 5 The m o d e l l i n g i s e x p l a i n e d i n more d e t a i l i n s e c t i o n 6.9. 126 l i n e s drawn t h r o u g h t h i s p o i n t and the t h r e e p o i n t s i n the (2,0,0) r e g i o n c o u l d account f o r most of the d a t a . 6 There was no need t o use the 1T model. The l a t t i c e was indexed on a 2 by 2 by 1 u n i t c e l l . The 2.23V, 2.56V, and 2.63V m a t e r i a l s , l a t t i c e s C, D, and E r e s p e c t i v e l y , had v e r y s i m i l a r d i f f r a c t i o n p a t t e r n s a n d . a l l showed two peaks i n a r e g i o n of 28 where no peaks c o u l d o c c u r i f the u n i t c e l l was 2 by 2 by 1 or s m a l l e r . The number of i t e r a t i o n s needed t o index t h e s e c r y s t a l s was reduced by assuming the e x t r a l i n e s i n each phase had the same M i l l e r i n d i c e s and t h a t the t h r e e l a t t i c e s c o u l d be indexed as a s e t . Samples of pure l a t t i c e C were produced v i a d e l i t h i a t i o n i n p r o p a n o l . The i n t e n s e d i f f r a c t i o n peaks were indexed by assuming t h a t the l a t t i c e had a 2 by 2 by 1 1T s t r u c t u r e , and the o t h e r peaks c o u l d be indexed by expanding the u n i t c e l l t o 2 by 2 by 2, t o 3 by 3 by 1 or t o 3 by 2 by 1. However, i t was d i f f i c u l t t o determine which s i z e u n i t c e l l was c o r r e c t j u s t by e xamining l a t t i c e C. The r e s u l t s of the l a t t i c e C i n d e x i n g were compared w i t h l a t t i c e s D and E d a t a t o t r y and f i n d an i n d e x i n g scheme c o n s i s t e n t w i t h a l l t h r e e l a t t i c e s . L a t t i c e s D and E were a l s o i n d e x e d by i n i t i a l l y assuming a 2 by 2 by 1 1T u n i t c e l l , and then g e n e r a t i n g a s e t of d i f f r a c t i o n peak p o s i t i o n s based on l a r g e r u n i t c e l l s . The e x t r a d i f f r a c t i o n l i n e s c o u l d o n l y be g i v e n the same i n d i c e s i n a l l t h r e e c a s e s i f the u n i t c e l l was 2 6 The t r i p l e degeneracy s u g g e s t s t h a t i t may be p o s s i b l e t o index t h i s phase on a u n i t c e l l t h a t has h i g h e r symmetry than t r i c l i n i c . I had p a r t i a l s u c c e s s i n d e x i n g the c r y s t a l oh o t h e r u n i t c e l l s , but I d i d not pursue t h i s because i t seemed more m e a n i n g f u l t o d e s c r i b e the m a t e r i a l i n terms of a n e a r l y hexagonal l a t t i c e . 1 27 by 2 by 2. T h i s suggested t h a t the c r y s t a l s may have been s t a g e d , and i f t h i s were so then t h e r e s h o u l d have been e x t r a (0,0,1) s p a c i n g s . S e v e r a l v e r y s m a l l d i f f r a c t i o n l i n e s were found a t the p o s i t i o n s e x p e c t e d f o r the two l a y e r u n i t c e l l , but they were o b s c u r e d by o t h e r peaks and i t was not p o s s i b l e t o unambiguously i d e n t i f y them as (0,0,1) peaks. In s p i t e of t h i s , a 2 by 2 by 2 u n i t c e l l b e s t e x p l a i n e d t h r e e of the t r i c l i n i c p hases. 6.8 L i x M o S 2 L a t t i c e Parameters The p r e c e e d i n g s e c t i o n s d e s c r i b e d the methods used t o deter m i n e the l a t t i c e p a r a m e t e r s . T h i s s e c t i o n summarizes the r e s u l t s . The g r a p h i c a l t e c h n i q u e was used t o i d e n t i f y s e t s of d i f f r a c t i o n peaks t h a t a l l had the same h and k i n d i c e s , but d i f f e r e n t / i n d i c e s . E s t i m a t e s of the l a t t i c e p arameters were made based on t h e s e l i m i t e d s e t s of d a t a , and t h e s e e s t i m a t e s o f t e n made i t p o s s i b l e t o index the o t h e r d i f f r a c t i o n l i n e s . Parameter r e f i n e m e n t based on a l l the in d e x e d peaks was done n u m e r i c a l l y . T a b l e I I I i s a l i s t of the l a t t i c e p a r a m e t e r s and c e l l volumes f o r the phases of L i x M o S 2 . The two l a y e r c e l l s a r e l i s t e d as 2 X the s i n g l e l a y e r s p a c i n g t o a l l o w e a s i e r comparisons of the d a t a . D i f f e r e n t b a t c h e s of powder had almost i d e n t i c a l l a t t i c e p a r a m e t e r s . The 1/d 2 v a l u e s f o r d i f f e r e n t samples o n l y v a r i e d i n the f o u r t h or f i f t h s i g n i f i c a n t f i g u r e s . The a and b d a t a i s r e p r o d u c i b l e t o ±0.002A, t h e c a x i s d a t a t o TABLE I I I L1 xMoS; L a t t i c e Parameters L a t t i c e C e l l x a (A) b (A) c (A) a C ) $ C) y (' ) Volume V o l t a g e (A 3 ) (V) A ' 1.70 1.00 6.798 6.754 6.262 90.0 90.0 121.2 4 X 61.48 B 1.93 0.75 6.691 6.655 6.337 90.4 89.8 120.3 4 X 60.91 C 2.23 0.50 6.605 6.598 2 X 6.346 92.4 87.4 120.4 8 X 59.56 D 2.56 0.27 6.592 6.592 2 X 6.354 92.2 87.8 120.8 8 X 59.23 E 2.63 0.12 6.575 6.574 2 X 6.346 90.8 88.0 120.7 . 8 X 58.92 F 2.75 0.00 n X 5.94 T h i s t a b l e i s a summary of the l a t t i c e parameters of 1T-Li xMoS;. The data i s a s s o c i a t e d with the " c h a r g i n g " h a l f of the h y s t e r e s i s c u r v e . The l e t t e r s used to d e f i n e the l a t t i c e s a r e d e f i n e d i n F i g u r e 19. The " c e l l v o l t a g e " was the p o t e n t i a l of the^ L i / L i M o S i c e l l when the data was recorded, and "x" i s the approximate x v a l u e of the ho s t . The a, b, c, a, 0, and y columns are the l a t t i c e parameters. The m u l t i - l a y e r u n i t c e l l s a r e l i s t e d as m u l t i p l e s of the s i n g l e l a y e r s p a c i n g . The u n c e r t a n t i e s a re 10.002A i n the a and b d a t a , ±0.001A i n the c d a t a , and ±0.1' i n the a n g u l a r d a t a . The f i n a l column i s the volume of the u n i t c e l l l i s t e d as a m u l t i p l e of the volume p e r MoSz f o r m u l a u n i t . The A and B l a t t i c e s have f o u r formula u n i t s per u n i t c e l l , and the C, D, and E l a t t i c e s have e i g h t f o r m u l a u n i t s per u n i t c e l l . The F l a t t i c e was not f u l l y indexed, but the (0,0.1) d i f f r a c t i o n peaks were i d e n t i f i e d . The 5.94A v a l u e i s the l a y e r s p a c i n g , but the number of l a y e r s i n the u n i t c e l l , n, was not determined. I t i s a l s o l i k e l y t h a t the F l a t t i c e i s t r i c l i n i c , and so the c a x i s may be longer than the l i s t e d l a y e r s p a c i n g . —* ro 129 ±0.001A, and the a n g u l a r data t o ±0.1°. The 2.8V l a t t i c e was not f u l l y i n d exed because the m a t e r i a l s l o w l y c o n v e r t e d t o alpha-phase and t h e r e were s e v e r a l o v e r l a p p i n g d i f f r a c t i o n p a t t e r n s . I t was p o s s i b l e t o i d e n t i f y two s e t s of (0,0,1) l i n e s , and i t would appear t h a t the 5.94A s p a c i n g c o r r e s p o n d e d t o the low x phase. The o t h e r (0,0,1) l i n e s were a s s o c i a t e d w i t h a 6.23A1 l a y e r s p a c i n g . The l a t t e r s p a c i n g w i l l be d i s c u s s e d l a t e r . The f i n a l column of Table I I I shows the c r y s t a l volume per MoS 2 f o r m u l a u n i t . The c e l l volumes d e c r e a s e m o n o t o n i c a l l y w i t h d e c r e a s i n g l i t h i u m c o n t e n t . The volume per f o r m u l a u n i t of the 1.7V m a t e r i a l i s o n l y 4% g r e a t e r than t h a t of the 2.63V m a t e r i a l . F o r compar i s o n , c o n v e r s i o n from 2H-MoS 2 t o 1T-LiMoS 2 causes a 15% ex p a n s i o n of the l a t t i c e . The h y s t e r e s i s noted i n the e l e c t r o c h e m i s t r y can a l s o be seen i n the s t r u c t u r a l d a t a . C e l l s t h a t had been f u l l y d i s c h a r g e d but never c y c l e d above ~2.6V have a l a y e r s p a c i n g of ~6.33A between ~2.0V and ~2.6V. I f a c e l l i s charged t o 2.8V then two m a t e r i a l s form. One of thes e has a 5.94A l a y e r s p a c i n g , and i s p r o b a b l y phase F. The o t h e r m a t e r i a l has a 6.23A1 s p a c i n g . I f t he c e l l i s then d i s c h a r g e d , the l a y e r s p a c i n g of L i x M o S 2 between 2.0V and 2.8V w i l l be 6.23A-6.25A. I f the c e l l i s f u l l y d i s c h a r g e d and then c h a r g e d , i t w i l l a g a i n have a ~6.33A l a y e r s p a c i n g between 2.0V and 2.6V. The ~6.23A" s p a c i n g i s a l s o c o n s i s t e n t w i t h the 0-phase s p a c i n g s measured by M a r c e l Py and shown here i n F i g u r e 23. A l l the d a t a shown i n T a b l e I I I a r e a s s o c i a t e d w i t h the c h a r g i n g of the c e l l . A f u l l a n a l y s i s of the 130 s t r u c t u r e s of the d i s c h a r g e d m a t e r i a l s was not performed, but t h e i r x - r a y p a t t e r n s were q u a l i t a t i v e l y s i m i l a r t o the p a t t e r n s a n a l y z e d above. I n t e r c a l a t i o n i s the r e v e r s i b l e i n s e r t i o n of an atom i n t o a host l a t t i c e w i t h o u t . major s t r u c t u r a l changes i n the h o s t . Chapter 5 e s t a b l i s h e d t h a t l i t h i u m c o u l d be i n s e r t e d and removed from the MoS 2. T h i s c h a p t e r p r e s e n t e d e v i d e n c e t h a t t h e r e were no major s t r u c t u r a l d i f f e r e n c e s i n t h e phases of L i x M o S 2 . The L i / 1 T - L i x M o S 2 system i s an i n t e r c a l a t i o n system. 6.9 C o n f i r m a t i o n of Near IT Symmetry X-ray s t u d i e s by Wainwright (1978) and by Py (1983) i n d i c a t e d the /3-phase might have a 1T s t r u c t u r e . S e v e r a l o t h e r t r a n s i t i o n m e t a l d i c h a l c o g e n i d e i n t e r c a l a t i o n h o s t s had 1T s t r u c t u r e s , and the a t o /3 phase t r a n s i t i o n c o u l d be q u a l i t a t i v e l y e x p l a i n e d by the 2H t o 1T c o n v e r s i o n model, but n e i t h e r x - r a y study was a b l e t o prove /3-phase was a 1T c r y s t a l . T h e i r s t r u c t u r a l a n a l y s i s d i d not t o t a l l y p r e c l u d e o t h e r p o l y t y p e s . I t was a l s o p o s s i b l e (but q u i t e u n l i k e l y ) t h a t t h e h i g h temperature grown m a t e r i a l would have a d i f f e r e n t symmetry. T h i s s e c t i o n w i l l d e s c r i b e how a method based on the t r i c l i n i c d i s t o r t i o n s i n c r y s t a l l i n e L i x M o S 2 was used t o examine the atomic arrangement. S e v e r a l models were t e s t e d and o n l y the 1T model a d e q u a t e l y e x p l a i n e d the d a t a . The b e s t s t r u c t u r a l a n a l y s i s of 0-phase had been done by Py (1983). He computed the i n t e n s i t y p a t t e r n s f o r f o u r models of 131 a t omic s t a c k i n g . The c a l c u l a t e d i n t e n s i t i e s f o r b o t h AbC s t a c k i n g and AbA s t a c k i n g were s i m i l a r t o the o b s e r v e d d a t a . The 1T p o l y t y p e (AbC s t a c k i n g ) was the b e s t f i t t o the o b s e r v a t i o n s , but n e i t h e r model was a p e r f e c t f i t t o the d a t a . I t was p o s s i b l e t h a t /3-phase had an AbA s t r u c t u r e . P r e f e r r e d o r i e n t a t i o n makes i t d i f f i c u l t t o make a m e a n i n g f u l comparison between c a l c u l a t e d and o b s e r v e d i n t e n s i t i e s f o r MoS 2 samples. S t a n d a r d i n t e n s i t y c a l c u l a t i o n s assume t h a t the i n d i v i d u a l c r y s t a l l i t e s i n the powder have a l l p o s s i b l e o r i e n t a t i o n s . I f the o r i e n t a t i o n s a r e not random then some d i f f r a c t i o n l i n e s w i l l be more i n t e n s e than p r e d i c t e d and o t h e r s w i l l be l e s s i n t e n s e . I t would be p o s s i b l e t o compensate i f the d i s t r i b u t i o n of p a r t i c l e o r i e n t a t i o n s was known, but t h a t i n f o r m a t i o n was not a v a i l a b l e . I w i l l show t h a t many s e t s of atomic p l a n e s a r e e q u a l l y l i k e l y t o d i f f r a c t the x - r a y beam i n s p i t e of the p r e f e r r e d o r i e n t a t i o n . U s u a l l y t h e s e s e t s of p l a n e s have degenerate d s p a c i n g s , but t h i s degeneracy i s l i f t e d i n the c r y s t a l s w i t h t r i c l i n i c d i s t o r t i o n s . The i n t e n s i t y of each of t h e s e n e a r l y - d e g e n e r a t e d i f f r a c t i o n peaks can be c a l c u l a t e d and m e a n i n g f u l l y compared t o the measured i n t e n s i t i e s even i f the degree of p r e f e r r e d o r i e n t a t i o n i s unknown. The MoS 2 c r y s t a l s formed as t i n y p l a t e l e t s which would o f t e n l i e w i t h the c a x i s p e r p e n d i c u l a r t o the s u b s t r a t e . The o r i e n t a t i o n was not p e r f e c t , and many c r y s t a l l i t e s would have t h e i r c a x i s t i p p e d a t an a n g l e 4> from the n o rmal, as shown i n F i g u r e 29. I t was e q u a l l y l i k e l y f o r a c r y s t a l l i t e i n a powder 132 Normal \ \ \ ft \ V \ Substrate \ F i g u r e 29 P r e f e r r e d O r i e n t a t i o n The MoS 2 p l a t e l e t s tended t o a l i g n with t h e i r c axes p e r p e n d i c u l a r t o the s u b s t r a t e . The o r d e r i n g was not p e r f e c t . I t was e q u a l l y l i k e l y f o r a g i v e n c r y s t a l l i t e to have i t s c a x i s anywhere on the cone d e f i n e d by the angle 133 t o have i t s c a x i s anywhere on t h i s cone. The a and 5 v e c t o r s can have any o r i e n t a t i o n , s u b j e c t t o the c o n s t r a i n t t h a t they a r e a t the prop e r a n g l e s t o the c a x i s . The degree of p r e f e r r e d o r i e n t a t i o n d e t e r m i n e s t h e p r o b a b i l i t y of f i n d i n g a c r y s t a l w i t h i t s c a x i s t i p p e d a t some a n g l e <t> away from the p r e f e r r e d d i r e c t i o n , and t h i s v a r i e s from sample t o sample. The c r y s t a l o r i e n t a t i o n was o t h e r w i s e random. The x-ray powder d i f f T a c t o m e t e r was d e s i g n e d so t h a t a d i f f r a c t i o n peak was o n l y o b s e r v e d i f the atomic p l a n e s and the s u b s t r a t e were p a r a l l e l . Suppose t h a t some hexagonal c r y s t a l was a l i g n e d w i t h i t s c a x i s a t some a n g l e 4> from the normal t o the s u b s t r a t e . That c r y s t a l would o n l y c o n t r i b u t e t o the obser v e d p a t t e r n i f some s e t of atomic p l a n e s i n the c r y s t a l happened t o be a l i g n e d p a r a l l e l t o the s u b s t r a t e . The a n g l e between the c a x i s and the normals t o t h e s e p l a n e s would a l s o be <p. F u r t h e r m o r e , s i n c e the o r i e n t a t i o n of the c r y s t a l s around the c a x i s i s random, any p l a n e a t an a n g l e 0 t o c i s e q u a l l y l i k e l y t o be found p a r a l l e l t o the s u b s t r a t e . A g i v e n c r y s t a l may have s e v e r a l s e t s of p l a n e s t h a t a r e a t the same a n g l e t o some r e f e r e n c e v e c t o r . For example, a hexagonal l a t t i c e g e n e r a l l y has s i x s e t s of p l a n e s t h a t c r o s s the c a x i s a t the same a n g l e , and thes e p l a n e s a l l have the same d s p a c i n g . The p l a n e s w i t h degenerate d s p a c i n g s may not a l l d i f f r a c t the same x- r a y i n t e n s i t y . F i g u r e 30a shows a p r o j e c t i o n of a u n i t c e l l w i t h two atoms. Two s e t s of p l a n e s w i t h s p a c i n g s d!=d 2 a r e i n d i c a t e d . Suppose the p a t h d i s t a n c e |ABC| or |DEF| was e q u a l t o one x-ray w a v e l e n g t h . C o n s t r u c t i v e i n t e r f e r e n c e would then o c c u r a l o n g both p a t h s and a s t r o n g d i f f r a c t i o n spot would 134 Figure 30 Interference from Different Atomic Planes a) The symmetric unit ce l l (solid lines) with two atoms (o) has two degenerate d spacings, d,=d 2. If the distance marked |DEF| ( ) is equal to an integer number of x-ray wavelengths, constructive interference can occur from the planes marked d 2 . The distance |ABC| is equal to the distance |DEF|, but constructive interference along this path wi l l only occur if the length corresponds to an even number of wavelengths. If |ABC| corresponds to an odd number of wavelengths then destructive interference wi l l occur with waves diffracted along the path |AGH|. If a powder sample is used, there is no way to determine if the planes d 2 contribute to the observed diffraction intensity but not planes d , . b) If the symmetric unit c e l l is distorted, then d,#d2 and the x-rays scattered from the different sets of planes can be investigated separately. 135 be e x p e c t e d . However, t h e r e would a l s o be s c a t t e r i n g from the o t h e r p l a n e of atoms a t s p a c i n g d,/2. |AGH| i s one h a l f of |ABC|. X-r a y s s c a t t e r e d a l o n g |AGH| would be 180° out of phase w i t h x - r a y s s c a t t e r e d a l o n g |ABC|. There would be d e s t r u c t i v e i n t e r f e r e n c e of the s e waves and the d i f f r a c t i o n l i n e would be e x t i n c t . T h e r e f o r e , atomic p l a n e s w i t h the same d s p a c i n g s need not have e q u a l i n t e n s i t i e s , s c a t t e r e d from them. I t i s i m p o s s i b l e t o use a random powder t o dete r m i n e the r e l a t i v e d i f f r a c t e d i n t e n s i t i e s of d i f f e r e n t p l a n e s i f a l l of the p l a n e s have degenerate d s p a c i n g s . However, i f the degeneracy i s l i f t e d then i t i s p o s s i b l e t o examine each d i f f r a c t i o n spot s e p a r a t e l y . The change i n d s p a c i n g due t o a d i s t o r t i o n i s i l l u s t r a t e d i n F i g u r e 30b. Most phases of L i x M o S 2 were t r i c l i n i c d i s t o r t i o n s of a hexagonal l a t t i c e . The p r e f e r r e d o r i e n t a t i o n of samples made w i t h t h e s e powders had the l a y e r s of the c r y s t a l p a r a l l e l t o the s u b s t r a t e . The c a x i s was not normal t o the l a y e r s , but the d i f f e r e n c e was s m a l l and w i l l have no e f f e c t on the f o l l o w i n g d i s c u s s i o n . The a n g l e between the (0,0,1) p l a n e s and the ( h , k , l ) p l a n e i s COS0 = —- { S 1 3 h + S 2 3 k + S 3 3 / } 6.2 These terms a r e d e f i n e d i n s e c t i o n A1.6. A l s o i t s h o u l d be mentioned t h a t t h e r e i s no o b s e r v a b l e d i f f e r e n c e between <p and IT - </>. For example the (1,0,1), (0,1,1), (1,-1,1), (1,0,-1), (0,1,-1), and (1,-1,-1) l i n e s were a l l degenerate i n a hexagonal l a t t i c e , but t h i s degeneracy was l o s t w i t h the t r i c l i n i c 136 d i s t o r t i o n . However, S 1 3 and S 2 3 were q u i t e s m a l l , so a l l s i x s e t s of p l a n e s were at a p p r o x i m a t e l y the same a n g l e from the (0,0,1) p l a n e s . P r e f e r r e d o r i e n t a t i o n c o u l d then be i g n o r e d when examining t h e s e s i x peaks. Thus, c a l c u l a t e d and o b s e r v e d i n t e n s i t i e s c o u l d be r e a d i l y compared even though the degree of o r i e n t a t i o n was unknown. Models of a l l known t r a n s i t i o n m e tal d i c h a l c o g e n i d e s t r u c t u r e s ( p l u s a few s t a c k i n g arrangements t h a t had never been observed) were examined. T a b l e IV c o n t a i n s the complete l i s t of s t a c k i n g arrangements i n v e s t i g a t e d . Only the 1T (AbC) s t a c k i n g was c o n s i s t e n t w i t h the d a t a . The i n t e n s i t y p a t t e r n s f o r the v a r i o u s models were c a l c u l a t e d w i t h the SPECTRUM program d e s c r i b e d i n Appendix I I . The u n i t c e l l used the a, B, and c v e c t o r s d e t e r m i n e d f o r l a t t i c e B (1.93V), and the a tomic p o s i t i o n s i n the u n i t c e l l were a p p r o x i m a t e d by i d e n t i f y i n g t h e l o c a t i o n of an A s i t e as — — 2 1 O a + O E + z c , l o c a t i o n of a B as -g- a + —j- B + z c, and 1 -. 2 — l o c a t i o n of a C as —^~ a + B + z c, The z v a l u e s were i n i t i a l l y chosen t o keep the Mo-S bond l e n g t h a t about the 2.5A l e n g t h used i n Py's work. The z v a l u e s were v a r i e d i n o t h e r t e s t s , as w i l l be e x p l a i n e d l a t e r . A one l a y e r u n i t c e l l had the s u l f u r s h e e t s a t z=0 and z=0.5, and the molybdenum sheet at z=0.25. L i t h i u m was put a t z=0.75, but had l i t t l e e f f e c t on the model. I f the model c a l l e d f o r a two l a y e r c e l l then the l e n g t h of the c a x i s was d o u b l e d and the s u l f u r s were put a t z=0, 0.25, 0.5, and 0.75 and the molybdenums were put a t z=0.125 and 0.625. The l i n e s c a l c u l a t e d w i t h index / on a 1 l a y e r u n i t c e l l had t o 137 TABLE IV Models of Atomic S t a c k i n g T e s t e d 1T AbC AbA AbC CbA 2H AbA BcB 2H AbC AcB 2Ha AbA CbC 2Hb AbA BaB AbA AcA AbA BcA AbC BaC 3R AbA BcB CaC 3R AbC BcA CaB 4H AbA BcB AbA CbC 4Ha AbA CbC AbA BcB 4Hb AbA CbA CbC AbC 4Hc AbA CbC AcA BcB 4Hdj AbA CbA CbC BaC 4H d : i AbC AcA CbA CbC 6R AbA BcA BcB CaB CaC AbC 6R AbC AbC BcA BcA CaB CaB 6R AbC AcA BcA BaB CaB CbC I n t e n s i t y c a l c u l a t i o n s were made f o r each of the ato m i c s t a c k i n g arrangements l i s t e d above. The n o t a t i o n s a r e d e f i n e d i n Chapter 1. The r i g h t column i s the atomic s t a c k i n g g i v e n i n the ABC n o t a t i o n , and the l e f t column i s the common a b b r e v i a t i o n f o r the s t a c k i n g . There i s no c o n v e n t i o n t o d i s t i n g u i s h t h e v a r i o u s 3R and 6R arrangements. 138 be compared w i t h l i n e s w i t h index 21 on a two l a y e r c e l l . L a r g e r u n i t c e l l s were hand l e d i n a s i m i l a r way. Some of the computed r e s u l t s a r e d i s p l a y e d i n F i g u r e 31 which i s a p l o t of the i n t e n s i t i e s of v a r i o u s l i n e s . The i n d i c e s shown c o r r e s p o n d t o a 2 by 2 by 1 u n i t c e l l . The a c t u a l d a t a had t h r e e e x t i n c t l i n e s . Most models p r e d i c t e d t h a t a l l s i x of the l i n e s near the (2,0,1) peak would have almost e q u a l i n t e n s i t y . Three models p r e d i c t e d t h a t t h r e e of the s i x l i n e s would be e x t i n c t : 1T-AbC; 2Hb-AbA BaB; and AbA AcA s t a c k i n g s . The AbA BaB and the AbA AcA s t a c k i n g s were not a c c e p t a b l e f o r two rea s o n s . F i r s t , they p r e d i c t e d s e v e r a l v e r y i n t e n s e l i n e s t h a t were not ob s e r v e d . Second, the p r e d i c t e d e x t i n c t i o n p a t t e r n s d i d not f i t the d a t a . The 1T AbC model a c c o u n t e d f o r a l l the major d i f f r a c t i o n peaks and d i d not p r e d i c t i n t e n s e peaks t h a t were not o b s e r v e d . The t e s t s d e s c r i b e d above would have been u s e l e s s i f minor v a r i a t i o n s i n the atomic p o s i t i o n s had a major e f f e c t on the peak i n t e n s i t i e s . V a r i o u s t e s t s were made w i t h 1T models t o see how c h a n g i n g the atomic p o s i t i o n s a l t e r e d t h e c a l c u l a t e d i n t e n s i t i e s . The f i r s t s e t of t e s t s examined how the p a t t e r n v a r i e d i f the Mo-S bond l e n g t h was changed. The s p a c i n g between the molybdenum and s u l f u r s h e e t s of atoms was v a r i e d a c r o s s the range of 0.2c t o 0.3c. T h i s a l s o changed the s i z e of the van der Waals gap. Some c a l c u l a t i o n s were made w i t h the gap empty, and o t h e r s were made w i t h l i t h i u m i n the /? s i t e s i n t h e c e n t r e of the gap. Changing the Mo-S bond l e n g h had no s i g n i f i c a n t e f f e c t 139 IOC-0 100 I o a> IOO CD > • M M ° 0 o: IOO IT-AbC AbA AbC CbA 2Hb-AbA fioB I •• I - I I I I I I I (2PJ) (0,2,1) (2,-2,0 (2,0,2) fc.2,2) (2,-2,2) (2,0,-1) (0,2,-1) (2,-2,-D (2,0,-2) (0,2,-2) (2,-2,-2) Miller Index F i g u r e 31 C a l c u l a t e d D i f f r a c t i o n P a t t e r n s T h i s f i g u r e shows a subset of the d i f f r a c t e d x-ray i n t e n s i t i e s c a l c u l a t e d f o r four d i f f e r e n t models of atomic s t a c k i n g . The i n t e n s i t i e s are measured r e l a t i v e t o the most i n t e n s e d i f f r a c t i o n peak i n each model. The M i l l e r i n d i c e s at the bottom of the f i g u r e a r e those of the AbC and AbA data. The / index f o r an n - l a y e r u n i t c e l l i s n times the / index shown. The 1T c r y s t a l has three n e a r l y e x t i n c t l i n e s i n the (2,0,nX7) f a m i l y , but the AbA and AbC CbA models p r e d i c t 6 e q u a l l y i n t e n s e peaks i n t h i s angular r e g i o n . The 2Hb model p r e d i c t s 3 of the 6 l i n e s near (2,0,nX;) t o be s i g n i f i c a n t l y s t r o n g e r than the o t h e r s . The (2,0,nXJ) r e g i o n (not shown) f o r the IT and 2Hb s t a c k i n g a l s o show a p a t t e r n of 3 strong and t h r e e weak l i n e s , but the p o s i t i o n s of the inte n s e l i n e s p r e d i c t e d by the 2Mb model are not c o n s i s t e n t with the observed d a t a . I t was not p o s s i b l e to e x p l a i n the L i x M o S 2 data with any l a t t i c e other than 1T-AbC. 140 on the r e l a t i v e i n t e n s i t i e s of the s e t s of s i x n e a r l y degenerate l i n e s , such as those near the (2,0,7) peak. The second s e t of t e s t s used a s i m p l e t e c h n i q u e t o d i s p l a c e the atoms from the i d e a l i z e d ABC s i t e s . The atoms were c o n s t r a i n e d t o move a d i s t a n c e D 0 i n any one of s i x d i r e c t i o n s a l o n g a, B, c, - a , -B, or - c . The d i r e c t i o n of the d i s p l a c e m e n t was chosen by a d i e r o l l . A d i f f r a c t i o n p a t t e r n was computed a f t e r a l l 12 atoms i n the 2 by 2 by 1 u n i t c e l l had been moved. The p r o c e s s was r e p e a t e d up t o f i v e t i m e s . 7 A l l of the r e s u l t s f o r a g i v e n d i s p l a c e m e n t were q u a l i t a t i v e l y the same. There was no major change i n the c a l c u l a t e d i n t e n s i t y p a t t e r n s f o r d i s p l a c e m e n t s of Do=0.006A, 0.03A, or 0.06A. Random d i s p l a c e m e n t s of 0.3A, and 0.6A d i d cause some v a r i a t i o n i n the r e l a t i v e i n t e n s i t i e s of a few l i n e s , however t h r e e of the s i x n e a r l y degenerate l i n e s remained c o n s i d e r a b l y weaker than the o t h e r t h r e e . The atoms had t o be d i s p l a c e d a p p r o x i m a t e l y 1A t o cause major changes i n the d i f f r a c t i o n p a t t e r n . The s e p a r a t i o n between the A, B, and C s i t e s was o n l y ~1 . 7A, so major changes i n t he d i f f r a c t i o n p a t t e r n o n l y o c c u r r e d when the atoms were moved so f a r t h a t they were b e s t d e s c r i b e d as b e i n g a t o t h e r s i t e s . The t h i r d s e t of t e s t s i n v e s t i g a t e d the e f f e c t of s y s t e m a t i c v a r i a t i o n s i n the atomic p o s i t i o n s had on the i n t e n s i t y p a t t e r n . Models of L i 3 M o 4 S 8 and L i 2 M O i , S 3 were i n v e s t i g a t e d . L i t h i u m was put i n t o /3 s i t e s and the s u r r o u n d i n g 7 N o r m a l l y a s i m u l a t i o n l i k e t h i s would be done w i t h a Monte C a r l o t e c h n i q u e . The p r o c e s s d e s c r i b e d here might b e s t be c a l l e d a Reno t e c h n i q u e . I t was not as s o p h i s t i c a t e d as the Monte C a r l o , but you c o u l d s t i l l l o s e your s h i r t i f you weren't c a r e f u l . 1 4 1 s u l f u r atoms moved away from the i n t e r c a l a n t . Molybdenum c l u s t e r i n g was examined i n o t h e r t e s t s . There were no major changes i n the d i f f r a c t i o n p a t t e r n s , t h u s minor v a r i a t i o n of a t o m i c p o s i t i o n does not have any q u a l i t a t i v e e f f e c t on the d i f f r a c t i o n p a t t e r n s . I t was p o s s i b l e t o g e n e r a t e d i f f r a c t i o n p a t t e r n s f o r 2 by 2 by 1 u n i t c e l l s t h a t had s i m i l a r i n t e n s i t i e s t o the o b s e r v e d d a t a , however t h e r e was no c o n v e n i e n t way of t e s t i n g t o see i f t h e s e models were s i g n i f i c a n t . P r e f e r r e d o r i e n t a t i o n p r e v e n t e d making any m e a n i n g f u l q u a n t i t a t i v e p r e d i c t i o n s of atomic p o s i t i o n s . The phases of L i x M o S 2 have a p p r o x i m a t e l y 1T-hexagonal symmetry. 142 CHAPTER 7 SUMMARY AND CONCLUSIONS So that we may say the door i s now opened, for the f i r s t time, to a new method fraught with numerous and wonderful r e s u l t s which in future years w i l l command t he at t ent i on of ot her mi nds. --Gal iIeo G a l i I e i 7.1 I n t e r p r e t a t i o n of R e s u l t s The p u r p o s e s of t h i s work were t o s t u d y l i t h i u m i n t e r c a l a t i o n i n MoS 2, and t o r e s o l v e a c o n t r o v e r s y i n the l i t e r a t u r e . These g o a l s were a c h i e v e d by the d i s c o v e r y and i n v e s t i g a t i o n of a c r y s t a l l i n e 1 T - L i x M o S 2 p o l y t y p e . There were c o n t r a d i c t o r y c l a i m s i n the l i t e r a t u r e r e g a r d i n g t h e b e h a v i o u r of the l i t h i u m / M o S 2 system. The p r e v a l e n t b e l i e f was t h a t c r y s t a l l i n e MoS 2 was not a s u i t a b l e i n t e r c a l a t i o n h o s t , as i t was thought t o d i s p r o p o r t i o n a t e i f i t s x v a l u e exceeded ~0.2. Amorphous MoS 2 was found t o be a good h o s t , but i t l o s t i t s c a p a c i t y i f t h e m a t e r i a l was made more c r y s t a l l i n e . Other work d i s c o v e r e d the /3-phase, which was a good i n t e r c a l a t i o n h o s t , and had an x - r a y p a t t e r n t h a t i n d i c a t e d an u n d e r l y i n g c r y s t a l s t r u c t u r e . The s t r u c t u r e was, however, d i s o r d e r e d , and t h e c a p a c i t y of /3-phase may have been r e l a t e d t o e i t h e r the c r y s t a l s t r u c t u r e or t o the d e f e c t s i n the c r y s t a l . 143 T h i s t h e s i s d e s c r i b e d the d i s c o v e r y of a c r y s t a l l i n e form of L i x M o S 2 which c o u l d be used as an i n t e r c a l a t i o n h o s t . I t c y c l e d w e l l over a l a r g e range of x and d i d not d i s p r o p o r t i o n a t e . The c l a i m t h a t c r y s t a l l i n e MoS 2 i s not a good i n t e r c a l a t i o n h o s t i s too g e n e r a l : i t i s the 2H-MoS 2 p o l y t y p e t h a t i s u n s u i t a b l e . The t h e o r i e s e x p l a i n i n g why L i x M o S 2 d i s p r o p o r t i o n a t e ^ were a l s o too g e n e r a l . They o n l y c o n s i d e r e d the f r e e energy of l i t h i a t e d 2H-MoS2 v e r s u s the f r e e energy of L i 2 S and, molybdenum. I t was i n d i c a t e d t h a t 2H-MoS 2 would p r o b a b l y not e x i s t w i t h a h i g h l i t h i u m c o n t e n t , but no o t h e r c r y s t a l p o l y t y p e of MoS 2 was c o n s i d e r e d . However, s e v e r a l r e f e r e n c e s c l a i m e d MoS 2 decomposed i f an x g r e a t e r than 0.2 was i n t e r c a l a t e d i n t o the c r y s t a l s . These r e f e r e n c e s a r e wrong. A g r e a t d e a l of p r i n t has been dev o t e d t o the i d e a t h a t o n l y n o n - c r y s t a l l i n e MoS 2 can be used as an i n t e r c a l a t i o n h o s t . T h i s t h e s i s e s t a b l i s h e s e x i s t e n c e of an i n t e r c a l a t i o n system u s i n g a c r y s t a l l i n e form of MoS 2. The h i g h t emperature grown, c r y s t a l l i n e L i x M o S 2 was shown t o be s i m i l a r t o the e l e c t r o c h e m i c a l l y produced /3-phase i n both e l e c t r o c h e m i c a l and s t r u c t u r a l b e h a v i o u r s , and a corres p o n d e n c e between the sharpness of the e l e c t r o c h e m i c a l f e a t u r e s and the c r y s t a l l i n i t y of the samples was n o t e d . T h i s t h e s i s i s the f i r s t p u b l i s h e d account of how the r e p e a t e d c y c l i n g of /3-phase c e l l s tended t o sharpen b o t h t h e e l e c t r o c h e m i c a l and d i f f r a c t i o n p a t t e r n s . A t h e o r y mentioned i n s e c t i o n 5.2 d e s c r i b e d a p r o c e s s by whic h the number of d e f e c t s i n 0-phase c o u l d be reduced. The 1 44 broad e l e c t r o c h e m i c a l f e a t u r e s of /3-phase were e x p l a i n e d i n terms of a v a r i a t i o n i n the l o c a l l i t h i u m environment caused by s t r a i n s and d e f e c t s i n the c r y s t a l . C y c l i n g the h i g h t emperature grown m a t e r i a l t h r o u g h a phase and back i n t o the 1T form produced a d i s o r d e r e d c r y s t a l w i t h e l e c t r o c h e m i s t r y v e r y s i m i l a r t o t h a t of /3-phase. The s i m i l a r i t i e s between h i g h t emperature grown m a t e r i a l and 0-phase im p l y t h a t the c r y s t a l l i n i t y of the samples had a major e f f e c t on the e l e c t r o c h e m i c a l b e h a v i o u r . I t was not p o s s i b l e t o prove t h a t the h i g h t emperature m a t e r i a l was i n f a c t a c r y s t a l l i n e form of /3-phase. The c o r r e s p o n d e n c e between /3-phase and c r y s t a l l i n e L i x M o S 2 may make i t p o s s i b l e t o u n d e r s t a n d b e t t e r the /3-phase b e h a v i o u r . The /3-phase had o r i g i n a l l y appeared t o be a s i n g l e phase, and bot h e l e c t r o c h e m i c a l and s t r u c t u r a l d a t a s u p p o r t e d t h i s i d e a . However, i t was d i f f i c u l t t o u n d e r s t a n d many of the /3-phase p r o p e r t i e s u s i n g a s i n g l e phase model. The i n f o r m a t i o n p r e s e n t e d here i n d i c a t e s t h a t i t would be more p r o d u c t i v e t o t h i n k of /3-phase c a p a c i t y as b e i n g due t o a c o l l e c t i o n of f i r s t o r d e r phase t r a n s i t i o n s broadened by c r y s t a l d e f e c t s . A h i g h l y c r y s t a l l i n e L i x M o S 2 was produced by the h i g h t e m p e r a t u r e r e a c t i o n d e s c r i b e d i n Chapter 3. T h i s i s the f i r s t r e c o r d e d p r o d u c t i o n of h i g h l y c r y s t a l l i n e MoS 2 w i t h a h i g h l i t h i u m c o n t e n t . A st u d y of t h i s m a t e r i a l would have been of i n t e r e s t even w i t h o u t the c o n t r o v e r s y i n the l i t e r a t u r e or the c o n n e c t i o n t o /3-phase. 145 L i x M o S 2 was found t o c o n s i s t of a v a r i e t y of d i f f e r e n t c r y s t a l phases, each w i t h a s l i g h t l y d i f f e r e n t s t r u c t u r e . Work p r e s e n t e d here showed t h a t i t was b e s t indexed by assuming an a tomic arrangement s i m i l a r t o t h a t of a hexagonal 1T l a t t i c e . The l i t h i u m i n s e r t i o n i n L i x M o S 2 was found t o be an i n t e r c a l a t i o n r e a c t i o n t a k i n g p l a c e i n d i s c r e t e s t e p s , which c o n v e r t e d th e h o s t from one l a t t i c e t o a n o t h e r . The d i f f e r e n t phases were v e r y s i m i l a r t o each o t h e r ; c o n v e r s i o n i n v o l v e d o n l y m i n i m a l v a r i a t i o n s i n the l a t t i c e p a r a m e t e r s . T h i s i n t e r c a l a t i o n seemed t o be i n t e r m e d i a t e between a s i n g l e phase c a p a c i t y w i t h a c o n t i n u o u s change i n parameters and a f i r s t o r d e r phase t r a n s i t i o n i n which t h e r e may be major changes i n the l a t t i c e . S i mple models f o r i n t e r c a l a t i o n assume the c h a l c o g e n - m e t a l - c h a l c o g e n sandwiches t o remain e s s e n t i a l l y r i g i d as t h e l i t h i u m i s i n t e r c a l a t e d i n t o t h e van der Waals gap. The i n t e r c a l a t i o n would cause l a t t i c e e x p a n s i o n , but no o t h e r major s t r u c t u r a l changes. The a ssumption of a t o t a l l y r i g i d l a t t i c e d i d not seem t o be a p p r o p r i a t e f o r c r y s t a l l i n e L i x M o S 2 . There seemed t o be many n e a r l y d egenerate c r y s t a l s t r u c t u r e s and the l a t t i c e changed from one t o a n o t h e r as the l i t h i u m c o n t e n t was v a r i e d . The d i f f e r e n t phases ar e low symmetry c r y s t a l s t h a t a r e a l most h e x a g o n a l . I t would appear t h a t t h e s e d i s t o r t i o n s lower the f r e e energy of the c r y s t a l . a n a l o g of a T h i s may be a t h r e e - d i m e n s i o n a l P e i e r l s d i s t o r t i o n . The s u p e r l a t t i c e s may r e s u l t from d i s p l a c e m e n t of the atoms i n the u n i t c e l l , or by a charge d e n s i t y wave. I f the s u p e r l a t t i c e i s a r e s u l t of a CDW then the wave i s commensurate t o w i t h i n e x p e r i m e n t a l e r r o r . 146 S u c c e s s f u l models of i n t e r c a l a t i o n systems must i n c l u d e the: major f a c t o r s d r i v i n g the r e a c t i o n . A s i m p l e l a t t i c e gas model c o n s i d e r s s i t e e n e r g i e s and some l i t h i u m - l i t h i u m i n t e r a c t i o n s . E l a s t i c energy f a c t o r s can be added t o e x p l a i n l a t t i c e e x p a n s i o n . I t i s not c l e a r how t o b e s t d e s c r i b e L i x M o S 2 . An i m p o r t a n t f e a t u r e seems t o be the energy g a i n e d by the many s m a l l d i s t o r t i o n s from hexagonal symmetry. E s t i m a t i n g the magnitude of these energy changes from f i r s t p r i n c i p l e s would be v e r y complex, and a d e t a i l e d e x p l a n a t i o n of c r y s t a l l i n e L i x M o S 2 i s p r o b a b l y not p o s s i b l e a t t h i s t i m e . However, much of the d a t a can be i n t e r p r e t e d q u a l i t a t i v e l y . I b e l i e v e t h a t the L i / L i x M o S 2 i n t e r c a l a t i o n system s t a g e s , and the f r e e energy of the system i s p e r t u r b e d by minor s t r u c t u r a l changes. There a r e two major v o l t a g e f e a t u r e s i n c e l l s made w i t h L i x M o S 2 : one near 2.6V, and the o t h e r near 1.9V. The l a t t i c e s a s s o c i a t e d w i t h the 2.6V d a t a a l l r e q u i r e a two l a y e r u n i t c e l l f o r adequate i n d e x i n g , but the two l a t t i c e s a s s o c i a t e d w i t h the 1.9V f e a t u r e s were i n d e x e d on one l a y e r c e l l s . The l a t t i c e s w i t h x<0.5 may form s t a g e 2 compounds w i t h s l i g h t l y unequal l i t h i u m c o n t e n t s i n a l t e r n a t e l a y e r s . The s p a c i n g of the two l a y e r s was not s u f f i c i e n t l y d i f f e r e n t t o produce s t r o n g (0,0,1) s u p e r l a t t i c e l i n e s , but enough t o produce a few e x t r a d i f f r a c t i o n peaks. The two l a t t i c e s w i t h x>0.5 were st a g e 1 compounds. I s u s p e c t the t r a n s i t i o n from s t a g e 1 t o s t a g e 2 causes t h e p o t e n t i a l t o r i s e from ~1.9V t o ~2.6V and v i c e v e r s a . S t r u c t u r a l p e r t u r b a t i o n s may cause a second o r d e r e f f e c t . The symmetry might change from hexagonal t o m o n o c l i n i c or t r i c l i n i c t o lower the f r e e energy of the c r y s t a l . The d e v i a t i o n changes the l o c a l environment f o r the l i t h i u m , and 147 t h a t a l t e r s the c h e m i c a l p o t e n t i a l of the l i t h i u m atom a t the s i t e . T h i s would e x p l a i n the s p l i t t i n g of the 1.9V c a p a c i t y i n t o two f e a t u r e s and some of the d i v i s i o n of the 2.6V c a p a c i t y i n t o t h r e e . The dominant f a c t o r would be r e l a t e d t o s t a g i n g , on which the s t r u c t u r a l m o d i f i c a t i o n s would be a p e r t u r b a t i o n . I a l s o s u s p e c t t h a t the h y s t e r e s i s i s a s s o c i a t e d w i t h the s t a g i n g t r a n s i t i o n and unequal f i l l i n g of the van der Waals gaps. T h i s i s c o n s i s t e n t w i t h the d a t a , but i s i n s u f f i c i e n t t o e x p l a i n a l l the e f f e c t s ; i t does n o t , f o r example, c o v e r the r e d u c t i o n i n the c a x i s l e n g t h a t the h i g h e s t l i t h i u m c o n t e n t . T h i s t h e s i s a l s o p r e s e n t e d a g r a p h i c a l t e c h n i q u e t o a i d i n i n d e x i n g d i f f r a c t i o n p a t t e r n s . Many t e c h n i q u e s d e s c r i b e d i n the l i t e r a t u r e a r e l i m i t e d t o s i m p l e l a t t i c e s or r e q u i r e d a t a from measurements of s i n g l e c r y s t a l s . The method d e s c r i b e d here i s a p p l i c a b l e t o any type of l a t t i c e , and can be used w i t h d i f f r a c t i o n d a t a from powder samples. The major l i m i t a t i o n of my t e c h n i q u e i s t h a t i t i s o n l y a p r o c e d u r e , t h e r e f o r e i n d e x i n g a complex p a t t e r n may t a k e a p r o h i b i t i v e amount of t i m e . However, the t e c h n i q u e i s a u s e f u l e x p e r i m e n t a l t o o l , as was shown i n t h i s t h e s i s by i t s use i n i n d e x i n g the d i f f r a c t i o n p a t t e r n s of s e v e r a l t r i c l i n i c l a t t i c e s . 148 7.2 S u g g e s t i o n s f o r F u t u r e Work I t i s common i n r e s e a r c h t h a t an experiment r a i s e s more q u e s t i o n s than i t s o l v e s . T h i s i s c e r t a i n l y t r u e h e r e . Many a s p e c t s of the L i / L i x M o S 2 i n t e r c a l a t i o n system a r e s t i l l unknown, and t h i s s e c t i o n l i s t s a few of the p o s s i b l e f u t u r e l i n e s of r e s e a r c h . The work c o u l d take two forms: p r e c i s i o n measurements t o e x t e n d what i s a l r e a d y known; and f u r t h e r e x a m i n a t i o n of b a s i c p r o p e r t i e s . Much of the h i g h l y p r e c i s e work w i l l r e q u i r e the growth of s i n g l e c r y s t a l s , as many q u e s t i o n s can o n l y be answered when exa c t atomic p o s i t i o n s a r e known. X-ray a n a l y s i s of the c r y s t a l s would p r o b a b l y be s u f f i c i e n t t o d e t e r m i n e the l o c a t i o n of the heavy atoms, but n e u t r o n d i f f r a c t i o n may be r e q u i r e d t o f i n d the l i t h i u m p o s i t i o n s . The e x a c t n a t u r e of the s t r u c t u r a l d i s t o r t i o n s i s not known. The l a t t i c e may d i s t o r t i n response t o the p h y s i c a l p r e s e nce of the L i + i o n , or i t may d i s t o r t because the e l e c t r o n s donated t o the bands f a c i l i t a t e symmetry b r e a k i n g . The l i t h i u m may be e i t h e r o r d e r e d or d i s o r d e r e d . Knowledge of the atomic p o s i t i o n s w i l l a l l o w an e s t i m a t e of the energy a s s o c i a t e d w i t h the c r y s t a l d i s t o r t i o n s , and a d e t a i l e d model f o r the H a m i l t o n i a n of the system r e q u i r e s such i n f o r m a t i o n . Even though much of the s t r u c t u r e r e l a t e d i n f o r m a t i o n would be b e s t o b t a i n e d by s i n g l e c r y s t a l work, much can be l e a r n e d from the powders. 149 The 1T phase w i t h the l o w e s t l i t h i u m c o n t e n t was not examined i n d e t a i l here because i t c o n v e r t e d t o a-phase. T h i s problem may be c i r c u m v e n t e d by u s i n g low temperature x-ray d i f f r a c t i o n : c o o l i n g the sample c o u l d s i g n i f i c a n t l y slow the r a t e of c o n v e r s i o n . However, the q u e s t i o n of temperature induced phase t r a n s i t i o n s would then have t o be a d d r e s s e d . I t was not e n t i r e l y c l e a r t h a t the i r r e v e r s i b i l i t i e s measured were due t o h y s t e r e s i s ; i t seemed the b e s t of t h r e e p o s s i b l e e x p l a n a t i o n s . D i f f u s i o n d i d not appear t o be the problem, because t h e r e was v e r y l i t t l e change i n the i r r e v e r s i b i l i t y a t v e r y low c u r r e n t s . S u p e r s a t u r a t i o n of a phase t r a n s i t i o n d i d not seem t o be an adequate e x p l a n a t i o n because some of the i r r e v e r s i b i l i t y seemed t o be a s s o c i a t e d w i t h the s i n g l e phase r e g i o n . H y s t e r e s i s may r e s u l t from t h e unequal l i t h i u m f i l l i n g of the l a y e r s i n a s t a g e d u n i t c e l l . These i s s u e s c o u l d be r e s o l v e d by v e r y slow c y c l i n g and d e t a i l e d s t r u c t u r a l work. The s t r u c t u r a l work p r e s e n t e d here c e n t r e d on one h a l f of the h y s t e r e s i s c u r v e . The o t h e r h a l f s h o u l d a l s o be examined. The e l e c t r o c h e m i c a l h y s t e r e s i s s h o u l d a l s o be d e a l t w i t h i n more d e t a i l , and t h e two t y p e s of measurement s h o u l d be c o r r e l a t e d . I p r e d i c t t h a t the s t r u c t u r e a s s o c i a t e d w i t h the s i n g l e phase r e g i o n w i l l be s l i g h t l y d i f f e r e n t on charge and d i s c h a r g e . The e x a c t Ax a s s o c i a t e d w i t h each f e a t u r e s h o u l d a l s o be d e t e r m i n e d . They w i l l be a f f e c t e d by the h y s t e r e s i s i n the samples. However, t h i s measurement s h o u l d o n l y be performed a f t e r a b e t t e r t e c h n i q u e f o r making cathodes under i n e r t 150 atmospheres has been d e v e l o p e d . I n a b i l i t y t o weigh a cathode a c c u r a t e l y was a major drawback i n the work d e s c r i b e d h e r e . I t i s a l s o p o s s i b l e t o e s t i m a t e the l i t h i u m d i f f u s i o n c o n s t a n t s by c y c l i n g c e l l s . Some of the p r e l i m i n a r y measurements i n d i c a t e d t h a t d i f f u s i o n was q u i t e slow i n a t l e a s t one phase. T h i s would s e v e r e l y r e s t r i c t the a p p l i c a b i l i t y of c r y s t a l l i n e L i x M o S 2 as a cathode i n a commercial c e l l . 1T t o a c o n v e r s i o n o c c u r s when enough e l e c t r o n s have been removed from the d bands t o make a-phase the e n e r g e t i c a l l y f a v o u r a b l e p o l y t y p e . I t may be p o s s i b l e t o make the 1T-MoS 2 s t a b l e a t x=0 by changing the number of e l e c t r o n s i n the bands. One t e c h n i q u e c o u l d be t o i n s e r t a n o t h e r m e t a l i n t o the van der Waals gap, but the metal may d e i n t e r c a l a t e or i n t e r f e r e w i t h the performance of the c e l l . Another t e c h n i q u e would be t o r e p l a c e some of the atoms i n the MoS 2 m a t r i x , f o r example, s u b s t i t u t e some s u l f u r atoms w i t h c h l o r i n e atoms. I have t r i e d making 1 T - M o 0 . 9 7 M n 0 .0 3 S 2 , but w i t h o u t s u c c e s s . Other t o p i c s were o n l y b r i e f l y mentioned i n the t h e s i s . There a r e no r e p o r t s i n the l i t e r a t u r e on the st u d y of 7-phase. which s h o u l d be f u r t h e r i n v e s t i g a t e d . T h i s m a t e r i a l does not behave l i k e t he amorphous MoS 2 mentioned i n the l i t e r a t u r e . I a l s o found a s m a l l amount of a-phase c a p a c i t y j u s t above the a t o (3 t r a n s i t i o n , and n o t i c e d t h a t t h i s t r a n s i t i o n seems t o ta k e p l a c e i n two d i s t i n c t s t a g e s . N e i t h e r of t h e s e o b s e r v a t i o n s has been examined i n d e t a i l . 151 7.3 Summary T h i s t h e s i s p r o v e d the e x i s t e n c e of an i n t e r c a l a t i o n system w i t h a c r y s t a l l i n e form of l i t h i a t e d MoS 2 as the h o s t . T h i s c o n f i r m e d t h a t many r e f e r e n c e s i n the l i t e r a t u r e a r e i n c o r r e c t . I t was a l s o found t h a t the s t r u c t u r e and e l e c t r o c h e m i c a l p r o p e r t i e s of the c r y s t a l l i n e m a t e r i a l were v e r y s i m i l a r t o t h a t of /3-phase. The /3-phase b e h a v i o u r may be u n d e r s t o o d i n terms of a d i s o r d e r e d c r y s t a l l i n e s t r u c t u r e . 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(1982) I n t e r c a l a t i o n C h e m i s t r y , Academic P r e s s , New York W i l d e r v a n c k , J . C , and J e l l i n e k , F. (1971) J . Less-Common Met. 24, 73 Woollam, J.A., and Somoano, R.B. (1976) Phys. Rev B J_3, 3843 157 Woollam, J.A., and Somoano, R.B. (1977) M a t e r i a l s S c i e n c e and E n g i n e e r i n g 3_1_, 289 — Z e t t l e m o y e r , A.C. (ed.) (1969) N u c l e a t i o n , M a r c e l Dekker, I n c . , New York 158 APPENDIX I X-RAY DIFFRACTION: THEORY AND GRAPHICAL. ANALYSIS TECHNIQUE The long chains of simple and easy reasoni ngs by means of which geometers are accustomeed to reach the conclusions of their most difficult demonstrations had led me t o imagine that all things to the knowl edge of which man is competent are mutually connected in the same way. --Rene Descartes A1.1 Overview T h i s appendix c o n t a i n s the d e t a i l s of the p r o c e d u r e I dev e l o p e d t o index the m o n o c l i n i c and t r i c l i n i c L i x M o S 2 l a t t i c e s . The t h e o r y of x- r a y d i f f r a c t i o n i s p r e s e n t e d i n s e c t i o n s A1.2 and A1.3 w i t h p a r t i c u l a r emphasis p l a c e d on i n f o r m a t i o n r e l e v a n t t o the l a t e r d i s c u s s i o n . The r e m a i n i n g s e c t i o n s p r e s e n t p r o c e d u r e s f o r the i n d e x i n g of p r o g r e s s i v e l y more complex c r y s t a l s . S e c t i o n A1.4 d i s c u s s e s a t e c h n i q u e f o r use on hexagonal c r y s t a l s . S e c t i o n A1.5 d e s c r i b e s a s p e c i a l case of a d i s t o r t i o n from hexagonal t o m o n o c l i n i c and how the t e c h n i q u e of the p r e v i o u s s e c t i o n can be extended t o index the new u n i t c e l l . 159 S e c t i o n A1.6 removes the l a s t symmetries and d i s c u s s e s a g e n e r a l t e c h n i q u e f o r t r i c l i n i c l a t t i c e s . The r e s e a r c h d e s c r i b e d i n the main body of the t h e s i s e x p l o i t e d the n e a r l y hexagonal symmetry of the c r y s t a l s , and the examples p r e s e n t e d i n t h i s appendix use d a t a v e r y s i m i l a r t o t h a t o b t a i n e d i n the x - r a y i n v e s t i g a t i o n s of L i x M o S 2 . I t i s i m p o r t a n t t o note t h a t a l t h o u g h the g r a p h i c a l t e c h n i q u e a p p l i e s t o a l l c r y s t a l s w i t h o u t r e s t r i c t i o n , i n p r a c t i c e some p r i o r knowledge of the c r y s t a l s t r u c t u r e i s almost e s s e n t i a l t o the s u c c e s s f u l c o m p l e t i o n of the i n d e x i n g . The t e c h n i q u e d e s c r i b e d i s o n l y a p r o c e d u r e and not an a l g o r i t h m , t h e r e f o r e t h e r e i s no guarantee i t w i l l p r o v i d e a s o l u t i o n i n any t ime l e s s than the age of the u n i v e r s e . A1.2 B a s i c s of C r y s t a l S t r u c t u r e and X-ray D i f f r a c t i o n Atoms can s c a t t e r x - r a y s . A r e g u l a r a r r a y of atoms can a c t as a d i f f r a c t i o n g r a t i n g f o r the s c a t t e r e d r a d i a t i o n . These a r e the b a s i c p r i n c i p l e s of x - r a y d i f f r a c t i o n . T h i s s e c t i o n w i l l d i s c u s s c r y s t a l s t r u c t u r e , a method of l a b e l l i n g the p e r i o d i c a r r a y s of atoms i n a l a t t i c e , and the most fundamental a s p e c t s of x - r a y d i f f r a c t i o n . A p e r f e c t c r y s t a l i s an i n f i n i t e r e g u l a r a r r a y of atoms. In p r a c t i c e , c r y s t a l s do not have i n f i n i t e e x t e n t , but they do have a p e r i o d i c s t r u c t u r e . A b a s i c c r y s t a l u n i t t h a t can be r e p e a t e d t o f i l l a l l space w i t h o u t v o i d s or o v e r l a p i s c a l l e d a u n i t c e l l . There a r e a f i n i t e number of s i n g l e u n i t c e l l s , each w i t h i t s own symmetry p r o p e r t i e s . Only t h r e e w i l l be d i s c u s s e d h e r e : 160 h e x a g o n a l , m o n o c l i n i c , and t r i c l i n i c . A u n i t c e l l i s commonly d e s c r i b e d e i t h e r i n terms of t h r e e v e c t o r s a, E, and c, or i n terms of the magnitudes of the s e v e c t o r s a, b, and c and the a n g l e s between them, a, 0, and 7 . a i s t he a n g l e between B and c, 0 i s the a n g l e between a and c, and 7 i s the a n g l e between a and B. C e r t a i n v a l u e s of l a t t i c e p arameters r e s u l t i n u n i t c e l l s w i t h h i g h symmetry. The hexagonal l a t t i c e has a=b, a=/3=90°, and 7=120°, as shown i n F i g u r e 32a. The c a x i s i s p e r p e n d i c u l a r t o the h i g h symmetry b a s a l p l a n e . The m o n o c l i n i c u n i t c e l l has two s l i g h t l y d i f f e r e n t d e f i n i t i o n s . T h i s t h e s i s w i l l e x c l u s i v e l y use the "second s e t t i n g " where a=/3=90° ( F i g u r e 32b). A l t h o u g h i t i s not as common as the " f i r s t s e t t i n g " where 0*90°, i t i s . a p p r o p r i a t e f o r the p r e s e n t work, where the m o n o c l i n i c c r y s t a l i s j u s t a p e r t u r b a t i o n of the h e x a g o n a l . The c a x i s i s s t i l l p e r p e n d i c u l a r t o t h e b a s a l p l a n e , but the p l a n e does not have any s p e c i a l shape. The t r i c l i n i c u n i t c e l l , shown in" F i g u r e 32c, i s the most g e n e r a l c r y s t a l form. Note t h a t the c h o i c e of a u n i t c e l l f o r a c r y s t a l i s not u n i q u e . The same c r y s t a l p e r i o d i c i t i e s can be d e s c r i b e d i n many ways s i m p l y by c h o o s i n g a d i f f e r e n t shape f o r the u n i t c e l l or c h a n g i n g the l a b e l l i n g of the l a t t i c e v e c t o r s . The x - r a y d i f f r a c t i o n p a t t e r n i s a f f e c t e d by e v e r y symmetry i n the c r y s t a l . H i g h e r symmetries u s u a l l y l e a d t o s i m p l e r p a t t e r n s . 161 C) F i g u r e 32 Three C r y s t a l L a t t i c e s a) Hexagonal l a t t i c e : a=b, a=0=9O°, 7=120° The dashed l i n e s t r a c e out t h e shape of a hexagon. b) M o n o c l i n i c L a t t i c e : a=/3=90° c) T r i c l i n i c L a t t i c e : no r e s t r i c t i o n s on the l a t t i c e p a r a m e t e r s . 162 Each atom of a c r y s t a l i s c e n t r e d at some l o c a t i o n i n the u n i t c e l l . The u n i t c e l l i s r e p e a t e d throughout space p r o d u c i n g , a r e g u l a r a r r a y of atoms. The p l a n e s of atoms can be l a b e l l e d i n terms of the u n i t c e l l . The M i l l e r i n d i c e s (h,k,l) a r e the most common way of l a b e l l i n g the c r y s t a l l o g r a p h i c p l a n e s . They a r e "the r e c i p r o c a l s of the f r a c t i o n a l i n t e r c e p t s which the p l a n e makes w i t h the c r y s t a l l o g r a p h i c a x e s . " ( C u l l i t y , 1959) Suppose a p l a n e was t o i n t e r c e p t the a a x i s a t ' - j p |a.|, the b a x i s a t —g-|E|, and was p a r a l l e l t o the c a x i s . The r e c i p r o c a l s of the f r a c t i o n a l i n t e r c e p t s a r e 2, 3, and °= r e s p e c t i v e l y . By c o n v e n t i o n , the i n f i n i t e number i s r e p l a c e d by z e r o . The M i l l e r i n d i c e s of the p l a n e j u s t d e s c r i b e d a r e (2,3,0). F i g u r e 33 shows s e v e r a l examples of M i l l e r i n d i c e s and the p l a n e s they d e f i n e . The p r e c e d i n g passage i m p l i e s t h a t the o r i g i n of the u n i t c e l l must be l o c a t e d b e f o r e a p l a n e can be i n d e x e d . In p r a c t i c e , o n l y t h e s i z e and shape of the c r y s t a l need be known. The q u a n t i t y of i n t e r e s t i s the s p a c i n g between two p l a n e s of atoms. I f an x - r a y l i n e i s g i v e n an index (h,k,l), then t h a t index r e f e r s t o an i n t e r - p l a n a r s p a c i n g . The i n d i c e s a r e d e t e r m i n e d by assuming t h a t one p l a n e p a s s e s t h r o u g h the o r i g i n of the u n i t c e l l , and a p a r a l l e l p l a n e i s i n dexed by the system d e s c r i b e d above. I t i s a l s o p o s s i b l e t o index a l l o b s e r v e d x - r a y l i n e s w i t h i n t e g e r i n d i c e s . At t h i s p o i n t , a b r i e f d i g r e s s i o n i s u s e f u l b oth t o f u r t h e r i l l u s t r a t e how M i l l e r i n d i c e s l a b e l p l a n e s , and a l s o t o s t r e s s a p o i n t r e l e v a n t t o the d i s c u s s i o n s i n Chapter 6 and some of the 163 1 64 examples i n t h i s Appendix. A h i g h symmetry c r y s t a l has s e v e r a l s e t s of p l a n e s t h a t a re not p a r a l l e l but do have the same s p a c i n g between them. C o n s i d e r the (1,0,0), (0,1,0), and (1,-1,0) p l a n e s of a hexagonal c r y s t a l . The upper p a r t of F i g u r e 34 shows them viewed a l o n g the c a x i s . The t h r e e p l a n e s appear as l i n e s i n t h i s p r o j e c t i o n . They a r e i l l u s t r a t e d a g a i n i n t h e lower f i g u r e which shows atoms r a t h e r than the more a b s t r a c t u n i t c e l l . The s p a c i n g between any one of t h e s e p l a n e s and a p a r a l l e l one through the o r i g i n i s 'd'. The t r i p l e degeneracy of s p a c i n g s i s l o s t i f 7*120° or i f a*b. These a r e t h e m o n o c l i n i c d i s t o r t i o n s t h a t w i l l be d i s c u s s e d i n A1.4 . The most e l e m e n t a r y e x p l a n a t i o n of x- r a y d i f f r a c t i o n i s g i v e n by the Bragg law, which r e l a t e s o b s e r v e d d i f f r a c t i o n a n g l e s t o i n t e r - p l a n a r s p a c i n g s . F i g u r e 35 shows a monochromatic x - r a y beam r e f l e c t i n g from a c r y s t a l . The Bragg f o r m a l i s m assumes t h a t the x - r a y s a r e p a r t i a l l y r e f l e c t e d from each p l a n e of atoms. I f the p a t h d i f f e r e n c e between two r a y s i s an i n t e g e r m u l t i p l e of the wav e l e n g t h , then the waves w i l l c o n s t r u c t i v e l y i n t e r f e r e . N o r m a l l y a c r y s t a l i s composed of many p l a n e s of atoms, and each r e f l e c t some of the o r i g i n a l i n c i d e n t beam. The m u l t i p l e r e f l e c t i n g l a y e r s of the c r y s t a l a c t as a d i f f r a c t i o n g r a t i n g and put the b u l k of the s c a t t e r e d i n t e n s i t y i n t o the p r i n c i p a l maximum. The r e s u l t i s t h a t a s t r o n g r e f l e c t e d beam i s ob s e r v e d o n l y when the Bragg c o n d i t i o n i s f u l f i l l e d . The Bragg law i s w r i t t e n as n X = 2 d s i n ( 0 ) A1 .1 165 \ \ b) \ \ 9 \ 0 / p — - © - - - - - f i \ .' \ G- - 0 - o F i g u r e 34 Atomic Planes i n a Hexagonal L a t t i c e a) Three s e t s of atomic planes (dashed l i n e s ) are shown. A l l have the same i n t e r - p l a n a r s p a c i n g , d. The s o l i d l i n e s r e present the b a s a l plane of the hexagonal u n i t c e l l . b) The r e g u l a r a r r a y of atoms (©) d e f i n e s the p o s i t i o n s of the atomic p l a n e s . The same three s e t s of planes as i n a) are shown. A d i f f e r e n t c h o i c e of l a t t i c e v e c t o r s a and E would change the inde x i n g of the c r y s t a l , but would represent the same c r y s t a l symmetries. 166 Incident Diffracted F i g u r e 35 Bragg Law of D i f f r a c t i o n An i n c i d e n t x - r a y beam can p a r t i a l l y r e f l e c t from any atomic p l a n e . C o n s t r u c t i v e i n t e r f e r e n c e can occur i f the p a t h d i f f e r e n c e , 2 d s i n 0 , i s e q u a l t o an i n t e g e r number of wa v e l e n g t h s . The i n c i d e n t beam i s a t an a n g l e 26 t o the d i f f r a c t e d beam. 167 X i s the w avelength of the monochromatic beam, 6 i s d e f i n e d i n F i g u r e 35, and d i s the s p a c i n g between p l a n e s . U s u a l l y the number of w a v e l e n g t h s , n, i s removed from the e q u a t i o n by d e f i n i n g a new d e q u a l t o d/n. T h i s merely changes the i n d i c e s of the o r i g i n a l s p a c i n g (h,k,l) t o a p a r a l l e l s e t of p l a n e s w i t h i n d i c e s ( n h , n k , n l ) . Atomic p l a n e s a r e u s u a l l y i d e n t i f i e d by the M i l l e r i n d i c e s r a t h e r than use the n u m e r i c a l v a l u e of d. A way of e x p r e s s i n g the Bragg law t h a t l e n d s i t s e l f t o the a n a l y s i s p r e s e n t e d here i s "4T~ = ~ T T ~ s i n 2 ( 2 0 / 2 ) A1.2 a A T h i s i s more c o n v e n i e n t because 26 i s the measured q u a n t i t y and 1/d 2 i s a u s e f u l v a r i a b l e . The Bragg law i s s u f f i c i e n t f o r u n d e r s t a n d i n g the b u l k of the work mentioned i n t h i s t h e s i s . The next s e c t i o n e x p r e s s e s the d i f f r a c t i o n c o n d i t i o n i n a more f o r m a l manner, and e x p l a i n s how the i n t e n s i t i e s of t h e d i f f r a c t i o n s p o t s can be c a l c u l a t e d . A1.3 I n t e n s i t y C a l c u l a t i o n s The shape of the u n i t c e l l d e f i n e s the p o s i t i o n s of the d i f f r a c t i o n l i n e s , but t h e i r i n t e n s i t i e s a r e d e t e r m i n e d by the p o s i t i o n s of the atoms. Each atom a c t s as a s c a t t e r i n g s i t e , and t h e i n t e r f e r e n c e of the s c a t t e r e d waves produces the d i f f r a c t i o n p a t t e r n . T h i s s e c t i o n w i l l g i v e an o v e r v i e w of the elements i n v o l v e d i n an i n t e n s i t y c a l c u l a t i o n . More d e t a i l e d i n f o r m a t i o n can be 168 found i n o t h e r s o u r c e s ( A s h c r o f t and Mermin, 1976; C u l l i t y , 1959; K i t t e l , 1976). The concept of the r e c i p r o c a l l a t t i c e i s an i m p o r t a n t m a t h e m a t i c a l c o n v e n i e n c e . R e c i p r o c a l space i s c l o s e l y r e l a t e d t o the t h r e e - d i m e n s i o n a l F o u r i e r t r a n s f o r m of the r e a l space l a t t i c e . An obse r v e d p e r i o d i c i t y i n r e a l space has a s i m p l e d e s c r i p t i o n i n . r e c i p r o c a l space. The r e c i p r o c a l l a t t i c e v e c t o r s , A, B, and C, a r e d e f i n e d as BXc . A = 2ir — a-(BXc) 1 = 2TT _ A1.3 a•(bXc) a-(BXc) S i t e s i n the r e c i p r o c a l l a t t i c e a r e co n n e c t e d by v e c t o r s of the form G " = A A + / t S + /C* A1.4 Note t h a t the r e g u l a r arrangement of atomic p l a n e s denoted by the i n d i c e s (h,k,l) has a d i r e c t t r a n s l a t i o n t o r e c i p r o c a l space v e c t o r s . I t can a l s o be shown t h a t exp { i 6 • 3 } = 1 A1.5 where R- i s any l a t t i c e v e c t o r of the form R" = n v a + n 2 B + n 3 c A1.6 where the n's a r e i n t e g e r s . There i s an a l t e r n a t e way of d e r i v i n g the d i f f r a c t i o n c o n d i t i o n . F i g u r e 36 shows two i n c i d e n t x - r a y s w i t h wave v e c t o r s 169 F i g u r e 36 Von Laue D i f f r a c t i o n C o n d i t i o n I n c i d e n t x - r a y s w i t h wave v e c t o r E, a r e e l a s t i c a l l y s c a t t e r e d i n t o x - r a y s w i t h wave v e c t o r k"2. C o n s t r u c t i v e i n t e r f e r e n c e o c c u r s i f t h e p a t h d i f f e r e n c e 8«n 2 ~ S^n, i s an i n t e g e r number of -wavelengths. 170 K , , s c a t t e r i n g o f f two atoms i n t o f i n a l s t a t e s w i t h wave v e c t o r £ 2 . C o n s t r u c t i v e i n t e r f e r e n c e o c c u r s i f the p a t h d i f f e r e n c e , (K*2 - K i ) • 5, i s r e l a t e d t o an i n t e g e r number of wavelengths. T h i s can be w r i t t e n as (¥ 2 - T<, ) • 3 = 2 TT n A1 .7 or exp { i (lc 2 - E, ) •. 5 } . = 1 • A1 .6 f o r an a r b i t r a r y l a t t i c e v e c t o r R\ T h i s has the same form as e q u a t i o n A1.5, so S = K 2 - l c 1 = A l c A1 .9 i s the c o n d i t i o n f o r d i f f r a c t i o n . The change i n wave v e c t o r must e q u a l a r e c i p r o c a l l a t t i c e v e c t o r . T h i s i s f o r m a l l y e q u i v a l e n t t o the Bragg c o n d i t i o n d e s c r i b e d i n the p r e v i o u s s e c t i o n . X - rays a r e s c a t t e r e d by e l e c t r o n s . Each type of atom has a d i f f e r e n t e l e c t r o n d e n s i t y a s s o c i a t e d w i t h i t . An atomic form f a c t o r , f ( A l c ) , can be c a l c u l a t e d f o r each type of atom. T h i s f a c t o r r e l a t e s charge d e n s i t y and the p r o b a b i l i t y t h a t an i n c i d e n t photon w i l l have i t s wave v e c t o r changed by Ale. N u m e r i c a l a p p r o x i m a t i o n s t o f ( A K ) can be found i n the I n t e r n a t i o n a l T a b l e s of X-ray C r y s t a l o g r a p h y V o l . I l l (1968). These f a c t o r s can be combined t o g i v e an e x p r e s s i o n f o r the x-ray s c a t t e r i n g from a c r y s t a l . The s t r u c t u r e f a c t o r i s S k = § fj ( A l c ) exp { i AE • F j } A L I O where F j i s the v e c t o r from the o r i g i n of the u n i t c e l l t o the j t n atom i n t h e c e l l . T h i s e x p r e s s i o n i s the sum of a l l the s c a t t e r e d a m p l i t u d e s from a l l the n atoms of the u n i t c e l l . 171 There i s an e q u a l p r o b a b i l i t y of, any d i f f r a c t i o n l i n e b e i n g o bserved i n a t o t a l l y random powder. However, s e v e r a l geometric f a c t o r s must be i n c l u d e d when computing the the i n t e n s i t y of a l i n e . The f i n a l r e s u l t i s I - I c. I2 r 1 + c o s 2 ( 2 0 ) > A1 11 I o " I Sk I P 1 s i n 2 ( 0 ) c o s ( 0 ) 1 A U 1 1 where I 0 i s the t o t a l i n t e g r a t e d i n t e n s i t y , and p i s the degeneracy a s s o c i a t e d w i t h the i n t e r - p l a n a r s p a c i n g . In p r a c t i c e t h e r e are f u r t h e r c o m p l i c a t i o n s i f the powder has a p r e f e r r e d o r i e n t a t i o n r a t h e r than a random one. A1.4 I n d e x i n g Hexagonal L a t t i c e s E a r l i e r s e c t i o n s e x p l a i n e d how the knowledge of the u n i t c e l l c o u l d be used t o l a b e l s e t s of p l a n e s i n the c r y s t a l , and the method of c a l c u l a t i n g the d i f f r a c t i o n a n g l e of an x-ray beam s c a t t e r e d from t h e s e p l a n e s . The remainder of t h i s appendix w i l l be devoted t o the r e v e r s e p r o c e s s , where o b s e r v e d d i f f r a c t i o n s p o t s a r e r e l a t e d back t o the u n i t c e l l . I n d e x i n g the x-ray l i n e s i s e q u i v a l e n t t o d e t e r m i n i n g the l a t t i c e p a r a m e t e r s . There a r e g e n e r a l p r o c e d u r e s f o r i n d e x i n g s i m p l e l a t t i c e s , such as hexagonal ones, but t h e r e i s no a n a l y t i c way of i n d e x i n g the more complex systems ( C u l l i t y , 1959). There a r e p u r e l y n u m e r i c a l t e c h n i q u e s f o r a r b i t r a r y l a t t i c e s , but t h e s e r e q u i r e a c e r t a i n amount of i n f o r m a t i o n about the c r y s t a l as i n p u t , and t h i s i n p u t i s most r e a d i l y o b t a i n e d from s i n g l e c r y s t a l work. Most books on x-ray c r y s t a l l o g r a p h y i g n o r e the problem of i n d e x i n g t h e powder p a t t e r n s of t r i c l i n i c c r y s t a l s because the 172 p r o c e s s i s t o o c o m p l i c a t e d . The u s u a l s u g g e s t i o n i s t h a t a t r i a l - a n d - e r r o r i n d e x i n g method may be the b e s t way of s o l v i n g the problem. C u l l i t y (1959) s u g g e s t s t h a t time spent growing s i n g l e c r y s t a l s i s more p r o d u c t i v e than time spent i n d e x i n g powder p a t t e r n s . There i s , however, a t e c h n i q u e i n the l i t e r a t u r e f o r i n d e x i n g a r b i t r a r y l a t t i c e s . I t i s a s e m i - a n a l y t i c a l method d e v e l o p e d by I t o (1950). The b a s i s of the p r o c e d u r e i s t o compute v a l u e s of 1/d 2 f o r the obser v e d l i n e s and guess a few M i l l e r i n d i c e s . The e n t i r e p a t t e r n can be p r e d i c t e d based on the guesses. The h e a r t of the t e c h n i q u e i s a proce d u r e f o r r e j e c t i n g bad g u e s s e s . The parameters can be r e f i n e d once the i n d e x i n g of the p a t t e r n i s a p p r o x i m a t e l y c o r r e c t . The t e c h n i q u e I d e v e l o p e d f o r t h i s t h e s i s i s f o r m a l l y e q u i v a l e n t t o I t o ' s work. The major d i f f e r e n c e s i n the two methods a r e i n the manner by which the i n i t i a l M i l l e r i n d i c e s a r e guessed and how the f i r s t bad c h o i c e s a r e r e j e c t e d . My t e c h n i q u e uses v i s u a l feedback t o determine p o s s i b l e i n d i c e s , whereas I t o ' s method i s p u r e l y n u m e r i c a l . N e i t h e r p r o c e s s i s i n h e r e n t l y s u p e r i o r t o the o t h e r . My g r a p h i c a l work u l t i m a t e l y r e l i e d on n u m e r i c a l t e c h n i q u e s t o r e f i n e the d a t a . However, the g r a p h i c a l t e c h n i q u e r e a d i l y showed t h a t the 1.7V L i x M o S 2 d a t a had a double s i z e d u n i t c e l l . A p p a r e n t l y i t i s d i f f i c u l t t o f u l l y i ndex l a t t i c e s u s i n g I t o ' s method i f t h e l o w e s t o r d e r d i f f r a c t i o n peaks a r e not o b s e r v e d . The f o l l o w i n g i s the g r a p h i c a l t e c h n i q u e I d e v i s e d f o r i n d e x i n g a r b i t r a r y l a t t i c e s . The p r o c e d u r e i s d e s c r i b e d i n t h r e e s e c t i o n s . F i r s t , a hexagonal l a t t i c e i s indexed t o i l l u s t r a t e t he t e c h n i q u e . Second, s e c t i o n A1.5 shows the e x t e n s i o n t o a 173 m o n o c l i n i c l a t t i c e . A m o n o c l i n i c d i s t o r t i o n of a hexagonal l a t t i c e i s p r e s e n t e d as an example. The parameters used i n the example a r e v e r y s i m i l a r t o those f o r L i , M o S 2 . T h i r d , s e c t i o n A1.6 p r e s e n t s a g e n e r a l i n d e x i n g scheme f o r a t r i c l i n i c l a t t i c e . I t i n c l u d e s an example based on a s m a l l t r i c l i n i c d i s t o r t i o n of a hexagonal l a t t i c e . The f i g u r e s p r o v i d e d w i t h t h i s t h e s i s a r e too s m a l l t o show the f i n e d e t a i l needed f o r d o i n g p r a c t i c a l work, but they can be used f o r i l l u s t r a t i o n . The a c t u a l work was c a r r i e d out on 40cm by 28cm paper. The f i r s t s t e p i n i n d e x i n g a l a t t i c e i s t o be aware of the r e l a t i o n s h i p between the o b s e r v e d d a t a and the u n d e r l y i n g c r y s t a l s t r u c t u r e . The f o r m u l a f o r t h e i n t e r - p l a n a r s p a c i n g of a h exagonal l a t t i c e i s 1 4 h2 + h k + k2 . I 2 The v a l u e of 1/d 2 i s c a l c u l a b l e from the o b s e r v e d d a t a . The u l t i m a t e g o a l i s t o e v a l u a t e a and c, the l a t t i c e p a r a m e t e r s . V a l u e s of h, k, and / must be a f f i x e d t o each d i f f r a c t i o n l i n e i n manner t h a t i s c o n s i s t e n t w i t h the d a t a . The b a s i s of the g r a p h i c a l t e c h n i q u e i s t o p l o t the d a t a i n such a manner t h a t l i n e a r r e l a t i o n s h i p s a r e found. These r e l a t i o n s h i p s put c o n s t r a i n t s on the p o s s i b l e ways t o index the 174 c r y s t a l . Formula A1.12 can be put i n t o the form of a s t r a i g h t l i n e , y = m x + b, by making the i d e n t i f i c a t i o n s x = / 2 y = 1/d 2 A1 . 1 3 m = 1/c 2 K 4 r h2 + h k + k2 x b = — { - 2 • } I f / were a c o n t i n u o u s v a r i a b l e , a p l o t of e q u a t i o n A1.12 would c o n s i s t of s t a i g h t l i n e s w i t h a s l o p e of 1/c 2 and i n t e r c e p t s d e f i n e d by a and the p e r m i t t e d h and k v a l u e s . In f a c t , / i s d i s c r e t e , but l i n e s drawn t h r o u g h the d i s c r e t e p o i n t s w i l l have the p r o p e r t i e s mentioned. The word " l i n e s " i s b e i n g used i n two senses h e r e . The r e g i o n s of h i g h d i f f r a c t e d i n t e n s i t y are c a l l e d " l i n e s " f o r h i s t o r i c a l r e a s o n s . The d i s c u s s i o n of the g r a p h i c a l t e c h n i q u e , however, r e f e r s t o " l i n e s " i n the g e o m e t r i c sense. The o b s e r v e d d a t a w i l l be r e f e r r e d t o as "peaks" (a more n a t u r a l word, c o n s i d e r i n g the appearance of the d i f f T a c t o m e t e r o u t p u t ) , " d i f f r a c t i o n l i n e s " , or " d i f f r a c t i o n s p o t s " i f t h e r e i s a m b i g u i t y . To i l l u s t r a t e my t e c h n i q u e , some d a t a has been g e n e r a t e d f o r a hexagonal l a t t i c e w i t h a=3.36A and c=6.294A. These a r e the p arameters of the IT-/3-phase d e t e r m i n e d by Py ( 1983). T a b l e V shows the 20 v a l u e s t h a t would be o b s e r v e d , the 1/d 2 v a l u e s c a l c u l a t e d , and the s t i l l unknown M i l l e r i n d i c e s . The f i r s t s t e p i n d e t e r m i n i n g the i n d i c e s i s t o assume t h a t any peak may have any / i n d e x . F i g u r e 37 shows a l l the d a t a of T a b l e V p l o t t e d assuming 1=0, then r e p l i c a t e d where l2=l, .175 TABLE V Hexagonal L a t t i c e Data 28 (°) 1/d 2 h k I 14.0593 0.02524 0 0 1 28.3361 0.10097 0 0 2 30.7000 0.11810 1 0 0 33.9118 0.14335 1 0 1 42.2659 0.21.908 1 0 2 43.0799 0.22719 0 0 3 53.8250 0.34529 1 0 3 54.5808 0.35431 1 1 0 56.6604 0.37955 1 1 1 58.6198 0.40389 0 0 4 62.6300 0.45528 1 1 2 63.9339 0.47241 2 0 0 65.8296 0.49765 2 0 1 67.6312 0.52200 1 0 4 71.3622 0.57338 2 0 2 71.9434 0.58150 J J 3 75.4564 0.63108 0 0 5 80.2233 0.69960 2 0 3 83.6289 0.74918 1 0 5 84.2454 0.75820 1 1 4 The t a b l e shows the d a t a used t o i l l u s t r a t e the g r a p h i c a l t e c h n i q u e f o r i n d e x i n g a hexagonal l a t t i c e . The d a t a i s f o r a m a t e r i a l w i t h a=3.36A and c=6.294A. These a r e the parameters f o r /3-phase r e p o r t e d by Py ( 1983). The f i r s t column of the t a b l e i s the Bragg a n g l e of a d i f f r a c t i o n peak, assuming the wav e l e n g t h of the x - r a y s i s 1.54056A. The second column i s the 1/d 2 ( 1 / A 2 ) v a l u e w h i c h i s p l o t t e d i n the g r a p h i c a l a n a l y s i s . The l a s t t h r e e columns of the t a b l e a r e the M i l l e r i n d i c e s of the d i f f r a c t i o n peaks t h a t a r e t o be d e t e r m i n e d from the g r a p h i c a l a n a l y s i s . 176 1.0 CM 0 . 8 0 . 6 0 . 4 0 . 2 -0 . 0 * r * I I * I I I * i r : x X X X X X X X —~ l-X X X X ; x X X X : ~X X X X l X I ar X X w X w i • M s s s fi -; x X X K X Iff — ; X X X X _x X X X : x X X X : - 6 fi fi fi -; x , T I I x I I i * 1 1 ' D 4 8 1 2 1 6 2 0 2 4 F i g u r e 37 G r a p h i c a l A n a l y s i s : Hexagonal L a t t i c e — I n i t i a l P l o t The graph shows the 1/d 2 v a l u e s of T a b l e V p l o t t e d assuming a l l the d a t a had 1=0, 1, 2, 3, 4, and J . S t r a i g h t l i n e s must be found i n t h e d a t a . I t i s u s u a l l y e a s i e s t t o f i n d l i n e s t h r ough the o r i g i n . 177 r e p l i c a t e d a g a i n f o r l2=4, and so on. The graph i s then s e a r c h e d f o r l i n e a r r e l a t i o n s h i p s . One of the e a s i e s t s t r a i g h t l i n e s t o see, and the most i m p o r t a n t t o f i n d , i s the (0,0,1) group. T h i s l i n e i s of the form y = m x and passes through the o r i g i n and s e v e r a l o t h e r p o i n t s . The (h,0,0) and (0,k,0) l i n e s have the same b e h a v i o u r , but can not be used i n the way d e s c r i b e d below. F i g u r e 38 i s a d u p l i c a t e of F i g u r e 37, except t h a t i t has s e v e r a l l i n e s drawn th r o u g h the d a t a . The lo w e s t of the l i n e s p a sses through what may be the (0,0,1) f a m i l y . I f the (0,0,1) group has been c o r r e c t l y i d e n t i f i e d , i t s h o u l d be p o s s i b l e t o use t h i s i n f o r m a t i o n t o a i d i n the i n d e x i n g of the o t h e r d i f f r a c t i o n p eaks. R e f e r r i n g back t o e q u a t i o n s A1.12 and A1.13, t h e r e a r e o t h e r p r o p e r t i e s which can be used t o h e l p i n d e x the l a t t i c e . I t s h o u l d be p o s s i b l e t o draw l i n e s of s l o p e 1/c 2 which account f o r a l l of the d a t a . The v a l u e of 1/c 2 i s o b t a i n e d from the s l o p e of the (0,0,1) l i n e . These l i n e s w i l l have i n t e r c e p t s (h,k,0). The 1/d 2 v a l u e s of the i n t e r c e p t s must be i n the r a t i o 1:3:4:7.... I f t he wrong s e t of d a t a p o i n t s had been i d e n t i f i e d as the (0,0,1) d a t a , t h e i n t e r c e p t s w i l l not have the r e l a t i o n s h i p s i n d i c a t e d . The p r o c e s s would have t o be r e p e a t e d f o r an o t h e r d a t a s e t t h a t might r e p r e s e n t the (0,0,1) peaks. The o t h e r l i n e s i n F i g u r e 38 a r e p a r a l l e l t o the f i r s t one drawn. These l i n e s meet a l l the r e q u i r e m e n t s imposed by e q u a t i o n A1.13. In F i g u r e 39, the same d a t a a r e r e p e a t e d , but w i t h the c l u t t e r removed. A l l of the da t a i s ac c o u n t e d f o r by the e x i s t i n g l i n e s . The / 2 v a l u e of each l i n e can be read o f f t h e graph, and the h 178 F i g u r e 38 G r a p h i c a l A n a l y s i s r H e x a g o n a l L a t t i c e — C a n d i d a t e L i n e s The graph shows f o u r l i n e s drawn through the d a t a of F i g u r e 37. The bottom l i n e passes t h r o u g h the o r i g i n and f i v e d a t a p o i n t s . I t may account f o r the (0,0,1) peaks. The o t h e r l i n e s a r e drawn p a r a l l e l t o the bottom one. I f the l a t t i c e i s hexagonal and i f the (0,0,1) peaks have been c o r r e c t l y i d e n t i f i e d , then the i n t e r c e p t s of t h e s e l i n e s w i l l be i n the r a t i o 1:3:4. 179 J 2 F i g u r e 39 G r a p h i c a l A n a l y s i s : Hexagonal L a t t i c e — F i n a l R e s u l t s A l l the d a t a from T a b l e V can be i n d e x e d on a hexagonal l a t t i c e . The I2 index f o r each d i f f r a c t i o n peak can be r e a d o f f the g r a p h , and the l i n e s c o n n e c t p o i n t s w i t h t h e same h and k i n d i c e s . T h i s t e c h n i q u e does not u n i q u e l y d e f i n e the M i l l e r i n d i c e s . Changing / t o nl, where n i s an i n t e g e r , a l s o g i v e s a v a l i d i n d e x i n g scheme. The i n d i c e s s h o u l d be chosen t o m i n i m i z e the s i z e of the u n i t c e l l . 180 and k i n d i c e s a r e c o n s t r a i n e d t o v a l u e s t h a t s a t i s f y the p r o p o r t i o n a l i t y of the i n t e r c e p t s . The p r e s e n c e of unindexed l i n e s ( t h e r e a r e none i n t h i s example) or d e v i a t i o n s from the r e l a t i o n s h i p s i m p l i e s t h a t the c r y s t a l i s not y e t s u c c e s s f u l l y i n d e x e d . There i s no unique s e t of v a l u e s f o r the M i l l e r i n d i c e s . R e p l a c i n g a l l i n d i c e s of the form (h,k,l) w i t h i n d i c e s (n,h, n2k,n3l ) where n, , n2, and n3 a r e p o s i t i v e i n t e g e r s i s s t i l l a v a l i d i n d e x i n g scheme. In the new scheme, however, the volume of the u n i t c e l l i s i n c r e a s e d by a f a c t o r of n,n2n3. There are a l s o d i f f e r e n t ways t o d e f i n e the shape of the u n i t c e l l . For example, i f the l a t t i c e v e c t o r s a, B, and c a r e r e p l a c e d w i t h a, a+E, and c, the c r y s t a l i s the same but the i n d i c e s a r e d i f f e r e n t . I t i s i m p o r t a n t t o r e c o g n i z e the r e s t r i c t i o n s of a g r a p h i c a l t e c h n i q u e . I t i s most p o w e r f u l when used i n c o n j u n c t i o n w i t h a d a t a p r o c e s s i n g system, and a l t h o u g h good f o r i d e n t i f y i n g l i k e l y i n d i c e s , i t i s t e d i o u s t o use i n d i s t i n g u i s h i n g between the c o r r e c t d i f f r a c t i o n spot and one v e r y c l o s e t o i t . Parameter r e f i n e m e n t and r e j e c t i n g i n c o r r e c t i n d i c e s a re b e s t done n u m e r i c a l l y . 181 A1.5 I n d e x i n g M o n o c l i n i c L a t t i c e s T h i s s e c t i o n w i l l g e n e r a l i z e the p r e v i o u s s e c t i o n ' s t e c h n i q u e t o the second s e t t i n g of m o n o c l i n i c l a t t i c e s . The example p r e s e n t e d w i l l be a m o n o c l i n i c d i s t o r t i o n of a hexagonal l a t t i c e . How t h i s s o r t of d i s t o r t i o n a f f e c t s the d i f f r a c t i o n p a t t e r n w i l l a l s o be d i s c u s s e d . The g e n e r a l form of the i n t e r - p l a n a r s p a c i n g of a m o n o c l i n i c l a t t i c e i s _ J _ 1 < h2 k2 _ 2 h k COS7 d 2 " s i n 2 7 1 a 2 b 2 a b + 1 2 p " ' 7 > A1 . 14 The hexagonal l a t t i c e i s a s p e c i a l c a s e of the m o n o c l i n i c where 7 = 1 2 0 ° and a = b. The m a t e r i a l s of i n t e r e s t t o t h i s t h e s i s are almost h e x a g o n a l . Two ways t o p e r t u r b the hexagonal symmetry w i l l be c o n s i d e r e d . In one, 7 i s changed from 1 2 0 ° , and, i n the o t h e r , a * b. The (1,0,0), (0,1,0), and (1,-1,0) peaks w i l l be d i s c u s s e d f o r i l l u s t r a t i o n . The v a l u e of 1/d 2 i s degenerate f o r a l l t h r e e of t h e s e l i n e s i n a hexagonal l a t t i c e . I f a=b and the a n g l e between a and B d i f f e r s from 1 2 0 ° , then t h e 1/d 2 v a l u e s 182 f o r (1,0,0) and (0,1,0) remain degenerate but the (1,-1,0) l i n e moves t o a new p o s i t i o n : 1 _ "' . 1 _ 1 d 2 ( 7 , 0 , 0 ) ~ d2(0,l,-Q) ~ a 2 s i n 2 7 * a2(1,-1,0) I f t he i n t e n s i t y of r a d i a t i o n s c a t t e r e d from the t h r e e s e t s of p l a n e s i s e q u a l , then t h i s s o r t of d i s t o r t i o n w i l l have a s p e c i f i c s i g n a t u r e . I n s t e a d of one l i n e b e i n g o b s e r v e d i n the (1,0,0) r e g i o n , two w i l l be v i s i b l e . One of the s c a t t e r e d l i n e s w i l l have t w i c e the i n t e n s i t y of the o t h e r . F i g u r e 40a and F i g u r e 40b show t h i s s p l i t t i n g s c h e m a t i c a l l y . In the d i s t o r t i o n where a * b but 7=120°, the degeneracy i s c o m p l e t e l y s p l i t : 1 4 d 2 ( 7 , 0 , 0 ) " 3 a 2 * d 2(0,7,O) ~ 3 b 2 . 1 _ 4 , _ J , 1 1 A 1 . r * a2(i,-i,o) ' T V U~ + ~b~2 a~b~ j A 1 * 1 6 The s i g n a t u r e of t h i s d i s t o r t i o n i s t h r e e r o u g h l y e q u a l - s p a c e d d i f f r a c t i o n l i n e s of about the same i n t e n s i t y ( F i g u r e 4 0 c ) . A d i s t o r t i o n can a l s o be a c o m b i n a t i o n of the two d e s c r i b e d . I t w i l l then show a complete l i f t i n g of the degeneracy, but a l s o an unequal s p a c i n g between l i n e s ( F i g u r e 40d). T h i s f i n a l d i s t o r t i o n was apparent i n the e x p e r i m e n t s d e s c r i b e d i n the main body of the t h e s i s . - 2 + 2 C O S ? A1.15 a 2 s i n 2 7 183 «0 fc. o w •mm < CO «> o) (1.0,0) (0,1,0) (1,'1,0) b) (1,'1,0) 1 | (7,0,0) I (0,7,0) c) (7,0,0) (7,-7,0) (0,7,0) 1 1 1 d) 3 0 (7,-7,0) _ J _ (7,0,0) (0,7,0) » I . 3 J 2 0 (°) 3 2 F i g u r e 40 Peak S p l i t t i n g Due t o M o n o c l i n i c D i s t o r t i o n s Lowering the symmetry of a c r y s t a l w i l l g e n e r a l l y i n c r e a s e the number of d i f f r a c t i o n peaks. a) Schematic of (7,0,0) peak f o r a hexagonal l a t t i c e . b) Peak s p l i t t i n g due t o 79*120° but a=b. c) Peak s p l i t t i n g due t o a*b but 7=120°. d) Peak s p l i t t i n g due t o l o s s of a l l symmetry of the ba s a l p l a n e . 1 8 4 The g e n e r a l case of t h e m o n o c l i n i c l a t t i c e , e q u a t i o n A1.14, a l s o has t h e form of a s t r a i g h t l i n e . The c o r r e s p o n d e n c e here i s x = / 2 y = 1/d 2 m = —T- A1 . 17 u 1 f h 2 ^ k 2 2 h k C O S T > The p l o t of 1 / d 2 v s . I2 w i l l produce s t r a i g h t l i n e s . There w i l l be a l i n e t h r o u g h the (0, 0,I) f a m i l y which p a s s e s t h r o u g h the o r i g i n , and a l l o t h e r l i n e s w i l l be p a r a l l e l t o i t . However, u s u a l l y t h e r e w i l l be a l i f t i n g of the t h r e e - f o l d b a s a l p l a n e degeneracy t h a t was seen w i t h the hexagonal l a t t i c e s . In g e n e r a l the m o n o c l i n i c l a t t i c e w i l l have t h r e e t i m e s the number of l i n e s drawn p a r a l l e l t o the ( 0 , 0 , 1 ) d a t a , and the r e l a t i o n s h i p between the i n t e r c e p t s w i l l no l o n g e r have a s i m p l e 1:3:4 r a t i o . I w i l l i l l u s t r a t e the t e c h n i q u e w i t h an example. T a b l e VI shows d a t a g e n e r a t e d from a l a t t i c e w i t h a = 3 . 3 9 9 A , b=3.377A >, C = 6 . 2 6 1 9 A , and 7 = 1 2 1 . 2 1 ° . T h i s d a t a i s v e r y s i m i l a r t o t h a t o b t a i n e d from c r y s t a l l i n e L i M o S 2 . F i g u r e 41 shows the d a t a of T a b l e VI w i t h 1 / d 2 p l o t t e d a g a i n s t v a l u e s of I2. The p r o c e d u r e f o r i d e n t i f y i n g l i n e s i s e x a c t l y the same as i t was f o r the hexagonal l a t t i c e . F i r s t a p o s s i b l e ( 0 , 0 , 1 ) f a m i l y of l i n e s i s i d e n t i f i e d . Second, a 185 TABLE VI M o n o c l i n i c L a t t i c e Data 26 (°) 1/d 2 h k I 14.1318 0.02550 0 0 1 28.4844 0.10201 0 0 2 30.2537 0.11477 1 -1 0 30.7303 0.11833 1 0 0 30.9355 0.11988 0 J 0 33.5359 0.14028 I -1 1 33.9711 0.14383 1 0 1 34.1589 0.14538 0 1 1 42.0338 0.21678 1 -1 2 42.3937 0.22034 J 0 2 42.5496 0.22189 0 1 2 43.3119 0.22953 0 0 3 53.7412 0.34430 1 -1 3 53.9123 0.34633 2 -I 0 54.0405 0.34786 1 0 3 54.1703 0.34940 0 1 3 54.3018 0.35097 1 -2 0 55.1902 0.36164 1 1 0 56.0300 0.37183 2 -I 1 56.4095 0.37647 1 -2 1 57.2761 0.38714 1 1 1 58.9498 0.40804 0 0 4 62.0975 0.44834 2 -1 2 62.4538 0.45298 1 -2 2 62.9215 0.45909 2 -2 0 63.2690 0.46365 1 1 2 64.0028 0.47332 2 0 0 64.4698 0.47951 0 2 0 64.8527 0.48460 2 -2 1 65.9166 0.49882 2 0 1 66.3766 0.50501 0 2 I 67.6917 0.52282 1 -1 4 67.9528 0.52637 1 0 4 68.0662 0.52792 0 1 4 186 TABLE VI c t d . 26 (°) 1/d 2 h k 7 0 . 4 7 8 7 7 1 . 5 0 1 7 7 1 . 5 3 9 3 71 . 8 7 1 6 7 1 . 9 4 4 9 7 2 . 6 3 3 2 7 5 . 9 1 1 6 7 9 . 4 6 4 7 8 0 . 4 4 6 9 8 0 . 8 7 3 3 8 3 . 8 4 4 8 8 3 . 9 8 3 6 8 4 . 0 8 7 9 8 4 . 1 9 3 7 8 4 . 3 0 0 9 8 5 . 0 2 9 5 0 . 5 6 1 1 0 0 . 5 7 5 3 3 0 . 5 7 5 8 6 0 . 5 8 0 5 0 0 . 5 8 1 5 2 0 . 5 9 1 1 6 0 . 6 3 7 5 7 0 . 6 8 8 6 2 0 . 7 0 2 8 4 0 . 7 0 9 0 3 0 . 7 5 2 3 4 0 . 7 5 4 3 7 0 . 7 5 5 9 0 0 . 7 5 7 4 4 0 . 7 5 9 0 2 0 . 7 6 9 6 8 2 2 2 1 0 J 0 2 2 0 1 2 1 0 1 1 2 0 1 2 2 1 0 2 0 2 1 1 0 1 2 1 2 2 3 3 2 3 5 3 3 3 5 4 5 5 4 4 T h i s t a b l e shows the d a t a used t o i l l u s t r a t e the g r a p h i c a l t e c h n i q u e f o r i n d e x i n g the second s e t t i n g of a m o n o c l i n i c l a t t i c e . The d a t a i s f o r a m a t e r i a l w i t h l a t t i c e p a r a m e t e r s : a = 3 . 3 9 9 A , b = 3 . 3 7 7 A , C = 6 . 2 6 1 9 A , and 7=121.211°. These parameters a r e s i m i l a r t o those f o r a s i n g l e f o r m u l a u n i t of the L i M o S 2 m a t e r i a l . The f i r s t column of the t a b l e i s the Bragg a n g l e of the d i f f r a c t i o n peaks, assuming the x- r a y wavelength i s 1.54056A. The second column i s the 1/d 2 v a l u e p l o t t e d i n the g r a p h i c a l a n a l y s i s . The l a s t t h r e e columns a r e the M i l l e r i n d i c e s t o be de t e r m i n e d i n t h e a n a l y s i s . 187 CM 0 . 7 0 . 6 0 . 5 0 . 4 0 . 3 0 . 2 0 . 1 0 . 0 1 1 1 t 1 1 1 1 1 1 * 1 1 1 1 1 1 1 r K fi _X X x x j i i : M M 1 ! M M i t 4 I _X X 4 1 I I X X _J i I J I H M J B M M B B : T i i T I i i i T i i i i i i T i i i i i i i i 3 0 2 4 6 8 1 0 1 3 1 6 1 9 2 2 2 5 F i g u r e 41 G r a p h i c a l A n a l y s i s : M o n o c l i n i c L a t t i c e — I n i t i a l P l o t T h i s graph shows the 1/d 2 v a l u e s shown i n Table VI p l o t t e d assuming a l l the data had 1-0, 1, 2, 3, 4, and 5. In g e n e r a l , a m o n o c l i n i c l a t t i c e has more d i f f r a c t i o n peaks than a hexagonal l a t t i c e . The 1/d 2 vs. I2 graph i s o f t e n c l u t t e r e d with data p o i n t s . A much l a r g e r graph than the one shown here i s r e q u i r e d t o r e s o l v e a l l the data p o i n t s . The steps i n the indexing p r o c e s s are s i m i l a r to those d e s c r i b e d f o r the hexagonal l a t t i c e . 188 v a r i e t y of p a r a l l e l l i n e s a r e drawn. T h i r d , the r e l a t i o n s h i p s between the l i n e s i s checked. F i g u r e 42 shows a p o s s i b l e set of l i n e s w i t h a l l redundant p o i n t s removed. E q u a t i o n A1.17 shows the f o r m u l a f o r the i n t e r c e p t s of the l i n e s drawn on the gr a p h . The a, b, and 7 parameters may be e s t i m a t e d i f t h r e e i n t e r c e p t s a r e p r o p e r l y i n d e x e d . These parameter v a l u e s can then be used t o p r e d i c t the p o s i t i o n s of the o t h e r d i f f r a c t i o n peaks. For example, t h e r e a r e t h r e e i n t e r c e p t s near 1/d 2 = 0.12. These 1/d 2 v a l u e s a r e 0.1184, 0.1199, and 0.1148. Assume t h e s e l i n e s a r e (1,0,0), (0,1,0), and (1,-1,0) r e s p e c t i v e l y . I f the i n d e x i n g scheme i s c o r r e c t , then one would p r e d i c t (1,1,0) a t 0.3617 (2,-1,0) a t 0.3464, and (-1,2,0) a t 0.3510. T h i s i s indeed where the o t h e r i n t e r c e p t s a r e l o c a t e d , which i m p l i e s t h a t p a r t i c u l a r c h o i c e of h and k may be a s u i t a b l e i n d e x i n g scheme f o r the c r y s t a l . I f the scheme i s v a l i d then i t s h o u l d be p o s s i b l e t o index a l l the ob s e r v e d l i n e s . O b t a i n i n g a s u i t a b l e i n d e x i n g scheme i s u s u a l l y an i t e r a t i v e p r o c e d u r e . A1.6 I n d e x i n g T r i c l i n i c L a t t i c e s T h i s s e c t i o n w i l l e x p l a i n how an a r b i t r a r y l a t t i c e may be i n d e x e d . Two t y p e s of graph must be drawn. The f i r s t i s i d e n t i c a l t o tho s e d e s c r i b e d i n the p r e v i o u s s e c t i o n , and the second i s d e s c r i b e d h e r e . The t r i c l i n i c l a t t i c e i s the l o w e s t symmetry c r y s t a l . F i g u r e 43 shows a top-down view of a t r i c l i n i c d i s t o r t i o n of a hexag o n a l c r y s t a l . The s o l i d l i n e i s the b a s a l p l a n e of t h e u n i t 189 F i g u r e 42 G r a p h i c a l A n a l y s i s : M o n o c l i n i c L a t t i c e — F i n a l R e s u l t s A l l the data of Table VI can be indexed on a m o n o c l i n i c l a t t i c e . The lowest of the l i n e s i s drawn through the (0,0,1) d a t a . The set i n t e r s e c t i n g the 1/d 2 a x i s at ""0.12 1/A2 i s a c t u a l l y a t r i p l e t , but the upper two l i n e s can not be r e s o l v e d on t h i s graph. The small m o n o c l i n i c d i s t o r t i o n from hexagonal symmetry i s very s i m i l a r t o the data f o r Li,MoS 2 shown i n F i g u r e 28. 190 F i g u r e 43 T r i c l i n i c D i s t o r t i o n — T o p View T h i s i l l u s t r a t e s a sma l l t r i c l i n i c d i s t o r t i o n from hexagonal symmetry. The c a x i s i s not p e r p e n d i c u l a r t o the plane d e f i n e d by i* and E. The d spacings of the planes (h,k,l ) and are not degenerate. The o r i g i n s O and O' are d e s c r i b e d i n the t e x t . 191 c e l l and the dashed l i n e i s the t o p of the c e l l . The d i s t o r t i o n has l i f t e d b o t h the t h r e e - f o l d b a s a l p l a n e degeneracy and the t w o - f o l d degeneracy of the r e f l e c t i o n symmetry. The g e n e r a l f o r m u l a f o r the i n t e r p l a n a r s p a c i n g of a t r i c l i n i c l a t t i c e i s £2 = y 2 { S 1 1 / i 2 + S 2 2 ^ 2 + S 3 3 / 2 + 2 S 1 2 H + 2 S 1 3 hi + 2 S 2 3 k I } A 1.1-8 where S 1 1 = b 2 c 2 s i n 2 a S 2 2 = a 2 c 2 sin 2/3 S 3 3 = a 2 b 2 s i n 2 7 5 1 2 = a b c 2 ( c o s a cos/3 - C O S 7 ) 5 1 3 = a b 2 c ( C O S 7 c o s a - cos/3 ) S 2 3 = a 2 b c ( cos/3 C O S 7 - c o s a ) V 2 = a 2 b 2 c 2 ( 1 - c o s a - cos/3 - COS7 + 2 c o s a cos/3 C O S 7 ) P l o t t i n g 1/d 2 v s . I2 i s no l o n g e r adequate t o index a l l the d a t a because the c r o s s terms i n v o l v i n g hi and kl make many d a t a s e t s 192 n o n l i n e a r . A s o l u t i o n i s t o p l o t 1/d 2 v s . I2 as b e f o r e and l o c a t e the (h,0,0), (0,k,0), and (0,0,1) f a m i l i e s of p o i n t s . These t h r e e s e t s of d a t a remain l i n e a r even i n the t r i c l i n i c c a se because no c r o s s terms a f f e c t the peaks. Knowledge of t h e s e groups a l s o d e f i n e s , the S n / V 2 , S 2 2 / V 2 , and S 3 3 / V 2 terms. The example below w i l l assume the (0,0,1) l i n e s have been i d e n t i f i e d . E q u a t i o n A1.18 can be w r i t t e n as a l i n e a r form ^2 ~ y 2 1 I 2 = y2 { 2 S 1 3 h + S2 3 k } / + -4rr- ( S n h2 + S 2 2 k2 + S 1 2 h k } A1.19 where x = / • J - -h- - -§r <* m = - ^ j — { 2 S 1 3 h + 2 S 2 3 k } b = - r U r ( S n h2 + S 2 2 k2 + S 1 2 h k } A graph of 1/d 2 - I2 S 3 3 / V 2 v s . / w i l l produce a s t r a i g h t l i n e . As was the case w i t h the hexagonal and m o n o c l i n i c l a t t i c e s , t he / inde x i s i n i t i a l l y unknown. F i r s t , t he graph i s p l o t t e d assuming / = 0 f o r a l l o b s e r v e d l i n e s . Second, / i s assumed t o have a v a l u e of 1, 1/d 2 - S 3 3 / V 2 i s computed f o r each peak, and t h e s e a r e p l o t t e d . Then 1/d 2 - 4 S 3 3 / V 2 v s . / = 2 i s p l o t t e d , 193 and so on. I t must be remembered t h a t . / can be p o s i t i v e , n e g a t i v e , or z e r o . The / i n d i c e s of the hexagonal and m o n o c l i n i c c r y s t a l s were always assumed t o be p o s i t i v e f o r c o n v e n i e n c e . T a b l e V I I shows d a t a g e n e r a t e d f o r a t r i c l i n i c l a t t i c e , and F i g u r e 44 shows a s m a l l amount of the d a t a p l o t t e d as d e s c r i b e d above. Note t h e f i n e s c a l e needed t o d i s t i n g u i s h the p o i n t s . F i g u r e 45 shows t h i s same d a t a w i t h s i x l i n e s drawn through the p o i n t s . These l i n e s would have had a z e r o s l o p e i n the o t h e r c r y s t a l systems mentioned. The s l o p e s of the l i n e s a r e r e l a t e d t o S 1 3 and S 2 3. There s h o u l d be one l i n e p a s s i n g t h r o u g h t h e (1,0,1) f a m i l y of p o i n t s , a n o t h e r p a s s i n g t h r o u g h the (0,1,1) and a t h i r d t h r o ugh the (1,-1,1) p o i n t s . Three of the s i x l i n e s shown must t h e r e f o r e be e l i m i n a t e d . T h i s i s done by r e a l i z i n g t h a t the t h r e e l i n e s must have s l o p e s of 2h S 1 3 / V 2 , 2k S 2 3 / V 2 , and (2h S 1 3 - 2k S 2 3 ) / V 2 . The d i f f e r e n c e between two of the s l o p e s must e q u a l the t h i r d i f |h|=|k\. The t h r e e l i n e s chosen as (1,0,1), (0,1,1), and (1,-1,1) a r e i n d i c a t e d on the graph. The o t h e r t h r e e l i n e s a r e an e q u a l l y v a l i d c h o i c e f o r the i n d e x i n g scheme. Choosing the o t h e r l i n e s would c o r r e s p o n d t o the p o i n t marked 0' i n F i g u r e 43 as the o r i g i n of the u n i t c e l l r a t h e r than the p o i n t marked 0. The v a l u e s of the c e l l parameters d e t e r m i n e d above can be used t o p r e d i c t the o t h e r d i f f r a c t i o n s p o t s . As w i t h the m o n o c l i n i c and hexagonal systems, i f any l i n e s a r e not i n d e x e d , then t h e r e i s a t l e a s t one e r r o r i n the system and the p r o c e s s must be r e p e a t e d . In p r a c t i c e , w o r k i n g on a t r i c l i n i c l a t t i c e 194 TABLE V I I Data f o r T r i c l i n i c L a t t i c e 28 (°) i / d 2 1/d 2 - h k I I2 S 3 3 / V 2 13.9525 0.02486 0 0 1 28.1173 0.09945 0 0 2 31.2877 0.12255 0.1226 . J -1 0 31.6476 0.12532 0.1253 0 1 0 31.6593 0.12541 0.1254 J 0 0 34.0347 0.14435 0.1195 1 -J 1 34.3284 0.14678 0.1219 1 0 I 34.6850 0. 14975 0.1249 0 1 1 34.7714 0.15048 0.1256 1 -1 -1 34.7868 0.15061 0.1257 0 1 -1 35.1605 0.15376 0.1289 J 0 -I 41.9422 0.21588 0.1164 1 -1 2 42.1452 0.21788 0.1184 1 0 2 42.7380 0.22377 0 0 3 42.7529 0.22391 0.1245 0 1 2 42.9236 0.22562 0.1262 0 1 -2 43.1731 0.2281 3 0.1287 1 -1 -2 43.5402 0.23183 0.1324 1 0 -2 53.1349 0.33714 0.1134 J -J 3 53.2685 0.33871 0.1149 1 0 3 54.0361 0.34780 0. 1240 0 1 3 54.2509 0.35036 0. 1266 0 1 -3 54.6803 0.35550 0.1317 1 -1 -3 55.0240 0.35964 0.1359 I 0 -3 55.9069 0.37033 1 -2 0 55.9291 0.37060 2 -1 0 56.6069 0.37889 1 1 0 57.4193 0.38892 2 -1 1 57.7129 0.39256 1 -2 1 58. 1338 0.39781 0 0 4 58.1357 0.39783 1 -2 -1 58.2967 0.39984 1 I 1 58.4700 0.40201 2 -1 -1 58.9202 0.40767 1 1 -1 195 TABLE V I I c t d . 26 (°) 1/d 2 1/d 2 - h k I I2 S 3 3 / V 2 62.7583 0.45696 2 -1 2 63.3348 0.46452 1 -2 2 63.7905 0.47052 1 1 2 64.1337 0.47505 1 -2 -2 64.7441 0.48315 2 -1 -2 64.9712 0.48617 1 1 -2 65.2738 0.49021 2 -2 0 66.0985 0.50127 0 2 0 66.1253 0.50163 2 0 0 66.6069 0.50812 0.1103 1 -J 4 66.6689 0.50895 2 -2 1 66.6918 0.50926 0.1115 1 0 4 67.4486 0.51951 2 0 1 67.5725 0.52120 2 -2 -I 67.5886 0.52142 0. 1236 0 1 4 67.8394 0.52483 0.1270 0 1 -4 67.8722 0.52528 0 2 1 67.9975 0.52698 0 2 - 1 68.4090 0.53260 0.1348 1 - 1 -4 68.4720 0.53347 2 0 - 1 68.7421 0.53716 0.1394 1 0 -4 71.4581 0.57472 2 -7 3 71.6514 0.57742 2 -2 2 72.2790 0.58620 1 -2 3 72.3454 0.58713 2 0 2 72.6156 0.59092 1 1 3 73.1915 0.59901 0 2 2 73.3970 0.60191 2 -2 -2 73.4037 0.60200 1 -2 -3 73.4338 0.60243 0 2 -2 74.2541 0.61401 2 -1 -3 74.2816 0.61440 1 1 -3 74.3258 0.61503 2 0 -2 196 TABLE V I I c t d . 26 (°) 1/d 2 1/d 2 - h k I I2 S 3 3 / V 2 74.7873 0.62157 0 0 5 79.9480 0.69561 2 -2 3 80.5588 0.70447 2 0 3 81 .7977 0.72247 0 2 3 82.1493 0.72759 0 2 -3 82.2336 0.72882 0.1072 1 -1 5 82.2829 0.72954 0.1080 1 0 5 82.4753 0.73234 2 -2 -3 83. 1517 0.74221 2 -I 4 83.3257 0.74475 0.1232 0 1 5 83.4327 0.74632 2 0 -3 83.6177 0.74902 0.1275 0 J -5 84.2043 0.75760 1 -2 4 84.3292 0.75943 0.1379 1 -I -5 84.4394 0.76104 1 1 4 84.6699 0.76442 0.1428 1 0 -5 85.6429 0.77868 1 -2 -4 86.5753 0.79236 1 1 -4 86.7282 0.79460 2 - 1 -4 T h i s t a b l e shows the d a t a used t o i l l u s t r a t e the g r a p h i c a l t e c h n i q u e f o r i n d e x i n g a t r i c l i n i c l a t t i c e . The d a t a p r e s e n t e d here has a=3.2875A\ b=3.287lA, c=6.3457A, a=90.8l°, 0=88.048°, and 7=120.753°. The d a t a was v e r y s i m i l a r t o t h a t of a s i n g l e f o r m u l a u n i t of L i x M o S 2 l a t t i c e E. The f i r s t column of the t a b l e i s t he Bragg a n g l e , assuming the x - r a y w a v e l e n g t h t o be 1.54056A. The second column i s the 1/d 2 v a l u e . The t h i r d column i s the 1/d 2 - I2 S 3 3 / V 2 v a l u e p l o t t e d i n F i g u r e s 44 and 45. The f i g u r e s were s i m p l i f i e d by o n l y p l o t t i n g the d a t a t h a t had h and k v a l u e s of 0 or ±7. P l o t t i n g a l l the p o i n t s would not change the system of a n a l y s i s . I t would j u s t i n c r e a s e the number of c a n d i d a t e l i n e s t o be i n v e s t i g a t e d . The l a s t t h r e e columns of the t a b l e a re the M i l l e r i n d i c e s t o be d e t e r m i n e d i n the a n a l y s i s . 197 0.15 0.14 -CM > r o if) 0 .13 I 0.12 -CN 0.11 -0.10 - 4 - 2 0 I F i g u r e 4 4 G r a p h i c a l A n a l y s i s : T r i c l i n i c L a t t i c e — S 3 3 / V 2 C o n t r i b u t i o n Removed The f i r s t s t e p i n the g r a p h i c a l a n a l y s i s of a t r i c l i n i c l a t t i c e i s t o p l o t a graph of 1/d 2 v s . I2. The (0,0,1) peaks a r e i d e n t i f i e d and used t o compute S 3 3 / V 2 . The v a l u e s of 1/d 2 - I2 S 3 3 / V 2 a r e then c a l c u l a t e d and p l o t t e d as i n t h i s g r a p h. The d a t a shown above i s a su b s e t of the d a t a shown i n T a b l e V I I : a l l the p o i n t s have h and k i n d i c e s of J, 0 or 0, 1 or J, -7. the d a t a chosen f o r t h i s i l l u s t r a t i o n i s v e r y s i m i l a r t o t h a t f o r L i x M o S 2 l a t t i c e E, except t h a t the u n i t c e l l was chosen t o c o n t a i n o n l y one f o r m u l a u n i t . 198 F i g u r e 45 G r a p h i c a l A n a l y s i s : T r i c l i n i c L a t t i c e — F i n a l R e s u l t s Six l i n e s c o u l d be drawn through the data shown i n F i g u r e 44. The s l o p e of the (1,-1,1) l i n e must equal the d i f f e r e n c e i n the s l o p e s of the (1,0,1) and (0,1,1) l i n e s . The c h o i c e of i n d i c e s i s not unique, and one p o s s i b l e c h o i c e i s shown on the graph. 199 u s u a l l y r e q u i r e s m u l t i p l e i t e r a t i o n s b e f o r e a c o m p l e t e l y s u i t a b l e scheme i s found. Q u i t e o f t e n t h e r e w i l l be many ways t o draw l i n e s t h r o u g h the l i m i t e d d a t a a v a i l a b l e . Each i n c o r r e c t l i n e drawn l e a d s t o an i n d e x i n g t h a t w i l l f a i l a t some l e v e l , and each must be r e j e c t e d i n t u r n . 200 APPENDIX I I Computer Programs "Open the pod bay doors, HAL. " --Arthur C. Clarke S e v e r a l computer programs were used t o a i d i n the a c q u i s i t i o n and p r o c e s s i n g of the x - r a y d a t a d e s c r i b e d i n t h i s t h e s i s . Most of the work was done w i t h s m a l l , s i m p l e computer programs w r i t t e n f o r s p e c i f i c s t e p s i n the d a t a p r o c e s s i n g , r a t h e r than w i t h l a r g e , g e n e r a l - p u r p o s e computer programs. T h i s appendix w i l l b r i e f l y o u t l i n e the c o n c e p t s used t o w r i t e the programs, but no source codes are i n c l u d e d . Data A c q u i s i t i o n : The s c a t t e r e d i n t e n s i t y of t h e x - r a y s as a f u n c t i o n of the d i f f r a c t i o n a n g l e , 26, had t o be r e c o r d e d . Automated d a t a c o l l e c t i o n was performed w i t h a P h i l i p s PW 1395 programmer and a Cromemco Z80 microcomputer. A s e r i e s of a n g u l a r i n t e r v a l s and c o u n t i n g t i m e s were s t o r e d as machine code i n s t r u c t i o n s i n the programmer. The x - r a y d e t e c t o r scanned the 26 a n g u l a r i n t e r v a l s i n 0.01° i n c r e m e n t s , and a t each p o s i t i o n the t o t a l number of x - r a y s d e t e c t e d i n some time i n t e r v a l was r e c o r d e d . The time i n t e r v a l was chosen t o o b t a i n good c o u n t i n g s t a t i s t i c s . A two second i n t e r v a l was s u f f i c i e n t f o r i n t e n s e l i n e s . Four seconds were s a t i s f a c t o r y f o r most d a t a , and 10 second i n t e r v a l s were used f o r weak or o v e r l a p p i n g peaks. I n t e r v a l s of 200 seconds 201 were used t o f i n d v e r y weak l i n e s . The da t a p a i r s (26, number of c o u n t s ) were r e c o r d e d on f l o p p y d i s k s . D e t e r m i n a t i o n of Peak P o s i t i o n s : PEAKFIT used a p a r a b o l i c a p p r o x i m a t i o n t o f i n d the c e n t r e s of the d i f f r a c t i o n peaks. L e a s t square f i t s t o a p a r a b o l a were made u s i n g s u b s e t s of n i n e a d j a c e n t d a t a p o i n t s from a s e t of N d a t a p a i r s ( 2 6 , i n t e n s i t y ) , and the narrowest of the N-9 p a r a b o l a s was assumed t o d e s c r i b e the d i f f r a c t i o n l i n e . The output l i s t e d t he parameters of t h i s p a r a b o l a . The 26 v a l u e s from d i f f e r e n t d a t a s e t s were u s u a l l y i d e n t i c a l t o a t l e a s t t h r e e d e c i m a l p l a c e s . PEAKFIT was w r i t t e n by J.R. Dutcher and i s e x p l a i n e d f u l l y i n h i s M.Sc. t h e s i s (1985). H i s t h e s i s a l s o showed t h a t t h e r e was v e r y good agreement between the observ e d d a t a and the p a r a b o l i c f i t s . Index E x a m i n a t i o n and Parameter Refinement: Indexed s e t s of peaks were examined b o t h t o check the q u a l i t y of t h e i n d e x i n g and t o d e t e r m i n e the l a t t i c e p a r a m e t e r s . There i s no a n a l y t i c a l way t o dete r m i n e t h i s i n f o r m a t i o n , t h e r e f o r e n u m e r i c a l t e c h n i q u e s were used t o f i n d a s e t of parameters t h a t m i n i m i z e d the d i f f e r e n c e s between the observ e d and c a l c u l a t e d peak p o s i t i o n s . The n u m e r i c a l f i t t i n g r o u t i n e s used the s o p h i s t i c a t e d m u l t i - p a r a m e t e r f u n c t i o n m i n i m i z a t i o n program "MINUIT" d e s c r i b e d by James and Roos (1971). The u s e r must p r o v i d e a command f i l e and a s u b r o u t i n e . The command f i l e l i s t s t he number of param e t e r s and e s t i m a t e s of t h e i r v a l u e s . The s u b r o u t i n e , which 202 must be c a l l e d FCN, computes a number based on the i n p u t d a t a and the computer g e n e r a t e d e s t i m a t e s of the p a r a m e t e r s . The main program uses a v a r i e t y of d i f f e r e n t i n t e r n a l r o u t i n e s t o f i n d the minimum v a l u e of FCN i n some u s e r - s p e c i f i e d domain. Many of the programs d e s c r i b e d below a r e FCN s u b r o u t i n e s , and a l l have the same b a s i c f o r m a t . An i n i t i a l i z a t i o n r o u t i n e reads i n the d a t a and does b a s i c m a n i p u l a t i o n s f o r l a t e r use. The d a t a always has one dependent v a r i a b l e , y, and t h r e e independent v a r i a b l e s , x. The d a t a i s assumed t o have some f u n c t i o n a l form y = F ( x | P ) A2.1 where P a r e the p a r a m e t e r s . The main l o o p computes a number / f o r a l e a s t squares f i t where f = Z { Yi ~ F(xi\P) }2 A2.2 I t i s the number / t h a t MINUIT m i n i m i z e s . The parameters P* t h a t m i n i m i z e / a r e summarized i n the output r o u t i n e s . The FCN s u b r o u t i n e s d e s c r i b e d here u s u a l l y d e t e r m i n e d the l a t t i c e p a rameters and o u t - o f - p l a n e d i s p l a c e m e n t by m i n i m i z i n g the d i f f e r e n c e between o b s e r v e d and c a l c u l a t e d Bragg a n g l e s . A l i s t of d i f f r a c t i o n peak p o s i t i o n s , yj_, and t h e M i l l e r i n d i c e s , x^, was r e a d i n t o the programs. The f u n c t i o n a l form F (x | P ) was based on the Bragg law ( e q u a t i o n A1.2), and the f o r m u l a s f o r i n t e r - p l a n a r s p a c i n g s . The former r e l a t e s a n g u l a r p o s i t i o n t o 1/d 2, and the l a t t e r r e l a t e s 1/d 2 t o the M i l l e r i n d i c e s and l a t t i c e p a r a m e t e r s . The i n t e r - p l a n a r s p a c i n g f o r m u l a s f o r h e x a g o n a l , m o n o c l i n i c , and t r i c l i n i c l a t t i c e s a r e g i v e n i n 203 e q u a t i o n s A1.12, A1.14, and A1.17. The Bragg a n g l e , 0 B, computed by the above p r o c e d u r e , d i f f e r e d from the measured a n g l e , m^' because of the o u t - o f - p l a n e d i s p l a c e m e n t , S. The main l o o p of the program i n c o r p o r a t e d t h i s d i s p l a c e m e n t i n t o the d a t a by u s i n g the e x a c t r e l a t i o n , e q u a t i o n 6.1, t o c a l c u l a t e the Bragg a n g l e from the measured a n g l e . The d i f f e r e n c e between the " c o r r e c t e d " measured a n g l e and the Bragg a n g l e c a l c u l a t e d from the l a t t i c e p arameters was m i n i m i z e d . The FCN o u t p u t r o u t i n e s c r e a t e d a t a b l e of the i n p u t d a t a and the p r e d i c t e d peak p o s i t i o n s . The p r e d i c t e d p o s i t i o n was computed by an a p p r o x i m a t i o n f o r m u l a t h a t e s t i m a t e d 0 m from 0 B and 6. The l a t t i c e p a r a m e t e r s , the o u t - o f - p l a n e d i s t a n c e , and the minimum v a l u e of / were a l s o d i s p l a y e d . I f any peaks were i m p r o p e r l y i n d e x e d , then / would be l a r g e and t h e r e would be poor agreement between the o b s e r v e d and c a l c u l a t e d peak p o s i t i o n s . CUBICFIT was a MINUIT "FCN" s u b r o u t i n e t h a t f i t the d a t a u s i n g the two parameters a and the o u t - o f - p l a n e d i s p l a c e m e n t , 6. The program had been d e s i g n e d t o a i d i n the i n d e x i n g of c u b i c l a t t i c e s , but i t c o u l d a l s o be used t o o b t a i n the (0,0,1) s p a c i n g and 6. HEXFIT was s i m i l a r t o CUBICFIT. I t was a t h r e e parameter MINUIT "FCN" s u b r o u t i n e used t o d e t e r m i n e the hexagonal a and c v a l u e s p l u s the o u t - o f - p l a n e c o r r e c t i o n , 5. TRIFIT and TRIPFIT were s i x parameter MINUIT "FCN" s u b r o u t i n e s . TRIFIT i s the t r i c l i n i c e x t e n s i o n of CUBICFIT and HEXFIT. The s i x parameters a r e a, b, c, a, /3, and 7 . The o u t - o f - p l a n e d i s t a n c e , 6, was r e q u i r e d as i n p u t t o the program. 204 T h i s program was slow and e x p e n s i v e t o run i f the o r i g i n a l e s t i m a t e s of the parameters were not a c c u r a t e . TRIPFIT was a s i m p l e r v e r s i o n of TRIFIT d e s i g n e d t o reduce the amount of i n t e r n a l p r o c e s s i n g . I t r e q u i r e d more pre and p o s t - p r o c e s s i n g of the d a t a , but was f a s t and cheap t o ru n . The i n p u t was of the form ( h , k , l , 1 / d 2 ) , and the v a l u e s of 1/d 2 were d e t e r m i n e d from the TRISLOPE program d e s c r i b e d below. The program m i n i m i z e d the d i f f e r e n c e between the "observed" and c a l c u l a t e d 1/d 2 v a l u e s . The p a r a m e t e r s , P", t h a t were v a r i e d t o m i n i m i z e the d i f f e r e n c e were the s i x S^j parameters d e f i n e d i n e q u a t i o n A1.17. Another program was used t o det e r m i n e the l a t t i c e parameters from the Sj^j p a r a m e t e r s . A s m a l l e r r o r was i n t r o d u c e d i n t o the r e s u l t s because a l l 1/d 2 v a l u e s were g i v e n e q u a l weight i n the m i n i m i z a t i o n r o u t i n e . The u n c e r t a i n t y i n s m a l l 1/d 2 v a l u e s i s l a r g e r than t h e u n c e r t a i n t y i n the l a r g e 1/d 2 v a l u e s , t h e r e f o r e the s m a l l 1/d 2 v a l u e s s h o u l d be g i v e n l e s s w e i g h t . However, the u n c e r t a i n t y i n 26 v a l u e s i s the same a t a l l a n g l e s . TRIPFIT was used t o r e j e c t bad guesses of the M i l l e r i n d i c e s , and TRIFIT was used i n the f i n a l r e f i n e m e n t . USGS i s a l a t t i c e parameter r e f i n e m e n t program w r i t t e n by Evans e t . a l . ( l 9 6 3 ) of the U.S. G e o l o g i c a l Survey. I t was used t o d o u b le-check t h e l a t t i c e p arameters I o b t a i n e d w i t h my programs. The USGS i n p u t i n c l u d e s the c r y s t a l t ype and e s t i m a t e s of a, b, c, a, /3, and 7. The d i f f r a c t i o n d a t a must i n c l u d e the M i l l e r i n d i c e s , the wavelength and the d i f f r a c t i o n a n g l e . The program m i n i m i z e s the d i f f e r e n c e between the obser v e d and c a l c u l a t e d Bragg a n g l e s . The program r e q u i r e s good e s t i m a t e s of the t r i c l i n i c p arameters or i t w i l l not converge. The o u t p u t i s a summary of the i n p u t d a t a and t h e c a l c u l a t e d 26 v a l u e s , p l u s a 205 l i s t i n g of a l l p o s s i b l e p o s i t i o n s f o r the d i f f r a c t i o n peaks. T h i s program r e q u i r e s a f i x e d - f o r m a t i n p u t , so a f r e e - f o r m a t program was used t o remove the o u t - o f - p l a n e d i s p l a c e m e n t s i n the da t a and generate a f i l e t h a t c o u l d be used by USGS. The l a t t i c e p a rameters d e t e r m i n e d f o r a p a r t i a l l y i n dexed c r y s t a l were used t o a i d i n t h e i n d e x i n g of the o t h e r d i f f r a c t i o n peaks. Programs used the l a t t i c e parameters e s t i m a t e d i n the f i t t i n g r o u t i n e s d e s c r i b e d above t o ge n e r a t e l i s t s of d i f f r a c t i o n peaks, the M i l l e r i n d i c e s , 1/d 2 v a l u e s , Bragg a n g l e s , and a n g u l a r p o s i t i o n s ( i n c l u d i n g o u t - o f - p l a n e c o r r e c t i o n s ) f o r a l l p o s s i b l e d i f f r a c t i o n peaks. SPECTRUM was w r i t t e n by R i c h a r d M a r s o l a i s and was used t o compute both the p o s i t i o n s and i n t e n s i t i e s of the d i f f r a c t i o n l i n e s . T h i s program r e q u i r e d an i n p u t f i l e t h a t i n c l u d e d : l a t t i c e p a r a m e t e r s ; type of atom and i t s p o s i t i o n i n the u n i t c e l l ; a p p r o x i m a t i o n s f o r the at o m i c form f a c t o r s ; w a v e l e n g t h s ; maximum h, k, I, and 28 v a l u e s ; and r e s o l u t i o n s and minimum i n t e n s i t i e s . The program output i n c l u d e d : the h, k, I v a l u e s ; 28; the s t r u c t u r e f a c t o r ; the m u l t i p l i c i t y of the l i n e s ; t he s t a n d a r d i n t e n s i t y ; t he i n t e n s i t y c o r r e c t e d f o r the a u t o m a t i c d i v e r g e n c e s l i t ; and the i n t e n s i t y c o r r e c t e d b o t h f o r the d i v e r g e n c e s l i t and f o r a b s o r p t i o n by the b e r y l l i u m window on the x - r a y c e l l . TRISLOPE was used t o a i d i n the i n d e x i n g of t r i c l i n i c l a t t i c e s . I t took as i n p u t the c and 5 v a l u e s ( d e t e r m i n e d by CUBICFIT) and a s e t of x-ray l i n e p o s i t i o n s , then computed 206 1 d 2 f o r / v a l u e s of 0 t o 5. The o u t p u t of t h i s program was used both t o p l o t the graphs d e s c r i b e d i n s e c t i o n A1.6 and t o determine the 1/d 2 v a l u e s needed as i n p u t f o r TRIPFIT. - / 1 3 3 V A2.3 207 APPENDIX I I I L i x M o S 2 X-Ray Data I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind: it may be the beginning of knowledge, but you have scarcely, in your t hought s, advanced to the stage of science, whatever t he mat I er may be. -William Thomson, Lord Kelvin T h i s appendix i s a summary of the x - r a y d a t a c o l l e c t e d f o r f i v e L i x M o S 2 l a t t i c e s . The o b s e r v e d d a t a are p r e s e n t e d i n the f i r s t two columns of the t a b l e s . The f i r s t column l i s t s the a n g u l a r p o s i t i o n of the d i f f r a c t i o n peaks t o ±0.0001° i f the c e n t r e was p r e c i s e l y measured, and t o ±0.01° i f o v e r l a p p i n g d i f f r a c t i o n l i n e s made i t d i f f i c u l t t o d e t e r m i n e the c e n t r e s . The second column i s the i n t e g r a t e d i n t e n s i t y of the d i f f r a c t i o n peak. The s t r o n g e s t peak i s g i v e n the v a l u e of 100, and a l l o t h e r i n t e n s i t i e s a r e measured r e l a t i v e t o t h i s . The p r e f e r r e d o r i e n t a t i o n i n the samples made t h e ( 0 , 0 , 1 ) l i n e s v e r y s t r o n g . A t i l d e (~) i n the i n t e n s i t y column i m p l i e s two or more d i f f r a c t i o n peaks o v e r l a p p e d , and the number g i v e n i s an e s t i m a t e of the i n t e n s i t y 208 a s s o c i a t e d w i t h t h a t d i f f r a c t i o n peak. The t h i r d column c o n t a i n s the 1/d 2 v a l u e s computed from the 26 a n g l e and the o u t - o f - p l a n e c o r r e c t i o n . The u n i t s of 1/d 2 a r e 1 / A 2 . I f a peak p o s i t i o n was not p r e c i s e l y d e t e r m i n e d , then the 1/d 2 v a l u e i s o n l y l i s t e d t o t h r e e s i g n i f i c a n t f i g u r e s . Columns f i v e t h r o u g h seven a r e the M i l l e r i n d i c e s a f f i x e d t o the d i f f r a c t i o n peaks. Column f o u r c o n t a i n s the 1/d 2 ( l / A 2 ) v a l u e s computed from the l a t t i c e p arameters and M i l l e r i n d i c e s . The l a t t i c e parameters were d e t e r m i n e d u s i n g both the complete d a t a s e t s and f o r o n l y t h o s e peaks t h a t were p r e c i s e l y d e t e r m i n e d . There was no s i g n i f i c a n t d i f f e r e n c e i n the f i n a l r e s u l t s . 209 TABLE V I I I L a t t i c e A (1.7V) X-Ray Data 26 (°) I 0 1/d 2 1/d 2 h k I obs. c a l c . 14.4656 1 00.0 0.0255 0.0255 0 0 1 15.3134 <0. 1 0.0287 0.0287 1 -I 0 20.9708 <0. 1 .0.0541 0.0542 1 -I 1 28.8137 10.8 0.1021 0.1020 0 0 2 30.5762 0.7 0.1148 0.1148 2 -2 0 31.0920 ~0.7 0.1186 0.1183 2 0 0 31 .20 ~0.7 0.119 0.1199 0 2 0 32.6709 1.2 0.1308 0.1307 1 -1 2 33.8669 2.3 0.1404 0. 1403 2 -2 1 34.2502 ~2. 1 0.1435 0. 1438 2 0 1 34.4800 ~2. 1 0.1454 0. 1454 0 2 1 42.3609 4.2 0.2170 0.2168 2 -2 2 42.80 ~5.8 0.221 0.2203 2 0 2 42.8708 ~5.8 0.2220 0.2219 0 2 2 43.6122 16.7 0.2294 0.2295 0 0 3 46.3881 3.3 0.2582 0.2582 1 -1 3 54.0222 ~3.3 0.3441 0.3443 2 -2 3 54.4173 ~3.3 0.3488 0.3478 2 0 3 54.48 ~3.3 0.349 0.3493 0 2 3 55.4866 ~0.6 0.3617 0.3615 2 2 0 57.67 <0. 1 0.388 0.3870 2 2 1 59.2332 25.0 0.4080 0.4080 0 0 4 61.4431 2.2 0.4362 0.4365 1 -1 4 67.92 ~0.6 0.523 0.5228 2 -2 4 68.23 ~0.6 0.527 0.5263 2 0 4 68.33 ~0.6 0.529 0.5279 0 2 4 76. 1 706 5.8 0.6375 0.6376 0 0 5 84. 1 5 ~1 .3 0.753 0.7523 2 -2 5 84.43 ~1 .3 0.757 0.7559 2 0 5 84.50 ~1 .3 0.758 0.7574 0 2 5 X-ray d i f f r a c t i o n d a t a r e c o r d e d f o r c e l l PMX-86. The cathode m a t e r i a l was 5p53 t h a t had been washed i n hot n - b u t y l l i t h i u m , and the p o t e n t i a l was 1.7V. The m a t e r i a l was L i M o S 2 . The o u t - o f - p l a n e s p a c i n g , 6, was 0.498mm. 210 TABLE IX L a t t i c e B (1.93V) X-Ray Data 20. (°) 1/d 2 1/d 2 h k I obs. c a l c . 14.1277 1 00.0 0.0249 0.0249 0 0 1 28.3022 10.0 0.0997 0.0996 0 0 2 31 .05 ~0.4 0.1196 0.1194 2 -2. 0 31 .05 ~0.4 0.1196 0.1199 2 0 0 31 .27 ~0.4 0.1213 0.1212 0 2 0 34.2672 ~3.3 0.1450 0. 1448 2 0 1 34.2672 ~3.3 0.1450 0.1450 2 -2 -1 34.2672 ~3.3 0.1450 0.1454 0 2 -1 42.38 ~3.0 0.219 0.2177 2 -2 2 42.38 ~2.0 0.219 0.2194 2 0 -2 42.38 ~2.0 0.219 0.2195 0 2 -2 42.4977 ~3.0 0.2199 0.2195 2 0 2 42.4977 ~3.0 0.2199 0.2204 2 -2 -2 42.9145 16.0 0.2241 0.2241 0 0 3 53.6338 2.0 0.3414 0.3416 2 -2 3 53.7685 ~2.5 0.3430 0.3439 2 0 -3 54.2194 ~3.5 0.3483 0.3473 0 2 3 54.9573 1 .0 0.3572 0.3574 4 -2 0 55.28 ~1 .0 0.361 0.3613 2 -4 0 55.4351 ~1 .0 0.3629 0.3626 2 2 0 57.50 0.2 0.388 0.3882 2 2 1 58.3164 20.0 0.3984 0.3985 0 0 4 67.25 ~0.6 0.5150 0.5152 2 -2 4 67.42 ~1 .8 0.5173 0.5183 2 0 -4 67.86 ~1 .2 0.5234 0.5223 0 2 4 72.0127 1 .0 0.5807 0.5815 2 -4 3 74.9808 6.0 0.6225 0.6226 0 0 5 83.05 0.2 0.739 0.7404 0 2 -5 83.41 ~0.5 0.744 0.7425 2 0 5 83.41 ~0.5 0.744 0.7453 •2 -2 -5 83.99 0.2 0.753 0.7532 4 -2 4 84.31 0.3 0.758 0.7584 2 2 -4 X-ray d i f f r a c t i o n d a t a r e c o r d e d f o r c e l l PMX-85. The cathode m a t e r i a l was 5p53, and the c e l l p o t e n t i a l was 1.93V. The m a t e r i a l was L i 0 > 7 5 M o S 2 . The o u t - o f - p l a n e s p a c i n g was 0.237mm. 21 1 TABLE X L a t t i c e C (2 .23V) X-Ray Data 28 (°) I o 1/d 2 1/d 2 h k / obs. c a l c . 13.9933 100 0.0249 0.0249 0 0 2 16.6816 1 0.0353 0.0354 1 -1 1 19.9450 3 0.0503 0.0526 1 -1 2 20.57 2 0.054 0.0542 1 0 2 28.1984 8 0.0997 0.0996 0 0 4 31 .2840 ~1 0.1222 0. 1222 2 -2 0 31 .4380 ~1 0.1234 0. 1234 2 0 0 31.4380 ~1 0.1234 0. 1236 0 2 0 34.2137 ~10 0.1455 0.1451 2 0 2 34.2137 ~10 0.1455 0.1460 0 2 -2 35.0532 8 0.1525 0.1526 2 -2 -2 38.1044 1 0.1792 0.1792 1 -J 5 39.5985 2 0.1930 0.1931 1 -1 -5 42.1499 ~10 0.2176 0.2167 2 0 4 42.1499 ~1 0 0.1276 0.2183 0 2 -4 42.7848 ~20 0.2239 0.2241 0 0 6 43.26 ~20 0.229 0.2280 0 2 4 43.26 ~20 0.229 0.2291 2 0 -4 43.67 ~20 0.238 0.2339 2 -2 -4 45.1783 1 0.2483 0.2477 1 1 5 45.1783 1 0.2483 0.2493 1 1 -5 49.9722 3 0.3003 0.3003 2 -3 4 52.1655 ~2 0.3254 0.3258 1 -1 7 52.4549 ~1 0.3288 0.3296 2 -2 6 54.7998 10 0.3565 0.3549 0 2 6 55.7162 ~4 0.3676 0.3675 4 -2 0 55.89 ~4 0.370 0.3682 2 -4 0 56.08 ~4 0.3721 0.3716 2 2 0 57.03 2 0.3836 0.3838 4 -2 2 58.2158 30 0.3984 0.3983 0 0 8 66.05 1 0.500 0.4983 2 -2 8 67.66 ~4 0.522 0.5121 0 2 -8 67.83 ~4 0.524 0.5240 0 4 2 67 .91 ~4 0.525 0.5222 1 -1 9 71.9561 1 0.5812 0.5805 4 0 4 72.8346 1 0.5932 0.5934 2 2 6 74.8792 10 0.6224 0.6224 0 0 10 X-ray d i f f r a c t i o n d a t a r e c o r d e d f o r sample " P r o p a n o l " . The m a t e r i a l was 5p53 washed i n p r o p a n o l . The m a t e r i a l was L i 0 . 5 M o S 2 C e l l s made wi t h t h i s m a t e r i a l had an i n i t i a l p o t e n t i a l of ~2.23V. The d a t a shown here i s not from a c e l l . The powder had been s p r i n k l e d onto an x- r a y c e l l s u b s t r a t e t h a t had been c o a t e d w i t h vacuum g r e a s e . T h i s sample had f a r l e s s p r e f e r r e d o r i e n t a t i o n than the o t h e r samples. The o u t - o f - p l a n e c o r r e c t i o n was 0.059mm. 212 TABLE XI L a t t i c e D (2.56V) X-Ray Data 28 (°) Io 1/d 2 1/d 2 h k I obs. c a l c . 14.0921 100.0 0.0248 0.0248 0 0 2 28.2413 8.0 0.0993 0.0993 0 0 4 31.3713 0.4 0.1221 0.1220 2 -2 0 31.7164 ~0.5 0.1247 0.1248 2 0 0 31.7164 ~0.5 0. 1247 0.1248 0 2 0 34.5239 ~2.0 0. 1472 0.1472 2 0 2 34.5239 ~2.0 0. 1472 0.1472 0 2 -2 35.0507 2.3 0.1516 0.1517 2 -2 -2 38.2435 0.9 0. 1795 0.1795 J -1 5 39.5383 1 .2 0.1915 0.1917 J -1 -5 42.4753 ~1 .2 0.2197 0.2192 2 0 4 42.4753 ~1 .2 0.2197 0.2192 0 2 -4 42.8297 6.0 0.2233 0.2234 0 0 6 43.3911 ~2.0 0.2289 0.2290 0 2 4 43.3911 ~2.0 0.2289 0.2290 2 0 -4 43.5362 ~2.0 0.2304 0.2310 2 -2 -4 45.3871 0.3 0.2494 0.2494 1 1 -5 46.8448 1 .0 0.2648 0.2649 3 -J 3 53.7432 1 .0 0.3427 0.3427 1 -1 7 54.8225 ~2.5 0.3556 0.3555 0 2 6 54.8225 ~2.5 0.3556 0.3555 2 0 -6 55.9163 ~0.3 0.3688 0.3688 2 -4 0 55.9163 ~0.3 0.3688 0.3688 4 -2 0 56.5839 0.9 0.3770 0.3773 2 2 0 58.0727 22.0 0.3970 0.3971 0 0 8 67.7477 1 .0 0.5219 0.5221 1 -1 9 68.46 ~1 .0 0.532 0.5316 2 0 -8 68.46 ~1 .0 0.532 0.5316 0 2 8 74.7979 4.8 0.6200 0.6204 0 0 JO X-ray d i f f r a c t i o n d a t a r e c o r d e d f o r c e l l PMX-85. The cathode m a t e r i a l was 5p53, and the c e l l p o t e n t i a l was 2.56V. The m a t e r i a l was L i 0 . 2 7 M o S 2 . The o u t - o f - p l a n e s p a c i n g was 0.237mm. 213 TABLE X I I L a t t i c e E (2.63V) X-Ray Data 26 (°). Io 1/d 2 1/d 2 h k I obs. c a l c . 14.1045 100.0 0.0249 0.0249 0 0 2 28.2681 8.0 0.0995 0.0995 0 0 . 4 31 .3904. 0.7 0.1222 0.1226 2 -2 0 31.7971 ~0.5 0.1253 0. 1252 0 2 0 31.7971 ~0.5 0.1253 0. 1253 2 0 0 34.4888 ~1 .9 0.1469 0. 1465 0 2 -2 34.8889 ~2.2 0.1502 0. 1506 2 -2 -2 34.8889 ~2.2 0.1502 0. 1506 2 0 2 38.5092 0.7 0.1820 0. 1822 1 -I 5 39.3676 0.8 0.1899 0.1890 1 -1 -5 42.2969 ~2.5 0.2180 0.2175 2 0 4 42.8816 12.0 0.2238 0.2238 0 0 6 43.3148 0.6 0.2281 0.2283 2 -2 4 43.6864 0.5 0.2319 0.2320 2 0 4 45.9259 0.5 0.2550 0.2552 1 1 - J 47.4266 0.5 0.271 1 0.2707 2 1 3 48.5832 0.5 0.2837 0.2845 3 -3 J 52.6232 0.7 0.3295 0.3298 1 -1 7 54.1732 ~2.0 0.3478 0.3475 0 2 6 55.1741 1 .7 0.3598 0.3599 2 0 -6 56.0603 ~0.5 0.3706 0.3703 2 -4 0 56.0603 ~0.5 0.3706 0.3706 4 -2 0 56.7347 0.6 0.3788 0.3783 2 2 0 58.2648 15.0 0.3978 0.3979 0 0 8 68. 10 ~1 .7 0.527 0.5272 1 -1 9 68.37 0.7 0.533 0.5332 4 0 -2 74.9021 5.0 0.6215 0.6217 0 0 10 X-ray d i f f r a c t i o n d a t a r e c o r d e d f o r c e l l PMX-85. The cathode m a t e r i a l was 5p53, and the c e l l p o t e n t i a l was 2.63V. The m a t e r i a l was L i 0 . 1 2 M o S 2 . The o u t - o f - p l a n e d i s t a n c e was 0.237mm. 214 APPENDIX IV Phenomena Ca u s i n g N o n - R e v e r s i b i l i t y Things are as they are because they were as they were. --Thomas Gol d S e v e r a l r e f e r e n c e s were made i n the main body of the t h e s i s t o asymmetries i n the charge and d i s c h a r g e f e a t u r e s of c e l l s . I f an i n t e r c a l a t i o n r e a c t i o n were p e r f e c t l y r e v e r s i b l e , then the l i t h i u m c o n t e n t would u n i q u e l y d e f i n e the c h e m i c a l p o t e n t i a l . However, a cathode m a t e r i a l may e n t e r a m e t a s t a b l e s t a t e on one or both of the charge and d i s c h a r g e h a l f - c y c l e s . I f t h i s happens, the l i t h i u m c o n t e n t does not u n i q u e l y d e f i n e the c e l l p o t e n t i a l . A l s o , the energy g a i n e d by d i s c h a r g i n g the c e l l i s not as l a r g e as the energy r e q u i r e d t o charge i t . T h i s appendix w i l l b r i e f l y d e s c r i b e t h r e e phenomena t h a t may cause a c e l l t o d i s p l a y i r r e v e r s i b i l i t y : h y s t e r e s i s and s u p e r s a t u r a t i o n , which a r e a s s o c i a t e d w i t h m e t a s t a b l e s t a t e s ; and d i f f u s i o n , which i s a s s o c i a t e d w i t h g r a d i e n t s i n the h o s t . D i f f u s i o n i s a p r o c e s s by which p a r t i c l e s a r e mixed as a r e s u l t of t h e i r random t h e r m a l m o t i o n s . T h i s can be viewed s t a t i s t i c a l l y as a random walk p r o c e s s . Thermodynamically t h i s i s r e p r e s e n t e d by a c u r r e n t of p a r t i c l e s d r i v e n by a g r a d i e n t i n the c h e m i c a l p o t e n t i a l . The m a c r o s c o p i c e f f e c t i s t h a t p a r t i c l e s w i l l seem t o f l o w from a r e g i o n of h i g h c o n c e n t r a t i o n t o a r e g i o n of low c o n c e n t r a t i o n . 215 The e x p e r i m e n t s d i s c u s s e d i n t h i s t h e s i s e s t a b l i s h e d a c o n c e n t r a t i o n g r a d i e n t i n the i n t e r c a l a t i o n h o s t s by c h a n g i n g the number of l i t h i u m p a r t i c l e s a t t h e s u r f a c e of the c r y s t a l s . A c e l l t h a t i s b e i n g c y c l e d a t c o n s t a n t c u r r e n t w i l l always have a c o n c e n t r a t i o n g r a d i e n t . However, a c e l l can be c o n s i d e r e d t o be i n a q u a s i - e q u i l i b r i u m s t a t e i f the g r a d i e n t s a r e s m a l l enough t h a t they do not i n f l u e n c e the r e s u l t s . The g r e a t e r the p r e c i s i o n r e q u i r e d i n the measurements, the s m a l l e r the c u r r e n t must be. I f t h e r e were an a p p r e c i a b l e c o n c e n t r a t i o n g r a d i e n t t h r ough the h o s t , then a V v s . x c u r v e measured w h i l e the c e l l was c h a r g i n g would not l o o k l i k e a V v s . x c u r v e measured w h i l e the c e l l was d i s c h a r g i n g . The d i f f e r e n c e i n the two c u r v e s might be i n c o r r e c t l y a t t r i b u t e d t o h y s t e r e s i s . D i f f u s i o n does not p r e v e n t a system from r e a c h i n g e q u i l i b r i u m , but a m a t e r i a l w i t h a l o n g d i f f u s i o n c o n s t a n t may be m i s t a k e n f o r a system w i t h a m e t a s t a b l e s t a t e . A more complete a n a l y s i s of the problem of d i f f u s i o n i n e l e c t r o c h e m i c a l c e l l s was p r e s e n t e d by McKinnon and H a e r i n g (1983). There w i l l always be an i r r e v e r s i b i l i t y a s s o c i a t e d w i t h a f i r s t o r d e r phase t r a n s i t i o n . For s i m p l i c i t y , assume t h a t t h i s t r a n s i t i o n i n v o l v e s c h a n g i n g one thermodynamic parameter, such as t e m p e r a t u r e , s t a t e A i s the h i g h t e m p e r a t u r e phase and s t a t e B i s the low t emperature phase. There i s some temp e r a t u r e a t which phase A and phase B a r e i n thermodynamic e q u i l i b r i u m . I f the t e m p e r a t u r e i s changed from the e q u i l i b r i u m v a l u e then one phase or the o t h e r i s f a v o u r e d . However, a system i n phase A 216 c o u l d be c o o l e d t o a t e m p e r a t u r e where phase B would be, t h e r m o d y n a m i c a l l y s t a b l e , but the system would remain i n a m e t a s t a b l e form of s t a t e A. T h i s i s t h e phenomenon of s u p e r s a t u r a t i o n or s u p e r c o o l i n g . The t r a n s i t i o n from phase A t o phase B i s i n h i b i t e d because t h e r e i s an energy a s s o c i a t e d w i t h the f o r m a t i o n of the i n t e r f a c e between the two phases. The phase t r a n s i t i o n w i l l not occur throughout the system u n t i l a s m a l l , s t a b l e amount of phase B forms. T h i s i s the phenomenon of n u c l e a t i o n . The s i z e of the phase B r e g i o n would then grow u n t i l i t consumed a l l the phase A. There a r e two g e n e r a l c l a s s e s of n u c l e a t i o n : homogeneous n u c l e a t i o n , and heterogeneous n u c l e a t i o n . Both p r o c e s s e s a r e f u n d a m e n t a l l y s i m i l a r : random motion of the p a r t i c l e s cause d e n s i t y f l u c t u a t i o n s t h a t may produce a r e g i o n of the new phase. I f the r e g i o n i s s u f f i c i e n t l y l a r g e , then i t i s more l i k e l y f o r the r e g i o n t o expand than s h r i n k . Homogeneous n u c l e a t i o n i s t h e f o r m a t i o n of one phase i n t h e b u l k of a n o t h e r phase. The c l a s s i c example i s the f o r m a t i o n of a s p h e r i c a l d r o p from a s u p e r s a t u r a t e d vapour. The f r e e energy a s s o c i a t e d w i t h the l i q u i d i s lower than the f r e e energy of t h e gas, w i t h the r e s u l t t h a t the system can lower i t s t o t a l energy by c o n v e r t i n g t o the l i q u i d . The f r e e energy change a s s o c i a t e d w i t h the f o r m a t i o n of the l i q u i d i s p r o p o r t i o n a l t o the cube of the r a d i u s of the drop. The s u r f a c e energy i s p r o p o r t i o n a l t o the square of the r a d i u s . There i s some c r i t i c a l r a d i u s , r c , a t which the s u r f a c e and b u l k e n e r g i e s sum t o z e r o . I f the r a d i u s 217 i s l e s s than r c then the drop w i l l s h r i n k and d i s a p p e a r . I f the r a d i u s i s g r e a t e r than r c . then the d r o p w i l l grow. The t r a n s i t i o n w i l l o n l y p r o c e e d i f the random t h e r m a l motion has c o n c e n t r a t e d enough p a r t i c l e s t o g e t h e r t o form a c l u s t e r of a t l e a s t the s i z e g i v e n by r c . The s i z e of r c i s a f f e c t e d by the d i f f e r e n c e i n f r e e energy between the two phases. I f the system i s brought f u r t h e r from e q u i l i b r i u m , then the f r e e energy d i f f e r e n c e between the two phases i s u s u a l l y i n c r e a s e d , and r c becomes s m a l l e r . The m e t a s t a b l e s t a t e s may be v e r y l o n g - l i v e d and may s u r v i v e f a r below the t h e o r e t i c a l e q u i l i b r i u m v a l u e . Pure water w i l l remain as a l i q u i d a t -10°C f o r days i f t h e c o n t a i n e r i s kept s t i l l . Germanium can be s u p e r c o o l e d by 227°C b e f o r e the s o l i d w i l l form. Heterogeneous n u c l e a t i o n , which s t a r t s from s u r f a c e s and i m p u r i t i e s , i s more l i k e l y t o o c c u r than homogeneous n u c l e a t i o n i n the b u l k . The d e n s i t y of p a r t i c l e s can be i n c r e a s e d a t an i n t e r f a c e i f they a d s o r b on t h e s u r f a c e . A s u r f a c e might a l s o impose an o r d e r on the p a r t i c l e s t h a t would f a v o u r the new phase. The shape of the s u r f a c e , the shape of t h e c l u s t e r , and s u r f a c e - c l u s t e r i n t e r f a c e energy are a l l r e l e v a n t t o m a t h e m a t i c a l d e s c r i p t i o n s of heterogeneous n u c l e a t i o n . N u c l e a t i o n phenomena a l s o a p p l y i n s o l i d s t a t e t r a n s i t i o n s . The c o n v e r s i o n from one c r y s t a l t y pe t o another i s i n h i b i t e d by the s t r a i n caused by the l a t t i c e parameter mismatch. Heterogeneous n u c l e a t i o n can o c c u r a t g r a i n b o u n d a r i e s , d i s l o c a t i o n s , and v a c a n c i e s . A l l m a t h e m a t i c a l t r e a t m e n t s of 218 n u c l e a t i o n phenomena i n v o l v e e s t i m a t i o n s of the e n e r g i e s a s s o c i a t e d w i t h the s u r f a c e i n t e r f a c e s , and the c o m p u t a t i o n s a s s o c i a t e d w i t h s o l i d systems a r e q u i t e complex. More d e t a i l e d d i s c u s s i o n s of n u c l e a t i o n i n v a r i o u s systems can be found i n the book e d i t e d by Z e t t l e n o y e r (1969). Any system i n which the s t a t e depends on the h i s t o r y of the sample i s s a i d t o e x h i b i t h y s t e r e s i s . S u p e r s a t u r a t i o n l e a d s t o h y s t e r e s i s because the t r a n s i t i o n from a s t a t e A t o a s t a t e B w i l l not occur a t the same v a l u e s of the thermodynamic parameters as the t r a n s i t i o n from s t a t e B t o s t a t e A. H y s t e r e s i s phenomena a r e not r e s t r i c t e d t o phase t r a n s i t i o n s . H y s t e r e s i s was examined i n d e t a i l i n a s e r i e s of papers by E v e r e t t and o t h e r s (1952, 1954, 1955, 1955). E v e r e t t uses t h e word " h y s t e r e s i s " t o d e s c r i b e systems i n which the p a t h from s t a t e A t o s t a t e B i s s t a b l e and r e p r o d u c i b l e . " S u p e r s a t u r a t i o n " d e s c r i b e s systems i n which the p a t h s between A and B a r e u n s t a b l e and may v a r y from experiment t o e x p e r i m e n t . M i n o r v a r i a t i o n s i n the sample do not a f f e c t t r u e h y s t e r e s i s , and the m a c r o s c o p i c s t a t e of the system w i l l be known i f the thermodynamic parameters and the h i s t o r y of the sample a r e known. S u p e r s a t u r a t i o n i s a f f e c t e d by minor p e r t u r b a t i o n s , such as i m p u r i t i e s and d e f e c t s , so the m e t a s t a b l e s t a t e s w i l l not always c o n v e r t a t the same v a l u e s of the thermodynamic p a r a m e t e r s . The d i s t i n c t i o n between s u p e r s a t u r a t i o n and o t h e r forms of h y s t e r e s i s i s not always c l e a r , but i t i s u s e f u l t o d i s t i n g u i s h between two p r o p e r t i e s . 219 The most common example of h y s t e r e s i s i s the m a g n e t i z a t i o n of a f e r r o m a g n e t i c sample. The domains i n t h e m a t e r i a l can be r e a r r a n g e d by an e x t e r n a l f i e l d . T h i s rearrangement i s not t o t a l l y r e v e r s i b l e , t h e r e f o r e the m a t e r i a l w i l l r e t a i n a remnant m a g n e t i z a t i o n a f t e r the e x t e r n a l f i e l d i s removed. The system can s e t t l e i n t o a s t a b l e c o n f i g u r a t i o n t h a t i s not the l o w e s t energy s t a t e . D i f f u s i o n , s u p e r s a t u r a t i o n , and h y s t e r e s i s can a l l cause an asymmetry i n the charge and d i s c h a r g e c u r v e s of e l e c t r o c h e m i c a l c e l l s , but i t i s p o s s i b l e t o d i s t i n g u i s h between the t h r e e e f f e c t s . I f d i f f u s i o n caused the asymmetry, then slo w e r c y c l e s would make the charge and d i s c h a r g e c u r v e s l o o k more symmetric. I t i s more d i f f i c u l t t o s e p a r a t e s u p e r s a t u r a t i o n and h y s t e r e s i s . I f s u p e r s a t u r a t i o n were t o cause the n o n - u n i f o r m i t y , then d i f f e r e n t samples would show d i f f e r e n t amounts of asymmetry because the phase t r a n s i t i o n s would not always o c c u r at the same o v e r p o t e n t i a l s . True h y s t e r e s i s would be p r e s e n t t o the same degree i n a l l samples of the same m a t e r i a l . 

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