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Low temperature x-ray diffraction studies of TaS₂ and LixTiS₂ Dutcher, John Robert 1985

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LOW TEMPERATURE X-RAY DIFFRACTION STUDIES OF TaS 2 AND L i T i S , z x 1 By JOHN ROBERT DUTCHER B . S c , Dalhousie U n i v e r s i t y , 1983 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Physics We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1985 © John Robert Dutcher, 1985 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 May 9 , 1 9 8 5 -6 (3/81) i i A B S T R A C T A l o w t e m p e r a t u r e x - r a y p o w d e r d i f f r a c t i o n a t t a c h m e n t f o r u s e o n t h e v e r t i c a l g o n i o m e t e r o f a d i f f T a c t o m e t e r i s d e s c r i b e d . We h a v e f o u n d t h a t d i f f r a c t i o n p a t t e r n s o b t a i n e d w i t h t h e a t t a c h m e n t m o u n t e d o n t h e g o n i o m e t e r a r e o f c o m p a r a b l e q u a l i t y t o t h o s e o b t a i n e d o n t h e g o n i o m e t e r i t s e l f . U s i n g t h i s a t t a c h m e n t , t h e l a t t i c e p a r a m e t e r d i s c o n t i n u i t i e s a s s o c i a t e d w i t h a c h a r g e d e n s i t y w a v e p h a s e t r a n s i t i o n i n 1 T - T a S 2 n e a r T = 2 0 0 K a r e m e a s u r e d w i t h a n a c c u r a c y g r e a t e r t h a n t h a t o f a n y p r e v i o u s r e s u l t s . E l e c t r o c h e m i c a l l y p r e p a r e d s a m p l e s o f L i T i S 2 n e a r x = 0 . 1 6 a r e s t u d i e d a t r o o m t e m p e r a t u r e a n d b e l o w . C l e a r e v i d e n c e f o r t h e f o r m a t i o n o f a 1 s t a g e t w o s u p e r l a t t i c e a t l o w t e m p e r a t u r e s w a s n o t o b t a i n e d . i i i TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES v LIST OF FIGURES v i ACKNOWLEDGEMENTS ix CHAPTER ONE INTRODUCTION 1 1.1 X-ray D i f f r a c t i o n 1 1.1.1 X-ray D i f f r a c t i o n Fundamentals 1 1.1.2 The Powder D i f f r a c t o m e t r i c 4 Method 1.2 Low Temperature X-ray D i f f r a c t i o n 7 CHAPTER TWO LOW TEMPERATURE X-RAY DIFFRACTION 10 SYSTEM DESCRIPTION 2.1 Apparatus 10 2.1.1 Low Temperature Attachment 10 2.1.2 X-ray D i f f r a c t i o n Apparatus 14 2.1.3 R e f r i g e r a t i o n Apparatus 14 2.2 Alignment Procedure 15 CHAPTER THREE PERFORMANCE CHARACTERISTICS OF 21 THE LOW TEMPERATURE ATTACHMENT 3.1 System Temperature C h a r a c t e r i s t i c s 21 3.2 Alignment Q u a l i t y E v a l u a t i o n 24 3.3 V i b r a t i o n C h a r a c t e r i s t i c s 32 3.4 Se q u e n t i a l Mounting Comparison 33 CHAPTER FOUR CHARGE DENSITY WAVE TRANSITION 38 STUDY IN 1T-TaS 2 4.1 I n t r o d u c t i o n 38 4.1.1 Charge Dens i t y Waves 38 4.1.2 Charge Dens i t y Wave Behavior 40 of 1T-TaS 2 4.2 Experimental 47 4.2.1 Sample P r e p a r a t i o n 47 4.2.2 The NC-C T r a n s i t i o n 51 i v A SEARCH FOR STAGING IN L i x T i S 2 CHAPTER FIVE 5.1 L i t h i u m I n t e r c a l a t i o n in T i S 2 5.2 Experimental 5.2.1 Sample P r e p a r a t i o n 5.2.2 Thermal Expansion Measurements 5.2.3 S u p e r l a t t i c e Peak Search 5.2.3.1 DJR18 (X=0.13). 5.2.3.2 DJR20 (x=0.20) 5.2.3.3 DJR22 (X=0.15) CHAPTER SIX CONCLUSION 6.1 Summary 6.2 Suggestions f o r Fu r t h e r Work BIBLIOGRAPHY APPENDIX A REFRIGERATION CYCLE DESCRIPTION APPENDIX B APPENDIX C A SIMPLIFIED LEAST SQUARES PEAK FITTING PROCEDURE ROOM TEMPERATURE CHARGE DENSITY WAVE STRUCTURE OF 1T~TaS„ Page 66 66 77 77 82 87 87 88 92 98 98 99 101 1 07 1 1 3 121 APPENDIX D ON THE RETENTION OF HIGH TEMPERATURE STRUCTURE UPON COOLING 124 V LIST OF TABLES Table Page 1. Comparison of measured and c a l c u l a t e d 29 27 values f o r a t y p i c a l step scan of the s i l i c o n standard on the theta s h a f t . 2. Comparison of measured and c a l c u l a t e d 26 28 values f o r a t y p i c a l step scan of the s i l i c o n standard on the c o l d f i n g e r s h a f t . 3. R e s u l t s of l e a s t squares f i t s to step 30 scan data taken on the theta and c o l d f i n g e r s h a f t s immediately f o l l o w i n g the s h a f t s ' alignment. 4. Observed room temperature s a t e l l i t e 49 peaks in a 1T-TaS2 s a m P l e « 5. Thermal h i s t o r y of the t h i n 1T-TaS2 5 2 sample. 6. Thermal expansion data f o r the NC-C 62 t r a n s i t i o n in 1T-TaS2» i n c l u d i n g those of Sezerman et a l . (1980). 7. x values f o r L i x T i S 2 e l e c t r o c h e m i c a l 80 c e l l s used i n low temperature s t u d i e s . 8. Approximate thermal expansion 86 c o e f f i c i e n t s f o r a l l low temperature samples of L i TiS2« v i LIST OF FIGURES F i g u r e Page 1. D i f f r a c t i o n of x-rays by a c r y s t a l . 2 2. The Bragg-Brentano f o c u s i n g geometry. 5 3. Schematic diagram of low temperature 11 x-ray d i f f r a c t i o n apparatus. 4. Exploded view of attachment. 12 5. Theta and c o l d f i n g e r s h a f t geometry. 17 6. D i r e c t comparison of d i f f r a c t i o n 26 p a t t e r n s of the s i l i c o n (311) peak obtained on the t h e t a and c o l d f i n g e r s h a f t s . 7. D i f f r a c t i o n p a t t e r n s of the s i l i c o n 34 (311) peak obtained with and without system v i b r a t i o n s . 8. The d i f f e r e n c e of the d i f f r a c t i o n 35 p a t t e r n s of F i g u r e 7. 9. The d i f f e r e n c e of the d i f f r a c t i o n 37 p a t t e r n s of the s i l i c o n (311) peak obtained f o r two d i s t i n c t mountings of the attachment on the goniometer. 10. The atomic s t r u c t u r e of 1T-TaS 2. 41 11. The Fermi s u r f a c e of 1T-TaS 2 and 44 i t s n e s t i n g p r o p e r t i e s . 12. The a- and c-axes versus temperature 46 data of Givens and F r e d e r i c k s (1977) . 13. Continuous x-ray scan data f o r the 55 1T-TaS 2 (005) peak near the NC-C t r a n s i t i o n obtained on c o o l i n g . 14. Continuous x-ray scan data f o r the 56 1T-TaS 2 (005) peak near the NC-C t r a n s i t i o n o b t a i n e d on h e a t i n g . 15. a-axis versus temperature data f o r the 58 t h i n 1T-TaS 2 sample. 16. c - a x i s versus temperature data f o r the 59 t h i n 1T-TaS 2 sample. Comparison of c - a x i s v e r s u s temperature d a t a o b t a i n e d f o r the t h i n sample from a l e a s t squares f i t t o ten Bragg peaks and t h a t o b t a i n e d from the (100) peak. Comparison of c - a x i s v e r s u s temperature da t a f o r the t h i n sample and t h a t o b t a i n e d f o r a sample 300u t h i c k e r . Schematic r e p r e s e n t a t i o n of the d i s c h a r g e of an e l e c t r o c h e m i c a l c e l l . -dx/dV v e r s u s x f o r a L i T i S 9 c e l l ( a f t e r Dahn 1982). x 1 V a r i a t i o n of the (004) Bragg peak w i d t h f o r L i TiS„ ( a f t e r Dahn 1982). x 2 dV/dT) v e r s u s x f o r a L i T i S c e l l ( a f t e r Dahn 1982). x 2 A phase diagram f o r L i T i S 9 ( a f t e r Dahn 1982). x 1 V a r i a t i o n of a- and c-axes w i t h t emperature f o r TiS 2« V a r i a t i o n of a- and c-axes w i t h t emperature f o r the DJR18-, sample ( X = 0 . 1 3 ) . 1 The (009/2) r e g i o n f o r the DJR18-sample (x=0.13) at 300K and 167KT The (009/2) r e g i o n f o r the DJR20 sample (x=0.20) a t 300K and 166K. The (009/2) r e g i o n f o r the DJR22-, sample (X=0.15) at 300K and 166KT The (009/2) r e g i o n f o r the DJR22 ? sample (x=0.15) at 300K and 165K f o r a c o u n t i n g time of 40 seconds per s t e p . The (009/2) r e g i o n f o r the DJR22 ? sample (x=0.15) a t 300K and 165K f o r a c o u n t i n g time of 400 seconds per s t e p . C r o s s - s e c t i o n a l diagram of the r e f r i g e r a t i o n u n i t . V I 1 1 F i g u r e Page 32. Cycle diagram of the r e f r i g e r a t i o n 109 un i t. 33. Comparison of a c t u a l and f i t t e d 118 data to (311) s i l i c o n peak. 34. Comparison of a c t u a l and f i t t e d 119 data to (004) peak of L i ioTiS2 sample. 35. R e c i p r o c a l space geometry of main 122 l a t t i c e and f i r s t order charge d e n s i t y wave s a t e l l i t e peaks f o r room temperature 1T-TaS2> ix ACKNOWLEDGEMENTS I would l i k e to thank my s u p e r v i s o r , Rudi Haering, f o r h i s help and ad v i c e d u r i n g the course of t h i s work. Many u s e f u l d i s c u s s i o n s with Doug Dahn, Peter Mulhern, J e f f Dahn and Rod McMillan are g r a t e f u l l y acknowledged. Alec R i v e r s -Bowerman, i n a d d i t i o n to p r o v i d i n g me with i n v a l u a b l e t e c h n i c a l a d v i c e , designed and c o n s t r u c t e d the low temperature attachment d e s c r i b e d i n t h i s t h e s i s and c o n t r i b u t e d F i g u r e 4. I t g i v e s me great p l e a s u r e to thank Heather Rokosh. Her strong support was g r e a t l y a p p r e c i a t e d . F i n a l l y , I would l i k e to thank the N a t u r a l Sciences and Eng i n e e r i n g Research C o u n c i l f o r f i n a n c i a l support. 1 CHAPTER ONE INTRODUCTION The purpose of t h i s chapter i s to introduce the b a s i c s of x-ray d i f f r a c t i o n and i t s a p p l i c a t i o n at low temperatures. The d i s c u s s i o n s w i l l concentrate on the techniques used i n t h i s t h e s i s ; they are b r i e f and are intended only to motivate a p h y s i c a l understanding of the p r i n c i p l e s i n v o l v e d . For more complete d i s c u s s i o n s of x-ray d i f f r a c t i o n p r i n c i p l e s and techniques, one should c o n s u l t standard r e f e r e n c e s such as C u l l i t y (1959), Warren (1969), and Klug and Alexander (1974). Rudman (1976) p r o v i d e s a good reference on techniques and d e v i c e s used i n low temperature x-ray d i f f r a c t i o n . 1.1 X-ray D i f f r a c t i o n 1.1.1 X-ray D i f f r a c t i o n Fundamentals The d i f f r a c t i o n of x-rays from planes of atoms in a c r y s t a 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 1, with p a r a l l e l planes of atoms separated by a d i s t a n c e d. A monochromatic beam of x-rays of wavelength X f a l l s on the c r y s t a l at an angle 6, where 8 i s measured between the i n c i d e n t beam and the p a r t i c u l a r c r y s t a l planes of i n t e r e s t . C o n s t r u c t i v e i n t e r f e r e n c e of the waves s c a t t e r e d by the d i f f e r e n t atoms of the c r y s t a l r e s u l t s i n the emergence of d i f f r a c t e d beams from the c r y s t a l . T h i s y i e l d s the well-known Bragg law, 2 F i g u r e 1. D i f f r a c t i o n of x-rays by a c r y s t a l . The s o l i d dots r e p r e s e n t atoms i n the c r y s t a l . 3 given by nX = 2d sine (n i n t e g e r ) , 1.1 where 6 i s c a l l e d the Bragg a n g l e . Many planes can be drawn through the atoms comprising the c r y s t a l . A convenient method of l a b e l l i n g the v a r i o u s planes i s to a s s i g n to each set of p a r a l l e l planes a set of three i n t e g e r s h, k and 1. For a given plane i n t h i s set with an adjacent plane passing through the o r i g i n , the values of h, k and 1 are i n v e r s e l y p r o p o r t i o n a l to the f r a c t i o n a l i n t e r c e p t s of the plane with the c r y s t a l axes a, b and c, r e s p e c t i v e l y . These i n t e g e r s are known as M i l l e r i n d i c e s . The M i l l e r i n d i c e s , the l a t t i c e parameters a, b and c and the angles a, 0 and 7 of the c r y s t a l ' s u n i t c e l l can be r e l a t e d to the plane spacing (Warren 1969). For example, f o r a cu b i c u n i t c e l l , one obta i n s ? 2 2 1 _ (h z +k z +l z ) -, 9 — 2 2 • L - Z d h k l and f o r a hexagonal u n i t c e l l , one obta i n s 1 _ 4 (h2+hk+k2) , l 2 1 o d h k l 3 Since the values of d ^ k i determine the allowed d i f f r a c t i o n d i r e c t i o n s , i t i s c l e a r from above that these d i r e c t i o n s are determined s o l e l y by the u n i t c e l l geometry. The e l e c t r o n s i n each atom are separated s p a t i a l l y and 4 hence phase d i f f e r e n c e s are present between the waves s c a t t e r e d by d i f f e r e n t e l e c t r o n s . T h i s e f f e c t can be represented by the atomic s c a t t e r i n g f a c t o r f, which i s roughly p r o p o r t i o n a l to the atomic number Z ( C u l l i t y 1 9 5 9 ) . The r e s u l t a n t wave s c a t t e r e d by a l l of the atoms of the u n i t c e l l i s c a l l e d the s t r u c t u r e f a c t o r F^]_ . I t can be w r i t t e n as where N i s the number of atoms per un i t c e l l , (UJ ,Vj ,Wj ) are th the f r a c t i o n a l c o o r d i n a t e s of the j atom and f j i s the til atomic s c a t t e r i n g f a c t o r of the j atom. A measure of the i n t e n s i t y of the d i f f r a c t e d waves i s given by |F| 2. Thus, although the allowed d i f f r a c t i o n d i r e c t i o n s are determined by the shape and s i z e of the u n i t c e l l , the i n t e n s i t i e s of the d i f f r a c t e d waves are determined by the atomic p o s i t i o n s i n the u n i t c e l l . 1 . 1 . 2 The Powder D i f f r a c t o m e t r i c Method A diffTactometer i s an x-ray instrument which c o n s i s t s of a source of u n p o l a r i z e d x-rays, a sample mount, and a d e t e c t o r of the d i f f r a c t e d r a d i a t i o n . A p a r t i c u l a r geometry, known as the Bragg-Brentano geometry and used in our s t u d i e s , i s shown i n Fi g u r e 2 . A c r y s t a l monochromator i s i n c l u d e d to transmit only a smal l range of wavelengths to the d e t e c t o r . In t h i s geometry, the source-to-sample and sa m p l e - t o - r e c e i v i n g s l i t d i s t a n c e s are kept equal and 1 . 4 monochromator source ' v divergence Radius of curvature • R Sample Figure 2. The Bragg-Brentano f o c u s i n g geometry. 6 constant, independent of the s c a t t e r i n g angle. To achieve t h i s , the angle between the i n c i d e n t beam and the f l a t sample, 8, i s maintained as one-half the angle between the i n c i d e n t beam and the r e c e i v i n g s l i t , 28, so that the sample surface remains tangent to the f o c u s i n g c i r c l e at a l l angles 28. The sample i s atta c h e d to the center of the goniometer by a r o t a t i n g theta s h a f t , and the monochromator and de t e c t o r are mounted on the p e r i p h e r y of the goniometer, where they r o t a t e at twice the rate of the theta s h a f t . The i n t e g r a t e d i n t e n s i t y I of the d i f f r a c t e d beam f o r a powdered sample i n the geometry of F i g u r e 2 can be w r i t t e n as (Warren 1969) I = I o m l F h k l | 2 6 ( > l + c o s 2 2 9 c o s 2 2 ( i ) ^ t 1.5 si n 6 s i n 2 9 where I 0 i s a constant, m i s the m u l t i p l i c i t y f a c t o r , F ^ ^ i s the s t r u c t u r e f a c t o r , 6 i s the angular divergence of the divergence s l i t , and the bracketed term i s p r o p o r t i o n a l to the Lorentz p o l a r i z a t i o n f a c t o r . The m u l t i p l i c i t y f a c t o r i s the number of v a r i a t i o n s i n p o s i t i o n and s i g n of the M i l l e r i n d i c e s h, k and 1 that y i e l d the same value of plane spacing d and s t r u c t u r e f a c t o r F ^ k i . The Lorentz p o l a r i z a t i o n f a c t o r i s a combination of geo m e t r i c a l and p o l a r i z a t i o n f a c t o r s , the l a t t e r r e s u l t i n g from a n i s o t r o p i c s c a t t e r i n g of x-rays by e l e c t r o n s . I t s general e f f e c t i s to decrease the d i f f r a c t e d beam i n t e n s i t i e s at angles between the forward and backward d i r e c t i o n s . T h i s r e l a t i o n f o r the i n t e n s i t y depends on random o r i e n t a t i o n of the c r y s t a l l i t e s . 7 Under c e r t a i n c o n d i t i o n s , e.g. a p p l i e d p r e s s u r e , p r e f e r r e d o r i e n t a t i o n may r e s u l t and the r e l a t i v e i n t e n s i t i e s w i l l d i f f e r from the values given by the above r e l a t i o n . 1.2 Low Temperature X-ray D i f f r a c t i o n The s t r u c t u r e of every m a t e r i a l at constant pressure changes as the temperature of the m a t e r i a l i s v a r i e d . When a c r y s t a l c o n t r a c t s , the c r y s t a l plane spacings decrease, r e s u l t i n g i n an i n c r e a s e i n the Bragg angles; the reverse i s true f o r the expansion of a c r y s t a l . In a d d i t i o n to s h i f t s in the Bragg angles i n response to changes i n temperature, there i s a l s o a temperature dependence of the i n t e n s i t y of the d i f f r a c t e d beam. T h i s temperature dependence i s given by the Debye-Waller f a c t o r , which f o r a monatomic c r y s t a l i s -2M e . T h i s f a c t o r m u l t i p l i e d by the temperature independent i n t e n s i t y y i e l d s the temperature dependent i n t e n s i t y . M i s r e l a t e d to the mean square amplitude of thermal v i b r a t i o n s , such that the d i f f r a c t e d beam i n t e n s i t y i s decreased as the temperature i s i n c r e a s e d , p a r t i c u l a r l y f o r high Bragg an g l e s . T h i s e f f e c t w i l l not be d i s c u s s e d f u r t h e r ; f o r a more d e t a i l e d e x p l a n a t i o n , one should c o n s u l t Warren (1969). The f i r s t low temperature device f o r x-ray d i f f r a c t i o n was re p o r t e d by Rinne (1917), in which the c r y s t a l was coo l e d with l i q u i d a i r i n s i d e a cork box through which the x-ray beam passed before and a f t e r reaching the c r y s t a l . Since that time, many t e c h n o l o g i c a l advances have been made in both x-ray d i f f r a c t i o n apparatus and c o o l i n g techniques; low temperature x-ray d i f f r a c t i o n has been performed with 8 synchrotron r a d i a t i o n (Skelton et a l . 1984) and at temperatures of 30 mK (Rudman 1976). Almost a l l low temperature apparatus has been designed for use on a l r e a d y e x i s t i n g room temperature x-ray d i f f r a c t i o n equipment. The attachment d e s c r i b e d i n t h i s t h e s i s was designed f o r use on a v e r t i c a l goniometer of a d i f f T a c t o m e t e r . Other s i m i l a r low temperature attachments have been developed p r e v i o u s l y ( J e t t e r et a l . 1957, Davis and Eby 1975, S p i n o l o et a l . 1979, Benedict et a l . 1979), but none have a l l of the advantages of the present d e v i c e . The design of a low temperature attachment w i l l depend to a l a r g e extent on the c h o i c e of sample c o o l i n g method. There are three b a s i c methods a v a i l a b l e f o r c o o l i n g the sample: passing a stream of c o l d gas over the sample (gas stream); immersing the e n t i r e x-ray instrument i n a c o l d l i q u i d or d r i p p i n g a c o l d l i q u i d onto the sample (immersion); and p l a c i n g the sample,'contained i n an evacuated chamber, in good thermal contact with a c o l d bath or r e f r i g e r a t o r ( c o n d u c t i o n ) . Rudman (1976) g i v e s a d e t a i l e d comparison of the d i f f e r e n t methods. The conduction c o o l i n g method i s used in the attachment d e s c r i b e d i n t h i s t h e s i s . The q u a l i t y of the low temperature x-ray data may be degraded with r e s p e c t to that of the room temperature data by s e v e r a l e f f e c t s : misalignment of the x-ray instrument because of f o r c e s or torques due to the attachment's s i z e and/or geometry or because of thermal g r a d i e n t s present in the x-ray d i f f r a c t i o n instrument; r e d u c t i o n i n the i n t e n s i t y of the x-ray beam by placement of m a t e r i a l i n the path of 9 the beam; and formation of f r o s t i n the path of the x-ray beam and on the sample due to condensation of vapours. The e f f e c t of the f i r s t two p o t e n t i a l problems can be reduced by proper mechanical design of the attachment; however, the sample chamber i n the conduction method causes a c e r t a i n amount of beam a t t e n u a t i o n . T h i s can be reduced by using chamber windows made from m a t e r i a l s which are poor absorbers of x-rays, such as b e r y l l i u m . The problem of f r o s t formation r e s u l t s i n a t t e n u a t i o n of the x-ray beam and p o s s i b l e misalignment of the sample, but i t i s avoided by evacuation of the sample chamber i n the conduction c o o l i n g method. Good temperature s t a b i l i t y of the sample i s r e q u i r e d dur i n g the time p e r i o d i n which the x-ray measurements are taken. T h i s i s obtained by the i n t r o d u c t i o n of heat to counterbalance the c o o l i n g . I t i s a l s o necessary to be ab l e to measure the sample temperature a c c u r a t e l y . E r r o r s in t h i s measurement are i n c u r r e d by displacements of the thermometer from the sample p o s i t i o n and by temperature g r a d i e n t s across the sample. The former i s necessary because the placement of the sample i n an x-ray beam would damage the thermometer, and can be c o r r e c t e d by c h a r a c t e r i z i n g the sample-thermometer temperature d i f f e r e n c e . For the conduction c o o l i n g method, poor thermal conduction i n the sample s u b s t r a t e w i l l r e s u l t in thermal g r a d i e n t s in the sample. The p a r t i c u l a r apparatus used f o r the s t u d i e s undertaken i n t h i s t h e s i s i s now d e s c r i b e d . 10 CHAPTER TWO LOW TEMPERATURE X-RAY DIFFRACTION SYSTEM DESCRIPTION 2.1 Apparatus The apparatus used for low temperature x-ray d i f f r a c t i o n measurements 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. 2.1.1 Low Temperature Attachment The attachment was designed for use on the v e r t i c a l goniometer of a d i f f T a c t o m e t e r with only minor m o d i f i c a t i o n s to the standard goniometer theta s h a f t A ( F i g u r e 4); the face of the theta s h a f t was recessed by 0.700 cm (by m i l l i n g ) and three p r o j e c t e d c o n i c a l p o i n t s B and three tapped holes C were added. The c o n i c a l p o i n t s B mate with corresponding r a d i a l grooves on the extension housing D. Spring-loaded screws hol d these p a r t s i n c o n t a c t . The -remaining components form a subassembly that h o l d s the sample. The c o l d f i n g e r E i s b o l t e d i n t o a c a r r i e r F that can be r o t a t e d .in the spacer G. The c o l d sample mount H i s b o l t e d to the c o l d f i n g e r , and the b e r y l l i u m can I ( w a l l t h i c k n e s s 0.25 mm) s l i p s onto the spacer to provide a chamber that can be evacuated through the vacuum port J . T h i s subassembly i s l o c a t e d to the extension housing by the same method of p o i n t s and grooves d e s c r i b e d above. S e a l i n g i s accomplished by indium wire squeezed between the c o l d f i n g e r and i t s c a r r i e r ; by an O-ring K p l a c e d 11 REFRIGERATION APPARATUS ROTARY VALVE LOW TEMPERATURE ATTACHMENT r DIFFRACTOMETER HIGH PRESSURE LOW PRESSURE DATA COLLECTION COMPRESSOR TEMPERATURE INDICATOR/ CONTROLLER VACUUM PUMPS CHART RECORDER PRINTER/ MICROCOMPUTER X-RAY DIFFRACTION APPARATUS F i g u r e 3. Schematic diagram of the low temperature x-ray d i f f r a c t i o n apparatus. F i g u r e 4. Exploded view of the low temperature x-ray d i f f r a c t i o n attachment (407o of a c t u a l s i z e ) . The l e t t e r s r e f e r to the f o l l o w i n g p a r t s : A- theta s h a f t ; B- p r o j e c t e d c o n i c a l p o i n t s ; C- tapped holes; D- extension housing; E- c o l d f i n g e r ; F- c a r r i e r ; G- spacer; H- c o l d sample mount; I- b e r y l l i u m can; J - vacuum p o r t ; K- 0 - r i n g ; and L- O-ring. 13 between the c a r r i e r and the spacer; and by an 0 - r i n g L placed between the b e r y l l i u m can and the spacer. E l e c t r i c a l feedthroughs (not shown) are i n s t a l l e d i n the c o l d f i n g e r c a r r i e r with epoxy. The presence of window s l o t s in the extension housing and the use of a t h i n b e r y l l i u m can f o r the evacuation chamber allow the i n c i d e n t and d i f f r a c t e d beams to reach the sample and d e t e c t o r , r e s p e c t i v e l y , with l i t t l e a t t e n u a t i o n . The c o l d f i n g e r sample mount i s made of copper to enable good thermal c o n t a c t between the cold' f i n g e r and the sample s u b s t r a t e . During the o p e r a t i o n of the r e f r i g e r a t o r , the e n t i r e attachment remains at room temperature, except f o r the c o l d f i n g e r and sample, which c o o l to cryogenic temperatures, and the c a r r i e r , which heats to s l i g h t l y above room temperature. The theta s h a f t of the goniometer i s not su b j e c t e d to l a r g e temperature g r a d i e n t s ; hence, the alignment of the goniometer i s not s i g n i f i c a n t l y a f f e c t e d by the o p e r a t i o n of the r e f r i g e r a t o r . Since the attachment i s f i x e d to the end of the goniometer's t h e t a s h a f t , u n d e s i r a b l e l o a d i n g i s p l a c e d on t h i s s h a f t which tends to m i s a l i g n the d i f f T a c t o m e t e r . To minimize these l o a d i n g e f f e c t s , c l o s e a t t e n t i o n was p a i d to the r e d u c t i o n of the attachment mass and the torques caused by the connected hoses. With the p o i n t and groove method of l o c a t i o n , the attachment may be q u i c k l y removed and r e i n s t a l l e d , without the need f o r realignment each time. No removal or 14 m o d i f i c a t i o n of the standard r a d i a t i o n s h i e l d i n g i s necessary. 2.1.2 X-ray D i f f r a c t i o n Apparatus X-ray d i f f r a c t i o n measurements were performed using a P h i l i p s powder d i f f r a c t o m e t e r comprised of an x-ray generator (PW 1730/10) which uses a copper tube (PW 2253/20), a v e r t i c a l goniometer (PW 1050/80), and a p r o p o r t i o n a l d e t e c t o r (PW 1965/60) (see F i g u r e 2 f o r the geometry). A g r a p h i t e monochromator i s p l a c e d between the sample and d e t e c t o r to prevent u n d e s i r a b l e x-ray wavelengths from reaching the d e t e c t o r . An automatic divergence s l i t (PW 1386/50) i s mounted between the x-ray generator and the sample to keep a constant area of the sample i l l u m i n a t e d over the e n t i r e angular range. R e c e i v i n g s l i t s p l a c e d between the sample and the monochromator with widths of 0.1 mm and 0.2 mm were used. The system has*a r e s o l u t i o n of 0.01° f o r the measured s c a t t e r i n g angle 26. 2.1.3 R e f r i g e r a t i o n Apparatus R e f r i g e r a t i o n i s suppied by a m o d i f i e d A i r Products and Chemicals, Inc. D i s p l e x system. T h i s c o n s i s t s of a c l o s e d -c y c l e , a d i a b a t i c expansion, cryogenic c o o l i n g system (model CS-1003), which f u n c t i o n s as a m o d i f i e d S o l v a y - c y c l e r e f r i g e r a t o r (Longsworth 1971b), and an APD-E d i g i t a l temperature i n d i c a t o r / c o n t r o l l e r . The l a t t e r uses a thermocouple, with i t s sample j u n c t i o n p l a c e d on the c o l d 15 f i n g e r , to determine the temperature. A heater wire, a l s o wrapped around the c o l d f i n g e r s h a f t , a l l o w s the i n t r o d u c t i o n of Joule heat. The r e f r i g e r a t o r c o n s i s t s of two main components: a compressor u n i t and a r e f r i g e r a t i o n u n i t . A d e t a i l e d d e s c r i p t i o n of the r e f r i g e r a t i o n c y c l e i s given in Appendix A. 2.2 Alignment Procedure A r e l i a b l e alignment procedure has been developed f o r the attachment, analogous to that performed on the goniometer ( P h i l i p s I n s t r u c t i o n Manual PW 1349/30 1974). In t h i s procedure, two angles of the d i f f r a c t i o n geometry, 28 and 8 (see F i g u r e 1), are zeroed a c c u r a t e l y . I n i t i a l l y , the zero of 28 i s set by a d j u s t i n g the d e t e c t o r p o s i t i o n u n t i l the i n t e n s i t y of the beam i s h a l f of i t s maximum va l u e . For t h i s procedure, a beam att e n u a t o r s u p p l i e d by P h i l i p s i s p l a c e d on the t h e t a s h a f t of the goniometer, s i n c e the theta s h a f t i s a c c u r a t e l y l o c a t e d on the goniometer a x i s of r o t a t i o n . T h i s d e v i c e attenuates the p o r t i o n of the x-ray beam below the d i f f r a c t i o n plane; i t i s important that i t s s u r f a c e , which l i e s i n the d i f f r a c t i o n p lane, be very f l a t . Next, a second a t t e n u a t o r i s placed on the c o l d f i n g e r s h a f t to set the zero of 8 (a stack of g l a s s s l i d e s was used). With the de t e c t o r p l a c e d at 0° 28 (which i s a c c u r a t e l y known from the p r e v i o u s procedure), the c o l d f i n g e r s h a f t i s r o t a t e d u n t i l a maximum i n t e n s i t y i s observed at the d e t e c t o r . T h i s corresponds to a zero value 16 of 6, s i n c e any non-zero 6 value would t i l t the a t t e n u a t o r s u r f a c e r e l a t i v e to the d i r e c t i o n of the x-ray beam, i n c r e a s i n g the amount of the beam i n c i d e n t on the a t t e n u a t o r , and hence reducing the i n t e n s i t y of the beam seen at the d e t e c t o r . The c o l d f i n g e r s h a f t r o t a t i o n i s e a s i l y e f f e c t e d by l o o s e n i n g the three screws which clamp the s h a f t to the spacer (see F i g u r e 4). Because the attachment i s r i g i d l y f i x e d to the goniometer t h e t a s h a f t , the c o l d f i n g e r s h a f t r o t a t e s about the c e n t r a l a x i s of the t h e t a s h a f t . However, the c e n t r a l axes of the two s h a f t s do not, i n g e n e r a l , c o i n c i d e . The geometry of the two s h a f t s i s shown in F i g u r e 5. There i s a displacement d between the s h a f t s . There i s a l s o a r o t a t i o n of one with respect t o the other, but t h i s i s minimized by the alignment procedure. In a d d i t i o n , on each s h a f t there i s a displacement of the sample from the s h a f t c e n t e r . T h i s i s denoted by 5g and f o r the theta and c o l d f i n g e r s h a f t s , r e s p e c t i v e l y . Hence f o r a sample mounted on the c o l d f i n g e r s h a f t , the displacement of the sample from the a x i s of r o t a t i o n i s e s s e n t i a l l y These displacements can be e l i m i n a t e d by m i l l i n g a p p r o p r i a t e amounts from the sample mounts; however, a small displacement of the d i f f r a c t i o n plane from the center of the theta s h a f t does not s i g n i f i c a n t l y a f f e c t the f o c u s i n g c o n d i t i o n of the d i f f T a c t o m e t e r . Each angular value can be a d j u s t e d f o r t h i s displacement by the exact r e l a t i o n 17 THETA SHAFT COLD FINGER SHAFT F i g u r e 5. Theta and c o l d f i n g e r s h a f t geometry. A d e t a i l e d view of the sample geometry i s a l s o given. The displacement d between the two s h a f t s has been purposely exaggerated. 18 tane = tane - — 2.1 a m „ A Rcos 0 m or by the approximate r e l a t i o n (Dahn et a l . 1982a) 3 - - cose , 2.2 m ^ m which i s true f o r small d i f f e r e n c e s between $ a and 8m, where 0 i s the ad j u s t e d Bragg angle, # m i s the measured Bragg angle, 6 i s the ou t - o f - p l a n e displacement and R i s the goniometer r a d i u s . C o r r e c t a l i g n m e n t of the c o l d f i n g e r s h a f t with respect to the d i f f T a c t o m e t e r geometry i s e s s e n t i a l to the attainment of optimum r e s o l u t i o n of the measured Bragg peaks, maximum i n t e n s i t y f o r t h i s degree of r e s o l u t i o n , and accurate angular r e a d i n g s . Peak r e s o l u t i o n and i n t e n s i t y i n f o r m a t i o n can be obtained by c a r e f u l s t u d i e s of a s i n g l e Bragg peak. To determine the accuracy of the angular readings, the out-of-plane displacements f o r both the the t a and c o l d f i n g e r s h a f t s must f i r s t be determined. C a r e f u l room temperature step scans of the Bragg peaks of a s i l i c o n standard sample were performed on both s h a f t s , the r e s u l t s of which are summarized i n S e c t i o n 3.2. These scans enable the deter m i n a t i o n of the out - o f - p l a n e displacement 6 and the l a t t i c e parameter "a" ( s i l i c o n has a cubic l a t t i c e ) by a l e a s t squares f i t to the Bragg peaks, as d e s c r i b e d below. 19 Although 6 can be used to a d j u s t each Bragg peak p o s i t i o n to ob t a i n the " t r u e " peak p o s i t i o n s , these " t r u e " peak p o s i t i o n s can be d i r e c t l y c a l c u l a t e d from "a" using equations 1.1 and 1.2. In f a c t , "a" can be used as a measure of the alignment q u a l i t y . I d e a l l y , with a s i n g l e s i l i c o n standard sample used f o r a l l scans, the same "a" value should be obtained f o r both s h a f t s a f t e r each alignment, and t h i s value should agree c l o s e l y with the accepted value. In each step scan, the i n t e n s i t y values about the c e n t e r s of the Bragg peaks are measured f o r a f i x e d counting time per angular step. The peak center p o s i t i o n s are obtained by f i t t i n g a q u a d r a t i c polynomial to the top of each peak, using nine p o i n t s about the top, and then s o l v i n g fo r the p o s i t i o n of the peak c e n t e r ( S a v i t s k y and Golay 1964). T h i s a l g o r i t h m i s o u t l i n e d i n Appendix B. The peak center p o s i t i o n s obtained from t h i s procedure w i l l be r e f e r r e d to as the measured peak p o s i t i o n s . Using the measured peak p o s i t i o n s and the M i l l e r i n d i c e s of each peak, the l a t t i c e parameter(s) and out-of-plane displacement are determined by a l e a s t squares procedure. In t h i s procedure, a f u n c t i o n F of the form , 29 (hkl) - 29 (hkl) 9 F = ( — ) Z ( — - ) 2.3 n"P h k l a i s minimized. In t h i s equation, n i s the number of x-ray peaks; p i s the number of parameters to be determined by the procedure;h, k and 1 are the M i l l e r i n d i c e s of each peak; 6 (hkl) are the Bragg angles c a l c u l a t e d from the l a t t i c e 20 parameters; 6 (hkl) are as d e f i n e d i n equation 2.1; and o i s a. the u n c e r t a i n t y of each measurement. T h i s f u n c t i o n i s a reduced c h i - s q u a r e s t a t i s t i c i f the parent d i s t r i b u t i o n of the measured angular values can be d e s c r i b e d by a and the angular values obtained from the l a t t i c e parameter(s) and 6, and i t serves as a measure of the "goodness" of the f i t (Bevington 1969). The u n c e r t a i n t y a i s determined by the q u a l i t y of the data and the peak f i t t i n g program, as d e t a i l e d i n Appendix B, and i n c l u d e s c o n t r i b u t i o n s from both s t a t i s t i c a l and systematic e r r o r s . The mi n i m i z a t i o n procedure i s done using the MTS system r o u t i n e TRMF:MINUITOB in c o n j u n c t i o n with the user s u p p l i e d s u b r o u t i n e s DCUBICFIT.S ( f o r c u b i c l a t t i c e s ) and DHEXFIT.S ( f o r hexagonal l a t t i c e s ) . Both subro u t i n e s are double p r e c i s i o n v e r s i o n s of subroutines w r i t t e n by P.J. Mulhern and are found under the UBC computer user account XBAT. 21 CHAPTER THREE PERFORMANCE CHARACTERISTICS OF THE LOW TEMPERATURE ATTACHMENT Experimental r e s u l t s of performance t e s t s on the low temperature attachment are presented i n t h i s chapter which i n c l u d e and expand upon r e s u l t s p r e v i o u s l y p u b l i s h e d (Rivers-Bowerman et a l . 1984). 3.1 System Temperature C h a r a c t e r i s t i c s With a vacuum of approximately 10" 5 T o r r surrounding the c o l d f i n g e r , the D i s p l e x system can c o o l the c o l d f i n g e r , loaded with a g l a s s s l i d e sample, from room temperature to l i q u i d n i t r o g e n temperature in approximately 35 minutes, and i t can reach i t s minimum temperature of 40K w i t h i n one hour. The c o o l i n g r a t e can be decreased, and intermediate temperatures maintained, by using the heater to counterbalance the c o o l i n g produced by the D i s p l e x system. Temperature f l u c t u a t i o n s around the set p o i n t temperature are t y p i c a l l y ±0.2K over extended p e r i o d s of time. Although the manufacturer does not s p e c i f y an upper temperature l i m i t f o r the D i s p l e x system, running the system at 360K s i g n i f i c a n t l y degraded the s e a l s w i t h i n the d i s p l a c e r -c y l i n d e r u n i t . The dominance of r a d i a t i o n heating over gas conduction h e a t i n g has been observed e x p e r i m e n t a l l y . D r a s t i c a l l y 22 d e c r e a s i n g the length of the vacuum pumping l i n e , which i n c r e a s e d the vacuum i n s i d e the evacuation chamber by at l e a s t an order of magnitude, d i d not reduce the system cooldown time or lower the minimum system temperature. As an i n d i c a t i o n of the magnitude of r a d i a t i o n h e a t i n g , the temperature d r i f t at low temperatures in the absence of c o o l i n g was measured. With a copper s l i d e mounted on the c o l d f i n g e r , the evacuation chamber under a vacuum of approximately 10~ 5 T o r r , and the temperature e q u i l i b r a t e d at 200K, the r e f r i g e r a t o r and heater were shut o f f and the temperature was observed as a f u n c t i o n of time. In the f i r s t ten minutes, the temperature i n c r e a s e d by 25K; i t had reached the i c e poin t (T=273K) a f t e r 50 minutes. Since heat t r a n s f e r from the warm end of the p i s t o n - c y l i n d e r arrangement i s small because of the t h e r m a l l y i n s u l a t i n g nature of the regenerator, t h i s l a r g e h e a t i n g rate can be a t t r i b u t e d to r a d i a t i o n h e a t i n g , at l e a s t when the c o l d f i n g e r temperature i s w e l l below the ambient temperature. No warming of the sample due to the presence of the x-ray beam of the d i f f T a c t o m e t e r has been o b s e r v e d . The temperature of the c o l d f i n g e r s h a f t , used f o r temperature c o n t r o l , was measured with a Au-0.07 a t . % Fe/KP (chromel) thermocouple. T h i s d i l u t e a l l o y of i r o n i n gold i s popular because of i t s r e l a t i v e l y high temperature s e n s i t i v i t y below 25K, but i s not r e a l l y necessary f o r these s t u d i e s s i n c e a l l system temperatures are above 40K. A system-supplied thermocouple was used u n t i l i t shorted to the case of the low temperature attachment. 23 Because the disassembly of the attachment was necessary to remedy the short, a second Au-0.07 a t . % Fe/KP thermocouple was i n s t a l l e d as a temporary s o l u t i o n , with the j u n c t i o n anchored to the c o l d f i n g e r s h a f t with GE 7031 v a r n i s h . A t h i n (6M) piece of Mylar was p l a c e d between the j u n c t i o n and the s h a f t to provide e l e c t r i c a l i s o l a t i o n . At l e a s t 5 cm of wire connected to the j u n c t i o n was anchored to the s u b s t r a t e to ensure that the j u n c t i o n was at the same temperature as the s u b s t r a t e (Kopp and Slack 1971). T h i s method of anchoring j u n c t i o n s was found to be q u i t e e f f e c t i v e and was used e x c l u s i v e l y for thermometry measurements. J u n c t i o n s anchored with Cry-Con grease, a copper-loaded, e l e c t r i c a l l y -i n s u l a t i n g grease made by A i r Products and Chemicals, Inc. tended to break away from the s u b s t r a t e with thermal c y c l i n g . The varnished j u n c t i o n s c o u l d be removed from the s u b s t r a t e s by d i s s o l v i n g the v a r n i s h i n a 50/50 mixture of methanol and toluene. It was necessary to p l a c e the j u n c t i o n of the second thermocouple a greater d i s t a n c e from the heater wire than that of the system-supplied thermocouple because of the i n a c c e s s i b i l i t y of the heater wire l o c a t i o n . A s i g n i f i c a n t d egradation in temperature r e g u l a t i o n was not observed, d e s p i t e the increased response time f o r the second conf i g u r a t i o n . Because the sample i s c o o l e d by conduction c o o l i n g and because of the lar g e amount of r a d i a t i o n h e a t i n g , temperature d i f f e r e n c e s e x i s t between the sample and the c o l d f i n g e r . These temperature d i f f e r e n c e s were measured 24 with a Au-0.07 a t . % Fe/KP d i f f e r e n t i a l thermocouple, r e f e r e n c e d to the sample temperature. The sample temperature was measured with a Au-0.07 a t . % Fe/KP thermocouple, r e f e r e n c e d to a l i q u i d n i t r o g e n bath. A systematic upward s h i f t i n the bath temperature i s expected due to condensation of a i r i n t o the l i q u i d n i t r o g e n . The magnitude of t h i s e f f e c t i s not known but was estimated to be 1K. D i f f e r e n t i a l thermocouple v o l t a g e s were measured using an hp 419A DC n u l l voltmeter; sample thermocouple v o l t a g e s were measured using a Fluke 895A d i f f e r e n t i a l v o l t m e t e r . The average v o l t a g e values f o r each thermocouple obtained from values f o r both thermocouple c u r r e n t p o l a r i t i e s , were used to determine the temperature by r e f e r e n c e to t a b l e s provided by Sparks and Powell (1972). The percentage e r r o r i ntroduced v i n t o the temperature values by the use of these t a b l e s i s estimated by Sparks and Powell to be 0.6% of the thermocouple v o l t a g e f o r temperatures between 75K and 280K. T h i s corresponds to an e r r o r i n temperature of ±1K f o r a sample temperature of 300K re f e r e n c e d to a l i q u i d n i t r o g e n bath. The e r r o r i n the measured temperature values w i l l be taken to be ±1K, with the p o s s i b i l i t y of an upward s h i f t of 1K, as d i s c u s s e d above. 3.2 Alignment Q u a l i t y E v a l u a t i o n A standard s i l i c o n sample was used f o r a l l measurements. T h i s sample was c o n s t r u c t e d by s p r i n k l i n g f i n e s i l i c o n powder ( p a r t i c l e s i z e < 38M) onto a pi e c e of double-s i d e d Scotch tape ( t h i c k n e s s of 85M) p l a c e d on a g l a s s 25 microscope s l i d e . D i r e c t comparisons were made between d i f f r a c t i o n p a t t e r n s obtained f o r the (311) peak of the s i l i c o n standard p l a c e d on the theta and c o l d f i n g e r s h a f t s , as shown in F i g u r e 6. These p a t t e r n s were obtained f o l l o w i n g the alignment of both s h a f t s . The peaks have been normalized to u n i t height to f a c i l i t a t e the comparison, even though the unnormalized peak h e i g h t s d i f f e r e d by l e s s than f i v e p e r c e n t . T h i s small d i f f e r e n c e i n measured i n t e n s i t i e s i s comparable to measured i n t e n s i t y d i f f e r e n c e s between s e q u e n t i a l scans on the theta s h a f t . The r e s o l u t i o n of the CuKa, and CuKa 2 peaks i s e q u a l l y good i n the two t r a c e s . Except f o r a small angular s h i f t between the peaks, the t r a c e s are e s s e n t i a l l y i d e n t i c a l . Step scans were performed on a l l Bragg peaks of the s i l i c o n sample i n the angular range 20=25° to 20=130°. Twenty s u c c e s s i v e angular values about each peak c e n t e r , with an angular step s i z e of 0.01°, were used to o b t a i n the peak cente r p o s i t i o n s from the peak f i t t i n g procedure d e s c r i b e d i n Appendix B. Tables 1 and 2 l i s t the measured peak center p o s i t i o n s 2# m of the s i l i c o n Bragg peaks f o r t y p i c a l t h e t a and c o l d f i n g e r s h a f t scans, r e s p e c t i v e l y , f o l l o w i n g alignment of the s h a f t s . A l s o i n c l u d e d are the c a l c u l a t e d p o s i t i o n s of the peak c e n t e r s 2# c a and the d i f f e r e n c e 26 -2d -, as w e l l as the counting time per step m ca c f o r each peak. The 26 values are c a l c u l a t e d from the C 3. l a t t i c e parameter determined by a l e a s t squares f i t and ad j u s t e d to i n c l u d e the out-of-plane displacement; 1.0 0 . 8 -8 0 . 6 Q Ld K l or O 0 . 4 -0 . 2 0 . 0 5 5 . 9 5 6 . 1 5 6 . 3 SCATTERING A N G L E 26 ( D E G R E E S ) 5 6 . 5 Figure 6. D i r e c t comparison of d i f f r a c t i o n p a t t e r n s of the s i l i c o n (311) peak o b t a i n e d on the thet a (X) and c o l d f i n g e r (+) s h a f t s . Both peaks have been normalized to u n i t h e i g h t . A counting time of 100 s per step was used f o r both scans. o> 27 Table 1 Comparison of measured and c a l c u l a t e d 2 9 values f o r a t y p i c a l step scan of the s i l i c o n standard on the t h e t a s h a f t . In the l e a s t squares f i t , A=1.54184A was used f o r the (111) peak; A=1.54056A was used f o r a l l other peaks. h k l Counting time 29 29 29 -29 ° m ca m ca per step (s) (degrees) (degrees) (degrees) 1 1 1 40 28. 4946 28. ,4936 +0. ,0010 2 2 0 ' 40 47. ,3246 47. .3211 +0. ,0035 3 1 1 100 56. . 1398 56, . 1360 +0, .0038 4 0 0 100 69. , 1422 69. . 1356 +0. , 0066 3 3 1 100 76. .3684 76. ,3769 - 0 , .0085 4 2 2 100 88. .0175 88, .0217 - 0 , ,0042 5 1 1 100 94. .9273 94, .9379 - 0 , ,0106 4 4 0 200 106. .6821 106, ,6815 +0. .0006 5 3 1 200 114. .0561 114, ,0560 +0, ,0001 6 2 0 200 127. .4910 127. .4866 +0, .0044 28 Table 2 Comparison of measured and c a l c u l a t e d 26 val u e s f o r a t y p i c a l step scan of the s i l i c o n standard on the c o l d f i n g e r s h a f t . In the l e a s t squares f i t . X=1.54184A was used f o r the (111) peak; X=1.54056A was used f o r a l l other peaks. h k 1 Counting time 29 26 26 -26 6 m ca m ca per step (s) (degrees) (degrees) (degrees) 1 1 1 40 28 .5853 28. ,5955 -0, .0102 2 2 0 40 47 .4281 47. ,4185 +0, .0096 3 1 1 100 56 .2309 56. .2304 +0. ,0005 4 0 0 100 69 .2256 69. ,2249 +0, ,0007 3 3 1 100 76 .4612 76, .4628 -0, ,0016 4 2 2 100 88 .1005 88, .1018 -0, .0013 5 1 1 100 95 .0152 95, .0141 +0, ,0011 4 4 0 200 106 . 7567 106, . 7511 +0. ,0056 5 3 1 200 114 . 1217 114, . 1212 +0, .0005 6 2 0 200 12 7 .5394 127, .5440 -0, .0046 29 t h e r e f o r e , the two s e t s of angular values are d i r e c t l y comparable. The number of p o s i t i v e and negative d i f f e r e n c e s are roughly equal and the mean d i f f e r e n c e i s roughly zero. Values near u n i t y f o r the "goodness of f i t " parameter F (see equation 2.3) were obtained f o r a=0.05°. For angular values measured to t h i s degree of p r e c i s i o n , c o r r e c t i o n s to the measured peak p o s i t i o n s can be d e s c r i b e d by an out-of-plane displacement. T h i s value i s comparable to the r e p e a t a b i l i t y found f o r s e q u e n t i a l mountings of the attachment, as d i s c u s s e d i n S e c t i o n 3.4, but i s a f a c t o r of ten l a r g e r than the standard d e v i a t i o n obtained from measurements on the theta s h a f t of a s i n g l e Bragg peak, as d i s c u s s e d i n Appendix B. Step scans with the s i l i c o n standard sample were performed both immediately f o l l o w i n g alignment of the s h a f t s and at times c o n s i d e r a b l y l a t e r . The former measures the r e p r o d u c i b i l i t y of the alignment procedure, while the l a t t e r i n d i c a t e s the degree of misalignment produced by c o n s i d e r a b l e use of the d i f f T a c t o m e t e r , with and without the attachment, f o l l o w i n g s h a f t alignments. Table 3 l i s t s the l a t t i c e parameter "a", out-of-plane displacement 6, and goodness of f i t parameter F f o r step scans on both the th e t a and c o l d f i n g e r s h a f t s performed immediately f o l l o w i n g the s h a f t s ' alignment. These measurements were done on alignments performed over a p e r i o d of 18 months. This data has been obtained from l e a s t squares f i t s to a l l ten Bragg peaks. Included i n the t a b l e are the mean valu e s of "a" and 6 f o r each s h a f t and the mean value 30 Table 3 Results of least squares f i t s to step scan data taken on both theta and cold finger shafts immediately following the shafts' alignment. Theta Shaft a(A) 6(u) i) 5 .43255 57 147 i i ) 5 .43245 67 267 i i i ) 5 .43231 38 78 Mean 5 .43244 54 -Standard Deviation 0 .00010 12 -Cold Finger Shaft i) 5 .43265 88 300 i i ) 5 .43228 213 128 Mean 5 .43246 150 -Standard Deviation 0 .00018 62 -Overall Mean 5 .43245 _ _ Overall Standard Deviation 0.00014 31 of "a" f o r the combined data. Although the q u a n t i t y of data i s i n s u f f i c i e n t to j u s t i f y r e l i a b l e s t a t i s t i c a l a n a l y s i s , standard d e v i a t i o n s are a l s o l i s t e d for the above data. The percentage d i f f e r e n c e i n the mean l a t t i c e parameter values f o r the two s h a f t s i s 0.0004%. T h i s very good agreement between the values i s f o r t u i t o u s because of the l i m i t e d q u a n t i t y of data, but the standard d e v i a t i o n of "a" f o r the combined data, 0.00014A, does not d i f f e r that much o from that f o r the theta s h a f t data alone, 0.00010A. Thus one o b t a i n s the same l a t t i c e parameter value f o r e i t h e r s h a f t to a high degree of p r e c i s i o n . The accuracy of the measurement i s not as good as the p r e c i s i o n , but i t i s s t i l l high; the percentage d i f f e r e n c e between the o v e r a l l mean l a t t i c e parameter v a l u e and the accepted value of 5. 43054±0.00017A ( P a r r i s h 1960, Klug and Alexander 1974) i s 0.04%. The f i t i s q u i t e i n s e n s i t i v e to 5 as can be seen from the r e l a t i v e l y l a r g e standard d e v i a t i o n , 12M, for 6 i n the theta s h a f t scans. A l a r g e d i f f e r e n c e (125M) e x i s t s between the 5 v a l u e s f o r the scans taken on the c o l d f i n g e r s h a f t , p o s s i b l y due to uneven wearing of the r a d i a l grooves on the brass e x t e n s i o n housing (see F i g u r e 4) as a r e s u l t of the many attachment mountings between the alignments. However, both scans y i e l d the same l a t t i c e parameter to w i t h i n 0.007%. Thus, changes of t h i s magnitude in the t h e t a - c o l d f i n g e r s h a f t geometry do not s i g n i f i c a n t l y a f f e c t the l a t t i c e parameter value o b t a i n e d . For the. s t e p scans performed many months a f t e r the s h a f t s ' alignment, l i t t l e d i f f e r e n c e was observed i n the 32 l a t t i c e parameter v a l u e . For one scan on the th e t a s h a f t and two scans on the c o l d f i n g e r s h a f t taken, on average, 11 months a f t e r alignment of the s h a f t s , the mean l a t t i c e parameter value i s 5.43212±0.00009A. T h i s i s lower than the o v e r a l l mean value of Table 3 by 0.00033A or 0.006%, which i s only twice the standard d e v i a t i o n of the value s l i s t e d in Table 3. Thus the q u a l i t y of the alignment i s not c o n s i d e r a b l y a f f e c t e d by many months of use of the goniometer and the low temperature attachment. A comparison was a l s o made between l a t t i c e parameter data o b t a i n e d f o r d i f f e r e n t subsets of Bragg peaks, corres p o n d i n g to d i f f e r e n t angular ranges, i n c l u d e d i n the l e a s t squares f i t f o r the scans l i s t e d i n Table 3. With the i n c l u s i o n of a l l higher angle peaks, the mean l a t t i c e parameter values and t h e i r corresponding standard d e v i a t i o n s were e s s e n t i a l l y the same as f o r the va l u e s o b t a i n e d with the i n c l u s i o n of a l l ten Bragg peaks. For t h i s reason, scans of a l l ten Bragg peaks were used for the e v a l u a t i o n of alignment q u a l i t y . 3.3 V i b r a t i o n C h a r a c t e r i s t i c s V i b r a t i o n damping between the vacuum pump and the attachment has been accomplished by i n s e r t i n g a s e c t i o n of copper t u b i n g in the vacuum l i n e , and clamping t h i s t u b i n g to a l e a d b l o c k . With t h i s device i n p l a c e , n e g l i g i b l e v i b r a t i o n t r a n s m i s s i o n from the pump to the attachment i s observed. The v i b r a t i o n s inherent i n the o p e r a t i o n of the 33 d i s p l a c e r - c y l i n d e r arrangement l o c a t e d w i t h i n the c o l d f i n g e r s h a f t , at a frequency of 7 Hz, are s m a l l . For c a r e f u l x-ray s t e p scans, t h i s r e l a t i v e l y high v i b r a t i o n frequency does not introduce spurious noise i n t o the d i f f r a c t i o n p a t t e r n s , but serves only to broaden the peaks. D i f f r a c t i o n p a t t e r n s obtained f o r step scans with a counting time of 100 seconds per step f o r the s i l i c o n (311) peak are shown in p a r t s (a) and (b) of F i g u r e 7, normalized to u n i t h e i g h t , with and without system v i b r a t i o n s r e s p e c t i v e l y . An e x c e l l e n t r e s o l u t i o n of the CuKa, and CuKa 2 peaks i s seen in both c a s e s . The l i n e broadening e f f e c t of system v i b r a t i o n s i s seen more c l e a r l y i n F i g u r e 8, i n which the p a t t e r n i n part (b) of F i g u r e 7 has been s u b t r a c t e d from that i n part (a ) . A low value of the d i f f e r e n c e i s obtained near both peak c e n t e r s because the s i g n a l / n o i s e r a t i o i s high at these angular v a l u e s . However, a p o s i t i v e d i f f e r e n c e i s observed away from the peak c e n t e r s . T h i s i s i n d i c a t i v e of a broadening of the peaks with the a d d i t i o n of system v i b r a t i o n s . T h i s broadening was measured to be 0.008° at the half-maxima of the peaks, which i s 6% of the f u l l width at half-maxima and h a l f of the peak width spread observed f o r samples on the theta s h a f t . Even with f a s t e r continuous scans, l i t t l e d i f f e r e n c e i s observed between d i f f r a c t i o n p a t t e r n s obtained with and without t h i s v i b r a t i o n . 3.4 S e q u e n t i a l Mounting Comparison In order that the attachment be convenient to use, i t must be p o s s i b l e to dismount and remount the u n i t , without 34 co < or o 1.0 0.8 -8 0-6 Q U l NJ 0.4 -0.2 -0.0 55.9 56.1 56.3 SCATTERING ANGLE 29 (DEGREES) (a) 56.5 1.0 0.8 co i— 8 0.6 Q Ld M _J < o 0.4 0.2 0.0 I I • .1 1 1 + 4- • — 4 * 4 • • 4-— 4* 4 4 4 + 4 4 4 + I 1 4 %  1 1 1 55.9 56.1 56.3 SCATTERING ANGLE 29 (DEGREES) (b) 56.5 Figure 7. D i f f r a c t i o n patterns of the s i l i c o n (311) peak obtained with (a) and without (b) system vibrations. Both peaks have been normalized to unit height; v e r t i c a l lines indicate peak center positions. A counting time of 100 s per step was used for both scans. The difference of these patterns i s shown in Figure 8. NORMALIZED COUNTS D I F F E R E N C E c I-I fD 0 0 cr ex a i-h a n H i H- q: rt) o cr i-t 03 rf (D 3 o (D rf fD 0) w n rf r r ft) O m H i r r O fD BV n m rt ft) ^ rt 3 v(t rt fD fD 3 CO H rr o fD i-h O co f t fD H-M (JQ rt C H- I-i O fD o 3 O i-h a m o m m I o b cn ro cn CD o o o NO O b CO o > m cn 2 cn o > CD r~ m CO cn cn cn cn Cn ~J T + + + + + T + 4-+ •t-+ + 36 the need f o r realignment each time. In between step scans of a s i n g l e s i l i c o n peak, the e n t i r e attachment was dismounted from the goniometer and then remounted. Each change was e a s i l y accomplished i n s e v e r a l minutes. A r e p r o d u c i b i l i t y f o r the peak cent e r p o s i t i o n s of ±0.0045° was obtained f o r e i g h t mountings of the attachment. A 100 second counting time per step was used for each scan. T h i s v a r i a t i o n in the peak p o s i t i o n s i s g r e a t e r than the p r e c i s i o n obtained f o r samples on the t h e t a s h a f t by a f a c t o r of .ten (see Appendix B), but i s of the order of the angular standard d e v i a t i o n s obtained from the l e a s t squares f i t s d e s c r i b e d i n S e c t i o n 3.2. The d i f f e r e n c e between two normalized peaks, each s i m i l a r to that shown in part (b) of F i g u r e 7, i s shown in F i g u r e 9. There i s a d i f f e r e n c e i n peak width, comparable i n magnitude to that observed i n the v i b r a t i o n comparison of S e c t i o n 3.3, as w e l l as a s l i g h t s h i f t i n the peak center p o s i t i o n . The s i z e of these d i f f e r e n c e s i s small enough that they may be ignored in most circumstances. These remarkably re p e a t a b l e r e s u l t s can be a t t r i b u t e d to the poin t and groove arrangements, which a c c u r a t e l y f i x the s p a t i a l r e l a t i o n s h i p s of the components of the attachment. H 0.04 f 0.02 Q UJ b! 0.00 _j < O - 0 . 0 2 1 1 • I I I ' + + + 4-+ + 4. + ++ + + -+ + + + + + ++ + + + + + + + + + + + -»• 1 1 1 1 1 5 5 . 9 56.1 5 6 . 3 5 6 . 5 SCATTERING A N G L E 26 ( D E G R E E S ) Figure 9. D i f f e r e n c e of d i f f r a c t i o n p a t t e r n s o f the s i l i c o n (311) peak, each n o r m a l i z e d to u n i t height, obtained f o r two d i s t i n c t mountings o f the attachment on the goniometer. The v e r t i c a l l i n e i n d i c a t e s the peak cente r p o s i t i o n f o r one of the p a t t e r n s . A u> counting time of 100 s per step was used. ^ CHAPTER FOUR 38 CHARGE DENSITY WAVE TRANSITION STUDY IN 1T~TaS 2 As a t e s t of the s e n s i t i v i t y of the low temperature x-ray d i f f r a c t i o n apparatus, the l a t t i c e parameter d i s c o n t i n u i t i e s a s s o c i a t e d with a charge d e n s i t y wave phase t r a n s i t i o n i n 1T-TaS 2 were s t u d i e d . 4. 1 I n t r o d u c t i o n 4.1.1 Charge D e n s i t y Waves A charge d e n s i t y wave i s a coupled p e r i o d i c d i s t o r t i o n in the conduction e l e c t r o n d e n s i t y and the u n d e r l y i n g c r y s t a l l a t t i c e . The l a t t i c e d i s t o r t i o n must be present to minimize the Coulomb energy of the conduction e l e c t r o n - i o n core c o n f i g u r a t i o n ( D i S a l v o 1977). T h i s mutual rearrangement of the e l e c t r o n s and the ions r e q u i r e s a s u f f i c i e n t l y l a r g e electron-phonon i n t e r a c t i o n , as demonstrated by Chan and Heine (1973). An i n s t a b i l i t y i n the conduction e l e c t r o n s i s necessary f o r the adoption of the charge d e n s i t y wave s t a t e . These i n s t a b i l i t i e s can be understood in terms of divergences in the g e n e r a l i z e d s t a t i c e l e c t r o n i c s u s c e p t i b i l i t y x(q) (Wilson et a l . 197 5-, D i S a l v o 1977) at wavevectors q 0 which j o i n many f i l l e d s t a t e s to many empty s t a t e s at the same energy. T h i s i s determined by the geometry of the Fermi s u r f a c e . D i S a l v o shows t h a t , i n the absence of i n t e r a c t i o n s 39 between conduction e l e c t r o n s , x(q) d i v e r g e s only at T=0K; he suggests that the i n c l u s i o n of i n t e r a c t i o n s may allow i n s t a b i l i t i e s to occur at f i n i t e temperatures. In g e n e r a l , the charge d e n s i t y wave wavelength X 0 = 27r/|q 0| w i l l not equal a l a t t i c e t r a n s l a t i o n , s i n c e q 0 i s determined by the Fermi su r f a c e geometry. In t h i s case, the charge d e n s i t y wave i s s a i d to be incommensurate with the l a t t i c e . I f X 0 does equal a l a t t i c e t r a n s l a t i o n , then the charge d e n s i t y wave i s commensurate with the l a t t i c e . To study the charge d e n s i t y wave, one can observe the accompanying l a t t i c e d i s t o r t i o n by d i f f r a c t i o n experiments. An i n f i n i t e s e r i e s of s a t e l l i t e peaks surround each main l a t t i c e peak i n the charge d e n s i t y wave s t a t e . The wavevectors o f . t h e s a t e l l i t e peaks,^k., are given by k = G + nq , 4.1 where G i s a r e c i p r o c a l l a t t i c e wave v e c t o r , q 0 i s the charge d e n s i t y wave wavevector, and n i s an i n t e g e r . The c a l c u l a t i o n of the expected d i f f r a c t i o n p a t t e r n i s e n t i r e l y analogous to that of the o p t i c a l d i f f r a c t i o n p a t t e r n obtained from a d i f f r a c t i o n g r a t i n g with a p e r i o d i c e r r o r of th r u l i n g (James 1965). The i n t e n s i t y of the n order s a t e l l i t e i s p r o p o r t i o n a l to the square of the n*"*1 order Bes s e l f u n c t i o n (Overhauser 1971). Changes i n the u n i t c e l l geometry may a l s o be produced with the adoption of a charge d e n s i t y wave s t a t e . 40 4.1.2 Charge Dens i t y Wave Behavior of 1T-TaS 2 The compound TaS 2 e x i s t s i n l a y e r e d s t r u c t u r e s i n which s t r o n g l y bonded sandwich l a y e r s are weakly bonded to each other by van der Waals f o r c e s . T h i s i s shown s c h e m a t i c a l l y in F igure 10a. Many d i f f e r e n t i n - l a y e r and i n t e r - l a y e r s t a c k i n g s are p o s s i b l e , g i v i n g r i s e to v a r i o u s polytypes for t h i s compound. The p a r t i c u l a r p o l y t y p e of i n t e r e s t i n the present study i s the 1T p o l y t y p e , which has the cadmium io d i d e s t r u c t u r e , as d e p i c t e d i n F i g u r e 10b. The f r a c t i o n a l c o o r d i n a t e s of the atoms with respect to the u n i t c e l l i n t h i s s t r u c t u r e are (0,0,0) f o r the tantalum atom and ±(1/3,2/3,1/4) f o r the sulphur atoms ( H u l l i g e r 1976). The hexagonal l a t t i c e parameters of 1T-TaS 2 at room temperature are a = 3.365A and c=5.897A (Brouwer and J e l l i n e k 1975). Although the 1T polytype i s s t a b l e only at c r y s t a l growth temperatures (above 1050K), i t can be r e t a i n e d as a metastable phase at room temperature by quenching the high temperature c r y s t a l s to room temperature, such that the r a t e of t r a n s f o r m a t i o n to the s t a b l e room temperature 2H po l y t y p e i s n e g l i g i b l e ( L i e t h and T e r h e l l 1977). Charge d e n s i t y waves are known to e x i s t in 1T-TaS 2 below 600K. Three d i f f e r e n t phases of t h i s polytype have been d e f i n e d - I, NC and C - to denote d i f f e r i n g charge d e n s i t y wave c h a r a c t e r i s t i c s (Nakanishi et a l . 1977). Above 600K, conver s i o n of the 1T to the 2H p o l y t y p e i s obtained. Upon c o o l i n g from 600K, there i s a phase t r a n s i t i o n near 350K from the I phase, i n which the charge d e n s i t y wave i s incommensurate with the l a t t i c e , to the NC phase, i n which 41 1T-TQS 2 - * I I 2 0 (b) F i g u r e 10. (a) The "sandwich" s t r u c t u r e o f TaS 2. H o r i z o n t a l l i n e s i n the f i g u r e represent planes of atoms, (b) The atomic s t r u c t u r e o f l T - T a S 2 . Tantalum atoms are d e p i c t e d by s o l i d c i r c l e s and sulphur atoms by open c i r c l e s . The p r e f i x IT r e f e r s to the 1 l a y e r u n i t c e l l w ith t r i g o n a l symmetry. 42 the charge d e n s i t y wave i s n e a r l y commensurate; near 200K, there i s another phase t r a n s i t i o n from the NC phase to the C phase, i n which the charge d e n s i t y wave i s commensurate and d e f i n e s a /l~3a X /T~3a s u p e r l a t t i c e (where a i s the b a s a l plane l a t t i c e s p a c i n g ) . D i f f e r e n t i a l thermal a n a l y s i s and d i f f e r e n t i a l scanning c a l o r i m e t r y performed by Thompson et a l . (1971) i n d i c a t e that both t r a n s i t i o n s are f i r s t o rder. In 1T-TaS 2 and other s i m i l a r l a y e r e d compounds, three charge d e n s i t y waves are c o n s i d e r e d i n each l a y e r because of the t h r e e - f o l d r o t a t i o n a l symmetry about the c - a x i s . These d i s t o r t i o n s are o r i e n t e d at 120° to each other and have the same wavevector magnitude |q 0|« Appendix C g i v e s a d e t a i l e d d e s c r i p t i o n of the b a s a l plane geometry for 1T-TaS 2 at room temperature. For the present d i s c u s s i o n , we w i l l c o n s i d e r only one of the wavevectors q 0 . In the I phase, q 0 i s o r i e n t e d along a r e c i p r o c a l l a t t i c e v e c t o r . In the I-NC t r a n s i t i o n , q 0 r o t a t e s by about 12° with respect to the r e c i p r o c a l l a t t i c e , stopping short of the commensurate angle of 13.9° (Scruby et a l . 1975). T h i s NC phase seems to be s t a b i l i z e d by secondary d i s t o r t i o n s , in p a r t i c u l a r , c o u p l i n g of the fundamental charge d e n s i t y wave to i t s t h i r d harmonic (Yamada and Takatera 1977). As the temperature i s lowered i n the NC phase, a f u r t h e r r o t a t i o n of q 0 i s obtained, u n t i l a commensurate charge d e n s i t y wave i s obtained by a f i n a l abrupt r o t a t i o n of q 0 at the NC-C t r a n s i t i o n . A c o n s i d e r a b l e thermal h y s t e r e s i s of approximately 50K has been observed upon c o o l i n g and h e a t i n g through the NC-C t r a n s i t i o n (see, 43 for example, Sezerman et a l . 1980). T h i s charge d e n s i t y wave behavior i s more complicated than that of a s i m i l a r m a t e r i a l 1T-TaSe 2, which proceeds d i r e c t l y from the I phase to the C phase v i a a f i r s t order t r a n s i t i o n at a temperature of 473K (Wilson et a l . 1975). However, both compounds have larg e l a t t i c e d i s t o r t i o n amplitudes; f o r 1T-TaSe 2 i t i s roughly 0.25A (Brouwer and J e l l i n e k 1976) and i s expected to be comparable for 1T-TaS 2 (DiSalvo 1977). The Fermi s u r f a c e of 1T-TaS 2 has been estimated by Wilson et a l . (1975) from band s t r u c t u r e c a l c u l a t i o n s by M a t t h e i s s (1973), and i s shown i n F i g u r e 11a, with a l l the e l e c t r o n s l y i n g i n one zone grouped around the s i x ML axes. The s u r f a c e i s n e a r l y two dimensional because of the very s l i g h t energy dependence of the s t a t e s i n the k z d i r e c t i o n . T h i s simple form of the Fermi s u r f a c e leads to l a r g e n e s t i n g areas f o r the proper choice of Fermi s u r f a c e spanning vec t o r q 0 . T h i s n e s t i n g s i t u a t i o n i s seen c l e a r l y i n F i g u r e 11b, in which the Fermi s u r f a c e has been d i s p l a c e d by the spanning v e c t o r i n the rM d i r e c t i o n . C o n t r i b u t i o n s to the n e s t i n g a r i s e from four out of the s i x Fermi s u r f a c e segments, which gi v e s u p e r i o r n e s t i n g r e s u l t s (Wilson et a l . 1975). However, c a l c u l a t i o n s of x(q) i n the absence of e l e c t r o n i n t e r a c t i o n s by Myron et a l . (1977) suggest that the Fermi s u r f a c e i s not as important in determining the charge d e n s i t y wave behavior in 1T-TaS 2 as might be thought from n e s t i n g arguments, and that the observed behavior i s a r e s u l t of i n t e r a c t i o n s between the e l e c t r o n s . D i s c o n t i n u i t i e s i n the l a t t i c e parameters at the NC-C (a) (b) Figure 11. (a) The Fermi surface of lT-TaS 2 > constructed from band-structure calculations of Mattheiss (1973) (after Wilson et a l . 1975). (b) The nesting of the IT Fermi surface for the spanning wavevector a p a r a l l e l to TM (after W i l s o n ^ t a l . 1975). 45 phase t r a n s i t i o n i n 1T-TaS 2 have been observed p r e v i o u s l y . Thompson et a l . (1971), i n an e a r l y study of the e l e c t r i c a l p r o p e r t i e s of l T - T a S 2 , r e p o r t e d low temperature x-ray powder d i f f r a c t i o n r e s u l t s of a d i s c o n t i n u o u s decrease i n the c-o a x i s l a t t i c e parameter of 0.02A upon h e a t i n g through the low temperature t r a n s i t i o n . No d i s c o n t i n u i t y in the a-axis l a t t i c e parameter was observed i n the temperature range T=80K to T=390K. No d e t a i l s of the experimental apparatus or procedure are g i v e n . Givens and F r e d e r i c k s (1977) performed x-ray powder d i f f r a c t o m e t r i c s t u d i e s of 1T-TaS 2 in the temperature range T=150K to T=500K. The sample h o l d e r , immersed in helium gas, was attached to a copper block through which c o l d n i t r o g e n gas was passed. The a and c l a t t i c e parameters were obtained from the angular v a l u e s of the (102) and (003) Bragg peaks at each temperature; p l o t s of t h e i r l a t t i c e parameter versus temperature data are shown in F i g u r e 12. They observed no d i s c o n t i n u i t y in the a - a x i s data, but t h e i r data i s very sparse i n the v i c i n i t y of the NC-C t r a n s i t i o n . The NOC t r a n s i t i o n i s c l e a r l y seen i n the c - a x i s data, with an e i n c r e a s e in the c - a x i s l a t t i c e parameter of 0.027A upon o c o o l i n g through the t r a n s i t i o n and a decrease of 0.020A upon h e a t i n g through the t r a n s i t i o n . A thermal h y s t e r e s i s loop of width 45K was observed. Sezerman et a l . (1980) measured the thermal expansion of 1T-TaS 2 s i n g l e c r y s t a l s i n a c a p a c i t a n c e d i l a t o m e t e r as a continuous f u n c t i o n of temperature between T=4K and T=360K. The temperature sweep r a t e was approximately 0.2K per 46 3.361 -3.359 -o 3.357 3.355 200 250 TEMPERATURE (K) 300 5.92 5.91 5.90 -5.89 150 200 250 TEMPERATURE (K) 300 F i g u r e 12. The a- and c-axis versus temperature data o f Givens and F r e d e r i c k s (1977). Data obtained on c o o l i n g (+) and h e a t i n g (dots) are shown. 4 7 minute. The p o s i t i o n of one end of the c r y s t a l was f i x e d and the other end was connected to one p l a t e of a v a r i a b l e c a p a c i t o r of c a p a c i t a n c e C by means of a s p r i n g - l o a d e d push rod. The push rod exer t e d a f o r c e e q u i v a l e n t of 10 grams on the c r y s t a l . With t h i s arrangement, the r e l a t i v e change in the. c r y s t a l ' s l e n g t h 1 i s given by Al/1 = (d/l)AC/C, where d i s the spacing of the c a p a c i t o r p l a t e s . Because the c r y s t a l was t h i n (^0.2mm), f o r the a-axis measurements i t was c o n s t r a i n e d a g a i n s t l a t e r a l movement by four t h i n w i r e s . They observed d i s c o n t i n u i t i e s i n both the a and c l a t t i c e parameters on both c o o l i n g and hea t i n g f o r the NC-C t r a n s i t i o n . The a - a x i s d i s c o n t i n u i t y i s c l e a r l y seen i n t h e i r data. T h e i r v a l u e s of Aa/a and Ac/c are given i n Table 6. A thermal h y s t e r e s i s loop of width 45K was observed, in agreement with Givens and F r e d e r i c k s . 4.2 Experimental 4.2.1 Sample P r e p a r a t i o n A s t o i c h i o m e t r i c mixture of tantalum and sulphur powders, with a s l i g h t excess (0.5% by weight) of sulphur, was p l a c e d i n a quartz ampoule to produce roughly 1Og of TaS 2 i n the 20cm 3 ampoule volume. The ampoule was then evacuated, s e a l e d o f f and heated at a rate of 20K per hour to T=1270K. A f t e r a n n e a l i n g i t at t h i s temperature f o r four days, the ampoule was quenched to room temperature by p l a c i n g i t i n a water bath. The quenching time was l e s s than f i v e seconds. The excess sulphur condensed out on the ampoule w a l l s . 48 Room temperature x-ray scans were performed to determine the q u a l i t y of the product. No evidence of poly t y p e s other than 1T was observed. In a d d i t i o n to the main l a t t i c e peaks, s u p e r l a t t i c e peaks were a l s o d i r e c t l y o bservable. Table 4 l i s t s these peaks, along with a suggested indexing scheme which i s e x p l a i n e d i n Appendix C. Most of the peaks c o u l d be matched to those observed by Wilson et a l . (1975). It was noted t h a t , f o r c r y s t a l s kept i n small a i r - t i g h t c o n t a i n e r s , the q u a l i t y of the x-ray l i n e s f o r the s u p e r l a t t i c e peaks degraded as the c r y s t a l s aged over a p e r i o d of months. For c r y s t a l s c o n t i n u a l l y exposed to a i r , t h i s degradation was not observed. Suppose, f o r some reason, that the sulphur i n the TaS 2 m a t e r i a l p r e f e r s t o leave the compound. With the compound exposed to a i r , the sulphur might react with hydrogen i n the moisture i n the a i r v i a the equat ion TaS 2 + 2H 20 + 2H 2S + Ta0 2 • 4.2 Although the s t a b l e form of tantalum oxide T a 2 + 0,. has y=0, y may be as l a r g e as 0.5 (Cotton and W i l k i n s o n 1972), as i n the above equation. T h i s r e a c t i o n w i l l proceed from the e x t e r i o r to the i n t e r i o r of each c r y s t a l l i t e , forming a tantalum oxide c o a t i n g . If t h i s c o a t i n g p r o h i b i t s f u r t h e r r e l e a s e of sulphur from the TaS 2 m a t e r i a l , then one o b t a i n s s t a b l e TaS 2 c r y s t a l l i t e s , coated with tantalum oxide. X-ray d i f f r a c t i o n , which examines the "bulk" s t r u c t u r e of a 49 Table 4 Observed room temperature s a t e l l i t e peaks i n lT-TaS2 A l l peaks in the angular ranges 26=21 to 26=25 and 28=35 to 28=46 have been l i s t e d . The angular values have been adjusted for the out-of-plane displacement. Also included are proposed peak indices; this indexing scheme i s explained i n Appendix C. 28 d h k 1 h' k' 1' (degrees) (A) 21. ,9 4. .06 22 . ,8 3, ,90 1 0 0 -1 0 0 24. 5 3. .63 30. , 7 2. ,91 1 0 0 35. , 1 2. ,56 1 0 0 1 -1 0 36. ,2 2. ,48 37. , 7 2. ,39 1 0 0 0 1 0 38. ,9 2. .32 40. ,0 2. ,25 1 0 0 1 0 0 40. .4 2. .23 41. , 1 2. ,20 42. .8 2, , 11 43. . 7 2. .07 1 0 2 44. ,2 2, , 05 1 1 0 -1 1 0 44. .9 2, .02 45, . 7 1, . 98 1 1 0 -1 0 0 54, .5 1. .68 1 1 0 50 c r y s t a l , would not detect the c o a t i n g s i f the c o a t i n g s were t h i n and the s u p e r l a t t i c e peaks would remain v i r t u a l l y unchanged. With a l i m i t e d exposure to moisture, the r e l e a s e of sulphur would be l e s s h indered, l e a v i n g tantalum atoms behind. T h i s would change the band s t r u c t u r e of the m a t e r i a l , a l t e r i n g the Fermi l e v e l and hence the Fermi su r f a c e geometry. If the charge d e n s i t y wave i s c r i t i c a l l y dependent on t h i s geometry, as d i s c u s s e d i n S e c t i o n 4.1.2, changes i n the charge d e n s i t y wave c h a r a c t e r i s t i c s would be expected. Large amounts of H 2S have been detected a f t e r opening a i r - t i g h t b o t t l e s c o n t a i n i n g 1T-TaS 2 c r y s t a l s with l i m i t e d exposure to a i r . T h i s i s a l s o observed when opening the se a l e d ampoules f o l l o w i n g c r y s t a l growth. Perhaps the f r e e sulphur present i n the c o n t a i n e r reacts with moisture to form H 2S. Another p o s s i b l e e x p l a n a t i o n for the e f f e c t . o n the charge d e n s i t y wave s t r u c t u r e i s conversion of the metastable 1T polytype to the s t a b l e 2H p o l y t y p e ; however, x-ray peaks corresponding to the 2H s t r u c t u r e are not seen in any of the samples. For the low temperature experiments, 1T-TaS 2 c r y s t a l s were s p r i n k l e d onto a t h i n l a y e r of vacuum grease p l a c e d on a copper s l i d e ( t h i c k n e s s of 1.4 mm). To improve thermal contact to the c o l d f i n g e r , the s l i d e s were backed with an L-shaped copper p l a t e which a l s o contacted the face of the c o l d f i n g e r s h a f t . Cry-Con grease was used as a bonding 5 1 agent between the p l a t e s and the c o l d f i n g e r s h a f t f a c e . The sample was h e l d to the c o l d f i n g e r s h a f t by s t a i n l e s s s t e e l c l i p s . With t h i s arrangement, the temperature d i f f e r e n c e between the c o l d f i n g e r and the middle of the sample v a r i e d approximately l i n e a r l y with temperature, from a zero value at 300K.to a value of 6K at a sample temperature of 56K. 4.2.2 The NC-C T r a n s i t i o n The thermal h i s t o r y of a p a r t i c u l a r sample, made from the 17/1/84 batch of 1T-TaS 2 c r y s t a l s , i s given i n Table 5. The c r y s t a l s were c o n t i n u a l l y exposed to a i r between t h e i r growth and the low temperature study. I n i t i a l l y , the t r a n s i t i o n temperatures f o r c o o l i n g and h e a t i n g through the NC-C phase t r a n s i t i o n were determined by s e v e r a l passes through each t r a n s i t i o n . Step scans were then performed to o b t a i n the l a t t i c e parameter d i s c o n t i n u i t i e s at each t r a n s i t i o n . Determinations of the t r a n s i t i o n temperatures are c o m p l i c a t e d by the c o e x i s t e n c e of the two (NC and C) phases over l i m i t e d temperature ranges about each t r a n s i t i o n . However, Thompson et a l . (1971) have shown that the t r a n s i t i o n i s f i r s t order, as d i s c u s s e d i n S e c t i o n 4*1.2; the observed phase c o e x i s t e n c e i s probably due to temperature g r a d i e n t s a c r o s s the sample or p o s s i b l e nonhomogeneity of the samples. The l a t t e r may give r i s e to domains w i t h i n the sample which have d i f f e r e n t c r i t i c a l temperatures, as d i s c u s s e d by E v e r e t t and Whitton (1952). I t was found that the temperature range over which each 52 Table 5 Thermal history of the thin lT-TaS 2 sample. The entries irt the table are l i s t e d in chronological order. Temperature (K) X-ray Scan Type 300 Step scan of nine peaks 184+156 Cont inuous scans of 005 peak 201+225 Continuous scans of 005 peak 175+174 Continuous scans of 005 peak 220+225 Continuous scans of 005 peak 177+176 Continuous scans of 005 peak 300 -182+173 Continuous scans of 005 (2 temperature c o n t r o l l f a i l u r e s ) peak er 221+224 Continuous scans of 005 peak 174 Temperature co n t r o l l e r f a i l u : 240 -180+175 Continuous scans of 005 peak 300 238+133 Step scans of nine peaks (results in Figures 15 and 16) 300 160+300 Step scans of ten peaks (results in Figures 15 and 16) 5 3 t r a n s i t i o n occurs c o u l d be markedly reduced by using a th i n n e r sample. T h i s suggests that sample temperature g r a d i e n t s are the primary reason f o r phase c o e x i s t e n c e . The r e s u l t s presented below are f o r the t h i n n e s t sample used, except where i t i s noted otherwise. With the above assumption, one can o b t a i n estimates of the t r a n s i t i o n temperatures. Temperature g r a d i e n t s e x i s t a cross the t h i c k n e s s and len g t h of the low temperature sample. When c o o l i n g through the t r a n s i t i o n , the c r y s t a l l i t e s c l o s e s t to the c o l d f i n g e r s h a f t and i n d i r e c t c o n t a c t with the s u b s t r a t e w i l l pass through the t r a n s i t i o n before a l l other c r y s t a l l i t e s ; when h e a t i n g through the t r a n s i t i o n , the same c r y s t a l l i t e s w i l l pass through the t r a n s i t i o n a f t e r a l l other c r y s t a l l i t e s . Since these c r y s t a l l i t e s are at the measured s u b s t r a t e temperature, one can determine the t r a n s i t i o n temperatures from t h e i r behav i o r . The (005) Bragg peak was monitored by continuous x-ray scans over the angular range 20=81° to 20=82° as the temperature was lowered and r a i s e d through the NC-C t r a n s i t i o n . Because of the l a r g e d i s c o n t i n u i t y i n the c - a x i s at the t r a n s i t i o n , a n o t i c a b l e s h i f t i n the higher angle (001) peaks was expected. Just above the t r a n s i t i o n obtained on c o o l i n g , a strong NC phase peak was observed at 20=81.52°. The temperature was decreased i n 1K steps toward the t r a n s i t i o n , while s e a r c h i n g f o r the appearance of the C phase peak at 20=81.15°. At each temperature, the sample was allowed to f u l l y e q u i l i b r a t e f o r times of up to t h i r t y 54 minutes. I f the t r a n s i t i o n had been passed, one would expect the C phase peaks to grow with time. Scans of the 20=81° to 20=82° angular range are shown in F i g u r e 13. The scan in p a r t (a) of the f i g u r e was taken j u s t a f t e r the temperature was lowered to 179K. A f t e r t h i r t y minutes at t h i s temperature, no f u r t h e r evidence of the growth of the C phase peak was observed. The scan i n part (b) at 178K shows c l e a r evidence of the formation of the C phase peak. It was concluded that the temperature of the c o o l i n g t r a n s i t i o n was T=179K. The a b s o l u t e e r r o r i n the temperature values was taken to be ±1K, as d i s c u s s e d i n S e c t i o n 3.1; however, the r e l a t i v e e r r o r between two c l o s e l y spaced temperatures was taken to be much smal l e r (±0.2K). J u s t below the t r a n s i t i o n obtained on h e a t i n g , a strong C phase peak i s observed at 20=81.05°. In F i g u r e 14 are shown a s e r i e s of scans of the 20=81° to 20=82° angular range f o r T=225K. The f i r s t and l a s t scans are separated by 50 minutes. The t r a n s i t i o n has a l r e a d y begun in the f i r s t scan, and i t proceeds with time with the temperature constant at T=225K. In the l a s t scan, the C phase peak has e s s e n t i a l l y disappeared. Scans taken previous to t h i s at T=224K over a 15 minute p e r i o d showed no si g n s of phase c o n v e r s i o n . Thus T=225K was taken to be the t r a n s i t i o n temperature on h e a t i n g . T h i s t r a n s i t i o n was observed to occur over a much narrower range of temperatures than the t r a n s i t i o n at 179K obtained on c o o l i n g . Since the temperature g r a d i e n t s a c r o s s the sample are smaller for sample temperatures c l o s e r to the ambient temperature, t h i s 55 1 1 MM CO NC 1— 2 005 15 >-cr Ka2 CD _ U " cr 1/\ < \ CO - / -r- c 1 I 005 / o / \ o ~J V 1 V l l 81.0 81.5 82.0 26 (°) (a) CO or CD s CO o o 81.0 81.5 82.0 26 (°) (b) Figure 13. Continuous x-ray scan data for the lT-TaS 2 (005) peak near the NC-C tr a n s i t i o n obtained on cooling. T=179K for (a) and T=178K for (b). 81.5 82.0 26 (°) ( c ) gure 14, (d) 00 fc: >-or fc= CD or s to O o 1 1 1 Ka, - J — / NC / 005 1 - 1 Ka2 A " . j " I -c / 005 J IV 1 1 t i 81.0 81.5 82.0 26 (°) Continuous x-ray scan data for the lT-TaS 2 (005) peak near the NC-C t r a n s i t i o n obtained on heating. T=225K for a l l traces. The scans are ordered chronologically, with the scan in parts (b), (c) and (d) taken 9, 24 and 50 minutes, respectively, after that i n part (a). 57 suggests that i t i s these g r a d i e n t s that determine the observed phase c o e x i s t e n c e about each t r a n s i t i o n , as d i s c u s s e d above. The t r a n s i t i o n temperatures determined above y i e l d a thermal h y s t e r e s i s loop of width 46K, i n agreement with Givens and F r e d e r i c k s (1977) and Sezerman et a l . (1980). Constant temperature step scans of at l e a s t nine Bragg peaks, performed as the sample was c o o l e d and heated through the t r a n s i t i o n s , were used to c a l c u l a t e the l a t t i c e parameters as a f u n c t i o n of temperature. T h i s data i s shown in F i g u r e s 15 and 16. The e r r o r estimate f o r each l a t t i c e parameter value was obtained by incrementing and decrementing each Bragg peak p o s i t i o n by 0.005° and c a l c u l a t i n g the l a t t i c e parameters obtained f o r these v a l u e s . T h i s leads to an u n c e r t a i n t y due to systematic e r r o r s i n each l a t t i c e parameter value of ±0.0004A. The l a t t i c e parameter d i s c o n t i n u i t i e s at each t r a n s i t i o n were obtained by e x t r a p o l a t i o n to the t r a n s i t i o n temperature of l i n e a r f i t s to the data of three scans on e i t h e r s i d e of the t r a n s i t i o n . The s e p a r a t i o n of data p o i n t s w i t h i n each three scan set was roughly 5K. For the t r a n s i t i o n obtained on c o o l i n g , Aa=0.0035±0.0002A and Ac=0.0294±0.0002A; f o r the t r a n s i t i o n obtained on h e a t i n g , Aa=0.0031±0.0005A and Ac=0.0249±0.0003A. The u n c e r t a i n t i e s a s s o c i a t e d with these q u a n t i t i e s are measures of the absolute e r r o r ; they were estimated from the observed average s c a t t e r of the data p o i n t s about the l i n e a r f i t s . A l l of the M i l l e r i n d i c e s of the Bragg peaks used to 3.366 -3.364 -3.362 3.360 -3.358 -3.356 100 Figure 15 150 200 250 TEMPERATURE (K) 300 a-axis versus temperature data f o r the t h i n lT-TaS2 sample on c o o l i n g ( c r o s s e s ) and heating (diamonds). A l s o shown are the l i n e a r f i t s to each three p o i n t set about the t r a n s i t i o n temperatures. The t r a n s i t i o n temperatures are i n d i c a t e d by the v e r t i c a l l i n e s . 5.93 5.92 5.91 5.90 -100 Figure 16, L _ 150 200 250 TEMPERATURE (K) 300 c-axis versus temperature data for the thin lT-TaS 2 sample on cooling (crosses) and heating (diamonds). Also shown are the linear f i t s to each three point set about the t r a n s i t i o n temperatures. The t r a n s i t i o n temperatures are indicated by the v e r t i c a l l i n e s . 60 determine the l a t t i c e parameters on the cooldown of the sample had non-zero 1 v a l u e s , and hence c o n t a i n e d a dependence on the c - a x i s . However, for the scans performed on the h e a t i n g c y c l e , the (100) peak was a l s o measured. The l a t t i c e parameter values c a l c u l a t e d from t h i s peak alone are compared with the values o b t a i n e d from the l e a s t squares f i t data of F i g u r e 15 i n F i g u r e 17. A comparable d i s c o n t i n u i t y i s observed i n the (100) peak data. Hence, the a - a x i s d i s c o n t i n u i t y obtained f o r the l e a s t squares f i t data on the h e a t i n g c y c l e i s r e a l and i s not an a r t i f a c t of the f i t t i n g program. S i m i l a r r e s u l t s are expected, but were not confirmed, on the c o o l i n g c y c l e . Values of Aa/a and Ac/c from the l e a s t squares f i t are given f o r both t r a n s i t i o n s i n Table 6, along with the corresponding data of Sezerman et a l . (1980). The e r r o r estimates quoted by Sezerman et a l . are only an i n d i c a t i o n of the r e p e a t a b i l i t y o b t a i n e d on thermal c y c l i n g of t h e i r s i n g l e c r y s t a l samples, and are not an i n d i c a t i o n of the a b s o l u t e accuracy of t h e i r measurements. The accuracy of t h e i r measurements i s reduced by the use of s i n g l e c r y s t a l samples, which do not c y c l e w e l l t h e r m a l l y , and the a p p l i c a t i o n of pressure to the c r y s t a l . T h e i r values f o r the a - a x i s d i s c o n t i n u i t y are c o n s i d e r a b l y lower than ours. Because t h e i r c r y s t a l was very t h i n , the compressive f o r c e a p p l i e d by the push rod d u r i n g the a-axis measurement may have caused the c r y s t a l to bow when passing through the t r a n s i t i o n . T h i s would lower t h e i r measured val u e s of Aa/a. Any c o n t r i b u t i o n to t h e i r a - a x i s measurement from the 3.366 -3.364 -3.362 -3.360 -3.358 3.356 150 200 250 TEMPERATURE (K) 300 Figure 17. Comparison of a-axis versus temperature data obtained f o r the t h i n l T - T a S 2 sample from a l e a s t squares f i t to a l l Bragg peaks (+) and that o b tained from the (100) Bragg peak (X). The v e r t i c a l l i n e i n d i c a t e s the temperature of the t r a n s i t i o n . 62 Table 6 Thermal expansion data for the NC-C t r a n s i t i o n in lT-TaS2- Included in the table are the results obtained in this study and those obtained by Sezerman et a l . (1980). The sign always refers to the change from the low to the high temperature side of the tr a n s i t i o n . NC-C Transition Our Data Data of Sezerman et a l . Aa/a Ac/c X10 4 Aa/a Ac/c X10 4 Cooling +10.5+0.6 -49.7+0.3 +6+1 -49+8 Heating +8+1 -42.0+0.5 +5.2+0.3 -32+4 63 r e l a t i v e l y l a r g e c - a x i s d i s c o n t i n u i t i e s would increase t h e i r v alues of Aa/a. T h i s e f f e c t was e v i d e n t l y s m a l l . For the c-a x i s data obtained f o r the h e a t i n g t r a n s i t i o n , the measured d i s c o n t i n u i t y f o r our data i s about 30 percent l a r g e r than t h e i r s and that of Givens and F r e d e r i c k s . T h i s t r a n s i t i o n may not be a simple C phase to NC phase t r a n s i t i o n , but r a t h e r a t r a n s i t i o n to an intermediate t r i c l i n i c phase (Tanda et a l . 1984). I t i s p o s s i b l e that the exact s t r u c t u r e of t h i s phase i s sample-dependent. T h i s c o u l d be r e l a t e d to the observed changes i n charge d e n s i t y wave behavior noted i n S e c t i o n 4.2.1, or i t may depend on the d e t a i l e d thermal h i s t o r y of the sample. L a t t i c e parameter data obtained on the i n i t i a l cooldown fo r a t h i c k e r sample made from the same batch of c r y s t a l s i s p l o t t e d i n F i g u r e 18, along with the data f o r the t h i n n e r sample, obtained a f t e r numerous thermal c y c l e s (as d e s c r i b e d in Table 5). The d i f f e r e n c e i n the out-of-plane displacement fo r the two samples i s roughly 300M, which i s q u i t e l a r g e . Although the data f o r the t h i c k e r sample i s sparse, the behavior of both samples i s e s s e n t i a l l y the same, d e s p i t e a s l i g h t o f f s e t of the l a t t i c e parameter v a l u e s . In p a r t i c u l a r , the same c - a x i s behavior i s observed on p a s s i n g through the h e a t i n g t r a n s i t i o n , with the Ac values w i t h i n roughly 5% of each o t h e r . The l a r g e r c - a x i s d i s c o n t i n u i t y obtained f o r the h e a t i n g t r a n s i t i o n in our data, as compared with that of Sezerman et a l . , i s r e p r o d u c i b l e f o r samples from the same batch of c r y s t a l s , and does not seem to depend on the thermal h i s t o r y of the sample. 5.93 5.92 5.91 5.90 I I I I I I — A * * % + A — to — — — A • • — i L + • I I I I I 1 100 150 200 250 300 TEMPERATURE (K) Figure 18. Comparison of c-axis versus temperature data obtained f o r the t h i n l T - T a S 2 sample (cooling:+; heating:diamonds) and that obtained f o r a sample 300u t h i c k e r (cooling:squares; h e a t i n g : t r i a n g l e s ) . The v e r t i c a l l i n e s i n d i c a t e the t r a n s i t i o n temperatures. 65 In summary, the NC-C charge d e n s i t y wave phase t r a n s i t i o n i n 1T-Tas 2 i s e a s i l y observed with our"low temperature x-ray d i f f r a c t i o n apparatus. The s e n s i t i v i t y of our apparatus i s such that l a t t i c e parameter d i s c o n t i n u i t i e s of one part i n 10 3 can e a s i l y be d e t e c t e d . We have obtained r e s u l t s s u p e r i o r to the p r e v i o u s l y r e p o r t e d low temperature x-ray d i f f r a c t i o n r e s u l t s of Givens and F r e d e r i c k s (1977) and c a p a c i t a n c e d i l a t o m e t e r measurements of Sezerman et a l . (1980). x. 66 CHAPTER FIVE A SEARCH FOR STAGING IN L i T i S , x i 5.1 L i t h i u m I n t e r c a l a t i o n i n T i S 2 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 guest atoms i n t o a host s o l i d such that the s t r u c t u r e of the host i s not s i g n i f i c a n t l y a l t e r e d i n the process (McKinnon and Haering 1983). The l a y e r e d compounds d i s c u s s e d i n S e c t i o n 4.1.2 are good host s t r u c t u r e s f o r i n t e r c a l a t i o n . The weak bonding of the van der Waals gaps allows the easy s e p a r a t i o n of the l a y e r s ; i n t e r c a l a n t atoms can then d i f f u s e i n t o and out of the gaps. The l a y e r e d i n t e r c a l a t i o n compounds which have r e c e i v e d the most study are the g r a p h i t e i n t e r c a l a t i o n compounds (Dresselhaus and Dresselhaus 1981). M a r s e g l i a (1983) reviews the i n t e r c a l a t i o n p r o p e r t i e s of a group of l a y e r e d host s t r u c t u r e s c a l l e d 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 . Both TaS 2 and T i S 2 belong to t h i s group. A remarkable f e a t u r e e x h i b i t e d by a number of l a y e r e d i n t e r c a l a t i o n systems is- that of s t a g i n g . A m a t e r i a l i s s a i d to be staged when each i n t e r c a l a t e d gap i s separated by a number n of u n i n t e r c a l a t e d gaps, where n d e f i n e s the stage of the m a t e r i a l . T h i s e f f e c t i s observed most commonly i n g r a p h i t e i n t e r c a l a t i o n compounds, but has a l s o been observed i n Na T i S 2 ( Z a n i n i et a l . 1981), Ag TaS 2 (Scholz and F r i n d t 1980) and L i NbSe 2 (Dahn and Haering 1982). X I n t e r c a l a t i o n can be accomplished by d i f f e r e n t methods (McKinnon and Haering 1983), but we w i l l be concerned only 67 with that using e l e c t r o c h e m i c a l techniques. An e l e c t r o c h e m i c a l c e l l i s formed by p l a c i n g the host and i n t e r c a l a n t s o l i d s , denoted by cathode and anode r e s p e c t i v e l y , i n t o an e l e c t r o l y t e which al l o w s i n t e r c a l a n t ions but not e l e c t r o n s to flow between the two e l e c t r o d e s . T h i s i s shown s c h e m a t i c a l l y i n F i g u r e 19 f o r the L i / L i T i S 2 i n t e r c a l a t i o n system. E l e c t r o n s are allowed to flow between the e l e c t r o d e s v i a an e x t e r n a l c i r c u i t . To maintain charge n e u t r a l i t y , the number of e l e c t r o n s and ions t r a n s f e r r e d between the e l e c t r o d e s are e q u a l . Thus by c o n t r o l l i n g the e x t e r n a l c u r r e n t flow, the i n t e r c a l a n t content of the host, x, can be a d j u s t e d . The q u a n t i t y x i s expressed in terms of the number of i n t e r c a l a n t atoms i n the cathode, n, r e l a t i v e to the number of metal atoms, N, i . e . x=n/N. The v o l t a g e of the c e l l , V ( x ) , i s given by v ( x ) = ^ ( y a - y c ( x ) ) , 5.1 where ze i s the charge of the i n t e r c a l a n t ions and M„ and u are the chemical p o t e n t i a l s of the anode and cathode, r e s p e c t i v e l y . Because the composition of the anode remains unchanged, u. i s constant and changes i n the c e l l v o l t a g e at a constant temperature can be a t t r i b u t e d to changes in M c ( x ) . Features i n the V(x) behavior can be seen more e a s i l y in dx/dV, and standard e l e c t r o c h e m i c a l techniques e x i s t for measuring t h i s q u a n t i t y (Thompson 1979, Dahn and Haering 1981, and Dahn and McKinnon 1984). We w i l l be i n t e r e s t e d i n the i n t e r c a l a t i o n of l i t h i u m 68 F i g u r e 19. Schematic r e p r e s e n t a t i o n of the discharge of a L i / L i T i S 9 e l e c t r o c h e m i c a l c e l l . 69 i n t o the l a y e r e d compound T i S 2 . T i S 2 has the same 1T s t r u c t u r e as 1T-TaS 2 (see S e c t i o n 4.1.2). The room o temperature hexagonal l a t t i c e parameters are a=3.407A and c=5.695A (Thompson et a l . 1975). I n t e r c a l a t i o n with l i t h i u m i s sometimes complicated by c o i n t e r c a l a t i o n of s o l v e n t molecules from the e l e c t r o l y t e which are c a r r i e d by l i t h i u m ions i n t o the host s t r u c t u r e . T h i s has been observed i n many e l e c t r o c h e m i c a l systems using propylene carbonate (PC) s o l v e n t in the e l e c t r o l y t e (Whittingham 1978, Dahn et a l . 1982a, McKinnon and Dahn 1985). Dahn et a l . noted that c o i n t e r c a l a t i o n c o u l d be prevented by i n i t i a l l y "quick d i s c h a r g i n g " the c e l l s . Recent work by McKinnon and Dahn showed that the use of a s a t u r a t e d s o l u t i o n of PC/LiAsF 6 e l e c t r o l y t e reduced the amount of c o i n t e r c a l a t i o n compared with that obtained f o r a 1M s o l u t i o n i n the L i / L i Z r S 2 , L i / L i Z r S e 2 and L i / L i TaS 2 X X X i n t e r c a l a t i o n systems. E a r l y work on L i / L i x T i S 2 centered on measurements by Thompson (1978) of peaks in -dx/dV at x values of 1/9, 1/4, and 6/7, and h i s suggestion that t h i s s t r u c t u r e in V(x) was due to the formation of ordered s t a t e s at these compositions ( C h i a n e l l i et a l . 1978, Thompson 1981, Hibma 1980, Dahn et a l . 1980, and K l e i n b e r g et a l . 1982). I t has s i n c e been shown that minima (not peaks) in -dx/dV are always expected at ordered compositions ( B e r l i n s k y et a l . 1979). The experimental and t h e o r e t i c a l i n v e s t i g a t i o n s by Dahn (1982) l e d to the c o n c l u s i o n that the three-dimensional a s p e c t s of t h i s system were important; i n p a r t i c u l a r , h i s 70 r e s u l t s p o i n t e d to the formation of a stage two s t r u c t u r e near x=0.16. For samples with t h i s composition, Dahn observed minima in -dx/dV, the (004) Bragg peak width, and the entropy (from measurements of dV/dT), a l l of which are i n d i c a t i v e of i n c r e a s e d order. These r e s u l t s are shown i n F i g u r e s 20, 21, and 22, r e s p e c t i v e l y . Peaks are observed i n the -dx/dV and peak width data at compositions s l i g h t l y higher and lower than x=0.16, c o n s i s t e n t with stage two to stage one c o n v e r s i o n as the composition i s changed from x=0.16. C a r e f u l room temperature x-ray scans over the angular range corresponding to the s t r o n g e s t unimpeded s u p e r l a t t i c e peaks d i d not show the presence of the peaks. I t was proposed by Dahn (1982) that the s t a g i n g i n L i x T i S 2 i s imperfect, g i v i n g r i s e to only short range order i n the c-d i r e c t i o n . For random s t a c k i n g of equal numbers of l a y e r s of two c h a r a c t e r i s t i c l i t h i u m c o n c e n t r a t i o n s x, and x 2 and c-a x i s parameters c, and c 2 , he c a l c u l a t e d the peak widths of the average l a t t i c e and s u p e r l a t t i c e (001) peaks as a f u n c t i o n of the s t a g i n g c o r r e l a t i o n l e n g t h £c, where c"=(c,+c 2)/2 i s the average c - a x i s l a t t i c e parameter. The i n t e n s i t y of the (001) l i n e s f o r l a y e r e d compounds i s u s u a l l y g r e a t e r than that of other l i n e s because the c r y s t a l l i t e s are p r e f e r e n t i a l l y o r i e n t e d with t h e i r c-axes p e r p e n d i c u l a r to the plane of the s u b s t r a t e . The f u l l width at h a l f maximum of the average l a t t i c e (001) peaks i s 71 Figure 20. -dx/dV versus x for a L i TiS2 c e l l . The triangles are the discharge and the diamonds are the charge (after Dahn 1982). 72 Figure 21. Variation of the (004) Bragg peak width for L i T i S 0 (after Dahn 1982). 73 Figure 22. dV/dT versus x for a L i T i S ? c e l l (after Dahn 1982). 7 4 A / 0 r t l N 4 i T s i n 9 t a n 6 / \2 . w 1 \ r o A ( 2 V = A c~f?j- (crc2) t a n h ^ 2 l > • 5 - 2 and the f u l l width at h a l f maximum of the s u p e r l a t t i c e (001) peaks i s A ( 2 0 1 ) = —,— E s i n h ( ^ ) , 5.3 v s y T T ( C ^ + C 2 ) C O S 6 where X i s the x-ray wavelength and 6 i s the Bragg angle. It can be shown from these equations t h a t , as £c i n c r e a s e s , the average l a t t i c e peaks sharpen much f a s t e r than the s u p e r l a t t i c e peaks. I f £c i s not l a r g e , the s u p e r l a t t i c e peaks w i l l be too broad to be seen i n a powder d i f f r a c t i o n experiment. Thus short range order s t a g i n g accounts f o r the o b s e r v a t i o n of v a r i a b l e average l a t t i c e (001) peak widths without the o b s e r v a t i o n of s u p e r l a t t i c e peaks. Because phase t r a n s i t i o n s from short range to long range order may occur as the temperature of the m a t e r i a l i s lowered, low temperature neutron d i f f r a c t i o n experiments were a l s o performed by Dahn (1982). C l e a r evidence f o r the formation of an (003/2) s u p e r l a t t i c e peak was observed in a L i 1 I ( T i S 2 sample at 100K and below, which was not observed in a L i < 2 5 T i S 2 sample. From the width of the s u p e r l a t t i c e peak, Dahn obtained an estimate f o r the s t a g i n g c o r r e l a t i o n o l e n g t h of £c^45A. However, the data s u f f e r e d from the high background l e v e l and contaminant peaks caused by e l e c t r o l y t e c o i n t e r c a l a t i o n . 75 A simple " s p r i n g and p l a t e " model (Dahn et a l . 1982b) based on the i n c l u s i o n of e l a s t i c energy i n the expression f o r the f r e e energy was shown to e x p l a i n the gross f e a t u r e s of the experimental r e s u l t s . I n c l u s i o n of a r e p u l s i v e i n t e r a c t i o n u' between nearest neighbour l a y e r s allowed the formation of a s t a b l e stage two s t r u c t u r e at low values of x. F i g u r e 23 shows a t y p i c a l phase diagram f o r J=0.4u'=0.7u and a=0.2, where J and a are r e l a t e d to the s t r e n g t h of the e l a s t i c energy c o n t r i b u t i o n to the free energy, and u i s an i n t e r a c t i o n between nearest neighbour i n t e r c a l a n t atoms i n the same l a y e r . It may be seen that the stage two s t r u c t u r e extends to higher x val u e s at lower temperatures. Monte C a r l o s i m u l a t i o n s of a Hamiltonian i n c l u d i n g both e l a s t i c energy and i n t e r - l a y e r r e p u l s i v e i n t e r a c t i o n s were done by Dahn (1982); t h i s technique i s capable of d e a l i n g with the short range o r d e r i n g expected i n L i _ 1 6 T i S 2 . Using a s i n g l e set of f i t t i n g parameters, the e x p e r i m e n t a l l y observed f e a t u r e s i n -dx/dV, dV/dT and the (004) peak width, as w e l l as the observed c - a x i s behavior, were reproduced to give q u a l i t a t i v e and, i n some cases, q u a n t i t a t i v e agreement. Dahn a l s o c a l c u l a t e d t h a t the c - a x i s thermal expansion c o e f f i c i e n t , a c , given by 1 d C s r , a c = c HT^x ' 5 - 4 would be peaked at x valu e s near 0.16 due only to the temperature dependence of the s t a g i n g p r o c e s s . In x-ray d i f f r a c t i o n experiments, e l e c t r o l y t e 76 Figure 23. A phase diagram for a l a t t i c e gas model of L i T i S 2 with e l a s t i c energy contributions to the free energy and repulsive interactions between nearest neighbour layers (after Dahn 1982). 77 c o i n t e r c a l a t i o n does not pose as great a problem as i t does in neutron d i f f r a c t i o n experiments. A l s o , strong i n t e n s i t y (001) s u p e r l a t t i c e peaks are expected in x-ray d i f f r a c t i o n experiments (Dahn 1982). These f a c t o r s , along with the wealth of i n f o r m a t i o n found by Dahn served as the m o t i v a t i o n for undertaking a search f o r s t a g i n g i n L i _ 1 6 T i S 2 using low temperature x-ray d i f f r a c t i o n . 5.2 Experimental 5.2.1 Sample P r e p a r a t i o n C r y s t a l s of T i S 2 were grown using a procedure i d e n t i c a l to that f o r the 1T-TaS 2 c r y s t a l s d e s c r i b e d in S e c t i o n 4.2.1, except that the c r y s t a l s were annealed f o r two days at 1020K. Room temperature x-ray scans showed only the presence of peaks due to T i S 2 . C r y s t a l s grown by J.R. Dahn were a l s o used i n these s t u d i e s . C r y s t a l l i t e s i z e s l e s s than 38M were obtained by g r i n d i n g the c r y s t a l s . I n t e r c a l a t e d samples-of L i ^ T i S 2 f o r use i n low temperature s t u d i e s were prepared i n e l e c t r o c h e m i c a l f l a n g e c e l l s , as d e s c r i b e d i n Dahn and Haering (1982) and Dahn (1982). Cathode masses were l a r g e (^200mg); t h i s allowed many low temperature samples to be taken from the same batch of cathode m a t e r i a l . The c e l l s were i n i t i a l l y d i s c h a r g e d to x=1 and then e q u i l i b r a t e d at a v o l t a g e corresponding to the d e s i r e d l i t h i u m composition with a P r i n c e t o n A p p l i e d Research (PAR) p o t e n t i o s t a t (model 173). When the c u r r e n t d e n s i t i e s had f a l l e n to a low valu e , i d e a l l y l e s s than 1 yA/cm 2, the c e l l s were disconnected from 7 8 the p o t e n t i o s t a t and allowed to f u r t h e r e q u i l i b r a t e for one or two days. The c e l l s were then disassembled i n an i n e r t atmosphere. The cathode powder was washed by p l a c i n g i t in propylene carbonate (PC), a g i t a t i n g the s o l u t i o n and a l l o w i n g the powder t o s e t t l e at the c o n t a i n e r bottom. The PC was then removed by decanting and then pumping on the s o l u t i o n . T h i s produced dry i n t e r c a l a t e d powdered c r y s t a l s which were s t o r e d i n an i n e r t atmosphere. C e l l s made with a 1M s o l u t i o n of PC / L i A s F 6 were "quick d i s c h a r g e d " immediately a f t e r c e l l c o n s t r u c t i o n by h o l d i n g the c e l l v o l t a g e at 1.6 V (Dahn 1982) to prevent e l e c t r o l y t e c o i n t e r c a l a t i o n . Those made with a s a t u r a t e d e l e c t r o l y t e s o l u t i o n d i d not r e q u i r e as f a s t an i n i t i a l d i s c h a r g e , since the r a t e of c o i n t e r c a l a t i o n i s slowed, as d i s c u s s e d i n S e c t i o n 5.1. For c e l l s d i s c harged at r a t e s f a s t e r than 120 h (f o r Ax=1), x-ray peaks due to c o i n t e r c a l a t e d m a t e r i a l were not observed; however, "quick d i s c h a r g i n g " these c e l l s seemed to cause e x t r a s i d e r e a c t i o n s , as evidenced by a darkening of the c e l l s e p a r a t o r s . For c e l l s d i s c h a r g e d at very slow r a t e s (^480 h), c o i n t e r c a l a t i o n peaks as t a b u l a t e d by Dahn (1982) were seen i n x-ray scans. A l l low temperature samples were o b t a i n e d from c e l l s u s i n g a s a t u r a t e d e l e c t r o l y t e s o l u t i o n , one of which (DJR22) had x-ray peaks due to c o i n t e r c a l a t e d m a t e r i a l . The x values of the e l e c t r o c h e m i c a l l y prepared L i T i S 2 samples can be o b t a i n e d i n s e v e r a l d i f f e r e n t ways: from the c e l l v o l t a g e j u s t b e f o r e c e l l disassembly; from the l a t t i c e parameters measured by x-ray d i f f r a c t i o n a f t e r c e l l 79 disassembly; and from the amount of charge t r a n s f e r r e d d u r i n g the i n t e r c a l a t i o n p r o c e s s . Since the c e l l v o l t a g e r e f l e c t s the chemical p o t e n t i a l of the l i t h i u m at the s u r f a c e of the i n t e r c a l a t e d c r y s t a l l i t e s (McKinnon and Haering 1983), use of the v o l t a g e as a measure of x r e l i e s on the establishment of a high degree of e q u i l i b r i u m w i t h i n the cathode. X-ray d i f f r a c t i o n , which probes the "bulk" of the m a t e r i a l , serves as a b e t t e r i n d i c a t i o n of x. Both c e l l v o l t a g e and l a t t i c e parameter data has been c a r e f u l l y measured as a f u n c t i o n of x by Dahn et a l . (1982a). Data f o r the c e l l v o l t a g e and l a t t i c e parameter methods i s shown in Table 7 f o r each c e l l used i n the low temperature s t u d i e s . Charge t r a n s f e r data was not a c c u r a t e l y c o l l e c t e d for these c e l l s ; t h e r e f o r e , the x values as determined by the l a t t i c e parameter method w i l l be used f o r each of the c e l l s . F i v e samples, obtained from the three e l e c t r o c h e m i c a l c e l l s of Table 7, were used f o r low temperature s t u d i e s ; they w i l l be l a b e l l e d by the c e l l from which they were obtained and, i f necessary, a numerical s u b s c r i p t . For each sample, a small (^lOmg) amount of cathode m a t e r i a l was pl a c e d i n t o an a i r t i g h t c e l l . P r e s s i n g of the powder a g a i n s t the s u b s t r a t e enhanced the p r e f e r r e d o r i e n t a t i o n which was u s e f u l f o r the (001) s u p e r l a t t i c e peak searches d e s c r i b e d i n S e c t i o n 5.2.3. The design of the a i r t i g h t c e l l was based on that of Dahn et a l . (1982a), using a b e r y l l i u m window to allow x-ray a n a l y s i s of the c e l l ' s c o n t e n t s . The c e l l mass was reduced by over a f a c t o r of two, with respect to that of Dahn et a l . , to allow e a s i e r c o o l i n g . The c e l l 80 Table 7 x values determined by c e l l voltage and l a t t i c e parameter methods for L i TiS2 electrochemical c e l l s used in low temperature studies. A l l values were measured at room temperature. C e l l V c e l l ( V ) x a(A) c(A) x DJR18 2. 355 0. 16 3. .4124 5. .8887 0. 13 DJR20* 2. , 344 0. , 19 3. .4150 5. . 9569 0. 20 DJR22*" 2. . 336 0. ,22 3. .4130 5. . 8950 0. , 15 crystals grown by J.R. Dahn cointercalation present 81 top i s made of brass and the c e l l base i s made of s t a i n l e s s s t e e l . For low temperature s t u d i e s , the c e l l base was backed by a copper p l a t e , as d e s c r i b e d in S e c t i o n 4.2.1. With t h i s arrangement, the temperature d i f f e r e n c e between the c o l d f i n g e r and the middle of the sample v a r i e d almost l i n e a r l y with temperature from a zero value at 300K t o approximately 15K at a sample temperature of 11 OK. Proper thermal anchoring of the thermocouple j u n c t i o n on the b e r y l l i u m p l a t e was d i f f i c u l t because of the l i m i t e d s u r f a c e area. A s i l i c o n rubber gasket was used i n s t e a d of the polypropylene gasket of Dahn et a l . to o b t a i n a b e t t e r c e l l s e a l . U n f o r t u n a t e l y , adequate s e a l s c o u l d not be c o n s i s t e n t l y o b tained. A l l low temperature samples, with the exception of DJR18,, were t r a n s p o r t e d from an i n e r t atmosphere to the c o l d chamber v i a an a i r t i g h t box. Exposure times of the c e l l s to a i r i s estimated to be l e s s than three minutes. With the c e l l i n s i d e the evacuated c o l d chamber, the s e a l q u a l i t y was not a problem. P r e l i m i n a r y s t u d i e s of the low temperature x-ray e l e c t r o c h e m i c a l c e l l DJRX11 showed that the c e l l c o u l d be charged and dis c h a r g e d with c o n s i d e r a b l e c a p a c i t y at room temperature a f t e r i t had been c o o l e d to 200K. T h i s a l s o showed that the c e l l s e a l c o u l d withstand the c o l d chamber vacuum. 82 5.2.2 Thermal Expansion Measurements Measurements of the thermal expansion of T i S 2 were performed. The low temperature sample c o n s t r u c t i o n was i d e n t i c a l to the copper p l a t e arrangement d e s c r i b e d i n S e c t i o n 4.2.1, except that i t was necessary to place a microscope cover s l i d e between the T i S 2 c r y s t a l s and the copper p l a t e to reduce the l a r g e background i n t e n s i t y l e v e l obtained from the copper p l a t e . With the cover s l i d e bonded to the copper p l a t e with Cry-Con grease, sample-cold f i n g e r temperature d i f f e r e n c e s were comparable to that of the copper p l a t e alone. X-ray scans, using twelve Bragg'peaks, were used to c a l c u l a t e the l a t t i c e parameters at each temperature. The data i s shown g r a p h i c a l l y i n F i g u r e 24, along with l i n e a r f i t s to the data. The data p o i n t s at 100K have been excluded from the f i t s . At t h i s temperature, a b u c k l i n g of the cover s l i d e was suspected because a sudden i n c r e a s e of 300tiV was observed i n the d i f f e r e n t i a l thermocouple v o l t a g e , corresponding to a temperature i n c r e a s e of roughly 15K. The data of F i g u r e 24 y i e l d s values f o r the thermal expansion c o e f f i c i e n t s a and a (see equation 5.4) of 14X10 _ 6/K and 22X10- S/K r e s p e c t i v e l y . These val u e s are r e l a t i v e l y c l o s e to the values of a =9.6X10"6/K a and a c=l9.4XlO' 6/K o b t a i n e d by Whittingham and Thompson (1975) between 20K and 300K. They a l s o measured the thermal expansion of L i T i S 2 and found a a=l3XlO" 6/K and a =16X10"6/K between 150K and 300K, with a ^0 below 100K. c c For a L i . 1 3 T i S 2 sample obtained from DJR18, the v a r i a t i o n of the a- and c-axes with temperature was 83 •< N ' D 3 . 4 0 9 3 . 4 0 7 h 3 . 4 0 5 3 . 4 0 3 h 3 . 4 0 1 h 3 . 3 9 9 100 2 0 0 TEMPERATURE (K) 3 0 0 5 . 7 0 5 . 6 9 h 5 . 6 8 h 2 0 0 TEMPERATURE (K) 3 0 0 F i g u r e 24. V a r i a t i o n of a- and c-axes with temperature f o r T i S The s o l i d l i n e s are l i n e a r f i t s to a l l of the data except that at 100K, 8 4 measured. The data i s shown g r a p h i c a l l y i n F i g u r e 25. The s o l i d l i n e s in the f i g u r e were obtained from l i n e a r f i t s to the three data p o i n t s above 200K; they give values f o r a„ Si and a of 17X10' 6/K and 32X10' 6/K r e s p e c t i v e l y . The a-axis c v a r i a t i o n i s roughly l i n e a r over the e n t i r e temperature range. The v a r i a t i o n of the c - a x i s with temperature i s roughly l i n e a r above 200K, but changes l e s s r a p i d l y with temperature below 200K. T h i s c - a x i s behavior i s i n q u a l i t a t i v e agreement with that of Dahn (1982) f o r h i s L i 1 u T i S 2 sample. Although h i s value of ac above 200K, 47X10~ 6/K, i s c o n s i d e r a b l y l a r g e r than the value of 32X10" 6/K obtained i n t h i s study, both are l a r g e compared with the values obtained f o r T i S 2 and L i T i S 2 by Whittingham and Thompson. These r e s u l t s are c o n s i s t e n t with the p r e d i c t i o n of a peak i n a Q near x=0.16 by Dahn. Rough estimates of a a and ac were a l s o obtained f o r the other low temperature samples, using only the l a t t i c e parameter data obtained near 300K and 160K, and are l i s t e d i n Table 8. Because the c - a x i s becomes l e s s s e n s i t i v e to temperature changes below roughly 150K, one can expect thermal expansion c o e f f i c i e n t s obtained i n t h i s way to be small e r than t h e i r v a l u e s f o r data above 200K, e.g. a decrease in a of 4X10" 6/K i s obtained f o r DJR18,. A l l c 1 values of a £ are approximately the same f o r a l l samples, c o n s i s t e n t l y higher than those f o r T i S 2 and L i T i S 2 ; t h i s i m p l i e s a r a t h e r broad peak i n a versus x. 85 100 200 300 TEMPERATURE (K) 100 200 300 TEMPERATURE (K) Figure 25. Variation of a- and c-axes with temperature for the DJR18 1 (x=0.13) sample. The s o l i d l i n e s are l i n e a r f i t s to the data above 200K. 86 Table 8 Approximate thermal expansion c o e f f i c i e n t s for a l l low temperature samples of L i T i S 2 . A l l values are calculated using only two data points: near 300K and 160K. a a X106/K a c X106/K D J R ^ DJR182 DJR20 DJR221 DJR222 16 18 16 16 17 28 27 30 28 30 0.13 0.13 0.20 0.15 0.15 87 5.2.3 S u p e r l a t t i c e Peak Search To determine i f a long range ordered stage two s t r u c t u r e was formed in the L i x T i S 2 samples at low temperatures, searches f o r the (009/2) s u p e r l a t t i c e peak were done using c a r e f u l x-ray step scans both at and below room temperature. At each temperature, the l a t t i c e parameters and out - o f - p l a n e displacement of the sample were determined by a step scan of at l e a s t seven Bragg peaks. T h i s enabled an a c c u r a t e c a l c u l a t i o n of the expected (009/2) peak p o s i t i o n and the p o s i t i o n s of nearby peaks. The data of Fig u r e 24 was used to c a l c u l a t e the expected peak p o s i t i o n s for T i S 2 m a t e r i a l , which i s present due to imperfect cathode u t i l i z a t i o n s i n the e l e c t r o c h e m i c a l c e l l s . The width of the (004) peak of the i n t e r c a l a t e d m a t e r i a l was a l s o monitored. The f u l l width at h a l f maximum (FWHM) was c a l c u l a t e d as twice the angular s e p a r a t i o n of the l e a d i n g peak edge at h a l f maximum and the c a l c u l a t e d peak cent e r p o s i t i o n . The temperature to which the samples can be c o o l e d before the l i t h i u m atoms are "f r o z e n i n " i s expected to be roughly 140K, as shown i n Appendix D, and i s not very s e n s i t i v e to changes i n the sample c o o l i n g r a t e . 5.2.3.1 DJR18 (X=0.13) R e l a t i v e l y l a r g e (004) peak widths (FWHM'vO . 2 0 0 ) were obtained at a l l temperatures with both DJR18 ^  and DJR18 2. T h i s i s i n c o n s i s t e n t with the value obtained by Dahn (see Figu r e 21) of roughly 0.16° (near the minimum i n the 88 f i g u r e ) . T h i s i n d i c a t e s a degradation of both DJR18 samples. Because the exposure of the DJR18 2 c e l l to a i r was very l i m i t e d , the cathode m a t e r i a l i t s e l f i s suspect. Because of the wide (004) peaks, the presence of s u p e r l a t t i c e peaks i s u n l i k e l y . F i g u r e 26 shows the r e s u l t s of scans about the expected (009/2) angular p o s i t i o n f o r DJR18 2 obtained at 300K and 167K. A 0.5K per minute c o o l i n g r a t e was used. The expected p o s i t i o n s of the (009/2) peak are 72.179° and 72.426°, r e s p e c t i v e l y . The expected i n t e n s i t y of the (004) peak i s 655000 counts on t h i s s c a l e ; the expected i n t e n s i t y of the (009/2) peak, for the same peak width, i s 13% of t h i s value, or 85000 counts. Although, there i s a l a r g e amount of s c a t t e r i n the data, the o r i g i n of which i s not known, there i s no s t a t i s t i c a l evidence f o r the presence of the (009/2) peak in e i t h e r scan. A c o o l i n g r a t e of 5K per minute was a l s o used on t h i s sample; no d i f f e r e n c e i n the x-ray i n t e n s i t i e s in the (009/2) peak v i c i n i t y was observed f o r the d i f f e r e n t c o o l i n g r a t e s . 5.2.3.2 DJR20 (x=0.20) The p o s s i b i l i t y of s t a g i n g at x values g r e a t e r than 0.16 was i n d i c a t e d i n S e c t i o n 5.1, with r e f e r e n c e to F i g u r e 23. Thus i t was c o n s i d e r e d u s e f u l to study t h i s sample at low temperatures. The DJR20 low temperature sample had l a r g e (004) peak widths (FWHM'vO. 22°) at both 300K and 166K. T h i s i s c o n s i s t e n t with the data o f Dahn (see F i g u r e 21). The r e s u l t s o f a search f o r the (009/2) s u p e r l a t t i c e to o o or CL tn o o 34000 33000 -32000 -31000 30000 -29000 71.5 **+* _!_ •* • * * « 71.7 71.9 72.1 72.3 SCATTERING ANGLE 26 (DEGREES) (a) 72. ZD o o 36000 35000 -IT) O ° 34000 h UJ CL in 33000 -32000 -31000 71.8 T T I **** J I L ;ure 26. 72.0 72.2 72.4 72.6 72.fi SCATTERING ANGLE 28 (DEGREES) (b) The (009/2) region for the DJR182 (x=0 sample at (a) 300K and (b) 167K. The v e r t i c a l arrows indicate the expected positions of the (009/2) peak. 90 peak at 300K and 166K i s shown i n F i g u r e 27. A 1K per minute c o o l i n g r a t e was used. The expected (009/2) i n t e n s i t y on t h i s s c a l e , r e f e r e n c e d to that of the (004) peak, i s 135000 counts, f o r peaks of the same width. In both scans, the main peak present i s the (102) l i n e due to the b e r y l l i u m p l a t e , with i t s Ka, and Ka 2 components c l e a r l y r e s o l v e d . In the room temperature scan, a n o t i c a b l e d i s t o r t i o n of the Ka 2 peak at 70.95° i s observed, with a c o n s i d e r a b l e i n c r e a s e i n i n t e n s i t y over i t s expected value of one-half of the Ka, component i n t e n s i t y . A secondary peak i s a l s o observed at 71.20°. The expected p o s i t i o n of the (009/2) peak i s 71.000°; the (202) peak of the i n t e r c a l a t e d m a t e r i a l i s very c l o s e to t h i s peak, at 70.956°. Because of the a n i s o t r o p y in the thermal expansion i n l a y e r e d compounds p a r a l l e l and p e r p e n d i c u l a r to the l a y e r s , as demonstrated i n S e c t i o n 5.4.2, c o o l i n g the sample w i l l separate these two peaks. T h i s i s observed in part (b) of F i g u r e 27. The d i s t o r t i o n of the b e r y l l i u m (102) Ka 2 peak has s h i f t e d r e l a t i v e to the b e r y l l i u m peak, but i s s t i l l present on i t s shoulder. T h i s d i s t o r t i o n i s once again c o i n c i d e n t with the expected (202) p o s i t i o n at 71.128°; the (009/2) peak has been s h i f t e d o f f the b e r y l l i u m peak completely to an expected p o s i t i o n of 71.298°. The small peak at 71.35° i s probably the (202) Ka 2 component. Thus i t appears that the observed d i s t o r t i o n i s due to the (202) peak and not the (009/2) peak. The problem of the presence of the b e r y l l i u m peak c o u l d be removed by using a copper s l i d e sample covered with Mylar. There are no peaks from t h i s sample c o n t a i n e r i n t h i s a ngular range. 91 140000 120000 -V) o ° 100000 C£ UJ CL m =3 O O 80000 -60000 -40000 70.5 70.7 70.9 71.1 71.3 SCATTERING ANGLE 28 (DEGREES) 71.5 (a) 150000 130000 in o ° 110000 cc - K a i •• Be (102) C L to 5 o o 90000 70000 -50000 i r Ka 2 J L i i i 1 1 r J L 70.7 70.9 71.1 71.3 71.5 SCATTERING ANGLE 28 (DEGREES) (b) Figure 27. The (009/2) region for the DJR20 (x=0.20) sample at (a) 300K and.(b) 166K. The s o l i d v e r t i c a l arrows indicate the expected positions of the (009/2) peak; the dashed v e r t i c a l arrows indicate the expected positions of the (202) peak. 92 Other l e s s o b s t r u c t e d s u p e r l a t t i c e peaks, i n p a r t i c u l a r the (007/2) peak, should be s t u d i e d f o r t h i s sample. 5.2.3.3 DJR22 (X=0.15) An i n i t i a l look at t h i s m a t e r i a l with the DJR22, sample r e v e a l e d small (004) l i n e widths of approximately 0.16° (FWHM) in agreement with the data of F i g u r e 21. The r e s u l t s of a search f o r the (009/2) s u p e r l a t t i c e peak for the DJR22, sample at 300K and 166K are shown i n F i g u r e 28. A 1K per minute c o o l i n g r a t e was used. The expected (009/2) i n t e n s i t y on t h i s s c a l e , r e f e r e n c e d to that of the (004) peak, i s 58000 counts, f o r peaks of the same width. The room temperature data shows no sig n of the (009/2) peak at 72.086° or the T i S 2 (202) peak at 71.965°. However, the low temperature data shows evidence of a broad peak at the expected p o s i t i o n of the (009/2) peak, 72.348°. The expected p o s i t i o n of the T i S 2 (202) peak has s h i f t e d to 72.089°. These r e s u l t s prompted a more d e t a i l e d study of t h i s angular region with a second sample, DJR22 2. Once again narrow (004) peaks (FWHM^0.16°) were observed at a l l temperatures with t h i s sample. F i g u r e s 29 and 30 show step scans at 300K and 165K f o r c o u n t i n g times of 40 seconds and 400 seconds per step, r e s p e c t i v e l y . At each temperature, data f o r both counting times was taken c o n s e c u t i v e l y . A c o o l i n g r a t e of 1K per minute was used. The expected (009/2)' i n t e n s i t y f o r the 40 second per s t e p data, r e f e r e n c e d to that of the (004) peak, i s 23000 counts, f o r peaks of the same width. The l a r g e peaks observed i n F i g u r e 29 are the 48000 tO § 46000 h CM 01 Ld CL to I 44000 o o i 1 r "i 1 1 r 42000 71.6 48000 trt 47000 o o CN or £ 46000 -to I -z ZD " 45000 h 44000 _L 4 T • • J L J I L 71.8 72.0 72.2 72.4 SCATTERING ANGLE 26 (DEGREES) (a) 72.I i r i r i r • J I I L I * I 71.9 72.1 72.3 72.5 72.7 SCATTERING ANGLE 20 (DEGREES) (b) gure 28. The (009/2) r e g i o n f o r the DJR22, (x=0. sample at (a) 300K and (b) 166K. The v e r t i c a l arrows i n d i c a t e the expected p o s i t i o n s o f the (009/2) peak. 94 to o •>*• or LU CL to 3 o o 17000 15000 -13000 -11000 70.8 T 1 /• (202) •+ + **• _L 71.2 71.6 72.0 72.4 SCATTERING ANGLE 29 (DEGREES) 72.8 (a ) 18000 to o or 16000 UJ to ZD 8 14000 12000 "I 1 1 1 (202) Ka, + • ~ (202) Ka2 • • • • • • _L ± Figure 29 71.2 71.6 72.0 72.4 72.8 SCATTERING ANGLE 26 (DEGREES) (b) The (009/2) region for the DJR22„ (x=0.15) sample at (a) 300K and (b) 165K for a counting time of 40 s per step. The v e r t i c a l arrows indicate the expected positions of the (009/2) peak. 95 130000 128000 -to o "+ 126000 h cc U J C L ^ 124000 -z> o o 122000 -120000 i r 145000 to g 140000 ec C L to W 135000 -o o 130000 • • — 71.6 71.8 72.0 72.2 SCATTERING ANGLE 26 (DEGREES) (a) 72.4 71.8 72.0 72.2 72.4 SCATTERING ANGLE 26 (DEGREES) 72.6 (b) Figure 30. The (009/2) region for the DJR22„ (x=0.15) sample at (a) 300K and (b) 165K for a counting time of 400 s per step. The v e r t i c a l arrows indicate the expected positions of the (009/2) peak. 9 6 Ka, and Ka 2 components of the (202) peak of the i n t e r c a l a t e d m a t e r i a l . At both temperatures, the Ka 2 component i s somewhat d i s t o r t e d . These d i s t o r t i o n s are f a r removed from the expected (009/2) peak p o s i t i o n s of 71.941°at 300K and 72.211° at 165K. The T i S 2 (202) peak i s expected at 71.874° at 300K and at 72.014° at 165K. The room temperature data does not show s t a t i s t i c a l evidence for e i t h e r the (009/2) or the T i S 2 (202) peak fo r both the 40 second and 400 second per step data. However, at 165K, a broad peak i s observed i n the v i c i n i t y of the (009/2) peak for both counting times. Attempts to f i t the (202) Ko, and Ka 2 components and the broad peak to Gaussian and L o r e n t z i a n p r o f i l e s produced poor r e s u l t s , e s p e c i a l l y i n the v i c i n i t y of the (202) Ka 2 component. To account f o r the observed i n t e n s i t i e s , other peaks must be p r e s e n t . I t i s d i f f i c u l t to i n f e r a background l e v e l f o r t h i s low temperature data from that at room temperature, s i n c e a l l of the peak p o s i t i o n s s h i f t as the temperature i s lowered. The absence of a s i m i l a r broad peak in the low temperature data f o r DJR18 2 (see F i g u r e 26) i m p l i e s that the peak i s not due to the c e l l case, but r e s u l t s from the DJR22 2 sample i t s e l f . If one assumes a s l o p i n g background, as i n d i c a t e d by the dotted l i n e i n F i g u r e 29(b), a f u l l width at h a l f maximum of roughly 0.5° i s obtained f o r the peak. The i n t e g r a t e d i n t e n s i t y of the peak i s roughly 65000 counts. T h i s i s f a r below the expected i n t e g r a t e d i n t e n s i t y of 700000 counts estimated from a comparison to the (004) peak i n t e n s i t y . 97 Because of the low q u a l i t y of the data, the broad peak cannot be unambiguously i d e n t i f i e d with any one e f f e c t . The l a r g e width of the peak i m p l i e s that the u n d e r l y i n g s t r u c t u r e must be d i s o r d e r e d . I t i s h i g h l y u n l i k e l y that u n i n t e r c a l a t e d T i S 2 m a t e r i a l present i n the sample would have t h i s degree of d i s o r d e r . I t i s p o s s i b l e that the peak i s due to i m p u r i t i e s or e l e c t r o l y t e c o i n t e r c a l a t e d m a t e r i a l ; x-ray peaks c h a r a c t e r i s t i c of c o i n t e r c a l a t e d m a t e r i a l were observed in t h i s sample. If one assumes that the peak i s due to a d i s o r d e r e d stage two s t r u c t u r e , an estimate fo r the — o s t a g i n g c o r r e l a t i o n l e n g t h (see equation 5.3) of fcc'WOA i s o b t a i n e d . Further s t u d i e s of the temperature dependence of t h i s peak and the search f o r other s u p e r l a t t i c e peaks would have to be done to determine i f the staged s t r u c t u r e e x i s t s . 98 CHAPTER SIX CONCLUSION 6. 1 Summary The low temperature x-ray powder d i f f r a c t i o n attachment d e s c r i b e d i n Chapter Two, when mounted on the v e r t i c a l goniometer of a d i f f T a c t o m e t e r , was shown to give d i f f r a c t i o n p a t t e r n s comparable to those obtained with the goniometer i t s e l f . Accurate c h a r a c t e r i z a t i o n of the t h e t a -c o l d f i n g e r s h a f t geometry by l e a s t squares f i t s to measured x-ray Bragg peaks a l l o w s d i r e c t comparison of r e s u l t s o btained on these two s h a f t s . System v i b r a t i o n s and remountings of the attachment were shown to c o n t r i b u t e only minor d i s t o r t i o n s and s h i f t s to the measured peaks. The sample temperatures were measured as a f u n c t i o n of the c o l d f i n g e r temperature f o r v a r i o u s s u b s t r a t e s used i n low temperature s t u d i e s . Measurements of the l a t t i c e parameters of 1T-TaS 2 between 300K and 130K were performed. The NC-C charge d e n s i t y wave phase t r a n s i t i o n was observed by the corre s p o n d i n g l a t t i c e parameter d i s c o n t i n u i t i e s obtained on both c o o l i n g and h e a t i n g through the t r a n s i t i o n . Measurements of these d i s c o n t i n u i t i e s are judged to be more accurate than those of previous measurements (Givens and F r e d e r i c k s 1977, Sezerman et a l . 1980), p a r t i c u l a r l y f o r the a - a x i s measurement. A thermal h y s t e r e s i s of 46K was observed on c o o l i n g and h e a t i n g through t h i s t r a n s i t i o n , i n agreement 99 with previous r e s u l t s . A p o s s i b l e e x p l a n a t i o n f o r the observed degradation of the room temperature charge d e n s i t y wave s a t e l l i t e peaks in terms of sulphur r e l e a s e from the compound was proposed. E l e c t r o c h e m i c a l l y prepared L i v T i S 2 samples were s t u d i e d at room temperature and below. Thermal expansion measurements of T i S 2 gave roughly q u a n t i t a t i v e agreement with the r e s u l t s of Whittingham and Thompson (1975). Thermal expansion measurements of L i . 1 3 T i S 2 were c o n s i s t e n t with the measurements and c a l c u l a t i o n s of Dahn (1982). A search was performed f o r the (009/2) s u p e r l a t t i c e peak, i n d i c a t i v e of a stage two s t r u c t u r e , i n samples with x near 0.16. A broad peak near the expected (009/2) angular p o s i t i o n was observed in a sample with X=0.15 at T=165K but not at room temperature. The exact nature of t h i s peak was not determined. 6.2 Suggestions f o r Further Work The low temperature x-ray d i f f r a c t i o n attachment can be used to study the s t r u c t u r e of powdered c r y s t a l samples of any m a t e r i a l at temperatures below ambient. The use of the low temperature x-ray c e l l i n c o n j u n c t i o n with the attachment allows the study of a i r - s e n s i t i v e m a t e r i a l s . A r e l i a b l e s e a l f o r t h i s c e l l should be developed. The phase diagrams (T versus x) f o r i n t e r c a l a t i o n systems c o u l d be "mapped out" by v a r y i n g T and x. Because x can only be v a r i e d at room temperature, the sample would be t h e r m a l l y c y c l e d many times. However, measurements made with 100 a s i n g l e sample over the e n t i r e range of the phase diagram would g i v e r e p e a t a b l e and r e l i a b l e r e s u l t s . By lowering the c e l l temperature, the l i t h i u m s a l t may c r y s t a l l i z e out of the e l e c t r o l y t e s o l u t i o n . Because of slow d i s s o l u t i o n of the s a l t back i n t o the e l e c t r o l y t e at room temperature, the charge t r a n s p o r t through the e l e c t r o l y t e may be l i m i t e d a f t e r numerous thermal c y c l e s . Measurements of t h i s e f f e c t should be performed. Furt h e r searches should be made for low temperature s t a g i n g i n L i T i S 2 near x=0.l6 using the low temperature attachment; the o b s e r v a t i o n of s e v e r a l s u p e r l a t t i c e peaks would be strong evidence f o r s t a g i n g . The r e l e a s e of sulphur from 1T-TaS 2 should be s t u d i e d as the exposure of the c r y s t a l s to a i r i s v a r i e d . Storage of the c r y s t a l s i n an i n e r t atmosphere and use of the x-ray d i f f r a c t i o n c e l l f o r s t r u c t u r a l measurements would allow c a r e f u l c o n t r o l of t h e i r exposure to a i r . To f a c i l i t a t e and improve the accuracy of low temperature experiments, the temperature c o n t r o l l e r should be m o d i f i e d to allow one to sweep the temperature at a user-d e f i n e d r a t e . 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(1984) Rev. S c i . Instrum. 55, 1590. Rudman, R. (1976) Low-Temperature X-ray D i f f r a c t i o n , Plenum. S a v i t s k y , A. and Golay, M.J.E. (1964) Anal. Chem. 36, 1627. Scholz, G.A. and F r i n d t , R.F. (1980) Mater. Res. B u l l . J_5, 1 703. Scruby, C.B.; W i l l i a m s , P.M. and Parry, G.S. (1975) P h i l . Mag. 3J_' 2 5 5 -Sezerman, 0.; Simpson, A.M. and J e r i c h o , M.H. (1980) S o l i d State Commun. 36, 737. Skel t o n , E.F.; Webb, A.W.; Quadri, S.B.; Wolf, S.A.; Lacoe, R.C; Feldman, J.L.; Elam, W.T.; C a r p e n t e r , J r . , E.R. and Huang, C Y . (1984) Rev. S c i . Instrum. 55, 849. Sparks, L.L. and Powell, R.L. (1972) J . Res. N a t l . Bur. Stand. A76, 263. Sp i n o l o , G.; M a s s a r o t t i , V. and Campari, G. (1979) J . Phys. E 12, 1059. Tanda, S.; Sambongi, T.; T a n i , T. and Tanaka, S. (1984) J . Phys. Soc. Jap. 5_3, 476. Thompson, A.H.; Gamble, F.R. and R e v e l l i , J.F. (1971) S o l i d S t ate Commun. 9, 981. 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APPENDIX A 107 REFRIGERATION CYCLE DESCRIPTION In F i g u r e 31 i s shown a d e t a i l e d diagram of the r e f r i g e r a t i o n u n i t . I t i s c h a r a c t e r i z e d by a p i s t o n - c y l i n d e r arrangement. The c y l i n d e r can be separated i n t o three r e g i o n s : a c o l d head volume at the bottom, a warm head volume i n the middle, and a space at the top of the c y l i n d e r l e a d i n g to a surge volume. The three regions are separated by p i s t o n s with t i g h t s e a l s between each p i s t o n s ' outer diameter and the inner c y l i n d e r w a l l . Gas flow i s allowed between the warm and c o l d head volumes through the lower p i s t o n . T h i s p i s t o n i s more p r o p e r l y termed a d i s p l a c e r because i t moves helium gas between the two volumes with l i t t l e pressure d i f f e r e n c e a c r o s s i t s l e n g t h . I t c o n t a i n s a regenerator which i s a s i n g l e channel heat exchanger through which the gas flows; i t accepts heat from gas at a higher temperature and r e j e c t s heat to gas at a lower temperature. The warm head-surge volume p i s t o n w i l l be c a l l e d the d i s p l a c e r cap; i t moves the d i s p l a c e r d u r i n g the op e r a t i o n of the r e f r i g e r a t o r . Consider the s t e a d y - s t a t e o p e r a t i o n of t h i s r e f r i g e r a t i o n system. I n i t i a l l y , the d i s p l a c e r i s l o c a t e d at the bottom of the c y l i n d e r (see F i g u r e 32a), with helium gas at low pressure p^ w i t h i n the warm head volume. Helium gas at an intermediate pressure p i i s present i n the volume above the d i s p l a c e r cap. 108 SURGE VOLUME ORIFICE DISPLACER CAP HIGH/LOW PRESSURE INLET PRESSURE SEAL DISPLACER PRESSURE SEAL WARM HEAD VOLUME REGENERATOR COLD HEAD VOLUME Figure 31. Cross-sectional diagram of the r e f r i g e r a t i o n unit. 109 Figure 32. Cycle diagram of the r e f r i g e r a t i o n unit. The arrows in parts (b) and (d) indicate the gas flow directions. 110 The high pressure v a l v e i s then opened, r a i s i n g the warm head pressure from r j ^ to p^, the hig h p r e s s u r e . T h i s i n c r e a s e i n pressure f o r c e s gas i n t o the regenerator. Because of the pressure d i f f e r e n c e between the warm head volume and the volume above the d i s p l a c e r cap, the d i s p l a c e r cap and the attached d i s p l a c e r are moved upward toward the surge volume (see F i g u r e 32b). T h i s movement f o r c e s gas through the regenerator to the c o l d head volume; the gas c o o l s as i t passes through the regenerator, d e c r e a s i n g the gas volume, and more gas i s drawn i n t o the warm head volume at constant p r e s s u r e . The gas present i n the volume above the d i s p l a c e r cap i s compressed during t h i s motion of the d i s p l a c e r , while gas bleeds at a constant r a t e through an o r i f i c e i n t o the surge volume. J u s t before the d i s p l a c e r reaches the top of i t s c y c l e (see F i g u r e 32c), the high pressure v a l v e i s c l o s e d and the low pressure valve i s opened. T h i s causes the warm and c o l d head volume pressures to f a l l from p^ to p-^. T h i s i s accompanied by an expansion of gas w i t h i n these volumes i n t o the low pressure l i n e . The gas i n the c o l d head volume flows back through the regenerator, thus p r e c o o l i n g the regenerator f o r the next c y c l e . T h i s r e d u c t i o n in the e x t e r n a l l i n e pressure r e s u l t s i n a l a r g e pressure d i f f e r e n c e between the warm head volume and the volume above the d i s p l a c e r cap. T h i s pushes the d i s p l a c e r cap and d i s p l a c e r downward and the r e s i d u a l expanded gas i n the c o l d head volume i s forc e d through the regenerator (see F i g u r e 32d) . The low pressure v a l v e i s shut j u s t before the I l l d i s p l a c e r reaches the c y l i n d e r bottom, and the c y c l e i s repeated. The usual method of d e s c r i b i n g a r e f r i g e r a t i o n c y c l e , that of a temperature-entropy (T-S) diagram, i s not u s e f u l f o r t h i s c y c l e . A d i f f e r e n t T-S diagram i s needed f o r each small f r a c t i o n of gas i n the system ( G i f f o r d 1966) because the motion of the d i s p l a c e r causes gas to be t r a n s f e r r e d i n t o and out of the warm head volume at d i f f e r e n t times i n the c y c l e . T h i s r e f r i g e r a t i o n c y c l e i s a v a r i a t i o n of an o r i g i n a l design by Solvay, who i n 1887 c o n s t r u c t e d a r e f r i g e r a t o r u s i ng a r e g e n e r a t i v e p i s t o n - c y l i n d e r arrangement ( C o l l i n s and Cannaday 1958). H i s design has s i n c e been improved by G i f f o r d and McMahon ( G i f f o r d and McMahon 1959, G i f f o r d 1960, McMahon and G i f f o r d 1960, G i f f o r d 1966) and Longsworth (1971a,b). Longsworth's design i s used i n the D i s p l e x system. In the Longsworth c y c l e , the major improvement over the Gifford-McMahon c y c l e i s that the d i s p l a c e r movement i s c o n t r o l l e d by v a r i a t i o n s in gas pressure across the d i s p l a c e r cap, as d i s c u s s e d above. T h i s s i m p l i f i e s the mechanical design and improves the c y c l e t i ming over that of G i f f o r d and McMahon, who use a motor-driven d i s p l a c e r . I n c o r p o r a t i o n of the surge volume o r i f i c e i n t o the design a l s o p r o v i d e s more e f f e c t i v e damping of the d i s p l a c e r motion near the extrema of i t s displacement, but i t complicates the thermodynamics of the c y c l e because of the accompanying p r o d u c t i o n of heat. The c o e f f i c i e n t o f performance 0 o f a r e f r i g e r a t o r i s 112 d e f i n e d as the r a t i o of the heat removed Q-^  from the load at temperature T-^  to the work performed W to remove t h i s heat, i . e . (5 = — = — , A . l w Q h - Q x where i s the amount of heat r e j e c t e d to a body at high temperature . For the i d e a l Carnot c y c l e , one o b t a i n s (Van Wylen and Sonntag 1978) T l 6 = — . A.2 T - T . lh 1 The " e f f i c i e n c y " of a r e f r i g e r a t o r i s u s u a l l y d e f i n e d with r e s p e c t to the Carnot c y c l e as the r a t i o of i d e a l work to a c t u a l work. T y p i c a l e f f i c i e n c i e s f o r the m o d i f i e d S o l v a y - c y c l e r e f r i g e r a t o r s are between 3% and 5% (Longsworth 1971b, O'Hanlon 1980). 113 APPENDIX B A SIMPLIFIED LEAST SQUARES PEAK FITTING PROCEDURE As d i s c u s s e d i n Chapter Two, the l a t t i c e parameters of powdered c r y s t a l samples are obtained u s i n g the measured Bragg angles and t h e i r corresponding M i l l e r i n d i c e s . Thus i t i s necessary that one be able to o b t a i n a c c u r a t e values of the ce n t e r of each measured Bragg peak. To accomplish t h i s , the peak f i t t i n g a l g o r i t h m of S a v i t s k y and Golay (1964), a l e a s t squares procedure, was used. In the l e a s t squares procedure a set of 2m+1 data p o i n t s y^ i s f i t to a polynomial of degree n (n<2m+1) of the form n k f. = I b . l . B. 1 1 k=0 n k The b ^'s are c a l c u l a t e d by minimizing the sum of the squares of the d i f f e r e n c e between the computed values f a n d the measured values y , i . e . m 2 3b i=-m nr . 1 ^ i ' 5 ^ = 0 ; r = 0 ' • ' • * n B.2 E q u i v a l e n t l y , m _ n , I t f o l l o w s that 114 n m , , m E b , E i = E 1 y. B.4 k=0 n R i=-m i=-m 1 or where and n E k=0 1 b n k S r + k F r , B.5 m r+k S ,, = E i k B .6 r+k i=-m m F = E i r y . . B.7 r J I i=-m The S r + k terms can be c a l c u l a t e d d i r e c t l y from m and r+k (S . ,=0 f o r a l l odd valu e s of r+k). The F terms are r+k r expressed i n terms of the y^. Hence, one o b t a i n s two se t s of simultaneous equations f o r the b . (one f o r even values of nk k, the other for odd v a l u e s of k), which can be s o l v e d for each of the b n k_ in terms of the measured y^ v a l u e s . One o b t a i n s , i n each case, an equation of the form m b 1 = izzIB , B.8 nk where the c ^ ^ ' s form s e t s of i n t e g e r s and the N n k ' s are the corresp o n d i n g n o r m a l i z i n g f a c t o r s ( a l s o i n t e g e r ) . In 115 g e n e r a l , a d i f f e r e n t set of c ^ n ^ ' s and must be used f o r each b n ^ ; however, f o r n and k both even or both odd, bnk = bn-rl,k-To f a c i l i t a t e the c a l c u l a t i o n of the b ^ c o e f f i c i e n t s , S a v i t s k y and Golay have compiled t a b l e s of the c ^ n ^ i n t e g e r s and N f o r d i f f e r e n t values of n and k (n determines the nk order of the f i t t i n g polynomial f^ and k determines the p a r t i c u l a r term i n f ^ (see equation B.1)), using from 5 to 25 measured data v a l u e s . T h e i r t a b l e s are a c t u a l l y used to i c a l c u l a t e a . c o e f f i c i e n t s , where nk a k f . ank " < - T * > i - 0 " k l b n k > B - 9 di one must d i v i d e the a values by k! to o b t a i n the d e s i r e d nk b . v a l u e s . nk I f one chooses c l o s e l y spaced data p o i n t s i n a small range about the top of a smooth peak, one can a c c u r a t e l y d e s c r i b e the data p o i n t s with a q u a d r a t i c polynomial of the form f . = b o n + b 0 , i + b 0 0 i 2 B.10 l 20 2 1 22 The peak cent e r i s given by the value of i corresponding to a zero value of d f ^ / d i , i . e . i ' - ^ 2 1 B . l l 2 b „ center 22 S i m i l a r r e s u l t s w i l l be obtained f o r subsets of 2m+1 116 data p o i n t s chosen immediately about the peak ce n t e r , but for good r e p r o d u c i b i l i t y , a c r i t e r i o n i s necessary f o r choosing a p a r t i c u l a r subset. We have noted that a qu a d r a t i c polynomial f i t t e d to an x-ray Bragg peak using only data p o i n t s about the peak cent e r y i e l d s a peak p r o f i l e which i s narrower than the measured peak p r o f i l e , as i l l u s t r a t e d in F i g u r e s 33 and 34. T h i s means that f i t t i n g polynomials c o n s t r u c t e d from data p o i n t s d i s p l a c e d from the peak center have l e s s c u r v a t u r e than those c o n s t r u c t e d from data p o i n t s about the c e n t e r . Thus, we have decided to choose the subset corresponding to the minimum second d e r i v a t i v e of the c a l c u l a t e d polynomials, i n d i c a t i v e of maximum c u r v a t u r e . For a q u a d r a t i c polynomial, d 2 f ^ / d i 2 i s p r o p o r t i o n a l to b 2 2> which i s a l r e a d y c a l c u l a t e d i n the f i t . T h i s c r i t e r i o n i s best f o r i d e a l , symmetric Bragg peaks. For assymmetric peaks such as those obtained at low Bragg angles because of ins t r u m e n t a l f a c t o r s , the' true peak center i s not obtained by the above procedure. However, t h i s e r r o r w i l l not a f f e c t changes i n peak p o s i t i o n s unless the peak shape changes. The computer program PEAKFIT.S i n UBC user account VOLT uses t h i s a l g o r i t h m to c a l c u l a t e the center p o s i t i o n s of Bragg peaks measured i n an x-ray scan. Quadratic polynomials are f i t to subsets of nine p o i n t s from the measured values, with the peak center c a l c u l a t e d from the subset with the minimum second d e r i v a t i v e of the polynomials. Because subsets of only nine p o i n t s are used i n the f i t , data need be measured only about the top of each peak. G r a p h i c a l methods of o b t a i n i n g the peak c e n t e r , e.g. f i n d i n g 117 the c e n t r o i d of the peak, r e q u i r e one to measure i n t e n s i t y values over the e n t i r e width of the peak. Hence, the above peak f i t t i n g procedure a l l o w s one to ob t a i n the d e s i r e d i n f o r m a t i o n with fewer measurements. As input, PEAKFIT.S r e q u i r e s the spacing of the data p o i n t s w i t h i n each peak, the number of peaks, and the measured-angular values and t h e i r corresponding i n t e n s i t i e s . As output, i t returns the peak center p o s i t i o n f o r each peak 26, and the corresponding c a l c u l a t e d i n t e n s i t y v a l u e s , as w e l l as a d d i t i o n a l i n f o r m a t i o n to enable one to g r a p h i c a l l y compare the measured and f i t t e d v a l u e s . In F i g u r e s 33 and 34, the measured and c a l c u l a t e d i n t e n s i t y values are shown f o r two d i f f e r e n t peaks. The peak c e n t e r s , as determined by PEAKFIT.S, are i n d i c a t e d by the v e r t i c a l l i n e s . F i g u r e 33 was obtained from a c a r e f u l scan of the (311) peak of a s i l i c o n sample, u s i n g a counting time of 100 seconds per point and an angular step s i z e of 0.01°. The peak i s 350000 counts above background and the peak center p o s i t i o n i s c a l c u l a t e d to be 56.1410° (a measure of the e r r o r of t h i s q u a n t i t y i s given below). The s c a t t e r i n the measured values i s very s m a l l and the c a l c u l a t e d q u a d r a t i c f u n c t i o n , the continuous curve i n the f i g u r e , f i t s the measured values w e l l . I t i s worthwhile to note that d e v i a t i o n s between the measured and c a l c u l a t e d values are observed immediately o u t s i d e the nine p o i n t i n t e r v a l used to c a l c u l a t e the f i t . T h i s i l l u s t r a t e s that the shape of Bragg peaks cannot be a c c u r a t e l y d e s c r i b e d by t h i s simple f u n c t i o n 4 0 0 0 0 0 3 0 0 0 0 0 2 0 0 0 0 0 100000 -I I I I 1 1 1 1 1 1 1 ~ / 1 V ++++* — / \ + + — \ + + + I + + \ + + 1 \ + + / I + +1 + — + + + + + + + + 1 1 1 1 i i i i i i i 0 55.90 56.00 56.10 56.20 5 6 3 0 5 6 . 4 0 5 6 . 5 0 SCATTERING A N G L E 26 ( D E G R E E S ) Figure 33. Comparison of actual and f i t t e d (311) s i l i c o n peak. A counting 100 s per step was used. data time for of 2 0 0 0 0 15000 10000 5 0 0 0 6 2 5 0 62.70 62.90 63.10 6 3 . 3 0 SCATTERING A N G L E 26 ( D E G R E E S ) 6 3 . 5 0 Figure 34. Comparison of a c t u a l and f i t t e d data f o r (004) peak of L i j ~ T i S 2 s a m p l e - A counting time o f 20 s per'step was used. 120 over more than a small angular range. F i g u r e 34 was obtained from a room temperature scan of the (004) peak of a L i . 1 3 T i S 2 sample, obtained from the e l e c t r o c h e m i c a l c e l l DJR18 (see Chapter F i v e ) . A counting time of 20 seconds per po i n t and an angular step s i z e of 0.01° were used. The peak i s 12500 counts above background; t h i s i s t y p i c a l of peaks observed i n most x-ray s t u d i e s . Because of the s h o r t e r counting time, there i s more s c a t t e r in the measured p o i n t s , but the c a l c u l a t e d q u a d r a t i c f u n c t i o n f i t s the data reasonably w e l l near the peak c e n t e r . Both s t a t i s t i c a l and systematic e r r o r s are present i n the peak center c a l c u l a t i o n s . To o b t a i n an estimate of the combined e r r o r , repeated step scans of the (311) peak of the s i l i c o n sample were performed. A step s i z e of 0.01° and a cou n t i n g time of 20 seconds per p o i n t was used. A t y p i c a l peak was 70000 counts above background. For f i f t y measurements of t h i s peak, the standard d e v i a t i o n of the peak cent e r p o s i t i o n s was a=0.00090°. F i f t y measurements made with twice the counting time gave a=0.00065°. These v a l u e s are comparable to those of Dahn and McKinnon (1984) who obtained v a l u e s of o=0.00060° and a=0.00044°, r e s p e c t i v e l y , f o r a peak which was only 15000 counts above background. T h e i r use of twice as many data p o i n t s i n the f i t and a step s i z e smaller by a f a c t o r of two p o s s i b l y e x p l a i n s t h e i r b e t t e r r e s u l t s . 1 2 1 APPENDIX C ROOM TEMPERATURE CHARGE DENSITY WAVE STRUCTURE OF 1T~TaS 2 The s a t e l l i t e nature of the d i f f r a c t i o n peaks formed by the presence of a charge d e n s i t y wave (see equation 4.1) i s most c l e a r l y seen i n r e c i p r o c a l space. F i g u r e 35 shows the room temperature r e c i p r o c a l l a t t i c e of 1T-TaS 2, as determined from the data of Scruby et a l . (1975). The main l a t t i c e d i f f r a c t i o n peaks form a hexagonal l a t t i c e with b a s i s v e c t o r s b,, b 2 and b 3 . The f i r s t order s u p e r l a t t i c e d i f f r a c t i o n peaks are t r i g o n a l l y disposed about each main l a t t i c e peak. Higher order s a t e l l i t e s are not i n d i c a t e d i n the f i g u r e . At room temperature, q 01=0.245b,+0.068b 2; the angle between q 0 , and the r e c i p r o c a l l a t t i c e v e c t o r b, i s 11.9°. Scruby et a l . have suggested a convenient indexing scheme f o r the s a t e l l i t e peaks. Each s a t e l l i t e peak i s l a b e l l e d by i t s corresponding main l a t t i c e peak with M i l l e r i n d i c e s ( h k l ) . I t i s f u r t h e r s p e c i f i e d by i n d i c e s ( h ' k ' l ' ) r e f e r r e d to the coo r d i n a t e system determined by the charge d e n s i t y wavevectors q 0 1 and q 0 2 , as we l l as b 3 , such that nq 0 i n equation 4.1 i s given by nq Q = h ' q 0 1 - k ' q 0 2 + 31'^ . C.l T h e r e f o r e , the wavevector of each s a t e l l i t e peak can be s p e c i f i e d as 122 F i g u r e 35. R e c i p r o c a l space geometry of main l a t t i c e and f i r s t order charge d e n s i t y wave s a t e l l i t e peaks f o r room temperature lT-TaS2" R e c i p r o c a l l a t t i c e v e c t o r s and b_2 and charge d e n s i t y wavevectors q n 1 and q n 9 are shown. 123 k = h b 1 + kb 2 + h ' q 0 1 -k'q 0 2 + (1 + 31')b 3 C.2 For powder d i f f r a c t i o n experiments, we are i n t e r e s t e d i n twice the Bragg angle 2 0 , which i s obtained from the wavevector k by the r e l a t i o n - l M k | 29 = 2 s i n 1 (——-) , C.3 where X i s the wavelength of the r a d i a t i o n . Table 4 l i s t s the r e s u l t s of indexing the observed room temperature s a t e l l i t e peaks i n 1 T-TaS 2 to t h i s scheme. 124 APPENDIX D ON THE RETENTION OF HIGH TEMPERATURE STRUCTURE UPON COOLING In the temperature v a r i a t i o n s t u d i e s d e s c r i b e d i n Chapter F i v e , i t i s important that the s t a t e of the m a t e r i a l at a given temperature corresponds to the e q u i l i b r i u m s t a t e at that temperature. S t u d i e s of quenched and annealed a l l o y s (Bragg and Wi l l i a m s 1934) show that t h i s i s not true at a l l temperatures. If one c o o l s the m a t e r i a l f a s t enough, one can "f r e e z e i n " a s t r u c t u r e c h a r a c t e r i s t i c of a higher temperature. The s t r u c t u r e of the m a t e r i a l i s c h a r a c t e r i s t i c of e q u i l i b r i u m at temperature 6, where 0 i s g r e a t e r than or equal to the a c t u a l temperature T of the m a t e r i a l . If 0>T, with T f i x e d , the s t r u c t u r e r e l a x e s toward that c h a r a c t e r i s t i c of e q u i l i b r i u m at temperature T ac c o r d i n g to the r e l a t i o n de = -Ce-T) D 1 dt x where T i s the r e l a x a t i o n time of the m a t e r i a l . I f the exchange of atoms between s i t e s i n the m a t e r i a l can be d e s c r i b e d as a hopping process, the r e l a x a t i o n time can be w r i t t e n as 125 T-= A exp(W/kT) , D.2 where A i s e f f e c t i v e l y a constant ( u s u a l l y of the order of 10" 1 2 ) and W i s the a c t i v a t i o n energy r e q u i r e d to surmount the p o t e n t i a l b a r r i e r between two s i t e s . Bragg and W i l l i a m s f i n d i t u s e f u l to d e f i n e a c h a r a c t e r i s t i c temperature T, such that the r e l a x a t i o n time i s one second, i . e . r , = 1=Aexp(W/kT,). At high temperatures, T i s very small and 8-T i s s m a l l , so that the s t r u c t u r e of the m a t e r i a l corresponds c l o s e l y to that of the a c t u a l temperature of the m a t e r i a l . As T i s lowered toward zero, at some temperature, 60, r becomes so l a r g e that the s t r u c t u r e of the m a t e r i a l i s e f f e c t i v e l y " f r o z e n " in a s t a t e corresponding to t h i s temperature. Suppose that the m a t e r i a l i s c o o l e d at a r a t e a such that dT/dt=-o. Using t h i s equation and equation D .1, we can express the d i f f e r e n c e between 6 and T as 0_T = O-T^ . D.3 dT i t may be noted that the r e l a t i o n 0=T+aT, which agrees with equation D.3 at high temperatures, has i t s minimum with respect to temperature at a temperature very c l o s e to 60. T h i s i s because f o r T j u s t above 8Qt although 0-T i s f i n i t e , the slope of 8 versus T i s s t i l l roughly u n i t y . At t h i s minimum, d0/dT=O, which g i v e s dr/dT=-1/a. Using t h i s 126 equation and equation D.2, we can w r i t e 60 i n terms of o, A and T, as - T l e = * =j =j— • D.4 ° l + ( l n C e o Z / a T 1 l n A " 1 ) / l n A " i ) Since A" 1 i s of the order of 1 0 1 2 , the denominator of the bracketed term i s c o n s i d e r a b l y l a r g e r than the numerator of that term. T h i s means that 60 i s not very s e n s i t i v e to changes in the c o o l i n g r a t e o. Bragg and W i l l i a m s show that i t i s i m p o s s i b l e to quench so r a p i d l y as to preserve the m a t e r i a l i n a s t a t e f o r which f9 0>1.2T,, or anneal so slowly as to r e a l i z e the s t a t e f o r which (9 o<0.7T 1. The motion of l i t h i u m i n L i T i S 2 has been measured by K l e i n b e r g (1982) using 7 L i NMR techniques; f o r L i 3 3 T ' i S 2 he found T=(1.9X10" 1 1s)exp(3370K/T) and f o r L i . 7 0 T i S 2 he found T=(4.9X10" 1 1s)exp(3370K/T). For x=0.33, we f i n d that the c h a r a c t e r i s t i c temperature i s T 1=137K. Assuming that the values of A and W/k f o r x values near 0.16 are not that d i f f e r e n t from those f o r x=0.33, t h i s simple theory t e l l s us that the samples can be c o o l e d at reasonable r a t e s to temperatures below 180K before one has to worry about quenching i n a s t r u c t u r e c h a r a c t e r i s t i c of a higher temperature. Temperatures below 100K w i l l e x h i b i t l i t h i u m s t r u c t u r e corresponding to temperatures above 100K. T h i s i s c o n s i s t e n t with the neutron d i f f r a c t i o n data of Dahn (1982) i n which he found that the (003/2) s u p e r l a t t i c e peak at 100K d i d not sharpen when the temperature was lowered to 17K. 

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