UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Lithium intercalation in titanium disulfide Dahn, Jeffery Raymond 1980

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1980_A6_7 D34.pdf [ 4.27MB ]
Metadata
JSON: 831-1.0085282.json
JSON-LD: 831-1.0085282-ld.json
RDF/XML (Pretty): 831-1.0085282-rdf.xml
RDF/JSON: 831-1.0085282-rdf.json
Turtle: 831-1.0085282-turtle.txt
N-Triples: 831-1.0085282-rdf-ntriples.txt
Original Record: 831-1.0085282-source.json
Full Text
831-1.0085282-fulltext.txt
Citation
831-1.0085282.ris

Full Text

LITHIUM INTERCALATION IN TITANIUM DISULFIDE by JEFFERY RAYMOND DAHN B.Sc. Hon. Dalhousie Un i v e r s i t y , 1978 A-THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of Physics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1980 © JEFFERY RAYMOND DAHN, 1980 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V6T 1W5 D a t e DE-6 BP 75-51 I E ABSTRACT The behaviour of Li/LiC10.,PC/Li TiS„ electrochemical c e l l s has been 4 x 2 investigated. E l e c t r o l y t e i n s e r t i o n ( i r r e v e r s i b l e c o - i n t e r c a l a t i o n ) occured i n some L i x T i S 2 cathodes and was found to be dependent on the c r y s t a l l i t e s i z e of the cathode powder. In cases where el e c t r o l y t e , i n s e r t i o n was small, the voltage as a function of state of discharge was found to agree with that reported by Thompson(1978). A neutron d i f f r a c t i o n study on L i TiS„ with x = 0, .12, .33, .66 and x 2 1.0 was performed. The l o c a t i o n of the l i t h i u m atoms i n the octahedral s i t e s of the host HS2 l a t t i c e i s established. The d e t a i l e d c r y s t a l s t r u c t -ure of Li^TiS2 i s reported. No evidence for l i t h i u m ordering i s observed i n any of the samples at room temperature. S u p e r l a t t i c e peaks were not detected at 106K f o r the L i 23T1S2 sample. The existence of three dimen-s i o n a l l i t h i u m ordering can be excluded by the data. TABLE OF CONTENTS i i i Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES .v LIST OF FIGURES v i ACKNOWLEDGEMENTS v i i i II INTRODUCTION 1 I I . CRYSTAL STRUCTURE OF T i S 0 AND L i T i S 0 4 2 x 2 2.1 TiS 4 2.2 L i TiS„ 9 x I . . I I I . L i / L i TiS„ ELECTROCHEMICAL CELLS 12 x 2 3.1 Introduction 12 3.2 Preparation of T i S 2 14 3.3 Cathode preparation 15 3.4 E l e c t r o l y t e preparation ,• . 16 3.5 Separators 16 3.6 Electrochemical c e l l s 17 3.7 C e l l c y c l i n g techniques 17 3.8 Experimental r e s u l t s and discussion 20 3.8.1 TIS^ with an average c r y s t a l l i t e diameter of 40 um 22 3.8.2 TIS^ with an average c r y s t a l l i t e diameter of 15 um 24 3.8.3 T i S 2 with an average c r y s t a l l i t e diameter of 2 um 27 3.9 Summary 30 IV. THEORY OF NEUTRON DIFFRACTION 31 V. THE NEUTRON SPECTROMETER 38 VI. THE NEUTRON DIFFRACTION EXPERIMENT 43 i v Page 6.1 Introduction . 43 6.2 The "dynamic" neutron d i f f r a c t i o n study .... 43 6.3 The " s t a t i c " neutron d i f f r a c t i o n study 48 6.3.1 Sample preparation 50 6.3.2 Preferred o r i e n t a t i o n 53 6.3.3 T i S 2 54 6.3.4 L i 1 T i S 2 56 6.3.5 L i TiS„ 62 x 2 VII. CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK . 74 BIBLIOGRAPHY 75 APPENDICES 77 I. Incoherent neutron scattering and sca t t e r i n g from a randomly occupied l a t t i c e 77 I I . The Debye-Waller factor 81 I I I . Absorption and background of L i ^ T i S ^ samples . 83 IV. The Rietveld p r o f i l e refinement program 86 V. Sup e r l a t t i c e e f f e c t s 91 LIST OF TABLES Table Page I. The x-ray d i f f r a c t i o n p r o f i l e of the cathode 25 material of c e l l JD-63 I I . The x-ray d i f f r a c t i o n p r o f i l e of the cathode 28 material of c e l l JD-57 I I I . Least squares parameters f o r HS2 55 IV. Least squares parameters f o r Li^TiS2 61 V. b.(p) versus 9 94 v i LIST OF FIGURES Figure Page 1. Schematic representation of a L i / L i TiS„ c e l l 2 x 2 2. The "sandwich", structure of T i S 2 4 3. The atomic structure of HS2 5 4. The van der Waals gap of TiS^ 6 5. Proposed guest atom l o c a t i o n i n TiS^ 8 6. Dependence of l a t t i c e parameters of L i TiS„ on x 10 7. The quasi-open c i r c u i t voltage of a L i / L i TiS c e l l 13 8. The s u p e r l a t t i c e decomposition of the octahedral 15 s i t e s i n a s i n g l e van der Waals gap 9. Exploded view of an electrochemical c e l l 18 10. Several cycles of a Li/LiClO^,PC/Ni c e l l 21 11. The V(x) curve of c e l l JD-49 22 12. An'inverse d e r i v a t i v e .curve of c e l l JD-40 23 13. The constant current c y c l i n g behaviour of c e l l JD-63 26 14. The inverse d e r i v a t i v e curve of c e l l RM-12 29 15. A s i n g l e Debye-Scherrer cone for scattering from a 36 powder 16. A l t e r n a t i v e forms of the specimen f o r the ... 37 Debye-Scherrer method 17. The major components of the C5 spectrometer 39 18. Relative abundance of neutrons as a function of 40 t h e i r wavelength 19. Exploded cut away view of the dynamic neutron d i f f - 44 r a c t i o n c e l l 20. Observed room temperature neutron d i f f r a c t i o n p r o f i l e 45 fo r the f u l l y charged c e l l 21. C i c u i t used to hold the c e l l at 2.1 v o l t s 46 V X 1 Figure Page 22. Current measured while holding the c e l l at 2.1 v o l t s 47 23. Observed neutron d i f f r a c t i o n p r o f i l e of the c e l l at 49 2.1 v o l t s 24. Aluminum sample holder 52 25. Neutron d i f f r a c t i o n p r o f i l e of TiS2 55 26. Preferred o r i e n t a t i o n measurement on Li^TiS2 57 27. Observed and calculated room temperature neutron 58 d i f f r a c t i o n p r o f i l e s f o r L i ^ T i S ^ 2 28. x a n d the nuclear R factor versus percentage of 60 l i t h i u m occupying tetrahedral s i t e s i n Li^TiS2 29. Observed room temperature neutron d i f f r a c t i o n pro- 63,64 f i l e s f o r L i 1 0 T i S „ , L i 0 0TiS„ and L i ,,TiS„ . lz Z . j j z . oo z 30. V a r i a t i o n of the l a t t i c e parameters of L i ^ T i S ^ as 65 measured by neutron d i f f r a c t i o n 31. Ratio of the i n t e n s i t y of the 101,101 peak to the 68 i n t e n s i t y of the 100 peak of L i TiS„ as a function f x z X 32. The observed neutron d i f f r a c t i o n p r o f i l e of L i ^^T±S^ 71 between 19 and 24 33. The observed neutron d i f f r a c t i o n p r o f i l e of L i TiS 72 at 106K - J J 34. The function v(R" ) 78 —m 35. Contributions to the incoherent cross-section of 85 L i TiS„ as a function of x x 2 36. The spectrometer r e s o l u t i o n function 87 v i i i ACKNOWLEDGEMENTS I am indebted to my supervisor, Rudi Haering, f o r h i s help and guid-ance throughout the course of t h i s i n v e s t i g a t i o n . I benefitted from many us e f u l discussions with my co-workers Ross McKinnon, Jim Chiu, U l r i c h Sacken and L. Abello. The assistance given to me by the s t a f f and s c i e n t i s t s of the Atomic Energy of Canada l i m i t e d plant i n Chalk River Ont. was invaluable. I t i s not possible to thank everyone involved here but the help of B i l l Buyers, Brian Powell, Peter Martel, Harald Nieman and Mel Potter i s g r a t e f u l l y acknowledged. Discussions, i f at times heated, about the e f f e c t s of l i t h i u m ordering on neutron d i f f r a c t i o n p r o f i l e s with A. J . Berlinsky were extremely b e n e f i c i a l . The friendship and camaraderie extended to me by the Deep River Ont. soccer team helped to preserve my sanity while I was at A.E.C.L. . I would l i k e to thank Rosanna Chui for typing t h i s t h e s i s . The author i s g r a t e f u l f o r f i n a n c i a l support from the Natural Sciences and Engineering Research Council i n the form of a postgraduate scholarship. 1 Chapter I INTRODUCTION In t e r c a l a t i o n i s the i n s e r t i o n of a guest species into a host structure without d r a s t i c a l l y a l t e r i n g the structure of the host, so that upon removal of the guest, the host reverts back to i t s o r i g i n a l structure. In order f o r the i n t e r c a l a t i o n process to proceed i t i s clear that there must be s i t e s a v a i l a b l e f or occupa-t i o n by the guest species. For i n t e r c a l a t i o n of any s i g n i f i c a n t quantity of guest, these s i t e s must be ac c e s s i b l e from the surface of the material. Layer compounds l i k e TiS2 are good candidates for i n t e r c a l a t i o n because of t h e i r van der Waals gaps i n which i t seems l i k e l y that a c c e s s i b l e s i t e s a v a i l a b l e f o r guest atoms could be found. In f a c t , TiS^ has been found to i n t e r c a l a t e a l k a l i metal atoms (Rouxel et a l . 1971, Whittingham 1976a, Whittingham 1978), Copper (Lazzari et a l . 1976, Whittingham 1974), ammonia ( C h i a n e l l i et a l . 1975a) and many other guest species. The review a r t i c l e by Whittingham (1978)' gives an overview of many i n t e r c a l a t i o n hosts. Electrochemical c e l l s based on the i n t e r c a l a t i o n process can be constructed with TiS2 as the cathode and one of the a l k a l i metals as the anode. In the case of a l i t h i u m anode the half c e l l reactions are x L i <— x L i + + xe at the anode and x L i + + x e 7 + TiS„ L i TiS„ 2 —*- x 2 at the cathode. The electron makes the journey from the anode to cathode v i a an external e l e c t r i c c i r c u i t and the L i + ion i s transported by an 2 e l e c t r o l y t e . Figure 1 shows a schematic diagram of a t y p i c a l electrochemical c e l l . In the absence of k i n e t i c hindrance, the i n t e r c a l a t i o n process w i l l proceed as long as i t i s thermodynamically favorable f o r a guest T.S< cathode Lithium metal Electrolyte (LiCI0 4 /PC) o * e~ • - L i + • = CIO4 Figure 1. Schematic representation of the discharge of a L i / L i ^ T i S ^ c e l l . atom to reside i n the host rather than i n the external environment. Intercalated species w i l l i nteract with the host and also with them-selves. These int e r a c t i o n s can lead to many i n t e r e s t i n g physical phenomena (e.g. McKinnon 1980, Thompson 1979 or Hooley 1977). For instance, repulsive interactions between guest atoms could lead to ordering of the guest to maximize the nearest neighbor distance between guest atoms. A t t r a c t i v e i n t e r a c t i o n s could lead to condensation. Strong in t e r a c t i o n s between guest and host could lead to s t r u c t u r a l phase t r a n s i t i o n s i n the 3 host. The e l e c t r o n i c properties of the i n t e r c a l a t e d substance can be very d i f f e r e n t from the unintercalated host. This thesis i s s p e c i f i c a l l y concerned with TiS^ i n t e r c a l a t e d with lith i u m . I n i t i a l investigators of t h i s system reported a l i m i t i n g composition of L i ^ T i S ^ - Subsequent work (Murphy and Carides 1979) has shown that L i ^ T i S ^ cathodes can cycle r e v e r s i b l y between 0 < x < 2 i n electrochemical c e l l s . Recent work (Dahn and Haering 1979) indicates that T i S 2 can accommodate three l i t h i u m atoms per titanium atom and that electrochemical c e l l s can be cycled over 0 < x < 3. Lithium ordering i n T i S 2 has been suggested by Thompson (1978) to explain the structure of the experimental voltage-charge, V(x), curve of L i / L i T i S 0 c e l l s . Various t h e o r e t i c a l models of i n t e r c a l a t i o n have x 2 been examined i n an attempt to understand the observed V(x) behaviour (Berlinsky et a l . 1979, McKinnon 1980). To check the hypotheses of these models i t i s of paramount importance to e s t a b l i s h the l o c a t i o n of the l i t h i u m atoms i n the host T i S 2 l a t t i c e . The s t r u c t u r a l changes of the host l a t t i c e as a function of x i n L i TiS„ have been measured by x-ray d i f f r a c t i o n techniques (Bichon et a l . 1973, C h i a n e l l i et a l . 1978). However, x-rays give l i t t l e information on the l o c a t i o n of the i n t e r c a l a t e d l i t h i u m atoms due to the low atomic scattering factor of l i t h i u m . The neutron scattering cross-section of l i t h i u m i s comparable to those of s u l f u r and titanium. Therefore a neutron d i f f r a c t i o n p r o f i l e w i l l be s e n s i t i v e to l i t h i u m atom p o s i t i o n . A neutron d i f f r a c t i o n experiment such as the one described i n t h i s thesis was deemed to be i n order. 4 Chapter II CRYSTAL STRUCTURE OF T i S 0 AND L i TiS„ . 2 x 2 2.1 TiS, Titanium d i s u l f i d e i s a material which i s not found i n nature and must be produced s y n t h e t i c a l l y . The existence of stoichiometric TiS^ has been established (Thompson et a l . 1975) and i t s e l e c t r o n i c character c l a s s i f i e d as semimetallic. I t has a s p e c i f i c g r a v i t y of 3.22 and i s golden brown i n color. Single c r y s t a l s have been prepared by iodine vapor transport and grow i n the form of p l a t e l e t s . The c r y s t a l structure of i s the well known cadmium iodide — 3 structure (Wyckoff 1963) . The space group i s P3ml - T>^ which has a t r i g o n a l point group (Henry and Lonsdale 1952). This i s a layered structure which consists of S-Ti-S "sandwiches" stacked upon each other and held together by r e l a t i v e l y weak van der Waals forces as depicted i n f i g u r e 2. s Ti S S Ti S Van der Waals gap Figure 2. The "sandwich" structure of T i S 2 Horizontal l i n e s i n the f i g u r e represent planes of atoms. Each "sandwich" consists of hexagonal close packed atoms, formed by a plane of titanium atoms between two planes of s u l f u r atoms. The atomic structure of the "sandwiches" i s shown i n f i g u r e 3. The c r y s t a l l o g r a p h i c indexing of the TiS^ structure can be done on the basis of a hexagonal o o unit c e l l containing one molecular unit with a = 3.407 A and c = 5.695 A ( C h i a n e l l i et a l . 1975b). The f r a c t i o n a l coordinates of the atoms i n the unit c e l l are titanium at (0,0,0) and s u l f u r atoms at +(1/3, 2/3, .250). l T - T i S 2 Figure 3. The atomic structure of TiS^. Titanium atoms are depicted by s o l i d c i r c l e s and s u l f u r 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 layer unit c e l l with t r i g o n a l (T) symmetry. We now consider the l o c a t i o n of s i t e s a v a i l a b l e for l i t h i u m occupation i n TiS2« Figure 4 shows a projection of the van der Waals gap of TiS^ in the 001 plane. We see that there are three types of s i t e s , the f i r s t coordinated octahedrally^ by three s u l f u r atoms top and bottom, the second '''The coordination i s a c t u a l l y not p e r f e c t l y octahedral or tetrahedral, but i s elongated along the c-axis. 6 Figure 4. A projection of the van der Waals gap of T i S 2 i n the 001 plane. Large open c i r c l e s represent s u l f u r atoms from the plane located at z = .75 and large dashed c i r c l e s represent s u l f u r atoms with z = .25. The small s o l i d c i r c l e s give the positions of octahedral s i t e s and the t r i a n g l e s give the positions of tetrahedral s i t e s . 7 coordinated t e t r a h e d r a l l y by three s u l f u r atoms on top and one on bottom and the t h i r d merely the inversion of the second. A drawing of the s i t e s and the postulated guest atom l o c a t i o n i s given i n figure.5. The f r a c t i o n a l coordinates of the s i t e s i n the un i t c e l l are (0,0,1/2) for the octahedral s i t e and +(1/3, 2/3, z) with z = 5/8 f o r the t e t r a -hedral s i t e s . One observes that there are two tetrahedral s i t e s and one octahedral s i t e per titanium atom a v a i l a b l e f o r guest atom occupation. One fa c t o r important i n the i n t e r c a l a t i o n process i s the phys i c a l s i z e of the s i t e s . We consider a model where the s u l f u r atoms are repre-sented by close packed hard spheres and determine the s i z e of the largest hard sphere that can be inserted into each type of s i t e . .'.Awsp.her.ecof. radius o .72 A can be inserted into the octahedral s i t e and l i m i t i n g radius of 0 + .40 A i s found f o r the tetrahedral s i t e s . The radius of a L i ion i s o o approximately equal to .60 A and that of a L i atom approximately 1.5 A (Ashcroft and Mermin 1976). This suggests that occupation of the octa-hedral s i t e s by l i t h i u m i n i o n i c form i s favorable. (In the r i g i d band theory, the outer l i t h i u m electron i s donated to the host titanium d-batnd) . However, simple models of t h i s type do. not work as i t has been found that copper i n Cu TiS2 resides i n tetrahedral s i t e s (Le Nagard et a l . 1975) ++ 0 f o r x = 0.7. The Cu ion radius i s .72 A. We must therefore consider the .TiS-j l a t t i c e as.'.not ".being completelyl'.rigid .but allow, some local:::' d i s t o r t i o n s of the l a t t i c e around the i n t e r c a l a t e d species. A p r i o r i assignments of li t h i u m atom (ion) l o c a t i o n i n L i x T i S 2 cannot be made. It i s of in t e r e s t to c a l c u l a t e the various i n t e r s i t e distances i n o TiS2» The nearest neighbor octahedral-tetrahedral distance i s 2.11 A o and the nearest neighbor tetrahedral-tetrahedral distance i s 2.47 A. (This d i f f e r e n c e i s due to the tetrahedral s i t e s being displaced by ~+c/8 from the center of the gap). The octahedral-octahedral nearest Figure 5. Proposed guest atom (small spheres) l o c a t i o n i n a) tetrahedral s i t e s and b) octahedral s i t e s . The p r i m i t i v e l a t t i c e vectors of the host are give i n c ) . Large spheres represent s u l f u r atoms. neighbor distance i s given by the a axis of TiS^ and i s equal to 3.407 A. These distances must be considered when one t r i e s to i n s e r t l i t h i u m ions (atoms) into adjacent s i t e s . I f the s i t e s are too close t h i s may be d i f f i c u l t to do because of the f i n i t e s i z e of the ions (atoms). Naively then, one would expect a l i m i t i n g composition of x = 3 i n L i TiS„ provided that i t i s possible to f i l l a l l the s i t e s , x 2 2.2 L i TiS„ x 2 The c r y s t a l structure of L i TiS„ has been investigated for 0 < x < 1 x 2 - -(Whittingham 1976b, Bichon et a l . 1973). X-ray d i f f r a c t i o n techniques were used and i t was found that over the range 0 < x < 1 the c r y s t a l could s t i l l be described in terms of a hexagonal unit c e l l . The depend-ence of the l a t t i c e constants as a function of x i s shown i n f i g u r e 6. The major e f f e c t i s an expansion of the c axis of the structure as more li t h i u m i s inserted. The a axis i s r e l a t i v e l y i n s e n s i t i v e to l i t h i u m concentration (Aa/a ^ .01). There are no s t r u c t u r a l phase t r a n s i t i o n s of the host l a t t i c e between 0 < x < 1. Lithium atom l o c a t i o n cannot be determined from these studies due to the low x-ray scattering factor of li t h i u m . A more c a r e f u l study by C h i a n e l l i et a l . (1978) using dynamic x-ray d i f f r a c t i o n techniques reveals f i n e structure i n the l a t t i c e constants as a function of x. C h i a n e l l i measures the 101,101 plane spacing as a function of x and his data shows a plateau between .30 < x < .50 which i s suggested to be due to l i t h i u m ordering i n the TiS^ layers. Unfortunately he only measures the position.of t h i s p a r t i c u l a r powder peak as a function of x, so i t i s not possible to a s c e r t a i n how the l a t t i c e constants are behaving independently. I w i l l discuss the phenomena 6.2 i 1 1 1 T~ o o o i o o 6.1 — 0 o A ~ 6.0 •< o -°5.9 5S A • 5.7 " O 1 1 1 1 1 0 .2 X .4 .6 .8 in L i x T i S 2 1.0 3.46 3.44,. 0 < H3.42 3.40 Figure 6. Dependence of the l a t t i c e parameters c and a on x i n LI TiS ' 2 C i r c l e s are c axis data points and t r i a n g l e s are a axis data points. Note the difference i n the scales f o r a and c. (After Whittingham 1976b) of l i t h i u m ordering i n more d e t a i l l a t e r . The d e t a i l e d c r y s t a l structure of L i ^ T i S ^ f o r x > 1 has not yet been published, although Murphy and Carides (1979) claim that f o r '.  1 < x < 2 the structure i s a mixture of Li^TiS2 and L i ^ T i S ^ phases. No s t r u c t u r a l data has been reported f o r x > 2 although the c e l l voltage of L i / L i x T i S 2 c e l l s versus x for x > 2 has been measured (Dahn and Haering 1979). CHAPTER H i : L i / L i T i S 0 ELECTROCHEMICAL CELLS x I  3.1 Introduct ion Electrochemical c e l l s can be constructed using TiS^ as the cathode material, an aprotic solvent i n which a l i t h i u m s a l t i s dissolved as the e l e c t r o l y t e and a l i t h i u m metal anode. (A common choice for the e l e c t r o l y t e i s propylene carbonate (PC) as the solvent and l i t h i u m per-chlorate as the s a l t ) . A large quantity of data has been published on the experimental voltage versus state of discharge p r o f i l e s of i n t e r c a l a -t i o n c e l l s , but T i S 2 has undoubtedly received the most attention i n the l i t e r a t u r e l a r g e l y due to the vast research e f f o r t on TiS^ at Exxon. Figure 7a shows the quasi-open c i r c u i t voltage, V(x), of a L i / L i x T i S 2 c e l l as a function of x as measured by A.H. Thompson (1978). A p l o t of -Ax/AV vs x also measured by Thompson i s shown i n f i g u r e 7b. One notices the presence of f i n e structure on the c e l l discharge curve which becomes very apparent i n the inverse d e r i v a t i v e curve. Thompson associates the peaks at x = 1/4, 1/9 and 6/7 i n the inverse d e r i v a t i v e curve with the r e a l i z a t i o n of states with long' range l i t h i u m order which supposedly form due to long range Coulomb repulsion. For instance, Thompson asso-c i a t e s the peak at x = 1/4 with an ordered l i t h i u m s u p e r l a t t i c e with a-axis, a', equal to twice the a axis of the T i S 2 host. Octahedral s i t e occupation i s assumed throughout t h i s discussion and i s based on unpublished work by A.J. Jacobsen which supposedly shows that the l i t h i u m ions reside i n octahedral s i t e s at x = 1. Theoretical c a l c u l a t i o n s of the V(x) curve by Berlinsky et a l . (1979), 13 T 1 1 1 r Q I i i • l i l 1 l U . 1 0 0.2 0.4 0.6 0.8 1.0 X IN Li xTiS 2 Figure 7. a)'The quasi-open c i r c u i t voltage of a L i / L i TiS^ c e l l , b) The inverse d e r i v a t i v e of a. (After Thompson 1978) assuming only octahedral s i t e s a v a i l a b l e f o r L i + ion occupation, show that short range nearest neighbor repulsive l i t h i u m - l i t h i u m i n t e r a c t i o n s can lead to phase t r a n s i t i o n s exhibiting long range l i t h i u m order. I t i s shown that i n the case of the nearest neighbor repulsion equal to 4kl that the li t h i u m atoms w i l l undergo a phase t r a n s i t i o n at x = .25 to a state e x h i b i t i n g long range order. The long range ordered state i s one where a l l the l i t h i u m ions are located on one of the three super-l a t t i c e s of octahedral s i t e s as shown i n f i g u r e 8. The V(x) curve exhibits a sharp drop at x = 1/3 which i s due to the introduction of f i l l e d nearest neighbor s i t e s f o r x > 1/3. The -9x/8V curve shows a peak at x - 1/4 and a minimum at x - 1/3 corresponding to the phase t r a n s i t i o n to and the f i l l i n g df the ordered state r e s p e c t i v e l y . The above des c r i p t i o n i s i n c o n f l i c t with that presented by Thompson, who associated the peak at x = 1/4 i n the -9x/9V curve with the f i l l i n g of an ordered state with a' = 2a. The model c a l c u l a t i o n of Berlinsky et a l . predicts that the peak should be associated with a phase t r a n s i t i o n to an ordered state, but the subsequent minima i n -3x/8V predicted by the theory at x = 1/3 i s not seen i n the data. It i s not c l e a r whether li t h i u m ordering can be used to explain the f i n e structure of the voltage curve between 0 < x < 1. Electrochemical c e l l s were constructed to tr y to duplicate Thompson's measurements. These experiments w i l l now be described. 3.2 Preparation of T i S 2 T i S 2 can be prepared i n a v a r i e t y of ways and i n these experiments T i S 2 formed in two d i f f e r e n t ways was used. The f i r s t method of pre-paration involved placing stoichiometric r a t i o s of highly pure T i and S powders i n a s i l i c a tube which was then evacuated and sealed. The sample was then heated to 300?C and the temperature raised at approximated 25°C/hr to between 550 C and 800 C (depend ing on the s p e c i f i c batch) where i t was held f o r approximately two days. The product of an 800°C run consisted of p l a t e l i k e c r y s t a l l i t e s , golden brown i n c o l o r , of an average diameter of approximately 15 ym. When the heating was c a r r i e d out at 5 0 0 ° C 6 0 0 ° C , the average c r y s t a l l i t e diameter was approximately 1 um.to 2 ym. Single 2 c r y s t a l s prepared by iodine vapor transport of titanium and s u l f u r i n "2 Thanks to U. Sacken who grew these c r y s t a l s . B C A C A B C A B C A B B C A B C A A B C A B C A B C Figure 8. The l o c a t i o n of octahedral s i t e s i n a s i n g l e van der Waals gap and the s u p e r l a t t i c e (A,B, or C) decomposition of these s i t e s . the standard way were washed i n chloroform and ground into powder. The average c r y s t a l l i t e diameter of t h i s material was approximately 40 um. 3.3 Cathode preparation 2 Nickel f o i l substrates, of surface area ~2 cm , were cleaned by etching with concentrated n i t r i c acid and then r i n s i n g i n d i s t i l l e d water. A s l u r r y of TiS2 powder i n propylene g l y c o l was then spread t h i n l y over the n i c k e l f o i l . This assembly was then baked at 200°C under flowing nitrogen gas to remove the propylene g l y c o l . The mass of TiS2 on the cathode was measured by weighing the n i c k e l f o i l before spreading the s l u r r y and a f t e r 2 baking. Ty p i c a l T i S ? mass densities achieved were =5 mg/cm . (Weighing was done using a Sartorius 2434 balance and cathode masses were accurate to + .1 mg). The r e s u l t was a f a i r l y w e ll adhering, uniform thickness powder on a mechanically stable n i c k e l f o i l substrate. Nickel f o i l was used due to i t s low capacity f o r a l l o y i n g with, l i t h i u m . This i s discussed in d e t a i l l a t e r . 3.4 E l e c t r o l y t e Preparation The e l e c t r o l y t e used i n a l l experimental c e l l s discussed i n t h i s t hesis was IM LiC10^/PC. Several cleanup procedures were used to p u r i f y the PC and involved several vacuum d i s t i l l a t i o n s . In a l l cases; and i n some cases t h i s was supplemented by passing the PC through, columns of activated alumina. Gas chromatography showed a maximum impurity l e v e l of 6 ppm.of propylene g l y c o l , the major impurity, i n the p u r i f i e d PC, Hydrous li t h i u m perchlorate was vacuum dried at 130°C and the stored under argon u n t i l use. The e l e c t r o l y t e was mixed i n a Vacuum Atmospheres- dry box under argon and stored there u n t i l use. 3.5 Separators Celgard #3501 microporous fi l m s (separators) were used most frequently in the c e l l s I 'constructed. In a few cases, Celgard #25Q0 separators were used. #3501 separators were cut into one inch square sheets ( t y p i c a l l y ) , given two soaks i n p u r i f i e d PC (to remove a surfactant used as a wetting agent i n these separators) under argon and then f i n a l l y soaked i n the IM LiC10^/PC e l e c t r o l y t e . #2500 separators were wetted by a soak i n reagent grade tetrahydrofuran (THF) and then rinsed i n PC. Residual THF was boiled off under vacuum. The separators at t h i s stage were then soaked twice i n p u r i f i e d PC under argon and then once i n the e l e c t r o l y t e . Separators soaked i n e l e c t r o l y t e were then ready to use i n c e l l construction. I t should be noted that #3501 and #2500 separators prepared by the above methods performed equally well in electrochemical c e l l s . 3.6 Electrochemical c e l l s Electrochemical c e l l s were constructed as shown i n f i g u r e 9. S t a i n l e s s s t e e l flanges were greased with s i l i c o n high vacuum grease which was found to be i n e r t with respect to the l i t h i u m and e l e c t r o l y t e . This was done to protect the s t a i n l e s s s t e e l and also to prevent any a l l o y i n g of l i t h i u m with the s t a i n l e s s s t e e l . The cathode, two separators and l i t h i u m metal f o i l were stacked as shown i n f i g u r e 9 and placed i n the c e l l which was then sealed by the 0-ring when tightened. C e l l assembly was performed i n a Vacuum Atmospheres dry box under argon. Water and oxygen l e v e l s i n t h i s system are estimated to be le s s than 3-5 ppm. 3.7 C e l l c y c l i n g techniques Two c y c l i n g methods were u t i l i z e d to study the c e l l s constructed for these experiments. The f i r s t involved charging and discharging the c e l l s at constant current between f i x e d upper and lower voltages. In the second method, the voltage on the c e l l was swept l i n e a r l y i n time and the current monitored. When a c e l l i s discharged at constant current, the amount of charge that has flowed i s simply the current m u l t i p l i e d by the discharge time. By knowing the mass of the i n t e r c a l a t i o n cathode i n the c e l l i t i s a simple matter to ca l c u l a t e the value of x. We f i n d , x = «M 3 1 96,500 zm ' where I i s the el e c t r o n i c current, t i s the time the current has been flowing, z i s the valence of the metal ions being transported by the e l e c t r o l y t e , m i s the cathode mass, and M i s the molecular weight of the cathode material. We can, i n theory, control the amount of i n t e r c a l a t e d species i n the cathode by simply discharging the c e l l at constant current f o r a s p e c i f i e d time i n t e r v a l (Chemists c a l l t h i s a coulometric t i t r a t i o n ) . In p r a c t i c e , eqn. 3.1 does not s t r i c t l y hold due to the presence of various sinks f o r lithium. The i n t e r a c t i o n of the l i t h i u m with the c e l l components must be considered. A l l o y i n g with the n i c k e l substrate and reactions with the e l e c t r o l y t e are two examples of these processes which we c a l l "side reactions". Another problem i s that of "cathode u t i l i z a t i o n " ; the e n t i r e cathode may not be connected e l e c t r i c a l l y to the substrate f o r one reason or another. In general cathodes that have been properly prepared show cathode u t i l i z a t i o n near 100%. Currents-drawn by side reactions are t y p i c a l l y very much smaller than those drawn by the cathode material and to a f i r s t approxima-t i o n , equation 3.1 can be used to determine x. Constant current c y c l i n g tests were performed on a constant current c y c l e r b u i l t by the U.B.C. Physics E l e c t r o n i c s shop. Cycling c e l l s between f i x e d voltage l i m i t s at constant current i s a technique that was used to determine the coarse features of the V(x) curve. The cycle l i f e of c e l l s can also be q u a l i t a -t i v e l y determined i n t h i s way. More detai l e d examination of the voltage, V(x), curve i s f a c i l i t a t e d by measuring the inverse d e r i v a t i v e , dx/dV, of the V(x) curve. This can be done by sweeping the c e l l voltage at a constant rate and measuring the current supplied to the c e l l . The current, I, i s dt dV dt ^odV dt ' where Q q i s the charge corresponding to x = 1 for the p a r t i c u l a r cathode of i n t e r e s t . We see that f o r constant dV/dt, the current i s d i r e c t l y proportional to dx/dV. In deriving equation 3.2 we have neglected c e l l resistances and overvoltages of a l l kinds. The e f f e c t s of series r e s i s -tances due to the anode, e l e c t r o l y t e and surface processes and d i f f u s i o n overvoltages on eqn. 3.2 has been discussed by McKinnon (1980). I t i s shown that when dV/dt i s small, equation 3.2 becomes a good approximation. The v a l i d i t y of t h i s technique was demonstrated by McKinnon (1980) who obtained good agreement with the data reported by Thompson (1979) for 0 < x < 1 i n L i / L i ^ T i S ^ c e l l s . Current voltage curves generated i n t h i s way w i l l be c a l l e d inverse d e r i v a t i v e curves. A Princeton Applied Research (PAR) model 173 potentiostat/galvanostat and a PAR 175 u n i v e r s a l programmer were used to perform these t e s t s . 3.8 Experimental Results and Discussion Figure 10 shows the experimental voltage versus charge curve f o r a c e l l with no cathode material, j u s t a bare n i c k e l substrate. The surface 2 area of the substrate was -2 cm . From t h i s data we see that n i c k e l sub-2 strates have capacity of order 50 millicoulombs/cm on the f i r s t discharge to 0.3 v o l t s . Recharge and subsequent discharge capacity i s reduced by a factor of 6. Note that a L i TiS„ cathode has a capacity of .86 coulombs/mg x 2 to x = 1. For reasonable mass cathodes, the e f f e c t of the n i c k e l substrate i s small. It was found i n these studies that d i f f e r e n t batches of TiS2 behaved i n d i f f e r e n t ways i n electrochemical c e l l s and that i n c e r t a i n cases, c e l l behaviour was dependent on past h i s t o r y . This i s due to the i n t e r c a l a t i o n of l i t h i u m ions along with t h e i r PC sol v a t i o n cloud. The word i n t e r c a l a t i o n should not be used i n t h i s context because the changes i n the host l a t t i c e are quite d r a s t i c and appear to be i r r e v e r s i b l e . I w i l l use the term PC i n s e r t i o n when r e f e r r i n g to t h i s process. This process has been mentioned previously by Whittingham (1976b) ... The sol v a t i o n cloud ^ a p p a r e n t l y bound... t i g h t l y enough to the L i + ion for t h i s to occur. PC i s a highly polar mole-cule with d i e l e c t r i c constant equal to 65.0 ( J a s i n s k i 1971). It i s believed that solvent i n s e r t i o n may be avoided w i t h other le s s polar solvents. This 21 3 TIME (hrs) Figure 10. Several cycles of a Li/LiC10^, PC/Ni electrochemical c e l l . The constant current cycles were measured at a current of 10 pA. appears to be the reason f o r the use of l e s s polar solvents l i k e dioxolane by some investigators (eg. C h i a n e l l i et a l . 1978). Dioxolane i s more v o l a t i l e and more d i f f i c u l t to handle than PC. The i n s e r t i o n of PC causes large changes i n the c-axis of the TiS^ host and i t i s not yet c l e a r whether the PC can be removed from the host. Features on the battery discharge curve associated with PC i n s e r t i o n are e a s i l y recognizable and w i l l be discussed l a t e r . We w i l l now discuss the differences i n the behaviour of the HS2 samples with d i f f e r i n g p a r t i c l e s i z e . Material with an average c r y s t a l l i t e diameter of 40 ym, prepared by crushing s i n g l e c r y s t a l s w i l l be denoted 40 ym TiS^-The powders which were baked at 550°C and 800°C, having average c r y s t a l l i t e diameters of 2 pm and 15 ym re s p e c t i v e l y w i l l be c a l l e d 2 ym T i S 2 and 15 ym T i S 2 . 3.8.1 1 i S 2 with an average c r y s t a l l i t e diameter of 40 ym V(x) curves generated by c e l l s with cathodes of 40 ym T i S 2 were i n good agreement with those published i n the l i t e r a t u r e (Thompson 1979) i n the range 0 < x < 1. The f i r s t c ycle of c e l l JD-49 i s shown i n fi g u r e 11. We show the region of the voltage curve corresponding to 0 < x < 2 f o r reader i n t e r e s t although we .are concerned only with 0 < x < 1 i n t h i s t h e s i s . An inverse d e r i v a t i v e curve of c e l l JD-40 i s depicted i n f i g u r e 12. One notices Figure 11. Voltage of c e l l JD-49, containing 1.8 mg TiS , cycled at a constant current of 75 yA. 5.8 hours corresponds to x = 1 i n L i T i S 2 . Note that the x scale applies to the discharge only. x that some of the features present i n Thompson's (1979) data, e s p e c i a l l y the small peak at x - 1/9, (fig u r e 7b) have not been resolved. The c h a r a c t e r i s t i c d i f f u s i o n time, T , for l i t h i u m i n a c r y s t a l l i t e of radius R i s of order T - R2/D 3.3 where D i s the d i f f u s i o n constant of l i t h i u m in-the host (Carslaw and Jaeger -9 2 1959). D i f f u s i o n constants of order 10 cm /sec have been measured by severa Figure 12. An inverse d e r i v a t i v e curve of c e l l JD-40 taken at a voltage sweep rate of 7.04 yv/sec. The c e l l contained 3.0 mg of TiS,,. The-relative s h i f t of the discharge and recharge curves i s due to the non-zero sweep rate. researchers f o r l i t h i u m i n T i S 2 (Nagelberg 1978, Chiu and Haering 1979). We expect a r e s o l u t i o n in-.-volt age, AV, of AV - T dV/dt . 24 For the data i n fi g u r e 12, dV/dt = 7.04 yv/sec which y i e l d s AV = 30 mv. There-fore i t i s reasonable that the small peak i n Thompson's data near x - 1/9 i s smeared into a shoulder at the sweep rate u t i l i z e d . Slower sweep rates were not t r i e d on t h i s material, although i t i s f e l t that slower rates would resolve the peak at x - 1/9. Evidence f o r PC i n s e r t i o n was not observed i n 40 ym TiS^ cathodes. 3.8.2 TiS^ with an average c r y s t a l l i t e diameter of 15 ym Interesting e f f e c t s were observed i n t h i s material that are not yet f u l l y understood. 15 ym TiS2 cathodes, when discharged at low rates on the f i r s t discharge, were observed to intercalate, l i t h i u m i n the usual way and p a r t i c i p a t e i n the PC i n s e r t i o n process. However, when c e l l s of t h i s type were discharged at high rates on the f i r s t c y c l e , PC i n s e r t i o n was not detected. Figure 13 shows the constant current c y c l i n g behaviour of c e l l JD-63 between 1.6 v o l t s and 2.7 v o l t s . One notices that the f i r s t discharge i s r e l a t i v e l y f l a t i n the region near 2.32 v o l t s . The subsequent recharge, although s i m i l a r to a t y p i c a l 40 ym T i S 2 recharge, has very small capacity. After the t h i r d discharge, t h i s c e l l was dismantled i n an argon f i l l e d dry box and a Debye-Scherrer x-ray photograph was taken of the cathode material i n the manner described by Wainwright (1978). The observed x-ray pattern i s given i n table I. This pattern appears to be made up of l i n e s correspond-ing to two d i s t i n c t structures with one set corresponding to those L i T i S ^ . The largest plane spacing c a l c u l a t e d from the other set of r e f l e c t i o n s , d = 17.8 A, suggests that solvent i n s e r t i o n could be taking place. Whitting-ham (1976b) states that PC i n s e r t i o n can cause a sub s t a n t i a l l a t t i c e expansion, o o about 18 A to 24 A in TiS2> We w i l l see l a t e r (Chapter 6) that solvent i n -ser t i o n d e f i n i t e l y occurs and r e s u l t s i n large expansion p a r a l l e l to the c ax i s . TABLE I The observed x-ray d i f f r a c t i o n pattern from the cathode material of c e l l JD-63. The r e f l e c t i o n s corresponding to those of' L i T i S ^ and t h e i r M i l l e r indices; are-indicated. Intensity abbreviations are as follows, v=very, s=. st rong , m=med ium, and w= weak. Plane spacing co-rn 17.8 8.93 6.173 5.946 2.974 2.862 2.703 2.536 2.347 2.151 2.047 1.983 1.888 1.714 1.659 1.452 11345 Relative i n t e n s i t y m s m m WW vw m m w m vvw vvw vvw m vw w w o o LiTiS„ a=3.46A, c=6.19A -CI { { 0 0 0 0 1 0 0 0 0 0 0 1 T: o 3_ 3 1 T 2 1 Plane spacing o (A) ' 6.19 2.70 2.15 1.73 1:70 1.456 1.348 26 TIME (hrs) Figure 13. The constant current c y c l i n g behaviour of c e l l JD-63. The c e l l contained 5.8 mg of 15 ym TiS„ and was cycled at 275 yA. The time corresponding to A x = 1 i n L i TiS„ i s depicted i n the f i g u r e , x 2 Many c e l l s with 15 ym TiS^ were tested and when discharged at rates s i m i l a r to c e l l JD-63 showed capacity much less than x = 1 on the f i r s t d i s -charge. In a l l cases f l a t regions at 2.32 v o l t s corresponding to PC i n s e r t i o n were observed. The small capacity i§ due to the greater volume needed to accommodate PC molecules i n the portion of the host which they occupy. Subsequent cycles between 1.6 v o l t s and 2.7 v o l t s showed V ( x ) curves of the type reported by Thompson (1979) but with capacity much less than A x = 1. This suggests that the solvent i n t e r c a l a t i o n process i s i r r e v e r s i b l e and portions of the cathode that have taken up PC are no longer a c t i v e above 1.6 v o l t s . More tangible evidence w i l l be-given l a t e r . C e l l JD-57, which had a 15 ym T i S 2 cathode, was held at 1.6 v o l t s by the PAR potentiostat immediately a f t e r assembly. The c e l l current was i n i t i a l l y about 10 mA and a f t e r several hours had decayed to a few microamps. The c e l l was then dismantled and an x-ray was taken of the in t e r c a l a t e d cathode material. 27 The observed and ca l c u l a t e d i n t e n s i t i e s (which correspond to those of Li^TiS2) are shown i n table I I . The i n t e n s i t y c a l c u l a t i o n included the geometrical structure f a c t o r , Lorentz factor and the p o l a r i z a t i o n f a c t o r ( C u l l i t y 1959). The data shows Bragg r e f l e c t i o n s corresponding to a hexagonal l a t t i c e with , o o a = 3.46 I .01 A and c = 6.19 I .01 A. This i s i n good agreement with o o Whittingham (1976b) who reports a = 3.455 A and c = 6.195 A f o r L i j T i S ^ We note the absence of any r e f l e c t i o n s corresponding to solvent i n s e r t i o n . C e l l s discharged i n t h i s way showed no evidence of PC i n s e r t i o n and when cycled a f t e r the i n i t i a l "quick" discharge displayed V(x) curves i n good agreement with those of Thompson (1979) . The PC i n s e r t i o n process i s very dependent on the rate of the i n i t i a l discharge i n c e l l s with 15 um HS^ cathodes. Further study i s required to determine the exact dependence of t h i s process on discharge rate. The fact that a f t e r the i n i t i a l discharge, very l i t t l e e l e c t r o l y t e i n s e r t i o n occurs i s not understood and also requires further study. 3.8.3 TiS2 with an average c r y s t a l l i t e diameter of 2 ym It was found that the PC i n s e r t i o n process occured i n c e l l s with 2 ym TiS2 cathodes i r r e s p e c t i v e of discharge rate. X-ray patterns ind i c a t e a mixture of two structures, one of which i s i d e n t i c a l to that of Li^TiS2-These x-ray patterns are s i m i l a r to the one given i n table I. The constant current c y c l i n g behaviour of c e l l s with 2 ym TiS2 cathodes i s s i m i l a r to that shown i n f i g u r e 13. Inverse d e r i v a t i v e curves taken on 3 c e l l RM-12 a f t e r the i n i t i a l discharge are shown i n f i g u r e 14. Thompson's (1979) -Ax/AV data i s normalized to the capacity of our c e l l and i s replotted 3 This c e l l was prepared and tested by Ross McKinnon. TABLE II Observed and calculated plane spacings and r e l a t i v e i n t e n s i t i e s f o r L i T i S ^ obtained from c e l l JD-57 as described i n the text. Observed plane spacing [ Relative i n t e n s i t y I Calculated plane spacing (i) Re-la t ive i n t e n s i t y 6.17 2.699 2.154 1.729 1.700 1.664 1.547 1.455 1.348 m wm w w w 0 0 K { o l 2 2 .2 2 .2 1 0 1 0 2 0 0 0 1 0 T 0 2 0 1 0 3 1 0 0 3 0 7 1 1 0 4 1 2 0 0 0 1 0 T 0 2 0 "Z 1 3 6.19 3.10 3.00 2.697 2.153 2.063 1.730 1.699 1.666 1.547 1.510 1.498 1.456 1.348 1.326 66 5 1 165 : 97 4 51 42 14 12 3 <1 25 21 6 V ( v o l t s ) Figure 14. The inverse d e r i v a t i v e curves of c e l l RM-12. Thompson's (1979) data i s normalized to our cell's capacity and i s rep l o t t e d here against voltage ( s o l i d dots). The cathode mass was 4.9 mg and the voltage sweep rate was 17.1 yV/sec. 30 i n f i g u r e 14 against voltage f o r comparison. In t h i s case the product of the c h a r a c t e r i s t i c d i f f u s i o n time and the voltage sweep rate i s small enough f o r the peak at x - 1/9 to be resolved. The agreement of our data with that of Thompson's i s good, with the peaks corresponding to x - 1/4 and x - 1/9 well reproduced. The s i m i l a r i t y of our data and Thompson's (which were obtained with a dioxolane based e l e c t r o l y t e ) suggest that the material which has p a r t i c i p a t e d i n the PC i n s e r t i o n process i s i n a c t i v e a f t e r the f i r s t discharge f o r cycles above 1.6 v o l t s . 3.9 Summary of L i x T i S 2 c e l l s We must conclude on the basis of the preceeding experiments that PC i n s e r t i o n into TiS^ cathodes i s an e f f e c t that i s dependent on discharge rate and on average c r y s t a l l i t e s i z e . The i n t e r c a l a t i o n of l i t h i u m appears to be k i n e t i c a l l y f a s t e r than the solvent i n s e r t i o n process and would explain the success of the "quick" discharge method i n 15 um.TiS2. The e f f e c t s of p a r t i c l e s i z e are not well understood, although s t r a i n e f f e c t s may be more pronounced i n larger c r y s t a l l i t e s , making them les s amenable to PC i n s e r t i o n . I t i s c l e a r that further study i s required. The features i n the V(x) curve reported by Thompson (1979) have been confirmed and i t i s now of paramount importance to understand t h e i r o r i g i n . A neutron d i f f r a c t i o n study was undertaken to attempt:;to answer t h i s question. 31 Chapter IV THEORY OF NEUTRON DIFFRACTION The wave-like q u a l i t i e s of the neutron cause interference e f f e c t s which p a r a l l e l those observed f o r electromagnetic r a d i a t i o n . Neutrons with de Broglie wavelengths comparable to atomic distances i n s o l i d s can be used to study c r y s t a l structures. In f a c t , e l a s t i c neutron scattering from c r y s t a l s and x-ray d i f f r a c t i o n are very s i m i l a r . Neutrons which have come to thermal equilibrium at temperature T have a root mean square v e l o c i t y , v, given by } m v 2 = f V , 4.1 where m i s the neutron mass and k^ i s Boltzmann's constant. The de Br o g l i e wavelength of such neutrons i s or 4.2 4.3 For neutrons at 0 WC and 100 UC, the wavelengths corresponding to root mean o o square v e l o c i t i e s are 1.55 A and 1.33 A re s p e c t i v e l y . These wavelengths are of j u s t the ri g h t magnitude f o r studying c r y s t a l structures. Moderated neutrons of t h i s type are e a s i l y obtained when they have come to thermal equilibrium at the moderator temperature i n a reactor. The i n t e r a c t i o n of a neutron with a c r y s t a l l i n e s o l i d i s very complex. For our purposes i t i s not necessary to consider magnetic or i n e l a s t i c s c a t t e r -ing so we w i l l neglect these e f f e c t s . It should beinoted that since L i ^ T i S ^ i s paramagnetic, (Thompson et a l . (1972), Murphy et a l . (1976)) magnetic scattering w i l l be incoherent i n the absence of a magnetic f i e l d . An .excel-32 lent treatment of these e f f e c t s i s given i n the book by Marshall and Lovesy (1971). Neglecting the magnetic scattering.and, f o r the moment, i n t e r a c t i o n s between nuclear and neutron spins we assume that the target i s a s i n g l e nucleus. If a plane wave of thermal neutrons with wave vector k = k 1 i s incident on the nucleus the scattered wave w i l l be s p h e r i c a l l y symmetric of the form iji = — exp(ikr) . sc r In t h i s notation r i s the distance from the point of measurement to the nucleus and b i s a complex quantity with units of length, b = a + i3 , c a l l e d the scattering length of the nucleus. We can neglect p-wave and higher order scattering because of the low energy of thermal neutrons. For most n u c l e i , the imaginary part of b can be neglected and we can define the scattering cross-section of the nucleus, a, as outward current of scattered neutrons incident neutron f l u x In our case a reduces to a = 4irb We note that the scattering length, b, plays a r o l e i n thermal neutron s c a t t e r -ing s i m i l a r to the atomic scattering f a c t o r i n x-ray s c a t t e r i n g . Let's consider a r i g i d c r y s t a l l i n e s o l i d with atomic or molecular u n i t s centered at the points of a Bravais l a t t i c e . If a plane wave of neutrons i s incident on t h i s object, the scattering amplitude, f ( q ) , i s given by N f ( .1 > = I F n ( £L ) e x P ( i S . 1 ^ ) • 4.4 n=l In t h i s expression, q i s the momentum transfe r (q = k. . , ^ - k ^ , ) , f » ^ — xncxdent — s c a t t e r e d R i s the p o s i t i o n of the n*"*1 atomic or molecular unit which must be a Bravais —n l a t t i c e vector and N i s the number of these u n i t s i n the s o l i d . The structure 33 f a c t o r , F (c[) , of the n t b atomic or molecular unit i s V ^ } = I b m expCig^.^) 4.5 m where m i s the number of n u c l e i i n the unit and b^ i s the scattering length of the nucleus at p o s i t i o n d . The d i f f e r e n t i a l s c a ttering cross-section, —m da . •da-' 1 S J _ 9 = I I. Fn(il) F n,(^) expd^.CP^-^,)). 4 > ? F n ( A ) I * n n For a c r y s t a l with a r i g i d three dimensional monatomic Bravais l a t t i c e with a nucleus of scatteringMength b at each l a t t i c e point , do" 2 n v dfi - b XI, a c p C i a . C ^ , ) ) . . 4.8 n n If N i s large, i t i s c l e a r that, 11 exp(i£.(R -R ,)) = I 6(^-G.)-constant n n' i where _G i s a r e c i p r o c a l l a t t i c e vector defined i n the usual way. We integrate both sides over one u n i t c e l l of the r e c i p r o c a l l a t t i c e to f i n d the constant and obtain da _ N(2tf) 3 K2 Y . 0 1 where v 0 i s the volume of the unit c e l l i n r e a l space. Isotopic and spin e f f e c t s lead to several values of b f o r a s i n g l e element. I t i s shown i n appendix I that equation 4.9 generalizes to 3 da N(2ir) I T I 2 v *, „ x . „ I r, I 2 .-|2 b| Z I 6(^-G ±) + N | |b|- |b|| 4 - 1 0 dtt v„ _ 0 1 where b denotes the average of b over the c r y s t a l . The f i r s t term of 4.10 i s referred to as the coherent d i f f e r e n t i a l cross-section and the second as the incoherent. We w i l l denote the coherent cross-section by a and the incoherent J c by a_^ . The incoherent cross-section leads to i s o t r o p i c scattering upon which the Bragg peaks a r i s i n g from coherent s c a t t e r i n g are found. The g e n e r a l i z a t i o n of equation 4.9 to the case with more than one atom per unit c e l l , but each c e l l i d e n t i c a l i s 34 0 1 where F(G) i s defined i n equation 4.5. By introducing the r e c i p r o c a l l a t t i c e vectors (Ashcroft and Mermin 1976) b^ •>••'*-J^  a n ^ ^.3 w e ^ i t 6 F(G) = 7 b exp(2Tri(hx +ky +lz )) . l o — L m m m m 4.12 m = F(hk£) In t h i s expression, h, k and £ are integers, and G = hb +kb +£b d = x a . + y b + z c —m m— nr - m— where a_, b and c_ are the r e a l space l a t t i c e vectors. Equation 4.12 i s con-venient for doing r e a l c a l c u l a t i o n s of structure f a c t o r s , b i s generally l i s t e d i n tables as the coherent scattering length. In the Debye-Scherrer method, a powder of randomly oriented c r y s t a l l i t e s i s studied i n the geometry of f i g u r e 15. Under these conditions, the. require-ment that c[ = G must be averaged over a l l o r i e n t a t i o n s of the vector G. There-fore cones of d i f f r a c t e d neutrons are observed whenever [_qj = |GJ and the angle (j> shown i n f i g u r e 15 i s given by q = sIn(cf>/2) . 4.13 A From 4.10 q - |G| = | h b 1 + k b 2 + £ b 3 | - ^ L . 4 > 1 4 where d ^ £ i s the distance between the l a t t i c e "planes" leading to the r e f l e c -t i o n . Using 4.13 and 4.14 we obtain the f a m i l i a r Bragg condition, 2 d h k £ = sin(c|>/2) • 4 , 1 5 The allowed values of <f> can be determined from equation 4.15 i n terms of the r e c i p r o c a l l a t t i c e coordinates h, k and L. For a hexagonal c r y s t a l l i k e L i ^ T i S ^ with l a t t i c e parameters a and c, 1 = 4 r h 2+hk+k 2 \ . ^_ 4.16 3 (• ? J.'T 9 ,2:. 3 >- 2 J ' ~ 2 ( C u l l i t y 1959). hk£ d,^ _o a There are two common geometrical arrangements which permit a ready comparison 35 of the experimentally measured i n t e n s i t i e s with the r e s u l t s of a c a l c u l a t i o n . These are depicted i n f i g u r e 16. In the configuration shown i n 16a, a c y l i n -d r i c a l sample i s bathed i n the neutron beam. The number of neutrons d i f f r a c t e d into the detector per unit number of incident neutrons at scattering angle <j> due to e l a s t i c coherent scattering only, P(<f>) i s 3_.,T | n / t ^ l v I 2 A / 1T.7\ X t i. J, (Bacon 1975). In t h i s expression, Q i s a p r o p o r t i o n a l i t y constant, j i s the m u l t i p l i c i t y of the hk£ r e f l e c t i o n , A ^ ^ i s the absorption f a c t o r and exp(-2W) i s the Debye-Waller f a c t o r . The m u l t i p l i c i t y i s the number of r e c i p r o c a l l a t t i c e vectors with d i f f e r -ent hk£ indices having the same length and structure f a c t o r . This factor a r i s e s due to the f a c t that the Debye-Scherrer method i s only s e n s i t i v e to the lengths of r e c i p r o c a l l a t t i c e vectors. The absorption f a c t o r , A^^, a r i s e s due to the f i n i t e s i z e of the sample and depends i n a complicated way on uR g and cf>, where R g i s the radius and u i s the l i n e a r absorption c o e f f i c i e n t of the sample. The angular v a r i a t i o n of A^k£ i s small for values of uR^ l e s s than 0.4 (Bacon 1975) and i s usually neglected i n these cases. The Debye-Waller factor i s due to the n o n - r i g i d i t y of r e a l c r y s t a l l a t t i c e s . A discussion of the o r i g i n of the Debye-Waller fa c t o r i s given i n appendix I I . The Debye-Waller factor i s generally written as ( i n case of cubic symmetry). exp(-2W) = exp(-2Bsin 2 ( ( J)/2)/X 2) , where B i s c a l l e d the temperature parameter. In cases of cubic symmetry i t i s shown i n appendix II that B = Tu^uT , *-18 where u. i s the displacement of an atom from i t s l a t t i c e s i t e . For cases of . 36 Figure 15. A sing l e Debye-Scherrer cone for scattering from a powder. lower symmetry the Debye-Waller factor i s not i s o t r o p i c and the fa c t o r i n equation 4.17 should be modified. However as an approximation an i s t r o p i c Debye-Waller f a c t o r i s often used and w i l l be used here. The f a c t o r ( s i n <j> sin(<(>/2)) * i n 4.17 i s the well known Lorentz f a c t o r . This factor a r i s e s p a r t l y from the geometry of the Debye-Scherrer method. It should be noted that i n a r e a l measurement a c a l c u l a t i o n of P(<t>) corresponds to the integrated i n t e n s i t y of the Bragg peak at angle ^k^-Using equation 4.17 i t i s a simple matter to c a l c u l a t e the expected inten-s i t i e s of the various powder peaks. Their p o s i t i o n s can be predicted using equations 4.15 and 4.16 f o r a hexagonal c r y s t a l . Figure 16. A l t e r n a t i v e forms of the specimen f o r the Debye-Scherrer method. 38 Chapter V THE NEUTRON SPECTROMETER A l l neutron d i f f r a c t i o n data were measured with the C5 spectrometer at the Atomic Energy of Canada l i m i t e d NRU reactor. The C5 spectrometer i s a t y p i c a l t r i p l e axis neutron spectrometer ( F i g . 17). I t consists of six major components, collimators, monochromator, monitor, analyser, detector and a neutron source, i n t h i s case the NRU reactor. The collimators used in the C5 spectrometer are S o l l e r s l i t s . These consist of many thi n metal plates which are plated with one thousandth of an inch of"cadmium and placed p a r a l l e l to each other. The spacing between plates i s -0.1" and the plates are about 12 inches long. The d i r e c t i o n of the wave-vector of neutrons passing through such a system w i l l be defined to within ±0.2°. Neutrons can be monochromated using a large s i n g l e c r y s t a l , i n our case the 113 planes of germanium. Neutrons passing through the f i r s t collimator of f i g u r e 17 s t r i k e the c r y s t a l and those scattered coherently w i l l obey the Bragg condition, nX = 2d,, nsin0 , 5.1 113 m where d.,~ i s the plane spacing of the 113 planes and 6 i s defined i n f i g u r e 113 m 17. By placing a second collimator at angle 26^, only neutrons obeying equation 5.1 w i l l be passed. We w i l l obtain neutrons of wavelengths X , \ Q / 2 , . XQ/3 where X n = 2d., 0sinG . 5.2 0 113 m To minimize the number of f r a c t i o n a l wavelength neutrons i n the beam, we consider the r e l a t i v e abundances of neutrons i n thermal equilibrium with the 39 MONOCHROMATOR NEUTRON SOURCE Figure 17. A schematic diagram of the major components of the C5 spectrometer. The symbol C designates a collimator. 40 Figure 18. Relative abundances of neutrons as a function of wavelength (After Bacon 1975) moderator as a function of t h e i r wavelength ( F i g . 18). We pick a \ 0 so that the abundances of AQ/2 and XQ/3 neutrons are small. By making a sui t a b l e choice of monochromator planes, we can pick a r e f l e c t i o n so that the structure factor f o r AQ/2 neutrons i s zero. The germanium 111 and 113 planes s a t i s f y t h i s condition. Another method used to remove the f r a c t i o n a l wavelength neutrons i s to place a f i l t e r (eg. p y r o l y t i c graphite), which has a higher absorption f a c t o r f o r low wavelength neutrons, i n the beam. This reduces the r a t i o of f r a c t i o n a l wavelength neutrons at the cost of some i n t e n s i t y i n the main X = \ Q beam. Neutrons e x i t i n g from the second collimator w i l l have a small spread i n wavelength and a small v a r i a t i o n i n d i r e c t i o n . The monitor i s a device which counts a f r a c t i o n of the neutrons which pass 41 through i t . A f t e r a c e r t a i n number.of monitor counts, the t o t a l number of neutrons that have passed through the monitor i s e s s e n t i a l l y a constant (within s t a t i s t i c s ) . The operation of the spectrometer i s d i r e c t l y linked to the monitor as follows. The detector i s activated f or a c e r t a i n number of monitor counts, then deactivated while the detector i s repositioned at a new scattering angle. The monitor i s necessary because the number of neutrons emitted by the reactor per unit time i s not constant over the long periods of time needed f o r a t y p i c a l powder p r o f i l e (=24 h r s ) . The neutron f l u x incident 14 _2 - -1 on the f i r s t collimator i s t y p i c a l l y 3 x 10 cm sec -at the NRU reactor. The f l u x i s g r e a t l y reduced by the monochromating process and fluxes measured 6 2 1 a f t e r the monitor are t y p i c a l l y 10 cm sec The analyser i s a second s i n g l e c r y s t a l and i s used when the energy d i s -t r i b u t i o n of the scattered neutrons i s to be measured. The analyser i s often omitted when doing e l a s t i c s c attering and was omitted f o r the measurements described l a t e r . There are several standard methods of detecting neutrons, a l l of which are dependent on a nuclear reaction of some type. The C5 spectrometer uses 3 a He detector i n which the reaction of i n t e r e s t i s ^He + *n —v *H + ^H + .76 Mev. 3 These hydrogen and t r i t i u m t r a v e l through the He gas producing ion p a i r s which accelerated to the cathode and anode by an e l e c t r i c f i e l d of approximately 150 volts/cm. This produces a current pulse which i s i n d i c a t i v e of a neutron entering the detector. The pulses produced by gamma rays are much smaller than neutron i n i t i a t e d pulses and can be discriminated against. The form of the output from the spectrometer i s a table of neutron counts vs. scattering angle. This data i s then entered into a large computer to be analysed and/or p l o t t e d . More de t a i l e d descriptions of the operation of neutron spectrometers can be found i n Bacon (1975), Iy.enger (1965) and Cocking and Webb (1965) A d e s c r i p t i o n of the mechanical construction of the C5 spectrometer can found i n the a r t i c l e by McAlpin (1964). 43 Chapter VI THE NEUTRON DIFFRACTION EXPERIMENT 6.1 Introduction Two experiments were designed which would hopefully give information about lit h i u m atom l o c a t i o n and be s e n s i t i v e to l i t h i u m ordering in:the TiS^ host. The f i r s t was to be a "dynamic" study, i n which a s p e c i a l l y designed L i / L i TiS^ electrochemical c e l l would be discharged while mounted i n the neutron beam of the C5 spectrometer. I t was hoped that i n t h i s way the neutron d i f f r a c t i o n p r o f i l e of L i x T i S 2 could be measured as a function of x. The second experiment involved " s t a t i c " measurements on several powdered samples prepared v i a the n-butyl l i t h i a t i o n technique. Measurements were taken on samples of TiS^, L i 1 0 T i S „ , L i 0 0 T i S 0 , L i ,,TiS„ and L i , T i S _ . .12 2 .33 2 .66 2 1 2 6.2 The "Dynamic" Neutron D i f f r a c t i o n Study S p e c i a l l y designed c e l l s were constructed f o r t h i s experiment and are depicted i n f i g u r e 19. Consideration was given to the neutron absorption and the sca t t e r i n g cross-sections of the c e l l components i n t h i s design. I t was decided that 6 separators would be necessary to ensure good c e l l operation even though t h i s would create a large background due to the e l e c t r o l y t e (PC contains hydrogen) incoherent scattering cross-section. Because of t h i s and the f a c t that PC was inserted into the TIS^ c r y s t a l l i t e s , the experiment proved only s l i g h t l y i n t e r e s t i n g . Two cathodes of grown at 550°C were prepared by pressing 12.0 grams of powder for each cathode into 2 inch diameter discs with a force of 30 tons. 44 Cell top (A) J J J J J , J J 7-y. y hole pressure plate ( A l ) 3 separators L i anode r z z z z z z z 7 z n ^ Z separators T i S 2 cathode 0 - r i n g R threaded \ hole cell base(AI) anode connecting tab (Ni) ( insulated from cell base) Figure 19. "Exploded" eut away view of the c e l l used f o r the "dynamic" neutron d i f f r a c t i o n experiment. E l e c t r i c a l connection to the L i anode was made by crimping the anode connecting tabs to the L i . The c e l l case formed the con-nection to the cathode. The anode connecting tabs were e l e c t r i c a l l y insulated from the c e l l case. Steel bolts were used to hold the c e l l together. The c e l l was "topped" up with e l e c t r o l y t e when assembled. 4 5 65^ ro o * . 55 ; Al III CO H 1 0 0 1 1 0 0 o 4 5 - ^ ^ p * * ^ ^ 3 bJ 2 or 0 0 2 ^ 35 i L i 101 0 10 20 3 0 4 0 5 0 6 0 S C A T T E R I N G A N G L E ( D E G R E E S ) Figure 20. Observed Room Temperature neutron d i f f r a c t i o n p r o f i l e f o r the f u l l y charged (x=0) c e l l . 46 The cathodes were given a one day soak i n the IM LiClO^/PC e l e c t r o l y t e i n the dry box p r i o r to c e l l assembly. Six large #2500 separators were prepared as described e a r l i e r and the c e l l was assembled using 5 two inch diameter li t h i u m sheets f o r the anode. Each l i t h i u m sheet had a thickness of .44 mm. The c e l l was then taken to Chalk River i n a f u l l y charged state. The c e l l was mounted i n the neutron beam i n the geometry of f i g u r e 16b. Data was c o l l e c t e d o using neutrons of wavelength 1.75709 A and i s shown i n f i g u r e 20. The high background i s due to the large incoherent cross-section of the e l e c t r o l y t e and the decrease of the background with scattering angle i s due to the absorption factor a r i s i n g from the geometry of the sample. Even so, peaks due to the TiS2, the l i t h i u m anode and the aluminum case are seen i n the data. I t was hoped that some q u a l i t a t i v e r e s u l t s at l e a s t could be obtained by discharging the c e l l . + 2.1 V D. C. POWER S U P P L Y 5 a 10 W Cell Figure 21. C i r c u i t used to hold the c e l l at 2.1 v o l t s . The c e l l was held at 2.1 v o l t s using the c i r c u i t shown i n f i g u r e 21 f o r a period of 4 1/2 days while monitoring the current. A plot of the current as a 15 10 < E 0 5 0 1 0 0 1 5 0 TIME (hrs) Figure 22. The current as a function of time while holding the dynamic neu-tron d i f f r a c t i o n c e l l at 2.1 v o l t s . The o r i g i n i n time corresponds to the closing of the switch i n f i g u r e 21. The arrows i n the f i g u r e i n d i c a t e b r i e f periods of open c i r c u i t i n g the c e l l . The arrow designated N indicates the s t a r t of the neutron d i f f r a c t i o n measurement. 48 function of time i s shown i n f i g u r e 22. The anomalous drops i n the current are spontaneous, while the sharp increases are due to b r i e f p e r i o d s . ( t y p i c a l l y 60 seconds) of open c i r c u i t i n g the c e l l . The t o t a l charge passed corresponded to x - .06 i n L i x T i S 2 by weight. The c e l l was transferred to the neutron spec-trometerrwhile connected to the 2.1 v o l t holding c i r c u i t and a neutron d i f f r a c -t i o n p r o f i l e was measured. This measurement was not completed due to a reactor shut down but the data that was r e t r i e v e d i s shown i n f i g u r e 23. We observe o that a peak at <J> = 5.7 has appeared which corresponds to a plane spacing of o o 17.67 A. The 001 peak of the TiS^ at <j> ^ 17.0 has disappeared completely. o The scan was interrupted at <$> - 25 by a reactor t r i p due to a thunderstorm. Aft e r t h i s measurement, the c e l l was dismantled and upon opening I found that the cathode material has swelled to occupy a l l the i n i t i a l l y u n f i l l e d volume of the c e l l and that v e r y i l i t t l e e l e c t r o l y t e was l e f t . The cathode material weighed 41 grams compared to i t s o r i g i n a l 24 grams. Due to the large l a t t i c e expansion and the missing e l e c t r o l y t e , I conclude that l i t h i u m ions solvated by PC were inserted into the cathode during discharge of the c e l l . The volume of the cathode increases by a factor of about three, so large stresses must have been present which may have caused the spontaneous drops i n current as shown i n f i g u r e 22. As we have seen e a r l i e r , t h i s p a r t i c u l a cathode material seems amenable to solvated L i + ion i n s e r t i o n (This was not known at the time t h i s c e l l was designed). The r e s u l t s of t h i s experiment allow us to conclude that the e l e c t r o l y t e i n s e r t i o n process occurs at voltages above 2.1 v o l t s . This i s i n agreement with the r e s u l t s discussed i n chapter three. 6.3 The S t a t i c Neutron D i f f r a c t i o n Experiment This experiment, u n l i k e the l a s t , yielded much us e f u l information. The l o c a t i o n of the l i t h i u m ions (atoms) i n the van der Waais ga,p of the host could 49 10 15 20 SCATTERING A N G L E ( d e g r e e s ) Figure 23. Observed room temperature neutron d i f f r a c t i o n p r o f i l e gf the c e l l at 2.1 v o l t s . Note the intense peak which has appeared at <j> = 5.7 . 50 be determined f o r each of the samples measured and no evidence for l i t h i u m ordering i n L i TiS„ was observed, x 2 6.3.1 Sample Preparation TIS^ was prepared at 800°C as described i n chapter 3. The n-butyl l i t h i a -t i o n technique (Dines 1975) was used to add l i t h i u m to these samples. The i r r e v e r s i b l e reaction of n-butyl l i t h i u m , C^H^Li, with T i S 2 i s as follows, xC.H^Li + T i S n —> L i T i S 0 +. 1/2x0-11, „ 6.1 4 9 2 x 2 8 18 producing Li xTiS2~arid i t s endpoint corresponds to approximately 1 v o l t versus l i t h i u m as discussed by Murphy and Carides (1979) . The endpoint of the n-butyl l i t h i a t i o n r eaction i s L i ^ T i S ^ which has been measured by chemical techniques (Dines 1975). We see from f i g u r e 11 that the endpoint at x = 1 i n Li xTiS2 m a ^ e s sense due to the large drop i n voltage of a L i / L i TiS„ c e l l at x = 1. I produced Li^TiS2 by adding an excess amount of 2.42 M n-butyl l i t h i u m i n hexane to 20. grams of TiS2> This sample was made i n si x 30 ml bot t l e s which were each f i l l e d with approximately 3.4 grams of HS2 i n the dry box and stoppered with septurns. The b o t t l e s were removed from the dry box and approximately 30 ml of the n-butyl l i t h i u m s o l u t i o n was added to each b o t t l e . ( T h i s i s an excess of 17 ml of n-butyl l i t h i u m per b o t t l e ) . The rate of reaction i s very f a s t , so the n-butyl l i t h i u m had to be added slowly, over a period of twenty minutes. A f t e r three days i n the n-butyl l i t h i u m s o l u t i o n the samples were decanted and rinsed twice with hexane, using a syringe to move the l i q u i d s through the septums. Excess hexane remaining a f t e r the second r i n s e was removed by pumping with a fore pump on each b o t t l e through a syringe needle inserted through the septum. The bot t l e s were then transferred to the dry box, the sample packed into a c y l i n d r i c a l aluminum sample holder and sealed with a t e f l o n plug and epoxy. Electrochemical c e l l s were constructed using t h i s material to obtain a check 51 on l i t h i u m concentration. Cathodes were constructed by simply s p r i n k l i n g some of the powdered L i TiS„ on a n i c k e l f o i l sheet. These were then used to x 2 construct c e l l s i n the manner described e a r l i e r . The open c i r c u i t voltage of these c e l l s ranged from l".80 to 1.83 v o l t s versus l i t h i u m which corresponds to .98 < x < 1.00 according to Thompson (1978). Constant current cycles of these c e l l s led to behavior i n d e n t i c a l to that of 15 ym TiS2 that had been discharged "quickly" to 1.6 v o l t s . The f r a c t i o n a l x-value samples were made while I was at Chalk River and 4 were mailed to mer.there . These samples were made i n a manner s i m i l a r to that used i n the production of L i ^ T i S ^ except that equation 6.1 was u t i l i z e d to obtain the';desired endpoint. Because the reaction i s i r r e v e r s i b l e o c a r e was taken to produce homogeneous samples. Five to ten m i l l i l i t e r s of hexane was added to approximately 11 grams of TiS^ i n an argon f i l l e d b o t t l e stoppered with a septum. The b o t t l e was kept at i c e temperature while a stoichiometric amount of n-butyl l i t h i u m s o l u t i o n was slowly dripped into the continuously s t i r r e d sample. This was done to hopefully slow down the reaction to enable each c r y s t a l l i t e i n the powder to contact an equal amount of n-butyl l i t h i u m . After several days'.:the sample was rinsed, evacuated and sealed i n an aluminum sample holder as before. The preparation of samples with x = .25, .33 and .66 was attempted. The aluminum sample holders ( f i g u r e 24). were designed to make the product, yR g, of the absorption constant, y, and the holder radius le s s than .4. This leads to an absorption f a c t o r , A.^^, whose dependence on scattering angle i s n e g l i g i b l e (Bacon 1975). Cal c u l a t i o n of the absorption constant y and the eff e c t of multiple and incoherent scattering i n these samples are considered i n Appendix I I I . A diameter, 2Rg = .475" and a wall thickness of .020" were decided upon. Aluminum was used because of i t s low incoherent and absorption 7+ Thanks to R.R. Haering who made these samples. 52 Figure 24. Cutaway view of an aluminum sample holder. Shaded areas are s o l i d . The scale i s 1:1. 53 cross-section. 6.3.2 Preferred Orientation Neutron d i f f r a c t i o n p r o f i l e s of the TiS2 s t a r t i n g materials and the l i t h i a t e d samples were measured and where possible a p r o f i l e refinement using the R i e t v e l d program was performed. A b r i e f discussion of the Rietveld program can be found i n appendix IV. As stated e a r l i e r , the Li^TiS2 c r y s t a l l i t e s comprising the samples were p l a t e l e t s of approximately 15 ym i n diameter. Because of t h i s , preferred 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 due to packing was anticipated and was l a t e r observed. The Rietveld p r o f i l e refinement program includes a cor r e c t i o n f o r preferred o r i e n t a t i o n . In order to correct the data by means of the p r o f i l e refinement program, the preferred o r i e n t a t i o n was measured. This was done by measuring the i n t e n s i t i e s of 100 and 002 peaks (which are separated by about one degree i n scattering angle) i n two positions of the sample holder. In the f i r s t p o s i t i o n the axis of the cylinder was perpendicular to the plane formed by the incident neutron beam and the detector. This i s c a l l e d the scattering plane. The axis of the cyl i n d e r was placed i n the scattering plane and normal to the incident neutron beam f o r the second measurement. Note that the e n t i r e p r o f i l e was measured i n the former geometry. The Rietveld program deals with preferred o r i e n t a t i o n i n the following manner. When the preferred o r i e n t a t i o n i s small, the preferred o r i e n t a t i o n c o r r e c t i o n can be written as I = I ,exp(-G a 2) 6.2 corrected observed where a i s the acute angle ( i n radians) between the r e c i p r o c a l l a t t i c e vector, hk£;, and the normal to p l a t e l i k e c r y s t a l s ( i n t h i s case 001). G i s the preferred o r i e n t a t i o n parameter and i s a measure of the halfwidth of the assumed Gaussian 54 d i s t r i b u t i o n of the normals about the preferred o r i e n t a t i o n d i r e c t i o n . G can be r e f i n e d by the program or measured and held f i x e d . 6.3.3 T i S 2 Figure 25 shows the neutron d i f f r a c t i o n p r o f i l e of the TiS^ used for the s t a r t i n g material of the n-butyl l i t h i a t i o n process. The calculated p r o f i l e 2 ( s o l i d curve) i s seen to be i n good agreement, x =1.5, R = 2.5 with the data. Peaks due to the aluminum sample holder were not included i n the f i t and are indicated i n the f i g u r e . Regions of the p r o f i l e included i n the f i t are i n -dicated by the bracketed regions i n the f i g u r e . A t o t a l of t h i r t e e n independent r e f l e c t i o n s have been included i n the f i t . It should be noted that the hO-c and h0£ r e f l e c t i o n s have d i f f e r e n t structure factors due to the t r i g o n a l symmetry of the unit c e l l although they appear at the same scattering angle. Regions of the data near the aluminum peaks were excluded from the refinement. This resulted i n the exclusion of the 102, 102 peak which i s located very close to the aluminum 200 peak. The 200, 112 aggregate peak i s half masked by the ex-c l u s i o n of the aluminum 202 peak. The least squares parameters included i n / the f i t were the l a t t i c e constants, a..and c, the scale f a c t o r , an o v e r a l l i s o -t r o p i c temperature parameter Q, and the preferred o r i e n t a t i o n parameter G. Descriptions of these-parameters can be found i n appendix IV. The preferred o r i e n t a t i o n f o r t h i s p a r t i c u l a r sample was not measured but was f i t t e d . The values of these parameters a f t e r refinement are shown i n table I I I . The o v e r a l l temperature f a c t o r , Q, i s i n d i c a t i v e of an average i s o t r o p i c v i b r a t i o n for a l l the atoms regardless of type or l o c a t i o n and i s defined as where u i s the displacement of an atom from i t s l a t t i c e point. The errors i n 55 5000 h- T iS, - 4000| cr o z O 3000\ or U J a. ,„ 2000 1-o o 1000 X •- 1.75709 A T = 300 K 001 101, 101 100 200. 112 AI 202 A l } M 1 1 0 4 , 104 I . . A l 222 10 20 30 40 50 60 70 SCATTERING ANGLE (DEGREES) 80 90 100 Figure 25. Observed (points) and calculated room temperature neutron d i f f r a c -t i o n p r o f i l e f o r TiS„. The regions included i n the f i t are enclosed by brac-kets. The background was f i t t e d by hand. Table III Least Squares parameters f o r HS2 o o o a(A) c(A) 'Q(A) G 3.4079±.0004 5.6989-.0006 .57±.06 -.0281.006 the table are calculated by the program and represent errors due to the s t a t -i s t i c s of the data only. They do not represent errors due to the sharpness 2 of the minimum i n X . The l a t t i c e parameters obtained are i n good agreement with those reported by C h i a n e l l i et a l . (1975b). The atomic pos i t i o n s used i n the c a l c u l a t i o n were those of C h i a n e l l i as "given i n chapter 2. I conclude that the TiS^ s t a r t i n g material was of good q u a l i t y . 56 6.3.4 L i 1 T i S 2 Preferred o r i e n t a t i o n was measured for the L i ^ T i S ^ sample i n the way des-cribed e a r l i e r . Figure 26 shows the measured data for the 002 and 100 peaks i n the two pos i t i o n s of the sample holder. The data taken with the axis of the c y l i n d e r normal to the s c a t t e r i n g plane i s shown i n f i g u r e 26a .and with the axis of the cylinder normal to the scattering plane i n f i g u r e 26b. Because the peaks are separated by only one degree i n scattering angle, we can neglect any scattering angle dependent absorption e f f e c t s when considering the r e l a t i v e i n t e n s i t i e s of these peaks. The halfwidths of the peaks are the same^. so the r a t i o of peak heights w i l l be equal to the r a t i o of integrated i n t e n s i t i e s . Using equation 6.2 and the fa c t that the angle a between the 100 and 002 r e c i p r o c a l l a t t i c e vectors i s ir/2 we f i n d I (100) I (002) , „ . , 0.2. —p n = exp(-2G(iT/2) ) 6.3 I (002)"I (100) P n where I (hk£) and I (hk£) are the measured integrated i n t e n s i t y of the hk£ p n peak with the sample axis p a r a l l e l and normal to the scattering plane respect-i v e l y . A value of G = .031 ± .002 i s calculated for our data. This parameter was held f i x e d at t h i s value throughout p r o f i l e refinement. Figure 27 shows the measured neutron d i f f r a c t i o n p r o f i l e f o r L i ^ T i S 2 and the calculated p r o f i l e based on l i t h i u m i n : a) tetrahedral s i t e s at (1/3, 2/3, 5/8) and b) octahedral s i t e s . I t i s c l e a r that the octahedral s i t e model produces a better f i t to the data. The same number of l e a s t squares parameters were used i n both cases except thermal parameters could not be used i n the tetrahedral s i t e model as they were refined to negative values which i s unphysical. The thermal parameters 57 a) o 8 8000 or o O 6000 or UJ Q. <o b) o or o 3 O u 4000 3 8 2000 8 4000 3000 2j 2000 o_ 1000 100 0 0 2 0 0 2 1 0 0 31 32 33 34 35 36 37 SCATTERING ANGLE (degrees) Figure 26. Preferred o r i e n t a t i o n measurement on the Li^TiS2 sample, (see text) a) 7 8 0 0 O O CvJ or o \-z o 5 or LU 0. l/l \~ z o o b) o o ( U or O z o a: a. 6 5 0 0 5200 3 9 0 0 2 6 0 0 L i , Ti S 2 X = I 75709 A T E T R A H E D R A L COORDINATION OF L I T H I U M 0 0 2 . 101, I0T I 12, 2 0 0 A I 2 0 2 7 8 0 0 h 6 5 0 0 5 2 0 0 3900 rj g 2 6 0 0 A l 2221 X = I 75709 A OCTAHEDRAL COORDINATION OF L ITH IUM 10 20 30 4 0 50 60 70 SCATTERING ANGLE (DEGREES) too Figure 27. Observed (points) and calculated room temperature neutron d i f -f r a c t i o n p r o f i l e s f o r L i ^ T i S ^ . The regions!included i n the f i t are enclosed by brackets. The background was f i t t e d by hand. Lithium i n a) tetrahedral s i t e s and b) octahedral s i t e s 59 were set to zero i n the tetrahedral case. The models compared as follows: 2 2 octahedral, R = 4.50, x = L 4 ; tetrahedral, R = 51.2, x = 10.4. In a 2 si m i l a r study on LiCrS^, (Van Laar and Ijdo 1971) x values of 10.3 and 6.0 fo r l i t h i u m in'.tetrahedral and octahedral s i t e s r e s p e c t i v e l y were used to state that the l i t h i u m atoms are located i n the octahedral s i t e s of the CrS2 2 host. Although the differ e n c e i n the x values I have obtained i s greater than the dif f e r e n c e reported by Van Laar and Ijdo, i t isLnot c l e a r that such a statement can be made without q u a l i f i c a t i o n . Any system w i l l e x i s t i n i t s lowest free energy state and i n the case of LiTiS2 i t may be favorable f o r some of the l i t h i u m atoms to reside i n tetrahedral s i t e s due to the large gain i n entropy t h i s disorder would cause. Calculations were made on the assumption that some l i t h i u m atoms could reside 2 in tetrahedral s i t e s . Figure 28 shows a plot of R and x v s percentage of li t h i u m i n tetrahedral s i t e s (holes). Least squared parameters included i n these f i t s were the l a t t i c e constants, a and c, the scale f a c t o r , an o v e r a l l temperature parameter Q, the octahedral s i t e l i t h i u m temperature parameter B . J_J1 and the s u l f u r atom z coordinate ( p a r a l l e l to the c- a x i s ) . The p o s i t i o n of the minimum was determined by the use of a constraint function in';the Rietveld program which allowed the percentages of l i t h i u m atoms i n octahedral and t e t r a -hedral s i t e s to vary subject to the condition that they sum to 100%. The min-imum occurs when 7% of the l i t h i u m occupies tetrahedral s i t e s . The values of the least squares parameters, assuming a l l the t e t r a h e d r a l l y coordinated l i t h i u m i s i n s i t e s at (1/3, 2/3, 5/8) obtained i n these r e f i n e -ments are given i n table IV. The p o s i t i o n of the minimum i n f i g u r e 28 i s un-changed by the assumption of equal l i t h i u m occupation of the tetrahedral s i t e s at +(1/3, 2/3, 5/8) and the values of the parameters are only s l i g h t l y d i f f e r -ent and therefore are not given. 0 I 1 1 1 u 0 10 2 0 30 % Li in tetrahedral holes Figure 28. a) X and b) nuclear R f a c t o r versus percentage of l i t h i u m occupying tetrahedral s i t e s (holes). Note the d i f f e r e n c e i n scale f o r the two curves. 61 Table IV Least Squares parameters f o r L i ^ T i S ^ % L i i n tetrahed-r a l s i t e a (X) c &) k P B L i P Sulfur Z coord. 0% 3.4590±.0003 6.18791.0006 .331.07 2.61.3 .238 1 .001 7% 3.45891.0003 6.18791.0006 .581.09 1.11.4 .239 1 .001 10% 3.4589±.0003 6.18791.0006 .701.08 .51.3 .240 1 .001 20% 3.45861.0004 6.18751.0008 .731.08 0 .246 1 .002 Two values f o r the l a t t i c e parameters of Li^TiS2 have been reported o (Whittingham and Thompson 1975 and Whittingham 1976b) : c = 6.187 A, a = 3.454 o o o o A and c = 6.195 A, a = 3.455 A. C l e a r l y my r e s u l t s , c = 6.188 A and a = 3.459 o A, are i n good agreement with these. We note that had the "sandwich" thickness (the thickness of an S-Ti-S layer) been the same i n TiS^ and L i ^ T i S ^ then the s u l f u r atom z coordinate would have been 0.229. The f a c t that the z coordin-ate i s 0.239 t. .001 i s due to the a t t r a c t i v e force between the l i t h i u m and s u l f u r atoms i n the material. One notices that the background of the Li^TiS2 data ( f i g u r e 27) i s much higher than that of the TiS2 s t a r t i n g material ( f i g u r e 25). This cannot be accounted f o r simply by the increased monitor s e t t i n g used to measure the Li^TiS2 sample. Also the increases i n incoherent cross-section and multiple scattering e f f e c t s due to the added l i t h i u m are too small to produce t h i s 62 change. Residual hexane l e f t i n the sample i s believed to be the cause of t h i s increase i n background. The L i ^ T i S 2 r e s u l t s show that octahedral s i t e s are d e f i n i t e l y favored with a maximum probable tetrahedral s i t e occupation of about 12%. I t should be 2 noted that the shallow minimum i n x a n Q R a t 7% l i t h i u m i n tetrahedral s i t e s may be due to the presence of one more adjustable parameter i n the Rietveld program. The f a c t . t h a t only 15 independent peaks are included i n the r e f i n e -ment f a c i l i t a t e s convergence of the program i n the case of large numbers of parameters. Decreasing the neutron wavelength may increase the confidence with which maximum tetrahedral s i t e occupancy can be stated because more peaks would be included i n the p r o f i l e . However, peaks separated by Acj> = 1 ° may not be resolved at lower wavelengths. Single c r y s t a l data could be used to a s s i s t us i n determining the per-centage of l i t h i u m i n tetrahedral s i t e s , but l i t h i u m i n t e r c a l a t e d s i n g l e cry-s t a l s of good q u a l i t y have yet to be prepared. This i s because s t r a i n s caused by l a t t i c e expansion are great enough to break up large c r y s t a l s . 6.3.5 L i TiS„ x 2 These samples were prepared by the n-butyl l i t h i a t i o n process so some error i n x i s expected. The dependence of the l a t t i c e constants as a func-t i o n of x has been measured by Whittingham (1976b) and by C h i a n e l l i et a l . (1978). This data was used as a check on the values of x obtained. I t was found .that the l a t t i c e constants of the L i „„TiS„ and L i ,,TiS„ samples were .33 2 .66 2 i n good agreement with those reported previously. The l a t t i c e constants of the L i sample, however, corresponded to those reported for L i ^ ^TiS^. To check t h i s , an electrochemical c e l l was discharged to 2.315 v o l t s versus l i t h i u m which corresponds to x = 1/4 and the cathode material x-rayed as 64 c) 7000 o If) - 5600 cc o t-z o cc QL 4200 W 2800 t-z O o 1100 L i , T i S ?  2 /3 2 X = I 75709 A T = 300 K 001 101, loi ' 102, .. .. 102 JL A I 202 112 *200* V . A l 311 * 103, " ' • . I01- • 203, '03 . • '24 ",3 005 " 203 " o . ZIM °°,4. •.; . 5. 202./ \:-< 10 20 30 40 50 60 70 S C A T T E R I N G A N G L E ( D E G R E E S ) 222 80 90 100 Figure 29. Observed room temperature neutron d i f f r a c t i o n p r o f i l e s f o r a) M 1 2 T i S 2 b) L i > 3 3 T i S 2 c) L i ^ T i S , , . 65 described e a r l i e r . L a t t i c e constants were i n good agreement with those of C h i a n e l l i et a l (1978) and Whittingham (1976b) for x = .25. I t was concluded that our L i ^^TiS^ sample had somehow only reached a l i t h i u m concentration corresponding to L i ^2^132' The reason f o r t h i s i s not c l e a r . The data obtained for x = .12, .33 and .66 samples i s shown i n f i g u r e 29a, b and c r e s p e c t i v e l y . This data was taken at room temperature with the axis of the c y l i n d r i c a l sample holder normal to the scattering plane. A neutron o wavelength of 1.75709 A was used again. In a l l cases the p r o f i l e s could be indexed c r y s t a l l o g r a p h i c a l l y on the basis of a unit c e l l containing one L i x T i S 2 u n i t . The dependence of the l a t t i c e constants on x i s shown i n f i g u r e 30. Figure 30. V a r i a t i o n of l a t t i c e parameters a(o) and c(o) vs. l i t h i u m con-centration as measured by neutron d i f f r a c t i o n . 66 Preferred o r i e n t a t i o n was checked on the samples and values obtained fo r the x = .33 and x = .66 samples were G = .043 ± .005 and G = .009 t .002 res p e c t i v e l y . In the case of the L i ^ ^T±S^ sample the;overlap of the 002 and 100 peaks made i t impossible to u t i l i z e the standard preferred o r i e n t a t i o n check. The r a t i o of the 100, 002 aggregate peak to the 101, 101 p a i r was found to be unchanged i n the two orientations of the sample holder. Preferred o r i e n t a t i o n f o r t h i s sample was therefore small. Differences i n preferred o r i e n t a t i o n are due i n part to the packing pressure that was used. Samples were generally l o o s e l y packed by l i g h t l y tamping them into the sample holder. Close examination of the data reveals that the 00-£ peaks are excessively wide. This i s due to the inhomogeneity of the samples a r i s i n g from the n-butyl l i t h i a t i o n process. From f i g u r e 30 we see that a range i n x produces a large v a r i a t i o n i n the c axis and only a small v a r i a t i o n i n a. We expect peaks strongly dependent on c to have halfwidths greater than'that given by the re s o l u t i o n function of the neutron spectrometer. The width of these peaks i s a d i r e c t measure of the degree of inhomogeneity i n the samples since we know the behaviour of c vs. x. Assuming a Gaussian d i s t r i b u t i o n i n x about o the mean we f i n d f o r the L i ^ 2^132 sample that the observed 0.9 halfwidth o of the 001 peak, combined with the 0.55 instrumental r e s o l u t i o n at <j> = 17° y i e l d s a concentration range of x = .33 i .1. Similar degrees of inhomo-geneity appear i n the other f r a c t i o n a l x samples. It turns out that t h i s inhomogeneity makes i t impossible to use the Rietveld program on the data. This i s because the re s o l u t i o n function of the spectrometer i s no longer the only f a c t o r c o n t r o l l i n g the widths of the Bragg peaks. Even i n the L i ^ ^TIS^ sample, where the range of c i s smallest, i t was found that the Rietveld program could not produce a good f i t to the data. However, the f i t did produce the best value f o r the mean l a t t i c e constants and t h i s i s why the error bars on a and c f o r L i c c T i S 0 6 7 are small i n f i g u r e 30. The Rietveld program was t r i e d on the the L i ^^TiS^ data but poor r e s u l t s were obtained. L a t t i c e constants were determined by hand f o r the x = .12 and x = .33 samples. By considering the r a t i o of the integrated i n t e n s i t i e s of peaks strongly dependent on l i t h i u m l o c a t i o n we can make a statement as to the p o s i t i o n of the l i t h i u m i n octahedral or tetrahedral s i t e s . As i s discussed i n appendix I, the i n t e n s i t i e s of the Bragg peaks f o r an L i TiS„ sample with l i t h i u m atoms randomly occupying octahedral s i t e s , say, are the same as those of the average structure (x l i t h i u m at each octahedral s i t e ) . As i s discussed l a t e r and i n appendix V the e f f e c t of l i t h i u m ordering i s to create super-l a t t i c e r e f l e c t i o n s . Peaks corresponding to the o r i g i n a l l a t t i c e w i l l have unchanged i n t e n s i t i e s . The r a t i o of the i n t e n s i t i e s of the 101, 101 p a i r to the 100 peak i s very dependent on l i t h i u m atom l o c a t i o n . Figure 31 shows calculated values of t h i s r a t i o as a function of x assuming li t h i u m i n octahedral or tetrahedral s i t e s . The measured values of the r a t i o are also plotted i n f i g u r e 31. The c a l c u l a t i o n assumed the s u l f u r atom z coordinate to be .250 and included the geometrical structure f a c t o r , Lorentz factor and the m u l t i p l i c i t y . The L i ^^T±S^ and L i ^^TiS^ data was handled by separating the combined i n t e n s i t y of the 002, 100 overlapping peaks into proportions according to the c a l c u l a t e d r a t i o f o r the i n t e n s i t i e s of the peaks using both octahedral and tetrahedral models. The octahedral s i t e model i s consistent while the tetrahedral s i t e model i s not. I t i s c l e a r from f i g u r e 31 that most of the l i t h i u m i s r e s i d i n g i n the octahedral s i t e s at these l i t h i u m concentrations. The error bars i n the f i g u r e are due to the i n a b i l i t y to p r e c i s e l y measure the integrated i n t e n s i t i e s and the e f f e c t s of preferred o r i e n t a t i o n . The data i n f i g u r e 29 shows no evidence of s u p e r l a t t i c e peaks. We must x in L i x T i S 2 Figure 3 1 . Ratio, I-^Q^ / ^ Q Q » ° ^ 1 0 1 , 1 0 1 pair to 1 0 0 peak as a func-t i o n of X i n L i TiS„: ' x 2. a) Lithium i n octahedral s i t e s b) Lithium i n tetrahedral s i t e s at ( 1 / 3 , 2 / 3 , 5 / 8 ) c) Lithium divided equally between tetrahedral s i t e s at i ( l / 3 , 2 / 3 , 5 / 8 ) . , Points shown are. the. values of the ratio:, f o r our data. 69 consider the expected i n t e n s i t i e s of such peaks and whether or not they should be v i s i b l e . As we have seen l i t h i u m p r e f e r e n t i a l l y occupies octahedral s i t e s f o r the f r a c t i o n a l x samples measured. The ordered state that the l i t h i u m atoms should f i l l at x = .33 i s given by f i l l i n g one of the three s u p e r l a t t i c e s of octahedral s i t e s , A, B or C as shown i n f i g u r e 8. At x = .66 two of these s u p e r l a t t i c e s would be f i l l e d . The p o s s i b i l i t y of staging should be considered as we l l . I t can be shown that the s u p e r l a t t i c e peaks generated for x = 1/3 and x = 2/3 w i l l have the same i n t e n s i t i e s due to the symmetry about x = 1/2 of f i l l e d and u n f i l l e d s i t e s . I w i l l consider the x = 1/3 case i n d e t a i l . Obviously there are many stacking sequences of f i l l e d s u p e r l a t t i c e s that we must consider and either two or three dimensional order i s possible. An example of three dimensional order i s the stacking sequence AAAA*•• . Random stacking i s the sole p o s s i b i l i t y f o r two dimensional order. The i n -t e n s i t i e s of the s u p e r l a t t i c e peaks created by simple stacking sequences exhibiting three dimensional order and the e f f e c t s of two dimensional order are calculated i n appendix V. Calculations show that f o r the stacking sequence AAAA*•• , the strongest s u p e r l a t t i c e peak should have an i n t e n s i t y equal to about 3% of the i n t e n s i t y of the strongest peak (100) i n fi g u r e 29b. This i s the 100 peak of the en-larged unit c e l l . This corresponds to a peak of height approximately 200 o o o counts at (j) = 19.7 f o r X = 1.75709 A, assuming a halfwidth of Ac)) = .5 . (Note that t h i s peak has no dependence on the c axis so the halfwidth should not be excessively wide). In the case of ABCABC**" stacking, which creates a three layer u n i t c e l l , the strongest r e f l e c t i o n should have a s i m i l a r i n -te n s i t y but be located at <J> = 20.5 . (This i s the 101 peak of the enlarged unit c e l l ) . Similar i n t e n s i t i e s are expected f o r other simple stacking 70 sequences e x h i b i t i n g three dimensional order. The case of random stacking, i n which allowed c[ vectors l i e along l i n e s i n r e c i p r o c a l space (Appendix V), creates peaks of almost t r i a n g u l a r shape. The 10 peak, which would be the most intense, would l i e at <J> = 19.7 and should have a peak height correspond-o o ing to approximately 80 counts. The L i ^ T±S^ data between 19 and 24 as well as calculated s u p e r l a t t i c e peaks assuming AAAA*•* and random stacking are shown i n f i g u r e 32. I t i s clear that no evidence for ordering i s seen i n the data. Similar conclusions are reached f o r the x = .12 and x = .66 data. The p o s s i b i l i t y that L i a n < i L i 66^"^2 a r e s t a § e t n r e e compounds 0 may be excluded due to the absence of a peak near cp = 5.6 corresponding to the 001 peak of the enlarged (3c) unit c e l l . One must be c a r e f u l about making a statement l i k e : "There i s no li t h i u m ordering i n the L i x T i S 2 samples studied'.'. F i r s t l y because there i s a spread i n x i n the samples, a reduction i n the i n t e n s i t i e s of the s u p e r l a t t i c e peaks could r e s u l t due to only a portion of the sample being i n the ordered phase. Secondly the standard deviation, a, of the background of the L i ^ T i S 2 data i s about 35 counts, so the height of the s u p e r l a t t i c e peak corresponding to two dimensional order i s only s l i g h t l y greater than 2a. Thus any statement made about the exclusion of l i t h i u m order must be made c a r e f u l l y . I believe that three dimensional l i t h i u m ordering i s excluded by my data but that better data i s needed to make a si m i l a r statement about the two dimensions a l l y ordered, randomly stacked case. To obtain better data i t w i l l be necessary to decrease the background and to increase the homogeneity of the samples. The background can be dee creased by using i s b t o p i c a l l y pure titanium, a major component of the ine:. coherent s c a t t e r i n g . Residual hexane i n the f r a c t i o n a l x-value samples i s believed to be present but i n smaller quantities than i n the L i TiS„ sample. 71 SCATTERING ANGLE (degrees) Figure 32. Q The observed (points) neutron d i f f r a c t i o n p r o f i l e of L i -joTiS between 19 and 24 . The s o l i d curve corresponds to the expected peak i f AAAA-•• stacking of ordered s u p e r l a t t i c e s occured. The dashed curve cor-responds to two dimensional order (random stacking). 72 Li, T iS , '3 X = 1.75709 A T = I 06 K Al 202 101, 101 I I 2, 200 102, 102 • 103, 103 in / 004 '. ' v 203 AI 222 1 0 20 30 40 50 60 70 S C A T T E R I N G A N G L E ( D E G R E E S ) 80 90 100 Figure 33. Observed neutron d i f f r a c t i o n p r o f i l e of L i T i S , at 106K. If removal of the hexane cannot be accomplished, deuterated hexane could be used. The samples could be made more homogeneous by.rfirst placing them i n an electrochemical c e l l where upon a l l c r y s t a l l i t e s would achieve the same -value of x. It i s believed that as the temperature i s lowered, eventually the l i t h -ium w i l l order at l i t h i u m concentrations l i k e 1/3. A neutron d i f f r a c t i o n p r o f i l e ( f i g u r e 33) of the L i . ^ T i ^ sample was taken at 106K. As we can see there i s again no evidence f o r l i t h i u m ordering. Further runs at low temperatures were not performed due to my return to U.B.C. but i t i s hoped that low temperature data can be c o l l e c t e d at a l a t e r date on better samples As the reader can see, the s t a t i c neutron d i f f r a c t i o n study y i e l d e d l o t of i n t e r e s t i n g r e s u l t s . The d e t a i l e d c r y s t a l structure of Li^TiS2 was i n -vestigated. Occupation of octahedral s i t e s by l i t h i u m atoms between 0 < x < i s c l e a r l y preferred over tetrahedral s i t e s with maximum tetrahedral s i t e occupation probably le s s than 10% of the i n t e r c a l a t e d l i t h i u m . No evidence for l i t h i u m ordering was observed and three dimensional l i t h i u m ordering can be excluded f o r the L i TiS„ samples measured. 74 Chapter VII CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK The experiments discussed i n chapter 3 force us to conclude that the PC i n s e r t i o n process i n electrochemical c e l l s i s dependent on TiS^ c r y s t a l -l i t e s i z e and discharge rate. The reasons f o r t h i s are not clear and further work i s needed to understand t h i s phenomenon. The l o c a t i o n of the l i t h i u m atoms i n the octahedral s i t e s of the T i S ^ host has been established f o r 0 £ x £ 1. No evidence of l i t h i u m ordering i n L i TiS„ was observed and three dimensional l i t h i u m ordering can be discounted x 2 on the basis of my data. The peak i n the inverse d e r i v a t i v e curve at x = 1/4 i s s t i l l not under-stood. The p r o b a b i l i t y that i t i s caused by l i t h i u m ordering i s low, although to discount t h i s e n t i r e l y , further work should be performed. Further work i s needed at low temperature to see whether or not l i t h i u m ordering does eventually occur as the temperature i s lowered. A s t r u c t u r a l study using neutron ans x-ray d i f f r a c t i o n techniques of L i x T i S 2 f o r x > 1 i s also i n order. 75 BIBLIOGRAPHY Ashcroft, N.W. and Mermin, N.D. (1976), S o l i d State Physics, Holt, Rinehart and Winston. Bacon, G.E. (1975), Neutron D i f f r a c t i o n , 3rd ed., Clarendon Press, Oxford. Berlinsky, A.J.; Unruh, W.G.; McKinnon, W.R. and Haering, R.R. (1979), S o l i d State Comm. 3_1_, 135. Bichon, J . ; Danot, M. and Rouxel, J . (1973) C.R. Acad. S c i . P a r i s C, 276, 1283. Blech, I.A. and Averbach, B.L. (1965), Phys. Rev. A 137_, 1113 Carslaw, H.S. and Jaeger, J.C. (1959), Conduction of Heat i n S o l i d s , 2nd ed., Clarendon Press, Oxford. C h i a n e l l i , R.R.; Scanlon, J . C ; Whittingham, M.S. and Gamble, F.R. (1975a) Inorg. Chem. 1A, 1691. C h i a n e l l i , R.R.; Scanlon, J.C. and Thompson, A.H. (1975b) Mat. Res. B u l l . 10, 379. C h i a n e l l i , R.R.; Scanlon, J.C. and Rao, B.M.L. (1978) J . Electrochem. Soc. 125, 1563. Chiu, J.C.H. and Haering, R.R. (1979) Paper presented at the 1979 Canadian Association of P h y s i c i s t s Conference. Cocking, S.J. and Webb, F.J. (1965) i n Thermal Neutron Scattering, E g e l s t a f f , P.A. ed., Academic Press. C u l l i t y , B.D. (1959) Elements of X-ray D i f f r a c t i o n , Addison-Wesley. Dahh, J.R. and Haering, R.R. (1979) Mat. Res. B u l l . _14, 1259. Dines, M.B. (1975) Mat. Res. B u l l . K), 287. Guinier, A. (1963) X-ray D i f f r a c t i o n , W.H. Freeman and Co. Henry, N.F.M. and Lonsdale, K. (1952) International Tables for X-ray C r y s t a l - lography, Vol. 1, Kynoch Press. Hewat, A.W. (1973) Harwell Report 73/239. Hooley, J.G. (1977) Mat. S c i . and Eng. 31_» 1 7 -Iyengar, P.K. (1965) i n Thermal Neutron Scattering, E g e l s t a f f , P.A. ed., Academic Press. J a s i n s k i , R. (1971) i n Advances i n Electrochemistry and Electrochemical Engineering, Vol. 8. ed. Tobias, C.W., Wiley & Sons. Jones, R.C. (1949) Acta Cryst. _2» 252. Laz z a r i , M.; Razzini, G. and S c r o s a t i , B. (1976) J . Power Sources 57. Le Nagard, N.; Gorochov, 0. and C o l l i n , G. (1975) Mat. Res. B u l l . J L O , 1287. Marshall, W. and Lovesy, S.W. (1971) Theory of Thermal Neutron Scattering, Clarendon Press, Oxford. McAlpiri,. W. (1964) Nuc. Inst, and Meth. 25, 208. McKinnon, W.R. _(1980) P M - Thesis, The Univ e r s i t y of B r i t i s h Columbia, Vancouver, Canada. Murphy, D.W.; DiSalvo, F.J.; H u l l , G.W. and Waszczak, J.V. (1976) Inorg. Chem. JL5, 17. Murphy, D.W. and Carides, J.N. (1979) J . Electrochem. Soc. 126, 349. Nagelberg, A.S. (1978) Phd. Thesis, The Univ e r s i t y of Pennsylvania. Rietveld, H.M. (1969) J . Appl. Cryst. 2, 65: Rouxel, J . ; Danot, M. and Bichon, J . (1971) B u l l . Soc. Chim. France 11, 3930. Thompson, A.H.; Pisharody, K.R. and Koehler, R.F. (1972) Phys. Rev. L e t t . 29_, 163. Thompson, A.H.; Gamble, F.R. and Symon, C.R. (1975) Mat. Res. Bull.110, 915. Thompson, A.H. (1978) Phys. Rev. L e t t . 40, 1511. Thompson, A.H. (1979) J . Electrochem. Soc. 126, 608. Van Laar, B. and Ijdo, D.J.W.((1971) J . S o l i d State Chem. J3, 590. Wainwright, D.S. (1978) MSc. Thesis, U n i v e r s i t y of B r i t i s h Columbia, Van-couver, Canada. Warren, B.E. (1941) Phys. Rev. 59, 693. Whittingham, M.S. (1974) J. Chem. Soc. Chem. Comm. p. 328. Whittingham, M.S. and Thompson, A.H. (1975) J . Chem. Phys..^62, 1588. Whittingham, M.S. (1976a) Science 1^2, 1126. Whittingham, M.S. (1976b) J. Electrochem. Soc. 123, 315. Whittingham, M.S. (1978) Prog. S o l i d State Chem. JjZ, 41. Wyckoff, R.G. (1963) C r y s t a l Structures V o l . 1, Wiley and Sons. 77 Appendix I INCOHERENT NEUTRON SCATTERING AND SCATTERING FROM A RANDOMLY OCCUPIED LATTICE For a c r y s t a l with N atoms and volume V, a r i g i d three dimensional l a t t i c e and one nucleus of scattering length b at each l a t t i c e point, R^, the d i f f e r e n t i a l scattering cross-section, do , i s N N d ^ ^ = b I J exp (ia-CR^ - R^,)) . 3.8 n=1 n =1 If we set R = R - R , we can rewrite 3.8 as —ui —n —n da 2 N d i T b £ I e x P ' A K 1 m=l n . The l i m i t on the sum over n must now be considered c a r e f u l l y . The vector R i s the displacement from R to R .. Thus only points of the c r y s t a l , —m —n —n J ' R , such that R - R i s also i n the c r y s t a l should be summed i n the sum over —n —n —m J n. These points can be expressed by the function v(R ) defined by f i g u r e 34. (This i s the function defined by Guinier (1963J). The shaded volume, V*v(R m) , i n the f i g u r e i s the.volume common to the c r y s t a l and i t s "ghost" displaced by R . The number of l a t t i c e points i n the volume V*v(R ) i s simply —m —m V-v(R ) v ^ = N v(R ) where V Q i s the volume of the unit c e l l . Equation A l . l becomes da 2 N — = b N T v(R ) exp(iq-R ) . A1.2 du i —m - 1 —m m=l The reason f o r introducing V(R ) w i l l become c l e a r shortly. —m Now we consider a c r y s t a l of volume V with N atomic or molecular units with structure factor F (^ ) located at each l a t t i c e point. Equation 4.7 gives Figure 34. The function V(R ). The c r y s t a l i s depicted by the s o l i d l y outlined sphere and i t s "ghost" by the dashed sphere. n=l n -1 and setting R = R - R , —m —n —n as before we obtain f = 11 ( F n ( i ) F * _ m ( ^ ) ) e x p ( i ^ . V m n m Defining y (q) = F (q) F* (q) m - 1 n n-m we have § = N I V C I O y ^ ) e x p d ^ ) . m We write F (3) = F (£) + <j) (3) n n n where ^ ( 3 ) i s the perturbation to the average value of the structure factor. F^Cg) . Using the d e f i n i t i o n s of <j> (cj) above, we obtain 4 (3) = 0 and ym(a) = l-V I^ ^ n ^ n - m ^ where the average i s taken over n. Denoting y<i) = • n(£)+ n^(£) . equation A1.3 becomes , -2 i = N [ v ( R ) |T (a) I exp(ig/R ) + du  u —m n —m m rf N T v(R ) $ (q) exp(iq-R ). u —m m ^  - 1 —m m If there i s no c o r r e l a t i o n between § (q) and d> (q) , (random v a r i a t i o n of n -J* n-m -*• the structure f a c t o r from s i t e to s i t e ) then * (3) = 6 . m mo Using the d e f i n i t i o n of $ m(g) We note that v(R =0) = 1 —m so N I v Q O |Fn(s>| e x p d ^ ) + + N!(F (3) F (3) - F-.(a) ) A1.4 n n n -jt The f i r s t term corresponds to the scattering from the average l a t t i c e and i s termed the coherent d i f f e r e n t i a l s c a t t e r i n g cross-section. The second termf.is referred to as the incoherent d i f f e r e n t i a l s c a ttering cross-section. For the case of F (3) = b , a monatomic r i g i d Bravais l a t t i c e , equation A1.4 reduces r e a d i l y to equation 4.10 using 4.8 and 4.9. For most elements the incoherent cross-section i s appreciable due to the existence of many isotopes and the fac t that the nucleus can have non-zero spin. Equation A1.4 can also be used to describe the case of a randomly f i l l e d c r y s t a l l a t t i c e . For instance, consider a r r i g i d l a t t i c e of N s i t e s of which a:":fraction x are f i l l e d with n u c l e i of scattering length" b. We f i n d that j „ N N •77- - N (xb) I I exp(iq-(R -R )) + N x(l-x) b . ail 1 , , —n —m n=l m=l The coherent portion of the scattering i s again given by the sca t t e r i n g from the average l a t t i c e , while randomness introduces an incoherent portion to the cross-section. When we want to c a l c u l a t e the coherent cross-section from a.rraridomly occupied l a t t i c e , we need only consider the average l a t t i c e Appendix II THE DEBYE-WALLER FACTOR For a r i g i d monatomic Bravais l a t t i c e the coherent d i f f e r e n t i a l scatt-ering cross-section i s || = (b) 2- | f expUi-R^I? . A2.1 n=l This i s equivalent to | | = ( b ) 2 | I d 3 r p(r) exp(i^.r) | 2 A2.2 where p(r) = I 6(r-R^) . n Lets consider the case where the atoms can be displaced from t h e i r l a t t i c e points by u . We learned that the coherent e l a s t i c c ross-section r J —n i s the same as that of the average lattice(Appendix I ) . Therefore | | = ( b ) 2 | / d 3 r ~p~M expCicL-R^) f A2..3 where ~PM = I SCr-P^-u^) n Using the Fourier representation f o r the de l t a function, p(r) = - I / d 3k exp(ik-(r-R )) exp(-ik-u ) . A2.4 (2 T T ) 3 n -n n Now we use Bloch's i d e n t i t y ( M a r s h a l l and Lovesy 1971), exp(Q) = exp(- ( Q )) , 82 where Q must be a l i n e a r combination of Bose operators, can be expressed i n terms of phonon modes (see equation 4.16 i n Marshall and Lovesy 1971). Therefore -1 exp(-ik-u ) = e x p ( - ~ (k>u ) ) n z n Inserting t h i s back into equations A2.3 and A2.4 we f i n d ( a f t e r doing some easy i n t e g r a l s ) , = ( b ) 2 exp(-(£-u n) 2: ) | I exp(i£-R n) | 2 n This i s simply the r e s u l t f o r a r i g i d l a t t i c e (A2.1) m u l t i p l i e d by the Debye-Waller f a c t o r , 2 exp( -(ci/v^) ) = exp(-2W) In cases of cubic symmetry, Because and f - - Y 2 1 2 2 -*- ^ n 3 —-n 2 16TT 2 . 2.,._, 1 = - 2 s i n (t|)/2) \ = 2 B s i n 2 ( ^ / 2 ) x 2 we obtain 8ir 2 ~2 B = — - — u 3 —n i n cases of cubic symmetry. B i s the temperature parameter defined i n chapter IV. 83 Appendix III ABSORPTION AND BACKGROUND OF L i TiS„ SAMPLES x 2 The mass absorption c o e f f i c i e n t s , u/p, for l i t h i u m , titanium and s u l f u r o at a neutron wavelength of 1.08 A are 3.5, .044 and .0055 re s p e c t i v e l y i n 2 units of cm /gram (Bacon 1975). The l i n e a r absorption c o e f f i c i e n t of a * compound, i s given by y c - P C I ( y / p ) ^ X th where i s the density of the compound, w^  i s the proportion of the i element in the compound by weight and the sum i s taken over a l l elements in.the compound. Using t h i s r e l a t i o n we f i n d that the absorption c o e f f i c i e n t , o u(x), of L i TiS„ at X = 1.08 A i s given by (assuming the density of L i TiS„ X X z. i s the same as that of T i S 2 ) y(x) = .070 + .71x cm"1 The absorption c o e f f i c i e n t i s proportional to wavelength (Bacon 1975) o so at X = 1.75709 A we have u(x) = .11 + 1.15x cm - 1 A3.2 This i s the absorption c o e f f i c i e n t for s o l i d Li^TiS,-,. The c y l i n d r i c a l sam-ples were not s o l i d but loosely packed c r y s t a l l i t e s with a packing f r a c t i o n of approximately .5. The packing f r a c t i o n i s determined by the volume of the sample holder, the mass of the sample and the density of Li^TiS.^. Thus the absorption c o e f f i c i e n t of the sample w i l l be one half that of the s o l i d . This leads to a product, y ( l ) R = .38 s for our sample holders f i l l e d with L i ^ T i S 2 - The angular dependence of the absorption f a c t o r , A^^s i s small for t h i s value of ryR^. For example, an 84 o o error of 2% i n the r e l a t i v e i n t e n s i t i e s of peaks near <j> = 0 and <f> = 90 i s introduced by neglecting when uR g = .4 (Bacon 1975). The background observed in the data a r i s e s p r i m a r i l y from the incoherent and multiple s c a t t e r i n g cross-section of the sample and incoherent scattering from any r e s i d u a l hexane present. Other sources include thermal d i f f u s e s c a t t e r i n g , paramagnetic scattering and the normal background found i n a reactor environment. The incoherent and multiple scattering cross-section are easy to c a l -culate. The incoherent cross-section of one molecular unit of the L i TiS„ x 2. sample, o\(x) , i s simply the sum of the incoherent cross-section f o r each element. We f i n d a.(x) = 3.4 + .6x barns. l M u l t i p l e scattering of neutrons i n c y l i n d r i c a l samples has been treated by Blech and Averbach (1965). I t i s shown that the multiple scattering cross-section, a , i s given by m J a as(.°s/a+)S A3.3 m l - ( a s / a + ) S where a i s the s c a t t e r i n g cross-section (coherent and incoherent), a + i s the t o t a l cross-section (scattering and absorption) and 6 i s a f a c t o r de-pendent on u(x) and the sample geometry. For one molecular unit of L i TiS„ we f i n d that a (x) = 6.8 + 1.2x barns, s A3.4 a (x) = 10.9 + 41x barns and 6(x) =-u(x)R g for the sample geometry used. Using equations A3.2, A3.3 and A3.4 we can compute a (x) m As we have seen i n appendix I, there i s a contribution to the incoherent 85 cross-=section due to the randomness of s i t e occupation by the l i t h i u m . This cross-section, a , i s given by 0 = x ( l - x ) 0 • • . A3.5 r c L i where' 0 i s the coherent scattering cross-section of the l i t h i u m . c L i i r -P l o t s of 0., 0 and 0 as a function of x are given i n f i g u r e 35. We 1 m r see that the incoherent cross-section of the titanium contributes about 75% to the t o t a l incoherent cross-section ( a . m . = 3.0 barns) i f the e f f e c t s 1T1 of r e s i d u a l hexane, thermal d i f f u s e and paramagnetic scattering are neglected. Figure 35. Contributions to the incoherent cross-section of L i TiS„ as a function of x. The curve f o r 0 i s the dashed curve. x 86 Appendix IV THE RIETVELD PROFILE REFINEMENT PROGRAM Rietveld's (1969) p r o f i l e refinement program modified by A.W. jHewat (1973) i s the state of the art i n structure refinement from neutron powder p r o f i l e s . This method does not use the integrated i n t e n s i t i e s of the powder peaks, si n g l e or overlapping, but u t i l i z e s the d e t a i l e d structure of the act u a l p r o f i l e obtained from the spectrometer. A l e a s t squares refinement i s performed on many parameters i n the usual way. It i s an empirical f a c t that the shape of powder peaks i s almost ex-a c t l y Gaussian f o r scattering angles above 30 degrees. Rietveld convolutes the delta function Bragg r e f l e c t i o n s with a Gaussian, representative of the spectrometer r e s o l u t i o n function. Because the spectrometer i s step scanned, the number of neutron counts expected at the i t b counting point, y , i s the quantity of i n t e r e s t . Rietveld writes 2 , , 2 / E n 2 " k ' i = I t <Fk> j-kL* k tl vn k x exp[-4£n2{((|>i-cf,k)/Hk}2] A4.1 where t = step width of the detector, t i l F^ = structure f a c t o r of k r e f l e c t i o n (including Debye-Waller f a c t o r ) , = m u l t i p l i c i t y df k t b r e f l e c t i o n , L^ = Lorentz f a c t o r at scattering angle, corresponding to the k*"*1 r e -f l e c t i o n , = calculated p o s i t i o n of k r e f l e c t i o n , th <!> = the s c a t t e r i n g angle of the i counting point, and H^ = the f u l l width at half height of the Gaussian. No c o r r e c t i o n i s 87 Figure 36. The spectrometer r e s o l u t i o n function. The points are the measured values from f i g u r e 25 and the s o l i d l i n e i s the Rietveld program's f i t to these points. 88 made for the absorption of neutrons i n the sample, so samples of MR £ .5 s must be used. At low scattering angles the peak begins to deviate s l i g h t l y from Gaussian. Rietveld includes a semi-empirical co r r e c t i o n f a c t o r i n equa-t i o n A4.1 which gives a good approximation to the asymmetric peak p r o f i l e . The widths of the peaks as a function of scattering angle can be written as H k = U tan2(<}>k/2) + V tan((f> /2) + W , \ A4.2 where U, V and W are the half width parameters (Rietveld 1969). A plot of the measured half widths of the TiS^ sample versus scattering angle and the f i t by the Rietveld program from A4.2 i s shown i n f i g u r e 36. In general the ha l f width parameters were f i t to each set of data by the l e a s t squares refinement program. The program allows for co r r e c t i o n of preferred o r i e n t a t i o n a r i s i n g from the packing of p l a t e l i k e c r y s t a l l i t e s . This i s discussed i n Chapter 6. The peak posit i o n s are determined by the l a t t i c e parameters, a, b, c, a, 3, y» t n e neutron wavelength and the zero point of the detector. These parameters were r e f i n e d (except the neutron wavelength) i n a l l cases, even i f they were well known as i n the case of TiS2, because they have a large e f f e c t on the p r o f i l e . The s t r u c t u r a l parameters describe the contents of the unit c e l l and are used to c a l c u l a t e the structure f a c t o r . They include : Q = the o v e r a l l i s t o t r o p i c temperature parameter, x , y , z = the f r a c t i o n a l coordinates of the m*"*1 atom i n the unit m m m c e l l , B = the i s o t r o p i c temperature parameter of the ra~^ atom, m n^ = the average occupancy of s i t e m th and B m i >, = i j .. component of the anisotropic temperature tensor f o r t tl the m atom. The coherent scattering lenghts of the atoms i n the unit c e l l are f i x e d and input to the program by the user. The structure f a c t o r , F., i s written as (neglecting anisotropic temperature parameters which were not u t i l i z e d ) F ± = exp(-Qsin2(<t> ,./2) / \2) x x Y b n exp["2iri(hx + ky + £z )1 x L m m L m Jm m J m x exp(-B sin2((f>./2) / A 2) . A4.3 m 1 The program allows l i n e a r and quadratic constraint functions to be placed on the parameters i f desired. This option was used at one point to allow the percentage of l i t h i u m atoms i n octahedral and tetrahedral s i t e s to vary'subject to the constraint that they sum to 100%. The background, g^, of the data i s f i t by l i n e a r i n t e r p o l a t i o n between c h a r a c t e r i s t i c background points input to the program by the user and i s then subtracted from the data. Certain user defined regions of the p r o f i l e can be masked out (not included i n the refinement) which i s u s e f u l i n ridding the p r o f i l e of peaks due to the aluminum sample holder. A l e a s t squares f i t to the data (minus background) i s performed by min-imizing the quantity 2 1 2 X = I w.|y.(obs) - — y . ( c a l c ) | / N-P A4.4 i where : c i s a scale f a c t o r used to scale the calculated values to the observed (c i s constant f o r a l l i ) , til w_^  i s the s t a t i s t i c a l weight assigned to the i counting point, N i s the number of counting points and P i s the number of least squares parameters. The s t a t i s t i c a l weight.is where a i s the standard deviation of the observed neutron count, y\(obs) + P . . 1 The program calculated quantities which are i n d i c a t i v e of the q u a l i t y of the f i t to the data. The nuclear R f a c t o r , R = 100 I 11 (obs)' - I ( c a l c ) | * 7 I. (obs) k k i s c a l c u l ated by the program. In t h i s expression I i s the integrated i n -t e n s i t y of the k'"'1 peak. A nuclear R f a c t o r of 10 i s i n d i c a t i v e of 10% d i f -ference i n observed and calculated peak i n t e n s i t i e s on average. R factors o l e s s than 5 are considered i n d i c a t i v e of reasonable f i t s to powder p r o f i l e s by researchers i n t h i s f i e l d (Rietveld 1969 and Van Laar and Ijdo 1971). A copy of t h i s program i s on permanent f i l e at U.B.C. i n my permanent f i l e -space. 91 Appendix V SUPERLATTICE EFFECTS Lithium ordering i n the host van der Waals gaps may occur i n L i x T i S 2 . I w i l l now consider the magnitude of s u p e r l a t t i c e peaks which are the con-sequence of presumed ordering i n L i ^2TiS2» Ordering at x = 1/3 w i l l consist of a l l l i t h i u m atoms located on one of the three s u p e r l a t t i c e s of octahedral s i t e s , A, B or C ( f i g u r e 8) i n each van der Waals gap. There are two cases we must consider, three dimensional order and two dimensional order. A5.1 Three Dimensional Order In the case of AAA* • • stacking«!..the a axis of the unit c e l l i s enlarged to /3a. A t o t a l of ten atoms, 1 l i t h i u m , 3 titanium and 6 s u l f u r are con-tained i n the unit c e l l . The geometrical structure f a c t o r becomes, / b L ± exp(iir£) + 3 F T ± S h + 2k = 3s F(hk£) =1  2 [b exp(i7r£) h + 2k 4 3s where F T i S 2 " b T i + 2 b s ( - s f (h +k)) , s i s an integer and b i s the coherent scattering length of atom r . The integers h, k and £ are the coordinates of _£ with respect to the recipror-c a l l a t t i c e vectors of the enlarged, /Ja, unit c e l l i n r e a l space. It can be shown that the condition h + 2k = 3s i s equivalent to the vector hk£ being a vector of the r e c i p r o c a l l a t t i c e of the host l a t t i c e . In t h i s case the structure factor i s the same as three times the structure f a c t o r 92 f o r the average l a t t i c e , i . e . F(hk£) = 3[F T._ + 1/3 b exp(iTPc)] . A5.1 We observe that ordering does not a f f e c t r e f l e c t i o n s corresponding to r e -c i p r o c a l l a t t i c e vectors of the host. The case h + 2k 4 3s produces s u p e r l a t t i c e peaks. In a l l cases |F(hk£)| 2 = b L . 2 f o r these r e f l e c t i o n s . The most intense peak w i l l be the one with the lowest scattering angle due to the Lorentz f a c t o r , i n t h i s case the 100 peak. Neglect-ing Debye-Waller terms we f i n d that the integrated i n t e n s i t y , 1(100) of the 100 peak i s 6 x b 2 1(100) a L l s l n ( W s i n ( W 2 ) where 6 i s the m u l t i p l i c i t y of the 100 r e f l e c t i o n and <J>^ QQ i s the scattering angle. We would l i k e to compare t h i s with the integrated i n t e n s i t y of an intense peak of the host l a t t i c e . The 110 peak of the enlarged unit c e l l i s the 100 peak of the host l a t t i c e . We f i n d 9(l/3b T . + b T . - b ) 2 x 6 sm(<f>110) sm((f> 1 1 0/2) where ^-QQ i s the scattering angle and 6 i s m u l t i p l i c i t y . The f a c t o r of 9 a r i s e s from equation A5.1. Inserting values f o r the relevent scattering lengths and c a l c u l a t i n g the scattering angles from the l a t t i c e parameters and the neutron wavelength.we_find that <f> = 19 7 ioo i y , / i> =34.6 110 1(100) a 4.73 and The expected i n t e n s i t y of the 100 peak i s 3.0% that of the 110 peak (100 of the host). The case of ABCABC*'' stacking, which creates a three layer unit c e l l , i s s i m i l a r . It turns out that we can separate the structure factor of the host from that of the l i t h i u m as before. We f i n d that the 101 peak of the enlarged unit c e l l i s the strongest s u p e r l a t t i c e peak with an i n t e n s i t y 3.0% that of the 110 peak (100 of the host). Other simple stacking sequences produce s i m i l a r e f f e c t s . In a l l cases of three dimensional order the strongest possible s u p e r l a t t i c e peak i s less than or equal to 3% of the 100 peak of the host l a t t i c e . A5.2 Two Dimensional Order In t h i s case the f i l l e d s u p e r l a t t i c e i n each van der Waals gap i s chosen at random from A, B or C.' As before i t turns out that the host does not con-t r i b u t e to s u p e r l a t t i c e peaks so we w i l l neglect the host and simply consider the e f f e c t s of the l i t h i u m . v We pick a unit c e l l with 3 octahedral s i t e s , one from each s u p e r l a t t i c e , A, B or C, i n the basal plane but with only one s i t e f i l l e d . The f i l l e d s i t e i s the same i n any given plane, but varies randomly from plane to plane. The f i l l e d s i t e s are located at r = ma + nb + pc + r -mnp - - r- - -p., where a., b_ and c^  are the Bravais l a t t i c e vectors corresponding to the unit c e l l discussed above. The dimensions of the c r y s t a l are given by 0 < m < N^, 0 < n < N 0 and 0 < p < N- and r i s either ( i n f r a c t i o n a l atomic coordinates) r x = (0,0,0) or r = (1/3, 2/3, 0) ~P 2 Ip3 " ( 2 / 3 ' 1 / 3> 0 ) - -^,2' The occupied s i t e i n each layer i s picked at random from r ,, r _ and r „. — p i ~p2 —'pi We write = I I I I V P ) eXp(i_q_-r ,) .:x m n p j = l -PJ x exp(icL' (ma + nb + pc)) where . b.(p) = 1/3. b_ . 11 + 2 c o s ( i l j + e ) | . J L i 1 3 J p y 1 6^ va r i e s randomly among the three values 0, 2TT/3 and 4ir/3. The values of K ( p ) as a function 0^ are given i n table V. Table V K (p) vs 0 p e p br(pH - b 2(p) b 3( P) 0 b L i 0 0 2TT/3 0 b L i 0 4TT/3 0 0 b T . L i sum Rewriting bj-(p) i - n terms of exponentials, _q_ i n r e c i p r o c a l space (g_ = hb^ + kb_2 + ^ 3 with h, k and £ not ne c e s s a r i l y integers) , carrying out the over j and doing a few steps we f i n d F(hkc) =H 1/3 b exp(2iri(hm + kn)) ,'x L i l ' m n x I exp(2i:i£p) [f (h + 2k) + f ( h + 2k+1) exp(iOp) + P + f (h + 2k -1) exp(-iOp)'] . In t h i s expression f(x) = 1 + 2cos(2ux/3) The d i f f e r e n t i a l s c a ttering cross-section i s , neglecting Debye-Waller terms, 2 1 da c = 1_ 2 dfi 9 L i . s i n T r h 2 • s i n ^ T T k ^ J s i n i T k X 11 p p' f 2 ( h + 2k) + f 2 ( h + 2k + 1) e x p ( i 0 p - i 9 p I ) + + f (h + 2k - 1) exp(i6 - ie ,) + P P + f ( h + 2k) f ( h + 2k + 1) [exp(i9 p) + exp(i0 ,)] + + f ( h + 2k) f ( h + 2k - 1) [exp(-;ie ) + exp(i8 ,)] + + f ( h + 2k +• 1) f ( h + 2k -1) fexpde - 18 ,) ' + P P ' + exp(-iO - i6 ,)1 P P 1 where I have used equations 45 and 511 i n chapter 9 of Crawford (1968) We observe the following points. 1) If N i s large, 9 rsin N T T X A5.2 c s m 1 7 -1 . t • J ~ 0 > sinux unless x i s i n t e g r a l . 2) If t i s an integer, then r o f ( t ) = t 4 3s t = 3s, for s i n t e g r a l . Thus for a given i n t e g r a l p a i r of h and k only one of f ( h + 2k), f ( h + 2k + 1) and f ( h + 2k - 1) w i l l be non-zero. 3) The average values, exp(i6 ) = exp(ie ,) = exp i(6 +6 ,) = 0 P P P P and exp i(9 - e ,) = $ , P P PP hold i n t h i s case. The cross terms i n equation A5.2 can eliminated using the above ob-servations and i n the case when h and k are i n t e g r a l and h + 2k = 3s we obtain da 2 — = b dft L i This i s the standard form f o r a three dimensional r e f l e c t i o n from a f i n i t e l a t t i c e (Warren 1941). When h + 2k 4 3s we obtain 2 sinN ^ i T h 0 /_ smN^irk 2 'sinN 3 T r £ ' ^ • simrh -1 simrk ' sinTr£ ) da c T.2 ,T dJT = b L i N3 sinN^Trh sinfrh fsinN T r k l A5.3 simrk There i s no dependence on Z i n the above equation, so allowed r e f l e c t i o n s are found on l i n e s i n r e c i p r o c a l space corresponding to h + 2k 4 3s with h and k i n t e g r a l . I t can e a s i l y be shown that the condition h + 2k = 3s i s equivalent to the r e c i p r o c a l l a t t i c e vector being a r e c i p r o c a l l a t t i c e vector of the host l a t t i c e (This assumes 3, the angle between a. and JD i s chosen to o be 120 ). This means that random stacking produces s u p e r l a t t i c e peaks of two dimensional character when h + 2k 4 3s and delta function Bragg peaks when h + 2k = 3s. When h + 2k = 3s, the scattering i s the same as that from a l a t t i c e with the l i t h i u m atoms placed at random i n the octahedral s i t e s . Therefore there i s no change i n i n t e n s i t y of peaks corresponding to r e c i -procal l a t t i c e vectors of the host at the onset of two dimensional order. Equation A5.3 i s the r e s u l t obtained by Warren (1941) f or d i f f r a c t i o n from two dimensional l a t t i c e s which were stacked with random s h i f t s para-l l e l to the layer s . Warren's r e s u l t i s computed f o r the case of x-ray d i f -f r a c t i o n and with small changes can be u t i l i z e d f o r neutron d i f f r a c t i o n . The i n t e n s i t y as a function of scattering angle f o r measurements using the Debye-Scherrer method i s f Q N l N 2 N 3 ^ 2 c b L i J ? ? Try v > sin("<fr/2) ( s i n (<f,/2) - s i n ((J.^/2)) L U h k P(<!>) = where A5.4 * K *hk |hb + kb | X sin(fl), . /2) = - > • hk 4ir j = m u l t i p l i c i t y of hk r e f l e c t i o n , c = c l a t t i c e constant, A- = the neutron wavelength and Q i s a p r o p o r t i o n a l i t y constant. This i s equivalent to equation 39 i n Warren (1941). The integrated i n t e n s i t y along the hk l i n e i n r e c i p r o c a l space i s equal to the sum over £ of the i n d i v i d u a l integrated i n t e n s i t i e s of the hk peaks had the ordering been three dimensional. The i n t e n s i t y of the o three dimensional peaks i s "smeared" over the range (|> ^  - $ 1 180 when the order i s of a two dimensional nature. Using equation 4.17 we note that i n the case of AAA*•• stacking (ne-gle c t i n g Debye-Waller terms) Q A ^ N ^ b J T . j P ( < ) , ) = sin<4>)sin(4,/2) 6 ( *~W ) where the symbols are defined e a r l i e r and Q i s the same constant as i n A5.4. We plot P(<f>) f o r the cases of two and three dimensional order with Q normal-ized to our data i n f i g u r e 32. In both cases a Gaussian of half width o A<|>' = .5 has been convoluted with P(if>) . The convolution was done numerically i n the case of two dimensional order. The s i g n i f i c a n c e of t h i s graph i s discussed i n the text. A discussion of the e f f e c t s of one,-two and three dimensional order given by Jones (1949) and i s noted here f or reader i n t e r e s t . 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0085282/manifest

Comment

Related Items