LATTICE DYNAMICS OF GRAPHITE, ITS INTERCALATION COMPOUNDS AND
ALKALI HALIDES
By
R. S. NARAYANAN
THESIS SUBMITTED TO THE
INDIAN INSTITUTE OF TECHNOLOGY, DELHI IN FULFILMENT OF THE REQUIREMENTS
FOR THE AWARD OF THE DEGREE OF
DOCTOR OF PHILOSOPHY
DEPARTMENT OF PHYSICS
INDIAN INSTITUTE OF TECHNOLOGY, DELHI (INDIA)
JUNE, 1984
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* MI GiMffZ? PARENTS *
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The a u t h o r w i s h e s t o e x p r e s s h i s deep
s e n s e o f g r a t i t u d e t o P r o f e s s o r B . B 0 T r i p a t h i f o r h i s g u i d a n c e and e n c o u r a g e m e n t a t v a r i o u s s t a g e s d u r i n g t h e c o u r s e o f t h i s w o r k .
The a u t h o r i s d e e p l y i n d e b t e d t o Dr „ H, C . Gup t a f o r h i s c o n s t a n t h e l p , p a i n s t a k i n g g u i d a n c e , s i m u l a t i n g and f r u i t f u l d i s c u s s i o n s and h i s u n t i r i n g e f f o r t s
t h r o u g h o u t t h e p r e s e n t i n v e s t i g a t i o n s .
The v a l u a b l e s u g g e s t i o n s o f
P r o f e s s o r M . S . D r e s s e l h a u s ( M a s s a c h u s e t t s I n s t i t u t e o f T e c h n o l o g y , U . S . A . ) and P r o f e s s o r S , A , S o l i n ( M i c h i g a n S t a t e U n i v e r s i t y , U . S . A . ) a r e a c k n o w l e d g e d . Thanks a r e a l s o due t o D r . J . S h a n k a r , D r . ( M r s . ) Nee l i ma Rani and Mr, S . P . B h a t n a g a r f o r t h e i r c o mme n t s , I am a l s o t h a n k
f u l t o a l l t h e o t h e r members i n t h e g r o u p .
I t i s a g r e a t p r i v i l e g e t o r e c o r d t he
f o r b e a r a n c e and t o l e r a n c e o f my p a r e n t s a p a r t f r o m t h e i r c o n s t r u c t i v e c r i t i c i s m s and s u p p o r t . The f r i e n d l y
e n c o u r a g e m e n t r e n d e r e d by my u n c l e s Mr, N . K r i s h n a s w a m i and Mr„ R. S a mp a t h Kumar and t h e i r w i v e s i s g r a t e f u l l y a e k n o w l e d g e d„
I t i s a p l e a s u r e t o t h a n k my s i s t e r Ms „ K . N a n d i n i f o r h e r p o s i t i v e and p h i l o s o p h i c a l a d v i c e s , n o t t o m e n t i o n t he mo r a l s u p p o r t r e c e i v e d f r o m h e r .
The f i n a n c i a l a s s i s t a n c e r e c e i v e d f r o m the C o u n c i l o f S c i e n t i f i c and I n d u s t r i a l R e s e a r c h , Go v e r nme nt o f I n d i a i s t h a n k f u l l y a c k n o w l e d g e d . The s e r v i c e s o f
ACKNOWLEDGE MENTS
Mr . N , S „ Gupt a f o r t h e e f f i c i e n t d r a w i n g s and Mr. H , C . B h a t i a f o r t he n e a t t y p i n g a r e a l s o a p p r e c i a t e d .
L a s t l y b u t n o t t h e l e a s t , t h e a u t h o r w i s h e s t o t h a n k a l l o t h e r s who o f f e r e d i n d i r e c t s u p p o r t by way o f t h e i r b e s t w i s h e s , ,
June 2 5 ‘ 19 8 4 0 ( R » S . N a r a y a n a n )
PREFACE
The t h e o r e t i c a l and experimental study of l a t t i c e dynamics of s o l i d s has been o f great i n t e r e s t to p h y s i c i s t s as i t now forms an important part o f any course in s o l i d s t a t e p h y si c s. The v i br at io n s of the consti tue nt atoms at l a t t i c e s i t e s play a s i g n i f i c a n t role in determining the e l a s t i c , e l e c t r i c a l , magnetic and thermal p rop er tie s o f s o l i d s . The v i b r a t i o n s also govern phenomena l i k e d i f f u s e s c a t t e r i n g o f x - r a y s , neutron s c a t t e r i n g , spin l a t t i c e r e l a x a t i o n e t c .
The present t h e s i s has been devoted to the study of a newly developing group o f compounds, viz graphite and i t s i n t e r c a l a t i o n compounds, which comprises the f i r s t s e c t i o n . In the second part a study has been attempted on the a l k a l i h al id es family based on a new quantum mechanical approach.
Graphite is a very good example o f an element having layered structure Cl], The key property of
layered struc tur es i s t h e ir anisotropy, making them s u i t a b l e for a wide range o f a p p l i c a t i o n s . An added
feature o f such compounds i s t h e ir quasi two dimensional character that can be strengthened or weakened by
i n t e r c a l a t i o n with organic acids or metals r e s p e c t i v e l y .
l-LJ-J
Graphite has a hexagonal layered st r u c t u r e , with the layers tending to form weakly coupled hexagonal planar s h e e t s [ 2 ] . The l a t t i c e belongs to the point
group 3] , with the unit c e l l containing four atoms.
The forces between adjacent b a s a l planes are nearly two orders of magnitude le s s than those between neighbouring atoms in the same p l a n e [ 4 ] . Along the c - a x i s , the
bonding is o f the van der Waals type .and very weak[53.
The e a r l i e r . , studies in graphite have been re vo lv in g around the Born-von Karman m o d e l [ 2 , 4 , 6 - 8 ] . However, d e f i c i e n c i e s were observed while ex pl ai ni n g the phonon spectrum or s p e c i f i c heat data. More r e c e n t l y , experimental st ud ie s have revealed for the f i r s t time, the occurence o f a Raman a c t i v e A? mode at 868 cm**1C9].
/ ^ u
Nemanich and S o l i n [ 1 0 ] pointed out th a t the key features o f the phonon spectrum in graphite could be b e s t explained only on the li n e s o f Nicklow et a l [ 4 ] , but even that
1
pred icte d the A., mode at ~ 1400 cm inst ea d o f at
r 2u
868 cm"1 [ 11 ].
An attempt was made to ex pla in t h i s mode by Maeda et a l [ 1 2 ] and AhJishi and Dresse lhaus [ 1 3 ]. In both these models, the mode i t s e l f was taken as
2u
input data for determination o f force constants e t c . , which in a way makes the force constants biase d towards
the experimental f req uen ci es .
(iii)
In the present work, the A? mode at 86 8 cm L U.
has been explained s a t i s f a c t o r i l y without taking i t as an input parameter and by using a l e s s e r number of force consta nts , by employing an angular force model on the li n e s o f deLauney[ 14] .
In the present a n al ys is i n t e r a c t i o n s upto two neighbours in the b a s a l plane and the nearest neighbour along the c - a x i s have been considered. While both the ce n t ra l and angular force constants have been employed for the nearest neighbours along the plane and along
c - a x i s , only the c e n t r a l force constant has been employed for the second neighbour along the b a sa l pla ne. Further, the f iv e force constants thus used, are reduced to four by using the r e l a t i o n a^=5a^' where a^* are
r e s p e c t i v e l y , the f i r s t neighbour ce nt ra l and n on- ce ntr al force co nst ant s.
Though in the structure o f graphite, only the al t er n at e layers along the c - a x i s are i d e n t i c a l as far as the p o s i t i o n o f carbon atoms in a hexagon are concerned, the present approach assumes a l l layers as i d e n t i c a l , which i s observed to occur upon i n t e r c a l a t i o n C 1 5 ]. Thereby, the in-plane force constants of graphite could be extended to the i n t e r c a l a t i o n compounds[ 1 6 ] . The symmetry along
c - a x i s f a c i l i t a t e s the usage o f the sum and diffe ren ce -I
mode an al ys is E17] in the present approach to solve the dynamical matrix;. .
Expanding the dynamical matrix in the long wavelength l i m i t the expressions for the e l a s t i c
constants C ^ , and are obtained which are taken as input data together with the E^u mode at 1575 cm"'*'.
- 1
The A~ , mode is observed at 909 cm and the other branches in the phonon spectrum also ind icate a good
agreement with the av ai lab le experimental r e s u l t s , though the quadratic nature o f the transverse ac ous tic branch in the (qoo) d ire ct io n at q + o i s not observed in the
present study.
The study o f the graphite i n t e r c a l a t i o n
compounds, is primarily based on the stacking sequence employed along the c - a x i s which in turn, depends on the stage o f the compounds. The s t a g e , being the number o f carbon layers between two in t e rc a l a n t layers i . e . a c - a x i s ordering is s e e n [ 1 8 ] .
There have been only a few t h e o r e t i c a l approaches to study these compounds [ 16 ,19 , 2 0 ] , the v a r ia ti o n in these approaches being due to the
d i f f e r e n t stacking sequences adopted. But in these models, the number of force constant employed are large
(V)
and the force constants have been obtained by f i t t i n g to experimental freq uen cie s, with the b as al plane force constants being extended from graphite which tends to make the model more dependent on experimental phonon f r e q u e n c ie s .
In the present work, the f i r s t , second and the t h i r d stage graphite - a l k a l i metal i n t e r c a l a t i o n compounds have been studied. A stacking sequence o f CoiCaCa..,,
CCaCCa..., CCCaCCCa... . r e s p e c t i v e l y have been adopted for the f i r s t , second and the t h i r d stage compounds; where C,a denote r e s p e c t i v e l y the carbon and i n t e r c a l a t e la y e r s .
Along with the force constants obtained in graphite, a d d it io na l ce nt ra l and non-ce ntr al force constants have been employed to account for the carbon-metal in t e ra c t io n
on the lin es o f Horie et alt 16]. The dynamical matrix obtained is solved to get the phonon dispersion r e l a t i o n s in the (qoo) and (ooq) d i r e c t i o n s , comparison with the a v a i la b le experimental r e s u l t s ind icate a f a i r l y good agreement.
Expansion o f the dynamical matrix in the long wavelength l i m i t y i e l d s e l a s t i c constants and C44<
While the agreement in the case of i s s a t i s f a c t o r y , when compared with the av ai la b le experimental va lu es ,
those for are only q u a l i t a t i v e .
The second part o f the present study is devoted to the thermal p ro p er ti es o f a l k a l i h a l i d e s . The various phenomenological models based on the p r i n c i p l e s in c l a s s i c a l theory applied to the a l k a l i halides have been numerous, s t a r t i n g from the e a r l i e s t Born and von Karman theory[21] though the s h e l l
modelC223, deformation dipole model[23] upto the modified r i g i d ion model [ 2 4 , 2 5 ] which was based on the r i g i d ion model developed by Ke llermannC 26 , 2 7] .
In the present work, a newly developed form o f overlap p o t e n t i a l based on the f i r s t p r i n c i p l e s of quantum mechanics has been employed to study the a l k a l h a l id e s fam il y. The io n ic a l k a l i h a l id e s family are widely used to t e s t the v a l i d i t y o f the i n t e r a c t i o n p o t e n t i a l used, with s p e c i a l a t t e n t i o n being paid to the phonon spectrum thus obtained, whose knowledge, helps gain i n s i g h t into various p r o p e r t i e s [ 2 8 ] .
In the present study, the quantum mechanical model that has been adopted i s based on the t i g h t bind ing theory developed by HarrisonC 29 ] , wherein the Mace lung term has been replaced by the band ener gy and the Born Mayer term by a quantum mechanical poten
t i a l . Deviations are seen in the r e s u l t s obtained by
(vii)
HarrisonC 2 9 ], which could probably be due to the r e p l a c e ment o f the Made lung term by a band energy.
The present formulism while r e ta in in g the Made lung term, adopts a quantum mechanical overlap p o t e n t i a l instead o f the Born Mayer term on the li n e s o f Harrison. The t o t a l p o t e n t i a l energy w i l l be the sum of the co ntributions from anion-anion, an i on -c a ti on and c a t i o n - c a t i o n i n t e r a c t i o n s together with the Made lung term.
The separation energies have been studied on the above li n es and comparison with experimental r e s u l t s [ 3 0 ] i s seen to indicate a good agreement.
The pressure deriva tive o f bulk modulus has been evaluated and the r e s u l t s obtained in the present study are found to be in confirmity with those obtained e x p e r i mentally [ 3 1 - 3 3 ] , and those got by other workers based on the c l a s s i c a l th e o ry [ 3 4 ] .
The phonon disp ers io n r e l a t i o n s have been
studied using an e l e c t r o s t a t i c c o n t r i b u t io n on the li n e s o f KeHermannE2 6 ] , by including an e f f e c t i v e charge[35]
to take into account the p o l a r i s a b i l i t y o f ions and a short range overlap i n t e r a c t i o n e f f e c t i v e upto the second neigh
bour .within the framework o f the medified r i g i d ion model[ 2 4 ] , employing thus only four force constants - a ^, &2 the c e n t r a l force constants for the f i r s t and
(vi i i)
second neighbour and the non ce n t ra l component.
The dynamical matrix elements have been expanded in the long wavelength l i m i t to give the e l a s t i c constants
C1]> and ^4 4*
The r e s u l t s indicate a s a t i s f a c t o r y agreement in the disp ers io n r e l a t i o n s when compared with the
r e s u l t s obtained experimentally and those got by Karo and Hardy[36] based on the r i g i d ion model. The shear e l a s t i c constant is a l s o seen to e x h i b i t a f a i r degree o f agreement when compared with the experimental va lu e s.
The present t h e s i s has been divided into two p a r t s . The f i r s t part is devoted to the study of
graphite and i t s i n t e r c a l a t i o n compounds. S t a r t i n g from the s t r u c t u r e , the study traces the various approaches adopted to study graphite and i t s compounds with li t h i u m , potassium, rubidium, and cesium for the stages one, two and thr ee. It also describes in depth the present
formulism to study graphite and i t s i n t e r c a l a t i o n compounds.
The second part o f the t h e s i s i s devoted to study the a l k a l i hal ide s fam il y. The e a r l i e r models that have been employed to study the l a t t i c e dynamical proper
t i e s have been c i t e d . The Ha rri son ’ s p o t e n t i a l , a
modified form of which has been used in the present study has also been described. The r e s u l t s obtained for the properties li k e binding energy, pressure d eri va ti ve of bulk modulus, phonon d isp ers io n and shear e l a s t i c
constant have b e e n . discussed.
The e n t ir e work has been summed up in four chapters. The chapterwise summary is given below:
Chapter one deals with the l a t t i c e dynamics o f graphite; s t a r t i n g from the structure o f grap hit e, the chapter reviews the e a r l i e r work done and gives in d e t a i l the present an al ys is for the phonon d is p er si o n and e l a s t i c constants of graphite. An appendix has been given on the deLauney’ s angular force model.
Chapter two deals with the graphite i n t e r c a l a t io n compounds, t h e i r structure and the various s t a g e s . Apart from the e a r l i e r studies undertaken, i t gives an account of the formulism adopted in the present study to in d i v i d u a l l y deal with the phonon d isp ers io n r e l a t i o n s and e l a s t i c constants for the f i r s t , second, and the t h i r d stages of the g r a h i t e - a l k a l i metal i n t e r c a l a t i o n compounds.
Chapter three is mainly for the review of the a l k a l i halide family, the sodium chloride and cesium
chloride s t r u c t u r e s . Elaboration has been done on the various models developed and adopted for t h e i r study.
An introduction has been given to the Harrison’ s poten
t i a l , a modified form o f which has been used in the present a n al ys is for the study on the a l k a l i - h a l i d e s .
Chapter four gives the H a r r i s o n 's formulism in d e t a i l and also- the modifications e f f e c t e d in the present study. This new form o f the p o t e n t i a l has been employed to study the pr op er tie s li k e binding energy, pressure deri vat iv e of bulk modulus, phonon disp ersion and e l a s t i c constant C ^ , The mathematical formulism employed have been derived and discussed.
The present study has r e s u l t e d in the f ol lo w in g p ub li ca ti o n s :
(1) Theory of Inter i o n i c forces in a l k a l i halid e c r y s t a l s , Phys . S tat . S o l . 113(b) , 339 ( 198 2) . (2) Lattice dynamical c a l c u l a t i o n of the phonon
dis persion in sodium halide c r y s t a l s , In d. J.
Pure.Appl.Phys. 20 , 7 7 7 ( 1 9 8 2 ) .
(3) Phonon dis persion in graphite, Presented at the D.A.E.Symposium in Nuclear Physics and S o l i d State Physics in 1982.
(X )
Phonon dis per si on r e l a t i o n s o f graphite and f i r s t stage g r a p h i t e - a l k a l i metal i n t e r c a l a t i o n compounds, Syn.Metals _7 9 3 4 7 ( 1 9 8 3 ) .
The phonon disp ers io n in t h i r d stage a l k a l i metal-graphite i n t e r c a l a t i o n compounds,
Presented at the D.A.H. Symposium in Nuclear Physics and S o l i d State Physics in 1983.
Phonon dispersion r e l a t i o n s o f second and
t h i r d stage a l k a l i metal-graphite i n t e r c a l a t i o n compounds - accepted for p re se nt at io n at the I nt er na ti on al Carbon Conference to be held at Bordeaux, France in 19 84.
Phonon spectrum in f i r s t stage g r a p h i t e - l i t h i u m i n t e r c a l a t i o n compound - accepted fo r p u b l i c a t i o n in Carbon ( 1984).
CONTENTS
PAGE
PART - I
1 ON THE LATTICE DYNAMICS OF GRAPHITE
1 .1 INTRODUCTION 1
1. 2 STRUCTURE OF GRAPHITE 4
1 . 3 EARLIER STUDIES 6
1.4 PRESENT APPROACH ' 12
1.5 RESULTS AND DISCUSSION 19
1.6 REFERENCES 2 2
1 .7 APPENDIX 25
2 GRAPHITE INTERCALATION COMPOUNDS
2 . 1 INTRODUCTION 27
2 .2 STRUCTURE OF GRAPHITE - 29
INTERCALATION COMPOUNDS
2 .3 EARLIER STUDIES 34
2.4 THE PRESENT APPROACH AS APPLIED 39 TO STAGE ONE COMPOUNDS
2.5 EXTENSION TO STAGE TWO COMPOUNDS 5 5
2.6 PRESENT STUDIES ON STAGE THREE 6 7
COMPOUNDS
2 .7 REFERENCES 91
PART - 11
PAGE
3 INTRODUCTION TO HARRISON’ S POTENTIAL
3. 1 INTRODUCTION TO NaCl AND CsCl 95
STRUCTURES
3 .2 EARLIER MODELS 9 8
3 . 3 HARRISON’ S POTENTIAL 10 7
3.4 REFERENCES 109
4 APELICATION OF HARRISON'S POTENTIAL TO ALKALI HALIDES
4 . 1 HARRISON'S FORMULATION 113
4 . 2 MODIFICATIONS IN PRESENT STUDY 115 AND APPLICATION TO BINDING
ENERGY
4 . 3 PRESSURE DERIVATIVE OF BULK MODULUS 125 4 . 4 ( a ) ON THE PHONON DISPERSION AND 128
ELASTIC CONSTANTS
(b) RESULTS AND DISCUSSION 131
4 .5 REFERENCES 170
