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ATOMIC AND MOLECULAR MOTION IN COVALENT SOLIDS
R. Herber
To cite this version:
R. Herber. ATOMIC AND MOLECULAR MOTION IN COVALENT SOLIDS. Journal de Physique
Colloques, 1979, 40 (C2), pp.C2-386-C2-388. �10.1051/jphyscol:19792134�. �jpa-00218506�
JOURNAL DE PHYSIQUE
Colfoque C2, supplement au n o 3, Tome
40,mars 1979, page C2-386
ATOMIC AND M3LECULAR MOTION I N COVALENT S O L I D S
R.H. Herber
Department of Chemistry, R u t g e r s U n i v e r s i t y , New Bmozswick,
N.
J., 08903, U. S. A.R6sumb.- L ' a n i s o t r o p i e de v i b r a t i o n d e s atomes m g t a l l i q u e s dans d e s composds l a m e l l a i r e s du type SnS, e t SnSe, peut Gtre mesurge ?I p a r t i r de l a dzpendance a n g u l a i r e de l a f r a c t i o n d ' e f f e t sans r e c u l s u r d e s 6 c h a n t i l l o n s m o n o c r i s t a l l i n s . L ' e f f e t ~ 6 s s b a u e r de " ' ~ n montre que l e s atomes d 1 6 t a i n e f f e c t u e n t 21 295 K un mouvement p a r a l l s l e aux p l a n s de s o u f r e avec une amplitude deux f o i s p l u s grande que pour l e mouvement p e r p e n d i c u l a i r e 1 ces p l a n s . On prCsente un c a l c u l de l a f r a c t i o n d ' e f f e t sans r e c u l q u i t i e n t compte
B
l a f o i s du nombre d'atomes s u r l e chemin o p t i q u e e t de l ' a i r e sous l a courbe de r6so- nance MEssbauer en f o n c t i o n de l a d i r e c t i o n d ' o b s e r v a t i o n .Abstract.- The a n i s o t r o p y i n t h e v i b r a t i o n a l amplitudes axecuted by t h e m e t a l atom i n s t r o n g l y aniso- t r o p i c van d e r Waals l a y e r compounds of t h e type SnS, and SnSe, can be o b t a i n e d from a n g u l a r dependen- ce measurements of t h e r e c o i l - f r e e f r a c t i o n , u s i n g o r i e n t e d s i n g l e c r y s t a l a b s o r b e r s . Such s t u d i e s have been c a r r i e d o u t u s i n g t h e 23.8 keV Mijssbauer t r a n s i t i o n i n '19sn, and show t h a t t h e t i n atom executes a motion p a r a l l e l t o t h e sulphur l a y e r p l a n e which i s approximately a f a c t o r of 2 l a r g e r than t h e motion p e r p e n d i c u l a r t o t h i s p l a n e a t 295 K. A formalism i s p r e s e n t e d which p e r m i t s t h e c a l c u l a - t i o n of t h e a n g u l a r dependence of t h e r e c o i l - f r e e f r a c t i o n t a k i n g i n t o account both t h e a n g u l a r depen- dence of t h e number of atoms i n t h e o p t i c a l p a t h and t h e a n g u l a r dependence of t h e a r e a under t h e M6ssbauer resonance curve.
Temperature dependent Mgssbauer e f f e c t s t u d i e s provide an e x c e p t i o n a l l y s e n s i t i v e t o o l f o r t h e in- v e s t i g a t i o n of atomic and molecular motion i n s o l i d s and can provide i n f o r m a t i o n which i s n o t r e a d i l y ac- c e s s i b l e by o t h e r physico-chemical techniques. I n p a r t i c u l a r , t h e d e t e r m i n a t i o n of t h e temperature dependence of t h e r e c o i l - f r e e f r a c t i o n can be used t o c h a r a c t e r i z e t h e phonon spectrum of t h e s o l i d and t o provide a measure of t h e l a t t i c e temperature of t h e s u b j e c t m a t e r i a l i n terms of thef'Mgssbauer tem- p e r a t u r e " , QM. On t h e o t h e r hand, t h e a n i s o t r o p y of t h e motion of t h e Mgssbauer a c t i v e atom ( o r t h e mo- l e c u l e i n which i t i s i n c o r p o r a t e d ) can be e l u c i d a - t e d f o r a p o l y c r i s t a l l i n e a b s o r b e r when t h e r e i s a r e s o l v a b l e quadrupole h y p e r f i n e i n t e r a c t i o n , by means of t h e Gol'danskii-Karagin e f f e c t , a n d - i n t h e
absence of such an i n t e r a c t i o n
-
f o r a s i n g l e crys- t a l sample, by d e t e r m i n a t i o n of t h e a n g u l a r depen- dence of f a t c o n s t a n t temperature.Experimental d a t a on t h e a n i s o t r o p y of atomic and molecular motion i s e s p e c i a l l y u s e f u l i n t h e s t u d y of s t r o n g l y a n i s o t r o p i c i n s u l a t o r s and semi- conductors, f o r which t h e r e l a t i o n s h i p between t h e microscopic dynamics and t h e macroscopic (bulk) p r o p e r t i e s may be expected t o lead t o a c l e a r e r un- d e r s t a n d i n g of t h e t r a n s p o r t p r o p e r t i e s of such ma- t e r i a l s .
Recently, a d e t a i l e d s t u d y / I / of t h e system SnS,-xSex (05x52) has been c a r r i e d o u t w i t h t h e p r i - mary aim of understanding t h e v a r i a t i o n of w i t h
composition. Although d e t a i l e d c r y s t a l s t r u c t u r a l d a t a a r e a v a i l a b l e o n l y f o r t h e compositions x=O, 1 and 2 , a l l members of t h e s e s e r i e s a r e assumed t o be i s o s t r u c t u r a l and t o have t h e CdI, s t r u c t u r e of SnS,
/ 2 / . The l a t t i c e parameters of t h e s o l i d s
-
preparedfrom 6N5 Sn, 6N S and 4 N Se by an evacuated t u b e - high temperature process,followed by s e v e r a l 600°C a n n e a l i n g c y c l e s
-
were determined from powder X-ray p a t t e r n s and show t h a t 5 i s l i n e a r w i t h x; t h e varia- t i o n of 5 w i t h x shows pronounced c u r v a t u r e ; andc/a
has a maximum a t xz0.8. Both i o n microprobe mass spectrometry and " ' ~ n Mijssbauer e f f e c t s t u d i e s in- d i c a t e t h a t t h e S and Se d i s t r i b u t i o n s i n t h e s e sam- p l e s a r e completely random. The i n t e r l a y e r coupling i s a minimum a t x-0.8. The v a r i a t i o n of w i t h x
M
f o r p o l y c r y s t a l l i n e a b s o r b e r s has been r e p o r t e d e a r - l i e r / ] / .
Since t h e ' 1 9 s n resonance i n t h e s e samples g i v e s ri- s e t o a s i n g l e a b s o r p t i o n maximum, no i n f o r m a t i o n concerning t h e v i b r a t i o n a l a n i s o t r o p y of t h e t i n atom w i t h r e s p e c t t o t h e van d e r Waals l a y e r p l a n e s can be e x t r a c t e d from measurements on p o l y c r y s t a l l i n e a b s o r b e r s . Such d a t a a r e , however, a c c e s s i b l e by means of o r i e n t e d s i n g l e c r y s t a l s t u d i e s , and have been c a r r i e d o u t a t 295
*
2 K. The a b s o r b e r s f o r t h e s e s t u d i e s were prepared by u s i n g a vapour t r a n s - p o r t t e c h n i q u e employing molecular i o d i n e a t a con- c e n t r a t i o n of -5 mg. cm-j, and a thermal g r a d i e n t of -50". Mosaics of s i n g l e c r y s t a l s were o r i e n t e d s o t h a t t h e c r y s t a l l o g r a p h i c c-axis was normal t o t h eArticle published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19792134
p l a n e of t h e a b s o r b e r h o l d e r , and t h e samples were examined i n normal t r a n s m i s s i o n geometry, u s i n g a c o n s t a n t a c c e l e r a t i o n spectrometer and a Ba " 9 m ~ n 0 3 Msssbauer source. The angle between t h e o p t i c a l
(gamma r a y ) a x i s and t h e c r y s t a l l o g r a p h i c c-axis could be v a r i e d from 0 t o 60" i n a h o r i z o n t a l l y ro- t a t i n g goniometer head.
It i s w e l l known / 3 / t h a t t h e a r e a under t h e resonance curve can be expressed a s a f u n c t i o n of absorber t h i c k n e s s , t a , i n u n i t s of r 0 / 2 by S = a ? ' r f s ~ ( t a )
where
i n which I. and I1 a r e z e r o and f i r s t o r d e r B e s s e l f u n c t i o n s of imaginary argument. The v a r i a t i o n of t h e observed a r e a under t h e resonance curve w i t h ro- t a t i o n a l a n g l e depends b o t h on t h e a n g u l a r dependen- ce of f and t h e a n g u l a r dependence of t h e number of atoms p e r u n i t a r e a of a b s o r b e r along t h e o p t i c a l a x i s . The l a t t e r i s given simply by
na(e) = na(0)
.
cose-
I (3)The a n g u l a r dependence of f a i s given by f,(B) = e -k2<x2>(8)
( 4 )
where the a n g u l a r l y dependent mean square amplitude of v i b r a t i o n , <x2> (8) i s / 4 /
< x 2 > ( 8 ) = <x:>
+
[ < x ~ ~ ~ >-
<x2,->] C O S ~ ~ (6) ( t h e I and 1 1 s u b s c r i p t s r e f e r , of c o u r s e , t o t h e angle between t h e c r y s t a l l o g r a p h i c symmetry a x i s and t h e o p t i c a l a x i s of t h e experiment).Equation (5) i s more simply w r i t t e n a s
= fL R c ~ ~ 2 e = f
,,
Rcos2b-l (7)i n which R = f , , /f,. Using t h i s symbolism, t h e reso- n a n t a b s o r b e r t h i c k n e s s parameter becomes
c0s2e-1
taw
= t l l R-cose
where t , , i s measured f o r 0 = 0. Then e q u a t i o n (1) be comes
see) = aTfSK [ t a ( e ) l = u T ~ ~ K
C~,,R~OSIB-II
(9)cose
and t h e a n g u l a r dependence of t h e normalized a r e a i s given by
and can be r e a d i l y computed a s a f u n c t i o n of a n g l e f o r a given t , ! and R /5/. For a f i n i t e f l l and R<1, A(@) i n c r e a s e s w i t h i n c r e a s i n g 8 up t o
emax,
and t h e n d e c r e a s e s i n t h e range < 85
T / ~ . The va-max
-
l u e of
em,,
obtained by s e t t i n g t h e d e r i v a t i v edA(B)/d cose = 0 , i s g i v e n i m p l i c i t l y by 2 cos28 Rnr =
-
I 1 Eta (5max)lmax I. Ct a (emax'l
-
2
which can be solved e x p l i c i t l y f o r given tll and R.
To permit c a l c u l a t i o n of a b s o l u t e r e c o i l - f r e e f r a c t i o n (and hence a b s o l u t e mean s q u a r e amplitudes of v i b r a t i o n ) parameters from t h e experimental d a t a , i t i s n e c e s s a r y t o measure A(0) and na(0) f o r t h e ex- p e r i m e n t a l sample and f o r a r e f e r e n c e sample under c o n d i t i o n s of i d e n t i c a l geometry. Under such condi- t i o n s
Ax(0)nrfr x , - A r ( 0 ) n x
where t h e x and r s u b s c r i p t s r e f e r t o t h e sample and the r e f e r e n c e m a t e r i a l , r e s p e c t i v e l y . For ' ~ n &ss- bauer s p e c t r a , an a p p r o p r i a t e c h o i c e of r e f e r e n c e m a t e r i a l i s m e t a l l i c
(B)
t i n f o r which a c c u r a t e f v a l u e s have been determined over a wide temperature range by Hohenemser / 6 / .The normalized a r e a r a t i o v a l u e s , p l o t t e d a s a f u n c t i o n of o r i e n t a t i o n a l a n g l e , determined f o r a s i n g l e c r y s t a l mosaic of SnS,, i s summarized i n f i - gure 1 .
Fig. I : Angular dependence of t h e a r e a under t h e Msssbauer e f f e c t resonance curve, normalized t o t h e zero degree r o t a t i o n a l angle v a l u e . The open c i r c l e s r e f e r t o t h e experimental v a l u e s a s measured; t h e f u l l c i r c l e s r e f e r t o t h e r e s i d u a l a n g u l a r dependen- ce a f t e r c o r r e c t i o n f o r t h e number of atoms p e r u n i t a r e a i n t h e o p t i c a l p a t h .
The b e s t agreement between experimental and calcula- ted v a l u e s i s obtained f o r R = 0.06. Using t h i s va- l u e and t h e c a l c u l a t e d v a l u e of f l l a t 8 = 0 l e a d s to t h e mean v i b r a t i o n a l amplitudes of t h e m e t a l atom i n SnS, a s summarized i n t a b l e I. This t a b l e a l s o l i s t s t h e corresponding v a l u e s and hyperfine i n t e r -
C2-388
JOURNAL DE PHYSIQUE a c t i o n p a r a m e t e r s f o r a s i n g l e c r y s t a l m o s a i c o fSnSe,, a s w e l l a s t h e Mgssbauer l a t t i c e t e m p e r a t u r e s of t h e s e compounds.
T a b l e I :
V i b r a t i o n a l a n i s o t r o p y d a t a f o r some a n i s o t r o p i c van d e r Waals l a y e r compounds
Sample I S ( 7 8 K ) ~ -d I n A ( T ) / A ( ~ ~ ' ) b
dT OM < X ~ ~ ' X b
-
< X I [ >bmm s-' deg.-' deg.
8, ii
<x*>SnS2 1.066
+
0.006 5.025 k 0.060 190 L 3 ( p o l y c r y s t a l l i n e a b s o r b e r )SnS, 1.062 k 0.010 4.595 f 0.054 197 f 3 0.155 0.069 2.25
SnSe, 1.397 f 0.010 3.951 f 0.060 213
*
5 ( p o l y c r y s t a l l i n e a b s o r b e r )SnSe, 1.371 f 0.025 5.256 ? 0.080 185 5 5 0.147 0.093 1.58
SnSSe 1.306 ? 0.010 3.688 k 0.040 220
*
5 ( p o l y c r y s t a l l i n e a b s o r b e r )a ~ e l a t i v e t o BaSnO, a t 295 ? 2 K b ~ t 295
*
2 KThese d a t a r e f l e c t t h e l a r g e v i b r a t i o n a l a n i - s o t r o p y of t h e m e t a l atom motion i n v a n d e r Waals l a y e r compounds h a v i n g t h e Cd12 s t r u c t u r e . F u r t h e r s t u d i e s of r e l a t e d a n i s o t r o p i c semiconductor and co- v a l e n t s o l i d s a r e c u r r e n t l y under way i n t h e s e labo-.
r a t o r i e s .
T h i s work was s u p p o r t e d by a g r a n t from t h e N a t i o n a l S c i e n c e Foundation (DMR 76-00139-A01) and
from t h e C e n t e r f o r Computer and I n f o r m a t i o n S e r v i - c e s , R u t g e r s U n i v e r s i t y . The a u t h o r i s i n d e b t e d t o Drs.M. Katada and G.S. C o l l i n s , a s w e l l a s M s . A.
Smelkinson and "Ir. R.F. Davis f o r numerous h e l p f u l d i s c u s s i o n s and a s s i s t a n c e w i t h t h e measurements r e p o r t e d .
R e f e r e n c e s
/ I / H e r b e r , R.H., Smelkinson, A.E., Sienko, M.J. and Schneemeyer, L . , J. Chem. Phys.
68
(1978) 3705./ 2 / W e l l s , A.F., " S t r u c t u r a l I n o r g a n i c Chemistry", (Oxford U n i v e r s i t y P r e s s , Oxford) 1962, p. 512;
Au-Yang,
M.Y.
and Cohen, M.L., Phys. Rev.178
(1969) 1279.
/ 3 / Bykov, G.A. and Hien, P.Z., Sov. Phys. JETP
16
(1973) 646.
/ 4 / C o l l i n s , G.S. and Benczer-Koller, N . , Phys. Rev.
B 1 7 (1978) 2085.
/ 5 / The a u t h o r i s i n d e b t e d t o D r . G.S. C o l l i n s f o r c a r r y i n g o u t t h e s e c a l c u l a t i o n s .
/ 6 / Hohenemser, C . , Phys. Rev. (1965) 185.
