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ANHARMONIC CONTRIBUTIONS TO THE THERMAL DIFFUSE X-RAY SCATTERING
INTENSITY FROM SINGLE CRYSTALLINE SODIUM
G. Fritsch, W. Dietz, R. Häring, E. Lüscher
To cite this version:
G. Fritsch, W. Dietz, R. Häring, E. Lüscher. ANHARMONIC CONTRIBUTIONS TO THE THER- MAL DIFFUSE X-RAY SCATTERING INTENSITY FROM SINGLE CRYSTALLINE SODIUM.
Journal de Physique Colloques, 1981, 42 (C6), pp.C6-143-C6-145. �10.1051/jphyscol:1981643�. �jpa- 00221579�
JOURNAL DE PHYSIQUE
Colloque C6, suppZ6ment au nolZ, Tome 42, &'cembre 1982 page C6- 143
ANHARMONIC CONTRIBUTIONS T O T H E T H E R M A L DIFFUSE X-RAY SCATTERING INTENSITY FROM SINGLE CRYSTALLINE SODIUM
G. ~ritsch*, W. Dietz, R. Hsring and E. Lcscher
Physik-Department, Teehnische Universitat MUnchen, 0-8046 Garching, F. R. G.
'zWE Physik, Hochschule der Bundesuehr Wnchen, 0-8014 Neubiberg, F. R. G.
Abstract.- We report on measurements of t h e thermal diffuse X-ray s c a t t e r i n o from single c r y s t a l l i n e Na.Data were taken with t h e momentum t r a n s f e r being aligned along t h e ( l o o ) , (110) and (111) directions a s a function of tempera- t u r e . They are discussed in terms of one, two and higher multi-phonon contri- butions. In addition, a part asymmetric with respect t o a reciprocal l a t t i c e vector i s determined and analyzed.
1. Introduction.- The X-ray i n t e n s i t y scattered diffusely from a metal s i n g l e cry- s t a l i s t h e sum of one, two and higher order phonon processes.') A t temperatures T > el) interference terms caused by anharmonic interactions will also contribute. 1) This picture may be complicated i f deformations of t h e ion cores2) o r e l a s t i c defect scattering3) e x i s t s . In t h i s communication we would 1 i ke t o discuss our r e s u l t s on
4-6) Na and compare them with other work i n t h i s f i e l d ,
2. Data Analysis.- The experimental set-up used was a modified version of t h e one described elsewhere.6) After subtracting from t h e measured cross section
t h e calculated one-phonon p a r t 6 ) , t h e multi-phonon contribution higher than of order two4) ( d o l d ~ ~ ) ~ ~ and t h e Compton cross section, ve a r r i v e a t ( d ~ / d f i ) ~ . The l a t t e r i s interpreted as a sum of t h e two-phonon and an asymmetric interference contribution.
t (doldn),=f 2 e x p ( - 2 ~ ) [ e ~ ~ ( 2 ~ ) - 1 - 2 ~ - 2 ~ ~ 3 , where exp(-2W) i s t h e Debye-Wal l e r fac- t o r 4, and f t h e ionic s c a t t e r i n g f a c t o r . ] 2
In t h e main crystallographic directions t h e two phonon term can be written a s 1 ) :
(do/dNp =br4 expl-ZW(.~)I ~ ( $ 4 ) . f 2 ( x ) ,
where A(:/~) i s symmetric with respect t o the reciprocal l a t t i c e vector
&B. Iae = 3% + q]. Furthermore, we ,assume t h a t only t h e lowest order interference term1) contributes:
( d o / d - ~ : ~ ~
,
,2 = c 3 exp L-ZW(=)I - B ( G I ~ ) . f2*)with B ( ; / ~ ) asymmetric w i t h respect t o .jeg. There i s an additional symmetric term (do/d":nh 1,2 which has t h e same@- and ~ - d e ~ e n d e n c e s I ) a s does s (do/dn)2.
Using data above and below % t h e q u a n t i t i e s ( d c ~ / d ~ ) ~ + ( d ~ / d n ) , , ~ as well as ( d d d n ) t n h , 2 may be determined. The r e s u l t s a r e given in Figs 1 and 2.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981643
C6- 144 JOURNAL DE PHYSIQUE
X ~ l l ~ n Brill
Fig.1 Results f o r t h e asymmetric an- Fig.2 Results f o r t h e symmetric c o n t r i - harmonic c o n t r i b u t i o n s ( d o / d ~ ) $ h . b u t i o n ( d o / dR)2
a) e: 30°C; o : 40%; A : 9 5 0 ~ an9 a ) e: 30°C; a: ~ P O C ; A : 9 5 0 ~ and o: see r e f . 6 : 30°C 0: d i r e c t c a l c u l a t i o n 30°C b) c r y s t a l 1 : U: 63.6OC; c r y s t a l 2: b ) c r y s t a l 1: 0: 63.6OC; c r y s t a l 2:
0: 52.6OC;A: 69.8OC. 0: 52.6OC;A : 69.g0c.
c ) d : 26.8OC; 0: 6 5 . 0 ~ ~ ; U: 94.5OC.
The l i n e s a r e o n l y guides t o t h e eye.
3. Discussion.- The asymmetric p a r t i n t h e ( 1 1 0 ) - d i r e c t i o n (Fio. l a ) agrees w i t h t h e one determined e a r l i e r . 6 ) I n t h i s paper we showed t h a t t h e T-dependence i s t h e one t o be expected f o r anharmonic i n t e r f e r e n c e e f f e c t s between t h e one and two phonon process. However, t h e symmetric p a r t i s s y s t e m a t i c a l l y h i g h e r t h a n t h e two phonon p a r t c a l c u l a t e d d i r e c t l y (Fig. 2a). Close t o t h e r e c i p r o c a l l a t t i c e vectoraeg t h e discrepancy i s probably due t o t h e approximations used i n e v a l u a t i n g t h e h i g h e r o r - der many phonon processes. I n between t h e t w o a B values t h e q u a n t i t y (do/dn)inh might c o n t r i b u t e .
For t h e (100) d i r e c t i o n , t h e r e s u l t s f o r t h e asymmetric term (Fig. I b ) a r e r a t h e r puzzling, since t h e y apparent7y do not show a zero ataeBril. T h i s f a c t may p o i n t towards a wrong ae-dependence o f t h e h i g h e r o r d e r many phonon c o n t r i b u t i o n s ( o r i e n t a t i o n a l dependence). The symmetric p a r t (Fig. 2b) i s c l o s e t o t h e (110) r e - s u l t , c a l c u l a t i o n s o f t h e two phonon term i s i n progress.
The asymmetric r e s u l t s (Fig. I c ) i n t h e ( I l l ) - d i r e c t i o n a r e s i m i l a r t o t h e ones i n t h e (110)-case. However, t h e magnitude i s much l a r g e r , i n c r e a s i n p w i t h temperature. The comments f o r t h e symmetric p a r t ( F i g . 2c) a r e t h e same as i n t h e (100)-case.
I n view o f t h e temperature dependence o f t h e asymmetric p a r t and i t s ae-depen- dence, we conclude t h a t these a r e anhamonic c o n t r i b u t i o n s and deformation s c a t t e r - i n g i s n o t present. a rink ha us^) has shown t h a t these d e f e c t s i n thermal e q u i l i b r i u m g i v e a symmetric p a r t N q-' and an asymmetric p a r t N q-I. An e s t i m a t i o n using a r e l a t i v e a c t i v a t i o n v o l u m e ~ v / v = -0.6 f o r t h e vacancy y i e l d s 0.16 and -0.14 r e s p e c t i v e l y i n t h e most favourable case. These values a r e r e l a t i v e t o ( d d d d , . L i t e r a t u r e
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