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TRANSLATIONAL - ROTATIONAL COUPLING IN s-TRIAZINE CRYSTAL
T. Luty
To cite this version:
T. Luty. TRANSLATIONAL - ROTATIONAL COUPLING IN s-TRIAZINE CRYSTAL. Journal de Physique Colloques, 1981, 42 (C6), pp.C6-590-C6-592. �10.1051/jphyscol:19816172�. �jpa-00221251�
JOURNAL DE PHYSIQUE
CoZloque C6, suppldment au n012, Tome 42, de'cembre 1981 page C6-590
T R A N S L A T I O N A L - R O T A T I O N A L C O U P L I N G I N s - T R I A Z I N E CRYSTAL
T. Luty
I n s t i t u t e of Organic and PhysicaZ Chemistry, TechnicaZ University, 50-370 Wroc2aw, Poknd
Abstract.- An anomalous d i s p e r s i o n of t r a n s v e r s e acoustic pho- nons, r e l a t e d t o t h e f e r r o e l a s t i c phase t r a n s i t i o n i n s-tria- z i n e c r y s t a l , have been analyzed. The d i s p e r s i o n r e l a t i o n has been found from t h e harmonic dynamical matrix which i n c l u d e s t r a n s l a t i o n a l - r o t a t i o n a l coupling. For t h e n e a r e s t neighbours i n t e r a c t i o n s , t h e r e l a t i o n contains two compstive terms follow- ed from d i r e c t and i n d i r e c t t r a n s l a t i o n a l - t r a n s l a t i o n a l coup- ling. When t h e i n d i r e c t i n t e r a c t i o n s a r e assumed t o be tempera- t u r e dependent, t h e r e l a t i o n reproduces t h e observed d i s p e r s i o n curves almost exactly.
1. Introduction.- It has long been known t h a t i n molecular c r y s t a l s t h e r e should be a s t a t i c coupling between r o t a t i o n s and t r a n s l a t i o - ns of molecules. Recently well studied s - t r i a z i n e c r y s t a l i s a good example t o i l l u s t r a t e t h e problem of t h e coupling i n ordered molecular c r y s t a l s . A ;r~e&ly 2 i r s t order t r a n s i t i o n a t 200 K i s from
a high temperature, t r i g o n a l phase / R ~ C / t o a low temperature, mono- c l i n i c phase /C2/c/ [I] The f e r r o e l a s t i c d i s t o r t i o n of s - t r i a z i n e i s a s s i s t e d by a pronounced softening of t h e t r a n s v e r s e acoustic mo- des i n t r i g o n a l Z and
h
d i r e c t i o n s C21, causing an i n s t a b i l i t y of t h e c~~ e l a s t i c constant. The anomalous behaviour of t h e e l a s t i c constant has been r e c e n t l y i n t e r p r e t e d [j] i n terms of t h e mean f i e l d theory a s a r e s u l t of coupling between t h e shear s t r a i n e, /e / and molecular r o t a t i o n s R/Rd.
The a i m of t h e present t r e a t -YZ Y
men% i s t o i n t e r p r e t e t h e anomalous and extended i n t h e B r i l l o u i n zone d i s p e r s i o n of t i e t r a n s v e r s e a c o u s t i c modes i n terms of trans- l a t i o n a l - r o t a t i o n a l coupling. A s t h e phonons i n s - t r i a z i n e have been found t o be well defined and underdamped down t o 'PC, t h e ana- l y s i s i s based on harrconic dynanical matrixo
2. Dynamical matrix.- The harmonic p o t e n t i a l energy of a m o l e c u l ~ c r y s t a l can be ivritten i n t e r n s of d y n m i c a l matrix elenents a s ,
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19816172
where 2 (qk) and (qk) a r e t r a n s l a t i o n a l and r o t a t i o n a l displace- ments of t h e s u b l a t t i c e k , r e s p e c t i v e l y , The enrgy i s decoupled in- t o noninteracting p a r t s , each designated by a s p e c i f i c d i r e c t i o n of wave-vector q. A f u r t h e r decoupling can be done by a group-the+
r e t i c a l analysis. Por t h e high temperature, t r i g o n a l phase of s-tri- m i n e , t h e 12 d i s p e r s i o n curves i n t h e h d i r e c t i o n can be c l a s i f i e d as: 2
A1
@ 2A2
84Aj
withA
being double degenerate mode. A s t h e t r a n s v e r s e acoust.ic mode which shows an anomalous d i s p e r s i o n transforms according t oA 3
i r r e d u c i b l e r e p r e s e n t a t i o n /as well a s 3 o p t i c a l branches/ we have t o extiract a p a r t of t h e c r y s t a l poten- t i a l energy which corresponds t o this symmetry. It i s easy done when a new s e t of coordinates / a s follow from p r o j e c t i o n operators/i s introduced,
v j (91 =
3
[ u j (ql) + u j (q22,
w J.
(q) = 7 1 [ u j ( s 1 )-
Uj(92.g, yj
(q) = 2 1tej
[ql) +ej
(qa]4j
(9) =2
[ ~ j(91.' - ej
(~211f o r j = 1,2, Now, t h e i r r e d u c i b l e p a r t of t h e c r y s t a l energy cor- responding t o t h e two-dinensional r e p r e s e n t a t i o n
A3
contains two p a r t s with i d e n t i c a l eigenvalues and f o r f u r t h e r discussion we can use only one of them, reducing t h e problen t o 4-dimensional one.We have,
with c o e f f i c i e n t s being proper combinations of dynamical matrix elements, The t r a n s l a t i o n a l - r o t a t i o n a l coupling i s given by D'S and A'S c o e f f i c i e n t s . It has t o be s t r e s s e d t h a t i n t h e l i m i t q
-+
0 t h e above expression reduces t o t h e harmonic p a r t of t h e in- t e r n a l energy i n t h e mean f i e l d theory.To analyze d i s p e r s i o n of t h e t r a n s v e r s e a c o u s t i c mode we have t o solve an equation of n o t i o n f o r t h e coordinates and, with an assumption t h a t i n t e r n a l s t r a i n s given by coordinates w2,
y2
'anda,,
respond s t a t i c a l l y t o t h e s t r a i n given by v,, /the acoustic mode/one g e t s ,
C6-592 JOURNAL DE PHYSIQUE
c4'
Cq,0) i s t h e s t a t i c phomn s u s c e p t i b i l i t y corresponding t o t h e o p t i c a l phonons which a r e coupled /the same symmetry! t o t h e acous- t i c phonon. An exact a n a l y s i s of t h e s u s c e p t i b i l i t y shows t h a t t h e element ,Gz2 1 (q,O) follows from an e f f e c t i v e o r i e n t a t i o n a l p o t e n t i a l : a sum of r o t a t i o n a l - r o t a t i o n a l s e l f term and i n d i r e c t o r i e n t a t i o n a l p o t e n t i a l given by t r a n s l a t i o n a l - t r a n s l a t i o n a l i n t e r a c t i o n s , a- (q)and t r a n s l a t i o n a l - r o t a t i o n a l coupling A- (q)
.
3. &esults.- I n general, t h e r e l a t i o n (3) i s r a t h e r complicated but i n t h e case of s - t r i a z i n e c r y s t a l , where t h e s h o r t range i n t e r - a c t i o n s a r e domilrant we can l i m i t ourselves t o t h e n e a r e s t neigh- bours and t h e n t h e r e l a t i o n can be w r i t t e n i n a s i m p l i f i e d form as,
with t h e phase f a c t o r
(hcq).
The f i r s t term d e s c r i b e s d i s p e r s i o n of t h e acoustic phonon due t o t h e d i r e c t t r a n s l a t i o n a l - t r a n s l a t i o n a l coupling, while t h e second term, responsible f o r an anomalous dis- persion, follows from i n d i r e c t i n t e r a c t i o n s aue t o t r a n s l a t i o n a l - r o t a t i o n a l coupling, When t h e r e l a t i o n i s generalized by making t h e i n d i r e c t i n t e r a c t i o n s temperature dependent /as i n t h e mean f i e l d theory, f o r example/ it does reproduce t h e observed d i s p e r s i o n curves [ 2 ] , a t d i f f e r e n t temperatmes, almost exactly. A s i t i s seen from r e l a t i o n (4) a s t a b i l i t y l i m i t i s acbieved TorC ( T ) = - q B q . 1
q c
References.
513
J.H. Smith and A.I.bI. Rae, J. Phys., C u , 1761 (1978),
[23 I.U. Eeilman, V.D. Fillenson and J. E c k e r t , J. 3% s., C 1 2 , 118.5 8 9 7 9 ) ,
[3] J.C. Raich and 5.3. S e r n s t e i n , J. Chem. Phys.,
