RAMAN SCATTERING TENSORS FOR ICE Ih

HAL Id: jpa-00226233

https://hal.archives-ouvertes.fr/jpa-00226233

Submitted on 1 Jan 1987

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

RAMAN SCATTERING TENSORS FOR ICE Ih

L. Ziemczonek

To cite this version:

L. Ziemczonek. RAMAN SCATTERING TENSORS FOR ICE Ih. Journal de Physique Colloques, 1987, 48 (C1), pp.C1-15-C1-21. �10.1051/jphyscol:1987103�. �jpa-00226233�

JoURNAL DE PHYSIQUE

Colloque C1, supplement au n o 3, Tome 48, mars 1987

RAMAN SCATTERING TENSORS FOR I C E I,

L . ZIEMCZONEK

Department of Physics, Pedagogical University of S?upsk, Arciszewskiego 2 2 B , PL-76-200 Sfupsk, Poland

R6suin6 . Les coefficients de Clebsch-Gordan pour les reprksentations irrkductibles du groupe d'espace ~4~~ ont 6tk calcul6s au point central de la zone de Brillouin hexagonale. Les tenseurs de diffusion pour la diffusion Raman du premier ordre de la lumihre pour la glace Ih ont QtQ ainsi Qtablis.

Abstract, Clebsch-Gordan c o e f f i c i e n t s f o r t h e 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 s of t h e space group D:~ have been c a l c u l a t e d a t t h e c e n t r e point

r

1, Introduction

The s t r u c t u r e of a c r y s t a l , i n t e r n s applicable t o d i f f r a c t i o n r e s u l t s , i s s p e c i f i e d by giving i t s space group and giving, i n addition, t h e p o s i t i o n s of t h e equivalent s c a t t e r i n g cenkres i n t h e u n i t c e l l . The u n i t c e l l of t h e i c e c r y s t a l has f o u r molecules i n t h e space g r o w DGh 4 (P6 /mmc) [IL-31 with f o u r oxygen atoms i n

3

p o s i t i o n s 2(1/3, 2 / 3 , z ) , f ( 2 / 3 , 1/3, 1/2 + z ) , where z i s c l o s e l y equal t o 1/16,

Because t h e molecules i n i c e a r e o r i e n t a t i o n a l l y disordered a l l t h e nuclear v i b r a t i o n s a r e a c t i v e i n Raman Spectra. I n [3] i s shown t h a t t h e disorder-allowed v i b r a t i o n s should b$ r e l a t i v e l y weak i n t h e spectrum and t h e order-allowed v i b r a t i o n s ( t h e zero-wave v e c t o r

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987103

JOURNAL DE PHYSIQUE

v i b r a t i o n s ) should be r e l a t i v e l y s t r o n g .

I n next s e c t i o n s we have constructed t h e f i r s t o r d e r Raman s c a t t e r i n g t e n s o r s f o r t h e o r d e r a l l o w e d v i b r a t i o n s . This i s t h e f i r s t s t e p i n construction of Rman t e n s o r s f o r i c e ,

Although i c e Ih appears t o be f u l l y disordered within t h e

r e s t r i c t i o n s of t h e i c e r u l e s t h e r e i s s t i l l a good deaf of order[4].

From [4] we have t h a t i t seems l i k e l y t h a t t h e s p e c t r a of

o r i e n t a t i o n a l l y ordered and disordered i c e Ih a r e c l o s e l y r e l a t e d t o one another. This i s so i n t h e case of i c e s V I I I and V I I which a r e r e s p e c t i v e l y ordered and disordered forms of t h e same s t r u c t u r e and where both Raman s p e c t r a a r e similar, The d i s o r d e r i n i c e r e s u l t s i n broadened bands,

To understand t h e i c e spectrum i n d e t a i l i t must be approached from t h e point of view of s o l i d s t a t e spectroscopy, not i s o l a t e d - molecule spectroscopy, I n terms of t h e group theory: from t h e space group of a c r y s t a l , not t h e symmetry of a molecule, The d i s o r d e r reduces t h e symmetry of t h e c r y s t a l . S t a r t i n g from D:~ we have t o c o n s i d e r t h e subgroups of t h i s space group and t h i s can be predicted i n Birman and Berenson method of c o n s t r u c t i o n of Raman t e n s o r s ,

2 , Clebsch-Gordan c o e f f i c i e n t s f o r t h e space group

For t h e i r r e d u c i b l e space group r e p r e s e n t a t i o n DkL contained i n k U l ' t h e d i r e c t product of t h e 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 s &" ^{and D-}

w i t h wave v e c t o r s s a t i s f y i n g t h e wave v e c t o r s e l e c t i o n r u l e s

p6'C ⁺ vd"kU= yd& ( 1 t h e b a s i s f u n c t i o n s are l i n e a r combinations of t h e b a s i s

k'l 4 gL" with t h e Clebsch-Gordan c o e f f i c i e n t s f u n c t i o n products -,

d :a 6" a"

where 8 i s m u l t i p l i c i t y ^index. We o f t e n w r i t e

, , &"" a r e t h e i r r e d u c i b l e r e p i e s e n t a t i o n s induced from t h e

kL ~ U L U

so c a l l e d small 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 s d- , d g f ' , d- o f t h e wave v e c t o r groups G(k), G(&' ), G(kl') r e s p e c t i v e l y ,

It w i l l be of l a t e r i n t e r e s t t o demonstrate how t h e matrix U of CGCs transforms whea t h e r e p r e s e n t a t i o n s , & , &'" undergo s i m i l a r i t y transformations A, A', A". It has been shown /5]1 t h a t

a ^{= (A' g,} ) u A - ~ (4 1

where ri i s t h e m a t r i x of CGCs a f t e r transformation,

To compute C G C s we use a method given by Berenson and Birman[5].

D e t a i l s a r e given i n [ 6 ] , Table I gives C G C s f o r

f@r.

For

r

r

r

2+ 2+ 1t

I n t h e t a b l e l a b e l s of r e p r e s e n t a t i o n s t h a t c o n t r i b u t e t o t h e symmetrized square a r e enclosed i n square b r a c k e t s , The number a t t h e bottom of each r e g r e s e n t a t i o n column d i v i d e s each element of t h a t one, Labels and g e n e r a t o r s of t h e 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 s , t h e canonical wave v e c t o r , numbering of symmetry o p e r a t i o n s a r e as

given i n [ 7 ] .

Table I . CGCS f o r r@

r

CI-18 JOURNAL DE PHYSIQUE

Table 11. CGCs f o r re9

r

3 , Rman s c a t t e r i n g t e n s o r s

R a m a n s c a t t e r i n t ; i n v o l v e s t h e o p t i c a l p o l a r i z a b i l i t y which 2 2 z2

t r a n s f o m s as t h e b i l i n e a r components x , ^y , , ^xz and y z , ~t was shown by gilagavarltm [8] t h a t t h e c h a r a c t e r of t h e r e d u c i b l e r e p r e s e n t a t i o n s of t h e s e t r a n s f o m s can be w r i t t e n as

r%(i ) = 2 ^{~ C O S} 0; (1+2cos 4l ) ( 5 ) where $i i s t a e r o t a t i o n angle and t h e p l u s s i g n i s f o r r o t a t i o n s while t h c a i n u s s i g n i s f o r r o t o i n v e r s i o n s , Squation ( 5 ) can be w r i t t e n i n terms of t h e t r a c e of t h e 3x3 r o t a t i o n m a t r i c e s ~ ( i ) as

I n o r d e r f o r Ranan s c a t t e r i n g t o t a k e g l a c e symmetry r e q u i r e s t h a t t h e i r r e d u c i b l e r e d r e s e n t a t i o n s &' of t h e ground s t a t e be i n t h e same subspace as t h e r e d u c i b l e r e p r e s e n t a t i o n of t h e p h y s i c a l process involved, I n terms of c h a r a c t e r s t h e c o n d i t i o n i s

where g i s t h e o r d e r of t h e space group,

It should be mentioned t h a t t h e r e d u c i b l e r e p r e s e n t a t i o n by which t h e o p t i c a l p o l a r i z a b i l i t y transforms i s t h e symmetrized product of

t h e v e c t o r r e p r e s e n t a t i o n s [DY@

31.

w e h a v e : [ I I ~ B D ~ ] = 2 L t @ & + @ c + .

Let us consider f i r s t o r d e r Raman s c a t t e r i n g f o r l i g h t i n c i d e n t i n t h e a d i r e c t i o n and s c a t t e r e d i n t h e p d i r e c t i o n , producing

a piionon of symmetry

I:) ^.

" P

e XI"

p&i)g)

w r ~ e r e c ( A ) degends on r e p r e s e n t a t i o n DL (DCz&rL i n previous n o t a t i o n ) and U i s t h e CGC m a t r i x which reduces t h e symmetrized product of v e c t o r represent a t i o n s DV] i n t o 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 s D L , I f t h e v e c t o r r e p r e s e n t a t i o n i s not

i r r e d u c i b l e but i s a sum of 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 s D" i t i s convenient t o use double i n d i c e s and w r i t e : DLIalZ= v 6 _,I,, D ,p. ^4' (9)

Then t h e s c a t t e r h i n y t e n s o r a l s o has double i n d i c e s and we have 197

The i n d i c e s A ' a and k"p r e f e r t o rows of t h e i r r e d u c i b l e represen- t a t i o n s DL' and D~'' r e s p e c t i v e l y , and 3" and DL' a r e contained i n . DV, These i n d i c e s correspond t o t h e usual C a r t e s i a n i n d i c e s s i n c e t h e

basis f u n c t i o n s belonging . t o row a. of D"' and row p of DL" a r e e i t h e r (x, y, z ) o r a r e r e l a t e d t o (x, y, z ) by a u n i t a r y transformation.

For t h e hexagonal-close-packed s t r u c t u r e we have:

where

G+

c+

.

r e p r e s e n t a t i o n can be transformed t o one witn b a s i s ( x , y , z )

-

transforming

L+

G+

The C G C s f o r h , c , p , s t r u c t u r e , using t h e transformed bases when r e l e v a n t , a r e obtained f rorn t h o s e given i n t a b l e I a s follows ( 4 ) :

f l d ' @ ~ " = (jfBB")U"e*U B-l (11)

where B: BB(: ^{B= A} o r u n i t y matrix, and a r e given i n t a b l e 11.

'21-20 JOURNAL DE PHYSIQUE

Nov~ we can c o n s t r u c t Baman s c a t t e r i n g t e n s o r s using (9-10).

- -

For a =x, y and p =x, y we have 1= ^A"= r ^{and 1 =}

r

- ^{6+ l+J+}

For a =x, y and p = z we have /'<+, ^,f"=

r

For a = z and p = x , y w e have A'&+,

/=G+

For a = z and p = z we have A ' = ~ " =

r2+

r

These t e n s o r s a r e presented i n t a b l e I11 which i s i n accordance

i t h [LO]. 4

Table 111. Raman s c a t t e r i n g t e n s o r s f o r D,,: PL'?/!)

4 , References

b]

[2 1 F l e t c h e r , N.H., The Chemical Ehysics of I c e ( U n i v e r s i t y Yress Cambridge) 1970.

[ 5 ] WOW, P.T.T., K ~ w , D . D . and Whalley, E., The Raman Spectrum of

t h e T r a n s l a t i o n a l L a t t i c e Vibrations of I c e I in: Physics h

and chemistry of I c e , Ed. E. Whalley, S.J. Jones and L.W. Gold, Royal S o c i e t y of Canada, Ottawa, 1973, p. 87-92,

[4I Whalley, E., Can. J. Chem. 3 (1977) 3429-3441.

b] ^{Berenson, R.} and Biman, J.L., J. Math. B y s , 16 (1975) 227-235.

16 1 Ziemczonek, L. and Suff czyliski,M., J.&ys.A 18 (1985) 1627-1635.

[ 7 ] C r a c h e l l , A . P . , Davies, B.L., M i l l e r , S.C. and Love, w.F.,

~roneckerc Product Tables, vol. 1-4 (New York: IPI-Plenum) 1979.

181 a q a v a n t m , s., e o c , I n d i a n cad. S c i . , Sect.A 2 (1941) 443, 19 1 B i m a n , J.L. and Berenson, R., Phys. Rev. B 2 (1974) 4512-4517.

k O ] Loudon, H., Adv. Phys. 13 (1964) 423-482,

COMMENTS

W.B. HOLZAPFEL

Do you think that your considerations are relevant for disordered forms of ice ? Answer :

Now, I am not in a position to say yes.

However the symetry describes also the small vibrations from the equilibrium states.

From this point of view we can use a Berenson and Birman method in case when the disorder is not very high.