HAL Id: jpa-00226447
https://hal.archives-ouvertes.fr/jpa-00226447
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.
THE POSSIBLE ORDERED STRUCTURES OF ICE Ih
R. Howe
To cite this version:
R. Howe. THE POSSIBLE ORDERED STRUCTURES OF ICE Ih. Journal de Physique Colloques, 1987, 48 (C1), pp.C1-599-C1-604. �10.1051/jphyscol:1987182�. �jpa-00226447�
JOURNAL D E PHYSIQUE
C o l l o q u e C1, s u p p l 6 m e n t au n o 3, Tome 48, m a r s 1987
THE POSSIBLE ORDERED STRUCTURES OF I C E Ih
R . HOWE
D e p a r t m e n t of P h y s i c s , U n i v e r s i t y of B i r m i n g h a m , G B - B i r m i n g h a m B15 ZTT, re at-~ritain
R6sum6 - En considerant l a r e c e n t e decouverte de l a g l a c e X I , l a forme ordonnee de l a g l a c e I h , il e s t i n t e r e s s a n t d S 6 t u d i e r l e s p o s s i b i l i t e s dlarrangement protoniques dans c e t t e phase Ih. Ce t r a v a i l p r e s e n t e une analyse syst6matique des 17 s t r u c t u r e s ordonn6es q u l i l e s t p o s s i b l e de former & p a r t i r de l a g l a c e Ih en gardant l a c e l l u l e de base & 8 mol6cules. Les p o s s i b i i t 6 s avec une c e l l u l e de 12 mol6cules ef quelques c a s p a r t i c u l i e r s d l o r d r e p a r t i e l s o n t a u s s i envisag6s.
Abstract - I n view of t h e r e c e n t discovery of i c e X I , t h e ordered form of i c e I h , it i s of i n t e r e s t t o consider t h e possible arrangements of protons i n t h e phase. This paper p r e s e n t s a systematic a n a l y s i s of t h e 17 ordered s t r u c t u r e s which it i s p o s - s i b l e t o form from i c e I h assuming a u n i t c e l l of 8 molecules. The p o s s i b i l i t i e s w i t h a c e l l of 12 molecules and s p e c i a l cases of p a r t i a l ordering a r e a l s o considered.
INTRODUCTION
The arrangement of t h e protons i n i c e I h has been of i n t e r e s t f o r many y e a r s , and s e v e r a l attempts have been made t o propose ordered s t r u c t u r e s , e i t h e r a s an a l t e r n a t i v e t o t h e s t a t i s t i c a l model a t high temperatures, o r a s a d i s t i n c t phase which should be a t t a i n e d a t low temperatures. , t h i s i n t e r e s t has been s t i m u l a t e d by t h e observation by c a l o r i m e t r i c me%zt%] of a f i r s t - o r d e r t r a n s i t i o n a t 72 K, c a t a l y s e d by doping with KOH. The b e l i e f t h a t t h i s i s t h e long-sought t r a n s i t i o n t o t h e ordered phase, i c e X I 121, i s supported by d i e l e c t r i c measurements (31 and a l s o by a neutron d i f f r a c t i o n study 141 of p a r t i a l l y
transformed D20 i c e t h e r e s u l t s o f which a r e c o n s i s t e n t with a p o l a r s t r u c t u r e of space group Cmcll a s proposed by Kamb [5] f o r t h e ordered phase.
The aim o f t h i s paper i s t o p r e s e n t a systematic a n a l y s i s of t h e p o s s i b l e ordered s t r u c t u r e s with u n i t c e l l s o f f o u r , e i g h t , and ( i n summary) twelve molecules. The Bernal-Fowler r u l e s [6] - two protons p e r oxygen and one p e r bond
- a r e assumed t o be s t r i c t l y obeyed, b u t no c o n s i d e r a t i o n s of energy o r s t a b i l i t y have been taken i n t o account.
THE FOUR-MOLECULE CELL
I f t h e protons a r e ignored, t h e s t r u c t u r e of i c e I h has t h e hexagonal symmetry o f t h e space group ~ 6 ~ / m m c , with f o u r oxygen atoms i n t h e u n i t c e l l . I n t h e s t a t i s t i c a l model, t h i s symmetry i s n o t broken by t h e a d d i t i o n o f t h e protons, a s t h e two proton s i t e s on each bond a r e c r y s t a l l o g r a p h i c a l l y e q u i v a l e n t and each is considered t o be half-occu&ed. Any attempt t o o r d e r t h e protons i n t h e four- molecule c e l l w i l l r e s u l t i n t h e l o s s both o f t h e c e n t r e of symmetry and, u n l e s s o r d e r i n g occurs i n t h e c-axis d i r e c t i o n only, of t h e hexagonal symmetry of t h e d i s o r d e r e d s t r u c t u r e .
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987182
C1-600 JOURNAL DE PHYSIQUE
It i s p o s s i b l e t o f i n d arrangements of protons which r e t a i n t h e t r a n s l a t i o n a l symmetry of t h e four-molecule c e l l while s a t i s f y i n g t h e Bernal-Fowler r u l e s . There i s only one arrangement of the protons i n one puckered (0001) l a y e r of t h e c e l l which maintains t h e t r a n s l a t i o n a l symmetry. I n such an ordered l a y e r , t h e u n i t c e l l can be chosen i n any of three o r i e n t a t i o n s , a s shown i n Figure 1. '?'he a l t e r n a t e l a y e r can a l s o have any of t h r e e o r i e n t a t i o n s . There a r e t h e r e f o r e two d i s t i n c t s t r u c t u r e s possible with the four-molecule c e l l , of which one, with t h e mirror-axes of t h e l a y e r s a n t i p a r a l l e l , has t h e orthorhombic space group CmcZ1. The o t h e r , where t h e mirror axes of the l a y e r s make an angle of 60°, i s monoclinic, with space group Cc. Both these s t r u c t u r e s a r e conventionally represented on an eight-molecule centred c e l l (Figure 2 ) .
It i s i n t e r e s t i n g t o note t h a t t h e four-molecule c e l l permits only p o l a r ordered s t r u c t u r e s . I t i s t h e r e f o r e misleading t o attempt, a s has been done f o r example by Minagawa [7], t o p r e d i c t on the b a s i s of c a l c u l a t i o n s using t h i s c e l l the conditions f o r s t a b i l i t y of an a n t i f e r r o e l e c t r i c s t r u c t u r e .
Figure 1 Ordered l a y e r of four-molecule s t r u c t u r e with t h r e e choices of u n i t c e l l .
Figure 2 Oxygens and bonds involved i n t h e e i g h t - molecule c e l l .
THE EIGHT-MOLECULE CELL
I f t h e orthorhombic t r a n s l a t i o n a l symmetry of t h e eight-molecule c e l l i s preserved, o t h e r p o s s i b l e ordered s t r u c t u r e s can be found, a s shown i n Figure 3 , i n which each rectangle represents t h e upper (001) l a y e r . For t h e purposes of t h i s a n a l y s i s , it i s convenient t o consider t h e c-axis a s v e r t i c a l and t h e puckered s h e e t s of t h e i c e s t r u c t u r e a s horizontal l a y e r s , of which t h e r e a r e two i n t h e u n i t c e l l . Here, and subsequently, t h e oxygen'site a t t h e l e f t of t h e 'upper' l a y e r i s considered t o be a t t h e f o o t of a c-axis bond connected t o t h e l a y e r above, while t h e oxygen a t t h e r i g h t i s a t t h e upper end of a c-axis bond connected t o t h e l a y e r below. I n t h e 'lower' l a y e r t h i s i s reversed and t h e s l a n t of each oblique bond i s opposite t o t h a t of t h e bond immediately above it. Those protons which a r e
independently f i x e d a r e shown i n Figure 3 by small f i l l e d c i r c l e s : t h e small open c i r c l e s represent protons whose p o s i t i o n s a r e f i x e d a s t h e r e s u l t of t h e choice of t h e o t h e r s . The f i r s t column shows t h e choice of a proton on the c-axis bond s i t e a t l e f t : t h e second column shows t h e t h r e e possible choices f o r t h e p o s i t i o n of t h e second occupied c-axis s i t e i n t h e l a y e r . In t h e t h i r d column t h e p o s s i b l e ways of completing t h e molecule on t h e l e f t a r e shown f o r each configuration of t h e c-axis bonds. I n t h r e e cases, l a b e l l e d 3 , 6 and 9, t h e Bernal-Fowler r u l e s and t h e boundary conditions of t h e c e l l f i x t h e p o s i t i o n s of a l l t h e remaining protons i n t h e l a y e r . I n t h e o t h e r cases, two o r i e n t a t i o n s of t h e molecule a t r i g h t a r e possible: once one of these has been chosen a l l o t h e r protons a r e fixed. The configurations r e s u l t i n g from these choices a r e shown i n t h e fourth column. The diagram is s o arranged t h a t each of t h e t h r e e s e c t i o n s defined by t h e choice of c-axis bonds has complete mirror symmetry.
There a r e thus 13 f i n a l arrangements of a s i n g l e layer: t h e r e a r e of course a f u r t h e r 13 p o s s i b l e , each corresponding t o one of those shown, with a l l occupied proton s i t e s exchanged with empty s i t e s . These 'converse' arrangements do not however increase t h e number of unique arrangements of t h e extended l a y e r , of which
there a r e five: 2 , 3 and 4 a r e a l l equivalent by r o t a t i o n through 120°; 1 and 5, 10 and 1 3 , and 11 and 1 2 a r e mirror-image p a i r s ; 6 , 7 , 8 and 9 a r e a l l r e l a t e d by r o t a t i o n and r e f l e c t i o n .
10
11
12
13
Figure 3 Generation of 1 3 upper layers of t h e eight-molecule c e l l .
For each upper layer shown, a lower layer may be constructed i n projection by the same process as the converse layers, remembering however t h a t i n three dimensions the bonds s l a n t i n the opposite sense. Each lower layer so constructed w i l l be designated by t h e same number a s i t s corresponding upper layer. Provided t h a t t h e v e r t i c a l bonds a r e compatible, any upper layer may be combined with any lower layer.
There a r e thus 5 x 5 + 4 x 4 + 4 x 4 = 57 possible combinations of t h e upper layers shown with the corresponding lower layers. However, r o t a t i o n s and
Cl-602 JOURNAL DE PHYSIQUE
r e f l e c t i o n s s e v e r e l y r e s t r i c t t h e number o f unique s t r u c t u r e s : f o r example, t h e combination o f upper l a y e r s 2, 3 and 4 w i t h t h e i r a p p r o p r i a t e lower l a y e r s a l l g i v e t h e same s t r u c t u r e , denoted by F i n Table 1. S i m i l a r arguments apply t o o t h e r groups o f combinations: t h e 1 7 unique s t r u c t u r e s and t h e ways i n which t h e y may b e formed a r e shown i n Table 1, where t h e upper l a y e r s a r e shown a l o n g t h e t o p l i n e and t h e lower l a y e r s down t h e l e f t - h a n d column. The s t r u c t u r e s a r e denoted by l e t t e r s A-S. No combinations o f l a y e r s shown i n F i g u r e 3 w i t h lower l a y e r s from t h e
'converse' s e t a r e p e r m i t t e d , a s by t h e n a t u r e o f t h e s e l a y e r s t h e c - a x i s bonds w i l l n o t be compatible.
Table 1 Combinations o f t h e upper and lower l a y e r s 1-13 o f t h e eight-molecule c e l l t o q i v e t h e 1 7 s t r u c t u r e s A-S.
1 A B C D E 6 H J K L l O M N P Q
2 B F G G D 7 J H L K 1 1 N R S P
3 C G F G C 8 K L H J 1 2 P S R N
4 D G G F B 9 L K J H 1 3 Q P N M
5 E D C B A
The 1 7 s t r u c t u r e s , w i t h t h e i r space groups and p o l a r o r a n t i f e r r o e l e c t r i c n a t u r e , a r e summarized i n Table 2 , where t h e l e t t e r s A-S r e f e r t o Table 1.
F, t h e s i m p l e s t and most symmetrical o f t h e s e s t r u c t u r e s , i s t h e C m ~ 2 ~ s t r u c t u r e d i s c u s s e d above. G i s t h e o n l y o t h e r p o s s i b l e c e n t r e d s t r u c t u r e .
M i s t h e s t r u c t u r e proposed by Owston [8] on t h e grounds t h a t t h e p r o t o n s o f each molecule a r e a s f a r a s p o s s i b l e from t h o s e o f t h e neighbouring molecules. I n t h a t p a p e r , t h e s t r u c t u r e was i n c o r r e c t l y a s s i g n e d t o t h e monoclinic space group PC.
The same s t r u c t u r e , w i t h i t s c o r r e c t space roup Pnaal, h a s a l s o been proposed on s i m i l a r grounds by Davidson and Morokuma [9(1.
Q and R a r e i n t e r e s t i n g i n t h a t , a l t h o u g h n o t centrosymmetric, t h e y a r e g e o m e t r i c a l l y non-polar : a l l t h e o t h e r s t r u c t u r e s belong t o p o l a r space groups.
THE TWELVE-MOLECULE CELL
The o r d e r e d s t r u c t u r e proposed by Bernal and Fowler [63 h a s a c e l l o f twelve molecules, w i t h t h e hexagonal space group PE3cm. This i s t h e s m a l l e s t c e l l on whichan o r d e r e d s t r u c t u r e w i t h hexagonal symmetry can be found. I t i s p o s s i b l e t o c a r r y o u t a n a n a l y s i s o f o t h e r s t r u c t u r e s , w i t h t h e t r a n s l a t i o n a l b u t n o t t h e r o t a t i o n a l symmetryof t h e twelve-molecule c e l l , i n a s i m i l a r manner t o t h a t g i v e n above f o r t h e e i g h t - molecule c e l l . This y i e l d s 5 t y p e s o f l a y e r , o f which one i s i d e n t i c a l t o 2/3/4 i n Figure 3 and one i s t h a t used by Bernal and Fowler. The remaining t h r e e l a y e r s havetwo c-axis bonds i n one o r i e n t a t i o n f o r every one i n t h e o p p o s i t e o r i e n t a t i o n : two a r e l a y e r s with no symmetry while t h e t h i r d h a s a m i r r o r a x i s . The p o s s i b l e combinationsof t h e s e l a y e r s g i v e a t o t a l o f 1 5 d i s t i n c t s t r u c t u r e s i n c l u d i n g F, G and t h e Bernal- Fowler s t r u c t u r e : most o f t h e r e s t a r e o f low symmetry.
PARTIAL ORDERING
T h e p o s s i b i l i t y o f p a r t i a l o r d e r i n g a l s o h a s t o b e t a k e n i n t o a c c o u n t . T h i s can t a k e a v a r i e t y o f forms. Most simply, and probably most r e a l i s t i c a l l y , t h e r e i s t h e p o s s i b i l i t y t h a t , w h i l e some s i t e s which would b e occupied i n a f u l l y o r d e r e d s t a t e a r e v a c a n t , and v i c e v e r s a , t h e r e i s a b i a s towards t h e ' c o r r e c t ' s i t e s .
C r y s t a l l o g r a p h i c a l l y , t h i s would r e s u l t i n any s u p e r l a t t i c e r e f l e c t i o n s a p p r o p r i a t e t o t h e f u l l y o r d e r e d s t a t e b e i n g p r e s e n t b u t weak.
I t i s a l s o c o n c e i v a b l e t h a t some s i t e s should be completely o r d e r e d , f o r example t h e c - a x i s bonds o n l y , w h i l e t h e o t h e r s a r e d i s o r d e r e d s u b j e c t t o t h e Bernal-Fowler r u l e s . Such a c a s e was proposed by Rundle [lo], i n an a t t e m p t t o
account f o r t h e apparently polar behaviour of some i c e specimens. In a s i m i l a r ' fashion, a d i r e c t i o n i n t h e plane may be chosen t o be completely ordered while leaving d i s o r d e r i n t h e c-axis bonds and i n t h e o t h e r d i r e c t i o n s i n t h e plane. Such configurations would have much lower r e s i d u a l entropy than t h e f u l l y disordered s t a t e .
Table 2 Symmetryandpolar p r o p e r t i e s of t h e 17 eight-molecule s t r u c t u r e s .
Dipole Dipole moment moment p a r a l l e l perpendicular t o c t o c
A Pcazl Yes No
P1 Yes Yes
C PC Yes yes
Pl Yes Yes
E PnaZl Yes Yes
F CmcPl Yes No
G Cc yes yes
H P21 No No
J P21 No Yes
K PC NO yes
L PC No Yes
N P21 No Yes
P P21 No Yes
S PnaZl No Yes
F i n a l l y , i t i s p o s s i b l e t o imagine a s t r u c t u r e i n which t h e bonds i n two d i r e c t i o n s a r e f u l l y ordered, dividing t h e l a t t i c e i n t o an a r r a y of l i n e a r chains each of which can be independently o r i e n t e d i n e i t h e r of two senses. Such a s i t u a t i o n would leave only a very small r e s i d u a l entropy. Some, though n o t a l l , such s t r u c t u r e s have a higher symmetry than t h e corresponding f u l l y ordered s t a t e . CONCLUSION
The possible arrangements of protons i n i c e X I $ssuming c e r t a i n u n i t c e l l s of four, e i g h t o r twelve molecules have been i n v e s t i g a t e d and c l a s s i f i e d . The simplest f u l l y ordered s t r u c t u r e , denoted by 'F' i n t h i s paper, i s t h a t proposed by Kamb [6].
However, t h e p o s s i b i l i t y t h a t some of t h e o t h e r s t r u c t u r e s suggested might have lower energy cannot y e t be ruled out.
ACKNOWLEDGEMENTS
This work was c a r r i e d out during t h e tenure of a research studentship from t h e Science and Engineering Research Council. The author wi,shes t o acknowledge many h e l p f u l discussions with D r R. W. Whitworth, D r A. I . M. Rae, D r R. C. Ward and Professor A. J. Leadbetter.
JOURNAL DE PHYSIQUE
REFERENCES
[l] TAJIMA, Y . , MATSUO, T., and SUGA, H . , Nature, Lond. 299 (1982) 811-812.
TAJIMA, Y . , MATSUO, T., and SUGA, H., J. Phys. Chem. S o l i d s 5 (1984) 11351144.
11
MATSUO, T., TAJIMA, Y., and SUGA, H., J. Phys. Chem. S o l i d s 5 (1986) 165-173- KAWADA, S., and DCHATA, H . , J. Phys. Soc. Japan 54 (1985) 477-479LEADBETTER, A. J . , WARD, R. C., CLARK, J. W., TUCKER, P. A., MATSUO, T., and SUGA, H., J. Chem. Phys. 82 (1985) 424-428.
[s] KAMB, B., ' P h y s i c s and Chemistry o f I c e ' , Eds E. Whalley, S. J. Jones and L. W. Gold, Royal S o c i e t y o f Canada, Ottawa f1973) 28-41.
BERNAL, J. D., and FOWLER, R. H., J. Chem. Phys. 1 (1933) 515-548.
MINAGAWA, I . , J. Phys. Soc. Japan 50 (1981) 3669-3676 OWSTON, P. G., J. Chim. Phys. 50 (1953) C13-~18.
DAVIDSON, E. R., and MOROKUMA, K . , J. Chem. Phys. 81 (1984) 3741-3742.
RUMILE, R. E., J. ~ h y s . Chem. 59 (1955) 680-682.
COMMENTS
Q u e s t i o n of T. MATSUO
It seems t h a t an i n f i n i t e l y l o n g c h a i n of d i s p l a c e d hydrogen atoms is r e q u i r e d i f one wants t o i n t r o d u c e p a r t i a l d i s o r d e r i n t h e ordered s t r u c t u r e s . Is it p o s s i b l e t o do t h a t without changing t h e i r s p a c e groups ?
Answer :
I n t h e case of some of t h e s t r u c t u r e s d e s c r i b e d , it would be p o s s i b l e t o i n c r e a s e t h e symmetry of t h e u n i t c e l l by having series of much c h a i r s a l l s t r a i g h t and p a r a l l e l t o one s i d e of t h e c e l l , w i t h a random series of o r i e n t a t i o n s . I n t h e c a s e o f structure "FW, however, t h e space group would n o t be changed as its symmetry is a l r e a d y as high as p o s s i b l e f o r t h i s u n i t cell.
A c h a i n o r c h a i n s of r e v e r s e d bonds which w a s n o t s t r a i g h t would have no e f f e c t on t h e o v e r a l l s y m m e t r y o f t h e s t r u c t u r e e x c e p t s l i g h t l y t o w h e a k e n a n y c r y s t a l l o g r a p h i c r e f l e c t i o n s due t o t h e lowering o f symmetry caused by t h e o r d e r i n g .