PHONONS IN SUPERIONIC CONDUCTORS

HAL Id: jpa-00221586

https://hal.archives-ouvertes.fr/jpa-00221586

Submitted on 1 Jan 1981

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.

PHONONS IN SUPERIONIC CONDUCTORS

W. Hayes

To cite this version:

W. Hayes. PHONONS IN SUPERIONIC CONDUCTORS. Journal de Physique Colloques, 1981, 42

(C6), pp.C6-167-C6-174. �10.1051/jphyscol:1981650�. �jpa-00221586�

JOURNAL DE PHYSIQUE

CoZZoque C6, suppZc5ment au n012, Tome 42, dgcembre 1981 page C6-167

PHONONS IN SUPERIONIC CONDUCTORS

W. Hayes

CZarendon Laboratory, U n i v e r s i t y o f Oxford, United Kingdom

A b s t r a c t

-

Detai l e d i n f o r m a t i o n about t h e n a t u r e o f d i s o r d e r i n s u p e r i o n i c conductors can be o b t a i n e d from t h e study o f v i b r a t i o n a l e x c i t a t i o n s u s i n g Raman, B r i 1 i o u i n and n e u t r o n s c a t t e r i n g techniques. The p r e s e n t review w i l l be concerned w i t h t h e a p p l i c a t i o n o f these techniques t o m a t e r i a l s such as Agl, CaF2 and sodium 6-alumina, w i t h emphasis on t h e l a t t e r .

I. I n t r o d u c t i o n - .

-

Most i o n i c s o l i d s have values o f t h e e l e c t r i c a l c o n d u c t i v i t y o immediately below t h e m e l t i n g temperature TFq about f o u r o r d e r s o f magnitude smal l e r t h a n i n t h e me1 t. i n L i F, f o r example, a increases from 10-1°(ncm)-I a t

room temperature t o l ~ - ~ ( Q c m ) - l j u s t below TM = 1140 K. On m e l t i n g a increases d i s c o n t i n u o u s l y t o 1 0 ( Q c m ) - ' . However, some i o n i c s o l i d s have values o f o i n t h e c r y s t a l l i n e s t a t e comparable t o t h a t i n molten s o l i d s and a r e r e f e r r e d t o as s u p e r i o n i c s o r f a s t - i o n conductors

r .

T h i s h i g h v a l u e o f o i s due t o i o n i c t r a n s - p o r t and i s a consequence o f e x t e n s i v e d i s o r d e r i n a component s u b l a t t i c e o f t h e s o l i d . The study o f i o n i c v i b r a t i o n s i n d i s o r d e r e d systems has an i n t r i n s i c

i n t e r e s t o f i t s own and i n s u p e r i o n i c s such s t u d i e s throw l i g h t on t h e mechanisms o f c o n d u c t i o n 2.3

.

The s u p e r i o n i c m a t e r i a l s t h a t have been s u b j e c t t o t h e most e x t e n s i v e study a t a fundamental lever i n r e c e n t years may be d i v i d e d i n t o 3 groups:

( I ) S i l v e r and copper based compounds e . g . Agl, RbAg4I5 and Cul, i n which d i s o r d e r occurs i n t h e s i l v e r and copper s u b l a t t i c e s . Here t h e p r o t o t y p e m a t e r i a l i s Agl

.

A t room temperature Agl has t h e hexagonal w u r t z i t e s t r u c t u r e and i s r e f e r r e d t o as 6 Agl. A t T = 147C a f i r s t - o r d e r phase change occurs t o a body-centred c u b i c s t r u c t u r e , r z f e r r e d t o as a Agl

.

The v a l u e o f o j u s t below Tc i s s 3 x 1 9 - ~ ( n c m ) - l ;

i t increases a b r u p t l y a t T, t o r 1.3(Qcm)-' and fa1 I s s l i g h t l y from t h i s v a l u e on m e l t i n g . A v a r i e t y o f s t r u c t u r a l s t u d i e s ' i n d i c a t e s t h a t i n t h e a phase t h e i o d i n e

ions form a f a i r l y r i g i d bcc l a t t i c e and t h a t t h e s i l v e r ions a r e randomly d i s t r i - buted between them i n t e t r a h e d r a l i n t e r s t i t i a l s i t e s . I t a l s o appearsf t h a t t h e motion o f A ~ + i o n s between nearest-neighbour t e t r a h e d r a l s i t e s i s t h e b a s i c s t e p i n i o n conduction. Phonon s t u d i e s on Agl and o t h e r m a t e r i a l s i n t h i s group have been reviewed r e c e n t l y 3 and w i l l n o t be discussed f u r t h e r here. However, t h e attempts t h a t have been made t o o b t a i n i n f o r m a t i o n about t h e dynamics o f t h e m o b i l e ions u s i n g q u a s i e l a s t i c l i g h t s c a t t e r i ng4*5'6 and quasie l a s t i c neutron s c a t t e r i ng 7 should be mentioned.

( i i M a t e r i a l s w i t h t h e f l u o r i t e s t r u c t u r e . These w i 1 1 be discussed i n § 2.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981650

C6- 168 JOURNAL DE PHYSIQUE

( i i i ) M a t e r i a l s w i t h t h e 6 alumina s t r u c t u r e . These w i I I be discussed i n § 3.

2. C r y s t a l s w i t h t h e F l u o r i t e s t r u c t u r e .

-

A pronounced h e a t anomaly occurs i n ha1 ides w i t h t h e f l u o r i t e s t r u c t u r e a t a temperature Tc we1 l below TM;for CaF2 Tc =

1430 K and T,,, = 1633 K. T h i s anomaly i s due t o development o f e x t e n s i v e d i s o r d e r i n t h e a n i o n s u b l a t t i c e and i s a s s o c i a t e d w i t h t h e o n s e t o f h i g h i o n i c c o n d u c t i v i t y . 2 A s t u d y o f t h e e f f e c t s o f anharmonicity ( T < Tc) and l a t t i c e d i s o r d e r (T > Tc) on t h e Raman spectrum o f CaF2, SrF2, BaF2, SrCl ₂ and RbF2 was made by E l l i o t t e t a l 8

.

I n t h e s e m a t e r i a l s t h e T Raman-allowed phonon broadens w i t h i n c r e a s i n g T and t h e d e t a i l e d shape o f t h e Raman band below Tc can be e x p l a i n e d q u i t e w e l l u s i n g t h i r d 29 and f o u r t h - o r d e r anharmoni c i t y

.

^{Addi -ti} ona l s c a t t e r i n g develops on t h e low energy s i d e o f t h e T phonon f o r T > T and t h i s can be accounted f o r by a t h e o r y o f

29

defect-induced s c a t t e r i n g , i n c l u d i n g e f f e c t s o f b o t h a n i o n vacancies and a n i o n i n t e r s t i t i a l s .

E f f e c t s o f d i s o r d e r i n t h e s u p e r i o n i c phase o f f I u o r i t e s (T

2

Tc) on e las- t i c c o n s t a n t s was s t u d i e d by B r i I l o u i n s c a t t e r i n g techniques9 showing a dramatic fa1 l o f CII a t Tc. N e i t h e r C 1 2 n o r C44 i s a p p r e c i a b l y a f f e c t e d by t h e d i s o r d e r . S i m i l a r r e s u l t s have been o b t a i n e d by n e u t r o n s c a t t e r i n g techniques1'. The d i f f e r e n t behaviour o f t h e e l a s t i c c o n s t a n t s may be r a t i o n a l i s e d from t h e f a c t t h a t t h e c o n t r i b u t i o n s t o CII from Coulomb and short-range f o r c e s have t h e same s i g n whereas f o r C12 and C44 t h e y have o p p o s i t e s i g n s . I t seems t h a t e f f e c t s o f d e f e c t s on C12 and Cq4 i s smal l because t h e change i n Coulomb and short-range forces l a r g e l y cancel each o t h e r .

I t should be emphasised t h a t t h e t h e o r i e s used t o account f o r e f f e c t s o f d i s o r d e r on t h e Raman and B r i I l o u i n s p e c t r a o f f l u o r i t e s 8 " a r e n o t s e n s i t i v e t o t h e p r e c i s e c o n f i g u r a t i o n s o f vacancies and i n t e r s t i ti a l s . However, very u s e f u l

i n f o r m a t i o n has been o b t a i n e d about t h e s t r u c t u r e o f t h e s u p e r i o n i c s t a t e o f

F i g I : Model o f a 2:2:2 d e f e c t i n f l u o r i t e showing two a n i o n i n t e r - s t i t i a l s g , t w o r e l a x e d a n i o n s 0 and two a n i o n vacancies 0

.

f l u o r i t e s from t h e q-dependence o f t h e i n t e n s i t y o f quas i e la s t i c n e u t r o n s c a t t e r - i n g ' ' . These r e s u l t s may be e x p l a i n e d q u i t e we1 l assuming t h e e x i s t e n c e o f a t r a n s i e n t v a c a n c y - i n t e r s t i t i a l complex o f t h e s o - c a l l e d 2:2:2 t y p e ( F i g u r e I ) , i n v o l v i n g two a n i o n vacancies, t w o a n i o n i n t e r s t i t i a l s and two r e l a x e d anions.

The neutron l i n e w i d t h s suggest t h a t these complexes s u r v i v e i n t h e s u p e r i o n i c s t a t e f o r t i m e s o f ^Q, I ps. T h i s t y p e o f complex i s very s t a b l e and i t s f o r m a t i o n

c o n t r i b u t e s t o t h e l a r g e r e d u c t i o n i n a n i o n Frenkel energy needed t o account f o r t h e o n s e t o f c o o p e r a t i v e d i s o r d e r a t T 12

C

-

B r i I lo u i n s c a t t e r i n g techniques have a l s o been used t o study e f f e c t s on Tc o f doping w i t h t r i v a l e n t c a t i o n s 13

.

^{I t} was found, f o r example, t h a t doping CaF2 w i t h 9 mole p e r c e n t o f YF3 reduces T from 1430 K t o

*

1200 K. i n t h e doped m a t e r i a l 2:2:2 t y p e complexes form, s i m i l a r t o t h a t shown i n F i g u r e I, b u t i n v o l - v i n g y3+ i n c a t i o n s i t e s , r a t h e r t h a n anion vacancies. C a l c u l a t i o n showed 13 t h a t such complexes a c t as t r a p s f o r t h e r m a l l y generated anion i n t e r s t i t i a i s , t h u s r e d u c i n g t h e energy o f f o r m a t i o n o f a n i o n Frenkei p a i r s and a l s o o f Tc.

3. Compounds w i t h t h e $-alumina s t r u c t u r e .

-

Melt-grown c r y s t a l s o f sodium B-al umi na have t h e f o n u l a ( I +x)Na20: I I A1 203 where x

*

0.3 r e p r e s e n t s d e p a r t u r e from s t o i c h i o m t r y . The i o n i c c o n d u c t i v i t y a t room temperature i s l a r g e (0 % 0.04 (acm)-') due t o nonstoichiometry. The c r y s t a l s have a c e n t r o s y m m t r i c s t r u c t u r e w i t h hexagonal symmetry. They c o n s i s t o f s p i n e l - l i k e ( A l I l O i 6 ) b l o c k s separated by m i r r o r planes c o n t a i n i n g sodium and oxygen ions ( F i g u r e 2 ) . The s p i n e l b l o c k s

F i g . 2: Model o f t h e B alumina s t r u c t u r e .

a r e 11.26: t h i c k a l o n g t h e c a x i s . The oxygen i o n i n t h e m i r r o r p l a n e (O(5) i n F i g u r e 2) a r e c o o r d i n a t e d by Al 3+ i o n s i n t h e s p i n e l blocks, j o i n i n g t h e b l o c k s t o g e t h e r . I n e f f e c t , each m i r r o r p l a n e i s a hexagonal network o f O(5) i o n s i n t e r - spersed w i t h c a t i o n s i t e s r e f e r r e d t o as Beevers-Ross (BR) and anti-Beevers-Ross

(aBR), which have s i t e symmetry D6hr and mid-oxygen (mo) which has s i t e symmetry

JOURNAL DE PHYSIQUE

C2" ( F i g u r e 2 ) .

Neutron d i f f r a c t i o n s t u d i e s by Roth e t a1 l 4 on sodium 6-alumina a t room temperature i n d i c a t e t h a t

*

66% o f t h e ~ a + i o n s a r e near BR s i t e s , ^Q 30% a r e near mo s i t e s and

*

4% a r e near aBR s i t e s . These s t u d i e s a l s o led t o t h e suggestion t h a t charge compensation f o r t h e excess sodi um i n t h e mi r r o r p lane occurs through i n t e r s t i t i a l oxygens, 0:-, i n rno s i t e s bound by two aluminium i o n s ( A I ( I ) i n F i g u r e 2 ) d i s p l a c e d from t h e i r normal p o s i t i o n s towards t h e Oi 2-

.

S t u d i e s o f t h e Raman spectrum o f sodium B alumina showed E peaks a t 61 29

and 100 cm- I 15' 16. The 61 cm-I peak was assigned t o v i b r a t i o n s o f Na' i o n s i n BR s i t e s and t h e 100 cm-I peak t o s h e a r i n g o f s p i n e l b l o c k s . The i n f r a r e d a b s o r p t i o n spectrum measured w i t h t h e e l e c t r i c v e c t o r i n t h e m i r r o r p l a n e i s much more complex 17

.

However, i n t e r p r e t a t i o n i s a s s i s t e d by measuring t h e absorp- t i o n o f m a t e r i a l approximating t o s t o i c h i o m e t r y ( F i g u r e 3a). Here e x c i t a t i o n s a r e

- I

sharp (FWHM

*

5 cm ⁾ compared t o t h e w i d t h s o f t h e o v e r l a p p i n g bands found f o r t h e

F i g . 3: I n f r a r e d a b s o r p t i o n w i t h E A c o f n e a r l y s t o i c h i o r n e t r i c sodi um 6 alumina a t 2K, ( a ) , and o f n o n s t o i c h i o m e t r i c sodium B alumina a t 2K, (b), and 300K, ( c ) .

n o n s t o i c h i o m e t r i c m a t e r i a l (FWHM

*

25 c m - ' ) ( F i g u r e 3 b ) . Since we e x p e c t t h e occupation o f BR s i t e s t o be predominant i n t h e more s t o i c h i o r n e t r i c m a t e r i a l we a s s i g n t h e 59 crn-I l i n e i n F i g u r e 3a t o Elu v i b r a t i o n s o f Na* ions near BR s i t e s , c o n s i s t e n t w i t h t h e assignment o f t h e Raman peak I 5 ' l 6 . The o t h e r l i n e s i n F i g u r e 3a a r e due t o complexes o f NaC ions i n v a r i o u s degrees o f a s s o c i a t i o n w i t h 02- 17

and t h e r e i s a one-to-one correspondence w i t h t h e bands o f F i g u r e 3b.

F i g u r e s 3b and 3c show t h e change i n t h e i n f r a r e d a b s o r p t i o n o f n o n s t i o c h i o - m e t r i c m a t e r i a l on g o i n g from 2 t o 300 K. The i n t e n s i t y o f t h e 102 cm-I band i s very s e n s i t i v e t o temperature, decreasing w i t h i ncreasi ng temperature w i t h an

a c t i v a t i o n energy o f

-

^{0.02 eV.} T h i s may be taken t o be t h e energy requi r e d t o remove an e x t r a Na' i o n from t h e v i c i n i t y o f 0;-, making i t avai l a b l e f o r long- range conduction. The 85 cm-' band decays w i t h i n c r e a s i n g temperature w i t h a s i m i l a r a c t i v a t i o n energy. The i n t e n s i t y o f t h e 135 cn-I band i s less tempera- t u r e s e n s i t i v e and t h e i n t e n s i t y o f t h e 176 cm-I band shows l i t t l e change between 2 and 300 K. I f , i n comparison w i t h F i g u r e 3a, we assume t h a t t h e ~ a ' ions g i v i n g r i s e t o t h e main peak i n F i g u r e 3c a r e near BR s i t e s we conclude, i n agreement w i t h Roth e t a1 14, t h a t 66% o f t h e Na+ i o n s occupy such s i t e s a t room temperature.

However, it wou I d seem t h a t t h e assignment by Roth e t a1 l 4 t o mo s i t e s cannot, i n o u r case, r e f e r t o a unique complex s i n c e a t l e a s t f i v e d i f f e r e n t c e n t r e s a r e i nvol ved. The percentage assigned by Roth e t a 1 l 4 t o aBR s i t e s i s t o o sma l l t o be r e a d i l y r e c o g n i s a b l e i n o u r s t u d i e s , b u t it seems l i k e l y t h a t ions i n aBR s i t e s w i I I have a v i b r a t i o n a l frequency comparab l e t o t h a t o f BR s i t e s , Our r e s u l t s a r e i n general agreement w i t h t h e s u g g e s t i o n o f Wol f18 t h a t t h e s o d i um i o n s s h o u l d be considered as two major groups, those bound w i t h d i f f e r e n t s t r e n g t h s t o Oi 2- i.e.

i n a s s o c i a t e d areas and those n o t a p p r e c i a b l y p e r t u r b e d by 0:- i .e. i n unassoci a- t e d areas.

We have c a r r i e d o u t s i m i l a r s t u d i e s on s i l v e r I 9 and p o t a s s i um B-alumi na 20 and more r e c e n t 1 y on s o d i um B" alumi na and i t s isomorphs2'. We s h a l l g i v e a

p r e l i m i n a r y account o f o u r work on sodi um B" alumina here. T h i s m a t e r i a l i s s i m i l a r i n s t r u c t u r e t o s o d i um B-alumina. I t has t h e rhombohedra1 space group D3d and i s 5 composed o f s p i n e l - l i k e b l o c k s o f aluminium oxide, b u t ordered i n a t r i p l e s t a c k i n g sequence a l o n g t h e c a x i s . These b l o c k s a r e separated by conduction s l a b s c o n t a i n - i ng sodium and oxygen ions. The i d e a l s t r u c t u r e i s represented by t h e formula u n i t Na20.MgO.5AI2O3 w i t h M~'' ions d i s s o l v e d p r i m a r i l y i n t e t r a h e d r a l l y - c o o r d i nated A I ~ + s i t e s . F i g u r e 4 shows a p r o j e c t i o n o f t h e p o s i t i o n s o f ~ a + ions i n t h e con- d u c t i n g s lab on t o t h e p l a n e d e f i n e d by t h e b r i d g i n g 0'- ions. The s i t e s which

4: Conduction s l a b o f sodi um

%mi na show i ng oxygen i o n s ( l a r g e c i r c l e s ) , sodium i o n s

( s m s l l c i r c l e s ) and a sodium vacancv (square).

C6- 172 JOURNAL DE PHYSIQUE

correspond t o t h e BR and aBR p o s i t i o n s i n sodium 0 alumina ( F i g u r e 2 ) a r e i n t h i s case e q u i v a l e n t . I t i s general l y assumed t h a t t h e c r y s t a l s a r e n o t s t o i c h i o m e t r i c , c o n t a i n i n g about 15% o f ~ a + vacancies, and t h a t t h e h i g h i o n i c c o n d u c t i v i t y ( %

0.06 ( 0 cm)-I a t room temperature) i s due t o vacancy motion.

F i g u r e 5a shows t h e i n f r a r e d a b s o r p t i o n a t 2K o f a c r y s t a l o f sodi um B"

alumina p r o v i d e d by W. L. Roth, w i t h t h e e l e c t r i c v e c t o r o f t h e r a d i a t i o n perpen-

F i g . 5 : i n f r a r e d a b s o r p t i o n o f sodium 6"

a l uml na w i t h EEc, a t 2K and 300K.

d i c u l a r t o t h e c a x i s . The spectrum i s complex, showing some s i m i l a r i t y t o t h a t o f sodium 6 alumina (Figure 3 ) . An approximate d e c o n v o l u t i o n o f o v e r l a p p i n g bands i s shown i n F i g u r e 5. The i n t e r p r e t a t i o n o f these bands i s a t p r e s e n t t e n t a t i v e a w a i t i n g t h e outcome o f m d e l c a l c u l a t i o n s and f u r t h e r experiments. The bands a t

155 and 176 cm-I a r e probably due t o ~ a + i o n s i n a s s o c i a t i o n w i t h Oi 2-

.

^{The band}

a t 90 cm-' loses about ha1 f o f i t s i n t e n s i t y on warming t o room temperature ( F i g u r e 5b) and by analogy t o sodium 6 alumina may be due t o ~ a + ions weakly bound t o 0 2-

i s

The peak a t 65 cm-I increases i n i n t e n s i t y on warming t o room temperature and i s probably due t o ~ a + i o n s on normal l a t t i c e s i t e s w i t h o u t any o t h e r d e f e c t s near- by. The bands a t 30 and 45 cm-' decrease on warming t o room temperature and may be due t o sodium-vacancy complexes. These a r e a l s o i n d i c a t i o n s o f a weak band a t

% 78 cm-

.

The d i s c u s s i o n g i v e n above suggests t h a t a t room temperature about 45% o f t h e sodium ions a r e on l a t t i c e s i t e s w i t h o u t o t h e r d e f e c t s nearby and t h a t t h e r e s t o f -the sodium i o n s form a v a r i e t y o f complexes. T h i s behaviour i s s i m i l a r t o t h a t o f sodium 6 alumina, but, again, t h e s i t u a t i o n i s more complex than t h e d i f f r a c t i o n

resu l t s f o r sodi um B" a l umi na14 suggest.

4. Conclusions.

-

The s u p e r i o n i c m a t e r i a l s a l r e a d y discussed d i v i d e i n t o two c a t e g o r i e s from t h e p o i n t o f view o f Raman and i n f r a r e d s t u d i e s o f phonons:

( I A s u b l a t t i c e o f t h e m a t e r i a l i s e f f e c t i v e l y t o t a l l y d i s o r d e r e d as i n a Agl o r p a r t l y d i s o r d e r e d as i n f l u o r i t e s . The d i s o r d e r h e r e i s i n t r i n s i c and i s a coopera- -ti ve, thermal l y - i nduced phenomenon. The c h a r a c t e r i s t i c f e a t u r e s o f these m a t e r i a I s

a r e a Raman spectrum corresponding t o a defect-induced single-phonon d e n s i t y o f s t a t e s and a q u a s i e l a s t i c peak a s s o c i a t e d w i t h d i f f u s i v e motion.

( I I ) N o n s t o i c h i o m e t r i c m a t e r i a l s , such as sodium 6 alumina, where t h e d i s o r d e r i s p r e s e n t i n as-grown c r y s t a Is. The ions r e s p o n s i b l e f o r c o n d u c t i v i t y i n these m a t e r i a l s g i v e r i s e t o r e l a t i v e l y sharp Raman and i n f r a r e d l i n e s , g i v i n g d e t a i l e d

i n f o r m a t i o n about envi ronment. There i s a need f o r q u a s i e l a s t i c l i g h t - s c a t t e r i n g s t u d i e s o f these m a t e r i a l s .

References

I . Salamon, M.B., E d i t o r , Physics o f Superioni c Conductors, S p r i nger-Verlag 1979.

2. Hayes, W., Cont. Phys.

19,

469 ( 1978).

3 . Hayes, W., L i g h t S c a t t e r i n g i n Sol ids, Vol

.

^I I, E d i t o r s , M. Cardona and G. Guntherodt, 1982, t o be published.

4. W i n t e r l i n g , G., Senn, W., Grimsditch, M. and K a t i y a r R., L a t t i c e Dynamics (Ed. M. Bal kanski Flammarion Press, 553 ( 1977).

5. F i e l d , R.A., Gallagher, D.A. and K l e i n , M.V., Phys. Rev. B E , 2995 (1978).

6. Nemanich, R.J. M a r t i n R.M. and Mikkelsen, J .C. Jr., F a s t I o n T r a n s p o r t i n S o l i d s (EdS P. Vashishta, J.N. Mundy and G.K. Shenoy), N o r t h H o l l a n d Pub.

Co. 547 (1979).

Eckold, G., Funke, K., Kalus, J. and Lechner R., J. Phys. Chem. Sol.

z,

1097 (1976).

E l l i o t t , R.J., Hayes W., Kleppmann W.G., Rushworth, A.J. and Ryan J.F., Proc.

Roy. SOC. A=, 317 (1978).

Catlow, C.R.A., Comins, J.D. Germano, F.A., Harley, R.T., and Hayes, W., J . P ~ Y S . C. SOI.

s t .

P ~ Y s . ~ , 3 1 9 7 ( 1 9 7 8 ) .

Dickens, M.H., Hayes, W., Hutchings, M.T. and Kleppmann, W.G., J . Phys. C.

Sol. S t . Phys.

12,

17 (1979).

Clausen, K., Hayes, W., Hutchings, M.T., Kjems, J.K., Schnabel, P., and Smith, C. Proc. I n t . Conf on Superionics, G a t l i n b u r g , 1981, t o be published.

Catlow, C.R.A., and Hayes, W., J. Phys. C. Sol. St. Phys., 1981, t o be pub l i shed.

Catlow, C.R.A., Comins, J.D., Germano, F.A. Harley, R.T., Hayes, W. and Owen, I.B., J. Phys. C. Sol. St. Phys.

s,

329 (1981).

Roth, W.L., Reidinger, F. and La Placa S., S u p e r i o n i c Conductors (EdS G.D. Mahan and W.L. Roth), Plenum Press 223 ( 1976).

Chase, L.L., Hao, C.H. and Mahan, G.D., Sol. St. Comm.,

18,

401 (1976).

Hao, C.H., Chase, L.L. and Mahan, G.D., Phys. Rev. B c , 4306 (1976).

Hayes, W. and Holden L. J. Phys. C, S o l . St. Phys., t o be published.

18. Wolf, D., J. Phys. Chem. Sol.

40,

757 (1979).

C6- 174 JOURNAL DE PHYSIQUE

19. Hayes, W., Holden, L. and T o f i e l d , B.C., 1981, J. Phys. C., Sol. St. Phys 14, 511.

-

20. Hayes W., Holden, L., and Hopper G.F., J . Phys. C. Sol. St. Phys.

E,

L317 (1980).

21. Hayes, W. and Hopper, G.F., J . Phys. C., Sol. S t . Phys., t o be p u b l i s h e d .