Jump relaxation in solid ionic conductors

JUMP ~V.tTION

IN S O L m IONIC C O ~ U C l O R S

lOans F L ~ R E I~,~tu,J~t Phyz~ltsclw Cherale#erWe~l~l~h~n ~ ithd~Uni~r#t, ~4aO0M ~ t ~

~m~r ofth~maxau~ ~sdr

in a s~mpl~medclwhlcbyieldsinpaaicuhr thef ~ q ~ g y sp~tmm of~hrn ~ i a s ran,an.

i~odLngt~ewell-kno~ ~cs in Iheo m ~ plan~ productivity~d ~ i n l v R y ,the~ d a w

f~u~

depend~ oCIhe

L lnix@da~ion

~rJl~d in t e ~ s o f a dlstfib~llono f relaxationtlmes, ~e, e.g. refs. [10,11]. Tais g ~ a [ behax4our has

The t~nsport of io~ in ~tids is ~nveni~fly probed by m e a s ~ e n t of the lo~f~qnency ~ndur and oft~er-diffosion~ fi~cients,The m-

~meflmes been termed ~ivemal dy~mie r ~ sponse"[3],whilethenotation"unive~aldielectric ~spon~" [I ] ~fersto the f~quency dependenceo f

suitsa ~ n ~ a l l y interpreted ~ i f the ~obi]e charged defects - ions or e f f ~ t i ~ l ychaTged v a ~ n r - perf o ~ a random jump diffusion.This simple~ n ~ p t is often found to furl, h o o v e r , i f one applies techniques that provide a r ~ l a t l o B on the frequency ~ l e , like the dynamic r qaasielaSlie Peutmn s~ttesin~ and m ~ s u r ~ e n l o f lhe NMR spectra[ density, Aetmdly, frequeocy-zesolved spott m differ from lhe predielionso f the random-h~pplug model i n a specific way, s ~ below, i fl h e mobile ddcets am cha~ctelS~d by (i) stmelur,d disord~ and (il) a repaIsi~ m e t u a L i n ~ t i o a , e.g, the Coulomb inleraelion, Fast lad eondu~o~ likea-AgI. Na-~-alumi~a, and many o t h ~ , g l ~ s . ~ d molten ~aRs faro] the ~ q u i ~ m ~ t s (i) ~ d (it), Pop--ale ~ n d ~ l o n ~ e exampleswhe~ the defects are not ~onic, ~ut e l ~ tron[o. In all o f the Rbovr kiuds o f materials, f ~ qaency-~solved measu~mr h~ve provided evidence for the exist~ce o f a particularrelaxation pro~ss [1-11 ]. Its m ~ t pramin~nt f e a t ~ is a slowing dovm i n time. which can be f o ~ a l l y de-

the p c ~ i t tivity and c ~ d u c f i v i t yonly, Let us brieflyr ~ the experlmentalfindings. (i) I f the ~ n d u c t i v i t y and p e ~ i t l i v i t ylhat a ~ due to the ~laxation are plotted i n their eompt~ planes.theslochn~do~insimesho~upinllle~J]kno~a d i s p l a c ~ e n | o f the ~ n t ~ s o f the almostclrcolor ares b e l ~ the ~ a l a ~ s , ~ e.g. ~f. [6]. Carrespondingly, theslopenflogIa'(o)--~'(O)} ~ u s Iog~ is not close to 2 at small~ , but always s m a l l ~ t h a n l [ 1,2]. Hem a ' denotes the ~ a l part o f the ~ m p l e x d ~ t r i e a l ~nduetivity,#, and w ig the angularfrequency. F u r t h e ~ , the ~tivalion~erg~ o f ~ ' T i s always found t a I~elarg~ i~ t h e l i m i to f Iow than i n ;he I/mit o f ~igh frequencies[ 12,131. (it) H c u l ~ n s~tteringsp~tr~ o f s t r a a u ~ t t y disordered fast ionle ~nduetors and gIas~s generally ~ n t a i n b ~ d quasielastie ~ l ~ n e n t s which ~ n not be unde~tood i n t e ~ s o f sta~isfl~l hopping [ 8,9]. Rather, they s ~ to be ~ u ~ d hy some Inealised hoppingprocess, since their ~ i g h t deceases with d ~ a s i n g momenlumlransf~, ~ [g,9]. Intemstlngly,shape and width o f the b ~ a d ~ m p f f

o 167-273818815 03.50 9 FA~i~r Science P u b l i s h ~ B.V. (North-Holland PhysicsFabhshingDivision)

~ents are found to be prac~i~ny independent of Q [8 I. (ill) In fast ion conducto~ and glas~s the spinlatti~ ~laxation time, T1, is often obeyed to d~ pearl on t e m p e ~ t u ~ ~ d frequency in ~ unexpetted ~r way [3,5,7]. I n ~ n t r a s t to the B l ~ m b t 'gen-Pu~clI-Pound model [ 14], one t y ~

respect to the n ~ l y occuplvd sil~ This is csscntiaby achieved by the hopping of individual defectsin the cloud. AS ~ ~suLt, the ~g~ffect pot~fial is beins shiRcd towards tilen~ sit~ where a n~ ab~[ute potentJalnnnim~isthus~cingfo~ed.Inthis~ the initial f ~ a r d hop h ~ ev~tual]y proved ~essfuL I n our t ~ t m ~ t o f section 2, the p r o ~ of jump ~laxation will bc described in t c ~ s of t h ~ rely r a n t funftlong (i) W(t) d~otes the p~babflity that the c o ~ lated backward hop h ~ ~ot yet been p e r f o ~ r at time t after t1~ initia; f o ~ a r d hop. (ii) The normaJi~d distan~ b e r g e n the c a g ~ f feet potential minimum a n d the newly ~ p i e d site,

icallyobtainsdlffe~ntsloOesoflog(l/Tn)ve~vsl/ T on the high. ~ d Iow-Iempecatu~ side, the hight e m p e r a ~ activa~on energy being the larg~ one. At low t ~ p e r a l u ~ , rep]a~ment of the N M R an* gular f~quency ~ b y 2to results in a ~ u c t i o n of 1/ T, b y a factor c 1 ~ to 2 n t h ~ than 4. In the following, the phenomenon of j u m p ~[axation will be s h ~ to give r i ~ to all of the above e x p ~ i m ~ t a l findivgs. The basic i d ~ of the juml~ ~lLxatlon mode] was a l ~ d y p i n . t e d in ~ f . [ 15], ~ d its leitmotif is now repeated in fig. 1. In fig I, the edngl~particle pc~ential felt by and acting on a n individual charged defect (ion or v a ~ n c y ) is ~ n stmeted b y superposltlon of the periodic lettlee 13otentizl and ~ ~ge-cffect potential due Io t he ~putsivr interaction between defects. After each ( t h e ~ a l ] y activated) "initial f o ~ r d " hop two competing p ~ ~ e s may be conceived: (i) T h e defect may hop back. In this e ~ a ~ r related fo~td~bacl(wa,xI hopping sequen~ has been p e r f o ~ e d . (ii~ The surrounding " d e f ~ l cloud" relaxes with

x(t)/xo, s~fig, l, iscalledg(t). (iii) The n o ~ a I i s e a ~ m p l ~ frequency s p ~ t ~ m o f the hopping motion, ~ (w),is the ~nnccting link b c t w ~ n the model ~ d the experim~t al dynamic spectra. Befo~ deriving r in section 2, let ~s briefly point out t ~ consequences of the shlfting of the cageeffect potential O ~ is that it taings about a n ~ - z e o dc c~ductivity. The other is 1he non-Debye cha~ acter of the rel~ation. This is ~ e n with the help of the i~ght.hand side of fig. 1. The backward barrier height, ~ _ ~ ( t ) , incl~ases ~ the defect stays at B. T h e ~ f o ~ , the ~]axation time of the back-hop p ~ ~ s s incres~s with time a n d the ~sultthg spectra

oee~eeoe oooooo~oe oe~eeoooo

--~r

oooo|

oooo@oooo .........

t>0

ooooooooo

.

.

___t~ 8

....

.

.

X

.

T~-

.

I ~ e ~ c

Fig t. (a) ~ deltas ( O ) on a periodic $ublatti~. (b) ~nstmaion of the s l n # e - ~ l e ~tenti~ eeli~on the d ~ t a site A. (r ~ o p m c m ofth~ def~'s ~tentml ~t~ time, after an 'qnitlaI fo~ard"~p ~ A to Bat t=0.

h v n ~ l ~ k as i f ~ l ~ adon times".

w e ~ 9 "dlstrlbullon of

relax-

2. Constrdc~on d the f ~ q u ~ e y spr Accox~ilng to the definition of W(t), thc ~ lated back.hop rote is - W(/), Neglecting the finltc du~tion of hops ~ may now f o ~ the wl~ity aut o ~ l a t i o n function ofth~ hopping motion, (v(0),r

+ ~'(0.

(I)

In eq, ({), J(t ) is the delta function. Fouri~ t ~ s fo~atlon r t~gl~-h~dsid~of~4. (I ~~ieldslhe n o ~ M i ~ d ~mpl~x f~qu~cy s p ~ t m m of the hop ping motinn: ~*(~) = q,'(w) + i ~ " ( ~ )

= ~ [J(t) + #(t)] ~p(-i~t) dt = ~l+fW(tlexp(--i~t)dl.

(2)

The function ti'( 0 required f ~ insetqon into eq. (2) is obtained b y the a n ~ t z

slnhgr } - ' imteM o f a single r e l ~ l a o n l i m c This fealu~ c a u l s the Slowing d t ~ of the ~ I ~ llon p ~ s . The ~ n t i a l properties of W(I) ~ a i ~ a d u be read from eq. (7), if ~ ~ p l a ~ the hypvrbolic sine f~ncUon by its awament and at the mine time put

g(t)=elP(--~t}" T~AT~I" W(t)f~p{--2,~ll-exp(-IFo/A)]}

which is itself t h e ~ a l l y activated. The mlutlon for IV(t) is ~ l y slightly changed, if hyperbolic sin~ function is properly taken into a e ~ n t . The r~ult is n ~

W(t)=exp{-A~[l-~(t)]},

(11)

~th ~=Ei(~)--Ei(~O

(12)

~d

S(t)ffi~_l[Ei(~exp(_tlt))_Ei(_~r (13)

- ~vr -Sa-r (3) w h ~ e ~v is an atlempt f ~ q a ~ c y . The approximate relations

areeasilyv~.i~*dwiththehelpoffig, abbreviations Po =re~ p ( -~lkT}

(4) l. Withthe (5)

and

r

(~)

we no~ obtain the rote equation

-#(O=2F~W(l) dnh[(g(r

(9)

Hence~ t heinirial f o ~ a r d hop e ~ n t ually p ~ v ~ s u ~ cessfaiwith the p~babillly W(~)=cxp[-2A(4-~)IkT], (IO)

-#(t)~IV(O{voexp[--$~^(t)lkT]

~S~A(t}~--(~--J)~t), Js~c(t) •zJ+ (d--~)g(I)

(8)

w h ~ A isa ~nstant of the orderof~e, In this we obtain

(7) which is eq. (23) of rcf. [ 15 ]. Note that eq. ( 7 } ~ n rains the ta'ae.dep~ndent e x p ~ i ~ {2to •

lnscrtion of W(t) into eq. (2) 9elds th* ;~queacy sputum r Fig. 2 is a plot of the ~ p l e x ptane disp[aying a ~ o f constant lempe~lum arai ]itvs of constant frequency. The set o f ~ n s m n t - t m p e m t u ~ arcs may be characterlsed by only one patamely, which is A(3--8)/kZ Nolo that

*(O)=W(~) and t~(~)ffil (14) at all t c m p e r a m ~ . The ~ a r ~ a i r ~ d y d e ~ l y ssmblc experim~tal ~ d u c t i v i t y arcs; they ~ e almost ci~ular, with ~trcs displacedbelow the real ~ i s , ~ d they apprVach the real ~ i s a l ~ y s verXie,llly, We have ~ far used the ~ s ~ -~g(~)=g(t)=g(l)'~(t)' g(t)~clP(-#t)'

(IS) whlch m ~ n s that the r~p~nse f~cfion of thc ~d~

f~tctoud"isa~um9 fuaoion, In r

(15), the asterisk d~notes a canvo-

,O3 ~nduet[vity of the defects,which is connected to their~If
oso,.,,~,

om-~o

a~

fig 2- ~ c ~ m a l i ~ ( ~ ( ~ l = I I c~ f ~ ~" cram oflhr hvp~d~ mvfi~n,~ ~ ~" +i~'- (a) A ~ e r ~ v a n t u~ser.:~.=50Gnz,d=kxa~eIc.,~=~• t.(b) ~ , .

or~n~nzs

ilhA=I,

fillip,AClU~}y,'m~I.(15) p m v ~ to ~ S U e ~ f'al,{n perils]dr with ~g~rd to the r o f r~alislie a ~ in the complex p]a~r see flg~ 2-4. Am long times, howcv~, w h r n g(t) h ~ m z s much smMler t h ~ one, only very w m k fo~:es Ire r x e r t ~ on the "cisfeel cloud'. Now the cloudshould ~ c t according to the line~ ~s~on~ iheolT:

-,g,(l)=g(t),[a(t)d-f~(r)]. (16) lnasep~atepaperll6lwewillsh~lhataE~adual transition from eq, (I ~) to ~q. (16) with time does inderd i m p w v e the Iow4empc~mre, low-fr~ueney results. The i m p f o ~ e n l is evident in the log-log plot o f ~ ' ~ u s frequency, but is berdty s ~ n ~ the l i n o r scale o f fig. 2. 3. ]Predictions ~ exl~r~nt

me ~

and ~ m p e d ~

io

3 l, Conductivity ~ d perrn~tttv~y Consider defects, D, ~rl~iug ehalge, q~. The d ~ leers p e r f o ~ a hopping ruction oR 9 sublatlise, el. fig. L where each site has n n e ~ t - a e ~ g h b o ~ sites a n d the distan~ b e t ~ n t h e n is xo. Let t ~ e ~ ~e SV~ defects within a volume 1~ The "slngl~p~l~icle"

(}7)

~dm

B=q~nx~NDl~kdV.

(18)

in eq. (18), dis the almensionalily of a defects padl. From eq. 0 7 } il is immediately evident that the cal~lated function, do ( ~ , T), does indeed have the characteristic pmlmrti~ o f eXl:.etimental ~ n d u r ti,dfies: (i) At T~cd t 9 eq, (17) yields ~ m p l ~ ~nduaivity ~ h a v i ~ the shape o f the 9 arcs o f fig 2. in agreement wilh eXl~ri~r (il) From ca, (t 7). e ' is found to have diffe~nt ~ t ivation energies at i ~ ~ d hi~a fre~uenci~. T h ~

high-frequency~ndnet~vityiss~mp[yaetivateawith n, while ~he low-f~cluea~ aetW~tioa energ~ i~ d+~(A--a), ~f. ~s, (io) z~ct (I4). Fill, 3 { h o e eomplex~nduct{vi~y ~cm as wellas

alog.-logplozofa'Tvenmsftequ~cy,

lnlhelatter.

the slope is a l ~ . s s~al~cr than one. ~nd the ~ l l k n o ~ power-law bchaviour is obtained at 1 ~ lempem~u~m The compl~ pc~it~ivity ~ y be dedvedfrom the e~plex cond~ctlvlt?~by ~(o, 2) = C ( ~ T) - i ~ ' ( m . T) =~(~, T)(imeo)-'.

(19)

SO thejump~elaxation model ~ n be put to a f u r t h ~ test by ~ m p a l l n g ~leulated a n d ~ p e r i m ~ l a l Fr mittivitles. Ind~ed, a reatistie set of ~ is obtained in the plane of e" --o'(0)/e~o v e ~ s e' - e ' ( ~ ), s ~ fig~ 4. T h e a g ~ m ~ t b e l ~ n model a n d experiment thus ~ m s to be independent o f the pallieular way of data rep~sentation,

3.2. Qu~iel~tic n~tron ~atteri~g Tfarslatioaal j u m p dlfFazsion~uses quasielastic s~nerin4g. QuasieIastie l i n ~ have ~o far been ohsetve~ in a number of fast ion ~ n d u e t o ~ , s ~ e.g. reis, [ 8,9]. Their widths, shapes. ~ d anlsotropiss are generally ~11 accounted for by simple jump-dl ffusion models [ 17]. However, in addition to the

" 0005

" m" 0s

~ [f~c

a01

OD2 f/lcm~l

,O~

K

]

Fig~3~(")~sid~ands5aN~f~mp~x~ndu~i~tya~a~di~nttcmpr

K , s , era-,. At snmcivady] ~ ~ a t

,e

u~ the ceo=s of the circler ap~xima,i~, to ~he, r ~ closelyfollow a ~ i g h t [i~& (b)

L~-~1~T~nust~uen~`Pa~m~t 9162

k ~

i

-0 5 10 ~(v~-r~{~} Fig. 4. A~S of coasl~t te~p~t~re and lines of coa~ant f ~ i~ 1he complex.~it tiviw pl~e parame~ values as i~ fig 3- N~e the Ufiking~ m e ~ ~the ~ P ~ a l a t i ~ i n the q ~

lion which includes the ne~est-nclghbour sites, The width a n d shape of the b ~ a d e ~ p o n e n t a ~ found to be virt~lly indcpcnd~t of Reading the experimental quasir spectra as ~ n v o l u t i ~ s on the energy scale, cf, fig. 5, amounts to postulating the existence o f both su~essful hops

and a Iocalised hopping molion, In fact, the incoheront ~ t torlng function c~structed o~ the basis o f ~ e l a t e d fo~ard-hackward hopping sequ~ces t ~ S OUt to have the same specific p r o p e l l i ~ as the ~ p e r i m ~ t a [ broad ~mponent. In the ~ of a Bmvais jump latti~ this funainn is found to be [ 13]:

S,~.(O,e ) ~ [ l - M ( Q ) I J ( e ) + M ( Q ) ~ ( ~ ) , qu~iclastic line due to an u n ~ l a t e d hoppin 8 metion, experimental n ~ t r o a s~;tering sPeCtra o f stmelurally disordered ~olid electrolytes zlso ~ n l a i n

(20) I a e q , (20),O(w)isabell-shapedfunctionwhich~s normalised to unity:

a much b r ~ d e t quasiela~tic ~mpone~t, see fig. 5

~(~)=(~C~)_~[~,(e~)_~,(0)l

[6,8,9], The (~ dependen~ of its weight suggests that

'

(21}

N

ol

[&t.& &3, Spin-/atti~ rel~ation lime

valb c = 3 [w ( 0 - w(~)] dr.

(22)

The w~ight function,M(Q), is$iven by M(~) -O[ 1-cos(Q'x,)] §

~ cos[(/.(x,-x~)]}.

(23)

I~ eq. (23), m e b ~ denote lh* fo~at~o~ ofave~ a g ~ over the j ampvr x, a n d x1, ~nd the meaa~ ing o l d and E i~ D=2nI'~C(I+nFoC) -2 ,

(24)

E = ( n r 0 C ) = ( l +nFo C) -2 .

(25)

Note that shape a n d width o f ~ ( ~ ) do not d e p ~ d on Q, while the ~ i g h t function starts out from M ( 0 ) = 0 and inc~ascs with increasing Q, umll a m ~ i m u m is attained near Qm~lxo or beyond. The p~di~ions of the jump-rela~tion model arethus in ~cordance with the expcrimentaI ~sult s. Note also that highly damped o~inamry m o v ~ menlo of the toys may comribute furth~ i ntensit F to the outer flanges o f the broad qnaslelastic corn ponent~

In st ~ l n ~ l l y disordered solid ionic ~ n d n e t o ~ , experimental spin-lattice relaxation times have been found to display ~ ~ymmetric l / T d e p c n d e n = and a non-BPP-type frequency depend~oee. Both deviations from the standard model m easily ~plalned, if ~ takejump-~laxation p m e e ~ into a ~ n t , Consider an ion ~siding at a ccnain site A at tlmc 0, being s u ~ u n d e d by n v a m a t nr sites9 The p~babilhy to find the inn rustle A al a later tlme i is ~ t I ~ w~(t), w a f t ) is a good approximaIion the autoco~latlon function, a ( t ) , which is conn o t e d with the (~mplex) spectral den~ityj(~, T), via Fourier t r a n s ~ a t i o n . The ~pin-latti~ ~laxalton time is ~[ated to the real p a ~ o f the spectral density, j( ~ , T), b y TT'(~,7")=const.[j(o,T)+4j(2m, T~ ]

(26)

and thus depends on ~ and T i n a similar way as the sp~tral dcmity doe~ If the ion at sile A peH'o~s s u c ~ f u l hops only, wo have

WA(t, ~ = e x p [ - z . r ~ ( T)I and

(27)

106 ~ o ~ n g d o ~ o f the jump-~laxatlonp ~ s ~tb tlm~ ]n fig, 6 we pmsem the ~ a l parts o f the ~pectral den.~ti~ o f eqs. (29) and (28) as well a~ calculated spin-lattice ~ l ~ a ~ o n mte~ The latter ~ e asymmetric on the 1/T ~ l e , i n agreem~t vA*hexperiment, and they also display Ihe ch~*ac~cd~tie nonBPP-tTpe low-ternpezalum frequencydependence txosn-A&~o~l~gc~.ts I should like to thank W, M t l l l e p W a ~ u t h for call. lag m y attentionto the ~omBPP-typr behaviouto f spin-latti~~laxationtimes,Financials u p p o , from the Fonds der C h ~ i ~ h e n lndustrie is gratefully acknowledged, Refe~n~

f~m ~ (2~) ~ d (29). Parame~ v~lur ~ in f ~ 3, 4, b~t ~ 5 GHz. Note the a~mm~W a~d ~ Iow-i~f~t~e ff~ querier0epe~en~ ( b) LOgACgpZotof ~e ~ 1 pan ofthes ~ eq (29) P~meter valuesa~ 3Si. (~), (c) ~g-l~g plot ~ j

[ [ ~A,~ J ~ h e r , Nature2fi7(1977)6731Diele~ic ~laxat i ~ i* solids( ~ e / ~ Diel~d~ e ~ London,1983). [2 }IK~L" i i. N~L C~ S~ Sine PhYr9 (tgt9) 127;911980) [3] K L N~L ~lld ~ale Ion[~ 5 (1981)27. [4] R Kchl~r Ann,P ~ K (~ipz~) 12 (1847)393; G~Williamsand D,C.Wat~ Tfan~Fa~d~y~ , 56 (1970) [5 ] &L. Bjork s ~ P. F~r~ni and M. Villa, J. Chem.~ y s 73

. i.ro(/3]-' J ( ~ , 13 ~ 1+ i ~ [ n F d T)] - , ,

(28)

which is the BPP r ~ u l L I f t h ~ is a p~ferred o ~ n ~ o f backward hops, however, W~(t) is ~o longer a singleexponeafial. Rather, w~(t) is el~e 1o ex~(-nFot) at sholl tilaes, when backward hops ~ e slill vnlx~po~ant, and c l ~ t o e x p l - ~ F o W ( ~ ) t ] a t l o n g t i m e s , whentbebackward hops have to he consid~ed- Startingo . t wilh a rate cgualion for W^( O, we ~owfind that eq. { 28 ) has 1o be ~p~a~d by [ t31 r (nFo(T~ ~ ( ~ , TI] - ' 2(r 73 = ~ [ n / , o ( T ) ~ ( ~ , T) l.,,

t6l K, Fonke,im ~p~onie solidsand ~lld electS.ca; ,~ [ 9}w. M S ~ - ~ m h andW.~t~ J Chem ~ , 7 2 figs0) 174~. R e ~ , ~6 (19r9) 37I; G B~zecJ~l~enb~LHA~..O.KaaenandD.Wo~Pt~. am~teCo~s,,ZOdeh~$e~tembc, 19g4).

(29)

which m e ~ s that the conelat~on time has become frequ~ey dependent, It is ( n / ' o ) - i at highfrequencies, r to short tlm~, and [ n F o W ( ~ ) ] - ' at ~ a ] l ftequenci~, ~ s p ~ d i n g long times:This variationonce again refl~ts the

[9] M, Taehcz,R. Mere;dr,LP. Maluganisad A / Dias~ax,

~ l i d s 48 (1~87) 57, ] [0I J.R. M~edo~l& Soli~ Stale l o a i = t 5 (tPg5) I S9; l . A . D i ~ d o snd R.M. Hill. Solid S~I9 l o n h 22 (1987)

(~er,

Ncw Yor~, t~76) 9 I I .

[ t 2 l u. Sh~m and K L. Ngai. Solid Stal9 I o n i ~ 5 (1981) 167.

( 131K, F ~ k ~ , ~ Ph~sik.~ .

(NF} ~54 11987) 251.

[ 15] K . ~ k ~ ~ l i a $ ~ l e l o n l ~ I ~ 1 9 (19561 I g3. [ 1fi] i C Funke ,rid l~ H o p ~ . Solid S~te l ~ i c ~ t o bc publishe~

[ ] 71 C.T. Chudlm/a.d R3. Ellio., P ~ Phys So~. ?? (L~61) r

.r

32 (197L) 41,

[ 18] tO*Funk~ Pro6, Solid S~,.e Chem I 1 (]978) 345