The relationship between measured activation enthalpy and pre-exponential factor: Rate processes in ionic crystals in the intrinsic-extrinsic region

Solid State Ionics 8 (1983) 297-304 North-Holland Publishing Company

THE RELATIONSHIP BETWEEN MEASURED ACTIVATION ENTHALPY AND PRE-EXPONENTIAL FACTOR: RATE PROCESSES IN IONIC CRYSTALS IN THE INTRINSIC-EXTRINSIC REGION T. DOSDALE and R.J. BROOK • Department of Ceramics, University of Leeds, Leeds LS2 9JT, UK Received 3 September 1982

In measurements of activated processes, such as diffusion, ionic conduction, creep and sintering, it is common to use Arrhenius plots of the measured quantity,A (T), to give the logarithm of the pre-exponential factor, log A o, and the experimental activation enthalpy, h. It is shown that, in the cases of conduction and diffusion in ionic crystals, well-defined relationships can exist between the values of log A 0, log A and h, measured on similar samples with differing doping or impurity levels. These relationships are derived for ionic crystals showing either Frenkel, Schottky, or interstitial disorder. In the resulting plots of logAo versus h, approximately linear regions are shown to exist, similar to those described by the compensation law for glasses and by the Meyer-Neldel rule for semiconductors.

1. Introduction There are many experiments in the physical sciences where the natural logarithm o f the measured quantity may be plotted against the reciprocal o f the absolute temperature in the form o f an Arrhenius diagram. Such experiments include measurement o f rates in chemical reactions, o f electronic conduction in semiconductors, o f dielectric relaxation, of viscosity in glasses, and o f conduction and diffusion in ionic conductors; the common element in all these cases is the presence of some physical barrier to the process involved and the existence o f a Boltzmann-type energy distribution for the species involved in the process. If the experiments are repeated under conditions where the factors (other than temperature) which affect the process are varied (one possible factor is the dopant concentration; another is the degree o f non-stoichiometry), then a series o f Arrhenius plots may be made, one for each set o f values for the factors; fitted with a straight line, each plot will yield a slope and an intercept on the rate-constant axis at infinite temperature. Linear relationships between the * To whom all correspondence should be addressed. 0 167-2738/83/0000-0000/$ 03.00 © 1983 North-Holland

slope and intercept for such a series of measurements have been observed for the electronic conductivity in semiconductors (the Meyer-Neldel rule [ 1 - 3 ] ) , for ionic diffusion in glasses (the compensation law [4]), and for gas diffusion in polymers [5]. In an earlier paper [6], it has been shown that a linear relationship should apply to cation diffusion in ionic crystals under certain conditions; in addition to the alkali halide data (LiF) discussed there, other ionic systems have been found where the relationship applies, e.g. Li4_xPxSil _xO4 [7] and the fluorite structures PbF 2, BaF2, SrF 2 and CaF 2 [8]. In the present paper, the general form o f the relationship is developed for ion movements in ionic crystals. The relationship can be used in a number of ways [9] ; an example lies in its value for materials for which complete experimental data are not available or for which a wide spread of activation energies has been reported; under these conditions, the form o f representation is helpful in identifying the inconsistencies in the published data.

T. Dosdale, R.J.

298

Brook/Measured activation enthalpy and pre-exponential factor

2. Arrhenius plots and measured quantifies An Arrhenius plot is formed when the logarithm of the measured quantity (A) is plotted as a function of reciprocal absolute temperature (T), as exemplified in fig. 1. At any temperature it is possible to draw a tangent to the experimental curve from which the values of In A 0 (T) (here and below the subscript zero indicates the value for 1/T ~ 0) and h(T) (the experimental activation enthalpy) can be found, as shown. In many published Arrhenius plots, a straight line is fitted to the measured values. Where the values are obtained over a limited temperature range, this may hide curvature in the In A versus lIT relationship so that the values of In A 0 and h only approximate to the true values over most of the measurement range. In such a case, the values ofln A 0 and h may be taken as referring (approximately) to the centre of the 1/T range specified.

treatments will be similar, although not always iden. tical, for each case considered. Formulae for conductivity and diffusion coefficients due to defect carriers may be written as: / (all ion l(atl species) mechanisms)

ent/ z t/t~l] ,

(la)

or aT= ~

/ (all ion 1 (all species) mechanisms)

enl]Zl]#l]T

(lb)

where n is the defect carrier concentration, z the charge on the defect carrier and/a the defect carrier mobility, and

(nli/N)lgliTk/eZli'

D / = l~ ( a l l

(2)

mechanisms) where D~ is the diffusion coefficient for one ionic species and N is the number of molecules per unit volume. Consequently it is possible to write

3. Ionic conduction and diffusion The present calculations will be illustrated by reference to ionic conduction and diffusion, so that A will be either oT or D. Since these two quantities are related by a constant factor for one defect species in any material, assuming no correlation effects, the

(3)

DI = ~1 CljDll' where Cli is the defect molar fraction and Dtl is the defect diffusion coefficient for mechanism l. If it is

assumed that the diffusion is thermally activated then

Dli may be written Dlj = (Do)l~ exp[-(Ahm)l//kT] ,

(4)

where Ah m is the enthalpy for defect movement, and aT may be written

__i \\

oT= (Ne2/k) ~)'-~,Otlz~iCli . / t

(5)

4. Frenkel disorder

.N

\ x

Fig. 1. The determination ofA ,Ao and h at a particular temperature T.

For a crystal in which Frenkel disorder predominates, the relationship between vacancy concentration, CV1 = [Vzvi], and the interstitial concentration, Cij = L/Z~/], wl~ere j can represent either the anion (a) or cation (c) and where the concentrations are expressed as mole fractions, is given by CvlZ1ii/1('lz ~i] V/I =

exp(--AgF/kT)"

(6)

T. Dosdale, R.Z Brook/Measured activation enthalpy and pre-exponential factor Here Zv] ,z y are the effective charges for the vacancy and interstitial, respectively. The modified free energy of formation of Frenkel disorder isden9ted by Ag F; this term includes the conversion factors from the corresponding equation expressed in terms of site fractions [10]. In an impure or non-stoichiometric crystal, dopants may be either intentionally introduced or present as impurities. The charge-neutrality condition for fully ionised dopants may be expressed as

Cviz W + Cvz V + CdZ d

=

0,

(7)

where z a is the effective charge of the dopant and Ca is the dopant concentration represented as a mole fraction.

and CIZ 1 + C 2 z

2 +

Cdg d

(11)

= 0,

where the correspondance with eqs. (6), (7), (8) and (9) is clear. The mathematical solutions of these equations for - k d In Cu/d(1/T ) and In Cu, for both the case in which Izll = Iz21 and the more general case, are required in order to find the relationship between In D or In oT and h. Here u = 1, 2.

6.1. The case where IzI I = Iz21 For the case in which Izll = Iz21 =z, eq. (10) may be reformed as

C1C2 = exp(-Ag/zkT) = C2p ,

(12)

where Cp is the defect concentration for each defect type in the pure, stoichiometric crystal. Combining eqs. (11) and (12) gives

5. Schottky disorder The equations for Schottky and interstitial disorder closely resemble those for Frenkel disorder. In a crystal with Schottky disorder, = [V.ZV/] or with in_ C,,, v! ] terstitial disorder, Ci/= ~ V ] , where j = a, c, the defect equilibrium is given by

cl:lC~lt~lal = exp(-Ags,iIkT

299

) .

(8)

Here, Zla, Zle are the effective charges for the anion and cation defects respectively and Ags, I is the free energy of formation of Schottky (interstitial) disorder; the latter term again includes the appropriate conversion factors [10]. In an impure or non-stoichiometrie crystal, with dopants either intentionally introduced or present as impurities, the charge-neutrality condition is

ClaZla + ClcZlc + CdZ d = 0 ,

(9)

where z a is the effective charge of the fully ionised dopant and Ca is the dopant concentration (mole fraction).

CI(C 1 (C 2 -

CdZd/Z2)

(13a)

= C2 ,

(13b)

CdZd/Zl)C 2 = C 2

(since z 1 and z 2 must have opposite signs). The solutions to eqs. (13a) and (13b) are

ll[Zd~c

Cl-- It 1

F[Zd\2 2 +4c2]l/2}

a+Lt )

(14a)

and

_ 1 [[Zd~c

FlZd\2 2

2ql/2~

c2- tt 1 a +Eta) q +4c ,j J.

(14b)

Rewriting r 1 = Zd/Z 2 and r 2 = Za/Z 1 gives the general solution

Cu =~[ruCa +(r2uC2d+ 4C2)1/2] ,

(15)

which leads to

lnCu =--ag/2zkT+lnIx u +(x 2 + 1)1/21

(16)

and 6. Mathematical solution

-k

The equilibrium and charge neutrality conditions for both Frenkel and Schottky may be written quite generally as CI~2I CI~ , I = exp(-Ag/kT) (10)

equilibrium

d In Cu ~ --z d(1/T~

(17)

(x2 + 1)1/2ix u +(x2 + 1)1/2]

where x u = ru Ca/2C p a n d Ah is the enthalpy of formation (this term also includes the appropriate conversion factors [10]). It can be seen that there is a parametric relationship between On Cu + Ag/2zk T) and - ( k z / A h ) d In Cu/d(1/T); this relationship is

I". Dosdale, R.J. Brook/Measured activation enthalpy and pre-exponentiai factor

300 5

Making the substitution q I leads to

I I I I I I I I I

t~

IZll

[z21 In C 1 +

- z 1/12 - ( C d/ C 1 )Zd/Z 2

=

ln(qlC1) --- - A g / k T

(19)

3

and differentiating eq. (18) with respect to 1/T gives

2

d l n C 1 IZliZ 1 d l n C 1 Iz21 dO/T) - ql z2 d ( 1 / - ~ -

Ah k '

(20)

so that the eqs. for in C u and - k d In Cu/d(1/T ) are

ln Cu = ÷ X

Ag + IPul , I I '~ ~ in - (Izll+lz21)kT l+lPul [qu)

(21)

and "I

d l n C u z~lPul

1

(22)

- k a ( l m = iZut (l

"2

where Pl = Izll/z2, P2 = Iz21/11 and the moduli have been used in eq. (22) since z 1 and z 2 have opposite signs. As was found for the case in section 6.1, there is a parametric dependence of In Cu on - k d In Cu/d(1/T); for Izll = Iz21, this reverts to the same form, since

"3

t I I I I I I I .0 0.1 0.2 03 0U 0S 0.6 07 0B

O.SltV;J?Z';T( ~

I

09 10

+x})

5

Fig. 2. Parametric relationship for Izzl = Iz2 I.

merely shifted and scaled along the axes when In C u and - k d In Cu/d(1/T ) are used in the expressions for In D (or In(oT)) and h respectively, so that it is useful now to see the functional relationship between ln[(x 2 + 1) 1/2 +Xu] and (2(x 2 + 1)1/2[(x~+ 1) 1/2 + Xu ]-}- 1, as shown in fig. 2.

I I I I I I I I I

3 2

6.2. The general case For the case in which IZl] :/= Iz2] the solution of eq. (10) and (11) for C u is no longer straightforward. However, we are primarily interested in the relationship between In C u and - k d In Cu/d(1/T), which has a parametric solution as in the special case, but not directly in terms of C d. Taking the logarithm of both sides of eq. (10) and substituting for C 2 from eq. (11) gives

e-,

=-2 -3

I

Iz211nCl +lZll In,

Cl g 2

Zl C 1

-Ag/kT.

(18)

I

t

I

I

I

I

i

I

"50.0 0.1 0,2 0 , 3 0 A OS 0.6 0.7 0.6 0.9 1.0 1/(1 ÷ p2/q)

Fig. 3. Parametric relationships for IP[ = 2, 3/2, 1, 2/3, 1/2.

Z Dosdale, R~I. Brook/Measuredactivation enthalpy andpre-exponentialfactor

q~-I = [(x 2 + 1)1/2 + x u ] 2

(23)

in this instance. The range of qu is O <~q u <~oo

301

nates eq. (26), then the results for the diffusion coefficient due to the vacancies or interstitials separately are given by hl = (Ahm)/c + AhF ~.

and in the pure material ql = -Zl/Z2, q2 = - z 2 / z l , or qu = IPul since z I and z 2 are of opposite sign. For doping which promotes defect type u, qu < Pu" The functional relationship between [In Cu + Ag/ (Izll + Iz21)kr ] and - ( k l z u I/~hlPul)d In Cu/d(1/T ) is shown in fig. 3 for various values of Pu given by integer values of z u . It is convenient to define the "centres" of the curves as being at qu = 1, but it must be remembered that these points do not refer to the pure material except in the special case of IZll = Iz21.

+ 1) 1/2 + X / ] J '

(27a) In Dc = ln(Dohc - (ahmhJ

T-

ZF/2ZckT

+ln[(xt 2 + 1) 1/2 +Xll ,

(27b)

where l = i or V, x V = (Zd/Zic)Cd/2C p and x i = (Zd/ZVc)Ca/2C p. In both cases, the resulting graphs, as shown in the lower part of fig. 4 (plotted as log D e = In Dc/in 10), are similar to the universal curve o f fig. 2, centred at (Ahm)tc + Z~F/2Z c and ln(D0)/c -

7. Application of the solution to ionic conduction and diffusion In order to exemplify the results we have obtained, two specific cases will be considered and the results for ionic diffusion and conduction derived. The results for the specific cases will refer only to the case where Izll = Iz21. The calculations for the diagrams refer to the particular case o f l z l l = Iz21 = 1 and As F = AS s = 0 (As is the entropy of formation; it includes the conversion factors [10]); in all cases the maximum value of C d for which values have been calculated is about 0.0025 for Iz d - z/I = 1.

1

1

Zc k2(x2 + 1 ) 1 / 2 [ ( X 2

1 0 I l l 1 , 1

I

I

I

I

I

I

I

I

/ 0

/ ,/i :

0

-10 ..... .,

7.1. Cation Frenkel disorder

!

l

When Frenkel disorder dominates the defect equilibrium, the total cation diffusion coefficient is, from eq. (3), D c = C v c D v c + CicDic and since we are considering the case when [Zic] = z c, oTis given by o r = (Ne2z2/k )Dc .

(24)

Izvcl = (25)

The diffusion coefficient may be written more fully as D c = C v c ( D 0 ) v c exp [-(~d~m)Vc/kT] + Cic(D0)ic exp [--(ZShm)ie/kT ] .

(26)

If we suppose for the moment that one term domi-

I

I

I

f

fl

I

I

I

20;0 0.2 0/* 0.60~B tO 1.2 1.400 0.2 0.4. h~

l

0.6

l

l

l

~.8 1.0 1.2 1./, h~

Fig. 4. L o g D c and log(Do) c as functions o f h for Frenkel disorder. For all curves, Ah F = 0.7 eV, (~Lhm)Vc = 0.55 eV, (Ahm)ic = 0.35 eV and k T = 0.03 eV have been used. For the left-hand diagram, (Do)Vc = (Do)ic = 10 .4 m 2 s - l ; for the right-hand diagram, (Do)Vc = 100 , (Do°)ic = 10 --4 m 2 s "I • In the left-hand diagram, t h e lower d o t t e d curve for l o g D c is the universal curve (fig. 2) for t h e cation vacancy. The universal curve for the cation interstitial ties above it, part d o t t e d and part solid (where it forms part o f the total s u m m e d cation diffusion coefficient). In the right-hand diagram, owing to the different a s s u m p t i o n s for t h e Do values, the lower d o t t e d curve is that o f the interstitial. T h e solid curves represent the total cation diffusion behaviour, i.e. the s u m o f the vacancy and interstitial contributions.

T. Dosdale, R.Z Brook /Measured activation enthalpy and pre-exponentiai factor

302

~ caoitn .,,,

~ x ' , ~

-

itiat

(ii) i

"-,

(i}

"

~\

camion

\,\

I

I

""

~(i)

I

\" {i)

T-1 Fig. 5. The effect on cation vacancy diffusion of additions of dopant with effective positive charge. As the dopant concentration rises, h falls and In D c rises. Curve (ii) represents a higher dopant concentration than curve (i).

(ii) ~\ T-1 Fig. 6. The effect on the contributions to cation diffusion from vacancies and interstitials in the presence of increasing concentrations of dopants with effective negative charge. Curves (ii) represent higher dopant concentrations than curves (i).

(Ahm)lc/kT - AgF/2ZckT , and scaled by AhF/Z c on the h axis. If the dopant has effective positive charge, then cation vacancies will outnumber interstitials, h for the vacancies will fall towards the vacancy extrinsic value, and In D c will increase, as may be seen from eq. (27), where x V is positive (see also fig. 5). Similarly, when the number of dopants with negative effective charge is increased, h due to vacancies rises above the "pure" intrinsic value and In D v c falls, whereas the value o f h due to interstitials falls to the interstitial extrinsic value and In Die rises (fig. 6). When both interstitial and vacancy mechanisms are effective simultaneously, the total diffusion coefficient is the sum o f the two contributions, as shown by the solid line in fig. 4 for two different sets of values o f the constants (the crossover between the two regions has been computed separately, since the contributions to In D e a n d h are not merely additive). If the measurements Were made at a higher temperature, similar curves would be obtained, shifted by [(Ahm)/c + ~V/~l A(1/kT) up the In D c axis (1 = V, i); this applies to each of the component curves separately but, since these will usually move at different rates, the shape o f the composite curve will change with temperature. The separate component curves obtained at two different temperatures on impure materials can yield not only, for heavily doped

materials, (~llm)ic and (~t/m)Vc , but also z ~ F, even though it may be impossible to make measurements on very pure crystals. The graphs of In(D0) c against h are obtained by simply adding h/kT to the curves shown in fig. 4; the results o f this operation are shown in the same figure, plotted as log(D0) c = ln(D0)c/ln 10. Because the centre portions of the lnD c versush curves in fig. 4 are approximately linear, with slope ~-2Zc/£dz F, and since Z~ F >~ k T in many cases, then the corresponding parts of the In(D0) c versus h curves will be approximately straight with slope ~ l / k T - 2zc/Ah F. This leads to a linear relationship o f In(D0) c with h over a dopant concentration range of order Cp, where Cp is the intrinsic carrier concentration at that temperature.

7.2. Schottky disorder When Schottky disorder dominates the defect equilibrium, the anion and cation diffusion coefficients are given separately by Oa = CvaDva,

D c = CvcDvc ,

(28)

and considering the case in which Izval = Izvcl = z v ,

T. Dosdale, R.Z Brook/Measuredactivation enthalpy and pre-exponential factor

303

oT is given by oT = (Ne2z2/k)(Da + D c ) .

(29)

This time the two defect species have (chemically) separate diffusion coefficients, each of which is given by

dopantwith positive I effectivill ~harge

In D~ = In(Do)v~ - ( A h m ) v / / k T - Z~gS/2zv k T + ln[(x: + 1)1/2 +x/] ,

(30a)

dopant with negative effective charge i (i)

with activation enthalpy ~d~S [ 1 1" h: =(Z~rZm)V]+-~VL(X2~/+ 1) 1/2 [(X2 + 1) 1/2 +X/] (30b) The resulting graphs are shown in fig. 7 (plotted as log D~ = In Dj/ln 10), the independent parameters beingx a (Zd/ZVc)Cd/2C p andx c = (Zd/ZVa)Cd/2C p. An increase in the concentration of dopants with positive effective charge will thus cause an increase in D c and a drop in h c towards a more extrinsic value, whereas an excess of dopants with negative effective charge leads to a low value for D e and a value o f h c in excess

10

I

I

I

I

I

I

i

i

/

0

I

i

i

i

i ,

i

/

Fig. 8. The effect on cation vacancy diffusion o f increasing concentrations o f dopants. Curves (i.i) represent a higher dopant concentration than curves (i).

of that in the pure material (fig. 8); the latter changes are associated with a rise in the value o f D a and a drop in the value of h a. The shift of the curves up the In D axis in fig. 7 with increasing temperature is given by [(zahm)v/+ (Ahs/2Zv) ]/x(1/kT); h e r e / = a, c. The graphs of ln(D0) a and In(D0) e obtained by adding hj/kT to the lower curves of fig. 7 are shown as the upper curves in the same figure, plotted as log(D0)/ = ln(Do)//ln 10. The approximate slope of the central portion is 1 / k T - 2Zv/Z3h s, over a dopant concentration range of order Cp. The curves for oT are a combination of the results for D a and D c and are similar to those shown in fig. 4 for Frenkel disorder.

7.3. Iz I [ =/: Iz2l

-10

c~~ c~ o

-20

T-1

I

I

I

I

I

i

i

02" 0/* 0.6 0.8 1.0 1.2 1/, hc

t

t

t

i

t

t

0.0 0.20.t~ 0.6 0.8 1.0 t 2 h,

t

1/,

Fig. 7. L o g / 9 / a n d log(Do)/ as a function of h~ for Sehottky disorder. For all curves, ~d: S = 0.7 eV and k T = 0.03 eV have been used. For the cation results (left-hand plot), (~/~rn)Vc = 0.65 eV and ( D o ) v c = 101 m= s - t ; for the anion results (right-hand plot), (&hm)Va = 0.25 eV and (Do)Va = 10 - s m2 s-l.

In the case where the two types of defect involved have unequal charges, the arguments are similar to those set out in sections 7.1 and 7.2 above, using the results ofeqs. (21) and (22) rather than the simpler results of eqs. (16) and (17). The "centres" of the curves (as defined previously) occur at (~d~m)u + (Ah IPu I/I Zu 1)/(1 + pu2) and ln(D0)u - (~hm)u/kT - 6 g / k T ( I z l l + Iz21);here u = 1,2. The shift of the curves up the In D axis with increasing temperature is given by [(ZXhm)u + ~ / ( I z l l + Iz21)]A(1/kT ). Combining the diffusion coefficients to give oT is

304

T. Dosdale, R.£ Brook/Measured activation enthaipy and pre-exponential factor

also slightly more involved, since the two defect types have different chargeswhich weight the diffusion coefficients in the summation.

8. Conclusions It has been shown that for impure ionic materials, for which either Frenkel or Schottky equilibria are appropriate, there are fLxed relationships between In D (or In(aT)), h and the concentration of dopant (or impurity). From the temperature dependence of the In D versus h curves, it is possible to evaluate the Frenkel and Schottky formation enthalpies without making measurements on very high purity materials. In the case of In D O versus h (or In(aT)0 versus h), plots, there is a range of dopant concentrations, of the order of the defect concentration in the pure material, over which an approximately linear relationship between these two quantities may be observed.

OI

I

I

I

I

These results can explain the phenomenon in which series of measurements on "pure" material by different workers lead to widely differing estimates of In D O and h. A plot of In A 0 versus h may well lead to a linear interdependence. Such plots, or the corresponding In D versus h plots, can give estimates of the defect formation energies in the pure material. Of the two types of plot, the In D versus h type is often a more sensitive test for any correlation between In D O and h, since the range of In D is commonly less than that of In D 0. The In D versus h plot may, in the cases covered here, be fitted with the standard functions shown and the rate of movement of the curves up the In D axis as a function o f l I T can give information on defect formation energies. Such plots may also be used to distinguish between two different mechanisms, for example cation vacancy and interstitial diffusion, providing the curves for the two mechanisms are reasonably well separated. The validity of the procedure has already been verified in the instance of LiF [6], where experimental data for ionic conductivity are found to obey the proposed relationships for the particular case o f cation conduction in a material containing a dopant with an effective positive charge and having Schottky disorder as its intrinsic disorder type. Fig. 9, which is based on data given in the review by Wuensch [11], provides an additional example of the linear In D O versus h behaviour that has been discussed in the paper.

References

I

o

Mg

=0

.201 / C)

I

1

I

I

2

3

I

4

•

5

h/eV

Fig. 9. Literature data for cation and anion diffusion in MgO. Each point represents one set of diffusion data for which the results have then been reported in the form D = Do exp(-h/RT).

[1 ] W. Meyer and H. Neldel, Z. Tech. Phys. 18 (1937) 588. [2] F.L. Weighman and R. Kuzel, Can. J. Phys. 48 (1970) 63. [31 G.G. Roberts and D.G. Thomas, J. Phys. C7 (1974) 2312. [41 P. Winchell, High Temp. Sci. 1 (1969) 200. IS] R. McGregor and B. Milicevie, Nature 211 (1966) 523. [61 T. Dosdale and R.J. Brook, J. Mat. Sci. 12 (1977) 167. [71 Y.-W. Hu, I.D. Raistrick and R.A. Huggins, J. Electrochem. Soc. 124 (1977) 1240. [8] J. Schoonman, in: Fast ion transport in solids, eds. P. Vashishta, J.N. Mundy and G.K. Shenoy (North-Holland, Amsterdam, 1979) p. 631. I91 T. Dosdale and R.J. Brook, J. Am. Ceram. Soc. 66 (1983). [lO1 F.A. Kroger, The chemistry of imperfect crystals, Vol. 2 (North-Holland, Amsterdam, 1974) p. 240. [11] B.J. Wuensch, Mat. Sci. Res. 9 (1975) 211; S. Shixasaki and M. Hama, Chem. Phys. Letters 20 (1973) 361.