Determination of sro parameters of α-AgAl from resistivity measurement

Acra merall. Vol. 32, No. 7, pp. 1053-1060, 1984 Printed in Great Britain. All rights reserved

Copyright 0

OOOI-6160/84 $3.00 + 0.00 1984 Pergamon Press Ltd

DETERMINATION OF SRO PARAMETERS OF a-AgAl FROM RESISTIVITY MEASUREMENT Institut

W. PFEILER, fur Festkiirperphysik,

P. MEISTERLE and M. ZEHETBAUER Universitat Wien, Strudlhofgasse 4, A-1090 Vienna, Austria

(Received 11 October 1983; in revised form 1 December 1983) Abstract-Results of resistivity measurements on Ag-7.5,1 I .5,15.5 at.% Al show the features of statistical SRO in the Warren-Cowley sense. Therefore the quasichemical concept of mixtures is used to derive SRO parameters from resistivity measurement assuming a linear relation between residual electrical resistivity and first SRO parameter with the proportionality coefficient being independent of temperature. By comparing the change of equilibrium values of resistivity and the change of a, both as a function of temperature absolute values of a, are calculated, which are in excellent accordance with values from diffuse X-ray scattering. Further, values of the enthalpy of mixing evaluated from calculated SRO parameters are in satisfying correspondence with calorimetric measurement. R&un&-Les r&hats de mesures de resistivite dans des alliages d’argent avec 7,5, II,5 et I5,5 at.%Al presentent les caracteristiques de l’ordre P courte distance statistique au sens de Warren et Cowley. Nous utilisons done le concept quasichimique de melange pour obtenir les parambtres d’ordre a courte distance P partir de mesures de resistiviti en faisant I’hypothese d’une relation lin&aire entre la resistivite Clectrique residuehe et le premier paramitre d’ordre a courte distance, le coefficient de proportionalite ttant independant de la temperature. En comparant les variations des valeurs d’equihbre de la resistivite et celles d’a, en fonction de la temperature, nous pouvons calculer les valeurs absolues d’a,, valeurs qui sont en excellent accord avec celles que I’on obtient a partir de la diffusion diffuse de rayons X. De plus, les valeurs de I’enthalpie de melange 6valu&es a partir des parametres d’ordre a courte distance, presentent un accord satisfaisant avec les mesures calorimetriques.

Zusammenfaasung-Ergebnisse von Widerstandsmessungen an Ag-7,5;11,5; 15.5at.% Al stimmen mit der statistischen Interpretation der Nahordnung im Sinne von Warren und Cowley iiberein. Es wurde daher das quasichemische Konzept der Mischungen dazu beniitzt, Nahordnungsparameter aus den Widerstandsmessungen unter der Annahme abzuleiten, daD der elektrische Restwiderstand linear vom ersten Nahordnungsparameter abhangt und die Proportionalitatskonstante temperaturunabhiingig ist. Aus dem Vergleich der binderung der Widerstandsgleichgewichtswerte bei Temperaturiinderung mit der Temperaturabhangigkeit von a, werden Absolutwerte von a, berechnet, die in sehr guter &xeinstimmung mit Werten der diffusen Rontgenstreuung sind. Weiters stimmen die Werte der Mischungsenthalpie, die aus den berechneten Nahordnungsparametem ermitteit wurden, gut mit kalorimetrischen Messungen iiberein.

1. INTRODUCTION

Microscopically an alloy can be understood as a dynamic system: thermal energy tends to distribute the alloy atoms at random over the possible lattice positions; given small physical differences between the components, however, the free enthalpy of the system is minimized if special lattice positions are occupied preferentially by one type of alloy atoms. Even small differences in the interaction potentials between equal and unequal atoms lead to deviations from the random occupation of lattice positions. Therefore a great number of concentrated alloys show the features of short-range ordering (SRO) or short-range clustering (SRC) depending on the sign of the interaction potential: the probability for an atom of component A to have a B-atom as a neighbour is greater (SRO) or smaller (SRC) than the value corresponding to the mean alloy concentration. Since ordering or clustering competes with thermal agitation this gives rise to a temperature dependence AM

3217-E

of the microstructure of solid solutions, which in fact has been observed by various experimental methods. The local atomic configuration is usually described by the Warren-Cowley parameters (WCP), which show a simple relation to the diffuse scattering intensity (X-rays, neutrons): r.Xi = 1 -(J&/c,)

(1)

where ai = WCP for the ith coodination shell, pir, = probability to find a B-atom in the ith coordination shell around an A-atom, cg = concentration of Batoms (c~ + err = 1). The description of SRO with WCP implies the homogeneity of microstructure because of the statistical interpretation of SRO. Besides, a lot of experimental observations done since the early sixties for several alloys hint at deviations from the mean alloy concentration extending to several atomic distances as an origin of SRO-effects [l--S]. It seems questionable whether a description with the WCP will be

PFElLER

1054

ef cd.: SHORT-RANGE

adequate in the case of such a heterogeneous microstructure. However, recent measurements of resistivity kinetics give evidence that a-AgAl alloys are short-range ordered in the Warren-Cowley sense (6): singleprocess relaxation kinetics for the adjustment of a new degree of SRO have been observed contradicting a heterogeneous microstructure (quite in contrast to other materials, e.g. CuZn [5)). Therefore a-AgAl seems to be suited to try a derivation of the SRO parameters from measured changes of electrical resistivity, if it is possible to make a reasonable assumption for the relation between the change in resistivity and the change in the WCP’s. In Section 2 we discuss the proportionality of electrical resistivity and SRO parameters. Then (Section 3) the quasichemical concept of SRO is briegy described. Results of experimental investigation are given in Section 4, in Section 5 the calculation of absolute values a, of the first SRO parameter is shown. A comparison of values calculated this way with measurements of diffuse X-ray scattering and calorimetric measurements is also given in Section 5.

2. RELATION RESISTIVITY

BETWEEN ELECTRICAL AND SRO PARAMETERS

Although the functional relation between residual electrical resistivity and the degree of SRO is still a matter of discussion a lot of theoretical and experimental work suggests that resistivity should depend on SRO parameters in a linear way. Gibson [7] derived an expression for the residual resistivity of a short-range ordered binary alloy on the basis of Nordheim’s theory [S], which shows such a linear dependence of resistivity on SRO parameters, at least for the case of monovalent solid solutions. Asch and Hall [9] developed a quantum mechanical theory which considers changes in the Fermi volume with concentration. They also provide a method of calculating the first WCP from the composition dependence of absolute vaiues of electrical resistivity. Rossiter and Wells [lo] arrive at a proportionality similar to that of Gibson using a screened Coulomb scattering potential. The sign of the additional resistivity due to SRO depends on the number of conduction electrons per atom. With the formalism of [lo], but using a simple delta function for the scattering potential, Wagner et af. [I I] found a good agreement between changes of resistivity as measured on NiCu during appropriate temperature treatment, and changes in resistivity as calculated from changes in the WCP which had been simultaneously measured on the same samples by diffuse neutron scattering. They further found that for the special case of NiCu the main contribution to residual resistivity results from the atomic distribution in the first coordination shell. Vigier and Pelletier [12] point out that the value of

ORDER

IN a-AgAl

calculated resistivity is strongly influenced by the choice of scattering potential. Taking the point ion potential of Harrison a proportionality between the resistivity variation induced by SRO (SRC) and the first WCP arises in a good approximation. Using the results of m~suremen~ of electrical resistivity and thermopower Pelletier et al. 1131arrive at reasonable values of ai for a-CuZn assuming the discussed proportionality. However, a characterization of a-CuZn with usual SRO parameters seems questionable because resistivity kinetics [S] suggest a heterogeneous microstructure. In a recent detailed paper Kohl et al. [14] report on SRO formation in Au-15 at.% Ag detected by resistivity measurement. This alloy similar to a-AgAl exhibits one-process kinetics during the adjustment of equilibrium values of SRO which is in accordance with a homogeneous mi~ostru~ture. The investigation gives evidence that a single parameter is sufficient in describing the state of SRO. Using the theory of Rossiter and Wells [lo] the proportionality constant between the resistivity change due to SRO and the first SRO parameter is calculated. In summary one can say that a linear dependence of the SRO-induced electrical resistivity change on changes of the first SRO parameter seems to be a good approximation, especially for an alloy which behaves in conformity with the statistical interpretation of SRO. However, an influence of changes in the degree of SRO (SRC) on the Fermi-surface may lead to a more complicated relation between resistivity and SRO parameter: but this effect obviously is only of second order. 3. QUASICHEMICAL

TREATMENT

OF SRO

Within the quasi~hemical theory of mixtures strain energy effects and coulombic effects are neglected and only chemical pair interactions are considered assuming a statistical distribution of interacting pairs over the volume of the crystal. We start from a chemical reaction AA + BB+ZAB

(2)

of an alloy, composed of A and B atoms differing in their interaction potential only. For the law of mass action we can write [15] N%[(%ZN,

- NAB)(caZNr# - NAJ = exp (- 2W/RT)

(3)

where NAB= number of AB-pairs per mol, Z = N,, = Avogadro number, coordination number, w-v,,(VA,+,+ V&/2, ordering energy, k = Boltzmann constant, T = absolute temperature. Equation (3) is a quadratic equation for N,, with the solution N,,, = [ZN,/2(w2 - t)] [- 1 f Jl with wz = exp (2 W/kT).

+4c,ca(w2

- l)]

(4)

PFEILER et al,:

Using pAB = N,,/ZN,c, p,qj = [-

1 + Jl

SHORT-RANGE

one gets (16)

+ 4c,c,(w2

- I)]/ (4’)

[2c,(w2 - l)l.

Considering (1) and confining to next-nearest neighbours (ai = a,) one gets for the first WCP and its temperature dependence a,(T)=

1 +(l -Jl

+4cAce[w2(T)-

IN

1055

a-AgAl

(ii) Above this temperature an increasing deviation from linearity is observed, which can be attributed to an increasing influence of the quenching procedure: during the quench from the annealing to the measuring temperature ordering sets in; it is no longer possible to freeze in the equilibrium value, which belongs to the actual annealing temperature.

11)

/(2cAcrI]wz(7-) - 11). The enthalpy of mixing in the approximation quasichemical theory reads as H,,, = WNA, = WZN,c,c,(l

ORDER

-CL,).

(5)

of the

(6)

4. RESULTS OF RESISTIVITY MEASUREMENTS

In a recent experimental investigation (6) changes in the residual electrical resistivity of a-AgAl were measured during isochronal and isothermal temperature treatment. After quenching in liquid nitrogen the resistivity changes were detected with an accuracy of + 3 x 10 - 5 by a standard potentiometric method relative to a dummy specimen. In contrast to usual experiments observing SRO kinetics after a great change AT in annealing temperature (typically AT > 1OOC) the adjustment of a new equilibrium state of SRO here was studied after a small and sudden change of annealing temperature (typically AT = 10°C) leading from one stable state to another. This way SRO kinetics could be studied under the conditions of a constant vacancy concentration. The main results are: (i) The establishment of SRO in a-AgAl is accompanied by a reduction of electrical resistivity. (ii) There exist equilibrium values of resistivity corresponding to specific annealing temperatures. (iii) The equilibrium values of SRO are adjusted by one single exponential process. Figure 1 shows a series of isothermal anneals on Ag-7.5,l l&15.5 at.% Al demonstrating the change of equilibrium resistivity with annealing temperature. In Fig. 2 these equilibrium values Q, of resistivity are plotted together with those obtained by isochronal annealing. The equilibrium curve proves to deviate from a straight line the contrary of which was observed in a-CuZn over a wide temperature interval [17, 181. The curvature is better resolved in Fig. 3, where the step height of 10°C temperature steps (change of equilibrium resistivity after a temperature step of 10°C) is shown as a function of annealing temperature. Two ranges can be distinguished: (i) Up to a critical temperature (ccI, = 7.5 at.%: 250°C; c,, = 11.5 at.%: 220°C; c~, = 15.5 at.%: 2Oo”C), which decreases with increasing alloy concentration, the step height decreases as linear with temperature.

5. CALCULATION OF SRO PARAMETERS FROM RESISTIVITY MEASUREMENT Assuming a linear relation between the first SRO parameter and the resistivity change it seems easy to calculate values for a, from resistivity measurement. However, because of experimentally limited quenching rate it is impossible to measure the resistivity value of the completely disordered alloy. In the following we avoid this difficulty: we give a simple method for calculating absolute values of a, from measured changes in resistivity on the example of a-AgAl. The method is based on two assumptions: (i) There is a linear relation between the contribution of SRO to electrical resistivity and first SRO parameter: psRo = A ‘aI.

(7)

For the case of SRO the first WCP is negative; therefore for positive A a decrease in resistivity with increasing degree of order will result, which is in accordance with experimental observation (6). (ii) The coefficient A is assumed to depend on the alloy concentration only and to be independent of temperature. Because the temperature during the measurement was constant 77 K, only small changes of resistivity were observed. They exclusively can be attributed to configurational changes of atoms in the rather small temperature interval AT investigated (AT < 1OOC). The quasichemical theory gives (Section 3)

W(T)]=f[w(T)I

UI =flc,,,

for a fixed concentration [see equation (5)]. Assuming the Matthiessen rule to be valid the resistivity of a sample quenched to liquid nitrogen temperature reads as P =

PSRO +

PNz

being the fictitious sample resistivity at 77 K without SRO induced resistivity psRo. Using equation (7) we can write for the measured relative change in resistivity pN2

(P, - PJPO

= AP,/P,

= (A /PO)&

(9)

with PO= PLO + Plv2 and

piRo

= A .a!.

The index 0 of the symbols po, pzRO and mp stands for the initial temperature taken as a reference for relative resistivity measurement. Differentiation with respect to the annealing

PFEILER et al.: SHORT-RANGE ORDER IN a-AgAl

1056

-0.2

- 0.4

0

5

10

15

30

35

40

45

t 1 hr)

Fig. 1. A series of isothermal anneals at various temperatures showing the adjustment of stable equilibrium values of resistivity. temperature

gives aiaw~,i~d

= wpdda,iaT.

(10)

Now the ordering

energy W has to be chosen appropriately, so that for all annealing temperatures

a/aT(dp,/p,)/(aa,/aT)= A/p, = const.

(IO’)

Because a, and aa,/dT are steady and diflerentiable functions of the annealing temperature if W is assumed to be independent of temperature, a, (T) and A can easily be obtained once W has been determined: for a given ordering energy W a,(T) is calculated from equation (5); then A is evaluated as the quotient of the change of equilibrium values p, of resistivity and the slope of a,, both taken as a

function of temperature. The correct value of the ordering energy W has to be determined from the requirement that A in equation (10’) be constant for all temperatures. This condition holds only for temperatures below the critical value (see Section 4), that means as long as the slopes of p, and a, differ in the coefficient A only. Evaluating W from equation (9) also is possible but not so accurate as using equation (10): besides a higher accuracy considering changes of Ape/p, with temperature instead of the measured (Ape/p,)(T) itself, values outside the allowed temperature interval lead to large deviations of the coefficient A from the constant value, evaluated for lower temperatures. Therefore equation (10) was used to determine Wand subsequently a,.

PFEILER Ed ul.:

SHORT-RANGE

ORDER

1057

IN a-AgAl

35

CA=11

501

%Al

1.0 2 -0

0

% s 25

F 9”

05

8

Y -1700

J/mot

0 - 2050

J/mol

+ - 2400

0 140

160

180

200

220

240

20

260

r 1°C)

6. RESULTS OF CALCULATION

AND DISCUSSION Figure 4 gives an example of a search for the coefficient A for the case of c,, = 11.5 at.%. For the calculation the equalized values of Fig. 3 were used. It is demonstrated that A is a sensitive indicator for a definite ordering energy: there is only a very small interval of ordering energies W for which A is really constant up to the critical temperature already mentioned. For pO, which is necessary in equation (lO’),to obtain A, but does not influence the calculation ‘of a,(T), values of Borelius and Larson [19] were extrapolated to the temperature of liquid nitrogen (Table

240 r

1

0

200

250

r(T)

Fig. 2. Equilibrium values of resistivity won by isothermal (open dots) and isochronal annealing (solid dots) as a function of annealing temperature. 0,. 7.5 at.% Al; A,A 11.5 at.% Al; q,m 15.5 at.% Al.

740

J /mol

Fig. 4. Change of coefficient A with change of parameter W (ordering energy) as a function of annealing temperature. c,, = 11.5 at.% Al.

1). As the contribution of SRO to the equilibrium values of residual resistivity is as small as 2.5 x lo-* for a change of the annealing temperature of 100°C the influence of thermal pre-treatment is neglegible for the measurement of pO. Because of the low accuracy in estimating our sample geometry (all measurements were done relative to a dummy speci-

35

30

E0

260

I’C)

Fig. 3. Step height 100 [A/AT’][(p, - pO)/p,Jfor steps of 10°C as a function of annealing temperature. Extrapolation of linear part is indicated by broken line. 0 7Sat.%Al; A 11.5 at.% Al; 0 15.5 at.% Al.

T(‘cl

Fig. 5. Values of coefficient A as a function of annealing temperature for best fitting ordering energy W. 0 7.5 at.% Al, W = -2100 J/mol; A Il.5 at.% Al, ;V = -2050 J/mol; 0 15.5 at.% Al, W = -2200 J/mol.

PFEILER et al.:

1058

SHORT-RANGE ORDER IN a-AgAl 013

Table 1. Used values of pO (extrapolated from [19] lo 77 K), the proportionality coefficient A and A/p,

r

Alp, (:r$J 7.5

Il.5 15.5

ta2mJ 1.60 2.17 2.52

&L-d 2.11 2.87 3.33

1.31

I .32 I .32

men), it was preferred to use the absolute values of [191t. In Fig. 5 the values of A evaluated in the way mentioned above are plotted as a function of annealing temperature for the three concentrations investigated. As required A remains constant over a wide interval of annealing temperatures using the indicated values for the ordering energy W. The method obviously is only correct for annealing temperatures below a critical value as already shown in Section 4: above’ this temperature the ordering process during the quench to the measuring temperature falsifies the true step height. Therefore above this critical temperature no value for the ordering energy W can be found that keeps A constant. It seems to be a proof for the consistency of the present calculation that the coefficient A shows the same dependence on composition as p0 (Table 1) leading to equal values of A /p. for all concentrations investigated. Figure 6 gives the thus obtained first WCP a, as a function of temperature for the concentrations investigated. There is only a very small curvature in the temperature dependence of a, reflecting the very small curvature of the step height of SRO-induced equilibrium resistivity with varying temperature. This suggests a possible critical temperature for long-range order, if any, to be far away at low temperatures. For an estimation of a possible transition temperature the Bragg-Williams approximation can be used [20]. Assuming an ordered phase Ag,Al, which‘seems rather improbable because of the known structure of the AgAl p-phase (isotypic with B-Manganese), this yields T,(15.5 at.%Al) = -35”C, T,(11.5 at.% Al) = -75°C and T,(7.5 at.% Al) = - 134°C. The observed decrease of the step height with increasing temperature for a-AgAI is in contrast to other alloys like CuZn [5]. This difference may be attributed to the comparably low ordering energy of the present alloy: according to equation (5) the higher the ordering energy involved, the more linear a,(T) and subsequently p,(T) results (Fig. 7). The values of a, for 200°C are shown in Table 2 together with values extrapolated to other concentrations (14.3 at.% Al and 18.5 at.% Al) and higher

aI

100

200

300

T ( “C )

Fig. 6. SRO parameters a, of the first coordination sphere as a function of temperature, calculated for 7.5, 11.5 and 15.5at.%Al. W= -2lOOJ/mol.

temperatures (450 and 500°C) for a comparison with literature. No essential influence of concentration on ordering energy W could be observed (Fig. 5) and the fitted values of W for all concentrations are within the uncertainty of about k 100 J/mol, which results from the limited accuracy of the slope of the resistivity equilibrium curve. Therefore all values of a, are calculated using a mean ordering energy of - 2100 J/mol. In Table 2 three values from literature are included. The correspondence of these values obtained by diffuse X-ray scattering [21,22] with those extrapolated from SRO-induced resistivity changes is quite excellent. The difference for 14.3 at.% Al is most likely caused by influences of the poor quenching process from 500°C into oil of room temperature used in [22]. Once a,(T) and A/p, are known the hypothetical increase of SRO-induced resistivity can be calculated.

tit has to be emphasized that the value p0 used here does not correspond to the completely disordered state of the alloy, which would be necessary for an evaluation of absolute SRO parameters from relative changes in electrical resistivity. In our case absolute values of a, are obtained by the artifice of comparing the slopes of p,(T) and q(T).

o/o

T(K)

Fig. 7. Change of temperature dependence varying ordering energy W.

of a, with

1059

PFEILER ef al.: SHORT-RANGE ORDER IN rx-AgAI Table 2. SRO parameter a, for 200°C calculated from resistivity measurement a, CA1 (at.“/.)

200°C

1.5 11.0 Il.5 14.3 15.5 18.5 Accuracy: f 5%

500°C

450°C

- 0.050 - 0.074 - 0.078 - 0.097 -0.105 - 0.125

-

0.038 0.055 0.057 0.071 0.076 0.090

- 0.090 [2 I]

-

0.036 0.052 0.054 0.067 0.072 0.085

-__

~

- 0.050 [22] -0.10

[22]

The values for 450 and 500°C as well as those for 14.3 and 18.5 at.% Al are extrapolated for a comparison with data of difluse X-ray scattering [21,22]. All values are calculated using an ordering energy of -2lOOJ/mol.

Table 3. Enthalpies of mixing as calculated using the SRO parameters of Table 1 H,

CA1 (at.%) 1.5 11.5 15.5

H, calorimetry [23]

calculated

(J/mot)

(J/mot)

- 1810 - 2700 - 354b

- 1270 - 1900 - 2500

For a comparison with calorimetric measurements of Wittig and Schilling 1231these values are extrapolated to 500°C.

Figure 8 shows the increase of ~sso between 0 and 800 K for the concentrations investigated. The temperature range accessible to experimental investigation is indicated by the hatched area. From equation (6) the enthalpies of mixing can be estimated from the calculated a, values. Table 3 shows the result of this evaluation (extrapolated to 500°C) together with experimental results of calorimetric measurements at 500°C [22]. Apart from a systematic deviation by a factor of about 1.4 probably arising from influences of the measuring method, the relative variation as a function of concentration agrees well. This way the present method of evaluation is supported by a quite different experimental measurement. 7.

CONCLUSIONS

(i) a-AgAl obviously is an alloy which behaves in accordance with the usual statistical interpretation of SRO. This suggests an application of the quasichemical concept of mixtures in order to calculate

SRO parameters from measured SRO-induced resistivity changes. (ii) By assuming a linear relation between residual electrical resistivity and first SRO parameter with a proportionality coefficient independent of temperature the ordering energy can be determined by comparing the slopes of pp and a, with temperature. This subsequently leads to the values of the first SRO parameter involved. (iii) The result of a,(T) is in excellent accordance with the rather few values from diffuse X-ray scattering. The enthalpies of mixing evaluated from calculated SRO parameters are in satisfying correspondence with calorimetric measurements. Acknowledgement-The continued interest of Professor Dr K. Lintner is gratefully acknowledged. The authors are indebted to Dr K. Siebinger for valuable discussion and to Dr W. Piischl for a critical reading of the manuscript. The work was financially supported by the Austrian “Fonds zur Fiirderung der wissenschaftlichen Forschung” grant number 4 134.

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3. L. Trieb and G. Veith, Acfa metall. 26, I85 (1978). 4. F. Adunka, M. Zehetbauer and L. Trieb, Physica s[aIus solidi. (a) 62, 213 (1980). 5. D. Trattner and W. Pfeiler, J. Phys. F. Mefall. Phys. 13, 739 (1983). 6. P. Meisterle and W. Pfeiler, Acta metnll. 31, 1543 (1983). 7. J. B. Gibson, J. Phys. Chem. SolidF 1, 27 (1965). 8. L. Nordheim, Ann. Physik. 9, 607, 641 (1931). 9. A. E. Asch and G. L. Hall, Phys. Rev. 132, 1047 ( 1963). IO. P. L. Rossiter and P. Wells, J. Phys. C. Solid Sr. Phys. 4, 354 (1971). W. Wagner, R. Poerschke and H. Wollenberger, Phil. Mug. B43, 345 (198I). G. Vigier and J. M. Pelletier, Acra mefall. 30, 1851 (1982). J. M. Pelletier, G. Vigier and R. Borrelly, Scripia mefall. 16, 1343 (1982).

0

100

200

300

400

500

600

7ccl

800

T(K)

Fig. 8. Resistivity increase as calculated from the temperature dependence of a, using the values of Table 1 for A/p,. c,,=7.5,11.5,15.5at.%AI. W was used for Fig. 5.

W. Kohl, R. Scheffel, H. Heidsiek and K. Liicke, Acfa metall. 31, 1895 (1983).

15. E. A. Guggenheim, Proc. R. Sot. A183, 213 (1944). 16. S. Radelaar, J. Phys. Chem. So/ids 31, 219 (1970). 17. A. C. Damask, J. appl. Phys. 27, 610 (I 956).

1060

PFEILER er al.:

SHORT-RANGE

18. R. Poerschke and H. Wollenberger, J. nucl. Mater. 74, 48 (1978). 19. G. Borelius and L. E. Larson, Arkiu Fysik 11, 137 (1956). 20. J. L. Boquet, in Solid State Phase Transformalionsin Me& and Alloys (edited by de Fontaine), p. 1. Les Editions de Physique, Orsay (1978).

ORDER

IN wAgA

21. P. S. Rudman, SC. D. thesis, MIT (1955); value cited by P. A. Flinn, Whys. Rec. 104,350 (1956). 22. R. I. Bagdasaryan et al., Izw. Akad. Nauk. Arm. SSR Fiz. 10,372 (1975). 23. F. E. Wittig and W. Schilling, Z. MetaNk. SO, 610 (1959).