ELECTRONIC CONFIGURATION AND LATTICE COLLAPSE IN Sm COMPOUNDS

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ELECTRONIC CONFIGURATION AND LATTICE

COLLAPSE IN Sm COMPOUNDS

T. Penney, R. Melcher

To cite this version:

T. Penney, R. Melcher. ELECTRONIC CONFIGURATION AND LATTICE COLLAPSE

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IOURNAL DE PHYSIQUE Colloque C4, supplkment au no 10, Tome 37, Octobre 1976, page C4-275

ELECTRONIC CONFIGURATION AND LATTICE COLLAPSE IN Sm COMPOUNDS

T. PENNEY and R. L. MELCHER

IBM T. J. Watson Research Center Yorktown Heigts, N. Y. 10598 U. S. A.

RBsumB.

-

Nous presentons un mod6le analytique simple qui illustre la physique de la transition de configuration apparaissant dans SmS sous pression ainsi que dans SmS-YS et SmS-As substitues chimiquement. Le modde n'inclut que les interactions les plus fortes et sert a mettre en evidence les param6tres necessaires pour avoir une transition de configuration vers un Btat de valence interme- diaire.

Abstract. - A simple analytic model is presented which illustrates the physics of the configurational phase transition occuring in SmS under pressure and in the SmS-YS and SmS-SmAs systems with chemical substitution. The model includes only the largest interactions and is intented to demonstrate the requirements for a configurational phase transition to a stable intermediate valent state.

1. Introduction. - The SmS lattice collapses with pressure or with chemical substitution, such as Y or As. There are many unusual characteristics of this collapse. For example, the volume change is very large

(- 10

%)

and there is no change in the cubic symme- try. Further, in the collapsed state the Sm ion exists in an intermediate valent state which is a combination of divalent (f6) and trivalent (f5 d) configurations.

Figures 1-4 shows the experimental data for the lattice collapse of SmS with pressure [l, 21 and chemical substitution. Also shown are the results of our simple theory. In this paper we will first examine the important physical processes involved and then describe the model which utilizes these processes. The general phenomena occuring in SmS and its alloys have been reviewed recently by Varma [3] and

by Von Molnar et al. [4]. Although the intermediate

0 0 . 2 0 4 0.6 0.8 I .O

COMPOSITION x

FIG. 1.

-

Volume strain, e, vs. composition x, for Sml-zYxS : Experiment, 80 K ( 0 ) ; theory (LO-). Data from Ref. [4].

valence state was deduced by Maple and Wohlleben 151

from magnetic susceptibility data, the strongest evidence to date is from Mossbauer isomer shift [6, 71

X-ray photoemission 18, 91 and lattice constant data [5, 10, 111.

COMPOSITION x

FIG. 2a. - Volume strain e vs. composition x for Sml-,YXS : Experiment, 300 K ( 0 ) ; theory (--Om). Data from Ref. [4].2b) Configuration parameter z vs. composition x : theory (-0-). Valence = 2

+

z. 2c) Bulk modulus B vs. composition x : Experiment, 300 K ( O ) , theory (-0-). Data from Refs [13,14].

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C4-276 T. PENNEY AND R. L. MELCHER - SmS -0.2 - 0 2 0 4 0 6 0 8 0 100 PRESSURE p (kbar)

Fro. 3.

-

Volume strain e vs. pressure p for SmS : Experimen- tal data (0) is a composite from Refs [I] and [2] ; theory (-0-).

0 0 . 2 0 . 4 0 6 0 . 8 1.0 COMPOSITION x

FIG. 4.

-

Volume strain e vs. composition x for SmSl-xAsx : Experiment 300 K (0) ; theory (-0-). Data from Ref [4].

2. Physical properties.-- Electrical transport [12] bulk modulus [13, 141 and specific heat [15, 161 results indicate that in the collapsed state there is a large electronic density of states degenerate with a lower density of states at the Fermi level, E,. The

situation is shown schematically in figure 5. The large density of states is primarily f6 like. The broad density of states arises from the d band of the f 5 d configuration. On the other hand these same measure-

'i..

Ed-

FIG. 5. - Schematic density of states of S ~ I - ~ Y ~ S , (a) x = 0 pure SmS ; (b) x < 0.15, black phase ; (c) x > 0.15, gold collap-

sed phase.

ments indicate that in the uncollapsed state the configu- ration is predominantly f6 like with E, lying above the f 6 states (Fig. 2b).

It is apparent that the lattice collapse and the electronic configuration are closely related. This strong coupling occurs because of the crystal field acting on the d states of the Sm f 5 d configuration. In the free ion the f d state is 2-3 eV above the f 6 but with the crystal field the lowest f 5 d(t2,) is nearly degenerate with the f 6 [2,17,18] Franck-Condon relaxation effects may be expected to reduce the equilibrium f 6-f d separation below the optical value so that there is a large density off d states within about 0.1 eV of the f 6 states.

Since the near degeneracy is caused by the crystal field, the relative positions of the levels are sensitive to pressure or volume. Optical studies 118, 19, 201 show that in SmS the f6-f d(tZg) energy difference decreases with pressure at 10 f 2 mevlkbar. Compressibility [21 and sound velocity measurements [14] give a bulk modulus, B, of 475 kbar. The deformation potential, y, which is the change in this energy difference per volume strain, is the product of these two numbers, 4.75 eV.

A volume strain of 15

%

will then reduce the f 5 d energy by about 0.7 eV which will put the lowest f 5 d state well below the f states instead of

--

0.1 eV above as they were initially. If the density of states of the d levels is of order 1 statelev-atom than a large fraction of an electron per ion could leave the f 6 configuration and occupy the f 5 d.

This explanation of the phase transitions in terms of the deformation potential, y, does not depend on an unusually large y, but rather on an unusually large strain (-- 15

%)

and the near degeneracy between the levels.

Usually a large volume strain occurs only at very high pressure since it involves large lattice energy. However, the lattice constant of the trivalent rare earth sulfides such as NdS and GdS are much smaller

(-- 15

%)

than those of the divalent SmS and EuS. Interpolation between the NdS and GdS values gives a volume for trivalent SmS 16

%

smaller than divalent SmS. The reason for the difference is that in the.Sm3+ configuration there is one less f electron screening the 5 s and 5 p valence electrons from the nuclear charge than there is in the Sm2' case.

The large lattice collapse is clearly related to this ionic collapse and should be included in the expression for the lattice energy.

The final physical effect which we wish to include is the dependence of the bulk modulus on electronic configuration. For SmS, the bulk modulus, B, between 20 and 80 kbar is about 1 100 kbar [I], over twice the 475 kbar value of uncollapsed Sm2

's.

This large value is probably representative of pure trivalent SmS [21]. The bulk moduli for YS is 1 000 kbar, and for SmAs, 785 kbar, are considerably larger than for sm2+S.

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ELECTRONIC CONFIGURATION AND LATTICE COLLAPSE IN Sm COMPOUNDS C4-277

divalent to trivalent Sm which stabilizes the intermediate valence, Neglecting hybridization effects it might be expected that if the lattice collapse is first order it would go all the way to pure trivalent f5d. However, as we have argued before [14], if the lattice stiffens as the collapse takes place it could begin as a discontinuous transition but stop short of pure f 'd, due to the increased rigidity.

We have discussed phenomena which are large and which must be important for the phase transition. They are :

1) The near degeneracy (-- 0.1 eV) of the f 6 and f 5d configurations.

2) The deformation potential which measures the strain dependence of the f 5d-f energy difference

(- 5 eV).

3) Volume renormalization due to the smaller size of the Sm f 5 core compared to the f 6 core. G ~ ~ + S is smaller (- 16

%)

than SmZ+S.

4) The bulk moduli of collapsed Sm3 + S and YS are about twice as large as that for Sm2+S.

In order to test whether these effects are sufficient to describe the lattice collapse to an intermediate valence we have constructed the following model using only these effects.

3. Model. - It is assumed that the properties of the integral valent materials Sm2+S, Sm3 +S, Y + S and

Sm3 +As are known. The theory is essentially a way of interpolating between these known constituents while including the four important physical effects of the previous section. The reason for this approach is that most of the parameters can be determined directly from experiment.

Essentially the same model is used for the pressure collapse of SmS and the chemical collapse of Sm,-,Y,S and SmS,-,As,. The following explicitly describes the case of SmYS and SmS including pressure. The simplified electronic density of states is shown in figure 5. For SmS (Fig. 5a) the f5d states with a moderate density of states, Nd, occur at a small energy, A2, above the f6. states. These f 6 states have a large density of states, N,, and small width, A,. It is unders- tood in this one electron like picture that only one electron per Sm may be removed from the f 6 states and put in the f5d at which point the f 6 states would be

considered empty. That is,

The value of A, is unimportant for this theory provided that it is much smaller than the d band width so that

N, % Nd except as x -+ 1. Kasuya [21] has suggested that a finite A , could result from mixing with the sulfur p bands.

For the sake of analytical simplicity we have used a constant density of states and zero temperature. A further simplyfying assumption is that the d electrons

from the YS and the d electrons from the SmS f5d form a common band with density of states N d , regardless of composition. Figure 5b shows electrons in the d band due to Y substitutions but with all of the Sm still in the f 6 state. However, figure 5c shows the situation after the collapse. The d band contains electrons from both the Y(d) and Sm(f5d) configuration. The f 6 states areshown partially empty.

The bottom of the d band is taken to be :

Ed = ye

+

A,, (2)

where q is the deformation potential discussed earlier,

e = AV/V is the volume strain relative to SmS at zero pressure

gives the movement of the d band for reasons other than the strain, e. According to eq. (2), if the lattice collapses and y

>

0, Ed decreases and f -+ f 5d transitions occur.

In order to determine the equilibrium strain and configuration we find the electronic energy and the lattice energy and then minimize their sum. The electronic energy is given by

which is a function of strain e through Ed and the Fermi level, EF. The zero of energy is taken as the top

of the f 6 states. EF is determined by the number of

electrons in the d band per cation

Here

x

is the contribution of one d electron per Y. z is the Sm configuration parameter and z is the fraction of Sm in the f 5d configuration. Another expression for z is given by the holes in the f 6 states,

Because of the assumption of constant density of states and zero temperature the integrals are trivial. Eq. (6) and (5) may be used to eliminate z and EF from eq. (1) leaving

U,,

a function of strain only. The expressions are given in the appendix. The lattice energy for a simple system in the harmonic approximation can be written 77, = B,(V

-

V0)2/2 Vo

+

p(V

-

V,) where Bo is the bulk modulus and V, is the equilibrium volume at zero pressure, p,. However, we have shown above that B, and Vo are not constant. Both depend on configuration and composition. Instead we use

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C4- 278

and

T. PENNEY AND R. L. MELCHER

e = (V

-

Vx)/V2 ex = (1

-

x) ze,

+

xe,

e3 = (V3 - V2)/V2 (9)

el = (Vl - Vz>/Vz e2

-

0

B,, B, and B, are, respectively, the bulk moduli of divalent SmS, trivalent SmS, and YS. The bulk modulus for the lattice, B,, neglecting electronic effects, is taken to be a linear interpolation of these values (eq. 8). Similarly e,, e,, el, and ex are the lattice volume strains at p = 0 neglecting electronic effects

and defining e2 0. We have also multiplied B and p, which have the units of pressure, by the normalization volume V2 to get Bx and p in energy and U, in energy per formula unit. This manner of writing the lattice energy is the simplest analytic form which allows for the different bulk moduli of the constituents and the different ionic core volumes. U, is a function of e and z but the z dependence can be removed using eq. (5) and (6) as before.

Having written U, = Uel

+

U, as a function of e only, the equilibrium strains may be found analytically from dU/de = 0, in this T = 0 approximation. Furthermore, the bulk modulus'including the electron lattice coupling is given by B = d2u/de2. There are three possible cases, each of which is treated separately. In case A, EF lies above f6 as in figure 5a, b and z = 0. That is, there is no f5d present. In case B, EF lies in the f 6 states (Fig. 5c) and both f 6 and f 5d are present (0

<

z

<

1). In case C, EF lies below the f 6 states, only f 5 d exists and z = 1. In practice, solutions to the equations are found for all cases (A, B, C) but then EF is determined, compared with the assumption and only consistent solutions kept.

In case A, z = 0, eq. (5) becomes

and

U,, = x(qe

+

A,)

+

x2/2 Nd

-

(1

-

x) Af/2 ₍₁₂₎ after using eqs. (I), (2), and (10) the lattice energy eq. (7) is

with

and

ex = xe,

.

(15) The solution for case A is then

and

B = d2U/dez = B,

.

(17) The simplicity of this result is due to the fact that EF

does not cross the f 6 states and the configuration does not change. The solution is shown as the straight line starting at the origin in figure 1. In the same fashion, if EF is below the f states (Case C) so that only f5d exists ( z = 1) we find e = ex

-

- dB, (1 8) and B = B, (19) with ex = (1 - x) e,

+

xe, ₍₂₀₎ and B, = (1 - x) B,

+

xB1

.

(21) Case A is just a binary alloy of Sm2+S and YS while

Case Cis an alloy of Sm3 ' S and YS.

In the expressions for e, ex is just a linear interpolation which gives the volume renormalization. The

-

xy/B, term is a volume decrease which lowers the d band energy. It is proportional to the electrons in the d band, x for Case A and 1 for Case C. If B, were a constant then for both Cases A and C, e would be linear in x (Vegard's law) as is often observed. However, if B, changes with x there will be deviations from straight line behaviour. In both cases there is no contribution of the electronic terms to the bulk modulus. B is simply the weighted average of the constituents. A completely different type of behaviour is found for the intermediate valence situation (Case B) where

0

<

z

<

1. Eq. (6) and eq. (5) become

z = - EF/Af (22)

d = x

+

(1

-

X) z _(EF_-_Ed)_Nd ₍₂₃₎ which together determine

EF = Ed Nx/Nf

+

xNX/Nd Nf (24)

for

Since Ed = ye

+

A,, as before, EF and z are linear functions of e which may be substituted into Uel and

U,. For example

and

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ELECTRONIC CONFTGURATION AND LATTICE COLLAPSE I N Sm COMPOUNDS C4-279 From eq. (7),

with ex = (1 - x) ze,

+

xe, a linear function of e via z. The resulting quadratic equation, which yields the equilibrium strain, is given in the appendix.

A second differentiation of the above equations gives

with

In this expression e and ex = (1

-

x) ze,

+

xe, must be found from the equilibrium e and z (see appendix). Comparison of this expression for B with B = Bx found for Cases A and C shows the softening effect of the intermediate valence on the bulk modulus. In the first place Bx is reduced by the renormalization factor R2 = (1

+

e3 N, Y ) ~ .

This term contai,ns the softening which occurs because the f 5 core is smaller than the f 6 (e,

<

0) as mentioned in the discussion. This renormalization effect depends on the band dropping as the lattice is compressed and so depends on e, and y being of oppo- site sign.

A second softening effect is the

-

q2 N, term which is independent of the sign of y. This term is electronic and is independent of the renormalization effect. It represents the fact that if there are two bands (or any sets of states) degenerate at EF which move relative to one another with strain, then a redistribution of electrons will take place between them. Under stain one band will move up with respect to the other, but instead of this raising the electronic energy, the elec-

trons spill over into the other band. Since the electronic energy is not raised as much as it would be if these transitions did not take place, the lattice is softer than it would be otherwise. This term is very general and could be important in other systems whenever different bands are degenerate at EF. Its magnitude, however,

depends on the effective density of states N,. The last term in B is a cross term including both effects. From this discussion of the bulk modulus, B, we have tried to show why in the intermediate valence case

B is soft. Naturally the same effects which cause a soft B can produce a lattice collapse.

4. Choice of parameters.

-

Comparisons of the calculated equilibrium volume strains with the experimental data are shown in figures 1, 2. Since the data is somewhat temperature dependent we have shown results of calculations of the T = 0 theory with slightly different parameters. The agreement is very good given the simplicity of the model.

The purpose in setting the model up in this form was to see if known parameters could produce the observed effects. The parameters we have used and their values as determined from other experiments are given in table I. The bulk moduli of the constituents were taken equal to the measured values quoted earlier. These values, converted to eV/fomula unit as shown in the appendix are given in table I. The renormalization parameters el and e, were chosen so that the equilibrium strains for YS and Sm3+S would be equal to their experimental values. These equilibrium strains are given by eqs. (18) and (20) as e(x = 1) = el - q/Bl and e(X = 0, 3 +) = e, - y/B3 and are listed in the table.

Values of the three key parameters y, Nd and A, are close to their experimental values (Table I). The value

Experimental values Model parameters

SmYS SmYS SmS SmSAs SmS YS SmAs (Fig la) (Fig. 1 b) (Fig. 3) (Fig. 4)

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C4-280 T. PENNEY AND R. L. MELCHER of the deformation potential, y = 4.8 eV, for SmS has

already been discussed. The d band density of states,

N, = 1.6 eV-' is taken from y = 3.9 mJ/mole K2, the coefficient of the electronic contribution to the low temperature specific heat of YS [4]. There is no very good measure of A,, but 10-I eV is consistent with the optical results [2, 181, We have measured the resistivity of a number of SmS crystals and found activation energies ranging from 0 to 0.04 eV. Zhuze et al. [22,23]

have found values from 0 to 0.2 eV at high temperatu- res. These experimental values are given in table I.

A blank in table I indicates that the values is unknown

while a dash indicates that the value is not applicable. The only parameters that we can not measure are A, and A,. Since it is N, (eq. (25)) which enters into the equations, the value of A, has almost no effect as long as N, 9 N,. The value l/A, = 20 eV-' is taken from the specific heat data '[4].

A , adjusts the bottom of the Y d band with respect to the Sm f levels, eq. (2). It is not within this model to predict where the Sm f 6 states lie when a small amount of Sm is substituted into YS. The fact that the Sm is in the intermediate valence indicates that the f 6 states must lie at EF. A , is chosen so that

satisfies

-

A,

<

EF

<

0.

The agreement of the theory with experiment (Figs. 1-4) supports the general features of the model. As shown in the table, the parameters chosen are all very close to independently determined experimental values. However, because of its simplicity, the model should not be considered as a way of determining the parameters.

5. SmS and SmS-SmAs. - The expressions for SmS under pressure are found simply by setting x =

0.

The parameters used are the same as those used for SmYs except for minor changes in A, and e3 and the elimination of the YS parameters B,, el and A,. The theory (Fig. 3) shows a first order lattice collapse a t 6 kbar to an intermediate valence state in general agreement with experiment [I, 21. The theory does not do a very good job of predicting the second transition to the pure Sm3+ state. For these parameters the second transition occurs at about 60 kbar while the data appears to show the transition at about 20 kbar. Probably it is the assumption of constant density of states, N,, which causes this difficulty.

The theory for the Sm SAs system is identical in spirit to that described above. The equations are given in the appendix and the results in figure 4. The parameters are chosen generally as described earlier and are compared with the experimental values in table I. As before A , is used to adjust the position of the Sm f 6 levels with respect to the SmAs band structure. This choice of A , causes the Sm to be completely trivalent for large x. The theory describes the general situation. With

increasing x, there is first a lattice collapse and discontinuous transition to an intermediate valence state and then another transition to the 3

+

state. In this case we changed e3 somewhat from the previous value in order to give better agreement in the purely 3

+

state.

6. Configuration and bulk modulus. - The fraction of f 5 d configuration, z, and the bulk modulus, B, are given for the SmYS systems in figures 2b, c. The theoretical bulk modulus is only slightly changed up to the phase transition, it falls to less than half its original value at the phase transition and then climbs to the YS value as the Sm is removed by substitution. This behaviour is similar to the experimental data also shown, but the softening effect is smaller.

The theoretical values for z given in figure 2b show that the first order transition is from the divalent to an intermediate valent state. The actual values given for z by the theory are not important, they would be changed by a more realistic density of states, for example. It is of interest to note that for the black phase,

x < 0.15, before the lattice collapse, this theory gives z = 0, that is the pure 2

+

state. On the other hand the isomer shift [7] indicates that a small amount of 3

+

is present (z

-

0.15). This theory, however, will never give anything but z = 0 before the transition. The reason is that the theory assumes :

1) zero temperature,

2) homogeneity, all Sm equivalent regardless of their actual environment,

3) no explicit f 6-f 5d hybridization.

The breakdown of any of these assumptions would cause some deviation from z = 0 in the black phase. For example, for T # 0 the f5d state would be ther- mally occupied before the transition and with hybridi- zation the large f 6 density of states, N,, would contain some f5d character. Kasuya [21] has suggested that the behaviour in the black phase is primarily determined by clustering, that is by the actual number of Y or As which are nearest neighbours of each ~ m . Such inhomogeneous environmental effects would cause individual Sm ions with more than the average number of Y or As neighbours to collapse before the collective collapse of the whole crystal. The result would be an increase in z from z = 0 and an additional strain in the black phase near the phase transition. In fact, from figures 1, 2, 4, the largest deviation between theory and experiment is just before the transition where clustering effects should be largest.

Naturally there are no alloy clustering effects in SmS with pressure similar to those in SmYS. Interestingly the isomer shift [7] shows no Sm3 + in SmS at one half the transition pressure while there is 10-1 5

%

Sm" in SmYS at one half the transition concentration.

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ELECTRONIC CONFIGURATION AND I ,ATTICE COLLAPSE IN Sm COMPOUNDS C4-281

ning the valence from the lattice constant also indicates that there is an intermediate valence in the black phase of SmYS [4, 121. However, the actual value may be different since the relationship between the valence and lattice constant should be determined self consis- tently.

The homogeneous model presented here should be good away from the transition. It explains why SmS is divalent and SmYS is collapsed in an intermediate valence state at large x

7. Other work.

-

The model described here is similar in several respects to that of Hirst [24] for SmS under pressure. The basic mechanism, which we call the deformation potential, is his compression shift mechanism. His model, as ours, is a zero temperature, zero hybridization model which does not include the Falicov Kimball [25], electron correlation parameter, G. Hirst finds that the intermediate valence is stabi- lized by using an anharmonic bulk modulus of the Birch form. Our choice is similar but allows us to insert the actual measured values of B. In addition, the simpler form for B allows us to find an analytic solution. The major extensions of our theory beyond Hirst's is to include the alloy systems and the lattice renormalization. Wio, Alascio and Lopez [26, 271 have included the renormalization effect for SmS under pressure, and have used an expression for the electron energy which may include both the Falicov-Kimball G and the deformation potential. Varma and Heine [28] use the deformation potential and lattice renormalization. However, they allow only a small change in the bulk modulus so it is necessary to include a large anharmonic term in the lattice renormalization. Recen- tly, Jefferson [29] has extended the Hirst work to the alloys and to finite temperature but restricted his theory to the case of donors for which the conduction band is degenerately occupied. He is able to explain the anomalous thermal expansion of the SmGdS sys- tem [I 11 due to changing valence.

8. Conclusion. - The simple theory presented here is intended to illustrate the physics of the configurational phase transition. The approach is that if the properties of the pure constituents Sm2+S, Sm3+S, YS and SmAs are known then the pressure and composition dependence can be described. The important parameters for the phase transition, i. e. the f6-f5d deformation potential, the d band density of states and the SmS band gap, are close to their experimental values. The intermediate valence is stable because the bulk modulus of Sm3+S is about twice as stiff as that for Sm2+S.

Acknowledgements.

-

We would like to thank F. Holtzberg, T. Kasuya, J. Kiibler, E. Pytte, D. Sher- rington and S. Von Molnar for valuable discussions.

Appendix 1 : Units and Normalization.

-

The calculation is normalized to one formula unit. The reference volume for the strain is

the volume associated with each Sm in black SmS at zero pressure with lattice constant a, = 5.966

A.

The measured values of the bulk moduli are given in table I. They are converted from kbar to eV per formula unit using 30.18 kbar = 1 eV/formula unit. This number is the inverse of the product of the reference volume V2 and the conversion factor

6.242 x lo2' ev/cm3 = 1 kbar

.

Appendix 2 : Sm,-,Y,S.

-

The equation for the equilibrium strain in the intermediate valence state (Case B) of Sm, -,Y,S is 0 = auli3e = A, e2

+

A, e f A, (Al) where and R = 1 + e 3 q N x (A51 Bxx = B2 - xBS1

-

N,6B32 (A6) B31 = B 3 - B 1 , B 3 2 = B 3 - B 2 (A7) Nx 6 = Nx Ax

-

xNX/Nf (As)

Q

= Nx 6e3

+

x(e3 - e,)

.

(A9) The bulk modulus is

B = d2U/de2 = Bx, R2

-

q2 N

-

- B32 qNx(2 RQ -t 3 R2 e,) (A10) where the e, which appears is the equilibrium strain determined by the eq. (Al) above.

Appendix 3 : SmS, -,As,.

-

The final equations for the volume strain e and bulk modulus B for the As acceptor system are similar to those for the Y donor system. The equations and definitions which are different are noted here

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(3-282 T. PENNEY AND R. L. MELCHER

In case A , z = x and

0 = aU/ae = B,(e

-

ex)

+

p B = d2 u / d e 2 = B, with ex = xe, and B, = (1 - x ) B,

+

xBl

.

In case C, z = 1

0 = aU/ae = B,(e - ex)

+

p

+

(1

-

x) q

B = d2 U/de2 = B,

with

ex = (1

-

x) e ,

+

xe,

and

(A1 5 ) B, = (1

-

X ) B3

+

x B , .

( ~ 1 6 ) In case B, with z = d

+

x we get a quadratic equation

for e which is identical t o that given in appendix 2 except that the following definitions a r e changed ('417) B,, = B,

+ xB12

-

N, 6B3, (A23)

N, 6 = NA, x xNX/Nf _('424)

(A181 Q = N , 6e3 - xe, (A25)

B , , = B ,

-

B, (A26)

and although

(A20) l / N x = l / N f

+

l / N d (A27) is defined as before, it is n o w a constant, since N, = l / A f (A21) is constant.

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[2] KALDIS, E. and WACHTER, P., Solid State Commun. 11

(1972) 907.

[3] VARMA, C. M., Rev. Mod. Phys. 48 (1976) 219.

[4] VON MOLNAR, S., PENNEY, T. and HOLTZBERG, F., J. Phy-

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