Concentrated alloys and mixed crystals

In this section we go beyond the dilute limit of impurities or defects that was considered above, and discuss (one-phase) concentrated al-loys and mixed crystals (e.g. NacKi_cCl). The lattice vibrations are assumed to be harmonic, but since there is no translational invariance, the eigenstates are no longer plane waves, and the eigenfrequencies cannot be mapped as sharp dispersion curves &>(q, s). Even if the dispersion curves are strongly modified, or give a completely inade-quate description, there is still a well-defined density of states F(OJ).

In the long-wavelength limit, &>(q, s) = CS0Und(q> s)\q\9 implying that F(co) ~ co² for small co. A strict Debye model, which assumes that the elastic limit is extrapolated to the frequency eo^, may be an equally good approximation in a pure element as in a concentrated alloy or a mixed crystal. In, say, a substitutional alloy with a bcc or fee lattice, F(co) usually retains the general shape with two humps, corresponding to longitudinal and transverse modes. However, the sharp structures in F(co) characteristic of a perfect periodic lattice are smoothed out.

Consider a compound or a solid solution with the composition AcBi_c. Here 0 < c < 1, but it is not required that c or 1 — c is small.

The masses of the constituents are MA and MB. Then the first equalities in eqs. (9.16)—(9.18) hold exactly, within the mass-defect model (i.e.

without allowance for changes in the interatomic forces). Thus,

0 D ( - 3 ; C) = 0£(-3)[(l - c) + cMB/MAV^l/\ (9.22)

#D(0; C) = ^ ( 0 ) [ M_A/ M_B]^{c / 2}, (9.23)

0D(2; c) = # ( 2 ) [ ( 1 - c) + cMA/MB]V². (9.24) These relations can also be written as interpolation formulae between

the Debye temperatures 0£ and 6^ of the pure components:

[ 0 D ( - 3 ; C ) ] -² = (1 - c ) [ ^ ( - 3 ) ] "² + c [ ^ ( - 3 ) ] "², (9.25)

#D(0; C) - [^(0)]^{( 1}-^{c )}[^(0)]^c, (9.26)

0 D ( 2 ; C)]² = (1 - c)[fl£(2)]² + c[9g(2)]². (9.27) Within the mass-defect model (i.e. without force constant changes), the

interpolation formulae are excact. They are also exact if the masses MA

Concentrated alloys and mixed crystals 163

240

220

g

co 200

180

"~~0 0.2 0.4 0.6 0.8 1.0

KCl ^c KBr

Fig. 9.2. The measured Debye temperature 0 D ( - 3 ) = 0^(T = OK) in KBrcCl!_c

(symbols), the interpolation formula (9.25) and an extrapolation from 0D(—3) of pure KCl using the mass-defect model only (eq. (9.22)).

and M_B are equal but the properly averaged interatomic force constants vary as l/f(c) = c/ /B+ ( l - c ) / /B ineq. (9.25), as f(c) = [fB]^c[fA]^l'^c in eq. (9.26) and as f(c) = cfB + (1 - c)fA in eq. (9.27). Thus, eqs.

(9.25)-(9.27) go beyond the mass-defect model and may give a good account of the concentration dependence of the Debye temperatures in many real systems, but there are also exceptions (cf. the two examples below).

Example: interpolation formulae in KBr_cCl\-_c. Figure 9.2 shows the interpolation formula (9.25) for KBrcCli_c, fitted at the two ends of pure compounds, and compared with experimental 6D(—3) from low temperature heat capacity experiments by Karlsson (1970). Also shown is 9D(—3) as extrapolated from pure KCl, using only the mass-defect model (eq. (9.22)). It is seen that the extended mass-defect model (the interpolation formula) gives a good description. This is expected be-cause the bonding (ionic in character) is not very different in KCl and KBr. This should be contrasted with the following example of Nb-Mo alloys.

Example: the Debye temperature #D(—3) in Nb-Mo alloys. Niobium and molybdenum form a solid solution with bcc lattice structure over the entire composition range of Nb_cMoi__c alloys. The atomic masses

1 1 1 KBrc Cl-j .c

extrapolated mass-defect model

interpolation formula

164 Ch. 9. Atomic vibrations in defect lattices 500

Fig. 9.3. The measured Debye temperature 0 D ( - 3 ) in the alloy Nb-Mo. Data points, from White et al. (1978), lie scattered in the shaded region. The lower solid curve is the

interpolation formula (9.25).

of Nb and Mo differ by only 3%, and thus have almost negligible effect on the concentration dependence of the phonon frequencies. In spite of the fact that Nb and Mo are located next to each other in the Periodic Table, there are large variations in the interatomic forces. Figure 9.3 shows experimental results for 0 D ( - 3 ) , from White et al. (1978). The data are scattered in the shaded band. The interpolation formula (9.25), given as a solid curve, fails to give a good account of the variation in

# D ( - 3 ) .

6. Vacancies

The atoms surrounding a vacant site are more loosely bound than those in the bulk and therefore give an increased vibrational entropy. At high temperatures, we have

= *B /

Jo

f

ln(k_BT/ho))AF_Y3LC(a)) dco

= ~h

where we have used the result that ln(k_BT/h) f AF(co) dco = 0, by eq.

(9.2). The physical dimension (the unit) of co in \n(co) in the last part of eq. (9.28) is of no concern since by eq. (9.2), ln(co/coi)AF(co) dco is independent of the choice of the frequency unit co\.

Vacancies 165

We crudely estimate Svac as follows. If a certain vibration mode has the frequency a^uik in the perfect crystal and &>vac when a vacancy has been introduced, the vibrational entropy is changed by k_B ln(o>t>uik/&>vac)-Consider a simple cubic lattice with central nearest-neighbour interac-tions. The 6 atoms adjacent to the vacancy have their force constants reduced by a factor of 2, for vibrations towards the vacant site. Because vibration frequencies vary as [force constant]¹/², we reduce the corre-sponding frequencies by a factor of 1/V2. Then the vibrational excess entropy in the high temperature limit becomes (per vacant site)

Svac ^ ^6kB ln(A/2) = 3*8 In 2 - 2kB. (9.29)

This is, of course, a much too simplified picture, but it gives the right order of magnitude of 5vac. Similar bond-cutting models have been ap-plied to an fee lattice (Stripp and Kirkwood 1954), a simple cubic lattice (Mahanty et al. 1960) and fee Cu (Huntington et al. 1955). They all give SVac ^ ^1.7/:B to ^2.0*B per vacant site. Experimental values of SVac/^B*

compiled by Brudnoy (1976) and Wollenberger (1996) usually lie in the range 1-3, but with a large scatter between different measurements on the same element. The data given in the Landolt-Bornstein tables (Ullmaier 1991) confirm the picture of a large uncertainty in Svac /&B-See also Harding and Stoneham (1981) and Sahni and Jacobs (1982) for similar data in ionic crystals.

An accurate calculation of 5vac must include several features, in ad-dition to the bond-cutting approach. First, the atoms near the vacancy will relax to new equilibrium positions which changes the effective force constants acting on them. Then there is a dilatation of the lattice even far from the vacant site. The correponding shifts can be handled using the Gruneisen model (see, for example, Mott and Gurney 1940, Vineyard and Dienes 1954, Huntington et al. 1955 and §4.4 above). The dilatation term may also be obtained (Huntington et al. 1955) from the macroscopic relations

(dS/dV)T = (dp/dT)v = KTp = CVYG/V. (9.30) If we take the values C_v = 3Nk_B, V = NQ_a, y_G = 1.5 and AV =

Vvac = 0.5£2a, the dilatational term gives 5vac ~ 2&B, per vacancy.

7. Dislocations

The dislocation core has a more open structure than the perfect lattice.

One therefore expects a softening of the atomic vibrations near the core.

Simple estimates (Friedel 1982) show that, for the core of a dislocation, the vibrational entropy S^s\ ~ 0.5&B (or less) per atom in the core. The strain field surrounding a dislocation is of long range and has regions of compression as well as expansion, where the Gruneisen description should be applicable. There seems to be no estimation of the overall effect of dislocations on the vibrational spectrum.

Vibrations of the dislocations as such should also be considered.

When a dislocation line is pinned at its ends, it can vibrate much like a string under tension. Granato (1958) and Ohashi and Ohashi (1980) developed a theory for the contribution of such vibrations to the heat capacity of a solid, and Bevk (1973) performed experiments on copper. The corresponding heat capacity Cdisi, which varies lin-early with the temperature 7\ is exceedingly small compared to the lattice part of the total heat capacity C_p, except at very low tem-peratures. Theory (Granato 1958) and experiments (Bevk 1973) show that CdiSi/Cp ~ 10~³ at T/6D ~ 10"² in heavily cold-worked samples (dislocation density 10¹⁵ m~²).