Point imperfections 1. The mass-defect model

The mass defect is the simplest point imperfection in a vibrating lattice.

One then assumes that the mass of a particular atom in the perfect lattice is altered from M to M', without any change of the force constants. That is the case if an atom is replaced by one of its isotopes, and one therefore also speaks of an isotope defect. The important parameter characterising the impurity is the relative mass difference e\

M-M^f _ _

e = ———. (9.10) M

In the case of a light impurity (M^r < M; s > 0), there may be a localised mode, with a frequency &>i_mp > &>_max> i.e. above the highest frequency of the host lattice. When the impurity is much heavier than the host atoms (M' 3> M; e < 0) there is a pronounced resonance at a frequency COJ which is "embedded" in the quasicontinuous spectrum of the host lattice.

Starting from a general expression for &>; (Kagan and Iosilevskii 1962,1963, Lifshitz 1956, Brout and Visscher 1962, Dawber and Elliott 1963, Mannheim 1968, Dederichs and Zeller 1980), we restrict the dis-cussion to the case of cubic symmetry and one atom per primitive cell.

The impurity-mode frequencies cot are threefold degenerate (equivalent x, y and z directions) and obtained from

Jo ^{Q)J — CO1}

where F{co) and &>max refer to the unperturbed vibrations of the host.

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For a light impurity, coi > &>max. We then expand the integrand in eq.

(9.11) in powers of co/cot and consider s « 1 (i.e. M' <g; M). Keeping only the first two terms in the expansion gives

(Of ^ (1 - sy^l/2co(2) = (M/M^f)^l/2a)(2), (9.12) where co(2) is the unperturbed second-moment frequency (eq. (6.21)).

When the impurity is very heavy (|e| ^> 1; s < 0) there is a res-onance mode a* = coi, where co[ has a small imaginary part. A Debye model for the host lattice, F(co) = 9co²/a>l_>, in eq. (9.11) gives, to lowest order in the limit that coD 3> &>,,

^ = I r ^¹ - '(a>,-M>)0r/2)]. (9.13) We now see explicitly that the heavy impurity gives rise to a frequency

with an imaginary part, i.e. a damped mode. When coi/co^ « 1 we can neglect the imaginary term in eq. (9.13) and treat the resonance as if it were a true eigenstate with a frequency [3|£|]~^1/2

CL>D-4.2. Thermal displacement in the mass-defect model

The theory for the thermal displacement, (u^ef), of impurity atoms is complicated, but in an approximate theory one just scales the displacement of the replaced host atom;

<«def>/<«Lt) = (M/M')^{1 / 2}, T « 0D; (9.14)

(»def)/(«host> = 1. T»9D. (9.15)

Equations (9.14) and (9.15) may be compared with the result in Chapter 7 (§5.2), that (u²) in the high temperature limit does not depend on the atomic mass M, and varies as M^{_ 1 / 2} at 0 K. Calculations by Dawber and Elliott (1963), using a Debye model for the host lattice, showed eq.

(9.15) to be exact and eq. (9.14) to be correct to within about 5%. These authors also give an expression for the velocity of the defect as (v^ef).

Both (ujkf) and (v^ef) can be measured in Mossbauer experiments.

4.3. Debye temperature in the mass-defect model

Consider a crystal with a low concentration, c, of impurities. When their vibrations are described by the mass-defect model, i.e. with all force constants unchanged, the Debye temperatures 0_D(n; c) are related to 6v(n) of the pure host lattice as

0D( - 3 ; c) = 0D(-3)[1 - ceV^l/2 « 0 D ( - 3 ) [ 1 + ce/2], (9.16)

#D(0; C) = 0D(O)[1 - s]~^c/2 « 0D(O)[1 + ce/2], (9.17) 0D(2; c) = ^0D(2)[1 + (ce)/(l - e)]^{1 / 2} « ^0D(2)[1 + ce/2]. (9.18)

These relations easily follow from the fact that 0v(n; c) ~ [Meff(n)]~^1/2, where Afeff(—3) varies as the mass density of the spec-imen (cf. eq. (6.16)), Meff(0) is the logarithmic average of the atomic masses (cf. eq. (6.30)), and Meff(2) follows from £&>² = TrD (Appendix C) as 1/M_eff(2) = (1 - c)/M + c/M^f.

4.4. Force constant changes

The equation (9.11) for the localised or resonance-mode frequencies cot can be generalised to include force constant changes at the impurity (Kagan and Iosilevskii 1962, Mannheim 1968). Work by Tiwari and Agrawal (1973a, b, c) and Tiwari et al. (1981) exemplify theoretical calculations of resonance states, with allowance for both mass and force constant changes.

Let the force constants associated with an impurity atom be changed by a relative amount Af/f (cf. Chapter 4, §10). The elastic-limit Debye temperature &>D(—3) depends predominantly on the shear modulus, as coD(-3) ~ G^1/2 (cf. Chapter 6, §4). The relations in Chaper 4 (§10) for the elastic shear moduli c\\ — cyi and C44 now give, approximately,

£_D( - 3 ; c) « 0_D(-3) ^{1 +} 2c(A///) ^-,1/2

l + A//(3/)J

« 0D( - 3 ) [ l + c ( A / / / ) ] . (9.19)

Then it has been implicitly assumed that the lattice coordinates are unaltered. However, there are two kinds of relaxations in the atomic positions; an overall change of the volume of the specimen and a "local"

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change in the vicinity of an impurity atom. We write the first type of volume change AV as AV/V = cAQa/Qa, where AC2a is the defect formation volume. With a Griineisen parameter approach we get, as a crude estimate for the effect of the overall volume change,

^dilated = #undilated[l ~ CyG(AQJ fta)]. (9.20)

With the realistic values YG = 1.5 and A£2a/£2a = 0.2, we see that this type of relaxation effect may be significant (cf. Tiwari et al. 1981, on Cu-Sn).

4.5. Heat capacity

We now turn to the relative change, A Char/ Char, in the vibrational heat capacity associated with a low concentration of impurity atoms. Con-sider first heavy impurities. Very roughly, one obtains a contribution from the 3Nc localised or resonance modes, superimposed on a change in the heat capacity of the 3N(l — c) extended modes. Heavy impu-rities lead to a A Char/Char that is peaked at low temperatures (below

T = [3|£|r^{1 / 2}#D, cf. eqs. (9.13) and (9.3)). This effect has been ob-served, e.g. for Pb in Mg (Panova and Samoilov 1965, Cape et al. 1966).

To achieve a quantitative account of such measurements one must go beyond our simple idea of a sharp resonance mode (e.g. Tiwari and Agrawal 1973a, b, c).

The effect AChar of a light impurity will be difficult to see directly in heat capacity measurements since the heat capacity of the host has already reached its (large) classical value at the temperatures when the localised modes of the impurity start to be significantly excited.

Although it is very difficult to obtain a precise expression for AChar>

there is an integral relation which links AChar to the excess entropy AShar- By using (7.24) we can write for the high temperature limit

Shar(00):

AShar(00) = / d7

Jo l

= 3NkBln[9D(0)/9D(0',c)l (9.21)

where #D(0) refers to the logarithmically averaged phonon frequencies of the host material.