Kosevich A.M. The crystal lattice (2ed., Wiley, 2005)(ISBN 3527405089)(342s)_PSa_

262 11 Localization of Vibrations

If all Tαβ are determined from this set of linear algebraic equations, the Green

Equations (11.3.7) are simple, but writing their solution for an arbitrary number of defects is rather cumbersome, and their quantitative analysis is complicated. Therefore we restrict ourselves to a qualitative characterization of the Green function (11.3.8).

If the distance between all neighboring defects rαβ greatly exceeds the dimension l of the localization region of vibrations near an isolated defect, there are values of the parameter ε such that U0 Gε0(nα − nβ) D(ε) for α = β. In the relation (11.3.7), we can then omit the terms with γ = α. Subsequently, it is simple to solve (11.3.7):

The solution obtained does not, in fact, differ from (11.3.2) and the Green function is written analogously (11.3.4)

If the defects are distributed randomly but uniformly, (11.3.10) yields an approximate expression for the Green function, which is valid under the assumption that the average distance between the defects greatly exceed the length l. The expression (11.3.10) is inapplicable near the boundaries of the continuous spectrum of an ideal crystal and in the vicinity of a local frequency.

Coming back to an analysis of Fig. 11.4a we see that (11.3.10) takes into account the excitation transfer from site n to site n through a simple use of all the defects. This situation resembles a kinematic X-ray scattering theory taking into account only simple X-ray scattering by each of the crystal atoms, but even with two defects available (at sites n1 and n2) there exists an excitation transfer channel (n → n1 → n2 → n) disregarded in (11.3.10) and shown in Fig. 11.4b.

Assuming the contribution of two defect processes to be small, it is easy to describe this. We find a correction to the solution (11.3.9) by setting

A small correction tαβ is easily found by substituting (11.3.11) into (11.3.7)

266 11 Localization of Vibrations

We now return to the initial formula (11.4.1) as applied to a scalar model

In a linear approximation with respect to the concentration c, the expression (11.4.6) determines the variation of the density of vibrational states due to point defects. The applicability of the final formula (11.4.6) is not limited to a scalar model. The presence of three vibrational polarizations in a cubic crystal does not affect (11.4.6). It is sufficient to take account of a simple replacement G0(0) → (1/3)G0ll (0) in the definition of D(ε). For a monatomic lattice having no cubic symmetry D(ε) is expressed through the determinant appearing in (11.1.14):

If the value of ε lies outside the continuous spectral band for an ideal crystal, then g0(ε) = 0, and Gε0(0) is a real function of the real variable ε:

lim Gε0−iγ(0) = Gε0(0) ≡ Re{Gε0(0)}.

γ→0

However, in this case the density of vibrations for a crystal with point defects can be nonzero outside the continuous spectrum only at the points where the argument of the logarithm in (11.4.6) vanishes: D(ε) = 0, i. e., for the values of ε coincident with the squares of local frequencies. We represent in close proximity of the point ε = εd, the function D(ε − iγ) as

D(ε − iγ) = (ε − εd − iγ) dD(εd) , dεd

remembering that εD is the solution to the equation D(ε) = 0. We then have from (11.4.6)

However, this limit is equivalent to a δ-function. Therefore, in a corresponding frequency range the density of vibrational states has a δ-like character

showing that the local frequency is discrete.

The result contained in (11.4.8) has already been discussed, so that we shall now pass to analyzing the density of “defect” crystal vibrations in the continuous spectrum range of an ideal lattice (ε1 < ε < ε2).

11.5 Quasi-Local Vibrations 267

In writing (11.2.24), we have taken into account that in the continuous spectrum region

where P.V. means the principal value of the integral.

In order to separate real and imaginary parts of the derivative (d/dε)Gε0 (0) it is sufficient to differentiate (11.4.9) with respect to ε for all ε except for those getting into near the edges of the continuous spectrum interval and the van Hove specific point in which the function g0(ε) loses its analyticity. Thus

lim D(ε − iγ) = R(ε) − iπU0 g0 (ε),

γ→+0

The first of these relations is consistent with (11.2.25).

Performing the operations indicated in (11.4.6), it is easy to obtain

The expression (11.4.11) describes the deformation of the continuous spectrum of frequency squares under the influence of point defects. It has nonphysical singularities at the points where the function g0(ε) loses its analyticity and has root singularities. Thus, (11.4.11) is applicable only away from the vicinity of these points. The appearance of singularities in (11.4.11) results from the fact that an approximation that is linear in concentration is not sufficient for examining the spectral deformation near its singular points.

11.5

Quasi-Local Vibrations

We have mentioned that (11.4.11) has nonphysical singularities, because the linear approximation with respect to the defect concentration is inapplicable when describing δg(ε) in the vicinity of certain values of ε. However, the function δg(ε) may have another singularity having a physical meaning. Thus, with a defect-isotope present in the crystal, the real part D(ε) can vanish for a certain value of ε = εq lying inside the continuous spectrum:

Generally, at ε = εq the function δg(ε) has no singularities in a formal mathematical sense, but if the point ε = εq is near the edges of the continuous spectrum band where

268 11 Localization of Vibrations

the density of states δg0(ε) is very small, then the denominator in (11.4.11) becomes anomalously small in the vicinity of εq and the function δg(ε) increases.

This situation corresponds to the appearance of a quasi-stationary vibration with

ε = εq.

To determine the form of the resonance peak of the function δg(ε), we use an expansion of the type (11.4.7): R(ε) = (ε − εq)dR(εq)/dεq ) and restricting ourselves to a small region near the point considered we set ε = εq in the numerator of the r.h.s. of (11.4.11)

The formula (11.5.2) has the characteristic form of an expression describing the frequency distribution of a quasi-stationary state of the system. The parameter Γq gives the half-width of a corresponding resonance curve and may be associated with the imaginary part of ε in the conventional notation of (11.2.7). If Γq is small, the expression (11.5.2) results in a sharp peak in the vibration density. The frequency ωq = √εq is then called a quasi-local frequency and the vibration corresponding to it a quasi-local vibration.

Analysis of (11.5.2) is of most interest in the case when the quasi-local vibration frequencies are near the long-wave edge of an acoustic spectrum (such vibrations will be shown to be generated by heavy isotopes with ∆m m). If ε ωD2 , where ωD is the Debye frequency, the main part of the expansion of R(ε) in powers of ε is obtained from (11.2.26)

We note that negativeness of the derivative R (ε) and positiveness of the parameter Γq for ∆m > 0 follow from (11.5.3).

Substituting (11.5.3) into (11.5.1), we see that for a large enough value of ∆m/m

It follows from (11.5.4) that with increasing ∆m/m, the quasi-local frequency position approaches the long-wave edge of the vibration of the continuous spectrum (εq → 0). Simultaneously, the peak in the spectral density narrows. Indeed, since at small ε we may use the estimate (11.2.26) for g0 (ε), the resonance peak half-width is

11.5 Quasi-Local Vibrations 269

Thus, the relative half-width of the Lorentz curve (11.5.2) for heavy isotopes is determined by

confirming the resonance peak narrowing with increasing ∆m/m. Hence, with increasing ∆m/m the resonance peak acquires δ-function character. The possible appearance of a sharp peak in the function δg(ε) in the region of low crystal vibrational frequencies with heavy impurity isotopes was first noted by Kogan and Iosilevskii (1962), and Brout and Visher (1962).

It is natural that the Lorentz-type peak (11.5.2) also arises in a plot of the frequency spectrum

where ωq2 = εq. A typical plot of the function ν(ω) = ν0(ω) + δν(ω) for ω ωD is shown in Fig. 11.5.

Fig. 11.5 The frequency spectrum near a quasi-local frequency (the curve 1: ν0 (ω) = constω2).

It is easy to estimate the impurity concentration c at which the peak height (Fig. 11.5) calculated by (11.5.6) is comparable with the frequency distribution function ν0 (ω) of an ideal crystal at a quasi-local frequency ω = ωq. From the relation

we get for the concentration to be estimated:

270 11 Localization of Vibrations

We substitute in (11.5.7) the estimates (11.5.4), (11.5.5) to obtain

Since, by assumption, the parameter m/∆m is very small, the quasi-local vibration density peak may already be equal in order of magnitude to the vibration density of an ideal crystal at small concentrations of heavy impurities (c c 1). However, the derivation of (11.5.2), (11.5.6) and the calculation scheme were based on using the linear approximation (c 1) with respect to concentrations for which additions (proportional to c) to the quantities studied are assumed to be small. Since for c c 1 the last assumption is not justified, it is necessary to analyze more consistently the influence of heavy isotopic impurities on the spectrum of crystal vibrations even at their smallest concentrations.

Quasi-local vibrations affect the thermodynamic and kinetic properties of a crystal. The singularities in the amplitudes of elastic wave scattering near quasi-local frequencies ωq lead to resonance anomalies in the ultrasound absorption. Coherent neutron scattering (with emission or absorption of a simple phonon) in the presence of the corresponding impurity in a crystal has certain specific features at frequencies of emitted (or absorption) phonons close to ωq. The differential cross section of this neutron scattering has an additional characteristic factor of the type (11.5.6) increasing anomalously near a quasi-local frequency. It is natural that similar peculiarities should be observed in the infrared absorption spectrum of crystals with impurities that

produce quasi-local vibration.

Finally, much interest has been generated in quasi-local vibrations, related to the Mössbauer effect for the nuclei of impurity atoms. The Mössbauer phenomenon in impurities is connected with the specific relation between the momentum and the energy are transferred to an impurity nucleus. This relation is determined by those possible motions in which an impurity atom is capable of participating, i. e., by the expansion of the impurity displacement vector with respect to normal modes of the defect crystal. Among the vibration modes there is a large group of vibrations with very close frequencies (quasi-stationary wave packets of these oscillations cause the quasi-local vibrations). This leads to the fact that in expanding the displacement vector of an impurity atom with respect to normal vibrations the relative contribution of a quasi-local vibration greatly exceeds the relative contribution of ordinary crystal vibration modes with frequencies of a continuous spectrum. Thus, quasi-local crystal vibrations should manifest themselves in the spectrum of phonon transitions of the Mössbauer effect.

It should be noted that in phenomena such as the Mössbauer effect or neutron scattering the contribution of true local vibrations is quite pronounced and larger than the contribution of the quasi-local vibrations (reduced to the same frequency). However, in the range of low frequencies (considered, for instance, while studying the influence of relatively heavy impurities on the crystal properties) there exist quasi-local vibrations only: local vibrations cannot appear near the low-frequency boundary of an acoustic spectrum.

11.6 Collective Excitations in a Crystal with Heavy Impurities 271

11.6

Collective Excitations in a Crystal with Heavy Impurities

The Green tensor G0 of an ideal crystal in the site representation is a function of the difference in the numbers of sites n and n . It follows from (11.3.18) that the averaged matrix G for a crystal with randomly distributed point defects is also a function of this difference. This reflects the macroscopic homogeneity of a crystal with defects that appear as a result of averaging over random distributions of impurity atoms.

For the Green function of a homogeneous system the long-wave k-representation is introduced in the usual way. Thus, we just write the averaged Green function (11.3.18) of a crystal with point defects in (ε, k)-representation.

1

G(ε, k) = . (11.6.1)

ε − ω02(k) − c ∏(ε)

We first show that the validity of (11.6.1) is not really restricted by the linear approximation with respect to c. We focus on a scalar model and assume that the defectisotope introduces a point perturbation (11.3.5). Discussing the Green function of the defect crystal, it is necessary to find the matrix (11.3.6) as a solution to (11.3.7). In studying the long-wave vibrations when the distances rα − rβ a/c1/3 give the main contribution, the quantity

U0 ∑ G0(nα − nγ)Tγβ,

γ=α

on the l.h.s. of (11.3.7) at fixed α is determined by the total contribution from a large number of terms. To a first approximation, this contribution can be regarded as being averaged over a large number of impurities and be calculated using (11.3.14). However, averaging transforms a crystal into a homogeneous macroscopic medium for which

where rα is the α-th impurity coordinate that can now be considered to be varying continuously.

Substituting (11.6.2) into (11.3.7) and replacing the Kronecker delta symbol δαβ on the r.h.s. by a δ-function normalized in the appropriate way we obtain

We introduce the Fourier transformation for the function T(x)

and rewrite (11.6.3) in k-representation