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

11.6 Collective Excitations in a Crystal with Heavy Impurities 273

Let us analyze (11.6.1) or (11.6.10) intending, first, to consider eigenvibrations such as plane waves, i. e., collective excitations of a crystal with point defects. These vibrations are characterized by the wave vector k and the frequency ω. The dispersion law ω = ω(k) connecting the real ω and k is the primary characteristic of elementary excitations and determined by the Green function poles of a crystal in the (ε, k)-representation. Therefore, we first discuss the problem of the poles of the function (11.6.10), i. e., of the roots of the equations

We come back to the above procedure of regularizing the Green function of stationary vibrations of an ideal lattice and note that the function Π(ε) is complex.

Hence it follows that for a real wave vector k the solutions to the dispersion equation (11.6.11) for ω will be complex. The presence of an imaginary part of the vibration frequency is evidence that the wave with fixed k gets damped in time. Thus, a plane wave of displacements in a crystal with impurities has the form

where the damping time τ is determined by the imaginary part of the function Π(ε). It is natural that separation of collective excitations of a crystal such as (11.6.12) is

Let the condition ωτ 1 be satisfied, i. e., the damping is weak. Writing the solution to (11.6.11) in the form

ε = (ω − i/τ)2 ω2 − 2iω/τ ω2 − 2iω0(k)/τ,

it is then easy to calculate ω(k) and τ(k) in the main approximation in ωτ.

The dependence ω = ω(k) (see the problem) obtained in this way plays the role of a dispersion law of crystal vibrations with point impurities, but the corresponding excitations (11.6.12) prove to be damped, i. e., “living” for a finite time. The lifetime of these excitations τ can be directly associated with a correction δg(ε) for the vibration density (11.5.2) near a quasi-local frequency

It is clear that τ has a minimum at ω = ωq. A small lifetime of the collective excitation with ω = ωq can be explained easily. The energy of a “homogeneous” plane wave (11.6.12) is spent to excite the continuous spectrum vibrations whose frequencies are close to a quasi-local one. However, the coordinate dependence of quasilocal vibrations differs from (11.6.12). Therefore, the collective vibration is damped.

For the maximum of (11.6.13) we have the estimate

274 11 Localization of Vibrations

We introduce the concentration c0 at which the average distance between impurities has the order of magnitude of a characteristic wavelength of a single quasi-local vibration λ 2πs/ωq. To an order of magnitude,

Consequently, the estimate (11.6.14) can be rewritten as ωτ c0 /c; hence it follows that the condition ωτ 1 is satisfied for c c0. Thus at small enough concentrations of impurity atoms (c c0), ordinary vibrational excitations (such as plane waves) are weakly damped at all frequencies.

11.7

Possible Rearrangement of the Spectrum of Long-Wave Crystal Vibrations

Now consider the dynamic properties of a crystal where the concentration of heavy impurities is not restricted by the inequality c c0 (but the condition c 1 remains). If c >c0 then in the resonance region ωτ <1, and the concept of collective excitations (11.6.12) with frequencies close to ωq is physically meaningless. The wave (11.6.12) is damped practically in one period of vibrations.

In this frequency region, for c0 <c 1 the spectrum of crystal eigenvibrations can be characterized by the quantity Im G(ε, k) as a function of ε and k, because it can be measured experimentally. We consider Im G(ε, k) as a function of k with a given ε (this is typical for ordinary optical experiments) and examine how the position of the maximum of this function changes depending on the value of ω = √ε. The maxima we are interested in are in the space ω, k on hypersurfaces whose points are given by a straightforward condition

We restrict ourselves to the long-wave isotropic approximation when ω0 (k) = s0 k. As a result of simple calculations, we get the following frequency dependence of the modulus of a wave vector providing the maximum Im G:

The plot of k = k(ω) (Fig. 11.6) can be considered as the dependence of the wave vector on the frequency of the crystal eigenvibrations in this case.

The function k = k(ω) has extrema kmax and kmin between which the “anomalous dispersion” region is situated. The difference in heights of the maximum and minimum in Fig. 11.6 is equal, in order of magnitude, to

11.7 Possible Rearrangement of the Spectrum of Long-Wave Crystal Vibrations 275

With increasing concentration the minimum in Fig. 11.6 decreases and, at concentrations c c0, kmin may reach zero. For long-wave vibrations, one should then expect phenomena such as total internal reflection in optics. In fact, for c c0, in addition to ω = 0, there appear two more frequency values corresponding to k = 0. These frequencies lying somewhat higher than ωq limit the range of frequencies at which the crystal has no collective excitations described by the wave vector.

Therefore, in the case of large concentrations of impurities with pronounced quasilocal frequencies the spectra of long-wave crystal vibrations change significantly.

Fig. 11.6 The dependence k = k(ω) that provides the maximum of

Im G(ε, k).

We consider Im G(ε, k) as a function of ε at fixed k (just this dependence is generally studied in neutron experiments). The maximum value of this function is determined by the condition (11.7.1), so that the frequency range where vibrations such as plane waves are absent is wide enough and satisfies the condition

The inequality (11.7.3) enables us to omit Γ in the real part of the function Π(ω2 ) and the condition (11.7.1) will be replaced by a similar one:

It follows from (11.7.4) that Im G as a function of ε at any fixed k has two maxima instead of one in an ideal crystal. The dependence of the frequencies on k providing the maximum Im G (Fig. 11.7a) resembles a typical diagram of frequency splitting near the resonance point, i. e., in the vicinity of the intersection of the dispersion curves ω = s0 k and ω = ωq = const (Fig. 11.7b). The choice of the parameters ∆m/m, ωk , and c in the plot is matched to the experiment of Zinken et al. (1977). A hypothetical degeneration of frequencies arising in Fig. 11.7b is removed and the

12

Localization of Vibrations Near Extended Defects

12.1

Crystal Vibrations with 1D Local Inhomogeneity

In any macroscopic specimen of a real crystal there are dislocations. A dislocation, being a quasi-one-dimensional structure, breaks the lattice regularity only in a small region near a certain line – its axis. Vibrations localized near the dislocation have the form of waves running along the dislocation line.

We are interested in small atomic vibrations near new equilibrium positions and thus in a zero approximation the dislocation axis may be considered to be fixed. We assume that the dislocation is a straight-line and its axis is perpendicular to the crystal symmetry plane. We direct the z-axis along the dislocation line and denote by ξ a 2D radius vector in the plane xOy: r = (ξ, z), ξ = (x, y). Assume also that the dislocation does not change the substance density along its axis. The perturbation introduced by a dislocation is then connected with a local change in the matrix of atomic force constants. To analyze the long-wave crystal vibrations (λ a), this perturbation is assumed to be concentrated on the dislocation axis. Using a scalar model we consider the displacement as a continuous function of the coordinates: u = u(ξ, z). The equation of long-wave lattice vibrations near the linear singularity can be represented as

where δ(ξ) is a two-dimensional δ-function (δ(ξ) = δ(x)δ(y)) and the summation over the lattice sites can be replaced by the integration

The Crystal Lattice: Phonons, Solitons, Dislocations, Superlattices, Second Edition. Arnold M. Kosevich Copyright c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ISBN: 3-527-40508-9

12.1 Crystal Vibrations with 1D Local Inhomogeneity 281

If we set ξ = 0 in (12.1.11), we obtain an equation for the vibration frequencies

The upper limit of integration in (12.1.12) can be estimated as κ0 1/a. The fact that the integration limit in (12.1.12) is determined by the order of magnitude only, and has the character of some “cut off” parameter, is connected with a model assumption of the point-like character of a perturbation in (12.1.5). The assumption (12.1.1) of a delta-like localization of the perturbation on the dislocation axis as well as the dispersion law (12.1.10) are valid for the long-wave vibrations only (aκ 1).

At the same time the integration in (12.1.11), (12.1.12) should be extended to the whole interval of a continuous spectrum of frequencies. Since the integrand in (12.1.11) does not exhibit a decrease at infinity necessary for the integral to converge, we have to take into account the natural limit of integration over the quasi-wave vector (aκ0 π).

Simplifying (12.1.12) we obtain

by omitting small terms of the order of magnitude

As in Chapter 11, (12.1.13) has a solution for eigenfrequency squares ω2 with a definite sign of W0 only, namely, for W0 > 0, so that necessarily ω2 < s2 k2. If the condition W0 > 0 is satisfied, (12.1.13) always has a solution (Lifshits and Kosevich, 1965)

We recall that in (12.1.14), W0 > 0 and s20 κ02 ωD.

The frequencies (12.1.14) have an exponential dependence on the perturbation intensity, i. e., on Uk that is characteristic for two-dimensional problems. Their definition has no critical value of the perturbation intensity at which the local vibration frequency starts splitting off and that is typical for point defects. The existence of the vibrations localized near a dislocation requires a definite sign of the perturbation (W0 > 0), and the corresponding frequency is always separated by a certain finite gap δω the origin of the spectrum of bulk crystal vibrations

282 12 Localization of Vibrations Near Extended Defects

We have considered elastic waves traveling along the dislocation. Their existence is due to changes in the elastic moduli in the dislocation core, but the vibrations localized near the chain of impurity atoms also have frequencies such as (12.1.14). Such a chain is also a linear defect. The long-wave vibrations near this defect are described by the same equation (12.1.1), but with a different perturbation matrix

It is clear that the problem of crystal vibrations with the perturbation (12.1.15) is reduced to finding a solution to the two-dimensional equation (12.1.5) in which the replacement Uk → U0 is necessary. Thus, the equation for local frequencies remains the same, replacing k2W0 → (∆m/m) ω2,

For ∆m > 0, (12.1.16) has the following solution in a frequency region of the spectrum (ω2 (m∆m) ω2D),

It is clear that (12.1.17) is not qualitatively different from (12.1.14).

Let us note that the frequencies (12.1.14), (12.1.17) do correspond to vibrations localized near the linear defect. According to (12.1.11), we perform the integration

×cos(κρ cos φ)dφ,

where ρ2 = x2 + y2 and the upper limit of integration over κ is shifted to infinity since for ρ = 0 the integral in (12.1.18) converges.

The integrals on the r.h.s. of (12.1.18) can be taken from any reference book; therefore, we give only the final expressions for the vibration amplitude

where K0 (x) is a cylindrical zero-order Hankel function (of an imaginary argument); κ is the inverse radius of localization of vibrations near the dislocation line