[Modern Problems in Condensed Matter Sciences] Elastic Strain Fields and Dislocation Mobility Volume...

CHAPTER 11

The Phonon-Dislocation Interaction and its Role in Dislocation Dragging and

Thermal Resistivity V.l. ALSHITS

Institute of Crystallography Academy of Sciences of the USSR

Leninsky Pr., 59, Moscow, 117333, USSR

Elastic Strain Fields and Dislocation Mobility Edited by

Elsevier Science Publishers B.V., 1992 V.L. Indenbom and J. Lothe

625

Contents 1. Introduction * 628 2. The Hamiltonian of the phonon subsystem in a crystal with a dislocation 629 3. The theory of the flutter mechanism 635

3.1. Some preliminary remarks 635 3.2. The one-particle phonon Green function 636 3.3. The cross-section of phonon scattering by dislocations 638 3.4. The contribution of the flutter effect to dislocation dragging 642

4. The non-linear mechanism of phonon scattering 645 4.1. Statement of the problem 645 4.2. The cross-section of phonon scattering by a dislocation 646 4.3. The contribution of the non-linear mechanism to dislocation drag 647 4.4. The drag of a straight dislocation 649 4.5. The drag of a dislocation loop 650 4.6. Comparison of contributions from the flutter mechanism and the phonon wind 651

5. The relative roles of phonon scattering and relaxation processes in dislocation dragging 652 5.1. Statement of the problem 652 5.2. The temperature retarded Green function and the kinetic equation for phonons 653 5.3. The solution of the kinetic equation in the long-wavelength region qlph < 1 . . . 657 5.4. Phonon viscosity and thermoelastic damping 660 5.5. The kinetic equation in the region qlph > 1 664 5.6. Discussion of phonon-wind estimates 667 5.7. Relaxation of 'slow' phonons 670 5.8. The role of optical phonons; the hierarchy of anharmonic mechanisms in

dislocation drag 674 6. Dynamic dragging of dislocations in the Peierls relief 675

6.1. Raman scattering of phonons by dislocations 675 6.2. Radiation dragging of dislocations 677

7. Electron drag of dislocations in metals 679 8. Analysis of experimental data 684

8.1. The influence of dislocations on thermal conductivity; flutter effect or non-linear mechanism? 684

626

The phonon-dislocation interaction 627

8.2. The temperature dependence of the dislocation damping constant 686 8.3. The influence of impurities and other defects 688 8.4. Electron manifestations in the plasticity of metals 689

9. Conclusion 690 References 691

1. Introduction

Depending on the level of stress applied to the crystal, the resistance for dislocation motion and, accordingly, the kinetics of plastic deformation are determined by the competition between thermal fluctuations and dynamic dissipative processes. At low stresses, the dislocation mobility is limited by various potential barriers, which are always present in crystals (Peierls relief, elastic fields of different defects of the lattice). The higher the stress level is, the broader is the spectrum of obstacles that can be overcome without thermal activation and the shorter is the average time of being held up by the other barriers. With increasing stresses, the dislocation motion becomes more and more continuous and at some critical stress level, it transforms into 'overbarrier' motion. The dislocation drag attains now a viscous character and becomes determined mainly by energy dissipation in the system of elementary excitations in the crystal (phonons, electrons, magnons, etc.).

The viscous drag of dislocations manifests itself directly in the amplitude-independent internal friction which is due to damping of dislocation segments oscillating between immovable pinning points in the ultrasound field. Moreover, there are experimental and theoretical indications that the probability of thermally activated surmounting of barriers, such as involved in amplitude dependent internal friction and in thermally activated mobility in general, is to some extent also connected with the level of purely viscous dissipation and in fact roughly inversely proportional to the viscous drag coefficient B.

The modern level of understanding of the relative roles of the different mechanisms of dislocation drag is high enough that we can be confident of our theoretical interpretation of accumulated experimental data and may design new experiments with clear physical purpose. The progress in this field is due to theoretical and experimental contributions from many investigators. Errors and delusions have been overcome by joint efforts. The number of references at the end of this chapter gives some idea about the activity in the dislocation dynamics field during the last decades. One can follow the evolution of ideas in this field by reading consecutively review articles (Indenbom and Orlov 1962, Lothe 1962, Gilman and Johnston 1962, Dorn et al. 1965, Seeger and Schiller 1966, Nabarro 1967, Indenbom and Orlov 1968, Gilman 1969, Christian and Vitek 1970, Kaganov et al. 1973, Alshits and Indenbom 1975a, b, Ninomiya 1975, 1981, Galligan 1982, Hirth and Lothe 1982, Anderson 1983, Alshits and Indenbom 1986) in the order they appeared.

As a rule, in the dynamic drag of dislocations a predominant role is played by dissipative processes in the phonon subsystem. Phonon scattering on dislocations also gives a deformation component to thermal conductivity, which makes the problem of the phonon-dislocation interaction even more important. Accordingly, in this chapter the main attention will be concentrated on the phonon component of the energy dissipation.

628


The aim of this chapter is not so much to give a review of the literature as to present a consistent theory of the phonon-dislocation interaction as a mechanism for energy dissipation. We will strive for a unified presentation with all results derived from one common Hamiltonian.

2. The Hamiltonian of the phonon subsystem in a crystal with a dislocation

In a crystal with a dislocation the thermal displacements of the lattice, ue, are superimposed on the displacement field, wd, of the dislocation vibrating around its equilibrium form (fig. 1). One may consider the thermal vibrations of the dislocation line to be confined mainly to the glide plane, since dislocations are usually much more rigid with respect to displacements in the perpendicular direction. When expressing the vector of the total displacements of the medium,

U = ul + Ua (1) it is convenient to include the oscillating part of the dislocation field in the lattice term, defining ul relative to the displacement field of a 'smoothed-out' dislocation of equilibrium form. Let the dislocation apart from thermal oscillations be at rest. Then, the value of ud is independent of time and the total kinetic energy of the vibrating crystal with a dislocation in it is determined by ul only;

dVpl^{r,t)V (2)

Here, p is the crystal density and the integral is over the volume of the crystal. Below, for simplicity, we shall assume a cube of unit volume. Such an assumption does not reduce the generality of the final results, which do not contain external dimensional effects.

Fig. 1. Thermal oscillations of the dislocation line in the XZ glide plane.

630 V.l. Alshits

The distortions uij9 determining the elastic energy of the medium (retaining anharmonic terms to the lowest order),

£.1 = dVF{u) = \ dViCijuUijUu + ÂiÛijUuU^), (3)

are not simply the sum of lattice distortions u{j = 9wf/9x7· and dislocation distortions ufj. Taking our way of describing the total displacements, given in eq. (1), into account, we see that it is necessary to subtract an intrinsic distortion ufj corresponding to the momentary form of the vibrating dislocation,

uu(r9 t) = u{j(r91) + ufj(r) - ug(r, t). (4)

The intrinsic distortion ufj is non-vanishing only on the surface S (which is shaded in fig. 1), stretched out between the momentary form of the dislocation line and the 'smoothed-ouf dislocation. It is defined as

ufj(r, t) = b,njS(C), (5)

where b is the Burgers vector of the dislocation, ζ is a coordinate referring to an axis normal to S and with origin at the surface S, and δ(ζ) is the Dirac delta-function. For example, if the 'smoothed-ouf dislocation is a straight line lying in the XZ glide plane along the z-axis (fig. 2), then ufj equals

ufj(r91) = btnjSMMx) - η{χ - ξ(ζ9 ί)]}. (6)

Here, ^(z, ί) is the deviation of the dislocation line from its equilibrium form and η(χ) is the Heaviside step-function

η(χ) = 1 , x ^ 0,

= 0, x < 0.

%(&)

Fig. 2. Straight dislocation oscillating in the glide plane.


When substituting eq. (4) into eq. (3) it is permissible to neglect contributions from the local field ufj in comparison with contributions from the long-range field ufj in anharmonic terms. However, one should retain ufj contributions in quadratic terms, since here it turns out that the static-field terms ufj disappear from dislocation-lattice cross terms so that, in a harmonic crystal, the phonon-dislocation interaction (flutter effect) will be exclusively connected with the intrinsic distortion field w?(r, ί).

Thus, as explained above, substitution of eq. (4) into eq. (3) gives

£el = êl + £el + dKC y wug(Ki-4/- t«i)

+ i dVA$2HuiJuilU*m + dVuijafj.

Here,

£el = dVF{u{j)9 7d - dVF(ufj),

and afj is the stress field of a static dislocation,

d _ VFyUjj) _ c d , l A J l n j . d d Gij ~ pj d — îjklUkl T- 2îkmUklUmn' CUij

(7)

(8)

(9)

It is simple to show that the last term in eq. (7) vanishes. Indeed, integrating by parts, one has

[dVuijaf, = (dSjuiafj - (dVuiofu. (10)

The first term on the right-hand side of eq. (10) is equal to zero because of the condition of the free surface, σ^η, = 0. The second term also vanishes since for the equilibrium dislocation cfjj = 0 throughout the body, including the dislocation line.

Adding up eqs. (2) and (7) for the kinetic and the elastic energy and omitting the constant term £{j„ one obtains the expression for the Hamiltonian governing the thermal vibrations of a crystal with a dislocation,

/ / = H0 + Hpp + Hfl + Hw. (11)

Here, H0 is a harmonic term, describing a subsystem of non-interacting phonons,

Ho — 2 dVlp{u\)2 + Cijmnu{jU^; (12)

632 V.l. Alshits

Hpp is an anharmonic term describing phonon-phonon interaction,

Hpp = ±[dVAiWiju{iuiy, (13)

Hn is a Hamiltonian for the phonon-dislocation interaction, which describes the flutter effect, i.e., processes of reradiation of phonons by dislocations vibrating in a field of thermal lattice oscillations,

Hn dVCijklu?jtiu0kl - 4 , - «£); (14)

and Hw is a Hamiltonian of interaction describing phonon scattering by dislocations (phonon "wind") via elastic non-linearity (anharmonicity),

dVAJlluljuijui,. (15)

Strictly speaking, in the derivation of the above expressions the anharmonicity should be treated within the framework of the general theory of non-linear elasticity (Kröner and Seeger 1959, Kröner 1960). We shall not embark on such a cumbersome procedure here. We only note that the general theory (Eckhardt and Wasserbäch 1978, Kotowski and Alshits 1983) leads to the same expressions, except that the tensors A{l

kn

m in eqs. (13) and (15) are replaced by

Aikm = A i k m + Cijlnokm + Ckljnoim + Cmnjidik9 (16)

and

Aikm = Aikm — (îjkl^mn + ^mnjlîk + îjln^km

+ CijknVlm + Ckljnoim + Cklin°jm> U ')

respectively, where öu is the Kronecker delta. Apart from that, one should multiply the integrand in eq. (2) by (1 — u^\ renormalizing the crystal density. Later on, it will be assumed that such a renormalization of eqs. (2), (13) and (15) has been performed. The isotropic representation of the tensors Äjl

km and Aj£m can be found in Bross (1962).

Let us now transform the Hamiltonian H to the canonical form convenient for carrying out a second quantization. As usual, first one should expand the displacement vector ul into Fourier series:

«i(î)=4=Zeêi*r' (18)

where a = (k, s) is a combination of the wave vector k and the polarization index s, which indicates the sth wave branch whose dispersion law is ωα = cos(k). Qa is the generalized coordinate and la is the polarization vector of the partial


wave a. Analogously, the Fourier expansions of ufj(r) and w? (r, t) in eq. (6) are

4-w = Z 4 e i ^ ' (19)

<(Μ)ΛΛΣ^ Γ d/e-^1-^^'^0 3 . (20) k J-oo l^x

As will be shown, the fluttering mechanism is essential only in the low-temperature range, when the long wavelength terms in eq. (20) dominate. Thus, we may limit ourselves to a calculation where the integrand in eq. (20) is replaced by the first non-vanishing term in an expansion in terms of the parameter /cx£(z', t). In this approximation, eq. (20) becomes

«&('. t) = fein,X4,(i)ei*·', (21) k

where ξΙί are coefficients in the Fourier expansion

ξ(ζ, t) = Σ ^tWkz- (22) k

Substituting eqs. (18), (19) and (21) into eqs. (13)—(16), one obtains

ίίο = ΐ Σ ( β ? δ α + ω«0?0«); (23)

a,

#PP = - ^7372 Σ ^ l" lßk lymKjkßi kyn Qa Qß Qy öka + kß + krol (24) <x,P,y

H™ = Y Σ (^smKjkpt - PMaMßöisSmn)lJ%ulnQaQ$öka_kß + qi0; (25)

***=γνϊΣ^&Λΐ + *?ö?i*j+^Σ®ζ* £ ΐφ«/ω«ι2; (26)

wnere

= 0 Ä: Φ 0* lîjpq°injl*PKq· \Δ')

The dynamic amplitudes £k are not independent variables and they may be expressed in terms of the generalized coordinates Qa by means of the condition of minimum energy,

kz = k fcz = 0

8£? ρ'/2-Σ*.β« + ^

Γ\Ί-Ι 1 "Z ."* 1 "- "

Ϊ = ^ Σ Φ « Ο « + & - Σ ΐ φ * / ω ° ι 2 = ° » (28)

634 V.l. Alshits

which gives kz = k

Σ φ .β . ξ ^ - ρ 1 1 2 ^ · (29)

α

Accordingly, instead of eq. (26), one has

* . ß

where kz = 0

κ-ι= Σ I*«/««!2· (3i) a

Before proceeding to secondary quantization of the Hamiltonian, let us suppose for the sake of generality that the dislocation moves translationally in the crystal with a velocity v small in comparison with the sound velocity c. Obviously, eqs. (23) and (24) are independent of the dislocation motion and they should not be changed. The Hamiltonian terms in eqs. (25) and (30), which describe phonon-dislocation interactions, will keep their form (up to some 'relativistic' corrections unessential at v < c) only in a coordinate system moving together with the dislocation. Before quantization, it is necessary to bring eq. (25) and (30) into the rest coordinate system, since the Born-Karman boundary conditions are implied to refer to the crystal surfaces. To that end, one should introduce in eqs. (25) and (30) the Doppler shift frequency into the generalized coordinates,

Q.- 'Q.e**' , Qk = k-v. (32)

The next standard step in secondary quantization (Ziman 1960) is transformation from coordinates Qa and Qa to creation and annihilation operators αα

+

and aa,

Qa = yfifixoM* + <*&)> Qa = iy/ha)J2(a? + αά), (33)

where ä = ( —Ac, s) and h is Planck's constant. Substituting eqs. (32) and (33) into eqs. (23)-(25) and (30), one obtains

H = X ha>.(a: a, + ±) + \h £ (Γ% + Γ ^ ξβ<Ρ»< a α,β

+ 5 f E r « » W r (34)


rfl -1 äß — r w — 1 äß -

Γαβγ =

-vi \Aism

- (

<vßK, ikajkßt

, kßz 9 "a y/tC/2(Da

- Pâ^ßSisömn)lail^su 2ρ^/ωαωβ

' h \3 /2 ÄJtn 1 1 1 n \ s\ismlciilßslym

2pJ ν /ω α ω /

L· k k âj Kßt Kyr\

Φ 9 mn A( Δ\

'Δ(Κ

-*«

+ kt

+ kß + q),

s + ft,)·

Here, the following notation has been introduced:

ξ. = «α + 4 Ω«β = (*/> - ft«)»» (35)

(36)

(37)

(38)

The Kronecker deltas in eqs. (24) and (25) express the law of momentum conservation in three-phonon processes. In fact, as is well-known, in a discrete crystal the momentum conserves only to within an arbitrary vector K of the reciprocal lattice. This result certainly cannot be obtained within the framework of the above continuum description. Therefore, the continuum expressions, eqs. (37) and (38), were generalized by replacement of ö_k+k +¥>0 and öka + kß + k7,o by

A(-ka + kß + q) = S_ka + kß + qtK and

A(ka + kß + ky) = Skm + kß + ky,K, (39)

respectively. The study of the mechanisms in dislocation dragging will start below (sec

tion 3) with the theory of the flutter effect at very low temperatures, in which case the anharmonic terms Hw and Hpp in the Hamiltonian, given in eq. (34), may be ignored. As will be shown, with increasing temperature the phonon wind becomes more effective than flutter. This happens at temperatures which are still fairly low, before relaxation processes become essential. The theory of the phonon wind developed in section 4 and based on the Hamiltonian H = H0 + Hn + i/w applies just in this temperature region. Finally, passing on to the region of higher temperatures, when it becomes necessary to take phonon-phonon processes into account, we shall have to include the term Hpp in the Hamiltonian.

5. The theory of the flutter mechanism

3.1. Some preliminary remarks

As we have seen in the derivation of the Hamiltonian, the non-linear properties of the medium (anharmonicity of the crystal) are not the only reason for a phonon-dislocation interaction. Having its own degrees of freedom, the dislocation vibrates in the thermal field of the lattice and radiates elastic waves, as

636 V.l. Alshits

any vibrating source of internal stresses would do. Thus, in a crystal with a dislocation the phonon wave function should be a superposition of a plane wave and an outgoing wave radiated from the dislocation, and this may be described in terms of phonon scattering.

This effect was originally pointed out by Nabarro (1951). Lothe (1960, 1962) carried out some first estimates of the role of this mechanism in dislocation dragging*. Ninomiya (1968, 1970) later developed a consistent theory for the flutter effect, which was used in subsequent calculations (Alshits and Sandier 1974, Ninomiya 1974) of dislocation drag. Ninomiya's theory is basically classical. Below, a quantum-mechanical theory of flutter effects will be presented, which appears to be a convenient alternative to the Ninomiya theory.

3.2. The one-particle phonon Green function

As has already been noted in the development of a theory for the flutter mechanism, we may neglect anharmonic effects, keeping only two terms in the Hamiltonian, given in eq. (34),

H = H0 + Hn. (40)

The non-diagonal second term on the right-hand side of eq. (40) does not contain any small parameters. Generally speaking, it may not be considered as a small perturbation relative to the background harmonic term H0. Thus, we have to solve a general quantum-mechanical problem without making recourse to ordinary perturbation theory. Fortunately, a solution may be obtained by a method making use of a factorized form of the coefficients Γ ^ , eq. (36).

The probability Waß of a phonon transition from the state a to the state /?, per unit time, may be found by means of the one-particle phonon Green function Daßit);

Waß=Hm(\Daß(t)\2/t). (41) f->oo

Here,

Ώαβ(ί)=-ΚΤ{ξβ(ΐ)ξ:(0)}}, (42)

A(t) are operators in the Heisenberg representation,

A(t) = eiHt/hAe-iHt/\ (43)

T is a time ordering operator, demanding that the operators following it be arranged in the order of decreasing times. The brackets < > mean a quantum-mechanical average over the basic state. Passing over to the interaction repre-

* The contribution from the flutter effect to the drag of dislocation kinks has been estimated by Eshelby (1962) and Brailsford (1970).


sentation A(t) = &HotßAe-iH0t/h9

in eq. (42), one obtains (Abrikosov et al. 1962)

(44)

Daßit) = X n = 0 n\hn at,·· di„

x<f{^( t ) e (0 )H f l ( t 1 ) - - -H f l ( i „ ) }> c , (45)

where the index 'c' implies a definite rule for the pairing of the operators, which in diagram language corresponds to summation of connected diagrams only.

The first term in eq. (45) (at n = 0) gives the unperturbed Green function

D ^ ( i ) = - i < f { ^ ( i ) C ( 0 ) } >

= -ϊ<Τ{αβε-^α;})-ΚΤ{α^<αά}}

= -ϊϊ-[ω^δΛβ = ϋα{ΐ)δαβ. (46)

The next term of eq. (45) represents the first correction to the function D$;

y,ö J - 0 0

= * Σ 1% Γ dt^^lD^t-t^Df^ + Dfsit-t^DfXt,)-] y,& J - 0 0

= r%[m dt^'D^timt-h). (47) J — 00

In the calculation of the function Daß(t\ it is convenient to use a Fourier transformation,

Daß(a>)= f°° dte°*Daß(t) = £ Dt$(a>). J - 0 0 „ = 0

In particular, for the first term in eq. (48), one has

D%(CD) = 2ω„

δΛβ = Όα{ω)δΛβ,

(48)

(49)

where ε is an infinitesimal positive quantity. The main advantage of calculating in Fourier space consists in the existence of simple rules for writing out any correction ί)$(ω). For example, application of Fourier transformation to the function D$(i) , eq. (47), brings out the expression

β # ( ω ) = Γΐχβϋα{ω + aap)Dß(a>)9 (50)

638 V.l. Alshits

or, in a more symmetrical form,

Ό^{ω + β,) = Da(co + Ωα)Γ^ϋβ(ω + ß„), (51)

where Ωα = ι? · Λ. Analogously,

Dg)(G) + β / | ) = θα(ω + ß j £ Γ ^ ( ω + β7)Γ$2>„(ω + ß„), (52) y

and generally

0$(ω + Ωβ) = ϋα{ω + ß«) Σ 1%07(ω + Α7)Γ£ y,<5 , . . . , σ

χ β,(ω + Ο,) · · · D„(Ö> + βσ)Γ^-β,(ω + Ωβ). (53)

The simple well-established algorithm, eq. (53), for construction of arbitrary terms of eq. (48) allows one to write the Dyson equation,

Dxß(a> + Ωβ) = Dx(co + Ωχ)δχβ + Dx(co + β.) £ I%Dyt{(o + β,) , (54)

for the Green function ϋαβ{ω). Because of the factorized form of the coefficients Γ{*β in eq. (36), this equation may be solved without any approximations. Indeed, multiplying eq. (54) by the factor Γ% and summing the result over a, keeping the relation

r f l r f l _ r f l r f l e 1 δά1 ay — * Λα1 Sy°kaz,kÖ2

in mind, one has

L, Γαγ^β(ω + Ωβ) = j ^ £ .

1 - £ Γ -ογ(ω + 07)

(55)

(56)

Finally, substituting eq. (56) into eq. (54) with a frequency shift ω + Ωβ-+ω gives

Α*θ(ω) = < δαβ + Γ%Ρ«(ω + Ωαβ)

" 7 ζ "ΊΧΖ

1 - £ Γ^-0,(ω + 0,?)<

β.(ω). (57)

5 J . 77ie cross-section of phonon scattering by dislocations

The knowledge of the Green function, eq. (57), allows one in principle to find the most essential characteristics of the phonon subsystem. In particular, the poles of the function ϋΛβ{ω) determine a renormalized phonon spectrum. For example, the eigenfrequencies of a crystal containing a dislocation at rest (v = 0)


are determined by the equation

F(CO2)=1-K ' £ ΙΦ, ω; ω

= 0, (58)

where the sum is a sum in the discrete sense. Equation (58) coincides naturally with the corresponding equation of Ninomiya (1968, 1970, 1975). Figure 3 shows qualitatively the graph solution of eq. (58) presented by Ninomiya (1968). One can see how a local dislocation mode arises in a way that does not violate the law of conservation of the total number of modes. The properties of the local dislocation modes have been investigated in many papers (Lifshitz and Kosevich 1966, Maradudin 1970, Ninomiya 1970, Ohashi et al. 1985a).

Consider now the task of calculating the probability Waß9 eq. (41), for phonon scattering from state a into state ß. Obviously, this problem is equivalent to the problem of determining the differential cross-section of scattering σαβ, which differs from Waß only by a factor ca, namely the phonon phase velocity in the state a;

™aß — caaaß.

Applying the Fourier conversion,

(59)

(60)

to the Green function, eq. (57), using eq. (49) and considering only those terms which do not vanish in the limiting transition, eq. (41), one obtains

^ f l C 6 ΐ (ώβ - ώβ)ί/2 e - ί(ώβ - <bp)t/2 Daß(t) = ir «ß βϊ(ώ« + ώβ)ί/2 ^ e J

ω„-ωβ G(a>a) G(&ß)

(61)

Fig. 3. Renormalization of eigenfrequencies of a crystal with dislocation; ωί is a frequency of the local mode.

640 V.l. Alshits

Here, ώα = ωα + Ωα, and

k = k 0(ώΛ) = 1 - " £ ΛΖ / ^ ( ώ . - Oy).

Substituting eq. (61) into eq. (41) and employing the relation

- 2 - = <5(ω), sin2 ωί

lim

(62)

(63)

one finds the sought expression for the probability per unit time, Waß9 of a phonon transition from state a into state /J,

Waß = 2π 1 *β

0(ώ β ) <$(ώβ - ώ,)

Φ«Φ* <?(ώβ) ^ "Κ2Λβ2ΗωΛ -ωβ + Ωαβ).

The factor

ο(ωα) - Σ Φν

ωΛ)

| φ ν 1 2

ω2 — ω2 + ίε

(64)

(65)

has been calculated in the isotropic approximation by Ninomiya (1970), with Φγ defined as

Φν= -ΐμ i(b -ky){n ·/,) + (*· ly)(n-ky)l (66)

where μ is the shear modulus. The result may be expressed in the form

G(a)a) (pbcoa)2 mfctk

az CtÔ 8π Ψ ωΆ ωα

(67)

Here, c, = ~Jμ)ρ is the velocity of shear waves in the isotropic medium, kD is the Debye limit in the phonon spectrum, Ψ(χ, y) is a dimensionless function logarithmically dependent on y,

Ψ{χ, y) = ax2 + b(

+ c(x) Γι \χ2~χ2

+ \πη{\

4- \πη(λ — x) , (68)

where λ = cjc(, c( being the velocity of longitudinal phonons. For a screw dislocation,

a = 1, b{x) = 1 - 3x2 + 4x4, c(x) = 4χ2{λ2 - x2); (69)


(*,y

W 0,8

0,6

0,H

0,2

)Γ

-

-

a

X

I I I

2

Γ—1—J

0,2 0,H 0,6 0,8 1,0 x

m^y)\~]

0,2 0,H 0,6 Οβ 1,0 x

Fig. 4. Plot of the function | Ψ(χ^\~2 for (1) a screw dislocation and (2) an edge dislocation at (a) y = 10 and (b) y = 100.

for an edge dislocation,

a = 1 + λ\ b(x) = l - x \ c(x) = (λ2 - x2)2. (70)

Figure 4 shows a plot of the function | Ψ(χ, y)\~2 made on the assumption that the Cauchy relation λ2 = | is satisfied.

Calculate now the differential cross-section for phonon scattering from the state a = (k, s) into the cylindrical polar interval 0(0 + d0) in a plane perpendicular to the dislocation*;

d<7a(0) = d0 Snk

k[ = kz Op = ωα

4>ks<t>ts G(a>a) cscs,

The corresponding total cross-section of scattering

Cs β Jo

(71)

(72)

* In the analogous expression for the differential cross-section άσα in the papers by Ninomiya (1968, 1970) there is a misprint which has led to a wrong formula for the total cross-section σα. Corrections have been made by Alshits and Sandier (1974) and Ninomiya (1975).

642 V.l. Alshits

in the same approximation as in eq. (67), has the form

k[ = K

(73)

where <Pks = <Pks/ßbk. Thus, as was first noted by Nabarro (1951), the scattering cross-section σα, having the dimension of length, turns out to be proportional to the phonon wavelength.

3.4. The contribution of the flutter effect to dislocation dragging

The number of phonon transitions per unit time from state a into state β is determined by the product between the probability Waß and the phonon density in state a,

wa = [ 6χρ (^ω α /Γ ) - 1] \ (74)

Here, T is temperature in energy units. Taking into account that in each scattering act the momentum h(k — k') is transferred, the phonon drag force on the dislocation may be expressed as

F = ^ h(k - k')na Waß = ^ hk(na - nß) Wn aß Λ,β Λ,β

« £ hk^-Wî(k - A')»] = £ hk^W^k-v),

or in the form

Fi=-Bl}v„ where Btj is an effective viscosity tensor

8n, dn„ Σ οη„ h v^ on„ <*,P

'θω, ζ,β

5ω„

= -^k'kJ*tc*°°

(75)

(76)

(77)

One may check that the non-diagonal components of Btj are equal to zero.


Therefore, instead of a tensor Bij9 we shall have a scalar damping constant

which is the quantity directly involved in comparison with experiments. Substituting into eq. (78) the expression for the scattering cross-section, σα,

eq. (73), one has after some calculations

hk3

Βίι = 2^ίφΑΤ/θι) + φ((Τ/θ()1 (79)

Here, θ8 = hcskO are modified Debye temperatures (in energy units), and the functions (pt(x) and φι{χ) describing the temperature dependence have the form

%1/x d i i 3 e ' cps(x) = x - v-3

o ( e ' - l ) : y.(*f), (so)

where Ψι()>) and W((y) are logarithmically slow functions, which are different for screw and edge dislocations. For a screw dislocation

<Pt(y) = π P

ΨΛν) = 4*λ4

dx(l - x2) (1 - x2 + 4x4) Im {1/f *(x, y'1)},

dxx 2 ( l - x 2 ) 2 I m { l / f * ( x , > ' - 1 ) } . (81)

For an edge dislocation

Wt(y) = 2n r d x ( l - x 2 ) ( l - x 4 ) I m { l / « P * ( x , y _ 1 ) } ,

Wt{y) = 2nXA dx(l - x2)3 Im{l/!P*(x,>'-1)}. (82)

It is easy to demonstrate that at low temperatures

<Pt(T/et)/<pt{T/et) ~ (cjc(y < 1. (83)

Therefore, we shall later on neglect the second term in eq. (79)*,

2π2 Βη^^φι(Τ/θι). (84)

* For the same reason, in real crystals, at low temperatures, the parameter 1/c] gives a sharp differentiation between the contributions from different phonon modes to the dislocation drag (and to the dislocation part of thermal resistivity) and it gives the predominant role of the branch of slowest phonons.

644 V.l. Alshits

%(T/St) 1,0

0,8

0,6

0,1

0,2

0 . OJ 0,8 1,2 T/0

Fig. 5. Plot of the function (pt(T/6t) for (1) a screw dislocation and (2) an edge dislocation.

According to Alshits and Sandier (1974), *Ft(y), eq. (82), for an edge dislocation may be satisfactorily approximated by the simple function

Ψί(γ) * 1+(1/π2)1η2(>>-2 + 1)·

Figure 5 shows plots of the function <pt(x) for screw and edge dislocations. Thus, as follows from eqs. (80), (84) and fig. 4, at low temperatures Bn oc Γ3 to an accuracy of logarithmic corrections. At very low temperatures, the drag of screw dislocations is somewhat more intensive than for edge dislocations. However, with increasing temperature this predominance is diminishing and at T> 10" 2 Θ the relative roles are changed. Further increases of temperature bring Bn(T) into a range of linear temperature dependence. However, as we shall see, in this temperature region the flutter mechanism gives weak effects compared with the phonon wind mechanism.

The calculations carried out above were based on an exact solution of the Dyson equation. It is easy to show that a calculation of the Green function to a first-order perturbation theory would give qualitatively incorrect results. Indeed, a non-diagonal term in the solution, eq. (57), differs from the first correction, eq. (51), of perturbation theory by a factor |Ga |~2 °c (ωβ/ωα)4 . In the low-temperature region where low-frequency phonons are of predominant importance, this factor is obviously very essential. In the region T <ζ 0, perturbation theory gives a dependence Bn oc Γ7 , instead of the correct dependence Bn cc T 3, as obtained above.


4. The non-linear mechanism of phonon scattering

4.1. Statement of the problem

Let us now include a non-linear mechanism of phonon scattering in our considerations. We add the term //w from eq. (11) to the Hamiltonian, eq. (40),

H = H0 + Hn + Hw. (85)

Leibfried (1950) noted that any mechanism of phonon scattering by a dislocation should contribute to the drag of the dislocation by purely aberrational effects-the asymmetry of the phonon distribution relative to the moving dislocation. The result is as if a phonon 'wind' blows past the dislocation impeding its motion. This drag should obviously be proportional to the density of thermal energy £ th , and according to Leibfried (1950) it may be estimated to be

Nabarro (1951) pointed out that one should distinguish between two different mechanisms of phonon scattering - the flutter effect and non-linear mechanisms of anharmonic nature. The last mechanism was later called 'phonon wind' in some of the literature. Lothe (1960, 1962) showed that at room temperature the primary estimate, eq. (86), describes equally well both effects, when the anharmonicity is described in terms of the Grüneisen constant yG.

In the same approximation, Klemens (1955, 1958) and Nabarro and Ziman (1961) carried out the first calculations of the contribution of non-linear mechanisms to the cross-section of phonon scattering by dislocations. Unfortunately, the Grüneisen constant has proved to be an unsatisfactory measure of anharmonicity in these kinds of problems. The calculations (Klemens 1955, 1958, Nabarro and Ziman 1961) showed a substantial discrepancy with experimental data on the dislocation part of thermal resistivity. This problem stimulated a series of theoretical investigations starting with explicit consideration of the crystal anharmonicity, in an isotropic approximation in the papers by Bross (1962), Bross et al. (1963a, b), Ohashi (1968), Grüner and Bross (1968), Grüner (1970) and with regard for anisotropy in the paper by Eckhardt and Wasserbäch (1978). On the basis of the Hamiltonian obtained by Bross (1962), the first quantum-mechanical calculations of the non-linear mechanism of kinks (Seeger and Engelke 1968, 1970) and dislocations (Alshits 1969b, Grüner 1970, Alshits and Kotowski 1975, Alshits et al. 1979) were accomplished.

As a rule, such calculations were carried out to the first order of perturbation theory, which appears to be reasonable, at least at low temperatures when the small parameter of the perturbation theory is determined by the strain field of

646 V.l. Alshits

the dislocation at a distance from the core of the order of the average wavelength of phonons. However, the flexibility of dislocations was always ignored, i.e., the second term in eq. (85) was omitted. The knowledge obtained of the Green function for the flutter-effect allows us to estimate the role of non-linear mechanisms against a background of fluttering.

4.2. The cross-section of phonon scattering by a dislocation

If one considers the function Ώαβ(ω\ eq. (57), as a zero-order approximation, then, correspondingly, the Green function for the Hamiltonian, eq. (85) to the first order of perturbation theory for the effect of iiw becomes

ϋ%^\ω) = Daß(w) + X Day(co + Ω7β)Γ$0Λβ{ω). (87)

Substituting into eq. (87) the expression ϋαβ(ω\ eq. (57), and using the standard procedure described above in connection with the derivation of eq. (64), one obtains that the probability of phonon scattering per unit time from state a into state β equals

WnR = In G(a>J + /Ί& δ(ώα - ώβ\

where

r w . _ r w _ , V J r w υγ(ω*) Γ Π r w Dy(Üß) r f l G(&.)

(88)

(89)

Accordingly, the generalization of eq. (71) for the differential cross-section of phonon scattering into a region άθ of polar angles has the form

άσα(θ) =

K = K du γ^ OJ„ ωη

Ink (cscs,)2

rfl_ 1 aß G(coa)

+ Πβ (90)

Let us introduce now a transport cross-section of scattering σ%, which will be of use in the establishing of a natural relation between the damping constant B and the dislocation component of thermal conductivity /cd. Let u be a unit vector directed along the dislocation velocity v or (in the thermal conductivity problem) along the temperature gradient. Then, by definition,

<{u) = - 2 , Waß -2 = J l{k-k')-uV άσα(θ).

In accordance with eqs. (77), (78),

h »--?Σ* 2 - ' £"■■««

(91)

(92)


On the other hand, it is known that in a certain approximation a 'harmonic formula' for the thermal resistivity is valid (Ziman 1960),

κΛ C2T L do», v d Y^k2^csa^{VTI\VT\\ (93)

where Nd is the dislocation density and C is the specific heat of the crystal. Comparing eqs. (92) and (93) for directions VT\\ t>, one has

According to eq. (94), additivity of the damping constant B with respect to different mechanisms of phonon scattering is equivalent to Mathiessen's rule for thermal conductivity (Ziman 1960). In fact, one should remember that eq. (93) was obtained under a number of simplifying assumptions, which are not universal properties and which may be invalid in some cases*. Accordingly, the above-established interrelation, eq. (94), between two different fundamental kinetic characteristics of the crystal appear as well to be somewhat idealized. Still, we will use the simple and convenient formula (94) for a number of semiqualitative speculations. The relation between Kd and B was considered more rigorously by Grüner (1970) for the case of non-linear mechanisms of phonon scattering.

4.3. The contribution of the non-linear mechanism to dislocation drag

Combining eqs. (92), (91) and (88), it is easy to obtain

β = - π » Σ q*v x2 rfl-+ 1 <xL 0(ωβ)

2 %!£*(».-<»,)■ (95)

The cross-term in eq. (95) containing the product Γ^β Γ™β vanishes after summation over a, /}, since, as can be checked, Γ™β = — Γjfc. As a result, the damping constant, eq. (95), is determined by the sum

B = Bn + Bw, (96)

where the first term, due to the flutter effect, is already calculated in eq. (84), above, and the second term,

Bw = -nh X h^X\r-f ?j±δ(ωχ - ω„\ (97)

* For instance, at very low temperatures, when the mean wavelength of phonons becomes much larger than the average distance between pinning points along dislocations, the flutter mechanism is obviously unimportant in thermal resistivity whereas it is still the main mechanism of dislocation drag at overbarrier motion (see section 4.6).

648 V.l. Alshits

describes the contribution to dislocation drag from the non-linear mechanism. Expression (97) differs from the usually used one (Seeger and Engelke 1968, 1970, Alshits 1969b, Grüner 19J0, Alshits and Kotowski 1975, Alshits et al. 1979) by a renormalized factor Γ%ρ, eq. (89), instead of the unrenormalized Γ™β. Thus, the thermal vibrations of the dislocation not only give rise to the flutter effect, but also exert an influence on the non-linear mechanism.

Below, we shall estimate by means of eq. (97) the drag of straight dislocations and dislocation loops. The elastic fields, given in eq. (19), of such dislocations are necessary for derivation of explicit forms for the factor Γφ eq. (37), and they are given by Fourier amplitudes (Alshits et al. 1979)

ul = -Sfjkl(q/q)bkm^(qX (98)

where

Sfjki(n) = öiköß - Ck^npA-în)^; (99)

4μ(Ό = npCpqijnP (100) m = txq/q, t is a unit vector tangent to the dislocation line and orthogonal to the vector q\ L is a dislocation length which, for a straight dislocation, is everywhere taken to be of unit length*, and which for a loop is equal to 2nR; the function (p(q) is defined as

<f>«(4) = - i ^ , . o , <Pt(9) = MqPR), (101) for a straight dislocation and dislocation loop, respectively, [ ^ (x ) is the Bessel function.]

Let us write | Γ ^ | 2 in the form

l^-|2 = ^ J | ^ ) | 2 L F a , . (102)

Here Faß is a dimensionless function of directions. The order of magnitude of this function is determined by the ratio:

Faß~\A/ß\\ (103)

where A is a characteristic value of elastic moduli of the third order. Substituting eq. (102) into eq. (97) and changing the summation over ka, kß in eq. (97) to integration over ka, q = kß — ka, one obtains

(104)

* This assumption is made in eq. (97) as well. Accordingly, in subsequent calculations, for a dislocation loop we shall substitute into eq. (97) |. . .ujJ|2/L instead of |. . .ujj|2.


4.4. The drag of a straight dislocation

When carrying out the integration in eq. (104) for a straight dislocation parallel to the z-axes, note that in this case the factor

L\<p(q)\> = 2n6{qz) (105)

limits the change of the vector q to be in a plane perpendicular to the dislocation. Let us express the function Faß in coordinates q>q, cpk, 6k,

Faß = /V(</V <Pk, cos ek\ (106)

where (pq is the azimuthal angle of the vector q counted from the direction of velocity t>, q>k is the azimuthal angle of the vector k in the plane perpendicular to the vector q and 6k is the polar angle of the vector k counted from the direction of the vector q. Substituting eqs. (105) and (106) into (104), one has after some calculations*

^w = ^ ( ^ - ) f[w)> (107)

where « C1/x d i f V m" . w=w· m)

9 = 2* d(pq cos2 cpq Jo

d<pk\ άί^Ρ3Αφ9,Ψ^ΐ)^\Α/μ\2. (109)

Formulae (107) and (108) allow one to make judgements about the temperature dependence of the drag mechanism. This dependence was first studied in detail by Alshits (1969b). At low temperatures, T < 0, the phonon wind is 'frozen-out' much more rapidly than the flutter effect: ßw ~ Γ5 . With increasing temperature, this sharp dependence changes into a linear dependence: ßw ~ T. The linear regime starts already in the region Τ<θ.

The absolute magnitude of the constant B depends on the magnitude of the parameter g, eq. (109), which may be found only after explicit and fairly cumbersome calculations. For screw and edge dislocations, such calculations were carried out (Alshits 1969b, Grüner 1970, Alshits et al. 1979) in an isotropic approximation describing anharmonicity in terms of Murnaghan's moduli n, m and /**. For a screw dislocation,

0sc = 4 + Ι(η/μ) + 6] 2 , (110)

* In the derivation of eq. (107), contributions from longitudinal phonons were neglected. The accuracy of this approximation is the same as in the derivation of eq. (83) in the flutter effect theory.

** All calculations (Alshits 1969b, Grüner 1970, Brailsford 1972) were carried out without the renormalization of eq. (89), Γ™β -> f™ß.

650 V.l. Alshits

according to Alshits (1969b). The more bulky expression for an edge dislocation is given by Alshits et al. (1979).

4.5. The drag of a dislocation loop

Consider now eq. (104) in the case of a prismatic dislocation loop moving in the direction of the Burgers vector b perpendicular to the loop plane. Integration over k in eq. (104) is as earlier conveniently carried out in the spherical coordinates φ^ 0*, counting the polar angle 6k from the direction of the vector q. In this case, the integration over q will be also carried out in spherical coordinates with an azimuthal angle cpq in the loop plane and a polar angle 0q counted from the direction of the velocity v. Accordingly, the function Faß will be assumed to be expressed in these coordinates as

Faß = Fss((pq, cos θν (pk, cos 0k). (111)

The calculation is quite analogous to the previous one in section 4.4 and leads to the following expression for the damping constant of the dislocation loop (Alshits and Kotowski 1975, Alshits et al. 1979);

b*\2n) 7U, BV=^(-^)f(wl (112)

with the temperature dependence determined by the function

C1/x d i i V -f(x) = x5\ j-r-^Vißxt). (113)

Jo l e — l) Here, ß = 2kDR and the function Ψ{γ\ distinguishing eq. (113) from eq. (107), has the form

v(y) = i*y

■i

1 pi duu2 dvvJ\(yv^/T

- l Jo )

- 1

acpk άφ9ΥίΡ^(φ9,η,φ/(9ν). (114)

Formulae (112)—(114) allow one to analyse the temperature dependence of the drag of dislocation loops at low and high temperatures. When T < 0t/j3, the main contribution to the integral in eq. (113) is determined by the magnitude t ~ 1. Therefore, in the calculation of the temperature dependence of/( T/9t) in this temperature region it is sufficient to know the behaviour of the function Ψ(γ) for small arguments. Taking into consideration that for y <ζ 1

j\(yVJ\-u2) * iy2v2(l - u2\ (115)

the function Ψ{γ) may be expressed in the form

$(y)~giy\ (116)


in this region. Here,

f1 f1 Γ2π Γ2π ν ^ 9ι=^π duu2(l-u2) dvv3 d<pk dcpq 2^FM,(<jof, u, % , t;).

J - i Jo Jo Jo s s, (117)

With eq. (116) the function f(T/0t) becomes

0 J 60 " i r \θχ

in the range Γ < 9Jß. Thus, at very low temperatures, when the average wavelength of phonons exceeds the radius R so that they 'perceive' the loop as a point, the damping constant B^ freezes out much more rapidly than the damping constant, eq. (107), for a straight dislocation.

In the same way, it can be shown that when T$> 6Jß, the main part in eq. (113) stems from asymptotics of the function Ψ{γ) in the region y > 1,

1 f1 duu2 f1 f2

yM*2 ~7Γ=Λ dv\ L J - i J \ — uz Jo Jo

dq>, k

'2π

d(pq o

x X FSsi<Pq, u> <Pk, v) = g2 = const. (119) s, s'

In other words, at high temperatures, T > 0t/jS, the loop drag has the same temperature dependence as the drag of a straight dislocation, as should be expected. The drag is described by the same formula, eq. (107), with g replaced by 02· This lack of full coincidence has a simple physical reason. Due to anisotropy of the drag of loop segments of different orientation, a g2 that represents an average of g over all possible directions of the vector t in the loop plane must be used.

Note that β = 2kOR > 10, even for a crowdion which may be considered as a small prismatic loop with a radius of the order of the lattice parameter. As regards the drag of dislocation loops with radius R>l0b, the temperature region T<9Jß, in which one should take into account the curvature of such dislocations proves to be negligibly narrow. Accordingly, we shall subsequently be interested in the drag of straight dislocations only.

4.6. Comparison of contributions from the flutter mechanism and thephonon wind

The relative roles of the two above-discussed mechanisms of phonon scattering on dislocations is, according to eqs. (84) and (107), determined by the ratio

* w _ 9 fkDb\2 f(T/et) Bn An \ In ) φχ{ T/6t)

As regards the explicit form of the functions /(T/9 t), eq. (108), and ^ t(7/ö t), eq. (80), it is easy to demonstrate that at low and high temperatures this ratio

652 V.l. Alshits

tends asymptotically to

Bw (T/T0)2 i r r ^ x Λ Β

Bn !Pt(i0r/flt) v l " Bn

respectively, where T0 and d§ are constants,

_ 5.40, _ _0_ (/cDb/27i)2

(T 4 0t), and -^-~d0 = const. ( Γ > 0t), (121)

0

(122)

t0 is a numerical parameter of the order of unity (ί0 ~ 2.4). For example, an estimate for a screw dislocation in copper gives T0 ~ 14 K, ά^ ~ 18. These figures are obtained when one uses eq. (110), i.e., without regard for the renor-malization Γ%β -► Γ%β. Also, it should be remembered that the comparison made above was based on expressions for damping constant B which do not contain a specific contribution from the dislocation core. Therefore, the value of T0 which relates to the region of low temperatures, where the problem is insensitive to the precise size of the dislocation core, is estimated more reliably by eq. (122) than the parameter ^ . Nevertheless, the basic conclusion that increasing temperatures give a crossover between different mechanisms apparently remains valid. Comparison between the same mechanisms in the drag of dislocation kinks (Eshelby 1962, Seeger and Engelke 1968,1970, Brailsford 1970) leads to qualitatively similar conclusions (Seeger and Engelke 1968, 1970, Brailsford 1972).

5. The relative roles of phonon scattering and relaxation processes in dislocation dragging

5.1. Statement of the problem

Dislocation dragging may be described as the damping of a moving dislocation packet of elastic waves [eq. (19)];

iiy(r,i) = Z 4 e i ( f , r " f l f 0 · (123) q

The energy dissipation per unit time in the packet is simply obtained as the sum of the separate partial wave contributions,

ö = Zfl§^«(^°fKtt«f· (124) q

Here, x\(q, Qq) is an effective viscosity tensor for the partial wave with wave vector q and frequency Qq.

In this consideration, the Hamiltonian term Hpp, which describes the phonon-phonon interaction process, was omitted, i.e., we did not take into


account the relaxation of the phonon gas by its scattering by the dislocation. This is permissible only as long as the phonon energy change hQq in each act of scattering is high in comparison with the width of the energy level, h/rph. The inequality Qqxph > 1 may typically be satisfied only for a short wavelength part of the packet given in eq. (123), q > (c/v)l~^ (Zph = rrph is the mean free path of phonon). In spite of this, the developed theory remains roughly valid for fast dislocations at low temperatures. However, note that the calculations should also be revised for the region of low velocities and high temperatures, when (c/v)l~^ > qm ~ To *. Generally speaking, an estimate of the contribution to the energy dissipation D from partial waves of the packet given in eq. (123) satisfying the inequality QqTph < 1 is needed. This calls for a solution of the kinetic equation for phonons relaxing in the field of the moving dislocation.

Below, we shall consider phonon drag of dislocations in the high-temperature region (Τ>θ) and retrace how new dissipative mechanisms caused by an increasing intensity of the phonon-phonon interaction come into existence. Unfortunately, some additional new difficulties arise, since with increasing temperature, when the average wavelength of phonons becomes comparable with the dislocation core size and the dispersion law more and more appreciably deviates from the Debye model, the elastic field behaviour near the dislocation core and the specific features of the real phonon spectrum should be taken into account.

5.2. The temperature retarded Green function and the kinetic equation for phonons

A description of the different phonon mechanisms in dislocation drag within one single formalism may in principle be accomplished by the solution of a classical kinetic equation, as was suggested by Brailsford (1972). However, for the purposes of this chapter, including our aim to derive all effects from one common Hamiltonian given in eqs. (12) and (34), the quantum approach developed by Alshits and Malshukov (1972) appears to be preferable. In addition, that approach allows control of all approximations in the derivation of the kinetic equation and if necessary improvement beyond the limits of the usual Boltzmann equation. Distinct from the paper by Brailsford (1972), where the collision integral is written in the relaxation time approximation, we obtain and analyse below the kinetic equation with the total collision integral retained.

Starting our analysis of dissipation processes in the temperature region Τ>θ, where, as we have seen, the flutter mechanism apparently is much less effective than the phonon wind, and phonon-phonon 'collisions' have come appreciably into play, we shall describe the behaviour of the phonon subsystem by the Hamiltonian

H = H0 + Hpp + Hw = MT + HJt). (125)

654 V.l. Alshits

One should, however, take into consideration that flutter effects contribute to the renormalization Γζβ -► Γζβ through a non-linear mechanism which couples phonon scattering with the oscillating part of a dislocation field. Accordingly, we shall below use a model Hamiltonian ifw, in which the replacement Γζβ -> Γζβ already is carried out. This will later on allow us to obtain a series of results, which are a natural generalization of formulae derived above.

When studying the dissipative properties of the system described by the Hamiltonian given in eq. (125), it is convenient to apply the quantum-mechanical techniques of non-equilibrium statistical thermodynamics. The basis for such an approach consists in an analysis of the evolution in time of the density matrix p, which satisfies the equation

! + > , ρ ] = 0 , (126)

where the square brackets mean the operation of commutation of operators. The deviation Aß of the density matrix from its equilibrium value,

p ( -oo) = p0 = e - ^ / T r j e - ^ } , (127)

determines the basic kinetic and dissipative properties of the system. In particular, the dissipation of energy D per unit time in the system equals

It is easy to check formally that the solution of eq. (126) which satisfies the initial condition, eq. (127), may be expressed in the form

P = Po + l emt -ο/Λ[/5) Hw(t')] e - i j r " ' - '>/* at'. (129) — oo

Of course, eq. (129) is nothing more than an integral equation, but it allows one to build up a series of successive approximations for the density matrix p.

It may be shown that, at high temperatures, the Hamiltonian Hw can be characterized as small*, and then it is possible to use a first-order approximation in eq. (129),

df e*V" '>/* [po, # w ( 0 ] e " ^ ' - <>/* . (130) Δ/5 = β - ßo = j

Substituting eq. (130) into eq. (128) and employing the rule Tr{[A, B] C} = Tr{A [B, C]}, one obtains

D = -l- Γ di'TrIpoU^'-WHUOe-^'- t)/h 8HW(Q dt

(131)

* To be more precise, the ratio of the second-order correction of the perturbation theory expansion in terms of Hw to the first-order term is small.


By means of the explicit form, eq. (34), of the operator Hw, eq. (131) may be expressed in the form

D=-\ih^QqGR{q,Qq\ (132)

where GR(q, ω) is the Fourier transformation of the retarded two-particle temperature Green function,

GR(f,o>) = i Γ dte'-j-Mi) J f:ßffs J - 0 0 ^ α,β,γ,δ

x T r C p o i e ^ ^ i . ^ e - ' " / * , ^ ^ ) ] ! . (133)

The symbol q above the summation sign means that in the sum one should maintain ka + kß = ky + kd = q = const.

Comparing eqs. (124) and (132), it is easy to see that the effective viscosity, f7i</k/, and the introduced Green function, GR, satisfy the relation

Im GR(q, Qf) = ? ß f f/w(f, ß f Κ - κ ΰ ' . (134)

Keeping the first non-vanishing term in the power series expansion in v/c of eq. (132), one obtains the expression for the viscous component of the energy dissipation,

D^-frZt^^l ■ (135) However, this formula should be used with some care, since the function GR(q, ω), as will be seen, may have a logarithmic singularity at small ω.

Thus, the dissipative properties of the system considered are directly determined by the Green function GR(q, ω). In practice, the procedure for calculating the retarded Green function GR usually consists of two stages. First, one calculates the appropriate causal Green function, in our case

rh/τ c * „ _ G{q,'^n) = \ \ d ie 1 «"] X r:ßrJö-Tr(ß0Te^h

J ° ^α,β,γ,δ

χξαξβ*-*τ'Ηξϊξί)\. (136)

Then, from the coincidence of the functions GR and G on the discrete set of points \ωη = IninT/h (n = 1, 2, . . .) (Abrikosov et al. 1962), one extends the function G(q, \ωη) analytically to frequencies in the upper halfplane, \ωη -> ω + ίε, i.e.,

GR(q9 ω) = G{q9 ω + iß). (137)

656 V.l. Alshits

Calculation of the causal Green function has the advantage that it allows one to use a well-developed diagram technique (Matsubara 1955, Abrikosov et al. 1959, Fradkin 1959, Abrikosov et al. 1962).

As a matter of fact, the above-mentioned procedure is just a method for calculation of the perturbation theory series in Hpp. The zero approximation of such an expansion corresponds to neglect of relaxation effects describing the scattering processes discussed in section 4. However, as has already been noted, such an approximation is only valid as long as the change of the phonon energy during scattering is greater than the width of phonon energy level hyph = h/rph i.e., Ωςτρϊϊ Ρ 1. In this case, the next terms in the expansion give only small corrections of the order yph/coa. However, at not too low a temperature and for velocities v<\0~2 c (usual in experiments), even the shortest waves in the dislocation packet given in eq. (123), with wave vectors of magnitude qm ~ l/r0, are characterized by parameters QqTph ~ (v/c)(lph/r0) which are small rather than large. Accordingly, below we shall assume that all waves in the packet given in eq. (123) satisfy Qqxph < 1. This means that we have to describe phonon transitions between energy levels which overlap due to their finite width;

\ωΛ-ωβ\<\ΙτνΥί. (138)

Certainly, under these circumstances, the zero approximation mentioned above is completely unsatisfactory.

It may be shown that under the condition given in eq. (138) the expansion of G(q9 Ίωη) contains an infinite set of singular (so-called ladder') diagrams which have to be summed. The special role of ladder diagrams in the expansion of the phonon and electron Green functions was noted by a number of authors (Eliashberg 1961, Holshtein 1964, Sham 1967). In particular (Eliashberg 1961, Sham 1967), it was shown that the procedure of summation of ladder diagrams leads to the Boltzmann kinetic equation. Omitting fairly cumbersome calculations, we put down only the result,

G*{q, ω) = - \ Qq £ H, (j£- + ^fj U«ß(l, ω) +fße(q, ω)], (139)

where the function faß(q9 ω) is determined by the kinetic equation

-ϊ(ωβ - ωα + ω + 2iy)fafi(q, ω)

~ π ν^ = Γ*β + ρ" Lu ΓσβαΓδρβ/σδ(ς, ω)

σ,δ,ρ

X {(ηρ - ησ)[δ(ωσ - ωρ - ωΛ) + δ(ωσ - ωρ + ωα)]

+ (ησ + ηρ + Ι)δ(ωσ + ωρ - ωα) + (ηρ - ηδ)

X [<5(ω,5 - ωρ - ωα) + δ(ωδ -ωρ + ωα)]

+ (ηδ + ηρ+ Ι)δ(ωδ + ωρ - ωα)}. (140)


Here,

V = 2ft2 Lu \Γ^δ\2{{ησ ~ ηδ)ίΗωα + ωσ - ωδ) - δ(ωα - ωσ + ω5)] σ,δ

+ (η„ + na + 1)<5(ωα - ω , - ω,)}. (141)

Inserting eq. (139) into eq. (132), one obtains an energy dissipation of the form

D = - X hΩ\ | ^ f:ß Re faß(q, flf). (142)

Thus the problem is generally reduced to finding the funct ion^, i.e., to the solution of the kinetic equation, eq. (140), which determines the transport and dissipative properties of the phonon subsystem disturbed by a moving dislocation. Analysis of eq. (140) will be presented in the next sections. In particular, we shall see that the usual Boltzmann equation results only in the region qlph< 1, i.e., only for the long-wavelength part of the packet, eq. (123).

5.3. The solution of the kinetic equation in the long-wavelength region qlph< 1

For qlph < 1, the inequality, given in eq. (138), is obviously satisfied for any states a and β belonging to the same polarization branch (s = s') and obeying condition fta + kß = q, since, for small q and when s = s',

|9ω„ , = \W\ ~ <?WTPh < lAph ~ 7· (143) ωΛ - ω„

8ft '

On the other hand, when s φ s' and qlph < 1, the opposite inequality (Ια^ — ωα| > y) is as a rule valid. This means that transitions between different branches of the spectrum do not markedly influence the behaviour of the function faß(q, ω) and may be described by first-order perturbation theory. It is easy to check that, in the region qlph < 1, such transitions make an insignificant contribution to the energy dissipation.

In view of this it is natural to make the substitution

faß(l><0)=fa(9,<1>)8s9s- ( 1 4 4 )

in eqs. (139) and (140). Taking into account the closeness of the states a « ß, the properties of the factors Γζβ [eq. (89)] and Γσβά [eq. (38)] and the assumed diagonality [eq. (144)] of the function^, one obtains

fw _ p _ pq — riJuH. 1aßÎ<xßn^1oc — * a uip

2ρωα

δπσ, ωδπ ωσ9 ηδ « ησ, Γσβόί Γδρβ ~ \ Γσβα\2.

658 V.l. Alshits

Accordingly, eqs. (139) and (140) simplify to

G*(q, ω) = -2ϊω £ Γ^/Μ ω), (146)

-i(vaq + ω)ηα(ηα + l)fa(q, ω) = na(na + 1)Γ"« + J[^(ft ω)], (147) with the collision integral in the classical Boltzmann form (Peierls 1955),

JUM, ω)] = j£ Σ {lr«tfäl2»,(«, + 1) («« + 1)

x (X ~/p ~Χ)^(ωσ - ωρ - ωα) + Ί\Γσρα\2ησηρ(ηα+ 1)

x(X +/ρ -/«)5(ωσ + ωρ - ωβ)}. (148)

When solving the kinetic equation, eq. (147), we shall limit ourselves to the analysis of the limiting case qlph <| 1. Knowledge of the precise behaviour of the function fa(q, ω) at qlph ~ 1 turns out to be unessential, since, as will be seen, the main contribution to the energy dissipation is connected with partial waves in the packet [eq. (123)] with q ~(v/c)l~k, and the integral determining the energy dissipation in the region 0 < q < l~^ depends only logarithmically on the upper limit.

As suggested by Balagulov et al. (1970), it is convenient to search for a solution of eq. (147) in the form of an expansion,

fa(9>«>)= Σ * ) Λ (149) n = 0

in terms of the eigenfunctions of the collision integral,

J(<fi) = Vn<Pl (150) One may show that the functions φη

α form a complete system. We shall suppose that they are orthonormal,

I ^ > « = t , (151) a

and numbered in the order of increasing eigenvalues yn > 0. The minimum eigenvalue, y0 = 0, corresponds to an eigenfunction (Balagulov et al. 1970)

φ°α = const. ωα. (152) It turns out that the first term makes the basic contribution to the expansion in eq. (149), at least for not too low a temperature when umklapp processes are not negligible and so that the eigenvalues y{ corresponding to the eigenfunctions


(Balagulov et al. 1970)

ΦΪ = *,Α; ( U = l , 2 , 3 ) (153)

are not small. In the temperature range under consideration, Γ>0, the eigenvalues yn(n ^ 1) can be supposed to have finite values of an order which is determined by the average level of damping in the phonon subsystem, i.e., 7 η ~ τ ρ ν ·

Let us introduce the notations

(154)

rv = riju^ Γυ = — V Η(Χ wmrij

α α

and consider the long-wavelength part of the packet [eq. (123)], where

Wmn^yrm ^mn < 7m· ( 1 5 5 )

At ω = Qq, the second inequality in eq. (155) is less restrictive than the first one. It is satisfied in all of the region qlph < 1,

GqSmn/ym ~ *Vph ~ ~^'ph < <?'ph < L (156)

It is easy to check that under condition (155) substitution of the expansion [eq. (149)] into the kinetic equation, eq. (147), and use of the orthogonality, eq. (151), lead to the following expressions for the coefficients α%(ω) of the expansion, of eq. (149),

r0-<+ia)Y^sOHr;</yH

-ia>S00 + Z J V n=l Yn

a, v i f l o ( y P m O + Q > S w o ) + r w ^ α%{ω) = , m ^ 1. (157)

Thus, the first term on the right-hand side of eq. (149) indeed has a singularity for (ω, q) -► 0 and in the region qlph < 1 the contribution of this singularity to eq. (149) is the predominant one. Taking into consideration that the normalizing constant in eq. (152) disappears from the final result, we shall below put it equal to unity. Combining eqs. (157) and (149), one obtains an explicit form for the

660 V.l. Alshits

solution of the kinetic equation, eq. (147), r-q

1 n f*(q>o>)= X -JL- <P« In

(ro-q + ™ts0nr;Vy^ 7n

• „ , y (g ' t>o , , ) 2 WSg„ -ιωΞ00 + }

«=ι ?« (158)

Certainly, in this expression not all terms are equivalent. As was already mentioned, the first term on the right-hand side plays the main role in the expansion, eq. (149), and it should be quite sufficient to put

fx(q,o})âl(w)9°x. (159)

However, we also want to discuss the in principle important problem of the relation between phonon viscosity and thermoelastic losses of energy in the same picture of damping of plane waves in the packet given in eq. (123). There was much debate in the literature on this problem. It would appear to be appropriate to discuss the problem here on the basis of the general theory developed, using eq. (158).

5.4. Phonon viscosity and thermoelastic damping

The damping of plane elastic waves in the packet given in eq. (123) is evidently determined by dissipative processes of the same type as cause phonon attenuation of ultrasound. In the long-wavelength region considered, qlph < 1, it is natural to expect that the damping should be a combination of thermoelastic damping and an Akhiezer phonon viscosity effect.

The first mechanism is of a thermoconductivity origin. The elastic wave motion is accompanied by heating and cooling of volume elements undergoing rapid compression and expansion, respectively. Thus, heat fluxes between hot and cooler parts of the crystal arise and, accordingly, thermoelastic dissipation of energy takes place.

Phonon viscosity is a damping mechanism which was first noted by Akhiezer (1939). It is related with the process of the redistribution of phonons in the elastic wave field, and the interchange of energy between phonons leads to energy dissipation. Phonon viscosity manifests itself also in shear wave fields, where thermoelastic damping is absent. Akhiezer has shown that in the process of restoration of equilibrium in a phonon subsystem disturbed by a sound wave, phonons behave as a gas with an effective viscosity Âkh ~ y ^ t h T

Ph-It seems quite natural to estimate the dislocation drag from the expressions

for the effective viscosity ηίΜ known from the theory of sound attenuation. Such


estimates have been made by Mason (1960), who substituted into a dissipation expression of the type of eq. (124) a constant Akhiezer viscosity f7Akh. Unfortunately, such a simple and, at first sight, reasonable estimate is incorrect and leads to results which are wrong even in order of magnitude. We shall see that in contrast to the attenuation of long-wavelength ultrasound which is well-described by a constant viscosity, the damping of the plane waves in the packet given in eq. (123) is characterized by a very pronounced time and spatial dispersion even in the region qlph < 1. This basically changes both the quantitative and qualitative features of the phenomenon.

Actually, we are here dealing with a rather typical change of situation, with the appearance of a new physical parameter (in the case considered - the phase velocity of a plane wave v <ζ c). Accordingly, along with the parameter qlph, new parameters (v/c)qlph and (c/v)qlph also appear. In particular, as we have already seen, the condition {v/c)qlph > 1, equivalent to the inequality Qqxph > 1, was a criterium for the applicability of the concept of phonon scattering on dislocations. It will be shown below that the inequality (c/v)qlph < 1 is the condition for a dispersionless effective viscosity, ηί]Μ, instead of the usual criterium qlph < 1 that can be applied to ultrasound where v = c.

The solution given in eq. (158) of the kinetic equation allows us to calculate explicitly the expression for the tensor ηίΜ^, Qq) with complete account of both time and spatial dispersion. Combining eqs. (134), (146), (154) and (158), the form of the tensor ηίΜ can easily be established. Furthermore, in order to obtain an expression applicable in the description of ultrasound damping as well, we shall keep along with terms of the order qlph terms Qqrph, omitting as before terms of the order (qlph)2 and higher. The result may be expressed in the form

Mthel _i Mphvs

%Μ(9,Ω,) = η'*ΙΙη/' · (160) Here, p = x^qj/Qq is the adiabaticity parameter, χ0 is thermal diffusivity tensor:

Xij ^ Σ ^ . <i«>

η^ξι and η^Β are effective viscosities, describing respectively thermoelastic damping and Akhiezer damping for a sound wave with wave vector q and frequency Qq. The analytical expressions for these quantities,

h2 rlJ rkl v π η „thel _ n l 01 0 XmnHmHn (\c~>\ η^'--~ΤΞ^~Ω^~' ( 1 6 2 )

phvs " V 1 WOnÔ _ ^00*n ) WnO * 0 — ^00-* n ) (AC~>\

lim = -ψ Z J ^2~z > (163)

662 V.l. Alshits

coincide with the corresponding formulae in the microscopic theory of damping of long-wavelength ultrasound (Balagulov et al. 1970). This comparison illuminates the physical meaning of the tensor χυ [eq. (161)].

In the theory of ultrasound attenuation, the fact that when qlph <| 1 a sound wave is adiabatic, p <ζ 1, is used (indeed, since χ ~ clph and Qq ~ cq one has P ~ <z'Ph ^ 1)· Then it is permissible to neglect the dispersion of the phonon viscosity and the thermoelastic damping. However, when it comes to the analysis of the damping of the dislocation wave packet given in eq. (123), such neglect is permissible only for the longest waves, since the condition p < 0 now is equivalent to the inequality qlph <ζ v/c.

The other important difference from ultrasound consists in a relation between the two terms in eq. (160). In the case of a sound wave, thermoelastic losses of energy and phonon viscosity give comparable contributions to the damping. As was noted by Balagulov et al. (1970), at high temperatures, phonon viscosity will be somewhat more effective since the first term in eq. (160) contains an additional small parameter proportional to the square of the average ratio between the phonon group velocity and the phase velocity. For the partial waves in the packet given in eq. (123), the situation is basically different since this parameter becomes large ~(c/v)2. Accordingly, thermoelastic damping turns out to be approximately (c/v)2 times more effective than phonon viscosity in this case.

Summarizing, one may say that when qlph < 1 a specific feature of the damping of the dislocation waves in the packet given in eq. (123) is an overwhelming predominance of thermoelastic dissipation with a very pronounced time and spatial dispersion. The effective viscosity will be

Wf.0,)= Tsc (χΜγ + Ωγ <1 6 4>

When writing eq. (164), we made use of the relation between S00 and the specific heat C

T^r 0 dna (T 2

soo=^ää=-^-ic The possible role of thermoelastic processes in dislocation drag was first

noted by Eshelby (1949), and other estimates in various approximations followed (Weiner 1958, Lothe 1962). The contribution of the thermoelastic mechanism to energy dissipation is determined by the combination of eqs. (124) and (164). Replacing the summation by an integration in eq. (124) and introducing a cut-off qm = lph

l in the integral, one has

D = h 4

T3C 4 r\l\ Tîi 9\2 K.mnQ.mQn Ω>\Π4\>^ Τ:™"η2- (165)

? < / ρ ν ( 2 π ) 2 " ° «' (Xmnqmqa)2 + Ω\


It is simple to see that the basic contribution to the integral, eq. (165), stems from the region q ~ {v/c)l~^, which is not close to the upper limit. Accordingly, the precise choice of this limit is not very important; the result will only depend logarithmically on the upper limit. The integral in eq. (165) also turns out to be insensitive to the dislocation-core dimensions since it describes dissipative processes in parts of the crystal which are further away from the dislocation than the mean free path /ph.

Confining ourselves to a calculation of the integral in eq. (165) in an isotropic approximation, we put xu = χδ^ (χ « ic/ph). By eqs. (154), (145) and (17), Γ(/ has in this approximation the form

(166)

where ηί and ηΧ are the relative contributions to the specific heat from longitudinal and transverse phonons, respectively, v is the Poisson ratio, K is the bulk modulus, m, n and / are the Murnaghan moduli.

The diagonal form of the tensor Γ% shows, as expected, that a non-vanishing contribution to the dissipation is given only by the dilatational part of the dislocation field ug

lh which produces a local change of the density of the crystal. For an edge dislocation,

l - 2 v \ 2 fr2sin2(? l"7/l2 = ^ — - — ^ > (167)

where the angle φ is counted from the direction of the Burgers vector b. Substituting eqs. (166) and (167) into eq. (165), one obtains

Blhel = D/v2 = ^f — ln-, (168) 64π χ ν

where K + m-^n „ f l -2v[~l -2v/5 fc + m + Λ (\

+ 3 μ (169)

Note that if g is replaced by y^ in eq. (168), we would find Lothe's result (Lothe 1962).

Expression (168) shows that with decreasing temperature the effect considered rapidly 'freezes out', Bthcl ~ r4/ 'Ph( T). It should be remembered, however, that the above solution of the kinetic equation was obtained under the condition of not too low a temperature. It may be shown (Alshits and Malshukov 1972), that in the region of low temperatures where umklapp processes practically are absent, the value Bthel(T) 'freezes out' even more rapidly than predicted by eq. (168). On the other hand, in this region, thermoelastic damping is in any case

664 V.l. Alshits

certainly weaker than damping due to other phonon mechanisms. Thus, any refinement of eq. (168) for the low-temperature region seems to be unwarranted.

At high temperatures, T > 0, when the specific heat practically becomes constant and the mean free path /ph decreases proportionally to Γ - 1 , thermoelastic damping increases with increasing temperature according to the low Bthel ~ Γ2 , i.e, more rapidly than linearly with temperature such as does the phonon.-wind damping. In this temperature region, the parameters r\i and η{ may be replaced by ^ and f, respectively, and the value g turns out to be temperature independent. Numerical estimates for copper gives g « 2. A comparison of the relative roles of thermoelastic damping and a background of other anharmonic effects will be presented below.

5.5. The kinetic equation in the region qlph > 1

When analysing the kinetic equation, eq. (140), in the region qlph > 1, we shall, as before, suppose that the inequality Qqxph < 1 is satisfied for any q. In that case, as has already been mentioned, the non-vanishing contribution to the energy dissipation is exclusively related with phonon transitions between 'quasi-degenerate' states, eq. (138). In contrast to the long-wavelength region, where the inequality (138) is satisfied practically for any state a [see eq. (143)], when qlph > 1, states a = (k, s) and β = (q — k,s') cannot be quasi-degenerate in the sense of eq. (138) at all points of the Brillouin zone (even when 5 = s'). The geometrical image of states satisfying eq. (138) in the short-wavelength region, qlph > 1, for a fixed frequency ωα is the vicinity of the intersection of two isofrequency surfaces ωα = ωβ = const, shifted relative to each other with a vector q. In the absence of phonon spectrum singularities, the surfaces intersect along a line which, with a change of ωα, gives rise to some surface S. In this case, eq. (138) is fulfilled in a thin layer over the surface 5, the layer thickness being of the order Z^1, and the phase volume of the layer will roughly be Ωχ ~ &B/kDlph (ΩΒ is the phase volume of the first Brillouin zone).

Assume now that, on the isofrequency surface ωα = const., there is a flat part containing q and which is longer than q along q. Then, the common part of surfaces ωα = ωβ = const, at s = s' evidently contains not only a line but also a portion of flat surface and this gives rise with a change of ωα to an increase Qs of the phase volume where eq. (138) is satisfied. It may, more precisely, be shown that eq. (138) is fulfilled on quasi-flat parts of surface with a Gaussian curvature \K\ < 1/σ3/2/ρ1ϊ (σ is a phase area of flattening) with a corresponding expansion of the phase volume Ω5. We shall call such parts of the Brillouin zone the singular regions Ks (in contrast to the rest of phase space, which will be denoted as the K{ region), and we shall assume that the sizes of the regions are determined mainly by the structure of the phonon spectrum and that they depend only weakly on the magnitude of rph.


Of course, as a rule κ = Ω3/ΩΒ <ζ 1. However, it is quite obvious that the physical importance of the singularities of the phonon spectrum is determined by the value of Qs relative to Ωι rather than relative to the total volume ΩΒ. On the other hand, we shall also need the small parameters Ω5/ΩΒ and Ωι/ΩΒ to obtain an asymptotic solution of the kinetic equation, eq. (140), when qlph > 1. Fortunately, knowledge of these asymptotics turn out to be sufficient for our purposes, just as we earlier similarly could limit ourselves to solving the same equation in the opposite asymptotic region qlph <ζ 1. Indeed, as we shall see, the partial waves in the packet given in eq. (123) which have wave vectors q ~ lph

x

give a relatively small contribution to dissipation and the corresponding integral over the region qlph > 1 is practically independent of the lower limit.

Let us represent the kinetic equation, eq. (140), in the form

1-ϊ(ωβ - ωα + flf) + 2 y ] / „ = Γζβ + £ J\\U (170) σ,δ

Here, the last term on the right-hand side is an ingoing term in the collision integral. Explicit coefficients Ja

aö

ß may easily be found by the comparison between eqs. (170) and (140).

Unfortunately, in the region qlph > 1, it is impossible to transform the kinetic equation, eq. (170), to a usual Boltzmann type equation, eq. (147), since now the quasi-degenerate character of initial (a) and final (ß) states does not at all ensure proximity of these states. Accordingly, in the region qlph > 1, even when eq. (138) is fulfilled, the input part of the collision integral in eq. (170) contains transitions between states ωσ and ωδ, which are not necessarily quasi-degenerate, i.e.,

When qlph > 1, the situation is made easier because of the small value of the input term of the collision integral. There are good reasons to neglect this term. Divide the sum ^ / ^ / σ ί in eq. (170) into terms corresponding to conditions |ω^ — ωσ\ > Tph1 and \ωδ — ωσ\ ^ τ " ^ , respectively. In the first region, the function fad may be estimated to the lowest order by perturbation theory,

/.*(*, Of)*> ϊΓ]' Μ , · · (171) H ω^ — ωσ + üq + 2ιε

Here, ε is an infinitesimal positive damping constant. This region, which when qlph> 1 occupies a predominant part of the volume of the first Brillouin zone, gives a small contribution in the collision integral because the value of the function/^ itself turns out to be small. In the second region, this function is not small and it may be estimated as/ff<5 ~ Γ™δτρϊί. Nevertheless, the second term of the input part of the collision integral is also small because of the small value of the phase volume of the region \ωδ — ωσ\ < τ~^. In a rough estimate Σ ^αβ ~ l/T

Ph a n d \ωδ — ωσ\ ~ qC (f°r t n e Kf region) which, combined with the

666 V.l. Alshits

above considerations, give

».a

In view of the last inequality, when qlpb P 1 the solution of eq. (170) is given by eq. (171) with the infinitesimal damping ε replaced by a finite value y = γ{ω„), eq. (141),

This shows that in the region qlph > 1, independently of what the relation between the values of |ωα — ωβ\, Qq and y is, one may use the τ-approximation for the solution of the kinetic equation. Note, however, an important specific feature of the solution given in eq. (172). Instead of a phenomenological relaxation time, it contains an analytically given quantity l/y(coa), eq. (141), which has the clear physical meaning as the average time between phonons going out of state a.

However, in the K{ region of reciprocal space, the dependence offaß on y still turns out to be inessential* since up to small corrections, ocy, the function

R e / . ( , . a , ) - w < _ m ^ / + v (173)

is actually proportional to the Dirac delta-function, reflecting the law of the conservation of energy for phonon scattering from state a into state ß in an external field oscillating with frequency Qq;

Re/^(», ß f ) * πΓ:βδ(ωα - ωβ - Qq). (174)

In the Ks-region, we have, according to eq. (173), a quite different behaviour of the same function,

**ί*β(9,Ωι)~δ,,Μ.Γ:β/2γ. (175)

Combining eqs. (142) and (173) and keeping eqs. (174) and (175) in mind, we derive from the main-order term in an expansion in terms of v/c an energy

* Indeed, the sum over ka in eq. (142) is obviously proportional to the product of Re/a/? = F£p/y and the phase volume Ωι ~ y, and, consequently, does not depend on y in a first approximation.


dissipation per unit time;

^ · ' "*' 8ωα l ( e ) , - f l ) , ) 2 + 4 y 2 j

2π .|κ5(2π)3 8ωα / J (176)

Before passing on to explicit estimates of the dislocation drag by eq. (176), which will involve a number of simplifying model assumptions, we should note that eq. (176) is generally characterized by high-temperature asymptotics of the form AT + const., since when T > Θ the quantities 9ηα/9ωα and γ are proportional to the temperature. Clearly, this will be an important point in comparisons with experimental data.

5.6. Discussion of phonon-wind estimates

Evaluation of the integral in eq. (176) with full account of anharmonicity, anisotropy and other specific features of real phonon spectra requires computer calculations. Such calculations have been accomplished by Martin and Paetsch (1975,1976) and Paetsch (1975,1976), where the general expressions obtained by Brailsford (1972) were tabulated for a number of crystals. However, for the purposes of this chapter, an analytical estimate that reveals general features (Alshits and Indenbom 1974, 1975a, b,c) seems more appropriate.

The first term in eq. (176), obtained earlier, eq. (97), in the Born approximation, describes phonon wind and may be estimated in the usual Debye model. Because of the small magnitude of the parameter /c, the integral over the Kf region may be expanded to an integral over the total phase volume within the Debye sphere. The lower cut-off limit, q ~ /"h1, is also unimportant. However, it would be incorrect to use the estimate given in eqs. (107)-(109) in its literal form obtained above for the region of low temperatures.

With increasing temperatures, when the average phonon wavelength becomes comparable with the size of the dislocation core, a more accurate description of the elastic field near the dislocation is required. In the continuum theory, a dislocation is considered as a linear singularity giving rise to an elastic field, which increases with 1/r as one approaches the line. Certainly, in a real crystal, discreteness makes the continuum model inapplicable in the near vicinity of a dislocation core. In principle, the presence of a core may be allowed for within the framework of the model description by a smoothly varying factor which

668 V.l. Alshits

removes the singular line behaviour. For example, one may as a quite reasonable model (Lothe 1962) use

utjir) = u?/r)(l - e " " n (177)

where u^r) is the strain tensor of a dislocation in the continuum approximation. The model radius r0 for the dislocation core must be of the order of several lattice parameters. It can be shown (Alshits et al. 1979), that for a straight dislocation the Fourier transform of the elastic field, eq. (177), is

M- υ (178) ij y i + ^ o ) 2 '

where uf. is the Fourier transform of the function u^r). Substitution of ttf as given by eq. (178) into the first term of eq. (176) [or into

eq. (97)] gives a modified estimate of the damping constant (Alshits et al. 1979) similar in form to eq. (107),

* -4 (£ ) ' * ( Ϊ ) · Here, the factor g is, as before, given by eq. (109), and the temperature dependence of the drag is determined by the function

5 [llx dttV ~X Jo (e 7 ^?

arctan ßxt ßxt fßW = χ5 I 7-Γ-.ΓΤ2 ^ 7 ^ > (180)

where ß = 2/cDr0. Compared with eq. (108), eq. (180) contains the additional factor arctan ßxt/ßxt which reflects smooth behaviour, eq. (177), of the elastic field at the dislocation core. Of course, with r0 = 0, this factor turns into unity and the function fß(x) then coincides with/(x). As should be expected, at low temperatures (T < 9t/ß\ this renormalization is inessential, since then the main contribution to the integral in eq. (180) is related to magnitudes t ~ 1 so that ßxt <ζ 1 and arctan ßxt % ßxt. Such low-temperature behaviour of the function fß(T/6t) reflects in a quite natural way the physical property of long-wavelength phonons that they are quite insensitive to small scale changes of the disturbing field, changes within distances small in comparison with the wavelength.

With increasing temperature, the function^ (x) fairly rapidly passes from the fß(x) ~ x5 dependence to a linear dependence with a slope depending on the parameter ß;

f,(x) * ^ [arctan/? - ^ ~ ^ f + 1} 1 » fr, / U 1; (181)

«x /2ß , ßpl.


Fig. 6. Plot of the function fß(x) for different values of the parameter ß.

Fig. 7. Plot of the function fß(x)/fß(l) for different values of the parameter ß.

For usual estimates of kD and r0 ~ (l-3)ft, the magnitude of ß comes out to be of the order 10-30. Therefore, it follows from eq. (181) that at high temperatures (T >0 t) allowance for a finite size of the dislocation core reduces the estimate of the phonon-wind contribution to dislocation drag with a dividing factor \ß = /cDr0, i.e., a reduction of about one order of magnitude*. Figure 6 shows plots of the function^ (x) for different values of the parameter ß. Note that the ratio fß(x)/fß(l) is fairly weakly dependent on ß (fig. 7).

* This eliminates the contradiction between theory and experiment, which was noted by Brailsford (1972) for the case of copper crystals.

670 V.l. Alshits

5.7. Relaxation of'slow'phonons

The second term in eq. (176) is sensitive to details of the phonon spectrum and only very crude analytical estimates are possible. We shall, below, try to formulate a reasonable model for the singular region Ks which allows a rough analytical estimate.

As already explained, the region Ks is that part of phase A>space where there are flat regions on the isofrequency surfaces of the phonon spectrum which are orthogonally oriented relative to the dislocation. It turns out that the presence of such flat regions is a fairly typical feature of phonon spectra of many crystals (Walker 1956, Wolfe 1980, Eisenmeger 1981, Northrop et al. 1983, Chernozatonskii and Novikov 1984). It is clear that the 'flat surface' relaxation mechanism of dragging must be strongly anisotropic with respect to dislocation orientation. This effect must intensify when the dislocation comes into an orientation where it is parallel to the group velocity of the phonons belonging to one of the flat regions of the isofrequency surface. We are dealing with a dissipation of energy which is determined by phonons which move 'slowly' away from the dislocation. Hence, the term 'relaxation of slow phonons' for this mechanism. This term was introduced by Alshits and Indenbom (1974, 1975a, b,c), where also the possible special importance of that part of phase space which is associated with the vicinity of the Brillouin zone boundary where the group velocity is reduced and where the phonons, thus, literally are slow, was discussed. In this latter region of the spectrum, the condition given in eq. (138) could in principle be fulfilled independently of the dislocation orientation.

Let the 'slow' phonon region Ks be relatively small (κ <ζ 1) and sufficiently well distinguished from other regions, so that one may neglect transitional regions and may assume that \ωβ — ωα\ <ζ y throughout all the Ks region. Suppose further that the area of the flat regions has a weak frequency dispersion so that the deviations from the average value σ are insignificant in the range of frequencies in question, lying between comin = ω8 — Δ^ω and comax = ω8 + Δ^ω. Finally, we neglect dispersion in the phonon group velocity vg and estimate it as vg « Δω/Δ&ζ, where Akz is the size of the Ks region along the dislocation.

Substituting eq. (102) with a renormalized strain, eq. (178) into the second term of eq. (176), one obtains

' * d3kx δη, ωΐ Cd2q {q-vf Ks (2πγ 8ωα 7(ω.) J (2π)2 q\l + q*rl) **■ K™1)

Summation over s is omitted in eq. (182), since for the sake of simplicity it is assumed that for a given direction of the dislocation a singular Ks region occurs in only one phonon branch. The limits of integration in eq. (182) should be chosen so that the vectors ka and kx + q are kept inside the Ks region. It is convenient to carry out the integrations over ka as well as the integration over q

D = hbz


in a cylindrical coordinate system where the z-axis is directed along the dislocation and is perpendicular to the flat region of the isofrequency surface in the Ks region;

Λ hb2v2 Γ dna ω2 Γ2π J fMw)

D = T\dL·—-—— dwk d 8i27r)5J z8coay(c»a)J0

φ* ]0

r 8ωαν(ωα

'2π d ^ c o s 2 ^

(<?*) d^fi 2^,2* (183)

'y/ki + klJJo l+rlq

Next, we pass from integration over kz to integration over ωα according to the recipe

Δω ίΗ dωα, (184)

and, finally, make the approximation of replacing the variable kz in the last argument of the function Faß with an average value cos/cs. This gives

hb2v2Qsln(l + dr\ D = g

Δω , ω2 9n Δω ?(ω) θω 16(2π)5Γ2Δω

where the following notation has been introduced,

(185)

'2π d<p. Γ d/cp/cp

'2π dq>9cos2 q>qFaß[ φ,,φ*, ω.

' A 2 + W 2

xln[ l + τ^ΐΧφΛ] = gäln(l + στ20). (186)

The value of g, as well as the value of g, eq. (109), is of the order of magnitude determined by the ratio \Α/μ\2. However, depending on the actual anisotropy of the crystal, it may differ considerably from g to either side. We are not going to make here any explicit calculations of this factor. In the remainder of this chapter, we will keep it as a phenomenological parameter.

Since the Ks region can be supposed to be of small extension, dispersion in the damping term y(a>) can be neglected and γ can be replaced by an average value y~Tph- This gives the following expression of the damping constant Bs = D/v2,

B* = Out h ( kDb

2π ΜΛΤ/θ,),

where

] 9 ^ ^ l n ( l + i r J ) , Te = Tph(0s), 0, = h(os; 9 P

2 Λ(1+«)/χ Att2e·' f(x\ = h.?-\ att e

MX) r e 2 a j ( 1 _ , ) / ; c ( e < - l ) 2 a = -

Δω 2ω'

(187)

(188)

(189)

672 V.l. Alshits

Thus, the magnitude of the damping constant Bs and its temperature dependence are related to the relaxation time rph, which may be found from experimental data on the thermal diffusivity χ of the crystal from

l p h {T)*= 3χ(Γ) (190)

Taking into account that the functionfa(T/es) decreases sharply at low temperatures, a simple approximation,

*P*(T): ■r (191)

may be quite satisfactory in many cases since in the temperature region where the dependence τρ11(Γ) begins to deviate markedly from eq. (191), the relaxation of 'slow' phonons is often already 'frozen out' and inessential compared with a background of phonon wind. Then the temperature dependence in the relaxation component of the energy dissipation may be characterized by a more simple function (fig. 8),

x Π1+*>/* d i t V (192)

instead of£(x) [eq. (189)]. Furthermore, γ(ω) may be determined by means of the analytical expression,

eq. (141), which allows one to take into account a frequency dispersion in the relaxation time. Such calculations, carried out in the Debye approximation, lead to replacement of φΛ(χ) by another function φα which is shown in fig. 9. It is easy to see that for sufficiently small values of a the curves φα(χ) and φα(χ) are similar both qualitatively and quantitatively. However, for larger values of a the function φΛ(χ) turns out to be appreciably more sensitive to a change of a' = (1 - a)/(l + a) = wmin/a;max and when a' < 0.3 it begins to show non-

Ψ-Μ

Fig. 8. Plot of the function φΛ(χ) for different values of the parameter a' = (1 - a)/(l + a) = wmin/wmax.


Fig. 9. Plot of the function φΛ(χ) for different values of the parameter α': (1) 0.1; (2) 0.2; (3) 0.3; (4) 0.4; (5) 0.6; (6) 0.9.

monotonic behaviour with the formation of a low-temperature maximum. Certainly, we should not pretend that our model, based on the assumption of a small volume for Ks region, gives a quantitative description of this maximum. Nevertheless, the obtained result does indicate the principal possibility of the appearance of low-temperature anomalies in dislocation drag when the Ks region stretches to sufficiently low frequencies.

However, the situation that the boundary frequencies comin and ct)max do not differ much, seems to be a more typical situation. Taking into account a weak independence of the function φα{χ\ it is reasonable to replace it by the simpler function φ0(χ) corresponding to the limiting value of φα(χ) when a -» 0,

φ0(χ) = 1 Λ/χ

(193) x2 (ellx - l ) 2 '

As shown by Alshits and Indenbom (1974, 1975a, c), even the simplest function

/β(Τ/θ) + λθΨο(Τ/θ), (194)

with reasonable values of λθ, describes quite well the experimental data on the temperature dependence of B(T) for a number of crystals. Alshits and Petchenko (1976), in processing experimental data, used the function /o(x) = l imôX(x) ,

„es/r fo(T/et) = x(T) e.

χ(θ.) r ( e e ^ - l ) 2 ' (195)

instead of φ0(χ), eq. (194). Below, we shall confine ourselves to the simple variant, eq. (194), in the discussion of experimental data.

674 V.l. Alshits

5.8. The role of optical phonons; the hierarchy of anharmonic mechanisms in dislocation drag

So far, in all estimates and calculations we have only considered acoustical phonons to be present in crystals. However, phonon spectra of crystals containing more than one atom per elementary cell also contain optical modes where ω(0) ΞΞ ω0 = Θ0/Η Φ 0.

At low temperatures, T < θ0, the density of optical phonons is exponentially small and their contribution to the energy dissipation may be neglected. In the high-temperature region, Τ>θ0, however, the dissipative processes in the system of optical phonons are quite analogous to the processes involving acoustical phonons considered above. Optical phonons can simply be taken into account by an appropriate renormalization of the constants g, g and g. This renormalization is not very important - the contrary conclusion maintained by Martin and Paetsch (1975, 1976), Paetsch (1975, 1976), is based on a misunderstanding. The authors (Martin and Paetsch 1975, 1976, Paetsch 1975, 1976), following Brailsford (1972), use the factor Γχ ~ yGcoa as a measure of the interaction of phonons with a dislocation, instead of the more correct factor Γ2 ~ {A^)c2kk'/y/ojaojß. Of course, by employing such a description, they overestimate the role of optical modes by a multiplication factor Ι Γ 7 / / Τ Ι 2 : \Γγ/Γ?|2 - (co0/cfc)4, i.e., at high temperatures by a factor (θ0/θ)4

and at low temperatures by a factor (0Ο/Γ)4. A more essential contribution to dislocation dragging is given by 'soft' optical

modes arising in some crystals in the vicinity of phase transition. Unfortunately, the resulting singularities in the temperature dependence of B(T) predicted and theoretically investigated by Alshits and Mitlianskij (1980), Levanjuk and Shchedrina (1983) seem hardly observable.

The estimates of thermoelastic damping, eq. (168), the phonon wind, eq. (179), and the relaxation of 'slow' phonons, eq. (187), allow us to carry out a comparison of their relative roles in dislocation drag. At low temperatures, when relaxation mechanisms give an exponentially small contribution to energy dissipation, scattering processes prevail. At high temperatures, when the function fß(T/6) is practically linear with temperature and when the heat capacity C(T) and the function fa(T/0) tend to constant values, the total damping constant B = B thel + Bw + Bs has the structure (three components with the same meanings and which occur in the same order)

B = B0IA(JY + J +2βλθ (196)

Here, A ~ 10-1(r0//e)ln(c/i;) < 1, which means that thermoelastic losses of energy usually are hidden by stronger losses due to phonon wind. At high temperatures, due to their square temperature dependence, thermoelastic losses


may in principle emerge and be separately observable. However, experiments of higher accuracy are required for such observations.

If the Ks region occupies a relative volume κ ~ ro/l0 in the first Brillouin zone, 2βλθ becomes of the order unity, i.e., the 'slow' phonon relaxation then gives a contribution to dislocation drag comparable with that of scattering. Such conditions are not unrealistic.

Generally speaking, the determination of λθ calls for fairly complex computations. Let us alternatively consider λθ, like the Debye temperature, as a phenomenological parameter to be determined by comparison with experiment. As regards the temperature 0S, in most cases it is apparently sufficient to put 0S = 0, since the 'slow' phonon regions usually are close to Brillouin zone boundaries.

6. Dynamic dragging of dislocations in the Peierls relief

6.1. Raman scattering of phonons by dislocations

Due to lattice discreteness, the atomic configuration of the dislocation core and the elastic energy change periodically as the dislocation moves through the crystal. The corresponding potential relief is called the Peierls relief (Peierls 1940) in the literature. The oscillations of the elastic field of the dislocation gliding in the Peierls relief should provide an additional channel of energy dissipation: inelastic (Raman) scattering of phonons in the alternating field should take place. This effect was predicted and estimated by Alshits (1969a).

Consider a straight dislocation parallel to the z-axis moving in the Peierls relief in the glide plane XZ with an average velocity v. Then the dislocation energy E (per unit length) oscillates with a frequency proportional to the dislocation velocity v,

E = E0 + £P sin2 π (v/a)t. (197)

Here, E0 is the constant part of the energy, EP = aPab/n is the Peierls energy, σΡ is the Peierls stress, and a is a lattice period in the direction of motion. Oscillating terms will be present in the strain field of the dislocation as well, and accordingly in the interaction Hamiltonian between phonons and the dislocation.

One should distinguish between two types of oscillations of essentially different origins - dynamic oscillations and configurational oscillations.

Dynamic oscillations of the elastic field are related with non-uniformity of the dislocation motion in the potential relief. As was shown by Alshits (1969a), in the dynamic region of velocities (v/ct)2 > σΡ/μ, corresponding to a kinetic dislocation energy much higher than the Peierls energy, the considered oscillations are small and may be neglected.

676 V.l. Alshits

Configurational oscillations due to periodical changes of the dislocation core structure might be more essential. The corresponding periodical disturbances in the strain field may be estimated within the framework of the following model; represent the pulsating dislocation as a superposition of a stationary dislocation and a dislocation dipole oscillating in the glide plane XZ with the frequency Ω,

utJ(r91) = iig(r, t) + Cb dUij^ t] sinQt. (198)

Here, ufj(r, t) is the strain field, given in eq. (123) of the uniformly moving dislocation [renormalized according to eq. (178)] and ζ is a small dimensionless parameter. Substitute eq. (198) into the expression for the dislocation energy,

d3r((jyiiy + pu2). (199)

Calculation of the integral in eq. (199) for a screw dislocation in the isotropic approximation shows that the dynamic component in eq. (199) may be omitted up to terms of the order (ν/ο)2(μ/σΡ)112. The result of the integration agrees with eq. (197) for

C*£(°^Y ' 2 , Ω = πν-, (200) b \ μ ) a

The corresponding Fourier amplitude in eq. (123) contains, in this case, an oscillating component,

w?. = w?(l+i ib4*sini2i) . (201)

The interaction Hamiltonian between phonons and the dislocation may be obtained by replacement of u^n by w^n, eq. (201), in eq. (37). Then, the first term in the renormalized Hamiltonian describes, as before, phonon wind, but the second term gives a new effect, Raman scattering,

HRam(t) = iift Cb X ^ Γ ^ { β % β " ν s i n Ot. (202)

Further, in analogy with eq. (97), one has

ß R » m — £ ( Ω 2 + β 2 ) ( ^ ) 2 | Γ ^ | 2 ^ 5 ( ω β - ωβ). (203) α,ρ

The first component in eq. (203) gives less dissipation than the second component, since at low temperatures this term contains a factor ~(Τ/Θ)2, and at high temperatures it differs from the second term by a numerically small multiplying factor (a/2nr0)2. Neglecting for simplicity the first term in eq. (203), we see that BRam differs from eq. (97) only by a constant factor

BRam = i(nC)2Bw. (204)


Thus, Raman scattering of phonons leads to a viscous drag of dislocations, which might be appreciable compared with the background of phonon wind in crystals with a sufficiently high Peierls relief, σΡ/μ > 10" 3. In crystals with a low Peierls relief, the effect can be ignored.

6.2. Radiation dragging of dislocations

The potential relief due to discreteness of the atomic structure of the crystal prevents free motion of dislocations in the glide plane. In order to set a dislocation lying in a valley of the relief in motion, one must apply some 'starting' stress to it, the Peierls stress σΡ. The quasi-static properties of the relief may be described within the framework of the well-known models of Peierls-Nabarro (Peierls 1940, Nabarro 1947, Foreman et al. 1951) and Frenkel-Kontorova (Kontorova and Frenkel 1938, Indenbom 1958, Hobart 1965a, b, Dubnova and Indenbom 1966). One should take into account, however, that in stationary motion of the dislocation the influence of the relief does not at all reduce to a static resisting Peierls stress σΡ. Actually, in the absence of dissipative processes, the dislocation motion through the relief will be accompanied by successive transformations from potential energy to kinetic energy and vice versa. External stress would not be necessary at all to maintain such motion. However, completely dissipation-free motion is an idealization. Even at absolute zero temperature, when the phonons are completely 'frozen out', radiation friction remains as a channel of dissipation. Elastic waves will be radiated due to configurational oscillations of the dislocation core and due to dynamic oscillations of the dislocation velocity.

Radiation related to periodic changes in the dislocation core configuration was first considered in a simple way by Kontorova and Frenkel (1938). Next, the same model and some of its modifications [see, e.g., Maradudin (1958)] were used as basis for a number of more detailed investigations (Weiner 1964, Atkinson and Cabrera 1965, Rogula 1967, Celli and Flytzanis 1970, Ishioka 1971, Flytzanis and Celli 1972) of the radiation effect. In particular, Atkinson and Cabrera (1965), Rogula (1967), Celli and Flytzanis (1970), Ishioka (1971) predicted a series of radiation resonances, which become infinitely closely spaced as the velocity decreases to zero, while they found no singularities in the vicinity of the sound velocity. It will be recalled that in continuum theory the dislocation energy tends to infinity as the velocity approaches the sound velocity.

The above results (Atkinson and Cabrera 1965, Rogula 1967, Celli and Flytzanis 1970, Ishioka 1971) were not corroborated by subsequent papers by Ishioka (1973), Earmme and Weiner (1973, 1974). One of the most important conclusions of Ishioka (1973) was that the piece-wise harmonic approximation for the atomic relief that had been used was unsuitable in the description of motion at low velocities because the model did not allow stationary solutions in

678 V.l. Alshits

that velocity region. On the other hand, Earmme and Weiner (1973,1974) noted in their revision of the results of Atkinson and Cabrera (1965) a failure of the stationarity conditions for dislocation motion at near-sonic velocities v > vc (vc< c). According to Ishioka (1973), there are no difficulties with smooth potentials, which have a continuous first derivative. Computations for a sinusoidal potential and for two types of smoothly joined together, piece-wise parabolic potentials led to a radiation dragging monotonically decreasing with decreasing velocity v. According to Ishioka (1973), at near-sonic velocities there is a sharp dynamic wave zone in the dislocation field. In the process of motion, the dislocation radiates elastic waves intensely. Then, radiation dragging greatly exceeds the static Peierls stress σΡ. With a decrease of the dislocation velocity, 'relativistic' effects weaken and the level of radiation friction becomes considerably reduced and the dislocation field will be more like a quasi-static field, eq. (123), uniformly moving together with dislocation. It is essential that motion of such quasi-static character starts at fairly high velocities where the level of the kinetic energy considerably exceeds the Peierls energy so that the deviations from uniformity of the dislocation motion still may be supposed to be negligible. This is the reason why we are allowed to divide the radiation friction into configurational contributions and dynamic contributions.

The first attempt to estimate the role of dynamic oscillations in the radiation drag of dislocations was made by Hart (1955) in a single-mode approximation. It later became clear that only the high-velocity asymptotics of radiation friction may be obtained this way. A correct solution of the problem requires a self-consistent determination of the dislocation law of motion in the periodic potential field, with the effect of the radiation reaction included. An approach along these lines, outlined by Alshits (1968), was carried out by Alshits et al. (1971).

According to Alshits (1968), Alshits et al. (1971), at high velocities, when the kinetic energy of the dislocation considerably exceeds the Peierls energy, the presence of the relief insignificantly disturbs the uniformity of the dislocation motion. The dislocation mainly radiates in the basic mode and, in accordance with the Hart estimate (Hart 1955), the radiation friction decreases with increasing velocity proportionally to v~2. With lowering of the velocity, v, the dislocation motion becomes more and more non-uniform. Then, radiation in higher harmonics becomes more and more pronounced and the radiation losses of energy increase. Decrease of the average velocity v is possible only down to some critical value v0 ~ Cy/σρ/μ, which corresponds to a motion where the dislocation has zero kinetic energy on the top of the relief.

Alshits et al. (1971) considered the same problem in a more general formulation where an additional force of viscous drag/vs = Bx acting on the dislocation was taken into account. According to Alshits et al. (1971), the critical velocity decreases with increasing viscosity B and, from some threshold magnitude Bc on, vanishes. Stationary motion of any velocity can be realized when B > Bc.


When viscosity is ignored, the dynamic drag, being determined only by radiation friction, is a decreasing function of velocity, implying instability of stationary motion. Viscous dissipation stabilized the motion by adding a term linear in the velocity to the drag force. When B > Bc, a 'dry-friction' type effect should set in: the total drag stress σ approaches the static Peierls stress σΡ as velocity tends to zero. When B > Bc, the function σ(ν) is asymptotically described by a simple expression,

σ(υ) = σΡ coth [ίχτΡ/Βι;], (205)

which exhibits the 'dry-friction' phenomenon clearly*.

7. Electron drag of dislocations in metals

In metals, at low temperatures, when the phonons are 'frozen out', the interaction between dislocations and conduction electrons may be an essential factor. Discussion of this problem will give us occasion to further illustrate some of the ideas and methods employed in the preceding theory for phonon dragging of dislocations.

The properties of the electrons in the field of a moving dislocation will be described by the Hamiltonian

HC = H°C+Hint(t\ (206)

where H° is the Hamiltonian of an electron subsystem in a crystal without dislocations, Hmi is the Hamiltonian for the interaction between the electrons and the dislocation,

Xtj is the deformation potential tensor (the components of Xtj have a magnitude of the order of the Fermi energy eF), and a£ and ak are creation and annihilation operators of an electron with momentum p = hk (the spin variable which is inessential in the following considerations, is omitted).

The energy dissipation De per unit time in the system described by the Hamiltonian, eq. (206), may be expressed (Alshits and Indenbom 1973) in a form

* The analogous problem of the tangential motion along a dislocation of a kink interacting with a secondary Peierls relief was solved by Alshits et al. (1972). Qualitatively, the results are the same as for a dislocation.

680 V.l. Alshits

quite analogous to eqs. (132) and (135),

q k,k'

--ϊΣΣ°ϊΐνΓβ<*·(*ω) hrti q ,J ,J 9ω "-- (208)

where G ^ ( ? , ω) is the Fourier transformation of the two-particle retarded Green function,

GUl,a>) = dxeiax

— oo

χ{-ί^(χ)<[β^/^++^6-^^,ν^ + , ] > } . (209) The brackets < > mean average over the grand canonical Gibbs ensemble.

An effective viscosity tfJnm(q, Qq) of the electron gas analogous with eq. (124), can be introduced. In accordance with eq. (208), we write

(210) nijmnW* Uq) ~ ^ Z J AijAmn 8 β ) k,k' ω = 0

It is interesting that the same Green function determines the dynamic electro-conductivity of metals (Eliashberg 1961) as well,

. »)-»(;)·Σ<»-*«^*ν^*ν-Η· (211)

Here, e and m are the charge and the mass of an electron, respectively. As in the case of the analogous phonon problem, in the long-wavelength region, qle < 1 (/e is an electron mean free path), calculation of the function G$k,(q, ω) calls for summation of singular 'ladder' diagrams, which, according to Eliashberg (1961) is equivalent to solving the kinetic equation for the electrons. It may be shown that this cumbersome procedure leads to the constant Mason electron viscosity (Mason 1955) for the region qlc< 1,

η' - N eeF i e , (212)

where Nc is the density of conduction electrons and te is the relaxation time of electrons. This expression is quite applicable for describing damping of a low-frequency ultrasound. It may as well be used for estimating the contribution to the dissipation, De, from the damping of the long-wavelength part (qle < 1) of the packet given in eq. (123). In this estimate, one must use a cut-off in the integral over q at the upper limit qm ~ Z"1, where roughly speaking the low-frequency regions ends. Apparently, Mason (1964, 1965, 1966) was the first one who formulated the problem of the electron drag of dislocations. Unfortunately,


he groundlessly cut-off the integration at qm ~ l/r0 (as in his theory of phonon viscosity). For a correct choice of qm, the Mason estimate must be multiplied by a factor (r0//e)2 , which is very small at low temperatures. The conclusion is that the contribution of the long-wavelength part of the packet given in eq. (123) to the dissipation, De, may be neglected.

The effective viscosity */e(0, ω), with general time and spatial dispersion taken into account was obtained by Akhiezer et al. (1957), who developed the kinetic theory of electron attenuation of ultrasound in metals. One of the principally important results of Akhiezer et al. (1957) was the proof that the damping of short-wavelength ultrasound (qle > 1) is described by the same expression both in the quantum region (ωτβ > 1) and in the kinetic region (ωτ6 <0 ) . The ultimate reason is the fact that for a sound wave, which moves relatively slowly compared with electrons (vjc ~ 103),

ωτβ = — qle<^ qle. (213)

In the example of phonon damping of waves with short wavelengths (qlp > 1) in the packet given in eq. (123), when an analogous inequality is valid,

Ω,τρ ~ % U < «/ph. (214)

we have seen a similar situation. This is the reason why Kravchenko (1966a), who solved the kinetic equation

for electrons in the field of a moving dislocation, and Holstein (1966), who considered electron scattering at dislocations, independently ended up with the same estimate of the electron damping constant of a dislocation

Be = DJv>J-^, (215)

where vF = y/2sF/m. For ordinary metals, eq. (215) predicts a temperature independent value Be ~ 10"5 P which is by 1-2 orders less than typical magnitudes of the damping constant measured at room temperature and which are determined by the phonon component of the energy dissipation. Thus, as a rule, the electron drag is essential only at low temperatures, when phonons are 'frozen out'.

On the other hand, it can hardly be claimed that eq. (215) is more than an order of magnitude estimate of Bc. Indeed, when we obtained eq. (215), the Fermi surface was supposed to be spherical, the average value of a certain linear combination of the components of the tensor λ§ was replaced by ερ, the integral over q was cut-off at an upper limit qm ~ r$ *, and, finally, umklapp processes were not taken into account, although these should lead to some renormaliz-ation of Bc (Natsik 1976) (but within the same order of magnitude). In view of

682 V.l. Alshits

this, the suggestion of Kaganov et al. (1973) to consider Be as a phenomeno-logical parameter subject to determination by experiment seems to be a quite reasonable suggestion. In real metals, one may expect (especially in metals with a complex Fermi surface) a quite wide dispersion of magnitudes of Be. There is, e.g., experimental evidence (Lenz and Lücke 1969) that in lead crystals Be ~ 10 " 4 P, i.e., of the same order of magnitude as the phonon component of B at room temperature.

The electron contribution to energy losses during plastic deformation is in principle determined by the same dissipative processes in the electron subsystem of the metal which determine the electroconductivity. Some authors (Mason 1964, Huffman and Louat 1967a, b, 1968) have tried to relate the electron drag of dislocations with the macroscopic electroconductivity of metal, and subsequently many discussions arose (Kravchenko 1966a, Tittman and Bommel 1966, Brailsford 1969, Kravchenko 1970). All of these attempts turned out to be incorrect because the basic contribution to the energy dissipation takes place by processes of electron scattering in the short-wavelength part of the packet given in eq. (123), where a macroscopic description is not allowed and where it is required that the spatial dispersion of the electroconductivity is taken into account. There was a similar problem in the theory of phonon drag of dislocations.

The correct relation between dissipation, De, and the dynamic electroconductivity tensor, o^q, ω), was first established by Alshits and Indenbom (1973). The basic idea of this chapter is to take advantage of the possibility to eliminate the function Gfk, between eqs. (210) and (211) when qle > 1. The result is

».=-U-YT w m 1 w"i>(-t0) > <2 i6> 2\eJ g J fip-t* Ή« 1k<ll

where ΛΪ is some average value of the tensor λ\· along the contour L, which is defined as the intersection between two Fermi surfaces shifted relative to each other by the vector hq and mü * is a tensor with components of the order of m~*. The directly established relation between the two kinetic characteristics of the crystal offers, in principle, new possibilities for experimental studies of the deformation potential, which has never been investigated systematically.

Substituting into eq. (216) a known expression (Eliashberg 1961) for the short-wavelength asymptotics of the tensor okl(q, 0), one has

D =iyß2UiM? |2_L_ tef y% l (217)

Here, e* is the energy of an electron with momentum p = hk. Integration over the Fermi surface is implied. The resemblance between eqs. (217) and (176) leads naturally to the idea that there might be an analogue to the 'slow' phonon relaxation in the electron subsystem (Alshits 1974).


The resonant factor in brackets in eq. (217) is usually replaced by a delta-function nö(sk + q — sk), which selects the contour L on the Fermi surface. Thus, τ6 disappears from eq. (217) and the electron drag of the dislocations turns out to be temperature independent. Let now the contour L go through a flat region σ6 of the Fermi surface which is orthogonal to the dislocation line. Then, in all of the flat region σβ, sk+q = &k and the expression in brackets in eq. (217) is identically equal to re as long as hq is inside the region σ6. Accordingly, the integral in eq. (217) may be expressed in the form

άσρ ) h ^ ? - π άσΡ su \ , f d(JP

(218)

The first term on the right-hand side of eq. (218) is related to electron scattering and leads to an estimate of the type of eq. (215). The second term is determined by the relaxation of electrons whose group velocities happen to be parallel to the dislocation*.

Substituting eq. (218) into eq. (217), one may estimate the relative roles of the considered mechanisms,

Be = * o ( l + ? < £ ^ \ (219) r„ σ.

where σ¥ = 4πρ| ls the area of the 'equivalent' spherical Fermi surface, rm ~ max{r0, h/y/ä~c}, and B° is a drag constant determined by the estimate given in eq. (215). As follows from eq. (219), the relaxation contribution to the energy dissipation will be appreciable against a background of scattering only if the flat area σ6 > (r0//e)aF where r0 > ft/^/σ^", or σ6 > (h2GF/r0le)1/2 where ro £ h/yjae. These conditions do not seem so rigid that they could not be realized in some metals. For example, in tungsten a substantial area of the Fermi surface is flat (Kaganov 1971).

Taking into account that the value of /e grows with decreasing temperatures, the discussed relaxation mechanism should lead to low temperature anomalies in the dynamic drag of dislocations, i.e., increasing values of Be with decreasing temperatures. Such anomalies are actually sometimes observed (Mason and Rosenberg 1966, 1967, Mason 1968, Leroy and Offret 1971, Parameswaran and Weertman 1971, Parameswaran et al. 1972), but one cannot be confident that they should be identified with the above-mentioned mechanism.

With reference to the relation given in eq. (216) between the electron drag of a dislocation and the dynamic electroconductivity of a metal, one may expect analogous relaxation anomalies also in the temperature dependence of the

* A corresponding relaxation mechanism related to electrons moving slowly from the dislocation was named the 'slow' electron relaxation by Alshits (1974).

684 V.l. Alshits

tensor o^q, 0). In particular, electron relaxations of flat portions of the Fermi surface should manifest themselves in an anomalous skin effect for definite orientations of the surface of metal. A quantitative treatment would involve a renormalization of the type of eq. (219) in the usual expression for the surface impedance.

We shall not dwell on specific features of the electron-dislocation interaction which have no direct phonon analogies. Such problems as the electron drag of dislocations in external electric (Kravchenko 1966b) and magnetic (Kravchenko 1970, Natsik and Potemina 1974, Grishin et al. 1976,1978,1980) fields, the effect of the normal to superconducting phase transition on dislocation drag (Huffman and Louat 1970, Kaganov and Natsik 1970,1971, Natsik 1971, Barjakhtar et al. 1972), and all the converse problems, the influence of dislocations on the energy spectrum, on the thermodynamics and on the kinetics of the electron subsystem (Hunter and Nabarro 1953, Seeger and Bross 1960, Bonch-Bruevich and Glasko 1961, Kosevich and Tanatorov 1964, Kaner and Feldman 1968,1971, Gutnikov and Feldman 1972, Terwilliger and Higgins 1973, Kosevich 1978, Potemina 1979, Natsik and Potemina 1980), are discussed not only in the above-mentioned, original publications, but as well in a number of review articles (Kaganov et al. 1973, Alshits and Indenbom 1975b, 1986).

8, Analysis of experimental data

8.1. The influence of dislocations on thermal conductivity; flutter effect or non-linear mechanism?

The considerable discrepancies between the first theoretical estimates of the influence of dislocations on thermal conductivity (Klemens 1955,1958, Nabarro and Ziman 1961, Carruthers 1961) at low temperatures and experimental data (Taylor et al. 1956, Kemp et al. 1959, Lomer and Rosenberg 1959) were intriguing and they inspired to further serious theoretical research. Subsequent theoretical investigations revealed two different mechanisms of phonon scattering by dislocations and two different temperature dependences in the dislocation thermal resistivity according to which mechanism prevails. When the flutter effect prevails, according to eq. (94), the thermal resistivity κ^1 should be proportional to Bn (T)/C2Tcc Γ " 4 . However, when the non-linear mechanism determines the energy dissipation, then κ^1 oc By/(T)/C2Toz T~2. In some crystals, it is possible that the contributions from both mechanisms are comparable. In such cases, some intermediate temperature dependence κ^1

would be observed. Taking into account that κά depends on a dislocation density Nd which is

measured with rather low accuracy, the temperature dependence of κά(Τ) must


be the basic criterion for identification of the controlling mechanism in the thermal resistivity. However, sometimes a broader approach is required for theoretical interpretation of the experimental curve κά(Τ). For example, the observed dependence Kd(T)cc Ta (a < 2) in copper (Zeyfang 1967) was explained theoretically by Grüner (1970) as due to the non-linear mechanism of phonon scattering by dislocation dipoles, instead of in terms of the two scattering mechanisms for individual dislocations. On the other hand, according to Suzuki and Suzuki (1972), the dependence /cd( T) oc TA measured in LiF definitely shows the predominant role of the flutter effect in that case. The other experimental data (Anderson and Malinowski 1972, Roth and Anderson 1979, Roth et al. 1981) for the same crystal also appear as quite convincing. The authors observed the dislocation contribution to the thermal resistivity but the lack of experimental accuracy did not allow them to determine quantitatively the dependence of κά( T). In order to distinguish between possible mechanisms of phonon scattering, the y-irradiation was used. If the non-linear mechanism prevails over the flutter effect, the irradiation would only increase the thermal resistivity due to the Rayleigh scattering by additional point defects. According to Anderson and Malinowski (1972), Roth and Anderson (1979), and Roth et al. (1981), gradual irradiation of the crystal was accompanied by a decrease of the thermal resistivity down to predeformation values. This experimental fact is consistent with the concept of fluttering dislocations becoming under irradiation more and more immobile, which reduces flutter contribution to the phonon scattering.

Actually, thermal-conductivity measurements are a poor test of phonon-dislocation interaction, because the phenomenon is determined not by scattered phonons. Further investigation of the flutter mechanism in a LiF crystal was accomplished by Northrop et al. (1983), Alshits et al. (1989) by means of the non-equilibrium phonon technique which allows one to observe both ballistic phonon propagation and diffusive phonon transport. In particular, the average time of a diffusive transport of phonons through the sample is inversely proportional to the mean free path /ph, which gives a unique possibility of direct measurement of the temperature dependence l^(T) with specific resonant behaviour for flutter mechanisms (Alshits et al. 1989). The maximum of l~h

x in this case corresponds to the temperature at which the mean wavelength of phonons is close to the distance between pinning points along the dislocation.

Alshits (1969a, b) pointed out first the particularly strong contribution to dislocation drag (and consequently to thermal resistivity in general) from transverse phonons with small phase velocity. With allowance for the aniso-tropy of real crystals (see also the footnote on page 643), the predominant role in thermal resistivity should be played by the slowest phonon modes. This conclusion is corroborated by explicit calculations (Ohashi et al. 1985b), taking the anisotropy in LiF crystals into account, as well as by experimental data (Northrop et al. 1983, Anderson and Malinowski 1972).

686 V.l. Alshits

8.2. The temperature dependence of the dislocation damping constant

There are two basic methods of measuring the dislocation damping constant. One of them consists in direct observation of the mobility of separate dislocations during pulse loading of crystals. The first of such measurements of the damping constant, B, was carried out by Johnston and Gilman (1959) on LiF crystals. The second procedure is based on measurement of parameters of overdamped dislocation resonances in amplitude-independent internal friction in the region of high-frequency ultrasound (106-108 Hz)*. This method, first applied by Alers and Thompson (1961), allowed the first data on the temperature dependence of the damping constant in copper to be obtained. In the years that followed, a large number of experiments on the dynamic drag and its temperature dependence were carried out (Stern and Granato 1962, Hutchison and Rogers 1962, Gutmanas et al. 1963, Mitchell 1965, Sylwestrowics 1966, Thompson and Pare 1966, Parijskij et al. 1966, Parijskij and Tret'yak 1967, Lavrent'ev and Salita 1967, Greeman et al. 1967, Pope et al. 1967, Platkov et al. 1967, Khalilov and Agaev 1967, Ikushima and Kaneda 1968a, b, Lavrent'ev et al. 1968, Lubenets and Novikova 1968, Gosman et al. 1969a, b, Pope and Vreeland Jr 1969, Jassby and Vreeland Jr 1970, Lubenets and Novikova 1970, Ganguly et al. 1970, Kaneda 1970, Mason and McDonald 1971, Akita and Fiore 1971, Ermakov and Nadgornyi 1971, Jassby and Vreeland Jr 1971, Vreeland Jr and Jassby 1971, 1973, Korovkin and Soifer 1971, Hikata et al. 1972, Jassby and Vreeland Jr 1973, Gektina et al. 1974, Zuev et al. 1974, Petchenko and Startsev 1974, Andronov et al. 1975, Kaufmann et al. 1975, Darinskaya and Urusovskaya 1975, Urabe and Weertman 1975, Kobelev and Soifer 1975, 1976, Gektina et al. 1976, Gutmanas 1976, Pal-Val et al. 1976, Jassby and Vreeland Jr 1976, 1977a, b, Kurilov et al. 1977, Zuev et al. 1977, Boiko et al. 1978, Iwasa et al. 1979, Kobelev et al. 1979, Alshits et al. 1981, Paalanen et al. 1981, Darinskaya et al. 1981,1982a, b, 1983, Vreeland Jr 1984, Gektina and Kuzmenko 1985, Alshits et al. 1985).

In contrast to data on thermal conductivity, which are mainly confined to the low-temperature region, a predominant number of the measurements on the B(T) dependence have been performed in the temperature interval Θ/ 10 < T < 0, where the flutter effect, eq. (84), and the electron mechanisms, eq. (219), usually are already less important than phonon wind, eq. (179), and relaxation of'slow' phonons, eq. (187), while, on the other hand, thermoelastic damping, eq. (168), has not yet become important. In this region, in a simple variant of the theory [eq. (194)], B( T) can conveniently be expressed in the form

B( T) = Β(Θ)1 ψ ^ - [1 - Δ<ρ0(1)] + Δ<ρ0( Τ/θ) \ , (220)

* Methodical questions concerning measurement procedures are discussed in detail, e.g., by Alshits and Indenbom (1986) and we shall not dwell on them here.


where φ0(1) = e/(e — l)2

B(6) = g In

0.92, and

fßiX) A = λθ

Ι - Δ φ ο ( Ι ) ' Λ(1) + λβφ0(1)" (221)

Since the function fß(T/9)/fß(l) is practically independent of r0 (see fig. 7), the temperature dependence of the relative value Β(Τ)/Β(Θ) is determined by a single dimensionless parameter A, which is natural to consider as a phenomeno-logical constant subject to determination by experiment. Figures 10 and 11 show the results of a comparison of temperature dependence Β(Τ)/Β(Θ) with experimental points for a number of crystals. Apparently, there is quite reasonable agreement between theory and experiment.

Unlike the relative value B( Τ)/Β{θ\ the absolute magnitude B turns out to be sensitive to a model radius r0 of the dislocation core. Since usually the parameter ß = 2/cDr0 is of the order of ten, one has, in accordance with eq. (181), fß{\) % 1/2/J and, consequently, B ~ l/r0. In this connection, it seems to be more consistent to determine a value for the radius r0 from experimental values of the damping constant B. Then the theoretical situation should be considered

B(T)/B(B)

1,5 Τ/Θ

Fig. 10. Comparison of the temperature dependence B( T) as given by eq. (220) with experimental points for the crystals Zn, Cu [from data by Jassby and Vreeland Jr (1971)], KC1 [from data by

Andronov et al. (1975)] and Sb [from data by Pal-Val et al. (1976)].

Β(Τ)ΙΒ(Θ) 2,0

1.5

1,0

0,5

• CsL

^//·

y i i

•^

I

0,5 1,0 1,5 2 T/B

Fig. 11. Comparison of the temperature dependence B(T) as given by eq. (220) with experimental points for the crystals Al [from data by Gosman et al. (1969a, b)] and Csl [from data by Darinskaya

et al. (1981)].

688 V.l. Alshits

as satisfactory, if agreement is achieved with r0 ~ (\-3)a. This procedure applied to copper [according to Seeger and Buck (1960) η€η/μ~ — 33] leads to an experimental value (Jassby and Vreeland Jr 1970) B = 1.6 x 10"4 P at room temperature when the value r0 = 3a is substituted into eq. (220). Thus, allowance for the dislocation core removes the discrepancy noted by Brailsford (1972) between theory and experiment concerning absolute values of the damping constant B.

83. The influence of impurities and other defects

The question about what role other lattice defects play in the dynamic drag of dislocations has given rise to much discussion in the literature. Three independent channels for dissipation by dislocation-point-defect mechanisms were considered in the theory: radiation losses due to non-uniformity of dislocation motion in the elastic field of point-defect centres (Ookawa and Yazu 1963); excitation of local and quasi-local vibrations of impurity atoms (Takamura and Morimoto 1963, Lyubov and Chernizer 1965, Kosevich and Natsik 1966, 1968) by the moving dislocation; and, finally, diffusion processes accompanying the dislocation motion (Kosevich and Natsik 1968, Nabarro 1948, Cottrell and Jaswon 1949, Cottrell 1953, Schoeck and Seeger 1959, Hirth and Lothe 1968, Klemens 1968, Gutnikova et al. 1970, Yoshinaga and Morozumi 1971, Altundzhi and Lyubov 1973). However, existing experimental data appertain to a region of dislocation velocities and temperatures where all the above-mentioned mechanisms contribute negligibly to the dislocation drag. This conclusion is fully corroborated by experiments, e.g., according to Lavrent'ev et al. (1968), the dynamic drag of pyramidal dislocations in Zn crystals is practically insensitive to the impurity content. The same result is obtained by Darinskaya et al. (1981, 1983) who studied the mobility of edge dislocations in Csl and NaCl crystals.

Experimental investigations on the impurity influence on the dynamic drag require special methodical caution. The higher the defect concentration in the crystal the higher the probability is that before the end of the stress pulse the dislocation will meet a strong obstacle, which cannot be overcome dynamically. Consequently, the mean free path of the dislocations decreases when the impurity concentration grows and this might be wrongly interpreted as due to a reduced dynamic mobility. There are good reasons to suppose that earlier conclusions regarding the influence of irradiation (Ermakov and Nadgornyi 1971) and impurities (Zuev et al. 1974) on the magnitude of B(T) in NaCl crystals and its temperature dependence just have to do with this methodical problem. In order to avoid this difficulty, one should, as suggested by Darinskaya et al. (1981, 1983), collect statistics for only the longest paths and ignore dislocations which have only moved short distances. Interpreted this


way, the measurements by Darinskaya et al. (1983), Alshits et al. (1985) showed that B(T) did not depend on the concentration and the composition of impurities in NaCl crystals.

Analogous methodical perils exist as well in the methods of interpreting amplitude-independent internal friction. At present, one may consider to be established that data interpreted as the influence of impurities (Ikushima and Kaneda 1968a, Kaneda 1970), thermal pretreatment (Kaufmann et al. 1975) or irradiation (Kobelev et al. 1979, Alshits et al. 1981) on B(T) in copper actually have to do with a superposition of viscous processes and relaxation effects. As was shown by Araki and Ninomiya (1977), Kobelev et al. (1979), Alshits et al. (1981), Granato and Lücke (1981), in such experiments some effective value B* is measured, which, however, should not be interpreted as a true damping constant for viscous dragging of dislocations. According to Kobelev et al. (1979), Alshits et al. (1981), an appearance of a temperature dependence in the modulus defect is an indication of the presence of such parasitic effects. A special procedure for experimental data processing suggested by Kobelev et al. (1979), Alshits et al. (1981) allows one to exclude relaxation effects and to extract the true temperature dependence of the damping constant B.

Another separate problem is the role in the dynamic drag B of the moving dislocation played by vibrating segments of pinned dislocations belonging to the so-called 'forest' dislocations. The additional energy dissipation due to vibrating segments in the field of the moving dislocation is determined by the same mechanisms as the direct dislocation drag. Therefore, these additions to the damping constant B should have the same temperature dependence as the basic term. In other words, the above-mentioned 'forest' mechanism should lead only to a renormalization of the magnitude of B and no change in Β(Τ)/Β(Θ). This effect, first discussed by Indenbom and Orlov (1962), was quantitatively estimated in the article by Natsik and Minenko (1970), where it was shown that the renormalization could be noticeable at reasonable densities of 'forest' dislocations. The dependence of the damping constant B on the density of 'forest' dislocations was measured by Gektina et al. (1976) for Zn crystals.

8.4. Electron manifestations in the plasticity of metals

The electron component of the dislocation drag has been investigated directly by experiments only in the case of Cu (Jassby and Vreeland Jr 1973, Kobelev et al. 1979), Pb (Lenz and Lücke 1969, Kobelev and Soifer 1976) and Al (Hutchison and Rogers 1962, Hikata et al. 1970, Parameswaran et al. 1972) crystals. The contribution of conduction electrons manifests itself most convincingly at the normal superconducting transition, as shown in Kobelev and Soifer (1976) where an abrupt change of amplitude-independent internal friction in lead is reported. One might suppose that the same effect would reveal itself in many metals as a step-wise behaviour in creep, relaxation, and active loading

690 V.l. Alshits

at the normal-superconducting transition [see review articles by Kaganov et al. (1973), Startsev (1983)].

Likewise, observed oscillations in the low-temperature plasticity of metals with increases of the external magnetic field (Belessa 1973, Galligan 1985) [see also the theoretical discussions by Grishin et al. (1976, 1978)] appear to be manifestations of an electron component in dislocation dragging. On the other hand, the nature of another interesting phenomenon, namely increases in the plasticity of a metal as the result of electric current pulses [see, e.g., Troitskii (1969)] seems less clear. This phenomenon has first been wrongly interpreted as due to electron entrainment of dislocations. However, theory (Kravchenko 1966b) predicts an entrainment force many orders of magnitude smaller than what is necessary for the explanation of the observed effects. It is possible that the explanation of the phenomenon is a direct influence of the electric current on local defects of the crystal which lead to an effective decrease of the minimum stress for dislocation motion and a corresponding increase of the dislocation mobility. Another explanation is the polarization of the dislocation cores in the magnetic field accompanying an electric current and an attendant lowering of the potential barriers connected with the interaction between dislocations and other defects in the lattice*. However, it is also altogether possible that we are not dealing with an electric or magnetic phenomenon as such, but with thermal effects, e.g., with a local heating and subsequent dissociation of impurity complexes.

Whatever the correct interpretation may be, the phenomenon is real enough and has already led to important technical applications.

9. Conclusion

In the consideration of the theory of dynamic dragging of dislocations, the emphasis has been on the phonon mechanisms of energy dissipation. As has been shown, they usually play the main part in the viscous damping in the dislocation motion. At low temperatures, in metals, one should take into account also the electron component of dragging and, in magnetics, the spin wave contribution (Barjakhtar et al. 1977, 1982) as well. The main contribution to phonon dragging of dislocations comes from the attenuation of the short-wavelength part (qlph > 1) of the packet given in eq. (123), due to a combination of phonon scattering and relaxation of 'slow' phonons (quasi-particles with group velocities that are concentrated along the dislocation directions). Ther-moelastic attenuation of the long-wavelength part (qlph < 1) of the packet given

* Note the recently discovered magnetoplastic effect (Alshits et al. 1987, 1990) - the motion of individual dislocations in diamagnetic crystals (NaCl, Zn) in static external magnetic field without mechanical loading.


in eq. (123) is usually masked by other effects. It could manifest itself only in the high-temperature range because of the quadratic temperature dependence. Lastly, the characteristic feature of the packet given in eq. (123), namely its low phase velocity in comparison with the sound velocity, gives an infinitesimal contribution of Akhiezer phonon viscosity to dissipation in comparison with thermoelastic energy losses.

The experimental data agree quite well with the above-developed theory. Along with a simplified phenomenological description, it also allows calculation of the viscous damping of dislocations from first principles, i.e., with the allowance for the real phonon spectra and elastic lattice strains close to the dislocations core, but of course, with the aid of computers. The next steps of the theory would be expected to be in this direction.

Acknowledgement

The author is very much indebted to Prof. V.L. Indenbom for numerous valuable discussions and to Prof. J. Lothe for useful remarks and kind assistance with the improvement of the English translation of this chapter.

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