Diffusion of elastic waves in a two dimensional continuum with a random distribution of screw dislocations
aa r X i v : . [ c ond - m a t . m t r l - s c i ] A ug Diffusion of elastic waves in a two dimensional continuum with arandom distribution of screw dislocations
Dmitry Churochkin and Fernando Lund
Departamento de F´ısica and CIMAT,Facultad de Ciencias F´ısicas y Matem´aticas,Universidad de Chile, Santiago, Chile
Abstract
We study the diffusion of anti-plane elastic waves in a two dimensional continuum by many,randomly placed, screw dislocations. Building on a previously developed theory for coherent prop-agation of such waves, the incoherent behavior is characterized by way of a Bethe Salpeter (BS)equation. A Ward-Takahashi identity (WTI) is demonstrated and the BS equation is solved, asan eigenvalue problem, for long wavelengths and low frequencies. A diffusion equation results andthe diffusion coefficient D is calculated. The result has the expected form D = v ∗ l/
2, where l , themean free path, is equal to the attenuation length of the coherent waves propagating in the mediumand the transport velocity is given by v ∗ = c T /v , where c T is the wave speed in the absence ofobstacles and v is the speed of coherent wave propagation in the presence of dislocations. . INTRODUCTION The interaction of elastic waves with dislocations in elastic solids has been studied overseveral decades, since the pioneering work of Nabarro [1], Eshelby [2, 3], Granato and L¨ucke[4, 5] and Mura [6]. Nabarro [1] and Eshelby [2, 3] noted that the mathematics describ-ing the interaction of screw dislocations with anti-plane elastic waves in a two-dimensionalcontinuum is the same as that of electromagnetic waves in interaction with point charges,also in two dimensions. A translation of the well-studied electrodynamics allowed theseresearchers to obtain expressions for the elastic radiation emitted by a screw dislocation inarbitrary motion. The more difficult case of the radiation generated by an edge dislocationin arbitrary motion was solved by Mura [6], who also found expression for the radiationgenerated by a dislocation loop of arbitrary shape undergoing also arbitrary motion. Theconverse problem, the response of a dislocation loop to an incoming, time dependent stresswave, was solved by Lund [7] using arguments of energy and momentum conservation.Granato and L¨ucke [4, 5] formulated a theory for the propagation of an averaged acousticwave in the presence of many dislocation segments, using a string model for the dislocationpioneered by Kohler [8]. Their results have been widely used to interpret experimental resultsof mechanical damping and modulus change that are both frequency and strain-amplitudedependent. This theory is a scalar theory that cannot distinguish between longitudinal andtransverse waves, nor between edge and screw dislocations. Hence, it could not account forresults that depend on the polarization of the waves. Also, attempts at using it to explainthermal conductivity measurements did not take into account the attenuation due to loss ofcoherence nor the difffusive behavior of incoherent contributions [9, 10].In recent years, Maurel, Lund, and collaborators have revisited the theory of thedislocation-wave interaction. Using the equations of [7] they obtained explicit expressionsfor the scattering of an elastic wave by screw and edge dislocations in two dimensions [11],and by pinned dislocation segments [12] and circular dislocation loops [13] in three. In ad-dition, and using multiple scattering methods that go back to the work of Foldy [14], Karaland Keller [15] and Weaver [16], they obtained expressions for the effective velocity andattenuation of an elastic wave moving through a maze of randomly placed dislocations intwo [17] and three [18] dimensions. These results generalized the theory of Granato-L¨ucketo keep track of the wave polarization and vector character of the dislocations. In this way2reviously unexplained experimental results concerning the different response of materialsto longitudinal and transverse loadings found an explanation [19]. More interestingly, theysuggested a way to use ultrasound as a non-intrusive wave to measure dislocation density.This possibility has been recently shown to be feasible, and leads to measurements moreaccurate than those obtained with X-ray diffraction [20, 21]. The multiple scattering theoryof elastic waves has also been studied in relation to propagation in polycrystals [22–24] andcomposites [25, 26].Having studied the behavior of coherent waves, the question naturally arises as to thebehavior of incoherent waves. Diffusion techniques developed to deal with the localizationof de Broglie waves that describe the quantum mechanics of electrons in interaction withrandomly placed scatterers have been used to study the behavior of classical waves, a centralrole being played by the Bethe-Salpeter (BS) equation [27–29]. The behavior of elastic wavesin interaction with a variety of scatterers and within a variety of geometries has been studiedby Kirkpatrick [30], Weaver [31], and Van Tiggelen and collaborators [32–34]. Also, an earlyapproach that uses energy transport equations [35] has been widely applied.The diffusion of elastic waves, albeit in their quantized form, phonons, is central to ther-mal transport in materials, a topic of much current concern. Yet, surprisingly little appearsto be quantitatively known about the role played by the interaction of elastic waves withdislocations [36] in thermal transport. It stands to reason then that a detailed study of thediffusive behavior of elastic waves in interaction with many, randomly placed, dislocations,should be undertaken. Since elastic waves are vector waves and dislocations are linear ob-jects characterized by their tangent and Burgers vectors, the tensor algebra associated withthe proposed study appears daunting. A first step should be to consider a simplified settingthat captures the essence of the problem. This paper is devoted to precisely this aim: itstudies the diffusive behavior, in two dimensions, of an anti-plane elastic wave in interactionwith many, randomly placed, screw dislocations.
A. Organization of this paper
This paper is organized as follows: Section II recalls existing results for the interactionof anti-plane waves with screw dislocations. It includes the behavior of coherent waveswhen many such dislocations are present, as well as a recent result of Churochkin et al.337] concerning the summability to all orders of the perturbation expansion needed to makesense of the theory. Section III constructs the Bethe-Salpeter equation and establishes theWard-Takahashi identity that relates the coherent with the incoherent kernels. Section IVsolves the BS equation in the low frequency limit needed to obtain a diffusion behavior andan expression for the diffusion coefficient (Eqn. (97)) is obtained. Section V has a discussionand final comments. A number of computations are carried out in several appendices.
II. INTERACTION OF ANTI-PLANE ELASTIC WAVES WITH SCREW DISLO-CATIONS.
The interaction of an anti-plane wave with a single dislocation was studied by Eshelby[2], Nabarro [1] and by Maurel et al. [11]. The coherent behavior that emerges when ananti-plane wave interacts with many, randomly located screws, was elucidated by Maurel etal [17], using the following equation of motion in the frequency domain: (cid:0) ∇ + k β (cid:1) v ( ~x, ω ) = − V (0) ( ~x, ω ) v ( ~x, ω ) (1)where v is particle velocity as a function of two-dimensional position ~x and frequency ω , k β = ω /c T with c T = µ/ρ ,and the potential V (0) is V (0) ( x , ω ) = N X n =1 A n (0) ∂∂x a δ ( ~x − ~X n ) ∂∂x a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ~x = ~X n , (2)and A n (0) = µM (cid:18) b n ω (cid:19) . (3)Here the superscript n denotes the corresponding characteristics of the n-th screw dislocation, ρ is the (two dimensional) mass density, µ is the shear modulus and M = ( µb / πc T ) ln( δ/ǫ )is the usual mass per unit length of a screw dislocation with δ and ǫ a long- and short-distance cut-off, respectively. Eqns. (2,3) were obtained in [11] using an approximationthat neglected the Peierls-Nabarro (PN) force [38], and led to a scattering cross section thatdiverges at low frequencies. The origin of this divergence can be understood qualitatively:since the only length scale present in the problem is the wavelength, the scattering cross sec-tion will have to be proportional to it, thus diverging as the wavelength grows. The detailedcalculation bears out this qualitative reasoning [11]. Since in the present work we wish to4xplore a diffusion behavior associated with long wavelengths and low frequencies, we shallintroduce a Peierls-Nabarro restoring force as well as a viscous damping into the dislocationdynamics, whose equation of motion , for small oscillations around the PN minimum, willthen be M ¨ X b + B ˙ X b + γX b = µbǫ bc ∂u∂x c ( ~X ( t ) , t ) . (4)This dynamics leads to the following equation of motion, that is a generalization of (1): (cid:0) ∇ + k β (cid:1) v ( ~x, ω ) = − V ( ~x, ω ) v ( ~x, ω ) (5)with V ( x , ω ) = N X n =1 A n ∂∂x a δ ( ~x − ~X n ) ∂∂x a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ~x = ~X n (6)and A n = µb M ω + ıω ( B/M ) − ω (7)where ω ≡ γ/M . We shall consider all dislocations have a Burgers vector of the samemagnitude, but possibly different sign. A. Scattering by a single dislocation in the first Born approximation
Starting from Eqn. (5) and following the reasoning of [11] it is a simple matter to showthat the scattering amplitude for the scattering of an anti plane wave by a screw dislocationlocated at the origin of coordinates is, in the first Born approximation, f ( θ ) = − µb M e ıπ/ p πc T ω / ω − ω + ıωB/M cos θ (8)from which the total scattering cross section σ s ≡ R | f ( θ ) | dθ follows: σ s = µ b M c T ω ( ω − ω ) + ω ( B/M ) . (9)This cross section vanishes at low frequencies, that is, for ω ≪ ω : σ s ∼ ω ω c T ω . (10)5 . Coherent behavior The coherent behavior of the elastic wave in the presence of many dislocations is describedby the average Green function h G i , where G is the solution of ρω G ( ~x, ω ) + µ ∇ G ( ~x, ω ) = − µV ( x , ω ) G ( ~x, ω ) − δ ( ~x ) . (11)and the brackets denote an average over the random distribution of dislocations.The average Green function is obtained as the solution of the Dyson equation h G i − = ( G ) − − Σ (12)where G = 1 µk − ρω (13)is the free space Green function and Σ, the mass (or self-energy) operator, is given byΣ = h T i − h T i G Σ (14)with T the T-matrix, given in terms of the “potential” V by T = V + V G T . (15)In a perturbation approach this last equation is developed to give T = V + V G V + V G V G V + · · · (16)When the scatterers are statistically independent of each other (ISA, or IndependentScattering Approximation) it is easy to show thatΣ = n Z h t i d ~X n (17)where t is the T-matrix for scattering by a single object, as defined in [37]. Maurel et al.[17] computed the mass operator to second order in perturbation theory. It is shown inAppendix A that, due to the point-like nature of the interaction (6) the perturbation seriesis geometric and can be summed to all orders to obtain the following expression for theGreen function < G > + ( k , ω ) for the outgoing waves in the momentum space < G > + ( k , ω ) = 1 ρω (cid:8) k K − (cid:9) (18)6ith K = ωc T rh − Σ ρc T k i = ωc T √ n T (19)Σ = − nµ T k (20) T ≡ < A n µ A n I π > = µb M ω − ω R + i (cid:16) ω BM + µb ω Mc T (cid:17) (21)with 2 µI = Λ + iπk β and c T = µ/ρ , where Λ is a short distance (high wave number) cut-offthat is absorbed through a renormalization of the Peierls-Nabarro frequency: ω R ≡ ω − µb πM Λ , (22)see Appendix A for a discussion. The incoming waves, related to < G > − ( k , ω ) andΣ − ( k , ω ), are described by the complex conjugate form of Eqns. (18) and (20).Eqn. (19) provides an effective velocity v ≡ ωRe [ K ] = c T | n T | q nRe [ T ]+ | n T | (23)and attenuation length l ≡ Im [ K ] = c T | n T | ω q | n T |− (1+ nRe [ T ])2 (24) III. BETHE-SALPETER EQUATION AND WARD-TAKAHASHI IDENTITYA. BS equation
The diffusive transport regime is determined by the two-point correlation < G + G − > inthe low frequency, long wavelength limit. The objective now is to find, in Fourier space, adiffusive pole structure, < G + G − > ∼ ( ı Ω+ Dq ) − and to identify D as a diffusion coefficent[27]. The first step to be taken is the construction of Bethe-Salpeter (BS) equation. Thiscan be achieved with a reasoning analogous to the one that leads to the Dyson equation:7irst, the intensity < G + G − > is written as < G + G − > = < G + >< G − > + (cid:0) < G + G − > − < G + >< G − > (cid:1) = < G + >< G − > + < G + >< G − >< G + > − < G − > − × (cid:0) < G + G − > − < G + >< G − > (cid:1) < G + G − > − < G + G − > = < G + >< G − > + < G + >< G − > × (cid:0) < G + > − < G − > − − < G + G − > − (cid:1) < G + G − > . (25)Defining the irreducible part K as K = < G + > − < G − > − − < G + G − > − (26)and substitution into (25) gives the BS equation the well known form < G + G − > = < G + >< G − > + < G + >< G − > K < G + G − > , (27)where K plays a role similar to the role played by the self-energy Σ in Dyson’s equation (12).It is convenient to work with Eqn. (27) in Fourier space. Introducing the Fourier trans-formation as [27] G + ( x , x ′ ; ω + ) = Z k Z k ′ e ı k x G + ( k , k ′ ; ω + ) e − ı k ′ x ′ G − ( x , x ′ ; ω − ) = Z k Z k ′ e − ı k x G − ( k ′ , k ; ω − ) e ı k ′ x ′ , (28)we obtain < G + G − > = Z k Z k ′ Z q Φ( k , k ′ ; q , Ω) e ı ( kr − k ′ r ′ + q ( R − R ′ )) (29)where Φ( k , k ′ ; q , Ω) ≡ < G + ( k + , k ′ + , ω + ) G − ( k ′− , k − , ω − ) > (30)with k ± = k ± q , ω ± = ω ± Ω2 . x = R + r , x = R − r x ′ = R ′ + r ′ , x ′ = R ′ − r ′ k = k + = k + q , k ′ = k ′ + = k ′ + q k ′ = k ′− = k ′ − q , k = k − = k − q . Applying the inverse Fourier transform [39] Z d ( R − R ′ ) d r d r ′ e − ı ( kr − k ′ r ′ + q ( R − R ′ )) (32)to Eqn. (27) the BS equation in momentum space is obtained asΦ( k , k ′ ; q , Ω) = < G + >< G − > ( k ; q , Ω) δ k , k ′ + (33) Z k ′′ < G + >< G − > ( k ; q , Ω) K ( k , k ′′ ; q , Ω)Φ( k ′′ , k ′ ; q , Ω)with δ k , k ′ = (2 π ) δ ( k − k ′ ) and, as usual, the integration over the internal momentumvariables, i.e. k ′′ , is assumed, with R k ≡ π ) R d k .To proceed from Eqn. (33) to a kinetic form we use the following identity for the averagedGreen’s functions:( < G + > − − < G − > − ) < G + >< G − > = < G − > − < G + > (34)Multiplying from the left Eqn. (33) by the difference ( < G + > − − < G − > − ) and usingthe property (34) it is straightforward to obtain( < G + > − − < G − > − )Φ = ( < G − > − < G + > ) (35) × δ k , k ′ + Z k ′′ K ( k , k ′′ ; q , Ω)Φ( k ′′ , k ′ ; q , Ω) and, using Dyson’s equation (12) and the explicit form for the free space Green function(13)this can be rewritten as( µ (cid:0) ( k + ) − ( k − ) (cid:1) − ρ (cid:0) ( ω + ) − ( ω − ) (cid:1) + Σ − − Σ + )Φ = (36)( < G − > − < G + > ) δ k , k ′ + Z k ′′ K ( k , k ′′ ; q , Ω)Φ( k ′′ , k ′ ; q , Ω) . P ( k ; q ) ≡ ıρ (cid:0) L ( k − ) − L ( k + ) (cid:1) = µ k · q ıρ (37)∆ G ( k ; q , Ω) ≡ ıρ (cid:0) < G > − ( k − , ω − ) − < G > + ( k + , ω + ) (cid:1) ∆Σ( k ; q , Ω) ≡ ıρ (cid:0) Σ − ( k − , ω − ) − Σ + ( k + , ω + ) (cid:1) L ( k ± ) = − µk ± l k ± l . yields the kinetic form of the BS equation[ ıω Ω + P ( k ; q )] Φ( k , k ′ ; q , Ω) + Z k ′′ U ( k , k ′′ ; q , Ω)Φ( k ′′ , k ′ ; q , Ω) = δ k , k ′ ∆ G ( k ; q , Ω) (38)with U ( k , k ′ ; q , Ω) ≡ ∆Σ( k ; q , Ω) δ k , k ′ − ∆ G ( k ; q , Ω) K ( k , k ′ ; q , Ω) . (39) B. pre-WTI
The manipulations of the previous subsection are quite general and do not rely on thespecifics of the interaction (6). We now specialize to the case at hand. A Ward-Takahashiidentity (WTI) is needed to relate the mass operator Σ with the irreducible K . To this end,a preliminary identity (“pre-WTI”), Eqn. (48) below, will be obtained.In order to get the pre-WTI identity we start with the dynamical equation assuming themost general form of the source for the antiplane case s ( ~x, t ) [11]: ρ ∂ ∂t v ( ~x, t ) − µ ∇ v ( ~x, t ) = s ( ~x, t ) (40)where the source is defined as s ( ~x, t ) = µbǫ ab ˙ X b ∂∂x a δ ( ~x − ~X ) (41)It yields, Z ~x s ( ~x, t ) ≡ ~X alwaysbrings the source to the Fourier transformed form s ( ~x, ω ) = µV n ( ~x, ω, ~X ) v ( ~x, ω ) (43)with V n ( ~x, ω, ~X ) = A n ∂∂x a δ ( ~x − ~X n ) ∂∂x a (cid:12)(cid:12)(cid:12)(cid:12) ~x = ~X n . (44)Eqns. (42,43) lead to an identity for the corresponding Green function G ( ~x , ~x ′ , ω ) Z ~x s ( ~x , ω ) = Z ~x µV n ( ~x , ω , ~X ) v ( ~x , ω ) ≡ ⇒ Z ~x µV n ( ~x , ω , ~X ) G ( ~x , ~x ′ , ω ) ≡ ω Z ~x Z ~x µV n ( ~x , ω , ~X ) G ( ~x , ~x ′ , ω ) G ( ~x , ~x ′ , ω ) − ω Z ~x Z ~x µV n ( ~x , ω , ~X ) G ( ~x , ~x ′ , ω ) G ( ~x , ~x ′ , ω ) ≡ . (46)This last identity enables the construction of a pre-WTI following Refs. [40, 41]. Thisis achieved as follows: a system of equations is constructed, writing out Eqn. (11) twicefor G ( ) = G ( ~x , ~x ′ , ω ) and G ( ) = G ( ~x , ~x ′ , ω ) respectively. Next, the first and secondequations of the resulting system are multiplied by ω G ( ) and ω G ( ) respectively, andsubstracted from each other. Using the property (46) of the potential and integrating over ~x and ~x we come to the following identity, that does not explicitly involve the potential µV : Z x Z x (cid:0)(cid:0) ω L ( ) − ω L ( ) (cid:1) ( G ( ) G ( )) + G ( ) δ ( x − x ′ ) ω − G ( ) δ ( x − x ′ ) ω (cid:1) ≡ R x = R d x , L = L ( ı ∂∂ x ). The identity in Eqn. (47) must be fulfilled for arbitrary valuesof the external parameters such as x ′ , and ω , . This is possible only if the expression inthe brackets is equal to zero. Its subsequent averaging gives the pre-WTI:lim x → x x ′ → x ′ (( g L ( ) − g L ( )) < G ( ) G ( ) > ≡ < G ( ) δ ( x − x ′ ) g − G ( ) δ ( x − x ′ ) g > )(48)with g , = g ( ω , ) = ω , . 11 . WTI The WTI relates the vertex K and the self energy Σ, using Eqn. (48). To ob-tain this relationship we modify the BS Eqn. (33), multiplying it from the right with( g L ( k ′ + ) − g L ( k ′− )) and integrating over k ′ . This gives Z k ′ Φ( k , k ′ ; q , Ω) (cid:0) g L ( k ′ + ) − g L ( k ′− ) (cid:1) = (49) Z k ′ < G + >< G − > ( k ; q , Ω) δ k , k ′ + Z k ′′ < G + >< G − > ( k ; q , Ω) K ( k , k ′′ ; q , Ω)Φ( k ′′ , k ′ ; q , Ω) × (cid:0) g L ( k ′ + ) − g L ( k ′− ) (cid:1) Using the Fourier-transformed Eqn. (48) with arguments (resp ) replaced by “+” (resp“ − ”) in Eqn.(49) and then multiplying the obtained expression with < G + > − < G − > − ( k ; q , Ω) from the left we come to the result < G − > − ( k ; q , Ω) g − < G + > − ( k ; q , Ω) g = (cid:0) g L ( k + ) − g L ( k − ) (cid:1) (50)+ Z k ′′ ( K ( k , k ′′ ; q , Ω)) (cid:0) < G + > ( k ′′ ; q , Ω) g − < G − > ( k ′′ ; q , Ω) g (cid:1) . Introducing the explicit form of the averaged Green function, Eqns. (12,13), the followingWTI, relating Σ and K , is obtained:Σ + ( k ; q , Ω)) g − Σ − ( k ; q , Ω) g = (51) Z k ′′ ( K ( k , k ′′ ; q , Ω)) (cid:0) < G + > ( k ′′ ; q , Ω) g − < G − > ( k ′′ ; q , Ω) g (cid:1) It is convenient to rewrite this form of the WTI, Eqn. (51), in a different way. By presenting g and g as g = g + g g − g )2 , g = g + g − ( g − g )2 (52)and substituting them into Eqn. (51), the commonly used appearance for the WTI isobtained: (cid:0) Σ + ( k ; q , Ω) − Σ − ( k ; q , Ω) (cid:1) − Z k ′′ K ( k , k ′′ ; q , Ω) (cid:0) < G + > ( k ′′ ; q , Ω) − < G − > ( k ′′ ; q , Ω) (cid:1) = (53)( g − g ) g + g (cid:0) Σ + ( k ; q , Ω) + Σ − ( k ; q , Ω) (cid:1) + Z k ′′ K ( k , k ′′ ; q , Ω) (cid:0) < G + > ( k ′′ ; q , Ω)+ < G − > ( k ′′ ; q , Ω) (cid:1) Z k ′′ U ( k ′′ , k ′ ; q , Ω) = i A ( k ′ ; q , Ω) ( g + − g − ) (54)with g , = g ± = ω ± , U ( k , k ′ ; q , Ω) = ∆Σ( k ; q , Ω) δ k , k ′ − ∆ G ( k ; q , Ω) K ( k , k ′ ; q , Ω) , (55) A ( k ′ ; q , Ω) = 2 g + + g − R Σ( k ′ ; q , Ω) + Z k ′′ R G ( k ′′ ; q , Ω) K ( k ′′ , k ′ ; q , Ω) , R Σ( k ′ ; q , Ω) = 12 ρ (cid:0) Σ − ( k ′− , ω − ) + Σ + ( k ′ + , ω + ) (cid:1) , and the operation R , here defined for the self-energy Σ, acts in the same way on the Greenfunction G . It is remarkable that the essential point that enables the derivation of the WTIis the property (42) of the source (41). That is, a Peach-Koehler force that acts at theequilibrium position ~X of the dislocation, that in turns perform conservative (i.e., glide,not climb) motion. The specific dislocation dynamics embodied in (4) does not play a role.It does, however, become relevant when we take the long wavelength, low frequency and lowdensity limit, to which we now turn our attention. D. WTI in the low-density-of-dislocations, low frequency, and long wavelengthapproximations
The plan is to solve the BS equation (38) to obtain the two point correlation in Fourierspace, Φ( k , k ′ ; q , Ω), in the low frequency (Ω →
0) and long wavelength ( q →
0) limit.Hopefully a diffusion behavior will result. In order to do this we need information about theself energy Σ and vertex K . The mass operator Σ is given by (17) in the ISA approximation:scatterers are uniformly distributed through space with uncorrelated positions. This isvalid for low scatterer density n and, indeed, (17) is the leading order term in a low- n approximation scheme. Similarly keeping the lowest order term in dislocation density n forthe kernel K leads to its Boltzmann approximation K → K B , with K B ( k , k ′ ; q , Ω) = n h t ω + ( k + , k ′ + ) t ∗ ω − ( k ′− , k − ) i (56)where the frequency dependence of the T matrix has been made explicit [37]. In orderto obtain the diffusive behaviour of our system in this approximation, we need the low13requency and long wavelength asymptotics: the limit q →
0, Ω →
0. It can be easily seenfrom Eqns. (54) and (55) that, in this limit, the WTI prescribes the following constraint forthe corresponding part of the kernel K B ( p ′′ , k ′ ; , k ′ ) = Z p ′′ ∆ G ( p ′′ ) K B ( p ′′ , k ′ ; ,
0) = Z p ′′ K B ( k ′ , p ′′ ; , G ( p ′′ ) (57)The second equality in Eqn. (57) is due to the reciprocity of the K B ( p ′′ , k ′ ; Q , Ω), a conse-quence of the symmetries of the Green’s function G . It is shown in Appendix B that thisrelation holds within the ISA, in the abscence of viscosity in the dislocation dynamics; i.e.,when B = 0 in (4). IV. LOW FREQUENCY ASYMPTOTICS AND DIFFUSION BEHAVIOR
The following relations for the self energy and for the Green function will prove useful:∆Σ( k ; ,
0) = 12 ıρ (Σ ∗ ( k ) − Σ( k )) (58)= ρc T k ıρ (cid:18)(cid:18) − ω ( K ) ∗ c T (cid:19) − (cid:18) − ω K c T (cid:19)(cid:19) = − Im [ K ] ω k K ( K ) ∗ . ∆ G ( k ; ,
0) = 12 ıρ ( < G > ∗ ( k ) − < G > ( k )) (59)= 12 ıρ ω (cid:18) ( K ) ∗ k − ( K ) ∗ − K k − K (cid:19) = − k ρ ω Im [ K ]( k − ( K ) ∗ ) ( k − K ) ≈ − πk ρ ω δ (cid:0) k − Re [ K ] (cid:1) (60)where the last approximate equality holds in the limit | Im [ K ] | ≪ | k − Re [ K ] | . Themeaning of this inequality in terms of the dislocation parameters is explored in AppendixC. A. Perturbation approach to BS eigenvalue problem
In this section we follow the approach that was used in Refs. [39–42] to study thediffusion of electromagnetic and acoustic waves: The BS equation written in the form (38),14upplemented by the relation between mass operator Σ and kernel K implemented by theWTI (54), is solved for the intensity Φ, defined by (30), in the diffusive limit. To this end,the BS equation (38) is written in operator form: Z k ′′ H ( k , k ′′ ; q , Ω)Φ( k ′′ , k ′ ; q , Ω) = ∆ G ( k ; q , Ω) δ k , k ′ . (61)with the operator H defined by H ≡ [ ıω Ω + P ( k ; q )] δ kk ′′ + U ( k , k ′′ ; q , Ω) . (62)It is easy to see, using the explicit form of U , and the reciprocity of the tensor K , that H has the following symmetry: H ( k , k ′′ ; q , Ω)∆ G ( k ′′ ; q , Ω) = H ( k ′′ , k ; q , Ω)∆ G ( k ; q , Ω) . (63)The solution of the BS equation will be found in terms of the eigenvalues and eigenfunctionsof the operator H .The eigenvalue problem for H is set up as follows: Z k ′′ H ( k , k ′′ ; q , Ω) f r nkl ( k ′′ ; q , Ω) = λ n ( q , Ω) f r n ( k ; q , Ω) (64)where f r n ( k ′′ ; q , Ω) (resp. f l n ( k ′′ ; q , Ω)) are right (resp. left) eigenfunctions and λ n ( q , Ω)the corresponding eigenvalues. Following [39–42] we shall assume that the eigenfunctions in(64) satisfy completeness and orthogonality conditions: Z k f r m ( k ; q , Ω) f l n ( k ; q , Ω) = δ mn , (65) X n f r n ( k ; q , Ω) f l n ( k ′ ; q , Ω) = δ kk ′ . Furthermore, the symmetry restriction for the operator H from Eqn. (63) determines therelation between left and right eigenfunctions: f r n ( k ; q , Ω) = ∆ G ( k ; q , Ω) f l n ( k ; q , Ω) . (66)The eigenfunction properties (65,66) allow for a representation of the solution Φ of (61)as a series over the states n [39–42] :Φ = X n f r n ( k ; q , Ω) f r n ( k ′ ; q , Ω) λ n ( q , Ω) (67)15he existence of a diffusion regime assumes that in the limit q →
0, Ω → λ ( q → , Ω → → λ ( q , Ω) to second orderin q and first order in Ω around the point q = 0, Ω = 0. To do this, Eqn. (64) has tobe treated perturbatively, with the condition that Eqns. (54,63) hold at every order of theperturbation in q , and Ω [40–42].In order to solve Eq. (64), we write, in a small Ω and a small q approximation, H ( k , k ′′ ; q , Ω) = H ( k , k ′′ ; ,
0) + H ( k , k ′′ ; , Ω)+ H q ( k , k ′′ ; q ,
0) + H q ( k , k ′′ ; q ,
0) + . . . ,f r ( k ′′ ; q , Ω) = f ( k ′′ ; ,
0) + f ( k ′′ ; , Ω) (68)+ f q ( k ′′ ; q ,
0) + f q ( k ′′ ; q ,
0) + . . .λ ( q , Ω) = λ ( , Ω) + λ q ( q ,
0) + λ q ( q ,
0) + . . .
It is shown in Appendix D that substitution of the above expansions into Eqn. (64) leadsto the following set of coupled integral equations: Z k ′′ H ( k , k ′′ ) f ( k ′′ ) = 0 (69) Z k ′′ (cid:0) H ( k , k ′′ ) f ( k ′′ ) + H ( k , k ′′ ) f ( k ′′ ) (cid:1) = λ f ( k ) (70) Z k ′′ (cid:0) H ( k , k ′′ ) f q ( k ′′ ) + H q ( k , k ′′ ) f ( k ′′ ) (cid:1) = 0 (71) Z k BP ( k ; q ) f q ( k ) = λ q (72)Within Eqns. (69-72) we have used a shorthand notation for the quantities appearing in Eqn.(68), in which the arguments q and Ω are omitted. The superscript indicates the variableand the order of perturbation for the operator H , right eigenfunction f r , and eigenvalue λ .The disappearance in Eqns. (69-72) of contributions from some terms that appear in Eqn.(68) is due to the implementation of symmetry and conservation restrictions coming from16qns. (54) and (63), as shown in Appendix D. Importantly, the first-order-in-wavenumbercontribution to the eigenvalue vanishes: λ q = 0. Without this result there would be nodiffusion behavior.With the aid of Eqns. (54) and (69), the eigenfunction f r at q = 0, Ω = 0 can be foundat once: f ( k ′′ ) = B ∆ G ( k ′′ ; ,
0) (73)with B = 1 rR v ∆ G ( v ; , . (74)Note that ∆ G ( v ; ,
0) is negative, but only B appears in the expression for the diffusionconstant. Integrating Eqn. (70) over k along with the subsequent implementation of theWTI from Eqn. (54) at the corresponding order, leads to the expression for the eigenvalue λ ıω Ω (1 + a ) = λ (75)where we have introduced a parameter a defined by a = 1 R k f ( k ) Z k ′′ A ( k ′′ ; , f ( k ′′ ) . (76)This is an analogue of the well-known parameter that appears in the diffussion of light,which, being positive, renormalizes the phase velocity downwards to a lower value for atransport velocity [40, 41, 43, 44]. To see that our a is indeed positive, replace Eqns. (55)and (73) into Eqn. (76) to obtain a = − ρ ω R v ∆ G ( v ; , Z k Im [Σ + ( k ) G + ( k )] ≈ (cid:18) c T Re [ K ] ω − (cid:19) > F de-fined by Eqn. (F5) that enters the integrand of Eqn.(77) as a consequence of the calculationof Im [Σ + ( k ) G + ( k )]. 17 . Explicit form of the diffusion constant Within the spectral approach that we are using, Eqns. (67-72) lead to the followingexpression for the singular part of the intensity, Φ sing :Φ sing = f r0 ( k ; q , Ω) f r0 ( k ′ ; q , Ω) λ + λ q = f r0 ( k ; q , Ω) f r0 ( k ′ ; q , Ω) λ − ı Ω (cid:16) − ı Ω + − ı Ω λ q λ q q (cid:17) . (78)Then, with the assistance of Eqns. (75,78) the diffusion constant can be identified as D ≡ − ı Ω λ q q λ = − Bq ω (1 + a ) Z k P ( k ; q ) f q ( k ) (79)= B q ω (1 + a ) Z k P ( k ; q ) Z k Φ( k , k ) P ( q ; k ) − ∆ G q ( k ) (80) ≡ D R + D ∆ G q (81)with B given by (74). To obtain Eqn. (80), in which the diffusion constant is written asthe sum of two terms, we have substituted the values for λ q , λ given by Eqns. (72) and(75). The last one ensued from the form of f q ( k ) (See Appendix E). Thus, the expressionfor the diffusion constant in Eqn. (79) is the sum of two contributions, as defined in (81).While the second term in (81), D ∆ G q , can be calculated straightforwardly (See AppendixF), and it vanishes, D ∆ G q = 0, the calculation of the first term, D R , is more laborious.Indeed, it depends on the unknown function Φ. However, it is not the complete functionthat is needed, but an integrated form over one of its variables that, as we now show, canbe expressed as a function of the mass operator Σ and the kernel K using the BS equationand the WTI. In order to do this we apply, inspired by the treatment of light diffusion [46],the method that uses an auxiliary function Ψ ,s ( k ) defined by the relationΨ ,s ( k ) q s ≡ Z k ′ Φ( k , k ′ ) P ( q ; k ′ ) = − Z k ′ Φ( k , k ′ ) 12 ıρ ∂L ( k ′ ) ∂k s q s . (82)Then, from Eqns. (38-39) the following expression for Ψ ,s ( k ) immediately follows: P ( p )Ψ ,s ( p ) + ∆Σ( p )Ψ ,s ( p ) − Z p ′′ ∆ G ( p ) K ( p , p ′′ )Ψ ,s ( p ′′ ) = − ∆ G ( p ) 12 ıρ ∂L ( p ) ∂p s (83)Eqn. (83) can be substantially simplified if we recall the explicit form of the free mediumGreen’s function, Eq.(13), along with relations from Eqns. (12,37) [17]. Then,∆ G ( p ) = − iρ (cid:0) G ∗ ( p ) − − G ( p ) − (cid:1) G ( p ) G ∗ ( p ) = ( P ( p ) + ∆Σ( p )) G ( p ) G ∗ ( p ) (84)18s a next step, we define an angular entity Υ that is analogous to the coefficient relatingtransport mean free path and extinction length in the diffusion of electromagnetic waves[47, 48]: Ψ ,s ( p ) q s = G ( p ) G ∗ ( p )Υ( p , q ) , (85)and an integral equation for Υ follows straightforwardly from Eqns. (83-85):Υ( p , q ) − Z p ′′ K ( p , p ′′ ) G ( p ′′ ) G ∗ ( p ′′ )Υ( p ′′ , q ) = P ( p ; q ) . (86)Then, using Eqns. (82), (85) and (F9), the diffusion constant from Eq. (79) can be writtenas D = B q ω (1 + a ) Z k P ( k ; q ) ( G ( k ) G ∗ ( k )Υ( k , q )) (87)Furthermore, guided by the definition from Eqn. (85) we can make a conjecture that tensorΥ( p , q ) should be linear in q . Keeping in mind this property of Υ( p , q ) we seek for thecorresponding solution in the form Υ( p , q ) = αP ( p ; q ) (88)Then, by multiplying Eqn. (86) with P ( p ; q ) G ( p ) G ∗ ( p ) from the left and integrating overthe p we remain with the relation α = − R p R p ′′ P ( p ; q ) G ( p ) G ∗ ( p ) K ( p , p ′′ ) G ( p ′′ ) G ∗ ( p ′′ ) P ( p ′′ , q ) R k P ( k ; q ) G ( k ) G ∗ ( k ) P ( k ; q ) − (89)where the ratio of two integrals is the analog of the h cos θ i term in the diffusion of electro-magnetic waves [48]. As a consequence, the diffusion constant can be represented as D = B q ω (1 + a ) Z k αP ( k ; q ) G ( k ) G ∗ ( k ) P ( k ; q ) (90)It must be noted here that α included in the general expression for the diffusion constantfrom Eqn. (90) can be evaluated explicitly using the symmetry properties of the Greenfunction, the self-energy, and the WTI from Eqns. (18), (20),and (57), respectively, as wellas the reciprocity property of the kernel K . Indeed, those equations support the validity of19he following relations: ∆Σ( − k ′ ; ,
0) = (91) Z k ′′ ∆ G ( k ′′ ; , K B ( k ′′ , − k ′ ; ,
0) =∆Σ( k ′ ; ,
0) = Z k ′′ ∆ G ( k ′′ ; , K B ( k ′′ , k ′ ; ,
0) = Z k ′′ K B ( k ′ , k ′′ ; , G ( k ′′ ; ,
0) = Z k ′′ K B ( − k ′ , k ′′ ; , G ( k ′′ ; ,
0) = ∆Σ( − k ′ ; , . This yields K B ( k ′ , k ′′ ; ,
0) = K B ( − k ′ , k ′′ ; ,
0) = K B ( k ′ , − k ′′ ; ,
0) (92)so that K B ( k , − k ′′ ) G ( − k ′′ ) G ∗ ( − k ′′ ) P ( − k ′′ ; q ) = −K B ( k , k ′′ ) G ( k ′′ ) G ∗ ( k ′′ ) P ( k ′′ ; q ) . (93)In turn, as one can easily see from the Eq.(89), the odd character of the function from theEq.(93) immediately determines the value α = 1. Therefore, the diffusion constant from theEq.(90) can be brought into the form D = B q ω (1 + a ) Z k P ( k ; q ) G ( k ) G ∗ ( k ) P ( k ; q ) (94)Finally, exploiting the representation for G ( k ) G ∗ ( k ) through Eqn.(84), along with Eqns.(37),(59) and (77) we can write D = c T ω (1 + a ) K ( K ) ∗ Im [ K ] ≈ c T v (1 + a ) vl ≈ c T v vl v = ωRe [ K ] , l = 12 Im [ K ] , (96)and we have used the approximations K ( K ) ∗ ≈ ω /v , and Re [ K ] ≈ ω /v . It is easy tocheck that these approximations hold within terms linear in the density n .To sum up, we have the following relation for the diffusion coefficient of anti-plane wavestraveling through a maze of screw dislocations: D = 12 v ∗ l , (97)20he usual form of diffusion coefficients, in terms of a transport velocity v ∗ = c T /v , where c T is the “bare” wave velocity, v is the velocity of coherent waves, and l a transport mean freepath which in this case is equal to the attenuation length of the coherent waves. V. DISCUSSION AND CONCLUSIONS
We have computed, Eqn. (97), the diffusion coefficient for anti-plane elastic waves mov-ing incoherently through many, randomly placed, screw dislocations in two dimensions. Al-though the procedure is based on a standard Bethe-Salpeter approach, a number of featuresof the calculation deserve to be pointed out.The first is that we are studying the diffusive behavior of anti-plane waves of frequency ω in a two dimensional continuum. The limit ω → n , and all higher order terms in n have been omitted. Themass operator can then be computed to all orders in perturbation theory, in which theperturbation is carried out for weak dislocation-wave coupling. Given the nature of theinteraction, Eqn. (6), this is the case for long wavelengths. The summation is possible dueto the point-like nature of the interaction. Also, the fact that the interaction involves agradient of a delta function is responsible for the pre-WTI that is needed in order to obtainthe WTI. This, in turn, depends on the fact that the interaction between dislocation andelastic wave is given by the Peach-Koehler force. When the ISA is used, however, and the lowfrequency and long wavelength limits that are needed to make sense of a diffusion behaviorare applied, it becomes necessary to impose B = 0; that is, there is no viscous dampingassociated with the string dynamics. Since damping is associated with retardation effects,this restriction can be associated with the fact that the interaction of the string with the21lastic wave is evaluated at the equilibrium position of the string thus neglecting retardationeffects. It is conceivable that relaxing this condition could lead to a compensation with B = 0terms in the dynamics that has been considered through the frequency dependence of thepotential (6)In addition, the approximation | Im [ K ] | ≪ | k − Re [ K ] | has been used, where K isthe effective wave number of the coherent waves. As discussed in Appendix C, this places arestriction on the regions of ( ω, k ) space, where the function Φ is defined, in which a diffusionbehavior occurs when B = 0. Along the diagonal k ∼ k β = ω/c T , this is automaticallysatisfied for frequencies ω that are small compared to the natural frequency of the oscillatingdislocations, as well as small compared to the viscous damping. It is not satisfied forfrequencies around the resonant frequency, and it is again satisfied for high frequencies,both for small and high dampings.In terms of the dislocation parameters, the low density approximation has different impli-cations for low and high frequencies ω . For low frequencies, the distance between dislocationshas to be large enough so that the time it takes the bare wave to go from one to the nextis large compared with the period of oscillation around the PN potential well minimum.For high frequencies, the distance between dislocations must be large compared to barewavelength. A. Concluding remarks
A Bethe-Salpeter approach as been used to study the behavior of incoherent anti-planewaves inside a two-dimensional elastic continuum populated by a random distribution ofscrew dislocations. A diffusive limit has been identified and the corresponding diffusionconstant has been calculated. A natural next step would be to consider whether the diffusioncoefficient can vanish, leading to localization of the waves. Another, certainly, would beto use the techniques developed in this paper to address the more involved, but also morerealistic, case of elastic waves in a three-dimensional elastic continuum with many, randomlyplaced, vibrating dislocation segments. 22 cknowledgments
This work was supported by Fondecyt Grants 1130382, 1160823, and ANR-CONICYTgrant 38, PROCOMEDIA. A useful discussion with M. Riquelme is gratefully acknowledged.
Appendix A: Summation of the perturbation expansion for the mass operator
Following the ISA we evaluate the mass operator as Σ = n R < t > d ~X n , where theaverage is over the Burgers vector. We consider screw dislocations, with a Burgers vectorof fixed magnitude but randomly oriented. Dislocation position has been assumed to beuniformly distributed with density (number per unit surface) n . We introduce the definitionof the t -matrix in momentum space through [37] t ( ~k, ~k ′ ) = Z d~xd~x ′ e − ı~k · ~x t ( x, x ′ ) e ı~k ′ · ~x ′ (A1)and its Born expansion t ( ~k, ~k ′ ) = t (1) ( ~k, ~k ′ ) + t (2) ( ~k, ~k ′ ) + t (3) ( ~k, ~k ′ ) . . . (A2)The first Born approximation is easily computed: t (1) ( ~k, ~k ′ ) = Z d~xe − ı~k · ~x µV ( x ) e ı~k ′ · ~x = − µ A n ~k · ~k ′ e ı ( ~k ′ − ~k ) · ~X n . (A3)The second Born approximation is defined as t (2) ik ( ~k, ~k ′ ) = µ Z d~x d~x ′ e − ı~k · ~x V ( x ) G ( x − x ′ ) V ( x ′ ) e ı~k ′ · ~x ′ (A4)= ( − µ A n ) (2 π ) Z d~x d~x ′ d~q e − ı~k · ~x ∂∂x m δ ( ~x − ~X n )( − ıq m ) e − ı~q · ~X n G ( ~q ) e ı~q · ~x ′ × ∂∂x ′ p δ ( ~x ′ − ~X n )( ık ′ p ) e ı~k ′ · ~X n = ( − µ A n ) (2 π ) Z d~q k ′ p k m q m q p G ( ~q ) e ı ( ~k ′ − ~k ) · ~X n = ( − µ A n ) π I~k · ~k ′ e ı ( ~k ′ − ~k ) · ~X n . where ( k β ≡ ω/c T ) I ≡ Z dq q G ( ~q ) = 1 µ Z dq q ( q − k β ) (A5)23n turn, the third order Born approximation to the T matrix can be determined from therelation t (3) ( ~k, ~k ′ ) = µ Z d~x d~x ′ d~x ′′ e − ı~k · ~x V ( ~x ) G ( ~x − ~x ′ ) V ( ~x ′ ) G ( ~x ′ − ~x ′′ ) V ( ~x ′′ ) e ı~k ′ · ~x ′′ (A6)which yields, explicitly, t (3) ( ~k, ~k ′ ) = ( − µ A n ) (4 π ) I ~k · ~k ′ e ı ( ~k ′ − ~k ) · ~X n . (A7)It can be easily seen that the series (A2) for t ( ~k, ~k ′ ) is geometric and can therefore be summedto get t ( ~k, ~k ′ ) = t ( ~k, ~k ′ ) 11 + µ A n I π , (A8)Although I , given by (A5), diverges, it can be regularized with a scheme similar to the oneused in the case of a random ensemble of edge dislocations in 3D medium [37]. The precisedefinition of the integral I from Eqn. (A5) is I ≡ µ lim η → Z dq q ( q − k β − ıη )= 1 µ " P Z dq q ( q − k β ) + ıπ Z dq q δ ( q − k β ) (A9)= I P + I R . where P denotes the principal value. We have then I P = ℜ [ I ] ≡ µ P Z dq q ( q − k β ) (A10) I R = ı ℑ [ I ] ≡ ı πµ Z dq q δ ( q − k β ) . where we have used lim η → x − ıη = P x + ıπδ ( x ) . (A11)The second integral appearing in Eqn. (A10) can be evaluated at once. It is I R = ı πµ ∞ Z −∞ dq Θ( q ) q δ ( q − k β ) = ı πµ k β , (A12)24here Θ( q ) is Heaviside function. On the other hand, the first integral in Eqn. (A10) isdivergent and the introduction of a short wavelength (upper limit) cut-off Λ is needed. Asa consequence, we have the following expression for Λ > k β : I P ( k β , Λ) ≡ µ P Λ Z dq q ( q − k β ) = 12 µ " Λ + k β ln Λ k β − ! ≈ Λ µ . (A13)Finally, the self-energy Σ readsΣ = n Z < t > d ~X n = n< − µ A n µ A n I R π >k = − nµ b k M ω − ω R + i (cid:16) ω BM + µb ω Mc T (cid:17) . (A14)with ω R = ω − µb Λ / πM .The integral (A13) nominally diverges because in continuum mechanics there is no intrin-sic length scale. Consequently, all wave vectors, even very high ones, have to be integratedover in (A13). But this cannot be completely correct, since we are dealing with an ap-proximation to a material that is made of atoms and molecules, and it does have a naturallength scale, the interatomic distance. We take this indeterminacy into account through arenormalization of the frequency ω to ω R .In order for the geometric series to converge, it is needed that | µ A n I | < π . For lowfrequencies this means that ( µb / πM ) / Λ < ω . This is the only possibility, actually, forthe vanishing viscosity case, B = 0, that is needed in order to have a WTI in the ISA. Appendix B: Optical theorem
We need to show ∆Σ( k ′ ) = Z p ′′ ∆ G ( p ′′ ) K B ( p ′′ , k ′ ; ,
0) (B1)with K B ( k , k ′ ; ,
0) = n < t ( k , k ′ ) t ∗ ( k ′ , k ) > , (B2)a limiting form of (56). 25onsider the right-hand-side of (B1), together with Eqns. (60) and (A8): Z k ∆ G ( k ) K B ( k , k ′ ; ,
0) = V Z k k δ (cid:0) k − Re [ K ] (cid:1) ( k · k ′ ) (B3)= ( k ′ ) V π ( Re [ K ]) ≈ ( k ′ ) V ω πc T (B4)= − nk ′ ω A |D| (B5)with V = − nπµ h|T | i /ρ ω . In the third line we have used ( Re [ K ]) ≈ ω /v ≈ ω /c toleading order in the density n , as discussed in Appendix C. For convenience, we reproducehere Eqn. (19) K = ω c T (1 + n T ) (B6)Let us write T ≡ A / D , with D ≡ µ A I R ) / π . For the left-hand-side of (B1) we have,from (60) and (B6) ∆Σ( k ; ,
0) = − Im [ K ] ω k K ( K ) ∗ (B7)= − nk c T A|D| Im [ D ]= − nµk c T π A |D| Im [ I R ] (B8)which is equal to (B5) provided B = 0 since, by (A12), Im [ I R ] = πω / µc T . ConsequentlyEqn. (B1) is satisfied to leading order in n , for small n , as needed. Notice the importanceof being able to sum the complete series for the t matrix in order to establish the result ofthis Appendix. Appendix C: Discussion of the approximations used in this work
This paper involves waves in interaction with scatterers. The waves are characterized bytheir frequency ω , and the scatterers by their density n , resonant frequency ω and viscousdamping B/M . The computations that have been carried out involve approximations thatrestrict the values these parameters are allowed to have. This Appendix discusses thissituation. 26 central assumption of this paper is that there is a random distribution of scatterersand that each is an independent random variable (the Independent Scattering Approxima-tion, ISA). This leads to expressions (17) and (56) for the mass operator Σ and kernel K ,respectively, that are linear in the density of dislocations n . Keeping correlations would leadto higher order terms in n . As a consequence, only terms linear in n must be kept in allexpressions throughout. This has consequences that are explored below.In addition, the inequality | Im [ K ] | ≪ | k − Re [ K ] | , (C1)has been used in Appendix B, to verify that the WTI holds within the ISA, and in AppendixF to compute part of the diffusion constant, with K = ωc rh − Σ ρc k i = ωc √ n T = k β √ n T , (C2)Σ = − nµ T k , T ≡ µb M ω − ω R + i µb Mc T ω , where M = ( µb / πc T ) ln( δ/ǫ ) ≈ µb /c T , and we shall use the last approximation for es-timates. These expressions coincide with (21) when B = 0, as required by the ISA tothe WTI. The average that is implicit in (C2) can be ignored since it involves b and theonly remaining randomness (after having averaged over position to get proportionality todislocation density n ) is in the sign of b .We have, from (C2) and for small densities n , K = k β
11 + n T ≈ k β (1 − n T ) . (C3)Rewriting the inequality (C1) in terms of the attenuation length l = 1 / (2 Im [ K ]) and coher-ent velocity v = ω/Re [ K ], one obtains that the frequency ω and wavenumber k are restrictedby either ω ≪ vl r k l + 12 − ! (C4)or ω ≫ vl r k l + 12 + 12 ! . (C5)27e expect the region of wavenumbers large compared to the inverse of the attenuationlength, kl ≫
1, to be of special interest. In this case these restrictions become ω ≪ kv , or ω ≫ kv .Along the diagonal k ∼ ω/c T = k β in ( ω, k ) space it is possible to be a little more precise.In this case we have k − Re [ K ] = n k β Re [ T ] (C6) Im [ K ] = − n k β Im [ T ] (C7)and the inequality (C1) translates into | Im [ T ] | ≪ | Re [ T ] | . (C8)The consequences of this restriction depend on the dislocation parameters M , B , γ , as wellas frequency ω . Notice that the dependence on the Burgers vector b has dropped out underthe approximation in (7). We consider various regimes for the frequency ω .In the case of small frequencies ω ≪ ω R the restriction (C8) is automatically satisfied,and the small density requirement | n T | ≪ nc /ω R ≪
1. That is, the period ofoscillation of the screw dislocation around its (renormalized) NP minimum is small comparedto the time the wave takes to go from one dislocation to the nearest one.Near resonance, ω ∼ ω R , T is pure imaginary and (C8) cannot be satisfied.For high frequencies ω ≫ ω R , we have | Im [ T ] || Re [ T ] | = µb M c ∼
18 (C9)and inequality (C8) is satisfied, to the extent that 1 ≪
8. The condition | n T | ≪ n ≪ k , or, the distance between dislocations must be large compared to wavelength. Appendix D: Perturbation scheme for the spectral problem
To build up a system of equations for the determination of the diffusive pole structurewe have to substitute the series from Eqn. (68) into Eqn. (64) and gather together allterms of the same order, both in Ω and q . Moreover, we assume that at every order of theperturbation scheme both WTI from Eqn. (54) and symmetry constraints from (63) are28alid. Omitting the Ω and q arguments for brevity, this yields Z k ′′ ( H ( k , k ′′ ) + H ( k , k ′′ ) + H q ( k , k ′′ ) + H q ( k , k ′′ ) + . . . ) (D1) × ( f ( k ′′ ) + f ( k ′′ ) + f q ( k ′′ ) + f q ( k ′′ ) + . . . ) =( λ + λ q + λ q + . . . )( f ( k ) + f ( k ) + f q ( k ) + f q ( k ) + . . . ) . At first order in Ω and zero order in q Eqn. (D1) easily leads to Eqns. (69) and (70) in thetext. In a similar manner, collecting the first order in q terms from Eqn. (D1) we obtainthe following equation for λ q Z k ′′ ( H ( k , k ′′ ) f q ( k ′′ ) + H q ( k , k ′′ ) f ( k ′′ )) = λ q f ( k ) . (D2)Integrating (D2) over k cancels the contribution from the first term on its left hand sidebecause of the WTI. So that, using (73) we have Z k Z k ′′ H q ( k , k ′′ )∆ G ( k ′′ ) = λ q Z k ∆ G ( k ) (D3)The left hand side of (D3) is equal to zero because of the WTI written to first order in q ,as well as the odd in k character of the tensor P from (62). Therefore, we obtain λ q = 0.To complete the set of equations for the reconstruction of λ ( q , Ω) we need λ q . Tosecond order in q Eqn. (D1) gives Z k ′′ ( H ( k , k ′′ ) f q ( k ′′ ) + H q ( k , k ′′ ) f q ( k ′′ ) + H q ( k , k ′′ ) f ( k ′′ )) = λ q f ( k ) . (D4)Then, Eqn. (72) of the text is obtained integrating (D4) over k and using the explicit formof the WTI at corresponding orders. Appendix E: Solution for f q ( k ) To find the solution for f q ( k ) we have to modify accordingly Eqn. (71). Indeed, using(73) we can rewrite its second term as Z k ′′ H q ( k , k ′′ ) B ∆ G ( k ′′ ) = Z k ′′ B ( P ( q ; k ′′ ) δ k ′′ , k ∆ G ( k ) (E1) − H ( k , k ′′ )∆ G q ( k ′′ ) (cid:1) = B Z k ′′ H ( k , k ′′ )[ Z k Φ( k ′′ , k ) P ( q ; k ) − ∆ G q ( k ′′ )] . H q ( k , k ′′ ). The second equality is a result of substituting δ k ′′ , k ∆ G ( k )by its value given by (38). Hence, f q ( k ′′ ) = − B Z k Φ( k ′′ , k ) P ( q ; k ) − ∆ G q ( k ′′ ) (E2)and ∆ G q ( k ) = q · ∂ ∆ G ( k ; q ′ , ∂ q ′ | q ′ =0 = − ıρ q · ∂ ( Re [ G − ( k )]) ∂ k (E3) Appendix F: Calculation of D ∆ G q Here we show that, in the appropriate limit spelled out below, D ∆ G q = 0. From Eqn.(79) we have D ∆ G q = − B q ω (1 + a ) Z k P ( k ; q )∆ G q ( k ) (F1)= − µB ρ q ω (1 + a ) Z k q · k ∂ ( Re [ G − ( k )]) ∂ k · q (F2)where Eqn. (F1) has been obtained using Eqns. (37) and (E3).In Eqn. (F1) we have to deal with the following integral ( s, t = 1 , J st = Z k (cid:18) ∂ ( Re [ G − ]) ∂k t (cid:19) k s . (F3)Then, using (18) we can write J st = Z k (cid:18) k t k s (2 k F ( ω, k ) Im [ K ] − Re [ K ] F ( ω, k )) ρω Im [ K ] (cid:19) (F4)= ∞ Z −∞ δ st Θ( k ) k dk (cid:18) k F ( ω, k ) Im [ K ] − Re [ K ] F ( ω, k )2 πρω Im [ K ] (cid:19) . Where F ( ω, k ) = (cid:18) Im [ K ]( k − Re [ K ]) + Im [ K ] (cid:19) (F5)The expression for J st in (F4) contains an ill-defined term in the integrand, proportional to F ( ω, k ), that can be regularized as follows [45]: Introduce a new variable x ≡ ( k − Re [ K ]),30nd consider the following auxiliary integrals: ∞ Z −∞ (cid:18) Im [ K ] x + Im [ K ] (cid:19) dx = π, (F6) ∞ Z −∞ (cid:18) Im [ K ] x + Im [ K ] (cid:19) dx = π Im [ K ] . They show that it is possible to make the replacements (cid:18) Im [ K ] x + Im [ K ] (cid:19) −→ πδ ( x ) , (F7) (cid:18) Im [ K ] x + Im [ K ] (cid:19) −→ πδ ( x )2 Im [ K ] . which yield the same result after integration over x . Moreover, both replacements give theproper asymptotic behaviour in the low-density limit (our case) when Im [ K ] → | Im [ K ] | ≪ | k − Re [ K ] | . Therefore, on the basis of Eqns. (F5)-(F7) wehave F ( ω, k ) = lim Im [ K ] → πδ (cid:0) k − Re [ K ] (cid:1) , (F8) F ( ω, k ) = lim Im [ K ] → πδ ( k − Re [ K ])2 Im [ K ]Technically, Eqn. (F8) indicates that in all final expressions the limit Im [ K ] →
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