# Coherent propagation and incoherent diffusion of elastic waves in a two
dimensional continuum with a random distribution of edge dislocations

Dmitry Churochkin1 and Fernando Lund2 1Saratov State University, Saratov,
410012 Russia.
2Departamento de Física and CIMAT, Facultad de Ciencias Físicas y Matemáticas,
Universidad de Chile, Santiago, Chile.

###### Abstract

We study the coherent propagation and incoherent diffusion of in-plane elastic
waves in a two dimensional continuum populated by many, randomly placed and
oriented, edge dislocations. Because of the Peierls-Nabarro force the
dislocations can oscillate around an equilibrium position with frequency
$\omega_{0}$ . The coupling between waves and dislocations is given by the
Peach-Koehler force. This leads to a wave equation with an inhomogeneous term
that involves a differential operator. In the coherent case, a Dyson equation
for a mass operator is set up and solved to all orders in perturbation theory
in independent scattering approximation (ISA). As a result, a complex index of
refraction is obtained, from which an effectve wave velocity and attenuation
can be read off, for both longitudinal and transverse waves. In the incoherent
case a Bethe-Salpeter equation is set up, and solved to leading order in
perturbation theory in the limit of low frequency and wave number. A diffusion
equation is obtained and the (frequency-dependent) diffusion coefficient is
explicitly calculated. It reduces to the value obtained with energy transfer
arguments at low frequency. An important intermediate step is the obtention of
a Ward-Takahashi identity (WTI) for a wave equation that involves a
differential operator, which is shown to be compatible with the ISA.

##  I Introduction

The propagation of elastic (including acoustic) waves in complex media has
evolved into a rich variety of fields and sub-fields, ranging from the very
basic to the very applied. A basic tenet is that there is an elastic medium
endowed with an heterogeneous structure, for example inclusions or defects. A
problem that is simple to formulate is the computation of the scattering of
elastic waves in an elastic medium by a static spherical inclusion that has
different elastic properties. This problem is solved by use of the respective
wave equation and appropriate boundary conditions at the surface of a sphere.
A next step in complexity is taken by considering not one sphere but many.
Here the problem becomes unwieldy very quickly and approximation methods must
be used. The resulting phenomena can be very rich, depending on the relative
spacing of the spheres, the impedance contrast, and the wavelength considered.
Particularly important is the geometrical location of the spheres: At one end,
they can be considered to be located in a periodic lattice, and this leads to
phononic crystals. At the other end, they can be randomly distributed with,
say, a uniform distribution. And once randomness is allowed, there is no
reason to limit it to position: size and elastic properties can also be
randomly distributed. A rich variety of phenomena can ensue, depending on the
relative size of wavelength, sphere radius and inter-sphere distance. For
example, at appropriate wavelength resonances may occur that will considerably
influence the wave propagation. And instead of spheres, other obstacle
geometry can be considered. In any case, there are two major possible
approaches to any one of these problems: Either the properties of the
obstacles are known, and it is desired to understand the behavior of the
elastic waves therein, or the behavior of the waves is known, and from their
behavior it is desired to infer the properties of the heterogeneous medium.

When the obstacles to wave propagation have random intrinsic properties and/or
are randomly placed, three major phenomena can occur: A coherent, average
(“effective”), wave can exist. It being described by a complex index of
refraction, whose real part gives an effective phase velocity, and whose
imaginary part gives an attenuation. This attenuation in turn can have two
origins: one, the intrinsic losses within the medium and its inclusions, and
two, the loss of coherence due to the incoherent waves scattered by the random
obstacles. Since the coherent wave attenuates, it will die out after a while.
It may happen that the remaining, incoherent, radiation can be described in
terms of a diffusion equation, with a qualitatively different wave behavior.
And it may happen that the diffusion coefficient vanishes, in which case the
wave can be localized (“Anderson localization”) by the disordered medium.
Experiments concerning this last phenomenon have been conducted with
ultrasound traveling inside a ‘mesoglass’, an ensemble of aluminum balls
brazed together to form a disordered solid [1].

Randomly placed elastic heterogeneities in an otherwise elastic medium can be
described in two ways: On the one hand, specific obstacles can be considered,
as with the spheres used for illustration in a paragraph above. In this case
the randomness, so to speak, is “discrete”. But one can also consider a medium
described by a continuum random variable, for instance random, but
continuously varying, elastic moduli. One object of study is the time and
space evolution of the average field amplitude, also called the coherent wave.
This approach was pioneered by Foldy [2] and has been widely used ever since.
This coherent wave inevitably attenuates, even in the absence of inelastic
losses, because of the disorder that takes, so to speak, energy out of an
incoming beam. However, when this coherent wave dies out, there still remains
elastic energy moving around. As emphasized by Weaver [3], the dynamics of
this “diffusive” wave, is best described taking the average of the energy,
that involves the average of the square of the field amplitude, rather than
the square of the average of said field. In this way Weaver addressed the
diffusivity of ultrasound in polycrystals. The multiple scattering technique
used by Weaver has been further elaborated upon: For example, Thomson et al.
[4] have considered duplex microstructures; Yang et al. [5] have considered
the effect of elongated grains; Li et al. [6] have studied the effect of
macroscopic anisotropy; and, quite recently, Bai et al. [7] have compared the
behavior of two- and three-dimensional polycrystals. This methodology has been
further extended to study the behavior of composites [8, 9]. In all these
works the obstacles doing the scattering are static. In the present work, as
explained in detail below, we shall be concerned with a specific obstacle that
responds dynamically to the incident wave.

Rather that deriving the diffusive properties of an elastic field from those
of the wave amplitude, it is also possible to consider the evolution of the
energy density from the beginning, bypassing the field amplitude description.
In this way the specifics of wave interference arising from specific obstacles
is lost, but the resuting algebra is much simpler to deal with. This approach
has been describe by Ryzhik et al. [10] and is widely used in seismology,
where the nature of the scatterers that are obstacles to the propagation of
seismic elastic waves is largely unknown [11]

Topological defects, such as dislocations, present an obstacle of a different
nature to the propagation of elastic waves. For wavelengths longer than a few
interatomic spaces, a continuum approximation is appropriate, and this is the
approach we shall adopt in this work. The main difference between dislocations
and other obstacles to elastic waves is that dislocations have their own
dynamics: They can be treated as strings endowed with a mass and line tension,
loaded by the stress associated with the elastic wave according to the well-
known Peach-Koehler force [12]. Their response is described by the equation of
an elastic string, as pioneered by Granato and Lücke [13, 14] and revisited
more recently by Maurel at al. [15, 16, 17]. These strings oscillate in
response to an elastic wave and, as a result, generate secondary, scattered
waves. The general formalism for this aspect of the wave-dislocation
interaction was discussed clearly by Mura [18] and elaborated by Lund [19].
The wave behavior in the presence of many dislocations was studied by
Churochkin et al. [20], who computed the coherent wave behavior, providing an
expression for the (complex) index of refraction that describes such a wave.
These expressions have been used to develop new non-intrusive ultrasonic non-
destructive measurements of dislocation density in metals [21, 22, 23, 24].
The quantum theory has also been studied in some detail, leading to new
insights into the role played by dislocations in thermal transport [25, 26]. A
rather unexpected application of these results is to the understanding of the
acoustic properties of glasses in the THz range [27, 28], where the continuum
approach is helpful in getting around the fact that glasses have no crystal
structure. A natural evolution of this train of thought is to explore the
properties of elastic waves within a medium filled with randomly placed
dislocations, in the diffusive regime. In a previous paper we have treated the
case of a two-dimensional medium with screw dislocations [29]. In this work we
tackle the problem of a two-dimensional medium as well, but populated with
edge dislocations. This turns the problem from a scalar to a vector nature,
and is considerably richer, as will be seen in the text that follows.

###  I.1 Organization of this paper

Section II has the basic layout of the problem to be studied: Equations in two
space dimensions for the response of an edge dislocation loaded by an
external, time-dependent, stress are written down, as well as the wave
equation for dynamic elasticity when the source of elastic waves is a set of
$N$ edge dislocations. Section III sets the problem of the propagation of
waves in the presence of many dislocations as the problem of a wave in the
presence of a random external potential. Standard tools are used to solve this
problem to find a solution in terms of a coherent, average, wave propagating
through a medium with an effective, complex and frequency-dependent, index of
refraction. There appear longitudinal and transverse phase velocities, as well
as longitudinal and transverse attenuations. The problem of the diffusion
behavior of the waves in interaction with many edge dislocations is treated in
section IV. A Bethe-Salpeter (BS) equation is written down and a Ward-
Takahashi identity (WTI) is established. As in the coherent wave problem of
Section III, we use an independent scattering approximation (ISA). That is,
the random variables characterizing each dislocation (in practice here,
position and Burgers vector orientation) are statistically independent of each
other. It is established that the WTI is consistent with the ISA. The low
frequency and small wave number of said equations are considered, and the BS
equation, written in the form of an eigenvalue problem, is solved
perturbatively. A diffusion behavior is established, and an explicit
expression for the diffusion coefficient is obtained. Section V has
discussions and conclusions. As mentioned in the first part of this
Introduction, the problem of waves interacting with topological defects that
have their own dynamics has a number of different characteristics from the
problem posed by static obstacles. For this reason we write down our
calculations in some detail, relegating a number of the more cumbersome to
several appendices. Also in an Appendix we compare the approach used here to
obtain a WTI with the approach used in [29] for screw dislocations, and the
consequences for the diffusion constant of anti-plane elastic waves traveling
in a medium populated by a set of randomly located screw dislocations.

##  II Interaction of in-plane elastic waves with edge dislocations.

The problem of scattering of an in-plane wave on a single edge dislocation has
been extensively treated by Maurel et al. [30]. In turn, the coherent behavior
emerges when the interaction of an in-plane wave with an ensemble of randomly
located edge dislocations is considered, as it was elucidated by Maurel et al
[31], using the two-dimensional adaptation of the general equation of motion,
which takes the following form:

$\rho\frac{\partial^{2}v_{a}(\vec{x},t)}{\partial
t^{2}}-c_{abcd}\frac{\partial^{2}}{\partial x_{b}\partial
x_{c}}v_{d}(\vec{x},t)=s_{a}(\vec{x},t)$ (1)

where $\rho$ is the (two dimensional) mass density, $v_{a}$ is a vector of in-
plane particle velocity as a function of two-dimensional position $\vec{x}$
and time $t$,
$c_{abcd}=\lambda\delta_{ab}\delta_{cd}+\mu\left(\delta_{ac}\delta_{bd}+\delta_{ad}\delta_{bc}\right)$
is the two-dimensional elastic constants tensor with
$\left(\lambda,\mu\right)$ the Lamé constants,and the source
$s_{a}(\vec{x},t)$ is

$s_{a}(\vec{x},t)=\sum\limits_{n=1}^{N}c_{abcd}\epsilon_{ce}\dot{X}_{e}^{n}b_{d}^{n}\frac{\partial}{\partial
x_{b}}\delta(\vec{x}-\vec{X}^{n})$ (2)

Here the superscript n denotes the corresponding characteristics of the n-th
edge dislocation, $\epsilon_{ce}=\epsilon_{ce3}$ and $\epsilon_{ijk}$ is the
completely antisymmetric tensor with indexes $i,j,k,\dots=1,2,3.$, $\vec{b}$
and $\vec{X}$ are the Burgers vector and position of the edge dislocation,
respectively, with condition $\dot{\vec{X}}\|\vec{b}$ (that is, an infinite
straight edge dislocation line, which is directed for definiteness along
$x_{3}$, moves only along its Burgers vector, to which it is perpendicular)
and $a,b,c,\dots=1,2$. It should be also noted that within this model the edge
dislocation was originally treated as a point in two dimensions, which
performs a small free oscillation without damping around a certain resting
equilibrium position $\vec{X}_{0}$ (See [31] for details). Smallness of
oscillations means that the source from Eq.[2] should be always taken at
$\vec{X}=\vec{X}_{0}$, and their free character, that neglects the Peierls-
Nabarro (PN) force [32], leads to a scattering cross section diverging at low
frequencies [30]. Qualitatively, this divergence is due to the fact that the
scattering cross section must be proportional to the wavelength, which is the
only length scale presented in the problem. Thus, it diverges at low
frequencies when the wavelength grows. Since in the present work, a diffusion
behavior associated with long wavelengths and low frequencies is of our main
interest, we introduce a Peierls-Nabarro restoring force as well as a viscous
damping into the edge dislocation dynamics in a similar manner as for screw
dislocations in two dimensions [29]. It yields, for small oscillations around
the PN minimum, the following equation of motion

$M\ddot{X}_{b}+B\dot{X}_{b}+\gamma X_{b}=F_{b}\,.$ (3)

Where $M=(\mu b^{2}/4\pi
c_{T}^{2})\left(1+\frac{c_{T}^{4}}{c_{L}^{4}}\right)\ln(\delta/\epsilon)$ is
the mass per unit length of an edge dislocation[31] with
$c_{L}=\sqrt{\frac{\left(\lambda+2\mu\right)}{\rho}}$ and
$c_{T}=\sqrt{\frac{\mu}{\rho}}$; $\delta$ and $\epsilon$ are a long- and
short- distance cut-off, respectively. Parameters $B$ and $\gamma$ represent a
damping and the Peirls-Nabarro restoring force, correspondingly. And, finally
$F_{b}$ is a two-dimensional version of a Peach-Koehler force
$F_{k}=\epsilon_{kjm}\tau_{m}b_{i}\sigma_{ij}$ with $\vec{\tau}$ as the unit
tangent vector to the dislocation line ($\hat{x}_{3}$ in our case) and
$\sigma_{ij}$ is the stress tensor taken at the dislocation line, with
$\sigma_{ij}=c_{ijkl}\partial u_{k}/\partial x_{l}$, where the vector
$\vec{u}$ is the elastic displacement, with $\dot{u}_{a}=v_{a}$.

In distinction from the original consideration of the coherent wave
propagation problem formally presented by Eqs.(1,2,3)(See [31]), where an
auxiliary angle $\theta$ that the Burgers vector makes with a specified
direction was explicitly introduced, we will treat the problem in a general
way, i.e. by representing the source through a combination of tensors,
similarly to three-dimensional case [30], [20]. The natural question here is
to what sort of reduction of a 3D theory our 2D case corresponds to? To answer
this question, it should be recalled that in 3D, in terms of geometry, the
problem is characterized by a random vector triad
$(\vec{\tau},\vec{t},\vec{n}\equiv\vec{\tau}\times\vec{t})$ associated with
each edge dislocation, where $\vec{t}$ is the unit vector of the common
direction of $\dot{\vec{X}}$ and $\vec{b}$. Reduction to 2D means fixing the
equilibrium dislocation line direction, i.e. $\vec{\tau}$, and integrating out
along the line. In other words, geometrically, it is similar to the Granato-
Lucke configuration, in which dislocation segments of length $L$ all point
along the same direction (See Appendix F in [16]), resulting in two-
dimensional $\vec{n}$ and $\vec{t}$

Given that $\vec{\tau}=(0,0,1)$, as discussed above, we can write:

$n_{a}=\epsilon_{a3b}\tau_{3}t_{b}=-\epsilon_{ab}t_{b}\,.$ (4)

Then, using Eq. (4), the following simplifications, similar to 3D case [16],
become valid

$\displaystyle
c_{abcd}\epsilon_{ce}\dot{X}_{e}b_{d}=c_{abcd}\epsilon_{ce}\dot{X}t_{e}bt_{d}=-c_{abcd}n_{c}\dot{X}bt_{d}=-\mu{\mathsf{M}}_{ab}\dot{X}b.$
(5) $\displaystyle F_{a}t_{a}=\epsilon_{ab}t_{a}b_{c}c_{cbde}\partial
u_{d}/\partial x_{e}=n_{b}bt_{c}c_{cbde}\partial u_{d}/\partial x_{e}=\mu
b{\mathsf{M}}_{ed}\partial u_{d}/\partial x_{e}.$ (6)

With

${\mathsf{M}}_{ab}=n_{a}t_{b}+n_{b}t_{a}\,.$ (7)

In Eq. (6) we used the explicit form for 3D Peach-Koehler force as well as the
fact that the in-plane regime assumes two-dimensionality of both displacements
($\vec{u}$) and coordinates ($\vec{x}$). In addition, again smallness of
vibrations means that the right-hand side of Eq. (6) should be taken at the
equilibrium position of the point representing an edge dislocation, i.e. at
$\vec{X}_{0}$.

Employing Eq.(6) in Eq.(3) we come to the expression for the Fourier transform
of $\dot{\vec{X}}$

$\displaystyle\dot{X}(\omega)=\frac{\mu bM_{ed}\partial v_{d}/\partial
x_{e}|_{\vec{x}=\vec{X}_{0}}}{-\omega^{2}M-Bi\omega+\gamma}$ (8)

Eventually, from Eqs. (2,5,8) one can get for the Fourier transform of 2D
source

$s_{a}(\vec{x},\omega)=\sum\limits_{n=1}^{N}\mu{\cal
A}^{n}{\mathsf{M}}^{n}_{ab}\frac{\partial}{\partial
x_{b}}\delta(\vec{x}-\vec{X}_{0}^{n}){\mathsf{M}}^{n}_{ed}\frac{\partial}{\partial
x_{e}}|_{\vec{x}=\vec{X}_{0}^{n}}v_{d}$ (9)

and

${\cal A}^{n}=\frac{\mu
b^{2}}{M}\frac{1}{\omega^{2}{+}i\omega(B/M)-\omega^{2}_{0}}$ (10)

where $\omega_{0}^{2}\equiv\gamma/M$. Notice that this coefficient is the same
for all dislocations, since it depends on the magnitude, not the orientation,
of the Burgers vector. Accordingly, we shall drop its superscript “$n$”. The
tensor ${\mathsf{M}}_{ab}^{n}$, however, does depend on Burgers vector
orientation and is randomly distributed over the various dislocations. Then,
the 2D version of the tensorial perturbation potential for a single edge
dislocation can be introduced in the way similar to 3D case [16],[20]

$V^{n}_{ad}(\vec{x},\omega)=\mu{\cal
A}{\mathsf{M}}^{n}_{ab}\frac{\partial}{\partial
x_{b}}\delta(\vec{x}-\vec{X}_{0}^{n}){\mathsf{M}}^{n}_{ed}\frac{\partial}{\partial
x_{e}}|_{\vec{x}=\vec{X}_{0}^{n}}$ (11)

And the total perturbation potential is written as a superposition

$V_{ad}(\vec{x},\omega)=\sum\limits_{n=1}^{N}V^{n}_{ad}(\vec{x},\omega)$ (12)

It is worth mentioning that the 2D tensor from Eq. (11) can be obtained from
the 3D potential [20] by making use of the following formal reduction:

$\displaystyle i,j,k,\dots.$ $\displaystyle\Rightarrow$ $\displaystyle
a,b,c,\dots$ (13) $\displaystyle{\cal A}$ $\displaystyle\Rightarrow$
$\displaystyle\mu{\cal A}$

In what follows, we consider that all edge dislocations have a Burgers vector
of the same magnitude, but randomly oriented in 2D.

##  III Coherent propagation

The coherent behavior of the elastic wave in the presence of many edge
dislocations is described by the average Green function $\langle G\rangle$,
where $G$, according to [31] and Eqs. (1, 9,11,12), should be the solution of

$\rho\omega^{2}G_{ae}(\vec{x},\omega)+c_{abcd}\frac{\partial^{2}}{\partial
x_{b}\partial
x_{c}}G_{de}(\vec{x},\omega)=-V_{ad}(\vec{x},\omega)G_{de}(\vec{x},\omega)-\delta_{ae}\delta(\vec{x}).$
(14)

and the brackets denote an average over the random distribution (of both
position and Burgers vector orientation) of edge dislocations in 2D.

The average Green function is expressed as the solution of the Dyson equation
in 2D

$\langle G\rangle=\left[(G^{0})^{-1}-\Sigma\right]^{-1}$ (15)

where $G^{0}$ is the Green’s function of the medium without dislocations
(i.e., “free” or “bare” ) [31, 16, 3]. For an infinite, homogeneous and
isotropic 2D medium it obeys the equation in Fourier space

$\rho\omega^{2}G^{0}_{ae}(\vec{x},\omega)+c_{abcd}\frac{\partial^{2}}{\partial
x_{b}\partial x_{c}}G^{0}_{de}(\vec{x},\omega)=-\delta_{ae}\delta(\vec{x}).$
(16)

giving as a solution

$[(G^{0})^{-1}]_{ab}(\vec{k},\omega)=(-\rho\omega^{2}+\mu
k^{2})\delta_{ab}+(\lambda+\mu)k_{a}k_{b}$ (17)

or

${G^{0}_{ab}}(\vec{k},\omega)=\frac{1}{\rho
c_{T}^{2}(k^{2}-k_{T}^{2})}(\delta_{ab}-\hat{k}_{a}\hat{k}_{b})+\frac{1}{\rho
c_{L}^{2}(k^{2}-k_{L}^{2})}\hat{k}_{a}\hat{k}_{b}\,.$ (18)

where $\vec{k}$ is the unperturbed wave number of the 2D medium described by
$G^{0}$, $k_{T}\equiv\omega/c_{T}$, $k_{L}\equiv\omega/c_{L}$, and with an
implied small imaginary part in the denominator to insure causality. We note
that, as can be easily verified, the result in Eq. (18) agrees up to a
prefactor with that presented in [31]. The corresponding expression in
coordinate space is defined by

$G^{0}_{ab}(\vec{x}-\vec{x}^{\prime},\omega)=\frac{1}{(2\pi)^{2}}\int
d\vec{k}e^{-i\vec{k}\cdot\vec{x}}{G^{0}_{ab}}(\vec{k},\omega)e^{i\vec{k}\cdot\vec{x}^{\prime}}.$
(19)

Finally, the mass (or self-energy) operator $\Sigma$, is given by

$\Sigma=\langle T\rangle-\langle T\rangle G^{0}\Sigma$ (20)

with $T$ as the T-matrix, represented in terms of $V$ (given by Eqn. (12) by

$T={V}+{V}G^{0}T\,.$ (21)

In turn, the last equation can be easily transformed into a series in respect
of a perturbation potential

$T={V}+{V}G^{0}{V}+{V}G^{0}{V}G^{0}{V}+\cdots$ (22)

A statistical independence of the scatterers of each other is assumed in this
paper, that is an Independent Scattering Approximation (ISA) is used to treat
the coherent wave propagation problem analytically as it was done in 3D [20].
Assuming a uniform dislocation distribution through a surface $S$ it can be
easily shown that the self-energy $\Sigma$ within the ISA is evaluated as

$\Sigma=\frac{N}{S}\int\langle t^{n}\rangle d\vec{X}_{0}^{n}$ (23)

where $t^{n}$ is the T-matrix for scattering by a single object, similar to
that defined in [20], for 3D, and, in this instance, the brackets denote an
average over the Burgers vector orientation. Maurel et al. [31] computed the
mass operator to second order in perturbation theory. It is shown in B that,
due to the point-like nature of the interaction (12) the perturbation series
is geometric and can be summed to all orders to obtain the following
expression for the Green function ${<G^{+}>}({\bf k},\omega)$ for the outgoing
waves in the momentum space

${<G^{+}>_{ab}}({\bf
k},\omega)=\frac{(\delta_{ab}-\hat{k}_{a}\hat{k}_{b})}{\rho\omega^{2}\left\\{\frac{k^{2}}{K_{T}^{2}}-1\right\\}}+\frac{\hat{k}_{a}\hat{k}_{b}}{\rho\omega^{2}\left\\{\frac{k^{2}}{K_{L}^{2}}-1\right\\}}$
(24)

with

$\displaystyle K_{T,L}$ $\displaystyle=$
$\displaystyle\frac{\omega}{c_{T,L}\sqrt{\left[1-\frac{\Sigma}{\rho
c_{T,L}^{2}k^{2}}\right]}}=\frac{\omega}{c_{T,L}\sqrt{1+n\left(\frac{c_{T}}{c_{T,L}}\right)^{2}{\cal
T}}}$ (25) $\displaystyle\Sigma_{ab}$ $\displaystyle=$
$\displaystyle\Sigma\delta_{ab}=-n\mu{\cal T}k^{2}\delta_{ab}$ (26)
$\displaystyle{\cal T}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{{\mathcal{A}}}{1+\mu{\mathcal{A}}I}$ (27)

Here, we note that the mass operator from Eq.(26) suggests a mode insensitive
effect (See Eq.(25)) of an ensemble of randomly distributed vibrating edge
dislocations on the effective wave numbers $K_{T,L}$ of the coherently
propagating in-plane elastic waves irrespectively of their types. In this
regards, the problem under consideration is expected to be similar to that,
which has been recently formulated and treated in [29] for a purely scalar
case of screw dislocations in two dimensions. On the other side, the two-
dimensional character of the problem is still held and exhibited through a
tensorial structure of the mass operator, which makes the Green’s tensor from
Eq. (24) resembling the one characteristic for the three-dimensional problem
[20]. The above mentioned similarities are extensively used further when the
problem of diffusion of the elastic waves is treated.

##  IV Incoherent diffusion

We highlighted above an intermediate character (bearing scalar and tensorial
attributes) of the problem under consideration to proceed with the incoherent
(diffusive) transport of the elastic waves. Indeed, we naturally adopt some
aspects of a scalar version of the spectral approach, that was applied to a
medium with random scatterers of a screw dislocation type (See, [29] and
references therein for details), and generalize it to our 2d tensorial case
with randomly distributed vibrating edge dislocations. Meanwhile, to save the
space we use bold notations for 2d tensors or vectors from now on unless
indices are explicitly required to avoid misunderstanding.

###  IV.1 Bethe-Salpeter (BS) equation and Ward-Takahashi Identity

####  IV.1.1 BS equation

Following [29], to describe the diffusive transport regime, the two-point
correlation $<\bf{G}^{+}\otimes\bf{G}^{-}>$ needs to be determined in the low
frequency, long wavelength limit; the operation $\otimes$ is defined for
arbitrary second rank tensors $\bf{A}$ and $\bf{B}$ as:
($\bf{A}\otimes\bf{B}$)ab,cd=$A_{ac}B_{db}$. To this end, in Fourier space, a
diffusive pole structure,
$<\bf{G}^{+}\bf{G}^{-}>\,\sim(\imath\Omega+q_{a}D_{ab}q_{b})^{-1}$ should be
found, with $\bf{D}$ identified as a diffusion tensor [33]. The correlation,
for its part, obeys the Bethe-Salpeter (BS) equation, which can be constructed
in two steps: First, $<\bf{G}^{+}\otimes\bf{G}^{-}>$ is formally rewritten as

$\displaystyle<\mathbf{G^{+}}\otimes\mathbf{G^{-}}>$ $\displaystyle=$
$\displaystyle<\mathbf{G^{+}}>\otimes<\mathbf{G^{-}}>+\left(<\mathbf{G^{+}}\otimes\mathbf{G^{-}}>-<\mathbf{G^{+}}>\otimes<\mathbf{G^{-}}>\right)$
$\displaystyle=$
$\displaystyle<\mathbf{G^{+}}>\otimes<\mathbf{G^{-}}>+<\mathbf{G^{+}}>\otimes<\mathbf{G^{-}}>:<\mathbf{G^{+}}>^{-1}\otimes<\mathbf{G^{-}}>^{-1}:$
$\displaystyle\left(<\mathbf{G^{+}}\otimes\mathbf{G^{-}}>-<\mathbf{G^{+}}>\otimes<\mathbf{G^{-}}>\right):<\mathbf{G^{+}}\otimes\mathbf{G^{-}}>^{-1}:$
$\displaystyle<\mathbf{G^{+}}\otimes\mathbf{G^{-}}>$ $\displaystyle=$
$\displaystyle<\mathbf{G^{+}}>\otimes<\mathbf{G^{-}}>+<\mathbf{G^{+}}>\otimes<\mathbf{G^{-}}>:$
$\displaystyle\left(<\mathbf{G^{+}}>^{-1}\otimes<\mathbf{G^{-}}>^{-1}-<\mathbf{G^{+}}\otimes\mathbf{G^{-}}>^{-1}\right):<\mathbf{G^{+}}\otimes\mathbf{G^{-}}>$

where the tensor scalar product $:$ is defined as
$[(\bf{A}\otimes\bf{B}$):($\bf{C}\otimes\bf{D}$)]ab,ef=($\bf{A}\otimes\bf{B}$)ab,cd($\bf{C}\otimes\bf{D}$)cd,ef.
Note that $[(A\otimes B):(A^{-1}\otimes
B^{-1})]_{ab,ef}=\delta_{ae}\delta_{{fb}}$. Second, the pole structure can be
easily introduced from the last expression of the Eqn. (IV.1.1) by defining
the irreducible vertex $\mathbf{K}$ as

$\displaystyle\mathbf{K}\,{\equiv}<\mathbf{G^{+}}>^{-1}\otimes<\mathbf{G^{-}}>^{-1}-<\mathbf{G^{+}}\otimes\mathbf{G^{-}}>^{-1}$
(29)

and substituting it into Eqn. (IV.1.1), which result in the BS equation of the
well known form [33]

$<\mathbf{G^{+}}\otimes\mathbf{G^{-}}>=<\mathbf{G^{+}}>\otimes<\mathbf{G^{-}}>+\\\
<\mathbf{G^{+}}>\otimes<\mathbf{G^{-}}>:\mathbf{K}:<\mathbf{G^{+}}\otimes\mathbf{G^{-}}>\,,$
(30)

where $\mathbf{K}$ plays a role similar to the role played by the self-energy
$\Sigma$ in Dyson’s equation, Eqn. (15). If we define the two-sided Fourier
transforms as[33]

$\displaystyle\mathbf{G}^{+}(\mathbf{x}_{1},\mathbf{x}^{\prime}_{1};\omega^{+})=\int\limits_{\mathbf{k}_{1}}\int\limits_{\mathbf{k}^{\prime}_{1}}e^{\imath\mathbf{k}_{1}\mathbf{x}_{1}}\mathbf{G}^{+}(\mathbf{k}_{1},\mathbf{k}^{\prime}_{1};\omega^{+})e^{-\imath\mathbf{k}^{\prime}_{1}\mathbf{x}^{\prime}_{1}}$
(31)
$\displaystyle\mathbf{G}^{-}(\mathbf{x}_{2},\mathbf{x}^{\prime}_{2};\omega^{-})=\int\limits_{\mathbf{k}_{2}}\int\limits_{\mathbf{k}^{\prime}_{2}}e^{-\imath\mathbf{k}_{2}\mathbf{x}_{2}}\mathbf{G}^{-}(\mathbf{k}^{\prime}_{2},\mathbf{k}_{2};\omega^{-})e^{\imath\mathbf{k}^{\prime}_{2}\mathbf{x}^{\prime}_{2}}$

then

$<\mathbf{G^{+}}\otimes\mathbf{G^{-}}>={\int
d\left(\frac{\mathbf{R}+\mathbf{R}^{\prime}}{2}\right)\mathbf{G^{+}}\otimes\mathbf{G^{-}}=}\int\limits_{\mathbf{k}}\int\limits_{\mathbf{k}^{\prime}}\int\limits_{\mathbf{q}}\mathbf{\Phi}({\bf
k},{\bf k}^{\prime};{\bf
q},\Omega)e^{\imath\left(\mathbf{k}\mathbf{r}-\mathbf{k}^{\prime}\mathbf{r}^{\prime}+\mathbf{q}\left(\mathbf{R}-\mathbf{R}^{\prime}\right)\right)}$
(32)

where

$\displaystyle\mathbf{\Phi}({\bf k},{\bf k}^{\prime};{\bf
q},\Omega)\equiv<\mathbf{G}^{+}(\mathbf{k}^{+},\mathbf{k}^{\prime+},\omega^{+})\otimes\mathbf{G}^{-}(\mathbf{k}^{\prime-},\mathbf{k}^{-},\omega^{-})>$
(33)

with

$\mathbf{k}^{\pm}=\mathbf{k}\pm\frac{\mathbf{q}}{2},\quad\omega^{\pm}=\omega\pm\frac{\Omega}{2}\,.$
(34)

And space variables have been introduced as

$\displaystyle\mathbf{x}_{1}=\mathbf{R}+\frac{\mathbf{r}}{2},$
$\displaystyle\mathbf{x}_{2}=\mathbf{R}-\frac{\mathbf{r}}{2}$ (35)
$\displaystyle\mathbf{x}^{\prime}_{1}=\mathbf{R}^{\prime}+\frac{\mathbf{r}^{\prime}}{2},$
$\displaystyle\mathbf{x}^{\prime}_{2}=\mathbf{R}^{\prime}-\frac{\mathbf{r}^{\prime}}{2}$
$\displaystyle\mathbf{k}_{1}=\mathbf{k}^{+}=\mathbf{k}+\frac{\mathbf{q}}{2},$
$\displaystyle\mathbf{k}^{\prime}_{1}=\mathbf{k}^{\prime+}=\mathbf{k}^{\prime}+\frac{\mathbf{q}}{2}$
$\displaystyle\mathbf{k}^{\prime}_{2}=\mathbf{k}^{\prime-}=\mathbf{k}^{\prime}-\frac{\mathbf{q}}{2},$
$\displaystyle\mathbf{k}_{2}=\mathbf{k}^{-}=\mathbf{k}-\frac{\mathbf{q}}{2}\,.$

By applying the inverse Fourier transform[34]

$\displaystyle\int
d\left(\mathbf{R}-\mathbf{R}^{\prime}\right)d\mathbf{r}d\mathbf{r}^{\prime}e^{-\imath\left(\mathbf{k}\mathbf{r}-\mathbf{k}^{\prime}\mathbf{r}^{\prime}+\mathbf{q}\left(\mathbf{R}-\mathbf{R}^{\prime}\right)\right)}$
(36)

to Eqn. (30) the BS equation in momentum space is obtained as

$\displaystyle\mathbf{\Phi}({\bf k},{\bf k}^{\prime};{\bf q},\Omega)$
$\displaystyle=$ $\displaystyle<\mathbf{G}^{+}>\otimes<\mathbf{G}^{-}>({\bf
k};{\bf q},\Omega)\delta_{{\bf k},{\bf k}^{\prime}}$
$\displaystyle+\int\limits_{\mathbf{k}^{\prime\prime}}<\mathbf{G}^{+}>\otimes<\mathbf{G}^{-}>({\bf
k};{\bf q},\Omega):\mathbf{K}({\bf k},{\bf k}^{\prime\prime};{\bf
q},\Omega):\mathbf{\Phi}({\bf k}^{\prime\prime},{\bf k}^{\prime};{\bf
q},\Omega)$

with $\delta_{{\bf k},{\bf k}^{\prime}}=(2\pi)^{2}\delta({\bf k}-{\bf
k}^{\prime})$ and, as usual, the integration over the internal momentum
variables, i.e. ${\bf k}^{\prime\prime}$, is assumed, with
$\int\limits_{\mathbf{k}}\equiv\frac{1}{(2\pi)^{2}}\int d\mathbf{k}$.
Proceeding from Eqn. (IV.1.1) to a kinetic form requires using the following
auxiliary identity for the averaged Green’s functions:

$(<\mathbf{G^{+}}>^{-1}\otimes\mathbf{I}-\mathbf{I}\otimes<\mathbf{G^{-}}>^{-1}):(<\mathbf{G}^{+}>\otimes<\mathbf{G}^{-}>)=(\mathbf{I}\otimes<\mathbf{G^{-}}>-<\mathbf{G^{+}}>\otimes\mathbf{I})$
(38)

with $\mathbf{I}$ as a unit tensor. Now, acting with
$(<\mathbf{G^{+}}>^{-1}\otimes\mathbf{I}-\mathbf{I}\otimes<\mathbf{G^{-}}>^{-1})$
on the left of Eqn. (IV.1.1) and using Eqn. (38) it is straightforward to
obtain

$\displaystyle(<\mathbf{G^{+}}>^{-1}\otimes\mathbf{I}-\mathbf{I}\otimes<\mathbf{G^{-}}>^{-1}):\mathbf{\Phi}$
$\displaystyle=$
$\displaystyle(\mathbf{I}\otimes<\mathbf{G^{-}}>-<\mathbf{G^{+}}>\otimes\mathbf{I}):\left(\mathbf{I}\otimes\mathbf{I}\delta_{{\bf
k},{\bf k}^{\prime}}+\mathbf{K}({\bf k},{\bf k}^{\prime\prime};{\bf
q},\Omega):\mathbf{\Phi}({\bf k}^{\prime\prime},{\bf k}^{\prime};{\bf
q},\Omega)\right)$

and, using Dyson’s equation Eqn. (15) and the free space Green tensor Eqn.
(18) this can be eventually reduced to the kinetic form of the BS equation

$\left[\imath\omega\Omega\mathbf{E}+\mathbf{P}({\bf k};{\bf
q})\right]:\mathbf{\Phi}({\bf k},{\bf k}^{\prime};{\bf
q},\Omega)+\int\limits_{\bf{k}^{\prime\prime}}\mathbf{U}({\bf k},{\bf
k}^{\prime\prime};{\bf q},\Omega):\mathbf{\Phi}({\bf k}^{\prime\prime},{\bf
k}^{\prime};{\bf q},\Omega)=\delta_{{\bf k},{\bf k}^{\prime}}\mathbf{\Delta
G}({\bf k};{\bf q},\Omega)$ (40)

with

$\displaystyle\mathbf{U}({\bf k},{\bf k}^{\prime};{\bf
q},\Omega)\equiv\Delta\mathbf{\Sigma}({\bf k};{\bf q},\Omega)\delta_{{\bf
k},{\bf k}^{\prime}}-\Delta\mathbf{G}({\bf k};{\bf q},\Omega):\mathbf{K}({\bf
k},{\bf k}^{\prime};{\bf q},\Omega)$ (41)

and notations

$\displaystyle\mathbf{P}({\bf k};{\bf q})$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2\imath\rho}\left(\mathbf{I}\otimes\mathbf{L}(\mathbf{k}^{-})-\mathbf{L}(\mathbf{k}^{+})\otimes\mathbf{I}\right)$
(42) $\displaystyle\mathbf{\Delta<G>}({\bf k};{\bf q},\Omega)$
$\displaystyle\equiv$
$\displaystyle\frac{1}{2\imath\rho}\left(\mathbf{I}\otimes<\mathbf{G}>^{-}(\mathbf{k}^{-},\omega^{-})-<\mathbf{G}>^{+}(\mathbf{k}^{+},\omega^{+})\otimes\mathbf{I}\right)$
$\displaystyle\mathbf{\Delta\Sigma}({\bf k};{\bf q},\Omega)$
$\displaystyle\equiv$
$\displaystyle\frac{1}{2\imath\rho}\left(\mathbf{I}\otimes\mathbf{\Sigma}^{-}(\mathbf{k}^{-},\omega^{-})-\mathbf{\Sigma}^{+}(\mathbf{k}^{+},\omega^{+})\otimes\mathbf{I}\right)$
$\displaystyle\mathbf{E}$ $\displaystyle\equiv$
$\displaystyle\mathbf{I}\otimes\mathbf{I}\quad$
$\displaystyle\mathbf{L}(\mathbf{k}^{\pm})$ $\displaystyle\equiv$
$\displaystyle L_{ad}(\mathbf{k}^{\pm})=-c_{abcd}k^{\pm}_{b}k^{\pm}_{c}$

####  IV.1.2 pre-WTI

As is well-known (see, for example [35, 36]), a Ward-Takahashi identity (WTI),
that mathematically relates $\mathbf{\Sigma}$ to $\mathbf{K}$ and physically
reflects conservative laws, which are satisfied in a particular physical
system, and therefore, restricts classes of relevant scattering potentials,
should be established in order to properly treat Eqn. (40). To this end, a
preliminary identity (“pre-WTI”), Eqn. (55) below, will be obtained.

We start with two equations for Green tensors at two different set of
variables: $(\vec{x}_{1},\vec{x}^{\prime}_{1},\omega_{1})$ and
$(\vec{x}_{2},\vec{x}^{\prime}_{2},\omega_{2})$. It reads as follows

$\displaystyle\rho\omega_{1}^{2}G_{a_{1}e_{1}}(\vec{x}_{1},\vec{x}^{\prime}_{1},\omega_{1})+c_{a_{1}b_{1}c_{1}d_{1}}\frac{\partial^{2}}{\partial
x_{b1}\partial
x_{c1}}G_{d_{1}e_{1}}(\vec{x}_{1},\vec{x}^{\prime}_{1},\omega_{1})$
$\displaystyle=$ $\displaystyle-
V_{a_{1}d_{1}}(\vec{x}_{1},\omega_{1})G_{d_{1}e_{1}}(\vec{x}_{1},\vec{x}^{\prime}_{1},\omega_{1})-\delta_{a_{1}e_{1}}\delta(\vec{x}_{1}-\vec{x}^{\prime}_{1})$
$\displaystyle\rho\omega_{2}^{2}G_{a_{2}e_{2}}(\vec{x}_{2},\vec{x}^{\prime}_{2},\omega_{2})+c_{a_{2}b_{2}c2d2}\frac{\partial^{2}}{\partial
x_{b2}\partial
x_{c2}}G_{d_{2}e_{2}}(\vec{x}_{2},\vec{x}^{\prime}_{2},\omega_{2})$
$\displaystyle=$ $\displaystyle-
V_{a_{2}d_{2}}(\vec{x}_{2},\omega_{2})G_{d_{2}e_{2}}(\vec{x}_{2},\vec{x}^{\prime}_{2},\omega_{2})-\delta_{a_{2}e_{2}}\delta(\vec{x}_{2}-\vec{x}^{\prime}_{2}).$

Then, we take the inverse Fourier transform from both expressions according to
the rule

$\displaystyle\mathbf{F}(\vec{k},\vec{k}^{\prime};\omega)=\int\int
d\vec{x}d\vec{x}^{\prime}e^{-\imath\vec{k}\vec{x}}\mathbf{F}(\vec{x},\vec{x}^{\prime};\omega)e^{\imath\vec{k}^{\prime}\vec{x}^{\prime}}$
(45)

and substitute the advanced Green tensor Fourier transform at the terms, where
operators act on it, as follows

$\displaystyle\mathbf{G}(\vec{x},\vec{x}^{\prime};\omega)=\int\limits_{\vec{k}}\int\limits_{\vec{k}^{\prime}}e^{\imath\vec{k}\vec{x}}\mathbf{G}(\vec{k},\vec{k}^{\prime};\omega)e^{-\imath\vec{k}^{\prime}\vec{x}^{\prime}}$
(46)

It yields

$\displaystyle\rho\omega_{1}^{2}G_{a_{1}e_{1}}(\vec{k}_{1},\vec{k}^{\prime}_{1},\omega_{1})-c_{a_{1}b_{1}c_{1}d_{1}}k_{b_{1}}k_{c_{1}}G_{d_{1}e_{1}}(\vec{k}_{1},\vec{k}^{\prime}_{1},\omega_{1})$
$\displaystyle=$ $\displaystyle-\int\limits_{\vec{p}_{1}}\int
d\vec{x}_{1}e^{-\imath\vec{k}_{1}\vec{x}_{1}}V_{a1d1}(\vec{x}_{1},\omega_{1})e^{\imath\vec{p}_{1}\vec{x}_{1}}\int\limits_{\vec{p}^{\prime}_{1}}G_{d_{1}e_{1}}(\vec{p}_{1},\vec{p}^{\prime}_{1};\omega_{1})\int
d\vec{x}^{\prime}_{1}e^{-\imath(\vec{p}^{\prime}_{1}-\vec{k}^{\prime}_{1})\vec{x}^{\prime}_{1}}-\delta_{a1e1}\delta_{\vec{k}_{1},\vec{k}^{\prime}_{1}}$
$\displaystyle=$
$\displaystyle-\sum\limits_{n=1}^{N}\int\limits_{\vec{p}_{1}}\int
d\vec{x}_{1}e^{-\imath\vec{k}_{1}\vec{x}_{1}}V^{n}_{a1d1}(\vec{x}_{1},\omega_{1})e^{\imath\vec{p}_{1}\vec{x}_{1}}G_{d_{1}e_{1}}(\vec{p}_{1},\vec{k}^{\prime}_{1};\omega_{1})-\delta_{a_{1}e_{1}}\delta_{\vec{k}_{1},\vec{k}^{\prime}_{1}}$
$\displaystyle=$ $\displaystyle
g(\omega_{1})\sum\limits_{n=1}^{N}\int\limits_{\vec{p}_{1}}\frac{\mu^{2}b^{2}}{M}{\mathsf{M}}_{a_{1}h_{1}}^{n}k_{h_{1}}p_{g_{1}}{\mathsf{M}}_{g_{1}d_{1}}^{n}e^{i\left(\vec{p}_{1}-\vec{k}_{1}\right)\cdot\vec{X}^{n}_{0}}G_{d_{1}e_{1}}(\vec{p}_{1},\vec{k}^{\prime}_{1};\omega_{1})-\delta_{a_{1}e_{1}}\delta_{\vec{k}_{1},\vec{k}^{\prime}_{1}}$

with
$g(\omega_{1})=\frac{1}{\omega_{1}^{2}{+}i\omega_{1}(B/M)-\omega^{2}_{0}}$. To
underline the fact that frequency dependent part $g(\omega_{1})$ of
${\mathcal{A}}^{n}(\omega_{1})$ does not depend on $n$, we take it out of the
summation over $n$. Acting similarly for the second set of variables one can
come to the following system of equations

$\displaystyle-(G^{0})^{-1}_{a_{1}d_{1}}(\vec{k}_{1},\omega_{1})G_{d_{1}e_{1}}(\vec{k}_{1},\vec{k}^{\prime}_{1},\omega_{1})+\delta_{a_{1}e_{1}}\delta_{\vec{k}_{1},\vec{k}^{\prime}_{1}}$
$\displaystyle=$ $\displaystyle
g(\omega_{1})\sum\limits_{n=1}^{N}\int\limits_{\vec{p}_{1}}\frac{\mu^{2}b^{2}}{M}M_{a_{1}h_{1}}^{n}k_{h_{1}}p_{g_{1}}M_{g_{1}d_{1}}^{n}e^{i\left(\vec{p}_{1}-\vec{k}_{1}\right)\cdot\vec{X}^{n}_{0}}G_{d_{1}e_{1}}(\vec{p}_{1},\vec{k}^{\prime}_{1};\omega_{1})$
$\displaystyle-(G^{0})^{-1}_{a_{2}d_{2}}(\vec{k}_{2},\omega_{2})G_{d_{2}e_{2}}(\vec{k}_{2},\vec{k}^{\prime}_{2},\omega_{2})+\delta_{a_{2}e_{2}}\delta_{\vec{k}_{2},\vec{k}^{\prime}_{2}}$
$\displaystyle=$ $\displaystyle
g(\omega_{2})\sum\limits_{n=1}^{N}\int\limits_{\vec{p}_{2}}\frac{\mu^{2}b^{2}}{M}M_{a_{2}h_{2}}^{n}k_{h_{2}}p_{g_{2}}M_{g_{2}d_{2}}^{n}e^{i\left(\vec{p}_{2}-\vec{k}_{2}\right)\cdot\vec{X}^{n}_{0}}G_{d_{2}e_{2}}(\vec{p}_{2},\vec{k}^{\prime}_{2};\omega_{2})$

with

$\displaystyle(G^{0})^{-1}_{ad}(\vec{k},\omega)=-\left(\rho\omega^{2}\delta_{ad}-c_{abcd}k_{b}k_{c}\right).$
(49)

So far, we have played only with advanced Green tensors, whereas in order to
construct intensity we need to involve the equation for the retarded Green
tensor. To achieve this we make a complex conjugate of the second equation of
the system, which eventually goes to the form:

$\displaystyle-(G^{0})^{-1}_{a_{1}d_{1}}(\vec{k}_{1},\omega_{1})G_{d_{1}e_{1}}(\vec{k}_{1},\vec{k}^{\prime}_{1},\omega_{1})+\delta_{a_{1}e_{1}}\delta_{\vec{k}_{1},\vec{k}^{\prime}_{1}}$
$\displaystyle=$ $\displaystyle
g(\omega_{1})\sum\limits_{n=1}^{N}\int\limits_{\vec{p}_{1}}\frac{\mu^{2}b^{2}}{M}M_{a_{1}h_{1}}^{n}k_{h_{1}}p_{g_{1}}M_{g_{1}d_{1}}^{n}e^{i\left(\vec{p}_{1}-\vec{k}_{1}\right)\cdot\vec{X}^{n}_{0}}G_{d_{1}e_{1}}(\vec{p}_{1},\vec{k}^{\prime}_{1};\omega_{1})$
$\displaystyle-(G^{0*})^{-1}_{a_{2}d_{2}}(\vec{k}_{2},\omega_{2})G^{*}_{d_{2}e_{2}}(\vec{k}_{2},\vec{k}^{\prime}_{2},\omega_{2})+\delta_{a_{2}e_{2}}\delta_{\vec{k}_{2},\vec{k}^{\prime}_{2}}$
$\displaystyle=$ $\displaystyle
g^{*}(\omega_{2})\sum\limits_{n=1}^{N}\int\limits_{\vec{p}_{2}}\frac{\mu^{2}b^{2}}{M}M_{a_{2}h_{2}}^{n}k_{h_{2}}p_{g_{2}}M_{g_{2}d_{2}}^{n}e^{-i\left(\vec{p}_{2}-\vec{k}_{2}\right)\cdot\vec{X}^{n}_{0}}G^{*}_{d_{2}e_{2}}(\vec{p}_{2},\vec{k}^{\prime}_{2};\omega_{2})$

Now we act on the first and second equations of the system from the right by
$g^{*}(\omega_{2})(G)^{-1}_{e1f1}(\vec{k}^{\prime}_{1},\vec{k}^{\prime\prime}_{1};\omega_{1})$
and
$g(\omega_{1})(G^{*})^{-1}_{e2f2}(k^{\prime}_{2},\vec{k}^{\prime\prime}_{2};\omega_{2})$,
respectively. It leads to the following system

$\displaystyle-(G^{0})^{-1}_{a_{1}f_{1}}(\vec{k}_{1},\omega_{1})\delta_{\vec{k}_{1},\vec{k}^{\prime\prime}_{1}}g^{*}(\omega_{2})+g^{*}(\omega_{2})(G)^{-1}_{a_{1}f_{1}}(\vec{k}_{1},\vec{k}^{\prime\prime}_{1};\omega_{1})$
$\displaystyle=$ $\displaystyle
g(\omega_{1})g^{*}(\omega_{2})\sum\limits_{n=1}^{N}\frac{\mu^{2}b^{2}}{M}M_{a_{1}h_{1}}k_{h_{1}}k^{\prime\prime}_{g_{1}}M_{g_{1}f_{1}}e^{i\left(\vec{k}^{\prime\prime}_{1}-\vec{k}_{1}\right)\cdot\vec{X}^{n}_{0}}$
$\displaystyle-(G^{0*})^{-1}_{a_{2}f_{2}}(\vec{k}_{2},\omega_{2})\delta_{\vec{k}_{2},\vec{k}^{\prime\prime}_{2}}g(\omega_{1})+g(\omega_{1})(G^{*})^{-1}_{a_{2}f_{2}}(\vec{k}_{2},\vec{k}^{\prime\prime}_{2};\omega_{2})$
$\displaystyle=$ $\displaystyle
g(\omega_{1})g^{*}(\omega_{2})\sum\limits_{n=1}^{N}\frac{\mu^{2}b^{2}}{M}M_{a_{2}h_{2}}k_{h_{2}}k^{\prime\prime}_{g_{2}}M_{g_{2}f_{2}}e^{-i\left(\vec{k}^{\prime\prime}_{2}-\vec{k}_{2}\right)\cdot\vec{X}^{n}_{0}}$

The next step consists in subtraction at $B=0$ (so that
$g(\omega)=g^{*}(\omega)$) of the second equation from the first one and
evaluating at $a_{1}=f_{2}=a$, $f_{1}=a_{2}=f$;
$\vec{k}_{2}\rightarrow\vec{k}^{\prime\prime}_{1}$,
$\vec{k}^{\prime\prime}_{2}\rightarrow\vec{k}_{1}$. Otherwise, it is not
possible to eliminate the remaining parts of the potentials in both equations
that are subjected to the subtraction from each other, since the parts imply
not only summation over defects but also contain components of the second rank
tensor, i. e. to achieve identity of those parts between each other, the
components must be also identical. As a result, we have

$\displaystyle\lim\limits_{{\begin{subarray}{c}\vec{k}_{2}\rightarrow\vec{k}^{\prime\prime}_{1}\\\
\vec{k}^{\prime\prime}_{2}\rightarrow\vec{k}_{1}\end{subarray}}}\left(-(G^{0})^{-1}_{af}(\vec{k}_{1},\omega_{1})\delta_{\vec{k}_{1},\vec{k}^{\prime\prime}_{1}}g^{*}(\omega_{2})+g^{*}(\omega_{2})(G)^{-1}_{af}(\vec{k}_{1},\vec{k}^{\prime\prime}_{1};\omega_{1})+\right.$
$\displaystyle\left.(G^{*0})^{-1}_{fa}(\vec{k}_{2},\omega_{2})\delta_{\vec{k}_{2},\vec{k}^{\prime\prime}_{2}}g(\omega_{1})-g(\omega_{1})(G^{*})^{-1}_{fa}(\vec{k}_{2},\vec{k}^{\prime\prime}_{2};\omega_{2})\right)$
$\displaystyle\equiv$ $\displaystyle 0$

By acting on the obtained identity with
$\lim\limits_{{\begin{subarray}{c}\vec{k}_{2}\rightarrow\vec{k}^{\prime\prime}_{1}\\\
\vec{k}^{\prime\prime}_{2}\rightarrow\vec{k}_{1}\end{subarray}}}G(\vec{k}^{\prime\prime}_{1},\vec{k}^{\prime\prime\prime}_{1};\omega_{1})_{fg}G^{*}(\vec{k}^{\prime\prime}_{2},\vec{k}^{\prime\prime\prime}_{2};\omega_{2})_{ah}$
from the right, we get

$\displaystyle\lim\limits_{{\begin{subarray}{c}\vec{k}_{2}\rightarrow\vec{k}^{\prime\prime}_{1}\\\
\vec{k}^{\prime\prime}_{2}\rightarrow\vec{k}_{1}\end{subarray}}}\left(-G^{*}(\vec{k}^{\prime\prime\prime}_{2},\vec{k}^{\prime\prime}_{2};\omega_{2})_{ha}(G^{0})^{-1}_{af}(\vec{k}_{1},\omega_{1})G(\vec{k}_{1},\vec{k}^{\prime\prime\prime}_{1};\omega_{1})_{fg}g^{*}(\omega_{2})+g^{*}(\omega_{2})\delta_{\vec{k}_{1},\vec{k}^{\prime\prime\prime}_{1}}G^{*}(\vec{k}^{\prime\prime}_{2},\vec{k}^{\prime\prime\prime}_{2};\omega_{2})_{gh}\right.$
$\displaystyle\left.+G(\vec{k}^{\prime\prime\prime}_{1},\vec{k}^{\prime\prime}_{1};\omega_{1})_{gf}(G^{*0})^{-1}_{fa}(\vec{k}_{2},\omega_{2})G^{*}(\vec{k}_{2},\vec{k}^{\prime\prime\prime}_{2};\omega_{2})_{ah}g(\omega_{1})-g(\omega_{1})\delta_{\vec{k}_{2},\vec{k}^{\prime\prime\prime}_{2}}G(\vec{k}^{\prime\prime}_{1},\vec{k}^{\prime\prime\prime}_{1};\omega_{1})_{hg}\right)$
$\displaystyle\equiv$ $\displaystyle 0$

After making averaging and introducing notations :

$\displaystyle\vec{k}_{1}=\vec{k}^{+}\quad\vec{k}^{\prime\prime}_{2}=\vec{k}^{-}\quad\vec{k}^{\prime\prime}_{1}=\vec{k}^{\prime\prime+}\quad\vec{k}_{2}=\vec{k}^{\prime\prime-}\quad\vec{k}^{\prime\prime\prime}_{1}=\vec{k}^{\prime\prime\prime+}\quad\vec{k}^{\prime\prime\prime}_{2}=\vec{k}^{\prime\prime\prime-}$
(52) $\displaystyle\omega_{1}=\omega_{+}\quad\omega_{2}=\omega_{-}\quad
G=G^{+}\quad G^{*}=G^{-}\quad G^{0}=G^{0+}\quad G^{0*}=G^{0-}$

we come to the following result

$\displaystyle-{\int\limits_{\mathbf{k}}}<G^{-}(\vec{k}^{\prime\prime\prime-},\vec{k}^{-};\omega_{-})_{ha}(G^{0+})^{-1}_{af}(\vec{k}^{+},\omega_{+})G^{+}(\vec{k}^{+},\vec{k}^{\prime\prime\prime+};\omega_{+})_{fg}>g^{\ast}(\omega_{-})$
(53)
$\displaystyle+g^{\ast}(\omega_{-})<\delta_{\vec{k},\vec{k}^{\prime\prime\prime}}G^{-}(\vec{k}^{-},\vec{k}^{\prime\prime\prime-};\omega_{-})_{gh}>$
$\displaystyle+{\int\limits_{\mathbf{k}^{\prime\prime}}}<G^{-}(\vec{k}^{\prime\prime\prime-},\vec{k}^{\prime\prime-};\omega_{-})_{ha}(G^{0-})^{-1}_{af}(\vec{k}^{\prime\prime-},\omega_{-})G^{+}(\vec{k}^{\prime\prime+},\vec{k}^{\prime\prime\prime+};\omega_{+})_{fg}g(\omega_{+})>$
$\displaystyle-g(\omega_{+})<\delta_{\vec{k}^{\prime\prime},\vec{k}^{\prime\prime\prime}}G^{+}(\vec{k}^{\prime\prime+},\vec{k}^{\prime\prime\prime+};\omega_{+})_{hg}>$
$\displaystyle\equiv$ $\displaystyle 0$

If we use Eqn.(33) in Eqn.(53) and recall that

$\displaystyle(G^{0\pm})^{-1}_{af}(\vec{k}^{\pm},\omega_{\pm})$
$\displaystyle=$ $\displaystyle(G^{0\pm})^{-1}_{af}({\bf k};{\bf q},\Omega)$
(54)
$\displaystyle<G^{\pm}(\vec{k}^{\prime\prime\prime\pm},\vec{k}^{\prime\prime\prime\pm};\omega_{\pm})_{hg}>$
$\displaystyle=$ $\displaystyle<G>^{\pm}(\vec{k}^{\prime\prime\prime};{\bf
q},\Omega)_{hg}$

then, the pre-WTI reads as follows

$\displaystyle{\int\limits_{\mathbf{k}}}\left((G^{0-})^{-1}_{fa}({\bf k};{\bf
q},\Omega)g(\omega_{+})-(G^{0+})^{-1}_{fa}({\bf k};{\bf
q},\Omega)g^{\ast}(\omega_{-})\right)\Phi_{fa,gh}({\bf k},{\bf
k}^{\prime\prime\prime};{\bf q},\Omega)$ (55)
$\displaystyle+g^{\ast}(\omega_{-})<G>^{-}(\vec{k}^{\prime\prime\prime};{\bf
q},\Omega)_{gh}-g(\omega_{+})<G>^{+}(\vec{k}^{\prime\prime\prime};{\bf
q},\Omega)_{gh}$ $\displaystyle\equiv$ $\displaystyle 0$

This is a relation between the averaged Green’s function, and its two-point
correlations.

####  IV.1.3 WTI

We now turn the relation between averages obtained at the end of the last
subsection into a relation between their ”irreducible” parts. In this
subsection, based on Eqn. (55), we establish an explicit relationship between
the irreducible vertex $\mathbf{K}$ and the mass operator $\mathbf{\Sigma}$
which is the essence of the WTI. Indeed, by acting with
$\Phi^{-1}_{gh,bc}({\bf k}^{\prime\prime\prime},{\bf
k}^{\prime\prime\prime\prime};{\bf q},\Omega)$ on Eqn. (55) from the right ,
we obtain

$\displaystyle\left((G^{0-})^{-1}_{bc}({\bf k}^{\prime\prime\prime\prime};{\bf
q},\Omega)g(\omega_{+})-(G^{0+})^{-1}_{bc}({\bf
k}^{\prime\prime\prime\prime};{\bf q},\Omega)g^{\ast}(\omega_{-})\right)$ (56)
$\displaystyle+g^{\ast}(\omega_{-}){\int\limits_{\mathbf{k}^{\prime\prime\prime}}}<G>^{-}(\vec{k}^{\prime\prime\prime};{\bf
q},\Omega)_{gh}\Phi^{-1}_{gh,bc}({\bf k}^{\prime\prime\prime},{\bf
k}^{\prime\prime\prime\prime};{\bf q},\Omega)$
$\displaystyle-g(\omega_{+}){\int\limits_{\mathbf{k}^{\prime\prime\prime}}}<G>^{+}(\vec{k}^{\prime\prime\prime};{\bf
q},\Omega)_{gh}\Phi^{-1}_{gh,bc}({\bf k}^{\prime\prime\prime},{\bf
k}^{\prime\prime\prime\prime};{\bf q},\Omega)$ $\displaystyle\equiv$
$\displaystyle 0$

According to Eqn. (29), we have

$\Phi^{-1}_{gh,bc}({\bf k}^{\prime\prime\prime},{\bf
k}^{\prime\prime\prime\prime};{\bf
q},\Omega)=(<G>^{+})^{-1}(\vec{k}^{\prime\prime\prime};{\bf
q},\Omega)_{gb}(<G>^{-})^{-1}(\vec{k}^{\prime\prime\prime};{\bf
q},\Omega)_{ch}\delta_{\vec{k}^{\prime\prime\prime},k^{\prime\prime\prime\prime}}-K_{gh,bc}({\bf
k}^{\prime\prime\prime},{\bf k}^{\prime\prime\prime\prime};{\bf q},\Omega)$
(57)

It gives

$\displaystyle\left((G^{0-})^{-1}_{bc}({\bf k}^{\prime\prime\prime\prime};{\bf
q},\Omega)g(\omega_{+})-(G^{0+})^{-1}_{bc}({\bf
k}^{\prime\prime\prime\prime};{\bf q},\Omega)g^{\ast}(\omega_{-})\right)$ (58)
$\displaystyle+g^{\ast}(\omega_{-})\left((<G>^{+})^{-1}(\vec{k}^{\prime\prime\prime\prime};{\bf
q},\Omega)_{bc}-{\int\limits_{\mathbf{k}^{\prime\prime\prime}}}<G>^{-}(\vec{k}^{\prime\prime\prime};{\bf
q},\Omega)_{gh}K_{gh,bc}({\bf k}^{\prime\prime\prime},{\bf
k}^{\prime\prime\prime\prime};{\bf q},\Omega)\right)$
$\displaystyle-g(\omega_{+})\left((<G>^{-})^{-1}(\vec{k}^{\prime\prime\prime\prime};{\bf
q},\Omega)_{bc}-{\int\limits_{\mathbf{k}^{\prime\prime\prime}}}<G>^{+}(\vec{k}^{\prime\prime\prime};{\bf
q},\Omega)_{gh}K_{gh,bc}({\bf k}^{\prime\prime\prime},{\bf
k}^{\prime\prime\prime\prime};{\bf q},\Omega)\right)$ $\displaystyle\equiv$
$\displaystyle 0$

Eventually, using (15) the WTI (through which, the incoherent properties,
captured by $K$, are related to the coherent properties, described by $\Sigma$
and $<G>$) reads

$\displaystyle\left(\Sigma^{-}_{bc}({\bf k}^{\prime\prime\prime\prime};{\bf
q},\Omega)g(\omega_{+})-\Sigma^{+}_{bc}({\bf
k}^{\prime\prime\prime\prime};{\bf
q},\Omega)g^{\ast}(\omega_{-})\right)\equiv$ (59) $\displaystyle\hskip
40.00006pt{\int\limits_{\mathbf{k}^{\prime\prime\prime}}}\left(g^{\ast}(\omega_{-})<G>^{-}(\vec{k}^{\prime\prime\prime};{\bf
q},\Omega)_{gh}-g(\omega_{+})<G>^{+}(\vec{k}^{\prime\prime\prime};{\bf
q},\Omega)_{gh}\right)K_{gh,bc}({\bf k}^{\prime\prime\prime},{\bf
k}^{\prime\prime\prime\prime};{\bf q},\Omega)$

It is worth mentioning that the WTI immediately reduces to the optical theorem
when $B=0$, i.e. $g$ is real, ${\bf q},\Omega$ tend to zero, and the standard
ISA expressions for ${\bf\Sigma}$ and ${\bf K}$ tensors, Eqn. (IV.1.3) below,
are taken (See C).Explicitly, the WTI reads in this case

$\displaystyle\left(\Sigma^{\ast}_{bc}({\bf
k}^{\prime\prime\prime\prime})-\Sigma_{bc}({\bf
k}^{\prime\prime\prime\prime})\right)\equiv{\int\limits_{\mathbf{k}^{\prime\prime\prime}}}\left(G^{0\ast}(\vec{k}^{\prime\prime\prime})_{gh}-G^{0}(\vec{k}^{\prime\prime\prime})_{gh}\right)K_{gh,bc}({\bf
k}^{\prime\prime\prime},{\bf k}^{\prime\prime\prime\prime})$ (60)

with the following expressions, valid to leading order in $n$, the density of
scatterers:

$\displaystyle\Sigma_{bc}({\bf k}^{\prime\prime\prime\prime})=\Sigma_{bc}({\bf
k}^{\prime\prime\prime\prime};{\bf 0},0)$ $\displaystyle=$ $\displaystyle
n<t>_{bc}({\bf k}^{\prime\prime\prime\prime})$ $\displaystyle K_{gh,bc}({\bf
k}^{\prime\prime\prime},{\bf k}^{\prime\prime\prime\prime};{\bf
0},0)=K_{gh,bc}({\bf k}^{\prime\prime\prime},{\bf
k}^{\prime\prime\prime\prime})$ $\displaystyle=$ $\displaystyle n<t_{gb}({\bf
k}^{\prime\prime\prime},{\bf k}^{\prime\prime\prime\prime})t^{*}_{ch}({\bf
k}^{\prime\prime\prime\prime},{\bf k}^{\prime\prime\prime})>$
$\displaystyle<G>(\vec{k}^{\prime\prime\prime};{\bf
0},0)_{hg}=<G>(\vec{k}^{\prime\prime\prime})_{hg}$ $\displaystyle\rightarrow$
$\displaystyle G^{0}(\vec{k}^{\prime\prime\prime})_{hg}$ (61)

In terms of the general, i. e. symbolical, representation of the WTI there are
two differences compared to a well-known tensorial version of the WTI for
electromagnetic waves [36]: first, $g$ is a complex valued resonance like
function; second, a tensor rank of the WTI is two rather than four as in the
case of electromagnetic waves [36]. In our case, this is all we need to solve
the problem at hand.

We now change to a compact notation for the WTI by introducing

$\displaystyle
g(\omega_{+})=\frac{g(\omega_{+})+g^{\ast}(\omega_{-})}{2}+\frac{g(\omega_{+})-g^{\ast}(\omega_{-})}{2}$
(62) $\displaystyle
g^{\ast}(\omega_{-})=\frac{g(\omega_{+})+g^{\ast}(\omega_{-})}{2}-\frac{g(\omega_{+})-g^{\ast}(\omega_{-})}{2}$

and come to the WTI in the form

$\displaystyle\Sigma^{-}_{bc}({\bf k}^{\prime\prime\prime\prime};{\bf
q},\Omega)-\Sigma^{+}_{bc}({\bf k}^{\prime\prime\prime\prime};{\bf
q},\Omega)-{\int\limits_{\mathbf{k}^{\prime\prime\prime}}}\left(<G>^{-}(\vec{k}^{\prime\prime\prime};{\bf
q},\Omega)_{gh}-<G>^{+}(\vec{k}^{\prime\prime\prime};{\bf
q},\Omega)_{gh}\right)$ (63) $\displaystyle\times K_{gh,bc}({\bf
k}^{\prime\prime\prime},{\bf k}^{\prime\prime\prime\prime};{\bf
q},\Omega)\equiv\left(\frac{g^{\ast}(\omega_{-})-g(\omega_{+})}{g(\omega_{+})+g^{\ast}(\omega_{-})}\right)\left(\Sigma^{-}_{bc}({\bf
k}^{\prime\prime\prime\prime};{\bf q},\Omega)+\Sigma^{+}_{bc}({\bf
k}^{\prime\prime\prime\prime};{\bf q},\Omega)\right.$
$\displaystyle+\left.{\int\limits_{\mathbf{k}^{\prime\prime\prime}}}\left(<G>^{-}(\vec{k}^{\prime\prime\prime};{\bf
q},\Omega)_{gh}+<G>^{+}(\vec{k}^{\prime\prime\prime};{\bf
q},\Omega)_{gh}\right)K_{gh,bc}({\bf k}^{\prime\prime\prime},{\bf
k}^{\prime\prime\prime\prime};{\bf q},\Omega)\right)$

Eqn. (63) can be rewritten in a more compact form, as

$\int\limits_{\mathbf{k}^{\prime\prime}}\mathbf{\overline{U}}(\mathbf{k}^{\prime\prime},\mathbf{k}^{\prime};\mathbf{q},\Omega)=\frac{i}{2}\mathbf{\overline{A}}(\mathbf{k}^{\prime};\mathbf{q},\Omega)\left(g(\omega_{+})-g^{\ast}(\omega_{-})\right)$
(64)

with

$\displaystyle\mathbf{\overline{U}}(\mathbf{k}^{\prime\prime},\mathbf{k}^{\prime};\mathbf{q},\Omega)$
$\displaystyle=$ $\displaystyle U_{aa,cd}({\bf k},{\bf k}^{\prime};{\bf
q},\Omega)$ (65)
$\displaystyle\mathbf{\overline{A}}(\mathbf{k}^{\prime};\mathbf{q},\Omega)$
$\displaystyle=$ $\displaystyle
A_{bb,cd}(\mathbf{k}^{\prime};\mathbf{q},\Omega)$
$\displaystyle\mathbf{A}(\mathbf{k}^{\prime};\mathbf{q},\Omega)$
$\displaystyle=$
$\displaystyle\frac{2}{g(\omega_{+})+g^{\ast}(\omega_{-})}\left({\cal
R}\mathbf{\Sigma}(\mathbf{k}^{\prime};\mathbf{q},\Omega)+\int\limits_{\mathbf{k}^{\prime\prime}}{\cal
R}\mathbf{G}(\mathbf{k}^{\prime\prime};\mathbf{q},\Omega):\mathbf{K}({\bf
k^{\prime\prime}},{\bf k^{\prime}};{\bf q},\Omega)\right)\,,$
$\displaystyle{\cal R}\mathbf{\Sigma}(\mathbf{k}^{\prime};\mathbf{q},\Omega)$
$\displaystyle=$
$\displaystyle\frac{1}{2\rho}\left(\mathbf{I}\otimes\mathbf{\Sigma}^{-}(\mathbf{k}^{\prime-},\omega^{-})+\mathbf{\Sigma}^{+}(\mathbf{k}^{\prime+},\omega^{+})\otimes\mathbf{I}\right)\,,$

with tensor $U$ given by Eqn.(41) and the operation ${\cal R}$, here defined
for the self-energy tensor $\mathbf{\Sigma}$, acts in the same way on the
Green tensor $\mathbf{G}$.

###  IV.2 Low frequency asymptotics and diffusion behavior

The following relations for the self energy and for the Green function will
prove useful:

$\displaystyle\mathbf{\Delta\Sigma}({\bf k};{\bf 0},0)$ $\displaystyle=$
$\displaystyle\mathbf{\Delta\Sigma({\bf k})}=\Delta\Sigma_{ab,cd}({\bf
k})=\frac{1}{2\imath\rho}\left(\delta_{ac}\Sigma^{*}_{db}(\mathbf{k})-\Sigma_{ac}(\mathbf{k})\delta_{db}\right)$
$\displaystyle=$
$\displaystyle\frac{\delta_{ac}\delta_{db}}{2\imath\rho}\left(\Sigma^{*}(\mathbf{k})-\Sigma(\mathbf{k})\right)$
$\displaystyle=$ $\displaystyle\delta_{ac}\delta_{db}$
$\displaystyle\left(\frac{\rho
c^{2}_{T}k^{2}}{4i\rho}\left(\left(1-\frac{\omega^{2}}{(K_{T}^{2})^{*}c^{2}_{T}}\right)-\left(1-\frac{\omega^{2}}{K_{T}^{2}c^{2}_{T}}\right)\right)+\right.$
$\displaystyle\left.\frac{\rho
c^{2}_{L}k^{2}}{4i\rho}\left(\left(1-\frac{\omega^{2}}{(K_{L}^{2})^{*}c^{2}_{L}}\right)-\left(1-\frac{\omega^{2}}{K_{L}^{2}c^{2}_{L}}\right)\right)\right)$
$\displaystyle=$
$\displaystyle-\frac{\delta_{ac}\delta_{db}\omega^{2}k^{2}}{2}\left(\frac{Im[K_{T}^{2}]}{K_{T}^{2}(K_{T}^{2})^{*}}+\frac{Im[K_{L}^{2}]}{K_{L}^{2}(K_{L}^{2})^{*}}\right)\,.$
$\displaystyle\mathbf{\Delta G}({\bf k};{\bf 0},0)$ $\displaystyle=$
$\displaystyle\mathbf{\Delta G({\bf k})}=\Delta G_{ab,cd}({\bf
k})=\frac{1}{2\imath\rho}\left(\delta_{ac}G^{*}_{db}(\mathbf{k})-G_{ac}(\mathbf{k})\delta_{db}\right)$
$\displaystyle=$
$\displaystyle\frac{\delta_{ac}}{2\imath\rho^{2}\omega^{2}}\left(\frac{(\delta_{db}-\hat{k}_{d}\hat{k}_{b})}{\left\\{\frac{k^{2}}{(K_{T}^{2})^{*}}-1\right\\}}+\frac{\hat{k}_{d}\hat{k}_{b}}{\left\\{\frac{k^{2}}{(K_{L}^{2})^{*}}-1\right\\}}\right)-\frac{\delta_{db}}{2\imath\rho^{2}\omega^{2}}\left(\frac{(\delta_{ac}-\hat{k}_{a}\hat{k}_{c})}{\left\\{\frac{k^{2}}{K_{T}^{2}}-1\right\\}}+\frac{\hat{k}_{a}\hat{k}_{c}}{\left\\{\frac{k^{2}}{K_{L}^{2}}-1\right\\}}\right)\Rightarrow$
$\displaystyle\Delta G_{aa,cd}({\bf k})$ $\displaystyle=$
$\displaystyle\frac{1}{2\imath\rho^{2}\omega^{2}}\left(\frac{(\delta_{dc}-\hat{k}_{d}\hat{k}_{c})}{\left\\{\frac{k^{2}}{(K_{T}^{2})^{*}}-1\right\\}}-\frac{(\delta_{dc}-\hat{k}_{d}\hat{k}_{c})}{\left\\{\frac{k^{2}}{K_{T}^{2}}-1\right\\}}+\frac{\hat{k}_{d}\hat{k}_{c}}{\left\\{\frac{k^{2}}{(K_{L}^{2})^{*}}-1\right\\}}-\frac{\hat{k}_{d}\hat{k}_{c}}{\left\\{\frac{k^{2}}{K_{L}^{2}}-1\right\\}}\right)$
(69) $\displaystyle=$
$\displaystyle\frac{(\delta_{dc}-\hat{k}_{d}\hat{k}_{c})}{2\imath\rho^{2}\omega^{2}}\left(\frac{(K_{T}^{2})^{*}}{k^{2}-(K_{T}^{2})^{*}}-\frac{K_{T}^{2}}{k^{2}-K_{T}^{2}}\right)+\frac{\hat{k}_{d}\hat{k}_{c}}{2\imath\rho^{2}\omega^{2}}\left(\frac{(K_{L}^{2})^{*}}{k^{2}-(K_{L}^{2})^{*}}-\frac{K_{L}^{2}}{k^{2}-K_{L}^{2}}\right)$
$\displaystyle=$
$\displaystyle\frac{-(\delta_{dc}-\hat{k}_{d}\hat{k}_{c})k^{2}}{\rho^{2}\omega^{2}}\frac{Im[K_{T}^{2}]}{\left(k^{2}-(K_{T}^{2})^{*}\right)\left(k^{2}-K_{T}^{2}\right)}+\frac{-\hat{k}_{d}\hat{k}_{c}k^{2}}{\rho^{2}\omega^{2}}\frac{Im[K_{L}^{2}]}{\left(k^{2}-(K_{L}^{2})^{*}\right)\left(k^{2}-K_{L}^{2}\right)}$
$\displaystyle\approx$
$\displaystyle\frac{-\pi(\delta_{dc}-\hat{k}_{d}\hat{k}_{c})k^{2}}{\rho^{2}\omega^{2}}\delta\left(k^{2}-Re[K_{T}^{2}]\right)+\frac{-\pi\hat{k}_{d}\hat{k}_{c}k^{2}}{\rho^{2}\omega^{2}}\delta\left(k^{2}-Re[K_{L}^{2}]\right)$

with

$\Sigma=\rho\left(c_{T,L}^{2}-\frac{\omega^{2}}{K_{T,L}^{2}}\right)k^{2}\,,$
(70)

where we introduced the shorthand notations for $<G^{+}>_{db}({\bf
k},\omega)=G_{db}({\bf k})$ and $<G^{-}>_{db}({\bf k},\omega)=G^{*}_{db}({\bf
k}))$ and in the same way for $\Sigma^{\pm}_{db}({\bf k},\omega)$. It should
be emphasized that the last approximate equality in Eqn. (69) holds in the
limit $|Im[K_{T,L}^{2}]|\ll|k^{2}-Re[K_{T,L}^{2}]|$. The meaning of this
inequality in terms of the dislocation parameters is explored in Section V.1.

###  IV.3 Perturbation approach to BS eigenvalue problem

In this section we follow the approach that was used in Refs. [34, 35, 36, 37]
to study the diffusion of electromagnetic and acoustic waves: The BS equation
written in the form of Eqn. (40), supplemented by the relation between mass
operator $\mathbf{\Sigma}$ and kernel $\mathbf{K}$ implemented by the WTI Eqn.
(64), is solved for the intensity $\mathbf{\Phi}$, defined by Eqn. (33), in
the diffusive limit. To this end, the BS equation Eqn. (40) is written in
operator form:

$\int\limits_{\bf{k}^{\prime\prime}}\mathbf{H}({\bf k},{\bf
k}^{\prime\prime};{\bf q},\Omega):\mathbf{\Phi}({\bf k}^{\prime\prime},{\bf
k}^{\prime};{\bf q},\Omega)=\mathbf{\Delta G}({\bf k};{\bf
q},\Omega)\delta_{{\bf k},{\bf k}^{\prime}}\,.$ (71)

with the operator $\mathbf{H}$ defined by

$\mathbf{H}({\bf k},{\bf k}^{\prime\prime};{\bf
q},\Omega)\equiv\left[\imath\omega\Omega\mathbf{E}+\mathbf{P}({\bf k};{\bf
q})\right]\delta_{\mathbf{k}\mathbf{k}^{\prime\prime}}+\mathbf{U}({\bf k},{\bf
k}^{\prime\prime};{\bf q},\Omega)\,.$ (72)

It is easy to see, using the explicit form of $\mathbf{U}$, and the
reciprocity of the tensor $\mathbf{K}$, that $\mathbf{H}$ has the following
symmetry:

$H_{ab,cd}({\bf k},{\bf k}^{\prime\prime};{\bf q},\Omega)\Delta G_{cd,ef}({\bf
k}^{\prime\prime};{\bf q},\Omega)=H_{ef,cd}({\bf k}^{\prime\prime},{\bf
k};{\bf q},\Omega)\Delta G_{cd,ab}({\bf k};{\bf q},\Omega)\,.$ (73)

The solution of the BS equation will be found in terms of the eigenvalues and
eigentensors of the operator $\mathbf{H}$.

The eigenvalue problem for $\mathbf{H}$ is set up as follows:

$\int\limits_{\bf{k}^{\prime\prime}}H_{ab,cd}({\bf k},{\bf
k}^{\prime\prime};{\bf q},\Omega)f^{\mathbf{r}n}_{cd}({\bf
k}^{\prime\prime};{\bf
q},\Omega)=\lambda_{n}\left(\mathbf{q},\Omega\right)f^{\mathbf{r}n}_{ab}({\bf
k};{\bf q},\Omega)$ (74)

where $\mathbf{f}^{\mathbf{r}n}({\bf k}^{\prime\prime};{\bf q},\Omega)$ (resp.
$\mathbf{f}^{\mathbf{l}n}({\bf k}^{\prime\prime};{\bf q},\Omega)$) are right
(resp. left) eigentensors and $\lambda_{n}\left(\mathbf{q},\Omega\right)$ the
corresponding eigenvalues. Following [34, 35, 36, 37] we shall assume that the
eigentensors in Eqn. (74) satisfy completeness and orthogonality conditions:

$\displaystyle\int\limits_{\mathbf{k}}f^{\mathbf{r}m}_{ab}({\bf k};{\bf
q},\Omega)f^{\mathbf{l}n}_{ab}({\bf k};{\bf q},\Omega)$ $\displaystyle=$
$\displaystyle\delta_{mn}\,,$ (75)
$\displaystyle\sum\limits_{n}f^{\mathbf{r}n}_{ab}({\bf k};{\bf
q},\Omega)f^{\mathbf{l}n}_{ef}({\bf k}^{\prime};{\bf q},\Omega)$
$\displaystyle=$ $\displaystyle\delta_{{\bf k}{\bf
k}^{\prime}}\delta_{ae}\delta_{fb}\,.$

Furthermore, the symmetry restriction for the operator $\mathbf{H}$ from Eqn.
(73) determines the relation between left and right eigentensors:

$f^{\mathbf{r}n}_{ab}({\bf k};{\bf q},\Omega)=\Delta G_{ab,cd}({\bf k};{\bf
q},\Omega)f^{\mathbf{l}n}_{cd}({\bf k};{\bf q},\Omega)\,.$ (76)

The eigentensor properties Eqns. (75, 76) allow for a representation of the
solution $\mathbf{\Phi}$ of Eqn. (71) as a series over the states $n$ [34, 35,
36, 37] :

$\displaystyle\Phi_{ab,cd}=\sum\limits_{n}\frac{f^{\mathbf{r}n}_{ab}({\bf
k};{\bf q},\Omega)f^{\mathbf{r}n}_{cd}({\bf k}^{\prime};{\bf
q},\Omega)}{\lambda_{n}\left(\mathbf{q},\Omega\right)}$ (77)

The existence of a diffusion regime assumes that in the limit
$\mathbf{q}\rightarrow 0$, $\Omega\rightarrow 0$ the intensity $\mathbf{\Phi}$
has a pole structure given by the lowest eigenvalue asymptotics
$\lambda_{0}\left(\mathbf{q}\rightarrow 0,\Omega\rightarrow
0\right)\rightarrow 0$ that is separated from a regular part [35, 36, 37].
Therefore, the whole problem is reduced to the determination of coefficients
of a perturbative expansion for $\lambda_{0}\left(\mathbf{q},\Omega\right)$ to
second order in $\mathbf{q}$ and first order in $\Omega$ around the point
$\mathbf{q}=0$, $\Omega=0$. To do this, Eqn. (74) has to be treated
perturbatively, with the condition that Eqns. (64, 73) hold at every order of
the perturbation in $\mathbf{q}$, and $\Omega$ [35, 36, 37].

In order to solve Eq. (74), we write, in a small $\Omega$ and a small
$\mathbf{q}$ approximation,

$\displaystyle\mathbf{H}({\bf k},{\bf k}^{\prime\prime};{\bf q},\Omega)$
$\displaystyle=$ $\displaystyle\mathbf{H}({\bf k},{\bf k}^{\prime\prime};{\bf
0},0)+\mathbf{H}^{1\Omega}({\bf k},{\bf k}^{\prime\prime};{\bf 0},\Omega)$
$\displaystyle+\mathbf{H}^{1\mathbf{q}}({\bf k},{\bf k}^{\prime\prime};{\bf
q},0)+\mathbf{H}^{2\mathbf{q}}({\bf k},{\bf k}^{\prime\prime};{\bf
q},0)+\dots\,,$ $\displaystyle\mathbf{f}^{\mathbf{r}0}({\bf
k}^{\prime\prime};{\bf q},\Omega)$ $\displaystyle=$
$\displaystyle\mathbf{f}({\bf k}^{\prime\prime};{\bf
0},0)+\mathbf{f}^{1\Omega}({\bf k}^{\prime\prime};{\bf 0},\Omega)$
$\displaystyle+\mathbf{f}^{1\mathbf{q}}({\bf k}^{\prime\prime};{\bf
q},0)+\mathbf{f}^{2\mathbf{q}}({\bf k}^{\prime\prime};{\bf q},0)+\dots$
$\displaystyle\lambda_{0}\left(\mathbf{q},\Omega\right)$ $\displaystyle=$
$\displaystyle\lambda^{1\Omega}\left(\mathbf{0},\Omega\right)+\lambda^{1\mathbf{q}}\left(\mathbf{q},0\right)+\lambda^{2\mathbf{q}}\left(\mathbf{q},0\right)+\dots$

Substitution of the above expansions into Eqn. (74) leads to the following set
of coupled integral equations:

$\displaystyle\int\limits_{\bf{k}^{\prime\prime}}H_{ab,cd}({\bf k},{\bf
k}^{\prime\prime})f_{cd}({\bf k}^{\prime\prime})$ $\displaystyle=$
$\displaystyle 0$ (79)
$\displaystyle\int\limits_{\bf{k}^{\prime\prime}}\left(H_{ab,cd}({\bf k},{\bf
k}^{\prime\prime})f^{1\Omega}_{cd}({\bf
k}^{\prime\prime})+H^{1\Omega}_{ab,cd}({\bf k},{\bf
k}^{\prime\prime})f_{cd}({\bf k}^{\prime\prime})\right)$ $\displaystyle=$
$\displaystyle\lambda^{1\Omega}f_{ab}({\bf k})$ (80)
$\displaystyle\int\limits_{\bf{k}^{\prime\prime}}\left(H_{ab,cd}({\bf k},{\bf
k}^{\prime\prime})f^{1\mathbf{q}}_{cd}({\bf
k}^{\prime\prime})+H^{1\mathbf{q}}_{ab,cd}({\bf k},{\bf
k}^{\prime\prime})f_{cd}({\bf k}^{\prime\prime})\right)$ $\displaystyle=$
$\displaystyle 0$ (81)
$\displaystyle\int\limits_{\bf{k}}{\cal{B}}P_{aa,cd}({\bf k};{\bf
q})f^{1\mathbf{q}}_{cd}({\bf k})$ $\displaystyle=$
$\displaystyle\lambda^{2\mathbf{q}}$ (82)

Within Eqns. (79-82) we have used a shorthand notation for the quantities
appearing in Eqn. (IV.3), in which the arguments $\mathbf{q}$ and $\Omega$ are
omitted. The superscript indicates the variable and the order of perturbation
for the operator $\mathbf{H}$, right eigentensor $\mathbf{f}^{\mathbf{r}0}$,
and eigenvalue $\lambda_{0}$. The disappearance in Eqns. (79-82) of
contributions from some terms that appear in Eqn. (IV.3) is due to the
implementation of symmetry and conservation restrictions coming from Eqns.
(64) and (73). Importantly, the first-order-in-wavenumber contribution to the
eigenvalue vanishes: $\lambda^{1\mathbf{q}}=0$. Without this result there
would be no diffusion behavior. This algebra follows closely Ref. [29].

With the aid of Eqns. (64) and (79), the eigentensor
$\mathbf{f}^{\mathbf{r}0}$ at $\mathbf{q}=0$, $\Omega=0$ can be found at once:

$f_{cd}({\bf k}^{\prime\prime})={\cal{B}}{\Delta G_{cd,gg}({\bf
k}^{\prime\prime})}$ (83)

with

${\cal{B}}=\frac{1}{\sqrt{\int\limits_{\bf{v}}{\Delta G_{cc,gg}({\bf v})}}}.$
(84)

Note that $\int\limits_{\bf{v}}{\Delta G_{cc,gg}({\bf v})}$ is negative, but
only ${\cal{B}}^{2}$ appears in the expression for the diffusion constant.
Integrating Eqn. (80) over ${\bf k}$ along with the subsequent implementation
of the WTI from Eqn. (64) at the corresponding order, leads to the expression
for the eigenvalue $\lambda^{1\Omega}$

$\displaystyle i\omega\Omega\left(1+a\right)=\lambda^{1\Omega}$ (85)

where we have introduced a parameter $a$ defined by

$\displaystyle a=\frac{1}{\int\limits_{\bf{k}}f_{hh}({\bf
k})}\int\limits_{\bf{k}^{\prime\prime}}\frac{\left(A_{aa,cd}(\mathbf{k}^{\prime\prime};\mathbf{0},\Omega)\left(g(\omega_{+})-g^{\ast}(\omega_{-})\right)\right)^{1\Omega}}{2\omega\Omega}f_{cd}({\bf
k}^{\prime\prime})\,.$ (86)

This is an analogue of the well-known parameter that appears in the diffusion
of light, which, being positive, renormalizes the phase velocity downwards to
a lower value for a transport velocity [35, 36, 38, 39]. To see that our $a$
is indeed positive, replace Eqns. (65) and (83) into Eqn. (86) to obtain

$\displaystyle a$ $\displaystyle=$
$\displaystyle\frac{-1}{\rho^{2}\left(\omega_{0}^{2}-\omega^{2}\right)\int\limits_{\bf{v}}{\Delta
G_{cc,hh}({\bf
v})}}\int\limits_{\bf{k}}Im[{\Sigma_{ab}(\mathbf{k})G_{ab}}({\bf k})]$
$\displaystyle\approx$
$\displaystyle\frac{Re[K_{T}^{2}]\left(\frac{c_{T}^{2}}{\omega^{2}}Re[K_{T}^{2}]-1\right)+Re[K_{L}^{2}]\left(\frac{c_{L}^{2}}{\omega^{2}}Re[K_{L}^{2}]-1\right)}{\left(\frac{\omega_{0}^{2}}{\omega^{2}}-1\right)\left(Re[K_{T}^{2}]+Re[K_{L}^{2}]\right)}$

where the last approximate equality is obtained using expression Eqn. (150)
for the function $F$ defined by Eqn. (144) that enters the integrand of
Eqn.(IV.3) as a consequence of the calculation of
$Im[{\Sigma_{ab}(\mathbf{k})G_{ab}}({\bf k})]$. Clearly $a>0$ if
$\omega<\omega_{0}$.

####  IV.3.1 Explicit form of the diffusion constant

Within the spectral approach that we are using, Eqns. (77-82) lead to the
following expression for the singular part of the intensity,
$\mathbf{\Phi}^{sing}$:

$\Phi^{sing}_{ab,cd}=\frac{f^{\mathbf{r0}}_{ab}({\bf k};{\bf
q},\Omega)f^{\mathbf{r0}}_{cd}({\bf k}^{\prime};{\bf
q},\Omega)}{\lambda^{1\Omega}+\lambda^{2\mathbf{q}}}=\frac{f^{\mathbf{r0}}_{ab}({\bf
k};{\bf q},\Omega)f^{\mathbf{r0}}_{cd}({\bf k}^{\prime};{\bf
q},\Omega)}{\frac{\lambda^{1\Omega}}{-i\Omega}\left(-i\Omega+\frac{-i\Omega\lambda^{2\mathbf{q}}}{\lambda^{1\Omega}q^{2}}q^{2}\right)}\,.$
(88)

Then, with the assistance of Eqns. (85,88) the diffusion constant can be
identified as

$\displaystyle D$ $\displaystyle\equiv$
$\displaystyle-\frac{i\Omega\lambda^{2\mathbf{q}}}{q^{2}\lambda^{1\Omega}}=-\frac{{\cal{B}}}{q^{2}\omega\left(1+a\right)}\int\limits_{\bf{k}}P_{aa,cd}({\bf
k};{\bf q})f^{1\mathbf{q}}_{cd}({\bf k})$ (89) $\displaystyle=$
$\displaystyle\frac{{\cal{B}}^{2}}{q^{2}\omega\left(1+a\right)}\int\limits_{\bf{k}}P_{aa,cd}({\bf
k};{\bf q})\left(\int\limits_{{\bf k}_{2}}\Phi_{cd,gh}({\bf k},{\bf
k}_{2})P_{gh,ee}(\mathbf{q};{\bf k}_{2})-\Delta G_{cd,ee}^{1\mathbf{q}}({\bf
k})\right)$ (90) $\displaystyle\equiv$ $\displaystyle
D^{\mathcal{R}}+D_{\Delta G^{1\mathbf{q}}}$ (91)

with ${\cal{B}}$ given by Eqn. (84). To obtain Eqn. (90), in which the
diffusion constant is written as the sum of two terms, we have substituted the
values for $\lambda^{2\mathbf{q}}$, $\lambda^{1\Omega}$ given by Eqns. (82)
and (85). The last one ensued from the form of $f^{1\mathbf{q}}({\bf k})$ (See
D). Thus, the expression for the diffusion constant in Eqn. (89) is the sum of
two contributions, as defined in (91).

While the second term in Eqn. (91), $D_{\Delta G^{1\mathbf{q}}}$, can be
calculated straightforwardly (See E), the calculation of the first term,
$D^{\mathcal{R}}$, is more laborious. Indeed, it depends on the unknown tensor
function $\mathbf{\Phi}$. However, it is not the complete tensor function that
is needed, but an integrated form over one of its variables that, as we now
show, can be expressed as a function of the mass operator tensor
$\mathbf{\Sigma}$ and the kernel tensor $\mathbf{K}$ using the BS equation and
the WTI. In our case, in order to do this we apply, inspired by the treatment
of light diffusion [40], the method that uses an auxiliary tensor function
$\mathbf{\Psi}_{,s}({\bf k})$ defined by the relation

$\mathbf{\Psi}_{,s}({\bf k})q_{s}\equiv\Psi_{cd,s}({\bf
k})q_{s}\equiv\int\limits_{{\bf k}^{\prime}}\Phi_{cd,gh}({\bf k},{\bf
k}^{\prime})P_{gh,ee}(\mathbf{q};{\bf k}^{\prime})=-\int\limits_{{\bf
k}^{\prime}}\Phi_{cd,gh}({\bf k},{\bf
k}^{\prime})\frac{1}{2i\rho}\frac{\partial L_{gh}({\bf k}^{\prime})}{\partial
k^{\prime}_{s}}q_{s}\,.$ (92)

Then, from Eqns. (40-41) the following expression for $\mathbf{\Psi}_{,s}({\bf
k})$ immediately follows:

$\displaystyle P_{ab,cd}({\bf p})\Psi_{cd,s}({\bf
p})+\Delta\Sigma_{ab,c_{1}d_{1}}({\bf p})\Psi_{c_{1}d_{1},s}({\bf p})\hskip
30.00005pt$ (93) $\displaystyle-\int\limits_{\bf{p}^{\prime\prime}}\Delta
G_{ab,c_{2}d_{2}}({\bf p})K_{c_{2}d_{2},c_{1}d_{1}}({\bf p},{\bf
p}^{\prime\prime})\Psi_{c_{1}d_{1},s}({\bf p}^{\prime\prime})$
$\displaystyle=$ $\displaystyle-\Delta G_{ab,gh}({\bf
p})\frac{1}{2i\rho}\frac{\partial L_{gh}({\bf p})}{\partial p_{s}}$

Eqn. (93) can be substantially simplified if we recall the explicit form of
the free medium Green’s function, Eqn.(18), along with relations from Eqns.
(15, 42) [31]. Then,

$\displaystyle\Delta G_{ab,c_{2}d_{2}}({\bf p})$ $\displaystyle=$
$\displaystyle\frac{-1}{2i\rho}\left(G^{-1*}_{ac_{3}}({\bf
p})\delta_{d_{3}b}-\delta_{ac_{3}}G^{-1}_{d_{3}b}({\bf
p})\right)G_{d_{2}d_{3}}({\bf p})G_{c_{3}c_{2}}^{*}({\bf p})$ $\displaystyle=$
$\displaystyle\left(\Delta\Sigma_{ab,c_{3}d_{3}}({\bf
p})+P_{ab,c_{3}d_{3}}({\bf p})\right)G_{d_{2}d_{3}}({\bf
p})G_{c_{3}c_{2}}^{*}({\bf p})$

As a next step, we define a tensor angular entity $\mathbf{\Upsilon}$ that is
analogous to the coefficient relating transport mean free path and extinction
length in the diffusion of electromagnetic waves [41, 42]:

$\Psi_{cd,s}({\bf p})q_{s}=G^{*}({\bf p})_{cg}G_{hd}({\bf
p})\Upsilon_{gh}({\bf p},{\bf q})\,,$ (95)

and an integral equation for $\mathbf{\Upsilon}$ follows straightforwardly
from Eqns. (93-95):

$\Upsilon_{gh}({\bf p},{\bf
q})-\int\limits_{\bf{p}^{\prime\prime}}K_{gh,c_{1}d_{1}}({\bf p},{\bf
p}^{\prime\prime})G_{c_{1}g_{1}}^{*}({\bf
p}^{\prime\prime})G_{h_{1}d_{1}}({\bf
p}^{\prime\prime})\Upsilon_{g_{1}h_{1}}({\bf p}^{\prime\prime},{\bf
q})=P_{gh,ee}({\bf p};{\bf q})\,.$ (96)

Then, using Eqns. (92), (95), $D^{\mathcal{R}}$ from Eqn. (91) can be written
as

$D^{\mathcal{R}}=\frac{{\cal{B}}^{2}}{q^{2}\omega\left(1+a\right)}\int\limits_{\bf{k}}P_{aa,cd}({\bf
k};{\bf q})G_{cg}^{*}({\bf k})G_{hd}({\bf k})\Upsilon_{gh}({\bf k},{\bf q})$
(97)

Furthermore, guided by the definition from Eqn. (95) we can make a conjecture
that tensor $\mathbf{\Upsilon}({\bf p},{\bf q})$ should be linear in ${\bf
q}$. Keeping in mind this property of $\mathbf{\Upsilon}({\bf p},{\bf q})$ we
seek for the corresponding solution in the form

$\Upsilon_{gh}({\bf p},{\bf q})=\alpha P_{gh,ff}({\bf p};{\bf q})$ (98)

Then, by multiplying Eqn. (96) with $P_{e_{1}e_{1},g_{1}h_{1}}({\bf p};{\bf
q})G_{g_{1}g}({\bf p})G_{hh_{1}}^{*}({\bf p})$ from the left and integrating
over the ${\bf p}$ we remain with the relation

$\alpha^{-1}=1-\frac{\int\limits_{\bf{p}}\int\limits_{\bf{p}^{\prime\prime}}P_{ff,ab}({\bf
p};{\bf q})G_{ag}({\bf p})G_{hb}^{*}({\bf p})K_{gh,cd}({\bf p},{\bf
p}^{\prime\prime})G_{ca}^{*}({\bf p}^{\prime\prime})G_{bd}({\bf
p}^{\prime\prime})P_{ab,ee}({\bf p}^{\prime\prime},{\bf
q})}{\int\limits_{\bf{k}}P_{e_{1}e_{1},g_{1}h_{1}}({\bf k};{\bf
q})G_{g_{1}g}({\bf k})G_{hh_{1}}^{*}({\bf k})P_{gh,ee}({\bf k};{\bf q})}$ (99)

where the ratio of two integrals is the analog of the
$\langle\cos\theta\rangle$ term in the diffusion of electromagnetic waves
[42]. As a consequence, $D^{\mathcal{R}}$ can be represented as

$D^{\mathcal{R}}=\frac{{\cal{B}}^{2}}{q^{2}\omega\left(1+a\right)}\int\limits_{\bf{k}}\alpha
P_{aa,cd}({\bf k};{\bf q})G_{cg}^{*}({\bf k})G_{hd}({\bf
k})P_{gh,a_{1}a_{1}}({\bf k};{\bf q})$ (100)

It must be noted here that $\alpha$ included in the general expression for the
diffusion constant from Eqn. (100) can be evaluated explicitly using the
symmetry properties of the Green tensor, tensor $\mathbf{P}$, and the kernel
$\mathbf{K}$ from Eqns. (24), (42),and (IV.1.3), respectively. Indeed, those
equations support the validity of the following relations:

$\displaystyle K_{gh,c_{1}d_{1}}({\bf p},-{\bf
p}^{\prime\prime})G_{c_{1}g_{1}}^{*}(-{\bf
p}^{\prime\prime})G_{h_{1}d_{1}}(-{\bf
p}^{\prime\prime})P_{g_{1}h_{1},ee}(-{\bf p}^{\prime\prime},{\bf q})$
$\displaystyle=$ $\displaystyle-K_{gh,c_{1}d_{1}}({\bf p},{\bf
p}^{\prime\prime})G_{c_{1}g_{1}}^{*}({\bf
p}^{\prime\prime})G_{h_{1}d_{1}}({\bf p}^{\prime\prime})P_{g_{1}h_{1},ee}({\bf
p}^{\prime\prime},{\bf q})$

Hence

$\int\limits_{{\bf p}^{\prime\prime}}K_{gh,c_{1}d_{1}}({\bf p},{\bf
p}^{\prime\prime})G_{c_{1}g_{1}}^{*}({\bf
p}^{\prime\prime})G_{h_{1}d_{1}}({\bf p}^{\prime\prime})P_{g_{1}h_{1},ee}({\bf
p}^{\prime\prime},{\bf q})=0$ (102)

and consequently

$\alpha=1\,.$ (103)

Therefore, $D^{\mathcal{R}}$ from the Eqn.(100) can be brought into the form

$D^{\mathcal{R}}=\frac{{\cal{B}}^{2}}{q^{2}\omega\left(1+a\right)}\int\limits_{\bf{k}}P_{aa,cd}({\bf
k};{\bf q})G_{cg}^{*}({\bf k})G_{hd}({\bf k})P_{gh,a_{1}a_{1}}({\bf k};{\bf
q})$ (104)

Finally, using approximation from Eqn. (150) for Eqn. (141) in Eqn. (104) we
can write

$D^{\mathcal{R}}=\frac{-{\cal{B}}^{2}}{8\rho^{2}\omega^{5}\left(1+a\right)}\left(c_{L}^{4}\left(\frac{Re[K_{L}^{2}]^{3}}{Im[K_{L}^{2}]}+Re[K_{L}^{2}]Im[K_{L}^{2}]\right)+c_{T}^{4}\left(\frac{Re[K_{T}^{2}]^{3}}{Im[K_{T}^{2}]}+Re[K_{T}^{2}]Im[K_{T}^{2}]\right)\right)\,.$
(105)

Then, the total diffusion constant reads as

$\displaystyle D$ $\displaystyle=$ $\displaystyle D^{\mathcal{R}}+D_{\Delta
G^{1\mathbf{q}}}$ $\displaystyle=$
$\displaystyle\frac{-{\cal{B}}^{2}}{8\rho^{2}\omega^{5}\left(1+a\right)}\left(c_{L}^{4}\left(\frac{Re[K_{L}^{2}]^{3}}{Im[K_{L}^{2}]}+Re[K_{L}^{2}]Im[K_{L}^{2}]\right)+c_{T}^{4}\left(\frac{Re[K_{T}^{2}]^{3}}{Im[K_{T}^{2}]}+Re[K_{T}^{2}]Im[K_{T}^{2}]\right)\right)$
$\displaystyle\hskip
40.00006pt+\frac{{\cal{B}}^{2}\left(c_{L}^{2}-c_{T}^{2}\right)\left(Im[K_{L}^{2}]-Im[K_{T}^{2}]\right))}{16\rho^{2}\omega^{3}\left(1+a\right)}\,,$

and the leading term in the limit of small $Im[K_{T,L}^{2}]$ is

$\displaystyle
D^{lead}\approx\frac{-{\cal{B}}^{2}}{8\rho^{2}\omega^{5}\left(1+a\right)}\left(c_{L}^{4}\frac{Re[K_{L}^{2}]^{3}}{Im[K_{L}^{2}]}+c_{T}^{4}\frac{Re[K_{T}^{2}]^{3}}{Im[K_{T}^{2}]}\right)$
(106)

Explicitly, from Eqn.(84), we have

$\displaystyle-{\cal{B}}^{2}=\frac{4\rho^{2}\omega^{2}}{\left(Re[K_{T}^{2}]+Re[K_{L}^{2}]\right)}$
(107)

Then, using Eqns.(69, 84, IV.3, 106)

$D^{lead}\approx\frac{1}{\left(1+\frac{Re[K_{T}^{2}]\left(\frac{c_{T}^{2}}{\omega^{2}}Re[K_{T}^{2}]-1\right)+Re[K_{L}^{2}]\left(\frac{c_{L}^{2}}{\omega^{2}}Re[K_{L}^{2}]-1\right)}{\left(\frac{\omega_{0}^{2}}{\omega^{2}}-1\right)\left(Re[K_{T}^{2}]+Re[K_{L}^{2}]\right)}\right)}\times\frac{\left(c_{L}^{4}\frac{Re[K_{L}^{2}]^{3}}{Im[K_{L}^{2}]}+c_{T}^{4}\frac{Re[K_{T}^{2}]^{3}}{Im[K_{T}^{2}]}\right)}{2\omega^{3}\left(Re[K_{T}^{2}]+Re[K_{L}^{2}]\right)}$
(108)

In the low-frequency limit it reads as

$\displaystyle D^{lead}_{\omega\rightarrow
0}\approx\left(\frac{v_{T}^{2}c_{L}^{4}}{\left(v_{L}^{2}+v_{T}^{2}\right)v_{L}^{4}}\frac{v_{L}l_{L}}{2}+\frac{v_{L}^{2}c_{T}^{4}}{\left(v_{L}^{2}+v_{T}^{2}\right)v_{T}^{4}}\frac{v_{T}l_{T}}{2}\right)$
(109)

##  V Discussion

###  V.1 Approximations used in this work

There are two approximations that are central to our results: The independent
scattering approximation (ISA), and the restriction on the effective wave
numbers $K_{t},L$ given by

$|Im[K_{L,T}^{2}]|\ll|k^{2}-Re[K_{L,T}^{2}]|\,.$ (110)

The ISA means that each dislocation present in the two dimensional elastic
medium is characterized by a position and Burgers vector that are
statistically independent random variables. This leads to expressions (IV.1.3)
for the mass operator and for the irreducible kernel, that are linear in the
density of dislocations $n$, and involve the single-scattering $T$ matrix.
This $T$ matrix has been computed to all orders in a Born approximation
expansion.

The inequality (110) can be satisfied at fixed dislocation density by going to
low frequencies (smaller than the renormalized oscillation frequency
$\omega_{0R}$ defined in Eq. (129)), and at fixed frequency by going to low
dislocation densities. In the important case of resonance,
$\omega\sim\omega_{0R}$, the restriction (110) can be satisfied with
interparticle distances that are small compared to wavelength.

###  V.2 Nature of the wave-dislocation interaction and its consequences for
coherent wave behavior

In this paper we have considered the behavior of plane elastic waves in a two
dimensional elastic medium in interaction with many edge dislocations that can
oscillate around an equilibrium position, with a restoring force given by the
Peierls-Nabarro force. The fact that there is a natural frequency associated
with the dislocations implies the possibility of resonant scattering. The
equation that gives the response to an external loading are solved, and the
solution replaced into the wave equation for particle velocity in the presence
of dislocations, to give an overall equation for particle velocity only in the
presence of a random, frequency dependent, “potential”, that is of second
order in the space derivatives, provided by the dislocations. First, their
coherent behavior has been elucidated: there exists average transverse and
longitudinal waves, whose phase velocity and attenuation have been explicitly
calculated as a function of dislocation density and wave frequency. This has
been done solving for the mass operator in a Dyson equation, with the mass
operator calculated using an ISA, according to which it is the product of the
dislocation density and the averaged single-dislocation $T$-matrix. The latter
has been calculated to all orders in a Born approximation scheme. This part of
the work follows previous calculations of Maurel et al. [31], although the sum
to all orders is new.

###  V.3 Nature of the wave-dislocation interaction and its consequences for
diffusive wave behavior

The remaining bulk of this work has been devoted to the study of the diffusion
behavior of said waves. The reasoning that has been followed is in accordance
with procedures established in other contexts, but the nature of the
“potential” term introduces unique characteristics. A Bethe-Salpeter equation
is written for the irreducible kernel of the two-point field amplitude
correlations. Next a Ward-Takahashi identity, needed as a link between the
irreducible kernel and the mass operator, is established. This is done
following the methods of Barabanenkov [35, 36]. That is, establishing first a
pre-WTI that provides a link between the irreducible kernel and the averaged
Green’s functions. Although the BS equation is written out in terms of rank-4
tensors, such as the tensor product of two Green’s functions, both the pre-WTI
and the WTI itself are written in terms of rank-2 tensors. It turns out this
is enough to slove the BS equation. In order to establish, or not, diffusive
behavior, it is necessary to study the behavior of the BS equation at low
frequencies and wavenumbers. As for the coherent waves, an ISA is introduced,
and it is verified that this approximation is consistent with the WTI. Next
the BS equation is set up, following [35, 36] as an eigenvalue problem.
Eigenfunctions and eigenvalues are found to leading order in low frequency and
wavenumber. This portion of the work has a straightforward logic but the
algebra is rather cumbersome. At length, a diffusion behavior is established
and an explicit expression for the (frequency dependent) diffusion
coefficient, Eq. (108). is obtained.

###  V.4 Relation of the obtained diffusion behavior to other work

The obtained diffusion coefficient appears to vanish when the wave frequency
coincides with the resonant frequency of the dislocations. The vanishing of
the diffusion constant of electrons, treated as de Broglie waves, in the
presence of randomly placed impurities was linked to their (“Anderson”)
localization [43] and spurred a large body of research into analogous
properties of classical (i.e. non-quantum) waves. The role of resonant
interaction between classical electromagnetic radiation and matter has been
reviewed by Lagendijk and van Tiggelen [44]. The effect of resonances in the
transport of light in a random system of dielectric spheres has been studied
by Sapienza et al. [45], by Cowan et al. [46] in the transport of ultrasound
in a system of disordered plastic spheres immersed in water, and by Tallon et
al. [47] in emulsions. In our case, however, since the formalism we have
developed holds only in the loss less ($B=0$) case, we cannot set
$\omega=\omega_{0}$ in Eq. (10) because it would lead to a singular potential.
But we can ask ourselves about the behavior of the problem as $\omega$
approaches $\omega_{0}$ from below. And the answer to this question is that
the diffusion coefficient becomes smaller and smaller, until the behavior
determined by internal losses, as encapsulated by the $B$ coefficient, takes
over. The role of internal losses in experiments looking for the localization
of acoustic waves has been recently discussed by Cobus et al. [1]. It will be
of interest to see what is the behavior of elastic waves, in three dimensions,
in the presence of dislocation segments of finite length, with a resonant
frequency in the THz range [28]. At very low frequency, our diffusion constant
becomes a two-dimensional version of the diffusion constant found by Ryzhik et
al. [10] using energy transfer theory for elastic waves interacting with
generic obstacles.

##  VI Concluding remarks

We have studied the properties of elastic waves in interaction with many,
randomly placed, edge dislocations both in the coherent and incoherent
regimes. The basic scattering mechanism has the novel feature, compared with
the more usual studies of wave propagation in random media, that dislocations
are dynamical objects. This leads to properties of the “potential” term in the
wave equations that are wave-vector dependent, also at variance with previous
examples. The present study is part of an ongoing project to explore the
consequences of such an interaction for coherent and incoherent wave behavior,
both as a matter of principle and in light of possible applications. We have
noted in the Introduction that in the coherent, low frequency regime, these
studies have led to novel tools for the non-destructive evaluation of the
plastic behavior of metals and alloys.

As noted in the Introduction, there is an increasing amount of experimental
evidence pointing to a role played by dislocations in thermal transport, to an
extent that has not been fully understood to date. From this point of view, it
is important to have as complete an understanding of the elastic wave-
dislocation interaction, and of its consequences, as possible. The approach we
have taken in this work is to consider dislocations embedded in a continuum
elastic solid, and use the tools of continuum solid mechanics. It is of
interest to note that, from this point of view, dislocations are just
topological line defects (“Volterra dislocations” [48]) that do not need an
underlying crystal structure for their existence. This has the consequence
that the formalism can be appplied to any solid, not necessaarily crystals, as
long as they can be described through continuum mechanics. This approach has
led to a renewed understanding of elastic anomalies in glasses [27, 28].
Recently, the various mechanisms at play in the thermal transport of amorphous
materials has been studied in some detail [49], indicating that both
propagating (coherent) and diffusive (incoherent) modes are present, in
various degrees depending on the phonon frequency. In any case, a scattering
process appears to be essential, and it does not appear to be completely clear
what exactly is doing the scattering. It will be of interest to see if the
mechanisms found in the present paper carry over to three dimensions, and the
consequences they have, if any, for the thermal transport in glasses.

##  VII Acknowledgements

This work has been supported by Fondecyt Grant 1191179.

##  Appendix A Averages of rotation in 2D

In contrast to 3D case, where Euler angles need to be introduced for
description of angular averaging [16], in the 2D case only polar angle
$\theta$ determines the corresponding averages.

We define the average of an arbitrary function $f(\theta)$ over the polar
angle as

$<f>=\frac{1}{2\pi}\int\limits_{0}^{2\pi}f(\theta)d\theta$ (111)

According to combinatorics rules [50], the tensor integral, comprised of the
product of radial unit $n$-dimensional vectors $\hat{\vec{r}}$
($\hat{\vec{r}}^{2}=1$)

$<\hat{r}^{i_{1}}\cdots\hat{r}^{i_{k}}>_{\hat{\vec{r}}}=\frac{1}{\Omega^{(n)}}\int
d\Omega^{(n)}_{\hat{\vec{r}}}\hat{r}^{i_{1}}\cdots\hat{r}^{i_{k}}$ (112)

with $\Omega^{n}$ as a $n$-dimensional solid angle, is equal to zero when
there is an odd number of unit vectors in the product, and reduces to the
totally symmetric isotropic tensor $\tilde{{\cal L}}^{i_{1}\cdots
i_{2k}}_{(2k)}$ by the rule

$<\hat{r}^{i_{1}}\cdots\hat{r}^{i_{2k}}>_{\hat{\vec{r}}}=\tilde{{\cal
L}}^{i_{1}\cdots i_{2k}}_{(2k)}$ (113)

with $\tilde{{\cal L}}^{i_{1}\cdots i_{2k}}_{(2k)}$ obeying the recurrence
relation

$\tilde{{\cal L}}^{i_{1}\cdots
i_{2k}}_{(2k)}=\frac{1}{n+2k-2}\left(\delta_{i_{1}i_{2}}\tilde{{\cal
L}}^{i_{3}\cdots i_{2k}}_{(2k-2)}+\delta_{i_{1}i_{3}}\tilde{{\cal
L}}^{i_{2}i_{4}\cdots i_{2k}}_{(2k-2)}+\cdots+\delta_{i_{1}i_{2k}}\tilde{{\cal
L}}^{i_{2}\cdots i_{2k-1}}_{(2k-2)}\right)$ (114)

and satisfying to the condition $\tilde{{\cal L}}_{0}=1$.

Recalling that $\Omega^{(2)}=2\pi$, $d\Omega^{(2)}_{\hat{\vec{r}}}=d\theta$,
one can easily see that definitions from Eqn.(111) and Eqn.(112) coincides
with each other if $f=\hat{r}^{i_{1}}\cdots\hat{r}^{i_{k}}$.

Based on the above, we construct a list of averages of our particular interest

$\displaystyle\mbox{$k=1$:}\hskip 40.00006pt<\hat{r}^{i}\hat{r}^{j}>$
$\displaystyle=$ $\displaystyle\frac{\delta_{ij}}{n}$
$\displaystyle\mbox{$k=2$:}\hskip
23.99997pt<\hat{r}^{i}\hat{r}^{j}\hat{r}^{k}\hat{r}^{l}>$ $\displaystyle=$
$\displaystyle\frac{1}{n+2}\left(\delta_{ij}\tilde{{\cal
L}}^{kl}_{(2)}+\delta_{ik}\tilde{{\cal L}}^{jl}_{(2)}+\delta_{il}\tilde{{\cal
L}}^{jk}_{(2)}\right)$ (115) $\displaystyle=$
$\displaystyle\frac{1}{n(n+2)}\left(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}\right)\,,$

where $n=2$ in Eqn. (115) for two dimensions, the case of interest here.

In addition, since relation from the Eqn. (4) holds in 2D, then, the following
important average can be evaluated using Eqn. (115)

$\displaystyle<{\mathsf{M}}_{ab}{\mathsf{M}}_{cd}>$ $\displaystyle=$
$\displaystyle<(n_{a}t_{b}+n_{b}t_{a})(n_{c}t_{d}+n_{d}t_{c})>$ (116)
$\displaystyle=$
$\displaystyle<(\epsilon_{ae}t_{e}t_{b}+\epsilon_{bf}t_{f}t_{a})(\epsilon_{cg}t_{g}t_{d}+\epsilon_{dh}t_{h}t_{c})>$
$\displaystyle=$
$\displaystyle(\delta_{ac}<t_{b}t_{d}>+\delta_{ad}<t_{b}t_{c}>+\delta_{bc}<t_{a}t_{d}>+\delta_{bd}<t_{a}t_{c}>-4<t_{a}t_{b}t_{c}t_{d}>)$
$\displaystyle=$
$\displaystyle\frac{\delta_{bc}\delta_{ad}+\delta_{bd}\delta_{ac}-\delta_{ab}\delta_{cd}}{2}$

We shall need this result in the next B

##  Appendix B Summation of the perturbation expansion for the mass operator

We introduce the definition of the $t$-matrix of $n$-th scatterer in momentum
space through (in what follows we avoid frequency argument as a matter of
convenience) [20]

$t^{n}_{ab}(\vec{k},\vec{k}^{\prime})=\int
d\vec{x}d\vec{x}^{\prime}e^{-i\vec{k}\cdot\vec{x}}t^{n}_{ab}(x,x^{\prime})e^{i\vec{k}^{\prime}\cdot\vec{x}^{\prime}}$
(117)

and its Born expansion

$t^{n}_{ab}(\vec{k},\vec{k}^{\prime})=t^{n(1)}_{ab}(\vec{k},\vec{k}^{\prime})+t^{n(2)}_{ab}(\vec{k},\vec{k}^{\prime})+t^{n(3)}_{ab}(\vec{k},\vec{k}^{\prime})\ldots$
(118)

The first Born approximation is specified as:

$\displaystyle t^{n(1)}_{ab}(\vec{k},\vec{k}^{\prime})$ $\displaystyle=$
$\displaystyle\int
d\vec{x}e^{-i\vec{k}\cdot\vec{x}}V^{n}_{ab}(x)e^{i\vec{k}^{\prime}\cdot\vec{x}}$
(119) $\displaystyle=$ $\displaystyle{\int
d\vec{x}e^{-i\vec{k}\cdot\vec{x}}\mu{\cal
A}{\mathsf{M}}_{ac}^{n}\frac{\partial}{\partial
x_{c}}\delta(\vec{x}-\vec{X}_{0}^{n}){\mathsf{M}}_{eb}^{n}\frac{\partial}{\partial
x_{e}}|_{\vec{x}=\vec{X}_{0}^{n}}e^{i\vec{k}^{\prime}\cdot\vec{x}}}$
$\displaystyle=$
$\displaystyle{-\mu{\mathcal{A}}{\mathsf{M}}_{ac}^{n}k_{c}k^{\prime}_{e}{\mathsf{M}}_{eb}^{n}e^{i\left(\vec{k}^{\prime}-\vec{k}\right)\cdot\vec{X}^{n}_{0}}.}$

Here, we used the expression for perturbation potential of a single edge
dislocation from Eqn. (11).

The second order Born approximation is given by

$\displaystyle t^{n(2)}_{ab}(\vec{k},\vec{k}^{\prime})$ $\displaystyle=$
$\displaystyle\int
d\vec{x}\,d\vec{x}^{\prime}e^{-i\vec{k}\cdot\vec{x}}V^{n}_{ac}(x)G^{0}_{cd}(x-x^{\prime})V^{n}_{db}(x^{\prime})e^{i\vec{k}^{\prime}\cdot\vec{x}^{\prime}}$
$\displaystyle=$ $\displaystyle\frac{(\mu{\mathcal{A}})^{2}}{(2\pi)^{2}}\int
d\vec{x}\,d\vec{x}^{\prime}d\vec{q}\,{\mathsf{M}}_{ac_{1}}^{n}{\mathsf{M}}_{c_{2}c}^{n}{\mathsf{M}}_{dd_{1}}^{n}{\mathsf{M}}_{d_{2}b}^{n}$
$\displaystyle\hskip 10.00002pt\times
e^{-i\vec{k}\cdot\vec{x}}\frac{\partial}{\partial
x_{c_{1}}}\delta(\vec{x}-\vec{X}^{n}_{0})(-iq_{c_{2}})e^{-i\vec{q}\cdot\vec{X}_{0}^{n}}G_{cd}^{0}(\vec{q})e^{i\vec{q}\cdot\vec{x}^{\prime}}\frac{\partial}{\partial
x^{\prime}_{d_{1}}}\delta(\vec{x}^{\prime}-\vec{X}_{0}^{n})(ik^{\prime}_{d_{2}})e^{i\vec{k}^{\prime}\cdot\vec{X}_{0}^{n}}$
$\displaystyle=$ $\displaystyle\frac{(\mu{\mathcal{A}})^{2}}{(2\pi)^{2}}\int
d\vec{q}\,{\mathsf{M}}_{ac_{1}^{n}}{\mathsf{M}}_{c_{2}c}^{n}{\mathsf{M}}_{dd_{1}}^{n}{\mathsf{M}}_{d_{2}b}^{n}k^{\prime}_{d_{2}}k_{c_{1}}q_{c_{2}}q_{d_{1}}G_{cd}^{0}(\vec{q})e^{i\left(\vec{k}^{\prime}-\vec{k}\right)\cdot\vec{X}^{n}_{0}}$
$\displaystyle=$
$\displaystyle(\mu{\mathcal{A}})^{2}{\mathsf{M}}_{ac_{1}}^{n}{\mathsf{M}}_{c_{2}c}^{n}{\mathsf{M}}_{dd_{1}}^{n}{\mathsf{M}}_{d_{2}b}^{n}k^{\prime}_{d_{2}}k_{c_{1}}e^{i\left(\vec{k}^{\prime}-\vec{k}\right)\cdot\vec{X}^{n}_{0}}\frac{1}{2\pi}\int
dq\,q^{3}$ $\displaystyle\left(\frac{\delta_{cd}}{\rho
c_{T}^{2}(q^{2}-k_{T}^{2})}\frac{1}{2\pi}\int\limits_{0}^{2\pi}\hat{q}_{c_{2}}\hat{q}_{d_{1}}d\theta+\left(\frac{1}{\rho
c_{L}^{2}(q^{2}-k_{L}^{2})}-\frac{1}{\rho
c_{T}^{2}(q^{2}-k_{T}^{2})}\right)\frac{1}{2\pi}\int\limits_{0}^{2\pi}\hat{q}_{c_{2}}\hat{q}_{d_{1}}\hat{q}_{c}\hat{q}_{d}d\theta\right)$
$\displaystyle=$
$\displaystyle(\mu{\mathcal{A}})^{2}{\mathsf{M}}_{ac_{1}}^{n}{\mathsf{M}}_{c_{2}c}^{n}{\mathsf{M}}_{dd_{1}}^{n}{\mathsf{M}}_{d_{2}b}^{n}k^{\prime}_{d_{2}}k_{c_{1}}e^{i\left(\vec{k}^{\prime}-\vec{k}\right)\cdot\vec{X}^{n}_{0}}\frac{1}{2\pi}\int
dq\,q^{3}$ $\displaystyle\left(\frac{\delta_{cd}\delta_{c_{2}d_{1}}}{2\rho
c_{T}^{2}(q^{2}-k_{T}^{2})}+\left(\frac{1}{\rho
c_{L}^{2}(q^{2}-k_{L}^{2})}-\frac{1}{\rho
c_{T}^{2}(q^{2}-k_{T}^{2})}\right)\frac{1}{8}\left(\delta_{c_{2}d_{1}}\delta_{cd}+\delta_{c_{2}c}\delta_{d_{1}d}+\delta_{c_{2}d}\delta_{d_{1}c}\right)\right)$
$\displaystyle=$
$\displaystyle(\mu{\mathcal{A}}){\mathsf{M}}_{ac_{1}}^{n}{\mathsf{M}}_{d_{2}b}^{n}k^{\prime}_{d_{2}}k_{c_{1}}e^{i\left(\vec{k}^{\prime}-\vec{k}\right)\cdot\vec{X}^{n}_{0}}\frac{1}{4\pi}\int
dq\,q^{3}\left(\frac{1}{\rho c_{T}^{2}(q^{2}-k_{T}^{2})}+\frac{1}{\rho
c_{L}^{2}(q^{2}-k_{L}^{2})}\right)$ $\displaystyle=$
$\displaystyle(\mu{\mathcal{A}})^{2}{\mathsf{M}}_{ac_{1}}^{n}k_{c_{1}}I{\mathsf{M}}_{d_{2}b}^{n}k^{\prime}_{d_{2}}e^{i\left(\vec{k}^{\prime}-\vec{k}\right)\cdot\vec{X}^{n}_{0}}\,,$

where

$\displaystyle I=\frac{1}{4\pi}\int dq\,q^{3}\left(\frac{1}{\rho
c_{T}^{2}(q^{2}-k_{T}^{2})}+\frac{1}{\rho c_{L}^{2}(q^{2}-k_{L}^{2})}\right).$
(121)

We notice that, to get the result of Eqn.(B) we utilized explicit expressions
for the potential and the free Green function from Eqns. (11) and (18),
respectively, along with Eqn.(115) from A and properties of ${\mathsf{M}}$:

${\mathsf{M}}_{mm}=0\,,\hskip
20.00003pt{\mathsf{M}}_{ab}{\mathsf{M}}_{ab}={\mathsf{M}}_{ba}{\mathsf{M}}_{ba}=2\,.$
(122)

.

We notice that the integral (121 diverges due to the use of continuum
mechanics: there is no intrinsic length scale associated with the elastic
medium and all wavelengths, including infinitely short , are included. To
regularize this situation we proceed as in [29] and introduce a short distance
cut-off $\Lambda$. With this regularization procedure we obtain

$I=\frac{1}{8\pi\rho}\Lambda^{2}\left(\frac{1}{c_{L}^{2}}+\frac{1}{c_{T}^{2}}\right)+\frac{i}{8\rho}\omega^{2}\left(\frac{1}{c_{L}^{4}}+\frac{1}{c_{T}^{4}}\right)$
(123)

The third order Born approximation is given by

$t^{n(3)}_{ab}(\vec{k},\vec{k}^{\prime})=\int
d\vec{x}\,d\vec{x}^{\prime}d\vec{x}^{\prime\prime}\;e^{-i\vec{k}\cdot\vec{x}}V_{ac}(\vec{x})G^{0}_{cd}(\vec{x}-\vec{x}^{\prime})V_{de}(\vec{x}^{\prime})G^{0}_{ef}(\vec{x}^{\prime}-\vec{x}^{\prime\prime})V_{fb}(\vec{x}^{\prime\prime})e^{i\vec{k}^{\prime}\cdot\vec{x}^{\prime\prime}}\,.$
(124)

Proceeding in the same way as for the second order yields:

${t^{n(3)}_{ab}(\vec{k},\vec{k}^{\prime})=-(\mu{\mathcal{A}})^{3}{\mathsf{M}}_{ac}k_{c}I^{2}{\mathsf{M}}_{db}^{n}k^{\prime}_{d}e^{i\left(\vec{k}^{\prime}-\vec{k}\right)\cdot\vec{X}^{n}_{0}}\,.}$
(125)

Based on Eqns.(119, B, 125), summation of the corresponding geometrical series
for $t^{n}_{ab}(\vec{k},\vec{k}^{\prime})$ can be easily performed with the
result

$\displaystyle t^{n}_{ab}(\vec{k},\vec{k}^{\prime})$ $\displaystyle=$
$\displaystyle
t^{n(1)}_{ab}(\vec{k},\vec{k}^{\prime})+t^{n(2)}_{ab}(\vec{k},\vec{k}^{\prime})+t^{n(3)}_{ab}(\vec{k},\vec{k}^{\prime})+....$
$\displaystyle=$
$\displaystyle{\frac{-\mu{\mathcal{A}}}{1+\mu{\mathcal{A}}I}{\mathsf{M}}_{ac}^{n}k_{c}{\mathsf{M}}_{db}^{n}k^{\prime}_{d}e^{i\left(\vec{k}^{\prime}-\vec{k}\right)\cdot\vec{X}^{n}_{0}}\,}.$

Given that Eqns.(23, B, 116), we have for the mass operator

$\displaystyle\Sigma_{ab}(\vec{k})$ $\displaystyle=$ $\displaystyle
n\int<t^{n}_{ab}(\vec{k},\vec{k}^{\prime})>d\vec{X}^{n}_{0}$ $\displaystyle=$
$\displaystyle-n\mu\frac{{\mathcal{A}}}{1+\mu{\mathcal{A}}I}k_{c}k_{d}\frac{\left(\delta_{ad}\delta_{cb}+\delta_{ab}\delta_{cd}-\delta_{ac}\delta_{db}\right)}{2}$
$\displaystyle=$
$\displaystyle-n\mu\frac{1}{2}\frac{{\mathcal{A}}}{1+\mu{\mathcal{A}}I}k^{2}\delta_{ab}\hskip
20.00003pt$ $\displaystyle=$
$\displaystyle-\frac{n\mu^{2}b^{2}}{2M}\frac{k^{2}\delta_{ab}}{\omega^{2}-\omega^{2}_{0R}+i(\omega
B/M+\omega^{2}\kappa^{2})}$ (128)

where

$\omega_{0R}^{2}\equiv\omega_{0}^{2}-\frac{\mu b^{2}}{8\pi
M}\left(\frac{c_{T}^{2}}{c_{L}^{2}}+1\right)\Lambda^{2}$ (129)

is a renormalized frequency of dislocation oscillation about its equilibrium
position, and

$\kappa^{2}\equiv\frac{\mu
b^{2}}{8Mc_{T}^{2}}\left(\frac{c_{T}^{4}}{c_{L}^{4}}+1\right)\,.$ (130)

is a dimensionless constant. In arriving at (128) we have used (123).

##  Appendix C Optical theorem

Here we show that Eqn. (60)

$\displaystyle\left(\Sigma^{\ast}_{bc}({\bf k})-\Sigma_{bc}({\bf
k})\right)={\int\limits_{\mathbf{k}_{1}}}\left(G^{0\ast}(\vec{k}_{1})_{gh}-G^{0}(\vec{k}_{1})_{gh}\right)K_{gh,bc}({\bf
k}_{1},{\bf k})$

holds, in the ISA, when $B=0$ . That is, when the mass and irreducible vertex
operators are given in terms of the $t$ matrix by (IV.1.3), and the $t$ matrix
itself is given by (B). Consider the left-hand-side first:

$\displaystyle\left(\Sigma^{\ast}_{bc}({\bf k})-\Sigma_{bc}({\bf k})\right)$
$\displaystyle=$ $\displaystyle
Im\left[\frac{n\mu{\mathcal{A}}}{1+\mu{\mathcal{A}}I}\right]k^{2}\delta_{dc}$
(131) $\displaystyle=$ $\displaystyle
Im\left[\frac{n\mu{\mathcal{A}}\left(1+\mu{\mathcal{A}}I^{\ast}\right)}{\left(1+\mu{\mathcal{A}}I\right)\left(1+\mu{\mathcal{A}}I^{\ast}\right)}\right]$
(132) $\displaystyle=$
$\displaystyle-\left[\frac{n\mu^{2}{\mathcal{A}}^{2}}{\left(1+\mu{\mathcal{A}}I\right)\left(1+\mu{\mathcal{A}}I^{\ast}\right)}\right]Im[I]$
$\displaystyle=$
$\displaystyle-\left[\frac{n\mu^{2}{\mathcal{A}}^{2}}{\left(1+\mu{\mathcal{A}}I\right)\left(1+\mu{\mathcal{A}}I^{\ast}\right)}\right]Im\left[\frac{1}{4\pi}\int
dq\,q^{3}\left(\frac{1}{\rho c_{T}^{2}(q^{2}-k_{T}^{2})}+\frac{1}{\rho
c_{L}^{2}(q^{2}-k_{L}^{2})}\right)\right]$ $\displaystyle=$
$\displaystyle-\left[\frac{n\mu^{2}{\mathcal{A}}^{2}}{\left(1+\mu{\mathcal{A}}I\right)\left(1+\mu{\mathcal{A}}I^{\ast}\right)}\right]\frac{1}{4}\int
dq\,q^{3}\left(\frac{\delta\left(q^{2}-k^{2}_{T}\right)}{\rho
c^{2}_{T}}+\frac{\delta\left(q^{2}-k^{2}_{L}\right)}{\rho c^{2}_{L}}\right)$
(134)

And now the right-hand-side:

$\displaystyle{\int\limits_{\mathbf{k}_{1}}}\left(G^{0\ast}(\vec{k}_{1})_{gh}-G^{0}(\vec{k}_{1})_{gh}\right)K_{gh,bc}({\bf
k}_{1},{\bf k})$ (136)
$\displaystyle=-\frac{n}{2\pi}\int\limits_{k_{1}}\int\limits_{\hat{\bf
k}_{1}}k^{3}_{1}\left(\frac{\delta\left(k^{2}_{1}-k^{2}_{T}\right)}{\rho
c^{2}_{T}}\delta_{ab}\hat{k}_{1g}\hat{k}_{1f}+\left(\frac{\delta\left(k^{2}_{1}-k^{2}_{L}\right)}{\rho
c^{2}_{L}}-\frac{\delta\left(k^{2}_{1}-k^{2}_{T}\right)}{\rho
c^{2}_{T}}\right)\hat{k}_{1a}\hat{k}_{1b}\hat{k}_{1g}\hat{k}_{1f}\right)$
$\displaystyle\times\frac{\mu{\mathcal{A}}}{1+\mu{\mathcal{A}}I}\left(\frac{\mu{\mathcal{A}}}{1+\mu{\mathcal{A}}I}\right)^{\ast}<{\mathsf{M}}_{ag}^{n}{\mathsf{M}}_{fb}^{n}{\mathsf{M}}_{hc}^{n}{\mathsf{M}}_{de}^{n}>k_{h}k_{e}$
$\displaystyle=-n\int\limits_{k_{1}}k^{3}_{1}\left(\frac{\delta\left(k^{2}_{1}-k^{2}_{T}\right)}{\rho
c^{2}_{T}}\delta_{ab}\frac{\delta_{gf}}{2}+\left(\frac{\delta\left(k^{2}_{1}-k^{2}_{L}\right)}{\rho
c^{2}_{L}}-\frac{\delta\left(k^{2}_{1}-k^{2}_{T}\right)}{\rho
c^{2}_{T}}\right)\frac{\delta_{ab}\delta_{gf}+\delta_{ag}\delta_{bf}+\delta_{af}\delta_{bg}}{8}\right)$
$\displaystyle\times\frac{\mu{\mathcal{A}}}{1+\mu{\mathcal{A}}I}\left(\frac{\mu{\mathcal{A}}}{1+\mu{\mathcal{A}}I}\right)^{\ast}<{\mathsf{M}}_{ag}^{n}{\mathsf{M}}_{fb}^{n}{\mathsf{M}}_{hc}^{n}{\mathsf{M}}_{de}^{n}>k_{h}k_{e}$
$\displaystyle=-n\int\limits_{k_{1}}\frac{k^{3}_{1}}{2}\left(\frac{\delta\left(k^{2}_{1}-k^{2}_{T}\right)}{\rho
c^{2}_{T}}+\frac{\delta\left(k^{2}_{1}-k^{2}_{L}\right)}{\rho
c^{2}_{L}}\right)\frac{\mu{\mathcal{A}}}{1+\mu{\mathcal{A}}I}\left(\frac{\mu{\mathcal{A}}}{1+\mu{\mathcal{A}}I}\right)^{\ast}<{\mathsf{M}}_{hc}^{n}{\mathsf{M}}_{de}^{n}>k_{h}k_{e}$

which, using (116) is easily seen to be equal to (134).

##  Appendix D Solution for $f^{1\mathbf{q}}({\bf k})$

To find $f^{1\mathbf{q}}({\bf k})$ we need (81). Using (83) its second term
can be rewritten as

$\displaystyle\int\limits_{\bf{k}^{\prime\prime}}H^{1\mathbf{q}}({\bf k},{\bf
k}^{\prime\prime}){\cal{B}}\Delta G({\bf k}^{\prime\prime})$ $\displaystyle=$
$\displaystyle\int\limits_{\bf{k}^{\prime\prime}}{\cal{B}}\left(P(\mathbf{q};{\bf
k}^{\prime\prime})\delta_{\mathbf{k}^{\prime\prime},\mathbf{k}}\Delta G({\bf
k})-H({\bf k},{\bf k}^{\prime\prime})\Delta G^{1\mathbf{q}}({\bf
k}^{\prime\prime})\right)$ $\displaystyle=$
$\displaystyle{\cal{B}}\int\limits_{\bf{k}^{\prime\prime}}H({\bf k},{\bf
k}^{\prime\prime})[\int\limits_{{\bf k}_{2}}\Phi({\bf k}^{\prime\prime},{\bf
k}_{2})P(\mathbf{q};{\bf k}_{2})-\Delta G^{1\mathbf{q}}({\bf
k}^{\prime\prime})]\,.$

Where the first equality is a consequence of the symmetry property from Eqn.
(73) and application the WTI to $H^{1\mathbf{q}}({\bf k},{\bf
k}^{\prime\prime})$. The second equality is a result of substituting
$\delta_{\mathbf{k}^{\prime\prime},\mathbf{k}}\Delta G({\bf k})$ by its value
given by (40). Hence,

$f^{1\mathbf{q}}({\bf k}^{\prime\prime})=-{\cal{B}}\left(\int\limits_{{\bf
k}_{2}}\Phi({\bf k}^{\prime\prime},{\bf k}_{2})P(\mathbf{q};{\bf
k}_{2})-\Delta G^{1\mathbf{q}}({\bf k}^{\prime\prime})\right)$ (137)

with

$\displaystyle\Delta G^{1\mathbf{q}}_{cd,ee}({\bf k})$ $\displaystyle=$
$\displaystyle{\bf q}\cdot\frac{\partial\Delta G_{cd,ee}({\bf k};{\bf
q}^{\prime},0)}{\partial{\bf q}^{\prime}}|_{{\bf q}^{\prime}=0}$ (138)
$\displaystyle=$ $\displaystyle-\frac{1}{2\imath\rho}{\bf
q}\cdot\frac{\partial\left(Re[G_{cd}(\mathbf{k})]\right)}{\partial{\bf k}}$
(139) $\displaystyle=$
$\displaystyle-\frac{q_{h}Re[G_{L}-G_{T}]}{2\imath\rho}\frac{\partial
P_{\textbf{\^{k}}}}{\partial
k_{h}}-\frac{q_{h}}{2\imath\rho}\left(\frac{\partial\left(Re[G_{T}]\right)}{\partial
k_{h}}\left(\textbf{I}-P_{\textbf{\^{k}}}\right)+\frac{\partial\left(Re[G_{L}]\right)}{\partial
k_{h}}P_{\textbf{\^{k}}}\right)$ (140)

and

$\displaystyle\frac{\partial P_{\textbf{\^{k}}}}{\partial k_{h}}$
$\displaystyle=$
$\displaystyle\frac{\partial\left(\frac{k_{c}k_{d}}{k^{2}}\right)}{\partial
k_{h}}=\left(\frac{k_{d}\delta_{ch}+k_{c}\delta_{dh}}{k^{2}}\right)-\frac{2k_{c}k_{d}k_{h}}{k^{4}}$
(141) $\displaystyle Re[G_{T,L}]$ $\displaystyle=$
$\displaystyle\frac{F_{T,L}(\omega,k)}{\rho\omega^{2}Im[K_{T,L}^{2}]}\left(Re[K_{T,L}^{2}]\left(k^{2}-Re[K_{T,L}^{2}]\right)-Im[K_{T,L}^{2}]^{2}\right)$
(142) $\displaystyle\frac{\partial\left(Re[G_{T,L}]\right)}{\partial k_{g}}$
$\displaystyle=$
$\displaystyle\frac{2k_{g}F_{T,L}(\omega,k)}{\rho\omega^{2}Im[K_{T,L}^{2}]}\left(2k^{2}F_{T,L}(\omega,k)Im[K_{T,L}^{2}]-Re[K_{T,L}^{2}]\right)$
(143) $\displaystyle F_{T,L}(\omega,k)$ $\displaystyle=$
$\displaystyle\left(\frac{Im[K_{T,L}^{2}]}{\left(k^{2}-Re[K_{T,L}^{2}]\right)^{2}+Im[K_{T,L}^{2}]^{2}}\right)$
(144)

##  Appendix E Calculation of $D_{\Delta G^{1\mathbf{q}}}$

From Eqn. (89) we have

$\displaystyle D_{\Delta G^{1\mathbf{q}}}$ $\displaystyle=$
$\displaystyle\frac{-{\cal{B}}^{2}}{q^{2}\omega\left(1+a\right)}\int\limits_{\bf{k}}P_{aa,cd}({\bf
k};{\bf q})\Delta G_{cd,ee}^{1\mathbf{q}}({\bf k})$ $\displaystyle=$
$\displaystyle\frac{{\cal{B}}^{2}}{4\rho^{2}q^{2}\omega\left(1+a\right)}\int\limits_{\bf{k}}q_{h}\frac{\partial
L_{dc}({\bf k})}{\partial
k_{h}}\frac{\partial\left(Re[G^{-}_{cd}(\mathbf{k})]\right)}{\partial
k_{g}}q_{g}$ $\displaystyle=$
$\displaystyle\frac{-{\cal{B}}^{2}q_{h}q_{g}\left(c_{L}^{2}-c_{T}^{2}\right)}{2\rho
q^{2}\omega\left(1+a\right)}\int\limits_{\bf{k}}\left(Re[G_{L}-G_{T}]\left(\delta_{hg}-\frac{k_{h}k_{g}}{k^{2}}\right)\right)$
$\displaystyle\hskip 20.00003pt+\frac{-B^{2}q_{h}q_{g}}{2\rho
q^{2}\omega\left(1+a\right)}\int\limits_{\bf{k}}\left(c_{T}^{2}\frac{\partial\left(Re[G_{T}]\right)}{\partial
k_{g}}+c_{L}^{2}\frac{\partial\left(Re[G_{L}]\right)}{\partial
k_{g}}\right)k_{h}$

where Eqn. (E) has been obtained using Eqns. (42) and (138, 141).

In Eqn. (E) we have to deal with the following two types of integrals

$\displaystyle\mathbb{I}^{hg}_{T,L}$ $\displaystyle=$
$\displaystyle\int\limits_{\bf{k}}Re[G_{T,L}]\left(\delta_{hg}-\frac{k_{h}k_{g}}{k^{2}}\right)=\frac{\delta_{hg}}{4\pi}\int\limits_{-\infty}^{\infty}k\Theta(k)Re[G_{T,L}]dk$
(146) $\displaystyle\mathbb{J}^{hg}_{T,L}$ $\displaystyle=$
$\displaystyle\int\limits_{\bf{k}}\left(\frac{\partial\left(Re[G_{T,L}]\right)}{\partial
k_{g}}\right)k_{h}$

Then, using Eqn. (141) we can write

$\displaystyle\mathbb{I}^{hg}_{T,L}$ $\displaystyle=$
$\displaystyle\int\limits_{-\infty}^{\infty}\frac{\delta_{hg}k\Theta(k)F_{T,L}(\omega,k)\left(Re[K_{T,L}^{2}]\left(k^{2}-Re[K_{T,L}^{2}]\right)-Im[K_{T,L}^{2}]^{2}\right)}{4\pi\rho\omega^{2}Im[K_{T,L}^{2}]}dk$
(147) $\displaystyle\mathbb{J}^{hg}_{T,L}$ $\displaystyle=$
$\displaystyle\int\limits_{\bf{k}}\frac{2k_{g}k_{h}\left(2k^{2}F_{T,L}^{2}(\omega,k)Im[K_{T,L}^{2}]-Re[K_{T,L}^{2}]F_{T,L}(\omega,k)\right)}{\rho\omega^{2}Im[K_{T,L}^{2}]}$
$\displaystyle=$
$\displaystyle\int\limits_{-\infty}^{\infty}\delta_{gh}\Theta(k)k^{3}dk\left(\frac{2k^{2}F_{T,L}^{2}(\omega,k)Im[K_{T,L}^{2}]-Re[K_{T,L}^{2}]F_{T,L}(\omega,k)}{2\pi\rho\omega^{2}Im[K_{T,L}^{2}]}\right).$

The expression for $\mathbb{J}^{hg}_{T,L}$ in Eqn. (147) contains an ill-
defined term in the integrand, proportional to $F_{T,L}^{2}(\omega,k)$, that
can be regularized as follows [51]: Introduce a new variable
$x_{T,L}\equiv\left(k^{2}-Re[K_{T,L}^{2}]\right)$, and consider the following
auxiliary integrals:

$\displaystyle\int\limits_{-\infty}^{\infty}\left(\frac{Im[K_{T,L}^{2}]}{x_{T,L}^{2}+Im[K_{T,L}^{2}]^{2}}\right)dx_{T,L}$
$\displaystyle=$ $\displaystyle\pi,$ (148)
$\displaystyle\int\limits_{-\infty}^{\infty}\left(\frac{Im[K_{T,L}^{2}]}{x_{T,L}^{2}+Im[K_{T,L}^{2}]^{2}}\right)^{2}dx_{T,L}$
$\displaystyle=$ $\displaystyle\frac{\pi}{2Im[K_{T,L}^{2}]}.$

They show that it is possible to make the replacements

$\displaystyle\left(\frac{Im[K_{T,L}^{2}]}{x_{T,L}^{2}+Im[K_{T,L}^{2}]^{2}}\right)$
$\displaystyle\longrightarrow$ $\displaystyle\pi\delta(x_{T,L}),$ (149)
$\displaystyle\left(\frac{Im[K_{T,L}^{2}]}{x_{T,L}^{2}+Im[K_{T,L}^{2}]^{2}}\right)^{2}$
$\displaystyle\longrightarrow$
$\displaystyle\frac{\pi\delta(x_{T,L})}{2Im[K_{T,L}^{2}]}.$

which yield the same result after integration over $x_{T,L}$. Moreover, both
replacements give the proper asymptotic behaviour in the low-density limit
(our case) when $Im[K_{T,L}^{2}]\rightarrow 0$ or, more precisely, when
$|Im[K_{T,L}^{2}]|\ll|k^{2}-Re[K_{T,L}^{2}]|$. Therefore, on the basis of
Eqns. (141, 149) we have

$\displaystyle F_{T,L}(\omega,k)$ $\displaystyle=$
$\displaystyle\pi\delta\left(k^{2}-Re[K_{T,L}^{2}]\right),$ (150)
$\displaystyle F_{T,L}(\omega,k)^{2}$ $\displaystyle=$
$\displaystyle\frac{\pi\delta\left(k^{2}-Re[K_{T,L}^{2}]\right)}{2Im[K_{T,L}^{2}]}$

Technically, Eqn. (150) indicates that in all final expressions the limit
${Im[K_{T,L}^{2}]\rightarrow 0}$. The implications of this restriction are
discussed in Section V.1.

From Eqns. (E), (146), (147), and (150) it follows that

$\displaystyle\mathbb{I}^{hg}_{T,L}$ $\displaystyle=$
$\displaystyle\frac{-\delta_{hg}Im[K_{T,L}^{2}]}{8\rho\omega^{2}},\quad\mathbb{J}^{hg}=0\Rightarrow$
$\displaystyle D_{\Delta G^{1\mathbf{q}}}$ $\displaystyle=$
$\displaystyle\frac{{\cal{B}}^{2}\left(c_{L}^{2}-c_{T}^{2}\right)\left(Im[K_{L}^{2}]-Im[K_{T}^{2}]\right))}{16\rho^{2}\omega^{3}\left(1+a\right)}\,.$
(151)

##  Appendix F Improved diffusion constant for the case of screw dislocations

The application of the technique for WTI derivation, which is presented in
this paper, to the case of screw dislocations requires respective modification
of the result [29] for WTI, and, consequently, diffusion constant. Indeed,
given that Eqns. (6, 7, 11) from [29], for screw dislocations, the system of
equations similar to Eqns. (IV.1.2, IV.1.2) can be written as

$\displaystyle\rho\omega_{1}^{2}G(\vec{x}_{1},\vec{x}^{\prime}_{1},\omega_{1})+\mu\nabla^{2}_{1}G(\vec{x}_{1},\vec{x}^{\prime}_{1},\omega_{1})$
$\displaystyle=$ $\displaystyle-\mu
V(\vec{x}_{1},\omega_{1})G(\vec{x}_{1},\vec{x}^{\prime}_{1},\omega_{1})-\delta(\vec{x}_{1}-\vec{x}^{\prime}_{1})$
(152)
$\displaystyle\rho\omega_{2}^{2}G(\vec{x}_{2},\vec{x}^{\prime}_{2},\omega_{2})+\mu\nabla^{2}_{2}G(\vec{x}_{2},\vec{x}^{\prime}_{2},\omega_{2})$
$\displaystyle=$ $\displaystyle-\mu
V(\vec{x}_{2},\omega_{2})G(\vec{x}_{2},\vec{x}^{\prime}_{2},\omega_{2})-\delta(\vec{x}_{2}-\vec{x}^{\prime}_{2})\,.$
(153)

with

$\left.V({\bf x},\omega)={\cal
A}\sum\limits_{n=1}^{N}\;\frac{\partial}{\partial
x_{a}}\delta(\vec{x}-\vec{X}^{n}_{0})\;{\frac{\partial}{\partial
x_{a}}}\right|_{\vec{x}=\vec{X}^{n}_{0}}$ (154)

and

${\cal A}=\frac{\mu
b^{2}}{M}\frac{1}{\omega^{2}{+}i\omega(B/M)-\omega^{2}_{0}}=\frac{\mu
b^{2}}{M}g(\omega)$ (155)

where $g(\omega)\equiv\frac{1}{\omega^{2}{+}i\omega(B/M)-\omega^{2}_{0}}$.
Subscripts $1,2$ at $\nabla^{2}_{1,2}$ indicate action of operator
$\nabla^{2}$ on components of vectors $\vec{x}_{1}$ and $\vec{x}_{2}$,
respectively. Again, we consider all dislocations have a Burgers vector of the
same magnitude, but possibly different sign. Finally, we explicitly introduced
$g(\omega)$ here, which is formally the same as for edge dislocations (see
notation right after Eqn. (IV.1.2)), but it is completely different from
$g(\omega)$ for screw dislocations that was introduced in [29].

In fact, the way of the potential subtraction is the reason for such
difference. Namely, in [29], we initially integrated out the potential (see
Eqns. (45, 46) in [29]) instead of following the substraction procedure, that
is outlined in Section IV.1.2 for the case of edge dislocations and getting
somewhat similar to Eqn. (55). The former approach is independent of a
particular form of frequency dependence of the potential, i. e., $g(\omega)$
is reduced to $\omega^{2}$, which, in fact, is a frequency dependence of the
free medium Green function, whereas the latter approach is still capturing a
particular frequency dependence of the potential, so that $g(\omega)$ is
identified through it. Therefore, the second approach is considered to be more
precise, say differentiated, whereas the first one is integral and, as a
consequence of that, it is losing a crucially important information that
specifies frequency dependence of a particular potential. And this Appendix is
aimed to get an improved WTI and, consequently, diffusion constant within the
second approach adapted for the case of screw dislocations.

The adaptation is essentially passing through steps similar to Eqns.
(IV.1.2-55), but with system of Eqns. (152, 153) that includes a scalar
potential $\mu V(\vec{x},\omega)$ and Green function $G$. To be more specific,
with definitions from Eqns. (45, 46) used in Eqns. (152, 153) one can get

$\displaystyle-(G^{0})^{-1}(\vec{k}_{1},\omega_{1})\delta_{\vec{k}_{1},\vec{k}^{\prime\prime}_{1}}g^{*}(\omega_{2})+g^{*}(\omega_{2})(G)^{-1}(\vec{k}_{1},\vec{k}^{\prime\prime}_{1};\omega_{1}$
$\displaystyle=$ $\displaystyle
g(\omega_{1})g^{*}(\omega_{2})\sum\limits_{n=1}^{N}\frac{\mu^{2}b^{2}}{M}k_{h_{1}}k^{\prime\prime}_{h_{1}}e^{i\left(\vec{k}^{\prime\prime}_{1}-\vec{k}_{1}\right)\cdot\vec{X}^{n}_{0}}$
$\displaystyle-(G^{0*})^{-1}(\vec{k}_{2},\omega_{2})\delta_{\vec{k}_{2},\vec{k}^{\prime\prime}_{2}}g(\omega_{1})+g(\omega_{1})(G^{*})^{-1}(\vec{k}_{2},\vec{k}^{\prime\prime}_{2};\omega_{2})$
$\displaystyle=$ $\displaystyle
g(\omega_{1})g^{*}(\omega_{2})\sum\limits_{n=1}^{N}\frac{\mu^{2}b^{2}}{M}k_{h_{2}}k^{\prime\prime}_{h_{2}}e^{-i\left(\vec{k}^{\prime\prime}_{2}-\vec{k}_{2}\right)\cdot\vec{X}^{n}_{0}}$

with

$\displaystyle(G^{0})^{-1}(\vec{k},\omega)=-\left(\rho\omega^{2}-\mu
k^{2}\right).$ (158)

It should be noted that Eqns. (F) is formally a scalar version of Eqns.
(IV.1.2). Then, the next step is the substraction at $B=0$
($g(\omega)=g^{*}(\omega)$) of both equations from each other keeping the
following limit: $\vec{k}_{2}\rightarrow\vec{k}^{\prime\prime}_{1}$,
$\vec{k}^{\prime\prime}_{2}\rightarrow\vec{k}_{1}$. This results in the
expression

$\displaystyle\lim\limits_{{\begin{subarray}{c}\vec{k}_{2}\rightarrow\vec{k}^{\prime\prime}_{1}\\\
\vec{k}^{\prime\prime}_{2}\rightarrow\vec{k}_{1}\end{subarray}}}\left(-(G^{0})^{-1}(\vec{k}_{1},\omega_{1})\delta_{\vec{k}_{1},\vec{k}^{\prime\prime}_{1}}g^{*}(\omega_{2})+g^{*}(\omega_{2})(G)^{-1}(\vec{k}_{1},\vec{k}^{\prime\prime}_{1};\omega_{1})+\right.$
$\displaystyle\left.(G^{*0})^{-1}(\vec{k}_{2},\omega_{2})\delta_{\vec{k}_{2},\vec{k}^{\prime\prime}_{2}}g(\omega_{1})-g(\omega_{1})(G^{*})^{-1}(\vec{k}_{2},\vec{k}^{\prime\prime}_{2};\omega_{2})\right)$
$\displaystyle\equiv$ $\displaystyle 0$

Again, by acting on the identity with
$\lim\limits_{{\begin{subarray}{c}\vec{k}_{2}\rightarrow\vec{k}^{\prime\prime}_{1}\\\
\vec{k}^{\prime\prime}_{2}\rightarrow\vec{k}_{1}\end{subarray}}}G(\vec{k}^{\prime\prime}_{1},\vec{k}^{\prime\prime\prime}_{1};\omega_{1})G^{*}(\vec{k}^{\prime\prime}_{2},\vec{k}^{\prime\prime\prime}_{2};\omega_{2})$
from the right, averaging out the random variables, and introducing notations
from Eqn. (52), we get

$\displaystyle-<G^{-}(\vec{k}^{\prime\prime\prime-},\vec{k}^{-};\omega_{-})(G^{0+})^{-1}(\vec{k}^{+},\omega_{+})G^{+}(\vec{k}^{+},\vec{k}^{\prime\prime\prime+};\omega_{+})>g^{\ast}(\omega_{-})\hskip
20.00003pt$
$\displaystyle+g^{\ast}(\omega_{-})<\delta_{\vec{k},\vec{k}^{\prime\prime\prime}}G^{-}(\vec{k}^{-},\vec{k}^{\prime\prime\prime-};\omega_{-})>$
$\displaystyle+<G^{-}(\vec{k}^{\prime\prime\prime-},\vec{k}^{\prime\prime-};\omega_{-})(G^{0-})^{-1}(\vec{k}^{\prime\prime-},\omega_{-})G^{+}(\vec{k}^{\prime\prime+},\vec{k}^{\prime\prime\prime+};\omega_{+})g(\omega_{+})>\hskip
20.00003pt$
$\displaystyle-g(\omega_{+})<\delta_{\vec{k}^{\prime\prime},\vec{k}^{\prime\prime\prime}}G^{+}(\vec{k}^{\prime\prime+},\vec{k}^{\prime\prime\prime+};\omega_{+})>$
$\displaystyle\equiv$ $\displaystyle 0$ (159)

Eventually, this expression provides us with a final form for pre-WTI

$\displaystyle\left((G^{0-})^{-1}({\bf k};{\bf
q},\Omega)g(\omega_{+})-(G^{0+})^{-1}({\bf k};{\bf
q},\Omega)g^{\ast}(\omega_{-})\right)\Phi({\bf k},{\bf
k}^{\prime\prime\prime};{\bf q},\Omega)+$ (160) $\displaystyle
g^{\ast}(\omega_{-})<G>^{-}(\vec{k}^{\prime\prime\prime};{\bf
q},\Omega)-g(\omega_{+})<G>^{+}(\vec{k}^{\prime\prime\prime};{\bf q},\Omega)$
$\displaystyle\equiv$ $\displaystyle 0,$

with notations

$\displaystyle(G^{0\pm})^{-1}(\vec{k}^{\pm},\omega_{\pm})$ $\displaystyle=$
$\displaystyle(G^{0\pm})^{-1}({\bf k};{\bf q},\Omega)$ (161)
$\displaystyle<G^{\pm}(\vec{k}^{\prime\prime\prime\pm},\vec{k}^{\prime\prime\prime\pm};\omega_{\pm})>$
$\displaystyle=$ $\displaystyle<G>^{\pm}(\vec{k}^{\prime\prime\prime};{\bf
q},\Omega)$ $\displaystyle\Phi({\bf k},{\bf k}^{\prime};{\bf q},\Omega)$
$\displaystyle\equiv$
$\displaystyle<G^{+}(\mathbf{k}^{+},\mathbf{k}^{\prime\prime\prime+},\omega^{+})G^{-}(\mathbf{k}^{\prime\prime\prime-},\mathbf{k}^{-},\omega^{-})>$

The form of pre-WTI for screw dislocations presented by Eqn. (160) is formally
considered to be a scalar version of pre-WTI for edge dislocations Eqn. (55)
and would have reduced to the case of pre-WTI for screw dislocations published
in [29] if $g(\omega)$ had been equal to $\omega^{2}$. This connection
suggests the following logic to get an expression for diffusion constant $D$:
Scalar versions of Eqns. (64, 89) for WTI and diffusion constant for the case
of edge dislocations, are identical to corresponding relations published in
[29] (see, Eqns. (54, 79) therein), but with a new $g(\omega)$ presented in
this paper, and, consequently, with a new $a$ defined by a scalar version of
Eqn. (86). Consequently, the diffusion constant can be explicitly written as
follows

$\displaystyle D$ $\displaystyle\equiv$
$\displaystyle-\frac{i\Omega\lambda^{2\mathbf{q}}}{q^{2}\lambda^{1\Omega}}=-\frac{{\cal{B}}}{q^{2}\omega\left(1+a\right)}\int\limits_{\bf{k}}P({\bf
k};{\bf q})f^{1\mathbf{q}}({\bf k})$ (162) $\displaystyle=$
$\displaystyle\frac{{\cal{B}}^{2}}{q^{2}\omega\left(1+a\right)}\int\limits_{\bf{k}}P({\bf
k};{\bf q})\left(\int\limits_{{\bf k}_{2}}\Phi({\bf k},{\bf
k}_{2})P(\mathbf{q};{\bf k}_{2})-\Delta G^{1\mathbf{q}}({\bf k})\right)$ (163)
$\displaystyle\equiv$ $\displaystyle D^{\mathcal{R}}+D_{\Delta
G^{1\mathbf{q}}}$ (164)

with

$\displaystyle a$ $\displaystyle=$
$\displaystyle\frac{1}{\int\limits_{\bf{k}}f({\bf
k})}\int\limits_{\bf{k}^{\prime\prime}}\frac{\left(A(\mathbf{k}^{\prime\prime};\mathbf{0},\Omega)\left(g(\omega_{+})-g^{\ast}(\omega_{-})\right)\right)^{1\Omega}}{2\omega\Omega}f({\bf
k}^{\prime\prime})$ (165) $\displaystyle g(\omega)$ $\displaystyle\equiv$
$\displaystyle\left.\frac{1}{\omega^{2}{+}i\omega(B/M)-\omega^{2}_{0}}\right|_{B=0}.$
(166)

The definition of remaining entities in Eqns. (162-165) is the same as in
[29]. It means that the replacement of $g(\omega)$ does not affect the form of
results for $D^{\mathcal{R}}$, $D_{\Delta G^{1\mathbf{q}}}$ up to replacement
of $a$, providing us with a final result for diffusion constant of screw
dislocations

$D=\frac{{\cal{B}}^{2}}{q^{2}\omega\left(1+a\right)}\int\limits_{\bf{k}}P({\bf
k};{\bf q})G({\bf k})G^{*}({\bf k})P({\bf k};{\bf q})$ (167)

In order to analyze the modified result for screw dislocations we have to find
the explicit expression for $D$. It is established on the basis of the result
for the diffusion constant (we denote it as $D_{Isc}$) from [29], which reads
as

$D_{Isc}=\frac{{\cal{B}}^{2}}{q^{2}\omega\left(1+a_{Isc}\right)}\int\limits_{\bf{k}}P({\bf
k};{\bf q})G({\bf k})G^{*}({\bf k})P({\bf k};{\bf q})$ (168)

where $a_{Isc}$ is the corresponding version of $a$ parameter. In other words,

$D=\frac{\left(1+a_{Isc}\right)}{\left(1+a\right)}D_{Isc}$ (169)

According to Eqn.(95) from [29], we have

$D_{Isc}=\frac{c_{T}^{4}}{2\omega^{3}\left(1+a_{Isc}\right)}\frac{K^{2}(K^{2})^{*}}{Im[K^{2}]}\approx\frac{c_{T}^{4}}{v^{4}\left(1+a_{Isc}\right)}\frac{vl}{2}\approx\frac{c_{T}^{2}}{v^{2}}\frac{vl}{2}$
(170)

Substituting Eqn.(170) into Eqn.(169) we come to the result

$D=\frac{c_{T}^{4}}{2\omega^{3}\left(1+a\right)}\frac{K^{2}(K^{2})^{*}}{Im[K^{2}]}\approx\frac{c_{T}^{4}}{v^{4}\left(1+a\right)}\frac{vl}{2}$
(171)

with $a$ from Eqn. (165). Based on Eqn. (165) and Eqn. (76) from [29] it can
be proven the following relation

$a=\frac{\omega^{2}}{\omega^{2}_{0}-\omega^{2}}a_{Isc}$ (172)

This provides us with the approximation (see Eqn.(77) in [29])

$a\approx\frac{\omega^{2}}{\omega^{2}_{0}-\omega^{2}}\left(\frac{c_{T}^{2}Re[K^{2}]}{\omega^{2}}-1\right)\approx\frac{\left(\frac{c_{T}^{2}}{v^{2}}-1\right)}{\frac{\omega^{2}_{0}}{\omega^{2}}-1}$
(173)

Eventually, we get for $D$

$D\approx\frac{c_{T}^{4}}{v^{4}}\frac{1}{\left(1+\frac{\left(\frac{c_{T}^{2}}{v^{2}}-1\right)}{\frac{\omega^{2}_{0}}{\omega^{2}}-1}\right)}\frac{vl}{2}$
(174)

There are two specific instances of interest for this result: The first is
that $D=0$ when $\omega=\omega_{0}$, raising the possibility of elastic wave
localization by scattering from dislocations. As discussed in Section V.4,
however, a strict equality cannot hold because in that case the scattering
problem becomes singular. But the diffusion constant does decrease as the
frequency approaches the resonance frequency from below. The second is the low
frequency, or long wavelength, limit. Now, there are two length scales to
compare the wavelength $\lambda=c_{T}/\omega$ against: the mean distance
between dislocations, $1/\sqrt{n}$, and the Peierls-Nabarro frequency divided
by the speed of shear waves $c_{T}/\omega_{0}$. When $\lambda$ is small
compared to both, use of Eqs. (21) and (24) of [29] leads to (when $B=0$,
which is the case for which we have derived a WTI)

$\lim_{\omega\to 0}D=\frac{1}{2}c_{T}\ell$ (175)

with

$\ell=\frac{\alpha}{n}\frac{\omega}{c_{T}}\frac{\omega_{0R}^{4}}{\omega^{4}}$
(176)

with $\alpha=\left(\ln(\delta/\epsilon)\right)^{2}/2\pi^{2}$ a dimensionless
constant of order one. This is of the form $\ell\sim 1/n\sigma$ where
$\sigma\propto\omega^{3}$ is a scattering cross section corresponding to
Rayleigh scattering in two dimensions.

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