#  Anisotropic electron transport in the nuclear pasta phase

M. R. Pelicer1 , M. Antonelli2 , D. P. Menezes1 , F. Gulminelli2

1Depto de Física - CFM - Universidade Federal de Santa Catarina Florianópolis
- SC - CP. 476 - CEP 88.040 - 900 - Brazil
2CNRS and ENSICAEN, Laboratoire de Physique Corpusculaire, 14050 Caen, France
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The presence of nuclear pasta is expected to modify the transport properties
in the mantle of neutron stars. The non-spherical geometry of the pasta
nuclear clusters leads to anisotropies in the collision frequencies, impacting
the thermal and electrical conductivity. We derive analytical expressions for
the anisotropic collision frequencies using the Boltzmann equation in the
relaxation time approximation. The average parallel, perpendicular and Hall
electrical conductivities are computed in the high-temperature regime above
crustal melting, considering incoherent elastic electron-pasta scattering and
randomly oriented pasta structures. Numerical values are obtained at different
densities and temperatures by using the IUFSU parametrization of the non-
linear Walecka model to determine the crustal structure. We find that the
anisotropy of the collision frequencies grows with the length of the pasta
structures and, independently of the magnetic field, the presence of rod and
slab phases decreases the conductivity by more than one order of magnitude.
Our numerical results indicate that, even if the pasta structures might
survive above the crustal melting point, no strong anisotropies are to be
expected in the conduction properties in this temperature regime, even in the
presence of a very high magnetic field.

###### keywords:

dense matter – anisotropic transport – nuclear pasta – stars:neutron

††pagerange: Anisotropic electron transport in the nuclear pasta
phase–References††pubyear: 2022

##  1 Introduction

Observations related to the thermal, magnetic and spin evolution of neutron
stars can provide us with indirect information on the transport properties of
ultra-dense matter, e.g. Horowitz et al. (2015); Montoli et al. (2020);
Potekhin & Chabrier (2021). In principle, the observations must be compared
with simulations by properly modelling the coupled magneto-thermal evolution.
Hence, models are necessary for the microscopic processes that give rise to
the thermal and electric conductivities and viscosity throughout the star
(Page & Reddy, 2012; Chamel & Haensel, 2008; Schmitt & Shternin, 2018), which
are then used as inputs to the macroscopic simulations, see Bransgrove et al.
(2018); Pons & Viganò (2019); Camelio et al. (2022).

In the crust, transport properties are determined by the scattering of
electrons by other electrons, ionic impurities and phonons in the crystal
lattice. Electron-ion scattering dominates at the lowest densities and has
been extensively studied (Flowers & Itoh, 1976; Yakovlev & Urpin, 1980;
Nandkumar & Pethick, 1984; Baiko et al., 1998; Potekhin et al., 1999; Chugunov
& Yakovlev, 2005; Aguilera et al., 2009). In the inner crust at temperatures
$T<10^{7}\,$K, thermal conductivity due to degenerate electron-electron
Coulomb scattering dominates over the contribution due to electron-phonon
scattering (Shternin & Yakovlev, 2006) and becomes competitive with the
electron conductivity due to the scattering of electrons by impurity ions
(Chamel & Haensel, 2008).

The situation gets more complicated in the innermost part of the crust, where
it might be energetically favourable for the ions composing the crystal
lattice to deform in complex structures known as “pasta” (Ravenhall et al.,
1983; Hashimoto et al., 1984; Oyamatsu, 1993). Classical molecular dynamics
simulations suggest that this matter is disordered and amorphous and that
different shapes might coexist at a given depth of the star, due to the small
energy barriers between them (Schneider et al., 2014; Horowitz et al., 2015;
Caplan et al., 2021; Newton et al., 2022). This shape coexistence has been
validated by relativistic mean field (RMF) calculations (Pelicer et al.,
2021). In the case of a disordered and amorphous inner crust with randomly
distributed nuclear clusters of different sizes (Carreau et al., 2020;
Potekhin & Chabrier, 2021) and geometries (Pelicer et al., 2021), the main
mechanism of charge and heat transport is given by uncorrelated scattering
processes between the electrons and the clusters, which play a role similar to
one of the lattice impurities in a crystallized phase.

Regarding the possible astrophysical consequences, a high impurity parameter
in the inner crust raises the electrical resistivity of the star, decreasing
steeply the magnetic field after a certain age and thus the spin-down. This
may explain the very small number of isolated X-ray pulsars with spin periods
larger than 12s (Pons et al., 2013; Newton, 2013; Hambaryan et al., 2017; Tan
et al., 2018). The high impurity also lowers the thermal conductivity, leading
to a better fit of the late-time cooling of the binary MXB 1659-29 (Horowitz
et al., 2015; Deibel et al., 2017). Gravitational waves (Horowitz, 2010),
quasi-periodic oscillations (Sotani, 2012), quasi-persistent sources of SXRTs
and giant flares due to the relaxation of the crust after heat deposition and
neutrino emissivity (Alloy & Menezes, 2011; Horowitz et al., 2004; Lin et al.,
2020) are also influenced by the presence of an amorphous layer in the inner
crust. Furthermore, the presence of the pasta layers modifies the so-called
mutual friction force between the nuclear clusters and the neutron superfluid,
with consequences on the pulsar glitch phenomenon (Antonelli & Haskell, 2020).

In the presence of a strong $B$ field, electron transport is anisotropic, as
the field bends the electron trajectories in the orthogonal plane and
suppresses electron transport across the direction of $\bm{B}$, e.g. (Chamel &
Haensel, 2008). This argument considers that the only source of anisotropy is
given by the $\bm{B}$ direction. However, the spherical symmetry of nuclear
clusters is spontaneously broken in the pasta layers, leading to additional
anisotropies already at the level of the microscopic scattering process: in
particular, Yakovlev (2015) has shown that, even in the case of random
orientation of the pasta structures, anisotropic scattering can modify the
transport properties.

In the analysis of Yakovlev (2015), the scattering rates along and across the
pasta symmetry axis were taken as free parameters. While molecular dynamics
has been able to provide estimates of the transport properties in the inner
crust, by taking the angular average of the effective structure factor of the
charge distribution (Horowitz & Berry, 2008; Horowitz et al., 2015; Nandi &
Schramm, 2018), to our knowledge no estimation of the different collision
frequencies that arise due to the pasta anisotropic shapes has been performed
to date.

The existing microscopic simulations of the finite temperature pasta
(Schneider et al., 2014; Horowitz et al., 2015; Caplan et al., 2021; Newton et
al., 2022; Nandi & Schramm, 2018) are typically done at fixed proton fraction
and high temperatures $T\geq 10^{10}$ K, thermodynamic conditions that are
especially aimed at the description of proto-neutron stars formed in supernova
events. In these conditions, it appears from those calculations that the
distribution of baryonic matter is strongly disordered, and one might expect
that anisotropies should not have a strong effect on the transport properties.
On the other hand, in the case of neutron star binaries and soft X-ray
transients, the inner crust is close to $\beta$-equilibrium and temperatures
are one or two orders of magnitude lower, which might preserve both the
peculiar pasta geometrical shapes and the lattice quasi-long range order,
potentially leading to a strong anisotropy of the scattering rates, as assumed
by Yakovlev (2015).

In this paper, we show how the anisotropic collision frequencies can be
calculated from the Boltzmann equation in the relaxation time approximation,
in the case of elastic scattering of ultra-relativistic degenerate electrons
off pasta structures. We limit ourselves to the hypothesis of incoherent
scattering sources following the Matthiessen rule (Schmitt & Shternin, 2018;
Heiselberg & Pethick, 1993; Shternin & Yakovlev, 2006). Based on the behaviour
of the static structure factor, we argue that this hypothesis should be valid
in the high-temperature regime above crustal melting.

The paper is organized as follows. In Sec. 2 we calculate the general
anisotropic collision frequencies. The collision integral and the transition
matrix elements are first expanded in the spherical harmonics basis in Sec.
2.1. Then, to extract the physical real collision frequencies, in Sec. 2.2 we
consider the lowest order (dipole) deviation from equilibrium and take
advantage of the axial symmetry of the pasta phase. The contribution of the
collision integral to the conductivity is given in terms of axial and
perpendicular collision frequencies, in agreement with the analysis of
Yakovlev (2015). Analytical expressions for the conductivity matrix are given
in Sec. 3 for the case of a liquid, disordered, pasta phase. In Sec. 3.1, the
transition matrix is numerically evaluated in the temperature domain of
validity of our approximations. The conductivity tensor with and without
magnetic field is finally obtained in Sec. 3.2. To illustrate the formalism
and give quantitative estimations of the transport coefficients, in Sec. 4 we
present numerical calculations for the collision frequencies and conductivity
for different densities and $B$ values in the high-temperature regime.
Conclusions are presented in Sec. 5.

All the numerical estimates reported in this paper are obtained using the
IUFSU parametrization of the RMF approach for the crustal composition, see
Fattoyev et al. 2010; Avancini et al. 2012, but our expressions can be
employed with any nuclear physics model that gives the static structure of the
crust. In particular, while our quantitative numerical results might be model
dependent, the qualitative conclusions remain valid for any other realistic
equation of state model.

We use natural units $\hbar=c=k_{B}=1$ all over the paper.

##  2 Relaxation time approximation for anisotropic elastic collisions

The thermal and electrical electron conductivities due to electron-ion
scattering have been calculated in a wide range of temperatures $T$ and
electron densities $n_{e}$, see e.g. Potekhin (1999): for homogeneous media,
and in the absence of a magnetic field, they are expressed in terms of the
effective collision frequencies $\nu_{\sigma,\kappa}$ as

${\sigma}=\frac{e^{2}n_{e}}{m_{e}^{\ast}\nu_{\sigma}}\qquad\qquad{\kappa}=\frac{\pi^{2}Tn_{e}}{3m_{e}^{\ast}\nu_{\kappa}}\,,$
(1)

where $m_{e}^{\ast}$ is the effective electron mass, and in the liquid regime
$\nu_{\sigma}=\nu_{\kappa}\equiv\nu$, with the collision frequency $\nu$
defined as the inverse of the relaxation time, $\nu=1/\tau$. Because of the
isotropy assumption, the collision frequencies only depend on the modulus of
the momentum transfer according to the general expression (Flowers & Itoh,
1976; Yakovlev & Urpin, 1980; Nandkumar & Pethick, 1984):

$\nu=\frac{4\pi
n_{i}e^{4}Z^{2}}{v_{F}p_{F}^{2}}\int_{0}^{2p_{F}}\frac{dq}{q}\left(1-\frac{q^{2}}{4\upepsilon_{F}^{2}}\right)\frac{F^{2}(q)}{\varepsilon^{2}(q)}S(q)\,,$
(2)

where $F(q)$ is the ion form factor, $\epsilon(q)$ is the dielectric function,
$S(q)$ is the effective structure factor that accounts for ion correlations,
and $v_{F}$, $p_{F}$, and $\upepsilon_{F}$ are the Fermi velocity, momentum
and energy respectively. Unfortunately, eqs (1) and (2) cannot be
straightforwardly generalized to the case of anisotropic scatterings. To
derive the anisotropic collision frequencies, we consider a multipole
expansion of the Boltzmann equation in the relaxation time approximation, as
we detail below.

###  2.1 Anisotropic case: expansion in spherical harmonics

We consider a strongly degenerate relativistic electron gas with position-
dependent temperature and chemical potential fields $T(\bm{r})$ and
$\mu(\bm{r})$ in a constant external magnetic field $\bm{B}$ and a weak
electric field $\bm{E}$. Assuming that the gas is only slightly out of
equilibrium, we can write its distribution function as
$f(\bm{r},\bm{p},t)=f_{0}(\bm{r},\upepsilon_{p})+\delta f(\upepsilon_{p})$,
where $\bm{r}$, $\bm{v}$ and $\bm{p}$ are the electron position, velocity and
momentum, respectively, with the latter given by
$\bm{p}=\upepsilon_{p}\bm{v}$. The Fermi-Dirac function $f_{0}$ is given by

$f_{0}(\bm{r},\upepsilon_{p})=\left[1+\exp\left(\frac{\upepsilon_{p}-\mu(\bm{r})}{T(\bm{r})}\right)\right]^{-1}.$
(3)

The deviation from equilibrium can be found by solving the linearized
Boltzmann equation (Heiselberg & Pethick, 1993; Shternin & Yakovlev, 2006)

$\begin{split}\left(-\frac{\partial
f_{0}}{\partial\upepsilon_{p}}\right)\bm{v}\cdot\Bigg{[}\nabla\mu+e\bm{E}&+\frac{\upepsilon_{p}-\mu}{T}\nabla
T\Bigg{]}\\\ &-e(\bm{v}\times\bm{B})\cdot\frac{\partial\delta
f}{\partial\bm{p}}=I[f]\,,\end{split}$ (4)

where $I[f]$ is the collision integral that can be written as

$\begin{split}I[f]=\int\frac{d^{3}\bm{p}\,^{\prime}}{(2\pi)^{3}}&\big{[}\Gamma_{p^{\prime}\rightarrow
p}f(\bm{p}\,^{\prime})\left(1-f(\bm{p})\right)\\\ &-\Gamma_{p\rightarrow
p^{\prime}}f(\bm{p})\left(1-f(\bm{p}\,^{\prime})\right)\big{]}\,.\end{split}$
(5)

Here, $\Gamma_{p\rightarrow p^{\prime}}$ is the transition rate from an
initial momentum $\bm{p}$ to a final momentum ${\bm{p}}\,^{\prime}$,
introduced to account for electron scattering with any generic potential, and
we have omitted the position dependencies as they do not affect the
calculation. We restrict ourselves to elastic scatterings with a localized
source for the potential, such that the following simplification applies:

$\Gamma_{p\rightarrow p^{\prime}}=\Gamma_{p^{\prime}\rightarrow
p}=2\pi\delta(\upepsilon_{p}-\upepsilon_{p^{\prime}})W_{pp^{\prime}},$ (6)

where $W_{pp^{\prime}}$ is the transition matrix element, which we will write
explicitly for the case of electron-pasta scattering in the next section.
Taking into account that deviations from equilibrium are small, we can rewrite
the collision integral, eq. (5), as

$I[f]=-2\pi\int\frac{d^{3}\bm{p}\,^{\prime}}{(2\pi)^{3}}\delta(\upepsilon_{p}-\upepsilon_{p^{\prime}})W_{pp^{\prime}}\left[\delta
f(\bm{p})-\delta f(\bm{p}\,^{\prime})\right],$ (7)

where the Fermi-Dirac terms coming from the different momenta have cancelled
out due to the elasticity assumption in eq. (6).

In isotropic scatterings, $W_{pp^{\prime}}$ is a function of
$q=|\bm{p}-{\bm{p}}^{\prime}|$ and of the electron energy only. Since in this
work we are dealing with general anisotropic scatterings, we will assume the
matrix elements to be functions of the solid angles of both incoming and
outgoing electron momenta ($\bm{p}$ and ${\bm{p}}^{\prime}$), as well as the
energy $\upepsilon_{p}$, so there is no assumption of symmetry for the source
of potential. The transition matrix can be expanded in the basis of spherical
harmonics as

$W_{p\,p^{\prime}}\left(\Omega_{p},\Omega_{p^{\prime}},\upepsilon_{p}\right)=\sum_{lm\,l^{\prime}m^{\prime}}\mathcal{W}_{lm\,l^{\prime}m^{\prime}}(\upepsilon_{p})Y_{l}^{m}(\Omega_{p})Y_{l^{\prime}}^{m^{\prime}}(\Omega_{p^{\prime}}),$
(8)

whereas the assumption of elasticity implies that $\Omega_{p}$ and
$\Omega_{p^{\prime}}$ are interchangeable, such that

$W_{pp^{\prime}}=W_{p^{\prime}p}\;\;\Longrightarrow\;\;\mathcal{W}_{lm\,l^{\prime}m^{\prime}}=\mathcal{W}_{l^{\prime}m^{\prime}\,lm}.$
(9)

This is a generalization of the Legendre expansion used for scattering with
isotropic targets – see Sec. 3 in Pines & Nozières (2018). In App. A we show
how the isotropic limit can be recovered from our calculation.

Figure 1: Cylindrical (rod and tube) and planar (slab) geometries of the
nuclear pasta with $z$ as the symmetry axis. The transferred momentum vector
$\bm{q}$ is drawn arbitrarily and the magnetic field $B$ lies in the
$xz$–plane.

The deviation from equilibrium of the electron distribution is expanded as:

$\delta f(\bm{p})=\sum_{lm}\delta
f_{lm}(\upepsilon_{p})Y_{l}^{m}(\Omega_{p}).$ (10)

Substitution of eqs (8) and (10) into eq. (7) allows us to use the
orthogonality of spherical harmonics and the contraction rule

$\displaystyle Y_{l}^{m}(\Omega)$ $\displaystyle
Y_{l^{\prime}}^{m^{\prime}}(\Omega)=\sum_{L\,M}(-1)^{M}\sqrt{\frac{(2L+1)(2l+1)(2l^{\prime}+1)}{4\pi}}$
$\displaystyle\times\begin{pmatrix}l&l^{\prime}&L\\\
0&0&0\end{pmatrix}\begin{pmatrix}l&l^{\prime}&L\\\
m&m^{\prime}&-M\end{pmatrix}Y_{L}^{M}(\Omega)$ (11)

to rewrite the collision integral as

$\begin{split}I[f]=-\frac{p^{2}}{4\pi^{2}v}\sum_{lm,l^{\prime}m^{\prime}}\delta
f_{lm}\bigg{[}\mathcal{W}_{l^{\prime}m^{\prime}\;00}&\sum_{LM}(-1)^{M}\sqrt{(2l+1)(2l^{\prime}+1)(2L+1)}\\\
&\times\begin{pmatrix}l&l^{\prime}&L\\\
0&0&0\end{pmatrix}\begin{pmatrix}l&l^{\prime}&L\\\
m&m^{\prime}&-M\end{pmatrix}Y_{L}^{M}(\Omega_{p})-(-1)^{m}\mathcal{W}_{l^{\prime}m^{\prime}\,l-m}Y_{l^{\prime}}^{m^{\prime}}(\Omega_{p})\bigg{]}.\end{split}$
(12)

The 3-j Wigner symbols $\begin{pmatrix}l_{1}&l_{2}&l_{3}\\\
m_{1}&m_{2}&m_{3}\end{pmatrix}$ are invariant under even permutations of the
columns and non-zero only if $m_{1}+m_{2}+m_{3}=0$,
$|l_{1}-l_{2}|<l_{3}<l_{1}+l_{2}$ and $l_{1}+l_{2}+l_{3}$ is an integer (Brink
& Satchler, 1968; Edmonds, 2016).

We define the anisotropic collision frequencies by expanding the collision
integral linearly in $\delta f$,

$I[f]=-\sum_{lm,l^{\prime}m^{\prime}}\delta
f_{lm}\left(\upepsilon_{p}\right)\,\left[{\nu}\left(\upepsilon_{p}\right)\right]_{lm}^{l^{\prime}m^{\prime}}Y_{l^{\prime}}^{m^{\prime}}(\Omega_{p}),$
(13)

Integration of eqs (12) and (13) in $\Omega_{p}$ yields

$\left[\nu\right]_{lm}^{l^{\prime}m^{\prime}}=\frac{p^{2}}{4\pi^{2}v}\Bigg{[}(-1)^{m^{\prime}}\sqrt{(2l+1)(2l^{\prime}+1)}\sum_{LM}\mathcal{W}_{LM\;00}\sqrt{2L+1}\begin{pmatrix}l&l^{\prime}&L\\\
0&0&0\end{pmatrix}\begin{pmatrix}L&l&l^{\prime}\\\
M&m&-m^{\prime}\end{pmatrix}-\;(-1)^{m}\,\mathcal{W}_{l^{\prime}m^{\prime}\,l-m}\Bigg{]}.$
(14)

We can obtain a more compact form of this expression by using the
Wigner–Eckart theorem and the spherical harmonics representation of
irreducible tensor operators of rank $l$ (Racah, 1942a, b),

$C^{l}_{m}=\sqrt{\frac{4\pi}{2l+1}}Y_{lm}(\Omega)\,,$ (15)

such that eq. (14) becomes

$\begin{split}\left[\nu\right]_{lm}^{l^{\prime}m^{\prime}}=\frac{p^{2}}{4\pi^{2}v}\Bigg{[}\sum_{LM}\mathcal{W}_{LM\;00}&\sqrt{2L+1}\langle
l^{\prime}m^{\prime}|C_{M}^{L}|lm\rangle\\\
&-\;(-1)^{m}\,\mathcal{W}_{l^{\prime}m^{\prime}\,l-m}\Bigg{]}\,.\end{split}$
(16)

###  2.2 Derivation of the collision frequencies

To evaluate the collision frequencies in the pasta phase, we consider
idealized rod and slab-like geometries, as expected in the basic liquid-drop
modelling of the inner crust (Ravenhall et al., 1983; Hashimoto et al., 1984).
These geometries and the definitions entering the calculations are sketched in
Fig. 1.

Equation (16) is not yet a multipole expansion of the collision rates because
the different expansion coefficients of the collision integral
$\left[\nu\right]_{lm}^{l^{\prime}m^{\prime}}$ are complex numbers. This is
due to the fact that both the electron distribution function and the collision
integral are written on the basis of complex spherical harmonics. To relate
eq. (16) to the physical quantities, we must rewrite eqs (10) and (13) in
terms of real coefficients.

To do so, we notice that the coefficients
$\mathcal{W}_{l^{\prime}m^{\prime}\,lm}$ in eq. (8) are constrained by the
symmetries of rods and slabs. Both geometries are invariant under inversion of
the $z$-axis ($z\rightarrow-z$), implying that the only non-zero
$\mathcal{W}_{l^{\prime}m^{\prime}\,lm}$ are the ones having the sum
$l+l^{\prime}$ that is even. The sum $m+m^{\prime}$ is constrained by the
$xy$–plane symmetries: cylinders are invariant under arbitrary rotations: the
non-zero $\mathcal{W}_{l^{\prime}m^{\prime}\,lm}$ are only those with
$m+m^{\prime}=0$; slabs are invariant over $\pi/2$ rotations, so the sum
$m+m^{\prime}$ must be a multiple of 4. To summarize, the
$\mathcal{W}_{lm\,l^{\prime}m^{\prime}}$ are not zero if and only if:

$\text{Rods}\,\left.\begin{cases}l+l^{\prime}=2k\\\
m+m^{\prime}=0\end{cases}\right.\qquad\text{Slabs}\,\left.\begin{cases}l+l^{\prime}=2k\\\
m+m^{\prime}=4k^{\prime}\\\ \end{cases}\right.$ (17)

with $k,k^{\prime}\in\mathds{Z}$. To further progress, we restrict ourselves
to the case of electric and thermal conductivities. Accounting for spin
degeneracy, the electric current and heat flow are given by

$\bm{j}=-2e\int\frac{d^{3}\bm{p}}{(2\pi)^{3}}\bm{v}\delta
f\qquad\bm{q}=2\int\frac{d^{3}\bm{p}}{(2\pi)^{3}}\bm{v}(\upepsilon-\mu)\delta
f\,,$ (18)

so that only the odd $l$ terms in the expansion (10) contribute to the
integrals in eq. (18). Moreover, in the relaxation time approximation, the
collision integral, eq. (7), is linear in $\delta f$. The left-hand side of
eq. (4) is linear in $\bm{p}$, implying that only the coefficient
$l^{\prime}=1$ in the expansion of the collision integral, eq. (13)
contributes to the currents111 The multiplicity $l$ of the spherical harmonics
coincides with the power of $\bm{p}$ in an equivalent expansion in homogeneous
harmonic polynomials since they are isomorphic (Gallier, 2013; Freire, 2022).
. This is also discussed in depth in the case of isotropic scattering in Sykes
& Brooker (1970), and mentioned in the case of pasta in Schmitt & Shternin
(2018).

In the isotropic case, there is no mixing between the different terms of
$I[f]$ in eq. (13) and those of $\delta f$ in eq. (10). However, in the
anisotropic case, the collision frequencies can mix the $l^{\prime}=1$
contributions in eq. (13) with the $l=2,3...$ components of $\delta f_{lm}$ in
eq. (13). For simplicity, we neglect such mixing and restrict ourselves to the
most important contribution (see also Schmitt & Shternin 2018) by writing
$\nu_{lm}^{l^{\prime}m^{\prime}}=\nu_{1m}^{1m^{\prime}}\delta_{l1}\delta_{l^{\prime}1}$
in eq. (13). This approximate approach is probably good in the case of pasta,
due to the symmetry rules in (17).

We will show that the axial symmetry of the problem limits the number of
physical collision frequencies to two: an axial frequency $\nu_{a}$, and a
perpendicular one $\nu_{p}$. To do so, we need to rewrite the expansions eqs
(10) and (13) in terms of real coefficients. We introduce the real spherical
harmonics:

$\displaystyle{\cal
Y}_{lm}=\begin{dcases}\frac{i}{\sqrt{2}}\left(Y_{l}^{m}-(-1)^{m}Y_{l}^{-m}\right)&m<0\\\
Y_{l}^{0}&m=0\\\
\frac{1}{\sqrt{2}}\left(Y_{l}^{-m}+(-1)^{m}Y_{l}^{m}\right)&m>0\\\
\end{dcases}$ (19)

and rewrite the $l=1$ term of eq. (10), $\delta f_{1m}$, as:

$\delta f(\bm{p})\Big{|}_{l=1}={\cal Y}_{11}\delta f_{x}+{\cal Y}_{1-1}\delta
f_{y}+{\cal Y}_{10}\delta f_{z}$ (20)

where the coefficients are given by

$\delta f_{x}=\frac{\delta f_{1-1}-\delta f_{11}}{\sqrt{2}},\quad\delta
f_{y}=\frac{\delta f_{1-1}+\delta f_{11}}{\sqrt{2}i},\quad\delta f_{z}=\delta
f_{10}.$ (21)

Since the electron distribution function is real, so are the coefficients
defined above. Substituting eq. (19) and eq. (21) in the collision integral
eq. (13), we get

$\displaystyle I[f]=\begin{pmatrix}{\cal Y}_{11}&{\cal Y}_{1-1}&{\cal
Y}_{10}\\\ \end{pmatrix}\begin{pmatrix}\nu_{xx}&\nu_{xy}&\nu_{xz}\\\
\nu_{yx}&\nu_{yy}&\nu_{yz}\\\ \nu_{zx}&\nu_{zy}&\nu_{zz}\\\
\end{pmatrix}\begin{pmatrix}\delta f_{x}\\\ \delta f_{y}\\\ \delta f_{z}\\\
\end{pmatrix}$ (22)

with the physical collision frequencies given by:

$\hat{\nu}=\begin{pmatrix}\frac{1}{2}\left(\nu_{11}^{11}+\nu_{1-1}^{1-1}-\nu_{11}^{1-1}-\nu_{1-1}^{11}\right)&\frac{i}{2}\left(\nu_{11}^{11}-\nu_{1-1}^{1-1}+\nu_{11}^{1-1}-\nu_{1-1}^{11}\right)&\frac{1}{\sqrt{2}}\left(\nu_{1-1}^{10}-\nu_{11}^{10}\right)\\\
\frac{i}{2}\left(-\nu_{11}^{11}+\nu_{1-1}^{1-1}+\nu_{11}^{1-1}+\nu_{1-1}^{11}\right)&\frac{i}{2}\left(\nu_{11}^{11}+\nu_{1-1}^{1-1}+\nu_{11}^{1-1}+\nu_{1-1}^{11}\right)&\frac{i}{\sqrt{2}}\left(\nu_{1-1}^{10}+\nu_{11}^{10}\right)\\\
\frac{1}{\sqrt{2}}\left(\nu_{10}^{1-1}-\nu_{10}^{11}\right)&\frac{-i}{\sqrt{2}}\left(\nu_{10}^{11}+\nu_{10}^{1-1}\right)&\nu_{10}^{10}\\\
\end{pmatrix}$ (23)

The constraint of elasticity eq. (9) implies that the collision frequency
matrix is symmetric, $\nu_{ij}=\nu_{ji}$. Moreover, we can see from the pasta
symmetries in eq. (17) that the off-diagonal terms vanish and that the $xx$
and $yy$ terms are equal. This is valid for slabs because
$L_{1x}=L_{1y}=L_{1}$. Thus,

$\hat{\nu}=\begin{pmatrix}\nu_{xx}&\nu_{xy}&\nu_{xz}\\\
\nu_{yx}&\nu_{yy}&\nu_{yz}\\\ \nu_{zx}&\nu_{zy}&\nu_{zz}\\\
\end{pmatrix}=\begin{pmatrix}\nu_{p}&0&0\\\ 0&\nu_{p}&0\\\ 0&0&\nu_{a}\\\
\end{pmatrix}$ (24)

where $\nu_{p}=\nu_{11}^{11}$ and $\nu_{a}=\nu_{10}^{10}$. Writing, without
any loss of generality, $\delta
f_{1m}=\sqrt{4\pi/3}\Phi_{1m}(\upepsilon_{p})|\bm{v}|$, the collision integral
expansion, eq. (13), can be simply rewritten as

$I[f]=-v_{z}\Phi_{z}\nu_{a}-\bm{v}_{p}\cdot\bm{\Phi}_{p}\nu_{p},$ (25)

where $\bm{\Phi}$ is a vector that can depend on $\upepsilon_{p}$, and the
collision frequencies $\nu_{a}$ and $\nu_{p}$ are defined parallel and
perpendicular to the pasta symmetry axis. This result exactly coincides with
the generalization of the relaxation time approximation proposed by Yakovlev
(2015) on symmetry arguments to include the effect of the anisotropic medium.

The axial and perpendicular collision frequencies can be calculated from eq.
(14):

$\displaystyle\nu_{a}(\upepsilon_{p})$
$\displaystyle=\frac{p^{2}}{4\pi^{2}v}\bigg{[}\mathcal{W}_{00,00}-\mathcal{W}_{10,10}$
$\displaystyle\hskip
71.13188pt+\frac{1}{\sqrt{5}}\left(W_{20,00}+\mathcal{W}_{00,20}\right)\bigg{]}$
(26) $\displaystyle\nu_{p}(\upepsilon_{p})$
$\displaystyle=\frac{p^{2}}{4\pi^{2}v}\bigg{[}\mathcal{W}_{00,00}-\frac{1}{2\sqrt{5}}\left(\mathcal{W}_{20,00}+\mathcal{W}_{00,20}\right)$
$\displaystyle\hskip
71.13188pt+\frac{1}{2}\left(\mathcal{W}_{11,1-1}+\mathcal{W}_{1-1,11}\right)\bigg{]}.$
(27)

To rewrite $\nu_{a}$ and $\nu_{p}$ in terms of the transition matrix
$W_{pp^{\prime}}$, we invert eq. (8) using the orthogonality of spherical
harmonics:

$\mathcal{W}_{lm\,l^{\prime}m^{\prime}}=\int
d\Omega_{p}d\Omega_{p^{\prime}}W_{pp^{\prime}}{{Y}_{l}^{m}}^{\ast}(\Omega_{p}){{Y}_{l^{\prime}}^{m^{\prime}}}^{\ast}(\Omega_{p^{\prime}}).$
(28)

This leads to the final expression of the collision rates, for an arbitrary
interaction preserving axial symmetry, and assuming a dipole-like deviation
from the equilibrium of the electron distribution function:

$\displaystyle\nu_{a}\left(\upepsilon_{p}\right)$
$\displaystyle=\frac{3}{32\pi^{3}v}\int
d\Omega_{p}d\Omega_{p^{\prime}}W_{pp^{\prime}}\,q^{2}\,\cos^{2}\theta_{q}$
(29) $\displaystyle\nu_{p}\left(\upepsilon_{p}\right)$
$\displaystyle=\frac{3}{32\pi^{3}v}\int
d\Omega_{p}d\Omega_{p^{\prime}}\,W_{pp^{\prime}}\,q^{2}\frac{1}{2}\sin^{2}\theta_{q}.$
(30)

To get the generalization of eqs (1) and (2) to the physical problem of
electron-pasta scattering, we now turn to evaluate the transition matrix
$W_{pp^{\prime}}$.

##  3 Conductivity tensor for rods and slabs

###  3.1 Elastic scattering matrix in the incoherent scattering limit

Figure 2: Linear and Wigner-Seitz radius of the pasta as a function of density
in dashed and full curves for $T=0.5,1,3$ and 5 MeV (panels a, b c and d,
respectively). The different pasta geometries are indicated with different
colours. The results for $T=0$ and $T=0.5$ MeV are indistinguishable.

In the case of isotropic scattering, the transition matrix (8) depends solely
on the absolute value of the transferred momentum
$\bm{q}=\bm{p}-\bm{p}\,^{\prime}$. This is equivalent to a dependence on the
angle between the incoming and outgoing electron momentum because one can
write $q^{2}=p^{2}(1-\hat{p}\cdot\hat{p}\,^{\prime})$. In the case of
anisotropic scattering, the transition matrix can depend separately on the
angles of both incoming and outgoing momenta. However, because of the axial
symmetry of the pasta structures, it can only depend on the projections of the
transferred momentum in the axis perpendicular and parallel to the symmetry
axis of the pasta. We can assume without loss of generality that the pasta
symmetry axis coincides with the $z$ axis, such that the transition matrix
becomes a function of the vector $\bm{q}$ itself, see Fig.1.

The transition matrix $W_{pp^{\prime}}$ for the elastic scattering in the Born
approximation is given by (see, e.g., equation 81.5 of Berestetskii et al.
1971)

$\displaystyle W_{pp^{\prime}}(\bm{q},\upepsilon_{p})=$
$\displaystyle\frac{1}{2}\sum_{s,s^{\prime}}\left|\frac{e}{2\upepsilon_{p}}\bar{u}_{p^{\prime},s^{\prime}}\gamma^{0}u_{p,s}\int
d^{3}\bm{x}A_{0}(\bm{x})e^{-i\bm{q}\cdot\bm{x}}\right|^{2}$
$\displaystyle=e^{2}\left(1-\frac{q^{2}}{4\upepsilon_{p}^{2}}\right)\left|U(\bm{q})\right|^{2}S(\bm{q})$
(31)

where in the first line $u_{p,s}$ is the electron spinor, $A_{0}(\bm{x})$ is
the electric potential generated by nuclei and $\gamma^{0}$ is the Dirac
matrix. In the second line, the Fourier transform of the potential $U(\bm{q})$
is introduced, and the static structure factor is defined as (Flowers & Itoh,
1976):

$\displaystyle S(\bm{q})$ $\displaystyle=\langle
n_{p}(\bm{q})n_{p}(-\bm{q})\rangle_{T}$ $\displaystyle=\frac{1}{V}\int
d^{3}\bm{r}d^{3}\bm{r^{\prime}}e^{i\bm{q}\cdot\left(\bm{r}-\bm{r^{\prime}}\right)}\langle
n_{p}(\bm{r})n_{p}(\bm{r^{\prime}})\rangle_{T},$ (32)

where $n_{p}(\bm{q})$ is the charge density of the scatterer in momentum
space, $\langle...\rangle_{T}$ is the thermal average that accounts for
correlations between protons, and the integral covers the entire thermodynamic
system of volume $V$. The structure factor carries the whole information
regarding the anisotropy of the system both through the anisotropic shape of
the pasta and through the lattice arrangement.

Figure 3:  Left panel: correlation function decay parameter $\eta$ for slabs.
Middle (right) panel: estimation of the correlation between neighbouring slab
structures as a function of density (temperature), see text for details.
Curves for $T=$0.1, 0.5, 1, 3, and 5 MeV are shown on the left and center, and
for $n_{B}=$ 0.07, 0.075 and 0.08 fm-3 on the right.

A complete calculation of the static structure factor for the pasta has not
been performed to our knowledge. Still, variational theory in Wigner-Seitz
(WS) cells of different geometries with energy densities obtained from the RMF
approach, is routinely used by nuclear physicists to obtain a microscopically
funded estimation of the optimal average charge distribution $\langle
n_{p}(\bm{r})\rangle_{T}$, see e.g. Avancini et al. (2009, 2008, 2012) and
Haensel et al. (2007); Chamel & Haensel (2008) for reviews. In the simplest
liquid-drop modelling of Fig. 1, the pasta structures are characterized by
constant density profiles. We can write for rods ($d=2$) and slabs ($d=1$),
respectively:

$\begin{split}&\langle
n_{p}(\bm{r})\rangle_{T}^{d=2}=n_{p}\sum_{m}\Theta(R_{2}-r-mR_{W2})\\\
&\langle
n_{p}(\bm{r})\rangle_{T}^{d=1}=n_{p}\sum_{m}\Theta(R_{1}-z-mR_{W1})\end{split}$
(33)

where the average linear size of the cluster $R_{d}$, its internal proton
(neutron) density $n_{p}$ ($n_{n}$) and average WS cell radius $R_{Wd}$ are
variationally obtained for any given temperature $T$ and baryon density
$n_{B}$, as well as the (uniform) electron density, and the density of the
dripped neutrons222 This last quantity does not play any role in the
calculations concerned by the present paper. . The sum in (33) runs over the
parallel structures in the lattice. For this application, we use the
relativistic mean-field approach of Avancini et al. (2012). The mean field
Lagrangian is given by the non-linear Walecka model, parametrized by the IUFSU
force (Fattoyev et al., 2010) with a surface tension fitted to reproduce a
Thomas-Fermi simulation, see Avancini et al. (2012) for details. In Fig. 2 we
show the linear and WS cell radii as a function of the density, for some
representative temperatures that will be considered below. We can see that
non-spherical geometries are expected in the innermost part of the inner
crust, and are found to persist even at high temperatures well above crustal
melting, that occurs around $T_{m}\approx 1$ MeV in this density region
(Carreau et al., 2020). The different colours correspond to the geometries
that are associated, for a given baryon density, with the minimal free energy
density. It is known that the pasta properties are model dependent (Dinh Thi
et al., 2021), mainly the densities at which the different geometries appear,
but the qualitative behaviour shown in Fig.2 is obtained in all realistic
nuclear models found in the literature (Dinh Thi et al., 2021; Parmar et al.,
2022).

In the case of a perfect lattice, electron band structures suppress the
scattering rates and charge transport occurs only through electron-phonon
interactions. However, thermal fluctuations disturb the lattice periodicity
and destroy the long-range order. In particular, in the disordered limit
expected to dominate with increasing temperature, the correlation function
drops to zero on a length scale of the linear size of the cluster, and the
different pasta structures are fully uncorrelated and act as incoherent
impurity scatterers. These fluctuations have been calculated for the pasta by
Pethick & Potekhin (1998) within the classical approach of the Landau-de
Gennes model of liquid crystals (de Gennes & Prost, 1993; Chandrasekhar,
1992). For slabs, the thermal displacement presents a logarithmic divergence
with the linear dimension of the sample, reflecting the well-known Landau-
Peierls instability (Landau & Lifshitz, 1969). Concerning the rod phase, the
thermal displacement is finite and the long-range order in the transverse
plane is in principle preserved. A critical temperature for the long (or
quasi-long) range order was estimated by Watanabe et al. (2000) as the
temperature at which the thermal displacement becomes comparable to the cell
radius. Such a temperature was shown to strongly decrease with increasing
baryonic density and, for fiducial values of the elastic constants, to be of
the order of a few MeV both for slabs and for rods (Watanabe et al., 2000).
Above such temperatures, it is reasonable to expect that the conventional
pasta phase is fully destroyed by thermal fluctuations, even if complex
deformed disordered cluster structures may still be present, as suggested by
molecular dynamics simulations (Horowitz et al., 2015; Schneider et al., 2014;
Newton et al., 2022; Nandi & Schramm, 2018). Below the critical temperature,
the consequence of the reduced dimensionality of the pasta phase is that the
long-range order (or quasi-long in the case of slabs) is only preserved in the
directions corresponding to the lattice periodicity (that is along
$\bm{u}_{z}$ for the slab phase, and $\bm{u}_{\perp}$ for the rod one),
potentially leading to strong anisotropies in the collision frequencies.

Figure 4: Left: Effective length of the pasta transverse to the symmetry axis,
normalized to the Wigner-Seitz radius (see text). Right: Proton number of the
pasta as a function of density. The different pasta geometries are indicated
with different colours. The temperature is fixed to $T=3$ MeV.

Interestingly, limiting behaviours can be obtained for the density correlation
of slabs (Poniewierski et al., 1998; de Gennes & Prost, 1993) $\langle
n_{p}(\bm{r})n_{p}(\bm{r^{\prime}})\rangle_{T}\equiv\langle\delta
n^{2}(\bm{r}-\bm{r}^{\prime})\rangle_{T}$ showing the power law behaviour
characteristic of the quasi-long range order of the smectic (slab) phase:

$\displaystyle\langle\delta n^{2}(\bm{r})\rangle_{T}$ $\displaystyle\propto$
$\displaystyle|z|^{-\eta}\;\;|z|\to\infty$ (34) $\displaystyle\propto$
$\displaystyle r_{\perp}^{-2\eta}\;\;r_{\perp}\to\infty,$ (35)

where

$\eta=\frac{q_{0}^{2}T}{8\pi\lambda C_{0}},$ (36)

with $q_{0}=\pi/R_{W1}$, $\lambda^{2}=R_{W1}^{2}(1+2f-2f^{2})/45$ where
$f=R_{1}/R_{W1}$ is the average volume fraction of the cluster,and
$C_{0}=6E_{C}$, where $E_{C}$ is the average Coulomb energy density in the
cell. The numerical values of the $\eta$ parameter as a function of the
density, as numerically obtained from the average pasta configuration
predicted by the RMF model, are displayed in the left part of Fig. 3 for
different temperatures. As expected, the correlation decreases with
temperature and density.

In the absence of a complete calculation of the correlation function, we limit
ourselves in this paper to temperatures high enough for the hypothesis of
uncorrelated scatterers to be realistic. To this aim, we plot in the center
and right parts of Fig. 3 the quantity $2^{-\eta}$ as an estimation of the
ratio between the correlation function at $z=2R_{W1}$, corresponding to a
distance containing two different slabs, and the same quantity at $z=R_{W1}$,
such that a single slab is accounted for,

$2^{-\eta}\approx\frac{\delta n^{2}(2R_{W1})}{\delta n^{2}(R_{W1})}.$ (37)

Even if these distances might be small to justify the use of the asymptotic
behaviour given by eq. (34), the quantity $2^{-\eta}$ can be taken as an
estimation of the correlation reduction due to thermal effects. From Fig.3 we
can see that only at very high temperatures above 1 MeV the hypothesis of
incoherent scattering appears justified. For the following numerical
applications, we will focus on $T=3$ MeV as a representative temperature
value.

Since the correlation asymptotically follows the same power law in the
transverse as in the longitudinal plane see eq. (35), we define the effective
length of the slab $L_{1}$ from the same order-of-magnitude consideration:

$\frac{\delta n^{2}(L_{1}^{eff})}{\delta n^{2}(L_{W1})}\approx 2^{-\eta}$ (38)

where $L_{W1}$ is defined by normalizing the slab WS volume to the droplet
volume at identical thermodynamic conditions. Comparing eqs (37) and (38), we
consider that the effective length of the slabs is

$L_{1}^{eff}=\sqrt{2}L_{1W}.$ (39)

For the rods, in the absence of an exact calculation of $S(\bm{q})$, we assume
the length of interest to be equal to the slabs one if they were dominant at
the same density, $L_{2}^{eff}=L_{1}^{eff}$. The resulting numerical values of
the pasta length and proton number at $T=3$ MeV are shown in Fig. 4.

Within the hypothesis of incoherent scatterings, the structure factor integral
is limited to a single cell. Charge fluctuations within the cell being
negligible we can write $\langle
n_{p}(\bm{r})n_{p}(\bm{r^{\prime}})\rangle_{T}=\langle
n_{p}(\bm{r})\rangle_{T}\langle n_{p}(\bm{r^{\prime}})\rangle_{T}$, leading
to:

$S(\bm{q})\approx Z^{2}n_{i}|F(q)|^{2},$ (40)

with the form factor $F(\bm{q})$ defined as the Fourier transform of the
charge density normalized by the number of protons composing the cluster
($Z$):

$F(\bm{q})=\frac{1}{Z}\int_{\rm
WS}d^{3}\bm{r}e^{i\bm{q}\cdot\bm{r}}n_{p}(\bm{r}),$ (41)

and the number density of targets within the medium as $n_{i}=1/V$.

Analytic expressions can be found for the form factor by direct integration of
eq. (41) for spherical, cylindrical, and planar geometries (labelled
$d=3,2,1$, respectively):

$F_{d}(\bm{q})=\left.\begin{dcases}\frac{3}{(qR_{3})^{3}}\left[\sin(qR_{3})-qR_{3}\cos(qR_{3})\right],&\text{d=3
}\\\
\frac{2}{q_{z}L_{2}}\sin\left(\frac{q_{z}L_{2}}{2}\right)\frac{2}{q_{\perp}R_{2}}J_{1}\left(q_{\perp}R_{2}\right),&\text{d=2}\\\
\frac{2}{L_{1}q_{x}}\sin\left(\frac{L_{1}q_{x}}{2}\right)\frac{2}{L_{1}q_{y}}\sin\left(\frac{L_{1}q_{y}}{2}\right)\frac{1}{R_{1}q_{z}}\sin\left({R_{1}q_{z}}\right),&\text{d=1}\end{dcases}\right.$
(42)

Here, $q_{x}=q\sin\theta_{q}\cos\phi_{q}$,
$q_{y}=q\sin\theta_{q}\sin\phi_{q}$, $q_{z}=q\cos\theta_{q}$,
$q_{\perp}=\sqrt{q_{x}^{2}+q_{y}^{2}}$, and $J_{1}$ is the cylindrical Bessel
function,

$J_{1}(x)=\frac{1}{i\pi}\int_{0}^{\pi}d\phi e^{ixcos(\phi)}cos(\phi).$ (43)

Finally, using the Fourier transform of the electric potential

$U(\bm{q})=\frac{4\pi e}{q^{2}\varepsilon(\bm{q})},$ (44)

the transition matrix element is written as:

$W_{pp^{\prime}}(\bm{q},\upepsilon_{p})=n_{i}e^{2}\left(1-\frac{q^{2}}{4\upepsilon_{p}^{2}}\right)\left|\frac{4\pi
eZF_{d}(\bm{q})}{q^{2}\varepsilon(q)}\right|^{2},$ (45)

where the dielectric function $\varepsilon(q)$ is introduced, regularizing the
divergence at $q=0$, to account for electron screening. A complete calculation
within relativistic quantum mechanics, in the random phase approximation,
gives (Jancovici, 1962; Haensel et al., 2007):

$\displaystyle\epsilon(q)$
$\displaystyle=1+\frac{k_{TF}^{2}}{q^{2}}\Bigg{\\{}\frac{2}{3}-\frac{2}{3}\frac{y^{2}x_{r}}{\gamma_{r}}\ln(x_{r}+\gamma_{r})$
$\displaystyle+\frac{x_{r}^{2}+1-3x_{r}^{2}y^{2}}{6yx_{r}^{2}}\ln\left|\frac{1+y}{1-y}\right|$
$\displaystyle+\frac{2y^{2}x_{r}^{2}-1}{6yx_{r}^{2}}\frac{\sqrt{1+x_{r}^{2}y^{2}}}{\gamma_{r}}\ln\left|\frac{y\gamma_{r}+\sqrt{1+x_{r}^{2}y^{2}}}{y\gamma_{r}-\sqrt{1+x_{r}^{2}y^{2}}}\right|\Bigg{\\}},$
(46)

where $y=q/2p_{F}$, $x_{r}=p_{F}/m_{e}$, $\gamma_{r}=\sqrt{1+x_{r}^{2}}$,
$k_{TF}$ is the Thomas-Fermi momentum, defined as:

$k_{TF}=\sqrt{4\pi e^{2}\partial
n_{e}/\partial\mu_{e}}=2\sqrt{\alpha_{em}\gamma_{r}/(\pi x_{r})}k_{F},$ (47)

and the second equality supposes a strongly degenerate electron gas (Haensel
et al., 2007). Though in this work we assume that the screening is isotropic,
it is important to observe that strong magnetic fields lead to anisotropic
behaviour and can produce Friedel oscillations (Horing, 1969; Sharma & Reddy,
2011). We leave the consideration of this source of further anisotropies to a
future study.

Figure 5: Form factor squared of rods at the representative density
$n_{B}=0.06$ fm-3 (left, yellow) and slabs at $n_{B}=0.08$ fm-3 (middle and
right, blue) as a function of the azimuthal angle, for different transferred
momenta $q=k_{F}/2,k_{F}$ and $2k_{F}$ shown as continuous, dashed and dotted
curves, respectively. For slabs we fix $\phi_{q}=0$ (middle) and $\pi/4$
(right). The effective lengths $L$ are taken from Figure 4. The temperature is
fixed to $T=3$ MeV.

###  3.2 Collision frequencies and conductivities

The expression of the transition matrix in eq. (45) allows us to get the final
result for the collision rates. We substitute this expression into eqs (29)
and (30) and make the change of variables

$\int
d\Omega_{p}d\Omega_{p}^{\prime}=\frac{2\pi}{p^{2}}\int\frac{d^{3}\bm{q}}{q}\,,$
(48)

to get the collision frequencies

$\displaystyle\nu_{a}\left(\upepsilon_{p}\right)$ $\displaystyle=\frac{12\pi
n_{i}e^{4}Z^{2}}{vp^{2}}\int_{0}^{2p}dq\frac{1}{q\varepsilon^{2}(q)}\left(1-\frac{q^{2}}{4\upepsilon_{p}^{2}}\right)\int\frac{d\Omega_{q}}{4\pi}\left|F_{d}(\bm{q})\right|^{2}\cos^{2}\theta_{q}$
(49) $\displaystyle\nu_{p}\left(\upepsilon_{p}\right)$
$\displaystyle=\frac{12\pi
n_{i}e^{4}Z^{2}}{vp^{2}}\int_{0}^{2p}dq\frac{1}{q\varepsilon^{2}(q)}\left(1-\frac{q^{2}}{4\upepsilon_{p}^{2}}\right)\int\frac{d\Omega_{q}}{4\pi}\left|F_{d}(\bm{q})\right|^{2}\frac{1}{2}\sin^{2}\theta_{q}\,.$
(50)

Eqs (49) and (50) are the main results of the paper. Under the assumption of
incoherent scatterings among the different pasta structures, and using the
analytical expressions eq. (42) for the form factors, they allow calculating
the thermal and electric conductivity of the pasta phase (see eq. (55) below)
at any arbitrary temperature, density, proton fraction and magnetic field
value from a given nuclear physics model providing the composition of the
matter, namely the values of $L$, $R$ and $Z$ for the dominant pasta geometry.
Some representative results will be given in the next section.

To compute the transport coefficients, the collision frequencies eqs (49) and
(50) must be calculated at the Fermi energy $\upepsilon_{p}=\upepsilon_{F}$,
since in the strongly degenerate electron gas transport occurs close to the
Fermi surface. They can be written compactly as

$\nu_{K}=\frac{12\pi n_{i}e^{4}Z^{2}}{v_{F}p_{F}^{2}}\Lambda_{K},$ (51)

where $p_{F}$ ($v_{F}$) is the Fermi momentum (velocity). We have also defined
the axial ($K=a$) and perpendicular ($K=p$) Coulomb logarithms as

$\Lambda_{K}=\int_{0}^{2p_{F}}\frac{dq}{q}\frac{1}{\varepsilon^{2}(q)}\left(1-\frac{q^{2}}{4\upepsilon_{F}^{2}}\right)\langle
F^{2}\rangle_{K}\,,$ (52)

and the averages $\langle F^{2}\rangle_{K}$ as

$\langle
F^{2}\rangle_{a}=\int\frac{d\Omega_{q}}{4\pi}\left|F_{d}(\bm{q})\right|^{2}\cos^{2}\theta_{q}\,,$
(53)

and

$\langle
F^{2}\rangle_{p}=\int\frac{d\Omega_{q}}{4\pi}\left|F_{d}(\bm{q})\right|^{2}\frac{1}{2}\sin^{2}\theta_{q}\,.$
(54)

The calculation of the conductivities with the anisotropic collision
frequencies has been worked out in Yakovlev (2015), so here we only report the
main equations. The magnetic field is assumed, without loss of generality, to
lie in the $xz$–plane. By defining the unit vector
$\bm{b}=\bm{B}/B=b_{x}\hat{x}+b_{z}\hat{z}$ and the cyclotron frequency for
electrons $\omega=eB/\upepsilon_{F}$, the electric conductivity tensor can be
written as

$\hat{\sigma}=\frac{e^{2}n_{e}}{m_{e}^{\ast}\Delta}\begin{pmatrix}\nu_{a}\nu_{p}+\omega^{2}b_{x}^{2}&-\omega
b_{z}\nu_{a}&\omega^{2}b_{x}b_{z}\\\ \omega
b_{z}\nu_{a}&\nu_{a}\nu_{p}&-\omega b_{x}\nu_{p}\\\
\omega^{2}b_{x}b_{z}&\omega
b_{x}\nu_{p}&\nu_{p}^{2}+\omega^{2}b_{z}^{2}\end{pmatrix},$ (55)

where
$\Delta=\nu_{p}^{2}\nu_{a}+\omega^{2}b_{x}^{2}\nu_{p}+\omega^{2}b_{z}^{2}\nu_{a}$.
The thermal conductivity can be obtained by the Wiedemann-Franz law
$\kappa_{ij}=\sigma_{ij}(\pi^{2}T/3e^{2})$ which is valid for strongly
degenerate electrons (Ziman, 2001). We do not consider neutron superfluidity,
which results in corrections to the thermal conductivity, but not to the
electrical (Aguilera et al., 2009). For $B=0$, the conductivity becomes

$\hat{\sigma}_{0}=\frac{e^{2}n_{e}}{m_{e}^{\ast}}\begin{pmatrix}\nu_{p}^{-1}&0&0\\\
0&\nu_{p}^{-1}&0\\\ 0&0&\nu_{a}^{-1}\end{pmatrix}.$ (56)

##  4 Numerical results

Figure 6: Left panels (a,b): axial (full) and perpendicular (dashed) square
form factor of rods (yellow) and slabs (blue) at $n_{B}=0.06$ (a) and $0.08$
(b) fm-3. Right panels (c,d): average square form factors of rods (yellow, c
panel) and slabs (blue, d panel) are compared to the ones of droplets
(magenta) at the same density. The representative temperature T=3 MeV is
chosen.

The source of anisotropy entering the collision frequencies eq. (3) lies in
the angular dependence of the form factors $F_{d}(\bm{q})$. The latter is
displayed for rods and slabs in Fig. 5, as a function of the azimuthal angle
$\theta_{q}$ (see Fig.1) for different values of $q=k_{F}/2,k_{F},2k_{F}$. In
these figures, the temperature is fixed to $T=3\,$MeV and the two
representative densities $n_{B}=$0.06, 0.08 fm-3 are chosen, where rods
(slabs) are expected to be dominant according to the results presented in Fig.
2. We can see from Fig. 5 that for the rods, the form factor is strongly
peaked at $\theta_{q}=\pi/2$, while for slabs it is peaked at $\theta_{q}=0$
and $\pi$. Such behaviour is expected from their geometries, as the form
factors are peaked in the elongated direction.

However, this dependence is smoothed out by the angular average implied by eqs
(53) and (54). This is shown in Fig. 6, that displays the averaged form
factors $\langle F^{2}\rangle_{a,p}$ defined in eqs (53),(54). The average
value, given by $2\langle F^{2}\rangle_{p}+\langle F^{2}\rangle_{a}$, is also
plotted, in the same thermodynamic conditions as in Fig. 5. As we can expect
from Fig.1 and eq. (42), the form factor is maximum (minimum) in the symmetry
axis direction in the case of slabs (rods). This difference is pronounced at
momentum transfers smaller than $q=k_{F}$, as afterwards, both axial and
perpendicular components tend to zero. When comparing the form factors to the
one of the equivalent spherical droplet (right side of Fig.6), we can see that
the difference is essentially seen at low momentum as well, where the form
factor is systematically smaller within a deformed shape than for the
equivalent spherical geometry.

Figure 7: The Coulomb logarithms (upper part) and the ratio of perpendicular
to axial collision frequency (lower part) are shown as a function of the pasta
length normalized to the Wigner-Seitz radius, $L_{d}/R_{W}$. Quantities for
rods (slabs) are calculated at $n_{B}=0.06$ fm-3 ($n_{B}=0.08$ fm-3) and
plotted in yellow (blue). In the upper panel, the perpendicular (axial)
component are displayed as dashed (continuous) lines. The points in the lower
plot indicate the estimated effective length $L_{1}^{eff}=\sqrt{2}\,L_{1W}$
(see text).The representative temperature $T=3$ is chosen.

In the previous figures, we have estimated the effective pasta length $L_{d}$
based on the asymptotic behaviour of the thermal density correlation function
of the smectic phase in the perpendicular plane (see Figure 3 and associated
discussion). Though qualitatively the physical origin of the electron-pasta
scattering is certainly the breaking of the long-range order due to thermal
fluctuations, our estimations are very rough and would deserve to be
confronted with microscopic molecular dynamics simulations (Horowitz et al.,
2015; Caplan et al., 2021; Newton et al., 2022; Pelicer et al., 2021; Nandi &
Schramm, 2018). To evaluate the effect of the uncertainty on the estimation of
$L_{d}$, in Fig. 7 we show the axial and perpendicular Coulomb logarithms
eq.(52) and the ratio of perpendicular to axial collision frequency eq.(51) as
a function of the ratio of the pasta length $L_{d}$ to the WS-radius $R_{Wd}$
for rods and slabs.

Figure 8: Axial (continuous) and perpendicular (dashed) Coulomb logarithms as
a function of baryon density, for $T=1$ MeV (left panel) and $T=3$ MeV (right
panel). The different pasta geometries are indicated with different colours.
Figure 9: Top: ratio of perpendicular to axial relaxation time. Bottom:
Average electric conductivity. Curves are shown as a function of density for
the representative temperatures of $T=$ 1 (dashed) and 3 (continuous) MeV. The
different pasta geometries are indicated with different colours.

For both geometries, the Coulomb logarithm decreases with the increasing
length of the pasta, and its value tends to zero as $L_{d}$ becomes
sufficiently large. This is consistent with the expectation that the lattice
order should suppress the electron-ion scattering and increase the
conductivity of matter.

For rods (slabs), the perpendicular component is larger (smaller) than the
axial one, and the difference between them increases with the growing length
of the pasta, varying up to 100 (0.01) when $L_{d}\approx 150R_{W}$. We can
see that a precise estimation of the length of the structures is important for
the quantitative determination of the collision frequencies, as it affects in
a considerable way the difference between the two scattering directions. In
particular, the deviation from an isotropic scattering is small only for small
values of $L_{d}/R_{W}$, corresponding to the high-temperature regime. At
smaller temperatures, as correlations become more important, a larger
transverse length will contribute to the scattering, so the difference in the
anisotropic frequencies will be more pronounced, likely reaching those
expected in Yakovlev (2015).

In Fig. 8 the Coulomb logarithms eqs(52) are shown as a function of the
density for T=1 and 3 MeV. In both cases, we can see that the abrupt change of
favoured geometry leads to slight discontinuities in the Coulomb logarithms,
and both overall decrease with density. This can be understood from the
increase in length, shown in Fig. 4, and from Fig. 7.

The ratio of perpendicular to axial collision frequency eqs (51) and the
average conductivity are shown in Fig. 9 for the same temperatures. The slight
increase in the ratio with density is due to the increasing length. It is
important to note that, at the temperatures and lengths we are considering,
the different collision frequencies differ by a factor smaller than two, so
there is only a small deviation from isotropic scattering at high
temperatures. In the average conductivity on the other hand, there is a larger
discontinuity when the abrupt change of geometry happens. This difference is
mainly due to the associated discontinuity in proton number, which can be seen
on the right side of Fig. 4. All in all, the anomalously high resistivity of
the pasta layer reported in the literature (Caplan et al., 2018) is nicely
reproduced by our calculations, and it is seen to be essentially due to the
high $Z$ value of the clusters close to the crust-core transition, more than
to the specific geometry of the pasta phases.

Figure 10: Components of the electric conductivity in units of the
perpendicular conductivity at $B=0$, as a function of the magnetic field for
rods at $n_{B}=0.06$ fm-3. The angle between the pasta symmetry axis and the
magnetic field is fixed at 0 (left), 45º (center) and 90º (right). In the top
axis, we show the variable $x_{p}=eB/(\upepsilon_{F}\nu_{p})$.  Figure 11:
Components of the electron conductivity in units of the axial conductivity at
$B=0$, as a function of the magnetic field for slabs at $n_{B}=0.08$ fm-3. The
angle between the pasta symmetry axis and the magnetic field is fixed at 0
(left), 45º (center) and 90º (right). In the top axis, we show the variable
$x_{a}=\omega/\nu_{a}$.

We now turn to the effect of the magnetic field on the conductivities. When
including the magnetic field, we show all components of the conductivity in
units of the dominant conductivity at zero magnetic field, i.e. the
perpendicular (axial) conductivity for rods (slabs). In Figs. 10 (11) we show
the conductivity components when the magnetic field forms an angle
$\theta_{b}=0,45$º, and $90$º with the symmetry axis of the pasta. Different
off-diagonal components appear depending on the angle of the magnetic field:
if it lies in the symmetry axis of the pasta, only the perpendicular $xy$
component is not zero, the $zz$ component depends only on $\nu_{a}$, and the
perpendicular $xx$ and $yy$ components are determined by both $\nu_{p}$ and
$\bm{B}$. If it lies perpendicularly to the symmetry axis, only $zy$ is not
zero, the $zz$ component is determined by $\nu_{a}$ and $\bm{B}$ and $xx=yy$
only by $\nu_{p}$. At magnetic fields $B<10^{18}$ G, the $zz$ component is not
drastically modified, transverse components increase (decrease) for rods
(slabs), and the off-diagonal terms increase steadily. At $\sim 10^{18}$ G,
the diagonal components parallel to the magnetic field are unaffected, but the
perpendicular and off-diagonal components start to decrease. A magnetic field
of this order is not far from the one expected at the very bottom of the inner
crust of magnetars, which is about $20\%$ of the field in the core (Chatterjee
et al., 2019; Fujisawa & Kisaka, 2014).

For the average conductivity, we follow Yakovlev (2015) once more, and assume
the pasta takes random orientations with respect to the magnetic field since
up to date there is no information regarding its orientation or prevalence of
domains. To calculate the average parallel, perpendicular, and Hall terms we
define a plane orthogonal to the magnetic field with the vectors $\bm{e}_{1}$,
$\bm{e}_{2}=\bm{e}_{1}\times\bm{b}$ and make the projections:
$\sigma_{\perp}=\bm{b}\cdot\hat{\sigma}\cdot\bm{b}$,
$\sigma_{\|}=\bm{e}_{1}\cdot\hat{\sigma}\cdot\bm{e}_{1}$ and
$\sigma_{H}=\bm{e}_{1}\cdot\hat{\sigma}\cdot\bm{e}_{2}$, such that averaging
the coefficients over all directions leads to:

$\begin{pmatrix}\langle\sigma_{\perp}\rangle\\\ \langle\sigma_{\|}\rangle\\\
\langle\sigma_{H}\rangle\end{pmatrix}=\frac{e^{2}n_{e}}{m_{e}^{\ast}\omega^{2}}\begin{pmatrix}(\omega^{2}+\nu_{p}\nu_{a})(\nu_{p}^{2}+\omega^{2})H-\nu_{p}\\\
\frac{1}{2}[\nu_{a}\nu_{p}(\omega^{2}-\nu_{p}^{2})H+\nu_{p}]\\\
\omega(1-\nu_{a}\nu_{p}H)\end{pmatrix}\text{ with
}H=\begin{cases}(sr)^{-1}\arctan(s/r)&\nu_{a}>\nu_{p}\\\
(sr)^{-1}\operatorname{arctanh}(s/r)&\nu_{a}<\nu_{p}\\\
(\nu_{a}^{3}+\omega^{2}\nu_{a})^{-1}&\nu_{a}=\nu_{p}\end{cases}$ (57)

$s=\omega\sqrt{|\nu_{a}-\nu_{p}|}$ and
$r=\sqrt{\nu_{p}(\omega^{2}+\nu_{a}\nu_{p})}$. For $B\rightarrow 0$ we get

$\langle\sigma_{\perp}\rangle=\langle\sigma_{\|}\rangle=\frac{e^{2}n_{e}}{m_{e}^{\ast}}\langle\nu^{-1}\rangle,\quad\langle\nu^{-1}\rangle=\frac{1}{3}\left(\frac{2}{\nu_{p}}+\frac{1}{\nu_{a}}\right)$
(58)

and the Hall parameters is zero $\langle\sigma_{H}\rangle\rightarrow 0$. One
must notice that the average conductivity is proportional to the average of
the inverse of $\langle\nu\rangle$, and not $\langle\nu\rangle$ itself,
therefore its calculation does not amount to averaging the matrix element over
$\Omega_{q}$.

Figure 12: Average parallel, perpendicular and Hall conductivities for
randomly oriented rods (left) and slabs (right) at $n_{B}=0.06$ fm-3 and
$n_{B}=0.08$ fm-3, respectively.  In the top axis, we show the variable
$x_{i}=\omega/\nu_{i}$, with $i=p\;(a)$ on the left (right).

In Fig. 12 we show the average conductivities for rods and slabs,
respectively, in units of the average conductivity with $B=0$. This figure can
be compared with Fig.3 of Yakovlev (2015), though the parallel component in
our calculation is not so different from the perpendicular one due to the
small difference we found between the anisotropic frequencies, unlike the
assumption of Yakovlev (2015). To conclude this discussion, it is important to
note that for all the calculations reported in this paper, the inner crust
structure was computed without accounting for the magnetic field. Numerous
studies exist in the literature addressing this point, using CLD or Thomas-
Fermi techniques with different nuclear models, see e.g. Nandi et al. (2011);
de Lima et al. (2013); Bao et al. (2021); Wang et al. (2022). The general
result of these works is that only extreme magnetic fields of the order of
$B=10^{18}$ G affect the density profiles of the Wigner-Seitz cells, with an
increased average proton fraction, particularly in the outer part of the inner
crust dominated by spherical nuclei, and an increase of the charge of the
pasta structures, that however does not exceed $\approx 10-20\%$. These
modifications would not affect the results presented in Figs. 11 and 12, and
would lead to an extra decrease of the conductivity in Fig. 9, since
$\sigma\propto Z^{-2}$, see eq.(51).

##  5 Conclusions

In this paper, we studied the collision frequencies for elastic scattering
between electrons and two different pasta phase structures. To do so, we
performed an expansion of the collision integral in spherical harmonics, which
allowed us to treat also the scattering with non-spherical targets. We applied
this framework to calculate the electrical conductivity tensor.

The form factor of the pasta structures was evaluated by direct integration,
although we neglected contributions from the structure factor, which is
equivalent to assuming that electron scatterings with different pasta targets
are completely incoherent. This is a reasonable first approximation at high
temperatures of the order of the MeV or above. More work is needed to evaluate
the anisotropy of the transport coefficients at lower temperatures where the
lattice long-range order along the pasta symmetry axis is likely to be
preserved.

We find that anisotropic collision frequencies are highly dependent on the
length of the pasta structures. In the high-temperature regime, where the
effective length that participates in the scattering is comparable to the WS
length, the anisotropy is small and affects mainly the components of the
conductivity perpendicular to the pasta symmetry axis. It should be emphasized
that neutron star properties are expected to be significantly impacted by the
presence of different and mixed pasta geometries (Caplan et al., 2021;
Schneider et al., 2016; Newton et al., 2022), and their (possibly disordered)
mesoscopic arrangement. Unfortunately, information is still lacking on how
pasta domains, defects, and impurities appear at larger scales, but the
presence of this kind of disorder is considered to be a likely feature of the
pasta layers (Caplan et al., 2021; Schneider et al., 2016; Newton et al.,
2022; Pelicer et al., 2021). Our treatment, at the moment, does not include
precise corrections coming from scattering with domain boundaries and mixed
geometries. Future investigations of these matters must be incorporated within
the present framework to improve it.

Our numerical results are based on the IUFSU force, a simplified modelling of
the pasta phase using a one-component liquid drop approach, and an estimation
of the pasta sizes based on the values of the correlation between neighbouring
WS cells. However, the analytical results are general and can be used to
calculate the transport properties of the inner crust, with and without a
magnetic field, by using any microscopic estimation of the pasta linear
dimensions and proton number from microscopic mean-field or molecular dynamics
simulations.

## Acknowledgements

This work is a part of the project INCT-FNA Proc. No. 464898/2014-5, and of
the Master project In2p3 NewMAC. D.P.M. is partially supported by Conselho
Nacional de Desenvolvimento Científico e Tecnológico (CNPq/Brazil) under grant
303490/2021-7 and M.R.P. is supported by Conselho Nacional de Desenvolvimento
Científico e Tecnológico - Brasil (CNPq) and with a doctorate scholarship by
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (Capes/Brazil). M.
R. P. also acknowledges partial support from LPC Caen. MA acknowledges partial
support from PHAROS, COST Action CA16214.

##  Appendix A Isotropic limit

In this appendix, we show that the isotropic limit is obtained from eq. (14)
when $W_{pp^{\prime}}$ is a function only of $|\bm{q}|$ The equation obtained
is equivalent to to eq. (3.135) of Pines & Nozières (2018). In the isotropic
case, there is no change in the $l$ index of spherical harmonics during the
collision, and the sum of $m$ and $m^{\prime}$ indexes are zero since
$W_{pp^{\prime}}$ is a function of the relative angle between $\bm{p}$ and
$\bm{p^{\prime}}$, therefore:

$\mathcal{W}_{lm\,l^{\prime}m^{\prime}}=\mathcal{W}_{lm}\delta_{ll^{\prime}}\delta_{m\,-m^{\prime}}.$
(59)

This can be understood from the expansion in eq. (8), where in the isotropic
case the pair of spherical harmonics must be replaced by the Legendre
polynomial. Eq. (14) can be rewritten as

$\displaystyle\left[\nu\right]_{lm}^{l^{\prime}m^{\prime}}=\frac{p^{2}}{4\pi^{2}v}\Bigg{[}(-1)^{m^{\prime}}\sqrt{(2l+1)(2l^{\prime}+1)}\mathcal{W}_{00\;00}\begin{pmatrix}l&l^{\prime}&0\\\
0&0&0\end{pmatrix}$ $\displaystyle\times\begin{pmatrix}0&l&l^{\prime}\\\
0&m&-m^{\prime}\end{pmatrix}-\;(-1)^{m}\,\mathcal{W}_{lm\,l-m}\delta_{ll^{\prime}}\delta_{mm^{\prime}}\Bigg{]}.$
(60)

We simplify this expression by utilizing the following property of the
3j-Wigner symbols:

$\begin{pmatrix}l&l^{\prime}&0\\\
0&0&0\end{pmatrix}\begin{pmatrix}0&l&l^{\prime}\\\
0&m&-m^{\prime}\end{pmatrix}=\frac{(-1)^{-2l+m}}{2l+1}\delta_{ll^{\prime}}\delta_{mm^{\prime}}$
(61)

and defining $\mathcal{W}_{lm}=\mathcal{W}_{lm\;l-m}$, such that:

$\left[\nu\right]_{lm}^{l^{\prime}m^{\prime}}=\frac{p^{2}}{4\pi^{2}v}\delta_{ll^{\prime}}\delta_{mm^{\prime}}\Bigg{[}\mathcal{W}_{00}-(-1)^{m}\mathcal{W}_{lm}\Bigg{]}.$
(62)

To recover the usual integral equation with the transition matrix element we
use eq. (28), with the aid of eq. (48)

$\mathcal{W}_{lm}=\frac{2\pi}{p^{2}}\int\frac{d^{3}\bm{q}}{q}W_{pp^{\prime}}(-1)^{m}{{Y}_{l}^{m}}^{\ast}(\Omega_{p}){{Y}_{l}^{m}}^{\ast}(\Omega_{p^{\prime}})$
(63)

into eq. (62) and average over the $m$ index:

$\nu_{l}=\frac{p^{2}}{4\pi^{2}v(2l+1)}\int\frac{d^{3}\bm{q}}{q}W_{pp^{\prime}}\sum_{m}\Bigg{[}\frac{1}{4\pi}-{{Y}_{l}^{m}}^{\ast}(\Omega_{p}){{Y}_{l}^{m}}^{\ast}(\Omega_{p^{\prime}})\Bigg{]}.$
(64)

Using the property of spherical harmonics

$\sum_{m}{{Y}_{l}^{m}}^{\ast}(\Omega_{p}){{Y}_{l}^{m}}^{\ast}(\Omega_{p^{\prime}})=P_{l}(cos\xi)$
(65)

where $\cos\xi=\bm{p}\cdot\bm{p^{\prime}}/(|\bm{p}||\bm{p^{\prime}}|)$ and
changing variables as $2qdq=-p^{2}d(\cos\xi)$, we obtain

$\nu_{l}=\frac{p^{2}}{4\pi
v}\int_{-1}^{1}d(\cos\xi)W_{pp^{\prime}}\Bigg{[}1-P_{l}(\cos\xi)\Bigg{]},$
(66)

which is equivalent to eq. (3.135) of Pines & Nozières (2018) for electron
scattering with isotropic targets. We use $q^{2}=p_{F}^{2}(1-\cos\xi)$ to
change variables and write, for $l=1$:

$\nu_{1}(\upepsilon_{p})=\frac{1}{4\pi
p^{2}v}\int_{0}^{2p}q^{3}\,dqW_{pp^{\prime}},$ (67)

Using eq. (31), we recover eq. (2). Likewise, the viscosity can be obtained
with the $l=2$ – see eqs (2) and (3) of Chugunov & Yakovlev (2005):

$\nu_{2}(\upepsilon_{p})=\frac{3}{4\pi
p^{2}v}\int_{0}^{2p}q^{3}\,dq\Big{(}1-\frac{q^{2}}{4p^{2}}\Big{)}W_{pp^{\prime}}.$
(68)

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# Towards A Sign Language Gloss Representation Of Modern Standard Arabic

Salma El Anigri & M. Majid Himmi
LIMIARF Laboratory
Faculty of Sciences
Mohammed V University
Rabat, Morocco
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
Abdelhak Mahmoudi
LIMIARF Laboratory
Ecole Normale Supèrieure
Mohammed V University
Rabat, Morocco
<EMAIL_ADDRESS>

###### Abstract

Over 5% of the world’s population (466 million people) has disabling hearing
loss. 4 million are children. They can be hard of hearing or deaf. Deaf people
mostly have profound hearing loss, which implies very little or no hearing.
Over the world, deaf people often communicate using a sign language with
gestures of both hands and facial expressions. The Sign Language is a full-
fledged natural language with its own grammar and lexicon. Therefore, there is
a need for translation models from and to sign languages. In this work, we are
interested in the translation of Modern Standard Arabic (MSAr) into Sign
Language. We generated a gloss representation from MSAr that extracts the
features mandatory for the generation of animation signs. Our approach locates
the most pertinent features that maintain the meaning of the input Arabic
sentence.

##  1 Introduction

The main impacts of deaf is on the individual’s ability to communicate with
others in addition to the emotional feelings of loneliness and isolation in
society. Consequently, they can not equally access public services, mostly
education and health and have not equal rights in participating at the active
and democratic life. This leads to a negative impact in their lives and the
lives of the people surrounding them.

Over the world, deaf people often communicate using a sign language with
gestures of both hands and facial expressions. The Sign Language is a full-
fledged natural language with its own grammar and lexicon. However, sign
languages are not universal where each natural language have its corresponding
sign language. As research in machine translation from one language to another
increased in recent years; mainly due to advances in deep learning field;
research in translation from and to sign languages increased as well.

Typically, a machine translation system from an input language to a sign
language can be divided into two stages. The first stage consists of finding a
mapping between the text of the input language and a sign representation. In
the second stage, the sign representation is matched to a visual sign using an
animated avatar (Fig. 1). The representation of the sign could be textual; one
talk about the gloss representation; or symbolic such as Hamburg
representation Hanke (2004).

In the literature, in the best of our knowledge, there is very few research on
the translation of Modern Standard Arabic (MSAr) into Sign Language. This is
mainly due to the morphological and grammatical complexity of the MSAr and to
the lack of data. In this work, we aim to generate a gloss representation from
MSAr that extracts the features mandatory for the generation of the animation
signs.

Figure 1: Translation of a natural language to a sign animation.

##  2 Problem Statement

Unlike the majority of the natural languages, the Modern Standard Arabic
Language is written from right to left. Furthermore, it contains ambiguities
in many aspects, such as problems to identify the right function or meaning of
a word, the absence of capital letters which complicates the differentiation
between proper nouns, adjectives, nouns, abbreviations and acronyms Farghaly &
Shaalan (2009). In statistical Arabic translation systems, dealing with those
ambiguities were shown to have significant effects Oudah et al. (2019). This
observation remains true when one want to translate from MSAr to a sign
language gloss representation Rayner et al. (2016); Aouiti & Jemni (2018);
Luqman & Mahmoud (2019). In this work, we want to generate gloss
representation from MSAr while taking into consideration both the pre-
processing phase and the usability of such a representation in the animation
stage.

##  3 Approach and Results

This study aims to extract a set of relevant features while preserving the
meaning of the input phrase, to perform understandable sign animations by the
deaf.

This can be achieved by extracting phrase tense and forms using the lemmas and
the part of speech tags of each token. Furthermore, words expressing emphasis
like ”very (جدا)” and ”often (مرارا)” have to be recognised. On one side,
depending on the part of speech tags, one can capture the past, present or
future tense of the phrase. By default, a verb is always in its past tense. In
the case where it begins with (س), it switches to the future tense. Otherwise,
if it begins with ت، ن، ي، أ, the verb can be considered to be in the present
tense. On the other side, a phrase form could be negative, interrogative or
conditional. In the case of negative form, the tense can be captured using
particles. Indeed, the negative particle ”la (لا)” and ”layssa (ليس)”
indicates the present tense, ”lan (لن)”, indicates the future tense and ”lam
(لم” indicates the past tense. In addition, usually, interrogative forms can
be detected using the question mark (?) punctuation. Each interrogative form
could be labeled depending the associated adverb (”is (هل)”, ”who (من)”,
”where (أين)”, ”when (متى), ”what (ماذا)” and ”how (كيف)). Lastly, phrases in
the conditional form contain the ”if” token (”if (إذا) and ”if (لو)”.

For the extraction of features we were based on the work of Aouiti & Jemni
(2018), especially the way they introduced the glossing rules by analogy to
the ASL gloss in Aouiti et al. (2015). Our contribution can be seen in the way
we extracted each feature; we combined more grammar rules of the Arabic
language. Such as, To detect the future tense, we see if the part of speech
tags contains a future particle sin ” (س )”, or the particle lan ” (لن )” is
identified. Figure 2 shows two gloss representations based on those
established rules.

Figure 2: Two examples of the gloss representation of two Arabic sentences.
Top, interrogative form and down, negative form.

##  4 Conclusion and Perspectives

In this work, we were interested in the first stage of the translation from
Modern Standard Arabic to sign language animation that is generating a sign
gloss representation. We identified a set of rules mandatory for the sign
language animation stage and performed the generation taking into account the
pre-processing proven to have significant effects on the translation systems.
The presented results are promising but far from well satisfying all the
mandatory rules. For example, the gloss representation we proposed don’t take
into account the pronouns to express genders, plurals or singulars.
Furthermore, the named entities expressing locations, people or times are not
included. In upcoming work, we aim to tackle these problems.

### Acknowledgments

This work has been funded by Artificial Intelligence 4 Development (AI4D)
programme as part of the 1st Call for Project Proposals on Artificial
Intelligence 4 Development Technologies.

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