ELECTRON–ION SCATTERING IN DENSE MULTI-COMPONENT PLASMAS: APPLICATION TO THE OUTER CRUST OF AN ACCRETING NEUTRON STAR

Electron–ion scattering impedes the transport of electronic momentum and energy. The resulting electrical conductivity σ and the thermal conductivity κ can generally be expressed in terms of the collision frequencies ν_σ and ν_κ:

$$\begin{equation} \sigma =\frac{n_{\rm e}e^{2}}{m_{\rm e}^{*}}\tau _{\sigma } \end{equation} \tag{ 7 }$$

$$\begin{equation} \kappa =\frac{1}{3}v_{\rm F}^{2}C_{\rm V}\tau _{\kappa }=\frac{\pi ^{2}k_{\rm B}^{2}Tn_{\rm e}}{3m_{\rm e}^{*}}\tau _{\kappa } \end{equation} \tag{ 8 }$$

where $m_{\rm e}^{*}=m_{\rm e}\sqrt{1+x_{\rm r}^{2}}$ is the effective electron mass, v_F = p_F/m*_e is the Fermi velocity, and C_V is the heat capacity of the free relativistic electron gas.

The quantities τ_σ and τ_κ describe the typical timescale over which electron–ion collisions degrade the electric and thermal currents. Their inverses, the frequencies ν_σ,κ = 1/τ_σ.κ, can conveniently be written as the product of a "fundamental" frequency ν₀:

$$\begin{eqnarray} &&\nu _{0}=\frac{4\pi \langle Z^{2}\rangle e^{4}n_{i}}{p_{\rm F}^{2}v_{\rm F}}=\frac{4\alpha ^{2} E_{\rm F}}{3\pi \hbar }\frac{\langle Z^{2}\rangle }{\langle Z\rangle }, \end{eqnarray} \tag{ 9 }$$

multiplied by a dimensionless quantity ln Λ_σ,κ known as the Coulomb logarithm:

$$\begin{eqnarray} &&\nu _{\sigma,\kappa }=\nu _{0}\ln \Lambda _{\sigma,\kappa }. \end{eqnarray} \tag{ 10 }$$

In Equation (9), 〈Z²〉 results form the appropriate normalization S_zz(k → ∞) → 1 of the structure factor discussed below in Equation (15), α = e²/ℏc is the fine-structure constant, and E_F = m*_ec² is the electron Fermi energy. The fundamental frequency ν₀ can be seen as a frequency of Coulomb collisions associated with the transfer of momentum between electrons and ions; it can be written as

$$\begin{eqnarray} &&\nu _{0}=n\left[4\pi \left(\frac{\langle Z^{2}\rangle ^{\frac{1}{2}}e^{2}}{m_{e}^{*}v_{\rm F}^{2}}\right)^{2}\right]v_{\rm F}, \end{eqnarray} \tag{ 11 }$$

in which the bracketed term can be interpreted as the cross-section for momentum transfer in a binary collision between an electron of velocity v_F and charge −e and a massive ion of charge $\langle Z^{2}\rangle ^{\frac{1}{2}}e$. The dimensionless Coulomb logarithm contains all the fine details of electron–ion collisions in a plasma, such as charge screening, collective modes, and quantum diffraction effects. In general, ln Λ_σ ≠ ln Λ_κ since the two transport coefficients measure different mechanisms, namely momentum and energy transport. However, when the dominant source of electron–ion scattering is elastic, as in the disordered fluid phase or the high-temperature crystalline phase, the two Coulomb logarithms are equal and the Wiedemann–Franz relation

$$\begin{eqnarray} &&\kappa /\sigma =(\pi ^{2}k_{B}^{2}/3e^{2})T \end{eqnarray} \tag{ 12 }$$

is satisfied.

A calculation of the Coulomb logarithm that self-consistently treats all the many-body effects occurring in a MCP is in general rather complicated. However, because of the high electron density in neutron stars, the typical electron kinetic energy is very large compared to the electron–ion interaction energy and the average electron–ion correlations are weak. Accordingly, a good approximation can be made by considering the ions and electrons in the plasma as two weakly interacting subsystems, namely an MCP neutralized by a homogeneous negatively charged background and an homogeneous interacting electron gas (i.e., relativistic quantum electronic jellium.) Electrons scatter off the electrostatic potential created by the ionic charge distribution,

$$\begin{eqnarray} &&n_{z}({\bf r},t)=\sum _{j=1}^{N_{\rm sp}}{Z_{j}\sum _{l=1}^{N_{j}}{\delta \left({\bf r}-{\bf R}_{l}(t)\right)}}, \end{eqnarray} \tag{ 13 }$$

where R_l(t) is the position of the lth nuclei of the jth species at time t, and N_j = n_jV is the number of particles of species j. In the (first) Born approximation¹ of the electron–ion interaction and using the fact that the electrons are much heavier than the ions, we obtain (Flowers & Itoh 1976)

$$\begin{eqnarray} &&\ln \Lambda _{\sigma,\kappa }=\int _{0}^{2k_{\rm F}}{dkk^{3}\Big |\frac{v(k)}{\epsilon _{e}(k)}\Big |^{2}\left[1-\frac{x_{\rm r}^{2}}{1+x_{\rm r}^{2}}\left(\frac{k}{2k_{\rm F}}\right)^{2}\right]{\cal {S}_{\sigma,\kappa }}(k)},\nonumber\\ \end{eqnarray} \tag{ 14 }$$

with v(k) = 1/k² and _e(k) is the static dielectric function of the electron jellium.

Equation (14) can be interpreted as follows. The Coulomb logarithm ln Λ_σ,κ sums all the contributions of collisions characterized by momentum transfer k. The range of integration is limited to 2k_F because the electrons are degenerate and only those electrons near the Fermi surface take part in exchanges of momentum and energy. The electrons behave as if they were scattered independently of each other by the electronically screened electron–ion potential. Because the electrons are much lighter than the nuclei, the energy exchanges ℏω in inelastic collisions occur in a range over which _e(k, ω) ≈ _e(k, 0) = _e(k). For relativistic, degenerate electrons, an analytical expression for _e was obtained by Jancovici (1962) within the random-phase approximation (see Appendix B).

The bracketed term, which describes the kinematic suppression of backward scattering of relativistic electrons, $\big[ 1-\frac{x_{\rm r}^{2}}{1+x_{\rm r}^{2}}\big(\frac{k}{2k_{\rm F}}\big)^{2} \big]$, varies between unity for x_r → 0 and [1 − (k/2k_F)²] for x_r → ∞.

The many-body physics of the ionic subsystem is encapsulated in the function ${\cal {S}_{\sigma,\kappa }}(k)$. ${\cal {S}_{\sigma,\kappa }}(k)$ is a functional of the spectrum of charged fluctuations δn_z(k, t) = n_z(k, t) − 〈n_z(k)〉_eq in the ionic system

$$\begin{eqnarray} &&S_{\rm zz}(k,\omega)=\frac{1}{nV\langle Z^{2}\rangle }\int {\frac{d\omega }{2\pi }\left\langle \delta n_{z}({\bf k},t)\,\delta n_{z}(-{\bf k},0)\right\rangle _{\rm eq} e^{i\omega t}}\nonumber\\ \end{eqnarray} \tag{ 15 }$$

where 〈 〉_eq denotes the average of a canonical ensemble at temperature T. In other words, ${\cal {S}_{\sigma,\kappa }}(k)$ self-consistently takes into account all of the separate electron–ion relaxation mechanisms at the level of the Born approximation; it is really an umbrella term for electron–ion, electron–impurity, electron–phonon scattering, and includes multi-phonon processes. We have, from Flowers & Itoh (1976),

$$\begin{eqnarray*} \!\!\!{\cal {S}}_{\sigma }(k)&=&\int _{-\infty }^{\infty }{S_{\rm zz}(k,\omega)zn(z)} \quad z=\hbar \omega /k_{B}T\\ \!\!\!{\cal {S}}_{\kappa }(k)&=&\int _{-\infty }^{\infty }{S_{\rm zz}(k,\omega)zn(z)\left[1+\left(\frac{3k_{\rm F}^{2}}{k^{2}}-\frac{1}{2}\right)\right]z^{2}} \end{eqnarray*}$$

where n(z) = 1/1 − e^−z.

In the special case of a periodic, crystalline structure, the term 〈n_z(k)〉_eq peaks at values of the wavevector k commensurate with the periodicity of the lattice and subtracts out the elastic Bragg peaks: electron–ion scattering involves Bloch electrons and Equation (14) is obtained by approximating their behavior by means of free particle wave functions.

When the scattering is predominantly elastic in nature, as is expected in the fluid and solid phases at T > T_p or in a disordered (non-periodic) lattice, we have

$$\begin{equation} \hspace{-8.8pc}{\cal {S}}_{\sigma,\kappa }(k)=\int _{-\infty }^{\infty }{S_{\rm zz}(k,\omega)} \end{equation} \tag{ 16 }$$

$$\begin{equation} =\frac{1}{nV\langle Z^{2}\rangle }\left\langle \delta n_{z}({\bf k},0)\,\delta n_{z}(-{\bf k},0)\right\rangle _{\rm eq} \end{equation} \tag{ 17 }$$

$$\begin{equation} \hspace{-9pc}\equiv S_{\rm zz}(k) \end{equation} \tag{ 18 }$$

where S_zz(k) is the (charge–charge) structure factor. This leads to

$$\begin{eqnarray} &&\ln \Lambda _{\sigma,\kappa }=\int _{0}^{2k_{\rm F}}{dkk^{3}\Big |\frac{v(k)}{\epsilon (k)}\Big |^{2}\left[1-\frac{x_{\rm r}^{2}}{1+x_{\rm r}^{2}}\left(\frac{k}{2k_{\rm F}}\right)^{2}\right]S_{\rm zz}(k)}.\nonumber\\ \end{eqnarray} \tag{ 19 }$$

In this limit, σ and κ are insensitive to the details of the spectrum of ionic charge fluctuations. These details (ion collective modes and single-particle effects) are present implicitly but in an integrated form via the sum-rule (18).

Until now, we have assumed that the ions are point particles. In this "point-charge" approximation, the only species-dependent term in the integrand of Equation (14) is the charge–charge structure factor S_zz(k). In order to allow for finite nuclear size, one can replace the density (13) with

$$\begin{eqnarray} &&n_{z}({\bf k},t)=\sum _{j=1}^{N_{\rm sp}}{Z_{j}F_{j}(k)\sum _{l=1}^{N_{j}}{e^{-i{\bf k}\cdot {\bf R}_{l}(t)}}}, \end{eqnarray} \tag{ 20 }$$

where F_j(k) is the nuclear form factor, which reflects the charge distribution within a nucleus of species j. For instance, assuming uniform proton density in the nuclei,

$$\begin{eqnarray} &&F_{j}(k)=\frac{3}{(kR_{j})^3}(\sin (kR_{j})-kR_{j}\cos (kR_{j})) \end{eqnarray} \tag{ 21 }$$

where R_j the charge radius of the nuclear species j which has mass number A_j; we use R_j = 1.15A^1/3_j fm as in Kaminker et al. (1999). Direct MD calculation of the structure factor S_zz(k) with and without the nuclear form factor (21) shows that the finite nuclear size corrections are completely negligible in the outer crust of interest in this paper. (Note that for instance (F_j(k))² = 0.9766 at the peak of S_zz(k), i.e. at (k/2k_F) ≈ 0.33, for a typical heavy nucleus having mass number A_j ≈ 64–100 or A^1/3_j ≈ 4–4.6). Accordingly, we neglect finite nuclear size effects in the remainder of this paper.

In NPB, the electrons are accelerated by the Coulomb field of the crust as in photon Bremsstrahlung but with the emission of a neutrino–antineutrino pair instead of a photon. In the first Born approximation, the neutrino emissivity is

$$\begin{eqnarray} &&\epsilon _{\nu }=\frac{8\pi G_{\rm F}\langle Z^{2}\rangle e^{4}C_{+}^{2}}{567\hbar ^{9}c^{8}}(k_{\rm B}T)^{6}\ln \Lambda _{\nu } \end{eqnarray} \tag{ 22 }$$

with the Coulomb logarithm for NPB,

$$\begin{eqnarray} \ln \Lambda _{\epsilon }&=&\frac{\hbar c}{k_{\rm B}T}\int _{0}^{2k_{\rm F}}{dk_{\rm t} k_{\rm t}^{3}\int _{0}^{\infty }{dk_{\rm r}\Big |\frac{v(k)}{\epsilon (k)}\Big |^{2}S_{\rm zz}(k)R_{\rm T}(k_{\rm t},k_{\rm r})}} \nonumber \\ &\approx &\int _{0}^{2k_{\rm F}}{dkk^{3}\Big |\frac{v(k)}{\epsilon (k)}\Big |^{2}a(k)S_{\rm zz}(k)} \end{eqnarray} \tag{ 23 }$$

where k = k_t + k_r, G_F is the Fermi weak coupling constant, C²₊ ≈ 1.675 from the vector and axial-vector constants of the weak interaction, and $a(k)= \left[ 1-\frac{2k^{2}}{4k_{\rm F}^{2}-k^{2}}\ln \left(\frac{k}{2k_{\rm F}}\right) \right]$. The function R_T describes the thermal broadening of the electron Fermi surface and from Equation (20) of Haensel et al. (1996) this can be approximated as unity for T ≲ 0.6 GK typical of models of the neutron star crust at accretion rates characteristic of superburst progenitors (Kaminker et al 1999).

Thus, in both Equations (14) and (23) the only dependence on the nuclear species present (i.e., on the composition) enters via S_zz(k). Therefore, to tabulate fits to both κ and _ν for every nuclear species prior to application of a mixing rule is extremely cumbersome (note that the papers Itoh et al. 2008, 1996 tabulate fit coefficients for only 11 species for conductivities and NPB emissivities, respectively) and also unnecessary from a computational standpoint. The fits to OCP structure factors in Young et al. (1991) are sufficient to cover every one of the N_sp > 100 species encountered in the neutron star crust. They can also be used to compute both κ and _ν for an MCP of arbitrary complexity using the microscopic mixing rule (MLMR) described in Section 4 which involves only the OCP structure factors and species abundances. For higher values of Γ>225 in the neutron star crust we present a simple model for the behavior of the Coulomb logarithm that reduces the evaluation of thermal and electrical conductivity to an analytic formula incorporating crust impurity in a very simple way. We do not present calculations of the NPB neutrino emissivity in this paper, rather we point it out to demonstrate that all such "integral" determinants (i.e., summations over momenta) of crust thermal profile are related to the crust composition via the quantity S_zz(k).

For simplicity, we drop the σ, κ, and ν subscripts in what follows.

The most common approach of electron–ion scattering in the neutron star crust relies on the belief that the MCP is fluid in the upper part of the crust where Γ_eff < 175 and consist of a solid, i.e. a crystalline lattice with impurities, in the lower part where Γ_eff > 175. Accordingly, in the solid phase, the collision frequency ν = ν_e-ph + ν_e-imp is split into the sum of the electron–phonon and the electron–impurity scattering contributions; in the liquid state ν = ν_ei where ν_ei is the electron–ion scattering inverse relaxation time discussed in the next section.

The splitting ν = ν_e-ph + ν_e-imp was initially introduced in Flowers & Itoh (1976) to deal with periodic crystals with a small fraction of the lattice sites occupied by impurities. The idea is illustrated in Figure 2. The case of binary mixtures (N_sp = 2) was discussed in detail in Itoh & Kohyama (1993) for a small concentration of impurity ions (say species "2"): the contribution of the impurity to electron–ion scattering is equal to its concentration times the binary electron–ion collision frequency between an electron and a "residual" ion of charge (Z₂ − Z₁). This idea was extended to deal with more complex MCPs containing N_sp ⩾ 3 species, in which case the system is treated as a perfect crystal of charge 〈Z〉 plus the contribution of uncorrelated impurities of charge (Z_i − 〈Z〉).

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Within this approximation and defining the impurity parameter as

$$\begin{eqnarray} &&Q_{\rm imp}=\sum _{j=1}^{N_{\rm sp}}{x_{i}(Z_{i}-\langle Z\rangle)^{2}}, \end{eqnarray} \tag{ 24 }$$

the relaxation frequency can be written as

$$\begin{eqnarray} &&\nu _{\rm imp}=\nu _{0,\rm imp}\ln \Lambda _{\rm imp} \end{eqnarray} \tag{ 25 }$$

where

$$\begin{eqnarray} &&\nu _{0,\rm imp} = \frac{4\alpha ^{2}\epsilon _{\rm F}}{3\pi \hbar }\frac{Q_{\rm imp}}{\langle Z\rangle } \end{eqnarray} \tag{ 26 }$$

and

$$\begin{eqnarray} &&\ln \Lambda _{\rm imp}=\int _{0}^{2k_{\rm F}}{dkk^{3}\Big |\frac{v(k)}{\epsilon (k)}\Big |^{2}\left[1-\frac{x_{\rm r}^{2}}{1+x_{\rm r}^{2}}\left(\frac{k}{2k_{\rm F}}\right)^{2}\right]}. \end{eqnarray} \tag{ 27 }$$

ln Λ_imp corresponds to Equation (20) with S_zz set to unity to correspond to independent (uncorrelated) electron–impurity collisions.

While the impurity parameter formalism was developed to treat small impurity concentration in an otherwise almost perfect crystal lattice (see Figure 2), it has also been used to characterize electron–ion scattering in complex fluid and amorphous MCPs when Q_imp is large, say Q_imp ∼ 100. Such is usually the case when electron–impurity scattering is used to set a lower bound on conductivity for the entire crust. However, since Λ_imp is of the order of magnitude as Λ, we have

$$\begin{equation} \nu /\nu _{\rm imp}=\frac{\langle Z^{2}\rangle }{Q_{\rm imp}}\frac{\ln \Lambda }{\ln \Lambda _{\rm imp}} \end{equation} \tag{ 28 }$$

$$\begin{equation} \hspace{1.8pc}\approx \frac{\langle Z^{2}\rangle }{Q_{\rm imp}}\sim 10 \end{equation} \tag{ 29 }$$

over the whole outer crust as shown in Figure 1, the impurity parameter can lead to conductivities much higher than the those that would be obtained using the actual structure factors employed in Equation (20). We thus anticipate that Q_imp is inappropriate for the outer crust (impure liquid metal approaching heterogeneous amorphous solid at high density), and instead we shall justify the usage of 〈Z²〉 from its presence in our expression for the fundamental frequency ν₀ in Equation (9). The differences in crust conductivity that would result from these differing prescriptions are discussed in Section 5.

In the liquid state, Potekhin et al. (1999) proposed a different approach to deal with mixtures. In this formalism, one writes

$$\begin{eqnarray} \ln \Lambda &\approx & \frac{1}{\langle Z^{2}\rangle }\sum _{j}^{N_{\rm sp}}{x_{j}Z_{j}^{2}\ln \Lambda ^{\rm OCP}(\Gamma _{j},Z_j,x_r)} \end{eqnarray} \tag{ 30 }$$

where Γ_j = Z^5/3_jΓ_e, x_j refers to the number fraction as defined in Equation (1), and ln Λ^OCP(Γ_j, Z_j, x_r) is the Coulomb logarithm (20) for an OCP of charge Z_j at coupling Γ_j and relativistic parameter x_r. This prescription for mixtures was also recommended recently in the works of Cassisi et al. (2007) and Itoh et al. (2008) and we corroborate its efficacy in Section 4.

In Sections 4 and 5, we propose and validate with MD simulations a mixing rule for |ln Λ that is slightly modified from Equation (30). We also show that the impurity parameter formalism is inappropriate at large Q_imp.

The previous expressions tell us that the conductivities and NPB emissivity depend on the scattering of electrons off the fluctuations δn_z(k, t) in the ionic charge density about the average value 〈n_z(k)〉_eq. However, this average charge density is dependent upon the thermodynamic phase of the system.

• 1.  
  If the system is fluid, the averaged charge density 〈n_z(k)〉_eq = n_e δ(k) is uniform.
• 2.  
  If the system is in a crystalline solid state, i.e., comprised of a periodic (Bravais) lattice with the same physical unit, electrons are in Bloch states and 〈n_z(k)〉 = e^−2W(k)n_eV∑_Kδ_k,K depends on k, where, assuming N_sp = 1, K are reciprocal lattice vectors and W(k) is the Debye–Waller factor: Bloch electrons do not scatter off the static, wavevector-dependent charge fluctuations (Bragg peak).

Because of this fundamental difference, the formulae of the previous section predict significant (by a factor of 3–5) discontinuities of the electric and thermal conductivities at the melting point (Itoh et al. 1993). Recently, Baiko et al. (1998) argued that this jump in the OCP electronic transport properties upon crystallization is not physical and that the electron transport properties in strongly coupled plasmas should be fairly insensitive to the thermodynamic state of the plasma. Accordingly, they advance that strongly coupled Coulomb liquids are characterized locally by the same long-range order as present in the crystalline solid and although the long-range order does not persist forever as in a solid (the liquid state is fluid), electrons keep traveling in those local crystal-like structures without any significant degradation of their mean velocity. Thus, Baiko et al. (1998) advocate that, in the liquid state, the contribution corresponding to elastic scattering off the incipient crystalline structure must be subtracted from the static structure factor. As intended, this prescription removes the jump in the conductivities at the solid–liquid transition—this prescription was later used in a series of papers to redefine practical fitting formulae for the electron conductivities and NPB (e.g., Cassisi et al. 2007 and references therein; also Horowitz et al. 2008).

The prescription of Baiko et al. (1998) contradicts our current understanding of solid and liquid metals. We instead agree with Itoh et al. (2008) since we believe that the disordered positions and the oscillating and diffusive movement of the ions cannot be ignored in a liquid.

The arguments and conclusions of Baiko et al. (1998) contradict both experimental findings and the contemporary theoretical understanding of liquids: for example, it has been known for a long time that the electrical resistance of most metals in the liquid state just above their melting points are about 1.5–2.3 times larger than those of solid metals (Cusak & Enderby 1960). Even the simplest metals, whose electronic structure do not change appreciably upon melting, show a discontinuity in transport properties at melting, e.g., for sodium σ_s/σ_l ≈ 1.45 and for aluminum σ_s/σ_l ≈ 2.2. Interestingly, similar to Baiko et al.'s approach but with the opposite purpose in mind, early theoretical attempts to explain the increase of electrical resistivity upon melting relied heavily on ideas that had proved successful for solid metals. These approaches overlooked the effect of disorder inherent to liquids: for instance, in the "quasi-crystalline" models, the local coordination just above the melting point was treated very similar to that which prevails in the solid phase just below. It was also believed that, as in a solid metal, the resistivity of liquid metals could be divided into two parts: (1) a thermal term, proportional to the mean square amplitude of vibration of the ions, and hence dependent on the temperature T and (2) a residual term, independent of T due to defects (e.g., vacancies, dislocations). In 1934, Mott (1934) presented an elegant theory to reconcile these solid-like picture to the increase of electrical resistivity on melting in terms of the entropy change on melting per particle Δs = (S_l − S_s)/nV,

$$\begin{eqnarray} &&\Delta s=1.5 k_{\rm B}\ln \left(\sigma _{\rm s}/\sigma _{\rm l}\right). \end{eqnarray} \tag{ 31 }$$

Using the latent heat of fusion to evaluate Δs, Mott's model yields surprisingly good values of σ_s/σ_l (1.68 for sodium instead of 1.45.) It was later realized that Mott's assumptions do not hold in the light of other facts. Hence Mott's theory assumes that all the melting entropy is connected with the thermal agitation of the ions and ignored the configurational entropy arising from disorder in the rest positions about which ions are instantaneously vibrating (the so-called "cage effect"). Unfortunately, this argument does not hold: it is indeed known that for all simple metals, the experimental Δs is approximately a universal constant, Δs ≈ 0.8 ± 0.1 (rms), and is chiefly due to the change in configurational entropy (Wallace 1997). The entropy change due to the thermal agitation, the anharmonicities in ionic motions, and the electronic structure are responsible for the small scattering in the universal value. As illustrated in Table 1, similar findings apply the OCP over a wide range of electronic screening, namely Δs ≈ 0.85.

Finally, we would like to add another argument in disfavor of Baiko et al.'s prescription. If crystallites do indeed exist in the liquid phase as assumed by Baiko et al. (1998), they must be extremely small, smaller than required for a local band structure to be established, because the presence of nuclei would make it impossible for a liquid to supercool. However, all liquid metals can be persuaded to supercool through as much as 20% of their melting temperature (Turnbull & Cech 1950) and MD simulations of the OCP show the same trend (Daligault 2007).

Interestingly, for the multi-component systems found in the crusts of accreting neutron stars, the jump may be much less important than suggested by the calculations of Itoh et al. (1993). However, this is likely not due to the reason proposed by Baiko et al. but because of the lack of long-range order for those complex mixtures and the high degree of impurity-like disorder that those systems possess.

In this section we continue the comparison of electron–ion scattering as occurring in terrestrial liquid metals as opposed to the crust of a neutron star. To make this comparison as quantitative as possible, we shall discuss the different components of the Coulomb logarithm in Equation (14). While this expression applies to both terrestrial liquid metals and to the crusts of accreting neutron stars (under the assumption of the first Born approximation), there are several interesting differences between both systems which we enumerate below.

• 1.  
  While in the former the number of species is typically of order unity (N_sp = 1 in simple metals; N_sp = 2–3 in alloys), the number of species in the latter can be several hundred (see Section 3.1) Nevertheless, as shall see, the charge–charge structure factors S_zz(k) is similar in both cases, i.e., characterized by damped oscillations originating from the short-range order of strongly coupled systems. The heights of the peaks increase with the strength of the coupling parameter Γ_eff and therefore with depth.
• 2.  
  While in terrestrial liquid metals electrons are nonrelativistic, electrons in an accreting neutron star crust are relativistic. The main effects of relativity are encapsulated in $\big[1-\frac{x_{\rm r}^{2}}{1+x_{\rm r}^{2}}\big(\frac{k}{2k_{\rm F}}\big)^{2}\big]$, which varies from 1 when x_r → 0 and 1 − (k/2k_F)² when x_r → ∞ and describes kinematic suppression of backward scattering of relativistic electrons. Relativity also modifies the electronic screening of the ions (see Equation (B2)). The overall effects of relativistic corrections is illustrated in Figures 3 and 4. Figure 3 shows the integrand of Equation (14) for both relativistic and nonrelativistic electrons (x_r = 0 in Equation (14)). Relativistic effects are noticeable in large momentum transfer collisions (close encounters) and, as a consequence of the suppression of back-scattering, result in lower scattering cross-sections and consequently higher conductivities. Figure 4 compares the relativistic versus nonrelativistic Coulomb logarithm as a function of depth using the compositional profile of Gupta et al. (2007) discussed in Section 3.1.
• 3.  
  While in the neutron star crust, ions are fully stripped and all electrons are delocalized electrons and participate in the electronic conduction, n_e = 〈Z〉n with 〈Z〉 ∼ 40, most electrons in a liquid metal are bound to the nuclei, n_e = 〈Z_eff〉n where Z_eff is the effective charge of the ions and Z_eff = 1–3. As a consequence, in Equation (14), v_ie(k) must, in principle, be replaced by a pseudopotential that mimics the effect of bound electrons on free-electron ion scattering (this is a major difficulty in condensed matter physics). In contrast, for neutron star crusts, v_ie is known: it is the pure Coulomb potential, slightly corrected with a form factor to describe the finite size of the nuclei and very close electron–ion collisions.
• 4.  
  A final effect of major importance arises from the magnitude of the electron Fermi wavevector k_F = (3π²n_e)^1/3, which increases like Z^1/3_eff in terrestrial liquid metals as opposed to 〈Z〉^1/3 in the neutron star crust. Therefore the ranges of integration 0 ⩽ k ⩽ 2k_F in Equation (14) greatly vary between the two cases. The situation is illustrated in Figure 3: the limit of integration 2k_F lies just to the left of the main peak in S(k) for monovalent liquid metals, just to its right for divalent ones, and progressively further to the right for metals which have a valency greater that two and three. In neutron stars, 2k_F can be quite large (and increases with depth), i.e., large momentum transfer collisions (close encounters) become much more important in determining the electron–ion scattering cross section.

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