Electrical conductivity of a warm neutron star crust in magnetic fields Arus Harutyunyan based on: PhysRevC.94.025805, arXiv:1605.07612 Goethe University, Frankfurt am Main, Germany collaboration: A. Sedrakian (Frankfurt am Main) ● Introduction & motivation ● Physical conditions ● Boltzmann kinetic theory ● Scattering amplitude & relaxation time 。 Results ● Summary & outlook
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● Introduction & motivation ● Physical conditions ● Boltzmann kinetic theory ● Scattering amplitude & relaxation time ● Results ● Summary & outlook
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Neutron stars
Neutron stars (NS) are compact stellar objects with masses M ～ M ω and radii R ～ 10 km. They are embedded in strong magnetic fields B ～ 10¹0 – 10¹5 G. Typical temperature after cooling is T ～ 10 K. The outer crust of the star consists of nuclei and conducting Fermi gas of relativistic electrons. Transport coefficients Transport coefficients play important role in several astrophysical problems: ● Cooling or heating of a NS ● Magnetic field decay ● Oscillations & various instabilities ● Dissipation of magneto-hydrodynamic waves in NS crust Transport properties of NS crust are well studied at low temperatures. Calculation of high-temperature transport coefficients is motivated by some problems such as: ● Merging in binary NS ● Supernova explosions ● Accretion on NS from a companion We calculate the electrical conductivity tensor of warm outer crust of a NS for non-quantizing magnetic fields and for arbitrary electron degeneracy in the parameter range < ρ < 10 11 g cm −3 ,   0.1 < T < 10 MeV,    10 < B < 10 14 G. We use the natural (Gaussian) units h = c = k b = k e = 1,e = √ α,α = 1,e 50 9.0
● Introduction & motivation ● Physical conditions ● Boltzmann kinetic theory ● Scattering amplitude & relaxation time ● Results ● Summary & outlook
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4つなど
Phase diagram of one-component plasma and electron scattering mechanism The electron scattering mechanism is controlled by the ionic plasma parameter Γ = Z 2 e 2 / α i T : ● If Γ < 1 - Boltzmann gas, the scattering is on individual, uncorrelated nuclei ● If l < Γ < Γ m ~ 160 - liquid state, the scattering is on correlated nuclei ● If Γ > Γ m - solid state, the scattering is on phonons and impurities Physical conditions
The ion structure factor and nuclear formfactor
The ion-ion correlations and finite size of nuclei are taken into account via 2-point structure function S(q) and nuclear formfactor F(q), respectively
We adopt the Monte-Carlo results from S. Galam, J. P. Hansen, Phys.Rev.A.14, 816 (1976) ( Γ ≥ 2 ) and the analytical expressions from M. N. Tamashiro, Y. Levin, M. C. Barbosa, Physica A 268 , 24 (1999) (Γ ≤ 2) for S(q). ● Introduction & motivation ● Physical conditions ● Boltzmann kinetic theory ● Scattering amplitude & relaxation time ● Results ● Summary & outlook
<ロ> <タ> <ミ> <ミ>
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The Boltzmann equation
● The kinetics of electrons is described by the Boltzmann equation for the distribution function f in the presence of electric and magnetic field
● The collision integral for electron-ion scattering has the form
● For small perturbations the solution of Eq. (1) can be searched in the form
● In the relaxation time approximation I[f] = −δf / τ the solution has the form
● The relaxation time is given by the formula

The electrical conductivity tensor
● The electrical conductivity tensor is defined as the coefficient of proportionality between the electrical current and electrical field
● If magnetic field is directed along z axis, then σ ij tensor has the form
● Longitudinal conductivity does not depend on magnetic field (isotropic conduction) ● Transverse (σ 0 ) and Hall (σ 1 ) conductivities depend on magnetic field via the Hall parameter ωετ , where ως is the cyclotron frequency of electrons.

Recovering limiting cases
Limiting cases of strongly degenerate and non-degenerate electrons ! At low temperature limit T ≪ T ξ  one comes to the famous Drude formulae ( ε → ε f ):
Limiting cases of isotropic and strongly anisotropic conduction ● If ω ς τ ≪ 1 (weak magnetic fields), the conduction is isotropic:
● If ωςτ ≫ 1 (strong magnetic fields), the transverse conduction is strongly suppressed:
Overview ● Introduction & motivation ● Physical conditions ● Boltzmann kinetic theory ● Scattering amplitude & relaxation time ● Results ● Summary & outlook
<ロ> <タ> <ミ> <ミ>
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Scattering amplitude & relaxation time The scattering probability The electron-ion scattering matrix element can be calculated according to the Feynman rules
J P , J P are electronic and ionic 4-currents respectively −e* ï¥²³ (p³)ﾍï¥ ι¥²¹ (p¹), J " = The relaxation time in the case of inclastic electron-ion scattering is given by the formula
Limit of elastic scattering Due to the suppression factor θ the scat- tering is more effective via exchange of virtual photons with energies and mo- menta lying inside the triangle ω q < θ ≪  1 on the plane (ω, q).  In the limit of infinite massive ions (θ  →  0) the electron-ion interaction is pure electro- static, and we come to the limit of elastic scattering using the formula:
In the limit of elastic scattering one obtains the well-known formula for the relaxation time
The photon polarization tensor
● The polarization tensor Π μ ν (ω, q) is decomposed into longitudinal and transverse modes Πμν (ω, q) = Πι (ω, q)P μν (ω, q) + Πι (ω, q) ● In the plasma rest frame projectors P i ν and P i ν have the components ( Q 2 ● Within the HTL (hard-thermal-loop) effective field theory (q ≪ ρ ) Π L , Π T are given by
x = ω/q, v = v F in degenerate and v = 1 in ultra-relativistic limits, respectively. ● If the contribution of positrons is neglected, for the Debye momentum we have
● We use low-frequency x = ω/q ≪ 1 expansions for Π z , Π z to o rder O(x )
● Introduction & motivation ● Physical conditions ● Boltzmann kinetic theory ● Scattering amplitude & relaxation time 。 Results ● Summary & outlook
4月つなど
<ロ> <ウ> <ミ> <ミ>
Relaxation time and the Hall parameter ωετ as functions of density Relaxation time scales as τ ∞ ε 2 Z−1 ρ−1 T−δ with 0.1 ≤ δ ≤ 0.3. Degenerate regime T ∞ p-1/3 ωςτ ος ρ=2/3 ερ << ρ 1/3 , Non-degenerate regimeミ~ 3T, T oc p-1, ωςτ ∞ ρ−−気にしRelaxation time as a function of temperature Relaxation time decreases with the temperature in the degenerate and increases in the non-degenerate regimes. The temperature dependence in the first case arises solely from the ion-ion correlation function. Degenerate regime T ox T- 6 wcr oc T-8 E → EF, Non-degenerate regime ε ~ 3T,    τ α T 7/4, ω τ α τ 3/4
● The scalar conductivity σ shows a power-law increase with density σ ∞ ρα. ● In the deg. regime α ~ 0.5, in the non-deg. regime α ~ 0.1. ● σ decreases with temperature in the deg. regime and increases in the non-deg. regime. ● The minimum as a function of temperature arises around "transition" temperature T* ~ T e /3 almost independent on density and type of nuclei. 599 The density dependence of transverse and Hall conductivities ● At high densities or small magnetic fields ωςτ ≪ 1 (isotropic region)
● At low densities or strong magnetic fields ω c τ ≫ 1 (strongly anisotropic region)
● σο ∞ ρα , if ωρτ ∞ 1 degenerate: α ~ 1/2, non-degenerate: α ~ 1/10 ● σο α ρβ , if ωρτ → 1 degenerate: β ~ 3/2, non-degenerate: β ~ 2 ● σ 1     ρ = 1, if ω ρ degenerate: γ ~ 1/5, non-degenerate: γ ~ 3/4 ● σ 1 oc ρ , if ω ε τ  → 1 ● 
● At high densities or small magnetic fields ωςτ ≪ 1 (isotropic region)
● At low densities or strong magnetic fields ω c τ ≫ 1 (strongly anisotropic region)
● σ 0 ~ σ, σ 1 ~ σ 2 as functions of temperature and nucleus charge number in isotropic region. ● The temperature and Z dependence of σ 0 is reversed to that of σ in anisotropic region. ● σ 1 is independent on temperature and type of nuclei in anisotropic region.
Dependence on magnetic field
● At high densities or small magnetic fields ω ετ ≪ 1 (isotropic region)
● At low densities or strong magnetic fields ω c τ ≫ 1 (strongly anisotropic region)
● σ 0 decreases with magnetic field: σ 0 ～ B − 2 in anisotropic region. ● σ 1 ～ B in isotropic and σ 1 ～ B − 1 in anisotropic regions. Maximum occurs at a <ロ> < 99 The temperature dependence of the crust anisotropy To characterize the anisotropy we consider the ratio σ 0 / σ . ● All curves have a maximum at T ~ T* independent of density, magnetic field and type of nuclei. ● At this maximum the anisotropy of the crust is the smallest. ● In the degenerate regime the anisotropy decreases with temperature. ● In the non-degenerate regime σ 0 /σ ∞ T− 372 , and the crust is strongly anisotropic at very high temperatures.
The composition of the crust depends on density and temperature. For not very high temperatures the β -equilibrium composition derived for T = 0 can be used. For the conductivity we have the scaling ● In the degenerate regime
● In the non-degenerate regime
The results of density dependent composi- tion differ from those of 56Fe less than by a factor 1.4. Composition from J. M. Pearson, S. Goriely, N. Chamel, PhysRevC.83.065810 (2011). It is interesting to study the conductivity of warm matter, which is composed of nuclei in statistical equilibrium, in which case the crust composition may become an important factor. 19, C Results Fit formulae for three components of the conductivity tensor We have performed fit to the longitudinal conductivity using the formula For the transverse and Hall components the following fit formulae can be used ● The form of the fit formulae reproduces the correct temperature, density and magnetic field dependence of the conductivity tensor in all limiting cases discussed above. ● The fit parameters C , a , b , c , d , g , h depend on the ionic structure of the material. ● The relative error of the fit formulae vary from 7% to 15% depending on the composition.
Summary
● The behavior of electrical conductivity mostly depends on the electron degeneracy. ● The scalar conductivity of NS crust increases with density, but the slope of increase is much weaker in the non-degenerated regime. ● It has a minimum as a function of temperature around T* ~ T F /3. ◆ The anisotropy in transport properties becomes sizable at low densities, high magnetic fields, high temperatures and small ion charge numbers. ● The crust is the most isotropic at the transition temperature T * .
Summary
● The behavior of electrical conductivity mostly depends on the electron degeneracy. ● The scalar conductivity of NS crust increases with density, but the slope of increase is much weaker in the non-degenerated regime. ● It has a minimum as a function of temperature around T* ~ T F /3. ◆ The anisotropy in transport properties becomes sizable at low densities, high magnetic fields, high temperatures and small ion charge numbers. ● The crust is the most isotropic at the transition temperature T*.
Future outlook
● Consider temperature-dependent crust compositions of multi-component plasma (including light nuclear clusters). ● Take into account properly the contribution of positrons. ● Calculate other transport coefficients using the same formalism. ● Implementation of conductivities in MHD NS-NS merger codes.
Summary
● The behavior of electrical conductivity mostly depends on the electron degeneracy. ● The scalar conductivity of NS crust increases with density, but the slope of increase is much weaker in the non-degenerated regime. ● It has a minimum as a function of temperature around T* ~ T F /3. ◆ The anisotropy in transport properties becomes sizable at low densities, high magnetic fields, high temperatures and small ion charge numbers. ● The crust is the most isotropic at the transition temperature T*.
Future outlook
● Consider temperature-dependent crust compositions of multi-component plasma (including light nuclear clusters). ● Take into account properly the contribution of positrons. ● Calculate other transport coefficients using the same formalism. ● Implementation of conductivities in MHD NS-NS merger codes.
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