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On the screening condition in the core of neutron stars
D. N. Kobyakov1, ∗
1Institute of Applied Physics of the Russian Academy of Sciences, 603950 Nizhny Novgorod, Russia
The screening condition in neutron star core has been formulated as equality of velocities of superconducting
protons and the electrons vp=ueat wavenumbers qλ−1(λis the London depth) and has been used to derive
the force between the electronic flow past the flux tube, which has astrophysical applications. By calculating
the current-current response, I find that vp6=uefor l−1<qλ−1(lis the electron mean free path) at typical
realistic parameters. Therefore, the momentum exchange between the electrons and the flux tubes in the core of
neutron stars remains an open question.
Introduction. In the core of neutron stars, the protons are
superconducting and the electrons are normal. Moreover, ob-
servable neutron stars host the magnetic field in their inte-
rior, which according to theoretical expectations induces a
flux tube lattice in the superconducting protons. Scattering
of the electrons by the flux tubes effectively couples the elec-
trons and the protons, and this coupling plays an important
role in modelling of observable phenomena in neutron stars.
Scattering of the electrons by a single flux tube in super-
conducting protons (or by a single superfluid neutron vortex
with magnetization) has been a long-standing problem. Al-
par, Langer and Sauls [1] have considered this problem for
the first time and have discussed some astrophysical implica-
tions. Their work has given rise to many publications where
this problem was addressed. A review of the literature on this
subject goes beyond the aims of this paper and is not essen-
tial for the further discussion below. A comprehensive liter-
ature review may be found in a recent work by Gusakov [2],
who also has done detailed calculations of the scattering of the
electrons by the proton flux tube in order to resolve some dis-
crepancies between different approaches to the physical prob-
lem, which have emerged since the appearance of the work by
Alpar, Langer and Sauls [1]. In the calculations of Gusakov
[2], an essential step was made in equation (16) of [2], when
the screening condition vp=uewas used. Lately, this condi-
tion was also used by Sourie and Chamel [3] in their studies
of superfluid neutron vortices in the core of neutron stars.
The condition vp=ueis expected to hold in the hydrody-
namic regime [4], which is defined, in the present context, for
wavenumbers qmuch smaller than the electron inverse mean
free path l−1[5]. In other related problems, one may define
the hydrodynamic regime as length scales much longer than
the superfluid vortex separation [6], but here we will be inter-
ested in the electron properties.
However, the flux tube core length scale λat typical tem-
perature T=108K is about 80 fm, and therefore it is not
clear whether the hydrodynamic approximation applies for
the problem of extraction of the momentum exchange in the
electron-flux tube scattering, because lλ. Actually, we will
see that lis typically somewhere between 105fm and 109fm,
while the flux tube separation is somewhere between 144 fm
(at the magnetic field H=1015 Oe) and 455 fm at (H=1014
Oe).
∗dmitry.kobyakov@appl.sci-nnov.ru
In this paper, I will show that when the flux tube size is re-
solved by the electron dynamics, the latter are in the particle-
hole regime and consequently the screening condition does
not always hold.
Screening of charge current by electrons. We will do cal-
culations with the material parameters corresponding to the
nuclear saturation density. The baryon number density is
n=0.16 fm−3, the proton fraction xp=0.05 and the pro-
ton number density is np=xpn, the electron number den-
sity is ne=npand the the electron Fermi wavenumber is
ke= (3πne)1/3≈0.6187 fm−1. The electron velocity ueis the
microscopic velocity averaged on length scale much longer
than k−1
e. Likewise, the proton superflow velocity vpis the
quantity averaged on length scale much longer than the pro-
ton coherence length ξ(T). At TTc, the superconducting
density is equal to np, where Tc=γ∆p/kBπ, with lnγ=0.577
is the Euler constant and ∆pis the proton s-wave supercon-
ducting energy gap, and we have
ξ(0) = ¯
hvF p
2γ∆p
≈7.21MeV
∆pfm.(1)
Here, vF p =¯
hkp/mpis the electron Fermi velocity, kp=ke
and mpis the proton mass. The number currents are defined
as je=neueand jp=npvpand the electric currents are Je=
−eneueand Jp=enpvp, where eis the proton electric charge.
Noting that memp, where meis the electron mass, is a
reasonable lowest-order approximation even for the relativis-
tic electrons, we may neglect the inertia of the electrons [4].
This implies that in order to study the transverse electromag-
netic response of the system it is sufficient to assume that the
proton velocity is given and then to calculate the equilibrium
electron velocity. The physical meaning of the screening con-
dition vp=ueis that for a given proton velocity, the electron
velocity is the same and the total electrical current Je+Jpis
zero, or Je=−Jp. Thus, we are free to choose the proton
phase as a constant, and we obtain
Jp=enp(−eA/c),(2)
where Ais the electromagnetic vector potential [7]. It will be
convenient to work in the Fourier representation, X(q,ω) =
Rd3xdt eiq·x−iωtX(x,t), where X(x,t)is any function of space
and time. From the relation E=−c∂A/∂twe observe that
to calculate the electron current for a given proton velocity, in
the lowest approximation we need to calculate the quantity
σ(q,ω) = δJe(q,ω)
δE(q,ω)=ec
iω
δje(q,ω)
δA(q,ω),(3)
arXiv:2305.12882v1 [nucl-th] 22 May 2023
2
which is the electrical conductivity of the electrons.
From the other hand, the electrical conductivity may be ex-
pressed through the dielectric function εt, for which one has
at least two equivalent choices. We shall work with the trans-
verse fields in this paper; for convenience, the subscript tis
added to stress that εtis the transverse dielectric function.
Choosing the definition of the dielectric function following
Lindhard [8] from the electric induction D, or
D=ε(L)
tE,(4)
we have the Maxwell equation
q2−ω2
c2ε(L)
t(q,ω)A(q,ω) = 4π
cJext(q,ω).(5)
Alternatively, one can follow Jancovici [9] and define the di-
electric function from the equation for the total self-consistent
vector potential A=Aext +Aind, which is a sum of the exter-
nal part Aext due to a source and the induced part Aind due to
the electrons:
A(q,ω) = 1
εt(q,ω)Aext(q,ω).(6)
Then we have the Maxwell equation
q2−ω2
c2εt(q,ω)A(q,ω) = 4π
cJext(q,ω),(7)
and we obtain
ε(L)
t(q,ω)−1=c2q2−ω2
ω2(1−εt(q,ω)).(8)
From the Maxwell equations, σis related to ε(L)
tas following
[10]:
σ(q,ω) = ω
i4πε(L)
t(q,ω)−1.(9)
It is convenient to introduce the response function ˜
χt(q,ω)
according to the definition
εt(q,ω) = 1+4πe2
q2˜
χt(q,ω).(10)
Combining the above equations we can write
˜
χt(q,ω) = cq2
e(ω2−c2q2)
δje(q,ω)
δA(q,ω).(11)
Using the approximation discussed above we associate the ex-
ternal current Jext with Jpand the total current with Jp+Je;
thus, we associate the electronic current Jewith the induced
current Jind. With this definition, we have
Je(q,ω) = 1−εt(q,ω)
εt(q,ω)Jp(q,ω).(12)
In the present system, the electrons are relativistic and
quantum degenerate. It is important to distinguish between
the regimes of the electron dynamics [10]: the hydrodynamic
regime is realized when ωνand ql−1; the particle-hole
regime results when ω>νand/or q>l−1. There are two
options in the particle-hole regime: with q>l−1and ω>ν
the collisions are unimportant; with q>l−1and ω<νthe
collisions are important.
Here, ν=c/lis the electron collision rate, which is defined
as a sum of the collision rates with all kinds of impurities that
scatter the electrons. In the present system,
ν=ν1+ν2,(13)
where ν1(or ν2) is the rate of collisions of the electrons with
the normal protons within the magnetic flux tubes (or with
the magnetic field lines within the magnetic flux tubes and the
superfluid neutron vortices).
Calculation of σ(q,ω). As the first step, we address the
time and length scales involved in the problem of calculation
of σ(or, equivalently, εt). In the present system, we are inter-
ested in the short length scales (q≥l−1) and slow frequencies
(ων). We notice that Jancovici has already calculated [9]
the dielectric function εt(q,ω)for q≥l−1, but completely
neglecting the electron collisions (for ων). Here, ωare
rather small and may take values from ω∼0 for phenomena
related to smooth evolution of the magnetic field in neutron
stars, to ω∼104for phenomena related to quasi-periodic os-
cillations in magnetars.
At temperatures TTcthe normal protons in the bulk of
the superconductor may be neglected and collisions of the
electrons with the normal protons occur only within the flux
tubes, therefore
ν1≈τ−1
tr
Hc2(0)
H,(14)
where Hc2(0) = Φ0/2πξ (0)2,Φ0=π¯
hc/e, is the upper crit-
ical magnetic field of the superconductor at T=0, His the
stellar magnetic field and τtr ≈2×10−14 s is the electron
transport relaxation time with normal protons evaluated by
Baym, Pethick and Pines [11]. With typical H=1015 Oe and
∆p=1 MeV, we find ν1≈3.166 ×1015 s−1. For scattering of
the electrons by the magnetic field in the flux tube, we use the
order of magnitude estimate
ν2=c
let
≈cσetnt,(15)
where let is the electron mean free path between consecutive
collisions of the electron with the flux tube, σet is the dif-
ferential cross-section for scattering of the electron with the
flux tube and nt=H/Φ0is the number of flux tubes per unit
area. Note that if His smaller than the lower critical mag-
netic field, the neutron vortices would provide the dominant
impurity scattering for the electrons. The cross section is
given by σet =αk−1
e, where α=α(keξ,λ/ξ). From equa-
tions (37) and (40) from Gusakov [2] we infer that α∼10−2,
depending on the dimensionless parameters keξand λ/ξ.
Thus, with typical H=1015 Oe and ∆p=1 MeV, we find
ν2≈α×2.343 ×1019 s−1and ν≈ν2. This coincides with
the standard theoretical expectation that the main source of the
3
electron-proton coupling in superconducting matter of neu-
tron star core is the electron interaction with the magnetic flux
tubes.
We have found, as expected, that ωνfor typical pro-
cesses in neutron stars. Thus, for the further calculations,
εt(q,ω)calculated by Jancovici [9] must be modified in order
to include the electron collisions. We will use the following
notation:
˜
χt(q,ω) = χt(q,ω)for ων,
χν
t(q,ω)for ων.(16)
As the second step, we turn to the kinetic equation, from
which the functional derivative δje(q,ω)/δA(q,ω)can be
calculated in both cases, when either ων(the collision
integral I[n1p]is nonzero) or ων(I[n1p] = 0). Here, n1p
is the departure of the distribution function from true equilib-
rium [10], [12]. In the relaxation time approximation (RTA),
the collision integral is written in the form
I[n1p] = −νn1p−nR
1p,(17)
where nR
1pis the so-called locally relaxed equilibrium distri-
bution function [12].
As Conti and Vignale have shown [12], in RTA the response
function with collisions (χν
t(q,ω)) can be obtained from the
response function without collisions with the frequency ωre-
placed by ω+iν(χt(q,ω+iν)):
χν
t(q,ω) = ω
ω+iνχt(q,ω+iν).(18)
Notice that Conti and Vignale [12] have worked with the
quantity χτ
t(the superscript 1999 is to refer to the quantities
used in [12]),
χτ
t(q,ω)≡δj1999
e(q,ω)
δA1999(q,ω)=−c
e
δje(q,ω)
δA(q,ω),(19)
which, as can be easily seen from Eqs. (11), (16) and (19), is
proportional to χν
t:
χν
t(q,ω) = q2
c2q2−ω2χτ
t(q,ω).(20)
By virtue of the proportionality between χν
tand χτ
tseen in
Eq. (20), the result obtained in RTA for the relation between
χτ
tand its collisionless counterpart by Conti and Vignale [12],
is applicable to the relation between χν
tand χt; this validates
the formula in Eq. (18).
This conclusion is useful because it allows to find χν
t(q,ω)
from χt(q,ω), which has been obtained by Jancovici. Equa-
tion (66) from Jancovici [9] gives:
χt(q,ω) = s
2
∂ne
∂ µes
1−s2+1
2log s+1
s−1,(21)
where µeis the electron Fermi energy, ∂ne/∂ µe=k2
e/π2¯
hc
and s≡s(q,ω) = ω
cq . Collecting the results we obtain the
main formula of this paper:
σ(q,ω) = ω2−c2q2
iω
e2
q2
ω
ω+iνχt(q,ω+iν).(22)
FIG. 1. The real part of the proportionality coefficient ζbetween
the induced current and the test current, Eq. (23), as function of the
collision frequency νwith wavenumber set as q=ν/c. From left to
right, the lines correspond to ω=10, 102, 103and 104.
FIG. 2. The imaginary part of the proportionality coefficient ζbe-
tween the induced current and the test current, Eq. (23), as function
of the collision frequency νwith wavenumber set as q=ν/c. From
left to right, the lines correspond to ω=10, 102, 103and 104.
From Eqs. (10), (16), (18) and (21) we will calculate the quan-
tity
ζ≡1−εt(q,ω)
εt(q,ω),(23)
which indicates effectiveness of the current-current screening,
see Eq. (12). In the limiting case, when ζ=−1 we infer that
the screening condition holds, and vp=ue. In another limiting
case, when ζ=0 we infer that the proton supercurrent is not
screened by the electron current, and the screening condition
does not hold.
Numerical results. We turn to evaluation of ζfor various
q,ωand ν. Figures 1-4 display ζ, Eq. (23), as function of
νfor four different frequencies (ω=10, 102, 103and 104)
and for two different length scales (q=ν/cand 10ν/c). The
chosen set is sufficient to reveal general patterns in behavior
of ζas function of q,ωand ν. Comparing the positions of
FIG. 3. The same as Fig. 1 but q=10ν/c.
4
FIG. 4. The same as Fig. 2 but q=10ν/c.
lines in Figs. 1 and 3 we observe that at fixed ν, decreasing
of qleads to improvement of screening; this implies that the
smaller qis, the closer ζis to minus unity, as expected. We
can say that decreasing of qat fixed ωmoves the curve ζin
Fig. 1 to the right.
If we have νfixed, for instance, ν=3×1018 Hz, then for
any ωbetween 0 and 104, the electrons do not screen the pro-
ton supercurrent at any length scale shorter than the electron
mean free path (q>ν/c≈108cm−1).
A somewhat nontrivial result is that increasing of ωim-
proves the screening effectiveness; for example, with the col-
lision rate ν=1017 Hz, we would have complete screening
(ζ=−1) at q=ν/c≈3×106cm−1only for ω≥108Hz
(this numerical result is not shown explicitly in the Figures
but could be seen in Fig. 1 as ultimate shift of the curve ζto
the right leaving in the plot only the “tail” with ζ=−1). We
can say that increasing of ωat fixed qmoves the curve ζin
Fig. 1 to the right.
The imaginary part of ζare shown in Figs. 2 and 4. We see
that the imaginary part is zero at either complete screening
(ζ=−1) or at screening absent (ζ=0), while in the interme-
diate case, which can be called an incomplete screening, the
induced current may have the phase-shifted (by π/2) compo-
nent with magnitude equal to that of the in-phase component
of the induced current.
Conclusions. Based on microscopic physics, we have de-
veloped the framework, which can be used to estimate the ef-
fectiveness of the electrical current response of the electrons
to a given proton supercurrent in presense of a lattice of flux
tubes. We have used typical parameters corresponding to the
core of neutron stars and studied the screening condition for
various values of the electron momentum-nonconserving col-
lision frequency. We have obtained that for typical frequen-
cies of change of the relative velocity between the electrons
and the superconducting protons (between about 0 and 104
Hz), the electrons are unable to screen the proton supercur-
rent. The implication is that one of the basic conditions used
in the earlier literature for calculation of the effective force
acting between the electrons and a localized magnetic flux
tube associated with a proton (or neutron) quantized magne-
tized vortex, namely the screening condition, which assumes
that the electron velocity is equal to the superflow velocity at
distance an order of magnitude larger than the linear size of
the flux tube cross-section, does not hold. As a result, the rate
of momentum exchange between the electrons and the flux
tube lattice in the superconducting and/or superfluid nuclear
matter in the core of neutron stars remains an open question.
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