About Magnetic Susceptibility of Dense Superfluid Neutron Matter with Spin-triplet p-wave Pairing
Pure neutron matter with the spin-triplet p-wave pairing is studied in the framework of the non-relativistic generalized Fermi-liquid theory at subnuclear and supranuclear densities (in the range 0.7n₀ ≤ n < nс(Skyrme) < 2n₀ where n₀ = 0.17 fm⁻³ is the saturation density of the symmetric nucle...
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irk-123456789-562092014-02-14T03:12:07Z About Magnetic Susceptibility of Dense Superfluid Neutron Matter with Spin-triplet p-wave Pairing Tarasov, A.N. Астрофізика і космологія Pure neutron matter with the spin-triplet p-wave pairing is studied in the framework of the non-relativistic generalized Fermi-liquid theory at subnuclear and supranuclear densities (in the range 0.7n₀ ≤ n < nс(Skyrme) < 2n₀ where n₀ = 0.17 fm⁻³ is the saturation density of the symmetric nuclear matter) at zero temperature and in the presence of a strong magnetic field. The Skyrme effective forces are used as interactions between neutrons. As a result, the general expression (valid for an arbitrary parametrization of the Skyrme forces) is obtained for the magnetic susceptibility of superfluid neutron matter, and it is specified then for three types of the Skyrme interaction with different power dependences on the density n. In particular, it is found for neutron matter with the so-called RATP, Gs, and SLy2 parametrizations of the Skyrme forces that the magnetic susceptibility diverges at the densities nC(RATP) ≈ 1.03n₀, nC(Gs) ≈ 1.33n₀ and nC(SLy2) ≈ 1.72n₀. These critical densities correspond to phase transitions from the superfluid paramagnetic state of neutron matter with triplet pairing to the ferromagnetic state which coexists with triplet superfluidity at densities higher than nC(Skyrme). Such phase transitions might occur in the liquid outer cores of pulsars and the so-called magnetars. Суто нейтронна матерiя зi спiн-триплетним p-спарюванням вивчається у межах нерелятивiстської узагальненої теорiї фермiрiдини при суб’ядерних та над’ядерних густинах (у дiапазонi густин 0,7n ≤ n < nC(Skyrme) < 2n₀, де n₀ = 0,17 фм⁻³ – це густина насичення симетричної ядерної матерiї) при температурi, що дорiвнює нулю, та за наявностi сильного магнiтного поля. Ефективнi сили Скiрма використовуються у ролi взаємодiї мiж нейтронами. У результатi отримано загальний аналiтичний вираз (справедливий для довiльної параметризацiї сил Скiрма) для парамагнiтної сприйнятливостi надплинної нейтронної матерiї як функцiї вiд густини при нульовiй температурi. Цей вираз далi конкретизовано для трьох типiв взаємодiї Скiрма з рiзними степеневими залежностями вiд густини n. А саме, знайдено, що для випадкiв нейтронної матерiї з так званими RATP, Gs та SLy2 параметризацiями сил Скiрма у магнiтної сприйнятливостi виникає розбiжнiсть при критичних густинах nC(RATP) ≈ 1,03n₀,, nC(Gs) ≈ 1,33n₀ та nC(SLy2) ≈ 1,72n₀. Цi критичнi густини вiдповiдають фазовим переходам з надплинного парамагнiтного стану нейтронної матерiї з триплетним спарюванням у феромагнiтний стан, який може спiвiснувати з триплетною надплиннiстю при густинах, бiльших за nC(Skyrme). Такi фазовi переходи можуть виникати у рiдких зовнiшнiх ядрах пульсарiв i так званих магнетарiв. 2010 Article About Magnetic Susceptibility of Dense Superfluid Neutron Matter with Spin-triplet p-wave Pairing / A.N. Tarasov // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 644-650. — Бібліогр.: 49 назв. — англ. 2071-0194 PACS 26.60.Dd, 67.10.Fj, 97.60.Gb, 97.60.Jd http://dspace.nbuv.gov.ua/handle/123456789/56209 en Український фізичний журнал Відділення фізики і астрономії НАН України |
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Астрофізика і космологія Астрофізика і космологія |
| spellingShingle |
Астрофізика і космологія Астрофізика і космологія Tarasov, A.N. About Magnetic Susceptibility of Dense Superfluid Neutron Matter with Spin-triplet p-wave Pairing Український фізичний журнал |
| description |
Pure neutron matter with the spin-triplet p-wave pairing is studied in the framework of the non-relativistic generalized Fermi-liquid theory at subnuclear and supranuclear densities (in the range 0.7n₀ ≤ n < nс(Skyrme) < 2n₀ where n₀ = 0.17 fm⁻³ is the saturation density of the symmetric nuclear matter) at zero temperature and in the presence of a strong magnetic field. The Skyrme effective forces are used as interactions between neutrons. As a result, the general expression (valid for an arbitrary parametrization of the Skyrme forces) is obtained for the magnetic susceptibility of superfluid neutron matter, and it is specified then for three types of the Skyrme interaction with different power dependences on the density n. In particular, it is found for neutron matter with the so-called RATP, Gs, and SLy2 parametrizations of the Skyrme forces that the magnetic susceptibility diverges at the densities nC(RATP) ≈ 1.03n₀, nC(Gs) ≈ 1.33n₀ and nC(SLy2) ≈ 1.72n₀. These critical densities correspond to phase transitions from the superfluid paramagnetic state of neutron matter with triplet pairing to the ferromagnetic state which coexists with triplet superfluidity at densities higher than nC(Skyrme). Such phase transitions might occur in the liquid outer cores of pulsars and the so-called magnetars. |
| format |
Article |
| author |
Tarasov, A.N. |
| author_facet |
Tarasov, A.N. |
| author_sort |
Tarasov, A.N. |
| title |
About Magnetic Susceptibility of Dense Superfluid Neutron Matter with Spin-triplet p-wave Pairing |
| title_short |
About Magnetic Susceptibility of Dense Superfluid Neutron Matter with Spin-triplet p-wave Pairing |
| title_full |
About Magnetic Susceptibility of Dense Superfluid Neutron Matter with Spin-triplet p-wave Pairing |
| title_fullStr |
About Magnetic Susceptibility of Dense Superfluid Neutron Matter with Spin-triplet p-wave Pairing |
| title_full_unstemmed |
About Magnetic Susceptibility of Dense Superfluid Neutron Matter with Spin-triplet p-wave Pairing |
| title_sort |
about magnetic susceptibility of dense superfluid neutron matter with spin-triplet p-wave pairing |
| publisher |
Відділення фізики і астрономії НАН України |
| publishDate |
2010 |
| topic_facet |
Астрофізика і космологія |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/56209 |
| citation_txt |
About Magnetic Susceptibility of Dense Superfluid Neutron Matter with Spin-triplet p-wave Pairing / A.N. Tarasov // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 644-650. — Бібліогр.: 49 назв. — англ. |
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Український фізичний журнал |
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| fulltext |
A.N. TARASOV
ABOUT MAGNETIC SUSCEPTIBILITY
OF DENSE SUPERFLUID NEUTRON
MATTER WITH SPIN-TRIPLET p-WAVE PAIRING
A.N. TARASOV
Akhiezer Institute for Theoretical Physics, National Science Center
“Kharkiv Institute of Physics and Technology”, Nat. Acad. of Sci. of Ukraine
(1, Akademichna Str., Kharkiv 61108, Ukraine)
PACS 26.60.Dd, 67.10.Fj,
97.60.Gb, 97.60.Jd
c©2010
Pure neutron matter with the spin-triplet p-wave pairing is studied
in the framework of the non-relativistic generalized Fermi-liquid
theory at subnuclear and supranuclear densities (in the range 0.7n0 ≤
n < nC(Skyrme) < 2n0, where n0 = 0.17 fm−3 is the saturation
density of the symmetric nuclear matter) at zero temperature and
in the presence of a strong magnetic field. The Skyrme effective
forces are used as interactions between neutrons. As a result, the
general expression (valid for an arbitrary parametrization of the
Skyrme forces) is obtained for the magnetic susceptibility of su-
perfluid neutron matter, and it is specified then for three types of
the Skyrme interaction with different power dependences on the
density n. In particular, it is found for neutron matter with the
so-called RATP, Gs, and SLy2 parametrizations of the Skyrme
forces that the magnetic susceptibility diverges at the densities
nC(RATP) ≈ 1.03n0, nC(Gs) ≈ 1.33n0 and nC(SLy2) ≈ 1.72n0.
These critical densities correspond to phase transitions from the
superfluid paramagnetic state of neutron matter with triplet pair-
ing to the ferromagnetic state which coexists with triplet superflu-
idity at densities higher than nC(Skyrme). Such phase transitions
might occur in the liquid outer cores of pulsars and the so-called
magnetars.
1. Introduction
Already fifty years have past since the idea of possible
manifestations of the pairing (which was originally in-
troduced in theory by J. Bardeen, L.N. Cooper, and
J.R. Schrieffer [3]) and the superfluidity phenomenon
in infinite nuclear matter and in finite nuclei and also
in dense neutron stars (A.B. Migdal [4]) was proposed
for the first time by N.N. Bogolyubov [1] and A. Bohr,
B. Mottelson, D. Pines [2]. Note that pulsars were dis-
covered later in 1967 by S.J. Bell and A. Hewish [5] (see
also [6]), and then they were identified by T. Gold [7]
as rapidly rotating neutron stars (NS). Pairing and su-
perfluidity play an important role in modeling the struc-
ture and properties of atomic nuclei and pulsars. But, in
spite of the enormous efforts of many investigators (see,
e.g., monographs [9–14] and reviews [8,15–18] and refer-
ences therein), there are still many unsolved questions
concerning the superfluidity in neutron stars.
As is commonly accepted, neutron stars consist of the
crust (with subnuclear densities n . n0/2, where n0 =
0.17 fm−3 is the saturation density of symmetric nuclear
matter), and the core which are composed of the so-
called outer crust and inner crust and of the outer core
and inner core, respectively. They are distinguished from
each other by composition and by densities. The deeper
the layer in the interior of a neutron star, the denser it
is. Note that neutrons are the prime constituent in the
outer core of NS with a small fraction of protons and
electrons.
Here, we will restrict ourselves by studying the equi-
librium properties of the superfluid phases of infinite
pure neutron matter (SPNM) with spin-triplet pairing
existing inside a liquid outer core of neutron stars at
subnuclear 0.7n0 . n . n0 and supranuclear n > n0
densities of neutrons. These superfluid phases of pure
neutron matter are examples of superfluid Fermi liquids
(SFLs) with spin-triplet pairing similar to 3He (see, e.g.,
[15,19,20] and references therein). Here, we have inves-
tigated theoretically dense SPNM with p-wave pairing
of the 3He − A1,2 type in a stationary homogeneous
magnetic field H and have used the generalized non-
relativistic Fermi-liquid approach [21] to derive nonlinear
integral equations for the order parameter (OP) and the
effective magnetic field (EMF) Heff inside SPNM [22–
24] which are valid at arbitrary temperatures from the
interval 0 ≤ T ≤ Tc (Tc is the normal-superfluid phase
transition (PT) temperature). The effective Skyrme in-
teraction between neutrons depending on the neutron
density (see reviews [25, 26]) has been used.
Here, we have found analytically the approximate so-
lution of the obtained integral equations at zero temper-
ature T = 0 for SPNM with triplet p-wave pairing in
a strong magnetic field H and have obtained the gen-
eral approximate expression for the effective magnetic
field Heff (at T = 0) on the Fermi surface to the first
644 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
MAGNETIC SUSCEPTIBILITY OF DENSE SUPERFLUID NEUTRON MATTER
order in the small parameter hext ≡ |µn|H/εF � 1
(for an arbitrary parametrization of the Skyrme inter-
action and with the self-consistent accounting of the
dependence of the neutron effective mass m∗n on the
NM density n ≡ yn0). The function Heff (at T = 0)
is linear in H up to sufficiently high magnetic fields
(but H � εF/|µn|, where εF(n) is the Fermi energy
of NM, and µn < 0 is the magnetic dipole moment
of a neutron) and a nonlinear function of n. As a
result, the general expression (valid for an arbitrary
parametrization of the Skyrme forces) is obtained for
the paramagnetic susceptibility χSkyrme of superfluid
neutron matter (at T = 0), and it is specified then
for three types of the Skyrme interaction with differ-
ent power dependences on the density n. In particu-
lar, it was found for neutron matter with the Sly2, Gs,
and RATP parametrizations [27–29] of the Skyrme forces
that the paramagnetic susceptibility is a monotonically
increasing function of the neutron density (the corre-
sponding figures were plotted), and it diverges at the
critical densities nC(Skyrme) in the range of densities
0.7n0 . n < nC(Skyrme) < 2n0 under consideration
(where the non-relativistic Fermi-liquid theory is still
valid). These critical densities nC(Skyrme) correspond
to phase transitions from the superfluid paramagnetic
state of neutron matter with triplet pairing to the ferro-
magnetic state which coexists with triplet superfluidity
at densities higher than nC(Skyrme). Such phase tran-
sitions might occur in the liquid outer core of neutron
stars.
Note that other authors have previously investigated
the existence (or absence) of phase transitions of NM
from the normal (nonsuperfluid) state to the ferromag-
netic state in the absence of a magnetic field (see, e.g.,
[30–37] and references therein) and with the effects of
a strong magnetic field (see, e.g., [38–40]) within other
approaches and using different nucleon-nucleon effective
and so-called realistic interactions in NM.
This paper is organized as follows. In the second sec-
tion, we outline the main steps and assumptions made
for the derivation of general equations for the order
parameter and the effective magnetic field for SPNM
with the Skyrme forces and spin-triplet pairing of the
3He− A1,2 type between neutrons. The third section is
devoted to the derivation of a general formula for the
paramagnetic susceptibility in SPNM (valid for an arbi-
trary parametrization of the Skyrme forces) in a strong
magnetic field at zero temperature, which is specified
then for Sly2, Gs, and RATP parametrizations. In Con-
clusion, the general and particular results for SPNM
with triplet pairing in a high magnetic field are briefly
discussed and compared with those from some other
works.
2. General Equations for the Order Parameter
and the Effective Magnetic Field for SPNM
with the Skyrme Forces and Triplet Pairing
The order parameter (OP) for the so-called non-unitary
phase (NU) of 3He − A2 type with spin-triplet p-wave
pairing has the form [20]
Δα
A2(p) ≡ (Δ+d̂α + ıΔ−êα)ψ(p̂),
ψ(p̂) ≡ (m̂j + ın̂j)p̂j , p̂ ≡ p
p
. (1)
Here, Δ±(T ) ≡ (Δ↑(T )±Δ↓(T ))/2; d̂ and ê are mu-
tually orthogonal real unit vectors in the spin space,
d̂ · ê = 0, d̂2 = ê2 = 1; m̂ and n̂ are mutually orthog-
onal real unit vectors in the orbital space, m̂ · n̂ = 0,
m̂2 = n̂2 = 1. The value η(p) ≡ |Δ(p)×Δ∗(p)| 6= 0 is
non-zero for the NU superfluid phases of pure neutron
matter with spin-triplet pairing, in particular. Note also
that the superfluid phase of 3He − A1 type is realized
under the condition, when Δ↓ = 0, Δ↑ 6= 0.
We have chosen the effective Skyrme forces as the in-
teraction between neutrons for SPNM with spin-triplet
p-wave pairing in a spatially uniform magnetic field H.
A system of coupled equations for the OP of the 3He−A2
type and the effective magnetic field Heff inside SPNM
is simplified (in comparison with an analogous super-
fluid phase of real helium-3) because, in the case of the
Skyrme interaction, the normal Fermi-liquid Landau’s
exchange amplitudes F al 6= 0 are non-zero only for l = 0
and l = 1. We also assumed that the quantization axes
of spin and orbital moment of the Cooper pairs (i.e. the
vectors [d × e] and [m × n]) and the magnetic field H
are collinear to one another as in the so-called 3P2 su-
perfluid state of a dense neutron liquid of neutron stars
(where the strong spin-orbit coupling is taken into ac-
count). As a result, using general formulas (obtained
by us previously [41,42]) for the anomalous and nor-
mal distribution functions of quasiparticles (neutrons)
for SPNM in a magnetic field, we have derived a sys-
tem of integral equations for ξ(p), ΔA2
↑ , and ΔA2
↓ . In
this case for SPNM, ξ(p) = ξ(p)H/H ≡ −µnHeff(p)
(µn ≈ −0.60308× 10−17 MeV/G is the magnetic dipole
moment of a neutron [43]), and we have the equation
ξ(p) = −µnH + (r + sp2)K2(ξ) + sK4(ξ). (2)
Here, r = t′0 +(t′3/6)nα and s = (t′1− t′2)/(4~2), n ≡ yn0
is the density of neutron matter; t′0 = t0(1 − x0), t′1 =
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 645
A.N. TARASOV
t1(1−x1), t′2 = t2(1+x2), t′3 = t3(1−x3) and 1/6 ≤ α ≤
1/3 are the parameters of the Skyrme interaction. The
functionals Kβ(ξ) (β = 2, 4) in Eq. (2) have the form
Kβ(ξ) =
1
8π2~3
pmax∫
pmin
dqqβ
1∫
0
dxκ(q, x), (3)
where
κ(q, x) =
z(q) + ξ(q)
E+(q, x2)
tanh
(
E+(q, x2)
2T
)
−
−z(q)− ξ(q)
E−(q, x2)
tanh
(
E−(q, x2)
2T
)
, (4)
E2
± = q2Δ2
↑(↓)(1− x
2) + (z(q)± ξ(q))2, (5)
z(q) = q2/2m∗n − µ (m∗n is the effective mass of a
neutron, and µ is the chemical potential). We have
taken into account that, for SPNM with pairing of the
3He−A2 type, the OP can be written as ΔA2
↑(↓)(T, ξ, q) =
qΔ↑(↓)(T, ξ), where the functions Δ↑(↓)(T, ξ) obey the
equations
Δ↑(↓)(T, ξ) = −Δ↑(↓)(T, ξ)
c3
8π2~3
×
×
pmax∫
pmin
dqq4
1∫
0
dx(1− x2)
tanh(E±(q, x2)/2T )
E±(q, x2)
, (6)
(pmax & pF and (pmax − pmin)/pF < 1, where pF is
the Fermi momentum). Here, c3 ≡ t2(1 + x2)/~2 < 0
is the coupling constant leading to the spin-triplet p-
wave pairing of neutrons which is expressed through the
parameters t2 and x2 of the Skyrme interaction. We
consider here a model of neutron Cooper pairing in a
shell symmetric with respect to the Fermi sphere (i.e.,
pmax − pF = pF − pmin).
This system of nonlinear integral equations (2) and (6)
for the EMF and OP gives us a possibility to describe
the thermodynamics of superfluid non-unitary phases of
the 3He − A1,2 type in dense SPNM with spin-triplet
p-wave pairing in a static uniform high magnetic field
at arbitrary temperatures from the interval 0 ≤ T ≤
Tc(H). In the general case, these equations cannot be
solved analytically, and it is necessary to use numerical
methods for their solving. But we can solve Eqs. (2) and
(6), by using analytical methods in the limiting case, at
zero temperature (T=0), and it is the theme of the next
section.
3. Solutions of Equations for EMF and OP for
Dense SPNM at T = 0
Let us consider SPNM at T = 0. In this case, we have
solved analytically the integral equation (2) (with re-
gard for Eq. (6)) for EMF on the Fermi surface us-
ing perturbation theory on the small parameter hext ≡
|µn|H/εF � a < 1. Here, a ≡ εmax/εF−1 = 1−εmin/εF
is the cutoff parameter which is connected to the maxi-
mal εmax = pmax
2/2m∗n and minimal εmin = pmin
2/2m∗n
energies of quasiparticles (neutrons) (where pmax and
pmin have been introduced above in (3) and (6)). Note
that the maximal energy is somewhat larger than the
Fermi energy, εmax > εF, so that 0 < a < 1. Thus, we
have obtained the following solution of Eqs. (2) and (6)
to the first order in the small parameter hext:
γ(H, y) ≡ |µn|Heff(pF, H)
εF(y)
=
hext(H, y)
1− (r + 2spF
2)νF/2
, (7)
where r and s (see the text after (2)) are combinations
of the Skyrme parameters, and the density of states νF
at the Fermi surface is
νF(y) = (m∗npF)/(π2~3) ≈
≈ 0.00419
m∗n(y)
mn
y1/3 MeV−1fm−3. (8)
Formulas (4)–(8) contain a free neutron mass mn ≈
939.56563 MeV/c2 [43] and the effective neutron mass
m∗n which depends on the density of NM n = yn0 ac-
cording to the formula
m
m∗n
= 1 +
myn0
4~2
[t1(1− x1) + 3t2(1 + x2)], (9)
where m ≡ (mp +mn)/2 ≈ 938.91897MeV/c2 is the
mean value of a free nucleon mass [27]; the parameters
t1, t2, x1, and x2 have specific numerical values for each
Skyrme parametrization. Note also that the Fermi en-
ergy εF ≡ p2
F/2m
∗
n of a pure NM with density n = yn0
is defined by the formula
εF = (3π2yn0)2/3
~2
2m∗n
≈ 60.8601y2/3mn
m∗n
MeV. (10)
It should be emphasized that the general approximate
formula (7) for Heff(pF, H) is valid for all parametriza-
tions of the Skyrme forces admissible for an NM, and
Heff is independent of the cutoff parameter a < 1 and
the energy gap (with accuracy of the first order) in the
energy spectrum of neutrons in SPNM.
646 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
MAGNETIC SUSCEPTIBILITY OF DENSE SUPERFLUID NEUTRON MATTER
Fig. 1. Ratio χFree/χSLy2(y) (see (12)) as a function of the reduced
density y = n/n0 for superfluid NM with Sly2 parametrization of
the Skyrme forces and spin-triplet p-wave pairing of the 3He − A
type in a magnetic field at T = 0
Now let us consider the Sly2, Gs, and RATP
parametrizations of the Skyrme forces (see [27–29]).
This specification gives us a possibility to plot the fig-
ures for the ratio of the Pauli susceptibility of the free
neutron gas χFree and the paramagnetic susceptibility
of SPNM with the Skyrme interaction χSkyrme(y). Note
that the inverse ratio of these functions (see (7))
χSkyrme(y)
χFree
=
1
1− (r + 2sp2
F)νF/2
(11)
describes a renormalization of the magnetic field inside
SPNM with triplet p-wave pairing of the 3He−A1,2 type.
Here, we represent, for the Sly2, Gs and RATP-
variants of the Skyrme interaction, the power indices
αSLy2 = 1/6, αGs = 0.30, and αRATP = 0.20 in their
density dependence. Then, for the SLy2 parametrization
of the Skyrme forces, we have obtained the required ex-
pression for the ratio of χFree and χSkyrme(y) from (11):
χFree
χSLy2(y)
= 1 +
2y1/3(0.5004y1/6 + 0.5461)
(1 + 0.659y)
−
− 2.7608y
(1 + 0.659y)
. (12)
Fig. 2. Function χFree/χGs(y) (see (13)) for superfluid NM with
the Gs parametrization of the Skyrme forces and spin-triplet p-
wave pairing of the 3He−A type in a magnetic field at T = 0
In a similar way for the Gs parametrization of the
Skyrme forces [28], we have found from (11) that
χFree
χGs(y)
= 1− 2y1/3(2.31247y3/10 − 2.80053)
(1 + 0.0810y)
−
− 1.29732y
(1 + 0.0810y)
. (13)
Finally for the RATP parametrization of the Skyrme
forces [29], we have obtained the expression
χFree
χRATP (y)
= 1− y1/3(1.1757y1/5 − 2.6318)
(1 + 0.235y)
−
− 2.6248y
(1 + 0.235y)
. (14)
Now, on the basis of formulae (12)–(14), we can repre-
sent Figs. 1–3 for the functions χFree/χSkyrme(y).
Note that the points of intersection of these three lines
(12)–(14) with the abscissa axis correspond to the criti-
cal densities nC(SLy2) ≈ 1.72n0, nC(Gs) ≈ 1.33n0, and
nC(RATP) ≈ 1.03n0, respectively. These critical densi-
ties correspond to phase transitions from the superfluid
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 647
A.N. TARASOV
Fig. 3. Function χFree/χRATP(y) (see (14)) for superfluid NM
with the RATP parametrization of the Skyrme forces and spin-
triplet p-wave pairing of the 3He − A type in a magnetic field at
T = 0
paramagnetic state of neutron matter with triplet pair-
ing to the ferromagnetic state which coexists, quite pos-
sibly, with the triplet superfluidity at densities higher
than nC(Skyrme). But this problem of coexistence of
spin-triplet superfluidity and ferromagnetism in a dense
pure neutron matter should be examined in more details
in a separate investigation. Such phase transitions might
occur in the liquid outer core of neutron stars.
We have also solved Eq. (6) for the OP Δ0(a; y) =
Δ↑(a; y,H = 0) = Δ↓(a; y,H = 0) of the SPNM at H =
0 and T = 0 with a pairing of the 3He−A type and have
derived the ratio of the maximal value of the reduced
anisotropic energy gap g(a; y) ≡ pF(y)Δ0(a; y)/εF(y) to
the PT temperature tc0 ≡ Tc0/εF (Tc0 is the tempera-
ture of PT for NM to the superfluid state with triplet
p-wave pairing without magnetic field) which is valid for
an arbitrary parametrization of the Skyrme forces (see
[24]) in the form
g(a; y)
tc0(a; y)
= 2 exp
(
5
6
− b0
2
)
≈ 2.0174, (15)
where
tc0 =
a
2
exp
(
`(a)
2
+
2
c3n0ym∗n
)
, (16)
`(a) ≈ b0 +
3a2
8
+
3a4
256
, (17)
b0 = 2
(
1− 1
9
+
2
75
)
+ 4
∞∑
k=1
(−1)k+1Ei(−2k) ≈
≈ 1.64932, (18)
and
Ei(−x) =
−x∫
−∞
et
t
dt (19)
[formulas (15) and (16) are valid for all Skyrme
parametrizations]. Here, c3n0m
∗
n < 0 is the dimension-
less value depending on the Skyrme parameters t2, x2
and t1, x1 (see the text after Eq. (6) and Eq. (9)).
Note that, at T = 0 in a sufficiently strong EMF such
that
γ(H, y)� a < 1 (20)
(see (7)), the approximate analytic expressions for
g↑ 6= g↓ (which are, by definition, g↑(↓)(a; y,H) ≡
pF(y)Δ↑(↓)(a; y,H)/εF(y)) can be found from the inte-
gral equations (6) for the OP. But here we only remark
that, according to our numerical estimates for the SLy2,
Gs, and RATP parametrizations [26–29] (proposed for
astrophysical purposes to describe NM properties in the
core of a neutron star at high densities), the values
|g↑(↓) − g|/g . 0.01 are small (in the 0.7 . y < 2.0 in-
terval studied here) even in sufficiently strong magnetic
fields Heff,Skyrme < 1017 G (see (20) and (7)) which are
realized very likely in the so-called magnetars [6,44,45],
i.e., strongly magnetized neutron stars. It is possible (as
it was argued in [46]) that magnetars constitute about
10% of the neutron star population.
4. Conclusion
Having solved the integral equations (2) and (6), we have
obtained the general analytic formulas (7) (see also (11))
and (15) for the EMF and OP valid at zero tempera-
ture T = 0 for arbitrary parametrizations of the Skyrme
forces in a dense SPNM (at subnuclear and supranuclear
densities in the range 0.7n0 . n < nC(Skyrme) < 2n0)
with anisotropic OP similar to those of 3He − A. We
have specified Eq. (11) for the specific parametrizations
SLy2, Gs, and RATP [26–29] of the Skyrme forces and
obtained formulae (12), (13) and (14) for the ratio of
648 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
MAGNETIC SUSCEPTIBILITY OF DENSE SUPERFLUID NEUTRON MATTER
paramagnetic susceptibilities χFree/χSkyrme(y) which are
valid even for sufficiently strong magnetic fields H (but
H � Hmax). For upper limitHmax,Skyrme(y) of magnetic
fields, we have (see (9) and (10))
Hmax,Skyrme(y) ≈ 1.0098× 1019y2/3(1 + βSkyrme y) G,
(21)
where βSLy2 ≈ 0.659, βGs ≈ 0.081, and βRATP ≈ 0.235.
Ultra-strong magnetic fields may approach 1018 G (see,
e.g., [47,48]) in the core region of magnetars.
Note that the critical densities for the onset of ferro-
magnetism nC(RATP) ≈ 0.17 fm−3 and nC(SLy2) ≈
0.29 fm−3 (which are very close to our results for
nC(Skyrme)) have been obtained also in [30] and [31],
but for the case of phase transitions in pure NM from
the normal (nonsuperfluid) state to the ferromagnetic
state. Such a proximity with our results (accounting the
triplet superfluidity) for nC(Skyrme) can be explained
by a very small value of superfluid corrections (which
are of the second order, i.e. they are proportional to
(Δ/εF)2 � 1, where Δ is the maximal anisotropic en-
ergy gap in the spectrum of quasiparticles (neutrons) in
the SPNM considered here) to the paramagnetic suscep-
tibility in superfluid NM with spin-triplet pairing. Simi-
larly, in the case of the phase transition between normal
liquid 3He and superfluid 3He−A [19, 20] in a magnetic
field, their paramagnetic susceptibilities are almost coin-
cide with each other (the difference is less than 1%, see
also [49]).
We note finally that the phenomena of superfluidity
and magnetism in NM at high densities n > 2n0 (in-
side the fluid cores of pulsars and magnetars [6,44,45])
should be studied in the framework of a relativistic ap-
proach and with various interpretations of the hadron
matter structure (including mesons, hyperons, quarks,
and other possible constituents).
The material of this paper was presented by the author
at the International Bogolyubov Kyiv Conference “Mod-
ern problems of theoretical and mathematical physics”
(Kyiv, September 15–18, 2009).
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Received 03.11.09
ПРО МАГНIТНУ СПРИЙНЯТЛИВIСТЬ
ГУСТОЇ НАДПЛИННОЇ НЕЙТРОННОЇ
МАТЕРIЇ ЗI СПIН-ТРИПЛЕТНИМ p-СПАРЮВАННЯМ
О.М. Тарасов
Р е з ю м е
Суто нейтронна матерiя зi спiн-триплетним p-спарюванням ви-
вчається у межах нерелятивiстської узагальненої теорiї фермi-
рiдини при суб’ядерних та над’ядерних густинах (у дiапазонi
густин 0, 7n0 ≤ n < nC(Skyrme) < 2n0, де n0 = 0, 17 фм−3
– це густина насичення симетричної ядерної матерiї) при тем-
пературi, що дорiвнює нулю, та за наявностi сильного магнi-
тного поля. Ефективнi сили Скiрма використовуються у ролi
взаємодiї мiж нейтронами. У результатi отримано загальний
аналiтичний вираз (справедливий для довiльної параметриза-
цiї сил Скiрма) для парамагнiтної сприйнятливостi надплин-
ної нейтронної матерiї як функцiї вiд густини при нульовiй
температурi. Цей вираз далi конкретизовано для трьох типiв
взаємодiї Скiрма з рiзними степеневими залежностями вiд гу-
стини n. А саме, знайдено, що для випадкiв нейтронної ма-
терiї з так званими RATP, Gs та SLy2 параметризацiями сил
Скiрма у магнiтної сприйнятливостi виникає розбiжнiсть при
критичних густинах nC(RATP) ≈ 1, 03n0, nC(Gs) ≈ 1, 33n0 та
nC(SLy2) ≈ 1, 72n0. Цi критичнi густини вiдповiдають фазо-
вим переходам з надплинного парамагнiтного стану нейтрон-
ної матерiї з триплетним спарюванням у феромагнiтний стан,
який може спiвiснувати з триплетною надплиннiстю при гу-
стинах, бiльших за nC(Skyrme). Такi фазовi переходи можуть
виникати у рiдких зовнiшнiх ядрах пульсарiв i так званих ма-
гнетарiв.
650 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
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