Magnetization of Dense Neutron Matter in a Strong Magnetic Field
Spin polarized states in neutron matter at a strong magnetic field up to ¹⁸ G are considered in the model with the Skyrme effective interaction. Analyzing the self-consistent equations at zero temperature, it is shown that a thermodynamically stable branch of solutions for the spin polarization para...
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| Цитувати: | Magnetization of Dense Neutron Matter in a Strong Magnetic Field / A.A. Isayev, J. Yang // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 515-523. — Бібліогр.: 41 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860237310664114176 |
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| author | Isayev, A.A. Yang, J. |
| author_facet | Isayev, A.A. Yang, J. |
| citation_txt | Magnetization of Dense Neutron Matter in a Strong Magnetic Field / A.A. Isayev, J. Yang // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 515-523. — Бібліогр.: 41 назв. — англ. |
| collection | DSpace DC |
| container_title | Український фізичний журнал |
| description | Spin polarized states in neutron matter at a strong magnetic field up to ¹⁸ G are considered in the model with the Skyrme effective interaction. Analyzing the self-consistent equations at zero temperature, it is shown that a thermodynamically stable branch of solutions for the spin polarization parameter as a function of the density corresponds to the negative spin polarization when the majority of neutron spins are oriented oppositely to the direction of the magnetic field. In addition, beginning from some threshold density dependent on the magnetic field strength, the self-consistent equations have also two other branches of solutions for the spin polarization parameter with the positive spin polarization. The free energy corresponding to one of these branches turns out to be very close to the free energy corresponding to the thermodynamically preferable branch with the negative spin polarization. As a consequence, at a strong magnetic field, the state with the positive spin polarization can be realized as a metastable state at the high density region in neutron matter which changes into a thermodynamically stable state with the negative spin polarization with decrease in the density at some threshold value. The calculations of the neutron spin polarization parameter, energy per neutron, and chemical potentials of spin-up and spin-down neutrons as functions of the magnetic field strength show that the influence of the magnetic field remains small at the field strengths up to 10¹⁷ G.
Розглянуто спiновi поляризованi стани в нейтроннiй матерiї у сильних магнiтних полях до 10¹⁸ Гс у моделi з ефективною взаємодiєю Скiрма. На пiдставi аналiзу рiвнянь самоузгодження при нульовiй температурi показано, що термодинамiчно стiйка гiлка розв’язкiв для параметра спiнової поляризацiї як функцiї густини вiдповiдає вiд’ємнiй спiновiй поляризацiї, коли бiльшiсть нейтронних спiнiв орiєнтується протилежно магнiтному полю. Крiм цього, починаючи з деякої граничної густини, що залежить вiд напруженостi магнiтного поля, рiвняння самоузгодження мають також двi iнших гiлки розв’язкiв для параметра спiнової поляризацiї з додатною спiновою поляризацiєю. Вiльна енергiя, що вiдповiдає однiй iз цих гiлок, виявляється дуже близькою до вiльної енергiї, що вiдповiдає термодинамiчно стiйкiй гiлцi з вiд’ємною спiновою поляризацiєю. Як наслiдок, у сильному магнiтному полi стан з додатною спiновою поляризацiєю може реалiзовуватися як метастабiльний стан за високими густинами нейтронної матерiї, який змiнюється на термодинамiчний стiйкий стан з вiд’ємною спiновою поляризацiєю при зменшеннi густини, починаючи з деякої граничної густини.
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| fulltext |
NUCLEI AND NUCLEAR REACTIONS
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 515
MAGNETIZATION OF DENSE NEUTRON MATTER
IN A STRONG MAGNETIC FIELD
A.A. ISAYEV,1 J. YANG2
1Kharkov Institute of Physics and Technology
(1, Academicheskaya Str., Kharkov 61108, Ukraine; e-mail: isayev@ kipt. kharkov. ua )
2Department of Physics and the Institute for the Early Universe,
Ewha Womans University
(Seoul 120-750, Korea; e-mail: jyang@ ewha. ac. kr )
PACS 21.65.Cd, 26.60.-c,
97.60.Jd, 21.30.Fe
c©2010
Spin polarized states in neutron matter at a strong magnetic field
up to 1018 G are considered in the model with the Skyrme effec-
tive interaction. Analyzing the self-consistent equations at zero
temperature, it is shown that a thermodynamically stable branch
of solutions for the spin polarization parameter as a function of
the density corresponds to the negative spin polarization when
the majority of neutron spins are oriented oppositely to the di-
rection of the magnetic field. In addition, beginning from some
threshold density dependent on the magnetic field strength, the
self-consistent equations have also two other branches of solutions
for the spin polarization parameter with the positive spin polar-
ization. The free energy corresponding to one of these branches
turns out to be very close to the free energy corresponding to the
thermodynamically preferable branch with the negative spin po-
larization. As a consequence, at a strong magnetic field, the state
with the positive spin polarization can be realized as a metastable
state at the high density region in neutron matter which changes
into a thermodynamically stable state with the negative spin polar-
ization with decrease in the density at some threshold value. The
calculations of the neutron spin polarization parameter, energy
per neutron, and chemical potentials of spin-up and spin-down
neutrons as functions of the magnetic field strength show that the
influence of the magnetic field remains small at the field strengths
up to 1017 G.
1. Introduction
Neutron stars observed in the nature are magnetized ob-
jects with the magnetic field strength at the surface in
the range 109 – 1013 G [1]. For a special class of neu-
tron stars such as soft gamma-ray repeaters and anoma-
lous X-ray pulsars, the field strength can be much larger
and is estimated to be about 1014 – 1015 G [2]. These
strongly magnetized objects are called magnetars [3] and
comprise about 10% of the whole population of neutron
stars [4]. However, in the interior of a magnetar, the
magnetic field strength may be even larger, reaching the
values about 1018 G [5, 6]. The possibility of the exis-
tence of such ultrastrong magnetic fields is not yet ex-
cluded, because what we can learn from the magnetar
observations by their periods and spin-down rates, or by
hydrogen spectral lines, is only their surface fields. There
is still no general consensus regarding the mechanism to
generate such strong magnetic fields of magnetars, al-
though different scenarios were suggested such as, e.g.,
a turbulent dynamo amplification mechanism in rapidly
rotating neutron stars [2] or the possibility of the sponta-
neous spin ordering in the dense quark core of a neutron
star [7].
Under such circumstances, the issue of interest is the
behavior of a neutron star matter in a strong magnetic
field [5, 6, 8, 9]. In the recent study [9], the neutron star
matter was approximated by a pure neutron matter in
the model considerations with the effective Skyrme and
Gogny forces. It has been shown that the behavior of the
spin polarization of neutron matter in the high density
region at a strong magnetic field crucially depends on
whether neutron matter develops a spontaneous spin po-
larization (in the absence of a magnetic field) at several
times nuclear matter saturation density, as is usual for
the Skyrme forces, or the appearance of a spontaneous
polarization is not allowed at the relevant densities (or
delayed to much higher densities), as in the case with
the Gogny D1P force. In the former case, a ferromag-
netic transition to a totally spin polarized state occurs.
A.A. ISAYEV, J. YANG
While in the latter case, a ferromagnetic transition is
excluded at all relevant densities, and the spin polariza-
tion remains quite low even in the high density region.
Note that the issue of the spontaneous appearance of
spin polarized states in neutron and nuclear matter is
a controversial one. On the one hand, the models with
the Skyrme effective nucleon-nucleon (NN) interaction
predict the occurrence of spontaneous spin instability
in nuclear matter at densities in the range from %0 to
4%0 for different parametrizations of the NN potential
[10–22] (%0 = 0.16 fm−3 is the nuclear saturation den-
sity). For the Gogny effective interaction, a ferromag-
netic transition in neutron matter occurs at the den-
sity larger than 7%0 for the D1P parametrization and
is not allowed for D1 and D1S parametrizations [23].
However, for the D1S Gogny force, an antiferromagnetic
phase transition happens in symmetric nuclear matter at
the density 3.8%0 [24]. On the other hand, for the models
with the realistic NN interaction, no sign of spontaneous
spin instability at any isospin asymmetry has been found
so far up to densities well above %0 [25–31].
Here, we will study the thermodynamic properties of
spin polarized neutron matter at a strong magnetic field
in the model with the Skyrme effective forces. As a
framework for consideration, we choose a Fermi liquid
approach for the description of nuclear matter [32–34].
Proceeding from the minimum principle for the thermo-
dynamic potential, we get the self-consistent equations
for the spin order parameter and the chemical potential
of neutrons. In the absence of a magnetic field, the self-
consistent equations have two degenerate branches of so-
lutions for the spin polarization parameter correspond-
ing to the case where the majority of neutron spins is
oriented along or oppositely to the spin quantization axis
(positive or negative spin polarization, respectively). In
the presence of a magnetic field, these branches are mod-
ified differently. A thermodynamically stable branch cor-
responds to the state with the majority of neutron spins
aligned oppositely to the magnetic field. At a strong
magnetic filed, the branch corresponding to the positive
spin polarization splits onto two branches with the pos-
itive spin polarization as well. The last solutions were
missed in the study of Ref. [9]. We perform a thermody-
namic analysis based on the comparison of the respec-
tive free energies and arrive at the conclusion about the
possibility of the formation of metastable states in neu-
tron matter with the majority of neutron spins directed
along the strong magnetic field. The appearance of such
metastable states can be possible due to the strong spin-
dependent medium correlations in neutron matter with
the Skyrme forces at high densities.
Note that we consider the thermodynamic properties
of spin polarized states in neutron matter at a strong
magnetic field up to the high density region relevant for
astrophysics. Nevertheless, we take into account the nu-
cleon degrees of freedom only, although other degrees of
freedom, such as pions, hyperons, kaons, or quarks could
be important at such high densities.
2. Basic Equations
The normal (nonsuperfluid) states of neutron matter are
described by the normal distribution function of neu-
trons fκ1κ2 = Tr %a+
κ2
aκ1 , where κ ≡ (p, σ), p is the
momentum, σ is the projection of a spin on the third
axis, and % is the density matrix of the system [20, 21].
Further, it will be assumed that the third axis is directed
along the external magnetic field H. The energy of the
system is specified as a functional of the distribution
function f , E = E(f), and determines the single-particle
energy
εκ1κ2(f) =
∂E(f)
∂fκ2κ1
. (1)
The self-consistent matrix equation for determining the
distribution function f follows from the minimum con-
dition of the thermodynamic potential [32, 33] and is
f = {exp(Y0ε+ Y4) + 1}−1 ≡ {exp(Y0ξ) + 1}−1
. (2)
Here, the quantities ε and Y4 are matrices in the space
of κ variables,
Y4κ1κ2 = Y4δκ1κ2 ,
Y0 = 1/T , and Y4 = −µ0/T are the Lagrange multipli-
ers, µ0 is the chemical potential of neutrons, and T is
the temperature.
Given the possibility for the alignment of neutron
spins along or oppositely to the magnetic field H, the
normal distribution function of neutrons and the single-
particle energy can be expanded in the Pauli matrices σi
in the spin space
f(p) = f0(p)σ0 + f3(p)σ3,
ε(p) = ε0(p)σ0 + ε3(p)σ3. (3)
Using Eqs. (2) and (3), one can express evidently the
distribution functions f0, f3 in terms of the quantities ε:
f0 =
1
2
{n(ω+) + n(ω−)},
516 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
MAGNETIZATION OF DENSE NEUTRON MATTER
f3 =
1
2
{n(ω+)− n(ω−)}. (4)
Here, n(ω) = {exp(Y0ω) + 1}−1 and
ω± = ξ0 ± ξ3,
ξ0 = ε0 − µ0, ξ3 = ε3. (5)
As follows from the structure of the distribution func-
tions f , the quantity ω±, being the exponent in the Fermi
distribution function n, plays the role of a quasiparticle
spectrum. The spectrum is twofold split due to the spin
dependence of the single-particle energy ε(p) in Eq. (3).
The branches ω± correspond to neutrons with spin up
and spin down.
The distribution functions f should satisfy the norma-
lization conditions
2
V
∑
p
f0(p) = %, (6)
2
V
∑
p
f3(p) = %↑ − %↓ ≡ Δ%. (7)
Here, % = %↑ + %↓ is the total density of neutron mat-
ter, %↑ and %↓ are the neutron number densities with
spin up and spin down, respectively. The quantity Δ%
may be regarded as the neutron spin order parameter. It
determines the magnetization of the system M = µnΔ%,
µn being the neutron magnetic moment. The magne-
tization may contribute to the internal magnetic field
B = H+4πM . However, we will assume, analogously to
Refs. [6, 9], that the contribution of the magnetization
to the magnetic field B remains small for all relevant
densities and magnetic field strengths, and, hence,
B ≈ H. (8)
This assumption holds true due to the tiny value of the
neutron magnetic moment µn = −1.9130427(5)µN ≈
−6.031×10−18 MeV/G [35] (µN being the nuclear mag-
neton) and is confirmed numerically by finding solutions
of the self-consistent equations in two approximations,
corresponding to preserving and neglecting the contri-
bution of the magnetization.
In order to get the self-consistent equations for the
components of the single-particle energy, one has to set
the energy functional of the system. In view of the ap-
proximation (8), it reads [21, 33]
E(f) = E0(f,H) + Eint(f) + Efield, (9)
E0(f,H) = 2
∑
p
ε0(p)f0(p)− 2µnH
∑
p
f3(p),
Eint(f) =
∑
p
{ε̃0(p)f0(p) + ε̃3(p)f3(p)},
Efield =
H2
8π
V,
where
ε̃0(p) =
1
2V
∑
q
Un0 (k)f0(q), k =
p− q
2
,
ε̃3(p) =
1
2V
∑
q
Un1 (k)f3(q). (10)
Here, ε0(p) = p 2
2m0
is the free single-particle spectrum,
m0 is the bare mass of a neutron, Un0 (k), Un1 (k) are the
normal Fermi liquid (FL) amplitudes, and ε̃0, ε̃3 are the
FL corrections to the free single-particle spectrum. In
this study, we will not be interested in the total energy
density and the pressure in the interior of a neutron star.
By this reason, the field contribution Efield, being the
energy of the magnetic field in the absence of matter,
can be omitted. Using Eqs. (1) and (9), we get the self-
consistent equations in the form
ξ0(p) = ε0(p)+ ε̃0(p)−µ0, ξ3(p) = −µnH+ ε̃3(p). (11)
To obtain numerical results, we utilize the effective
Skyrme interaction (SLy4 and SLy7 parametrizations).
Using the same procedure as in Ref. [33], it is possible to
find expressions for the normal FL amplitudes in terms
of the Skyrme force parameters ti, xi, β:
Un0 (k) = 2t0(1− x0) +
t3
3
%β(1− x3)+
+
2
~2
[t1(1− x1) + 3t2(1 + x2)]k2, (12)
Un1 (k) = −2t0(1− x0)−
t3
3
%β(1− x3)+
+
2
~2
[t2(1 + x2)− t1(1− x1)]k2 ≡ an + bnk2. (13)
Further, we do not consider the effective tensor forces
which lead to coupling of the momentum and spin
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 517
A.A. ISAYEV, J. YANG
degrees of freedom [36–38] and, respectively, to the
anisotropy in the momentum dependence of FL ampli-
tudes with respect to the spin quantization axis. Then
ξ0 =
p2
2mn
− µ, (14)
ξ3 = −µnH + (an + bn
p2
4
)
Δ%
4
+
bn
16
〈q2〉3, (15)
where the effective neutron mass mn is defined by the
formula
~2
2mn
=
~2
2m0
+
%
8
[t1(1− x1) + 3t2(1 + x2)], (16)
and the renormalized chemical potential µ should be de-
termined from Eq. (6). The quantity 〈q2〉3 in Eq. (15) is
the second-order moment of the distribution function f3:
〈q2〉3 =
2
V
∑
q
q2f3(q). (17)
In view of Eqs. (14) and (15), the branches ω± ≡ ωσ of
the quasiparticle spectrum in Eq. (5) read
ωσ =
p2
2mσ
− µ+ σ
(
−µnH +
anΔ%
4
+
bn
16
〈q2〉3
)
, (18)
where mσ is the effective mass of a neutron with spin up
(σ = +1) and spin down (σ = −1)
~2
2mσ
=
~2
2m0
+
%σ
2
t2(1 + x2)+
+
%−σ
4
[t1(1− x1) + t2(1 + x2)], %+(−) ≡ %↑(↓). (19)
For totally spin polarized neutron matter, we have
m0
m∗
= 1 +
%m0
~2
t2(1 + x2), (20)
where m∗ is the effective neutron mass in the fully polar-
ized state. Since t2 < 0 usually for Skyrme parametriza-
tions, we have the constraint x2 ≤ −1 which guarantees
the stability of totally polarized neutron matter at high
densities.
It follows from Eq. (18) that the effective chemical
potential µσ for neutrons with spin-up (σ = 1) and spin-
down (σ = −1) can be determined as
µσ = µ+ σ
(
µnH −
anΔ%
4
− bn
16
〈q2〉3
)
. (21)
Thus, with account of expressions (4) for the distribu-
tion functions f , we obtain the self-consistent equations
(6), (7), and (17) for the effective chemical potential
µ, spin order parameter Δ%, and second-order moment
〈q2〉3.
3. Solutions of Self-Consistent Equations at
T = 0. Thermodynamic Stability
Here, we directly solve the self-consistent equations at
zero temperature and present the neutron spin order
parameter as a function of the density and the mag-
netic field strength. In solving numerically the self-
consistent equations, we utilize SLy4 and SLy7 Skyrme
forces, which were constrained originally to reproduce
the results of microscopic neutron matter calculations
(pressure versus density curve) [39]. Note that the den-
sity dependence of the nuclear symmetry energy, cal-
culated with these Skyrme interactions, gives the neu-
tron star models in a good agreement with the observ-
ables such as the minimum rotation period, gravitational
mass-radius relation, the binding energy released in a su-
pernova collapse, etc. [40]. In addition, these Skyrme
parametrizations satisfy the constraint x2 ≤ −1 ob-
tained from Eq. (20).
We consider magnetic fields up to the values allowed
by the scalar virial theorem. For a neutron star with
massM and radiusR, equating the magnetic field energy
EH ∼ (4πR3/3)(H2/8π) with the gravitational binding
energy EG ∼ GM2/R, one gets the estimate Hmax ∼
M
R2 (6G)1/2. For a typical neutron star with M = 1.5M�
and R = 10−5R�, this yields the maximum value of the
magnetic field strength Hmax ∼ 1018 G. This magnitude
can be expected in the interior of a magnetar while the
recent observations report the surface values up to H ∼
1015 G, as inferred from the hydrogen spectral lines [41].
In order to characterize the spin ordering in neutron
matter, it is convenient to introduce a neutron spin po-
larization parameter
Π =
%↑ − %↓
%
≡ Δ%
%
. (22)
Fig. 1 shows the dependence of the neutron spin po-
larization parameter on the density normalized to the
nuclear saturation density %0, at zero temperature in
the absence of a magnetic field. The spontaneous po-
larization develops at % = 3.70%0 for the SLy4 interac-
tion (%0 = 0.16 fm−3) and at % = 3.59%0 for the SLy7
interaction (%0 = 0.158 fm−3), which reflects the insta-
bility of neutron matter with the Skyrme interaction at
such densities against spin fluctuations. Since the self-
consistent equations at H = 0 are invariant with respect
to the global flip of neutron spins, we have two branches
of solutions for the spin polarization parameter, Π+
0 (%)
(upper) and Π−0 (%) (lower) which differ only by sign,
Π+
0 (%) = −Π−0 (%).
518 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
MAGNETIZATION OF DENSE NEUTRON MATTER
3 4 5 6
-1.00
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
Π
0
-
Π
0
+
SLy4
SLy7
T=0
Π
Fig. 1. Neutron spin polarization parameter as a function of the
density at vanishing temperature and magnetic field
Figure 2 shows the neutron spin polarization parame-
ter as a function of the density for a set of fixed values
of the magnetic field. The branches of spontaneous po-
larization are modified by the magnetic field differently,
since the self-consistent equations at H 6= 0 lose the in-
variance with respect to the global flip of spins. At non-
vanishing H, the lower branch Π1(%), corresponding to
the negative spin polarization, extends down to the very
low densities. There are three characteristic density do-
mains for this branch. At low densities % . 0.5%0, the ab-
solute value of the spin polarization parameter increases
with decrease in the density. At intermediate densities
0.5%0 . % . 3%0, there is a plateau in the Π1(%) depen-
dence, whose characteristic value depends on H, e.g.,
Π1 ≈ −0.08 at H = 1018 G. At densities % & 3%0, the
magnitude of the spin polarization parameter increases
with the density, and neutrons become totally polarized
at % ≈ 6%0.
Note that the results in the low-density domain should
be considered as the first approximation to the real com-
plex picture, since, as discussed in details in Ref. [9], the
low-density neutron-rich matter in β-equilibrium pos-
sesses a frustrated state, “nuclear pasta” arising as a
result of the competition of Coulomb long-range interac-
tions and nuclear short-range forces. In our case, where
a pure neutron matter is considered, there is no mechan-
ical instability due to the absence of the Coulomb inter-
action. However, the possibility of the appearance of a
low-density nuclear magnetic pasta and its impact on the
neutrino opacities in the protoneutron star early cooling
stage should be explored in a more detailed analysis.
0 1 2 3 4 5 6
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Π
0
+
Π
0
-
Π
3
Π
2
Π
1
(b)
SLy7
Π
H = 0 G
10
17
G
5 x 10
17
G
10
18
G
Π
0
+
Π
0
-
Π
3
Π
2
Π
1
(a)SLy4
ρ
Fig. 2. Neutron spin polarization parameter as a function of the
density at T = 0 and different magnetic field strengths for SLy4
(top) and SLy7 (bottom) interactions
At a nonzero magnetic field, the upper branch of spon-
taneous polarization Π+
0 (%), corresponding to positive
values of Π, turns into two branches with different de-
pendences on the density. For one of these branches,
Π2(%), the spin polarization parameter decreases with
the density and tends to zero. For the other branch,
Π3(%), it increases with the density and is saturated. It
is important that these branches appear step-wise at the
same threshold density %th dependent on the magnetic
field and being larger than the critical density of sponta-
neous spin instability in neutron matter. For example,
for SLy7 interaction, %th ≈ 3.80 %0 at H = 5 × 1017 G,
and %th ≈ 3.92 %0 at H = 1018 G. The magnetic field,
due to the negative value of the neutron magnetic mo-
ment, tends to orient the neutron spins oppositely to
the magnetic field direction. As a result, the spin po-
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 519
A.A. ISAYEV, J. YANG
0 1 2 3 4 5 6
0
50
100
150
200
250
300
H = 0 G, Π
0
(ρ)
= 10
18
G, Π
1
(ρ)
= 10
18
G, Π
2
(ρ)
= 10
18
G, Π
3
(ρ)
SLy4
E
/A
[
M
e
V
/n
e
u
tr
o
n
]
Fig. 3. Energy per neutron as a function of the density at T = 0 for
different branches Π1(%)-Π3(%) of solutions of the self-consistent
equations at H = 1018 G for the SLy4 interaction, including a
spontaneously polarized state
larization parameter for the branches Π2(%), Π3(%) with
the positive spin polarization is smaller than that for
the branch of spontaneous polarization Π+
0 , and, vice
versa, the magnitude of the spin polarization parameter
for the branch Π1(%) with the negative spin polarization
is larger than the corresponding value for the branch of
spontaneous polarization Π−0 . Note that the impact of
even such strong magnetic field as H = 1017 G is small:
the spin polarization parameter for all three branches
Π1(%)-Π3(%) is either close to zero or close to its value
in the state with spontaneous polarization which is gov-
erned by the spin-dependent medium correlations.
Thus, at densities larger than %th, we have three
branches of solutions: one of them, Π1(%), with the neg-
ative spin polarization and two others, Π2(%) and Π3(%),
with the positive polarization. In order to clarify, which
branch is thermodynamically preferable, we should com-
pare the corresponding free energies. Because the results
of calculations with SLy4 and SLy7 Skyrme forces are
very close, we will present the obtained dependences for
one of these parametrizations. Figure 3 shows the en-
ergy per neutron as a function of the density at T = 0
and H = 1018 G for these three branches, compared
with the energy per neutron for a spontaneously polar-
ized state [the branches Π±0 (%)]. It is seen that the state
with the majority of neutron spins oriented oppositely
to the direction of the magnetic field [the branch Π1(%)]
has a lowest energy. This result is intuitively clear, since
the magnetic field tends to direct the neutron spins op-
positely to H, as mentioned above. However, the state,
described by the branch Π3(%) with the positive spin po-
larization, has the energy very close to that of the ther-
modynamically stable state. This means that, despite
the presence of a strong magnetic field H ∼ 1018 G, the
state with the majority of neutron spins directed along
the magnetic field can be realized as a metastable state
in the dense core of a neutron star in the model con-
sideration with the Skyrme effective interaction. In this
scenario, since such states exist only at densities % > %th,
under decreasing the density (going from the interior to
the outer regions of a magnetar), a metastable state with
the positive spin polarization at the threshold density %th
changes into a thermodynamically stable state with the
negative spin polarization.
The existence of a metastable state with the positive
spin polarization in a neutron star at a strong magnetic
field could lead to the following effect. As is apparent
from our consideration, the p-h interaction in the Skyrme
model for neutron matter becomes large and attractive
in the spin channel with increase in the density. As a
result, the neutrino cross sections are enhanced at the
high-density region, and this drastically reduces the neu-
trino mean free paths [14]. Since the magnitude of the
spin polarization parameter in a metastable state with
the positive spin polarization is less than that in the
thermodynamically stable state with the negative spin
polarization, one can expect that the above-mentioned
effects will be less pronounced in a neutron star possess-
ing a metastable state with the positive spin polariza-
tion. Hence, this could lead to the accelerated cooling
of a neutron star with a metastable positive spin polar-
ized state as compared to the cooling rates of a neutron
star possessing a stable thermodynamic configuration of
neutron spins at a nonzero magnetic field.
At this point, we note some important differences be-
tween our results and those in Ref. [9]. First, while
studying the neutron matter in a strong magnetic field
[9], only one branch of solutions for the spin polariza-
tion parameter was found in the model with the Skyrme
interaction (for the same SLy4 and SLy7 parametriza-
tions). However, in fact, we have seen that the degener-
ate branches of spontaneous polarization (at zero mag-
netic field) with the positive and negative spin polariza-
tions are modified differently by the magnetic field. As
a result, there are generally three different branches of
solutions of the self-consistent equations at a nonvanish-
ing magnetic field in the Skyrme model. In addition,
the only branch which was considered in Ref. [9] and
corresponds to our thermodynamically stable branch Π1
is characterized by the positive spin polarization, con-
520 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
MAGNETIZATION OF DENSE NEUTRON MATTER
10
16
10
17
10
18
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Π
1
(H), ρ = 4ρ
0
Π
2
(H), ρ = 4ρ
0
Π
3
(H), ρ = 4ρ
0
Π
1
(H), ρ = 2ρ
0
SLy7
Π
H [G]
Fig. 4. Spin polarization parameter as a function of the magnetic
field strength at T = 0 for different branches Π1(H)-Π3(H) of
solutions of the self-consistent equations at % = 4%0 and for the
branch Π1(H) at % = 2%0 for the SLy7 interaction
trary to our result with Π1 < 0. This disagreement
is explained by the incorrect sign before the term in-
volving the magnetic field in the equation describing the
single-particle spectrum in Ref. [9] (analog to Eq. (18)
in our case). Clearly, the majority of neutron spins in
the equilibrium configuration are aligned oppositely to
the magnetic field.
Figure 4 shows the spin polarization parameter as a
function of the magnetic field strength at zero tempera-
ture for different branches Π1(H)-Π3(H) of solutions of
the self-consistent equations at % = 4%0 compared with
that for the branch Π1(H) at % = 2%0. It is seen that,
up to the field strengths H = 1017 G, the influence of
the magnetic field is rather marginal. For the branches
Π1(H) and Π2(H), the value of the spin polarization
parameter increases with the field strength, while it de-
creases for Π3(H).
Figure 5 shows the energy of neutron matter per par-
ticle as a function of the magnetic field strength at T = 0
under the same assumptions as in Fig. 4. It is seen that
the state with the negative spin polarization [branch
Π1(H)] becomes more preferable with increase in the
magnetic field, although the total effect of changing the
magnetic field strength by two orders of magnitude on
the energy corresponding to all three branches Π1(H)-
Π3(H) remains small. It is also seen that the increase in
the density by a factor of two leads to the increase in the
energy per neutron roughly by a factor of three reflect-
ing the fact that the medium correlations play a more
10
16
10
17
10
18
45
60
75
90
105
120
135
SLy7
E
/A
[
M
e
V
/n
e
u
tr
o
n
]
H [G]
Π
1
(H), ρ = 4 ρ
0
Π
2
(H), ρ = 4 ρ
0
Π
3
(H), ρ = 4 ρ
0
Π
1
(H), ρ = 2 ρ
0
Fig. 5. Same as in Fig. 4 but for the energy per neutron
important role in building the energetics of the system
than the impact of a strong magnetic field.
Figure 6 shows the chemical potentials of spin-up and
spin-down neutrons as functions of the magnetic field
strength, where the chemical potentials are calculated
according to Eq. (21). It is seen that the splitting of the
spin-up and spin-down chemical potentials is very small
up to the field strengths 1017 G. Hence, we arrive at the
previous conclusion that the impact of a magnetic field
remains small for such field strengths. Another point
is that the spin-up and spin-down chemical potentials
for branches Π1 and Π2 decrease, as H increases. But,
for the branch Π3, they increase with H. Hence, their
dependence on the magnetic field strength is basically
determined by the dependence of the effective chemical
potential µ on H which is found from the normalization
condition (6). In addition, the comparison of the spin-up
and spin-down chemical potentials for the branch Π1 at
% = 2%0 and % = 4%0 implies that the in-medium effects
strongly increase the effective chemical potentials.
In summary, we have considered the spin polarized
states in neutron matter at a strong magnetic field in a
model with the Skyrme effective NN interaction (SLy4
and SLy7 parametrizations). The self-consistent equa-
tions for the spin polarization parameter and the chem-
ical potential of neutrons have been obtained and ana-
lyzed at zero temperature. It has been shown that the
thermodynamically stable branch of solutions for the
spin polarization parameter as a function of the den-
sity corresponds to the case where the majority of neu-
tron spins is oriented oppositely to the direction of the
magnetic field (negative spin polarization). This branch
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 521
A.A. ISAYEV, J. YANG
10
16
10
17
10
18
200
300
400
500
Π
1
(H), ρ = 4ρ
0
Π
2
(H), ρ = 4ρ
0
Π
3
(H), ρ = 4ρ
0
Π
1
(H), ρ = 2ρ
0
µ
up
µ
down
SLy7
µ u
p
,
µ
Fig. 6. Chemical potentials of spin-up (solid curves) and spin-
down (dashed curves) neutrons as functions of the magnetic field
strength at T = 0 for different branches Π1(H)-Π3(H) of solutions
of the self-consistent equations at % = 4%0 and for the branch
Π1(H) at % = 2%0 for the SLy7 interaction
extends from very low densities to the high-density re-
gion, where the spin polarization parameter is saturated,
and, respectively, neutrons become totally spin polar-
ized. In addition, beginning from some threshold den-
sity %th dependent on the magnetic field strength, the
self-consistent equations have also two other branches
(upper and lower) of solutions for the spin polarization
parameter corresponding to the case where the major-
ity of neutron spins is oriented along the magnetic field
(positive spin polarization). For example, for the SLy7
interaction, %th ≈ 3.80 %0 at H = 5 × 1017 G, and
%th ≈ 3.92 %0 at H = 1018 G. The spin polarization
parameter along the upper branch increases with the
density and is saturated, while it decreases and vanishes
along the lower branch. The free energy correspond-
ing to the upper branch turns out to be very close to
that corresponding to the thermodynamically preferable
branch with the negative spin polarization. As a con-
sequence, at a strong magnetic field, the state with the
positive spin polarization can be realized as a metastable
state in the high-density region in neutron matter which
changes with decrease in the density (going from the in-
terior regions to the outer ones of a magnetar) at the
threshold density %th into a thermodynamically stable
state with the negative spin polarization. The calcula-
tions of the neutron spin polarization parameter, energy
per neutron, and chemical potentials of spin-up and spin-
down neutrons show that the influence of the magnetic
field remains small at the field strengths up to 1017 G.
Note that the consideration also has been done in Ref. [9]
for the Gogny effective NN interaction (D1S and D1P
parametrizations) up to densities 4%0. Since there is no
spontaneous ferromagnetic transition in neutron matter
for all relevant densities for the D1S parametrization,
and this transition occurs for the D1P parametrization
at a density larger than 7%0 [23], no sign of a ferromag-
netic transition at a strong magnetic field was found in
Ref. [9] up to densities 4%0 for these Gogny forces. Ac-
cording to our consideration, one can expect that the
metastable states with the positive spin polarization in
neutron matter at a strong magnetic field could appear
at densities larger than 7%0 for the D1P parametrization,
while the scenario with the only branch of solutions cor-
responding to the negative spin polarization would be
realized for the D1S force.
A.I. acknowledges the support from the Organiz-
ing Committee of Bogolyubov Kyiv Conference “Mod-
ern Problems of Theoretical and Mathematical Physics”
(September 15–18, 2009, Kyiv, Ukraine), where this re-
port was presented. J.Y. was supported by grant R32-
2009-000-10130-0 from WCU project of MEST and NRF
through Ewha Womans University.
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Received 18.09.09
НАМАГНIЧЕНIСТЬ ГУСТОЇ НЕЙТРОННОЇ МАТЕРIЇ
В СИЛЬНОМУ МАГНIТНОМУ ПОЛI
О.О. Iсаєв, Дж. Янг
Р е з ю м е
Розглянуто спiновi поляризованi стани в нейтроннiй матерiї у
сильних магнiтних полях до 1018 Гс у моделi з ефективною вза-
ємодiєю Скiрма. На пiдставi аналiзу рiвнянь самоузгодження
при нульовiй температурi показано, що термодинамiчно стiйка
гiлка розв’язкiв для параметра спiнової поляризацiї як функцiї
густини вiдповiдає вiд’ємнiй спiновiй поляризацiї, коли бiль-
шiсть нейтронних спiнiв орiєнтується протилежно магнiтному
полю. Крiм цього, починаючи з деякої граничної густини, що
залежить вiд напруженостi магнiтного поля, рiвняння само-
узгодження мають також двi iнших гiлки розв’язкiв для пара-
метра спiнової поляризацiї з додатною спiновою поляризацiєю.
Вiльна енергiя, що вiдповiдає однiй iз цих гiлок, виявляється
дуже близькою до вiльної енергiї, що вiдповiдає термодинамi-
чно стiйкiй гiлцi з вiд’ємною спiновою поляризацiєю. Як на-
слiдок, у сильному магнiтному полi стан з додатною спiновою
поляризацiєю може реалiзовуватися як метастабiльний стан за
високими густинами нейтронної матерiї, який змiнюється на
термодинамiчний стiйкий стан з вiд’ємною спiновою поляри-
зацiєю при зменшеннi густини, починаючи з деякої граничної
густини.
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 523
|
| id | nasplib_isofts_kiev_ua-123456789-56192 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 2071-0194 |
| language | English |
| last_indexed | 2025-12-07T18:25:37Z |
| publishDate | 2010 |
| publisher | Відділення фізики і астрономії НАН України |
| record_format | dspace |
| spelling | Isayev, A.A. Yang, J. 2014-02-13T19:55:09Z 2014-02-13T19:55:09Z 2010 Magnetization of Dense Neutron Matter in a Strong Magnetic Field / A.A. Isayev, J. Yang // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 515-523. — Бібліогр.: 41 назв. — англ. 2071-0194 PACS 21.65.Cd, 26.60.-c, 97.60.Jd, 21.30.Fe https://nasplib.isofts.kiev.ua/handle/123456789/56192 Spin polarized states in neutron matter at a strong magnetic field up to ¹⁸ G are considered in the model with the Skyrme effective interaction. Analyzing the self-consistent equations at zero temperature, it is shown that a thermodynamically stable branch of solutions for the spin polarization parameter as a function of the density corresponds to the negative spin polarization when the majority of neutron spins are oriented oppositely to the direction of the magnetic field. In addition, beginning from some threshold density dependent on the magnetic field strength, the self-consistent equations have also two other branches of solutions for the spin polarization parameter with the positive spin polarization. The free energy corresponding to one of these branches turns out to be very close to the free energy corresponding to the thermodynamically preferable branch with the negative spin polarization. As a consequence, at a strong magnetic field, the state with the positive spin polarization can be realized as a metastable state at the high density region in neutron matter which changes into a thermodynamically stable state with the negative spin polarization with decrease in the density at some threshold value. The calculations of the neutron spin polarization parameter, energy per neutron, and chemical potentials of spin-up and spin-down neutrons as functions of the magnetic field strength show that the influence of the magnetic field remains small at the field strengths up to 10¹⁷ G. Розглянуто спiновi поляризованi стани в нейтроннiй матерiї у сильних магнiтних полях до 10¹⁸ Гс у моделi з ефективною взаємодiєю Скiрма. На пiдставi аналiзу рiвнянь самоузгодження при нульовiй температурi показано, що термодинамiчно стiйка гiлка розв’язкiв для параметра спiнової поляризацiї як функцiї густини вiдповiдає вiд’ємнiй спiновiй поляризацiї, коли бiльшiсть нейтронних спiнiв орiєнтується протилежно магнiтному полю. Крiм цього, починаючи з деякої граничної густини, що залежить вiд напруженостi магнiтного поля, рiвняння самоузгодження мають також двi iнших гiлки розв’язкiв для параметра спiнової поляризацiї з додатною спiновою поляризацiєю. Вiльна енергiя, що вiдповiдає однiй iз цих гiлок, виявляється дуже близькою до вiльної енергiї, що вiдповiдає термодинамiчно стiйкiй гiлцi з вiд’ємною спiновою поляризацiєю. Як наслiдок, у сильному магнiтному полi стан з додатною спiновою поляризацiєю може реалiзовуватися як метастабiльний стан за високими густинами нейтронної матерiї, який змiнюється на термодинамiчний стiйкий стан з вiд’ємною спiновою поляризацiєю при зменшеннi густини, починаючи з деякої граничної густини. A.I. acknowledges the support from the Organizing Committee of Bogolyubov Kyiv Conference “Modern Problems of Theoretical and Mathematical Physics” (September 15–18, 2009, Kyiv, Ukraine), where this report was presented. J.Y. was supported by grant R32- 2009-000-10130-0 from WCU project of MEST and NRF through Ewha Womans University. en Відділення фізики і астрономії НАН України Український фізичний журнал Ядра та ядерні реакції Magnetization of Dense Neutron Matter in a Strong Magnetic Field Намагніченість густої нейтронної матерії в сильному магнітному полі Article published earlier |
| spellingShingle | Magnetization of Dense Neutron Matter in a Strong Magnetic Field Isayev, A.A. Yang, J. Ядра та ядерні реакції |
| title | Magnetization of Dense Neutron Matter in a Strong Magnetic Field |
| title_alt | Намагніченість густої нейтронної матерії в сильному магнітному полі |
| title_full | Magnetization of Dense Neutron Matter in a Strong Magnetic Field |
| title_fullStr | Magnetization of Dense Neutron Matter in a Strong Magnetic Field |
| title_full_unstemmed | Magnetization of Dense Neutron Matter in a Strong Magnetic Field |
| title_short | Magnetization of Dense Neutron Matter in a Strong Magnetic Field |
| title_sort | magnetization of dense neutron matter in a strong magnetic field |
| topic | Ядра та ядерні реакції |
| topic_facet | Ядра та ядерні реакції |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/56192 |
| work_keys_str_mv | AT isayevaa magnetizationofdenseneutronmatterinastrongmagneticfield AT yangj magnetizationofdenseneutronmatterinastrongmagneticfield AT isayevaa namagníčenístʹgustoíneitronnoímateríívsilʹnomumagnítnomupolí AT yangj namagníčenístʹgustoíneitronnoímateríívsilʹnomumagnítnomupolí |