Analytical solutions of equation for the order parameter of dense superfluid neutron matter with anisotropic spin-triplet p-wave pairing at finite temperatures
The previously derived equations for the components of the order parameter (OP) of dense superfluid neutron matter (SNM) with anisotropic spin-triplet p-wave pairing and with taking into account the effects of magnetic field and finite temperatures are reduced to the single equation for the one-co...
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irk-123456789-1284862018-01-11T03:02:59Z Analytical solutions of equation for the order parameter of dense superfluid neutron matter with anisotropic spin-triplet p-wave pairing at finite temperatures Tarasov, A.N. Квантовые жидкости и квантовые кpисталлы The previously derived equations for the components of the order parameter (OP) of dense superfluid neutron matter (SNM) with anisotropic spin-triplet p-wave pairing and with taking into account the effects of magnetic field and finite temperatures are reduced to the single equation for the one-component OP in the limit of zero magnetic field. Here this equation is solved analytically for arbitrary parametrization of the effective Skyrme interaction in neutron matter and as the main results the energy gap (in the energy spectrum of neutrons in SNM) is obtained as nonlinear function of temperature T and density n in two limiting cases: for low temperatures near T = 0 and in the vicinity of phase transition temperature Tc₀(n) for dense neutron matter from normal to superfluid state. These solutions for the energy gap are specified for generalized BSk21 and BSk24 parametrizations of the Skyrme forces (with additional terms dependent on density n) and figures are plotted on the interval 0.1n₀ < n <2.0n₀, where n₀ = 0.17 fm⁻³ is nuclear density. 2016 Article Analytical solutions of equation for the order parameter of dense superfluid neutron matter with anisotropic spin-triplet p-wave pairing at finite temperatures / A.N. Tarasov // Физика низких температур. — 2016. — Т. 42, № 3. — С. 222–229. — Бібліогр.: 53 назв. — англ. 0132-6414 PACS: 21.65.Cd, 26.60.Dd, 67.10.Fj, 67.30.H– http://dspace.nbuv.gov.ua/handle/123456789/128486 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| topic |
Квантовые жидкости и квантовые кpисталлы Квантовые жидкости и квантовые кpисталлы |
| spellingShingle |
Квантовые жидкости и квантовые кpисталлы Квантовые жидкости и квантовые кpисталлы Tarasov, A.N. Analytical solutions of equation for the order parameter of dense superfluid neutron matter with anisotropic spin-triplet p-wave pairing at finite temperatures Физика низких температур |
| description |
The previously derived equations for the components of the order parameter (OP) of dense superfluid neutron
matter (SNM) with anisotropic spin-triplet p-wave pairing and with taking into account the effects of magnetic
field and finite temperatures are reduced to the single equation for the one-component OP in the limit of zero
magnetic field. Here this equation is solved analytically for arbitrary parametrization of the effective Skyrme interaction
in neutron matter and as the main results the energy gap (in the energy spectrum of neutrons in SNM) is
obtained as nonlinear function of temperature T and density n in two limiting cases: for low temperatures near
T = 0 and in the vicinity of phase transition temperature Tc₀(n) for dense neutron matter from normal to superfluid
state. These solutions for the energy gap are specified for generalized BSk21 and BSk24 parametrizations
of the Skyrme forces (with additional terms dependent on density n) and figures are plotted on the interval
0.1n₀ < n <2.0n₀, where n₀ = 0.17 fm⁻³ is nuclear density. |
| format |
Article |
| author |
Tarasov, A.N. |
| author_facet |
Tarasov, A.N. |
| author_sort |
Tarasov, A.N. |
| title |
Analytical solutions of equation for the order parameter of dense superfluid neutron matter with anisotropic spin-triplet p-wave pairing at finite temperatures |
| title_short |
Analytical solutions of equation for the order parameter of dense superfluid neutron matter with anisotropic spin-triplet p-wave pairing at finite temperatures |
| title_full |
Analytical solutions of equation for the order parameter of dense superfluid neutron matter with anisotropic spin-triplet p-wave pairing at finite temperatures |
| title_fullStr |
Analytical solutions of equation for the order parameter of dense superfluid neutron matter with anisotropic spin-triplet p-wave pairing at finite temperatures |
| title_full_unstemmed |
Analytical solutions of equation for the order parameter of dense superfluid neutron matter with anisotropic spin-triplet p-wave pairing at finite temperatures |
| title_sort |
analytical solutions of equation for the order parameter of dense superfluid neutron matter with anisotropic spin-triplet p-wave pairing at finite temperatures |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2016 |
| topic_facet |
Квантовые жидкости и квантовые кpисталлы |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/128486 |
| citation_txt |
Analytical solutions of equation for the order parameter of dense superfluid neutron matter with anisotropic spin-triplet p-wave pairing at finite temperatures / A.N. Tarasov // Физика низких температур. — 2016. — Т. 42, № 3. — С. 222–229. — Бібліогр.: 53 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT tarasovan analyticalsolutionsofequationfortheorderparameterofdensesuperfluidneutronmatterwithanisotropicspintripletpwavepairingatfinitetemperatures |
| first_indexed |
2025-07-09T09:10:30Z |
| last_indexed |
2025-07-09T09:10:30Z |
| _version_ |
1837159924054884352 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 3, pp. 222–229
Analytical solutions of equation for the order parameter
of dense superfluid neutron matter with anisotropic
spin-triplet p-wave pairing at finite temperatures
A.N. Tarasov
Akhiezer Institute for Theoretical Physics, National Science Center “Kharkov Institute of Physics and Technology”
Kharkov 61108, Ukraine
E-mail: antarasov@kipt.kharkov.ua
Received November 4, 2015, published online January 26, 2016
The previously derived equations for the components of the order parameter (OP) of dense superfluid neutron
matter (SNM) with anisotropic spin-triplet p-wave pairing and with taking into account the effects of magnetic
field and finite temperatures are reduced to the single equation for the one-component OP in the limit of zero
magnetic field. Here this equation is solved analytically for arbitrary parametrization of the effective Skyrme in-
teraction in neutron matter and as the main results the energy gap (in the energy spectrum of neutrons in SNM) is
obtained as nonlinear function of temperature T and density n in two limiting cases: for low temperatures near
T = 0 and in the vicinity of phase transition temperature Tc0(n) for dense neutron matter from normal to superflu-
id state. These solutions for the energy gap are specified for generalized BSk21 and BSk24 parametrizations
of the Skyrme forces (with additional terms dependent on density n) and figures are plotted on the interval
0.1n0 < n <2.0n0, where n0 = 0.17 fm–3 is nuclear density.
PACS: 21.65.Cd Asymmetric matter, neutron matter;
26.60.Dd Neutron star core;
67.10.Fj Quantum statistical theory;
67.30.H– Superfluid phase of 3He .
Keywords: superfluid Fermi liquid, spin-triplet pairing, dense neutron matter, generalized Skyrme forces, order
parameter.
1. Introduction
This article is a continuation of our works [1,2] devoted
to theoretical study of phase transitions in dense neutron
matter with generalized Skyrme forces [3,4] and aniso-
tropic spin-triplet p-wave pairing of the 3He–A type [5,6]
in strong magnetic field (see also [7]). Here we shall study
the same dense superfluid neutron matter (SNM) in the
limit of zero magnetic field ( = 0H ) and analytical so-
lutions will be found at finite temperatures of the single
equation for the order parameter (OP) which is a conse-
quence (at = 0H ) from the set of equations (see (9) in [2])
for the components of OP (at 0H ≠ ) of dense SNM.
Note that this study may be interesting in connection
with investigation of thermodynamic properties of dense
superfluid outer cores in a majority of ordinary isolated neu-
tron stars (non-accreting pulsars) which magnetic fields are
much less in comparison with extremely strong fields of mag-
netars (see, e.g., [8–14] and also [15] and references therein).
Moreover, recent discovery with the aid of the NASA’s
Chandra X-Ray Observatory of unusually fast cooling of
supernova remnant in Cassiopeia A (Cas A), which is the
youngest known neutron star (NS) in the Milky Way Ga-
laxy, has attracted great attention (see, e.g., [16–30] and
references therein). Several authors [18–23] explain such
rapid cooling of NS in Cas A during last years (since Au-
gust 1999, when Chandra found point x-ray source in the
Cas A, up to 2014) due to the existence of spin-triplet
superfluidity of neutrons inside high-density liquid outer
core of this NS. But alternative explanations for the ob-
served rapid cooling of Cas A have also been proposed (see,
e.g., [24–29] and the discussion of [24] in [20,21]). This
NS in Cas A is the first one whose cooling has been ob-
served in the real time. Note also that there is, to date, no
© A.N. Tarasov, 2016
Analytical solutions of equation for the order parameter of dense superfluid neutron matter
evidence for the presence of a significant magnetic field in
the Cas A neutron star [20,21].
This discovery has revived interest in the problem of
the correct theoretical description of neutron spin-triplet
superfluidity in cores of NSs and, in particular, in dense
neutron matter within different alternative theoretical me-
thods (see, e.g., [30–33] and reviews [21,34–37], refer-
ences therein and also the discussion at the end of [23]).
In the present work we follow the so-called generaliz-
ed Fermi-liquid approach (see, e.g., review [38] and also
[39–41] and references therein) which has been already
used in [42] and [1,2] to describe dense SNM with aniso-
tropic spin-triplet p-wave pairing in steady and homogene-
ous strong magnetic field. Previously in [42] we applied
conventional Skyrme forces (see, e.g., [43,44]) with only
one term dependent on density n and then in [1,2] we used
generalized BSk18 [3] and BSk20, BSk21 [4] Skyrme
forces (with additional density dependent terms which bet-
ter take into account effects of three-body forces and other
properties of nuclear matter important at high densities) as
interaction in SNM at sub- ( 0<n n ) and supra-saturation
0( >n n ) densities (where 3
0 = 0.17 fmn − is nuclear density).
Here we apply generalized BSk21 and BSk24 Skyrme
forces [47,48] which lead to sufficiently stiff equations of
state of dense pure neutron matter (NM) and are consistent
(see [45–47] for details) with the recently measured values
Sun(1.97 0.04)M± and Sun(2.01 0.04)M± for the masses of
the heaviest yet observed pulsars PSR J1614–2230 [49]
and PSR J0348–0432 [50]. Note that selected here BSk21
and BSk24 are most likely the best parametrizations
among other generalized parametrizations of the Skyrme
forces (see Conclusions in [47]) which are sufficiently ac-
curate in calculation of neutron effective mass (which is
strongly density dependent). It is particularly important
because the magnitude of the energy gap in SNM (in the
energy spectrum of neutrons in SNM) is very sensitive not
only to the strength of attractive forces but also to the ef-
fective mass of a neutron (see, e.g., review [34] and refer-
ences therein).
We write down below the equation for the OP which in
the limit of zero magnetic field is a particular case of the
set of equations for the components 0↓ ↑∆ ≠ ∆ ≠ (at 0;H ≠
see Eqs. (9) from [2]). Then we shall solve this single
equation (valid for arbitrary parametrization of the Skyrme
forces) by analytical methods in two limiting cases: for low
temperatures near = 0T and in the vicinity of phase tran-
sition (PT) temperature 0 ( )cT n for dense neutron matter
from normal to superfluid state (with anisotropic spin-trip-
let p-wave pairing of the 3He–A type). These solutions are
specified then for generalized BSk21 and BSk24 paramet-
rizations of the Skyrme forces and figures for the PT tem-
peratures and energy gap in SNM are plotted on the inter-
val 0 00.1 < < 2.0n n n . In conclusion we shall briefly dis-
cuss our main results.
2. General equation for the OP for SNM with
generalized Skyrme forces between neutrons and
anisotropic spin-triplet pairing in zero magnetic field
It is evident that in the absence of magnetic field
( = 0H ) the effective magnetic field in SNM equals to ze-
ro, = 0ξ (see notations in [2]). In this case the components
of the OP ( ) ( , = 0)T↑ ↓∆ ξ for SNM with spin-triplet aniso-
tropic p-wave pairing of the 3He–A type coincide to each
other:
( , = 0) = ( , = 0) = ( ).T T T↑ ↓∆ ξ ∆ ξ ∆
For brevity, here and below we shall not write down densi-
ty n explicitly as the second argument of the function ( ).T∆
It is obvious now that the set of two equations (see (9)
from [2]) for the components of the OP is reduced to the
following equation for determination of ( )T∆ :
3
2 3( ) = ( ) ( ).
8
c
T T J T∆ −∆
π
(1)
Here 2
3 2 ( ) / < 0c t n′≡ is coupling constant leading to
spin-triplet p-wave pairing of neutrons, which is expressed
through the generalized parameters dependent on density:
2 2 2 5 5( ) = (1 ) (1 )t n t x t x nγ′ + + + (2)
(see (5) and details from [2] and also [3,4]) of the Skyrme
interaction. Double integral ( )J T is defined as follows:
1max 2
4 2
2
0min
tanh( ( , ; ) / 2 )( ) = (1 ) .
( , ; )
p
p
E q x T TJ T dqq dx x
E q x T
−∫ ∫ (3)
Here max = 1Fp p a+ , min = 1Fp p a− with cutoff pa-
rameter 0 < < 1a , where = / ( )c Fa E nε ; cE is the cutoff
energy, 2( ) = /2F Fn p m∗ε and Fp are the Fermi energy and
momentum; m∗ is the neutron effective mass dependent on
density n of NM and on the generalized Skyrme para-
meters 1( )t n′ and 2 ( )t n′ according to general formula (10)
from [2] (see also (29) and (30) here below). The function
2( , ; )E q x T is the energy of quasiparticles (neutrons) in
SNM with anisotropic spin-triplet pairing of the 3He–A
type and it has the form
2 2 2 2 2( , ; ) = ( )(1 ) ( ),E q x T q T x z q∆ − + (4)
where
2( , ) / 2 ( ) ( ) ( ) ( )Fz q T q m T q n z q∗= −µ ≈ ε − ε =
( ( )Tµ is the chemical potential which is substituted ap-
proximately by the Fermi energy at low temperatures in
SNM, 00 ( ) ( )c FT T n n< < ε ).
It will be more convenient to use another integral j
which is related with ( )J T by the following formula:
3( ) ( , ( ); ).FJ T m p j T T a∗≡ δ (5)
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 3 223
A.N. Tarasov
Then Eq. (1) for the function ( )T∆ with account of (5)
gets the following final form:
3
31 = ( ) ( , ( ); ),
8
nc m n j T T a∗− δ (6)
where
2
F
2( ) = 1.
( )2 ( )
F
F
T
p Tm T∗
ε ε
δ ≡ >> ∆∆
(7)
Note that the function ( ) ( , )F Fp T G T n∆ ≡ is the maxi-
mal value of the anisotropic energy gap in the energy spec-
trum (4) of neutrons in SNM. In the next sections we shall
solve this basic nonlinear integral Eq. (6) for determina-
tion of reduced energy gap ( , ) ( , ) / ( )F Fg T n G T n n≡ ε =
1/ ( , )T n= δ by analytical methods in two limiting cases:
near = 0T and close to the PT temperature 0cT without
specifying of the parametrization of the generalized Skyrme
forces. Then we shall select parametrization of the Skyrme
forces in order to plot figures for obtained solutions.
3. Solution of equation for the OP for SNM near T = 0
Here we shall consider SNM at low temperatures, when
00 < ( ) ( )c FT T n n<< << ε . In this case integral ( , ( ); )j T T aδ
in (6) can be approximated as the difference:
0 1( , ( ); ) ( ( ); ) 2 ( , ( ); ),j T T a j T a j T T aδ ≈ δ − δ (8)
where integrals 0 ( ( ); )j T aδ and 1( , ( ); )j T T aδ have the
following explicit form:
1 2
0 2 2
0
1( ( ); ) ( ) (1 ) ,
( , ( ))
a
a
xj T a T dy y dx
b y T x−
−
δ ≡ δ +
δ −
∫ ∫ (9)
1
2
1
0
( , ( ); ) ( ) (1 ) (1 )
a
a
j T T a T dy y dx x
−
δ ≡ δ + − ×∫ ∫
2 2
2 2
exp ( , , ( )) ( , ( ))
.
( , ( ))
A y T T b y T x
b y T x
− δ δ − ×
δ −
(10)
Here functions ( , ( ))b y Tδ and ( , , ( ))A y T Tδ are defined as
follows:
2
2 ( , ( )) 1 ( ),
1
yb y T T
y
δ ≡ + δ
+
(11)
11( , , ( ) 1.
( ) ( )
F yyA y T T
T T T
+ε +
δ ≡ ≡ >>
δ η
(12)
As a result of analytical calculations we have obtained
expressions for the integrals 0j and 1j . Namely, for the 0j
the following exact formula is valid:
2
0
1( ; ) = 1 arcsin
2 2 3 ( , )
a a aj a
b a
δ δ + − δ + δ
2
211 arcsin ( 1 1 )
2 3 ( , ) 6
a a a a a
b a
δ
+ − − δ + + + − + − δ
1 5 5(11 ) 1 (11 ) 1 22
9 4 4
a a a a + + + + − − − +
5 (2 1 1 )
12
a a+ − + − − +
δ
2
2
1 7 5 11 1
3 24 496 ln
1 11 1 1
4 4
− + + − δ δδ + × − − − δ δ
1 11 1 1 1 1 1
2 4 2 4
1 11 1 1 1 1 1
2 4 2 4
a aa a
a aa a
+ − + − − − − −
δ δ × +
+ + + − − + − − δ δ
1 3 5 1 1arctan arctan .
4 24
a a
a a
+ − + − − δδ δ δ
(13)
The integral 1j is closely approximated by the following
formula which is valid at 00 < c FT T<< << ε :
1( , ( ); )j T T aδ ≈
2
2 2
2 1 2 6 1( ) (1 )exp ( )
a
a
cT dy y Ac c c
A A AA A−
δ + − + + + + ≈
∫
4 2 88 ( )[1 8 ( )] ( ),T T O≈ η + η + η (14)
where
2 | | ( )
( , ( )) ( , ( )) 1 = 0.
1
y T
c y T b y T
y
δ
δ ≡ δ − ≥
+
(15)
Thus, with account (8) and (14) we can write down now
general Eq. (6) in the following approximate form valid at
low temperatures, c00 < FT T<< << ε :
4 20
3 0
3
1 ( ) ( ) ( ( ); ) 16 ( )(1 8 ( )) .
8
n
c y ym y j T a T T∗ ≈ − δ − η + η
(16)
Here we have introduced reduced density 0/y n n≡ of
SNM (where 3
0 = 0.17 fmn − is nuclear density, which
plays role of the density scale factor). Function 0 ( ( ); )j T aδ
(see (13)) determines the solution of Eq. (16) in the limit of
zero temperature (at = 0H ) for the required reduced ener-
gy gap ( , ; ) 1g T y a << (see after (7)). Thus, because
(0) = 0η (see definition (12)), we obtain at = 0T from (16)
the following expression for (0, ; ) (0, ; ) / ( )F Fg y a G y a y≡ ε
(see also [42]):
( )
3 0
2(0, ; ) = exp ( ) ,
( ) ( )
sg y a M a
c y n ym y∗
+
(17)
224 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 3
Analytical solutions of equation for the order parameter of dense superfluid neutron matter
where 3( ) < 0c y (see note after (1) and also (2) for general-
ized parametrizations of the Skyrme forces in SNM). Func-
tion ( ) ( )sM a which depends only on the cutoff parameter
= / ( ) < 1c Fa E yε is determined by the formula
( ) 11 4 4( ) 2 ln 2 1 1
6 3 3
s a aM a a a+ −
≡ − + + + − +
1 ( 1 1)(1 1 )ln .
2 ( 1 1)(1 1 )
a a
a a
+ − − −
+
+ + + −
(18)
In view of Eq. (17), we get from (16) the transcendental
equation for the function ( , ; ) 1g T y a << (at 00 < cT T<< ):
( )
3 0
2( , ; ) = exp ( )
( ) ( )
sg T y a M a
c y n ym y∗
+ −
4 2
4 216 1 8 ,
( , ; ) ( , ; )g T y a g T y a
τ τ
− +
(19)
where / ( ) 1FT yτ ≡ ε << . Owing to the smallness of the
temperature correction we get from (19) the following so-
lution in the main approximation on the small T :
4
( , ; ) (0, ; ) 1 16
( , ; )F
Tg T y a g y a
G T y a
≈ − ×
2
1 8
( , ; )F
T
G T y a
× +
4 2
(0, ; ) 1 16 1 8 .
(0, ; ) (0, ; )F F
T Tg y a
G y a G y a
− +
(20)
Note that obtained here in (20) leading power-law of tem-
perature dependence 4~ T for the energy gap in SNM
(with anisotropic spin-triplet p-wave pairing) near = 0T is
in qualitative accordance with the similar result obtained
earlier for the superfluid 3 He–A (see, e.g., review [51]) but
it is quite different from the exponential temperature de-
pendence of the isotropic energy gap near = 0T in tradi-
tional superconductors with spin-singlet s-pairing [52,53].
4. Solution of equation for the OP for SNM near Tc0
Let us consider SNM in the region of temperatures
close to 0cT , when 0 0| |c cT T T− << . But at the beginning
we shall study the limiting case, 0cT T→ . It can be shown
that the Eq. (6) in this limit is reduced to the following
transcendental equation:
0
3
0
( ) 2
1 ( ) ln
2
c
c
n ym y E
c y
T
∗ γ
≈ − + π
2 4
3 3 ,
16 ( ) 512 ( )
c c
F F
E E
y y
+ + ε ε
(21)
where = e 1.781072418Cγ ≈ ( = 0.5772156649...C is Eu-
ler’s constant). Here in [...] we neglected by small terms
2
0( / )c FO T ε . We get from (21) the following approximate
solution for the PT temperature 0 ( ; )c cT y E of SNM:
0
3 0
2 2( ; ) exp
( ) ( )
c c cT y E E
c y n ym y∗
γ
≈ +
π
2 4
3 3 .
16 ( ) 512 ( )
c c
F F
E E
y y
+ + ε ε
(22)
Note that pre-exponential numerical factor here,
2 / 1.134γ π ≈ , is somewhat more refined in comparison
with analogous expressions [1,2] for PT temperature of
SNM.
Now we define reduced PT temperature 0 ( ; )ct y a of
SNM:
0
0
( ; )
( ; ) 1
( )
c
c
F
T y a
t y a
y
≡ <<
ε
and then using obtained expression (17) for the reduced
energy gap (0, ; )g y a we find as a result the following ratio
for these functions:
c0
(0, ; ) 5 5exp = exp .
( ; ) 2 6 2 6
g y a C
t y a
π π ≈ − γ
(23)
This ratio is “universal” because it does not depend neither
on the cutoff parameter < 1a nor on the nature of inter-
action in the Fermi superfluid with anisotropic spin-triplet
p-wave pairing (in particular (23) is valid for arbitrary
parametrizations of the Skyrme forces in SNM) and it ex-
actly coincides with analogous ratio for the superfluid
3 He–A phase (see, e.g., [5]). The ratio (23) depends only
on the symmetry of the OP of superfluid system.
Now in order to solve Eq. (6) for SNM at temperatures
0 0| ( ) | ( )c cT T n T n− << we rewrite it as follows:
1
20
0
3ln = ( ) (1 ) (1 )
4
a
c
a
T
T dy y dx x
T
−
δ + − ×
∫ ∫
2 2F
2 2
( )1 tanh tanh ( , ( ))
2 2 ,
( ) ( , ( ))
A yy y b y T x
T
T y b y T x
ε + δ − × −
δ δ −
(24)
where we have used (21) and neglected by the small terms
2
0( / )c FO T ε . Functions ( , , ( ))A y T Tδ , 2 ( , ( ))b y Tδ and
( ) 1Tδ >> are defined by formulas (12), (11) and (7), re-
spectively. From (24) we obtain in the main approximation
on small parameter 21/ ( ) ( ) 1T g Tδ ≡ << the following
approximate equation:
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 3 225
A.N. Tarasov
5/2 5/2
0
0
1 (1 ) (1 )ln
5 ( )
a
cT y ydy
T T y
− + + ≈ − × δ ∫
tanh .
2
Fd y y
Tdy
ε ×
(25)
Let us use the following expansion into a series [52,53]:
2 2 2
=0
1tanh = 4 .
2 [ (2 1) ]n
x x
n x
∞
π + +
∑ (26)
Substituting this in Eq. (25) we obtain as a result of calcu-
lations the final approximate equation (see also note after (7))
valid at 0 0| ( ) | ( )c cT T n T n− << :
2 2
0
2 2
( ) ( , )1 7 (3) 7 (3)ln =
( ) 10 10
c F FT n G T n
T T T T
ε ζ ζ ≈ δ π π
(27)
( ( )xζ is the Riemann zeta function). It is obvious from (27)
that the energy gap has the form
2
0 ( )10( , ) ln ,
7 (3)
c
F
T n
G T n T
T
π ≈ ζ
(28)
where 210 / [7 (3)] 3.4248π ζ ≈ . It is in accordance with
analogous result [5] for 3 He–A but at the same time (28) is
more accurate than in [5], where 0ln ( / )cT T T is approx-
imated by 0 01 /c cT T T− (note here that such temperature
dependence of the energy gap in the vicinity of 0cT is con-
sistent with Landau’s theory of second-order phase transi-
tions; see, e.g., Appendix II in [52]). Moreover, for SNM
(with spin-triplet anisotropic p-wave pairing and with gen-
eralized parametrizations of the Skyrme forces) density
profile of PT temperature c0 ( )T n is essentially different
than in 3 He–A and it will be evident in the next section.
5. Solutions of equation for the OP for SNM with
generalized Skyrme forces near T = 0 and close to Tc0
and their density and temperature profiles
Formulas (17), (19), (20), (22), (28) contain the effec-
tive mass of neutron nm∗, which depends on the density
0n yn≡ of NM as in [2]:
0
1 22= 1 [ ( ) 3 ( )],
4n
mynm t n t n
m∗
′ ′+ +
(29)
where 2( ) / 2 938.91897 MeV / cp nm m m≈ + ≈ is mean
value of free nucleon mass. Generalized parameters
1 1 1 4 4( ) = (1 ) (1 ) ,t n t x t x nβ′ − + − (30)
and 2 ( )t n′ (see (2)) have specific numerical values for each
Skyrme parametrization. For NM with the best BSk21 and
BSk24 generalized parametrizations [4,47] of the Skyrme
forces we have from (29) that
, 21( )n BSkm y∗ ≈
1/12 ,
1 (3.97930 0.0422618 3.89571)
m
y y y
≈
+ + −
(31)
, 24 ( )n BSkm y∗ ≈
1/12 ,
1 (3.97930 0.0422618 3.89025)
m
y y y
≈
+ + −
(32)
and the Fermi energies of NM for the BSk21 and BSk24
Skyrme forces have the following forms (which are close
to each other because the parameters of the two forces are
very similar; see Fig. 1):
2/3 1/12
, 21( ) [1 (3.97930F BSk y y y yε ≈ + +
0.0422618 3.89571)]·60.902 (MeV),y+ − (33)
2/3 1/12
, 24 ( ) [1 (3.97930F BSk y y y yε ≈ + +
0.0422618 3.89025)]·60.902 (MeV).y+ − (34)
In zero magnetic field = 0H from general formula (22)
(see also (29)–(34) and (2)) it follows as the particular re-
sults the expressions for PT temperatures of dense NM
(with BSk21 and BSk24 Skyrme parametrizations) to SNM
with anisotropic spin-triplet pairing of 3He–A type:
0, 21( ; )c BSk cT E y ≈
2 4
, 21 , 21
2 3 3exp
16 ( ) 512 ( )
c c
c
F BSk F BSk
E E
E
y y
γ ≈ + × π ε ε
1/12
1/12
1 (3.97930 0.0422618 3.89571)
exp ,
(2.65286 2.85028)
y y y
y y
+ + −
×
−
(35)
Fig. 1. Fermi energies for SNM (see (33) and (34)) with BSk21
(line) and BSk24 (points) Skyrme forces as the functions of re-
duced density 0= /y n n are close to each other.
160
140
120
100
80
60
40
20
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
ε F
, M
eV
y
226 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 3
Analytical solutions of equation for the order parameter of dense superfluid neutron matter
2
0, 24
, 24
2 3( ; ) exp
16 ( )
c
c BSk c c
F BSk
E
T E y E
y
γ
≈ + π ε
4
, 24
3
512 ( )
c
F BSk
E
y
+ × ε
1/12
1/12
1 (3.97930 0.0422618 3.89025)
exp
(2.65286 2.84870)
y y y
y y
+ + −
×
−
(36)
(here cE is the cutoff energy which is less than Fermi en-
ergies, , 24 ( )c F BSkE y< ε and , 21( )c F BSkE y< ε ). Compare
improved formula (35) (see note after (22)) with analogous
formula (16) from [2] for 0, 21 c( ; )c BSkT E y .
If for the definiteness, we select cutoff energy
eV= 10 McE (so that , 21( )c F BSkE y< ε and , 24 ( )c F BSkE y< ε ,
see Fig. 1) it is easy to plot figures (see Figs. 2 and 3) for
the PT temperatures (35), (36) of NM at sub- and supra-
saturation densities on the interval 0 00.1 < < 2.0n n n .
6. Conclusion
Thus, we can conclude that temperature dependence
4
0(~ ( / ) 1cT T << , see (20)) of the energy gap in superfluid
of the 3He–A type near = 0T and close to 0 ( )cT n (see (28))
is determined only by the symmetry of the OP and doesn’t
depend on the nature of interactions which lead to the spin-
triplet Cooper pairing in the system. But as we can see
from Figs. 2–7 the density dependences of the PT tempera-
ture 0, ( ; )c BSk cT E y and the energy gap in SNM are signifi-
cantly different than in the superfluid 3 He–A [5].
Note also that obtained here general formula (22) for
PT temperature 0, ( ; )c BSk cT E y of dense NM (in zero mag-
netic field) to superfluid state with anisotropic p-wave
pairing of 3 He–A type and with generalized Skyrme inter-
actions [4,47] depends on density in nonmonotone way
((35) and (36) exhibit a bell-shaped density profile, see
Fig. 2). Such behavior of these PT temperatures
0, 21(10; )c BSkT y and 0, 24 (10; )c BSkT y and their maximal
values are in qualitative agreement with results of recent
articles [18,19,30] and are of the same order in magnitude
at = 10cE MeV (namely, 0, 21max ( (10; )) 0.063c BSkT y ≈
MeV and 0, 24max( (10; )) 0.060c BSkT y ≈ MeV, see Fig. 3).
Fig. 2. PT temperatures of SNM with generalized BSk21 and
BSk24 Skyrme forces (see (35) and (36) at 10 MeVcE = ):
0; 21(10; )c BSkT y (upper curve); c0; 24(10; )BSkT y (lower curve).
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
T ,
M
eV
y
0.06
0.05
0.02
0.03
0.04
0.01
0
Fig. 3. The same PT temperatures of SNM as in Fig. 2 with
BSk21 and BSk24 forces at 10 MeVcE = near their maxima:
0; 21(10; )c BSkT y (upper curve); 0; 24(10; )c BSkT y (lower curve).
0.66 0.68 0.70 0.72
T c
0,
M
eV
y
0.063
0.061
0.062
0.060
Fig. 4. Energy gap of SNM ( , ;10)FG T y (see (28)) with BSk21
Skyrme force and with anisotropic spin-triplet p-wave pairing (in
zero magnetic field, = 0H ) as a function of reduced density
0= /y n n at three temperatures near 0, 21(10; )c BSkT y (see (35)
with cutoff energy 10 MeVcE = ): at 0, 21= 0.91 (10; )c BSkT T y
(upper curve), at 0, 21= 0.96 (10; )c BSkT T y (middle curve) and at
0, 21= 0.99 (10; )c BSkT T y (bottom curve).
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
G
F,
M
eV
y
0.06
0.05
0.02
0.03
0.04
0.01
0
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 3 227
A.N. Tarasov
Note finally, that results (20), (28) for energy gap are ge-
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SNM with BSk24 parametrization of the generalized
Skyrme forces (proposed recently in [47,48]) and they are
close with BSk21 (it is clear from (20), (28) and from
Figs. 1, 2 for Fermi energies (33), (34) and PT tempera-
tures (35), (36) for BSk21 and BSk24 which are close to
each other).
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Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 3 229
1. Introduction
2. General equation for the OP for SNM with generalized Skyrme forces between neutrons and anisotropic spin-triplet pairing in zero magnetic field
3. Solution of equation for the OP for SNM near T = 0
4. Solution of equation for the OP for SNM near Tc0
5. Solutions of equation for the OP for SNM with generalized Skyrme forces near T = 0 and close to Tc0 and their density and temperature profiles
6. Conclusion
|