1655
Progress of Theoretical Physics, Vol. 62, No. 6, December 1979
Study on Neutron 3Pz-Superf luidity with Two-Dim ensional
Character under 7f° Condensa tion
in Neutron Star Matter
Tatsuyuki T AKATSUKA and Ryozo T AMAGAKI*
College of Humanities and Social Sciences, Iwate University, i\1orioha 020
*Departme nt of Physics, Kyoto University, Kyoto 606
(Received August 17, 1979)
A model calculation is performed in order to study how the 'P,-superfluid state of neutrons
in neutron star matter is affected by a well-developed 7!" 0 -condensation. Because of the [ALS]
(Alternating-L ayer-Spin) structure of nucleons, the neutron 'P,-pair being in the same layer
with the maximum angular-mome ntum component plays an important role in the pairing
correlation arising from the two-dimesiona l Fermi gas state. It is shown that this 'P,-superfluidit y
is still existent even in the well-developed [ALS] (n- 0 -condensed) phase and its critical
temperature is of the same order of magnitude, compared with that realized from the ordinary
0
Fermi gas state. Coexistence of the 'P,-superfluid and the n- condensation in neutoron star
matter provides an example of low-dimension al superfluid in nuclear system.
§ I.
Introductio n
Possible occurrence of interesting phases in high density nuclear matter hrrs
been pointed out through the study in the past decade. Among them pion condensation has attracted much attention in recent years_n. 2l Althought no positi\ce
evidence has been found yet, this phase is expected to be realized at the density
(p) not far from the normal density of the ordinary nuclear matter Po- In such
density region, namely p0 ':Sp< several p0 , the occurrence of another interesting
phase, the 'P2 -superfluidit y of neutrons in neutron star matter, was noticed about
ten years ago, soon after the discovery of pulsars. 3l, 4J It is of particular interest
to see whether these two ph<1ses can coexist or not. Without any kind of nucleon
5
superfluidity , it is impossible to explain the pulsar "glitch" phenomena. l In this
respect, interrelation between pion condensation and superfluidity (coexistence or
competition) has an observationa l implication for pion condensation . The subject
studied in this paper, that is, whether the neutron 3 P 2 -superfluid state can survive
6
in the well-develop ed ;: 0-condensed phase or not, is along such a llne of approach. l
;:-N
the
In the potential description, pion condensation realized primarily by
P-wave interaction is the phase where the first-order effect of the OPEP tensor
force, which vanishes in the normal phase, plays a substantial role, as shown for
71
This is one of the
condensation by Tamiya, Tatsumi and the present authors.
possible spin-isospin orderings of nucleons which is favorable in energy especially
for the OPEP. 8l In contrast to this, the 'P2-superfluidit y is realized mainly due
7C 0
1656
T. Takats uka and R. Tamag aki
to the spin-or bit force to which heavy mesons contrib ute
signific antly. 31 The problem under conside ration means the interpl ay betwee n the
two charact eristic aspects
origina ting from such strong noncen tral compon ents of
nuclea r forces in high density nuclea r matter.
As far as energy gains by the phase transiti ons from the
Fermi gas are concerned , the energy gain due to pion conden sation is much
larger than that due
to superfl uidity. The former one, rangin g from several
MeV to several tens of
MeV per nucleo n in accorda nce with the growth of the
phase, is caused by the
change of all the nucleon ic states, while the latter one
with the order of L12/E:F
;S;0.01 MeV per nucleo n is associa ted with the behavi or
of nucleon s near the
Fermi surface , where L1 is the energy gap (<1 MeV)
and ep the Fermi kinetic
energy at the density of interes t (60"'-' 180 MeV). Theref
ore we can set up the
problem as follows : To what extent nucleo n superfl uidity
is influen ced by pion
conden sation.
In the presen t paper we conside r the problem for a typical 0
TC conden sation
with standin g wave field, where the nucleo n system has
the [ALS] (Altern atingLayer-S pin) structu re charact erized by one-dim ensiona l
localiza tion and alterna ting
spin orderin g in the directio n of conden sed momen tum
of TC 0 • 71 When the phase
is well develop ed, the Wanni er wave functio n to represe
nt the orthog onal localize d
single- particle state is well approx imated by the Gaussi
an form, the Fermi surface
of the Bloch states become s cylindr ical and the single- particle
spectru m has a band
structu re. 91 Theref ore nucleo n superfl uidity, in which
nucleon s near the Fermi
surface play an essenti al role, is inevita bly influen ced
by the transiti on to the
[ALS] (rr 0-conden sed) phase. Becaus e even in this case
the ordinar y Fermi gas
nature still holds in the two-dim ensiona l space perpen
dicular to the conden sed
momen tum, the pairing -type correla tion can work, althou gh
in a somew hat restrict ed
manner . Such aspect of high density nuclea r system , if
realize d, provide s a new
exampl e of low-dim ensiona l superfl uid states in nuclea r system
, which is interes ting
from the viewpo int of quantu m theory of many-b ody system
s.
Severa l models have been propos ed to unders tand the
pulsar "glitch " phenomena.51'101 Follow ing the two-co mponen t model propos
ed by Baym, Pethick ,
Pines and Ruderm an,lll which we conside r to be the most
plausib le one among
various propos als, both neutron s and protons should be
superfl uid in some part of
neutro n star interio r. The presen t authors showed that
such a region appear s at
the density p~0.7p 0 "'-'2.8p 0 , where neutron s, being the domina
nt compon ent of neutron star matter, are in the 3 P 2-superf luid and protons
are in the 1S 0-superf luid. 31
When the nucleo n system turns into the [ALS] phase
with the aligned spin
configu ration of like-nu cleons in a layer, for the 3P -pairing
of neutron s the nucleo n
2
pairs in the same layer play a domina nt role and for
the 1S 0-pairing of protons
the nucleo n pairs in the nearest layers contrib ute mainly
. In this paper, we study
how the 3 P 2 -superf luidity of neutron s, which depend s sensitiv
ely on the surrou nding
situatio n, is affected by the transiti on to the [ALS] phase.
Study on Neutron
3 P2 -SupeJ~!fuidity
with Two-Dimensional
1657
We study the problem in the well-developed [ ALS] phase of pure neutron
matter. Actually the isobar L1 (1236 MeV) plays an important role to realize the
pion condensation over the suppression effects due to short-range effects!1 As
the results of balance of the two effects, the actual situation of the [ALS]
structure of quasi-neutrons accompanying n° condensation is simulated fairly well
by the simple model, that is, by the [ALS] structure obtained only by the n-N
P-wave interaction in the mean field approximation, if we choose suitably the
parameters employed. 121 The neglection of proton component for studying the neutron 3 P 2-superfluidity is not serious because of large difference of the Fermi energies
of neutrons and protons. Merit of adopting the well-developed situation lies in
providing an intuitive understanding by virtue of the well-localized aspect. As a
theoretical framework to treat this superfluidity we adopt the generalized BCS-type
theory used in previous works, 31 the validity of which has been recently confirmed
by Slaus and Serene. 131 In addition we employ a simple nonlocal-separable 3 P 2 potential to reproduce the 3 P 2 scattering phase shifts and the effective mass approximation. Here modification of interactions due to the transition into the [ALS]
phase is not considered, because it appears mainly for tensor force while spin-orbit
force is the most responsible for the 3 P 2-pairing correlation. We give an estimate
for critical temperature from the energy gaps obtained under such simplifications.
In the next section, the model and model Hamiltonian are presented and the
gap equation after the Bogoliubov transformation is given. In § 3, solutions of
the gap equation are shown and the results obtained are discussed. The last section
is devoted to concluding remarks.
§ 2.
3
P 2 -gap equation in [ALS] phase
[ALS] model
2.1.
Here we briefly recapitulate the essential points of the [ALS] model presented
m the previous works which are necessary for later explanation. The [ALS]
structure of neutron matter is schematically illustrated in Fig. 1: Neutrons are
nucleon configuration density source fn. ?I 0 field
1 +1
i=O
i
=1
[ALS]
NEUTRON MATTER
Fig. 1. Profile of the well-developed
[ALS] structure and the associated
aspects; ordinary density, source function (spin density) and condensed n'field in neutron matter.
1658
T. Talwtsuka and R. Tamagaki
localized one-dimensionally to form the layers whose spin directions align perpendicularly to the layer plane and change alternately layer by layer.') The [ALS]
model is constructed from the following orthogonal basis functions {¢j (E)}:
(2 ·1)
vvith the spin-ordering o,x.t = (-) tX•r Here, j= {q.l, t} denotes the single-particle
state assingned by the [-th layer and q.l ~ {qro qy}. Also, g= {r, spin}, r .l == {.x, y},
SJ.l is the two-dimensional normalization volume and Ot denotes the spin of the
neutron in the [-th layer (oj2 = (- ) 1 /2). As a reasonable choice for ¢ 1 (z), we
take the Gaussian form
which is localized around the lattice cite [d with the layer-distance d. The orthogonality between these basis functions (especially, between the nearest parallelspin pair) is assured for T=acl2 >2 with T representing the measure of localization
compared with the layer-distance. Then, the ground state lrliALs) of the nucleon
system is given by the Slater determinant of {¢j (g)} where the two-dimensional
Fermi gas states are occupied up to Jq.lJ<q.lF= (41Tpd) 112 with p being the number
density. The relation between the two-dimensional Fermi momentum q.lF and the
usual (three-dimensional) one qF becomes
(2 · 3a)
because we have
(2. 3b)
with N.l being the total neutron number in a layer. Thus the [ALS] model has
two parameters d and a, which are determined variationally by the minimization
of the energy at a given density p.
The [ALS] structure, which produces the nonvanishing source function of
1T 0 field, brings about the 1T 0 condensate:
The expectation value of the static IT 0field <Cf"o (r)) obeys the following field equation in neutron matter:
(P 2 - m/) <e;"o (r)) = - (//mrr) P · S (r),
(2. 4)
where m, denotes the pion mass, f is the coupling constant of the u · J7 interaction
and S (r) =<0ALsl¢tuy;JrliALs), with tjJ being the nucleon-field operator, represents
the spin-density.
Due to the spatial-dependence of the source function as
S (r) =zS (r) "--'Z cos kcz, where z is the unit vector denoting the z-axis, the resulting <Cf?0 (r)) from (2 · 4) becomes the coherent state of 1T0 with their momentum
concentrated into kc = (IT/ d)
as sketched in Fig. 1. The ground state of the
system can be described by the direct product of this coherent state and lrliALs).l4l
Thus a new phase in nuclear medium like liquid crystal with the IT0-condensate
z,
Study on Neutron 3P 2 -Superfiuidity with Two-Dimensional
1659
( [ ALS] phase) can be well described by such a model. This mainly comes from
the consequences: The condensed n°-field ( (/Jo (r)) generates the periodic onebody potential with the periodicity 2d for neutrons in the same spin-state and, as
a result, neutrons lead to be described by the Bloch- or equivalently Wannier-wave
functions. Under the developed localization, the Wannier-wave function can be
substituted in an extermely good approximation by the [ALS] model wave function
(Eq. 2 · 2) and hence the self-consistency between n°- and nucleon-field equations
comes to be satisfied by the model wave functions adopted at the starting
point. g), 15)
The [ALS] model which makes the conventional potential description possible
provides us with a basic tool for further investigations on the properties of other
new phases such as superfluidity to be discussed here.
2.2.
Application of BCS-Bogoliubov theory to [ALS] phases
We begin with the assumption that the [ALS] p'hase with remarkable onedimensional localization (T>2) can be well realized beyond a certain density (critical density Pc). Under this situation, pairing interaction between two neutrons is
most effective when they interact in the same layer owing to the developed localization of every layer. Namely, the pairing of type (q1.£, -ql.£) is of primary importance and those of different layers (q1.£, -ql.t') with £=/=£' can be neglected,
because the matrix elements of the pairing interaction (q1.'£/, -ql.'t/lvlql.l~>
-ql.£2 ) involve the damping factor due to localization
which is not larger than exp (- 3T/8) '"'-'0.1 for T>5, unless all the ['s are the
same. As the neutron spins align for the pair in the same layer, the pair states
become the triplet-odd ones caPo, 3 P 1 , 3 P 2 , etc.), in which 3 P 2 -pair state is the most
attractive. Among possible components of this pair state, the most favourable one
is that the spin component of a pair (S= 1, m 8 = ± 1 according to m 8 = ( - )l) is
combined with that of relative orbital angular momentum (L = 1, mL = ± 1) to form
the (J=2, mJ=m 8 +mL= ±2) pair, namely, the maximum lmJl coupling. This
is due to kinematical factors coming from the angular-momentum coupling, i.e., the
Clebsch-Gordon coefficients c• (LmLSmsl JmJ) with L = 1, S = 1, J = 2 and m 8
= (- )l· This aspect allows us to treat a simplified model Hamiltonian. In
order to take account of such pairing with the particular three-dimensional partial
wave, we expand the z-part of relative wave functions exp ( -az 2/ 4) into Fourier
components and introduce the 3P 2-paring correlation with ImJl = 2 into the threedimensional plane wave states. Then we can start with the following model
Hamiltonian obtained after the Fourier expansion of the localized wave function
¢ t (z):
Hmodel
= Ho + Hpair,
(2·4)
1660
T. Takatsuka and R. Tamagaki
Fig. 2. Vector diagram to show the definition of q, q, and <{Jq for momentum q.
(2·5)
(2·6)
where q and q' are the magnitudes of q=qJ. +zq, and q'=qJ.' +zqz', respectively,
as shown in Fig. 2 and
(2·7)
e
Here v P 2 ) is an effective 3 P 2 -state potential to reproduce the experimental phase
shifts involving implicitly the tensor coupling effect with the "F2 state, although we
do not deal with the 3 P 2 + 3 F 2 coupled state. In the above expressions, 'ttqJ. is
the single-particle energy measured from the two-dimensional Fermi surface in the
[-th layer (etqJ.=s([qJ.) -s(tqJ.F)), and C/qJ.(CtqJ is the creation (destruction)
operator for the neutrons in the state described by the ¢j (E). They satisfy the usual
Fermion commutation relations;
(2·8)
Also Q (Qz) is the three (one)- dimensional normalization volume, ~q is defined
by the relation
The pa1r operator
b/ (q)
is represented as
1
b t t ( q ) =----=
~2
sd ~ -q,';ay
qe
lmL
( q~) Cttq Ctt-q
J.
J.
(2·9)
Study on Neutron 3 P 2 -Superfiuidity with Two-Dimensional
1661
with mL= (- )l. The factor exp ( -q/ja) comes from the Fourier component of
the relative wave function exp ( -az 2 /4) in the z-direction.
The transformation of the [ALS] state ltZiALs) into the super state ltZiALs)Bcs
(BCS- [ALS] state) can be done by the following Bogoliubov transformation in
a manner similar to that used for the case realized from the three-dimensional Fermi
gas ([FG]) aal,acl
Itli ALs)Bcs =
e;s Itli ALs) ,
(2 ·10)
(2 ·11)
By the use of b t t (q) in Eq. (2 · 9), this is rewritten as
(2·12)
(} (e, qJ.)=
-/2 :E e-q,z;aa ([, q) Y1mL (ij_)
(2·13)
q,
with l:q, = Q,f dq,/2rr.
in the forms:
Then quasiparticle operators
CXtqJ.
and a 1tqJ. are obtained
(2 ·14)
where
{u (e, qJ.) =cos eo (t, qJ.)'
V(t, qJ.) ={}([, qJ.)sin eo(t, qJ.)/(}o(t, qJ.)
with (} 02 ([, qJ.) =I(}([, qJ.)
satisfy
1
2•
(2 ·15a)
(2·15b)
The transformation coefficients U(t, qJ.) and V(t, qJ.)
u Ct, - qJ.) = u Ct, qJ.) = U* Ct, qJ.),
{V(t, -qJ.)=-V(e,qJ.),
IU(t, qJ.) + IVCt, qJ.) =1.
12
12
(2·16a)
(2·16b)
(2·16c)
By the use of these relations, it can be shown that the quasiparticle operators also
satisfy the Fermion commutation relations;
(2 ·17)
and the mverse transformation is given by
(2·18)
1662
T. Takatsuka and R. Tamagaki
Then, using Eq. (2 ·18), we rewrite Hmodel in terms of quasiparticle operators
and separate it into H 00 (constant terms), H 11 (at a terms), H 20 (atat, aa terms)
and higher order terms with respect to quasiparticle operators. Neglecting the
terms '"'-'0 (1/ S2 _]_) and the higher order terms, we obtain
H =Hoo+Hn +H2o,
(2 ·19)
(2·21a)
H2o=~~
t qj_
-
[e tq U* (l, qj_) V (l, qj_)
j_
~ {J*(l,qj_)U 2 (l,qj_)-J(l,qj_)V 2 (l,qj_)}]a~qj_a~-qj_ +h.c.,
(2 ° 21b)
where
(2 ° 22)
with
mL
= (- )1 •
The elimination of the dangerous term H 20 leads to the equation
From Eq. (2·23) together with Eqs. (2·13), (2·15) and (2·16), the U(l,qj_) and
V (l, q _]_)-factors are expressed as
u (t, qj_)
12 =
{1 + e tq/E (t, qj_)} /2,
(2 ·24a)
IV (l, qj_) 1 =
{1-e tq/E (l, qj_)} /2,
(2·24b)
1
2
2U* (l, qj_) V* (l, qj_) = ::1 (l, qj_) /E (l, qj_),
(2·24c)
where
(2·24d)
In the derivation of Eqs. (24), use is made of the fact that ::1 (l, qj_) {} (l, qj_)
is real, as seen by the introduction of Eqs. (2 ·13) and (2 ·15) for Eq. (2 · 22).
Then, with the aid of Eqs. (2 · 24), each part of Hmodel is represented as
Hoo=~~[etq {1-etq /E(l,qj_)}/2
t
qj_
j_
-IJ (l,
j_
qj_) I2/4E (l, qj_)]
(2. 25a)
Study on Neutron 3 P 2 -Superfluidity with Two-Dimensional
1663
and
(2·25b)
Equation (2·25b) means that E(t, qJ.) is the quasiparticle energy and IJ(.t', qJ.) I
is nothing but the energy gap under consideration. Inserting Eq. (2 · 24c) into
Eq. (2·22), we finally get the gap equation for 3 P 2-pairing in the [ALS]-phase;
(2 ·26)
From this equation, together with the [-independence of
the gap is known to have the properties;
e
tqJ_
in the [ALS] phase,
(2·27)
with mL = ( - )t and (f/q defined in Fig. 2, namely, IJ (.t', qJ.) I IS angle- and [independent, as expected. Thus the gap equation to be solved becomes
X e-(q,''+q,')fa _ _q_~--
} J (q/)
qJ.
-./q/2+i.;a -.;q_~2+q/ E(q/) '
(2·28)
where E(qJ.)=Veh+J 2 (qJ.). The energy gap in the [ALS] phase is a function
of qJ. instead of q and in this sense the super:fluidity of this phase becomes essentially of two-dimensional character. Once we have the energy gap function
J (qJ.), the correlated pair wave function is given by introducing the correlation
factor J (qJ.) / E(qJ.) into the 3 P 2 , mL = m 8 = ± 1 components of the relative wave
function exp (- az 2 / 4 + iq J. · r J.) :
(2. 29)
The corresponding gap equation arising from the spherical [FG] in three-dimensional space, the 3 P 2 gap equation with the same type (maximum lmJI-coupling),
IS given by
J (q)
= - _!_ [00 dq' q'"(q' Iv CS P)2 Iq)
'IT.
Jo
J (q') IYn Cii') 12
sdii' -./e~,+LI
2 (q')IYn(ii')l 2
(2· 30)
as shown in Ref. 3) .
§ 3.
Numerical results and
disc~ssion
The density region, where the [ALS] phase exists under realistic situation of
high density medium, and the resulting values of parameters (a, d) have not been
1664
T. Talwtsuka and R. Tamagaki
firmly determined yet. Apart from the details, however, the characteristics of
this new phase, such as the remarkable one-dimensiona l localization, the particular
spin (isospin) -order and the mechanism of phase transition, have been revealed by
focusing attention only on the n-N P-wave interaction and making use of the
mean field approximation.7),gJ Therefore, as a typical example, we adopt the sets
of (a, d) obtained within the framework mentioned above, i.e., from those in Ref.
7b) and 7c), and research the aspects of superfluidity in the [ALS] state with n°
condensate.
In order to estimate simply the 3 P 2 -gap in the [ALS] phase, we adopt one
of Mongan's one-term nonlocal-separa ble potentials; 16 J
(3 ·la)
given by
with L=l, n=l, bA=l.509fm- 1 and cA=5.349(Me V·fm)'12 • It has been shown
that this potential gives rise to the 3 P 2 -energy gaps in a consistent manner with
those obtained for more realistic potentials with the tensor coupling to 3 F 2 , if we
take a suitable density-depend ent effective mass of nucleons. 3 'J Because of the
separable character of this potential, the form of L1 (qJ.) is determined from Eq.
(2 · 28) as
(3·2)
We express the quantities in the following dimension-less form:
(3·2'a)
(3 ·2'b)
where xJ.=qJ./qJ.F, Xz=qz/qJ.F, a=a/q~F, c:J.F denotes the two-dimensiona l Fermi
energy and
Then the gap equation (2 · 28) is reduced to the dispersion relation;
(3·3a)
with
(3. 3b)
where for convenience' sake effective-mass approximation is taken for SqJ.;
1665
Study on Neutron 3P 2 -Super.fiuidity with Two-Dimensional
1.5
1.0
Dispersion Relation
at f=2f.. and for m* = 1.
0.5
2-dim. ·. [ALSJ
[81' (1.6)
fJ.F=131 MeV
Fig. 3. A typical example of F(o) defined in Eq. (3.3a) at p=2po and for m*=l.
The lower line labeled as 2-dim. shows the [ALS] case with F=ad 2 =8 and
d=l.4 fm (optimum values'l). The upper line labeled as 3-clim. shows the
results for the [mJ[ =2 coupling realized from the ordinary spherical Fermi gas. 3"l
EJ.F (cp) is the two- (three-) dimensional Fermi energy.
(3. 4)
with m* -J{v* / 21./v being the effective-mass parameter and i11.v the nucleon mass.
From
to satisfy Eqs. (3 · 3), the energy gap J (ciJ.) is obtained through Eqs.
(3. 2').
As a typical case, F (r)) at p = 2p 0 and for the optimum values of
and d
ad'= 8, d = 1.4 fm) is plotted in Fig. 3, vV here the results obtained previously
in the three-dimensional [FG] -case (i.e., F (o) for Eq. (2 · 30)) at the same densitl"> are also shown for comparison. The 3 P 2 -energy gap in the [ALS] phase
is reduced by a factor of 3~4 compared with that realized frmn the [FG] phase.
This reduction of F(r'J) is caused mainly by the fact that the damping factors clue
to the one-dimensional localization, i.e., the last two factors in Eq. (3 · 2b'), reduce
the effective pairing attraction near the Fermi surface. This can be seen by comparing fx(xJ.) = (4/rra) 112f(xJ.), tending asymptotically to hL (xl.), with hL (xJ.) at
x1.=l. 6 ) Variation of F(o) for the allowable par<1meter range as r=5~8 and
d= (1.4~1.6)fm does not alter the essential features. The resulting 3 P 2 gaps in
the [ALS] phase JCALSJ_J(qJ.F) are illustrated in Fig. 4 as functions of p, together
with those arising from the [FG] phase JCFGJ=J (qF), where the values of (a, d),
equivalently to (T, d), in Refs. 7b) and 7c) are used for each p.
The density dependence of JCALSJ is different from that of JCFGJ, namely, their
gradual decrease beyond p~2p 0 in the [ALS] phase. This is caused by the (d, T)-
o
cr
r
T. Takatsuka and R. Tamagaki
1666
3
P"- Energy Gap tn Neutron
Matter
['c
(~leY)
(\'leI '
m
-3
* ~1
m * ~I
-j
m
* ~o.s
I
I
'
- j(i
- - - : i"o'ALSi
m*
~0.6
10
0.6
0.8
1.0
1.2
1.4
cl (fm)
Fig. 4. 'P,-gaps m neutron matter for m''
=1 and m*=O.S; in the [ALS] cace
.:JCALSlc=.:J (ql.F) determined by Eq. (2.28)
(solid lines) are shown in comparison
with the [FG] case; .:JCFG1 c=.d (qp) determined by Eq. (2.30).
Fig. 5. Dependence of .d (q J.F) on r and d is shown
for p=po, 2po and 3po. The dots indicate the
optimum values of r and d determined in Ref. 7b)
and 7c).
dependences of the gap, as shown in Fig. 5, although the effects of the 3 P 2 -pairing
interaction become generally stronger for larger Fermi momentum because the
interaction adopted here shows no repulsive aspect in high energy region as shown
in Fig. 6 o£ Ref. 3a) .
.::JCALSJ for the density· region of interest is just below 1 MeV for m* = 1 which
is the most preferable case for the gap to be realized and to become much smaller
in order of magnitude when smaller m* is adopted. The values for m * = 0.8
and 0.6 are shown in Fig. 4.
In connection with the properties of neutron stars, it is useful to introduce
the critical temperature Tc[ALSJ beyond which the superfluidity disappears. Tc[ALSJ
closely related to .::JCALSJ is estimated by the following approximate relation:
tcBT /ALSJ~0.57 _d[ALSJ
·with
ICE
being the Boltzman constant.
In the [FG] case,
tcBT /FGJ::::::0.57 _d[FGJj r1~0.14JCFGJ
with ln
rl
= 1.201
(3 ·5)
arising from the angle dependence of the gap. 3 C)
(3 · 6)
Critical tem-
1667
Study on Neutron 3 P 2 -Superfiuidity with Two-Dimensional
peratures versus density in both cases are shown in Fig. 6. Tc[ALSJ are of the
almost same magnitude with T/FGJ; T/ALSJ = 10 9 ~ 10 (10 8 ~ 9) °K for tn* = 1 (0.8). This
critical temperature for m*~0.8, supposed
to be not so far from _realistic values, is
still higher than or of the same order
of magnitude with the internal temperature
expected to be about 10 7 ~ 9 °K for young
pulsars.m This means that the neutron
3 P -superfiuidity
111
the well-developed
2
[ALS] (7r 0-condensed) phase survives at
the temperature in the intererior of young
neutron stars, unless sudden decrease
of m* (such as m*::;0.6) occurs when
the [ALS] phase is set on. Therefore,
the determination of reliable values of
m* in the [ALS] phase is needed.
I.
Critical Temperature for
8
]I
10
P - Superfluidity
2
,0------
---r
'v
]()
m
* ~o.R
r
''
''
111
* =0.6
--: T
c
[AJ.S
-------: T CIFC]
Fig. 6. Critical temperatures for the 'P,-superfl.uidity in neutron matter; for the [ALS] case
(T/ALSl) and for the [FG] case (T,CFGl).
§ 4.
PIP,
10
Concluding remarks
The results obtained in the previous section show that the neutron 3 P 2 -superfiuidity realizable from the Fermi gas state at the density p;;;:;0.7p 0 ~2.8p 0 can
coexist with the vvell-developed [ ALS] (7r 0-condensed) state in the interior of young
neutron stars such as in the Crab and Vela pulsars. This means that the neutron
3 P -superfiuid in the 7r 0-condensed
phase provides an example of low-dimensional
2
superfiuid in nuclear system. We have investigated the problem for a typical case
of 7r 0 condensation, although there have been pointed out other possible types of
7r 0 condensation such as the one accompanying two- or three-dimensional localization181' 19> and the one accompanying charged pion condensation. 20 >' 2 D It is of interest
to extend the study to such various cases.
In this paper we have employed the formulation which is suitable to take
account of the pairing correlation in the particular partial wave in 3 P 2 with lmJl = 2
in the [ALS] phase. This enables us to directly compare the results with the
previous ones for three-dimensional case. Since the substantial aspect of the pairing
correlation under consideration is of two-dimensional character, we can develop a
suitable formulation with which this character and the difference from three-dimensional situation are described in a more direct way, although we depart from the
1668
T. Takatsuka and R. Tamagaki
usual partial-wave description. Such an approach by the use of a kind of realistic
local potentials, which will be reported in a succeeding paper, supports the conclusion of the present paper.
Ther are the following problems to be studied further in order to confirm the
conclusion mentioned above. It is needed to perform calculations by adopting the
Bloch-orbital basis instead of the Gaussian function basis as an approximation to the
Wannier-functi on basis. This point is indispensable to treat the situation close to
the critical point and to extend the study to the proton 1S 0-superfluidity mainly
induced by the pairs between the nearest layers when the localization is not so
remarkable. Furthermore another problem is to consider modification of the pairing interaction caused by the change from neutrons to quasi-neutrons clue to L1
(1236 MeV) mixing. The problem on whether or not the proton 18 0-superfluidity
can survive under the n° condensation has an important implication on the pulsar
"glitch" phenomena.
This work has been performed as a part o£ one of the 1978 annual research
projects organized by the Research Institute for Fundamental Physics, Kyoto University, "Structure of Baryonic System with Pion Condensation". The authors
wish to thank all the members of this project, especially to Mr. K. Tamiya, Mr.
T. Tatsumi and Mr. T. Kunihiro, for their stimulating discussions.
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