Superfluidity in asymmetric nuclear matter
A. Sedrakian* and T. Alm
Max-Planck-Gesellschaft, AG ‘‘Theoretische Vielteilchenphysik,’’ Universita ¨t Rostock, D-18051 Rostock, Germany
U. Lombardo
Dipartimento di Fisica and INFN, 57 Corso Italia, I-95129 Catania, Italy
Received 31 July 1996
The onset of superfluidity in isospin-asymmetric nuclear matter is investigated within the BCS theory. A
neutron-proton superfluid state in the channel
3
S
1
-
3
D
1
comes about from the interplay between thermal
excitations and separation of the two Fermi surfaces. The superfluid state disappears above the threshold
value of the density-asymmetry parameter =( n
n
-n
p
)/ n 0.35. For large enough shift between the two
Fermi surfaces =
1
2
(
n
-
p
) the transition to the normal state becomes a first-order transition and a second
gap solution develops. This solution, however, corresponds to a metastable superfluid state which is unstable
with respect to the transition to the normal state. S0556-28139750402-X
PACS numbers: 21.65.+f, 25.70.-z, 25.75.-q, 74.20.Fg
Microscopic calculations, based on the BCS theory for the
bulk nuclear matter, show that the isospin-asymmetric matter
supports Cooper-type pair correlations in the
3
S
1
-
3
D
1
partial-wave channel due to the tensor component of the
nuclear force. The energy gap for this pairing has been found
to be of the order of 11 MeV for the infinite nuclear matter at
the saturation density within simple approaches which do not
include medium-polarization effects 1–3. Because of rela-
tively large values of the gap, pairing in the
3
S
1
-
3
D
1
chan-
nel could have important implications for nuclear physics
and nuclear astrophysics. Models of neutron stars, which per-
mit pion or kaon condensation such as the nucleon star
model recently proposed by Brown and Bethe 4, could give
a major role to the neutron-proton superfluidity in the inter-
pretation of neutron star rotation dynamics and its thermal
evolution. Furthermore, experimental evidence on neutron-
proton pairing could be obtained from the disassembling
phase of the compound system formed in heavy-ion colli-
sions 5. In this case, one has a unique chance to study the
crossover from the BSC neutron-proton pairing to a Bose
condensate of deuterons 5,6.
The existence of the pair correlations crucially depends
upon the overlap between the neutron and proton Fermi sur-
faces. If the system is driven out of the isospin-symmetric
state, one expects a suppression of the pairing correlations.
At zero temperature, a small asymmetry is enough to pre-
vent, at least in the BCS model, the formation of Cooper
pairs of neutrons and protons with momenta k
and -k
. The
superfluidity may be restored either by thermal excitations
which smear out the two Fermi surfaces or by collective
motion of the pairs which results in a shift of the two Fermi
spheres with respect to each other. For most applications, the
question arises of how large an isospin asymmetry could a
superfluid neutron-proton state sustain.
In the present Rapid Communication we study the pairing
in an infinite asymmetric nuclear matter. The isospin-singlet
pairing is assumed to be decoupled from the isospin-triplet
one, because the SD coupled channels contain the dominant
part of the attractive pairing force. Within this approxima-
tion, an extension of the BCS theory to asymmetric nuclear
matter in the Gorkov approach is straightforward. Assuming
the superfluid state to be a unitary triplet state see also 3
the spin dependence is explicitly worked out. Then the
proton/neutron propagator can be cast in the form ( =1)
G
,
'
p / n
k
,
m
=-
,
'
i
m
+
k
k
i
m
+E
k
+
i
m
-E
k
-
, 1
while the neutron-proton anomalous propagator acquires the
form
F
,
'
†
k
,
m
=-
,
'
†
k
i
m
+E
k
+
i
m
-E
k
-
, 2
where
m
are the Matsubara frequencies, the upper sign cor-
responds to protons and the lower one to neutrons. Here the
quasiparticle energy spectrum is separated in two branches
E
k
=
k
2
+
k
2
k
, 3
where
k
1
2
k
n
+
k
p
,
k
1
2
k
p
-
k
n
,
and
k
( n , p )
are the single particle energies of neutrons and
protons. In the case of the free Fermi gas to be considered
below
k
( n , p )
=k
2
/2m -
( n , p )
,
( n , p )
being the chemical po-
tentials for neutrons and protons, respectively.
From Eqs. 1 – 3 we obtain the BCS gap equation for
asymmetric nuclear matter using the angle-averaging proce-
dure, which has been proved to be a quite good approxima-
tion 3. We find
*Present address: Center for Radiophysics and Space Research,
Cornell University, Ithaca, NY 14853.
PHYSICAL REVIEW C FEBRUARY 1997 VOLUME 55, NUMBER 2
55 0556-2813/97/552/5823/$10.00 R582 © 1997 The American Physical Society