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Superfluidity of Neutron and Nuclear Superfluidity of Neutron and Nuclear MatterMatter
F. PederivaDipartimento di Fisica
Università di Trento I-38050 Povo, Trento, ItalyCNR/INFM-DEMOCRITOS
National Simulation Center, Trieste, Italy
CoworkersCoworkers
S. Gandolfi (SISSA)A. Illarionov (SISSA)S. Fantoni (SISSA)K.E. Schmidt (Arizona S.U.)
Why is it of interest?Why is it of interest?
• ““Superfluidity” of nucleiSuperfluidity” of nuclei as been long known. Attractive components of the NN force induce a pairing among nucleons. A few outcomes of it are the even-odd staggering of binding energies or anomalies on the momentum of inertia.
• More recently superfluidity of bulk nuclear matter has been recognized to play a role in the cooling process of cooling process of neutron stars.neutron stars.
Superfulidity and cooling of neutron Superfulidity and cooling of neutron starsstars
Superfluidity of nucleons has essentially three effects on neutrino emission in neutron stars (see e.g. Yakovlev, 2002):
1.1. Suppresses neutrino processes involving nucleons Suppresses neutrino processes involving nucleons (e.g. direct URCA process)(e.g. direct URCA process)
2.2. Initiates a specific mechanism of neutrino emission Initiates a specific mechanism of neutrino emission associated with Cooper pairing of nucleons associated with Cooper pairing of nucleons
3.3. Changes the nucleon heat capacityChanges the nucleon heat capacity
Superfulidity and cooling of neutron Superfulidity and cooling of neutron starsstars
The equations of thermal evolution of a NS are due to Thorne (assuming the internal structure independent on the temperature):
Ter
erc
Gm
r
L
t
T
e
cQLe
rrc
Gm
er
r
vr
22
2222
21
4
21
4
1
Changes occur when T=Tc in a given pairing channel.
It is therefore necessary to know the critical temperature Tc as a function of the density of the nuclear matter.
Superfluid gapSuperfluid gap
The easier way to estimate Tc is through the evaluation of the pairing gap. For instance, in the BCS model we have
76.1
cT
The pairing gap has been estimated by various theories and in different channels (mainly 1S0 and 3P2-3F2).
WE USE AFDMC to estimate the pairing gap as a function WE USE AFDMC to estimate the pairing gap as a function of density.of density.
Nuclear HamiltonianNuclear Hamiltonian
The interaction between N nucleons can be written in terms of an Hamiltonian of the form:
ji
M
p
pijp
N
i i
i VjiOrvm
pH
13
)(
1
2
),()(2
where i and j label the nucleons, rij is the distance between the nucleons and the O(p) are operators including spin, isospin, and spin-orbit operators. M is the maximum number of operators (M=18 for the Argonne v18 potential).
Nuclear HamiltonianNuclear Hamiltonian
The interaction used in this study is AV8’ cut to the first six operators.
)(),,(6...1jijji
piSO ττσσ1
where
jijijiijijS σσσrσr ))((3
EVEN AT LOW DENSITIES THE DETAIL OF THE EVEN AT LOW DENSITIES THE DETAIL OF THE INTERACTION STILL HAS IMPORTANT INTERACTION STILL HAS IMPORTANT EFFECTS (see Gezerlis, Carlson 2008)EFFECTS (see Gezerlis, Carlson 2008)
DMC for central potentialsDMC for central potentials
The formal solution
0
τ)(00
τ)(
τ)(
)0,()0,(
)0,(τ),(
0
nnn
EEEE
EH
RceRce
ReR
TnT
T
converges to the lowest energy eigenstatelowest energy eigenstate not not orthogonal to orthogonal to (R,0)(R,0)
Auxiliary Fields DMCAuxiliary Fields DMC
The use of auxiliary fields and constrained paths is originally due to S. Zhang for condensed matter problems (S.Zhang, J. Carlson, and J.Gubernatis, PRL74, 3653 (1995), Phys. Rev. B55. 7464 (1997))
Application to the Nuclear Hamiltonian is due to S.Fantoni and K.E. Schmidt (K.E. Schmidt and S. Fantoni, Phys. Lett. 445, 99 (1999))
The method consists of using the Hubbard-Stratonovich transformation in order to reduce the spin operators appearing in the Green’s function from quadraticquadratic to linearlinear.
Auxiliary FieldsAuxiliary Fields
For N nucleons the NN interaction can be re-written as
ji
jjiisisdsi sAsVVVV;
ββα;
where the 3Nx3N matrix A is a combination of the various v(p) appearing in the interaction. The s include both spin and isospin operators, and act on 4-component spinors:
pdncpbna iiiii
THE INCLUSION OF TENSOR-ISOSPIN TERMS HAS BEEN THE MOST RELEVANT DIFFICULTY IN THE APPLICATION
OF AFDMC SO FAR
Auxiliary FieldsAuxiliary Fields
We can apply the Hubbard-Stratonovich transformation to the Green’s function for the spin-dependent part of the potential:
nnnn
n
O
N
n
OV
Oxx
dxe
ee
nn
nnsd
Δτλ2
expπ2
1 2Δτλ2
1
3
1
Δτλ2
1Δτ
2
2 Commutators neglected
The xn are auxiliary variables to be sampled. The effect of the On is a rotation of the spinors of each particle.
Nuclear matter Nuclear matter
The functions J in the Jastrow factor are taken as the
scalar components of the FHNC/SOC correlation operator which minimizes the energy per particle of SNM at saturation density r0=0.16 fm-1. The antisymmetric
product A is a Slater determinant of plane waves.
Wave Function
The many-nucleon wave functionmany-nucleon wave function is written as the product of a Jastrow factorJastrow factor and an antisymmetric mean field antisymmetric mean field wave functionwave function:
)...;...;...()()...;...;...( 111111 NNNji
ijJNNN Ar ττσσrrττσσrr
11SS00 gap in neutron matter gap in neutron matterAFDMC should allow for an accurate estimate of the gap in superfluid neutron matter.
INGREDIENT NEEDED: A “SUPERFLUID” WAVEFUNCTION.
Nodes and phase in the superfluid are better described by a Jastrow-BCS wavefunctionJastrow-BCS wavefunction
),()()( SRrfR BCSji
ijJT
where the BCS part is a Pfaffian of orbitals of the form
a
jii
k
kjiij sse
u
vss ij
a
a ),(),,( rkr
Coefficients from CBF calculations! (Illarionov)
Gap in neutron matterGap in neutron matter
The gap is estimated by the even-odd energy difference at fixed density:
)1()1(2
1)()( NENENEN
•For our calculations we used N=12-18 and N=62-68. The gap slightly decreases by increasing the number of particles.
•The parameters in the pair wavefunctions have been taken by CBF calculatons.
Gap in Neutron MatterGap in Neutron Matter
Gandolfi S., Illarionov A., Fantoni S., P.F., Schmidt K., PRL 101, 132501 (2008)
13
0101.2105.2
Pair correlation functionsPair correlation functions
Gap in asymmetric matterGap in asymmetric matter
ConclusionsConclusions
• AFDMC can be successfully applied to the study of superfluid gaps in asymmetric nuclear matter and pure neutron matter. Results depend only on the choice of the nn interaction.
• Calculations show a maximum of the gap of about 2MeV at about kF=0.6 fm-1
•Large asymmetries seem to increase the value of the gap at the peak
•A more systematic analysis is in progress.