Pairing Correlations and Thermodynamic Properties of Inner Crust Matter

0.21 MB·English

by  Jérôme Margueron

Pairingcorrelations andthermodynamic properties ofinnercrustmatter 1
Chapter1
2
1 PAIRING CORRELATIONS AND THERMODYNAMIC
0
2 PROPERTIES OF INNER CRUST MATTER

n
a
J Je´roˆmeMargueron1∗and NicolaeSandulescu2†
3 1 InstitutdePhysiqueNucle´aire, IN2P3-CNRS and Universite´ Paris-Sud,
1
F-91406Orsay CEDEX, France

] 2 InstituteofPhysicsand NuclearEngineering,76900Bucharest, Romania
h
t
- January 16,2012
l
c
u
n
[ PACS 05.45-a, 52.35.Mw, 96.50.Fm. Keywords: Neutron star crust, nuclear superfluid-

1 ity.
v
4 Abstract
7
7 In this review paper we discuss the effects of pairing correlations on inner crust
2 matterinthedensityregionwherenuclearclustersaresupposedtocoexistwithnon-
1. localisedneutrons. Thepairingcorrelationsaretreatedintheframeworkofthefinite
0 temperatureHartree-Fock-Bogoliubovapproachandusingzerorangenuclearforces.
2 Afterashortintroductionandpresentationoftheformalismwediscusshowthepairing
1
correlations affect the structure of the inner crust matter, i.e., the proton to neutron
:
v ratio and the size of Wigner-Seitz cells. Thenwe show how the pairingcorrelations
i influence, thoughthe specific heatof neutrons, the thermalizationof the crust in the
X
caseofarapidcoolingscenario.
r
a
∗[email protected]
†[email protected]
2 J.Margueron &N.Sandulescu
1. Introduction
Thesuperfluid properties oftheinnercrust ofneutron stars havebeen considered long ago
inconnection tothelargerelaxation timeswhichfollowthegiantglitches. Thus,according
to the present models, the glitches are supposed to be generated by the unpinning of the
superfluid vortex lines from the nuclear clusters immersed in the inner crust of neutron
stars [1, 47]. Later on the superfluidity of the inner crust matter was also considered in
relation to the cooling of isolated neutron stars [34, 12] and, more recently, in the thermal
after-burst relaxation ofneutron starsfromX-raytransients [56,13,29].
The superfluid properties of the inner crust are essentially determined by the non-
localized neutrons. For baryonic densities smaller than about 1.4 x 10−14 g cm−3 the
non-localized neutronsaresupposedtocoexistwithnuclei-typeclusters[11,43]. Athigher
densities,beforethenuclearmatterbecomesuniform,theneutronsandtheprotonscanform
otherconfigurations suchasrods,plates, tubesandbubbles [46].
A microscopic ab initio calculation of pairing in inner crust matter should take into
account the polarization effects induced by the nuclear medium upon the bare nucleon-
nucleoninteraction. Thisisaverydifficulttaskwhichisnotyetcompletelysolvedevenfor
theinfiniteneutronmatter. Thus,comparedtoBCScalculations withbarenucleon-nucleon
forces, most of variational or diagrammatic models predict for infinite matter asubstantial
reduction ofthepairing correlations duetothein-medium polarisation effects[36]. Onthe
otherhand, calculations basedonMonteCarlotechniques predictfordiluteneutron matter
resultsclosertotheBCScalculations (forarecentstudysee[23]).
A consistent treatment of polarization effects on pairing is still missing for inner crust
matter(forarecentexploratory studysee[6]). Therefore atpresent themostadvanced mi-
croscopicmodelappliedtoinnercrustmatterremainstheHartree-Fock-Bogoliubov (HFB)
approach. Pairing correlations have been also considered in the Quasiparticle Random
Phase Approximation (QRPA)(see Section 2.1 below) in relation to the collective modes
in inner crust matter [31]. However, a systematic investigation of the effect of collective
QRPAexcitations onthermodynamic properties ofinnercrustmatterisstillmissing.
The scope of this chapter isto show how the HFBapproach can be used to investigate
the effects of pairing correlations on inner crust matter properties. Hence, in the first part
of the chapter we will discuss the influence of pairing, treated in HFB approach at zero
temperature, onthe structure ofinner crust matter. Then, using the HFBapproach atfinite
temperature, we will show how the pairing correlations affect the specific heat and the
thermalization oftheinnercrustmatterinthecaseofarapidcoolingscenario.
In the present study we will focus only to the region of the inner crust which is sup-
posed to be formed by a bbc crystal lattice of nuclear clusters embedded in non-localized
neutrons. Thecrystallatticeisdividedinelementary cellswhicharetreatedintheWigner-
Seitzapproximation.
Pairingcorrelations andthermodynamic properties ofinnercrustmatter 3
2. Treatment of pairing in the inner crust of neutron stars
2.1. Finite-temperature Hartree-Fock-Bogoliubov approach
Inthissectionwediscussthefinite-temperatureHFBapproximationforaWigner-Seitzcell
whichcontainsinitscenteranuclearclustersurroundedbyaneutrongas. Thecellcontains
alsorelativistic electrons whichareconsidered uniformlydistributed.
Inprinciple,theHFBequationsshouldbesolvedbyrespectingthebbcsymmetryofthe
inner crust lattice. However, imposing the exact lattice symmetry in microscopic models
is a very difficult task (for approximative solutions to this problem see Refs. [18, 26] and
the references therein). We therefore solve the HFB equations for a spherical WS cell,
as commonly done in inner crust studies [43, 3]. Since we are interested to describe the
thermodynamic properties of the inner crust matter, we present here the HFB approach at
finitetemperature.
TheHFBequations foraspherical WScells have thesameform asforisolated atomic
nuclei. Thus, for zero range pairing forces and spherical symmetry, the HFB equations at
finitetemperature aredefinedas[24],
h (r) λ ∆ (r) U (r) U (r)
T∆,q (−r) q h T(,rq)+λ Vi,q(r) = Ei,q Vi,q(r) , (1)
T,q T,q q ! i,q ! i,q !
−
whereE isthequasiparticleenergy,E = (e λ )2+∆2 ,U (r)andV (r)are
i,q i,q i,q − q i,q i,q i,q
thecomponentsoftheHFBwavefunctionandqλq isthechemicalpotential(q = n,pisthe
index for neutrons and protons) . The quantity h (r) is the thermal averaged mean field
T,q
hamiltonian and∆ (r)isthethermalaveraged pairingfield.
T,q
In a self-consistent HFB calculation based on a Skyrme-type force, as used in the
present study, h (r) and ∆ (r) are expressed in terms of thermal averaged densities,
T,q T,q
i.e., particle density ρ (r), kinetic energy density τ (r), spin density J (r) and, re-
T,q T,q T,q
spectively, pairing density κ (r). The thermal averaged densities mentioned above are
T,q
givenby[53]:
1
2 2
ρ (r) = g V (r) (1 f )+ U (r) f , (2)
T,q i,q i,q i,q i,q i,q
4π | | − | |
Xi h i
2
1 dV (r) V (r) l (l +1)
i,q i,q i,q i,q 2
τ (r) = g + V (r) (1 f )
T,q 4π i i,q("(cid:18) dr − r (cid:19) r2 i,q # − i,q
X
2
dU (r) U (r) l (l +1)
i,q i,q i,q i,q 2
+ + U (r) f , (3)
"(cid:18) dr − r (cid:19) r2 i,q # i,q)
1 3
J (r) = g j (j +1) l (l +1)
T,q i,q i,q i,q i,q i,q
4π − − 4
i (cid:18) (cid:19)
X
2 2
V (r) (1 f )+ U (r) f , (4)
i,q i,q i,q i,q
× | | − | |
1 (cid:16) (cid:17)
κ (r) = g U∗ (r)V (r)(1 2f ), (5)
T,q 4π i,q i,q i,q − i,q
i
X
wheref = [1+exp(E /T)]−1 istheFermi-Diracdistribution ofquasiparticles, Tisthe
i,q i,q
temperature expressed inenergy units, andg = 2j +1isthedegeneracy ofthe state i
i,q i,q
4 J.Margueron &N.Sandulescu
withangular momentum j . Thesummations inthe equations aboveareoverthe spectrum
i
ofboundnucleonswhichformthenuclearclusterandofunboundneutronswhichformthe
neutron gas. Aconstant density atthe edge of theWScell is obtained imposing Dirichlet-
Von Neumann boundary conditions at the edge of the cell [43], i.e., all wave functions of
evenparityvanishandthederivatives ofodd-parity wavefunctions vanish.
The nuclear mean field has the same expression in terms of densities as in finite nu-
clei [19]. However, for a WS cell the Coulomb mean field of protons has an additional
contribution comingfromtheinteraction oftheprotonswiththeelectronsgivenby
1
upe (r)= e2 d3r′ρ (r′) . (6)
Coul − e r r′
Z | − |
Assumingthattheelectrons areuniformly distributed insidethecell,withthedensityρ =
e
3
3Z/(4πR ),onegets
WS
2 2
1 Ze r
upe (r) = 2πe2ρ R2 r2 = 3 (7)
Coul − e(cid:18) WS − 3 (cid:19) 2RWS (cid:18)RWS(cid:19) − !
It can be seen that inside the WS cell the contribution of the proton-electron interaction to
theprotonmeanfieldisquadratic intheradialcoordinate.
Thepairingfieldiscalculated withazerorangeforceofthefollowingform
VPair,q(ri,rj) = V0 gPair,q[ρT,n(r),ρT,p(r)](1 Pσ)δ(rij), (8)
−
wherePσ = (1+σˆ1 σˆ2)/2 isthespin exchange operator. Forthis interaction thepairing
·
fieldisgivenby
∆T,q(r)= V0 gPair,q[ρT,n(r),ρT,p(r)] κT,q(r). (9)
In the calculations presented here we use two different functionals for
gPair,q[ρT,n(r),ρT,p(r)]. The first one, called below isoscalar (IS) pairing force, de-
pends only on the total baryonic density, ρ (r) = ρ (r)+ρ (r). Its expression is
T,B T,n T,p
givenby
ρ (r) α
gPair,q[ρT,n(r),ρT,p(r)] = 1 η T,B , (10)
− ρ0
(cid:18) (cid:19)
whereρ0 isthesaturation densityofthenuclearmatter. Thiseffectivepairinginteractionis
extensively used in nuclear structure calculations and it was also employed for describing
pairing correlations in the inner crust of neutron stars [52, 53, 54, 40]. The parameters
are chosen to reproduce in infinite neutron matter two pairing scenarii, i.e., corresponding
to a maximum gap of about 3 MeV (strong pairing scenario, hereafter named ISS) and,
respectively, toamaximumgaparound1MeV(weakpairingscenario, calledbelowISW).
These two pairing scenarii are simulated by two values of the pairing strength, i.e., V0= -
570,-430 MeVfm−3. Theotherparametersaretakenthesameforthestrongandthewea{k
pairing, i}.e., α=0.45, η=0.7 and ρ0=0.16 fm−3. The energy cut-off, necessary to cure the
divergenceassociatedtothezerorangeofthepairingforce,isintroducedthroughthefactor
e−Ei/100 actingforE > 20MeV,whereE aretheHFBquasiparticle energies.
i i
The second pairing functional, referred below as isovector strong pairing (IVS), de-
pends explicitly onneutron andproton densities and hasthe following form intheneutron
Pairingcorrelations andthermodynamic properties ofinnercrustmatter 5
Symmetric Matter Neutron Matter
]
V
ISW
e 3
M ISS
IVS
[
n
∆
p
a 2
g
g
n
i
r
i
a
p
n 1
o
r
t
u
e
N
0
1×1012 1×1013 1×1014 1×1012 1×1013 1×1014
ρ [g cm-3] ρ [g cm-3]
Figure1. (coloronline) Neutronpairing gapfortheinteractions ISW(isoscalar weak),ISS
(isoscalar strong) and IVS (isovector strong) in symmetric nuclear matter and in neutron
matter.
channel[38],
ρ (r) αs ρ (r) αn
gPair,n[ρT,n(r),ρT,p(r)] = 1 ηs(1 I(r)) T,B ηnI(r) T,B , (11)
− − ρ0 − ρ0
(cid:18) (cid:19) (cid:18) (cid:19)
1
whereI(r) = ρT,n(r) ρT,p(r). Thisinteraction isadjusted toreproduce theneutron S0
−
pairinggapinneutronandsymmetricnuclearmatterprovidedbytheBCScalculationswith
the bare nucleon-nucleon forces [14]. In addition, the pairing strength V0 and the cut-off
energy are related to each other through the neutron-neutron scattering length according
to the procedure described in Ref. [9]. Therefore this interaction is expected to describe
properly the pairing for all the nuclear densities of the inner crust matter, including the
lowdensity neutron gas. AsshowninRefs.[39,10],this pairing functional describes well
thetwo-neutron separation energies andtheodd-even massdifferences innucleiwithopen
shells in neutrons. In the present calculations for this pairing functional we have used the
parameters V0= -703.86 MeV fm−3, ηs=0.7115, αs=0.3865, ηn=0.9727, αn=0.3906. The
cut-offprescription isthesameasfortheisoscalarpairing force.
Thepairinggapsinsymmetricmatterandneutronmatterpredicted bythethreepairing
forcesintroduced abovearerepresented inFig.1forawiderangeofsub-nuclear densities.
It can be seen that the isovector IVSinteraction gives amaximum gap closer tothe strong
isoscalar ISS force, and the ISW interaction predict asuppression of the pairing gap up to
saturation density.
To illustrate how the pairing correlations are spatially distributed in the Wigner-Seitz
cells and how they are affected by the temperature, in Fig. 2 are shown the pairing fields
forneutronsinthecells2and5(seeTable1). Itcanbenoticedthattheclustershaveanon-
trivial influence on the pairing of neutron gas. Thus, depending of the relative intensity of
6 J.Margueron &N.Sandulescu
0
MeV] −1 MeV] −0.5
d [ d [
el el
pairing fi −2 11880000 SSnn ((12)) pairing fi 950 Sn (1)
950 Sn (2)
−3 −1.5
0 4 8 12 16 20 0 4 8 12 16 20
r [fm] r [fm]
Figure 2. Neutron pairing fields for the Wigner-Seitz cells 2 and 5 (see Table 1) labeled,
1800 950
respectively, as Sn and Sn. The full and the long-dashed lines correspond to the
ISS and ISW pairing interaction. Here the calculations have been done with the strenghts
3
V0 = 430, 330 MeVfm andwithanenergy cut-offof60MeV.
{− − }
pairingintheclusterandthegasregion,thepresenceoftheclustercansuppressorenhance
thepairinginthesurface regionofthecluster.
2.2. QuasiparticleRandom PhaseApproximation(QRPA)
Pairingcorrelations affectnotonlythegroundstatepropertiesofinnercrustmatterbutalso
itsexcitationsmodes. Thenon-collectiveexcitationsarecommonlydescribedbythequasi-
particleenergiesobtainedsolvingtheHFBequations. Tocalculatethecollectiveexcitations
one needs to take into account the residual interaction between the quasiparticles. In what
follows we discuss briefly the collective modes of the inner crust matter in the framework
ofQRPA,whichtakesproperlyintoaccount thepairingcorrelations [31].
TheQRPAcanbeobtainedfromthetime-dependentHFBapproachinthelimitoflinear
response. In the linear response theory the fundamental quantity is the Green function
whichsatisfiestheBethe-Salpeter equation
G= (1 G V)−1G = G +G VG. (12)
0 0 0 0
−
Theunperturbed Green’sfunctionG hastheform:
0
α1(rσ)¯∗β1(r′σ′) α2(rσ)¯∗β2(r′σ′)
G αβ(rσ,r′σ′;ω) = Uij Uij Uij Uij , (13)
0
¯hω (E +E )+iη − ¯hω+(E +E )+iη
XZij − i j i j
whereE aretheHFBquasiparticleenergiesand are3by2matricesexpressedintermof
i ij
U
thetwocomponentsoftheHFBwavefunctions[32]. The symbolintheequationabove
indicatesthatthesummationistakenovertheboundandunboundquasiparticle states. The
PR
lattercorresponds heretothenon-localised neutronsintheWScell.
V is the matrix ofthe residual interaction expressed interms ofthe second derivatives
oftheHFBenergyfunctional, namely:
2
∂
Vαβ(rσ,r′σ′)= E , α,β = 1,2,3. (14)
∂ρβ(r′σ′)∂ρα¯(rσ)
Pairingcorrelations andthermodynamic properties ofinnercrustmatter 7
x 104
2000
1750
)
1
-V1500
e
M1250
4
m1000
f
( 750

)
*
E 500
(
S
250
0
0 2 4 6 8 10 12 14 16 18 20
*
E (MeV)
Figure 3. Strength distributions for the quadrupole mode in WS cell 2 of Table 1. The
full(dashed) linecorresponds totheQRPA(HFB)strength. TheresultsareforISSpairing
forcewiththesameparameters asusedinFigure1.
Inthe above equation ρ1,ρ2,ρ3 ρ,κ,κ∗ , where ρand κare, respectively, theparti-
{ } ≡ { }
cle(2)andpairing(5)densities; thenotation α¯ meansthatwheneverαis2or3thenα¯ is3
or2.
The linear response of the system to external perturbation is commonly described by
the strength function. Thus, when the external perturbation is induced by a particle-hole
externalfieldF thestrength function writes:
1
S(ω) = Im F∗(r)G11(r,r′;ω)F(r′)drdr′ (15)
−π
Z
whereG11 isthe(ph,ph)component oftheQRPAGreen’sfunction.
As an example in Fig. 3 it is shown the strength function for the quadrupole response
calculated for the WScell 2ofTable1 below [31]. Theresults correspond tothe isoscalar
pairing force with the strength V0=-430 MeV fm−3 and an energy cut-off of 60 MeV. It
canbeseenthattheunperturbed spectrum, distributed overalargeenergy region, becomes
concentrated almostentirely inthepeaklocated atabout3MeVwhentheresidual interac-
tionbetweenthequasiparticles isintroduced. Thepeakcollectsmorethan99%ofthetotal
quadrupolestrengthanditisextremelycollective. Anindicationoftheextremecollectivity
of this low-energy mode can be also seen from its reduced transition probability, B(E2),
3
whichisequalto25 10 Weisskopfunits. ThisvalueofB(E2)istwoordersofmagnitude
×
higher than in standard nuclei. This underlines the fact that in this WS cell the collective
dynamicsoftheneutrongasdominatesovertheclustercontribution. InRef.[31]itisshown
that similar collective modes appears for the monopole and the dipole excitations. A very
collective low-energy quadrupole modeitwasalsofoundinalltheWigner-Seitz cellswith
8 J.Margueron &N.Sandulescu
N ρ k N Z R
cell B F WS
[gcm−3] [fm−1] [fm]
13
1 7.910 1.12 1460 40 19.7
13
2 3.410 0.84 1750 50 27.7
13
3 1.510 0.64 1300 50 33.2
12
4 9.610 0.55 1050 50 35.8
12
5 6.210 0.48 900 50 39.4
12
6 2.610 0.36 460 40 42.3
12
7 1.510 0.30 280 40 44.4
12
8 1.010 0.26 210 40 46.5
11
9 6.610 0.23 160 40 49.4
11
10 4.610 0.20 140 40 53.8
Table 1. The structure of the Wigner-Seitz cells obtained from a density matrix expansion
(DME) [43] ρ is the baryon density, k = (3π2n )1/3 the Fermi momentum calculated
B F B
3
as in Ref. [3] where n is the number of baryons per fm , N and Z are the numbers of
B
neutronsandprotonswhileR istheradiusoftheWScells. ComparedtoRef. [43]here
WS
itisnotshownthecellwiththehighestdensitylocatedattheinterfacewiththepastaphase.
Z=50 [27]. However, a systematic investigation of the influence of these collective modes
onthethermodynamic properties ofinnercrustmatterisstillmissing.
3. The effect of pairing on inner crust structure
Thefirstmicroscopic calculation oftheinner-crust structure wasperformed byNegeleand
Vautherin in 1973 [43]. In this work the crystal lattice is divided in spherical cells which
are treated in the Wigner-Seitz (WS) approximation. The nuclear matter from each cell is
described intheframeworkofHartree-Fock(HF)andthepairingisneglected. Theproper-
tiesoftheWScellsfoundinRef.[43],determined foralimitedsetofdensities, areshown
in Table I. The most remarkable result of this calculation is that the majority of the cells
have semi-magic and magic proton numbers, i.e., Z=40,50. This indicates that in these
calculations therearestrongprotonshelleffects,asinisolated atomicnuclei.
Theeffect ofpairing correlations onthestructure ofWigner-Seitz cellswasfirstinves-
tigated in Refs. [3, 4, 5] within the Hartree-Fock BCS (HFBCS)approach. In this section
weshalldiscuss theresults ofarecent calculations basedonHartree-Fock-Bogoliubov ap-
proach [28]. Thisapproach offersbettergrounds thanHF+BCSapproximation fortreating
pairingcorrelations innon-uniform nuclearmatterwithbothboundandunboundneutrons.
AsinRef.[43],thelatticestructureoftheinnercrustisdescribedasasetofindependent
cellsofsphericalsymmetrytreatedintheWSapproximation. Forbaryonicdensitiesbelow
14 3
ρ 1.4 10 g/cm , each cell has in its center a nuclear cluster (bound protons and
≈ ×
neutrons) surrounded by low-density and delocalized neutrons and immersed ina uniform
gas of ultra-relativistic electrons which assure the charge neutrality. At a given baryonic
density the structure of the cell, i.e., the N/Z ratio and the cell radius is determined from
theminimization overNand Zofthe totalenergy under thecondition ofbeta equilibrium.
The energy of the cell, relevant for determining the cell structure, has contributions from
Pairingcorrelations andthermodynamic properties ofinnercrustmatter 9
thenuclearandtheCoulombinteractions. Itsexpression iswritteninthefollowingform
E = E +E +T +E . (16)
M N e L
ThefirsttermisthemassdifferenceE = Z(m +m )+(N A)m whereNandZare
M p e n
−
the number of neutrons and protons in the cell and A=N+Z. E is the binding energy of
N
thenucleons, whichincludes thecontribution ofproton-proton Coulombinteraction inside
the nuclear cluster. T is the kinetic energy of the electrons while E is the lattice energy
e L
whichtakesintoaccounttheelectron-electron andelectron-proton interactions. Thecontri-
bution to the total energy coming from the interaction between the WS cells [44] it is not
considered sinceitisverysmallcompared totheothertermsofEq.(1).
Weshallnowdiscusstheeffectofpairingcorrelations onthestructureoftheWScells.
TostudytheinfluenceofpairingcorrelationswehaveperformedHFBcalculationswiththe
threepairing interactions introduced inSection2.1.
7.8
77..25 HISFS IISVWS(Cell 1) 7.5 HISFS IISVWS(Cell 1)
6.9 7.2
6.6 6.9
44..67 5.0 (Cell 2)
4.5 4.8
44..43 (Cell 2) 4.6
4.2 4.4
3.0 3.2 (Cell 3)
2.9
2.8 3.0
2.7 (Cell 3)
222...643 222...468 (Cell 4)
2.2
]22..01 (Cell 4) ]2.2
V1.9 V2.0
Me1.8 (Cell 5) Me21..08 (Cell 5)
[1.6 [1.6
A 1.4 A 1.4
E/10..08 (Cell 6) E/1.0 (Cell 6)
0.8
0.6
0.6
0.6
0.4 (Cell 7) 0.4 (Cell 7)
0.2 0.2
0.0 0.0
0.0 (Cell 8) 0.0 (Cell 8)
-0.2 -0.2
-0.4 -0.4
-0.6 -0.4
-0.6 (Cell 9) -0.6 (Cell 9)
-0.8 -0.8
-1.0 -1.0
-1.0 -1.0
-1.2 (Cell 10) -1.2 (Cell 10)
-1.4 -1.4
-1.6 -1.6
-1.8 -1.8
12162024283236404448525660 12162024283236404448525660
Z Z
Figure4. (coloronline)TheHFBenergiesperparticleasfunctionofprotonnumberforthe
pairingforcesISW(dotted line),ISS(dashedline)andIVS(dashed-dotted line). Thesolid
lines represent the HF results. In the left pannel are shown the results obtained including
thefinitesizecorrections.
The structure of the WS cells obtained in the HFB approach is given in Table II while
inFig.4itisshownthedependenceofthebindingenergies, atbetaequilibrium,onprotons
10 J.Margueron &N.Sandulescu
number. The contribution of pairing energy, nuclear energy and electron kinetic energy to
the total energy are given in Fig. 5 for the cells 2 and 6. From this figure we can notice
thatthenuclearbindingenergyisalmostcompensatedbythekineticenergyoftheelectrons
which explains the weak dependence of the total energy on Z seen in Fig. 4. In Fig.4 we
observe also that pairing is smoothing significantly the variation of the HF energy with Z.
Forthis reason the HFBabsolute minimaare verylittle pronounced compared to theother
neighboring energy values.
N N Z Z
cell corr
HF ISW ISS IVS HF ISW ISS IVS HF ISW ISS IVS
2 40 40 22 42
3 318 514 442 554 16 24 20 24 54 40 28 38
4 476 534 382 570 28 28 20 28 40 40 40 28
5 752 320 328 344 46 20 20 20 46 48 44 20
6 454 428 374 346 50 48 36 34 50 50 50 34
7 316 344 324 240 50 50 50 36 50 50 50 50
8 174 220 186 174 36 50 38 36 36 50 38 36
9 120 112 128 116 38 36 38 36 38 36 38 36
10 94 82 90 82 38 36 38 36 36 36 36 36
Table 2. The structure of Wigner-Seitz cells obtained in the HF and HFB approximation.
The results corresponds to the isoscalar weak (ISW), isoscalar strong (ISS) and isovector-
isoscalar(IVS)pairingforces. Inthelast4columnsareshowntheprotonnumbersobtained
withthefinitesizecorrections. Inthetableareshownonlythestructures ofthecellswhich
couldbewell-definedbythepresentcalculations.
From Fig. 4weobserve that in the cells 1-2 the binding energy does not converge to a
minimum atlowvalues ofZ.Forthe cells3-4, although absolute minimacan befound for
HFor/and HFBcalculations, theseminimaareveryclosetothevalueofbinding energy at
thelowestvaluesofZwecouldexplore. Thereforethestructureofthesecellsisambiguous.
Thesituation isdifferent inthe cells 5-10 where the binding energies converge toabsolute
minimawhicharewellbelowtheenergies oftheconfigurations withthelowestZvalues.
From Table II it can be observed that the structure of some cells becomes very differ-
ent when the pairing is included. Moreover, these differences depend significantly on the
intensity of pairing (see the results for ISW and ISS). However, as seen in Fig. 4, for the
majorityofcellstheabsoluteminimaareverylittlepronounced relativetotheotherenergy
values, especially for the HFB calculations. Therefore one cannot draw clear quantitative
conclusions onhowmuchpairing ischanging theprotonfractioninthecells.
When theradius ofthecell becomes too smallthe boundary conditions imposed atthe
cellborder through theWSapproximation generate anartificiallarge distance between the
energylevelsofthenonlocalizedneutrons. Consequently,thebindingenergyoftheneutron
gas is significantly underestimated. An estimation of how large could be the errors in the
binding energyinduced bytheWSapproximation canbeobtainedfromthequantity
f(ρ ,R ) B (ρ ) B (ρ ,R ), (17)
n WS inf. n WS−inf. n WS
≡ −
wherethefirstterm isthebinding energy perneutron forinfiniteneutron matterofdensity
ρ and the second term is the binding energy of neutron matter with the same density
n