Pairing renormalization and regularization within the local density approximation

0.32 MB·English

by  P. J. Borycki

Pairing renormalization and regularization within the local density approximation
P.J. Borycki,1,2 J. Dobaczewski,1,3,4,5 W. Nazarewicz,1,3,4 and M.V. Stoitsov1,3,5,6
1Department of Physics & Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA
2Institute of Physics, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland
3Physics Division, Oak Ridge National Laboratory,
P.O. Box 2008, Oak Ridge, Tennessee 37831, USA
4Institute of Theoretical Physics, Warsaw University, ul. Hoz˙a 69, 00-681 Warsaw, Poland
5Joint Institute for Heavy-Ion Research, Oak Ridge, Tennessee 37831, USA
6Institute of Nuclear Research and Nuclear Energy,
Bulgarian Academy of Sciences, Sofia-1784, Bulgaria
(Dated: February 9, 2008)
We discuss methods used in mean-field theories to treat pairing correlations within the local
density approximation. Pairing renormalization and regularization procedures are compared in
6 sphericalanddeformednuclei. Bothprescriptionsgivefairlysimilarresults,althoughthetheoretical
0
motivation,simplicity,andstability oftheregularization proceduremakesitamethodofchoicefor
0
futureapplications.
2

n PACSnumbers: 21.60.-n,21.60.Jz,31.15.Ew,21.10.Dr
a
J
8 I. INTRODUCTION each s.p. space. Thus the energy cutoff and the pairing
1 strengthtogetherdefine the pairing interaction,andthis
definition can be understood as a phenomenological in-
One of the main goals of the low energy nuclear the-
1 troduction of finite range [8, 13]. Such a procedure is
ory is to build a comprehensive microscopic framework,
v usually referred to as the renormalization of the contact
in which nuclear bulk properties, excitations, and low
4
pairingforce. Itisperformedinthe spiritoftheeffective
5 energy reactions can be described. For medium-mass
field theory, whereupon contact interactions are used to
0 and heavy nuclei, self-consistent methods based on the
1 Density Functional Theory (DFT) [1, 2] have already describe low energy phenomena while the coupling con-
0 achieved a level of precision that allows for analysis of stants are readjusted for any given energy cutoff to take
6 into account neglected high energy effects.
experimental data for a wide range of properties of nu-
0
clei throughout the chart of the nuclides. For example, Therenormalizationprocedureforthezero-rangepair-
/
h theself-consistentHartree-Fock-Bogoliubov(HFB)mod- ing interactions has been explored in Ref. [8] using the
-t els based on the Skyrme energy functionals [3, 4, 5] are numerical solutions of the HFB equations. It has been
l able nowadays to reproduce nuclear masses with an rms shownthatbyrenormalizingthepairingstrengthforeach
c
u error of about 700keV [6, 7]. The development of a uni- value of the cutoff energy one practically eliminates the
n versal nuclear density functional, however, still requires dependence of the HFB energy on the cutoff parameter.
: a better understanding and improved description of the Recently, the issue of contact pairing force has been
v
i density dependence, isospin effects, pairing force, many- addressed in Refs. [14, 15, 16, 17, 18, 19, 20, 21, 22],
X body correlations,and symmetry restoration. suggestingthatthe renormalizationprocedurecanbere-
r Nuclear pairing is an important ingredient of the nu- placedbyaregularizationschemewhichremovesthecut-
a
clear density functional, and it becomes crucial for open off energy dependence of the pairing strength. In subse-
shell nuclei, in particular weakly bound systems, where quentpapers,thisregularizationschemehasbeenapplied
the effects of coupling to continuum become significant topropertiesoftheinfinite nuclearmatter[19],spherical
[8, 9]. In this case, the BCS model is not adequate [8] nuclei [21, 23], and trapped fermionic atoms [16, 24].
andthefullyself-consistentHFBapproachmustbeused. Inthisstudy,weinvestigatethestabilityoftheregular-
In most HFB applications, pairing interaction is as- izationschemewith respectto the cutoffenergyforboth
sumed to be either in the form of the finite range Gogny spherical and deformed nuclei. Differences between the
force [10] or the zero-range, possibly density-dependent, HFB results emerging from the pairing renormalization
delta force [8, 9, 11]. Gogny interaction in the pairing and pairing regularizationprocedures are analyzed.
channel can be viewed as a regularized contact interac- The HFB and Skyrme HFB formalisms have been ex-
tion, with regularization fixed through the finite range. plained in great detail in many papers (see, e.g., Refs.
The resulting pairing field is, however,nonlocal. [12, 25]). The notation used in the present paper is con-
Calculations using the contact interaction are numer- sistent with that of Refs. [8, 12, 26]. This work is orga-
ically simpler, but one has to apply a cutoff procedure nized as follows. Sec. II gives a brief introduction to the
withinagivenspaceofsingle-particle(s.p.) states[8,12]. pairing renormalization and regularization schemes. In
When the dimension of this space increases, the pairing Sect. III we explain the numerical framework used. The
gap diverges for any given strength of the interaction. comparisonbetweenpairingregularizationandrenormal-
Therefore, the pairing strength has to be readjusted for izationtechniques,studiedforalargesetofsphericaland
2
deformednuclei,isdiscussedinSec.IV. Finally,thesum- where f is a regulator which removes the divergence at
mary and conclusions are given in Sec. V. x=0.
For cutoff energies high enough, one can explicitly
identify [17, 18, 19] components generating divergence
II. THE CUTOFF PROCEDURES
in the abnormal density (see, e.g., Eq. (21) of Ref. [18]):
A. Pairing Renormalization Procedure
ih˜(r)M∗(r)k (r) h˜(r)
f(r,x)= F + G (r+x/2,r−x/2),
Within the HFB theory, the energy cutoff can be ap- 4π~2 2 µ
plied either to the s.p. or to the quasiparticle spectrum. (2.4)
The first option is used when the HFB equations are where G is the s.p. Green’s function at the Fermi level
µ
solved within a restricted s.p. space. However, the s.p. µ in the truncated space, M∗ is the effective mass, and
energies play only an auxiliary role in the HFB method, the Fermi momentum is
and the cutoff applied to the quasiparticle spectrum is
morejustified. This is done by using the so-calledequiv- 2M∗(r)
alent s.p. spectrum [12]: kF(r)= p ~ µ−U(r), (2.5)
p
e¯ =(1−2P )E +µ, (2.1)
n n n with U being the self-consistent mean-field potential.
where E is the quasiparticle energy and P denotes the The first term in Eq. (2.4) comes from the MacLau-
n n
normof the lowercomponentof the HFB wavefunction. rin expansion with respect to x; it guarantees that the
Due tothe similaritybetweene¯ andthe s.p.energies, regularizationproceduredoesnotintroduceanyconstant
n
one takesinto accountonly thosequasiparticlestates for termtotheabnormaldensityandthatf(r,x)solelyrep-
which e¯ is less than the assumed cutoff energy ǫ . resents the divergent part of ρ˜.
n cut
It was shown [8] that for a fixed pairing strength the Using the Thomas-Fermiapproximation,the locals.p.
pairing energies depend significantly on the energy cut- Green’s function G (r):=G (r,r) becomes [18, 19]
µ µ
off. Within the renormalizationscheme employedin this
work, we use the prescription of adjusting the pairing 1 kcut(r) k2dk
strength to obtain a fixed average neutron pairing gap G (r)= lim ,
[12], µ 2π2 γ→0Z0 µ− 2M~2∗k(2r) −U(r)+iγ
(2.6)
∆¯ =− 1 d3rd3r′ h˜(rσ,r′σ′)ρ(r′σ′,rσ), (2.2) where the cutoff momentum is given by:
N Z
Xσσ′
2M∗(r)
in 120Sn equal to the experimental value of 1.245MeV. kcut(r)= p ~ ǫcut+µ−U(r). (2.7)
In Eq. (2.2) N is the number of particles, ρ is the par- p
ticle density, and h˜ is the pairing Hamiltonian (see Ap-
The regularized pairing Hamiltonian and the pairing
pendix A).
energy density may be written, respectively, as [18]:
Such a procedurealmost eliminates the dependence of
the HFB energy on the cutoff [8].
h˜(r) = g(r)ρ˜ (r)=g (r)ρ˜(r) (2.8)
r eff
1
H (r) = g (r)ρ˜(r)2, (2.9)
B. Pairing Regularization Procedure pair 2 eff
Using the HFB equations and properties of the Bo- where the effective pairing strength [17, 18, 19],
goliubovtransformation(seeappendixAfordetails),one
concludesthatthelocalabnormaldensityρ˜hasasingular 1 G (r) iM∗(r)k (r) −1
behavior when ǫ → ∞. The standard regularization g (r)= + µ + F , (2.10)
cut eff (cid:18)g(r) 2 4π~2 (cid:19)
technique is to remove the divergent part and define the
regularized local abnormal density ρ˜ (r) as
r
after calculating integral (2.6), can be expressed in the
ρ˜ (r)= lim [ρ˜(r−x/2,r+x/2)−f(r,x)], (2.3) form:
r
x→0
3
1 − M∗(r)kcut(r) 1− kF(r) lnkcut(r)+kF(r) −1 k (r)2 ≥0
g (r)= hg(r) 2π2~2 (cid:16) 2kcut(r) kcut(r)−kF(r)(cid:17)i F . (2.11)
eff  1 − M∗(r)kcut(r) 1+ |kF(r)|arctan|kF(r)| −1 k (r)2 <0
 hg(r) 2π2~2 (cid:16) kcut(r) kcut(r)(cid:17)i F
In this regularization scheme, only the Green’s func-
ttaCiiroooennnd.seiesqtTeuchreamenlctidunlyele,andttsheistdeeileFfus-e,csriompnngoistimsettnhoetemniatTellnysh,touwammintadh(si2-ncF.h5et)erhmmdeeiicpHaaeFlnpBdppsrototohxenenimomtriaayi--l. netic (MeV)21712170 120Sn SmLixye4d 19-1018otal (MeV)
croscopic HFB quantities. According to the sign of k2, Ki9 10T
F 6 -
opFnreoIrenmpooiRfsleteehdfv.eet[l2eh.0xaH]ptoraiewnsdevsivioffoelenvrr,esestn(h2tter.r1uHe1ng)FcuaiBlsatircuoiasznleacdtbui.eolalnotwisocnhasenmidneathhbeaosvqeubaeteshine- ng (MeV)21-11 C4u0toff 6E0nerg80y (M1e0V0) C4u0toff 6E0nerg80y (M1e0V0) 476-3475me (MeV)
particle basis should be performed for a high cutoff en- airi -3kyr
ergy of 50MeV and higher [8]. Since the magnitude of P2 S
1
theself-consistentmeanfieldU isalsoabout50MeV,for - 40 60 80 100 V)
such a high cutoff energy both methods are equivalent. V)48 Cutoff Energy (MeV) 348Me
TheThomas-Fermiapproximationrequiresthat,inorder Me- b (
htoigohbetanionugrehsufoltrskicnudtetpoenbdeernetalofevǫceurty,withservea.lue should be E (S.O.-49 347ulom
o
Throughthedensitydependenceofgeff,kcut,andkF, C
there appear rearrangement terms in the self-consistent 40 60 80 100 40 60 80 100
mean-field potential: Cutoff Energy (MeV)
δH δg
pair = effρ˜2 =ρ˜2×
δρ δρ FIG.1: VariouscontributionstotheHFBenergyfor120Snas
∂g δg ∂g δk ∂g δk afunctionofǫcut. CalculationsareperformedusingtheSLy4
eff eff F eff cut
× + + .(2.12) Skyrmefunctional and mixed pairing interaction (3.1).
(cid:18) ∂g δρ ∂k δρ ∂k δρ (cid:19)
F cut
The first term in Eq. (2.12) is similar to the usual re-
ations of about 2MeV) and pairing term (with varia-
arrangement term, while the other two terms associated
tionsofabout1.3MeV).Also,themomentum-dependent
with the regularization procedure are entirely new. It is
spin-orbitterm,E ,hassignificantvariationsofabout
easy to check that all the terms appearing in Eq. (2.12) S.O.
are continuous at the classical turning point k (r)=0. 1MeV. On the other hand, Skyrme and Coulomb ener-
F
gies are fairly stable with respect to ǫ .
In Eq. (2.9), the pairing energy density is divergent cut
with respect to the cutoff energy. However, the pairing
energy itself is not an observable, and in order for the
III. NUMERICAL IMPLEMENTATION
energy density functional to be independent of the cut-
off, other terms have to cancel out this divergence. As
A. Numerical Framework
discussedinRefs. [14,15,17,18],thekineticenergyden-
sity τ has the same type of divergence as the abnormal
density ρ˜, and the sum Asthe pairingrenormalizationandregularizationpro-
cedures remove the divergent part of the abnormal den-
H (r)=− ~2 τ + 1g (r)ρ˜2(r) (2.13) sity in a different way, one can expect some numerical
kin+pair 2M∗(r) 2 eff differences between both methods. In order to compare
their results, we have performed numerical calculations
does converge.
using two numerical codes solving the HFB equations:
VariouscontributionstothetotalHFBenergyasfunc-
tions of the cutoff energy are shown in Fig. 1. The • HFBRAD [27] – solves the HFB equations in the
total energy is stable with respect to ǫ , although spherically symmetric coordinate basis. The max-
cut
some of the components of the total energy vary signifi- imum angular momenta used in calculations were
cantly. As expected from Eq. (2.13), two terms exhibit- j =39/2 for neutrons and j =25/2 for pro-
max max
ing large fluctuations are the kinetic term (with vari- tons.
4
V) 0 V) 120Sn
Me-1 0. Me
nergy (3-2 -0.1 nergy (-1018.4 Sly4, mixed pairing -1018.5
E- E
Total -4 120Sn -0.2 Total -1018.6 018.6
V) 6 V)0 -1
p (Me2.0 1.2 ap (Me51.3 harmonic o22s08c ilsslahhteeollllrss: box s12iz55e ff:mm 26
airing Ga1.5 volume 1.24 Pairing G1.201.2 1.241.
P mixed
40 60 80 100 40 60 80 100
80 100 120 80 100 120
Cutoff Energy (MeV)
Cutoff Energy (MeV)
FIG.3: Totalenergy(top)andneutronpairinggap(bottom)
FIG.2: Totalenergy(top)andneutronpairinggap(bottom)
in 120Sn without (left) and with (right) pairing renormaliza- in120SnforthetwovaluesofNosc(left)orusingtwoboxsizes
(right). Calculations wereperformedusingthemixedpairing
tion applied. Resultsareshown forvolume(gray) andmixed
interaction.
(black) pairing. The total energy is plotted relative to the
values obtained for thecutoff energy of ǫcut=60MeV.
neutrongapvarysignificantly with increasingdimension
of the quasiparticle space. In this case, the total energy
• HFBTHO [26] – diagonalizes the HFB problem in
changes by several MeV.
the axially symmetric transformed harmonic oscil-
lator (HO) basis. Unless stated otherwise, we use
N = 20 HO shells in the basis.
osc C. Pairing Regularization
In our calculations, we use the SLy4 [28] and SkP [12]
parameterizations of the Skyrme functional in the p-h The total energy and the average neutron pairing gap
channel and the contact density-dependent force in the in 120Sn are shown in Fig. 3 after applying the pairing
p-pchannel, whichleadsto the pairingenergydensity of regularizationprocedure. ThepairingstrengthV iskept
0
the form: constant;itreproducestheneutronpairinggapfor120Sn
at the cutoff energy of ǫ =60MeV.
1 cut
H (r) = g(r)ρ˜(r)2 = IntheleftpanelsofFig.3,weshowresultsobtainedin
pair
2
the HO basis, while the results from the solution of the
1 ρ(r)
= V 1−V ρ˜(r)2, (3.1) HFB equations in coordinate space are displayed in the
0 1
2 (cid:20) (cid:18) ρ0 (cid:19) (cid:21) rightpanels. Onecancorrelatethecoordinate-spaceand
HO representationsby introducing an‘effective boxsize’
where ρ0 = 0.16fm−3. For V1 = 0 the resulting pair- R≈ 2Nosc~/mω[8]. Usingthisformula,thebasisof20
ing interaction is called volume pairing, while V1 = 1/2 HO sphells corresponds to a box radius of about 14.5fm.
corresponds to the so-called mixed pairing prescription Figure 3 demonstrates that the regularizationprocedure
(Ref. [29] and references quoted therein). is stable with respect to the cutoff energy. Moreover,
one obtains reasonable results already for fairly low cut-
off energies of about 40MeV. The variations in the to-
B. Pairing Renormalization talenergyincoordinate-spacecalculationsdonotexceed
40keV, while they are about 150keV in the HO expan-
Figure2illustratestheimportanceofthepairingrenor- sion. The latter number does not decrease significantly
malization procedure in the case of 120Sn. Due to the with N .
osc
constraint (2.2) on the pairing strength, the neutron av- The differences in applying the pairing regularization
erage pairing gap stays by definition constant, while the procedure in the coordinate-space and HO calculations
resulting total energy changes with the cutoff energy by can be explained by the different way the quasiparticle
a few hundred keV. On the other hand, without pairing space is expanded inboth approaches. The particle den-
renormalizationapplied,thetotalenergyandtheaverage sity ρ is defined by the lower components of the quasi-
5
mixed pairing volume pairing 8
8 7.
1018.2 -1017. V)-101 120 Sn
- e Sly4, mixed pairing
0 M
) 8. (
gy (MeV-1018.4 120Sn -101 Energy -1018.2 GTF
er al Gfree
otal En893.72 110Zr rearransgtaenmdeanrdt: -893.62 Tot6
T- full 8.
4 1
6 0
74 93. -1 40 60 80 100 120 140 160 180 200 220
93. -8 Cutoff Energy (MeV)
8
-
FIG. 5: Two pairing regularization schemes applied to the
40 60 80 100 40 60 80 100 case of 120Sn: the Thomas-Fermi approximation [19] (black
Cutoff Energy (MeV)
line) and the free particle Green’s function [13] (gray line).
Coordinate-spacecalculationswereperformedina15fmbox.
FIG. 4: Total energy of spherical 120Sn (top) and deformed
110Zr (bottom) obtained with pairing regularization (black ergiesbelow30MeVresultsfromneglectingquasiparticle
lines)formixedpairing(left)andvolumepairing(right). The
stateswithsignificantoccupationprobability. Thiseffect
results obtained without the rearrangement terms resulting
is more severe for the mixed-pairing than for volume-
fromthevariationofkcut andkF inEq.(2.12)arealsoshown pairing calculations due to the surface-peaked character
using (gray lines).
of mixed pairing fields. On the other hand, the stability
withrespectto the cutoffenergyis similarinbothcases.
We have also tested the importance of the rearrange-
particle wave functions, which are localized within the
ment terms arising as a result of the regularization pro-
nuclear interior. On the other hand, the abnormal den-
cedure. The gray lines in Fig. 4 show results obtained
sity is defined by the products of the upper and lower
without taking into account the second and third term
components of the quasiparticle wave function. For the
of Eq. (2.12). These terms lead to changes in the total
quasiparticle energies that are greater than the modu-
energy of a few keV and can be safely neglected.
lus of the chemical potential, the upper components of
Finally, we have tested the Thomas-Fermiapproxima-
the quasiparticlewavefunction arenot localized. There-
tion used in the pairing regularization procedure. In-
fore, contrary to the normal density, the abnormal den-
steadofadoptingtheThomas-Fermiansatz,onecanper-
sitystronglydependsonthecompletenessofthes.p. ba-
form regularization using the free particle Green’s func-
sis outside the nuclear interior.
tion [13]. As illustrated in Fig. 5, the convergenceof the
Inthecoordinate-spacecalculations,theboxboundary
latter method is very slow;the Thomas-Fermimethod is
conditions provide discretization of the spectrum for the
clearly superior.
quasiparticlecontinuumstatesthatarenotlocalized. On
theotherhand,alltheHObasisstatesarelocalized. Re-
sults ofstability withrespecttothe cutoffenergyforthe D. A link between the pairing renormalization and
coordinate-spaceandHO calculationsare,therefore,dif- regularization procedures
ferent. As far as the description of nonlocalized states is
concerned, the coordinate-space method is superior over
The renormalized and regularized pairing calculations
the HO expansion method.
are based, in fact, on two different effective interactions.
Fluctuations in the total energy shown in Fig. 3 co-
Consequently,theirresultsshouldbecomparableonlyas
incide with 2j+1-folded degenerate angular-momentum
muchastheireffectivepairingstrengthsg aresimilar.
eff
multipletsofstatesinsphericalnucleithatenterthepair-
By expanding Eq. (2.11) at very high cutoff energies
ing window with increasing cutoff energy. This can be
(k /k <<1), one obtains:
F cut
confirmedbyperformingasimilaranalysisforadeformed
nucleuswherethemagneticdegeneracyislifted. Suchre- M∗(r)g(r) −1
sultsareshowninFig.4fordeformed110Zrincomparison geff(r)≈(cid:18)1− 2π2~2 kcut(r)(cid:19) g(r), (3.2)
with spherical 120Sn. One can see that the fluctuations
of the total energy in 110Zr are down to about 40keV. whichhastheformofg =αg. Forthevolumepairing,
eff
The steepincreaseofthe totalenergyatthe cutoff en- the proportionalityfactor αis ρ-dependentonly through
6
1.2 120Sn plied to a large number of nuclei. As representative re-
sults,wediscussresultsobtainedforthedrip-to-dripline
isotopic chains of spherical Sn nuclei as well as for de-
formed Dy nuclei. Calculations are performed for the
1.0 volume and/or mixed pairing interactions by using the
HFB+THO approach.
0.8
N
Reff SLy4, volume pairing
g

/
RG eff1.2 SLy4, mixed pairing A. Spherical Nuclei
g
1.0 Figure7displaysdifferencesbetweenthepairingrenor-
malization and regularization procedures for the Sn iso-
30 MeV topes. Calculationsareperformedwithbothvolumeand
60 MeV mixed pairing interactions. For the two-neutron sep-
0.8
100 MeV aration energies, the maximum difference between the
180 MeV renormalizationand regularizationschemes is about 100
(300)keVforthevolume(mixed)pairing. Intheneutron
2 4 6 8
gap, the corresponding difference is about 50 (100)keV,
radius (fm)
and in nuclear radii (not displayed) it is practically neg-
ligible (about 0.01fm). The largest differences show up
in pairing energies – about 1 (3) MeV for the volume
FIG.6: Ratiobetweentheeffectivepairingstrengthsforpair-
ingregularizationandrenormalization,gRG/gRN,forthevol- (mixed) pairing; however, total energy differences are
ume (upper panel) and mixed pairing (loefwfer epfafnel) in 120Sn much smaller – about 400 (800)keV.
for several values of ǫcut. Analyzing the total energies obtained in both meth-
ods, Fig. 7(a), one can see that the pairing renormaliza-
tion procedure gives systematically more binding. The
the weak density dependence of the effective mass M∗.
differencesarenegligibleforstablenucleiandnucleinear
Ontheotherhand,forthemixedpairing,italsodepends
the proton drip line. They increase in mid-shell nuclei
on ρ through the density dependence of g. Therefore,
near the two-neutron drip line where the pairing effects
while for the volume pairing the renormalization proce-
are the largest, and then decrease towards the closed-
dure may be considered as a fair approximation to the shell nucleus 176Sn located just at the two-neutron drip
regularization scheme, this is not the case for the mixed
line. In general, both procedures give more similar re-
pairing, or – more generally – for any density-dependent
sults in the case of volume pairing than in the case of
pairing. Still, this approximate equality of the effective
mixed pairing.
pairingstrengthsforthepairingregularizationandrenor-
malization is an explanation of the remarkable stability Recently,thepairingregularizationprocedurehasbeen
ofthetotalenergyinphenomenologicalpairingrenormal- analyzedinthe contextofrelativisticmean-fieldapprox-
ization treatment (see Fig. 2), and it also explains why imation [23]. In order to simulate the finite range con-
results obtained for the volume pairing are more stable tribution to the nuclear matter pairing gap coming from
than those in the mixed pairing variant. the Gogny pairing force, it was necessary to introduce
This effect can be clearly seen in Fig. 6. The ratio strong density dependence in the pairing strength of the
between the effective pairing strengths in the regulariza- contact interaction.
tionandrenormalizationmethodsismuchclosertounity
Usingtheregularizationprocedureandcalculatingthe
for the volume pairing than for the mixed pairing in the
Sn chain with both volume and newly constructed (sur-
regionofspace,wherethe pairingenergydensityismax-
face) contact interaction, the authors of Ref. [23] have
imal.
found differences in pairing energies of the order of
20MeV in the neutron-rich nuclei around 148Sn. In our
work,forthe samenuclei,thedifferencesinpairingener-
IV. COMPARISON BETWEEN PAIRING gies between volume and mixed pairing variants do not
RENORMALIZATION AND REGULARIZATION exceed2.6MeV.Thiscomparisonshowsthatthedensity-
PROCEDURES
dependent contact interaction proposed in Ref. [23] is
questionable for finite nuclei, despite its agreement with
In this section, we presentacomparisonbetweenpair- thefinite-rangeGognypairingforceintheinfinitenuclear
ing renormalization and regularization procedures ap- matter.
7

0.5
(a) V) mixed pairing (b)
volume pairing
e
0 M
(

G n
R S2 0.0

-
N n
R 2
) S
V
e -1
M -0.5
(

RG E mvoilxuemd ep apiariirnigng (c) 0.1

- )
N V
R E e
-2 M
(

G
R n
ET volume pairing -
N
ET mixed pairing R n
pair
En volume pairing 0.0
-3 pair
En mixed pairing
40 50 60 70 80 90 100110120130 40 50 60 70 80 90 100110120130
Neutron number Neutron number
FIG. 7: Differences between pairing renormalization (RN) and regularization (RG) procedures for (a) total energies and
neutron pairing energies, (b) two neutronseparation energies, and (c) theaverage neutron gaps. TheHFB+THO calculations
are performed for thechain of the spherical Sn isotopes using the SkPSkyrmeparameterization.
B. Deformed Nuclei self do not exceed 50keV for the cutoff energy as low as
30MeV.However,ifastilllowercutoffenergyisassumed,
Weappliedthepairingrenormalizationandregulariza- the Thomas-Fermi approximation to the s.p. Green’s
tion procedures for the chain of deformed Dy isotopes. function may no longer be valid.
Differences between both sets of results are shown in We found that the differences between pairing renor-
Fig.8. We show onlythe results withthe mixedpairing, malizationandregularizationproceduresforvolumeand
since, as in the spherical nuclei, the differences between mixed pairing are rather small. Therefore, we conclude
both procedures are larger in this case. that physical conclusions previously obtained within the
As seen in Fig. 8(c), most of the nuclei considered pairing renormalization scheme remain valid. Neverthe-
are well deformed, and the deformations are practically less, we believe that the theoretical motivation and sim-
the same within both procedures. Despite the fact that plicityoftheregularizationmethodispreferredtoaphe-
the maximum difference in the pairing energy is around nomenologicalrenormalization scheme.
3MeV (not shown), other quantities are very similar.
The maximum difference in the total energy is about
360keV,inthetwo-neutronseparationenergy–160keV,
in the pairing gaps – 110keV, and in the rms radii (not Acknowledgments
shown) – less than 0.005fm.
Discussions with Aurel Bulgac are gratefully acknowl-
edged. This work was supported in part by the
V. SUMMARY AND CONCLUSIONS U.S. Department of Energy under Contract Nos. DE-
FG02-96ER40963 (University of Tennessee), DE-AC05-
Inthiswork,weinvestigatedthepairingregularization 00OR22725withUT-Battelle,LLC(OakRidgeNational
method using the s.p. Green’s function in the Thomas- Laboratory), by the National Nuclear Security Admin-
Fermi approximation and found it to be very suitable istration under the Stewardship Science Academic Al-
for a description of spherical and deformed nuclei. We liancesprogramthroughDOEResearchGrantDE-FG03-
checked the stability of the method with respect to the 03NA00083; by the Polish Committee for Scientific Re-
cutoff energy and found fluctuations in the total energy search(KBN) under contract N0. 1 P03B 059 27 and by
below 200keV.Fluctuations coming fromthe methodit- the Foundation for Polish Science (FNP).
8

0.1 (a) (b)
)
) V 0.1
V
e 0.0 Me
M
(
( G n
RG T -0.1 R S2 0.0
E

-
- N n
RN T -0.2 R S2
E
-0.1
-0.3
0.4
(c) (d)
)
V 0.1
e
M
0.2
(

G
R
0.0
-
0.0 N
R
-0.2 -0.1
RG RN neutrons protons
70 80 90 100110120130140150 70 80 90 100110120130140150
Neutron number Neutron number
FIG. 8: Similar to Fig. 7 except for the deformed Dy isotopes. Quadrupole deformations are displayed in panel (c). Mixed
pairing interaction was used.
APPENDIX A: DIVERGENCE IN THE is the solution of the HFB equation:
ABNORMAL DENSITY
In the DFT-HFB approach, the starting point is the Xσ1 Z d3r1(cid:20)hh˜µ((rr22σσ22,,rr11σσ11)) −h˜h(µr(2rσ22σ,2r,1rσ11σ)1)(cid:21)×
EnergyDensity Functional(EDF)H[ρ,ρ˜],whereρ isthe
particle density and ρ˜is the abnormal density: × ϕ1i(r1σ1) =E ϕ1i(r2σ2) , (A5)
(cid:20)ϕ2i(r1σ1)(cid:21) i(cid:20) ϕ2i(r2σ2)(cid:21)
ρ(r2σ2τ2,r1σ1τ1)=hΦ|a†r1σ1τ1ar2σ2τ2|Φi, (A1) for a given quasiparticle energy E .
ρ˜(r2σ2τ2,r1σ1τ1)=−2σ1hΦ|ar1−σ1τ1ar2σ2τ2|Φi,(A2) The HFB equations are a resuilt of variational mini-
mizationoftheenergydensityfunctionalH[ρ,ρ˜]withthe
whereaanda† aretheparticleannihilationandcreation
constraint of the mean value of particles kept constant:
operators,respectively, and |Φi is the HFB state. In the
following,weassumethat|Φiisaproductoftheneutron
δH| =0. (A6)
and protonstates, |Φ i|Φ i. Therefore,the neutron and hNˆi=N
ν π
proton wave functions are not coupled, and in the nota-
This condition defines the s.p. Hamiltonian h and the
tion below we can, for simplicity, omit the isospin index µ
pairing Hamiltonian h˜ in the HFB equations (A5):
with the understanding that all equations are separately
valid for neutrons and protons.
δH[ρ,ρ˜]
sitFieosrctahnebHeFwBrsitttaetnea|Φsi[,1t2h]:eparticleandabnormalden- hµ(r2σ2,r1σ1)= δρ(r1σ1,r2σ2) −µ (A7)
~2
ρ(r2σ2,r1σ1) = ϕ2i(r2σ2)ϕ∗2i(r1σ1), (A3) = −∇r22M∗(r2σ2,r1σ1)∇r1 +U(r2σ2,r1σ1)−µ
EXi>0 δH[ρ,ρ˜]
ρ˜(r2σ2,r1σ1) = − ϕ2i(r2σ2)ϕ∗1i(r1σ1), (A4) ,h˜(r2σ2,r1σ1)= δρ˜(r1σ2,r2σ2) (A8)
EXi>0
whereM∗istheeffectivemassandU istheself-consistent
where the two-component quasiparticle wave function ϕ mean-field potential. In the following derivations, the
9
spin-orbittermisomittedasunimportantintheregular- By multiplying the HFB equations (A5) by vector
izationscheme, althoughit is, ofcourse,alwaysincluded [ϕ∗ ,−ϕ∗ ], integrating over coordinates and summing
2i 1i
in calculations. over all the positive energy HFB solutions, one obtains:
EiX>0,σ2EiZ d3r2(cid:2)ϕ∗2i(r2σ2), −ϕ∗1i(r2σ2)(cid:3)(cid:20)ϕϕ21ii((rr22σσ22)) (cid:21)=
=Ei>X0,σ1σ2ZZ d3r1d3r2(cid:2)ϕ∗2i(r2σ2), −ϕ∗1i(r2σ2)(cid:3)(cid:20) hh˜µ((rr22σσ22,,rr11σσ11)) −h˜h(µr(2rσ22σ,2r,1rσ11σ)1) (cid:21)(cid:20) ϕϕ21ii((rr11σσ11))(cid:21) (A9)
i.e.,
EiZ d3r1{ϕ∗2i(r1σ1)ϕ1i(r1σ1)−ϕ∗1i(r1σ1)ϕ2i(r1σ1)}=
EiX>0,σ1
= ZZ d3r1d3r2(cid:26)ϕ∗2i(r2σ2)hµ(r2σ2,r1σ1)ϕ1i(r1σ1)+ϕ∗2i(r2σ2)h˜(r2σ2,r1σ1)ϕ2i(r1σ1)+
Ei>X0,σ1σ2
+ϕ∗1i(r2σ2)hµ(r2σ2,r1σ1)ϕ2i(r1σ1)−ϕ∗1i(r2σ2)h˜(r2σ2,r1σ1)ϕ1i(r1σ1)(cid:27). (A10)
Since for every HFB solution ([ϕ ,ϕ ],E ) there exists also an orthogonal solution ([ϕ ,−ϕ ],−E ), the left-hand
1i 2i i 2i 1i i
side of Eq. (A10) vanishes as a sum over scalar products of orthogonal wave functions.
For local and spin-independent Hamiltonians h and h˜, Eqs. (A8) and (A8) read
µ
~2
hµ(r2σ2,r1σ1) = −∇r22M∗(r2)δ(r2−r1)δσ2,σ1∇r1 +(U(r2)−µ)δ(r2−r1)δσ2,σ1, (A11)
h˜(r2σ2,r1σ1) = h˜(r2)δ(r2−r1)δσ2,σ1. (A12)
Note that for an attractive pairing force, the local pairing potential ˜h(r) = −∆(r) is negative, where ∆(r) is the
standard position-dependent pairing gap. By defining function F as
ǫcut
[ϕ1i(r2σ)ϕ∗1i(r1σ)+ϕ2i(r2σ)ϕ∗2i(r1σ)]=Fǫcut(r2−r1). (A13)
EiX>0,σ
and using expression (A4) for the abnormal density, one obtains after integrating the kinetic-energy term by parts:
~2
0 = −Z d3r1d3r2δ(r2−r1)(cid:20)h˜(r2)[Fǫcut(r2−r1)−2ρ(r2,r1)]+(cid:18)2M∗∇r2∇r1 +U(r2)−µ(cid:19)2ρ˜(r1,r2)(cid:21)=
~2 1
= − d3rd3xδ(x) ˜h(r)[F (x)−2ρ(r,r)]+ ∇2−∇2 +U(r)−µ 2ρ˜(r−x/2,r+x/2) ,(A14)
Z (cid:20) ǫcut (cid:18)2M∗ (cid:18)4 r x(cid:19) (cid:19) (cid:21)
where Whenthesummationoverpositivequasiparticleenergies
r = r1+r2, (A15) itshaexttended to infinity, the completeness relation implies
2
x = r2−r1, (A16)
Fǫcut(r2−r1)=δ(r2−r1), (A19)
and
ρ(r2,r1) = ρ(r2σ,r1σ), (A17) and the only term in Eq. (A14) capable of canceling out
Xσ this singularity is ∇2xρ˜(r−x/2,r+x/2). Therefore, the
ρ˜(r2,r1) = ρ˜(r2σ,r1σ). (A18) Laplacianofthe abnormaldensity∇2xρ˜(r−x/2,r+x/2)
must be singular at x = 0. Moreover, using the expres-
Xσ
10
sion
1
∇2 =−4πδ(r), (A20)
|r|
it is clear that due to the zero-range pairing interaction
abnormal density ρ˜has an ultraviolet 1/x divergence:
h˜(r)M∗(r)
ρ˜(r−x/2,r+x/2)∼− . (A21)
4π~2|x| (cid:12)(cid:12)
(cid:12)x→0
(cid:12)
(cid:12)
[1] P.HohenbergandW.Kohn,Phys.Rev.136,B864(1964) (1999)
[2] W.Kohn and L.J. Sham,Phys. Rev.140, A1133 (1965) [16] G. Bruun, Y. Castin, R. Dum, and K. Burnett,
[3] T.H.R. Skyrme,Phil. Mag. 1, 1043 (1956) Eur. Phys. J. D 7, 433 (1999)
[4] D.Vauterinand D.MBrink,Phys.Rev.C 5,626 (1972) [17] A. Bulgac, nucl-th/9907088
[5] E. Perlin´ska, S.G. Rohozinski, J. Dobaczewski, and [18] A. Bulgac and Y. Yu,nucl-th/0109083
W. Nazarewicz, Phys. Rev.C 69, 014316 (2004) [19] A.BulgacandY.Yu,Phys.Rev.Lett.88,042504(2002)
[6] M. Samyn, S. Goriely, P.-H. Heenen, J.M. Pearson, and [20] A. Bulgac, Phys. Rev.C 65, 051305(R) (2002)
F. Tondeur,Nucl. Phys.A 700, 142 (2002) [21] Y.YuandA.Bulgac,Phys.Rev.Lett.90,222501(2003)
[7] S. Goriely, M. Samyn, P.-H. Heenen, J.M. Pearson, and [22] A.BulgacandY.Yu,Int.J.Mod.Phys.E13,147(2004)
F. Tondeur,Phys. Rev.C 66, 024326–1 (2002) [23] T. Nikˇsi´c, P. Ring, and D. Vretenar, Phys. Rev. C 71,
[8] J. Dobaczewski, W. Nazarewicz, T.R. Werner, J.F. 044320 (2005)
Berger, C.R. Chinn, and J. Decharg´e, Phys. Rev. C 53, [24] M. Grasso and M. Urban, Phys. Rev. A 68, 033610
2809 (1996) (2003)
[9] J. Dobaczewski, W. Nazarewicz, and P.-G. Reinhard, [25] P.RingandP.Schuck,TheNuclearMany-BodyProblem.
Nucl.Phys. A 693, 361 (2001) Springer-Verlag, New York (1980)
[10] J.Decharg´eandD.Gogny,Phys.Rev.C21,1568(1980) [26] M.V. Stoitsov, J. Dobaczewski, W. Nazarewicz, and
[11] R.R.Chasman, Phys.Rev.C 14, 1935 (1976) P. Ring, Comput. Phys. Commun. 167, 43 (2005)
[12] J. Dobaczewski, H. Flocard, and J. Treiner, Nucl. Phys. [27] K.BenneceurandJ.Dobaczewski,Comput.Phys.Com-
A 422, 103 (1984) mun. 168, 96 (2005)
[13] H. Esbensen, G.F. Bertsch, and K. Hencken, [28] E. Chabanat, P. Bonche, P. Haensel, J. Meyer, and
Phys.Rev.C 56, 3054 (1997) F. Schaeffer,Nucl. Phys.A 635, 231 (1998)
[14] M. Marini, F. Pistolesi, and G.C. Strinati, [29] J. Dobaczewski, W. Nazarewicz, and M.V. Stoitsov,
Eur. Phys.J. B 1, 151 (1998) Eur. Phys. J. A 15, 21 (2002)
[15] T.PapenbrockandG.F.Bertsch,Phys.Rev.C59,2052