Table Of ContentPairing versus quarteting coherence length
D.S. Delion 1,2,3 and V.V. Baran 1,4
1 ”Horia Hulubei” National Institute of Physics and Nuclear Engineering,
30 Reactorului, POB MG-6, RO-077125, Bucharest-Ma˘gurele, Romaˆnia
2 Academy of Romanian Scientists, 54 Splaiul Independen¸tei RO-050085, Bucharest, Romaˆnia
3 Bioterra University, 81 Gˆarlei RO-013724, Bucharest, Romaˆnia
4 Department of Physics, University of Bucharest, 405 Atomi¸stilor,
POB MG-11, RO-077125, Bucharest-Ma˘gurele, Romaˆnia
(Dated: January 22, 2015)
Wesystematicallyanalyzethecoherencelengthineven-evennuclei. Thepairingcoherencelength
inthespin-singletchannelfortheeffectivedensitydependentdelta(DDD)andGaussianinteraction
isestimated. Weconsiderinourcalculationsboundstatesaswellasnarrowresonances. Itturnsout
5
that the pairing gaps given by the DDD interaction are similar to those of the Gaussian potential
1
if one renormalizes the radial width to the nuclear radius. The correlations induced by the pairing
0
interactionhaveinallconsideredcasesalongrangecharacterinsidethenucleusanddecreasetowards
2
the surface. The mean coherence length is larger than the geometrical radius for light nuclei and
n approaches this value for heavy nuclei. The effect of the temperature and states in continuum is
a investigated. Strong shell effects are evidenced, especially for protons. We generalize this concept
J
to quartets by considering similar relations, but between proton and neutron pairs. The quartet
0 coherencelengthhasasimilarshape,butwithlargervaluesonthenuclearsurface. Weevidencethe
2 important role of proton-neutron correlations by estimating the so-called alpha coherence length,
which takes into account the overlap with theproton-neutron part of the α-particle wave function.
] It turns out that it does not depend on the nuclear size and has a value comparable to the free α-
h
particleradius. Wehaveshownthatpairingcorrelationsaremainlyconcentratedinsidethenucleus,
t
- while quarteting correlations are connected to thenuclear surface.
l
c
u PACSnumbers: 21.30.Fe,24.10.Cn,25.70.Ef
n Keywords: Coherence length, Density-dependent pairingpotential, Gaussianpairingpotential, Pairingcor-
[ relations,Quartetingcorrelations
1
v I. INTRODUCTION nucleus and decreases beyond the nuclear surface. This
4
picture of an extended di-nuclear cluster can be under-
8
stood in terms of the Pauli blocking, hindering nucleons
9 Theconceptofcoherencehasageneralcharacterbeing
to cluster together inside the nucleus and, therefore, the
4
connected to the linear superposition of quantum states.
0 clusterlosesbindingandbecomeslarger. Itisincontrast
Two-body coherence properties in nuclear structure are
. to the α-clustering phenomenon, which takes place in a
1 directly connected to the properties of low-lying collec-
narrowregionclosetothesurfacearea[11,12],beingcon-
0 tivestates. Collectiveexcitationsaremicroscopicallyde-
5 nected to the very large binding energy of an α-particle
scribedbyasuperpositionofcreationpairoperatorsact-
1 movinginalowdensityregion[13]. Thus,weexpectthat
ing on the ground state, described by a coherent state
: the corresponding correlation length estimated between
v within the Random Phase Approximation (RPA). The
proton and neutron pairs will have significantly smaller
i
X coherentstateinthiscontextisdefinedasanexponential value.
excitation of products between pair operators acting on
r
a the vacuum state [1]. It is well known that ground state The finiteness of nuclear systems also has important
properties of even-even nuclei are well reproduced by consequencesas far as thermal properties are concerned.
the pairing interaction [2–4]. The wave function within Pairingcorrelationsinfinitenucleidonotvanishatsome
theBardeen-Cooper-Schrieffer(BCS)pairingapproachis critical temperature, but they slowly decrease over sev-
also of a coherent type, i.e. an exponential excitation of eral MeV [14, 15]. This can be theoretically obtained by
the pair creation operators acting on the vacuum state. projecting the particle number in the BCS theory [16].
However,hintsaboutsuchbehaviourcanbeextractedin
The spatial distribution of the two-particle density
the unprojected BCS approach from the spatial proper-
is very important in understanding nuclear correlations
ties of the correlations.
[5, 6]. In particular, in Ref. [7] it was analyzed the rela-
tionbetween coherenceandchaoticproperties ofthe nu- In this paper we will perform a systematic analysis of
clearpairing. Thecoherencepropertyischaracterizedby thepairingcoherencelengthandacomparisontothesim-
the so-called coherence length, defined as the root mean ilar quantity defined for quartets. In Section II we give
squaredistanceaveragedoverthedensity. Forsuperfluid the necessary theoretical background concerning pair-
nuclei this average is usually performed over the pairing ing equations containing resonant states and coherence
density. InRefs. [8–10]itwasshownthatthisquantityis length. In Section III we perform a systematic analysis
relativelylarge,comparabletothenuclearsizeinsidethe of the coherence length and in the last Section we draw
2
Conclusions. the operator κˆ, connecting N + 2 with N systems, is
called pairing density operator. In this case the expan-
sion coefficient
II. THEORETICAL BACKGROUND 1
X = BCS aˆ† aˆ† BCS
ǫ 2h N+2| ǫ⊗ ǫ 0| Ni
II.1. Pairing equations
√2j+1 (cid:2) (cid:3)
= x , (2.3)
ǫ
2
In order to investigate two-body correlations we ex-
is given in terms of BCS amplitudes as follows
pandthe wavefunctionofN+2particlesintermsofthe
wave function of N particles as follows
x u(N+2)v(N) u(N+2)u(N)+v(N+2)v(N)
ǫ ≡ ǫ ǫ k k k k
|ΨN+2i=κˆ|ΨNi=Xǫ Xǫ(cid:2)aˆ†ǫ ⊗aˆ†ǫ(cid:3)0|ΨNi . (2.1) ≈u(ǫN)vǫ(N) ≈kYu6=(ǫǫNh+2)vǫ(N+2) . i(2.4)
Wewillconsiderinourcalculationsthesphericalapprox- We will consider in our basis bound sp states with neg-
imation. Thus, the operator aˆ† creates a single particle ative energy, as well as relatively narrow sp resonances
ǫ
(sp) eigenstate of the sphericalmean field potential with with positive energy. Relatively narrow resonances are
standard quantum numbers ǫ (ǫlj). In the configura- similar to bound states and can be normalized to unity
≡
tion representation one has inthe internalregion,butatlargedistancesthey behave
like outgoing waves
hr,s|aˆ†ǫm|0i≡ψǫm(r,s)= ϕǫ(r)⊗χ12(s) jm H(+)(r) G (r)+iF(r)
ϕ (r)=ϕ (r)ilY (rˆ)h fǫ(r)ilY (riˆ) , (2.2) ϕǫ(r)→r→∞ Mǫ l r ≡Mǫ l r l ,(2.5)
ǫµ ǫ lµ lµ
≡ r
in terms of spherical Hankel functions for neutrons and
where ϕǫ(r) is the radial wave function and the rest of Coulomb-Hankel functions for protons. The coefficients
the notation is standard. M are called scattering amplitudes and their squared
ǫ
The operator κˆ in Eq. (2.1) is called within the decay values are proportionalto sp partial decay widths.
theory two-particle formation amplitude. In the absence The states in continuum play an important role on
of two-body correlations, when the wave functions are pairingcorrelations,especiallyfornucleiclosetothedrip
Slater determinants, this relation is nothing else than lines [6, 17–21]. For nuclear structure calculations the
the Laplace expansion of the (N+2) (N+2) normalized background contribution is not relevant and only rela-
×
determinant in terms of N N times 2 2 normalized de- tively narrow resonant states are important [22, 23]. A
× ×
terminants. very good approximation for BCS calculations is to ne-
The most important two-body correlation beyond the glect the finite resonance width, i.e. to treat the reso-
mean field in even-evennuclei is given by the pairing in- nances as bound-like states [24]. We label bound states
teraction. Wewilldescribesuchsystemswithinthestan- by a and resonanceswith positive energy by r. We treat
dardBCSapproach,wheretheaveragedparticlenumber protonandneutronpairingseparately;foragivenisospin
is conserved,separately for protons and neutrons. Thus, index the generalized system of BCS equations for gap
both wave functions in Eq. (2.1) have a BCS ansatz and parameters ∆ , ∆ and number of particles N is
a r
1 ∆a′ 1 ∆r
∆a = ja′ + Va,a′ + jr + Va,r ,
Xa′ (cid:18) 2(cid:19) 2 (ǫa′ −λ)2+∆2a′ Xr (cid:18) 2(cid:19) 2 (ǫr −λ)2+∆2r
1 p ∆a′ 1 p ∆r′
∆r = ja′ + Vr,a′ + jr′ + Vr,r′ ,
Xa′ (cid:18) 2(cid:19) 2 (ǫa′ −λ)2+∆2a′ Xr′ (cid:18) 2(cid:19) 2 (ǫr′ −λ)2+∆2r′
1 p ǫ λ 1 p ǫ λ
a r
N = j + 1 − + (j + ) 1 − , (2.6)
a r
Xa (cid:18) 2(cid:19) − (ǫa−λ)2+∆2a! Xr 2 − (ǫr−λ)2+∆2r!
p p
where λ is the chemical potential and the potential ma- lous and normal densities, respectively
trix elements V are computed according to Eq. (2.6)
α,β
of [25]. βEǫ
a a =u v tanh
ǫ ǫ¯ ǫ ǫ
h i 2
We will investigate pairing in excited nuclei by us-
ing the temperature-dependent equations with anoma- ha†ǫaǫi=vǫ2+(u2ǫ −vǫ2)/(eβEǫ +1) . (2.7)
3
II.2. Pairing coherence length The coherence length is defined as follows
The two-body operator entering the pairing density I(2)(R)
ξ(R)=
(2.1) can be written in the configuration representation. sI(1)(R)
By using the recoupling from j-j to the L-S scheme, one
∞
obtains spin-singlet and spin-triplet components. Our
dr r2 w(r,R) , (2.16)
calculations have shown that the largest contribution is ≡sZ0
given by the spin-singlet component, given the following
in terms of the integrals
expression
∞
κ(r1,r2)= zǫ[ϕǫ(r1)⊗ϕǫ(r2)]0 I(p)(R)≡ dr r2p κ¯2(r,R)
Xǫ Z0
= zǫfǫ(r1)fǫ(r2) l(cosθ) , (2.8) = Φ(N2λβ)(R)Φ(N2β′λ)(R)
ǫ r1r2 Y λXNN′
X ∞
in terms of two-particle azimuthal harmonics × GnNλGn′N′λ dr r2pφ(nβλ/2)(r)φ(nβ′λ/2)(r) .
Yl(cosθ)= ilYl(rˆ1)⊗ilYl(rˆ2) 0 Xnn′ Z0 (2.17)
(cid:2)√2l+1 (cid:3)
= P(cosθ) , (2.9)
4π l Let us finally mention that the quantity xǫ, defined by
Eq. (2.4), is also called ”anomalous” density, while the
where θ is the angle between particle radii and the ex-
quantity
pansion coefficient is given by
y =v2 . (2.18)
1 11 1 1 ǫ ǫ
z =x j+ (ll)0, 0;0 l j, l j;0 ,
ǫ ǫ
r 2h (cid:18)22(cid:19) |(cid:18) 2(cid:19) (cid:18) 2(cid:19) i is called ”normal” density. Therefore κ, defined by Eq.
(2.10) (2.8), canbe called”anomalous”coherencelength, while
asimilarquantityκ definedby usingthe ”normal”den-
in terms of LS-jj recoupling brackets. By expanding the 0
sity is called ”normal” coherence length.
sp wave function with respect to the harmonic oscillator
(ho) basis
ϕ (r)= c φ(β)(r) II.3. Quarteting correlations
ǫµ nǫ nlµ
n
X
φ(β)(r)=φ(β)(r)ilY (rˆ) , (2.11) We will investigate quarteting correlations in medium
nlµ nl lµ and heavy α-decaying nuclei, where the valence protons
where β = M ω/~ is the standard ho parameter, and andneutronsoccupydifferentmajorshells. Thestandard
N
by using the Talmi-Moshinski transformation to rela- assumptionto build a quartetfrom two protonsand two
tive r=r1 r2 and center of mass (c.o.m.) coordinate neutronsinsuchnucleiistoconsiderprotonandneutron
R=(r1+r−2)/2 one obtains the following expansion pairingseparately[26,27]. ThereforethesystemofNπ+
2,N +2 nucleons can be expressed in terms of N ,N
ν π ν
κ(r,R,θ)= fλ(r,R) λ(cosθ) , (2.12) nucleons in a factorized way as follows
Y
λ
X
Ψ =κˆ κˆ Ψ , (2.19)
with expansion coefficients given by | Nπ+2,Nν+2i π ν| NπNνi
f (r,R)= φ(β/2)(r) Φ(2β)(R) , (2.13) where κτ is defined by Eq. (2.1). Thus, the quartet
λ GnNλ nλ Nλ wave function is a product between proton and neutron
nN
X two-body wave functions (2.12). Anyway, calculations
where
in infinite nuclear matter suggest that α-clusters can oc-
z c c nλNλ;0n ln l;0 . cur only at relative small nuclear densities compared to
GnNλ ≡ ǫ n1ǫ n2ǫh | 1 2 i
the equilibrium value and the proton-neutron correla-
Xǫ nX1n2
tions play an important role [13]. Thus, an α-particle
(2.14)
can be formed only in the surface region where the nu-
Here,thebracketdenotesthestandardTalmi-Moshinsky clear density diminishes and proton-neutroncorrelations
recoupling coefficient. By averaging over the angle θ we become relevant. This situation can be simulated by a
get proper modification of the single particle mean field by
1 1 adding a gaussian interaction in the surface region [28]
κ¯2(r,R)= κ2(r,R,θ)dcosθ and still by keeping the factorized ansatz (2.19). This
2
Z−1 can expain why an α-particle can be formed from two
1
= f2(r,R) . (2.15) protons and two neutrons lying in different major shells.
(4π)2 λ This additional ansatz of the single particle mean field
λ
X
4
was recently confirmed by microscopic calculations [29] in terms of Moshinsky brackets and the proton/neutron
and fission-like theory [30]. Anyway, this modification monopole G-coefficients (2.14)
is important in order to reproduce the absolute value of
G (N )= φ(β/2)(r )φ(βα/2)(r )
the half-life, but has a minor influence on the coherence τ τ GnτNτ0h nτ0 τ | 00 τ i
length. Xnτ
In order to describe quartets we introduce the relative τ =π,ν . (2.24)
and c.o.m. coordinates for proton, neutron and proton- It does not depend on angles and therefore one can de-
neutron systems, respectively: fine the quarteting coherence length ξ (R ) without any
q α
additionalangularaverage(2.15)by usinginEqs. (2.16)
r +r
rπ =r1−r2 , Rπ = 1 2 2 and (2.17) the quarteting density squared κ2q(rα,Rα).
r +r
rν =r3 r4 , Rν = 3 4 B. Alpha coherence length
− 2
R +R
rα =Rπ Rν , Rα = π ν , (2.20) The next step is to consider proton-neutron correla-
− 2
tions. They are described by the corresponding part in
where we labeled by 1, 2 proton and by 3, 4 neutron the α-particle wave function (2.21) given by φ(βα)(r ).
00 α
coodinates. The internal α-particle wave function is In order to account for the narrow proton-neutron spa-
givenbythe productbetweenthelowestproton,neutron tial distribution in the free α-particle one defines the so-
and proton-neutron ho orbitals called alpha coherence length ξ (R ) by using the alpha
α α
density
ψ =φ(βα/2)(r )φ(βα/2)(r )φ(βα)(r ) , (2.21)
α 00 π 00 ν 00 α κ (r ,R )=κ (r ,R )φ(βα)(r ) , (2.25)
α α α q α α 00 α
whereβ 0.5fm−2 isthe freeα-particlehoparameter
α in performing the integrals (2.17).
≈
measured by electron scattering experiments [26]. This
Let us finally mention that the integral of the alpha
parameter is about 2-3 times larger than the similar sp
density over the relative proton-neutron coordinate
ho parameter in heavy α-emitters, due to the fact that
∞
α-particle is a very bound object. (R )= κ (r ,R )r2dr , (2.26)
We will describe quarteting correlations between pro- F α Z0 α α α α α
tonandneutronpairsbyoverlappingtherelativecoordi- defines the formationamplitude and its square describes
nates to the corresponding components of the α-particle the probability to find an α-particle in the quartet wave
wave function (2.21). We will proceed in two steps. function [12, 26].
A. Quarteting correlation length (cid:7)
(cid:30)(cid:23)(cid:31) (cid:31)(cid:14)(cid:12)(cid:22)(cid:16)(cid:7)(cid:27)!(cid:23) (cid:9)(cid:10)(cid:11)(cid:12)(cid:12)(cid:13)(cid:10)(cid:14)(cid:15)(cid:16)(cid:17)(cid:1)(cid:18)(cid:6)(cid:7)(cid:2)(cid:7)(cid:16)(cid:19)(cid:20)(cid:21)(cid:16)(cid:22)(cid:16)(cid:23)(cid:1)(cid:16)(cid:18)(cid:16)(cid:5)(cid:16)(cid:24)(cid:25)
(cid:6)(cid:2)(cid:3) (cid:26)(cid:26)(cid:26)(cid:15)(cid:16)(cid:11)(cid:1)(cid:18)(cid:16)(cid:27)(cid:4)(cid:28)(cid:2)(cid:7)(cid:16)(cid:19)(cid:20)(cid:21)(cid:16)(cid:24)(cid:25)(cid:6)
(cid:9)(cid:10)(cid:11)(cid:12)(cid:12)(cid:13)(cid:10)(cid:14)(cid:15)(cid:16)(cid:17)(cid:1)(cid:18)(cid:4)(cid:3)(cid:2)(cid:29)(cid:16)(cid:19)(cid:20)(cid:21)(cid:16)(cid:22)(cid:16)(cid:23)(cid:1)(cid:16)(cid:18)(cid:16)(cid:7)(cid:2)(cid:3)(cid:16)(cid:24)(cid:25)
Let us first consider only the overlap with respect to
(cid:6)
protonand neutronrelative coordinatesr , r , by keep-
π ν
ingfreetheinternalproton-neutroncoordinaterα. Thus, (cid:5)(cid:2)(cid:3)
we consider independent from each other proton and (cid:6)(cid:7)
neutron pairs by neglecting proton-neutron correlations. (cid:4)(cid:5) (cid:5) (cid:2)
(cid:1)(cid:2)(cid:3)
Therefore we can define the quarteting density in anal- (cid:4)(cid:2)(cid:3)
ogy to the pairing density, but between the proton and
(cid:4)
neutron pairs (instead of fermions):
(cid:1)(cid:2)(cid:3)
κq(Rπ,Rν)=hκπ(r1,r2)|φ(0β0α/2)(rπ)i (cid:1) "
×hκν(r3,r4)|φ(0β0α/2)(rν)i . (2.22) (cid:8)(cid:6)(cid:1) (cid:8)(cid:5)(cid:3) (cid:8)(cid:5)(cid:1) (cid:8)(cid:4)(cid:3) (cid:8)(cid:2)(cid:3)(cid:4)(cid:8)(cid:4)(cid:5)(cid:1)(cid:6)(cid:7) (cid:8)(cid:3) (cid:1) (cid:3) (cid:4)(cid:1)
FIG.1. ParinggapdefinedbythefirsttwolinesofEq. (2.6)
By recoupling the product between proton and neutron versus ǫ in 48Cr for DDD potential (squares) and Gaussian
pairs (2.12) to the relative and c.o.m. pair coordinates
potentials with r0= 2 fm (circles) and r0 =RN (diamonds).
one obtains for the leading monopole componentthe fol-
lowing relation
κ (r ,R ) κ(0)(r ,R )= G (N ) G (N )
q α α ≈ q α α π π ν ν III. NUMERICAL APPLICATION
NXπ,Nν
n 0N 0;0N 0N 0;0 φ(β) (r )φ(4β)(R ) ,
× h α α | π ν i nα0 α Nα0 α We analyzed all even-even nuclei with 20 < Z < 100
Xnα andknownexperimentalpairinggaps,determinedbythe
(2.23) binding energies of neighbouring nuclei [31].
5
TABLEI.Proton quantumnumbers,sp spectrum,decay widthsandgap parameters fortheGaussian, renormalized Gaussian
and DDD interactions in 48Cr, given by the diagonalization of the Woods-Saxon mean field with universal parametrisation
[32].
No. l 2j ǫ (MeV) Γ (MeV) ∆2fm(MeV) ∆4.5fm(MeV) ∆DDD(MeV)
1 0 1 -28.911 - 3.114 1.354 0.724
2 1 3 -20.837 - 3.173 1.810 1.482
3 1 1 -18.638 - 3.121 1.739 1.436
4 2 5 -12.118 - 2.908 2.131 2.387
5 0 1 -8.349 - 2.454 1.795 1.728
6 2 3 -7.488 - 2.886 2.047 2.351
7 3 7 -3.079 - .261 2.224 2.246
8 1 3 0.322 0.000 1.349 1.356 1.076
9 1 1 2.403 0.046 1.149 1.133 0.962
10 3 5 4.101 0.024 2.114 2.003 2.139
11 4 9 5.874 0.055 1.389 1.893 0.996
Forthe nuclearmeanfieldwe useda standardWoods- It is defined by the following ansatz:
Saxon potential with universalparametrization[32]. We
consideredinourspbasisallboundstatesandresonances v(r12)= v0e−[r12/r0]2 , (3.1)
−
incontinuumuptoe =10MeVwithaspdecaywidth
max
dependingontherelativeradiusr . Here,thewidthpa-
Γ 1MeV.AsanexamplewegiveinTableItheproton 12
sp≤spectrumfor48Cr. Here,therearegivenlevelnumber, rameter r0=2 fm corresponds to the spin-singlet ”bare”
value in the free space. The corresponding value of the
angularmomentum,twicethetotalspin,spenergy,decay
effective potential strength v is determined by the gap
width of sp states in continuum and pairing gaps for the 0
parameter at the Fermi level, which should be equal to
interactions considered below.
the experimental value.
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)
(cid:5)(cid:5)(cid:2)(cid:2)(cid:3)(cid:3) (cid:5)(cid:2)(cid:3)
(cid:9)(cid:9)(cid:10)(cid:10)(cid:11)(cid:11)(cid:12)(cid:12)(cid:12)(cid:12)(cid:13)(cid:13)(cid:14)(cid:14)(cid:15)(cid:15)(cid:1)(cid:1)(cid:16)(cid:16)(cid:14)(cid:14)(cid:6)(cid:5)(cid:2)(cid:14)(cid:17)(cid:3)(cid:18)(cid:14)(cid:17)(cid:18) (cid:9)(cid:9)(cid:10)(cid:10)(cid:11)(cid:11)(cid:12)(cid:12)(cid:12)(cid:12)(cid:13)(cid:13)(cid:14)(cid:14)(cid:15)(cid:15)(cid:1)(cid:1)(cid:16)(cid:16)(cid:14)(cid:14)(cid:6)(cid:5)(cid:2)(cid:14)(cid:17)(cid:3)(cid:18)(cid:14)(cid:17)(cid:18) II. Density dependent delta (DDD) interaction
(cid:19)(cid:19)(cid:19)(cid:13)(cid:14)(cid:20)(cid:16)(cid:4)(cid:2)(cid:1) (cid:19)(cid:19)(cid:19)(cid:13)(cid:14)(cid:20)(cid:16)(cid:4)(cid:2)(cid:1)
(cid:5)(cid:5) (cid:5)
(cid:21)(cid:15)(cid:22)(cid:23)(cid:22)(cid:24)(cid:12)(cid:13)(cid:14)(cid:6)(cid:8)(cid:25)(cid:15) (cid:21)(cid:15)(cid:22)(cid:23)(cid:22)(cid:24)(cid:12)(cid:13)(cid:14)(cid:6)(cid:8)(cid:25)(cid:15) It is known that the strength of the effective pairing
interactiondependsuponthelocaldensity[17,18],given
(cid:4)(cid:4)(cid:2)(cid:2)(cid:3)(cid:3) (cid:4)(cid:2)(cid:3)
by the following phenomenologicalansatz [19]
(cid:6)(cid:7)(cid:8)(cid:7)(cid:9)(cid:10) (cid:7)(cid:6)(cid:7)(cid:8)(cid:7)(cid:9)(cid:10) (cid:7)
(cid:4)(cid:4) (cid:4) ρ (r) γ
v(r,r′)=u δ(r r′) 1 X N , (3.2)
0 − ( − " ρ(0) # )
(cid:1)(cid:1)(cid:2)(cid:2)(cid:3)(cid:3) (cid:1)(cid:2)(cid:3) N
in terms of the nuclear density ρ . The value X=1 cor-
(cid:26)(cid:10)(cid:27)(cid:14)(cid:24)(cid:22)(cid:15)(cid:18)(cid:10)(cid:28)(cid:14)(cid:29)(cid:30)(cid:24)(cid:12)(cid:31)(cid:23) (cid:26)!(cid:27)(cid:14)(cid:10)(cid:24)(cid:22)(cid:18)(cid:10)(cid:28)(cid:22)(cid:11)(cid:12)(cid:14)(cid:29)(cid:30)(cid:24)(cid:12)(cid:31)(cid:23) N
responds to the surface DDD interaction.
(cid:1)(cid:1) (cid:1)
(cid:1)(cid:1) (cid:1)(cid:1)(cid:2)(cid:2)(cid:5)(cid:5) (cid:1)(cid:1)(cid:2)(cid:2)(cid:6)(cid:6) (cid:1)(cid:1)(cid:2)(cid:2)(cid:7)(cid:7) (cid:1)(cid:1)(cid:2)(cid:2)(cid:8)(cid:8) (cid:4)(cid:4) (cid:4)(cid:4)(cid:2)(cid:2)(cid:5)(cid:5) (cid:1) (cid:1)(cid:2)(cid:5) (cid:1)(cid:2)(cid:6) (cid:1)(cid:2)(cid:7) (cid:1)(cid:2)(cid:8) (cid:4) (cid:4)(cid:2)(cid:5)
(cid:9)(cid:7)(cid:8)(cid:7)(cid:9)(cid:10) (cid:9)(cid:7)(cid:8)(cid:7)(cid:9)(cid:10) As an example, in Fig. 1 we plotted the pairing gap
FIG. 2. Proton coherence length divided by geometrical (2.6) versus sp energy for 48Cr, given in Table I. Here,
radius versus c.o.m. radius in 48Cr computed with ”normal” circlescorrespondto the Gaussianinteractioninthe free
(a) and ”anomalous” densities (b) for DDD potential (solid space with r =2 fm. Notice large values for states below
0
line) and Gaussian potentials with r = 2 fm (long dashes)
0 theFermilevel. ThegapsgivenbyDDDinteractionwith
and r0 =RN (short dashes). X =γ=1areplottedbysquaresandthevaluesbelowthe
Fermi level are significantly smaller that the Fermi gap.
It is interesting to point out that a very similar be-
We solvedthe BCS equations (2.6) separately for pro- haviour has the Gaussian interaction where the width
tonsandneutronswithtwowidelyusedtypesofnucleon- parameteris renormalizedto the geometricalnuclear ra-
nucleon pairing interactions: dius (in fm) r = R = 1.2A1/3. A di-nuclear cluster
0 N
inside nuclear matter has different properties with re-
I. Gaussian interaction spect to the free space. It considerably loses the binding
6
(cid:1)(cid:2)(cid:7) (cid:1)(cid:2)(cid:7)
(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:13)(cid:15)(cid:21)(cid:17)(cid:22)(cid:23)(cid:18)(cid:24)(cid:25)(cid:26)(cid:27) (cid:10)(cid:11)(cid:6)(cid:2)(cid:7)(cid:12)(cid:13) (cid:14)(cid:31)(cid:16)(cid:17)(cid:15)(cid:18)(cid:19)(cid:13)(cid:15)(cid:21)(cid:19) (cid:24)(cid:17)(cid:22)(cid:23)(cid:18)(cid:24)(cid:25)(cid:26)(cid:27) (cid:10)(cid:11)(cid:6)(cid:2)(cid:7)(cid:12)(cid:13) (cid:6)(cid:1) (cid:9)(cid:28)(cid:11)(cid:12)(cid:29)(cid:17)(cid:18)(cid:16)(cid:18)(cid:13)(cid:19) (cid:6)(cid:1) (cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:13)(cid:19)
(cid:10)(cid:11)(cid:4)(cid:2)(cid:7)(cid:12)(cid:13) (cid:10)(cid:11)(cid:4)(cid:2)(cid:7)(cid:12)(cid:13)
(cid:10)(cid:11)(cid:1)(cid:2)(cid:1)(cid:12)(cid:13) (cid:10)(cid:11)(cid:1)(cid:2)(cid:1)(cid:12)(cid:13)
(cid:1)(cid:2)(cid:6) (cid:1)(cid:2)(cid:6)
(cid:28)(cid:20)(cid:19)(cid:26)(cid:19)(cid:18)(cid:24)(cid:29)(cid:17)(cid:6)(cid:9)(cid:30)(cid:20) (cid:28)(cid:20)(cid:19)(cid:26)(cid:19)(cid:18)(cid:24)(cid:29)(cid:17)(cid:6)(cid:9)(cid:30)(cid:20) (cid:5) (cid:5)
(cid:7)(cid:8)(cid:1)(cid:3)(cid:4)(cid:5)(cid:6)(cid:9)(cid:2)(cid:1)(cid:1)(cid:2)(cid:2)(cid:4)(cid:5) (cid:3)(cid:7)(cid:8)(cid:1)(cid:3)(cid:4)(cid:5)(cid:6)(cid:9)(cid:1)(cid:1)(cid:2)(cid:2)(cid:4)(cid:5) (cid:1)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:9) (cid:4) (cid:1)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:2) (cid:4)
(cid:3) (cid:3)
(cid:1)(cid:2)(cid:3) (cid:1)(cid:2)(cid:3)
(cid:2) (cid:20)(cid:21)(cid:1)(cid:22)(cid:27)(cid:4)(cid:12)(cid:23)(cid:14)(cid:24) (cid:2) (cid:20)(cid:21)(cid:1)(cid:22)(cid:5)(cid:5)(cid:12)(cid:23)(cid:14)(cid:24)
(cid:1)(cid:1) (cid:4) (cid:6) (cid:8) (cid:9) (cid:3)(cid:1) (cid:1)(cid:1) (cid:4) (cid:6) (cid:8) (cid:9) (cid:3)(cid:1) (cid:20)(cid:20)(cid:21)(cid:21)(cid:12)(cid:25)(cid:1)(cid:2)(cid:12)(cid:22)(cid:23)(cid:27)(cid:4)(cid:14)(cid:12)(cid:24)(cid:23)(cid:14)(cid:24) (cid:20)(cid:20)(cid:21)(cid:21)(cid:12)(cid:25)(cid:1)(cid:6)(cid:12)(cid:7)(cid:23)(cid:22)(cid:26)(cid:14)(cid:8)(cid:24)(cid:12)(cid:23)(cid:14)(cid:24)
(cid:10)(cid:3)(cid:4)(cid:5)(cid:6)(cid:9) (cid:10)(cid:4)(cid:5)(cid:6)(cid:9) (cid:1) (cid:1)
(cid:1) (cid:6) (cid:2) (cid:7) (cid:3) (cid:8) (cid:1) (cid:6) (cid:2) (cid:7) (cid:3) (cid:8)
(cid:8)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7) (cid:8)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)
FIG.3. Theintegrandoftheprotoncoherencelengthversus
the relative radius radius in 48Cr, computed with ”normal” FIG. 4. (a) Proton coherence length versus c.o.m. radius
(a) and ”anomalous” densities (b) for different c.o.m. radii. for different chemical potentials λ=-2.96 MeV (solid line), 0
Here, we used the Gaussian interaction withn r0=2fm. MeV (long dashes) and 0.96 MeV (short dashes) in 48Cr.
(b) Neutron coherence length versus c.o.m. radius for dif-
ferent chemical potentials λ=-13.75 MeV (solid line), 0 MeV
(long dashes) and 0.88 MeV (short dashes) in 48Cr.
property due to the Pauliblocking, becoming larger and
therefore the effective pairing interaction has a more ex-
tended shape. Above the Fermi sea we obtained similar
(cid:6)(cid:2) (cid:6)(cid:2)
values in all cases. (cid:8)(cid:3)(cid:4)(cid:2)(cid:9)(cid:9)(cid:10)(cid:10)(cid:9)(cid:9)(cid:1)(cid:8)(cid:2)(cid:4)(cid:2) (cid:20)(cid:2)(cid:15)(cid:25)(cid:15)(cid:4)(cid:15)(cid:26)(cid:27) (cid:8)(cid:3)(cid:4)(cid:2)(cid:9)(cid:9)(cid:10)(cid:10)(cid:11)(cid:11)(cid:1)(cid:8)(cid:2)(cid:4)(cid:2) (cid:20)(cid:2)(cid:15)(cid:25)(cid:15)(cid:4)(cid:15)(cid:26)(cid:27)
In Fig. 2 (a) we plotted the proton coherence length (cid:5)(cid:3) (cid:4)(cid:4)(cid:8)(cid:4)(cid:9)(cid:9)(cid:10)(cid:10)(cid:9)(cid:9)(cid:3)(cid:4)(cid:2)(cid:8) (cid:5)(cid:3) (cid:4)(cid:4)(cid:8)(cid:4)(cid:9)(cid:11)(cid:10)(cid:10)(cid:11)(cid:11)(cid:3)(cid:4)(cid:2)(cid:8)
given by Eq. (2.16) divided by the nuclear radius R , (cid:12)(cid:23)(cid:14)(cid:15)(cid:24)(cid:20)(cid:21)(cid:19)(cid:21)(cid:16)(cid:22) (cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:21)(cid:16)(cid:22)
N (cid:5)(cid:2) (cid:5)(cid:2)
arasdaiusfuRnc/tRioNn ionf4t8hCer.raHtieorebewtweeuesnedc.toh.em”.noarnmdanl”ucdleeanr- (cid:1)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:2)(cid:4)(cid:3) (cid:3)(cid:1)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:2)(cid:4)(cid:3) (cid:3)
sitywhileinFig. 2(b)weusedthe”anomalous”density.
(cid:4)(cid:2) (cid:4)(cid:2)
Notice that that all cases, plotted by different symbols
explainedincaption,haveverysimilarshapes. Thus,the (cid:1)(cid:3) (cid:1)(cid:3)
coherencelengthisnotsensitivetotheradialshapeofthe (cid:1)(cid:2)(cid:4)(cid:2) (cid:6)(cid:2) (cid:7)(cid:2) (cid:8)(cid:2) (cid:1)(cid:2)(cid:2) (cid:1)(cid:4)(cid:2) (cid:1)(cid:6)(cid:2) (cid:1)(cid:7)(cid:2) (cid:1)(cid:2)(cid:4)(cid:2) (cid:6)(cid:2) (cid:7)(cid:2) (cid:8)(cid:2) (cid:1)(cid:2)(cid:2) (cid:1)(cid:4)(cid:2) (cid:1)(cid:6)(cid:2) (cid:1)(cid:7)(cid:2)
interaction. The ”normal” coherence length is equal to (cid:9) (cid:9)
the nuclear radius in the internal region and diminishes
by a factor 0.5 on the surface. The ”anomalous” coher- FIG. 5. Strength parameter of the Gaussian interaction,
ence length has a similar shape, but with twice larger correspondingtor0=2fm,versusneutronnumberforproton
(a) and neutron systems (b).
internal value. This picture is very different from the
dependence of the two-body wave function versus c.o.m.
radius, which is peaked on the nuclear surface [25].
In orderto better understand the behaviourof the co-
herence length we plotted in Fig. 3 (a) the integrand of Gogny force for Ni isotopes. The shape is similar, pre-
the ”normal” correlation length w (r,R), given by the dicting a mean coherence length of about 6 fm in the
0
second line of Eq. (2.16), versus the relative radius r for internalregionanddecreasingasone approachesthe nu-
three values of the c.o.m. radius R=4.5 fm (solid line), clear surface and reaching the value of 2 fm just outside
2 fm (long dashes) and 0 fm (short dashes). Here, we the nucleus.
usedthe ”bare”versionofthe Gaussianinteraction. No- Mostoftheexoticnucleiclosetothedriplineshavethe
tice that the three curves have a similar shape, strongly lastnucleonincontinuum. Thereforeweinvestigatedthe
peakedaround2fm. Weobtaincompletelydifferentplots dependenceofthecoherencelengthontheFermilevel,by
for the integrand of the ”anomalous” coherence length changingtherealpartoftheWoods-Saxonpotential. We
w(r,R). They are given in Fig. 3 (b). The distribution plotted in Fig. 4 (a) the proton coherence length versus
corresponding to the c.o.m. radius on surface R=4.5 fm c.o.m. radiusin48Cr,fordifferentvalues ofthe chemical
(solid line) is peaked around the free singlet value of the potential. One sees that it increases by increasing the
Gaussian width i.e. r=2 fm. On the contrary, the dis- chemical potential. This effect is stronger for neutrons,
tribution corresponding to a smaller radius R=2.5 (long as seen in Fig. 4 (b), due to the absence of the Coulomb
dashes) is peaked around a much larger value r=7 fm. barrier. Therefore,inexotic nucleicloseto driplines the
Our conclusions are in agreementwith Ref. [8], where nucleons become more correlated.
inFig. 5the”anomalous”coherencelengthofthepairing Then we performed a systematic analysis of the
interaction was estimated within the more sophisticated ”anomalous” coherence length (by simply calling it co-
Hartree-Fock-Bogoljubov (HFB) approach, by using the herence length) for even-even nuclei with 20<Z <100.
7
(cid:5)(cid:5)(cid:2)(cid:2)(cid:8)(cid:8) (cid:5)(cid:5)(cid:2)(cid:2)(cid:8)(cid:8) (cid:1)(cid:2)(cid:7) (cid:1)(cid:2)(cid:7)
(cid:5)(cid:5)(cid:2)(cid:2)(cid:7)(cid:7) (cid:1)(cid:7)(cid:2)(cid:2)(cid:2)(cid:6)(cid:1)(cid:2)(cid:3)(cid:3)(cid:3)(cid:3)(cid:4)(cid:4)(cid:4)(cid:4)(cid:3)(cid:3)(cid:3)(cid:3)(cid:5)(cid:1)(cid:7)(cid:2)(cid:6)(cid:2)(cid:6)(cid:1)(cid:6) (cid:13)(cid:6)(cid:11)(cid:18)(cid:11)(cid:2)(cid:11)(cid:19)(cid:20) (cid:5)(cid:5)(cid:2)(cid:2)(cid:7)(cid:7) (cid:1)(cid:7)(cid:2)(cid:2)(cid:2)(cid:6)(cid:1)(cid:2)(cid:3)(cid:3)(cid:3)(cid:21)(cid:4)(cid:4)(cid:4)(cid:4)(cid:21)(cid:21)(cid:21)(cid:21)(cid:5)(cid:1)(cid:7)(cid:2)(cid:6)(cid:2)(cid:6)(cid:1)(cid:6) (cid:13)(cid:6)(cid:11)(cid:18)(cid:11)(cid:2)(cid:11)(cid:19)(cid:20) (cid:1)(cid:2)(cid:6)(cid:4) (cid:11)(cid:24)(cid:22)(cid:25)(cid:13)(cid:14)(cid:26)(cid:23)(cid:14)(cid:19)(cid:27)(cid:20)(cid:14)(cid:18)(cid:9)(cid:20)(cid:2)(cid:15)(cid:10)(cid:21)(cid:14)(cid:28)(cid:8)(cid:29)(cid:4) (cid:1)(cid:2)(cid:6)(cid:4) (cid:11)(cid:24)(cid:12)(cid:25)(cid:13)(cid:26)(cid:14)(cid:15)(cid:14)(cid:16)(cid:27)(cid:17)(cid:14)(cid:9)(cid:18)(cid:19)(cid:2)(cid:1)(cid:20)(cid:14)(cid:15)(cid:28)(cid:21)(cid:14)(cid:8)(cid:29)(cid:10)
(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:14)(cid:16)(cid:17)
(cid:8)(cid:22)(cid:10)(cid:11)(cid:16)(cid:23)(cid:24)(cid:15)(cid:13)(cid:14)(cid:16)(cid:17)
(cid:5)(cid:5)(cid:2)(cid:2)(cid:6)(cid:6) (cid:5)(cid:5)(cid:2)(cid:2)(cid:6)(cid:6)
(cid:12)(cid:13)(cid:1)(cid:2)(cid:6) (cid:12)(cid:13)(cid:1)(cid:2)(cid:6)
(cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:2)(cid:5)(cid:2)(cid:6)(cid:2)(cid:7)(cid:5)(cid:5)(cid:2)(cid:2)(cid:5)(cid:5) (cid:2)(cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:2)(cid:5)(cid:2)(cid:6)(cid:2)(cid:7)(cid:5)(cid:5)(cid:2)(cid:2)(cid:5)(cid:5) (cid:2) (cid:1)(cid:2)(cid:3)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:4)(cid:5)(cid:1)(cid:2)(cid:5)(cid:4) (cid:6)(cid:1)(cid:2)(cid:3)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:4)(cid:5)(cid:1)(cid:2)(cid:5)(cid:4) (cid:6)
(cid:5)(cid:5) (cid:5)(cid:5)
(cid:1)(cid:2)(cid:5) (cid:1)(cid:2)(cid:5)
(cid:1)(cid:1)(cid:2)(cid:2)(cid:4)(cid:4) (cid:1)(cid:1)(cid:2)(cid:2)(cid:4)(cid:4)
(cid:1)(cid:1)(cid:2)(cid:2)(cid:3)(cid:3)(cid:6)(cid:6)(cid:1)(cid:1) (cid:8)(cid:8)(cid:1)(cid:1) (cid:10)(cid:10)(cid:1)(cid:1) (cid:3)(cid:3)(cid:1)(cid:1) (cid:9)(cid:9)(cid:5)(cid:5)(cid:1)(cid:1)(cid:1)(cid:1) (cid:5)(cid:5)(cid:6)(cid:6)(cid:1)(cid:1) (cid:5)(cid:5)(cid:8)(cid:8)(cid:1)(cid:1) (cid:5)(cid:5)(cid:10)(cid:10)(cid:1)(cid:1) (cid:1)(cid:1)(cid:2)(cid:2)(cid:3)(cid:3)(cid:6)(cid:6)(cid:1)(cid:1) (cid:8)(cid:8)(cid:1)(cid:1) (cid:10)(cid:10)(cid:1)(cid:1) (cid:3)(cid:3)(cid:1)(cid:1) (cid:7)(cid:5)(cid:5)(cid:1)(cid:1)(cid:1)(cid:1) (cid:5)(cid:5)(cid:6)(cid:6)(cid:1)(cid:1) (cid:5)(cid:5)(cid:8)(cid:8)(cid:1)(cid:1) (cid:5)(cid:5)(cid:10)(cid:10)(cid:1)(cid:1) (cid:1)(cid:2)(cid:3)(cid:4) (cid:8)(cid:2)(cid:3) (cid:8)(cid:2)(cid:6) (cid:1)(cid:2)(cid:9)(cid:3)(cid:4)(cid:5)(cid:14) (cid:9)(cid:2)(cid:9) (cid:9)(cid:2)(cid:10) (cid:1)(cid:2)(cid:3)(cid:4) (cid:8)(cid:2)(cid:3) (cid:8)(cid:2)(cid:6) (cid:1)(cid:2)(cid:9)(cid:3)(cid:4)(cid:5)(cid:14) (cid:9)(cid:2)(cid:9) (cid:9)(cid:2)(cid:10)
FIG.6. Ratiohξi/RN,correspondingtoaGaussian interac- FIG. 8. Logarithm of the ratio hξi/RN versus logarithm of
tionwithr =2fm,versusneutronnumberforproton(a)and themass numberfor protons (a) and neutrons(b).
0
neutron systems (b).
(cid:6)(cid:2)
(cid:5)(cid:2)(cid:8) (cid:5)(cid:2)(cid:8) (cid:7)(cid:8)(cid:1)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)
(cid:3)(cid:12)(cid:6)(cid:1)(cid:10)(cid:10)(cid:11)(cid:11)(cid:10)(cid:10)(cid:5)(cid:3)(cid:1)(cid:6)(cid:1) (cid:23)(cid:24)(cid:5) (cid:23)(cid:24)(cid:5) (cid:7)(cid:8)(cid:1)(cid:12)(cid:13)(cid:14)
(cid:5)(cid:2)(cid:7) (cid:6)(cid:3)(cid:10)(cid:11)(cid:10)(cid:12)(cid:1) (cid:5)(cid:2)(cid:7)
(cid:6)(cid:6)(cid:10)(cid:11)(cid:10)(cid:6)(cid:3)
(cid:6)(cid:1)
(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:19)(cid:21)(cid:22) (cid:15)(cid:16)(cid:17)(cid:18)(cid:17)(cid:19)(cid:20)(cid:21)(cid:22)(cid:2)(cid:2)(cid:1)(cid:23)(cid:24)
(cid:5)(cid:2)(cid:6) (cid:5)(cid:2)(cid:6)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:3)(cid:5)(cid:3)(cid:6)(cid:7)(cid:5)(cid:2)(cid:5) (cid:1)(cid:2)(cid:3)(cid:4)(cid:3)(cid:5)(cid:3)(cid:6)(cid:7)(cid:5)(cid:2)(cid:5) (cid:5) (cid:25)(cid:30)(cid:16)(cid:1)(cid:1)(cid:24)(cid:8)(cid:8)(cid:26)(cid:2)(cid:2)(cid:20)(cid:22)(cid:2)(cid:31)(cid:20) (cid:9)(cid:27)(cid:5)(cid:24)(cid:22)(cid:19)(cid:12)(cid:22)(cid:28)(cid:13)(cid:17)(cid:14)(cid:16)(cid:29)(cid:13)
(cid:5) (cid:5)
(cid:13)(cid:25)(cid:15)(cid:16)(cid:21)(cid:26)(cid:27)(cid:20)(cid:18)(cid:19)(cid:21)(cid:22)
(cid:1)(cid:2)(cid:4) (cid:1)(cid:2)(cid:4) (cid:3)(cid:6)(cid:10)(cid:11)(cid:10)(cid:5)(cid:1)(cid:1) (cid:5)(cid:6)
(cid:1)(cid:2)(cid:3) (cid:1)(cid:2)(cid:3) (cid:12)(cid:6)(cid:6)(cid:1)(cid:3)(cid:6)(cid:10)(cid:10)(cid:10)(cid:11)(cid:11)(cid:11)(cid:10)(cid:10)(cid:10)(cid:3)(cid:12)(cid:6)(cid:6)(cid:1)(cid:3) (cid:2)(cid:3)(cid:1)(cid:4) (cid:4) (cid:2)
(cid:6)(cid:1) (cid:8)(cid:1) (cid:9)(cid:1) (cid:3)(cid:1) (cid:5)(cid:1)(cid:1) (cid:5)(cid:6)(cid:1) (cid:5)(cid:8)(cid:1) (cid:5)(cid:9)(cid:1) (cid:6)(cid:1) (cid:8)(cid:1) (cid:9)(cid:1) (cid:3)(cid:1) (cid:5)(cid:1)(cid:1) (cid:5)(cid:6)(cid:1) (cid:5)(cid:8)(cid:1) (cid:5)(cid:9)(cid:1)
(cid:7) (cid:7)
(cid:3)
FIG.7. Ratiohξi/RN,correspondingtotheDDDinteraction
(3.2) with X =γ =1, versus neutron number for proton (a)
and neutron systems (b). (cid:2)
(cid:1)
(cid:1) (cid:2) (cid:3) (cid:4) (cid:5)
In Fig. 5 (a) we plotted the effective strength v as a
0
(cid:7)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)
function of the neutron number for protons correspond-
ing to Gaussian interaction with r =2 fm. The isotope
0
chains are connected by solid lines and magic numbers FIG. 9. Proton coherence length versus c.o.m. radius in
are indicated by vertical lines. Different regions are 220Ra computed for T=0 (solid line), T=0.5675 MeV . Tc
plotted by open squares (20 < Z < 28), filled squares (long dashes), for a Gaussian potential with r0= 2 fm.
(28 < Z < 50), open circles (50 < Z < 82) and filled
circles (82 < Z < 100). As a general trend we remark a
strongdecreasingbehaviourwiththeincreaseoftheneu-
tron number. We notice a remarkable feature, namely it for protons.
hasalmostthesinglet”bare”valueinthefreespacev Wetheninvestigatedthedensitydependentpairingin-
0
∼
35 MeV for very light nuclei. The strength strongly de- teraction given by Eq. (3.2) with X = γ=1 in Fig. 7.
creases up to v0 ∼ 20 MeV for heavy nuclei, except the It turns out that the ratio hξi/RN has similar gross fea-
regions around magic numbers. In Fig. 5 (b) we give a tures, but with more pronounced shell oscillations. The
similar plot for neutrons. Notice that in this case shell fact that the coherence length for neutrons is larger is
effects are stronger. confirmed. It is interesting to notice the linear correla-
In Fig. 6 (a) we analyzed the mean coherence length tion between log10 ξ and log10A, plotted in Fig. 8 for
h i
ξ for protons, corresponding to the Gaussian interac- the Gaussian pairing interaction with r0= 2 fm.
h i
tion with the free value of the width parameter r =2 fm In order to investigate the behaviour of the coherence
0
as a function of neutrons. The ratio of this quantity to lengthforexcitedstates,inFig. 9weanalyzedtheroleof
the nuclear radius decreases from 1.4 for light nuclei up thetemperature. Firstly,wegivefor220Rathecoherence
to around unity for heavy nuclei. In Fig. 6 (b) we give length versus the pair c.o.m. radius for T = 0 and just
similar results for neutrons. As a general trend, the co- belowthe’critical’temperatureT 0.57MeV(herethe
c
herence length is larger for neutrons due to the absence gap decreases below 10−3 MeV). T≈he pairing coherence
of the Coulombbarrier,but the shell effects are stronger length shows very little change in shape up to T . The
c
8
(cid:9) (cid:9) (cid:9)(cid:1) (cid:1)(cid:17)(cid:2)(cid:7)
(cid:8) (cid:8) (cid:12)(cid:13)(cid:15)(cid:14)(cid:12) (cid:3)(cid:3)(cid:1)(cid:10)(cid:11) (cid:8)(cid:1) (cid:10)(cid:10)(cid:11)(cid:11)(cid:12)(cid:12)(cid:3)(cid:2)(cid:13)(cid:13)(cid:14)(cid:15)(cid:14)(cid:2)(cid:1)(cid:1) (cid:1)(cid:17)(cid:2)(cid:5) (cid:10)(cid:10)(cid:22)(cid:22)(cid:12)(cid:12)(cid:3)(cid:2)(cid:13)(cid:13)(cid:14)(cid:15)(cid:14)(cid:6)(cid:1)
(cid:7)(cid:1) (cid:1)(cid:17)(cid:2)(cid:3)
(cid:7) (cid:7) (cid:12)(cid:16)(cid:13) (cid:12)(cid:23)(cid:13)
(cid:6)(cid:1) (cid:1)(cid:17)(cid:2)
(cid:6)(cid:7)(cid:6) (cid:6)(cid:7)(cid:6) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:1)(cid:5)(cid:1) (cid:1) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:1)(cid:1)(cid:17)(cid:1)(cid:9) (cid:1)
(cid:1)(cid:3)(cid:4)(cid:5)(cid:2)(cid:5) (cid:3)(cid:1)(cid:3)(cid:4)(cid:5)(cid:9)(cid:5) (cid:3) (cid:4)(cid:1) (cid:1)(cid:17)(cid:1)(cid:7)
(cid:4) (cid:4) (cid:3)(cid:1) (cid:1)(cid:17)(cid:1)(cid:5)
(cid:2)(cid:1) (cid:1)(cid:17)(cid:1)(cid:3)
(cid:3) (cid:3)
(cid:1) (cid:1)
(cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) (cid:8) (cid:9) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) (cid:8) (cid:9)
(cid:2) (cid:3)(cid:3)(cid:1)(cid:10)(cid:11) (cid:12)(cid:13)(cid:11)(cid:14)(cid:12) (cid:2) (cid:9)(cid:1) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6) (cid:1)(cid:17)(cid:2)(cid:7) (cid:7)(cid:8)(cid:2)(cid:9)(cid:10)(cid:6)
(cid:1)(cid:1) (cid:3) (cid:8)(cid:9)(cid:3)(cid:4)(cid:5)(cid:5)(cid:6)(cid:7) (cid:7) (cid:9) (cid:1)(cid:1) (cid:2) (cid:3) (cid:4) (cid:8)(cid:9)(cid:3)(cid:4)(cid:5)(cid:5)(cid:6)(cid:7) (cid:6) (cid:7) (cid:8) (cid:9) (cid:7)(cid:8)(cid:1)(cid:1) (cid:10)(cid:10)(cid:19)(cid:19)(cid:12)(cid:12)(cid:3)(cid:2)(cid:13)(cid:13)(cid:14)(cid:15)(cid:14)(cid:2)(cid:1)(cid:1) (cid:1)(cid:1)(cid:17)(cid:17)(cid:2)(cid:2)(cid:5)(cid:3) (cid:10)(cid:10)(cid:20)(cid:20)(cid:12)(cid:12)(cid:3)(cid:2)(cid:13)(cid:13)(cid:14)(cid:14)(cid:15)(cid:15)(cid:14)(cid:14)(cid:2)(cid:2)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
(cid:12)(cid:21)(cid:13) (cid:12)(cid:18)(cid:13)
(cid:6)(cid:1) (cid:1)(cid:17)(cid:2)
cF.IoG.m..10ra.diu(sa.) Quarteting coherence length in 220Ra versus (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:1)(cid:5)(cid:1) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:1)(cid:1)(cid:17)(cid:1)(cid:9) (cid:1)
(cid:4)(cid:1) (cid:1)(cid:17)(cid:1)(cid:7)
(b) Same as in (a) but for alpha coherence length.
(cid:3)(cid:1) (cid:1)(cid:17)(cid:1)(cid:5)
(cid:2)(cid:1) (cid:1)(cid:17)(cid:1)(cid:3)
(cid:1)(cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) (cid:8) (cid:9) (cid:1)(cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) (cid:8) (cid:9)
(cid:1)(cid:2)(cid:7) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6) (cid:2)(cid:7)(cid:3)(cid:4)(cid:5)(cid:6)
FIG. 12. The two terms I(p)(R), p=2 (solid line) p=1
(dashed line) given by Eq,. (2.17), defining the pairing co-
herence length for protons (a), neutrons (b), quarteting co-
(cid:1)(cid:2)(cid:6)
herence length (c) and alpha coherence length (d) versusthe
c.o.m. radius.
(cid:8)(cid:9)
(cid:4)(cid:5)(cid:6)(cid:7)(cid:1)(cid:2)(cid:5) (cid:5)
(cid:3)
(cid:2)
(cid:1)
Itturnsoutthattheproton-neutroncorrelations,given
by the overlap with the corresponding proton-neutron
(cid:1)(cid:2)(cid:4) part of the α-particle wave function (2.25), completely
change this picture. One sees from Fig. 10 (b), where
we plotted the alpha correlationlength versus c.o.m. ra-
dius, that the values oscillate around the value of the
(cid:1)(cid:2)(cid:3)
(cid:1)(cid:8)(cid:8) (cid:1)(cid:9)(cid:8) (cid:1)(cid:10)(cid:8) (cid:1)(cid:4)(cid:8) (cid:1)(cid:6)(cid:8) (cid:9)(cid:8)(cid:8) (cid:9)(cid:9)(cid:8) (cid:9)(cid:10)(cid:8) (cid:9)(cid:4)(cid:8) geometrical radius of the α-particle R . Thus, our anal-
α
(cid:10) ysis confirm the crucial role played by proton-neutron
correlations in the formation of the α-particle. Finally,
in Fig. 11 we plotted the mean value of the alpha co-
FIG.11. Averagedalphacoherencelengthversusmassnum-
ber. herence length for even-even α-emitters above A = 100.
It has a quasi-constant value around 1.7 fm. Small local
maximacorrespondtoregionsabovedoublemagicnuclei
132Sn and 208Pb.
strongest variation appears in the internal region, while In order to better understand the difference between
on the surface, where the pairs are strongly coupled [8], pairingandquarteting correlationswe plottedinFig. 12
there is indeed almost no change. As a measure of the thetwotermsI(p),p=2(solidline)andp=1(dashedline)
pairing correlations, the coherence lenght would appear given by Eq. (2.17) versus the c.o.m. radius. The two
to indicate a gradualtransition to the normalstate with termsreachtheirmaximalvaluesforthepairingcase(left
increasingtemperature,asitsbehaviourissimilartothat panels)atR=0,while forthe quartetingcase(rightpan-
ofthepairinggapintheparticlenumberconservingcase els) the maxima are centered around the surface region.
[16, 33]. The pairing coherence length for protons (a) and neu-
Our purpose is to compare the pairing and quarteting trons (b) is given by the ratio between solid and dashed
coherence lengths. First we analyzed the quarteting co- curves which obviously decreases with increasing c.o.m.
herencelength,byusingthequartetingdensity(2.22),for radius. Quarteting coherence length is given by the ra-
theα-emitter220Raasafunctionofc.o.m. radiusinFig. tio between solid and dashed lines in Fig. 12 (c) which
10 (a). One notices a similar qualitative behaviour com- haveslighly shifted broadmaxima locatedbelow the nu-
pared to the pairing coherence length, but the absolute clear surface. Although the two terms have completely
values are larger on the nuclear surface. Our calcula- differentshapescomparedtothe pairingcase,theirratio
tions have shown that the temperature practically does plotted in Fig. 10 (a) is also a decreasing function with
not change this dependence. respect to the c.o.m. radius.
9
The alpha coherence length, given by the ratio of the larger than the geometrical radius for light nuclei and
two curves in Fig. 12 (d), deserves special attention. approachesthis value for heavy nuclei. Our analysis evi-
These curves have very narrow maxima centered at the denced strong shell effects.
samepointonthenuclearsurface. Moreover,itturnsout Thepairingcoherencelengthslowlydecreaseswithin-
thatthetwocurvesarealmostproportionalandtherefore creasing temperature, indicating a gradual quenching of
theirratioleadstothequasiconstantvalueinFig. 10(b), pairingcorrelations,asisnaturalinfinitesystems. Inex-
close the α-particle geometricalradius R(0) =1.2 41/3 oticnucleiclosetodriplines,wheretheFermienergyhas
α
1.9 fm. Notice that the shape of the curves in Fig. 1≈2 positive values, the correlation length has larger values
(d), peakedonthe nuclearsurface,is similartothe stan- and therefore the spatial correlationincreases.
dard α-particle formation probability, given by the inte- Thequartetingcoherencelengthdescribescorrelations
gral (2.26) squared [12, 28]. between proton and neutron pairs, by overlapping their
relative parts to the corresponding pp and nn compo-
nents of the α-particle wave function. It has a similar
behaviour, but with larger values on the nuclear sur-
IV. CONCLUSIONS
face. We evidencedthe importantroleplayedbyproton-
neutron correlationsby considering in addition the over-
In conclusion, we have performed in this paper a sys- lap with the pn component of the α-particle wave func-
tematic analysis of the pairing coherence length in the tion. Theychangecompletelythe behaviourofthequar-
spin-singlet channel for various types of pairing interac- teting coherence length, namely the alpha correlation
tion. We compared the DDD potential to the Gaussian length has oscillating values around the α-particle geo-
interaction. We considered in our calculations bound metricalradius. Itsmeanvalue 1.7fmweaklydepends
≈
states as well as narrow resonances. on the nuclear mass. The analysis of the two terms en-
As a very important conclusion we have shown that, tering the definition of the coherence length reveales the
by considering the singlet ”bare” value of the width pa- maindifferencebetweenthepairingandquartetingcases.
rameter r =2 fm, the strength parameter reproducing Itturnsoutthatpairingcorrelationsarelargerinsidenu-
0
the gap parameter for light nuclei is close to the singlet cleus, while quarteting correlations are connected to the
value in the free space v 35 MeV and decreasesup to nuclear surface.
0
∼
v 20 MeV for heavy nuclei. We have shown that the
0
∼
”renormalized”Gaussianinteractionwith a largerwidth
parameter than its free value r =2 fm (equal to the nu- ACKNOWLEDGMENTS
0
clearradius)hassimilarpropertiestothecommonlyused
density dependent pairing potential. This work has been supported by the project PN-II-
It turns out that the pairing coherence length has ID-PCE-2011-3-0092and NuPNET-SARFEN of the Ro-
similar properties for all considered interactions. It is manian Ministry of Education and Research.
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