[Lecture Notes in Physics] Extended Density Functionals in Nuclear Structure Physics Volume 641 ||...

11 Pairing and Continuum Effectsin Exotic Nuclei

Nguyen Van Giai and Elias Khan

Institut de Physique Nucleaire, CNRS-IN2P3, 91406 Orsay Cedex, [email protected]

Abstract. Some recent developments in theoretical treatments of pairing correla-tions and continuum effects are discussed in this paper. We first present the methodsto deal with the continuous spectra of particles and quasiparticles when calculatingthe nuclear ground state properties. The coordinate representation is obviously themost suitable framework when the effective two-body interaction is a contact inter-action like Skyrme forces. The continuum Hartree-Fock-Bogoliubov (HFB) equa-tions are solved and the results are compared with discretized-continuum HFBcalculations, and with continuum Hartree-Fock-BCS (HF-BCS) approximations. Itis found that continuum HF-BCS can be a quite good approximation to continuumHFB even near the neutron drip line. We also show that coordinate space HFBcalculations can be performed even if the effective interaction is a finite range forcelike the Gogny force. Next, we discuss how to perform consistent quasiparticle ran-dom phase approximation (QRPA) calculations on top of the previously determinedmean field. In a first step, QRPA predictions in configuration space are comparedto the most recent data on sulfur isotopes. To treat exactly continuum states, thenatural approach is then to calculate the QRPA response function in coordinatespace. The method is illustrated on neutron-rich oxygen isotopes. The applicationof the continuum-QRPA method to two particle transfer calculations is also shown.

11.1 Introduction

In nuclei far from the β-stability line there are circumstances in which theresonant part of the particle continuum plays an important role. The inter-play between the resonant continuum and pairing correlations can also beimportant for the estimation of unbound processes like single-particle (s.p.)excitations lying above the particle emission threshold, which may be foundespecially in nuclei close to the drip line.

The pairing correlations in the presence of continuum coupling have beentreated both in HFB [11.1–11.8] and BCS [11.9–11.12] approximations. In theHFB approximation the continuum is generally included by solving the HFBequations in coordinate representation. The calculations are done either inthe complex energy plane, by using Green functions techniques [11.1,11.6], oron the real energy axis [11.3,11.4]. In the latter case the HFB equations areusually solved by imposing box boundary condition, i.e., that the HFB wavefunctions cancel beyond a given distance far from the nucleus.

N. Van Giai, E. Khan, Pairing and Continuum Effects in Exotic Nuclei, Lect. Notes Phys. 641,303–336 (2004)http://www.springerlink.com/ c© Springer-Verlag Berlin Heidelberg 2004

304 N. Van Giai and E. Khan

For deformed systems working in coordinate representation is much moredifficult [11.5]. In most of the deformed- HFB calculations the continuum isdiscretized by expanding the HFB wave functions in a s.p. basis. Usually aharmonic oscillator basis is taken, eventually corrected for a better descriptionof the asymtotic properties [11.7,11.8]. We will examine some aspects of thes.p. continuum in deformed potentials.

Excited features in nuclei are usually described within the Random PhaseApproximation (RPA) framework. Recently there is a renewed interest, gen-erated mainly by the studies of unstable nuclei close to the drip line. Foropen-shell nuclei, the majority of nuclei, pairing effects are known to play animportant role. The collective excitations of atomic nuclei in the presence ofpairing correlations are usually described in the quasiparticle-Random PhaseApproximation (QRPA) [11.13]. The QRPA was applied to nuclear physicsmore than 40 years ago [11.14–11.16]. In such microscopic models the prop-erties of the states depend on two main inputs, the s.p. spectrum and theresidual two-body interaction. In a consistent approach these two features arelinked: the same effective interaction determines the s.p. spectrum and theQRPA residual interaction. This approach has proved to be an efficient wayto predict properties of collective excitations like giant resonances [11.17] andit has also been used for calculating low-lying collective states in closed-shellnuclei [11.18] within the RPA framework. The long-range goal is to have onlyone effective nucleon-nucleon interaction as input for both ground and ex-cited states. To achieve this goal, it is necessary to test the model on a widevariety of nuclei, as closed shell, open shell, stable or drip-line nuclei. TheQRPA is general enough to embody all these situations in principle.

The present discussion is based on effective Skyrme interactions (see e.g.[11.19]). The derivation of the residual interaction is more convenient due tothe zero range of the force, and allows for instance to apply the QRPA equa-tions in coordinate space formalism. Note that very recently, self-consistentQRPA calculations using the finite range Gogny force have been derived[11.20].

11.2 BCS Theory with Particle Continuum

Resonant states are important for determining the pairing properties of theground state of bound nuclei far from the β-stability line. Although in suchcalculations one should consider in principle the complete particle continuum,the largest contribution to the pairing correlations is expected to come fromthe resonant continuum part [11.21].

In this section we examine a method introduced by Sandulescu et al.[11.10]for incorporating the effect of the resonant continuum in the HF+BCS ap-proximation with Skyrme forces. Recently, this method has also been ex-tended to the relativistic mean field approach [11.22]. More precisely we inves-tigate here the effect of the width of the resonances on the pairing properties

11 Pairing and Continuum Effects in Exotic Nuclei 305

of nuclei far from stability. As discussed below, this effect is more difficultto estimate in self-consistent Hartree-Fock-Bogoliubov (HFB) calculationswhich are presently used for describing pairing correlations in nuclei close tothe drip line.

In order to derive the continuum-BCS equations we first enclose the nu-cleus in a box of very large radius Rb. This is only a formal step, since it willbe seen that the parameter Rb does not appear in the final results. The leveldensity is given by [11.23]:

g(ε) =∑

ν

{gν(ε) + gfreeν (ε)} ≡∑

ν

gν , (11.1)

where gν(ε) ≡ (1/π)(2jν +1)(dδν/dε) is the so-called continuum level density[11.24], δν is the phase shift of angular momentum ν ≡ (lν , jν), gfreeν (ε) isthe level density in the absence of the mean field and is given by gfreeν (ε) ≡(1/π)(2jν + 1)(dk/dε)Rb, where k is the momentum corresponding to theenergy ε. The wave functions corresponding to the positive energy states arenormalized within the box, ψν(ε, r) ≡ N −1/2

ν (ε)ϕν(ε, r), where Nν(ε) is thenorm of the scattering state ϕν(ε, r) in the box volume. It can be easily shownthat for the scattering states selected by the box Nν(ε) = (2jν + 1)−1gν(ε).

The gap equations can be written as follows:

∆i =∑

j

Vii,jjujvj +∑

ν

∫

Iν

gν(ε)Vii,νενεuν(ε)vν(ε)dε , (11.2)

∆ν(ε) =∑

j

Vνενε,jjujvj +∑

ν′

∫

Iν′gν′(ε′)Vνενε,ν′ε′ν′ε′uν′(ε′)vν′(ε′)dε′ ,

(11.3)where the indices i, j run over the bound states and Iν is an energy in-terval associated with each partial wave (lν , jν). The matrix elements ofthe interaction involving states in the continuum are given by Vii,νενε ≡〈ψiψi|V |ψν(ε)ψν(ε)〉, Vνενε,ν′ε′ν′ε′ ≡ 〈ψν(ε)ψν(ε)|V |ψ′

ν(ε′)ψν′(ε′)〉. The rest

of the notations are standard [11.13].The largest contributions to the integrals above are provided by the re-

gions where the wave functions ψν(ε) have a large localization inside thenucleus, i.e., for ε near narrow s.p. resonances, or sometimes near thresh-old [11.25,11.26]. This helps to select the intervals Iν . In these intervals thepositive energy wave functions have the largest localization inside a sphere ofradius D, where D is of the order of the nuclear radius. Within this sphere thepositive energy wave funtions can be related to the scattering wave function atresonant energy εν through simple factorization formulas [11.25–11.28]. Fol-lowing [11.27,11.28], the wave function ψν(ε) inside D can be approximatedby

ψν(ε, r) ≈ g1/2ν (ε)g−1/2

ν (ε)φν(εν , r) ≡ τ1/2ν (ε)φν(εν , r) , (11.4)


where φν(εν , r) is the scattering wave function calculated at the resonantenergy εν and normalised within a sphere of radius D. This factorizationrelation is very useful for evaluating matrix elements of finite range inter-actions since it is sufficient to carry space integrals over the volume insidethe radius D only. For instance, one can use Vii,νενε ≈ τν(ε)Vii,νεννεν andVνενε,ν′ε′ν′ε′ ≈ τν(ε)τν′(ε′)Vνεννεν ,ν′εν′ν′εν′ , where on the right hand sides thematrix elements of the interaction are calculated using the wave functionsφν(εν , r). For a discussion of the accuracy of these approximations see [11.28].

With the help of this factorization the gap equations (11.2,11.3) become:

∆i =∑

j

Viijjujvj +∑

ν

Vii,νεννεν

∫

Iν

gν(ε)uν(ε)vν(ε)dε , (11.5)

∆ν ≡∑

j

Vνεννεν ,jjujvj +∑

ν′Vνεννεν ,ν′εν′ν′εν′

∫

Iν′gν′(ε′)uν′(ε′)vν′(ε′)dε′ ,

(11.6)with ∆ν(ε) = τν(ε)∆ν . The latter expression can be written as gν(ε)∆ν(ε) =gν(ε)∆ν and it gives the connection between the gaps calculated with thewave functions ψν(ε) and φν(ε). One can get the same relation if one writesthe gap equation (11.3) in terms of a local pairing field ∆(r) of finite rangewhich cuts the tail of the wave function ψν(ε, r) beyond the radius D:

∆ν(ε) =∫ Rb

0|ψν(ε, r)|2∆(r)dr ≈ τν(ε)

∫ D

0|φν(ε, r)|2∆(r)dr ≡ τν(ε)∆ν .

(11.7)A similar relation can be derived for the positive energy s.p. spectrum. Byusing these relations it can be seen that the gap equations (11.5,11.6) areindependent of the box radius.

Within the same approximations as for deriving the gap equations theparticle number is:

N =∑

i

v2i +∑

ν

∫

Iν

gν(ε)v2ν(ε)dε . (11.8)

Equations (11.5,11.6,11.8) are the extended BCS equations for a general (fi-nite range) pairing interaction including the contribution of the resonantcontinuum.

The above BCS equations are well suited to perform the approximationof constant pairing interaction since the resonant states φν(εν , r) and thebound states have rather similar localizations inside the nucleus. Then, onejust has to replace in (11.5,11.6) all matrix elements by a constant value G.The corresponding BCS equations are the same as those of [11.29].

We now extend the above treatment of the resonant continuum to HF+BCS.In the case of a Skyrme force this is done by including into the HF densitiesthe contribution of the positive energy states with energies in the selected


intervals Iν and by using the factorization relation (11.4). Thus, the resonantcontinuum contribution to the particle density inside the sphere of radius Dreads

ρc(r) ≈∑

ν

|φν(εν , r)|2∫

Iν

gν(ε)v2ν(ε)dε ≡

∑

ν

|φν(εν , r)|2〈v2〉ν . (11.9)

Similar expressions can be derived for the kinetic energy density T (r) andspin density J(r),

T (r) ≈∑

i

v2i |∇ψi(r)|2 +

∑

ν

〈v2〉ν |∇φν(εν , r)|2 , (11.10)

J(r) ≈ −i∑

i

v2i ψ

∗i (r)(∇ψi(r)Λσ)− i

∑

ν

〈v2〉νφ∗ν(εν , r)(∇φν(εν , r)Λσ) ,

(11.11)where the first terms represent the contribution of the bound states. Thesedensities are used in the coupled HF and BCS equations to determine self-consistently by an iterative process the s.p. states and occupation probabili-ties as in the usual HF+BCS calculations [11.30].

Let us briefly comment the relation between the HF+BCS equations de-rived above and the HFB approach. The advantage of the continuum-HFBapproach is that the particle and pairing densities acquire automatically aproper asymtotic behaviour [11.2,11.3]. In order to preserve the same be-haviour in the HF+BCS limit one should keep the physical condition of afinite range pairing field. As seen in (11.7), a finite range pairing field impliesa cut in the tail of the positive energy wave functions. Without this cut thesolution of a continuum-HF+BCS calculation would correspond to a nucleusin dynamical equilibrium with a nucleonic gas and not to the nucleus itself[11.23]. In general the cut-off radius may be ambiguous but if one restrictsoneself to the resonant continuum, there is a rather wide region far fromthe nucleus in which the resonant wave functions have a value close to zerobefore they start oscillating. In this case the continuum-HF+BCS results donot depend sensitively on the cut-off radius chosen in that region.

As a numerical illustration of the continuum-HF+BCS approach we es-timate the errors one can introduce in calculating the pairing properties ofnuclei far from stability if one neglects the width of the resonant states. Theestimation is made for the nucleus 84Ni, which was also treated in HFB ap-proach [11.5]. In order to compare the HF+BCS and HFB results we takehere the same interaction as in [11.5]. The detailed inputs can be foundin [11.10]. The resonant states included in the HF+BCS calculations, withenergies smaller than 5 MeV (the energy cut off used in [11.5]), together withthe last bound state are listed in Table 11.1. The energy (width ) of a givenresonance is extracted from the energy where the derivative of the phase shiftreaches is maximum (half of its maximum) value.

The numerical results discussed here do not depend sensitively on theprecise choice of D in the range (3 − 4)R, where R is the nuclear radius.


Table 11.1. Results of HF+BCS calculations for the nucleus 84Ni. ∆n and v2n are

the averaged gap and averaged occupation probability of the s.p. state n of energyεn and width Γn. The notations ∆n, v2

n, εn and Γn stand for the correspondingquantities calculated without including the effect of the widths of resonant statesin HF+BCS equations. The s.p. energies, their widths and the pairing gaps areexpressed in MeV .

n εn εn Γn Γn v2n v2

n ∆n ∆n

s1/2 -0.669 -0.665 — — 0.301 0.307 0.479 0.647d3/2 0.480 0.457 0.0932 0.0836 0.036 0.068 0.600 0.809g7/2 1.670 1.654 0.0102 0.0098 0.025 0.050 0.926 1.260h11/2 3.370 3.364 0.0184 0.0183 0.015 0.031 1.176 1.601

We thus obtained for the total averaged gap the value ∆= 0.471 MeV. Thecorresponding pairing field is shown in Fig. 11.1 while the averaged occu-pation probabilities and the averaged gaps of resonant states and the lastbound state 3s1/2 are given in Table 11.1. The change of the particle densitydue to pairing correlations is shown in Fig. 11.2. It can be seen that in thetail region the contribution of the bound states to the total density, givenmainly by the loosely bound state 3s1/2, is dominant. In order to see how theneutrons are distributed at large distances, we have calculated the numberof neutrons outside a sphere of radius 12 fm. Thus we find that the totalnumber of neutrons distributed in bound and resonant states up to 22 fm isequal to 0.069 and 0.034, respectively.

Next, similar calculations are performed without taking into account theresonance widths. The corresponding changes in the pairing field and particledensity are shown in Figs. 11.1-11.2. From Table 11.1 one can see that theoccupancy of the resonant states is almost doubled compared to the case whenthe effect of the widths is taken into account. Thus, one can see that the effectof resonance widths is to decrease the amount of pairing correlations.

For the particular case of the discretized HFB calculations of [11.5], per-formed in a box with a radius of the order of 15 fm, the resonant states aredescribed by a set of quasidegenerate states, which appear due to the fact thatthe spherical symmetry is not strictly preserved. To which extent these statescan simulate or not the width of resonant states is not yet clear, although, asseen in Fig. 11.1, the HFB pairing field is closer to the continuum-HF+BCSresult obtained with the effect of the widths included.

In order to compare the HF+BCS and HFB results, and eventually toconclude about the continuum contributions in these two approches, oneneeds to solve the continuum HFB equations without the boundary conditionsimposed by a box of finite radius. This is the issue we will examine in thenext section.


Fig. 11.1. Neutron pairing field as a function of the radius. The full (dashed) lineshows the results of the HF+BCS calculations with (without) the effect of the widthincluded. The line marked by crosses shows the HFB results of [11.5].

11.3 Hartree-Fock-Bogoliubovwith Quasiparticle Continuum

11.3.1 HFB Equations in Coordinate Representation

The HFB equations in coordinate representation read [11.3]:

∫d3r′∑

σ′

(h(rσ, r′σ′) h(rσ, r′σ′)h(rσ, r′σ′) −h(rσ, r′σ′)

)(Φ1(E, r′σ′)Φ2(E, r′σ′)

)=

(E + λ 0

0 E − λ

)(Φ1(E, rσ)Φ2(E, rσ)

),

(11.12)

where λ is the chemical potential, h and h are the mean field and the pairingfield, and (Φi) represents the two-component HFB quasiparticle wave functionof energy E. The mean field h is composed of the kinetic energy T and themean field potential Γ ,

h(rσ, r′σ′) = T (r, r′)δσσ′ + Γ (rσ, r′σ′) . (11.13)

The mean field potential Γ is expressed in terms of the particle-hole (p-h)two-body interaction V and the particle density ρ in the following way:


Fig. 11.2. Neutron density in 84Ni, calculated in HF (long dashed line) andHF+BCS. In the order of decreasing tail the results of HF+BCS correspond tothe following approximations: effect of the widths neglected; effect of the widthsincluded; contribution of the bound states to the density. The density is in fm−3.

Γ (rσ, r′σ′) =∫

d3r1d3r2∑

σ1σ2

V (rσ, r1σ1; r′σ′, r2σ2)ρ(r2σ2, r1σ1) , (11.14)

whereas the pairing field h is expressed in terms of the pairing interactionVpair and the pairing density ρ:

h(rσ, r′σ′) =∫

d3r1d3r2∑

σ1σ2

2σ′σ′2Vpair(rσ, r

′ − σ′; r1σ1, r2 − σ2)ρ(r1σ1, r2σ2) .

(11.15)The particle and pairing densities ρ and ρ are defined by the following

expressions:

ρ(rσ, r′σ′) ≡∑

0<En<−λΦ2(En, rσ)Φ∗

2(En, r′σ′)

+∫ Ecut−off

−λdEΦ2(E, rσ)Φ∗

2(E, r′σ′) , (11.16)


ρ(rσ, r′σ′) ≡∑

0<En<−λΦ2(En, rσ)Φ∗

1(En, r′σ′)

+∫ Ecut−off

−λdEΦ2(E, rσ)Φ∗

1(E, r′σ′) , (11.17)

where the sums are over the bound quasiparticle states, with energies |E| <−λ, and the integral is over the quasiparticle continuous states, of energies|E| > −λ. The HFB solutions have the following symmetries:

Φ1(−E, rσ) = Φ2(E, rσ)Φ2(−E, rσ) = −Φ1(E, rσ) (11.18)

We choose to work with the positive energies.The p-h and pairing interactions in (11.14) and (11.15) are chosen as

density-dependent contact interactions, so that the integro-differential HFBequations reduce to coupled differential equations. The zero-range characterof the interaction is the reason why we are obliged to fix a cut-off in theenergy, as it can be seen in (11.16) and (11.17).

Here, we consider systems with spherical symmetry. In this case the an-gular part of the HFB wave functions reads [11.3]:

Φi(E, rσ) = ui(njl, r)1rymj

lj (r, σ) , i = 1, 2 , (11.19)

where:ymj

lj (r, σ) ≡ Ylm(Θ,Φ)χ1/2(mσ)(lm12mσ|jmj) . (11.20)

In what follows we use for the upper and lower component of the radial wavefunctions the standard notation ulj(E, r) and vlj(E, r).

11.3.2 The Treatment of Quasiparticle Continuum:Asymptotic Behaviours

The asymptotic behaviour of the HFB wave function is determined by thephysical condition that at large distances the nuclear mean field Γ (r) andthe pairing field ∆(r) vanish. This condition requires an effective interactionof finite range and finite-range nonlocality. Outside the range of mean fieldsthe equations for Φi(E, rσ) are decoupled and one can easily see how thephysical solutions must behave at infinity [11.2]. For a bound system (λ < 0)there are two well separated regions in the quasiparticle spectrum. Between0 and −λ the quasiparticle spectrum is discrete and both upper and lowercomponents of the radial HFB wave function decay exponentially at infinity:

ulj(E, r) = Ah(+)l (α1r) ,

vlj(E, r) = Bh(+)l (β1r) , (11.21)


where h(+)l are spherical Haenkel functions, α2

1 = 2m�2 (λ+E) and β2

1 = 2m�2 (λ−

E). These solutions correspond to the bound quasiparticle spectrum. ForE > −λ the spectrum is continuous and the solutions are:

ulj(E, r) = C[cos(δlj)jl(α1r)− sin(δlj)nl(α1r)] ,

vlj(E, r) = D1h(+)l (β1r) , (11.22)

where nl are spherical Neumann functions and δlj is the phase shift corre-sponding to the angular momentum (lj). One can see that the upper compo-nent of the HFB wave function has the standard form of a scattering state.

The asymptotic form of the wave function should be matched with theinner radial wave function, which for r → 0 can be written as follows:

(ulj(E, r)vlj(E, r)

)= D2

(rl+1

0

)+ D3

(0

rl+1

), (11.23)

The HFB wave function is normalized to the Dirac δ-function of energy.This condition fixes the constant C to the value:

C =√

1π

2m�2α1

. (11.24)

The difficulty of an exact continuum calculation, i.e., with asymptoticsolutions given by (11.11), is to identify the energy regions where the lo-calization of the wave functions changes fast with the quasiparticle energy.These are the regions of quasiparticle resonant states.

In HFB the quasi-particle resonant states are of two types. A first typecorresponds to the s.p. resonances of the mean field. The low-lying resonancesof the mean field situated close to the particle threshold are very importantin the treatment of pairing correlations of weakly bound nuclei because theybecome strongly populated by pairing correlations. A second kind of resonantstates is specific to the HFB method and corresponds to the bound s.p.states which in the absence of pairing correlations have an energy ε < 2λ.In the presence of the pairing field these bound states are coupled with thecontinuum s.p. states and therefore acquire a width. The positions and thewidths of these resonances, which originate from the bound s.p. states, arerelated to the total phase shift in the following way [11.2]:

δ(E) � δ0(E) + arctgΓ

2(ER − E), (11.25)

where ER and Γ are the energy and the width of the resonant quasiparticlestate. The function δ0(E) is the phase shift of the upper component of theHFB wave function in the HF limit, i.e. h = 0, which satisfies the equation:

hΦ01 = (E + λ)Φ0

1 . (11.26)


If in the HF limit there is no s.p. resonance close to the energy E+λ, then theHF phase shift δ0 has a slow variation in the quasiparticle energy region. Inthis case the derivative of the total phase shift has a Breit-Wigner form, whichcan be used for estimating the position and the width of the quasiparticleresonance.

11.3.3 Results for Ni Isotopes

We now apply the continuum HFB method to the case of Ni isotopes, whichhave been investigated extensively both in non-relativistic [11.5] and rel-ativistic HFB approximation [11.31,11.32]. The present results are takenfrom [11.33].

In the p-h channel we use the Skyrme interaction SIII, while in the pairingchannel we choose a density-dependent delta interaction:

V = V0

[1−(

ρ(r)ρ0

)γ], (11.27)

with the following parameters [11.5]: V0 = 1128.75, ρ0 = 0.134 and γ = 1.

Table 11.2. Hartree-Fock s.p.energies ε and HFB quasiparticle resonance energies(E) and widths (Γ ) are shown in the third, fourth and fifth columns of the table forthe isotope 84Ni. In first and second columns the corresponding quantum numbersj and l are listed.

j l ε (MeV) E (MeV) Γ

1 0 -0.731 1.276-22.530 20.878 98 KeV-45.010 43.3917 300 eV

1 1 -9.540 7.965 338 KeV-34.709 33.444 102 KeV

3 1 -11.194 9.712 576 KeV-36.364 34.976 76 KeV

3 2 0.475 2.317 816 KeV-23.055 22.028 58 KeV

5 2 -1.467 1.845 44 KeV-26.961 25.628 3 KeV

5 3 -10.586 8.863 944 KeV7 3 -17.023 15.857 882 KeV7 4 1.604 3.598 24 KeV9 4 -6.837 5.674 3 KeV11 5 3.295 5.380 52 KeV

Let us examine the quasiparticle resonant states for the isotope 84Ni. InTable 11.2 we report the resonant quasiparticle energies and the widths cal-culated from the derivative of the phase shift. The quasiparticle states 2d3/2,


1g7/2 and 1h11/2 originate from s.p. resonances while all the others are re-lated to bound states. A more detailed discussion of the resonant continuumdue to quasiparticle resonances and loosely bound states (in the case l=0)can be found in [11.33].

Comparison between HFB and HF-BCS. The HF-BCS approximationis obtained by neglecting in the HFB equations the non-diagonal matrixelements of the pairing field. This means that, in the HF-BCS limit oneneglects the pairing correlations induced by the pairs formed in states whichare not time-reversed partners.

In the case of Ni isotopes the effect of the continuum is introduced in HF-BCS calculations through the first three s.p. resonances, d3/2, g7/2 and h11/2.These resonances form together with the bound states 2d5/2 and 3s1/2 theequivalent of the major shell N = 50− 82. For each resonance one considersin the resonant- HF-BCS equations above the scattering states with energy εdefined such that |ε− εν | ≤ 2Γν , where εν is the energy of the resonance andΓν is its width.

We first examine the calculated pairing correlation energies defined asthe difference between the HF total energy and the HFB (or HF-BCS) totalenergy:

EP = E(HF )− E(HFB or HF −BCS) (11.28)The results are shown in Fig. 11.3.

Fig. 11.3. Pairing correlation energies calculated in resonant continuum HF-BCSand in continuum HFB.

One can see that the HF-BCS values follow closely the exact HFB resultsup to the dripline. This shows that, for estimating the pairing correlations


one needs actually to include from the whole continuum only a few resonantstates, which should be treated with their width.

In order to see the effect of the widths of resonant states upon pairing, wecan replace in the resonant continuum HF-BCS equations the continuum leveldensity with delta functions. This means that the resonant state is replacedby a scattering state at the resonance energy and normalized in a volume ofradius R=22.5 fm. As it can be seen from Fig. 11.4, the pairing correlationsincrease when one neglects the widths of the resonances and the results followclosely those of the box-HFB calculations. Thus, the overestimation of pairingcorrelations due to the continuum discretization is similar in HF-BCS andHFB calculations.

Fig. 11.4. Pairing correlation energies calculated in the resonant continuum HF-BCS approximation assuming no widths, compared to box-HFB results.

As we have already mentioned, in the present HF-BCS calculations weneglect all the continuum contribution except for the three resonances men-tioned above. Although this model space seems suficient for a proper eval-uation of pairing correlation energies, the rest of the continuum, present inHFB calculations, has an influence on the details of particle density distri-bution. In order to get a particle density closer to the HFB results one needsto introduce in the resonant HF-BCS calculations additional relevant piecesfrom the continuum.


11.4 Pairing with Finite Range Interactions

Zero-range forces are widely used because the self-consistent equations canbe conveniently solved in coordinate space. On the other hand, a zero-rangepairing interaction has the well-known pathology of producing diverging con-tributions if no cut off is imposed on the quasiparticle space. This cut off mustbe an inherent part of the phenomenological zero-range interaction [11.4], butit is not clear which cut off value must be adopted for a given interaction anda given nucleus. We note that some regularization scheme has been proposedrecently [11.34] for dealing with the question of HFB equations with zero-range pairing interactions.

Finite range pairing interactions of course do not require in principle atruncation of the quasiparticle space, even though in practical calculations thesummations are restricted to quasiparticle energies below some cut off energyEc.o.. If Ec.o. is large enough the results no longer depend on its precise value,in contrast to the case of zero-range interactions. It is desirable therefore tohave a method which combines the advantages of solving HFB equations incoordinate space and of describing pairing correlations with a finite rangeinteraction. Such a method has been proposed and applied recently [11.35]to calculate the neutron-rich nucleus 18C using a Skyrme interaction for theHF mean field and a Gogny force for the pairing field. The continuum statesare treated as explained in Sect. 11.3.

We take for Vpair a Gogny force which contains a sum of two gaussians, azero-range density-dependent part and a zero-range spin-orbit part. Withinthe parametrisation D1S [11.36] that we adopt here the zero-range density-dependent part does not contribute. Let us explain the contribution of thegaussian terms. That of the spin-orbit part is also included in the calculations.The finite range part of the interaction is:

Vpair(r1− r2) =∑

α=1,2

(Wα +BαPσ −HαPτ −MαPσPτ )e−|r1−r2|2

µ2α , (11.29)

where Wα, Bα, Hα, Mα and µα are parameters, Pσ and Pτ are the spinand isospin exchange operators, respectively. Then, the pairing field (11.15)becomes [11.4]:

h(rσ, r′σ) =∑

α=1,2

e−|r−r′|2

µ2α [(Wα −Hα)ρ(rσ, r′σ)− (Bα −Mα)ρ(r′σ, rσ)] .

(11.30)In the following we will need the multipole expansions of the gaussian formfactors:

e−|r−r′|2

µ2α =

∑

LM

FαL (r, r′)YLM (r)Y ∗

LM (r′) . (11.31)

We restrict ourselves to the case of spherical symmetry, for simplicity.The multipole decomposition of the pairing field h(r, r′) ≡

∑σ h(rσ, r′σ) is:


h(r, r′) =∑

L1M1

hL1(r, r′)YL1M1(r)Y

∗L1M1

(r′) , (11.32)

where

hL1(r, r′) =

∑α=1,2

∑L Fα

L (r, r′) 2L+14π

∑nlj(2j + 1)

[(Hα −Wα)u2(nlj,r)

ru1(nlj,r′)

r′ − (Mα −Bα)u2(nlj,r′)r′

u1(nlj,r)r

]

(L l L10 0 0

)2

.

(11.33)

The summations over n in (11.33) become integrals over the energy for thecontinuum states.

For each partial wave (lj) one has to solve a system of two coupled integro-differential equations whose general structure is:

hu1(r) +∫

h(r, r′)u2(r′)r′2dr′ = (E + EF )u1(r) ,

∫h(r, r′)u1(r′)r′2dr′ − hu2(r) = (E − EF )u2(r) . (11.34)

In the context of HF equations with finite range interactions it has beenshown by Vautherin and Veneroni [11.37] that one can transform the HFintegro-differential equation into a purely differential equation by introducinga so-called trivially equivalent local potential. Here, we generalize this methodto the system of (11.34) by defining local equivalent potentials Ui(r) in thefollowing way:

∫h(r, r′)ui(r′)r′2dr′ =

1ui(r)

(∫

h(r, r′)ui(r′)r′2dr′)ui(r)

≡ Ui(r)ui(r) , i = 1, 2 . (11.35)

Then, (11.34) become formally a system of two coupled differential equa-tions where the potentials depend on the solutions and therefore they mustbe solved iteratively. This is not a major problem since the self-consistencyrequirement already leads to an iterative scheme.

An additional difficulty comes from the fact that the local potentials Ui(r)have poles at the nodes of the wave functions ui(r). In [11.37] a very sim-ple and efficient method was proposed to overcome this problem, based onthe linearization of the local equivalent potential around the poles. Anotherequivalent potential Ui(ε, r) is introduced; it is equal to Ui(r) everywhereexcept inside the intervals [r0 − ε, r0 + ε] where r0 denotes a pole of Ui(r).Inside these intervals Ui(ε, r) is chosen as a segment which joins the valuesof Ui(r0 − ε) and Ui(r0 + ε). The approximation is good if ε is small enough


not to wash out the shape of the potential. Thus, Ui(r) is replaced by thefollowing potential:

U(n+1)i (εn+1, r) = Rεn+1

{1

u(n)i (r)

∫r′2dr′h(r, r′)ui(r′)(n)dr′

}, (11.36)

where Rεn+1 indicates the linear interpolation procedure described above andn counts the iterations by which the HFB equations are solved. The parameterε depends on the iteration and it is chosen so that lim εn = 0. At eachiteration of the HFB scheme we evaluate the equivalent potentials of allquasiparticle states by using the wave functions of the previous iteration andwe repeat this procedure until convergence.

11.5 Continuous Spectra of Deformed Mean-Fields

A s.p. resonance structure is much more complicated for a deformed sys-tem, where several angular momentum components of the single-particle wavefunction are coupled to one another. A standard method to solve this prob-lem is to expand the wave function on a discrete basis, which may be de-formed harmonic oscillator wave functions [11.30,11.38] or eigenfunctions ofa spherical potential [11.39,11.40]. The positive energy states obtained bydiagonalizing the Hamiltonian in such basis can describe properly the mainproperties of the resonant states [11.41,11.42], but this may require a largenumber of the basis set. A more direct method to obtain resonance states ina deformed system is to setup the coupled-channels equations for the radialmotion and solve them for a complex energy by imposing the outgoing waveboundary condition at infinity for all the open channels [11.43]. Recently,this method has been successfully applied to the proton emission decay ofproton-rich nuclei [11.44,11.45]. This method has an advantage over the ma-trix diagonalization method that the wave functions are subject only to theradial mesh discretization error for the Schrodinger equation and that thedesired asymptotic boundary condition can be imposed independently of thesize of the mesh interval. However, the description of resonant states withsuch complex wave functions is generally difficult to use in nuclear models,since all the evaluated observables become complex quantities if the rest ofthe complex non-resonant states is not taken into account. It is thus prefer-able to keep the energy as a real variable. In this section we briefly report onrecent progress [11.46] which lead to the possibility of performing continuumHF-BCS and QRPA calculations for deformed systems [11.47].

The aim is to develop a practical method to obtain scattering wave func-tions in the vicinity of a multi-channel resonance state. A difficulty is thatthere are N linearly independent solutions of the coupled-channels equationsat a given energy E (N is the number of included channels), whose asymptoticbehavior are all different. For example, a physical scattering boundary con-dition is defined by requiring that asymptotically there is an incoming wave


only in the incident channel. In this case, the equations have N degeneratesolutions depending on which channel is the incident one. Since any linearcombination of these N solutions is also a solution of the coupled-channelsequations, a problem arises as to which combination one should take in orderto study efficiently a multi-channel resonance state. If the resonance width isextremely small, as in the case of proton decay, one can easily construct a res-onance wave function by matching to a standing wave solution [11.48–11.50].However, this procedure is not applicable to a broad resonance.

The method consists in using as starting point discretized states whosewave functions vanish at a given radius. This is an extension of the boxboundary condition for a spherical system into the deformed regime. As inthe spherical case, resonances can be identified with those states whose energyis stabilized against the box size. This method provides the most convenientbasis states, where the unperturbed states are already a good approximationto the true resonance states. In fact, as we will show below, the resonanceenergy thus estimated is close to the energy at which the first derivative of theeigenphase sum with respect to energy has a maximum. Here, the eigenphasesum is defined as the sum of the phase shifts for the eigenchannels, for whichthe S- matrix is diagonal [11.51]. In the field of electron-molecule scattering,it has been known that the eigenphase sum has the same energy dependenceas the elastic phase shift in a one-channel case [11.52,11.53]. Using theseunperturbed wave functions, we then employ the first order perturbationtheory to construct wave functions around a resonance state.

Let us start with the following single-particle (s.p.) Hamiltonian:

H = − �2

2m∇2 + V (r), (11.37)

where the potential V may also contain a derivative operator such as a spin-orbit interaction. When the potential V is non-spherical, the s.p. angularmomentum is not conserved and the total s.p. wave function has the form

Ψ(r) =∑

ljm

uljm(r)rYljm(r), (11.38)

withYljm(r) =

∑

ml,ms

〈l ml12ms|j m〉Ylml

(r)χms. (11.39)

Here, Ylmlare spherical harmonics and χms denotes the spin wave function.

Projecting the Schrodinger equation, HΨ = EΨ , on the spin-angular statesYljm, the coupled-channels equations for the radial wave functions then read,[− �

2

2md2

dr2 +l(l + 1)�2

2mr2 − E

]uljm(r) + r

∑

l′j′m′〈Yljm|V |Yl′j′m′〉ul

′j′m′

r= 0.

(11.40)


These equations are solved with certain boundary conditions which de-pend on the problem of interest. A standard way to solve them is to generateN linearly independent solutions, N being the number of channel states tobe included, and then to take a linear combination of these N solutions sothat the relevant boundary condition is satisfied [11.48,11.54]. The linearlyindependent solutions can be obtained by taking N different sets of initialconditions at r = 0. We denote these solutions by φLL′(r), where L refers tothe channels while L′ refers to a particular choice of initial conditions. Here,we use a shorthand notation, L = (ljm). A simple choice for the N initialconditions is to impose

φLL′(r)→ rl+1 δL,L′ for r → 0. (11.41)

The coupled-channels equations are solved outwards for each initial conditionL′. The wave functions uL(r) in (11.40) are written in terms of the φLL′ as

uL(r) =∑

L′CL′φLL′(r), (11.42)

where the coefficients CL′ are determined from the asymptotic behavior ofuL(r).

When the s.p. energy E is positive, the physical wave function is a scatter-ing wave with an incoming wave behaviour in some particular incident chan-nel L0 and outgoing waves in all channels L. Thus, the asymptotic boundarycondition of the wave function uL(r) is given by

uL(r)→√

k

πE

i

2

{e−i(kr−lπ/2)δL,L0 − SLL0e

i(kr−lπ/2)}

for r →∞,

(11.43)where k =

√2mE/�2 and SLL0 is the scattering S-matrix. The normal-

ization is chosen so that the total wave function satisfies 〈ΨE,L0 |ΨE,L′0〉 =

δ(E−E′)δL0,L′0. Here, |ΨE,L0〉 is given by (11.38), whose channel components

uL(r) satisfy the boundary condition (11.43). In practice, the S-matrix canbe obtained by decomposing φLL′(r) in the asymptotic region with sphericalHankel functions h

(±)l (kr) and comparing (11.42) with (11.43)[11.48,11.54].

The generalization of the concept of s.p. resonances from the spherical tothe multi-channel cases is most conveniently done by monitoring the energydependence of the eigenphase sum [11.52,11.53]. The eigenphases are relatedto the eigenvalues of the S-matrix through

(U†SU)aa′ = e2iδaδa,a′ . (11.44)

Hazi [11.53] has shown that, the sum of the eigenphases ∆(E) ≡∑a δa(E),

has the same energy dependence around a resonance as the phase shift in aspherical system, i.e.,

∆(E) = ∆0(E) + tan−1 Γ

2(ER − E), (11.45)


where the slowly-varying quantity ∆0(E) is the sum of the background eigen-phases, and ER and Γ are the multi-channel resonance energy and totalwidth, respectively. Therefore, the properties of multi-channel resonances canbe extracted directly by plotting ∆(E), or its energy derivative, as a func-tion of E. The quantity ∆(E) is called the eigenphase sum, and it has beenwidely used in the context of electron-molecule scattering (see, e.g., [11.55]for a recent publication).

We illustrate the above discussion by the following example. We use aWoods-Saxon parametrization for the potential V which is given by

V (r) = Vcent(r) +∇(Vls(r)) · (−i∇× σ), (11.46)

with

Vcent(r) = V0(r)−RβdV0(r)

drY20(r), (11.47)

Vls(r) = Vso(r)−Rso βdVso(r)

drY20(r), (11.48)

where V0(r) and Vso(r) have a Woods-Saxon shape:

V0(r) = −V0/[1 + exp((r −R)/a)], (11.49)Vso(r) = Vso/[1 + exp((r −Rso)/aso)]. (11.50)

For simplicity, we have expanded the deformed Woods-Saxon potential andkept only the linear order of the deformation parameter β[11.56]. Also, weassume an axial symmetry, where both the parity π and the spin projection Konto the z axis are conserved. The parameters of the Woods-Saxon potentialare taken to be V0=42.0 MeV, R = Rso = 1.27 ×441/3 fm, a = aso = 0.67 fm,and Vso= 14.9 MeV/fm2, to somehow simulate the neutron potential in the44S region. As an example we calculate the neutron s.p. levels with Kπ = 5/2+

at β=0.2. We include the d5/2, g7/2, g9/2, i13/2, and i11/2 states in the coupled-channels equations. Notice that the s1/2 and d3/2 states do not contribute tothe Kπ = 5/2+ levels. We have checked that the results are not significantlyaltered even if we include higher angular momentum components.

The thick solid line in Fig. 11.5 shows the eigenphase sum and its energyderivative as a function of energy.The contributions from three main eigen-channels are also shown by thin lines. One clearly sees a maximum at ER=3.44 MeV in the first derivative of the eigenphase sum. The total width canbe estimated to be Γ= 0.46 MeV.

Let us now compare this continuum result with the s.p. spectrum of a box-discretized calculation. From the structure (11.42) of the general solution, itis clear that one can impose the boundary condition that the total wavefunction Ψ(r) vanishes at a radius Rbox. This is equivalent to putting thenucleus in an impenetrable spherical box of radius Rbox. If the condition

det(φLL′(Rbox)) = 0 (11.51)


Fig. 11.5. The eigenphase sum (the upper panel) and its energy derivative (thelower panel) as a function of energy for neutron positive energy states of 44S at β= 0.2. The spin projection onto the symmetry axis and the parity are taken to beKπ = 5/2+. These are denoted by the thick solid line, while the contributions fromthree main eigenchannesl are also shown by the thin dashed, dotted and dot-dashedlines.

is satisfied, then (11.42) has a non-trivial solution for CL such that uL(Rbox) =0 for all the channels L. This is a natural extension of the well-known boxdiscretization method for a spherical system. We notice that Johnson [11.57]has advocated to use det(φ(r)) as a generalized concept of node for a multi-channel wave function. We have recently used this method to find all thebound state solutions of a deformed Skyrme-Hartree-Fock mean field.

Figure 11.6 shows s.p. energies obtained with the box discretizationmethod as a function of the box size Rbox. One can clearly see that thereare two classes of s.p. levels. One consists of those whose energy changessignificantly as the box radius Rbox increases, and the other contains thosewhose energy is almost constant as a function of Rbox. This behavior is sim-ilar to that of box-discretized s.p. energies for a spherical system where thestabilized levels correspond to resonances [11.3]. This example shows that,in the deformed case it is also possible to generate a complete set of box-


Fig. 11.6. Neutron single-particle energies of 44S obtained with the box discretiza-tion method as a function of the box radius Rbox. The axial symmetry is assumed,with the deformation parameter β of 0.2. The spin projection onto the symmetryaxis and the parity are taken to be Kπ = 5/2+. The potential parameters are thesame as in Fig. 11.5.

discretized states and that some of these states can be a good approximationto multi-channel s.p. resonances.

The states obtained with the box discretization method are the eigenstatesof a potential which is the same as the original potential for r < Rbox, andinfinite for r ≥ Rbox. They thus form a complete set, and any regular functiondefined in the domain r < Rbox can be expanded on this basis. It appearsconvenient to introduce the eigenchannel wave functions [11.51] defined interms of the unitary matrix U of (11.44),

ΨE,a(r) ≡∑

L0

ΨE,L0(r)UL0a. (11.52)

We refer to the channels L as the physical channels, in order to distinguishthem from the eigenchannels a. Substituting (11.43) and (11.44), one can findthat the eigenchannel wave functions behave asymptotically (r →∞) as

ΨE,a(r)→ 1r

∑

L

√k

πE

i

2

{e−i(kr−lπ/2) − e2iδaei(kr−lπ/2)

}ULaYljm(r).

(11.53)For each eigenchannel, the asymptotic radial wave functions thus behave inthe same way for all angular momentum components, and the coupling isgreatly simplified. A numerical study of the behaviour of the eigenchannelwave function components near a s.p. resonance can be found in [11.46].


11.6 QRPA on Top of BCSwith the Constant Gap Approximation

In this section we present an earlier attempt to derive the QRPA equationson top of the HF+BCS calculations, with a self-consistent approach in thep-h channel. The HF and HFBCS methods in spherical nuclei with Skyrmeinteractions are well-known [11.30,11.58]. For the pairing interaction it ispossible to simply choose a constant gap given by [11.59]:

∆ = 12.A− 12 MeV . (11.54)

In a more realistic treatment of the pairing, the gap would depend on thes.p. state considered and it would tend to zero when the subshell is far fromthe Fermi level. Thus, in the constant gap approximation it is necessaryto introduce a cut-off in the s.p. space. Above this cutoff subshells don’tparticipate to the pairing effect. For instance, in the case of oxygen isotopes,we choose the BCS subspace to include the 1s, 1p and 2s− 1d major shells.

The continuous part of the s.p. spectrum is discretized by diagonaliz-ing the HF hamiltonian on a harmonic oscillator basis [11.60]. To generalizethe HF-RPA to the QRPA model we follow the standard procedure [11.61].Denoting by c†

α, cα the creation and annihilation operators of a particle ina HF state α = (jα,mα) and by a†

α, aα the corresponding operators for aquasiparticle state, we have:

a†jαmα

= uαc†jαmα

− vα(−1)jα+mαcjα−mα , (11.55)

where uα and vα are the BCS amplitudes.One can then build the two-quasiparticle creation operators in an angular

momentum coupled scheme:

C†αβ(JM) = (1 + δαβ)−1/2

∑

mαmβ

(jαjβmαmβ |JM)a†αa

†β . (11.56)

In QRPA the nuclear excitations correspond to phonon operators which arelinear combinations of two-quasiparticle creation and annihilation operators:

Qν†(JM) =∑

α≥βXναβ(J)C†

αβ(JM) + (−1)MY ναβ(J)Cαβ(J −M) . (11.57)

Making use of the condition that the QRPA ground state |0〉 is a vacuum ofphonon:

Qν(JM)|0〉 = 0 , (11.58)

one can then derive the QRPA equations in configuration space, whose solu-tions yield the excitation energies Eν and amplitudes Xν

αβ , Yναβ of the excited

states. In the present QRPA calculations the pairing force is not taken intoaccount as a residual interaction between quasiparticles.


An important quantity that characterizes a given state ν = (Eν , LJ) isits transition density:

δρν(r) ≡ 〈ν|∑

i

δ(r− ri)|0〉 , (11.59)

and a similar definition of the neutron (proton) transition density δρνn (δρνp)with the summation in (11.59) restricted to neutrons (protons). In QRPAthe radial part of the transition density is:

δρν(r) =∑

α≥βϕα(r)ϕ∗

β(r) < β||YL0||α > {Xναβ(J)− Y ν

αβ(J)}{uαvβ + (−1)Jvαuβ} ,

(11.60)where ϕα(r) is the radial part of the wavefunction of the quasiparticle stateα.

The neutron and proton matrix elements M =< ν|rLYL0|0 > of a multi-pole operator are obtained by integrating the corresponding transition den-sities over r:

Mn,p =∫

δρνn,p(r)rL+2dr , (11.61)

and the reduced electric multipole transition probabilities are calculated as

B(EL)n,p = |Mn,p|2 . (11.62)

In order to evaluate the validity of the calculations, a direct comparisonof calculated and measured 32,34S charge ground state and proton transitiondensities [11.62–11.65] for the 2+

1 state is displayed in Fig. 11.7. The directmeasurement of these densities was obtained through electron scattering.The microscopic calculations are in excellent agreement with the experimen-tal data. This overall agreement points to the validity of the HF+BCS/QRPAmodel at least in the case of sd-shell stable isotopes such as Sulfur and Argonnuclei. It is moreover interesting to check the validity of our model over theentire isotopic chains. Experimental and calculated values of the 2+

1 energiesare displayed in Table 11.3 for the sulfur chain. The QRPA overestimatesthe excitation energies by around 700 keV, but the evolution along the iso-topic chain is very well reproduced. We have also investigated the integratedB(E2) quantity, which is directly measured using electromagnetic probes.Calculated B(E2) values are compared to experimental results in Table 11.3.The agreement with the experimental values is acceptable within the exper-imental errors bars, except for 36S, which corresponds to the N = 20 shellclosure. The QRPA calculations predict higher 2+

1 energies and lower B(E2)values for these nuclei than for their neighbors, giving a hint of this shelleffect, but the experimental variations are more pronounced than the pre-dictions. Additionally, it should be noted that the 36S nucleus is known topresent a puzzle [11.66] because several microscopic models such as configu-ration mixing and the shell-model, are unable to reproduce both its B(E2)value and its inelastic proton scattering angular distribution.


Table 11.3. Calculated proton and neutron r.m.s. radii, measured and calculatedE2+

1energies (MeV), measured and calculated B(E2) (e2.fm4) for the Sulfur iso-

topes. The r.m.s. values are given for HF+BCS and QRPA calculations whereas theE2+

1and B(E2) calculated values are given both for the QRPA and the configuration

mixing (CM) calculations. The measured values are taken from [11.67–11.69].

rp(fm) rn(fm) E2+exp E2+

QRPA E2+CM B(E2)exp B(E2)QRPA B(E2)CM

30S 3.15 3.02 2.24 2.79 2.46 320 ± 40 327 34432S 3.16 3.12 2.21 2.94 2.00 300 ± 13 294 30934S 3.18 3.21 2.12 2.65 2.34 212 ± 12 256 26136S 3.20 3.29 3.29 3.46 2.41 96 ± 26 241 28938S 3.23 3.37 1.29 2.19 2.17 235 ± 30 325 28740S 3.24 3.44 0.89 1.54 1.71 334 ± 36 431 33142S 3.26 3.50 0.89 1.75 1.26 397 ± 41 396 36844S 3.28 3.57 1.29 2.15 1.52 314 ± 88 331 343

Fig. 11.7. Comparison between experimental (solid lines) and calculated (dashedlines) charge (upper panels) and 2+

1 proton transition (lower panels) densities for 32S(left) and 34S (right). Experimental data are taken from [33–36]. The experimentalerrors are of the order of 10% for small radii (r < 2 fm) and of 1% for larger radii.

11.7 QRPA on Top of HFBwith Exact Continuum Treatment

Nuclei close to the drip-line are characterized by a small nucleon separationenergy, and the excited states are strongly influenced by the coupling withthe quasiparticle (qp) continuum configurations. Among the configurations ofparticular interest are the two-qp states in which one or both quasiparticlesare in the continuum. In order to describe such excited states within QRPAone needs a proper treatment of the continuum coupling, which is missing inthe usual QRPA calculations based on a discrete qp spectrum. In the pastyears several attempts [11.70–11.72] have been made to describe consistentlyboth the pairing correlations and the continuum coupling within QRPA.


Thus, in [11.71] a QRPA approach was recently developed in which the effectof the continuum is calculated exactly for the p-h excitations whereas in thep-h channel the active space is limited to the bound states close to the Fermilevel. A continuum qp linear response approach in which the continuum isincluded also in the p-h channel was studied in [11.72], but in the calculationsthe ground state mean field is fixed independently of the residual interaction.

The aim of this section is to present a model which preserves the self-consistency and includes the exact continuum treatment. The ground state iscalculated using the continuum HFB approach [11.33] depicted in Sect. 11.3,with the mean field and the pairing field described by a Skyrme interactionand a density dependent delta force, respectively. Based on the same HFBenergy functional we derive the QRPA response function in coordinate space.The QRPA response is constructed by using real energy solutions for thecontinuum HFB spectrum.

The coordinate space formalism is naturally adapted to treat properly thecoupling to the continuum states. The QRPA equations are derived in coor-dinate space as the small amplitude limit of the perturbed time-dependentHFB equations. We start from the time-dependent HFB (TDHFB) equations[11.13], which allows to derive the QRPA equations in an alternate way fromthe previous section. Only the main points of the demonstration will be em-phasized. The reader interested by the detailed demonstration should referto [11.72,11.73]. The aim is to calculate QRPA response functions by meansof Green functions formalism. Starting from the perturbed time-dependentHFB (TDHFB) equations [11.13]:

i�∂R∂t

= [H(t) + F(t),R(t)] (11.63)

where R, H are the time-dependent generalized density and HFB hamilto-nian respectively, and F the external oscillating field, we obtain in the smallamplitude limit:

�ωR′ = [H′,R0] + [H0,R′] + [F,R0] (11.64)

where ’ stands for the perturbed quantity.The generalized density variation R’ is expressed in term of 3 quantities,

namely ρ′, κ′ and κ′, and is rewritten as a column vector:

ρ′ =

ρ′

κ′

κ′

(11.65)

In the following we will denote ρ′ in bold face the column vector defined in(11.65).

The variations of the particle and pairing densities of (11.65) in coordinaterepresentation are defined by:

ρ′ (rσ) =⟨0|c† (rσ) c (rσ) |′

⟩(11.66)


κ′ (rσ) = 〈0|c (rσ) c (rσ) |′〉 (11.67)

κ′ (rσ) =⟨0|c† (rσ) c† (rσ) |′

⟩(11.68)

where c† (rσ) is the particle creation operator in coordinate space andc† (rσ)= −2σc† (r− σ) is its time reversed counterpart.

Instead of the variation of one quantity in RPA (ρ′), we therefore have toknow the variations of three independent quantities in QRPA. It should benoted that in this three dimensional space, the first dimension represents thep-h subspace, the second the particle-particle (p-p) one, and the third thehole-hole (h-h) one. The response matrix have 9 coupled elements in QRPA,compared to one in the RPA formalism.

Similarly the variation of the HFB hamiltonian is expressed in terms ofthe second derivatives of the HFB energy functional E [ρ, κ, κ] with respect tothe densities:

H′ = Vρ′ (11.69)

where V is the residual interaction matrix, namely:

Vαβ(rσ, r′σ′) =∂2E

∂ρβ(r′σ′)∂ρα(rσ), α, β = 1, 2, 3. (11.70)

Here, the notation α means that whenever α is 2 or 3 then α is 3 or 2.The quantity of interest is the QRPA Green function G, which relates the

perturbing external field to the density change:

ρ′ = GF . (11.71)

Inputing the 3 equations above in (11.64), we land on the so-called Bethe-Salpeter equation:

G = (1−G0V)−1 G0 = G0 + G0VG (11.72)

which is a set of 9x9 coupled equations.In (11.72) the unperturbed Green function G0 appears, namely:

G0αβ(rσ, r′σ′;ω) =

∑

ij

Uα1ij (rσ)U∗β1

ij (r′σ′)�ω − (Ei + Ej) + iη

−Uα2ij (rσ)U∗β2

ij (r′σ′)�ω + (Ei + Ej) + iη

(11.73)

where Ei are the single qp energies and Uij are 3 by 2 matrices calculated fromthe U and V HFB wave functions [11.73]. It should be noted that the exactcontinuum treatment is performed in the summation of (11.73): it becomesan integral if the quasiparticle belong to the continuous part of the singlequasiparticle spectrum.


In the case of transitions from the ground state to excited states within thesame nucleus, only the (ph,ph) component of G is acting. If the interactiondoes not depend on spin variables the strength function is thus given by:

S(ω) = − 1πIm

∫F 11∗(r)G11(r, r′;ω)F 11(r′)dr dr′ (11.74)

In the case of transition from the ground state of a nucleus with A nucleonto a state of a nucleus with A+2 nucleon, the (pp,pp) component of G is usedinstead (see next section).

We apply our formalism to the calculation of neutron-rich oxygen isotopes18,20,22,24O. The calculations are performed assuming spherical symmetry.The ground states are calculated within the continuum HFB approach [11.33]where the continuum is treated exactly. The HFB equations are solved incoordinate space with a step of 0.25 fm for the radial coordinate. In theHFB the mean field quantities are calculated by using the Skyrme interactionSLy4 [11.19], while for the pairing interaction we take a zero-range density-dependent interaction given by:

Vpair = V0

[1−(

ρ(r)ρ0

)α]δ (r1 − r2) (11.75)

where V0, ρ0 and α are the parameters of the force. Due to its zero-range thisforce should be used in the HFB calculations with a cutoff in qp energy. Tominimize the number of free parameters, we use a prescription which relatesthe energy cutoff with the V0 value in finite nuclei [11.73]. With this prescrip-tion the calculated HFB neutron pairing gap ∆n remains constant for eachcouple (V0, Ecutoff ). In the HFB calculations we choose a qp cutoff energyequal to 50 MeV. The parameter ρ0 is set to the usual saturation density, 0.16fm−3. The value of the parameter α is chosen so as to reproduce the trend ofthe experimental gap in neutron rich oxygen isotopes. Note that this trendis at variance with the empirical rule ∆ = 12/

√AMeV which underlines the

limited application domain of the constant gap formula, especially for lightnuclei.

In the QRPA calculations the residual interaction is derived from the in-teraction used in HFB. The residual interaction corresponding to the velocity-dependent terms of the Skyrme force is approximated in the (ph,ph) subspaceby its Landau-Migdal limit [11.74]. The strength distribution is calculateduntil ωMax=50 MeV, with a step of 100 keV and an averaging width η=150keV. In a fully consistent calculation the spurious center-of-mass state shouldcome out at zero energy. Because of the Landau-Migdal form of the inter-action adopted here the consistency between mean field and residual qp in-teraction is broken and the spurious state becomes imaginary. We cure thisdefect by renormalizing the residual interaction by a factor α. We find thatin all cases the spurious state Jπ = 1− comes out at zero energy for α=0.80.Note that recently a fully self-consistent QRPA model (without using theLandau-Migdal approximation) have been developed [11.75]. However, in the


calculations [11.75], the continuum is discretized with box boundary condi-tions.

The results are shown in the left panel of Fig. 11.8 for 22O. One can iden-tify a strong low-lying state and the giant quadrupole resonance (GQR). Boxdiscretization calculations have also been performed in order to test the boxboundary condition approximation. One can see that only the low-lying stateis nearly insensitive whereas the structure of the GQR is more affected by theway the continuum is treated. This shows the necessity of the exact contin-uum treatment in order to study the giant resonances in neutron-rich oxygenisotopes. In order to investigate the effect of the density dependence of thepairing interaction we have also calculated the strength distribution of 18Owith a density-independent interaction, i.e., ρ0 going to infinity in (11.75).In the HFB calculation, the V0 parameter has been chosen to reproduce theexperimental gap of 18O, V0=-220 MeV.fm3 (in this case the prescription of[11.73] is no longer applied). Fig. 11.8 compares the results in 18O calculatedwith the density-dependent and density-independent interactions. The effectof the density dependence is to increase the energies of the 2+ states, and toslightly lower the strength of the low-lying states.

Fig. 11.8. Left: Isoscalar strength function calculated in continuum-QRPA (solidline) and with a box discretization (dashed line) for the 22O nucleus. Right: Isoscalarstrength function calculated with a density-independent pairing interaction (solidline) and density-dependent pairing interaction (dashed line) with box boundaryconditions for the 18O nucleus.

11.8 The Particle-Particle Responseand Transfer Reactions

Due to the concept of quasiparticle, the QRPA unifies on the same groundthe p-h RPA and the p-p RPA with the inclusion of the pairing effects.


The quantity of interest is the strength function describing the two-particletransfer from the ground state of a nucleus with A nucleons to the excitedstates of a nucleus with A+2 nucleons. This strength function is, comparedto (11.74)

S(ω) = − 1πIm

∫F 12∗(r)G22(r, r′;ω)F 12(r′)dr dr′ (11.76)

where G22 denotes the (pp,pp) component of the Green’s function.Two-neutron transfer reactions such as (t,p) or (p,t) have been used for

many years in order to study the nuclear pairing correlations (for a recentreview see [11.76] ). The corresponding pair transfer modes are usually de-scribed in terms of pairing vibrations or pairing rotations [11.77,11.78]. Highenergy collective pairing modes, called giant pairing vibrations (gpv), werealso predicted [11.79,11.80], but they have not been observed yet. Recentlythere is a renewed interest for the study of two-neutron transfer reactionswith weakly bound exotic nuclei. These reactions would provide valuableinformation about the pairing correlations in nuclei far from stability. Theuse of two-neutron transfer reactions with exotic nuclei can also increase thechance of exciting the gpv mode, as discussed recently in [11.81].

The strength function for the two-neutron transfer is calculated using(11.76). For the radial function F 22(r) we take the form rL, which is equalto the unity for the L=0 pair transfer mode considered here [11.81]. Theunperturbed Green’s function is calculated with an averaging interval η equalto 0.15 (1.0) MeV for excitations energies below (above) 11 MeV.

The results for the strength function corresponding to a neutron pairtransferred to 22O are shown in Fig. 11.9 (left panel, dashed lines). Thesubshell d5/2 is essentially blocked for the pair transfer. Therefore in thisnucleus we can clearly identify two peaks below 11 MeV, corresponding toa pair transferred to the states 2s1/2 and 2d3/2. A broad resonant structurearound 20 MeV which is built mainly upon the s.p. resonant sate 1f7/2. Thistwo-quasiparticle broad resonance has the characteristics of a giant pairingvibration [11.79–11.81]. The continuum treatment affects the magnitude ofthe lowest state. This is due to the collective nature of this state, since un-bound configurations such as the (1d3/2)2 contribute to this low-lying state.This points to the necessity to use exact continuum calculations even to pre-dict transitions towards low-lying states. The state at 9.8 MeV on the 22O +2n spectrum is embedded in the continuum and it is naturally more affectedby the continuum treatment. The left panel figure also displays in solid linethe unperturbed G0 response. The first two peaks located at 2.1 MeV and10.8 MeV in the G0 response correspond to the addition of two neutron qp onthe (2s1/2)2 and (1d3/2)2 subshells, respectively. Note a sensitive change inthe spreading widths of the two-quasiparticle resonant states when the resid-ual interaction is turned on. Thus, due to the mixing of the configurations(1f7/2)2 and (1d3/2)2 by the residual interaction, the broad peak around18 MeV becomes narrower and the narrow peak around 10 MeV becomes


wider. This is a general effect which appears whenever in the two-body wavefunctions wide and narrow s.p. resonant states are mixed together [11.82].

Fig. 11.9. Left: the response function for the two-neutron transfer on 22O. Theunperturbed response is in solid line and the QRPA response in dashed line. Right:DWBA calculations for the states located at 1.6 MeV (up) and 9.8 MeV (down).The solid (dashed) line corresponds to the QRPA results obtained with box (exact)boundary conditions. The calculations corresponds to the system 22O+2n.

In the nuclear response theory the transition from the ground state to theexcited state |ν〉 of the same nucleus is determined by the transition densitydefined by:

ρν (rσ) =⟨0|c† (rσ) c (rσ) |ν

⟩(11.77)

where c† (rσ) is the particle creation operator in coordinate space.The corresponding quantity for describing pair transfer processes is the

pair transition density defined by:

κν (rσ) = 〈0|c (rσ) c (rσ) |ν〉 (11.78)

The pair transition density defined above determines the transition from theground state of a nuclei with A nucleons to a state |ν〉 of a nucleus with A+2nucleons. This quantity is the output of continuum-QRPA calculations.

The form factor for the pair transfer is obtained by folding the pair tran-sition density κν (11.78) with the interaction acting between the transferredpair and the residual fragment [11.76]. In the zero-range approximation usedhere the dependence of this interaction on the relative distance between thepair and the fragment is taken as a delta force. Therefore in this approx-imation the pair transition density (11.78) coincides with the form factor[11.83]. The 22O+t Becchetti and Greenlees optical potential [11.84] is usedfor the entrance channel and the 22O+p Becchetti and Greenlees [11.85] for


the exit channel, in order to calculate the DWBA cross section. The DWBAcalculations are performed with the DWUCK4 [11.86] code and using thezero-range approximation. In this approximation the two-neutrons and theresidual fragment are located at the same point and the range function isexpressed through a simple constant D0 [11.83]. For the (t,p) reaction wetake D0 = 2.43 104 MeV2 fm3 [11.87]. This value relies on measurements ofthe 2n+p system and may be subject to uncertainties [11.83]. As discussedin [11.78,11.88,11.83], the shape of the angular distribution is usually de-scribed correctly by the zero-range approximation, but not its magnitude,which is generally underestimated by a large amount. Therefore we focus ourdiscussion not on the absolute values of the cross sections, but rather on therelative values obtained using different form factors, with and without theexact continuum treatment.

In order to see the continuum effect on the cross section, the right panelof Fig. 11.9 shows the 22O(t,p) angular distribution at 15 MeV/nucleon. Asexpected, the effect is small for the transition towards the state located at1.6 MeV, built mainly upon bound qp states, and large for the state locatedat 9.8 MeV, which is built mainly upon narrow resonant qp states. The con-tinuum treatment has also some effect on the form factor corresponding tothe high energy mode around 18 MeV, especially at small values of the nu-clear radius. However its effect on the global cross section remains negligible,about 3%.

11.9 Outlook: QRPA Calculations for the Next Decade

The next major issue in QRPA is most probably to include the deformationdegree of freedom in such models, with still preserving the self-consistencyand the exact continuum treatment. Such developments are in progress (seeSect. 11.5), and will be surely available in the next decade. Large scale QRPAcalculations on the whole nuclear chart are also more and more feasible forastrophysical applications [11.89] such as r-process abundances predictions.Such microscopic approaches to astrophysical problems are called for growingin importance in the forthcoming years. Finally, let us mention phonon cou-pling calculations [11.90] which is a first step towards second-QRPA models.

Acknowledgments

These notes summarize the work that we have done in the recent years incollaboration with Nicu Sandulescu, Marcella Grasso, Kouichi Hagino andRoberto Liotta. We are much indebted to them for the numerous lively dis-cussions and for their essential contribution to the work.


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