Gamow-Teller Strength distributions in {}^{76}\textrm{Ge}, {}^{76,82}\textrm{Se}, and {}^{90,92}\textrm{Zr} by the Deformed Proton-neutron QRPA

# Gamow-Teller Strength distributions in 76Ge, 76,82Se, and 90,92Zr by the Deformed Proton-neutron QRPA

Eunja Ha 111ejha@ssu.ac.kr and Myung-Ki Cheoun 222cheoun@ssu.ac.kr Department of Physics, Soongsil University, Seoul 156-743, Korea
July 14, 2019
###### Abstract

We developed the deformed proton-neutron quasiparticle random phase approximation (QRPA) and applied to the evaluation of the Gamow-Teller (GT) transition strength distributions including high-lying excited states, which data becomes recently available beyond one or two nucleon threshold by charge exchange reactions using hundreds of MeV projectiles. Our calculations started with single-particle states calculated by a deformed axially symmetric Woods-Saxon potential. Neutron-neutron and proton-proton pairing correlations are explicitly taken into account at the deformed Bardeen Cooper Schriffer theory. Ground state correlations, and two-particle and two-hole mixing states are included in the deformed QRPA. In this work, we use a realistic two-body interaction given by the Brueckner -matrix based on the CD Bonn potential to reduce the ambiguity on the nucleon-nucleon interactions inside nuclei. We applied our formalism to the GT transition strengths for Ge, Se, and , and compared to available experimental data. The GT strength distributions were sensitive on the deformation parameter as well as its sign, i.e., oblate or prolate. The Ikeda sum rule, which is usually thought to be satisfied under the one-body current approximation irrespective of nucleon models, is used to test our numerical calculations and shown to be satisfied without introducing the quenching factor, if high-lying GT excited states are properly taken into account. Most of the GT strength distributions of the nuclei considered in this work turn out to have the high-lying GT excited states beyond one nucleon threshold, which are shown to be consistent with available experimental data.

###### pacs:
23.40.Hc, 21.60.Jz, 26.50.+x

## I Introduction

In the core collapsing supernovae (SNe), medium and heavy elements are believed to be produced by rapid and slow successive neutron capture reactions, dubbed as r-process and s-process, respectively. In these processes, many unstable neutron-rich nuclei are produced iteratively and decay to more stable nuclei at their turning points in the nuclear chart. These r- and s-processes play vital roles of understanding abundances of the medium and heavy nuclei in the cosmos Haya04 ().

Since most of the nuclei produced in the processes are thought to be more or less deformed, we need to explicitly take into account the deformation in the nuclear structure and their effects on relevant nuclear reactions in the network calculations of the processes. One interesting process associated with the deformed nuclei may be the rapid proton process (rp-process), which is thought to be occurred on the binary star system composed of a massive compact star and a companion star. Because of the strong gravitation on the massive star surface, one expects hydrogen rich mass-flow from the companion star. Since the high density and low temperature on the neutron star crust make electrons degenerated, nuclear beta decays of unstable nuclei may be blocked by the degeneration, while stable nuclei may become unstable with respect to the beta decay. This physical situation gives rise to the nuclear pycno-reactions Haen91 (), where the deformation could be of practical importance on the understanding of the rp-process.

Up to now, many theoretical approaches to understand the nuclear structure are based on the spherical symmetry Suhonen14 (). In order to describe the neutron-rich nuclei and their relevant nuclear reactions in the nuclear processes, one needs to develop theoretical frameworks including explicitly the deformation simkovic (); saleh (); sarriguren_a (); sarriguren_b (); sarriguren_c (); sarriguren_d (). Ref. simkovic () exploited the Nilsson basis for deformed quasiparticle random phase approximation (DQRPA). But two-body interactions inside nuclei were derived from the effective separable force. A realistic two-body interaction derived from the realistic nucleon-nucleon (N-N) force in free space is firstly applied to and decays within the DQRPA at Ref. saleh (), where neutron-neutron () and proton-proton () pairing correlations are considered at the BCS stage.

There are many calculations regarding the deformation effects on the GT strength distributions sarriguren_a (); sarriguren_b (); sarriguren_c (); sarriguren_d (). Ref. sarriguren_a () considered the effect with the HF+RPA model using the Skyrme force. These calculations were extended to the deformed QRPA by exploiting the effective separable force sarriguren_b () or various effective Skyrme forces sarriguren_c (); sarriguren_d (). But, for more ab initio calculations, it would be more desirable to start from the realistic N-N force in free space and solve the Bethe-Salpeter equation for the N-N interaction in nuclei, i.e. Brueckner -matrix, as used at Ref. saleh ().

In this work, we extend our previous spherical QRPA based on the spherical symmetry Ch93 () to the DQRPA Ha13_1 (); Ha13_2 (). The spherical QRPA Ch93 () has been exploited as a useful framework for describing the neutrino-induced reactions sensitive on the nuclear structure of medium-heavy and heavy nuclei ch10 (). For these nuclei, the application of the shell model may have actual limits because of tremendous increase of mixing configurations as the mass number increases.

This paper is organized as follows. In Sec. II, we introduce detailed formalism for the DQRPA and the Gamow-Teller (GT) strength. Numerical results and related discussions are presented in detail at Sec. III. Summary and conclusions are addressed at Sec. IV.

## Ii Formalism

### ii.1 Total Hamiltonian

We start from the following nuclear Hamiltonian

 H=H0+Hint , (1) H0=∑ρααα′ϵρααα′c†ρααα′cρααα′    , Hint=∑ραρβργρδ,αβγδ, α′β′γ′δ′Vρααα′ρβββ′ργγγ′ρδδδ′c†ρααα′c†ρβββ′cρδδδ′cργγγ′,

where the interaction matrix is the anti-symmetrized interaction with the Baranger Hamiltonian Baran63 () in which two factors, from J and T coupling, are included. Greek letters denote proton or neutron single particle states with a projection of the total angular momentum on the nuclear symmetry axis. The projection is treated as the only good quantum number in the deformed basis. The is a sign of the total angular momentum projection of the state. The isospin of particles is denoted as a Greek letter with prime , while the isospin of quasiparticles is expressed as a Greek letter with double prime as shown later.

Therefore, the operator () in Eq. (1) stands for a usual creation (destruction) operator of the real particle in a state of with the angular momentum projection and the isospin . Since we assume the time-reversal symmetry, our intrinsic states are twofold-degenerate, i.e. state and its time-reversed state . means the single particle (s.p.) state energies.

In the cylindrical coordinate, eigenfunctions of a s.p. state and its time-reversed state in the deformed Woods-Saxon potential are expressed as follows

 |αρα=+1>= ∑Nnz[b(+)NnzΩα |N,nz,Λα,Ωα=Λα+1/2> + b(−)NnzΩα|N,nz,Λα+1,Ωα=Λα+1−1/2>], |αρα=−1>= ∑Nnz[b(+)NnzΩα |N,nz,−Λα,Ωα=−Λα−1/2> − b(−)NnzΩα |N,nz,−Λα−1,Ωα=−Λα−1+1/2>],

where ( ) is a major shell number, and and are numbers of nodes of the deformed harmonic oscillator wave function in and direction, respectively. is a projection of orbital angular momentum onto the nuclear symmetric axis . Coefficients and are obtained by the eigenvalue equation of the total Hamiltonian in the Nilsson basis. The 2nd terms in Eq. (II.1) have the same projection value as the 1st term, but retain another orbital angular momentum because of a flipped spin. Particle model space is exploited up to for deformed basis. In the expansion of the deformed basis to a spherical basis, we considered up to in the spherical basis.

Single particle spectrum obtained by the deformed Woods-Saxon potential is sensitive on the deformation parameter defined as

 R(θ)=R0(1+β2Y20(θ)+β4Y40(θ)) , (3)

where fm for the sharp-cut radius Ring (), and and are spherical harmonics. The customary parameter, , used in the deformed harmonic oscillator is related to as at the leading order. In the cylindrical Woods-Saxon potential, we use the following nuclear and spin-orbit potentials damgaard ()

where l is a distance function of a given point to the nuclear surface represented by Eq. (3). a and are the diffuseness parameter and the strength of spin-orbit potential, respectively. In this work, we assume and use the cylindrical Woods-Saxon potential parameters by Nojarov Nojarov (). We transform the Hamiltonian represented by real particles in Eq. (1) to the quasiparticle representation through the Hartree Fock Bogoliubob (HFB) transformation,

 a†ρααα′′=∑ρβββ′(uαα′′ββ′c†ρβββ′+vαα′′ββ′cρβ¯ββ′), aρα¯αα′′=∑ρβββ′(u¯αα′′¯ββ′cρβ¯ββ′−v¯αα′′¯ββ′c†ρβββ′). (5)

Since our formalism is intended to include the neutron-proton () pairing correlations, we denote the isospin of quasiparticles as , while the isospin of real particles is denoted as . We assume the time reversal symmetry, which means and , and do not allow mixing of different single particle states ( and ) to the quaisparticle in the deformed state. But, in the spherical state, the quasiparticle states turn out to be mixed with different particle states because each deformed state (basis) is represented by a linear combination of the spherical state (basis). The HFB transformation for each is then reduced to the following form

 ⎛⎜ ⎜ ⎜ ⎜ ⎜⎝a†1a†2a¯1a¯2⎞⎟ ⎟ ⎟ ⎟ ⎟⎠α=⎛⎜ ⎜ ⎜ ⎜⎝u1pu1nv1pv1nu2pu2nv2pv2n−v1p−v1nu1pu1n−v2p−v2nu2pu2n⎞⎟ ⎟ ⎟ ⎟⎠α⎛⎜ ⎜ ⎜ ⎜ ⎜⎝c†1c†2c¯1c¯2⎞⎟ ⎟ ⎟ ⎟ ⎟⎠α (6)

and the Hamiltonian is expressed in terms of the quasiparticle as follows

 H′=H′0+∑ρααα′′Eαα′′a†ρααα′′aρααα′′+Hqp.int . (7)

Finally, using the transformation of Eq. (6), we obtain the following deformed HFB equation:

 ⎛⎜ ⎜ ⎜ ⎜⎝ϵp−λp0Δp¯pΔp¯n0ϵn−λnΔn¯pΔn¯nΔp¯pΔp¯n−ϵp+λp0Δn¯pΔn¯n0−ϵn+λn⎞⎟ ⎟ ⎟ ⎟⎠α⎛⎜ ⎜ ⎜ ⎜⎝uα′′puα′′nvα′′pvα′′n⎞⎟ ⎟ ⎟ ⎟⎠α=Eαα′′⎛⎜ ⎜ ⎜ ⎜⎝uα′′puα′′nvα′′pvα′′n⎞⎟ ⎟ ⎟ ⎟⎠α, (8)

where is the energy of a quasiparticle with the isospin quantum number in the state . Pairing potentials in a deformed basis are detailed in the next subsection B. In the present calculation, we neglect . This equation therefore reduces to the standard Deformed Hartree Fock Bogoliubov (DHFB) equation simkovic ().

### ii.2 Spherical and deformed wave functions for a single particle state

Since various mathematical theorems regarding quantum numbers may not be easily used in the deformed basis, it is more convenient to play in the spherical basis. In addition, we exploit the -matrix based on the Bonn potential in order to reduce plausible ambiguities on the N-N interaction inside a deformed nucleus. Since the -matrix is calculated on the spherical basis, we need to represent the -matrix in terms of the deformed basis. Here we present regarding how to transform the deformed wave function to the spherical one.

The deformed harmonic oscillator wave function, in Eq. (II.1) can be expanded in terms of the spherical harmonic oscillator wave function

 |NnzΛα> |Σ>=∑N0=N,N±2,N±4,...∑l=N0,N0−2,N0−4,...AN0l, nr=N0−l2NnzΛ |N0lΛα> |Σ>, (9) |N0lΛα> |Σ>=∑jCjΩαlΛα12Σ|N0lj Ωα> ,

where the spatial overlap integral is calculated numerically in the spherical coordinate system. is the Clebsch-Gordan coefficient of the coupling of the orbital and spin angular momentum ( to the total angular momentum with the projection . Therefore, the expansion of the deformed state into the spherical state can be simply written as

 |αΩα>=∑aBαa |aΩα> , Bαa=∑NnzΣCjΩαlΛ12ΣAN0lNnzΛ bNnzΣ , (10)

where is the expansion coefficient. Here indicates quantum numbers () of a nucleon state, where major quantum number is related to the radial quantum number . Naturally, the expansion coefficient depends on the deformation parameter . In order to figure out the dependence, in Fig. 1, we show an example of the expansion, where the for state is plotted in terms of spherical states (basis) for = 0.01, 0.1 and 0.3 cases. The state turns out to be composed mainly of and states in the spherical basis.

The pairing potentials in the DHFB Eq. (8) are calculated in the deformed basis by using the -matrix calculated from the realistic Bonn CD potential for the N-N interaction in the following way

 Δαp¯αp=−12∑J,cgppairFJ0αa¯αaFJ0γc¯γcG(aacc,J)(u∗1pcv1pc+u∗2pcv2pc) , (11)

where is introduced for the -matrix representation in the deformed basis. Here , which is a projection number of the total angular momentum onto the axis, is selected at the DHFB stage because we consider pairings of the quasiparticles at and states. is the two-body (pairwise) scattering matrix in the spherical basis taking into account all possible scattering of nucleon pairs above Fermi surface.

In this work, we include all possible values which have projection. is the same as Eq. (11) with replacement of by . In order to renormalize the -matrix, strength parameters, and are multiplied to the -matrix Ch93 () by adjusting the pairing potentials to the empirical pairing potentials, and . The empirical pairing potentials of protons and neutrons are evaluated by the following symmetric five term formula for neighboring nuclei

 Δpemp = 18[M(Z+2,N)−4M(Z+1,N)+6M(Z,N) −4M(Z−1,N)+M(Z−2,N)] ,
 Δnemp = 18[M(Z,N+2)−4M(Z,N+1)+6M(Z,N) −4M(Z,N−1)+M(Z,N−2)] ,

where signs in the +(–) stand for even(odd) mass nuclei. As for masses in Eqs. (II.2) and (II.2), we use empirical masses.

### ii.3 Description of an excited state by the DQRPA

We take the ground state of an even-even target nucleus as the DBCS vacuum for a quasiparticle. In the following, we show how to generate an excited state in a deformed nucleus. Since deformed nuclei have two different frames, laboratory and intrinsic frames, we need to consider a relationship of the two frames. The GT excited state in the intrinsic frame of even-even nuclei, which is described by operating a phonon operator to the QRPA vacuum , can be transformed to the wave function in the laboratory frame by using the Wigner function as follows

 |1M(K),m> = √38π2 D1MK(ϕ,θ,ψ)Q†m,K|QRPA>  (for K=0), (14) |1M(K),m> = √316π2 [D1MK(ϕ,θ,ψ)Q†m,K +(−1)1+KD1M−K(ϕ,θ,ψ)Q†m,−K]|QRPA>  (for K=±1).

Here is the correlated ground state in the intrinsic frame, and is the proton-neutron DQRPA wave function for the Gamow-Teller excited state in the laboratory frame. () is a projection of the total angular momentum onto the (the nuclear symmetry) axis, where the is accepted as only a good quantum number in deformed nuclei. The DQRPA phonon creation operator acting on the ground state is given as

 Q†m,K=∑ρααα′′ρβββ′′[Xm(αα′′ββ′′K)A†(αα′′ββ′′K)−Ym(αα′′ββ′′K)~A(αα′′ββ′′K)], (15)

with pairing creation and annihilation operators composed by two quasiparticles defined as

 A†(αα′′ββ′′K)=a†αα′′a†ββ′′,   ~A(αα′′ββ′′K)=aββ′′aαα′′. (16)

The quasiparticle pairs in two-particle states, and with the parity , are chosen by the selection rules

 K=0 Ωα−Ωβ=0,  παπβ=1 (17) K=1 ⎧⎪⎨⎪⎩Ωα−Ωβ=1,παπβ=1−Ωα+Ωβ=1,παπβ=1Ωα+Ωβ=1,παπβ=1.

We note that our our quasiparticle pairs include all combinations of particle states and their time reversed states and and modes are degenerated through the time reversal symmetry. Two-body wave functions in the deformed basis are calculated from the spherical basis as follows

 |α¯β> = ∑abJFJKαa¯βb|ab,JK>, (18) |¯αβ> = ∑abJFJK¯αaβb|ab,JK>, |αβ> = ∑abJFJKαaβb|ab,JK>,

where two body wave function in the spherical basis, , and transformation coefficient are given as

 |ab, JK>=∑JCJKjaΩajbΩb|aΩa>|bΩb> , FJK¯αaβb=Bαa Bβb(−1)ja−Ωα CJKja−ΩαjbΩβ , (19)

where the phase factor comes from the time-reversed state . The expansion coefficient for a single particle state, , is defined in Eq. (10).

### ii.4 Deformed QRPA equation

By taking the same approach as the derivation of the QRPA equation in Ref. Suhonen (), we obtain the Deformed QRPA (DQRPA) equation. But, in this paper, we present more general formalism, which includes the pairing correlations, for further study. They become important for the description neutron deficient (proton-rich) nuclei or light nuclei. Our DQRPA equation is finally given as in the deformed basis

 ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝ A1111αβγδ(K) A1122αβγδ(K) A1112αβγδ(K) B1111αβγδ(K) B1122αβγδ(K) B1112αβγδ(K) A2211αβγδ(K) A2222αβγδ(K) A2212αβγδ(K) B2211αβγδ(K) B2222αβγδ(K) B2212αβγδ(K) A1211αβγδ(K) A1222αβγδ(K) A1212αβγδ(K) B1211αβγδ(K) B1222αβγδ(K) B1212αβγδ(K)−B1111αβγδ(K)−B1122αβγδ(K)−B1112αβγδ(K)−A1111αβγδ(K)−A1122αβγδ(K)−A1112αβγδ(K)−B2211αβγδ(K)−B2222αβγδ(K)−B2212αβγδ(K)−A2211αβγδ(K)−A2222αβγδ(K)−A2212αβγδ(K)−B1211αβγδ(K)−B1222αβγδ(K)−B1212αβγδ(K)−A1211αβγδ(K)−A1222αβγδ(K)−A1212αβγδ(K)⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ (20) ×⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝~Xm(γ1δ1)K~Xm(γ2δ2)K~Xm(γ1δ2)K~Ym(γ1δ1)K~Ym(γ2δ2)K~Ym(γ1δ2)K⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠=ℏΩmK⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝~Xm(α1β1)K~Xm(α2β2)K~Xm(α1β2)K~Ym(α1β1)K~Ym(α2β2)K~Ym(α1β2)K⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ ,

where 1 and 2 denote isospins of quasiparticles i.e. quasiprotons and quasineutrons as denoted in previous sections. The amplitudes and , which stand for forward and backward going amplitudes from state to , are related to and in Eq. (20), where = 1 if and = , otherwise = Ch93 (). If we neglect the pairing, i.e. take only 1212 terms and in the matrices, Eq. (20) becomes the proton-neutron DQRPA at Ref. saleh ().

The A and B matrices in Eq. (20) are given by

 Aα′′β′′γ′′δ′′αβγδ(K)= (Eαα′′+Eββ′′)δαγδα′′γ′′δβδδβ′′δ′′−σαα′′ββ′′σγγ′′δδ′′ × ∑α′β′γ′δ′[−gpp(uαα′′α′uββ′′β′uγγ′′γ′uδδ′′δ′+vαα′′α′vββ′′β′vγγ′′γ′vδδ′′δ′) Vαα′ββ′, γγ′δδ′ − gph(uαα′′α′vββ′′β′uγγ′′γ′vδδ′′δ′+vαα′′α′uββ′′β′vγγ′′γ′uδδ′′δ′) Vαα′δδ′, γγ′ββ′ − gph(uαα′′α′vββ′′β′vγγ′′γ′uδδ′′δ′+vαα′′α′uββ′′β′uγγ′′γ′vδδ′′δ′) Vαα′γγ′, δδ′ββ′],
 Bα′′β′′γ′′δ′′αβγδ(K)= − σαα′′ββ′′σγγ′′δδ′′ × ∑α′β′γ′δ′[gpp(uαα′′α′uββ′′β′vγγ′′γ′vδδ′′δ′+vαα′′α′v¯ββ′′β′uγγ′′γ′u¯δδ′′δ′) Vαα′ββ′, γγ′δδ′ − gph(uαα′′α′vββ′′β′vγγ′′γ′uδδ′′δ′+vαα′′α′uββ′′β′uγγ′′γ′vδδ′′δ′) Vαα′δδ′, γγ′ββ′ − gph(uαα′′α′vββ′′β′uγγ′′γ′vδδ′′δ′+vαα′′α′uββ′′β′vγγ′′γ′uδδ′′δ′) Vαα′γγ′, δδ′ββ′],

where and coefficients are determined from DHFB calculation with the pairing strength parameters and adjusted to the empirical pairing gaps and , respectively. indicates the quasiparticle energy of the state with the quasiparticle isospin . The two body interactions and are particle-particle and particle-hole matrix elements of the residual N-N interaction , respectively, in the deformed state. They are calculated from the -matrix in the spherical basis as follows

 Vαα′ββ′, γγ′δδ′ = −∑J∑abcdFJKαaβbFJKγcδdG(aα′bβ′cγ′dδ′, J) , (23) Vαα′δδ′, γγ′ββ′ = ∑J∑abcdFJK′αaδdFJK′γcβbG(aα′dδ′cγ′bβ′, J) , Vαα′γγ′, δδ′ββ′ = ∑J∑abcdFJKαaγcFJKβbδdG(aα′cγ′dδ′bβ′, J) ,

where with . The -matrices are two body particle - particle matrix elements obtained in spherical basis as solutions of the Bethe - Goldstone equation from the Bonn potential Ho81 (). If we do not consider the pairing correlations, the A and B matrices can be expressed in the following simple form, which are the same as the those of Ref. saleh (),

 Apnp′n′αβγδ(K)= (Eαp+Eβn)δαγδpp′δβδδnn′ + 2 [gpp(uαpuβnuγp′uδn′+vαpvβnvγp′vδn′) Vαβ, γδ + gph(uαpvβnuγp′vδn′+vαpuβnvγp′uδn′) Vαδ, γβ],
 Bpnp′n′αβγδ(K)= − 2 [gpp(uαpuβnvγp′vδn′+vαpvβnuγp′uδn′) Vαβ, γδ − gph(uαpvβnvγ