Two-Octupole-Phonon States in 146,148Gd

Abstract

1121The so-called "two-octupole-phonon states" in 146. 148 Gd are theoretically analyzed by using the Dyson boson mapping method. In our analyses, the free ground state of 146Gd is treated as doubly closed shell since Z=64 is rather good subshell closure. The starting collective multi-phonon space that we take consists of collective octupole phonons and collective monopole pairing phonons for 146Gd. Additionally correlated particle-pair modes with J ' *' 0 are included in the multi-phonon space for 148Gd. The effective Hamiltonian used is constituted by a W oods-Saxon-type single-particle potential, an octupole-octupole force for particle-hole modes and a surface-delta interaction for particle-pair and hole-pair modes. The numerical calculation can considerably well reproduce the octupole collectivity shown by experiments for 146. 148 Gd. It is shown that the Dyson boson mapping method is useful for such a complicated system as 148Gd. § 1. IntroductionThe octupole correlation is, as known well, quite strong and very important in medium-heavy nuclei. Particularly, most magic number nuclei have low-lying 3-states, which have been considered as one-phonon states of elementary vibrational mode of octupole character. A typical example is the nucleus 208Pb, whose first excited state is a strongly collective 3-state. In our previous paperl) (referred to as I hereafter), we analyzed the so-called two-octupole-phonon states in 208Pb using the Dyson boson mapping method, and we had deep understanding of the low-lying collective states in 20BPb. It is interesting to see how the collective octupole excitations occur in different regions of nuclei and how the same method as that used for 208Pb works there. The nucleus 208Pb is a very simple system; i.e., a doubly-closed shell one. We can therefore perform a microscopic analysis by using the Dyson boson mapping method as shown in I. If there exist additional nucleons on top of doubly-closed shell, then microscopic analyses would become more difficult. It is interesting to see whether microscopic analyses are possible or not for such complicated nuclei by using the Dyson boson mapping method.Kleinheinz and Piiparinen et a1. 2 )-7) have long investigated the octupole vibration states in Gd region nuclei, and they have recently published detailed experimental results on the two-phonon E3 transition strength in 148Gd. This region seems to be a good battlefield to study the collectivity of octupole correlation and to test the applicability of the Dyson boson mapping method for study on collective states in more complicated nuclei.The first excited (1.579 MeV) state of 146Gd is a strongly collective 3-state (B(E3; 3t--->Ot)~37 Bw). This seems quite similar to the case of 208Pb. In fact, as seen later, 146Gd can be treated as a doubly-magic nucleus, since Z=64 is considered as a rather good closed-subshell. If this 3-state can, roughly speaking, be considered as a one-octupole-boson (or one-I-boson) state, it is natural to expect "two-boson"

Test of Validity of the Hermitian Treatment of the Dyson Boson Mapping

“…In addition we have applied the Hermitian treatment to other various calculations for realistic cases. 16), 17), 9), 10) In all these works, we have checked the validity of the approximation of the Hermitian treatment by comparing it with the direct treatment, and all the results have shown that the Hermitian treatment gives deviations less than 1% from the direct treatment on the average, or ≤ 2% ∼ 3% even in the worst case. This implies that the Hermitian treatment is a surprisingly good approximation.…”

Non-Unitary Boson Mapping and Its Application to Nuclear Collective Motions

“…Hermitian treatment to other various calculations for realistic cases. 43), 44), 23), 24) In all these works, we have checked the validity of the approximation of the Hermitian treatment by comparing it with the direct treatment, and all the results have shown that the Hermitian treatment gives deviations less than 1% from the direct treatment on the average, or ≤ 2% ∼ 3% even in the worst case. This implies that the Hermitian treatment is a surprisingly good approximation.…”

“…20) Thus the D-type mapping for the collective subspace was finally established and has successfully been applied to various realistic nuclei. 10), 21) - 24) One of the main purposes of this paper is to represent the full detail of the D-type non-unitary boson mapping for the multi-collective-phonon subspace. The general theory of boson mapping is represented in §2.…”

On the Approximation in the Hermitian Treatment of Dyson Boson Expansion Theory

“…Then how can we explain the fact that it is a good approximation to the exact treatment in some realistic cases as was shown numerically in Refs. [29] and [30]? In this subsection, we show that the result of the Hermitian treatment is approximately valid in spite of the breakdown of the basic assumption.…”

“…If this formula works well, we no longer need to solve the right and left eigenvalue problems so that the efforts of the numerical calculation is greatly reduced and besides we can take the advantage of the finiteness of expansion. With the help of the Hermitian treatment, D-type BET has been applied to many nuclei in the various range of the nuclear chart [23]- [30].…”