Microscopic study of the variable-moment-of-inertia model for rare-earth nuclei

Abstract

LBL-2347Based on the microscopic cr~king model, the moment-of-inertia parameter .. deviate from the I(I+l) rule as the spins increase. Recently, it was discovered that at very high angular momenta, the rotational energies of some nuclei may exhibit anomalous behavior, the so called back-bending.l,2,3 We shall not discuss the back-bending phenomenon in this paper, but shall limit our calculations only to those states with moderate high spins.There exist many two-parameter formulas which fit very well the energy levels up to spin :r. with the inclusion of both quadrupole (12) and hexadecapole ('4) deformation.Since it has been shown that most of the two-parameter formulas are related to each other,7,17,24 we shall calculate specifically the parameters associated with the VMI model and the B coefficient connected with the 1 2 (I+l)2 term.The following section will briefly review the formulations developed in (I), the detailed calculations and formulas are given in Sec. III, and the last two sections will give the results and discussions.. .' . The VMI model can be expressed as follows: where -·1 a > is the deformed single particle state with a denoting the appropriate quantum numbers, m a -iST the~gnetic :quantumnnuJnber. along the· symmetry axis, U a and Va are the probability amplitudes in the presence of pairing interaction and Ea is the quasi-particle energy.The Inglis and Belyaev cranking formula (6) is based on the independent quasi-particle approximation. However, a recent calculation by Meyer, Speth 27 and Vogeler showed that the two carrection terms arising from the particleparticle and particle-hole interactions nearly cancel each other. It has also been shown by Rich 28 that correction due to particle-number conservation is also small. Thus it seems that the use of the cranking formula ·(6) is rather t,Tell justified numerically.The second term of Eq. (5) represents the fourth order cranking correction which was first studied by Harris 6 and the fourth order cranking 6 constant C n can be expressed as It is obvious from Eq. (5) thatThus, the contribution of the fourth order cranking in Eq. (4) The value of the force constant C VM1 or the B coefficient indicates the degree to which the spectrum deviates from the 1(1+1) rule. Both Eqs. (4) and (9) show that the contributions from various degrees of freedom are all posi.tblel1Z-"add~d.• with plus sign for protons and minus sign for neutrons. They also put in a linear surface dependence of G, which may be important for l~rge deformation.The BCS equation is then solved by including (15Z)1/2 or (lSN)1/2 states above and below the proton or neutron Fermi level.6. thus obtained are given in Table I. The pairing gap parameters 6. ,_'aridThe energy of a quasiparticle can be expressed as (12) where e: k is the single-particle energy and A is the chemical potential.Following (I) we parametrize the probability amplitudes {Uk' V k } by introducing a pairing correlation parameter VIf V = ~ (the energy gap ~ is the equilibirium value of V at ground state), Eq. ...

Moment of inertia in the interacting boson model

“…Solving (5), (6) for N ~> 1 gives/~o = If2-Inserting this value for /~ it is seen that the Jo ~ and Jo c become equal to 4/(8~'-32K), consistent with the energy expression in the SU(3) limit.…”

“…They are not reducible to each other, have different dependence on IBM hamiltonian parameters and appear to have even different dependence on the equilibrium shape variable rio-It is therefore appropriate to see how the two formulae differ among themselves. We also compare these results with those of microscopic calculations [6,7]. For guidance, we take "experimental" values for Jo as determined in the VMI model [8] in which Jo appears as one of the phenomenological parameters estimated by fitting the experimental level energies.…”

“…For nuclei in the actinide region and for ~56Dy and 15SDy, microscopic values Jo M are calculated using (13) taking deformed potential parameters and pairing strength directly from the work of Nilsson et al [11] without any re-adjustment of parameters. For the rest of rare-earth nuclei in Table 2 microscopic values are from the work of Ma and Tsang [6]. As is the case for Gd isotopes, IBM estimates are larger than microscopic estimates and minor changes in the pairing strength parameter with little adjustment of neutron levels will bring microscopic values in better agreement which we have not considered necessary to do.…”

“…The results for Gd isotopes are presented in Table 1 together with the microscopic cranking results taken from [6]. The "experimental" values are from [8].…”

“…(2) For a given nucleus which means choosing appropriate values for the IBM hamiltonian (1) parameters, the equilibrium deformation rio is determined solving (5), (6) and then Jo G and Jo c can be obtained using (4) and (10)- (12).respectively. The results for Gd isotopes are presented in Table 1 together with the microscopic cranking results taken from [6].…”