Correlating Radii and Electric Monopole Transitions of Atomic Nuclei

Abstract

A systematic analysis of the spherical-to-deformed shape phase transition in even-even rare-earth nuclei from 58Ce to 74W is carried out in the framework of the interacting boson model. These results are then used to calculate nuclear radii and electric monopole (E0) transitions with the same effective operator. The influence of the hexadecapole degree of freedom (g boson) on the correlation between radii and E0 transitions thus established, is discussed.PACS numbers: 21.10. Ft, 21.10.Ky, 21.60.Ev, 21.60.Fw Electric monopole (E0) transitions between nuclear levels proceed mainly by internal conversion with no transfer of angular momentum to the ejected electron. For transition energies greater than 2m e c 2 , electronpositron pair creation is also possible; two-photon emission is possible at all energies but extremely improbable. The total probability for a transition between initial and final states |i and |f can be separated into an electronic and a nuclear factor, P = Ωρ 2 , where the nuclear factorwith R = r 0 A 1/3 (r 0 = 1.2 fm) and where the summation runs over the Z protons in the nucleus. The coefficient σ depends on the assumed nuclear charge distribution but in any reasonable case it is smaller than 0.1 and can be neglected if the leading term is not too small [1]. The charge radius of a state |s is given byIt is found experimentally that the addition of neutrons produces a change in the nuclear charge distribution, an effect which can be parametrized by means of neutron and proton effective charges e n and e p in the charge radius operatorT (r 2 ). This leads to the following generalization of Eq. (2):where the sum is over all nucleons and e k = e n (e p ) if k is a neutron (proton). An obvious connection between ρ and the nuclear charge radius is established in the approximation σ = 0 (which henceforth will be made). Again because of the polarization effect of the neutrons, one introduces an E0 operator of the form [2]The ρ defined in Eq.(1) with σ = 0 is then given by ρ = f|T (E0)|i /eR 2 . The basic hypothesis of this Letter is to assume that the effective nucleon charges in the charge radius and E0 transition operators are the same. If this is so, comparison of Eqs. (3) and (4) leads to the relationAt present, a quantitative test of the correlations between radii and E0 transitions implied by (5) cannot be obtained in the context of the nuclear shell model. The main reason is that E0 transitions between states in a single harmonic-oscillator shell vanish identically [3] and a non-zero E0 matrix element is obtained only if valence nucleons are allowed to occupy at least two oscillator shells. This renders the shell-model calculation computationally challenging (if not impossible), certainly in the heavier nuclei which are considered here. We have therefore chosen to test the implied correlations in the context of a simpler approach, namely the interacting boson model (IBM) of atomic nuclei [4]. In this model low-lying collective excitations of nuclei are described in terms of N b bosons distrib...

Intertwined quantum phase transitions in the Zr isotopes

“…where r 2 c is the square radius of the closed shell, N v is half the number of valence particles, and η is a coefficient that takes into account the effect of deformation [21,34,35]. The isotope shift depends on two parameters, α = 0.235, η = 0.264 fm 2 , whose values are fixed by the procedure of Ref.…”

“…The isotope shift depends on two parameters, α = 0.235, η = 0.264 fm 2 , whose values are fixed by the procedure of Ref. [34,35]. ∆ r 2 0 + 1 should increase at the transition point and decrease and, as seen in Figure 6 Similarly, the two-neutron separation energies S 2n can be written as [21],…”

Evolution ofE2transition strength in deformed hafnium isotopes from new measurements onHf172,Rüdigier1, Nomura2, Dannhoff3 et al. 2015Phys. Rev. C

“…Likely, the inclusion of β 4 will improve agreement with experiment for these nuclei. In fact, the previous IBM-1 calculation indicated [17] that the hexadecapole degree of freedom may be non-negligible, though in a different context of some ground-state properties.…”

“…In recent years the IBM has been used with considerable success in spectroscopic studies [15][16][17][18][19] of a large set of nuclei with proton and neutron numbers in the range Z = 50-82 and N = 82-126, respectively. However, those studies are rather phenomenological because the parameters of the IBM Hamiltonian are determined by a fit to known experimental data.…”

E0transitions in deformed nuclei

“…A recent study using a geometric model [6] has investigated their behavior as a function of the stiffness of the quadrupole potential. In addition, their correlation with nuclear radii has been investigated in [7]. In some models, such as the IBA model, there are definite selection rules and analytic predictions for the strengths of E0 transitions from excited 0 + state to the ground state in the cases of the dynamical symmetries.…”