Electromagnetic Properties of the I π = 11/2− isomers in the cadmium isotopes: What can we learn from them?

The electromagnetic features of the 11/2− isomers in the odd-mass Cd isotopes are shown to exhibit an an anomaly near N = 70. We report shell-model calculations of these isotopes aimed at describing these properties. We find a sudden phase change in the hexadecapole component of the wave functions precisely at N = 70, which gives rise to different linear relations between the Q and μ values before and after N = 70, as needed to reproduce the experimental data. The particle-hole transformation properties associated with a strong subshell closure at N = 70 followed by fairly close-lying neutron orbitals from N = 70 − 82 is suggested as a possible explanation for this phase change.


Introduction
A recent study of the I π = 11/2 − isomers in the Cd-isotope chain [1] indicates that the quadrupole and magnetic moments (denoted by Q and µ, respectively) both vary roughly as linear functions of neutron number. However, as was also noted, there is a noticeable difference in the linear relation for the µ values between 111−117 Cd and 119−127 Cd, separated by N = 70. Indeed, if one performs a linear fit to the quadrupole moments, as shown in fig. 1(a), a similar behavior is observed: the best-fit linear relation perfectly describes the quadrupol moments of 111−117 Cd, while residues from the same fit obviously exceed experimental errors for 119−127 Cd, as shown in fig. 1(b). Thus, the quadrupole moment data also exhibit a change in linear behavior before and after N = 70.
A density functional study [2] recently reported suggested that the Q linearity in the Cd isotopes has a different mechanism for N < 70 and N > 70. In this work, we study the mechanism for the behavior of both the quadrupole moments and magnetic moments at N = 70 in the framework of a simple shell-model treatment, to see whether we can explain the data and in doing so obtain new insight on the structural properties of these isotopes. Further details can be found in [3] where the results of this study were first presented.  Figure 1. Quadrupole moments of the 11/2 − isomers in the Cd isotopes. In panel (a), we compare experimental values [1], linear fitting results, and results from the SDG+ and SDG− calculation. Panel (b) presents residues of the linear fit. The N = 70 position is highlighted. For N < 70, the residues are smaller than the experimental errors, so that a linear relation follows. In contrast, the residues for N > 70 are larger than experimental errors, which means that the same linear behavior is not respected. The SDG+ and SDG− results are consistent with the N < 70 and N > 70 experimental data, respectively, as discussed in the text.

Calculations and Analysis
We begin by outlining the basic ingredients of our calculations: 1 We first assume simple, but reasonable, neutron and proton configurations from which to build the 11/2 − isomers. 2 We then calculate the quadrupole and magnetic-dipole matrix elements for the above configurations. 3 We next consider several trial wave functions for the isomers and fit their respective amplitudes to optimally describe the quadrupole-and magnetic-moment data for the 11/2 −  Figure 2. Schematic illustration of the single-particle levels of relevance to the Cd isotopes and their corresponding energies. isomers.
As we will see, the procedure leads to a set of wave functions that nicely describe the evolution of the quadrupole and magnetic moments of the 11/2 − isomers, including the change in behavior at N = 70, and furthermore suggests a fairly simple picture as to why it occurs. In fig. 2, we schematically show the relevant single-particle energy levels appropriate to our analysis of the N = 48 Cd isotopes. For protons, only the g 9/2 orbital (for holes) is relevant. For neutrons, the levels of the N = 50 − 82 shell are all in principle relevant. However, the g 7/2 orbital is fairly well isolated from the others, which are themselves fairly close in energy.
Since the Cd isotopes are nearly spherical, we assume minimal proton-neutron configuration mixing. We thus assume, following [1], the simplest possible configuration for neutrons, a seniority-one configuration involving a single unpaired h 11/2 particle and the remaining particles paired to spin-zero and distributed over the close-lying levels shown in fig. 2. We denote this neutron configuration as φ ν .
With the above assumptions, our basis space for treating the 11/2 − isomers consists of only five configurations: The Q and µ matrix elements for the neutron configurations can be calculated straightforwardly based on the seniority-one nature of φ ν , using the well-known seniority reduction formulae [4] that apply for either a single orbit or a set of degenerate orbits, as assumed to approximately apply here. The proton Q and µ matrix elements of the (g 9/2 ) −2 configurations can likewise be calculated simply based on the same formalism. We need to choose appropriate effective charges and Lande g factors for the analysis. The proton effective charge are taken from a fit to the experimental quadrupole moment of the 2 + 1 state of 112 Cd, which should have as predominantly D π structure. Some minor modifications to this simple prescription are described in [3]. The usual Lande g factors are assumed. The neutron effective charge will be obtained from a fit to the quadrupole moments, to be described below.
Since S π and D π configurations dominate the low-lying states of the Cd isotopes, we first consider as a trial wave-function C SS + C DD , where C S and C D are wave function amplitudes. The corresponding probabilities C 2 S and C 2 D can be obtained from the magnetic moment data on the 11/2 − isomers. Using the values that emerge, we then determine the neutron effective charge e ν so as to optimize the reproduction of the associated quadrupole moments. We consider both possibilities for the relative sign of C D with respect to C S , with the eventual choice determined by which yields results in closer accord to experiment. This is referred to as our "SD fit". We present the optimal e ν and the root-mean-square deviation (RMSD) of the SD fit in the top row of Table 1. Both the resulting e ν = 7.334e and the RMSD=0.456eb are much too large for this to be viewed as an acceptable fit. Thus we need to consider additional proton configuration. We thus consider three other trial wave-functions: (a) C SS + C DD + C GG , (b) C SS + C DD + C IĨ and (c) C SS + C DD + C KK , which we denote "SDG, SDI and SDK", respectively. Here too C G , C I and C K are wave function amplitudes to be determined analogously through optimization. The final best-fit values and the RMSD are also listed in Table 1. All three fits achieve reasonable levels of agreement, with RMSD values∼ 0.01eb. Therefore, in what follows we limit ourselves to these trial wave functions.
When comparing the various fits, we conclude that the SDG fit seems to be the best for two principal reasons: • The SDG fit provides the smallest RMSD.
• The SDI and SDK fits both yield an unusually large e ν ∼ 3e value, as did the SD fit discussed earlier.
Therefore, we only consider the SDG wave-function and their associated best-fit e ν value in the analysis to follow. For the SDG wave functions, the choice of the C G phase relative to that of C S can lead to different quadrupole moments. We classify the SDG wave functions according to the relative phase of C G with respect to C S (assumed positive), with SDG+ having C G > 0 and and SDG− having C G < 0.
In fig. 1, the calculated quadrupole moments associated with the SDG+ and SDG− wave functions are compared with the experimental quadrupole moments. The SDG+ results are compatible with the best-fit linear relation of the experimental quadrupole moments, and provide a better description for N < 70. In contrast, the SDG− results exhibit some curvature beyond N = 70, as in the experimental data. From these results, we conclude that there is a C G phase change across N = 70. This provides a plausible explanation for the change in Q linearity on the two sides of N = 70, as reflected in fig. 1.
The precise difference between the SDG+ and SDG− quadrupole moments is observable but not very pronounced. It is desirable therefore to use other electromagnetic data to more clearly isolate the C G phase in the various isotopes. In particular, we will consider the systematic strong B(E2, 7/2 − 1 → 11/2 − 1 ) values in 113−119 Cd [5], to see whether they can provide additional information. We will calculate these B(E2) values using both the SDG+ and SDG− wave-functions obtained above, and carry out an isotope-by-isotope comparison between the experimental and calculated results.
As necessary input to the B(E2) calculation, we must first identify an appropriate wave function to use for the initial 7/2 − 1 state. The strong E2 transition 7/2 − 1 → 11/2 − 1 suggests a similarity between the 7/2 − 1 state and the 11/2 − isomeric state We thus treat the 7/2 − 1 state as a simple proton-neutron recoupling of the 11/2 − isomer, namely as φ ν ⊗ D π (this should be lower in energy than φ ν ⊗ G π and thus more important) but coupled of course to J π = 7/2 − . Thus, in the spirit of the analysis to date, we assume a single configuration for the state. But of course we still need to see whether a good description of the data follows from such a simple choice. What we see from the figure is that the C S value is close to 1 for all the Cd isotopes. Thus, the 11/2 − isomer is constructed with a dominant h 11/2 single-particle configuration, with perturbations arising from D π and G π excitations. The second feature to note is the rapid change of phase exhibited by C G , precisely at N = 70. Lastly, note that that the D π contributions also changes phase near N = 70, but it does so very smoothly.
How might we understand these results? In perturbation theory, changes in C D and C G relative to C S reflect sign changes in the associated p − n interaction coupling matrix elements. The relevant interactions correspond to scalar products of multipole operators in the Q · Q channel (responsible for C D ) and the H · H channel (responsible for C G ) . Whereas the proton reduced matrix elements are simple and do not evolve with neutron number, the neutron reduced matrix elements do evolve with neutron number, and must therefore be responsible for the sign change. As is well known [6], the matrix elements of an even multipole operator, in either a single orbit or a series of degenerate orbits, changes sign at mid-shell due to a change from particle-like to hole-like behavior.
As noted earlier, the neutron orbits from N = 58−82 (the s 1/2 , d 3/2 , d 5/2 , and h 11/2 ) are fairly close in energy and well isolated from the lower g 7/2 . Thus, they seem to provide the needed well-isolated subshell. We would expect, therefore, that both the quadrupole and hexadecapole matrix elements change sign at mid-shell, which is precisely at N = 70. Since C D evolves very smoothly with N , the effect of the sign change shows up most dramatically in C G .
On the basis of these arguments, we conclude that the dramatic change in the sign of C G seems to be a manifestation of a fairly isolated N = 58 − 82 subshell with relatively close-lying orbits and the particle-hole transformation properties that arise in such circumstances.

Summary
Summarizing, we have extracted shell-model wave functions of the 11/2 − isomers in the odd-mass 111−129 Cd by fitting their electromagnetic features, including both electromagnetic moments and B(E2) values. A phase change in the hexadecapole component of the extracted wave functions explains the change in linear behavior of the quadrupole and magnetic moments for N < 70 and N > 70 as well as the BE2(7/2 − 1 → 11/2 − 1 ) behavior. According to perturbation theory, this phase change at N = 70 is related to a change in sign of the hexadecapole proton-neutron matrix elements as we pass through N = 70, if we assume a strong subshell effect taking place for this neutron number.