Spectroscopy of the Cadmium isotopes

The cadmium (Z = 48) isotopes have been a focus of interest in nuclear structure for many decades. This is because they exhibit collective excitations and are located adjacent to the closed shell at Z = 50. At the closed neutron shells, N = 50, 82, pairing dominates and is manifested as excitations with good seniority. Moving towards the mid-neutron shell at N = 66, collectivity emerges; but exactly how and what kind is an open question. In the vicinity of the mid-shell, the former view of the structure of the Cd isotopes was one of near-harmonic quadrupole collective vibrations: this view has been refuted [1]. However, exactly what is the collective character of the mid-shell Cd isotopes, remains an open question. The present paper discusses these issues.


Introduction
The cadmium (Z = 48) isotopes have been a focus of interest in nuclear structure for many decades.This is because they exhibit collective excitations and are located adjacent to the closed shell at Z = 50.
At the closed neutron shells, N = 50, 82, pairing dominates and is manifested as excitations with good seniority.Moving towards the mid-neutron shell at N = 66, collectivity emerges; but exactly how and what kind is an open question.In the vicinity of the mid-shell, the former view of the structure of the Cd isotopes was one of near-harmonic quadrupole collective vibrations: this view has been refuted [1].However, exactly what is the collective character of the mid-shell Cd isotopes, remains an open question.The present paper discusses these issues.

A global view of the Cd isotopes
Figure 1 presents seniority structures due to the ʌg9/2 -2 configuration as manifested in 98-130 Cd. Figure 2 presents seniority structures due to the Ȟg 7/2 +2 , Ȟg 7/2 Ȟd 5/2 , and Ȟh 11/2 -2 configurations as manifested in 100-128 Cd.These are characterized spectroscopically by one-nucleon transfer reactions, magnetic moments, and lifetimes.In the mid-shell region of the Cd isotopes shape coexistence is well-established and is interpreted as resulting from a proton-pair excitation across the Z = 50 shell gap [2].This is shown in Fig. 3.
The major open question is the nature of the collective structures built on the Cd ground states.Maps    near-harmonic quadrupole vibrational behavior, the B(E2) pattern does not support this.Undertaking a characterization of just what is the nature of the collective nuclear structure built on the Cd ground states has proven to be one of the most highly demanding tasks in nuclear spectroscopy, probably, that has ever been undertaken.Some details are given in the next section.
A further issue that arises in the Cd isotopes is the possibility of elucidating the emergence of collectivity from its incipient appearance.To this end, the spectroscopy of 100-108 Cd is of particular interest.This is the subject of the contribution by Andrey Blazhev to these Proceedings.

Detailed spectroscopic studies of the Cd isotopes
Detailed spectroscopic data for the Cd isotopes have begun to be acquired, especially by the technique of inelastic neutron scattering as carried out at the Univ. of Kentucky Accelerator Laboratory [5][6][7][8]  been combined with ultra-weak Ȗ-ray decay branch measurements following ȕ decay [9,10].An outcome of these studies is presented in Fig. 10, which summarizes pertinent data that are a first look beyond a vibrational interpretation of 110-116 Cd.
To move beyond the present status of the structure of the Cd isotopes will be even more demanding of spectroscopic techniques.An obvious direction is multistep Coulomb excitation.At present, such data only exist for 114 Cd [11].A less obvious direction is transfer reaction spectroscopy, using both one-and multi-nucleon transfer (see the paper by Paul Garrett in these Proceedings).A particular outcome of such measurements is that they reveal the non-collective states in weakly collective nuclei such as the Cd isotopes.
A rarely conducted type of spectroscopy that has been carried out for many decades is conversion electron spectroscopy, which, besides the familiar outcome of transition multipolarities, is uniquely able to quantify E0 transition strengths (given that the lifetime of the parent level is known).Such strengths are a sensitive and modelindependent view of shape coexistence and mixing (see, e.g., [12]).
An analysis for the E0 transition strengths is presented here for 114 Cd.It relies on having information for mixing amplitudes for 0 + states from two-neutron transfer data [13].The essential theory and the input data are summarized in Fig. 11.The extension of the mixing analysis to states with spin other than zero is presented in Fig. 12.In particular, note that the input quantity, ǻ<r 2 > Figure 12.Electric monopole transition strengths, ȡ 2 (E0)*10 3 in 114 Cd for all strong E0 transitions and the fine-tuning of its strength (see text).The data are taken from [12,14]. is taken to be spin independent and is fine-tuned to the largest observed value of ȡ 2 (E0)*10 3 and the presumption that this corresponds to Į = ȕ = 0.50, i.e., maximal mixing.From this, one can deduce mixing of pairs of configurations as shown in Fig. 13.Also shown in Fig. 13 are mixing strengths from an IBM-MIX calculation [15].Evidently, the IBM-MIX calculation (which was directed at fitting E2 strengths) seriously fails to describe the pattern of mixing (note that the IBM-MIX calculations are multi-state mixing).

Future work and conclusions
From the emerging pattern of E2 and E0 decay strengths in the mid-shell Cd isotopes it is evident that considerably more spectroscopic study is needed.A first major direction will be multi-step Coulomb excitation.Such data will identify which excited states are connected by strong E2 transitions.In addition, one-and multi-nucleon transfer data are needed to identify noncollective, i.e., broken pair states.
An issue that impacts nuclear structure more widely and involves the Cd isotopes is the question; "Do we understand excited 0 + states in nuclei?"An initial exploration of this was made in [16].It would appear that, from a shell model perspective, the location of shell and subshell gaps is important for answering this question.However, from a more global perspective, the symplectic shell model may provide a unified way forward (see, e.g., [17]).
The author wishes to acknowledge collaborations with Mitch Allmond (Oak Ridge National Lab), Paul Garrett (U.Guelph), Kris Heyde (U.Gent), and Steve Yates (U. of Kentucky) on the study of the Cd istopes.

Figure 1 .
Figure 1.Seniority structures in the Cd isotopes resulting from ʌg9/2 -2 configurations.The data are taken from Nuclear Data Sheets.

Figure 3 .
Figure 3. Deformed structures in 108-118 Cd resulting from a ʌ(2p-4h) configuration.The data are taken from Nuclear Data Sheets. of B(E2) values provide a powerful summary of the collective character of a series of isotopes.These are

Figure 9 .
Figure 9. Map of B(E2; 0def + Î 21 + ) expressed in W.u. for 0 + states that are deformed states.The data are taken from Nuclear Data Sheets

Figure 10 .
Figure 10.Electric quadrupole transition strengths in 110-116 Cd deduced from lifetime measurements and Ȗ-ray transition intensities.The data are taken from Nuclear Data sheets and references given in the text.