B(E2­) values in T=1 multiplets and isospin purity

P.D. Cottle 

Measuring and understanding the evolution of single particle energies in nuclei far from stability is one of the highest priorities for the nuclear structure community.  Continuing advances in the technology for producing beams of radioactive isotopes are making it possible to determine single particle energies for long sequences of isotopes of a particular element.  The evolution of single particle energies in neutron-rich nuclei has been highlighted in a number of mass regions [1-4].  A particular focus of this work has been the spin-orbit force, which in nuclei determines major shell closures [5,6].   

The energies of the proton orbits in the sd shell in neutron-rich nuclei have been highlighted recently in studies of calcium and silicon isotopes [2,7,8].  In particular, the evolution of the Z=14 and 16 shell closures plays a role in the development of collectivity in the neutron-rich silicon isotopes [2].  We have identified all available experimental information on single proton energies in the silicon isotopes and extracted these energies.  Mass information is critical for determining single particle energies.  However, information on excited states in the adjacent odd-proton nuclei is important as well.  The nature of the information on excited states changes from the stable isotopes ^28,30Si, where single proton pickup and stripping data are available, to neutron-rich nuclei such as the N=20 isotope ³⁴Si, which appears doubly-magic in structure and for which some γ-ray spectroscopic information is available in the odd-proton neighbor ³⁵P.

The available experimental information is summarized in Fig.1 (a table is available in Ref. 9).  The nuclear mass information that forms the basis of this study is taken from [10,11].  However, it is important from the outset to recognize that the sources of the information represented in Fig. 1 are quite different, and that the precision of the data points plotted there varies considerably.

The single proton energies in the stable isotopes ^28,30Si take into account the masses of the silicon (Z=14) isotopes themselves and the masses of the aluminum (Z=13) and phosphorus (Z=15) isotones, as well as data from the single proton transfer reactions ^28,30Si(t,α)^27,29Al and ^28,30Si(³He,d)^29,31P [12-14].  The transfer reactions allow the location of the single proton (or proton hole) strength, even when it is fragmented among a number of states.  When the strength is fragmented, the spectroscopic factors determined for the states allow the determination of the centroid.

The proton transfer data with ^28,30Si targets provide the ideal situation for determining single proton energies.  However, such data are not yet available for the exotic Si isotopes, so we must make the best use of the data that are available in these nuclei.  For example, for ³²Si we have the precisely-known masses of the isotones ³¹Al, ³²Si and ³³P. 

The ground state of ³¹Al has J^π=5/2⁺, as would be expected for the d_5/2 hole configuration.  The ground state of ³³P has J^π=1/2⁺, reflecting its s_1/2 origin.  The lowest J^π=3/2⁺ state in ³³P is located at 1.43 MeV.  Given the concentration of d_3/2 hole strength in the 1.27 MeV state in ³¹P, it seems reasonable to conclude that a similar amount of d_3/2 strength is located in the 1.43 MeV 3/2⁺ state in ³³P.

The ^28,30Si proton transfer data clearly demonstrate that uncertainties are introduced into the determination of single proton energies by the absence of the transfer results.  However, the same results for ^28,30Si can be used to devise a procedure for estimating single particle energies and specifying uncertainties in the absence of proton transfer data.  We have done this (see details in Ref. 9) and determined that to take this into account, we should shift the single proton energy by 0.35 MeV (half of 0.70 MeV) in the appropriate direction (more bound for of d_5/2, less bound for s_1/2 and d_3/2) and assign an uncertainty of 0.35 MeV to the result.  This uncertainty is added in quadrature with the uncertainty from the mass measurements.  This procedure is applied for the states shown in Fig. 1 for ^{26,32,34,36,38,40,42}Si.  In the heaviest isotopes, the uncertainties in the mass measurements become significant and the resulting uncertainties in the single proton energies become larger than 0.35 MeV, reaching 0.76 MeV in ⁴²Si.   

In the N=20 isotope ³⁴Si, which appears from its spectroscopy to be doubly-magic, we extract the d_3/2 single proton energy by noting the 3/2⁺ state at 2.4 MeV in the isotone ³⁵P observed in ³⁵Si β-decay and proton pickup reactions on ³⁶S.  No other 3/2⁺ state is presently known in ³⁵P.  Fig. 1 also depicts a data point for the d_3/2 single proton energy in the near-dripline nucleus ⁴²Si.  A 184 keV γ-ray was observed in a study of the single-proton knockout reaction on ⁴⁴S and was interpreted as connecting the d_3/2 and s_1/2 single proton states [2].  The comparison of the ratio of the observed cross sections for the two states to a theoretical calculation led to the assignment of the ground state as the s_1/2 state and the state at 184 keV as the d_3/2 state.

The most easily discerned issue in Fig. 1 is the size of the gaps between the d_5/2 proton orbit and the s_1/2 and d_3/2 orbits in the neutron-rich nucleus ³⁴Si, which has a neutron number (20) that is magic at the line of stability.  The calculation of the gap between the d_5/2 and of s_1/2 proton orbits in ³⁴Si by Brown [15] is 2.8 MeV, considerably less than the empirical gap of 7.2 MeV shown in Fig. 1 (the Brown value is shown in Fig. 1 as well).  The empirical spin-orbit splitting between the d_5/2 and d_3/2 proton orbits is approximately 9.5 MeV.  Shell structure in ³⁴Si is an important physics issue because this nucleus is a neighbor of the “island of inversion” [16-18].  The absence of the d_5/2 single proton energy for N=28 in Fig. 1 also highlights the importance of measuring the mass of ⁴¹Al.  The existence of ⁴¹Al was established by Sakarai et al. [19].

Obtaining more single proton energies in the entire sequence of Si isotopes would require the systematic application of experimental probes that select single proton strength, as with the use of the (t,α) and (³He,d) reactions with the stable isotopes ^28,30Si described above.  The recent dissertation work of Roeder [20] suggests that the (d,n) reaction in inverse kinematics would provide a practical probe for single proton strengths with fast radioactive beams.  Roeder and collaborators measured cross sections of 0.5-0.7 mb for the inverse kinematics reactions d(⁴⁰S,n)⁴¹Cl, d(⁴²S,n)⁴³Cl, and d(⁴⁸Ca,n)⁴⁹Sc.  These cross sections make spectroscopic studies with a wide range of proton- and neutron-rich beams practical.

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[20] B.T. Roeder, Florida State University dissertation (2006).

 

Figure 1

 

 

Experimental single proton energies in the silicon isotopes (top panel) and these energies relative to the d_5/2 orbit (bottom panel).  The dashed line in the top panel connects the d_3/2 energy at N=20 to the corresponding energy at N=28.  The d_3/2 energies for N=22-26 are not known.