Single Neutron States

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Figure 1. A typical energy distribution for protons from (d,p) reactions. Figures above peaks are the excitation energy in the final nucleus (in MeV), and the numbers in parenthesis are the l values of the neutron in the nucleus as determined from the angular distributions.

Fig. 1 shows an example of an energy spectrum of protons from a given nucleus measured at a given angle. From similar measurements at many angles, angular distributions were obtained for reactions leaving the final nucleus in its various states of excitation, exemplified by the peaks in Fig. 1 – the excitation energies of these states are specified by the numbers (in MeV) above these peaks. The shapes of these angular distributions gives a determination of the orbital angular momentum transfer, l, as shown for one nuclear mass region in Fig. 2. The theory lines shown there are calculated from the Distorted Wave Born Approximation (DWBA). The l-values thus obtained are shown in parentheses in Fig. 1. The ratio of the observed cross section to the DWBA prediction gives the spectroscopic factor, S, for each state.

The l values and spectroscopic factors S for each state are shown for one example in Fig. 3. In most cases, the l-value indicates for which shell theory state (or single-particle (SP) state), j, that energy level has a component corresponding to the target nucleus spin + j, so the j-value can be determined; the exception is where the j-value available from shell theory can be either l+1/2 or l-1/2, as for the l=2 states in Fig. 3. These situations can be resolved by doing similar experiments with (d,t) neutron pick-up reactions on the target nucleus that have two additional neutrons, exciting the same final states.

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Figure 2. Typical angular distributions of proton groups corresponding to the residual nucleus being left in given states. These data are from the zirconium region.

Thus a j-value can be assigned to each state in Fig. 3. This allows determination of the sum of S-values for each j, as shown in the table on the left side of Fig.3, and determination of the average energy (weighted by S-values) for each shell theory state, j, as shown by the down-pointed arrows across the top of Fig.3 labelled by the j-values. Note from Fig. 3, the first four j-states listed have the sum of S-values close to 1.0, which means that essentially all components of the j-state are represented and these states are empty of neutrons in the target nucleus for the (d,p) reactions studied. This means that the average energy for each j-state is the actual energy of that shell theory state.

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Figure 3. Results of a stripping analysis of the ⁹⁰Zr(d,p)⁹¹Zr reaction. The bars represent the energies of states in ⁹¹Zr, their heights give the S value for the reactions leading to those states, and the figures over the bars are the l values determined from the angular distributions. The sums of the S values for each single-partticle (SP) state are listed; for example, for the s_1/2 state, the heights of the two bars topped by “0” add up to 0.96. The locations of the SP states are shown at the top; they are obtained as the “center of gravity” of the bars belonging to that state.  The g_7/2 and d_3/2 SP states happen to be at the same energy.

This situation arises for target nuclei which have closed shells for neutrons. For such nuclei, studying (d,t) reactions excites states in the shell below that are completely full and hence give the energies for the shell theory states in that shell. By doing similar experiments on all nuclei with neutron closed shells, the energies of the various shell theory states can be traced through the table of nuclei. The results are shown in Fig. 4.

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Figure 4. Summary of experimental determinations of locations of SP states. Points corresponding to the same SP state are connected by straight lines.

The primary behavior there is a shifting downward to the right, which is easily understood as due to the fact that, as the width of the shell theory potential well increases, the wave length of a given wave function increases, which means that its energy becomes lower. The anomalous behavior between masses 40-48 (calcium isotopes) and 90-96 (zirconium isotopes) is due to the symmetry energy term in the shell theory potential.

Other interesting pieces of information derived and quantified from Fig. 4 are:

• A self-binding effect – neutrons in the same shell theory state experience a stronger interaction with one another than with the other nucleons contributing to the shell theory potential;
• An interaction between neutrons and protons of the same l-value – these have a stronger attraction than between other nucleons contributing to the shell theory potential; the form of the spin-orbit interaction – shown to be ;
• Information on the shape of the shell theory potential – much more like an oscillator potential than like a square well;
• The effective mass of nucleons (which represents the velocity dependence of the forces acting on them) at small binding energies – about 1.3 times their actual mass in contrast to the value applicable to much deeper binding, 0.5 times the actual mass, indicating that the velocity dependence is more complicated than originally proposed;
• and the depth of the shell theory potential – it decreases slowly with increasing mass.

Figure 5. Plot of the average of the renormalized U_j² and (1-V_j²) as a function of the mass number for the even-even Sn isotopes.  In pairing theory, V_j² is the occupation probability of SP state j, while U_j² = (1-V_j²) is the probability that SP state j is unoccupied.

In nuclei where the neutrons are not all in a closed shell, these techniques are useful in determining the degree to which the neutron shells are still unfilled – represented by the sum of the S-values (spectroscopic factors), and how the energies of the nuclear states are affected by this filling, described in pairing theory as the energy of the single quasiparticle states – represented by the average energy of the observed nuclear states weighted by their S-values. For the tin isotopes, the degree U_j² to which each of the various shell theory states is unfilled is plotted in Fig. 5 and the energies of the corresponding single quasiparticle states are plotted in Fig. 6.

These results were in good agreement with predictions of pairing theory, which provided a timely important boost to that theory.

Figure 6. The centers of gravity of the single-quasiparticle states in the 50<N<82 neutron shell are plotted against the mass number.

The tin isotopes provide an important simplification in that their protons are in a closed shell. For other nuclei, the single quasiparticle states are more fragmented but the process and information derived is the same. As an example, the angular distributions for one nucleus are shown in Fig. 7, and the results of the analysis for that nucleus are shown in Fig. 8. Many dozens of studies of this type were done for nuclei in all mass regions. These determined the single quasiparticle components in the wave functions for the low lying states of all these nuclei.

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Figure 7. Angular distributions of protons from Pd¹⁰⁶ (d,p)Pd¹⁰⁷ reactions. Groupings are in accordance with l-value assignments. Figures are excitation energies of corresponding levels in Pd¹⁰⁷ in MeV. Further information on these levels is listed in Table II, and discussions of the angular distributions are given in the text. (All excitation energies on this figure are too low by about 1%. See Table II for correct energies.)

Figure 8. Summary of results for Pd¹⁰⁴(d,p). Levels are grouped according to the single-particle states to which they are assigned. Single-particle states are shown at both left and right sides of figure. Horizontal scale gives energies of the states, and vertical scale gives spectroscopic factors with S=0.10 as full scale. In cases where S>0.10, the number which is multiplied by 0.10 to give the S value is shown along side. Sums of spectroscopic factors for each single-particle state are listed at the left side of the figure as Σ S=(). In cases where assignments to single-particle states are uncertain, the level is shown in the group where its assignment is most probable.