Shell structure , emerging collectivity , and valence p-n interactions

The structure of atomic nuclei depends on the interactions of its constituents, protons and neutrons. These interactions play a key role in the development of configuration mixing and in the onset of collectivity and deformation, in changes to the single particle energies and magic numbers, and in the microscopic origins of phase transitional behavior. Particularly important are the valence proton-neutron interactions which can be studied experimentally using double differences of binding energies extracted from high-precision mass measurements. The resulting quantities, called δVpn, are average interaction strengths between the last two protons and the last two neutrons. Focusing on the Z=50-82, N=82-126 shells, we have considered a number of aspects of these interactions, ranging from their relation to the underlying orbits, their behaviour near close shells and throughout major shells, their relation to the onset of collectivity and deformation, and the appearance of unexpected spikes in δVpn values for a special set of heavy nuclei with nearly equal numbers of valence protons and neutrons. We have calculated spatial overlaps between proton and neutron Nilsson orbits and compared these with the experimental results. Finally we also address the relation between masses (separation energies), changes in structure and valence nucleon number.


INTRODUCTION AND AVERAGE PROTON-NEUTRON INTERACTIONS AND EMERGING COLLECTIVITY
The numbers of protons and neutrons and/or valence protons and neutrons can guide us in understanding special properties about a nucleus or a nuclear region such as the onset of deformation, shape changes and so on.The importance of the number of valence particles is already embodied in concepts such as the N p N n scheme and the P-factor [1].Concerning actual observables, in particular for even-even nuclei, we as nuclear structure physicists often first look at the energy of the first excited 2 + state E(2 1 + ), R 4/2 which is a ratio of the first excited 4 + over the first excited 2 + level energies, B(E2;2 + → 0 + ), charge radii, masses (hence binding and separation energies), etc.To gain an idea about structure, Figure 1 illustrates experimental E(2 1 + ) (left) and R 4/2 (right) values for almost the entire nuclei chart.The colors change from red around magic numbers (magicity) through blue around mid-shell (deformation).High (small) E(2 1 + ) (the R 4/2 ratio) values are seen at and also close to magic numbers.Lower E(2 1 + ) (larger R 4/2 ) occur towards midshell as expected.In both panels in Fig. 1, this is clearly seen and the similarity in the two panels is spectacular.One of the most important challenges relating to Fig. 1 is whether this remarkable pattern will continue for nuclei far from Masses reflect all the nucleonic interactions and thus play a special role as an observable.However, the mass itself does not easily give information on nuclei directly due to its scale (~1 GeV per nucleon).Thus, one works with differences of binding energies (or masses), separation energies.One can then easily see, in particular, shell effects at the magic numbers, or the onset of deformation, for example in the N~90 region.In addition to differences of binding energies for separation energies, a specific double difference of binding energies offers a special filter for proton (p) -neutron (n) interactions between the last two protons and the last two neutrons [2][3][4][5].The p-n interaction is important because it helps to understand, for example, magic numbers, deformation, single particle energies, collectivity, and the development of configuration mixing.There are many studies in the literature on the p-n interaction [6].In this Proceedings, we will only talk about average valence p-n interactions, called δV pn , between Z th and (Z-1) th protons with N th and (N-1) th neutrons.Equation (1) shows this special filter for p-n interactions using binding energies, B, for even-even nuclei.Figure 2 (Left) reveals experimental δV pn values around the 208 Pb region [3,5].Since the nuclei presented in Fig. 2 are close to magic numbers, our simple interpretation here is based on Shell Model orbits, nl j .In the Shell Model, except for the unique parity orbits, each shell begins with high j-low n and ends with low j-high n (see Fig. 2 (Right)).If protons and neutrons fill similar orbits (nl j ), the expected overlap between p and n wave functions will be large.The easiest example is a doubly magic nucleus 208 Pb in which both protons and neutrons are just below the magic numbers Z=82 and N=126, respectively.Thus the last two protons and the last two neutrons occupy similar orbits (low j-high n), so we observe a large δV pn as expected.As soon as two additional neutrons are added to 208 Pb ( 210 Pb), these two neutrons will cross the N=126 magic number and fill the beginning of the next shell although protons are still at the end of the 50-82 shell.For this case, the valence protons and neutrons in 210 Pb will have dissimilar nl j so we see a sudden drop in δV pn at N=128 for Pb [5].A similar analysis can be done for other nuclei close to magic numbers, but this simple interpretation does not work for deformed nuclei because the orbits occupied by protons and neutrons are complex admixtures of Shell Model configurations.Another facet about the interpretation of δV pn is that some effects are much clearer in odd-A nuclei compared to even-even nuclei due to pairing effects (see below).Experimental and calculated δV pn values, using nuclear density functional theory (DFT), for eveneven nuclei in the Z=50-82, N=82-126 shells are presented in Fig. 3. Red and pink colors refer to large δV pn values and dark and light blue are for low δV pn values.The DFT results clearly have very good agreement with the experimental results.In general, the largest δV pn values are in the particle-particle (pp) region in both theory and data and the smallest δV pn s are hole-particle (hp).The largest δV pn s are located around the diagonal from Z=52, N=84 to Z=82, N=126.Recent Xe mass measurements [7] at CERN-ISOLTRAP provided the first empirical δV pn values for Ba at N=90 and 92.The DFT results for these recent δV pn values compare very well with these new data.Calculated δV pn values are also shown for all the nuclei within the expected driplines) in the Z=50-82, N=82-126 shells.Since there is no data in the lower right quadrant (ph), we cannot judge the theory there but clearly this and the right hand side of the (pp) quadrant are prime testing grounds for future exotic beam mass measurements.Thus, some projects are in progress for future mass measurements.
Figures 2 and 3 show heavy nuclei which occupy the large shells (see Fig. 2 (right)) typical of heavy nuclei.In contrast to heavy nuclei, light nuclei fill smaller shells and therefore collective effects appear/disappear quickly so changes in structure occur faster than in heavy nuclei.In addition, light nuclei have cases of nuclei with Z=N.In terms of δV pn , nuclei with equal numbers of proton and neutron fill the same orbits so maximal spatial-spin overlap of proton and neutron wave functions is expected.Not surprisingly Z=N nuclei exhibit large singularities in δV pn [8] which have been linked [9] to Wigner SU(4) supermultiplet theory [10].Figure 4  function of neutron number for light nuclei with sharp peaks at Z=N.This clear trend (see the scale) is not expected in heavy nuclei because of the Coulomb force and the spin-orbit force which brings unique parity orbits into each shell.However, for heavy nuclei, we have found that δV pn values also have sharp peaks, and that they occur when a nucleus has approximately the same number of valence (val) protons and neutrons [11,12].Figure 4 (Middle) introduces our point for odd-Z with both even and odd-N nuclei (see also Fig. 3 of Ref. [11] where δV pn in even-even nuclei also show peaks for Z val ~ N val but the peaks are suppressed and smoother due to pairing effects).The top axis in the middle panel gives the number of valence neutrons.For example, we see a peak in δV pn at N=95 (N val =15) for 65 Tb (Z val =15).Similar examples are seen for Ho and so on.The right panel of Fig. 4 shows the locus of the maximum δV pn values (white circles) on an Z-N contour plot for R 4/2 in which structure changes from spherical (blue) through deformed, ellipsoidal shapes (red).The black line is placed to indicate Z val =N val .As seen in this plot, the nuclei for which we have peaks in the experimental δV pn values, lie where the saturation of collectivity starts so these maxima in δV pn have a clear link to the evolution of collectivity.One of the most interesting points about these results is that these deformed nuclei where we have peaks at Z val =N val or Z val =N val ± 1 usually have a very specific relation between the last filled proton and neutron Nilsson orbits, namely that their quantum numbers are related by δK[δN, δn z , δΛ] =0[110] (for even-even nuclei, the best example is 168 Er which has 7/2[523] (K[N,n z ,Λ]) p Nilsson orbit and 7/2[633] n Nilsson orbit with Z val =N val =18).Is this maximum in δV pn with 0[110] where the saturation of collectivity starts accidental?In order to understand these interactions in a simple way, we calculated [12] spatial overlaps of proton and neutron Nilsson wave functions with three deformations, namely 0.05, 0.22, 0.3.The details of these calculations are given (and updated) elsewhere [13,14].Here we focus on the interpretation and results.
Figure 5 illustrates the Nilsson orbits for protons (Left) in the 50-82 shell and neutrons (Right) in the 82-126 shell.When one starts to fill the proton and neutron Nilsson orbits synchronously, the difference between proton and neutron Nilsson orbits is usually 0[110] such as in the first Nilsson orbits in both shells, 1/2[431] proton orbit with 1/2[541] neutron orbit (follow the same colors in both panels).Because the Nilsson diagrams for the protons and neutrons have similar shapes and crossings, this approximate synchronization actually persists as more nucleons are added and as deformation sets in.Since the proton and neutron shells in Fig. 5 do not have the same size, however, some neutron orbits will not have a proton partner related by 0 [110].These neutron orbits are shown by solid/dotted/dashed gray lines.Interestingly, each of these neutron orbits has n z =0 which does not contribute to prolate deformation but to oblate.(see text for details).Based on Ref. [13].
The calculated overlaps are compared with the experimental δV pn results in Fig. 6.A diagonal line is placed to show where the neutron and proton orbits fill similar fractions of their respective shells.One can fill 32 nucleons into the proton shell and 44 nucleons into the neutron shell.Thus, the figures are designed as square so that, that along the diagonal line similar orbits will be filled so large overlaps (large δV pn ) are expected along this line (see explanation above nl j ) [4].The upper black line is placed to show the Z val =N val nuclei.Overall agreement between the data and theory is very good.As mentioned in the DFT results above, the largest δV pn values are in the pp region and large overlaps along the diagonal lines are seen.There are some discrepancies in the upper right in Fig. 6.However, this is not so surprising because disagreement may happen due to a γ-soft structure that our calculations do not take into account [13].In addition, we observe large δV pn values with 0

CONCLUSION
The average valence proton-neutron interactions, δV pn , which can be studied experimentally using double differences of binding energies extracted from high-precision mass measurements, are discussed from both experimental and theoretical perspectives by focusing on the Z=50-82, N=82-126 shells.We noted the correlation between large δV pn values and the growth of collectivity and the unique behaviour of δV pn when crossing magic numbers.Reflecting the clear spikes in δV pn for light nuclei, studies of δV pn for heavy nuclei are discussed in terms of the number of valence nucleons.We obtained a systematic behavior (maxima in δV pn ) when a nucleus has equal (or almost equal) numbers of valence protons and neutrons and noticed a consistent difference between proton and neutron Nilsson orbit quantum numbers, namely 0 [110].Calculated spatial overlaps between proton and neutron Nilsson orbits are presented and compared these with the experimental results.Finally, we also discussed the relation between masses (separation energies) and E(2 1

+
) by taking into account the number of valence nucleons.

Figure 2 .Figure 3 .
Figure 2. (Left) Experimental δV pn values versus neutron number for the 208 Pb region.(Right) A toy scheme considering the Shell Model orbits to reveal how j and n change within a shell and the expected relation to δV pn values.Based on Refs.[3,5].
Figures2 and 3show heavy nuclei which occupy the large shells (see Fig.2(right)) typical of heavy nuclei.In contrast to heavy nuclei, light nuclei fill smaller shells and therefore collective effects appear/disappear quickly so changes in structure occur faster than in heavy nuclei.In addition, light nuclei have cases of nuclei with Z=N.In terms of δV pn , nuclei with equal numbers of proton and neutron fill the same orbits so maximal spatial-spin overlap of proton and neutron wave functions is expected.Not surprisingly Z=N nuclei exhibit large singularities in δV pn[8] which have been linked[9] to Wigner SU(4) supermultiplet theory[10].Figure4(Left) shows empirical δV pn values as a

Figure 4 .
Figure 4. Experimental δV pn values for light nuclei (Left) and heavy nuclei (Middle).(Right): Color coded R 4/2 values.White circles are shown for the nuclei for which there is a maximum in δV pn in the Middle panel.Based on Refs.[12,13].

Figure 8 .
Figure 8.The symmetry triangle of the IBA showing the three dynamical symmetries at the vertices.The colors indicate calculated collective contributions in MeV to binding energies for N B = 5 (Left) and N B = 16 (Right).Based on Refs.[14,16].
For this, future experiments, in particular in the new facilities such as FAIR, FRIB, RIKEN, are needed.