Correlations in Structure among Observables and Enhanced Proton-Neutron Interactions R.Burcu ÇAKIRLI Istanbul University International Workshop "Shapes and Dynamics of Atomic Nuclei: Contemporary Aspects" (SDANCA-15), 8-10 October 2015, Sofia - Bulgaria

Broad perspective on structural evolution The remarkable regularity of these patterns is one of the beauties of nuclear systematics and one of the challenges to nuclear theory. Whether they persist far off stability is one of the fascinating questions for the future

Change in slope Sudden deformation N~90 Shape transitions Nuclear Radii  Mean square of charge radius   fm 2 Deformed nuclei have larger radius  sudden increase in Another useful observables, charge radii, B(E2), masses (S 2n )

Differential observables (isotope shifts)  N = N - (N-2) N=90, onset of deformation Literature  comparison with mostly B(E2). But why not E(2 1 + ) ? Brix Kopfermann plot  `E(2 1 + ) N = E(2 1 + ) N – E(2 1 + ) (N-2)

However, there is a better correlation with E(2 1 + )  E(2 1 + ) N = [ E(2 1 + ) (N-2) – E(2 1 + ) (N) ] / [ E(2 1 + ) (N-2) + E(2 1 + ) N ] Onset of deformation at N=90 In both plots!!!! Sm, Gd and Dy have a higher peak than Ce and Er like a normalization

With R 4/2  R 4/2 = R 4/2 (N) – R 4/2 (N-2)

With B(E2;  )  B(E2) = B(E2) (N) – B(E2) (N-2) Neutron Number

With S 2n  S 2n = S 2n(N) – S 2n(N-2)

E(2 1 + ) R 4/2 B(E2) S 2n   E(2 1 + )  R 4/2  B(E2)  S 2n Each plot has a different trend Each plot has the same trend Other regions are similar too Cakirli, Casten, Blaum, PRC 82, (2010) (R)

S 2n  binding energy difference  two binding energy (masses)  S 2n  S 2n diffference  4 masses |  V pn (Z,N)| = ¼ [ {B(Z,N) - B(Z, N-2)} -{B(Z-2, N) - B(Z-2, N-2)} ] * For even-even nuclei *J.D.Garrett and J.-Y.Zhang, Cocoyoc, 1988, Book of Abtsracts J.-Y. Zhang, R. F. Casten, and D. S. Brenner, Phys. Lett. B 227, 1 (1989) R.B.Cakirli, D.S.Brenner, R.F.Casten and E.A.Millman, PRL 94, (2005) Extract a measure of the p-n interaction strength Average p-n interaction between the last 2 protons and the last neutrons  Z th and (Z-1) th protons with N th and (N-1) th neutrons Masses reflect all interaction in the nucleus Various combinations of masses can isolate specific interactions Another filter with 4 masses :  V pn

Empirical  V pn * p-n interaction is short range * Similar orbits give largest p-n interaction High j, low n Low j, high n    V pn can be interpreted  by considering the orbit(nl j ) occupations Deformed nuclei – should be considered in terms of Nilsson orbits K [N, n z,  ] -To try to understand this effect we calculated spatial overlaps of proton and neutron wave functions. - Similar trend is seen in light nuclei for N=Z but it will not be discussed now.

Odd-Z -- middle panel -- enhanced peaks without the muting effects of pairing New result, not recognized before, that shows the effect in purer form Heavy nuclei - empirical valence p-n interactions from masses Note peaks at N val ~ Z val Locus of peaks in p-n interactions. Relation to onset of collectivity D. Bonatsos, S. Karampagia, R.B.Cakirli, R.F.Casten, K.Blaum, L.Susam, Phys. Rev. C 88, (2013)

Go further with Nilsson orbits Note: the last filled proton-neutron Nilsson orbitals for the nuclei where  V pn is largest are usually related by   n z   ]=0[110] 168 Er 7/2[523] 7/2[633] R. B. Cakirli, K. Blaum, and R. F. Casten, Phys. Rev. C 82, (R) (2010)

Synchronized filling of 0[110] proton and neutron orbit combinations and the onset of deformation Similarity of Nilsson patterns as deformation changes, and high overlaps of 0[110] orbit pairs, leads to maximal collectivity near the N val ~ Z val line * 0[110] *D. Bonatsos, S. Karampagia, Cakirli, Casten, Blaum, Amon, Phys. Rev. C 88, (2013)

Some disagreements, esp. in upper right But note that calculations do not take into account  -softness Note: Large values, for example, Z~52-64 and N ~ /2[413] with 5/2[512] and 1/2[420] with 1/2[521] that do not satisfy 0[110] Measurement of masses in the future at FAIR, FRIB, and RIKEN Calculate spatial overlaps of proton and neutron wave functions. Compare to empirical  V pn values (top panel) Overall agreement is good Bonatsos, Karampagia, Cakirli, Casten, Blaum, Amon, PRC 88, (2013)

Conclusions The study is useful for future measurements :, spectroscopic observables, masses and future theoretical approaches Striking correlations of observables representing single particle motion, nuclear radii and collective observables -- to our knowledge, not recognized heretofore The separation energy is a good filter for studying structure using masses. Another filter,  V pn, gives insight into average p-n interactions Enhanced valence p-n interactions are closely correlated with the development of collectivity, shape changes, and the emergence of deformation in nuclei. Proton-neutron Nilsson orbits for largest p-n interactions satisfy 0[110]. Spatial overlaps confirm large interactions for such cases and agree reasonably well with  V pn Highly interacting 0[110] pairs fill almost synchronously in heavy nuclei even as the deformation increases: saturation in R 4/2, emergence of deformation. Extensive, realistic, Density Functional Theory calculations work nicely in predicting p-n interaction strengths (in many mass regions)

Thanks for your attention ! Collaborators R.F. Casten D. Bonatsos K. Blaum L. Amon

Typical nuclear structure observables for even-even nuclei such as the first excited 2$^+$ state energy, and the ratio between the 4$^+$ and 2$^+$ states (R$_{4/2}$), give us the information about the evolution of structure. In one part of this talk, such observables and their differentials, including spectroscopic data and masses and correlations among them will be discussed. In addition, since the separation energy is a good filter for structure using masses, another filter, $\deltaV_{pn}$, for heavy nuclei will also be presented and discussed in terms of spatial-spin orbit overlaps between proton and neutron wave functions. We will discuss that proton- neutron pairs of orbitals that fill almost synchronously in deformed medium mass and heavy nuclei, satisfy 0[110] differences in Nilsson quantum numbers and correlate with changing collectivity. These data and results will be discussed in terms of the growth of collectivity in nuclei as a function of the numbers of valence nucleons.

  E(2 1 + )  R 4/2  B(E2)  S 2n

Empirical valence p-n interactions from masses,  V pn *J.D.Garrett and J.-Y.Zhang, Cocoyoc, 1988, Book of Abtsracts, J.-Y. Zhang, R. F. Casten, and D. S. Brenner, Phys. Lett. B 227, 1 (1989), R.B.Cakirli, D.S.Brenner, R.F.Casten and E.A.Millman, PRL 94, (2005) |  V pn (Z,N)| = ¼ [ {B(Z,N) - B(Z, N-2)} -{B(Z-2, N) - B(Z-2, N-2)}] * For even-even nuclei  V pn has singularities for N = Z in light nuclei Van Isacker, Warner, Brenner PRL 74, 4607 (1995).

B-H: Their own results show that the p-n components in fact dominate Sph Trans. Def.

Why did we get interested in the region ( ~ 168 Er ) ?  V pn has singularities for N = Z in light nuclei Van Isacker, Warner, Brenner, PRL 74, 4607 (1995). * Wigner energy, related to SU(4), spin-isospin symmetry. Physics is high overlaps of the last proton and neutron wave functions when they fill identical orbits. * Expected to vanish in heavy nuclei due to: Coulomb force for protons, spin-orbit force which brings UPOs into different positions in each shell and protons and neutrons occupying different major shells. *  V pn has peaks for N val ~ Z val !!!! That is, equal numbers of valence protons and neutrons – similar to light nuclei * And the trajectory of these maxima coincides with the emergence of deformation (soon)

 V pn (Z,N) = -¼ [ {B(Z,N) - B(Z, N-2)} - {B(Z-2, N) - B(Z-2, N-2)} ] p n p n Interaction of last two n with Z protons, N-2 neutrons and with each other Interaction of last two n with Z-2 protons, N-2 neutrons and with each other Average p-n interaction between the last 2 protons and the last 2 neutrons  Z th and (Z-1) th protons with N th and (N-1) th neutrons

dV eepn (Z;N) = −1/4({BE(Z;N) −BE(Z;N − 2)}− {BE(Z − 2;N) −BE(Z − 2;N − 2)}); dV eopn (Z;N) = −1/2({BE(Z;N) −BE(Z;N − 1)}− {BE(Z − 2;N) −BE(Z − 2;N − 1)}); dV oepn (Z;N) = −1/2({BE(Z;N) −BE(Z;N − 2)}− {BE(Z − 1;N) −BE(Z − 1;N − 2)}); dV oopn (Z;N) = −1/1({BE(Z;N) −BE(Z;N − 1)}− {BE(Z − 1;N) −BE(Z − 1;N − 1)})