1. The proton-neutron interaction and the emergence of collectivity in atomic nuclei R. F. Casten Yale University BNL Colloquium, April 15, 2014 The field of play: Trying to understand how the structure of the nucleus evolves with N and Z, and its drivers.

2. Themes and challenges of nuclear structure physics – common to many areas of Modern Science Complexity out of simplicity -- Microscopic How the world, with all its apparent complexity and diversity, can be constructed out of a few elementary building blocks and their interactions Simplicity out of complexity – Macroscopic How the world of complex systems* can display such remarkable regularity and simplicity What is the force that binds nuclei? Why do nuclei do what they do? What are the simple patterns that emerge in nuclei? What do they tell us about what nuclei do? * Nucleons orbit ~ 10 21 /s, occupy ~ 60% of the nuclear volume; 2 forces The p-n Interaction Linking the two perspectives

3. 10004+4+ 2+2+ 0 400 0+0+ E (keV) JπJπ Simple Observables - Even-Even Nuclei Masses 14002+2+ J B(E2) values ~ transition rates

4. Geometric Potentials for Nuclei: 2 types For single particle motion and for the many-body system r UiUi  V   

5. r = |r i - r j |  V ij r UiUi Reminder: Concept of "mean field“ or average potential Simple potentials plus spin-orbit force  Clusters of levels  shell structure Pauli Principle (≤ 2j+1 nucleons in orbit with angular momentum j)  Energy gaps, closed shells, magic numbers. Independent Particle Model Inert core of nucleons, Concept of valence nucleons and residual interactions Many-body  few-body: each body counts Change by, say, 2 neutrons in a nucleus with 150 nucleons can drastically alter structure  =  nl, E = E nl H.O. E = ħ  (2n+l) E (n,l) = E (n-1, l+2) E (2s) = E (1d) 64

6. 0+0+ 2+2+ 6 +... 8 +... Vibrator (H.O.) E(I) = n (   0 ) R 4/2 = 2.0 n = 0 n = 1 n = 2 Rotor E(I)  ( ħ 2 /2 I )I(I+1) R 4/2 = 3.33 Doubly magic plus 2 nucleons R 4/2 < 2.0 A few valence nucleons Many valence nucleons Pairing (sort of…) l l With many valence nucleons Multiple configurations with the same angular momentum. They mix, produce coherent, collective states R 4/2 A measure of collectivity

7. Nuclear Shape Evolution  - nuclear ellipsoidal deformation (  is spherical) Vibrational Region Transitional Region Rotational Region Critical Point Few valence nucleons Many valence Nucleons New analytical solutions, E(5) and X(5) R 4/2 = 2R 4/2 = 3.33

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

9. Structural evolution Complexity and simplicity

10. BEWARE OF FALSE CORRELATIONS!

11. Rapid structural changes with N/Z – Quantum Phase Transitions

12. Quantum phase transitions as a function of proton and neutron number Vibrator Rotor Transitional

14. Nuclear Shape Evolution  - nuclear ellipsoidal deformation (  is spherical) Vibrational Region Transitional Region Rotational Region Critical Point Few valence nucleons Many valence Nucleons New analytical solutions, E(5) and X(5) R 4/2 = 2R 4/2 = 3.33

15. Bessel equation Critical Point Symmetries First Order Phase Transition – Phase Coexistence E E β 1 2 3 4  Energy surface changes with valence nucleon number Iachello, 2001 X(5)

17. Casten and Zamfir Remarkable agreement. Discrepancies can be understood in terms of sloped potential wall and the location of the critical point Param-free

18. Sn – Magic: no valence p-n interactions Both valence protons and neutrons What drives structural evolution? Valence p-n interactions in competition with pairing

19. Seeing structural evolution Different perspectives can yield different insights Onset of deformation as a quantum phase transition (convex to concave contours) mediated by a change in proton shell structure as a function of neutron number, hence driven by the p-n interaction mid-sh. magic Look at exactly the same data now plotted against Z

20.  Vpn (Z,N) = ¼ [ {B(Z,N) - B(Z, N-2)} - {B(Z-2, N) - B(Z-2, N-2)} ] p n p n Int. of last two n with Z protons, N-2 neutrons and with each other Int. of last two n with Z-2 protons, N-2 neutrons and with each other Empirical average interaction of last two neutrons with last two protons --- - Valence p-n interaction: Can we measure it?

21. Empirical interactions of the last proton with the last neutron  V pn (Z, N) = ¼{[B(Z, N ) – B(Z, N - 2)] - [B(Z - 2, N) – B(Z - 2, N -2)]}

22. p-n interaction is short range similar orbits give largest p-n interaction HIGH j, LOW n LOW j, HIGH n 50 82 126 Largest p-n interactions if proton and neutron shells are filling similar orbits

23.  82 50 82 126 Z  82, N < 126 1 1 1 2 Z > 82, N < 126 2 3 3 Z > 82, N > 126 High j, low n Low j, high n

24. 208 Hg

25. Empirical p-n interaction strengths indeed strongest along diagonal.  82 50 82 126 High j, low n Low j, high n Empirical p-n interaction strengths stronger in like regions than unlike regions.

26. Direct correlation of observed growth rates of collectivity with empirical p-n interaction strengths p-n interactions and the evolution of structure

27. W. Nazarewicz, M. Stoitsov, W. Satula Microscopic Density Functional Calculations with Skyrme forces and different treatments of pairing Realistic Calculations

28. My understanding of Density Functional Theory Want to see it again?

29. These DFT calculations accurate only to ~ 1 MeV.  Vpn allows one to focus on specific correlations. M. Stoitsov, R. B. Cakirli, R. F. Casten, W. Nazarewicz, and W. Satula PRL 98, 132502 (2007); D. Neidherr et al, Phys. Rev. C 80, 044323 (2009) Recent measurements at ISOLTRAP/ISOLDE test these DFT calculations Comparison of empirical p-n interactions with Density Functional Theory(DFT) with Skyrme forces and surface-volume pairing

30. The competition of pairing and the p-n interaction. A guide to structural evolution. The next slides allow you to estimate the structure of any nucleus by multiplying and dividing two numbers each less than 30 (or, if you prefer, you can get the almost as good results from 100 hours of supercomputer time.)

31. Valence Proton-Neutron Interactions Correlations, collectivity, deformation. Sensitive to magic numbers. N p N n Scheme Highlight deviant nuclei A refinement to the N p N n Scheme. P = N p N n / (N p +N n ) No. p-n interactions per pairing interaction

32.   NpNnNpNn p – n P N p + N n pairing p-n / pairing P ~ 5 Pairing int. ~ 1 – 1.5 MeV, p-n ~ 200 - 300 keV p-n interactions per pairing interaction Hence takes ~ 5 p-n int. to compete with one pairing int. Rich searching ground for regions of rapid shape/phase transition far off stability in neutron rich nuclei McCutchan and Zamfir Competition of p-n interaction with pairing: Simple estimate of evolution of structure with N, Z

33. Comparison with the data

34. Comparison with R 4/2 data: Rare Earth region Deformed nuclei R 4/2 data

35. Special phenomenology of p-n interactions: Implications for the evolution of structure and the onset of deformation in nuclei New result, not recognized before, that shows the effect in purer form

36. Shell model spherical nuclei Nilsson model Deformed nuclei Seems complex, huh? Not really. Very simple. Deformation

37. How to understand the Nilsson diagram Orbit K 1 will be lower in energy than orbit K 2 where K is the projection of the angular momentum on the symmetry axis Nilsson orbit labels (quantum numbers): K [N n z  ] n z  is the number of nodes in the z-direction (extent of the w.f. in z) z

38. This is the essence of the Nilsson model and diagram. Just repeat this idea for EACH j-orbit of the spherical shell model. Look at the left and you will see these patterns. There is only one other ingredient needed. Note that some of the lines are curved. This comes from 2-state mixing of the spherical w.f.’s (not important for present context)

39. Note: the last filled proton-neutron Nilsson orbitals for the nuclei where dV pn is largest are usually related by:  K[  N,  n z,  Λ]=0[110] 168 Er 7/2[523] 7/2[633]

40. 7/2[523] 7/2[633] Specific highly overlapping proton-neutron pairs of orbitals, satisfying 0[110], fill almost synchronously in medium mass and heavy nuclei and correlate with changing collectivity 1/2[431]1/2[541] 3/2[422] 3/2[532] 168 Er

41. 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]

42. Summary Remarkable regularity and simplicity in a complex system Structural evolution – evidence for quantum phase transitions Drivers of emergent collectiivity -- valence p-n interaction in competition with pairing Empirical extraction of valence p-n interactions – sensitivity to orbit overlaps Enhanced valence p-n interactions for equal numbers of valence protons and neutrons Key to structural evolution in heavy nuclei – synchronized filling of highly overlapping 0[110] orbits

43. Principal collaborators R. Burcu Cakirli Dennis Bonatsos Klaus Blaum Victor Zamfir Special thanks to Jackie Mooney for decades of collaboration and friendship.

44. Backups

45. Competition between valence pairing and the p-n interactions A simple microscopic interpretation of the evolution of structure Partial history: Goldhaber and de Shalit (1953); Talmi (1962); Federman and Pittel ( late 1970’s); Casten et al (1981); Heyde et al (1980’s); Nazarewicz, Dobacewski et al (1980’s); Otsuka et al( 2000’s); Cakirli et al (2000’s); and others.

46. Shape/ Phase Transition and Critical Point Symmetry

48. Discrepancies for excited 0 + band are one of scale Scaled energes Identifying the nature of the discrepancies. Isolating scale

49. Based on idea of Mark Caprio A possible explanation

50. Discrepancies for transition rates from 0 + band are one of scale

51. Minimum in energy of first excited 0 + state Li et al, 2009 E E β 1 2 3 4  Other signatures

52. Isotope shifts Li et al, 2009 Charlwood et al, 2009

53. Enhanced density of 0 + states at the critical point Meyer et al, 2006 E E β 1 2 3 4 

54. Where else? Look at other N=90 nulei But, more generally, where else? To understand this we need to discuss the drivers of structural evolution in nuclei with N and Z. This evolution is a titanic struggle between good and good -- the pairing interaction between like nucleons and the proton- neutron interaction.

55. The history of nuclear structure past and future ( A VAST oversimplification) 50’s 60’s 70’s – 00’s 10’s 1 0  N - Z   Z Degen. 1 RIBs: 2 mass units is a big deal Z N A Accels.  -ray det. Arrays Weak, contaminated beams, fewer levels Mass sep., scintillators, “Back to the Future”