Emergent Phenomena in mesoscopic nuclear systems

1. Emergent Phenomena in mesoscopic nuclear systems S. Frauendorf Department of Physics University of Notre Dame

2. Emergence means complex organizational structure growing out of simple rule. (p. 200) Protection generates exactness and reliability,… The universal properties of ordering of rigid bodies, the flow of superfluids, and even the emptiness of space are among the many concrete, well documented examples of this effect. (p. 144) Macroscopic emergence, like rigidity, becomes increasingly exact in the limit of large sample size, hence the idea of emerging. There is nothing preventing organizational phenomena from developing at small scale,…. (p. 170) 2

3. Emergent phenomena • Liquid-Gas Phase boundary • Rigid Phase – Lattice • Superconductivity (Meissner effect, vortices) • Laws of Hydrodynamics • Laws of Thermodynamics • Quantum sound • Quantum Hall resistance • Fermi and Bose Statistics of composite particles 3

4. Mesoscopic systems Emergence of a phenomenon with N can be studied Rigid crystal structure T. P.Martin Physics Reports 273 (1966) 199-241 4

5. Ca clusters: the transition to the bulk is not smooth fcc lattice: Close packing with translational symmetry Icosahedra: Close packing with small surface bulk 5 T. P.Martin Physics Reports 273 (1966) 199-241

6. Emergent phenomena - nuclei • The bulk liquid • Superfluidity, superconductivity • Shell structure • Spatial orientation • Temperature • Phases and phase transitions Extroplation to bulk Finite nuclei 6

7. Neutron stars Suprafluid, superconducting nuclear matter and more SGR 1806-20 Weber 7

8. Transition to the bulk liquid Astrophysics: What is the equation of state for nuclear matter? The liquid drop model – scaling laws. Binding energy of K clusters Coulomb energy Neutral –one component 8

9. Ionization energy of Na clusters 9

10. Nuclei: charged two component liquid How good is it? Symmetry energy ???? What is the bulk equation of state? For example: compressibility Steiner, Li 10

11. Superconductivity/Superfluidity Described by the Landau – Ginzburg equation for the order parameter Controlled by ( inside the superconductor) coherence length (size of Cooper pair) penetration depth of magnetic field G, , Fermi energy , and critical Temperature related by BCS theory. 11

12. Solid state, liquid He: Calculation of very problematic – well protected. Take from experiment. local BCS very good Nuclei: Calculation of D not possible so far. Adjusted to even-odd mass differences. highly non-local BCS poor Sumaryada, Papenbrock How to extrapolate to stars? Vortices, pinning of magnetic field?

13. Mesoscopic regime 13

14. Symmetry breaking Spontaneous symmetry breaking Emergence Periodic crystal structure rigidity, transverse sound Finite N: Localization Shell structure, center of mass motion Orientation rotational alignment, … rotational bands

15. SHE ? Khoo, Nazarewicz Shell structure Fermions in spherical Potential Nuclei: magnitude OK, damping with N and T OK. Clusters: More washed out. Dies out quicker. Not quantitatively understood. Frauendorf, Pashkevich

16. Supershell structure of Na clusters SHE N-dependent factor multiplied for compensating the too rapid damping with N! 16 Emergence of resistivity

17. Emergence of orientation Example for spontaneous symmetry breaking: Weinberg’s chair Hamiltonian rotational invariant Why do we see the chair shape? Tiniest perturbation mixes |IM> states to a stable-oriented wave packet: the symmetry broken state. 17

18. 3 2 1 Mesoscopic variant I: Molecules Can be kicked and turned like a chair. Quantal states |IM> can be measured: Rotational bands Classical moments of inertia of arrangement of point masses. 18 16

19. Well deformed Mesoscopic variant II: Nuclei Symmetry broken state described by the mean field. How is orientation generated? Deformed potential aligns the partially filled orbitals Riley Partially filled orbitals are highly tropic Nucleus is oriented – rotational band 19

20. Imax>20 Currents caused by nucleons on periodic orbits 20

21. 21 B

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23. Emergence of thermodynamics Region of high level density: important for astrophysics, nuclear applications, … Limits to predictability of quantal states: uncertainties in the Hamiltonian deterministic chaos Johnson Mitchell Zelevinsky Give up individual quantal states: 23

24. Crossover phenomena Phase transitions T=0 transitions between different symmetries in nuclei. Spherical deformed IBA symmetries solid-liquid superfluid-normal liquid-gas Caprio Fang Sumaryada Artificial limit by mean field approximation 24

25. Grand canonical Canonical Microcanonical Superfluid-normal transition 25

26. Grandcanonical ensemble Canonical ensemble

27. Microcanonical 26 M. Schmitd et al.

28. Melting of Na clusters 27

29. Conclusions Emergent phenomena abound in nuclear systems Calculating them microscopically seems a long way (at least). Looking on non-nuclear mesoscopic systems is instructive and fun. More can be found in: S. Frauendorf, C. Guet, Ann. Rev. Nucl. Part. Sci. 51, 219 (2001) 28