Emergence of phases with size S. Frauendorf Department of Physics University of Notre Dame, USA Institut fuer Strahlenphysik, Forschungszentrum Rossendorf Dresden, Germany

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 … 2

Mesoscopic systems Length characterizing the phase  size of the system Emergence of phases with N. Fixed particle number, heat bath  canonic ensemble Fixed particle number, fixed energy  micro canonic ensemble 3

Superconductivity/Superfluidity Macroscpic phase described by the Landau – Ginzburg equations for the order parameter Density of Cooper pairs G, , Fermi energy , and critical Temperature related by BCS theory. LG valid if: coherence length size of Cooper pair << size of system BCS valid if : pair gap >> level distance 4

Phase diagram of a macroscopic type-I superconductor normal super Meissner effect 5

Superfluidity/superconductivity in small systems Non-local Mean field marginal Nuclei metal (nano-) grains Non-local Mean field bad in porous matrix Non-local Mean field ok 6

Atttractive interaction between Fermions generates Intermediate state of Reduced viscosity Atttractive interaction between Fermions generates Cooper pairs -> Superfluid 7

Moments of inertia at low spin are well reproduced by rigid Moments of inertia at low spin are well reproduced by cranking calculations including pair correlations. irrotational Non-local superfluidity: size of the Cooper pairs larger than size of the nucleus. 8

Superfluidity If coherence length is comparable with size system behaves as if only a fraction is superfluid Nuclear moments of inertia lie between the superfluid and normal value (for T=0 and low spin) 9

Rotation induced super-normal transition at T=0 M. A. Deleplanque, S. F., et al. Phys. Rev. C 69 044309 (2004) (88,126) (72,98) (72,96) (68,92) (Z,N) Energy difference between paired and unpaired phase in rotating nuclei 10 Superconductor in magnetic field

Deviations of the normal state moments of inertia from the rigid body value at T=0 Transition to rigid body value only for T>1MeV M. A. Deleplanque, S. F., et al. Phys. Rev. C 69 044309 (2004) 11

Rotation induced super-normal transition at T=0 Rotating nuclei behave like Type II superconductors Rotational alignment of nucleons  vortices Strong irregularities caused by discreteness and shell structure of nucleonic levels Normal phase moments of inertia differ from classical value for rigid rotation (shell structure) 12

Canonic ensemble: system in heat bath Superconducting nanograins in porous matrix 13

Heat capacity in the canonic ensemble Bulk Bulk = mean field N particles in 2M degenerate levels Exact solution in Ag sinter, pore size 1000A coherence length 900A N. Kuzmenko, V. Mikhajlov, S. Frauendorf J. OF CLUSTER SCIENCE, 195-220 (1999) R. Schrenk, R. Koenig, Phys. Rev. B 57, 8518 (1998) 14

Mesoscopic regime The sharp phase transition becomes smoothed out: Increasing fluctuation dominated regime. 15

Temperature induced pairing in canonic ensemble (nanoparticles in magnetic field) Grand canonic ensemble mean field S. Frauendorf, N. Kuzmenko, V. Michajlov, J. Sheikh Phys. Rev. B 68, 024518 (2003) 16

Micro canonic ensemble In nuclear experiments: Level density within a given energy interval needed Replacement micro  grand may be reasonable away from critical regions. It goes wrong at phase transitions. 17

Micro canonic phase transition q latent heat phase transition temperature micro canonic temperature micro canonic heat capacity Convex intruder cannot be calculated from canonic partition function! Inverse Laplace transformation does not work. 18

Fluctuations may prevent more sophisticated classification. near critical critical q q E E E Fluctuations may prevent more sophisticated classification. 19

Critical level densities (caloric curve) M. Guttormsen et al. PRC 68, 03411 (2003) 20

T. Dossing et al. Phys. Rev. Lett. 75, 1275 (1995) 21 40 equidistant levels

half-filled, monopole pairing, exact eigenvalues, 12 equidistant levels, half-filled, monopole pairing, exact eigenvalues, micro canonic, smeared A. Volya, T. Sumaryada Restriction of Configuration space 2qp 4qp 22 From data by M. Guttormsen et al. PRC 68, 03411 (2003)

Really critical? Yes ! constant T at low E 23 T. v. Egidy, D. Bucurescu

Temperature induced super-normal transition Seen as constant T behavior of level density Some indication seniority pattern Melting of other correlations contributes? Evaporation of particles from HI reactions with several MeV/nucleon well accounted for by normal Fermi gas Where is the onset of the normal Fermi gas caloric curve? 24

Liquid-gas phase boundary Develops early for nuclei and metal clusters ( well saturated systems): surface thickness a (~ distance between nucleons/ions) < size  scaling with Coulomb energy Binding energy of K clusters 25

Nuclei: charged two-component liquid Strong correlation What is the bulk equation of state? For example: compressibility neutron matter Clusters allow us studying the scaling laws. 26

Nuclear multi fragmentation- liquid-gas transition Gas of nucleons Normal Fermi gas From energy fluctuations of projectile-like source in Au+Au collisions 27 J. Pochodzella et al. , PRL 75, 1042 (1995) M. DeAugostino et al., PLB 473, 219 (2000)

28 M. Schmitd et al.

Melting of mass separated Na clusters in a heat bath of T From absorption of LASER light From atom evaporation spectrum 29

Micro canonic phase transition q latent heat phase transition temperature micro canonic temperature micro canonic heat capacity Probability for the cluster to have energy E in a heat bath at temperature 30

31 M. Schmitd et al.

Solid/liquid/gas transition Boiling nuclei – multi fragmentation: indication for (surface energy of the fragments) no shell effects Melting Na clusters: in contrast to bulk melting Strong shell effects 32

Transition from electronic to geometric shells In Na clusters 36 T. P.Martin Physics Reports 273 (1966) 199-241

Solid state, liquid He: Calculation of very problematic – well protected. Take from experiment. local BCS very good Nuclei: Calculation of not possible so far. Adjusted to even-odd mass differences. highly non-local BCS poor 16 How to extrapolate to stars? Vortices, pinning of magnetic field?

12 equidistant levels, half-filled, monopole pairing, exact eigenvalues, microcanonic ensemble A. Volya, T. Sumaryada

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Physics Emergence means complex organizational structure growing out of simple rule. (p. 200) 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) Not only physics, biology, sociology 3