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

2. 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

3. 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

4. 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

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

6. 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

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

8. 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

9. 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

10. Rotation induced super-normal transition at T=0
M. A. Deleplanque, S. F., et al.
Phys. Rev. C (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

11. 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 (2004) 11

12. 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

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

14. 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, (1999)
R. Schrenk, R. Koenig,
Phys. Rev. B 57, 8518 (1998) 14

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

16. 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, (2003) 16

17. 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

18. 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

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

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

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

22. 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, (2003)

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

24. 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

25. 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

26. 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

27. 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. 28
M. Schmitd et al.

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

30. 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. 31
M. Schmitd et al.

32. 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

33. Transition from electronic to geometric shells In Na clusters36
T. P.Martin Physics Reports 273 (1966)

34. 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?

35. 12 equidistant levels,
half-filled, monopole pairing,
exact eigenvalues,
microcanonic ensemble
A. Volya, T. Sumaryada

36. 8

37. 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