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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 - PowerPoint PPT Presentation
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Emergence of phases with sizeS. Frauendorf
Department of Physics
University of Notre Dame, USA
Institut fuer Strahlenphysik, Forschungszentrum RossendorfDresden, 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• … • …
8.258122
h
e
2
Mesoscopic systems
Emergence of phases with N.
52 1010~ NLength characterizing the phase size of the system
Fixed particle number, heat bath canonic ensemble
Fixed particle number, fixed energy micro canonic ensemble
3
LG valid if: coherence length size of Cooper pair << size of system
Superconductivity/SuperfluidityMacroscpic phase described by the Landau – Ginzburg equations for the order parameter
RvF /0
G/)()( rr
d
G, , Fermi energy , and critical Temperature related by BCS theory.cT
2/2FF mv
4
2|)(| r Density of Cooper pairs
BCS valid if : pair gap >> level distance
T
H
normal
super
Phase diagram of a macroscopic type-I superconductor
5
Meissner effect
Superfluidity/superconductivity in small systems
MeVdMeV
fmRfm
2.01
7~300
NucleiNon-localMean field marginal
metal (nano-)grains meVdmeV
nmRnm
1.0~1.0
5~1000
Non-local
Mean field bad
in porous matrix
He3
meVdmeV
nmRnm6
0
1005.0
100~~100
Non-localMean field ok
6
2
1
Intermediate state ofReduced viscosity
Atttractive interaction between Fermions generates Cooper pairs -> Superfluid
He3
7
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 largerthan 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
Dy150Rotation induced super-
normal transition at T=0
0
0
Hc1
Hc2
Hc
normal
super
E
H
Type I Type II
normal
Superconductor in magnetic fieldEnergy difference between paired and unpaired phase in rotating nuclei
M. A. Deleplanque, S. F., et al.Phys. Rev. C 69 044309 (2004)
(88,126)
(72,98)
(72,96)
(68,92) (Z,N)
10
rgid
M. A. Deleplanque, S. F., et al.Phys. Rev. C 69 044309 (2004)
Deviations of the normal state moments ofinertia from the rigid body value at T=0
Transition to rigidbody value only forT>1MeV
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
•
)(
capacity heat
),()( curve caloric
states ofdensity )( )(
)(),(
0
dT
TEdC
dETEEPTE
ETZ
eETEP
T
E
in porous matrixHe3
13
Heat capacity in the canonic ensemble
N particles in 2M degenerate levelsExact solution
Bulk = mean field
N. Kuzmenko, V. Mikhajlov, S. Frauendorf
J. OF CLUSTER SCIENCE, 195-220 (1999) R. Schrenk, R. Koenig,Phys. Rev. B 57, 8518 (1998)
in Ag sinter, pore size 1000Acoherence length 900A
Bulk
He3
14
Mesoscopic regime
15
The sharp phase transition becomes smoothed out:Increasing fluctuation dominated regime.
Canonic ensemble
Grand canonic ensemble mean field
Temperature induced pairing in canonic ensemble (nanoparticles in magnetic 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
Bolzman ln
,..),,,,( ,..),,,,(
micro
IZNES
S
eIZNE micro
Replacement micro grand may be reasonable away from critical regions.It goes wrong at phase transitions. 17
Micro canonic phase transition
1
1
c
dE
dT
dE
dST
q latent heat
micro canonic temperature
micro canonic heat capacity
mT phase transition temperature
Convex intruder cannot be calculatedfrom canonic partition function! InverseLaplace transformation does not work. 18
E E E
q q
criticalnear critical
T
cT
Fluctuations may prevent more sophisticated classification.
19
M. Guttormsen et al.PRC 68, 03411 (2003)
o
oc
cc
ES
FT
EESTF
ln)(
)(
Critical level densities (caloric curve)
20
T. Dossing et al. Phys. Rev. Lett. 75, 1275 (1995) 0.9MeV Hg192
40 equidistant levels
MeVBCST
MeVT
c
c
51.0)(
55.0
21
cT
2
0 1 2 3 4 5 60.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
172Yb
Tc
2eo
T[M
eV
]
E[MeV]
MeV
MeVBCST
MeVT
eoc
c
4.0
45.076.1/)(
52.0
76.1/)( BCSTc
12 equidistant levels,half-filled, monopole pairing,exact eigenvalues,micro canonic, smearedA. Volya, T. Sumaryada
intervallover smeared
,ln ,1
o
SdE
dST
From data by M. Guttormsen et al.PRC 68, 03411 (2003)
Restriction ofConfigurationspace
2qp 4qp
22
Really critical?
0 1 2 3 4 5 60.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0 Texp Tbsfg Tct Tbsfged
172Yb
Tc
2eo
T[M
eV]
E[MeV]
T. v. Egidy, D. Bucurescu
)(
2
11
21),(
1~ ED-BSFG
)ln(5
4)(2 BSFG
CT
EE
o
oct
eEE
NZSCaa
SEEEEaS
ST
ES
constant T at low E
Yes !
23
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
Develops early for nuclei and metal clusters ( well saturated systems):surface thickness a (~ distance between nucleons/ions) < size scaling with
Liquid-gas phase boundary
3/1~ aNR
3/1 NaaN
ESV
B
Coulomb energyBinding energy of K clusters
3/1N
25
223/423/1 )( AZNaAZaAaaA
ESCSV
B
What is the bulk equation of state?
For example: compressibilityd
dE
Nuclei: charged two-component liquid
26
Strong correlation
Clusters allow us studying the scaling laws.
neutron matter
Nuclear multi fragmentation-liquid-gas transition
J. Pochodzella et al. , PRL 75, 1042 (1995) M. DeAugostino et al., PLB 473, 219 (2000)
From energy fluctuations of projectile-like source in Au+Aucollisions
1
c
dE
dTLGT
Normal Fermi gas
Gas of nucleons
27
M. Schmitd et al.
28
Melting of mass separated Na clusters
in a heat bath of T
29
From atom evaporation spectrum
Fro
m a
bsor
ptio
n of
LA
SE
R li
ght
Micro canonic phase transition
bb TEESTE eeEEP
dE
dT
dE
dST
/)(/
1
1
)()(
c
q latent heat
micro canonic temperature
micro canonic heat capacity
Probability for the cluster to have energy Ein a heat bath at temperature bT
mT phase transition temperature
30
M. Schmitd et al.
31
Solid/liquid/gas transition
Boiling nuclei – multi fragmentation: MeVTLG 5
indication for 0C (surface energy of the fragments)
no shell effectsshellLG TT
Melting Na clusters: KTKT bulkm 310250
0C in contrast to bulk melting
Strong shell effects
32
Transition from electronic to geometric shellsIn Na clusters
KT 250~
36T. P.Martin Physics Reports 273 (1966) 199-241
Solid state, liquid He:Calculation of very problematic – well protected.Take from experiment.cT
K
K
T
T
N
N
F
c
F510
1~~~
RmvF 15~/0 local
BCS very good
Nuclei: Calculation of not possible so far. Adjusted to even-odd mass differences.
fmRfmvF 5~40~/0 highly non-local
MeV
MeV
T
T
N
N
F
c
F 40
1~~~
BCS poor
How to extrapolate to stars?
Vortices, pinning of magnetic field?
16
12 equidistant levels,half-filled, monopole pairing,exact eigenvalues,microcanonic ensemble A. Volya, T. Sumaryada
2
8
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)
3
Physics