Emergence of phases with size

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)

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Physics