Emergence of phases with size

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15 Νοε 2013 (πριν από 3 χρόνια και 7 μήνες)

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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


Superconductivity (Meissner effect, vortices)


Laws of Hydrodynamics


Laws of Thermodynamics


Quantum sound


Quantum Hall resistance


Fermi and Bose Statistics of composite particles












8
.
25812
2
h
e
2

Mesoscopic systems

Emergence of phases with N.

5
2
10
10
~

N
Length 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/Superfluidity

Macroscpic phase described by the Landau


Ginzburg

equations for the order parameter

R
v
F



/
0


G
/
)
(
)
(
r
r



d


G, , Fermi energy , and

critical Temperature related by BCS theory.

c
T
2
/
2
F
F
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

MeV
d
MeV
fm
R
fm
2
.
0
1
7
~
30
0







Nuclei

Non
-
local

Mean field

marginal

metal

(nano
-
)

grains

meV
d
meV
nm
R
nm
1
.
0
~
1
.
0
5
~
100
0






Non
-
local

Mean field

bad


in

porous

matrix

He
3
meV
d
meV
nm
R
nm
6
0
10
05
.
0
100
~
~
100







Non
-
local

Mean field

ok

6

2
1
Intermediate state of

Reduced viscosity

Atttractive interaction between Fermions generates

Cooper pairs
-
> Superfluid

He
3
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 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

Dy
150
Rotation induced super
-
normal transition at
T=0

0
0
H
c1
H
c2
H
c
normal
super


E
H
Type I
Type II
normal

Superconductor in magnetic field

Energy 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 of

inertia from the rigid body value at
T=
0

Transition to rigid

body value only for

T
>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

of
density

)
(

)
(
)
(
)
,
(
0
dT
T
E
d
C
dE
T
E
EP
T
E
E
T
Z
e
E
T
E
P
T
E









in porous matrix

He
3
13

Heat capacity in the canonic ensemble

N particles in 2M degenerate levels

Exact 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 1000A

coherence length 900A

Bulk

He
3
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
I
Z
N
E
S
S
e
I
Z
N
E
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
dS
T



q
latent heat

micro canonic temperature

micro canonic heat capacity

m
T
phase transition temperature

Convex intruder cannot be calculated

from canonic partition function! Inverse

Laplace transformation does not work.

18

E

E

E

q

q

critical

near critical

T
c
T
Fluctuations may prevent more sophisticated classification.

19

M. Guttormsen et al.

PRC 68, 03411 (2003)






















o
o
c
c
c
E
S
F
T
E
E
S
T
F


ln
)
(
)
(
Critical level densities (caloric curve)

20

T. Dossing
et al.

Phys. Rev. Lett. 75, 1275 (1995)

0.9MeV

Hg
192


40 equidistant levels

MeV
BCS
T
MeV
T
c
c
51
.
0
)
(
55
.
0


21

c
T

2
0
1
2
3
4
5
6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
172
Yb
T
c
2

eo


T[MeV]
E[MeV]
MeV
MeV
BCS
T
MeV
T
eo
c
c
4
.
0
45
.
0
76
.
1
/
)
(
52
.
0






76
.
1
/
)
(


BCS
T
c
12 equidistant levels,

half
-
filled, monopole pairing,

exact eigenvalues,

micro canonic, smeared

A. Volya, T. Sumaryada





intervall
over

smeared

,
ln

,
1

















o
S
dE
dS
T
From data by M. Guttormsen et al.

PRC 68, 03411 (2003)

Restriction of

Configuration

space

2qp

4qp

22

Really critical?

0
1
2
3
4
5
6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Texp
Tbsfg
Tct
Tbsfged
172
Yb
T
c
2

eo


T[MeV]
E[MeV]
T. v. Egidy, D. Bucurescu


























)
(
2
1
1
2
1
)
,
(
1
~

ED
-
BSFG
)
ln(
5
4
)
(
2

BSFG


CT
E
E
o
o
ct
e
E
E
N
Z
SC
a
a
S
E
E
E
E
a
S
S
T
E
S

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
~
aN
R
3
/
1



N
a
a
N
E
S
V
B
Coulomb energy

Binding energy of K clusters

3
/
1

N
25

2
2
3
/
4
2
3
/
1
)
(








A
Z
N
a
A
Z
a
A
a
a
A
E
S
C
S
V
B
What is the bulk equation of state?

For example: compressibility


d
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+Au

collisions

1
c










dE
dT


LG
T
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

From absorption of LASER light

Micro canonic phase transition

b
b
T
E
E
S
T
E
e
e
E
E
P
dE
dT
dE
dS
T
/
)
(
/
1
1
)
(
)
(
c



























q
latent heat

micro canonic temperature

micro canonic heat capacity

Probability for the cluster to have energy
E

in a heat bath at temperature

b
T
m
T
phase transition temperature

30

M. Schmitd et al.

31

Solid/liquid/gas transition

Boiling nuclei


multi fragmentation:

MeV
T
LG
5

indication for

0


C
(surface energy of the fragments)

no shell effects

shell
LG
T
T

Melting Na clusters:

K
T
K
T
bulk
m
310
250



0


C
in contrast to bulk melting

Strong shell effects

32

Transition from electronic to geometric shells

In Na clusters

K
T
250
~
36

T. P.Martin Physics Reports 273 (1966) 199
-
241

Solid state, liquid He:

Calculation of


very problematic


well protected.

Take from experiment.

c
T
K
K
T
T
N
N
F
c
F
5
10
1
~
~
~



R
m
v
F





15
~
/
0

local

BCS very good

Nuclei:

Calculation of


not possible so far.

Adjusted to even
-
odd mass differences.


fm
R
fm
v
F
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