Continuum shell model: From Ericson to conductance fluctuations

Novosibirsk, May 23, 2008

Continuum shell model:

From Ericson to conductance fluctuations

Felix IzrailevInstituto de Física, BUAP, Puebla, México

Michigan State University, E.Lansing, USA

in collaboration with :G. Berman -- Los Alamos, USA

L. Celardo -- Puebla, Mexico

S. Sorathia -- Puebla, Mexico

V. Zelevinsky – E.Lansing, USA


• Continuum shell model• Resonance widths • Cross sections• Ericson fluctuations• Conductance fluctuations• Discussion

Overview

V.G.L.Celardo, F.M.Izrailev, V.G.Zelevinsky, G.P.Berman,

Phys. Rev. E, 76 (2007) 031119, Phys. Lett. B 659 (2008) 170.


Effective Hamiltonian approach to open systems

N

Wi

HH eff 2

MiEc,

N

γδAA cc'

ijc'j

ci

(E)Aci

intrinsic many-body states coupled to open channels

with transition amplitude

an effective non-Hermitian Hamiltonian

c

cj

ciij AAWwith and

Scattering matrix

can be described by

bj

N

ji,ijeff

ai

ab AHE

1A(E)

EiES ababab

where


Isolated versus overlapped resonances

The complex eigenvalues of iii Γ2

iωE effH

are poles of - matrix with(E)S ab

2ND

πγκ Control parameter of the coupling to continuum:

where is the mean level spacingD

At 1κ we have perfect coupling regime, 1T For 1κ the segregation of widths occurs, with the formation

of Msuperradiant (wide) resonances and M-N narrow ones

V.P.Kleinwächer and I.Rotter, Phys.Rev.C 32 (1985) 1742; V.V.Sokolov and V.G.Zelevinsky, Phys. Lett. B 202 (1989) 10; Nucl. Phys. A 504 (1989) 562.

c

2

1 ccc ST


Two-Body Interaction Model

rpqkqpr

kkqprk

m

kkk aaaaVaaH

2

1

r,p,q,k single-particle states

kqprV two-body matrix elements

m number of single-particle states

n number of particles (“quasi-particles”)

k energy of single-particle states

H is considered in the many-particle basis of k

M

kkk aaH

1

0

1148 00 nmnnn/dvv cr transition to chaos :

220 kqprVv

V.V.Flambaum and F.M.I. – Phys. Rev. E 64 (2001) 036220

12m

6n


Redistribution of widths

50M 924N cr0 v/dv 10Here for


Average width

κ1

κ1ln

π

M

D

Γ

M

1c

cT1ln2π

1

D

Γ

Moldauer-Simonius

for equivalent channels:

21

4c

ccT


Resonance width distribution

tail

-2 P


Typical (elastic) cross sections

1κ

1κ

1κ

50M 2EE baba


Average cross section

M

T

1MF

Tσ

1M

inefl

M

FT

1MF

FTσ

1M

elfl

301252

052

2

00

00

/d/vfor.F

d/vfor.F

GOEforF

Elastic enhancement factor inefl

elfl σσF


Dependence of elastic average cross

section on the interaction strength


Enhancement factor vs interacton

inefl

elfl σσF


Ericson Fluctuations

22

2

Γε

Γ0CεC

some of the Ericson assumptions:

1 Var

2

DΓ

some of the Ericson predictions:

fl

fl-x

σ

σxexP

Lorentzian form (for cross sections)

for


Cross section autocorrelation length vs average width

2π

MT

D

lσ

σlΓ

Weisskopf relation:

Contrary to Ericson prediction:


Conductance

M/2

1a

M/2

1M/2b

abσG

-- for “left” channels

-- for “right” channels

a

b


Universal Conductance Fluctuations

1for M

; 8

1GVar

From Random Matrix Theory

For uncorrelated cross sections :

4

1GVar

Correlations are important !


Correlations between different cross sections

corr1/4σσσσGVar ba

baab

abbaab

/4M1baba 2,,,where

Correlations are increasing with M , and they occur for both chaotic and regular intrinsic dynamics !

above, the total correction term for variance is shown,

that is due to all correlations neglected in the Ericson theory


Two types of correlations for cross sections

where stand for correlations between cross sections

having one joint channel, and -- for correlations between

cross sections with no joint channels

Λ

Σ


Analysis of correlations

CNNCN

MF

TMGVar c

**22

14

/4ML2MLN/2LLN c2* ; ; 1 where

for 1M we have the estimate :

43 ;

2MCMC

V.A. Garcia-Martin et al, Phys. Rev. Lett. 88 (2002) 143901

8

1

8

1

4

1

4

1GVar !!


1) For the first time the truly TBRE is considered in the framework of the Continuum Shell Model

Conclusions

2) The statistics of resonance widths are found to be very sensitive to the intrinsic chaos.

3) Contrary to Ericson expectations the fluctuations of resonance widths cannot be neglected even for large number of channels

4) The elastic enhancement factor strongly depends on the intrinsic interaction, thus the Hauser-Feshback formula must be modified

5) Universal conductance fluctuations are due to strong correlations between cross sections, they are different from Ericson fluctuations


www.felix.izrailev.com


Divergence of the width variance

2

22κ1ln

κ1

1

Γ

ΓVar


Resonance width variance vs interaction strength

10M


Distribution of correlations -GOE


Dependence on the degree of internal chaos


Resonance width variance vs coupling to continuum

2M


Dependence on the coupling to continuum


4

MT

1MF

T

4

MG

1M

2

for the GOE:

Mean conductance


Resonance residues-energies correlations


Distribution of correlations –TBRI model


Cross section distribution

Comparison with Ericson exponential distribution


Fluctuations

Black line: Analytical

Results for GOE from

E.D.Davis and D. Boose,

Phys.Lett. B 211, 379 (1988).


Condactance Fluctuations vs M


Cross section fluctuations

Ericson prediction:

2

flfl σσVar


Resonance width variance vs M

Expectation (due to Ericson) -

1M

Var2

1

Continuum shell model: From Ericson to conductance fluctuations

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