Post on 30-Dec-2015
description
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
Novosibirsk, May 23, 2008
• 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.
Novosibirsk, May 23, 2008
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
Novosibirsk, May 23, 2008
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
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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
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Redistribution of widths
50M 924N cr0 v/dv 10Here for
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Average width
κ1
κ1ln
π
M
D
Γ
M
1c
cT1ln2π
1
D
Γ
Moldauer-Simonius
for equivalent channels:
21
4c
ccT
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Resonance width distribution
tail
-2 P
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Typical (elastic) cross sections
1κ
1κ
1κ
50M 2EE baba
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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
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Dependence of elastic average cross
section on the interaction strength
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Enhancement factor vs interacton
inefl
elfl σσF
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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
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Cross section autocorrelation length vs average width
2π
MT
D
lσ
σlΓ
Weisskopf relation:
Contrary to Ericson prediction:
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Conductance
M/2
1a
M/2
1M/2b
abσG
-- for “left” channels
-- for “right” channels
a
b
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Universal Conductance Fluctuations
1for M
; 8
1GVar
From Random Matrix Theory
For uncorrelated cross sections :
4
1GVar
Correlations are important !
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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
Novosibirsk, May 23, 2008
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
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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 !!
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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
Novosibirsk, May 23, 2008
www.felix.izrailev.com
Novosibirsk, May 23, 2008
Divergence of the width variance
2
22κ1ln
κ1
1
Γ
ΓVar
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Resonance width variance vs interaction strength
10M
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Distribution of correlations -GOE
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Dependence on the degree of internal chaos
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Resonance width variance vs coupling to continuum
2M
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Dependence on the coupling to continuum
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4
MT
1MF
T
4
MG
1M
2
for the GOE:
Mean conductance
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Resonance residues-energies correlations
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Distribution of correlations –TBRI model
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Cross section distribution
Comparison with Ericson exponential distribution
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Fluctuations
Black line: Analytical
Results for GOE from
E.D.Davis and D. Boose,
Phys.Lett. B 211, 379 (1988).
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Condactance Fluctuations vs M
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Cross section fluctuations
Ericson prediction:
2
flfl σσVar
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Resonance width variance vs M
Expectation (due to Ericson) -
1M
Var2
1