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Collective Response of Atom Clusters and Nuclei: Role of
Chaos
Trento April 2010
Mahir S. Hussein
University of Sao Paulo
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Contents
Metal ClustersNucleiGiant dipole resonancesExit doorway modelDamping width and ChaosRandom matrix theory
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Metal Clusters
Aggregate of N atoms Excite plasmon-type Mie resonances Electrons oscillate out of phase with respect
to ions. Probes: Laser, electrons, other clusters. Multi-plasmon resonances
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PRL, 68, 3916 (1992).
PRL,70, 2036 (1993).
2max
2 2 2 20
[ ]( )
[( ) ( ) ] [ ]n n
hh
h h h
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Energy and width of cluster plasmon resonances
PRL,70, 2036 (1993).
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Multiphonon excitation in Na clusters
PRL, 80, 1194 (1998).
A perfect harmonic oscillator : n=4
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Plasmon excitation energy vs. size of Xe and Ar clusters
Phys. Rev. Lett. 67, 3290 (1991).
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Plasmon excitation energy and width vs. size of Hg clusters
Phys. Rev. Lett. 69, 3212 (1992).
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Nuclei Aggregate of N neutrons and Z protons Collective excitation of giant dipole and
quadrupole resonances Probes: photons, electrons, other nuclei Neutrons oscillate against protons (dipole) Collective state is damped to chaotic
configuration.
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J. G. Woodworth et al., Phys. Rev. C, 19, 1667 (1979); G. J. O’keefe et al., Nucl. Phys. A 649, 239 (1987).
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R. Schmidt et al. Phys. Rev. Lett. 70, 1767 (1993)
Single- and double-phonon excitation in Xe nucleus.
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Rev.Mod.Phys., 47, 713 (1975).
2max
2 2 2 20
[ ]( )
[( ) ( ) ] [ ]n n
hh
h h h
Shape is Lorentzian (spherical).
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Rev. Mod. Phys., 55, 287 (1983)
Widths of giant dipole and quadrupole resonances in nuclei
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Rev.Mod.Phys., 47, 713 (1975).
Excitation energy of giant dipole resonances in nuclei
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Pygmy giant resonances!
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Phys. Rev. Lett., 95, 132501 (2005).
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Split dipole state: Schrődinger cat?
Z. Phys. A 355. 165 (1996)
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Exit doorway model
Consider the many-body Hamiltonian with time-dependent interaction
0 ( )H H V t
0H
( )V t
0 | |i i iH
Intrinsic Hamiltonian of the system
External perturbation
System responds to action of and is excited to some collective state (not an eigenstate of )
0H( )V t
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/| ( ) ( ) |ni tn n
n
t a t e
( ) | ( ) | ( )H t t i tt
/0 0
/ ( ) /0 0
0
( ) ( )
( ) ( ) ( ) ( )
n
n n m
i tn n
n
i t i tn n nm m
m
i a e V t a t
i a e V t a t e V t a t
0( )n na t
2| ( ) |n nP a t
Calculation of excitation probabilities
Write
Use
0 0 Get (taking )
Initial conditions
Excitation probabilities
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Consider
2
1
( ) ( )
ˆ( ) ( )( )
N
ii
V t D E t
ZeE t R t
R t
D e r
: dipole operator
: electric field of external probe.
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| | |mi n nm
d m
0 0 0| | | |n n n n odV V V d V
0 0 0 0| | | |mm
H m m | | [| | . .]d m
m
d d m d H C
Excitation of collective state with finite lifetime can be treated using the exit-doorway (ED) model
Expanding 0H
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' '|n n n n
2
2
2
| |
( )
1| ( ) |
1 m
n m
n
m
2| |mn d
m n m
,m m ms
2
cot d
n d s s
2
2' 22
1| ( ) |
2 ( )dd
dn
n
'd d
Since , We get
Model Uniform spectrum
get
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Spreading or damping width, measures the degree of mixing of the collective state with the background (chaotic) states.
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( ) ( )V t D E t
0nmV
2( )( ')/002
( ) 1( ) ' [ ( ')]* ( ')
dd
t i i t tda tV t dt e V t a t
dt
0, 1
0 ( ) 1 ( )a t A t
Take
does not mix excited statesD
With ED hypothesis, get
Write
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2
2
( )( )
( ) 2
| ( ) |1 ( ) 0
dd
V t iiA A t
V t
V tA t
( ) 0A t
2 / * 2
, , 02 2
1| | | ( ) ( ) |
2 / 4i td
n d d
P a dte V t a t
Get three coupled differential equations
With
Excitation probabilities
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min,2 ( )
b
dbdbP b
d
And cross-section
2 / * 2
, , 02 2
1| | | ( ) ( ) |
2 / 4i td
n d d
P a dte V t a t
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Ann. Phys. (New York), 284, 178 (2000).
Calculation of excitation cross-section of a damped plasmon in sodium clusters
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Role of chaos
•Response function, spreading width
•Dynamical enhancement of multiphonon excitations.
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Dynamical enhancement of multiphonon excitation
Owing to , to go from ground state to 2 phonon state
(via 1 phonon) one may, if collision time , excite a single
phonon on top of background which gives rise to
0 1 2ph ph ph
1
1
c
1
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Excitation probability
2
exp[ ] ( )!
1exp[ ] ( )
n
n n
pP p P fluctuation
n
p dt i t V ti
Harmonic (Poissonian)
Dynamical (Brink-Axel)
Ann. Phys.(New York), 276, 111 (1999). Nucl. Phys. A 731, 163 (2004).
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Ann. Phys. (NY) 276,111 (1999);
Nucl. Phys. A 690, 382 (2001).
Coupling between collective and chaotic states.
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Evolution of density matrix with damping and dynamical chaos enhancement
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Phys. Rev. C, 60, 014604 (1999).
Dynamical chaos enhancement of cross-section
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Ann. Phys.(New York), 276, 111 (1999). Nucl. Phys. A 731, 163 (2004).
Dynamical enhancement could be as large as 80%.
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Conclusions•Collective response of finite many-body systems is affected by the degree of chaoticity of the internal degrees of freedom.
•The system acquires a damping width: Damped harmonic oscillator
•Chaos leads to an enhanced excitation owing to the damping.
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Thanks to my collaborators:
Brazil:
C. A. Bertulani, L. F. Canto, B. V. Carlson, R. Donangelo, J. X. de Carvalho, M. P. Pato, A. F. R. de Toledo Piza
Germany:
T. Aumann, H. Emling
USA:
H. Feshbach, A. K. Kerman, V. Kharchenko, E. Timermmans
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Thank you!