collective response of atom clusters and nuclei: role of chaos trento april 2010 mahir s. hussein...
TRANSCRIPT
Collective Response of Atom Clusters and Nuclei: Role of
Chaos
Trento April 2010
Mahir S. Hussein
University of Sao Paulo
Contents
Metal ClustersNucleiGiant dipole resonancesExit doorway modelDamping width and ChaosRandom matrix theory
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
PRL, 68, 3916 (1992).
PRL,70, 2036 (1993).
2max
2 2 2 20
[ ]( )
[( ) ( ) ] [ ]n n
hh
h h h
Energy and width of cluster plasmon resonances
PRL,70, 2036 (1993).
Multiphonon excitation in Na clusters
PRL, 80, 1194 (1998).
A perfect harmonic oscillator : n=4
Plasmon excitation energy vs. size of Xe and Ar clusters
Phys. Rev. Lett. 67, 3290 (1991).
Plasmon excitation energy and width vs. size of Hg clusters
Phys. Rev. Lett. 69, 3212 (1992).
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.
J. G. Woodworth et al., Phys. Rev. C, 19, 1667 (1979); G. J. O’keefe et al., Nucl. Phys. A 649, 239 (1987).
R. Schmidt et al. Phys. Rev. Lett. 70, 1767 (1993)
Single- and double-phonon excitation in Xe nucleus.
Rev.Mod.Phys., 47, 713 (1975).
2max
2 2 2 20
[ ]( )
[( ) ( ) ] [ ]n n
hh
h h h
Shape is Lorentzian (spherical).
Rev. Mod. Phys., 55, 287 (1983)
Widths of giant dipole and quadrupole resonances in nuclei
Rev.Mod.Phys., 47, 713 (1975).
Excitation energy of giant dipole resonances in nuclei
Pygmy giant resonances!
Phys. Rev. Lett., 95, 132501 (2005).
Split dipole state: Schrődinger cat?
Z. Phys. A 355. 165 (1996)
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
/| ( ) ( ) |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
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.
| | |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
' '|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
Spreading or damping width, measures the degree of mixing of the collective state with the background (chaotic) states.
( ) ( )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
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
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
Ann. Phys. (New York), 284, 178 (2000).
Calculation of excitation cross-section of a damped plasmon in sodium clusters
Role of chaos
•Response function, spreading width
•Dynamical enhancement of multiphonon excitations.
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
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).
Ann. Phys. (NY) 276,111 (1999);
Nucl. Phys. A 690, 382 (2001).
Coupling between collective and chaotic states.
Evolution of density matrix with damping and dynamical chaos enhancement
Phys. Rev. C, 60, 014604 (1999).
Dynamical chaos enhancement of cross-section
Ann. Phys.(New York), 276, 111 (1999). Nucl. Phys. A 731, 163 (2004).
Dynamical enhancement could be as large as 80%.
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.
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
Thank you!