Transcript Seminar: Statistical Decay of Complex Systems (Nuclei)

CoulEx
Coulomb Excitation (Semi-Classical)
R0  
1  1 sin  2 
Torque exerted on deformed target  excitation of
collective nuclear rotations (angular momentum I ):
:a
For axially symmetric e  e nuclei
Deformed Charge
ground state I  0
EI 
2
I(I  1)
Distribution
I  0,2,....
238
2
  moment of inertia
I
U
 r  R
spherical
deformed
Adiabaticity Condition:
Fast “kick” will excite nucleus.
Otherwise adiabatic
adjustment of deformed
nucleus to minimize energy
Collision time
 coll 
2R0
rel
 if1
1
R0
t(R0)

1

  Ef*  Ei* 


1
Adiabaticity parameter

1
 :
Ef*  Ei*   coll    1
if
rel
4


d
coll
rot. period

Vcoul(t)
For small energy losses
(weak excitations
i  f , Ei*  Ef*):
 
i f
d
t
 Pi  f 
d Ruth  
d
Excitation probability in perturbation theory:
Pi f   bi f
f
2
bi f  
i

 dt e
i
Ef* Ei* t
f H (t ) i

Transition amplitude  Fourier transform of transition ME
elm projectile - target interaction H     VCoul  j  A
Electric Coulomb Excitation
3
Consider target excitation only
T
e  T  r 
rP  t   r
 r t    1
 Z


 T

P

  t   eZP    
Hel
Y P , P     e  riY * i , i  
  i 1
2  1
 



: M  E  
bi  f
Classical Pot. Scattering
  t   eZP   d 3r
Hel

4


eZP     dt ei if t
i
  

 

Y

(
t
)
,

(
t
)


P
  P
 1
f M  E   i
 1

  2  1

 rP  t  


 

Radial integral f(Ecm, , )
“geometry” factor
Radial integral is calculated for Rutherford
trajectory rP  t  and transition frequency  if
Fit to angular distribution determines
nuclear matrix element.
W. Udo Schröder, 2007
“Expand in multipoles”
Multipolarity ,
Projection 
2: quadrupole
deformation
Nuclear matrix element
dynamics factor
Total cross section
2
 eZP  2  2
 
B(E )  f  
 a

 rel 
B(E )  reduced nuclear ME
f   1 for 
1
Collective Rotations
z
4  R
ab
R
; R  a  b
3 5 R
2
 := quadrupole deformation parameter
R( ,  )  R0 1   Y02 ( ,  )


b
a
 
3
2
4
Quadrupole moment Q0: Q0  eZ  d r (r )  r 
Q0 
3eZ
R02  1  0.16  
Nuclear Deformations
5
Rotational and inversion
symmetry  even I
EI 
2
2
3 cos
2

  1
Deexcitation-Gamma Spectra
2
2
 15  18keV
I  I  1
Rigid body mom. o. inertia :
2
rig  MR02 1  0.31 
5
Hydro  dynamical :
9
irr 
MR02 
8
W. Udo Schröder, 2007
Wood et al.,Heyde
W. Udo Schröder, 2007
Classical Pot. Scattering
5