Transcript Seminar: Statistical Decay of Complex Systems (Nuclei)

W. Udo Schröder, 2005
Rotational Spectroscopy
1
Rigid-Body Rotations
r  r ':decompose into 3 rotations
x
 x "
 x "' 
 x '
 
 


 
z
y"
z "'
y


y
"


y
"'





 
 


 y '
z
 z "
 z "' 
 z '
 
 


 
2
 x "
x
 
 
y
"

R
(

)
z
 
y 
 z "
z
 
 
Rotational Spectroscopy
Axially symmetric nucleus
W. Udo Schröder, 2005

 x "' 
 x "


 
y
"'

R
(

)
y'


 y "
 z "' 
 z "


 
 x '
 x '" 
 


y
'

R
(

)
y
'"
z
 


 z '
 z '" 
 


ˆH
ˆ
ˆ
ˆ
ˆ
ˆ
H
intr  Hrot  Hint  Hintr  Hrot
I  total spin
R
I
K
R  coll. rot
Rotational Wave Functions


I3 due to intrinsic
2


ˆ
2
2
2
ˆ
ˆ
Hamiltonian : H
I ˆ
I3 
I3 s.p. spins =


2 
independent d.o.f.
 23
2
 R

2
R
I
M
Energy eigen values E IK 
2
2
I
Wave function :  MK
( ,  , )  IMK 
Rotational Spectroscopy
K  I  I  K , K  1, K  2, K  3, ....
I
I
DMK
( ,  , )  e  iM e  iK dMK
( )
2I  1
I
dMK
( )  Norm(I, M, K )  S(I, M, K ,  )
Norm(I, M, K )  (I  M)!(I  M)!(I  K )!(I  K )!
SI  M  K    
W. Udo Schröder, 2005
min ( IK  IM)

s  max ( 0  MK)
   
( 1)   sin  
  2 
KM2 s
   
  cos  
  2 
I
DMK
( ,  , )
8 2
Conserved : K
I
DMI 0 ( ,  , )  e iM dMK
( ) 
s
2
 I(I  1)  K 
K2
 2
2 
3
3
K
2 IKM2 s
( I  K  s)  ( I  M  s)  ( s  K  M)  s
4
YMI ( )
2I  1
Rotational Spectroscopy
4
Example Wave Functions
W. Udo Schröder, 2005
I
I
dMK
( ), DMK
( ) are complete basis
Overl(I1 , M1 , K1 , I2 , M2 , K2 )  I1 , M1 , K1 I2 , M2 , K2

I
I
  d sin  dM1 K ( )  dM2 K 2 ( )
1 1
2
0
I1 , M1 , K1 I2 , M2 , K2   I1I2  M1M2  K1K2
R Invariance of Axially Symmetric Nuclei
3
2
R
I
I
I
 MK
(q,  )   K (q)  DMK
( )
ˆ (q) invariant :
Intrinsic Hamiltonian H
int
M
K
5
ˆ ,
ˆ intr ( )  0
Rot of q about 2 axis ( ) H
intr


Rotation of orientation coll  coll ( ) intr ( )
I
coll ( )DMK
( )  (1)I  K DMI ,  K ( )
Rotational Spectroscopy
Construct symmetric total wave function:
I
 MK
(q,  ) 
I
 MK
(q,  )

2
1

1
1  intr
coll
2I  1

2I  1
8 2
I
 K (q)  DMK
( )


I
 K (q)  DMK
( )  (1)I  K   K (q)  DMI , K ( )
16 2
For K  0 (e.g. gg  Nuclei ) :
I
M
0 (q,  ,  ) 
W. Udo Schröder, 2005
“signature”
s=(-1)I+K

1
0 (q)  YMI ( ,  ) 1  (1)I
4

 I  0,2, 4,....
Example: Rot Spectrum
238U
Even-I sequence I=0+, 2+, 4+,…
Erot (I ) 
Rotational Spectroscopy
2

I 
W. Udo Schröder, 2005
I(I  1)
2
  Erot (I  1  I )
6
E2
2
2

2
  rigid
2(I  1)
  const.
 1
 I 2 I I 2
Q

K
0
K

I 0
I
 Q(M  I )
I
3K 2  I(I  1)

Q(M  I )
(I  1)(2I  3)
Q0  const.
Effect of rotation on
nucleonic motion
even for Q0 = const.
E. Grosse et al., Phys. Scripta 24, 71 (1977)
K Bands in
168Er
7
Bohr & Mottelson, Nucl. Struct. II
Different intrinsic spins (K)
and parities (r)
Rotational Spectroscopy
Mainly E2 transitions within
bands
W. Udo Schröder, 2005
K forbiddenness
“Back Bending”
Bohr & Mottelson, J. Phys. Soc. Japan 44, Suppl. 157 (1977)
excited state band
Rotational Spectroscopy
8
  rigid
ground state band
At high spins  break up of J=0 pair, reduction of moment of inertia .
W. Udo Schröder, 2005
Super Deformation
152Dy
Twin et al., 1986, ARNS 38 (1988)
108Pd(48Ca,
xn)156-xnDy*
9
SD band: 19 transitions I≤ 60
E ≈ 47 keV
large Q0 = 19 eb
Rotational Spectroscopy
BE2 = 2660 s.p.(W.u.)
highly collective
W. Udo Schröder, 2005
Wood et al., Phys. Rep. 215, 101 (1992)
Deformation Energy Surfaces
Tri-axial nuclear shapes:
2


R( ,  )  R0 1   2 Y2 ( ,  )
  2


Ellipsoid : 2 1  0 22  2  2
Rotational Spectroscopy
10
20   cos 
22 
1
2
 sin 
 deformation par.
 shape par.


5
c  R0 1 
 cos  
4



5
b  R0 1 
 cos( 
4


5
a  R0 1 
 cos( 
4


2 
)
3 
semi
axes
4 
)
3 
PES : Minimize pot. energy of rotating
liquid drop with int ernal structure :
W. Udo Schröder, 2005
E( ,  , I )  ERLDM ( ,  , I )  EShell ( ,  , I )
Angular Distribution of Symmetry Axis
 (2I  1)
Rotational Spectroscopy
11
I
WMK
( )
W. Udo Schröder, 2005
I
DMK
(,  , )
2