w. udo schröder, 2005 rotational spectroscopy 1. w. udo schröder, 2005 rotational spectroscopy 2...
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W. Udo Schröder, 2005
Rota
tion
al S
pect
rosc
op
y 1
W. Udo Schröder, 2005
Rota
tion
al S
pect
rosc
op
y 2
Rigid-Body Rotations
" "'" "' '
" "' '
" "' '
z y zx x x x
y y y y
z z z z
': 3decompose into rotatr onsr i
"
"
)" (zR
x
y
x
y
z z
'
"
"
"'
"' ( )
"' "y
x
y
zz
x
yR
'"
( ) '"
'
'
'' "z
x
y
x
R y
zz
Axially symmetric nucleus
IR
K
ˆ ˆ ˆ ˆ ˆ ˆintr rot int intr rotH H H H H H
.
I total spin
R coll rot
W. Udo Schröder, 2005
Rota
tion
al S
pect
rosc
op
y 3
Rotational Wave Functions
S I M K max 0 M K( )
min I K I M( )
s
1( )s sin2
K M 2 s
cos2
2 I K M 2 s
I K s( ) I M s( ) s K M( ) s
04
( , , ) ( ) ( , , ) ( ) ( )2 1
( ) ( , , ) ( , , , )
( , , ) ( )!( )!( )!( )!
I iM iK I I iM I IMK MK M MK M
IMK
D e e d D e d YI
d Norm I M K S I M K
Norm I M K I M I M I K I K
2 22 2 2
3 332
2 22 2
3
2
ˆˆ ˆ ˆ:2 2
( 1)2 2
2 1: ( , , ) ( , , )
8, 1, 2, 3, . :. ..
R
IK
I IMK MK
C
Hamiltonian H I I I
Energy eigen values E I I K K
IWave function IMK D
K I I K K K onserved KK
I3 due to intrinsic s.p. spins = independent d.o.f.I
R
K
M
W. Udo Schröder, 2005
Rota
tion
al S
pect
rosc
op
y 4
Example Wave Functions
1 1 1 2 2 2 1 1 1 2 2 2
1 221 1 20
1 1 1 2 2 2 1 2 1 2 1 2
( ), ( )
( , , , , , ) , , , ,
sin ( ) ( )
, , , ,
I IMK MK
I IM K M K
I I M M K K
d D are complete basis
Overl I M K I M K I M K I M K
d d d
I M K I M K
W. Udo Schröder, 2005
Rota
tion
al S
pect
rosc
op
y 5
R Invariance of Axially Symmetric Nuclei
IR
K
M3
2
( , ) ( ) ( )
ˆ ( ) :
ˆ ˆ2 ( ) , ( ) 0
I IMK K MK
int
intr intr
q q D
Intrinsic Hamiltonian H q invariant
Rot of q about axis H
,
( ) ( )
( ) ( ) ( 1) ( )
coll coll intr
I I K Icoll MK M K
Rotation of orientation
D D
Construct symmetric total wave function:
12
,2
0 0
1 2 1( , ) 1 ( ) ( )
82
2 1( , ) ( ) ( ) ( ) ( )
16
1( , , ) ( ) ( , ) 1 ( 1) 0,2, 4,....
4
( 1)
0 ( . . ) :
I IMK intr coll K MK
I I IMK K MK K M K
I I IM
I K
M
For K e g gg Nu
Iq q D
Iq q D q D
cl
q q Y
ei
I
“signature”
s=(-1)I+K
W. Udo Schröder, 2005
Rota
tion
al S
pect
rosc
op
y 6
Example: Rot Spectrum 238U
Even-I sequence I=0+, 2+, 4+,…2
2
2
( ) ( 1)2( 1 )
2( 1)2
1
rot
rot
E I I I
E I I
I
I
.const
rigid
0 .Q const
Effect of rotation on nucleonic motion
even for Q0 = const.
2
2 2( )
0 0
3 ( 1)( )
( 1)(2 3)
I I I IQ Q M I
K K I I
K I IQ M I
I I
E. Grosse et al., Phys. Scripta 24, 71 (1977)
E2
W. Udo Schröder, 2005
Rota
tion
al S
pect
rosc
op
y 7
K Bands in 168Er
Bohr & Mottelson, Nucl. Struct. II
Different intrinsic spins (K) and parities (r)
Mainly E2 transitions within bands
K forbiddenness
W. Udo Schröder, 2005
Rota
tion
al S
pect
rosc
op
y 8
“Back Bending”
rigid
Bohr & Mottelson, J. Phys. Soc. Japan 44, Suppl. 157 (1977)
ground state band
excited state band
At high spins break up of J=0 pair, reduction of moment of inertia .
W. Udo Schröder, 2005
Rota
tion
al S
pect
rosc
op
y 9
Super Deformation 152Dy
Twin et al., 1986, ARNS 38 (1988)
108Pd(48Ca, xn)156-xnDy*
Wood et al., Phys. Rep. 215, 101 (1992)
SD band: 19 transitions I≤ 60 E ≈ 47 keV large Q0 = 19 eb
BE2 = 2660 s.p.(W.u.) highly collective
W. Udo Schröder, 2005
Rota
tion
al S
pect
rosc
op
y 1
0
Deformation Energy Surfaces
: .
int :
( , , ) ( , , ) ( , , )RLDM Shell
PES Minimize pot energy of rotating
liquid drop with ernal structure
E I E I E I
Tri-axial nuclear shapes:2 2
0 22
2 1 22 2 2
20
22
( , ) 1 ( , )
: 0
cos .
1sin .
2
R R Y
Ellipsoid
deformation par
shape par
0
0
0
51 cos
4
5 21 cos( )
4 3
5 41 cos( )
4 3
c R
b R
a R
semi axes
W. Udo Schröder, 2005
Rota
tion
al S
pect
rosc
op
y 1
1
Angular Distribution of Symmetry Axis
2( ) (2 1) ( , , )I I
MK MKW I D