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How do nuclei rotate?
1. The molecular picture
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The classical rotor
1
2
3
conserved : a.m. of valueabsolute
conserved 2
1
2
1 :energy
)( :inertia of moments
:momentumangular
axes principal 3,2,1 :locityangular ve
22
22
2,3
2,21
i
i
iii
nnnn
iii
i
JJ
JE
xxm
J
i
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sphere momentumangular 22iJJ
2
1 spheroidenergy
2
i
iJE
Axial rotorClassical motion of J
21
orbit
K
J
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Triaxial rotorClassical motion of J
Intermediate E Large E
Small E
wobblingmotion
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Euler angles
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Quantization
nspermutatio cyclic and ],[ :axes laboratory zyx JiJJ
nspermutatio cyclic and ],[ :axes fixedbody 321 JiJJ
zyxkJJ k ,, 0],[ 2
0],[ 3,2,1 0],[ 32 JJiJJ zi
)1( :numbers quantum 22 IIJMJ z
)1( :numbers quantum 223 IIJKJ
KMI ,,| :states
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The molecular rotor
3NH
1
2
3 21 Axial rotor
3
23
1
23
2
2
1 JJJH
3
2
1
2)1(
2
1 KKIIE
0],[0],[0],[ 23 JHJHJH z
3
23
1
22
21
2
1 JJJH
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KMI ,,| :seigenstate
function Wigner D iKIMK
iMIMK edeD )(),,(
orbitK
J
),,(8
12,,|,, :rotor ofn orientatiofor
amplitudey probabilit2/1
2
I
MKDI
KMI
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Centrifugal stretching
....))1(()1( 2 IIBIAIEStiff bonds
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1
2
3
OH2
Triaxial rotor
0],[2
13
3
23
2
22
1
21
JHJJJ
H
Small E
wobblingmotion
K
Kn KMIcnMI ,,|,,| ,
)2/1(2
)1(
1
nII
E W
2/1
32
3121
1
))((
I
W
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.
.
Born-Oppenheimer Approximation
Electronic motion
Vibrations
Rotations eVrot410~
eVel 1~
CO
eVvib110~
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rot
el
vib
Adiabatic approximation
vibRvibmm
mm
OC
OC || 2
elHHHelRV eenenn ||)(
),,()()( , rotrelnucel
rotvibel
Xx
EEEE
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HCl
)1()()1(
)1()(
IBIEIE
JIIBIIE
Microwave absorptionspectrum
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Band Spectrum
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Indistinguishable Particles
.
.
2O
2
Upper particles Lower particles
Restriction of orientation
.2 with identical is 0
signature
2 ruleselection nI
elenuci
elenuc e )(2R
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oddnI
e elenuci
elenucelenuc
211
)( 12
R
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The nuclear rotor
Most nuclei have a deformed axial shape.
Unified Model (Bohr and Mottelson):
The nucleus rotates as a whole. (collective degrees of freedom)
The nucleons move independentlyinside deformed potential (intrinsic degrees of freedom)
The nucleonic motion is much fasterthan the rotation (adiabatic approximation)
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Nucleons are indistinguishable
),,(),,()( rotKrotin
rotin
x
EEE
2
)1( 2KIIEE in
The nucleus does not have an orientation degree of freedomwith respect to the symmetry axis.
03
2
K
Axial symmetryin
iKin e )(3R
K
2/1
2),,(
8
12
IMKD
I
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)(2R symmetry
KK )(2 R
KK
2/1
2),,()(),,(
16
12
I
KMKII
MK DDI
00
2/1
2
00
),,(8
12
ruleselection signatureodd
evenI 0
IMD
I
K
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Dyin band medsuperdefor a from rays - 152
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2))1(()1(),( IIBIAIEKIE K
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Electromagnetic Transitions
Emitted photon has multipolarity E1, E2, E2, ... or M1, M2, ...
2112221 ||),)((||);)((
))((y probabilit transition
2
MIMEMIIIMEB
MEBET
M
M
Multipole moments of the nucleus ),)(( MEM
1),1( dipole electric rYqE vM
vv rYqE 2),2( quadrupole electric M
11
2/1
4
3),1( dipole magnetic sglgM slM
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2121 ,0,0 KKKK
Reduced transition probabilities in the Unified Model
'|'
|
)''),((
)),((
rules Alaga
2221211
2221211
21
21
KIKKKI
KIKKKI
IIMEB
IIMEB
||),(||
| )),((2
2122
222121121
inKKKinK
KIKKKIIIMEB
M
|t Coefficien Gordan-Clebsh 221211 KIKKKI
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20
2
2
2
45
)(
|202
|101
),2,2(
),1,1(
Q
Kgg
IKKI
IKKI
KIIKEB
KIIKMB RK
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Limitations of the molecular picture
What is rotating? The nuclear surface
HCl
Nucleons are not on fixed positions.
rigid rotor
)( :inertia of moments 2,3
2,21
nnnn xxm
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More like a liquid, but what kind of?
viscous
))((d
:inertia ofmoment body" rigid"
2,3
2,2rig1, nn xxm x
Ideal“irrotational flow” moment of inertia
1rig223
22
223
22
1irr )()(
)()(
RR
RR
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rigid
irrotational
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Breakdown of adiabatic approximation
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Summary
• Molecules are the protoype of quantal rotors.• Electronic and vibrational motion are much faster than rotation.• Rotational bands consist of states with different angular momentum and the same intrinsic state
(elec., vib.).• Indistiguishability leads to restrictions in the possible values of the angular momentum.• Nuclei at low spin are are similar to molecules. The nuclear surface is rotating.• Unified model: intrinsic states correspond to the motion of nucleons in the deformed potential. • Nuclei are liquid-like. The flow pattern is dominated by quantal effects.• Microscopic theory needed for calculating them.