How do nuclei rotate?

1. The molecular picture

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

sphere momentumangular 22iJJ

2

1 spheroidenergy

2

i

iJE

Axial rotorClassical motion of J

21

orbit

K

J

Triaxial rotorClassical motion of J

Intermediate E Large E

Small E

wobblingmotion

Euler angles

5/30

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

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

KMI ,,| :seigenstate

function Wigner D iKIMK

iMIMK edeD )(),,(

orbitK

J

),,(8

12,,|,, :rotor ofn orientatiofor

amplitudey probabilit2/1

2

I

MKDI

KMI

Centrifugal stretching

....))1(()1( 2 IIBIAIEStiff bonds

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

10/30

.

.

Born-Oppenheimer Approximation

Electronic motion

Vibrations

Rotations eVrot410~

eVel 1~

CO

eVvib110~

rot

el

vib

Adiabatic approximation

vibRvibmm

mm

OC

OC || 2

elHHHelRV eenenn ||)(

),,()()( , rotrelnucel

rotvibel

Xx

EEEE

HCl

)1()()1(

)1()(

IBIEIE

JIIBIIE

Microwave absorptionspectrum

Band Spectrum

Indistinguishable Particles

.

.

2O

2

Upper particles Lower particles

Restriction of orientation

.2 with identical is 0

signature

2 ruleselection nI

elenuci

elenuc e )(2R

15/30

oddnI

e elenuci

elenucelenuc

211

)( 12

R

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 the deformed potential (intrinsic degrees of freedom)

The nucleonic motion is much fasterthan the rotation (adiabatic approximation)

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

symmetry

00

2/1

2

002

),,(8

12

rule selection signatureodd

evenI )(0

IMD

I

K R

)(2R

Dyin band medsuperdefor a from rays - 152

20/30

KKK )(2R

No signature selectionrule

2))1(()1(),( IIBIAIEKIE K

Electromagnetic Transitions

Emitted photon with multipolarity E1, E2, E2, ... or M1, M2, ...

Reduced transition probability

))((y probabilit n transitio thehas MEBET

2112221 ||),)((||);)((

2

MIMEMIIIMEBM

M

contains the information about nuclear structure.

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

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

20

2

2

2

45

)(

|202

|101

),2,2(

),1,1(

Q

Kgg

IKKI

IKKI

KIIKEB

KIIKMB RK

25/30

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

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

rigid

irrotational

Breakdown of adiabatic approximation

Summary

• Molecules are the protoype of quantal rotors.• Electronic and vibrational motions 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.