How do nuclei rotate?

1

2

3symmetry axis

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)

3

1

2

2i i

iJEenergy spheroid

Oblate and prolate quadrupole deformation

Choosing the vertical axis as the 3-axis one obtains the oblate by R1 = R2 > R3 and the prolate by R1 = R2 < R3 axially-symmetric quadrupole deformations

prolate deformation (β>0)

,1, *

0 YRR

,,sin

2

1,cos1, 2222200 YYYRR

00for

oblate deformation (β<0)

The Euler angles

It is important to recognize that for nuclei the intrinsic reference frame can have any orientation with respect to the lab reference frame as we can hardly control orientation of nuclei (although it is possible in some cases).

One way to specify the mutual orientation of two reference frames of the common origin is to use Euler angles.

(x, y, z) axes of lab frame(1,2,3) axes of intrinsic frame

The rotation from (x,y,z) to (x´,y´,z´) can be decomposed into three parts: a rotation by about the z axis to , a rotation of θ about the new y axis to , and finally a rotation of ψ about the new z axis .

zyx ,,

y zyx ,,

z

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

Quantization

KMI ,,| :seigenstate

function Wigner D iKIMK

iMIMK edeD )(),,(

),,(8

12,,|,, :rotor ofn orientatiofor

amplitudey probabilit2/1

2

I

MKDI

KMI

3

z

Rotational motion of a deformed nucleus

For a deformed nucleus with axial symmetry one obtains the same rotational frequency for the 1- and 2-axis. The Hamiltonian is given by

States with projections K and –K are degenerated

This fact is expressed in the nuclear wave function as a symmetric product

KI

KMKI

KIKMIMK DD

I

116

122/1

2

For K=0, only even-J are allowed, so that the wave function is given by a single term

00

2/1

28

12

IMIM D

I

3

23

1

23

23

1

2

2

ˆ

2

ˆˆ

2

ˆ

RRRRH

i i

irot

If the total angular momentum results only from the rotation (J = R), one obtains for the rotational energy of an axially symmetric nucleus by

12

2

IIErot

J

3

21

33.3)2(

)4(

1

12/4

E

ER

The nucleus does not have an orientation degree of freedom with respect to the symmetry axis

3

z

Note the characteristic, repeated patterns

Broad per spective on structural evolution:

neutron number

prot

on

num

ber

)2(

)4(

1

12/4

E

ER

γ-rays from a superdeformed band in 152Dy

Rotational motion of a deformed nucleus

axial symmetry:rotational axis symmetry axis

kinematic moment of inertia

E

I

EE

I

II

24242

22

1

22

12

KIIErot

dynamic moment of inertia

2

822

2

IEIE

rotational frequency

2

2

IEIE

dI

dE

3

z

Moment of inertia

z

R()

00

00

05.1900

53

4

R

R

R

RR

3/10 2.1 AR 0RR 201 Y 200 1 YRR

Rigid body moment of inertia:

dddrrrrR sin22

Irrotational flow moment of inertia:

22

8

9 oF RM

)32.01(5

2 2 oR RM

rigid

irrotational

Moment of inertia

Reduced transition probability

dMM abm

abm

ˆ*expectation value

ntrˆˆ im

mmm

abm MDM

QReZ

M m

16

12

4

3ˆ0

intr

wave function 0020 8

12

IMIM D

I

m

IMmmm

IM

fi

iiabmff dDMDD

IIMIMMI i

i

f

f

00intr

002ˆ

8

12120ˆ0

3321213321213

2

||12

83

33

2

22

1

11MIMMIIMIMMII

IdDDD I

MMI

MMI

MM

0intr00

ˆ0|00|12

120ˆ0

MIIMIMMMI

I

IMIMMI fiffifii

f

iii

abmff

Reduced transition probability

0intr00

ˆ0|00|12

120ˆ0

MIIMIMMMI

I

IMIMMI fiffifii

f

iii

abmff

Wigner-Eckart-Theorem (reduction of an expectation value):

0ˆ0

12

|0ˆ0 i

abf

f

ffifiiii

abmff IMI

I

MIMMMIMIMMI

00intr0 |ˆ0|001200 MIIIIEMI fiiif

special case: E2 transition I→I-2

4

3

122

1320,2 2

20ZeR

I

IIIEMI

reduced transition probability:

2020

12

1; if

ifi IEMI

IIIEB

2216

5

12122

132;2 Q

II

IIIIEB

Reduced transition probability

22513 ;210225.11

beIIEBMeVEs fi Transition probability:

2ln2/1 T

1521422 110161.8;2 sMeVEbeIIEB

iIETfi

half-life:

Weisskopf estimate:

223/461094.5;2 beAIIEB gsi

Hydrodynamical model

Reduced transition probability: 2

40

22

16

920;2

ReZ

EB 2

Excitation energy:2

62

2

E

Moment of inertia:2

8

9oF RM

2

2

3/1

2

20;2EA

ZconstEB

Hydrodynamical model

Reduced transition probability: 2

2

24

4222

220 33.097.036.01

4

320;2

ZeR

EB

Excitation energy: 24

22

20

2

2 3/53

8

RMAE

first indication of a hexadecapole deformation