How do nuclei rotate?

3. The rotating mean field

The mean field concept

A nucleon moves in the mean field generated by all nucleons.

][ imfV The mean field is a functional of the single particle states determined by an averaging procedure.

The nucleons move independently.

ii

N

c

cc

state in nucleona creates

0|......|tion)(configura statenuclear 1

functions) (wave states particle single

energies particle single

ial)(potentent field mean energy kinetic

i

i

mf

iiimf

e

Vt

ehVth

Total energy is a minimized (stationary) with respect to the single particle states.

with the 12vtH

Calculation of the mean field: Hartree Hartree-Fock density functionals Micro-Macro (Strutinsky method) …….

.0|| HEi

.12v

Start from the two-body Hamiltonian

effective interaction

Use the variational principle

Spontaneous symmetry breaking

Symmetry operation S

.|||

energy same with thesolutions fieldmean are states All

1||| and but

HHE

hhHH

|SS

|S

|SSSSS

mfVth

Deformed mean field solutions

zJiz e )( axis-z about the Rotation R

.energy same thehave )( nsorientatio All

peaked.sharply is 1|||

.but

|R

|R

RRRR

zz

zzzz hhHHMeasures orientation.

Rotational degree of freedom and rotational bands.

Microscopic approach to the Unified Model. 5/32

Cranking model

Seek a mean field solution carrying finite angular momentum.

.0|| zJ

Use the variational principle

with the auxillary condition

0|| HEi

0||' zJHEi

The state |> is the stationary mean field solution in the frame that rotates uniformly with the angular velocity about the z axis. In the laboratory frame it corresponds to a uniformly rotating mean field state

symmetry). rotational (broken 1|||| if ||

zz tJitJi

eet

tency selfconsis mfi V

functions) (wave states particle single

)(routhians frame rotatingin energies particle single '

ial)(potentent fieldmean energy kinetic

(routhian) frame rotating in then hamiltonia fieldmean '

'' -'

i

i

mf

iiizmf

e

Vt

h

ehJVth

Can calculate |ˆ|)( zz JJ

molecule )(zJ )( 22 n

nnn yxm

Comparison with experiment

Very different from

The QQ-model

','2

2 '||5

4

basis

potential model shell spherical

kkkk

kkkksph

kkksph

sphsph

cckYrkQcceh

eh

Vth

operator quadrupole ),(5

4

2

202

2

2

YrQ

QQhH sph

Mean field solution

Qq

QqJhheh

QQJhE

zsphiii

zsph

tencyselfconsis

'''

variation

2'

2

2

2

2

Intrinsic frame

Principal axes

2/sincos

00

20

2211

KqKq

qqqq

,ˆ toparallel bemust

tencyselfconsis

cossinsincossin

)('

2200

321

22200332211

JJ

QqQq

QQqQqJJJhh sph

22

222

2220

220

222

|)],,0(),,0([),,0(|4

5

||4

5)2,2(

protonproton

LAB

QDDQD

QIIEB

211

211 |),,0(|

4

3||

4

3)1,1(

v

vLAB DIIMB

Transition probabilities

Symmetries

2

22'

QQJhH zsph

Broken by m.f. rotationalbands

Principal Axis CrankingPAC solutions

nIe iz 2||)( R

Tilted Axis CrankingTAC or planar tilted solutions

Chiral or aplanar solutionsDoubling of states

The cranked shell model

Many nuclei have a relatively stable shape.

090

)0(

o

constconst

diagram) (Spaghetti )('

routhians particle single of Diagram

,, ie

tionclassifica ),(),( signatureparity

Each configuration of particles corresponds to a band.

/2)

MeV4.7

),(

0

(-,1/2)

(-,-1/2)

(+,-1/2)

(+,1/2)

(+,1/2)

(-,1/2)

(-,1/2)

Experimental single particle routhians

holes )('),('),1,('

particles )('),('),1,('

h

p

eNEhNE

eNEpNE

excitation hole-particle )(')('),('),,,(' hp eeNEhpNE

experiment Cranked shell modelMeVo 4.7

Rotational alignment

001':090 QqJhh spho

Energy small Energy large

torque

001' QqJhh sph

1

'')('J

h

d

hd

d

de

“alignment of the orbital”

1

3

Deformation aligned

dominates 00Qq

constKJ 3

1

3

Rotational aligned

dominates 1J

constJ 1

Slope = 1J

Pair correlations

Pair correlations

Nucleons like to form pairs carrying zero angular momentum.

Like electrons form Cooper pairs in a superconductor.

Pair correlations reduce the angular momentum.

The pairing+QQ model

kkk

kkkk

zsph

ccP

cckYrkQ

JPGPQQhH

operatorpair

operator quadrupole '||5

4

2'

''20

2

2

2

N

PGQq

V

U

e

V

U

QqNJhP

PQqNJh

PPQqNJhh

PPGQQNJhE

i

i

i

i

i

zsph

zsph

zsph

zsph

number particle thecontrols

tencyselfconsis

)('

variation

2'

2

2

2

2

2

2

2

2

Mean field approximation (CHFB)

particle

hole

amplitudes

Configurations (bands)

)(')(')('),('

)(')('),('

),(',even || ion configurat qp two

),(', odd | | ion configurat qp one

)(',even | ion configurat (vacuum) qp zero

clesquasiparti ,,

EeeijE

EeiE

ijENij

iENi

EN

cVcU

ji

i

ji

i

kkkikkii

Double dimensional occupation numbers.Different from standardFermion occupation numbers!

states

'' conjugate ~ii ee

01

or 10

states all of 1/2occupy

:rule

~

~

ii

ii

nn

nn

[0]

[A]

[AB]

[AB]

backbending

[B]

The backbending effect

ground band [0] s-band [AB]

gJ gsssiJ

rigid

Moments of inertia at low spin are well reproduced by cranking calculations including pair correlations.

irrotational

Non-local superfluidity: size of the Cooper pairs largerthan size of the nucleus.

Summary

• The pairing+QQ model leads to a simple version of mean field theory.• The mean field may spontaneously break symmetries. • The non-spherical mean field defines orientation and the rotational degrees of

freedom.• There are various discrete symmetries types of the mean field. • The rotating mean field (cranking model) describes the response of the

nucleonic motion to rotation.• The inertial forces align the angular momentum of the orbits with the

rotational axis. • The bands are classified as single particle configurations in the rotating mean

field. The cranked shell model (fixed shape) is a very handy tool.• At moderate spin one must take into account pair correlations. The bands are

classified as quasiparticle configurations.• Band crossings (backbends) are well accounted for.