4. 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

occupiedii

mf

xx

xxxvxdxV

2

123

|)'(|)'(

density particle

)'()'(')(

potentialnuclear - fieldmean

:Hartree

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 (axial)

yzJiJi

z ee ),( Rotation R

.energy same thehave ),( nsorientatio All

peaked.sharply is 1|||

.but

|R

|R

RRRR

hhHH

Measures orientation.

Rotational degree of freedom and rotational bands.

Microscopic approach to the Unified Model. 5/32

2

)1( 2KIIEE in K

2/1

2),,(

8

12

IMKD

I

Cranking Model

Seek a mean field solution carrying finite angular momentum.

.0|| zJ

Use the variational principle

with the auxiliary 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) p. (s. 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

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.

states particle quasi

routhians) (q.p frame rotatingin energies particle quasi '

(routhian) frame rotating in then hamiltonia fieldmean '

- -'

i

i

zmf

e

h

NJVth

p h h p

tency selfconsis i

Pair potential

Can calculate |ˆ|)( zz JJ

molecule )(zJ )( 22 n

nnn yxm

Comparison with experiment ok.

Very different from

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.

The cranked shell model

Many nuclei have a relatively stable shape.

diagram) (Spaghetti )('

routhians particle single of Diagram

,, ie

tionclassifica ),(),( signatureparity

Each configuration of particles corresponds to a band.

nIe iz 2||)( R

Experimental single particle routhians

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

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

h

p

eNEhNE

eNEpNE

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

1

1

)('

)('

jd

de

Jd

dE

Slope = 1j

experiment Cranked shell modelMeVo 4.7

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

Pairing taken into account

2

)(''2Ee 214.63 MeV

band E band EAB

bandcrossing

band Aband B

Er163

Rotational alignment

10' JoverlapVhe sph

Energy small Energy large

torque

10' JoverlapVhe sph

1

1

)('

'')('

Jconste

Jh

d

hd

d

de

“alignment of the orbital”

1

3

Deformation aligned

constKJ 3

1

3

Rotational aligned

dominates 1J

constJ 1

dominates 0 overlapV

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

Summary

• The mean field may spontaneously break symmetries. • The non-spherical mean field defines orientation and the rotational

degrees of freedom.• 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.