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.
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