Post on 21-Jan-2016
description
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
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.