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valence shell excitations in even-even spherical nuclei within microscopic model
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Transcript of valence shell excitations in even-even spherical nuclei within microscopic model
valence shell excitations in even-even spherical
nucleiwithin microscopic model
Ch. StoyanovInstitute for Nuclear
Research and Nuclear Energy
Sofia, Bulgaria
The model Hamiltonian
the Woods-Saxon potential;
monopole pairing interaction;
separable multipole-multipole interaction
in the particle-hole channel
pai
av
r
ph
M
S
pa pai
phr M
pv
pM
h
SM
H
HH
H
H H H HHH
separable spin-multipole interaction
in the particle-hole channel
residual interaction in the particle-particle
cha e
nn l
ph
M
PP
MH
Central forces
*1 2 1 2 1 1 2
1
1 2 1 2 12 12
1
, , ,
with
, 2 , cos cos
l lm lml m
l l
V r r V r r Y Y
V r r V r r P
d
����������������������������
1 2
1 2
†
1 2
often used:
1: :
2
and
,
is multipole operator
ll
l lm lmlm
llm lm j j
j j
f r
V Q Q
Q k r Y k a a
Another option: l
dV rf
dr
l 1 2 l 1 l 2
separable ansatz:
V r,r =f r .f r
Quasiparticle RPA(collective effects)
†
†
,
†
12
1 1 , ; 1 , ; , ;22 1
, ; ; ,
Q
j j jj jjj j
j j j j
H Q Q
Q f u A j j A j j v B j j
A j j B j j
Jm denote a single-particle level of the average field for neutrons (or protons)
The neutron […]λμ means coupling to the total momentum λ with projection μ:
The quantity is Clebsch-Gordon coefficient
Bogoliubov linear transformation
Quasiparticle RPA (2)(quasiboson approximation)
† † † †j j jmj m jm j m
mm
C
1 2 1 2 1 2 1 2 1 2 1 21 1 1 1 2 2 2 2
,† † †
, ; , , ;
11 [ ]
2
j j j j j j j j
n pi i
jj j j jj j jjj
A j j A j j
Q
jmj mC
Phonon properties Phonons are not only collective
• Collective many amplitudes• Non-collective a few amplitudes• Pure quasi-particle state only one amplitude
Diverse Momentum and Parity Jπ spin-multipole phonons The interaction could include any kind of correlations
(particle-particle channel)
LARGE PHONON SPACE
† †ijj j j
jj
Quasiparticle RPA (3)(collective effects)
† † †1,2 3,4 1,2 3,4 1,2 1,2 3,4 3,4
1,2,3,4 ,
†1,2 3,4 1,2 3,4 1,2 1,2 3 4
1,2,3,4
. .2 1
; . .2 1
kk k i i i i
RPA i i i ik i i
kk k i i
QP PH i ik i
H f f u u Q Q Q Q h c
H f f u v Q Q B j j h c
Harmonic vibrations
†
,
has to be diagonalized in multiphonon basis
RPA i i ii
QP PH
H Q Q
H
To avoid Pauli principle problem
Microscopic description of mixed-symmetry states in nearly spherical
nuclei
Chavdar Stoyanov
and
N. Lo Iudice
Introduction
Low-lying isovector excitations are naturally predicted in the algebraic IBM-2 as mixed symmetry states. Their main signatures are relatively weak E2 and strong M1 transition to symmetric states.
A. T. Otsuka , A.Arima, and Iachello, Nucl .Phys. A309, 1 (1978)
B. P. van Isacker, K.Heyde, J.Jolie et al., Ann. Phys. 171, 253 (1986)
Definitions
The low-lying states of isovector nature were considered in a geometrical model as proton-neutron surface vibrations.
is in-phase (isoscalar) vibration of protons and neutrons.
is out-of-phase (isovector) vibration of protons and neutrons.
A. A.Faessler, R. Nojarov, Phys. Lett., B166, 367 (1986)B. R. Nojarov, A. Faessler, J. Phys. G, 13, 337 (1987)
12
22
Review paper
N. Pietralla, P. von Brentano, and A. F. Lisetskiy,
Prog. Part. Nucl. Phys. 60, 225 (2008).
Microscopic calculations
Within the nuclear shell model
A. F. Lisetskiy, N. Pietralla, C. Fransen, R. V. Jolos, P. von Brentano, Nucl. Phys. A677, 1000 (2000)
Within the quasi-particle-phonon model (QPM)
N. Lo Iudice and Ch. Stoyanov, Phys. Rev. C 62, 047302 (2000)
N. Lo Iudice and Ch. Stoyanov, Phys. Rev. C 65, 064304 (2002)
Definition In order to test the isospin nature of 2+
states the following ratio is computed:
This ratio probes:
1. The isoscalar ((2+)<1)
and
2. The isovector (B(2+)>1)
properties of the 2+ state under consideration
2
2 22 2
2
2 22 2
2 .
2
2 .
p n
k kk k
p n
k kk k
r Y k r Y k g s
r Y k r Y k g s
B
The dependence of M1 and E2 transitions on the ratio G(2)/k0
(2) in 136Ba.
2 2
+ +iv iv
e
2
20
g.s. 2 2 2
2 ( 1) 2
is
b
ivRPARPAB E B M
G
B
2
N
________________________________________________ 0 0.0032 0.042 0.58
0.85 0.011 0.24
22.6
Structure of the first RPA phonons (only the largest components are given) and corresponding B(2+) ratios for 136Ba
B(2+)
The values of B(2+) for 144Nd
Explanation of the method used
The quasi-particle Hamiltonian is diagonalized using the variational principle with a trial wave function of total spin JM
1 2
2 2 1 1 1 2 2 2
1 1
2 2
1 1 2 2
3 3 1 1 1 2 2 2 3 3 3
1 1
2 2 3 3
† † †
, ,,
† † †0
I, , ,, , ,
ii iJM i i i JM
i ii
i i Ii i i iIK JMi
i i
JM R J Q P J Q Q
T J Q Q Q
Where ψ0 represents the phonon vacuum state and R, P and T are unknown amplitudes; ν labels the specific excited state.
Energies and structure of selected low-lying excited states in 94Mo. Only the dominant components are presented.
94Mo level scheme./low-lying transitions/
E2 transitions connecting some excite states in 94Mo calculated within QPM.
M1 transitions connecting some excite states in 94Mo calculated within QPM.
The N=80 isotones
N. Pietralla et al., Phys. Rev. C 58, 796 (1998). G. Rainovski, N. Pietralla et al., Phys. Rev. Lett. 96, 122501 (2006). T. Ahn, N. Pietralla, G. Rainovski et al., Phys. Rev. C 75, 014313 (2007).
K. Sieja et al., Phys. Rev. C, v. 80 (2009) 054311.
Experimental results
Fermi energy as a function of the mass number
Results on QRPA level
QPM Results for N=80 isotones
134Xe
136Ba
138Ce
134Xe
138Ce
N=84: Experimental results
N=84: theoretical description
N. Pietralla et al., Phys. Rev. C 58, 796 (1998).G. Rainovski, N. Pietralla et al., Phys. Rev. Lett. 96, 122501 (2006). T. Ahn, N. Pietralla et al.,Phys. Rev. C 75, 014313 (2007).
Two quasiparticle poles138Ce 142Ce
E [MeV]
Two-quasipartcle
state
E [MeV]
Two-quasipartcle
state
2.19 (2d3/2)2 N 1.98 (2f7/2)
2 N
2.49 (1h11/2)2 N 2.56 (1g7/2)
2 Z
2.63 (1g7/2)2 Z 2.66 (1g7/22d5/2) Z
2.65 (3s1/2 2d3/2) N 3.42 (2f7/2 1h9/2) N
2.89 (1g7/2 2d5/2) Z 3.83 (2f7/2 3p1/2) N
3.16 (2d5/2)2 Z 4.30 (1g7/2 2d3/2) Z
N=84: theoretical description
Comparison to the experiment