valence shell excitations in even-even spherical nuclei within microscopic model ch. stoyanov...
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valence shell excitations in even-even spherical
nucleiwithin microscopic model
Ch. StoyanovInstitute for Nuclear
Research and Nuclear Energy
Sofia, Bulgaria
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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
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Woods-Saxon potential
,
0
01 exp
N ZVV r
r R
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Spin-orbital term
1.ls
dV rV r l s
r dr
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Coulomb potential
3
20
0 0
0
3 11 , if
2 2
1, if
c
r rZ e r R
V r R Rr
r R
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Constant pairing
22
22
† † † †
, , , , , ,
1
11
2
, ; , ,
†0
1
j
j
pair jm j m j m j m jm j m j m j mj j m m j j m m
E jN
C E j
G
C E j
H G jm j m j m j m a a a a G a a a a
N
j jm jmj
H а а
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1 2
1 2 1 21 2
is single-particle matrix element
The interaction generates a superposition
of pp-pairs
† †
,
Particle-particle channel
: . :
j jp
j j j jj j
V V VP p а а
Separable force and multipole expansion
1 2
1 2 1 21 2
†
is single-particle matrix element
The interaction generates a superposition of
ph-pairs
Particle-hole channel
: . :
j j
j j j jj j
f
V Q Q
Q f a a
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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
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Spherical case
Nguyen Van Giai, Ch. Stoyanov, V. V. Voronov,
Phys. Rev. C 57 1204 (1998)
Contribution of F0(r):
0 20
, ,
,
F
F p h p h
H ph p h I ph p h p Y h p Y h
drI ph p h F r u r u r u r u r
r
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Landau-Migdal form of the Skyrme interaction
-11 2 0 0 1 0 1 1 2 0 1 0 1 1 2 1 2 1 2( , ) resV r r N F r G r F r G r r r
20 0 0 3 1 2 2
20 0 0 0 3 3 1 1 2 2
20 0 0 0 3 3 1 1 2 2
20 0 0 3 1
3 1 11 2 3 5 4 ,
4 16 8
1 1 11 2 1 2 1 2 1 2 ,
4 24 8
1 1 11 2 1 2 1 2 1 2 ,
4 24 8
1 1 1
4 24 8
F
F
F
F
F N t t k t x t
F N t x t x k t x t x
G N t x t x k t x t x
G N t t k t
2t
Nguyen Van Giai, Sagawa, H., Phys. Lett B106 (1981) 379
1 2 3 4
1 2 3 4
11 2 3 4 0 0 0 1 2 2
0
11 2 3 4 0 0 0 1 2 2
0
j j j jM
j j j jS
drI j j j j N F r F r u r u r u r u r
r
drI j j j j N G r G r u r u r u r u r
r
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0 20
,F p h p h
drI ph p h F r u r u r u r u r
r
cutoff radius R
02
1
,n
kF k p k h k p k h k
k k
F rI ph p h R u r u r u r u r
r
Introducing the coefficient and the p-h matrix elements
1
,
kp k h k
nk k k
k
D ph u r u r p Y h
H ph p h D ph D p h
02
k kk
k
F rR
r
Gauss integration formula with abscissas and weights {rk, wk}.
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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
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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
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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
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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
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Harmonic vibrations
†
,
has to be diagonalized in multiphonon basis
RPA i i ii
QP PH
H Q Q
H
To avoid Pauli principle problem
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Microscopic description of mixed-symmetry states in nearly spherical
nuclei
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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)
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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
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Review paper
N. Pietralla, P. von Brentano, and A. F. Lisetskiy,
Prog. Part. Nucl. Phys. 60, 225 (2008).
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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)
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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
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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
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Structure of the first RPA phonons (only the largest components are given) and corresponding B(2+) ratios for 136Ba
B(2+)
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The values of B(2+) for 144Nd
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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.
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Energies and structure of selected low-lying excited states in 94Mo. Only the dominant components are presented.
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94Mo level scheme./low-lying transitions/
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E2 transitions connecting some excite states in 94Mo calculated within QPM.
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M1 transitions connecting some excite states in 94Mo calculated within QPM.
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92 Zr
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92 Zr Contribution of N and Z in the 2+
QRPA phonons
State Jπ E [MeV]
B(E2) ↓ [w.u.]
Structure [% ]
N Z
21+ 1.21 7.2 74 26
22+ 2.08 3.4 37 63
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E2 and M1 transitions connecting excited st. in 92 Zr
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QPM, EXP and SM g-fact. of low-lying excited st. in 92 Zr
g(Jπ) [μN] EXP SM QPM
g(21+) -0.18(1) -0.08 -0.11
g(41+) -0.5(1) -0.38 -0.32
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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.
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Experimental results
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Fermi energy as a function of the mass number
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Results on QRPA level
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QPM Results for N=80 isotones
134Xe
136Ba
138Ce
134Xe
138Ce
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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).
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N=84: theoretical description
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Comparison to the experiment
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Recent experimental results Sn
PRL 98, 172501 (2007) PRL 99, 162501 (2007) PRL 101, 012502 (2008)
LoI
Phys. Lett. B 695, 110 (2011).
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Experimental and theoretical B(E2) values for the Sn isotopes reported from Ref.[5]. The dashed and solid curves represent the results from shell model calculations using different cores (for details see Ref.[5]).
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Calculations
A. Ansari, Phys. Lett. B 623, 37 (2005). A. Ansari and P. Ring, Phys. Rev. C 74,
054313 (2006). J. Terasaki, Nucl. Phys. A 746, 583c (2004).
N. Lo Iudice, Ch. Stoyanov,and D. Tarpanov PRC 84, 044314 (2011)
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Selected proton s. p. states around the Fermi energy
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Selected neutron s. p. states around the Fermi energy
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Selected neutron s. p. states around the Fermi energy
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Experimental values of B(E2, g.s. -->2+1) and
calculated neutron gaps in tin isotopic chain
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B(E2) through the Sn isotopic chainwithout and with quadrupole pairing.
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Calculated versus Experimental energiesof 2+1 states.
The data are taken from [19].
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QPM versus experimental B(E2). The data are taken from [4, 19]
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B(E2) through the Sn isotopic chainwithout and with quadrupole pairing.
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Mass number
Calculation
B(E2) e2b2
% EWSR
Exp. 1
PRL 99 (2007)
Exp. 2
PRL 101 (2008)
Exp. 3
PRL 98 (2007)
Percent
of Z
in the str.
of 2+1
104 0.144
2.4 %
--- 4.44
106 0.214
3.4 %
0.240 0:195 (39) 6.4
108 0.234
3.7 %
0.230 0:222 (19) 7.2
110 0.269
4.2 %
0.240 0:220 (0:022)
8.1
112 0.274
4.4 %
0.240 8.6
104-112SnB(E2; g. st. 2+
1) [e2b2]
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Quasiparticle composition of the 21+ state in two typical Sn isotopes.
Nucleus (q1q2)ν W(ν) % (q1q2)pi W(π)% 112Sn 1g7/21g7/2 20.6 1g9/22d5/2 5 1h11/21h11/2 16 1 g9/21i13/2 0.6 1g7/22d3/2 17 2d5/23s1/2 11.7 2d5/22d5/2 5.6 3s1/22d3/2 6 2d3/22d3/2 2.5 2d5/22d3/2 2.5 126Sn 1h11/21h11/2 61 1g9/22d5/2 2.6 2d3/22d3/2 8.1 3s1/22d3/2 9.3 1g7/22d3/2 6.5 1h11/22f7/2 3.1
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Percentof 2+
1 phonon in the str. of 2+1 state
Mass number B(E2) e2b2 Percent 102 0.078 97 % 104 0.171 93 % 106 0.248 91 % 108 0.255 92 % 110 0.255 94 % 112 0.260 96 %
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106Sn
Full single-particle space Truncated single-particle space
eeff (N) eeff (Z) B(E2) [e2b2] eeff (N) eeff (Z) B(E2) [e2b2]
0.1 1+ eeff (N) 0.214 1.1 1+ eeff (N) 0.226
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Mass number Separable Skyrme
Skyrme potential
Experiment
Energy[MeV]
B(E2)[e2b2]
Energy[MeV]
B(E2)[e2b2]
Energy
[MeV]
B(E2)[e2b2]
108 1.231 0.283 1.206 0.205 1.206 0.222 (19)106 1.235 0.256 1.206 0.194 1.206 0.195 (39)104 1.266 0.192 1.260 0.184 1.260 ---
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Conclusions There are two modes in the low-lying
quadrupole excitations – isoscalar and isovector one.
The properties of these two modes are close to IBM-2 symmetric and mixed-symmetry states.
The coupling of the modes leads to variety of excited states. There are well pronounced regularities of E2 and M1 transitions connecting the states.
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