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