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

Woods-Saxon potential

,

0

01 exp

N ZVV r

r R

Spin-orbital term

1.ls

dV rV r l s

r dr

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

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

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

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

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

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

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}.

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

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.

92 Zr

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

E2 and M1 transitions connecting excited st. in 92 Zr

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

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: 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).

N=84: theoretical description

Comparison to the experiment

Recent experimental results Sn

PRL 98, 172501 (2007) PRL 99, 162501 (2007) PRL 101, 012502 (2008)

LoI

Phys. Lett. B 695, 110 (2011).

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]).

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)

Selected proton s. p. states around the Fermi energy

Selected neutron s. p. states around the Fermi energy

Selected neutron s. p. states around the Fermi energy

Experimental values of B(E2, g.s. -->2+1) and

calculated neutron gaps in tin isotopic chain

B(E2) through the Sn isotopic chainwithout and with quadrupole pairing.

Calculated versus Experimental energiesof 2+1 states.

The data are taken from [19].

QPM versus experimental B(E2). The data are taken from [4, 19]

B(E2) through the Sn isotopic chainwithout and with quadrupole pairing.

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]

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

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 %

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

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 ---

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.