Nuclear Collective Excitation in a Femi-Liquid Model

Bao-Xi SUN

Beijing University of Technology

2012.06.15 KITPC, Beijing

Content

Fermi-liquid Model based on Landau Theory.

Relation between isoscalar giant resonance and isovector giant resonance

Collective excitation in nuclear matter Collective excitation in finite nuclei Conclusion

Fermi-liquid Model based on Landau Theory

Xiao-Gang Wen,

Quantum field theory of many-body systems,

Oxford University Press, Oxford, 2004.

Boltzmann Equation of quasi-nucleons

Boltzmann equation of nucleons

where

Density of quasi-nucleons

The quasi-nucleon density near the Fermi-surface:

with

Vibrations of Fermi surface

Fk

Linearized liquid equation of motion in the momentum space

with

and

Potential between nucleons in the linear Walecka model

Fermi liquid function

Fermi liquid function

with

and

Fermi energy and Fermi velocity

C. J. Horowitz and B. D. Serot, Nucl. Phys. A368 (1981) 503

10.47, 13.80,

520 , 783 ,

939 .N

g g

m MeV m MeV

M MeV

The quasi-nucleon density can be expanded in spherical harmonics:

Liquid equation of motion in spherical harmonics

The stability of the Fermi liquid requires the diagonal matrix elements of M must be positive definite, and we can write M as M =W WT. Letting

Eigen-energy equation for the nuclear collective excitation

with the Hamiltonian

, , ,i u Hu l m q tt

,

1, 1, 1,

1 12 22 2

* *3 3

, ,(2 ) (2 )

Tl l

lm l l l m l l

F FF F F F

H q W KW

q a a

k kv f l l v f l l

and

Tu W

Eigenvalues of the Hamiltonian

Since the nucleon near Fermi-surface is easier to be excited, in the following calculation, we set the value of nucleon momentum

Collective excitation energy El .vs. effective mass M*N .

L=0,Dash;L=1,Solid;L=2,Dot.

Collective excitation

* 0.742N NM M

Relation between isoscalar and isovector giant resonances

The nuclear isovector giant resonances correspond to the nuclear collective excitation that the collective excitation of protons is creating with the energy ES(l), while the collective excitation of neutrons is annihilating with the energy ES(l), and vice versa.

Relation between isoscalar and isovector giant resonances

The energy of the nuclear isovector giant resonance is about twice of the corresponding isoscalar giant resonance in the nuclear matter, i.e.,

Giant resonances of finite nuclei

The proton and neutron densities can be written approximately

Giant monopole resonances of finite nuclei

L=0 M*/M E0(p) E0(n) E0(p) +E0(n)

ES EV

Pb208 0.742 16.28 7.05 23.33 14.17

+-0.28

26.0+-3.0

Sm144 0.742 15.26 9.00 24.26 15.39

+-0.28

_

Sn116 0.742 15.26 9.00 24.26 16.07

+-0.12

_

Zr90 0.717 17.57 13.13 30.7 17.89

+-0.20

28.5+-2.6

Ca40 0.717 15.58 15.58 31.16 31.1+-2.2

Giant dipole resonances of finite nuclei

The isovector giant dipole resonance of the nucleus is a shift of the center of mass, which

corresponds to the creation of the L=1 collective excitation of protons or neutrons.

Giant dipole resonances of finite nuclei

The isoscalar giant dipole resonance in Pb-208 with a centroid energy E=22.5MeV should be a compression mode, which corresponds to a creation of the L=1 collective excitation of protons or neutrons and an annihilation of the L=1 collective excitation of neutrons or protons simultaneously.

B. F. Davis et al., PRL 79, 609 (1997)

Giant dipole resonances of finite nuclei

l=1 M*/M E0(p) E0(n) E0(p) +E0(n)

ES EV

Pb208 0.755 15.53 6.57 22.1 22.5 13.5+-0.2

Zr90 0.742 15.56 11.37 26.93 _ 16.5+-0.2

Ca40 0.7 19.58 19.58 39.16 _ 19.8+-0.5

Giant quadrupole resonances of finite nuclei

l=2 M*/M E0(p) E0(n) E0(p) +E0(n)

ES EV

Pb208 0.742 15.02 5.84 20.86 10.9+-0.1 22.0

Zr90 0.742 13.16 8.27 21.43 14.41+-0.1 _

Ca40 0.69 18.54 18.54 37.08 17.8+-0.3 32.5+-1.5

O16 0.69 18.54 18.54 37.08 20.7 _

Mixture of different L state (M*/M=0.742, kF=1.36fm-1)

Conclusion

In the Fermi-liquid model, the exchange interaction between nucleons causes the nuclear collective excitation.

It is different from RMF+RPA.

Of course, we need not take into account the contribution from Dirac sea.

Thanks for your attention!