Nucleon-nucleon cross sections in symmetry and asymmetry nuclear matter

School of Nuclear Science and Technology, Lanzhou University, 730000, China

Hong-fei ZHANG (张鸿飞）

Collaborators:

• U. Lombardo

• Z.H. Li

• F. Sammarruca

• W. Zuo

• J. M. Dong

Papers on the work:

1. H.F. Zhang, Z.H. Li, U. Lombardo, P.Y. Luo, and W.Zuo,

Phys. Rev. C, Vol. 76, 054001 (2007).

2. H.F. Zhang, U. Lombardo, J.M. Dong, Z.H. Li, W. Zuo,

Nucleon-nucleon cross sections and nucleon mean free paths in

asymmetricnuclear matter

In preparation.

Outline

• Introduction

• BHF with microscopic three-body forces

• Nucleon-nucleon cross sections in symmetry

and asymmetry nuclear matter

• Summary

Ⅰ. Introduction

• Heavy-ion collisions are theoretically described by

transport-model simulations whose input data are

the in-medium cross sections and the nuclear mean

field. Being intimately related to each other through

the nuclear matter equation of state (EOS), they

must be consistently determined.

• In-medium cross sections are necessary to study the

mean free path of nucleons in nuclear matter and

thus nuclear transparency.

• Size of exotic nuclei

In asymmetry nuclear matter, one can define the isospin asymmetry parameter

where

In-medium effective

Interaction G matrix

V3eff is reduced to a

density-dependent two-body force

v+v3effv

12 ( , ) , | 12Q

r r r r Ge

Defect function

Ⅱ. BHF with Microscopic three-body forces

For a given total densityρand asymmetryβ.a bare two-body forcev as input, solve the Equs self-consistently：

BBG equation

s.p. energy

s.p. auxiliary potentials

BHF

Pauli operator

Ⅲ. Nucleon-nucleon cross sections

In Brueckner theory, the G matrix plays the role of thein-medium scattering amplitude, with medium effectsbeing introduced through the mean field and Pauli blocking.In the zero density limit, the G matrix reduces to the T martix,and the Brueckner-Beth-Goldstone (BBG) equation to theLippmann-Schwinger equation.

Beyond the scattering amplitude, nucleon-nucleon collisionsin nuclear matter are also driven by kinematic degree of freedom,i.e.,entrance flow and density of states in the exit channel. Bothare related to the nucleon effective mass, which, in turn, is relatedto the self-energy. The latter is modified by a 3BF, which alsogenerates quite large rearrangement terms, leading to a large reduction of the effective mass. Thus one can expect that 3BFsmight have a strong influence on the in-medium cross section,as they depend quadratically on the effective mass.

1. Real and imaginary parts of the 1S0

components of the G matrix

While 3BFs are negligible atlow density, they start to be noticable at saturation densityand become more and more effective as density increase.

The real part of the G matrixis reduced due to Pauli blockingand dispersive effects.

The imaginary part of the G matrix,which is related to the particle-holeexcitations, become larger becauseof the 3BF enhancement of the ground correlations.

2. Effective mass

In the medium, the additional contribution from theself-energy can be reasonablely approximated byreplacing the bare mass with the effective mass:

The effective mass becomes substantially smaller with the inclusion of the 3BF, an effectivewhich will impact the in-medium cross sections through the level density in the entrance andexit channels, along with the 3BF enhancement of the repulsive components in the effectiveinteraction.

3 Free-space cross sections

Argonne V14 is used

The total cross sections converge rapidly to the corresponding experimental valueswith increasing number of partial waves

4. Total cross sections for identical nucleons

Up to the saturation density, the effect of the 3BF is small, and the medium suppressionis mainly controled by the reduction of density of state due to Pauli blocking.

At the higher density, the 3BF produces a larger reduction of the cross section, which persistsup to high energy. The latter is mainly due to the strong 3BF renormalization of the effective mass.

The scattering amplitude is also affected by the 3BF

5. Differential cross section for identical nucleons

The reduction of the cross sections is more sizable in the forward and backward directions,since low momentum transfers are strongly suppressed by the Pauli principle. This effect leads to distributions that are almost flat at high density. This feature justifies the frequency practiceof adopting isotropic cross sections in HIC simulations.

6. Total cross sections for nonidentical nucleons

In scattering of distinguishable nucleons, the T=0 component of the interaction is also included.As a consequence, the free cross sections for unlike particle is larger than the one for like particles,a property which remains true in the medium.

The 3Bf effect on the cross section is evident, especially in high density.

7. Differential cross section for nonidentical nucleons

The differential cross section is strongly asymmetric. The in-medium valuesexhibit similar asymmetry although less pronounced.

8. Comparison with DBHF predictions

The cross sectios from 2BF+3BF are in goodagreement with thevalues from DBHF, with theexception of the highest density.

Energy and density dependent appear quiteconsistent among the two cases, althoughthe cross sectios from 2BF+3BF is somewhatlarger than the values from DBHF across thebroad.

Examination of the last column in theleft table clearly suggests that 3BF otherthan Z diagrams are the main cause ofthe discrepancies between the DBHF andBHF+3BF predictions of the EOF and,consequently, of the respective crosssections.

9. nucleon-nucleon cross section

in asymmetry nuclear

100 150 200 250 300 350 400

10

20

30

40

50 = 0.17 [ fm-3 ] = 0.80

NN [

mb

]

E [ MeV ]

pn

pp

nn

Bonn B potential and a new version of three-body Force are used, Dr. Z.H. li will give a talk on the improvement for the previous BHF with 3BF !

Isospin dependent of total nucleon-nucleon

cross sections

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

10

15

20

25

= 0.17 [ fm -3 ] E = 300 [ MeV ]

NN [

mb

]

pn

pp

nn

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

5

10

15

20

25

= 0.34 [ fm -3 ] E = 212 [ MeV ]

NN [

mb

]

pn

pp

nn

The lowering (rising) proton (neutron) Fermi mementum and the reduced (increased)proton (neutron) effective mass tend to move the cross section in opposite direction.With pauli blocking applied to intermediate and final states, the final balance is thatThe neutron-neutron effective cross section is more strongly suppressed.

Ⅳ. Summary

• The TBF provides a repulsive contribution to

the EOS and improves remarkably the

predicted

saturation properties, which suppress the

magnitude of cross sections.

• The TBF from the Z-diagram provides the

saturation mechanism and gives the main

relativistic effect in DBHF approach.