PHYSICAL REVIEW C 73, 014001 (2006)

Effective nucleon-nucleon cross sections in symmetric and asymmetric nuclear matter

F. Sammarruca and P. KrastevPhysics Department, University of Idaho, Moscow, Idaho 83844, USA

(Received 3 October 2005; published 18 January 2006)

We calculate nucleon-nucleon cross sections in the nuclear medium with unequal densities of protons andneutrons. We use the Dirac-Brueckner-Hartree-Fock approach together with realistic nucleon-nucleon potentials.We examine the effect of asymmetry in neutron and proton concentrations and find that it can be significantfor scattering of identical nucleons. Numerical results are included for potential applications in transportequations.

DOI: 10.1103/PhysRevC.73.014001 PACS number(s): 21.65.+f, 21.30.Fe

I. INTRODUCTION

A topic of contemporary interest in nuclear physics is theinvestigation of the effective nucleon-nucleon (NN) interactionin a dense hadronic environment. Such environment can beproduced in the laboratory via energetic heavy-ion (HI) colli-sions or can be found in astrophysical systems, particularly theinterior of neutron stars. In all cases predictions rely heavilyon the nuclear equation of state (EOS), which is one of themain ingredients for transport simulations of HI collisions aswell as the calculation of neutron star properties.

Transport equations, such as the Boltzmann-Uehling-Uhlenbeck (BUU) equation, describe the evolution of anonequilibrium gas of strongly interacting hadrons. In BUU-type models, particles drift in the presence of the meanfield while undergoing two-body collisions, which requiresthe knowledge of in-medium two-body cross sections. In amicroscopic approach, both the mean-field and the binarycollisions are calculated self-consistently starting from thebare two-nucleon force.

We present microscopic predictions of NN cross sectionsin isospin symmetric and asymmetric nuclear matter. In asym-metric matter, the cross section becomes isospin dependentbeyond the usual and well-known differences between the npand the pp/nn cases. It depends on the total density and therelative proton and neutron concentrations, which of coursealso implies that the pp and the nn cases will in general bedifferent from each other. In this article, we are concernedonly with the the strong interaction contribution to the crosssection (Coulomb contributions to the pp cross section are notconsidered).

Asymmetry considerations are of particular interest at thistime. The planned Rare Isotope Accelerator will offer theopportunity to study collisions of neutron-rich nuclei thatare capable of producing extended regions of space/timewhere both the total nucleon density and the neutron/protonasymmetry are large.

Isospin-dependent BUU transport models [1] includeisospin-sensitive collision dynamics through the elementarypp, nn, and np cross sections and the mean field (which isnow different for protons and neutrons). The latter is a crucialisospin-dependent mechanism and was our focal point in anearlier article [2].

In a simpler approach, the assumption is made that thetransition matrix in the medium is approximately the same

as the one in vacuum and that medium effects on thecross section come in only through the use of nucleoneffective masses in the phase-space factors [3,4]. Concerningmicroscopic calculations, some can be found, for instance, inRefs. [5,6], but asymmetry considerations are not includedin those predictions. However, it is important to investigateto which extent the in-medium cross sections are sensitiveto changes in proton/neutron ratio, the main purpose of thisarticle. In-medium cross sections are necessary to study themean free path of nucleons in nuclear matter and thus nucleartransparency. The latter is obviously related to the total reactioncross section of a nucleus, which, in turn, can be used to extractnuclear r.m.s. radii within Glauber-type models [7]. Therefore,accurate in-medium isospin-dependent NN cross sections canultimately be very valuable to obtain information about thesize of exotic, neutron-rich nuclei.

In the next section we provide some details on thecalculation. We then present and discuss our results in Secs. IIIand IV. Our conclusions are summarized in Sec. V.

II. IN-MEDIUM NN CROSS SECTIONS: GENERALASPECTS

Details of our application of the Dirac-Brueckner-Hartree-Fock framework to asymmetric matter can be found in Ref. [8].We choose the Bonn-B potential [9] as our model for thefree-space two-nucleon (2N ) force.

The nuclear matter calculation of Ref. [8] provides, togetherwith the EOS, the single-proton/neutron potentials as wellas their parametrization in terms of effective masses [2].Those effective masses, together with the appropriate Paulioperator (depending on the type of nucleons involved),are then used in a separate calculation of the in-mediumreaction matrix (or G matrix) under the desired kinematicalconditions.

Our calculation is controlled by the total density, ρ, and thedegree of asymmetry, α = (ρn − ρp)/(ρn + ρp), with ρn andρp the neutron and proton densities.

For the case of identical nucleons, the G matrix is calculatedusing the appropriate effective mass, mi , and the appropriatePauli operator, Qii , depending on ki

F , where i = p or n.For nonidentical nucleons, we use the “asymmetric” Paulioperator, Qij , depending on both kn

F and kp

F [8]. We recall thatknF and k

p

F change with increasing neutron fraction according

0556-2813/2006/73(1)/014001(6)/$23.00 014001-1 ©2006 The American Physical Society

F. SAMMARRUCA AND P. KRASTEV PHYSICAL REVIEW C 73, 014001 (2006)

to the relations

knF = kF (1 + α)1/3 (1)

kp

F = kF (1 − α)1/3, (2)

where kF is the average Fermi momentum. This may facilitatethe interpretation of results later on.

In the usual free-space scattering scenario, the cross sectionis typically represented as a function of the incident laboratoryenergy, which is uniquely related to the nucleon momentumin the two-body center-of-mass (c.m.) frame, q0 (also equal tothe relative momentum of the two nucleons), through the well-known formula Tlab = 2q2

0/m. In nuclear matter, though, thePauli operator depends also on the total momentum of the twonucleons in the nuclear matter rest frame. For simplicity, herewe use in-vacuum kinematics to define the total 2N momentumin the nuclear matter rest frame (that is, the target nucleon is,on the average, at rest).

Another issue to consider when approaching the conceptof in-medium cross sections is the nonunitary nature of theinteraction. Because of the presence of Pauli blocking, thein-medium scattering matrix does not obey the free-spaceunitarity relations through which phase parameters are usuallydefined and from which it is customary to determine the NNscattering observables. As done in Ref. [6] but unlike Ref. [5],we calculate the cross section directly from the scatteringmatrix elements, thus avoiding the assumption of in-vacuumunitarity and its consequences (such as the optical theorem).We calculate the differential cross section as

σ (q0, Ptot, ρ) =∫

dσ

d�Q(q0, Ptot, θ, ρ)d�, (3)

where dσ/d� is given by the usual sum of amplitudes squaredand phase-space factors and Q is the Pauli operator. Themomentum q0 has been previously defined, and Ptot is thetotal momentum of the two-nucleon system with respect tonuclear matter.

Ignoring Pauli blocking on the final momenta amounts tosetting Q = 1 in the integrand above, as done in previousworks [5,6] and in a preliminary calculation by us [10]. Theneed for such correction may be dependent on the detailsof the chosen BUU equation, as this effect is sometimesincorporated at the level of the transport calculation. In suchcase, though, a Pauli blocking function is usually appliedtogether with in-medium differential cross sections. However,total effective cross sections with properly Pauli-blockedfinal states should be a better indication of the degree ofsuppression the interaction actually undergoes in the medium.Furthermore, this correction must be included for a realisticcalculation of the nucleon mean free path in nuclear matter.(Note that the intermediate states are always Pauli-blocked inour microscopic calculation of the amplitudes.)

The presence of the Pauli operator in Eq. (3) restricts theintegration domain to

k2F − P 2 − q2

0

2Pq0� cos θ �

P 2 + q20 − k2

F

2Pq0, (4)

where kF is the Fermi momentum. The integral becomes zeroif the upper limit is negative, whereas the full angular range isused if the upper limit is greater than 1. (Following a previously

FIG. 1. (Color online) pp total effective cross section in symmet-ric nuclear matter at various densities as a function of the 2N relativemomentum.

used convention [11], P in Eq. (4) denotes one-half the totalmomentum.) Notice that the angle θ in Eq. (4), namely theangle between the directions of �q0 and �P , is also the colatitudeof �q0 in a system where the z axis is along �P and, thus, in suchsystem it coincides with the scattering angle to be integratedover in Eq. (3).

III. RESULTS FOR PP AND NP EFFECTIVE CROSSSECTIONS IN SYMMETRIC MATTER

We begin with discussing our predictions in symmetricmatter. These are shown in Fig. 1 (for pp scattering) andFig. 2 (for np scattering) as a function of q0 at selecteddensities. More predictions are displayed in Tables I and IIfor pp and np scattering, respectively. The given values ofq0 cover a range between approximately 20 and 350 MeV interms of in-vacuum laboratory kinetic energy. The range ofdensities goes from zero to about twice saturation density.Because of the presence of the Pauli operator in Eq. (3),

FIG. 2. (Color online) np total effective cross section in symmet-ric nuclear matter at various densities and as a function of the 2N

relative momentum.

014001-2

EFFECTIVE NUCLEON-NUCLEON CROSS SECTIONS IN . . . PHYSICAL REVIEW C 73, 014001 (2006)

TABLE I. pp total effective cross sections in symmetric matter calculated with the DBHF model and according to Eq. (3). Kinematics anddefinitions of variables are explained in the text.

kF (fm−1) σpp (mb) σpp (mb) σpp (mb) σpp (mb) σpp (mb) σpp (mb) σpp (mb)q0 = 100 MeV q0 = 150 MeV q0 = 200 MeV q0 = 250 MeV q0 = 300 MeV q0 = 350 MeV q0 = 400 MeV

0.0 171.2 69.33 39.86 28.93 23.96 21.47 20.270.2 157.1 61.71 34.97 25.56 21.70 20.05 19.470.4 133.4 52.91 29.90 22.32 19.63 18.80 18.820.6 64.34 42.58 24.97 19.39 17.85 17.78 18.310.8 0.00 29.06 20.11 16.71 16.30 16.92 17.910.9 0.00 19.86 17.61 15.42 15.57 16.52 17.711.0 0.00 8.980 14.99 14.14 14.86 16.13 17.511.1 0.00 0.00 11.98 12.70 14.07 15.72 17.311.2 0.00 0.00 8.602 11.14 13.24 15.29 17.131.3 0.00 0.00 5.025 9.478 12.37 14.89 17.011.4 0.00 0.00 1.583 7.793 11.54 14.56 16.981.5 0.00 0.00 0.00 6.123 10.77 14.34 17.071.6 0.00 0.00 0.00 4.466 10.05 14.20 17.251.7 0.00 0.00 0.00 2.698 9.286 14.04 17.40

it is apparent that the cross sections will drop to zero atcertain densities depending on the value of the momentum.Thus, cross sections calculated without this mechanism can bequite different, both quantitatively and qualitatively, especiallyat those densities/energies where Pauli blocking of the finalmomenta would suppress the cross section entirely. Wedemonstrate this fact in Fig. 3, where we compare both ppand np cross sections (near saturation density) calculated withand without Pauli blocking of the final states. In Fig. 3, the twohighest curves represent the predictions we obtain when theintegration domain is not restricted and resemble, for instance,those shown in Refs. [5,6].

A. pp cross sections

At very low energy, we observe strong sensitivity tovariations of the Fermi momentum when approaching thedensity where the cross section becomes identically zero.

At very low (fixed) density, the cross section decreases withenergy, a behavior similar to what happens in free space. Athigher densities, though, the cross section grows with energy.This is because of the fact that the Pauli operator in Eq. (3)becomes less effective at the higher energies.

For fixed momentum, the cross section typically decreaseswith increasing density. There is some tendency to rise againat high density and for the higher momenta.

This typical “Dirac effect” was already observed in previousDBHF calculations [5,6]. It is related to the presence of theeffective mass in the NN potential and is different in naturethan any of the “conventional” medium effects. We found it tobe particularly pronounced in isospin-1 partial waves and thusrelatively more important in the pp channel as compared to thenp (see comments on the np cross sections below). To betterdemonstrate its origin, we also show results obtained withthe conventional Brueckner-Hartree-Fock approach (BHF)(Table III and IV for pp and np, respectively). The differences

TABLE II. np total effective cross sections in symmetric matter calculated with the DBHF model and according to Eq. (3). Kinematics anddefinitions of variables as explained in the text.

kF (fm−1) σnp (mb) σnp (mb) σnp (mb) σnp (mb) σnp (mb) σnp (mb) σnp (mb)q0 = 100 MeV q0 = 150 MeV q0 = 200 MeV q0 = 250 MeV q0 = 300 MeV q0 = 350 MeV q0 = 400 MeV

0.0 453.3 174.1 86.41 55.02 41.65 34.60 30.150.2 459.7 159.6 75.48 47.35 36.23 30.78 27.520.4 508.3 144.7 63.55 39.26 30.68 26.85 24.720.6 421.6 128.8 52.49 31.98 25.73 23.40 22.260.8 0.00 101.3 42.49 25.71 21.51 20.45 20.180.9 0.00 72.45 37.58 22.91 19.61 19.14 19.231.0 0.00 31.84 32.37 20.28 17.85 17.89 18.351.1 0.00 0.00 25.82 17.45 15.96 16.56 17.391.2 0.00 0.00 17.91 14.50 14.06 15.19 16.391.3 0.00 0.00 9.578 11.44 12.14 13.81 15.391.4 0.00 0.00 2.628 8.512 10.35 12.52 14.481.5 0.00 0.00 0.00 5.899 8.763 11.41 13.691.6 0.00 0.00 0.00 3.761 7.454 10.52 13.081.7 0.00 0.00 0.00 2.018 6.386 9.810 12.60

014001-3

F. SAMMARRUCA AND P. KRASTEV PHYSICAL REVIEW C 73, 014001 (2006)

FIG. 3. (Color online) np and pp total effective cross sections insymmetric matter near saturation density as a function of the 2N

relative momentum. The two highest curves are obtained by settingQ = 1 in Eq. (1).

are most apparent when moving down to the higher densitiesin the last few columns of Table I and comparing with thecorresponding entries in Table III.

B. np cross sections

As in the pp case, very strong kF sensitivity can be seenat the lowest momenta just before the cross section is totallysuppressed.

For fixed momentum, the cross section typically decreaseswith increasing density. An interesting difference between ppand np cross sections is the fact that the latter rises with densityat very low density and for the lowest momenta, see Table II.We traced this effect to an enhancement of the 3S1 partialwave and did not observe a similar phenomenon in isospin-1partial waves. The presence of low-density enhancements wasdiscussed before as a possible bound-state signature [12,13].

Finally, we notice that the differences between correspond-ing entries in Table II and Table IV for high momentum andhigh density are less pronounced than in the pp case.

200 250 300 350 400

q0(MeV)

0

5

10

15

20

σ pp

/ nn(

mb) pp

nn

DBHF, α = 0.2, kF = 1.3fm-1

FIG. 4. (Color online) pp and nn total effective cross sectionsversus the 2N relative momentum at fixed density and asymmetry.

IV. RESULTS FOR PP AND NN EFFECTIVE CROSSSECTIONS IN ASYMMETRIC MATTER

A representative set of predictions for pp and nn totalcross sections in isospin-asymmetric matter, obtained withour DBHF model, are given in Table V for a medium levelof asymmetry (α = 0.4). Additional results are displayed inFigs. 4–6. We will ignore the α dependence in the np channelas we found it to be very weak (because of the “competing”roles of protons and neutrons).

In Fig. 4, pp and nn effective cross sections are shown asa function of q0 for fixed total density (close to saturationdensity) and asymmetry. Concerning the energy dependence,similar comments apply as those made with regard toFig. 1. Concerning the relative sizes of σpp and σnn, the nn crosssection is (almost always) smaller than the pp cross section.This is because of the additional Pauli blocking included inEq. (3), together with the fact that the neutron’s Fermimomentum is larger than the proton’s, see Eqs. (1)–(2). In

TABLE III. As in Table I, but with the BHF model.

kF (fm−1) σpp (mb) σpp (mb) σpp (mb) σpp (mb) σpp (mb) σpp (mb) σpp (mb)q0 = 100 MeV q0 = 150 MeV q0 = 200 MeV q0 = 250 MeV q0 = 300 MeV q0 = 350 MeV q0 = 400 MeV

0.0 171.2 69.33 39.86 28.93 23.96 21.47 20.270.2 156.8 62.36 35.68 26.06 21.90 19.95 19.120.4 131.4 53.62 30.85 22.91 19.69 18.33 17.860.6 61.28 42.74 25.72 19.71 17.49 16.72 16.610.8 0.00 28.39 20.36 16.53 15.33 15.15 15.390.9 0.00 19.02 17.57 14.95 14.27 14.37 14.781.0 0.00 8.431 14.68 13.36 13.21 13.60 14.171.1 0.00 0.00 11.72 11.83 12.21 12.87 13.601.2 0.00 0.00 8.404 10.15 11.08 12.04 12.931.3 0.00 0.00 4.900 8.353 9.831 11.08 12.151.4 0.00 0.00 1.526 6.526 8.526 10.06 11.301.5 0.00 0.00 0.00 4.714 7.175 8.973 10.371.6 0.00 0.00 0.00 3.011 5.823 7.836 9.3801.7 0.00 0.00 0.00 1.493 4.499 6.665 8.332

014001-4

EFFECTIVE NUCLEON-NUCLEON CROSS SECTIONS IN . . . PHYSICAL REVIEW C 73, 014001 (2006)

TABLE IV. As in Table II, but with the BHF model.

kF (fm−1) σnp (mb) σnp (mb) σnp (mb) σnp (mb) σnp (mb) σnp (mb) σnp (mb)q0 = 100 MeV q0 = 150 MeV q0 = 200 MeV q0 = 250 MeV q0 = 300 MeV q0 = 350 MeV q0 = 400 MeV

0.0 453.3 174.1 86.41 55.02 41.65 34.60 30.150.2 462.3 162.0 77.26 48.79 37.38 31.61 28.040.4 520.4 149.4 66.39 41.42 32.38 28.06 25.390.6 464.9 135.6 55.71 34.23 27.46 24.57 22.780.8 0.00 109.2 45.62 27.64 22.91 21.30 20.340.9 0.00 78.78 40.53 24.59 20.77 19.76 19.181.0 0.00 34.49 35.03 21.68 18.72 18.26 18.051.1 0.00 0.00 28.91 19.00 16.85 16.89 17.021.2 0.00 0.00 20.99 16.15 14.89 15.43 15.891.3 0.00 0.00 11.91 13.16 12.88 13.88 14.661.4 0.00 0.00 3.466 10.13 10.90 12.33 13.421.5 0.00 0.00 0.00 7.161 8.971 10.78 12.141.6 0.00 0.00 0.00 4.442 7.137 9.263 10.851.7 0.00 0.00 0.00 2.129 5.432 7.790 9.562

fact, as a consequence of Eq. (3), the nn effective cross sectionmust drop to zero more quickly (for the same average density).This is apparent from Table V and also from Figs. 5–6.

Although at low density the nn cross section tends to startlarger, Pauli blocking soon takes over. Particularly for largevalues of α, see Fig. 6, the pp cross section “survives” muchlarger densities than the nn. This is to be expected, because theproton Fermi momentum, kp

F = kF (1 − α)1/3, tends to remainsmall for large α.

All of the above suggests the following observation: theregion of the density/momentum phase space where σnn isnearly or entirely suppressed, whereas σpp is still considerablydifferent than zero should be a suitable ground to look for aclear experimental signature of their difference.

Some additional comments are in place concerning therelative sizes of σpp and σnn. Predictions based on scaling thefree-space cross section through the use of effective masses inthe phase-space factor will clearly have the following trend:

σnn > σpp, if m∗n > m∗

p [14];

σpp > σnn, if m∗p > m∗

n [15].

TABLE V. pp and nn total effective cross sections in isospin-asymmetric matter calculated with the DBHF model and accordingto Eq. (3). The asymmetry parameter is fixed to 0.4. Kinematics anddefinition of variables are given in the text.

kF σpp/σnn σpp/σnn σpp/σnn σpp/σnn

(fm−1) q0 = 100 MeV q0 = 200 MeV q0 = 300 MeV q0 = 400 MeV

0.2 151.8/162.9 33.73/36.32 21.09/22.39 19.25/19.730.4 128.6/138.5 28.38/31.65 18.88/20.53 18.63/19.090.6 88.33/29.26 23.85/26.33 17.26/18.61 18.29/18.460.8 17.92/0.0 19.97/20.36 16.06/16.69 18.11/17.841.0 0.0/0.0 16.43/13.34 15.09/14.74 17.94/17.191.2 0.0/0.0 12.12/4.353 14.00/12.51 17.85/16.511.4 0.0/0.0 7.110/0.0 12.96/10.13 18.12/15.961.6 0.0/0.0 2.416/0.0 12.38/7.680 19.02/15.66

Notice that our previous predictions [10] were qualitativelyconsistent with the first case above, because of the dominantrole of the effective masses in that calculation (we predictm∗

n > m∗p [2]). In summary, one must be careful with extracting

the sign of the neutron/proton effective mass splitting from theσnn/σpp ratio.

Obviously, empirical constraints that might shed lighton these issues can only be indirect, for instance, throughanalyses of carefully selected heavy-ion observables. Suchefforts are presently underway [14,16]. In fact, a recentstudy indicates that the transverse flow of neutrons andprotons may be a reliable probe of in-medium cross section[14]. As the present discussion suggests, when comparingtheoretical predictions with experimental constraints it isimportant to be clear about the nature of the extracted effectivecross section, namely what medium effects are includedin the quantity that is being constrained by the analysis.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

kF(fm-1)

0

10

20

30

40

σ pp

/ nn(

mb)

pp

nn

DBHF, α = 0.2, q0 = 200MeV

FIG. 5. (Color online) pp and nn total effective cross sectionsversus density for fixed asymmetry and momentum.

014001-5

F. SAMMARRUCA AND P. KRASTEV PHYSICAL REVIEW C 73, 014001 (2006)

0.0 0.5 1.0 1.5

kF(fm-1)

0

10

20

30

40

σ pp

/ nn(

mb)

pp

nn

DBHF, α = 0.6, q0 = 200MeV

FIG. 6. (Color online) As in Fig. 5, but for a larger value of α.

V. CONCLUSIONS

We have presented microscopic calculations of cross sec-tions for scattering of nucleons in symmetric and neutron-richmatter. Our predictions include all “conventional” mediumeffects as well as those associated with the nucleon Diracwave function. Pauli blocking of the final states is included inthe integration of the differential cross section.

First, we discussed the basic density/momentum depen-dence of pp and np cross sections in symmetric matter.Although they generally exhibit qualitatively similar behaviorwith changing energy and density, pp and np effective crosssections show some interesting differences in specific regionsof the phase space. At this time, different predictions disagreeconcerning the presence of large bound-state effects that mightbe associated with the onset of superfluidity.

The sensitivity to the asymmetry in neutron and protonconcentrations comes in through the combined effect ofPauli blocking and changing effective masses. The lower-ing(rising) of the proton(neutron) Fermi momentum and thereduced(increased) proton(neutron) effective mass tend tomove the cross section in opposite directions. With Pauliblocking applied to intermediate and final states, the finalbalance is that the nn effective cross section is more stronglysuppressed.

In summary, sensitivity to the asymmetry is nonnegligiblefor scattering of identical nucleons and clearly separates ppand nn scatterings. The degree of sensitivity depends on theregion of the energy-density-asymmetry phase space underconsideration. We conclude that the mean free path of anucleon could be affected in a significant way by the presenceof isospin asymmetry in the medium. This will be the focus ofa future work.

ACKNOWLEDGMENTS

The authors acknowledge financial support from the U. S.Department of Energy under grant DE-FG02-03ER41270.

[1] See, for instance and B. A. Li, Phys. Rev. Lett. 85, 4221 (2000),and references therein.

[2] F. Sammarruca, W. Barredo, and P. Krastev, Phys. Rev. C 71,064306 (2005).

[3] V. R. Pandharipande and S. C. Pieper, Phys. Rev. C 45, 791(1992).

[4] D. Persram and C. Gale, Phys. Rev. C 65, 064611 (2002).[5] G. Q. Li and R. Machleidt, Phys. Rev. C 48, 1702 (1993); 49,

566 (1994).[6] C. Fuchs, A. Faessler, and M. El-Shabshiry, Phys. Rev. C 64,

024003 (2001).[7] See, for instance, R. Crespo and R. C. Johnson, Phys. Rev. C 60,

034007 (1999), and references therein.

[8] D. Alonso and F. Sammarruca, Phys. Rev. C 67, 054301(2003).

[9] R. Machleidt, Adv. Nucl. Phys. 19, 189 (1989).[10] F. Sammarruca and P. Krastev, nucl-th/0506081.[11] F. Sammarruca, X. Meng, and E. J. Stephenson, Phys. Rev. C

62, 014614 (2000).[12] A. Bohnet, N. Ohtsuka, J. Aichelin, R. Linden, and A. Faessler,

Nucl. Phys. A494, 349 (1989).[13] T. Alm, G. Ropke, and M. Schmidt, Phys. Rev. C 50, 31 (1994).[14] B.-A. Li and L.-W. Chen, Phys. Rev. C 72, 064611 (2005).[15] Q. Li, Z. Li, and E. Zhao, Phys. Rev. C 69, 017601 (2004).[16] B.-A. Li, P. Danielewicz, and W. Lynch, Phys. Rev. C 71, 054603

(2005).

014001-6