DOI 10.1140/epja/i2014-14022-1

Review

Eur. Phys. J. A (2014) 50: 22 THE EUROPEANPHYSICAL JOURNAL A

Microscopic approach to the nucleon-nucleon effectiveinteraction and nucleon-nucleon scattering in symmetric andisospin-asymmetric nuclear matter�

F. Sammarrucaa

Physics Department, University of Idaho, Moscow, ID 83844-0903, USA

Received: 29 June 2013 / Revised: 15 July 2013Published online: 25 February 2014 – c© Societa Italiana di Fisica / Springer-Verlag 2014Communicated by A. Ramos

Abstract. After reviewing our microscopic approach to nuclear and neutron-rich matter, we focus on hownucleon-nucleon scattering is impacted by the presence of a dense hadronic medium, with special emphasison the case where neutron and proton densities are different. We discuss in detail medium and isospinasymmetry effects on the total elastic cross section and the mean free path of a neutron or a proton inisospin-asymmetric nuclear matter. We point out that in-medium cross sections play an important role inheavy-ion simulations aimed at extracting constraints on the symmetry potential. We argue that mediumand isospin dependence of microscopic cross sections are the result of a complex balance among variouseffects, and cannot be simulated with a simple phenomenological model.

1 Introduction

In this article, we will be concerned with hadronic inter-actions in the nuclear medium, an issue which goes to thevery core of nuclear physics. In fact, our present knowl-edge of the nuclear force in free space is, in itself, the resultof decades of struggle [1] which will not be reviewed here.The nature of the nuclear force in the medium is of coursean even more complex problem, as it involves aspects ofthe force that cannot be constrained through free-spacenucleon-nucleon (NN) scattering. Predictions of propertiesof nuclei are the ultimate test for many-body theories.

Nuclear matter is an alternative and convenient the-oretical laboratory to test many-body theories. By “nu-clear matter” we mean an infinite system of nucleons actedon by their mutual strong forces and no electromagneticinteractions. Nuclear matter is characterized by its en-ergy/particle as a function of density and other thermody-namic quantities, as appropriate (e.g., temperature). Suchrelation is known as the nuclear matter equation of state(EoS). The translational invariance of the system facili-tates theoretical calculations. At the same time, adopt-ing what is known as the “local density approximation”,one can use the EoS to obtain information on finite sys-tems. This procedure is applied, for instance, in Thomas-Fermi calculations within the liquid drop model, where an

� Contribution to the Topical Issue “Nuclear Symmetry En-ergy” edited by Bao-An Li, Angels Ramos, Giuseppe Verde,Issac Vidana.

a e-mail: fsammarr@uidaho.edu

appropriate energy functional is written in terms of theEoS [2–4].

Isospin-asymmetric nuclear matter (IANM) simulatesthe interior of a nucleus with unequal densities of protonsand neutrons. The equation of state of (cold) IANM isthen a function of density as well as the relative concen-trations of protons and neutrons.

The recent and fast-growing interest in IANM stemsfrom its close connection to the physics of neutron-richnuclei, or, more generally, isospin-asymmetric nuclei, in-cluding the very “exotic” ones known as “halo” nuclei. Atthis time, the boundaries of the nuclear chart are uncer-tain, with a few hundred stable nuclides known to existand a few thousand believed to exist. The future Facil-ity for Rare Isotope Beams (FRIB) is expected to deliverintense beams of rare isotopes, the study of which can pro-vide crucial information on short-lived elements normallynot found on earth. Thus, this new experimental programwill have widespread impact, ranging from the origin ofelements to the evolution of the cosmos. In the meantime,systematic investigations to determine the properties ofasymmetric nuclear matter and to constrain the symme-try energy are proliferating at existing facilities.

It is the focal point of this article to review and discussour approach to the development of effective NN interac-tions in IANM, with particular emphasis on its applica-tions to in-medium isospin-dependent NN cross sectionsand related issues.

The article is organized as follows: In sect. 2, we setthe stage by presenting a brief overview of facts and

Page 2 of 14 Eur. Phys. J. A (2014) 50: 22

phenomenology about IANM. We then proceed to de-scribe our microscopic approach for calculating the en-ergy/particle in IANM (sect. 3). Although the EoS per seis not the focal point of this article, this step is importantto elucidate how the self-consistent determination of theeffective interaction and the (isospin-dependent) single-particle potential is obtained. The latter is closely relatedto the nucleon effective masses, which then become a cru-cial ingredient in the calculation of the isospin-dependenteffective cross sections. Those will be confronted in sect. 4.In sect. 5 we will discuss the mean free path of nucleons innuclear matter and its relation to the effective cross sec-tions. A brief summary and conclusive remarks are con-tained in sect. 6.

2 Facts about isospin-asymmetric nuclearmatter

Asymmetric nuclear matter is characterized by the neu-tron density, ρn, and the proton density, ρp. In infinitematter, they are obtained by summing the neutron orproton states per volume (up to their respective Fermimomenta, kn

F or kpF ) and applying the appropriate degen-

eracy factor. The result is

ρi =(ki

F )3

3π2, (1)

with i = n or p.It is more convenient to refer to the total density

ρ = ρn+ρp and the asymmetry (or neutron excess) param-eter α = ρn−ρp

ρ . Clearly, α = 0 corresponds to symmetricmatter and α = 1 to neutron matter. In terms of α and theaverage Fermi momentum, kF , related to the total densityin the usual way,

ρ =2k3

F

3π2, (2)

the neutron and proton Fermi momenta can be expressedas

knF = kF (1 + α)1/3 (3)

andkp

F = kF (1 − α)1/3, (4)

respectively.Expanding the energy/particle in IANM with respect

to the asymmetry parameter yields

e(ρ, α) = e0(ρ) +12

(∂2e(ρ, α)

∂α2

)α=0

α2 + O(α4), (5)

where the first term is the energy per particle in sym-metric matter and the coefficient of the quadratic term isidentified with the symmetry energy, esym. In the Bethe-Weizsacker formula for the nuclear binding energy, it rep-resents the amount of binding a nucleus has to lose whenthe numbers of protons and neutrons are unequal. Thesymmetry energy is also closely related to the neutron β-decay in dense matter, whose threshold depends on the

proton fraction. A typical value for esym at nuclear mat-ter density (ρ0) is 30MeV, with theoretical predictionsspreading approximately between 26 and 35MeV.

To a very good degree of approximation, the energyper particle in IANM can be written as

e(ρ, α) ≈ e0(ρ) + esym(ρ)α2. (6)

The effect of a term of fourth degree in the asymmetry pa-rameter (O(α4)) on the bulk properties of neutron stars issmall, although it may impact the proton fraction at highdensity. More generally, non-quadratic terms are usuallyassociated with isovector pairing, which is a surface effectand thus vanishes in infinite matter [5].

Equation (6) displays a convenient separation betweenthe symmetric and the aymmetric parts of the EoS, whichfacilitates the identification of observables that may besensitive, for instance, mainly to the symmetry energy.Typically, constraints are extracted from heavy-ion colli-sion simulations based on transport models. Isospin dif-fusion and the ratio of neutron and proton spectra areamong the observables used in these analyses. For a re-cent review on available constraints the reader is referredto ref. [6].

Empirical investigations appear to agree reasonablywell on the following parametrization of the symmetry en-ergy:

esym(ρ) = 12.5MeV(

ρ

ρ0

)2/3

+ 17.5MeV(

ρ

ρ0

)γi

, (7)

where ρ0 is the saturation density. The first term is thekinetic contribution and γi (the exponent appearing in thepotential energy part) is found to be between 0.4 and 1.0.Recent measurements of elliptic flows in 197Au + 197Aureactions at GSI at 400–800MeV/nucleon favor a poten-tial energy term with γi equal to 0.9 ± 0.4. Giant dipoleresonance excitation in fusion reactions [7] is also sensitiveto the symmetry energy, since the latter is responsible forisospin equilibration in isospin-asymmetric collisions.

Isospin-sensitive observables can also be identifiedamong the properties of normal nuclei. The neutron skinof neutron-rich nuclei is a powerful isovector observable,being sensitive to the slope of the symmetry energy, whichdetermines to which extent neutrons will tend to spreadoutwards to form the skin.

Parity-violating electron scattering experiments arenow a realistic option to determine neutron distributionswith unprecedented accuracy. The neutron radius of 208Pbis expected to be re-measured at the Jefferson Labora-tory in the PREXII experiment planned for the near fu-ture. Parity-violating electron scattering at low momen-tum transfer is especially suitable to probe neutron den-sities, as the Z0 boson couples primarily to neutrons. Amuch higher level of accuracy can be achieved with elec-troweak probes than with hadronic scattering. With thesuccess of this program, reliable empirical information onneutron skins will be able to provide, in turn, more strin-gent constraint on the density dependence of the symme-try energy.

Eur. Phys. J. A (2014) 50: 22 Page 3 of 14

A measure for the density dependence of the symmetryenergy is the parameter defined as

L = 3ρ0

(∂esym(ρ)

∂ρ

)ρ0

≈ 3ρ0

(∂en.m.(ρ)

∂ρ

)ρ0

, (8)

where we have used eq. (6) with α = 1. Thus, L is sensitiveto the gradient of the energy per particle in neutron matter(en.m.), that is, the neutron matter pressure. As to beexpected on physical grounds, the neutron skin, given by

S =√〈r2

n〉 −√〈r2

p〉, (9)

is highly sensitive to the same energy gradient.Predictions of L by phenomenological models show

a very large spreading. Values ranging from −50 to+100MeV are found from the numerous parametrizationsof Skyrme interactions (see ref. [8] and references therein),all chosen to fit the binding energies and the charge radiiof a large number of nuclei.

In ref. [6], values for the symmetry energy and theL parameter centered around 32.5MeV and 70MeV, re-spectively, are obtained, both from nuclear structure andheavy-ion collision measurements, for densities rangingbetween 0.3ρ0 and ρ0.

Typically, parametrizations like the one given in eq. (7)are valid at or below the saturation density. Efforts toconstrain the behavior of the symmetry energy at higherdensities are being pursued through observables such asπ−/π+ ratio, K+/K0 ratio, neutron/proton differentialtransverse flow, or nucleon elliptic flow [9].

Another important quantity which emerges from stud-ies of IANM is the symmetry potential. Its definition stemsfrom the observation that the single-particle potentials ex-perienced by the proton and the neutron in IANM, Un/p,are different from each other and satisfy the approximaterelation

Un/p(k, ρ, α) ≈ Un/p(k, ρ, α = 0) ± Usym(k, ρ)α, (10)

where the +(−) sign refers to neutrons (protons), and

Usym =Un − Up

2α. (11)

Thus, one can expect isospin splitting of the single-particlepotential to be effective in separating the collision dy-namics of neutrons and protons. In a neutron-rich envi-ronment, the symmetry potential tends to expel neutronsand attract protons, thus providing the opportunity of de-tecting sensitivity to the symmetry energy in observablessuch as the yield ratios of ejected neutrons/protons or therate of isospin diffusion [6]. The splitting of the single-nucleon potentials in IANM as a function of the momen-tum is shown in fig. 1 for three different meson-theoreticpotentials [1].

Furthermore, Usym, being proportional to the gradientbetween the single-neutron and the single-proton poten-tials, should be comparable with the Lane potential [10],namely the isovector part of the nuclear optical potential.Optical potential analyses (in isospin-unsaturated nuclei)can then help constrain this quantity and, in turn, thesymmetry energy.

Fig. 1. (Color online) Momentum dependence of the single-nucleon potential in IANM, Ui (i = n, p), predicted with BonnA (a), Bonn B (b), and Bonn C (c). The total density is equalto 0.185 fm−3, and the isospin asymmetry parameter is equalto 0.4. The momentum is given in units of the (average) Fermimomentum, which is equal to 1.4 fm−1.

3 Our microscopic approach toisospin-asymmetric nuclear matter

3.1 The two-body sector

Our approach is ab initio in that the starting point ofthe many-body calculation is a realistic NN interactionwhich is then applied in the nuclear medium without anyadditional free parameters. Thus the first question to beconfronted concerns the choice of the “best” NN interac-tion. After the development of Quantum Chromodynam-ics (QCD) and the understanding of its symmetries, chiraleffective theories [11] were developed as a way to respectthe symmetries of QCD while keeping the degrees of free-dom (nucleons and pions) typical of low-energy nuclearphysics. However, chiral perturbation theory (ChPT) hasdefinite limitations as far as the range of allowed momentais concerned. For the purpose of applications in dense mat-ter, where higher and higher momenta become involvedwith increasing Fermi momentum, NN potentials basedon ChPT are unsuitable.

Page 4 of 14 Eur. Phys. J. A (2014) 50: 22

Relativistic meson theory is an appropriate frameworkto deal with the high momenta encountered in dense mat-ter. In particular, the one-boson–exchange (OBE) modelhas proven very successful in describing NN elastic datain free space up to high energies and has a good theo-retical foundation. Among the many available OBE po-tentials, some being part of the “high-precision genera-tion” [12–14], we seek a momentum-space potential de-veloped within a relativistic scattering equation, suchas the one obtained through the Thompson [15] three-dimensional reduction of the Bethe-Salpeter equation [16].Furthermore, we require a potential that uses the pseu-dovector coupling for the interaction of nucleons withpseudoscalar mesons. With these constraints in mind, aswell as the requirement of a good description of the NNdata, Bonn B [1] is a reasonable choice. As is well known,the NN potential model dependence of nuclear matterpredictions is not negligible. The saturation points ob-tained with different NN potentials move along the fa-mous “Coester band” depending on the strength of thetensor force, with the weakest tensor force yielding thelargest attraction. This can be understood in terms ofmedium effects (particularly Pauli blocking) reducing the(attractive) second-order term in the expansion of the re-action matrix. A large second-order term will undergo alarge reduction in the medium. Therefore, noticing thatthe second-order term is dominated by the tensor compo-nent of the force, nuclear potentials with a strong tensorcomponent will yield less attraction in the medium. Forthe same reason (that is, the role of the tensor force in nu-clear matter), the potential model dependence is stronglyreduced in pure (or nearly pure) neutron matter, due tothe absence of isospin-zero partial waves.

In closing this section, we wish to highlight the mostimportant aspect of the ab initio approach: namely, theonly free parameters of the model (the parameters of theNN potential) are determined by fitting the free-space NNdata and never readjusted in the medium. In other words,the model parameters are tightly constrained and the cal-culation in the medium is parameter free. The presenceof free parameters in the medium would generate effectsand sensitivities which are hard to control and reduce thepredictive power of the theory.

3.2 The Dirac-Brueckner-Hartree-Fock approach tosymmetric and asymmetric nuclear matter

The main strength of the DBHF approach is its inher-ent ability to account for important three-body forcesthrough its density dependence. In fig. 2 we show a three-body force (TBF) originating from virtual excitation ofa nucleon-antinucleon pair, known as “Z-diagram”. Themain feature of the DBHF method turns out to be closelyrelated to the TBF depicted in fig. 2, as we will arguenext. In the DBHF approach, one describes the positiveenergy solutions of the Dirac equation in the medium as

u∗(p, λ) =(

E∗p + m∗

2m∗

)1/2⎛⎝ 1

σ · pE∗

p + m∗

⎞⎠ χλ, (12)

Fig. 2. Three-body force due to virtual pair excitation.

where the nucleon effective mass, m∗, is defined as m∗ =m +US , with US an attractive scalar potential. (This willbe derived below.) It can be shown that both the descrip-tion of a single-nucleon via eq. (12) and the evaluationof the Z-diagram, fig. 2, generate a repulsive effect on theenergy per particle in symmetric nuclear matter which de-pends on the density approximately as

ΔE ∝(

ρ

ρ0

)8/3

, (13)

and provides the saturating mechanism missing fromconventional Brueckner-Hartree-Fock (BHF) calculations.(Alternatively, explicit TBF are used along with the BHFmethod in order to achieve a similar result.) Brown showedthat the bulk of the desired effect can be obtained as alowest order (in p2/m) relativistic correction to the single-particle propagation [17]. With the in-medium spinor asin eq. (12), the correction to the free-space spinor can bewritten approximately as

u∗(p, λ) − u(p, λ) ≈

⎛⎝ 0

−σ · p2m2

US

⎞⎠ χλ, (14)

where, for simplicity, the spinor normalization factor hasbeen set equal to 1, in which case it is clearly seenthat the entire effect originates from the modification ofthe spinor’s lower component. By expanding the single-particle energy to order U2

S , Brown showed that the correc-tion to the energy consistent with eq. (14) can be writtenas p2

2m (US

m )2. He then proceeded to estimate the correctionto the energy per particle and found it to be approximatelyas given in eq. (13).

The approximate equivalence of the effective-mass de-scription of Dirac states and the contribution from theZ-diagram has a simple intuitive explanation in the obser-vation that eq. (12), like any other solution of the Diracequation, can be written as a superposition of positiveand negative energy solutions. On the other hand, the“nucleon” in the middle of the Z-diagram, fig. 2, can beviewed as the superposition of positive and negative en-ergy states. In summary, the DBHF method effectivelytakes into account a particular class of TBF, which arecrucial for nuclear matter saturation.

Having first summarized the main DBHF philoso-phy, we now proceed to review our DBHF calculation ofIANM [18, 19]. In the end, this will take us back to thecrucial point of the DBHF approximation, eq. (12).

Eur. Phys. J. A (2014) 50: 22 Page 5 of 14

As mentioned in the previous subsection, we start fromthe Thompson [15] relativistic three-dimensional reduc-tion of the Bethe-Salpeter equation [16]. The Thompsonequation is applied to nuclear matter in strict analogy tofree-space scattering and reads, in the nuclear matter restframe,

gij(q′, q,P , (ε∗ij)0) = v∗ij(q

′, q) +∫

d3K

(2π)3v∗

ij(q′,K)

×m∗

i m∗j

E∗i E∗

j

Qij(K,P )(ε∗ij)0 − ε∗ij(P ,K)

×gij(K, q,P , (ε∗ij)0), (15)

where gij is the in-medium reaction matrix (ij = nn, pp,or np), and the asterix signifies that medium effects areapplied to those quantities. Thus the NN potential, v∗

ij , isconstructed in terms of effective Dirac states (in-mediumspinors) as explained above. In eq. (15), q, q′, and K arethe initial, final, and intermediate relative momenta, andE∗

i =√

(m∗i )2 + K2. The momenta of the two interact-

ing particles in the nuclear matter rest frame have beenexpressed in terms of their relative momentum and thecenter-of-mass momentum, P , through

P = k1 + k2 (16)

andK =

k1 − k2

2. (17)

The energy of the two-particle system is

ε∗ij(P ,K) = e∗i (P ,K) + e∗j (P ,K) (18)

and (ε∗ij)0 is the starting energy. The single-particle energye∗i includes kinetic energy and potential energy contribu-tions (see eq. (32) below). The Pauli operator, Qij , pre-vents scattering to occupied nn, pp, or np states. To elim-inate the angular dependence from the kernel of eq. (15),it is customary to replace the exact Pauli operator with itsangle-average. Detailed expressions for the Pauli operatorand the average center-of-mass momentum in the case oftwo different Fermi seas can be found in ref. [18].

With the definitions

Gij =m∗

i

E∗i (q′ )

gij

m∗j

E∗j (q)

(19)

and

V ∗ij =

m∗i

E∗i (q′)

v∗ij

m∗j

E∗j (q)

, (20)

one can rewrite eq. (15) as

Gij(q′, q,P , (ε∗ij)0) = V ∗ij(q

′, q) +∫

d3K

(2π)3V ∗

ij(q′,K)

× Qij(K,P )(ε∗ij)0 − ε∗ij(P ,K)

×Gij(K, q,P , (ε∗ij)0), (21)

which is our working equation and has the convenient fea-ture of being formally identical to its non-relativistic coun-terpart.

The goal is to determine self-consistently the nuclearmatter single-particle potential which, in IANM, will bedifferent for neutrons and protons. To facilitate the de-scription of the procedure, we will use a schematic nota-tion for the neutron/proton potential. We write, for neu-trons,

Un = Unp + Unn, (22)

and for protonsUp = Upn + Upp, (23)

where each of the four pieces on the right-hand side ofeqs. (22) and (23) signifies an integral of the appropriateG-matrix elements (nn, pp, or np) obtained from eq. (21).Clearly, the two equations above are coupled through thenp component and so they must be solved simultane-ously. Furthermore, the G-matrix equation and eqs. (22)and (23) are coupled through the single-particle energy(which includes the single-particle potential, itself definedin terms of the G-matrix). So we have a coupled systemto be solved self-consistently.

Before proceeding with the self-consistency, one needsan ansatz for the single-particle potential. The latter issuggested by the most general structure of the nucleonself-energy operator consistent with all symmetry require-ments. That is,

Ui(p) = US,i(p) + γ0U0V,i(p) − γ · pUV,i(p), (24)

where US,i and UV,i are an attractive scalar field and arepulsive vector field, respectively, with U0

V,i the timelikecomponent of the vector field. These fields are in generaldensity and momentum dependent. We take

Ui(p) ≈ US,i(p) + γ0U0V,i(p), (25)

which amounts to assuming that the spacelike componentof the vector field is much smaller than both US,i andU0

V,i. Furthermore, neglecting the momentum dependenceof the scalar and vector fields and inserting eq. (25) inthe Dirac equation for neutrons/protons propagating innuclear matter,

(γμpμ − mi − Ui(p))u∗i (p, λ) = 0, (26)

naturally leads to rewriting the Dirac equation in the form

(γμ(pμ)∗ − m∗i )u

∗i (p, λ) = 0, (27)

with positive energy solutions as in eq. (12), m∗i = m +

US,i, and(p0)∗ = p0 − U0

V,i(p). (28)

The subscript “i” signifies that these parameters are dif-ferent for protons and neutrons.

As in the symmetric matter case [20], evaluating theexpectation value of eq. (25) leads to a parametrizationof the single-particle potential for protons and neutrons

Page 6 of 14 Eur. Phys. J. A (2014) 50: 22

(eqs. (22) and (23)) in terms of the constants US,i andU0

V,i which is given by

Ui(p) =m∗

i

E∗i

〈p|Ui(p)|p〉 =m∗

i

E∗i

US,i + U0V,i. (29)

These are the single-nucleon potentials displayed in fig. 1.Also,

Ui(p) =∑

j=n,p

∑p′≤kj

F

Gij(p,p′), (30)

which, along with eq. (29), allows the self-consistent de-termination of the single-particle potentials displayed infig. 1.

From the Dirac equation, eq. (26), the kinetic contri-bution to the single-particle energy is

Ti(p) =m∗

i

E∗i

〈p|γ · p + m|p〉 =mim

∗i + p2

E∗i

, (31)

and the single-particle energy is

e∗i (p) = Ti(p) + Ui(p) = E∗i + U0

V,i. (32)

The constants m∗i and

U0,i = US,i + U0V,i (33)

are convenient to work with as they facilitate the connec-tion with the usual non-relativistic framework [21].

Starting from some initial values of m∗i and U0,i, the

G-matrix equation is solved and a first approximation forUi(p) is obtained by integrating the G-matrix over theappropriate Fermi sea, see eq. (30). This solution is againparametrized in terms of a new set of constants, deter-mined by fitting the parametrized Ui, eq. (29), to its val-ues calculated at two momenta, a procedure known as the“reference spectrum approximation”. The iterative proce-dure is repeated until satisfactory convergence is reached.

Finally, the energy per neutron or proton in nuclearmatter is calculated from the average values of the kineticand potential energies as

ei =1A〈Ti〉 +

12A

〈Ui〉 − m. (34)

The EoS, or energy per nucleon as a function of density,is then written as

e(ρn, ρp) =ρnen + ρpep

ρ, (35)

or

e(kF , α) =(1 + α)en + (1 − α)ep

2. (36)

Clearly, symmetric nuclear matter is obtained as a by-prod-uct of the calculation described above by setting α = 0,whereas α = 1 corresponds to pure neutron matter.

4 Medium- and isospin-dependent NN crosssections

4.1 General aspects

Transport equations describe the evolution of a non-equilibrium gas of strongly interacting hadrons. In modelsbased on the Boltzmann-Uehling-Uhlenbeck (BUU) equa-tion [22, 23], particles drift in the presence of the meanfield while undergoing two-body collisions, which requirethe 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 will present microscopic predictions of NN totalelastic cross sections in isospin symmetric and asymmet-ric nuclear matter. In asymmetric matter, the cross sectionbecomes isospin dependent beyond the usual and well-known differences between the np and the pp/nn cases.Here, we are referring to isospin dependence induced bymedium asymmetries, meaning that, even in the sameisospin state, the nn, pp, and np interactions are differentbecause of different relative proton and neutron concen-trations. Also, we are only concerned with the strong inter-action contribution to the cross section (Coulomb effectson the pp cross section or charge-symmetry and charge-independence breaking effects are not considered.)

As mentioned in the introduction, there is increasinginterest in studying isospin asymmetries in nuclear mat-ter. Collisions of neutron-rich nuclei are capable of produc-ing extended regions of space/time where both the totalnucleon density and the neutron/proton asymmetry arelarge. Isospin-dependent BUU transport models includeisospin-sensitive collision dynamics through the elemen-tary pp, nn, and np cross sections and the mean field,which is different for protons and neutrons.

The in-medium cross sections are driven by the scat-tering amplitude and also by kinematic factors, i.e., en-trance flow and density of states in the exit channel, bothof which are related to the effective mass (and thus, the nu-cleon self-energy). In-medium cross sections depend non-trivially on several variables, such as the relative momen-tum of the nucleon pair, the total momentum of the pairin the nuclear matter rest frame, and, in the case of asym-metric matter, two different densities. To facilitate appli-cations in reactions, these multiple dependences have beenhandled in different ways and with different levels of ap-proximations, with the result that predictions can be quitedifferent from one another. Model differences include, forinstance, whether or not medium effects are present bothin the G-matrix and the density of states; or whether Pauliblocking effects are taken into account in both the finaland intermediate configurations.

In a simpler approach, the assumption is made thatthe transition matrix in the medium is approximately thesame as the one in vacuum and that medium effects on thecross section come in only through the use of nucleon effec-tive masses in the phase space factors [24–26]. Concerningmicroscopic calculations, some can be found, for instance,

Eur. Phys. J. A (2014) 50: 22 Page 7 of 14

in refs. [27–29], but considerations of medium asymme-tries are not included in those predictions. In ref. [30],the Brueckner-Hartree-Fock method with the Argonn v14

potential including the contribution of microscopic three-body forces is employed. A brief review of empirical andtheoretical findings is given next.

In-medium cross sections can provide information onthe mean free path of nucleons in nuclear matter and thusnuclear transparency. The latter is obviously related tothe total reaction cross section of a nucleus, which, inturn, can be used to extract nuclear r.m.s. radii withinGlauber-type models [31]. Therefore, accurate in-mediumisospin-dependent NN cross sections can ultimately bevery valuable to obtain information about the size of ex-otic, neutron-rich nuclei. In summary, it is important toinvestigate to which extent the in-medium cross sectionsare sensitive to changes in the proton/neutron ratio, oneof this article’s main purposes.

4.2 Brief overview of findings and observations fromthe literature

Most theoretical studies have been conducted in symmet-ric matter and at zero temperature. As mentioned earlier,results differ considerably. In ref. [30], three-body forceswere found to induce a stronger suppression of the crosssection as compared with Brueckner calculations with two-body forces only. This effect originated from enhancementof the repulsive component of the effective interaction, butmostly from reduction of the density of states in the en-trance and exit channels due to the rearrangement termin the self-energy, which can also be traced back to thethree-body force [30].

Alm et al. [32,33] considered in-medium cross sectionsat finite temperature and observed a strong enhancementat low temperature which might be attributed to the onsetof superfluidity. Such enhancement was found to be cru-cially determined by the inclusion of hole-hole scatteringin the Pauli operator.

The predictions from ref. [27] are based on DBHF cal-culations of the (real) R-matrix in symmetric nuclear mat-ter. They can be parametrized as

σmednp =(31.5+0.092|20.2−E0.53

lab |2.9)1.0 + 0.0034E1.51

lab ρ2

1.0 + 21.55ρ1.34

(37)and

σmedpp =(23.5+0.00256(18.2−E0.5

lab)4)

1.0 + 0.1667E1.05lab ρ3

1.0 + 9.704ρ1.2.

(38)Rather different conclusions were reached in ref. [34],where the in-medium cross section for collision of two slabsof nuclear matter was found to increase with density.

In a phenomenological approach [24–26, 35] the NNcross sections in the medium are scaled with the factor

σ∗NN

σfreeNN

=(

μ∗NN

μNN

)2

, (39)

where μ∗NN and μNN are the reduced masses of the colliding

nucleon pairs in the medium and in vacuum, respectively.Before we proceed, some comments are in place con-

cerning the meaning of isospin dependence of the NN crosssection. Generally, isospin dependence is understood asthe mechanism by which the ratio σnp

σppchanges in the

medium. Since σnp and σpp differ in free space due tothe fact that only one isospin state is allowed in the lat-ter, the evolution of this ratio in the medium indicates towhich extent partial waves with different isospin exhibitdifferent behavior as a function of density.

In our work, of course we have carefully considereddifferences between σpp and σnp and their density depen-dence, but we have gone beyond that point and also ad-dressed differences among σpp, σnn, and σnp which are in-duced by the presence of different proton and neutron den-sities (i.e., isospin-asymmetric medium). Of course theseare more subtle and, accordingly, more difficult to dis-cern experimentally. Nevertheless, we will demonstratethat these effects can be non-negligeable and do provideadditional insight into how the medium separates the dy-namics of protons and neutrons. Furthermore, they shouldbe included when addressing the mean free path of a pro-ton or a neutron in IANM.

Experimentally, evidence has been reported for in-medium modification of NN cross sections based on heavy-ion collisions. In particular, studies of collective flow haveprovided strong indication that the cross section is re-duced in the medium [36–38]. The empirical relation

σmedNN =

(1 + a

ρ

ρ0

)σfree

NN , (40)

with a ≈ −0.2, was found to be in better agreement withflow data as compared to calculations that made use offree-space cross sections [39]. More recent studies of thestopping power and collective flow at SIS/GSI energiesprovided indications that the NN in-medium cross sec-tions are reduced at low energy but enhanced at high en-ergy [40].

Although the nuclear stopping power in heavy-ion col-lisions has been found to be sensitive to medium effectson the NN cross sections, it is insufficient to discernisospin dependence [41]. Isospin tracers such as the neu-tron/proton ratio of free nucleons or the ratio of mirror nu-clei [42] have been proposed as potential probes of isospindependence.

In conclusion, the current status can be summarizedas follows: considerable theoretical effort has been spenton the issue of the in-medium dependence of NN cross sec-tions, but much less on their isospin dependence. From theexperimental standpoint, there is evidence of in-mediumreduction of the NN cross sections, but observables thatcan unambiguosly resolve the isospin dependence have notclearly been identified. A likely candidate seems to bethe neutron/proton ratio of free nucleons, specifically atbackward rapidities/angles, in reactions involving radioac-tive beams in inverse kinematics. For instance, in ref. [42]isospin sensitivity is tested by comparing rapidity distribu-tions in heavy-ion collisions with different assumptions for

Page 8 of 14 Eur. Phys. J. A (2014) 50: 22

the NN cross sections, such as σnp = σpp = σfreepp , or σnp =

σfreenp and σpp = σfree

pp , or σnp = σpp = 0.5(σfreenp +σfree

pp ).Using different σnp/σpp ratios has impact on the transferof neutrons or protons from forward to backward rapiditi-ties, an effect which is opposite for neutrons and protonsand thus reflects on the isospin asymmetry in a measur-able way [41].

4.3 Our approach to in-medium NN cross sections

The nuclear matter calculation described in sect. 3 pro-vides, along with the EoS, the single-proton/neutron po-tentials as well as their parametrizations in terms of effec-tive masses, see eq. (29). Those effective masses, togetherwith the appropriate Pauli operator (depending on thetype of nucleons involved), are then used in a separatecalculation of the in-medium reaction matrix under thedesired kinematical conditions.

The medium effects which we include in the calculationof the G-matrix, see eq. (21), are: Pauli blocking of the in-termediate (virtual) states for two nucleons with equal ordifferent Fermi momenta; dispersive effects on the single-particle energies; density-dependent nucleon spinors in theNN potential (Dirac effect).

Our calculation is controlled by the total density, ρ,and the degree of asymmetry, α = (ρn − ρp)/(ρn + ρp).For the case of identical nucleons, the G-matrix is calcu-lated using the appropriate effective mass, mi, and theappropriate Pauli operator, Qii, depending on ki

F , wherei = p or n. For non-identical nucleons, we use the “asym-metric” Pauli operator, Qij , depending on both kn

F andkp

F [18]. We recall that knF and kp

F change with increasingneutron fraction according to eqs. (3) and (4).

In the usual free-space scattering scenario, the crosssection is typically represented as a function of the inci-dent laboratory energy, which is uniquely related to thenucleon momentum in the two-body c.m. frame, q (alsoequal to one-half the relative momentum of the two nu-cleons), through the well-known formula Tlab = 2q2/m. Innuclear matter, though, the Pauli operator depends alsoon the total momentum of the two nucleons in the nuclearmatter rest frame. For simplicity, here we use in-vacuumkinematics to define the total two-nucleon momentum inthe nuclear matter rest frame (that is, the target nucleonis, on the average, at rest).

Another issue to consider when addressing in-mediumcross sections is the non-unitary nature of the interaction.In free space, and in absence of inelasticities, the (real) R-matrix and the (complex) T -matrix formalisms are equiva-lent. However, due to the presence of Pauli blocking, whichrestricts the accessible spectrum of momentum states, thein-medium scattering matrix does not obey the free-spaceunitarity relations through which phase-shift parametersare defined and from which it is customary to determineNN scattering observables. Therefore, we believe that thein-medium cross section should be calculated from thecomplex G-matrix amplitudes (as obtained from eq. (21)),and that significant loss of information may result if theR-matrix is used instead.

0

40

80

120

160

σ pp (

mb

)

0 200 400 600q (MeV)

Fig. 3. (Color online) pp total elastic cross section in sym-metric nuclear matter at various densities as a function of theNN relative momentum. Predictions are obtained from eq. (41)integrated over the whole solid angle. The dashed (red), dash-dotted (green), and dotted (blue) curves correspond to valuesof the Fermi monentum equal to 1.1, 1.3, and 1.5 fm−1, respec-tively. The values in free space are also shown (solid black).

0

200

400

600σ n

p (

mb

)

0 200 400 600q (MeV)

Fig. 4. (Color online) As in fig. 3 for np scattering.

A first step to obtain the in-medium total elastic crosssection is to integrate the elastic differential cross section,

dσ

dΩ=

(m∗)4

4π2s∗|G(q, q, θ)|2, (41)

where G is the amplitude obtained by summing the usualpartial wave helicity matrix elements, m∗ = m + US (seedefinition below eq. (27)), and s∗ = 4((m∗)2 + q2).

Representative results from the procedure outlinedabove are shown in figs. 3 and 4. There, we display pp andnp cross sections as a function of the NN momentum inthe center of mass of the pair and for three different densi-ties of symmetric matter corresponding to Fermi momentaequal to 1.1 fm−1, 1.3 fm−1, and 1.5 fm−1, respectively.The free-space predictions are also included. For fixedmomentum, the cross section typically decreases with in-creasing density. Likewise, for fixed density, generally itdecreases as a function of momentum. There is a cleartendency, though, of the in-medium predictions to rise

Eur. Phys. J. A (2014) 50: 22 Page 9 of 14

again with density for the higher momenta. This effect wasalready observed in previous DBHF calculations [27, 28].We determined that it originates from the presence of theeffective mass in the NN potential and is different in na-ture than any of the “conventional” medium effects. Wefound it to be particularly pronounced in isospin-1 par-tial waves, and thus relatively more important in the ppchannel as compared to the np one, as is apparent from acomparison of figs. 3 and 4.

At the lowest energies and for the lower densities, thecross sections show some enhancement before starting todecrease monotonically, a feature that is more pronouncedin the np channel (fig. 4), suggesting a stronger contribu-tion from T = 0 partial waves. We found that the presenceand size of such enhancement is dependent on the choiceadopted for P , the total momentum of the pair. The struc-tures seen in the figures are most likely the result of com-petition among effects which would rise or lower the crosssection. Notice that medium effects applied only on theG-matrix amplitudes but not on the phase space factorwould increase, rather than lower, the cross section.

The predictions shown in figs. 3 and 4 are only a base-line as they do not yet include Pauli blocking of the finalstates, an important effect for a realistic consideration ofphysical scattering. For that purpose, the allowed solidangle into which the nucleon final momenta are allowedto scatter must be restricted. We will discuss this aspectnext.

In nucleus-nucleus scattering, two interacting nucleonswithin each colliding nucleus can have momenta that are,in general, off the symmetry axis defined by the relativemomentum of the centers of the colliding nuclei, k. Toconsider such case, one defines an average effective NNcross section as [43]

σNN(k) =1

VF1VF2

∫dk1 dk2

2q

kσNN(q)

∫Pauli

dΩ, (42)

where k1 and k2 (the momenta of the two nucleons relativeto their respective nuclei) are smaller than kF1 and kF2,respectively, and the angular integrations extend over allpossible directions of k1 and k2 allowed by Pauli blocking.The total and relative momenta of the two-nucleon pairare given as 2q = k2 +k−k1, and 2P = k1 +k2 +k [43].Figure 5 shows the momenta appearing in eq. (42) andthe geometry of Pauli blocking. Vectors k1 and k2 mustremain smaller than the radii of their respective Fermispheres, and vector 2q can rotate while keeping constantmagnitude (which defines the scattering sphere).

The NN cross section in the integrand of eq. (42) cor-responds to those shown in figs. 3 and 4 (or their free-space counterparts, a choice often encounered in the liter-ature). VF1 and VF2 are the volumes of the two (in generaldifferent) Fermi spheres. A corresponding expression canbe worked out which is suitable for nucleon-nucleus reac-tions [43], involving only one Fermi sphere.

With regard to nucleon-nucleus reactions in particu-lar, it should also be mentioned that the optical modelcan be a powerful tool to constrain single-particle prop-erties in nuclear matter. The microscopic optical model

k k

KKF1

F2

k

12

2

p

2q

Fig. 5. Geometrical representation of Pauli blocking in nucle-us-nucleus collisions.

potential (OMP), which is typically obtained by foldingthe microscopic G-matrix with the nuclear density, canbe compared with the volume term of the empirical op-tical potential obtained from fits to nuclei. (Spin-orbit,surface, and Coulomb terms do not play a role in infinitenuclear matter.) Such study was reported in ref. [44], usingthe BHF approach together with microscopic three-bodyforces. The single-proton and the single-neutron poten-tials in IANM (both elastic and absorptive parts) werecompared to the volume term of the fitted potential asa function of isospin asymmetry. Overall the agreementwas found to be good, although the empirical absorptivepart showed no clear evidence of isospin splitting, mostlikely due to isospin resolution being of the same orderof magnitude as the uncertainty in the fit. We close thisshort detour by noting that, with appropriate folding ofthe nucleon-nucleus OMP, one can build an OMP suitablefor applications in nucleus-nucleus collisions.

Back to our cross sections, equation (42) is what wehave applied in ref. [45] in preparation for applications toion-ion scattering. On the other hand, here our focus is onmedium and medium-induced isospin effects rather thana specific type of reaction. Thus, to demonstrate thoseeffects in a more transparent way, we will adopt a simplerdefinition and calculate the effective NN cross section as

σ(q, P, ρ) =∫

dσ

dΩQ(q, P, θ, ρ)dΩ, (43)

where dσdΩ is the differential cross section obtained from

the G-matrix elements as in eq. (41) and Q signifies thatPauli blocking is applied to the final configurations. Wetake q = P = k/2 and assume that the target nucleons areinitially at rest, as in a typical free-space scattering sce-nario. A connection with physical scattering can be madeconsidering a nucleon bound in a nucleus (or, more ideally,in nuclear matter, as in our case) through the mean field.If such nucleon is struck (for instance, as in a (e, e′) reac-tion), it may subsequently scatter from another nucleon.This is the scattering we are describing.

Page 10 of 14 Eur. Phys. J. A (2014) 50: 22

0

10

20

30

σ pp (

mb

)

0 100 200 300 400 500 600q (MeV)

Fig. 6. (Color online) pp total effective cross section in sym-metric nuclear matter at the same densities as in figs. 3 and 4and as a function of the NN relative momentum. The predic-tions are obtained from eq. (43).

The presence of the Pauli operator in eq. (43) restrictsthe integration domain to

k2F − P 2 − q2

2Pq≤ cos θ ≤ P 2 + q2 − k2

F

2Pq. (44)

The integral becomes zero if the upper limit is negative,whereas the full angular range is allowed if the upper limitis greater than one. Note that the angle θ in eq. (44),namely the angle between the directions of q′ (the relativemomentum after scattering) and P , is also the colatitudeof q′ in a system where the z-axis is along the (conserved)vector P and, thus, it coincides with the scattering angleto be integrated over in eq. (43).

Ignoring Pauli blocking on the final momenta amountsto setting Q = 1 in the integrand above, as done in pre-vious works [27, 28], and results in predictions such asthose shown in figs. 3 and 4. Notice that the cross sec-tions displayed in figs. 3 and 4 and those we will obtainfrom eq. (43) can be dramatically different, nor shouldone expect agreement, as the restriction eq. (44) can com-pletely suppress the cross section is some regions of thedensity-momentum phase space. This mechanism shouldbe included for a realistic calculation of the nucleon meanfree path in nuclear matter, which must approach largevalues as the scattering probabilty goes to zero.

4.4 Effective cross sections in symmetric andasymmetric matter

The effective cross sections shown in this Section are ob-tained from eq. (43). We begin with representative resultsin symmetric nuclear matter (SNM). For this purpose, weneed only to address pp and np cross sections, whereas inIANM we will also need to distinguish between the pp andthe nn cases, as anticipated in sect. 4.2.

pp (or nn) cross sections in SNM

These are shown in fig. 6 as a function of q at selected den-sities. The given range of q corresponds to values of thein-vacuum laboratory kinetic energy up to approximately800MeV. (In passing, we recall that a good quality OBEpotential is able to describe NN elastic scattering up to

0

10

20

30

σ np (

mb

)

0 100 200 300 400 500 600q (MeV)

Fig. 7. (Color online) As in fig. 6, for np scattering.

nearly 1000MeV. Thus, as long as one is not concernedwith pion production reactions, it should be reasonable tocalculate the in-medium cross section from the elastic partof the NN interaction as described by the OBE model.)The densities associated with the chosen Fermi momentaare equal to 0.067 fm−3, 0.148 fm−3, and 0.228 fm−3, re-spectively. Due to the presence of the Pauli operator ineq. (43), it is apparent that the cross sections will becomeidentically zero at certain densities depending on the valueof the momentum. Thus, cross sections calculated withthis mechanism can be quite different, both quantitativelyand qualitatively, from those shown in fig. 3.

Naturally, at the lower momenta there is strong sen-sitivity to any small variation of the Fermi momentumas one is approaching the region where the cross sectionvanishes.

At the densities considered in the figure, the cross sec-tions mostly grow with energy. This is to be expected anddue to the fact that the Pauli operator in eq. (43) becomesless effective at the higher energies. The underlying energydependence displayed in fig. 3, combined with the trendof the cross section (as given in eq. (43)) to increase withenergy due to reduced Pauli blocking, results into a broadlocal maximum which disappears as density increases.

Other mechanisms responsible for the tendency of thecross section to rise at the higher momenta were alreadypointed out in conjunction with fig. 3, and play the samerole in fig. 7.

np cross sections in SNM

These are shown in fig. 7. Similar comments apply asthose made above for the pp case. There is very strongkF -sensitivity at the lowest momenta. As discussed abovefor the pp case, the cross section first grows with energydue to the fact that the Pauli operator in eq. (43) becomesless effective the higher the energy. The np cross section,though, shows a much more pronounced peak structure(as compared to the pp one) at relatively low densities.The peak is washed out as density increases. As discussedpreviously, these structures are the result of the energy de-pendence shown in fig. 4 and the Pauli operator in eq. (43)cutting less of the solid angle at higher momenta.

Summary of pp vs. np in SNM

The comparison of fig. 6 with fig. 7 reveals the isospin de-pendence as discussed in sect. 4.2. Such comparison shows

Eur. Phys. J. A (2014) 50: 22 Page 11 of 14

1

2

3

rati

o

0 100 200 300 400 500 600q (MeV)

Fig. 8. (Color online) Ratio of the np and pp cross sectionsfrom the previous two figures as a function of the momentum.The solid (black) curve shows the free-space values.

610

650

690

730

m*

(MeV

)

0 0.2 0.4 0.6 0.8 1α

p

n

Fig. 9. (Color online) Neutron and proton effective masses inIANM as a function of the neutron excess parameter. The totaldensity is fixed and corresponds to a Fermi momentum equalto 1.3 fm−1.

that the relation between σpp and σnp can be altered con-siderably as a function of density as compared to the vac-uum, because the two cross sections exhibit different struc-tures in the medium, particularly at specific densities andmomenta. This indicates non-trivial isospin dependence inthe way partial waves are impacted by the medium as afunction of momentum, as demonstrated in fig. 8. Therewe display the ratio of the cross sections from fig. 7 andfig. 6, σnp

σpp, along with its value in free space. The behavior

of this ratio as a function of energy and density reflectsthe previously discussed trend of the np cross section to beenhanced in the medium at the lower momenta as well asthe tendency of the pp cross section to grow more rapidlyat high momenta.

pp and nn cross sections in IANM

In IANM, as the neutron population increases, the single-neutron and the single-proton potentials, eqs. (22) and(23), become more repulsive and more attractive, respec-tively. Figure 9 shows how the corresponding effectivemasses change as a function of the neutron excess param-eter for fixed total density As the cross section dependsstrongly on the effective mass, this can be insightful wheninterpreting the predictions.

Predictions for pp and nn total cross sections inisospin-asymmetric matter are shown in figs. 10 and 11,as a function of q and fixed total density and degree ofasymmetry. A smaller and a larger degree of asymmetryare considered in figs. 10 and 11, respectively. Since pp

0

10

20

30

σ (m

b)

0 100 200 300 400 500 600q (MeV)

pp nn

α = 0.2

Fig. 10. pp and nn total effective cross sections in IANMversus the NN relative momentum for fixed total density andasymmetry. kF = 1.3 fm−1.

0

10

20

30

σ (m

b)

0 100 200 300 400 500 600q (MeV)

pp

nn

α = 0.6

Fig. 11. As in fig. 10 for a larger value of α.

0

10

20

30

40

σ (m

b)

0 0.2 0.4 0.6 0.8 1 1.2 1.4kF (fm-1)

pp nn

α = 0.2

Fig. 12. Density dependence of the pp and nn total effectivecross sections in IANM at fixed momentum (q = 200 MeV) andasymmetry.

(nn) scattering is only impacted by the proton (neutron)Fermi momentum, these cross sections are calculated us-ing the proton (or neutron) Fermi momentum in the Paulioperator that would be appropriate for symmetric matterat that density.

Concerning the momentum dependence, similar com-ments apply as those made with regard to fig. 6. Note thatfor α > 0, the nn cross section is (almost always) smallerthan the pp cross section. This is due to the additionalPauli blocking included in eq. (43) and the fact that theneutron’s Fermi momentum is larger than the proton’s.Therefore, for the same total nucleon density, σnn may beentirely suppressed at momenta where σpp is still consid-erably larger than zero. (See, in particular, fig. 11.)

Additional results are displayed in figs. 12 and 13.There, we show some interesting trends of the pp andnn cross sections versus density for fixed momentum andasymmetry. Although at low density the nn and pp crosssections are nearly equal, Pauli blocking soon takes overand clearly separates the pp and nn cases. Particularly

Page 12 of 14 Eur. Phys. J. A (2014) 50: 22

0

50

100

150

200

(m

b)

0 0.2 0.4 0.6 0.8 1 1.2kF (fm-1)

pp

nn

= 0.6

Fig. 13. As in fig. 12 for α = 0.6 and q = 100 MeV.

for large values of α, the pp cross section “survives” largerdensities than the nn one, due to the fact that the protonFermi momentum, kp

F = kF (1 − α)1/3, is much smallerthan the neutron one for large α. The splitting is remark-able at low momenta and for high degree of isospin asym-metry, compare figs. 12 and 13.

np cross sections in IANM

The effective masses of neutron and proton change withincreasing degree of asymmetry as shown in fig. 9, whichclearly impacts the cross section in opposite ways (rela-tive to its value in SNM at the same total density). At thesame time, the neutron and proton Fermi momenta be-come larger and smaller, respectively, which also impactsthe cross section in opposite directions, through Pauliblocking. More precisely, the neutron increasing mass andlarger Fermi momentum compete with the proton decreas-ing mass and lower Fermi momentum. We observed that,due to these competing effects, the overall α dependenceof the np cross section is very weak and can be ignored. Inother words, we find that the np cross section in IANM canbe calculated as in SNM using the average Fermi momen-tum and the average of the neutron and proton effectivemasses.

Summary of observations

We have discussed two levels of isospin dependence of thein-medium NN cross sections. One concerns nucleon pairswith different total isospin, whereas the other refers topairs of identical nucleons with different z-component ofthe isospin. (The explicit α dependence of σnp was foundto be very weak and ignored.)

Interesting differences exist between the momentumand density dependence of σpp and σnp. These can alreadybe seen when comparing figs. 3 and 4, but they becomemore pronounced with the inclusion of the additional Pauliblocking in the final states, see figs. 6 and 7.

With regard to identical nucleons, the region of thedensity/momentum phase space where σnn is nearly or en-tirely suppressed whereas σpp is still considerably differentfrom zero should be a suitable ground to look for a signa-ture of their difference. Figures 11 and 13, compared withfigs. 10 and 12, respectively, suggest that reactions involv-ing low momenta and medium to high densities, together

with a high degree of isospin asymmetry in the collisionregion, has the potential to clearly separate pp and nnscattering and to discriminate between models which door do not distinguish amongst different nucleon pairs. Re-calling the comments made at the end of sect. 4.2, isospindependence is expected to impact neutron and proton ra-pidities in opposite ways. This will increase the isospinasymmetry, α, and, in turn, may generate new isospin de-pendence, including the one demonstrated in figs. 10–13(namely, the one which refers to pairs of identical nucleonswith different values of Tz).

Naturally, isospin-sensitive description of reactions re-quires accurate knowledge of proton and neutron densitiesin the target and projectile, so that the appropriate crosssection, σij , can be applied at each specific point in spacewhere the nuclei have local baryon densities ρi and ρj

(i, j = n, p). Determining neutron densities (through mea-surements of the neutron r.m.s. radius and skin) is partof the many coherent efforts presently going on to con-strain the symmetry energy and related observables. Thuswe close this section underlining the importance of bothempirical constraints and microscopic calculations (whichhave true predictive power) towards a better understand-ing of neutron-rich systems. Microscopic in-medium NNcross sections can play an important role in such endeavor.

5 Mean-free path of protons and neutrons inIANM

We present here a brief discussion of the mean free pathof nucleons in nuclear matter.

A simple and intuitive way to define the mean free path(MFP) in terms of the effective cross sections discussed inthe previous section is

λp =1

ρpσpp + ρnσpn, (45)

with an analogous definition for the neutron,

λn =1

ρnσnn + ρpσnp. (46)

Notice that the above expression can be easily interpretedas the length of the unit volume in a phase space definedby the effective scattering area (the cross section) and thenumber of particles/volume [24].

We begin with discussing the MFP in symmetric mat-ter, in which case λp = λn. This is shown in fig. 14, as afunction of the relative momentum and for different den-sities. Obviously, the mean free path approaches infinitywhen both cross sections in the denominator tend to zero.The higher the density, the higher the energy at which theMFP begins to drop rapidly. The lowest energy for whichλ is finite corresponds to the lowest momentum allowedby Pauli blocking of the final states.

At the lowest density shown in the figure, the MFP dis-plays some fluctuation. This corresponds to similar fluc-tuations in the cross sections, which we discussed earlier

Eur. Phys. J. A (2014) 50: 22 Page 13 of 14

0

4

8

12

16

20

(fm

)

0 100 200 300 400 500 600q (MeV)

Fig. 14. The mean free path of a nucleon as from eq. (45) insymmetric matter as a function of momentum for three differ-ent densities corresponding to Fermi momenta equal 1.0 fm−1

(solid red), 1.3 fm−1 (dashed green), and 1.5 fm−1 (dottedblue).

0

5

10

15

20

(fm

)

0 100 200 300 400 500 600q (MeV)

Fig. 15. The mean free path of a proton (solid red) and aneutron (dashed green) obtained from eqs. (45) and (46) inasymmetric matter at fixed density (kF = 1.3 fm−1) and neu-tron excess parameter (α = 0.2).

0

5

10

15

20

(fm

)

0 100 200 300 400 500 600q (MeV)

Fig. 16. As in the previous figure but for α = 0.6.

in terms of competing mechanisms. Otherwise, the MFPdecreases monotonically. There is considerable density de-pendence.

Figures 15 and 16 show the proton (solid red line) andneutron (dashed green) MFP in asymmetric matter forfixed total density and a low and high degree of asymme-try, respectively. Due to the higher neutron Fermi momen-tum, the nn cross section is more strongly Pauli-blockedat the lower momenta (and thus λn → ∞). As to beexpected, λn ≈ λp at the higher momenta, where (all)

0

1

2

3

4

(fm

)

0 100 200 300 400 500 600q (MeV)

Fig. 17. The mean free path of a nucleon in symmetric mat-ter at fixed density calculated without considerations of Pauliblocking of the final states. The Fermi momentum is equal to1.3 fm−1. See text for more details.

medium effects tend to become less important. Qualita-tively, similar tendencies are observed in figs. 15 and 16,although more pronounced in the latter. Overall, thereare significant differences between a neutron and a pro-ton MFP, which are almost entirely due to the differencesbetween σnn and σpp.

Reconnecting with the previous discussion which fol-lowed eq. (43), we also show, see fig. 17, a sample of MFPcalculations without considerations of Pauli blocking ofthe final states. In this case, λ becomes very small at lowincident energies, due to the large values of the cross sec-tion, which approaches the free-space result, rather thanzero, in that region.

6 Conclusions

We reviewed our microscopic approach to the developmentof the EoS of IANM and, self-consistently, the effectiveinteraction in the isospin-asymmetric medium. Withinthe DBHF method, the interactions of nucleons with themedium are espressed as self-energy corrections to the nu-cleon propagator. That is, the nucleons are regarded as“dressed” quasiparticles. Relativistic effects lead to an in-trinsically density-dependent interaction which is approx-imately consistent with the contribution from the three-body force arising from virtual pair excitations.

The focal point has been how the presence of a densehadronic medium impacts the scattering amplitude andthus the cross section, with particular attention to the casewhere neutron and proton concentrations are different. Tothat end, we presented microscopic calculations of totalelastic cross sections for scattering of nucleons in sym-metric and neutron-rich matter. Our predictions includeall “conventional” medium effects as well as those associ-ated with the nucleon Dirac wavefunction. Pauli blockingof the final states is included in the integration of the dif-ferential cross section.

One of the mechanisms driving the in-medium crosssections are the neutron and proton effective masses. Inturn, these are determined by the potentials experiencedby the neutron and proton in asymmetric matter, which

Page 14 of 14 Eur. Phys. J. A (2014) 50: 22

are part of the calculation leading to the EoS of IANM.Thus, in our philosophy, medium effects originating fromthe equation of state are consistently incorporated in themean field and the NN cross sections.

First, we discussed the basic density/momentum de-pendence of pp and np cross sections in symmetric mat-ter. Although they generally exhibit qualitatively similarbehavior with changing energy and density, pp and npeffective cross sections show some interesting differencesin specific regions of the phase space. This gives rise toisospin dependence.

With regard to identical nucleons, the sensitivity to theasymmetry in neutron and proton concentrations comes inthrough the combined effect of Pauli blocking and chang-ing effective masses. The lowering (rising) of the proton(neutron) Fermi momentum and the reduced (increased)proton (neutron) effective mass tend to move the cross sec-tion in opposite directions. With Pauli blocking applied tointermediate and final states, the final balance is that thenn effective cross section is more strongly suppressed.

In summary, sensitivity to the asymmetry is non-negligible for scattering of different pairs of identical nu-cleons, and clearly separates pp and nn scatterings. Thedegree of sensitivity depends on the region of the energy-density-asymmetry phase space under consideration.

We also considered the mean free path of a nucleonand determined that it is affected in a significant way bythe presence of isospin asymmetry in the medium.

In-medium two-body collisions are only part of the in-put needed for reaction calculations. Therefore, in closing,we reiterate the importance of coherent effort from theoryand experiment as well as the importance of calculationswith predictive power towards improved understanding ofneutron-rich systems, both reactions and structure.

Finally, we cannot stress enough that the behavior ofmicroscopic in-medium cross sections can be rather com-plex being the result of several, often competing, mecha-nisms. Therefore, microscopic predictions do not appearto validate a simple phenomenological ansatz, such as theeffective mass scaling model in eq. (39).

Support from the U.S. Department of Energy under Grant No.DE-FG02-03ER41270 is acknowledged. I am grateful to C.A.Bertulani for insightful discussions.

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