Density dependence of in-medium nucleon-nucleon cross sections

The lowest-order correction of the density dependence of in-medium nucleonnucleon cross sections is obtained from geometrical considerations of Pauliblocking effects. As a by-product, it is shown that the medium corrections imply an 1/E energy dependence of the density-dependent term. PACS: 25.70.-z,25.75.Ld The nucleon-nucleon cross section is a fundamental input in theoretical calculations of nucleus-nucleus collisions at intermediate and high energies (E/A > ∼ 100 MeV). One expects to obtain information about the nuclear equation of state by studying global collective variables in such collisions (see, e.g., [1]). Transport equations, like the BUU equation, are often used as tools for the analysis of experimental data and as a bridge to the information about the equation of state (see, e.g., [2]). The nucleon-nucleon cross sections are building blocks in these transport equations. In previous theoretical studies of heavy ion collisions at intermediate energies (E/A ≃ 100 MeV) the nucleon-nucleon cross section was multiplied with a constant scaling factor to account for in-medium corrections [3,4]. As pointed out in ref. [2], this approach fails in low density nuclear matter where the in-medium cross section should approach its free-space value. A more realistic approach uses a Taylor expansion of the in-medium cross section in the density variable. One gets [5] 1 σNN = σ free NN (1 + αρ̄) , (1) where ρ̄ = ρ/ρ0, ρ0 is the normal nuclear density, and α is the logarithmic derivative of the in-medium cross section with respect to the density, taken at ρ = 0, α = ρ0 ∂ ∂ρ (ln σNN) |ρ=0 . (2) This parameterization is motivated by Brückner G-matrix theory and is basically due to Pauli-blocking of the cross section for collisions at intermediate energies [6]. Values of α between −0.4 and −0.2 yield the best agreement with involved G-matrix calculations using realistic nucleon-nucleon interactions [6]. In this article we give a simple and transparent derivation of the lowest order expansion of the in-medium nucleon-nucleon cross section in terms of the nucleon density. In our approach the leading term of the expansion comes out as αρ with α proportional to 1/E. This energy-dependence agrees with experimental results on total nucleus-nucleus cross sections. We adopt the idea that the main effect of medium corrections is due to the Pauliblocking of nucleon-nucleon scattering. Pauli-blocking prevents the nucleons to scatter into final occupied states in binary collisions between the projectile and target nucleons. This is best seen in momentum space, as shown in figure 1. We see that energy and momentum conservation, together with the Pauli principle, restrict the collision phase space to a complex geometry involving the Fermi-spheres and the scattering sphere. In this scenario, the inmedium cross section corrected by Pauli-blocking can be defined as σNN (k,KF1, KF2) = ∫ d3k1d 3k2 (4πK F1/3)(4πK 3 F2/3) 2q k σ NN (q) ΩPauli 4π , (3) where k is the relative momentum per nucleon of the nucleus-nucleus collision (see figure 1), and σ NN (q) is the free nucleon-nucleon cross section for the relative momentum 2q = k1 −k2 −k, of a given pair of colliding nucleons. Clearly, Pauli-blocking enters through the restriction that |k′1| and |k ′ 2| lie outside the Fermi spheres. From energy and momentum conservation in the collision, q is a vector which can only rotate around a circle with center at p = (k1 −k2 −k)/2. These conditions yield an allowed scattering solid angle given by [7] 2 ΩPauli = 4π − 2(Ωa + Ωb − Ω̄) , (4) where Ωa and Ωb specify the excluded solid angles for each nucleon, and Ω̄ represents the intersection angle of Ωa and Ωb (see figure 1). The solid angles Ωa and Ωb are easily determined. They are given by Ωa = 2π(1− cos θa) , Ωb = 2π(1− cos θb) , (5) where q and p were defined above, b = k − p, and cos θa = (p 2 + q −K F1)/2pq , cos θb = (p 2 + q −K F2)/2pq , (6) The evaluation of Ω̄ is tedious but can be done analytically. The full calculation was done in ref. [7] and the results have been reproduced in the appendix of ref. [8]. To summarize, there are two possibilities: (1) Ω̄ = Ωi(θ, θa, θb) + Ωi(π − θ, θa, θb) , for θ + θa + θb > π (7) (2) Ω̄ = Ωi(θ, θa, θb) , for θ + θa + θb ≤ π , (8) where θ is given by cos θ = (k − p − b)/2pb . (9) The solid angle Ωi has the following values (a) Ωi = 0 , for θ ≥ θa + θb (10) (b) Ωi = 2 [