Approximation for nuclear densities to be used in multiple scattering problems

Abstract

We have investigated the possibility of describing nuclear one-body densities as a sum of step functions.The results show that a relatively limited number of bins is sufficient to fit the form factor up to 2-3 fm . Such approximate nuclear densities are useful to calculate various contributions in multiple scattering problems; two examples are presented.In the multiple scattering description of nuclear reactions at intermediate energies, situations often occur in which second and higher order corrections have to be calculated with some care. On the other hand, their effects on the main contributions are usually small and, consequently, they hardly justify sophisticated and time consuming computing.One solution would be to use one-body nuclear densities functions simple enough to carry out integrations analytically, at least partly, while preserving the main characteristics of the densities. Several simple analytical forms are known which lead to valuable qualitative results, but their domains of validity can be quite restricted. For example, the Gaussian function clearly possesses many useful properties but reproduces badly the nuclear densities of all but the very light nuclei. A two-parameter Fermi function does much better in this respect but requires, in general, numerical integrations or approximations in the evaluation of, for example, its Fourier transform or its moments. Similar difficulties are encountered when calculating scattering amplitudes in the impact parameter representation. The projection of the spherical symmetry into a cylindrical one is easily performed only in the case of variable separability.Consequently, in many problems it is desirable to meet a balance, keeping the computing to a reasonable level without losing too much information concerning the nuclear densities.Many analytic forms exist, ' which could represent nuclear densities and be used in the kind of problems we want to tackle, but all of them have the undesirable feature of leading immediately to numerical integrations.One attractive method consists of decomposing the nuclear density as a weighted sum of simple functions taken at different positions. This procedure has been widely used to analyze electron scattering data. Two famous examples are given by the sum of 5 functions and the sum of Gaussians introduced by Friedrich and Lenz and by Sick, respectively. However, it has been noticed that, whereas such approximations are very efficient in reproducing form factors, they lose their simplicity when multiple scattering integrals are handled.The purpose of the present work is to investigate the possibilities of a sum of Heaviside step functions: N p(r ) = p,p(r ) = g a"0(R"-r )(1) As a case of practical interest, we consider a heavy spherical nucleus with A =200 particles, assuming the true p(r) to be given by a two-parameter Fermi function r -Ro p(r ) = po 1+exp with R()=7 fm and a =0.5 fm, the constant po being fixed by the normalization condition to the particle number.We shall first discuss the fitting procedure, namely, t...

Phase variation of the nucleon-nucleon amplitudes and non-eikonal propagation

“…The potential V ( r , 0 ) is then introduced in a Schrodringer-type equation where w = (p'+ m2)2 is the total energy in the CM system. The particular choice made for f ( q ) (equation (3)) gives rise to a potential which has a spin-independent and a spin-orbit part. For proton scattering on a spin-zero nucleus; the corresponding scattering amplitude has two components, as is well known F ( q ) = F , ( q ) + m.fiF,(q).…”

“…%b ko In order to deal with a semi-realistic example, the parameter values were chosen in a way to simulate proton-'He elastic scattering at 1 GeV incident energy [3].…”

“…The slopes' parameters are chosen to be pc = 0.25 fmz and p, = 0.5 fm'. These values, as well as the coefficients of the potential, are derived from an average between proton-proton and proton-neutron amplitudes [3].…”

“…Their studies were dedicated to differential cross sections. Furthermore, it has been shown that spin observables are sensitive to this phase variation [3], and that it was essential to achieve agreement with experimental data in protow4He elastic scattering at 800MeV 141.…”

Study of Different Forms of Density Distributions in Proton–Nucleus Total Reaction Cross Section and the Effect of Phase in NN Amplitude

“…Franco and Yin [31] had earlier introduced the phase factor exp − γ nn q 2 2 in f NN (q) by defining a nn = β nn + iγ nn . Studies show that by treating γ nn as a free parameter, good agreement between theoretical and experimental cross sections could be achieved [31][32][33]. Recently, considering three different models for the interaction potential, Dedonder et al [34] have shown that the phase of the scattering amplitude is a sensitive function of the relative size of the radii of the real and imaginary parts of potential.…”