A separable potential formalism is used to describe the ππ and KK interactions in the I G (J P C ) = 0 + (0 ++ ) states in the energy range from the ππ threshold up to 1.4 GeV. Introduction of relativistic propagators into a system of Lippmann-Schwinger equations leads to a very good description of the data (χ 2 = 0.93 per one degree of freedom). Three poles are found in this energy region: f 0 (500) (M = 506 ± 10 MeV, Γ = 494 ± 5 MeV), f 0 (975) (M = 973±2 MeV, Γ = 29±2 MeV) and f 0 (1400) (M = 1430±5 MeV, Γ = 145 ± 25 MeV). The f 0 (975) state can be interpreted as a KK bound state. The f 0 (500) state may be associated with the often postulated very broad scalar resonance under the KK threshold (sometimes called σ or ǫ meson). The scattering lengths in the ππ and KK channels have also been obtained. The relativistic approach provides qualitatively new results * Unité de Recherche des Universités Paris 11 et Paris 6 associée au CNRS

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Recent results concerning the interaction radius of very neutron-rich He and Li isotopes renewed the interest in measuring the size of a nucleus by means of a strong interacting probe. We stress that various ratios of π+ and π- production can bring valuable information in this respect.

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...