  We study the variation of the mean cross section with the density of the
samples in the quantum scattering of a particle by a disordered target. The
target consists of a set of pointlike scatterers, each having an equal
probability of being anywhere inside a sphere whose radius may be modified. We
first prove that scattering by a pointlike scatterer is characterized by a
single phase shift ${\delta}$ which takes on its values in $]0 \, , {\pi}[$ and
that the scattering by ${\rm N}$ pointlike scatterers is described by a system
of only ${\rm N}$ equations. We then show with the help of numerical
calculations that there are two stages in the variation of the mean cross
section as the density of the samples (the radius of the target) increases
(decreases). Depending on the value of ${\delta}$, the mean cross section first
either increases or decreases, each one of the two behaviours being originated
by double scattering; it decreases uniformly for any value of ${\delta}$ as the
density increases further on, a behaviour which results from multiple
scattering and which follows that of the cross section for diffusion by a hard
sphere potential of decreasing radius. The expression of the mean cross section
is derived in the particular case of an unlimited number of contributions of
successive scatterings.
