  The $^{4}$He nucleus is a well bound and highly correlated four-nucleon
system that is also found in form of $\alpha$-clusters as substructure in
nuclear many-body systems. The standard single-particle shell model cannot
represent these four-body correlations. It is highly desirable to find a simple
way to identify and distinguish these fundamental structures without
large-scale computations. In this paper, we investigate with intrinsic Slater
determinants as trial states in how far these two competing pictures prevail in
the ground states of $^{12}$C and $^{16}$O. The trial states can describe both
the $j$-$j$ coupling shell-model and $\alpha$-cluster configurations in a
unified way. One-body density distributions of the trial states are calculated
and compared to elastic scattering cross section data. The parameters of the
trial state are chosen to reproduce the experimental charge radii. A well
developed cluster structure is characterized by an enhancement of the
differential elastic scattering cross sections, as well as the elastic charge
form factors around the first peak positions. The density profiles of the
internal regions can also be probed at higher momentum transfer regions beyond
the second minimum. With these trial states one can visualize the competition
between cluster and shell structure in an intuitive and simple way.
