  Energies and spectroscopic factors of the first $7/2^-$, $3/2^-$, $1/2^-$ and
$5/2^-$ states in the $^{35}$Si$_{21}$ nucleus were determined by means of the
(d,p) transfer reaction in inverse kinematics at GANIL using the MUST2 and
EXOGAM detectors. By comparing the spectroscopic information on the $^{35}$Si
and $^{37}$S isotones, a reduction of the $p_{3/2} - p_{1/2}$ spin-orbit
splitting by about 25% is proposed, while the $f_{7/2} -f_{5/2}$ spin-orbit
splitting seems to remain constant. These features, derived after having
unfolded nuclear correlations using shell model calculations, have been
attributed to the properties of the 2-body spin-orbit interaction, the
amplitude of which is derived for the first time in an atomic nucleus. The
present results, remarkably well reproduced by using several realistic
nucleon-nucleon forces, provide a unique touchstone for the modeling of the
spin-orbit interaction in atomic nuclei.
