Experimental Study of the Two-Body Spin-Orbit Force in Nuclei

Energies and spectroscopic factors of the first $7/2^{-}$, $3/2^{-}$, $1/2^{-}$, and $5/2^{-}$ states in the ${}^{35}{\mathrm{Si}}_{21}$ 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 ${}^{35}\mathrm{Si}$ and ${}^{37}\mathrm{S}$ isotones, a reduction of the $p_{3/2}\text{−}{p}_{1/2}$ spin-orbit splitting by about 25% is proposed, while the $f_{7/2}\text{−}{f}_{5/2}$ 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 two-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.

Figure 1

Top: excitation energy spectrum $E^{*}$ of ${}^{35}\mathrm{Si}$ obtained with the detection of protons in the angular domain 106°–115° in the laboratory. The fitting procedure of the peaks, as well as the origin of the asymmetric black curve, are described in the text. Bottom: Doppler-corrected $\gamma$ energy spectrum shows two peaks at 0.910 MeV and 1.134MeV. The latter peak disappears when gating on excitation energy $E^{*}>1.4\mathrm{MeV}$ (red shaded area), indicating that it comes from a level at 2044 keV.

Figure 2

Proton angular distributions of the states at $E^{*}=0,910$, 2044, and 5500 keV in ${}^{35}\mathrm{Si}$. The curves correspond to ADWA calculations assuming transfer to $\ell =1$ (dashed-dotted line) or $\ell =3$ (solid line) states.

Figure 3

Distribution of the major fragments of the single particle strength in ${}^{41}\mathrm{Ca}$ (top), ${}^{37}\mathrm{S}$ (middle), and in ${}^{35}\mathrm{Si}$ (bottom). SF values in ${}^{41}\mathrm{Ca}$ are taken from Ref. [29]. The centroid of the $5/2{}^{-}$ strength, obtained from a summed SF strength of 0.32, is indicated as $⟨f_{5/2}⟩$. The SF of the $5/2{}^{-}$ components in ${}^{37}\mathrm{S}$ are taken from [24], while all others SF are derived from the present work with error bars due to statistics and fit distributions.