Neutron single-particle strength in silicon isotopes: Constraining the driving forces of shell evolution

Shell evolution is studied in the neutron-rich silicon isotopes ${}^{36,38,40}\mathrm{Si}$ using neutron single-particle strengths deduced from one-neutron knockout reactions. Configurations involving neutron excitations across the $N=20$ and $N=28$ shell gaps are quantified experimentally in these rare isotopes. Comparisons with shell model calculations show that the tensor force, understood to drive the collective behavior in ${}^{42}\mathrm{Si}$ with $N=28$, is already important in determining the structure of ${}^{40}\mathrm{Si}$ with $N=26$. New data relating to cross-shell excitations provide the first quantitative support for repulsive contributions to the cross-shell $T=1$ interaction arising from three-nucleon forces.

Figure 1

Experimental level schemes for low-lying states in ${}^{35}\mathrm{Si}$, ${}^{37}\mathrm{Si}$, and ${}^{39}\mathrm{Si}$, compared with shell model calculations using the SDPF-MU effective interaction [10].

Figure 2

Momentum distribution of the ${}^{35}\mathrm{Si}$ residues produced in the ground state and any isomers following one neutron knockout from ${}^{36}\mathrm{Si}$. The curves show a fit using a linear combination of the calculated distributions for removal from the $f_{7/2}$ and $d_{3/2}$ orbits.

Figure 3

(a) Maximum likelihood fit to the lifetime of the state in ${}^{39}\mathrm{Si}$ decaying by a 172-keV $\gamma$ ray. Only statistical uncertainties are shown. The lower panel shows the influence of the uncertainty in lifetime upon the uncertainty in the number of decays, where the error band shows the uncertainty in the fit. (b) The resulting spectra in the front, middle, and back rings of GRETINA using the best-fit lifetime. The shaded gray area indicates the continuous background from target breakup, the blue (light gray) area indicates the Compton continuum of all higher-lying peaks, and the magenta (dark gray) curve indicates the total fit [10].

Figure 4

One-neutron knockout reaction partial cross sections to bound final states with $J^{\pi }=7/2^{-}$ and $3/2^{-}$, as a function of the neutron number of the silicon projectile. The arrow, for $N=26$, indicates a lower limit. Theoretical predictions using the SDPF-MU (solid lines) and SDPF-U (dashed lines) effective interactions are also shown. The dotted line (SDPF-MU-NT) results when the tensor component of the cross-shell $sd-fp$ interaction of SDPF-MU is set to zero. All theoretical curves have been scaled by the observed knockout reaction systematics (see text).

Figure 5

Energies of $3/2^{+}$ and $1/2^{+}$ intruder states in the silicon isotopes as a function of mass number. The points indicate experimental data, while the lines connect theoretical calculations. The vertical bars show the shell-model spectroscopic factors of the states.

Figure 6

Comparison of selected $T=1$ monopoles terms of the SDPF-U and SDPF-MU effective interactions, evaluated at $A=42$.