Single-particle structure of silicon isotopes approaching ${}^{42}\mathrm{Si}$

The structure of the neutron-rich silicon isotopes ${}^{36,38,40}\mathrm{Si}$ was studied by one-neutron and one-proton knockout reactions at intermediate beam energies. We construct level schemes for the knockout residues ${}^{35,37,39}\mathrm{Si}$ and ${}^{35,37,39}\mathrm{Al}$ and compare knockout cross sections to the predictions of an eikonal model in conjunction with large-scale shell-model calculations. The agreement of these calculations with the present experiment lends support to the microscopic explanation of the enhanced collectivity in the region of ${}^{42}\mathrm{Si}$. We also present an empirical method for reproducing the observed low-momentum tails in the parallel momentum distributions of knockout residues.

Figure 1

Doppler-reconstructed $\gamma$-ray spectrum detected in coincidence with ${}^{35}\mathrm{Si}$. The 908-keV transition is broadened by a lifetime effect of approximately 80 ps. The inset figures show background-subtracted projections of the $\gamma \gamma$ coincidence matrix, gated on the 908- and 1134-keV transitions. The Doppler reconstruction was performed with a velocity $v/c=0.43$.

Figure 2

Parallel momentum distributions for the population of levels at 1688, 2042, 2164, and 2377 keV in ${}^{35}\mathrm{Si}$ by neutron knockout from ${}^{36}\mathrm{Si}$. The distribution in (b) has been scaled by a factor of 6.

Figure 3

Proposed level scheme for ${}^{35}\mathrm{Si}$ from this work compared with shell-model calculations (see text for details). The widths of the arrows are proportional to the efficiency-corrected $\gamma$-ray intensity.

Figure 4

The left panel shows the 3611-keV $\gamma$-ray peak (indicated by an arrow) detected in coincidence with ${}^{35}\mathrm{Si}$, with an inset showing the gates used on the outgoing parallel momentum distribution. The right panel shows the $\gamma$-ray spectra gated on the main peak in the momentum distribution (blue hatches) and the low-momentum tail (solid red). The 3611-keV peak is clearly associated with the tail of the distribution.

Figure 5

Doppler-reconstructed $\gamma$-ray spectrum detected in coincidence with ${}^{35}\mathrm{Al}$. The section of the spectrum in the box labeled “$\times 4$” has been rebinned by a factor 4. The blue dashed line shows the fitted background, suggesting a peak at 4275 keV. The inset in the upper-right shows background-subtracted projections of the $\gamma \gamma$ coincidence matrix, gated on the 1003- and 2237-keV transitions. The Doppler reconstruction was performed with a velocity $v/c=0.42$.

Figure 6

Parallel momentum distributions for the population of levels at 1972, 3243, and 4275 keV in ${}^{35}\mathrm{Al}$ by proton knockout from ${}^{36}\mathrm{Si}$. The distribution in (c) has been scaled by a factor of 2.

Figure 7

Proposed level scheme for ${}^{35}\mathrm{Al}$ from this work compared with shell-model calculations (see text for details). The width of the arrows is proportional to the efficiency-corrected $\gamma$-ray intensity.

Figure 8

Doppler-reconstructed $\gamma$-ray spectrum detected in coincidence with ${}^{37}\mathrm{Si}$. The section of the spectrum in the box labeled “$\times 2$” has been rebinned by a factor of 2. The inset shows the background-subtracted projections of the $\gamma \gamma$ coincidence matrix, gated on the peaks at 156 and 903 keV. The Doppler reconstruction was performed with a velocity $v/c=0.40$.

Figure 9

Parallel momentum distributions for the population of the ground state and excited states at 692 and 717 keV in ${}^{37}\mathrm{Si}$ by neutron knockout from ${}^{38}\mathrm{Si}$. Note that the ground-state distribution includes both the $5/2_1^{-}$ and $7/2_1^{-}$ states. The distributions in (b) and (c) have been scaled by factors of 5 and 4, respectively.

Figure 10

Level scheme for ${}^{37}\mathrm{Si}$ from this work compared with shell-model calculations (see text for details). The width of the arrows is proportional to the $\gamma$-ray intensity. Fine dashed lines indicate tentative levels and transitions, while the thicker dashed line labeled $S_n$ indicates the neutron separation energy, with the gray-shaded area indicating the uncertainty.

Figure 11

Doppler-reconstructed $\gamma$-ray spectrum detected in coincidence with ${}^{37}\mathrm{Al}$. The section of the spectrum in the box labeled “$\times 2$” has been scaled by a factor of 2. The inset shows the background-subtracted projection of the $\gamma \gamma$ coincidence matrix, gated on the peak at 775 keV. The Doppler reconstruction was performed with a velocity $v/c=0.40$.

Figure 12

Proposed level scheme for ${}^{37}\mathrm{Al}$ from this work compared with shell-model calculations (see text for details). The width of the arrows is proportional to the efficiency-corrected $\gamma$-ray intensity.

Figure 13

Doppler-reconstructed $\gamma$-ray spectrum detected in coincidence with ${}^{39}\mathrm{Si}$. The Doppler reconstruction was performed with a velocity $v/c=0.38$. The inset shows the background-subtracted $\gamma \gamma$ coincidence matrix gated on the peaks at 172 and 879 keV, revealing no strong coincidences with either peak.

Figure 14

Parallel momentum distributions gated on $\gamma$-ray transitions in ${}^{39}\mathrm{Si}$ (no feeding subtraction). The distributions in (b) and (d) have been scaled by factors of 2 and 3, respectively.

Figure 15

The left side shows the proposed level scheme for ${}^{39}\mathrm{Si}$. The right side is the level scheme proposed by Sohler et al. [39]. In the left level scheme, the width of the arrows in proportional to the efficiency-corrected $\gamma$-ray intensity. As described in the text, the levels could have an offset if the lowest $3/2^{-}$ state lies below the $5/2^{-}$.

Figure 16

Doppler-reconstructed $\gamma$-ray spectrum detected in coincidence with ${}^{39}\mathrm{Al}$. The Doppler reconstruction was performed with a velocity $v/c=0.39$.

Figure 17

Proposed level in ${}^{39}\mathrm{Al}$ at 800 keV with dashed lines indicating the possible locations of other levels, assuming ground-state transitions. The center and right figures are shell-model calculations (see text for details).

Figure 18

Inclusive one-neutron and one-proton knockout cross sections from silicon isotopes to all bound final states as a function of the neutron number $N$ of the projectile. The measured values (black points) are compared with theoretical predictions (connected by the solid red line) and with these theoretical predictions scaled by a reduction factor $R\left(\Delta S\right)$ deduced from knockout systematics [30] and ranging from 0.83 to 0.91 for neutron knockout and from 0.31 to 0.39 for proton knockout (see Figs. 19, 20, 21, 22, 23, 24). The theoretical error bars indicate the sensitivity when varying the neutron separation energy by $\pm 500$ keV. The experimental value for ${}^{34}\mathrm{Si}$ is taken from Ref. [40].

Figure 19

One proton knockout cross section to final states in ${}^{35}\mathrm{Al}$. Experimental data are shown in the top panel, and theoretical predictions are shown in the bottom two panels.

Figure 20

One proton knockout cross section to final states in ${}^{37}\mathrm{Al}$. Experimental data are shown in the top panel, and theoretical predictions are shown in the bottom two panels.

Figure 21

One proton knockout cross section to final states in ${}^{39}\mathrm{Al}$. Experimental data are shown in the top panel, and theoretical predictions are shown in the bottom two panels.

Figure 22

One neutron knockout cross section to final states in ${}^{35}\mathrm{Si}$. Experimental data are shown in the top panel, and theoretical predictions are shown in the bottom two panels.

Figure 23

One neutron knockout cross section to final states in ${}^{37}\mathrm{Si}$. Experimental data are shown in the top panel, and theoretical predictions are shown in the bottom two panels.

Figure 24

One neutron knockout cross section to final states in ${}^{39}\mathrm{Si}$. Experimental data are shown in the top panel, and theoretical predictions are shown in the bottom two panels.

Figure 25

Experimental parallel momentum distributions for the indicated final states of selected knockout residues. The (symmetric) eikonal momentum distributions after folding with (i) the incoming momentum distribution (dashed blue lines) and (ii) the inelastic-scattering momentum distribution (solid red lines), as described in the text, are shown, all scaled to the total number of measured counts.

Figure 26

(a) The number of ${}^{36}\mathrm{Si}$ ions detected in the S800 after impinging the ${}^{36}\mathrm{Si}$ secondary beam on a beryllium target, as a function of their longitudinal momenta. (b) $\gamma$ Rays detected in coincidence with the outgoing ${}^{36}\mathrm{Si}$ fragments, gated on low-momentum (red filled) and high-momentum (blue hatches) regions of the distribution of panel (a). (c) The unreacted momentum distribution (no $\gamma$ rays detected) is shown in blue hatches and the momentum distribution in coincidence with a $\gamma$ ray with $E_{\gamma }>1$ MeV (scaled for comparison) is shown by the solid red filled area.