High-resolution measurement of the ${}^{118,124}\mathrm{Sn}$($p$,$t$)${}^{116,122}\mathrm{Sn}$ reactions: Shell-model and microscopic distorted-wave Born approximation calculations

The ${}^{118,124}\mathrm{Sn}$($p$,$t$)${}^{116,122}\mathrm{Sn}$ reactions have been investigated in high-resolution experiments at incident proton energies of 24.6 and 25 MeV, respectively. Angular distributions for 55 transitions to levels of ${}^{116}\mathrm{Sn}$ and 63 transitions to levels of ${}^{122}\mathrm{Sn}$, up to excitation energies of $~3.850$ and $~4.000$ MeV, respectively, have been measured. The spin and parity identification was carried out by means of a distorted-wave Born approximation (DWBA) analysis, performed by using conventional Woods-Saxon potentials. A shell-model study of ${}^{116}\mathrm{Sn}$ and ${}^{122}\mathrm{Sn}$ nuclei was performed using a realistic two-body effective interaction derived from the CD-Bonn nucleon-nucleon potential. The doubly magic nucleus ${}^{132}\mathrm{Sn}$ was assumed as a closed core, with the 16 and 10 valence neutron holes occupying the five levels of the 50-82 shell. The energy spectra have been calculated and compared with the experimental ones, and the theoretical two-nucleon spectroscopic amplitudes, evaluated in a truncated seniority space, have been used in the microscopic DWBA calculation of some cross-section angular distributions of both reactions.

Figure 1

Position spectrum of tritons measured at $\theta$ $=$ ${20}^{°}$ for the ${}^{118}\mathrm{Sn}$($p$,$t$)${}^{116}\mathrm{Sn}$ reaction, and at $\theta$ $=$ ${15}^{°}$ for the ${}^{124}\mathrm{Sn}$($p$,$t$)${}^{122}\mathrm{Sn}$ reaction. Some levels are labeled with their excitation energy in MeV.

Figure 2

Differential cross sections for the excitation of $0^{+}$ states by the ${}^{118}\mathrm{Sn}$($p$,$t$)${}^{116}\mathrm{Sn}$ reaction. The dots represent the experimental data, and the solid lines the theoretical estimates obtained with semimicroscopic DWBA calculations. The energies attributed to the observed levels are those given in the present work.

Figure 3

Differential cross sections for the excitation of $2^{+}$ states by the ${}^{118}\mathrm{Sn}$($p$,$t$)${}^{116}\mathrm{Sn}$ reaction. See Fig. 2 for details.

Figure 4

Differential cross sections for the excitation of $4^{+}$ states by the ${}^{118}\mathrm{Sn}$($p$,$t$)${}^{116}\mathrm{Sn}$ reaction. See Fig. 2 for details.

Figure 5

Differential cross sections for the excitation of $6^{+}$ states and three doublets at 3.453, 3.549, and 3.805 MeV by the ${}^{118}\mathrm{Sn}$($p$,$t$)${}^{116}\mathrm{Sn}$ reaction. See Fig. 2 for details.

Figure 6

Differential cross sections for the excitation of $1^{-}$ and $3^{-}$ states by the ${}^{118}\mathrm{Sn}$($p$,$t$)${}^{116}\mathrm{Sn}$ reaction. See Fig. 2 for details.

Figure 7

Differential cross sections for the excitation of $5^{-}$, $7^{-}$, and $9^{-}$ states by the ${}^{118}\mathrm{Sn}$($p$,$t$)${}^{116}\mathrm{Sn}$ reaction. See Fig. 2 for details.

Figure 8

Differential cross sections for the excitation of $0^{+}$ states by the ${}^{124}\mathrm{Sn}$($p$,$t$)${}^{122}\mathrm{Sn}$ reaction. See Fig. 2 for details.

Figure 9

Differential cross sections for the excitation of $2^{+}$ states by the ${}^{124}\mathrm{Sn}$($p$,$t$)${}^{122}\mathrm{Sn}$ reaction. See fig. 2 for details.

Figure 10

Differential cross sections for the excitation of $4^{+}$ states by the ${}^{124}\mathrm{Sn}$($p$,$t$)${}^{122}\mathrm{Sn}$ reaction. See Fig. 2 for details.

Figure 11

Differential cross sections for the excitation of $6^{+}$ states and two doublets at 2.654 MeV, and 3.704 MeV by the ${}^{124}\mathrm{Sn}$($p$,$t$)${}^{122}\mathrm{Sn}$ reaction. See Fig. 2 for details.

Figure 12

Differential cross sections for the excitation of $1^{-}$ and $3^{-}$ states by the ${}^{124}\mathrm{Sn}$($p$,$t$)${}^{122}\mathrm{Sn}$ reaction. See Fig. 2 for details.

Figure 13

Differential cross sections for the excitation of $5^{-}$, and $7^{-}$ states by the ${}^{124}\mathrm{Sn}$($p$,$t$)${}^{122}\mathrm{Sn}$ reaction. See Fig. 2 for details.

Figure 14

Calculated differential cross sections for the ${}^{118}\mathrm{Sn}$($p$,$t$)${}^{116}\mathrm{Sn}$ reaction for transitions to a hypothetical level of ${}^{116}\mathrm{Sn}$ with an excitation energy of 3 MeV and given $L$ transfer: (a) even $L$ transfers, (b) odd $L$ transfers. The cluster DWBA calculations have been made using the set of optical model parameters of Table .

Figure 15

Comparison of measured differential cross sections (dots) with calculations based on assumed $L$ transfers, relative to the 3.493-MeV level populated in the ${}^{118}\mathrm{Sn}$($p$,$t$)${}^{116}\mathrm{Sn}$ reaction. Solid line corresponds to $7^{-}$; dashed line to $8^{+}$.

Figure 16

Comparison of measured differential cross sections (dots) with calculations based on mixtures of assumed $L$ transfers, relative to the 3.549-MeV transition populated in the ${}^{118}\mathrm{Sn}$($p$,$t$)${}^{116}\mathrm{Sn}$ reaction. Solid line corresponds to $30\%$ $8^{+}$, $70\%$ $3^{-}$; dashed line to $30\%$ $9^{-}$, $70\%$ $3^{-}$.

Figure 17

(a) Measured differential cross sections for the population of the ground states by the ${}^{118}\mathrm{Sn}$($p$,$t$)${}^{116}\mathrm{Sn}$ and ${}^{124}\mathrm{Sn}$($p$,$t$)${}^{122}\mathrm{Sn}$ reactions. The error bars associated with the points are smaller than the symbols used to represent the points. (b) Microscopic calculations of the differential cross sections for the population of the ground states by the ${}^{118}\mathrm{Sn}$($p$,$t$)${}^{116}\mathrm{Sn}$ and ${}^{124}\mathrm{Sn}$($p$,$t$)${}^{122}\mathrm{Sn}$ reactions. The same arbitrary cross-section units are used for both reactions.

Figure 18

Comparison between experimental and calculated cross sections for $0^{+}$ and $6^{+}$ final states of ${}^{116}\mathrm{Sn}$. The lines represent results of the microscopic calculations of the differential cross sections (in $\mu$b/sr) as a function of center-of-mass angle (in degrees). The subscripts $1,2,...$ associated with the lines indicate the calculated energy ranking of the corresponding state, with 1 representing the lowest state of given $J$${\hspace{0.5em}}^{\pi }$. The error bars associated with the points are smaller than the symbols used to represent the points.

Figure 19

Comparison between experimental and calculated cross sections for $2^{+}$ final states of ${}^{116}\mathrm{Sn}$. Note that the data points correspond only to clearly resolved states. The doublets are not represented in the figure. See Fig. 18 for details.

Figure 20

Comparison between experimental and calculated cross sections for $4^{+}$ final states of ${}^{116}\mathrm{Sn}$. Note that the data points correspond only to clearly resolved states. The doublets are not represented in the figure. See Fig. 18 for details.

Figure 21

Comparison between experimental and calculated cross sections for $3^{-}$, $5^{-}$, $7^{-}$ and $9^{-}$ final states of ${}^{116}\mathrm{Sn}$. Note that the data points correspond only to clearly resolved states. The doublets are not represented in the figure. See Fig. 18 for details.

Figure 22

Comparison between experimental and calculated cross sections for $0^{+}$ and $6^{+}$ final states of ${}^{122}\mathrm{Sn}$. Note that the data points correspond only to clearly resolved states. The doublets are not represented in the figure. See Fig. 18 for details.

Figure 23

Comparison between experimental and calculated cross sections for $2^{+}$ final states of ${}^{122}\mathrm{Sn}$. Note that the data points correspond only to clearly resolved states. The doublets are not represented in the figure. See Fig. 18 for details.

Figure 24

Comparison between experimental and calculated cross sections for $4^{+}$ final states of ${}^{122}\mathrm{Sn}$. Note that the data points correspond only to clearly resolved states. The doublets are not represented in the figure. See Fig. 18 for details.

Figure 25

Comparison between experimental and calculated cross sections for $3^{-}$, $5^{-}$, and $7^{-}$ final states of ${}^{122}\mathrm{Sn}$. The lines represent results of the microscopic calculations of the differential cross sections (in $\mu$b/sr) as a function of center-of-mass angle (in degrees). The subscripts 1, 2,..., 6 associated with the lines indicate the calculated energy ranking of the corresponding state, with 1 representing the lowest state of given $J$${\hspace{0.5em}}^{\pi }$. Error bars are shown for some $7^{-}$ states; in all other cases, the error bars are smaller than the symbols representing the measured differential cross sections. Note that the data points correspond only to clearly resolved states. The doublets are not represented in the figure.