Low-lying quadrupole and octupole collective excitations in the ${}^{112,116,118,120,122,124}\mathrm{Sn}$ isotopes

Background: In recent years, considerable interest has been focused on the study of the effect of neutron skin on the collective properties of vibrational nuclei. This can be best evidenced by exclusively determining neutron and proton transition matrix elements involved in a particular excitation by investigating the Coulomb-nuclear interference feature of inelastic scattering.

Purpose: Measurement of angular distributions of the inelastic scattering cross sections for excitations to low-lying $2_1^{+}$ and $3_1^{-}$ states in ${}^{112,116,118,120,122,124}\mathrm{Sn}$ using ${}^7\mathrm{Li}$ beam as probe at $E_{\mathrm{lab}}=28\mathrm{MeV}$ and determination of neutron and proton transition matrix elements involved in each excitation.

Methods: Projectilelike fragments were detected using six sets of Si-surface barrier detector telescopes to measure the cross sections for elastic and inelastic scattering channels. Optical model analysis of elastic scattering data, coupled reaction channels, and continuum discretized coupled channels calculations were performed to understand the measured differential cross sections. An attempt has been made to extract the microscopic mass and charge deformation lengths.

Results: For the $2_1^{+}$ state, experimental $B\left(E2\right)$ values are in good agreement with existing results obtained by electromagnetic methods. Charge and mass quadrupole vibrations are homogeneous. Significant differences are observed for excitation to the $3_1^{-}$ state across the Sn isotopic chain. Available structural information for collective octupole vibrations could not reproduce the present data for this excitation. Results show much lower values of octupole mass deformation parameters.

Conclusions: Isoscalar nature of surface vibrations for the $2_1^{+}$ state in Sn isotopes is verified, with $M_n/M_p\sim N/Z$. For the $3_1^{-}$ state, damped mass vibration is the primary observation. On comparison with existing estimates, a significant deviation from isoscalar nature is conjectured for this excitation, when probed with ${}^7\mathrm{Li}$.

Figure 1

Typical two-dimensional ($\mathrm{\Delta }E$ versus $E_{\mathrm{total}}$) gain-matched spectrum showing the outgoing projectilelike fragments at ${\theta }_{\mathrm{lab}}={60}^{\circ }$ in the ${}^7\mathrm{Li}+{}^{122}\mathrm{Sn}$ system. Colour gradient scale given on the right side of the plot represents number of correlated events in both $\mathrm{\Delta }E$ and $E$ detectors. Inset: One-dimensional spectrum showing $Q$-value distribution of states identified in elastic and inelastic scattering.

Figure 2

Experimental cross sections (open squares) and the results of the CRC calculations (solid lines for WS potential, dash-dot-dot lines for DFM potential) for $\lambda =2$ inelastic scattering processes corresponding to the target excitations in ${}^7\mathrm{Li}+{}^{112,116,118,120,122,124}\mathrm{Sn}$ systems. Inset: Experimental elastic scattering angular distributions (circles) with calculation using WS potential (dashed lines).

Figure 3

Experimental cross sections (filled triangles up) and the results of the CRC calculations (lines) for $\lambda =3$ inelastic scattering processes corresponding to the target excitations in ${}^7\mathrm{Li}+{}^{112,116,118,120,122,124}\mathrm{Sn}$ systems. Calculations using WS potential with ${\delta }_3^{ch}$ values from existing Coulomb excitation measurement [11] (solid lines), electron scattering [13, 14] (dashed lines), and proton scattering [15] (dash-dot lines) are shown. Also shown are calculations using DFM potential with ${\delta }_3^{ch}$ from Coulomb excitation (dash-dot-dot lines).

Figure 4

Experimental cross sections (hollow squares) for inelastic scattering corresponding to bound excited state of ${}^7\mathrm{Li}$ in ${}^7\mathrm{Li}+{}^{120}\mathrm{Sn}$ system. The lines show CRC calculations for different pairs of ${\delta }^{ch}$ and ${\delta }^m$ values : ($\mathrm{i}){\delta }^{ch}=3.944\mathrm{fm}, {\delta }^m=1.993\mathrm{fm}$ (solid), ($\mathrm{ii}){\delta }^{ch}={\delta }^m=2.0\mathrm{fm}$ (dash-dotted), and (iii) CDCC-CRC calculations (dashed) with $B(E2$; $3/2^{-}\to 1/2^{-})=8.39{\text{e}}^2{\text{fm}}^4$ (see text). Calculations for (i) and (iii) are found to suitably agree with the data and these parameters are used for complete theoretical modeling in the CRC and CDCC-CRC frameworks, respectively.

Figure 5

Experimental cross section (hollow triangles) and calculation (dashed lines) for $\lambda =3$ inelastic scattering process in (a) p+${}^{116}\mathrm{Sn}$ [15], ($\mathrm{b}){}^3\mathrm{He}+{}^{116}\mathrm{Sn}$ [18], ($\mathrm{c})\alpha +{}^{116}\mathrm{Sn}$ [17], and ($\mathrm{d}){}^6\mathrm{Li}+{}^{116}\mathrm{Sn}$ [27] systems. Inset: elastic scattering angular distribution (circles) and respective calculation (solid lines).

Figure 6

Experimental cross sections (filled triangles up) and the results of the CDCC-plus-CRC calculations (solid lines) for $\lambda =3$ inelastic scattering processes corresponding to target excitations in ${}^7\mathrm{Li}+{}^{112,116,118,120,122,124}\mathrm{Sn}$ systems. Inset: Experimental elastic scattering angular distribution (circles) with calculation under CDCC-CRC formalism (dashed lines). The calculations for inelastic scattering with reduced values of ${\delta }_3^{ch}$ but equal to ${\delta }_3^m$ are also shown (dashed lines).

Figure 7

$M_n/M_p$ ratios for low-lying excitations in Sn isotopes corresponding to (a) $\lambda =2$ and (b) $\lambda =3$. The dotted line shows the homogeneous isoscalar value of $N/Z$. The dashed and dash-dot-dotted lines represent the results of QRPA calculations for $2^{+}$ and $3^{-}$ transitions respectively.