Excitation energy shift and size difference of low-energy levels in $p$-shell $\mathrm{\Lambda }$ hypernuclei

Structures of low-lying $0s$-orbit $\mathrm{\Lambda }$ states in $p$-shell $\mathrm{\Lambda }$ hypernuclei (${}_{\mathrm{\Lambda }}{}^{A}Z$) are investigated by applying microscopic cluster models for nuclear structure and a single-channel folding potential model for a $\mathrm{\Lambda }$ particle. For $A>10$ systems, the size reduction of core nuclei is small, and the core polarization effect is regarded as a higher-order perturbation in the $\mathrm{\Lambda }$ binding. The present calculation qualitatively describes the systematic trend of experimental data for excitation energy change from ${}^{A-1}Z$ to ${}_{\mathrm{\Lambda }}{}^{A}Z$, in $A>10$ systems. The energy change shows a clear correlation with the nuclear size difference between the ground and excited states. In ${}_{\mathrm{\Lambda }}{}^7\mathrm{Li}$ and ${}_{\mathrm{\Lambda }}{}^9\mathrm{Be}$, the significant shrinkage of cluster structures occurs consistently with the prediction of other calculations.

Figure 1

Total energy $[E\left(\epsilon \right)=E_N\left(\epsilon \right)+E_{\mathrm{\Lambda }}\left(\epsilon \right)]$ and $\mathrm{\Lambda }$ energy $\left[E_{\mathrm{\Lambda }}\left(\epsilon \right)\right]$ for polarized core $\mathrm{\Phi }\left(\epsilon \right)$ in ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}\left(0_1^{+}\right)$. The energies are plotted against the rms nuclear matter radius $R_N\left(\epsilon \right)$ in the top and middle panels. The nuclear energy $\left[E_N\left(\epsilon \right)\right]$ subtracted by 10 MeV is also shown. $E_{\mathrm{\Lambda }}\left(\epsilon \right)$ plotted to the sharp-cut density ${\rho }_{\mathrm{sharp}}-\mathrm{cut}=(3/4\pi ){(3/5)}^{3/2}A{R}_N^{-3}$ reduced from $R_N\left(\epsilon \right)$ for the uniform density ansatz is shown in the bottom panel. The calculated values obtained with ESC08a(DI) and ESC08a(DD) are shown.

Figure 2

Energy spectra calculated with ESC08a(DI) and the experimental spectra. The experimental data for ${}^{A-1}Z$ are from Refs. [65, 66, 67, 68], and those for ${}_{\mathrm{\Lambda }}{}^{A}Z$ are taken from Refs. [2, 48, 71, 72, 73, 74, 75, 76, 77, 78]. The excitation energy of ${}^8\mathrm{Be}\left(2^{+}\right)$ is calculated by the RGM calculation. The experimental data for ${}_{\mathrm{\Lambda }}{}^{A}Z$ are the spin-averaged values reduced from the excitation energies of spin doublets, $J^{\pi }=I^{\pi }\pm 1/2$, except for ${}_{\mathrm{\Lambda }}{}^{12}\mathrm{B}(1/2_1^{-},3/2_2^{-}), {}_{\mathrm{\Lambda }}{}^{12}\mathrm{C}(1/2_1^{-},3/2_2^{-})$, and ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}\left(2^{+}\right)$. For ${}_{\mathrm{\Lambda }}{}^{12}\mathrm{B}(1/2_1^{-}), {}_{\mathrm{\Lambda }}{}^{12}\mathrm{C}(1/2_1^{-})$, and ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}\left(2^{+}\right)$, the experimental values of $E_x\left(1^{-}\right), E_x\left(1^{-}\right)$, and $E_x(3/2^{+})$ are used, respectively. For ${}_{\mathrm{\Lambda }}{}^{12}\mathrm{B},\mathrm{C}(3/2_2^{-})$, the experimental values of $E_x\left(1^{-}\right)$ in ${}_{\mathrm{\Lambda }}{}^{12}\mathrm{B}$ and $E_x\left(2^{-}\right)$ in ${}_{\mathrm{\Lambda }}{}^{12}\mathrm{C}$ are averaged by assuming that the Coulomb shift in ${}_{\mathrm{\Lambda }}{}^{12}\mathrm{B}(3/2_2^{-})-{}_{\mathrm{\Lambda }}{}^{12}\mathrm{C}(3/2_2^{-})$ is the same value as that in ${}^{11}\mathrm{B}(3/2_2^{-})-{}^{11}\mathrm{C}(3/2_2^{-})$.

Figure 3

(Top) Excitation energy shift ${\delta }_{\mathrm{\Lambda }}\left(E_x\right)$ calculated with ESC08a(DI) and ESC08a(DD), and experimental values. The excitation energy shift without the core polarization calculated with ESC08a(DI) is also plotted by open circles. (Middle) Nuclear size difference $(R_N-R_{N,\mathrm{gs}})$ of excited states from that of the ground states obtained with ESC08a(DI) and ESC08a(DD). The result without the core polarization calculated with ESC08a(DI) is also shown by open circles. (Bottom) The excitation energy shift plotted against the nuclear size difference obtained with ESC08a(DI) and ESC08a(DD). Experimental energy shift is plotted against the theoretical size difference calculated with ESC08a(DI).

Figure 4

$\mathrm{\Lambda }$ density and nuclear density distributions in ${}_{\mathrm{\Lambda }}{}^7\mathrm{Li}, {}_{\mathrm{\Lambda }}{}^9\mathrm{Be}, {}_{\mathrm{\Lambda }}{}^{11}\mathrm{B}, {}_{\mathrm{\Lambda }}{}^{12}\mathrm{B}$, and ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}$ calculated with ESC08a(DI). In the left panels, $\mathrm{\Lambda }$ density obtained in the calculation with and without the core polarization (w/o cp) are shown. In the middle and right panels, nuclear densities and $r^2$-weighted densities are shown, respectively. In the middle panels, $\mathrm{\Lambda }$ density $\left[{\rho }_{\mathrm{\Lambda }}\left(r\right)\right]$ multiplied by 4 in the ground state of $A_{\mathrm{\Lambda }}Z$ is also shown for comparison.

Figure 5

Enhancement factor $\epsilon$ in the ESC08a(DI) calculation. The factor is plotted as a function of $A-1$.