Shape of $\Lambda$ hypernuclei in the ($\beta ,\gamma$) deformation plane

We study the shape of $\Lambda$ hypernuclei in the full ($\beta ,\gamma$) deformation plane, including both axially symmetric and triaxial quadrupole deformations. To this end, we use the constrained Skyrme Hartree-Fock $+$ BCS method on a three-dimensional Cartesian mesh. The potential energy surface is analyzed for carbon hypernuclei as well as for sd-shell hypernuclei such as ${}_{\Lambda }^{27,29}$Si and ${}_{\Lambda }^{25,27}$Mg. We show that the potential energy surfaces in the ($\beta ,\gamma$) plane of the hypernuclei and the corresponding core nuclei are similar to each other, although the addition of the $\Lambda$ hyperon makes the energy surface somewhat softer along the $\gamma$ direction.

Figure 1

The potential energy surfaces of (a) ${}^{10}\mathrm{C}$ and (b) ${}_{\Lambda }^{11}$C in the ($\beta ,\gamma$) deformation plane obtained with the SGII parameter set. Each contour line is separated by 0.07 MeV. The triangles indicate the absolute minima in the PES.

Figure 2

The energy curve for ${}^{10}\mathrm{C}$ and ${}_{\Lambda }^{11}$C along the axially symmetric deformation corresponding to Fig. 1. The energy surface for ${}_{\Lambda }^{11}$C is shifted by a constant amount as indicated in the figure.

Figure 3

Same as Fig. 1, but for ${}^{12}\mathrm{C}$ and ${}_{\Lambda }^{13}$C. Each contour line is separated by 0.2 MeV.

Figure 4

Same as Fig. 2, but for ${}^{12}\mathrm{C}$ and ${}_{\Lambda }^{13}$C.

Figure 5

Same as Fig. 1, but for ${}^{18}\mathrm{C}$ and ${}_{\Lambda }^{19}$C. Each contour line is separated by 0.15 MeV.

Figure 6

The energy curve as a function of the triaxial deformation parameter $\gamma$ for the optimum values of the $\beta$ deformation parameters shown in Fig. 5. The energy surface for ${}_{\Lambda }^{19}$C is shifted by a constant amount as indicated in the figure.

Figure 7

The energy difference between ${}_{\Lambda }^{19}$C and ${}^{18}\mathrm{C}$ nuclei in the $(\beta ,\gamma )$ deformation plane.

Figure 8

Same as Fig. 1, but for ${}^{28}\mathrm{Si}$ and ${}_{\Lambda }^{29}$Si. Each contour line is separated by 0.6 MeV.

Figure 9

Same as Fig. 1, but for ${}^{24}\mathrm{Mg}$ and ${}_{\Lambda }^{25}$Mg. Each contour line is separated by 0.5 MeV.

Figure 10

(a) The energy curve for ${}^{24}\mathrm{Mg}$ and ${}_{\Lambda }^{25}$Mg along the axially symmetric deformation corresponding to Fig. 9. (b) The energy curve as a function of the triaxial deformation parameter $\gamma$ for the optimum values of the $\beta$ deformation parameters shown in Fig. 9. The energy surfaces for ${}_{\Lambda }^{25}$Mg are shifted by a constant amount as indicated in the figures.

Figure 11

Same as Fig. 1, but for ${}^{26}\mathrm{Si}$ and ${}_{\Lambda }^{27}$Si. Each contour line is separated by 0.4 MeV.

Figure 12

Same as Fig. 1, but for ${}^{26}\mathrm{Mg}$ and ${}_{\Lambda }^{27}$Mg. Each contour line is separated by 0.4 MeV.

Figure 13

(a) The energy curve for ${}^{26}\mathrm{Mg}$ and ${}_{\Lambda }^{27}$Mg along the axially symmetric deformation corresponding to Fig. 12. (b) The energy curve as a function of the triaxial deformation parameter $\gamma$ for the optimum values of the $\beta$ deformation parameters shown in Fig. 12. The energy surfaces for ${}_{\Lambda }^{27}$Mg are shifted by a constant amount as indicated in the figures.

Figure 14

The overlap between the nucleon and the $\Lambda$ particle densities as a function of the triaxiality $\gamma$ for a fixed value of $\beta =0.3$. The $\Lambda$-particle density is assumed to be a spherical Gaussian function, while a deformed Woods-Saxon shape is considered for the nucleon density.