Potential energy surfaces for cluster emitting nuclei

Potential energy surfaces are calculated by use of the most advanced asymmetric two-center shell model that allows us to obtain shell-and-pairing corrections that are added to the Yukawa-plus-exponential model deformation energy. Shell effects are of crucial importance for the experimental observation of spontaneous disintegration by heavy-ion emission. Results for ${}^{222}\mathrm{Ra}$, ${}^{232}\mathrm{U}$, ${}^{236}\mathrm{Pu}$, and ${}^{242}\mathrm{Cm}$ illustrate the main ideas and show for the first time, for a cluster emitter, a potential barrier obtained by use of the macroscopic-microscopic method.

Figure 1

Shell and pairing corrections versus $(R-R_i)/(R_t-R_i)$ for a symmetric ($\eta =0$) fission of ${}^{236}\mathrm{Pu}$. Compared with the shell correction (dotted curve), the pairing correction (dashed curve) is out of phase and smaller in amplitude, leading for $\eta =$ constant to a smoother total curve $\delta E$ (solid curve).

Figure 2

Variation of the neutron and proton numbers of the two fragments with mass asymmetry at the touching point, $R=R_t$, for ${}^{222}\mathrm{Ra}$.

Figure 3

Variation of the neutron and proton numbers of the two fragments with mass asymmetry at the touching point, $R=R_t$, for ${}^{232}\mathrm{U}$.

Figure 4

Variation of the neutron and proton numbers of the two fragments with mass asymmetry at the touching point, $R=R_t$, for ${}^{236}\mathrm{Pu}$.

Figure 5

Variation of the neutron and proton numbers of the two fragments with mass asymmetry at the touching point, $R=R_t$, for ${}^{242}\mathrm{Cm}$. Every time a nucleon number reaches a magic value, the corresponding shell correction has a local minimum. The very deep valley around $\eta =0.719$ is produced when three magic numbers, $N_1=126,N_2=20$, and $Z_1=82$, are reached simultaneously.

Figure 6

PES of ${}^{222}\mathrm{Ra}$ vs. $\xi =(R-R_i)/(R_t-R_i)\ge 0$ and $\eta =(A_1-A_2)/(A_1+A_2)$. Y + EM (bottom), shell + pairing corrections (center), and total deformation energy (top). Apart from a rather shallow valley not far from symmetry that is due to a cold fission process with the ${}^{132}\mathrm{Sn}$ heavy fragment and the ${}^{90}\mathrm{Sr}$ light fragment, two valleys are very deep on the central figure: one around $\eta =0.87$ for ${}^{14}\mathrm{C}$ radioactivity with the ${}^{208}\mathrm{Pb}$ daughter and the other near $\eta =0.36$ for asymmetric fission with the ${}_{28}^{70}{\mathrm{Ni}}_{42}$ light fragment and the ${}^{152}\mathrm{Nd}$ daughter.

Figure 7

Contour plot of shell and pairing corrections for ${}^{222}\mathrm{Ra}$ vs. $(R-R_i)/(R_t-R_i)$ and η.

Figure 8

A cut through the PES of ${}^{222}\mathrm{Ra}$ at symmetry $\eta =0$ (top) and for ${}^{14}\mathrm{C}$ radioactivity with the ${}^{208}\mathrm{Pb}$ daughter (bottom).

Figure 9

Deformation energies at the touching-point configurations ($R=R_t$) of ${}^{222}\mathrm{Ra}$ vs. the asymmetry $\eta$: $E_{Y+\mathrm{EM}}$ and $E_{\mathrm{def}}$ (top), shell + pairing corrections, and only shell corrections (bottom).

Figure 10

PES of ${}^{232}\mathrm{U}$ vs. ξ and η. Y + EM (bottom), shell + pairing corrections (center), and total deformation energy (top). The deepest valley in the central figure at $\eta =0.8$ corresponds to ${}^{24}\mathrm{Ne}$ radioactivity with the ${}^{208}\mathrm{Pb}$ daghter. Not far from the symmetry is the valley shaped by the doubly magic heavy fragment ${}^{132}\mathrm{Sn}$ (the light fragment is ${}^{100}\mathrm{Mo}$). At an intermediate asymmetry $\eta \simeq 0.3$, the light fragment ${}_{32}^{82}{\mathrm{Ge}}_{50}$ has a neutron magic number (the corresponding heavy fragment is ${}^{150}\mathrm{Nd}$).

Figure 11

Contour plot of shell and pairing corrections for ${}^{232}\mathrm{U}$ vs. $(R-R_i)/(R_t-R_i)$ and η.

Figure 12

A cut through the PES of ${}^{232}\mathrm{U}$ at symmetry $\eta =0$ (top) and for ${}^{24}\mathrm{Ne}$ radioactivity with the ${}^{208}\mathrm{Pb}$ daughter (bottom).

Figure 13

Deformation energies at the touching-point configurations ($R=R_t$) of ${}^{232}\mathrm{U}$ vs. the asymmetry $\eta$: $E_{Y+\mathrm{EM}}$ and $E_{\mathrm{def}}$ (top); shell + pairing corrections and only shell corrections (bottom).

Figure 14

PES of ${}^{236}\mathrm{Pu}$ vs. ξ and η: Y + EM (bottom), shell + pairing corrections (center), and total deformation energy (top). The deepest two valleys in the central figure are due to the magicity of the daughter ${}^{208}\mathrm{Pb}$ for ${}^{28}\mathrm{Mg}$ radioactivity at about $\eta =0.75$ and of the heavy fragment ${}^{132}\mathrm{Sn}$ (light fragment ${}^{104}\mathrm{Ru}$) at $\eta \simeq 0.1$. An intermediate shallower one appears at $\eta \simeq 0.28$ for ${}^{236}\mathrm{Pu}$ $\to$${}^{152}\mathrm{Nd}$ + ${}_{34}^{84}{\mathrm{Se}}_{50}$.

Figure 15

Contour plot of shell and pairing corrections for ${}^{236}\mathrm{Pu}$ vs. $(R-R_i)/(R_t-R_i)$ and η.

Figure 16

A cut through the PES of ${}^{236}\mathrm{Pu}$ at symmetry $\eta =0$ (top) and for ${}^{28}\mathrm{Mg}$ radioactivity with the ${}^{208}\mathrm{Pb}$ daughter (bottom).

Figure 17

Deformation energies at the touching-point configurations ($R=R_t$) of ${}^{236}\mathrm{Pu}$ vs. the asymmetry $\eta$: $E_{Y+\mathrm{EM}}$ and $E_{\mathrm{def}}$ (top), and shell + pairing corrections and only shell corrections (bottom).

Figure 18

PES of ${}^{242}\mathrm{Cm}$ vs. ξ and η. Y + EM (bottom), shell + pairing corrections (center), and total deformation energy (top). There are two very deep valleys in the central plot that are due to ${}_{14}^{34}{\mathrm{Si}}_{20}$ radioactivity with the ${}^{208}\mathrm{Pb}$ daughter at $\eta =0.7$ and due to the cold fission with heavy and light fragments ${}^{132}\mathrm{Sn}$ and ${}^{110}\mathrm{Pd}$, respectively, at $\eta \simeq 0.06$. The shallower intermediate valleys at $\eta \simeq 0.3$ and $\eta \simeq 0.41$ are due to $N_2=50$ and $Z_2=28$.

Figure 19

Contour plot of shell and pairing corrections for ${}^{242}\mathrm{Cm}$ vs. $(R-R_i)/(R_t-R_i)$ and η.

Figure 20

A cut through the PES of ${}^{242}\mathrm{Cm}$ at symmetry $\eta =0$ (top) and for ${}^{34}\mathrm{Si}$ radioactivity with the ${}^{208}\mathrm{Pb}$ daughter (bottom).

Figure 21

Deformation energies at the touching-point configurations ($R=R_t$) of ${}^{242}\mathrm{Cm}$ vs. the asymmetry $\eta$: $E_{Y+\mathrm{EM}}$ and $E_{\mathrm{def}}$ (top) and shell + pairing corrections and only shell corrections (bottom).