Nascent fragment shell effects on the nuclear fission processes in semiclassical periodic orbit theory

Making use of the semiclassical periodic orbit theory (POT), we propose, for the first time, a method to exclusively evaluate the shell effects associated with each of the nascent fragments (prefragments) generated by the neck formation in nuclear fission processes. In spite of the strong indication of such shell effects in asymmetric fragment mass distributions, they could not have been accessed by any previous theoretical approach since most of the single-particle wave functions are delocalized. In the POT, we have found that the prefragment shell effects can be naturally and unambiguously identified as the contributions of the classical periodic orbits localized in each of the prefragments. For a numerical test, simple cavity potential models are employed with the shape described by the three-quadratic-surface shape parametrization. Deformed shell energies are studied with the trace formula for degenerate orbits in a truncated spherical cavity which was recently derived [K. Arita, preceding paper, Phys. Rev. C 98, 064310 (2018)]. In this simple model, it is shown that the prefragment shell effect dominates the total shell energy shortly after the neck formation, and the magicity of the heavier prefragment plays a significant role in establishing the fission saddle with asymmetric shape which leads to an asymmetric scission.

Figure 1

The shapes of the potential surfaces in the TQS parametrization, varying the elongation parameter ${\sigma }_1$ and the prefragment mass asymmetry parameter ${\alpha }_2$, with spherical prefragments $\left({\sigma }_3=1,{\alpha }_3=0\right)$ and fixed value of the neck parameter ${\sigma }_2=-0.6$. The broken lines represent the joints of the neighboring quadratic surfaces, and the cross symbols represent the centers of the left and right prefragments. Horizontal and vertical dotted lines indicate the symmetry axis and the position of the center of mass, respectively.

Figure 2

Liquid-drop model deformation energies $\mathrm{\Delta }{E}_{\mathrm{LDM}}$ for ${}^{236}\mathrm{U}$ nucleus plotted as functions of the elongation parameter ${\sigma }_1$ with fixed values of the mass asymmetry parameter ${\alpha }_2$. Solid, long-dashed and short-dashed lines represent the results for ${\alpha }_2=0$, 0.1 and 0.2, respectively.

Figure 3

Single-particle level diagram for the symmetric TQS cavity model as function of the elongation parameter ${\sigma }_1$. Solid (red) and broken (blue) lines represent the levels with positive and negative parities, respectively. The shapes of the potential surface at ${\sigma }_1=0$, 1.0, and 2.0 are shown at the top of the diagram.

Figure 4

Single-particle level diagrams as functions of the asymmetry parameter ${\alpha }_2$, with the elongation parameter ${\sigma }_1=1.0$ (a) and 2.0 (b). The short-dashed (red), long-dashed (green), and solid (blue) lines represent levels with the magnetic quantum numbers $K=0,1\le K\le 2$, and $K\ge 3$, respectively. The thick dotted (light blue) lines are drawn so as to be proportional to $1/R_j^2$, which would be helpful for recognizing the levels indicating localization to each of the prefragments. Shapes of the surface at ${\alpha }_2=0$ and 0.3 are displayed at the top of each panel.

Figure 5

Lengths of the symmetric classical periodic orbits in the TQS cavity model as functions of the elongation parameter ${\sigma }_1$, with spherical prefragments and the neck parameter fixed at ${\sigma }_2=-0.6$. Solid lines (red) represent the prefragment orbits $(p,t)$, broken lines (green) represent the meridian-plane orbits, and dotted lines (blue) represent the equatorial regular polygon orbits in the neck surface (see Ref. [25]).

Figure 6

Fourier transform of the single-particle level density. The squared amplitude $|F^{\mathrm{qm}}(L;{\sigma }_1){|}^2$ is plotted as a function of $L$ and the elongation parameter ${\sigma }_1$.

Figure 7

The relative amplitude factors $w_{j,pt}$, given by Eq. (26), of some short prefragment orbit families $(p,t)$ for symmetric shapes as functions of the elongation parameter ${\sigma }_1$. The wave number is put to the Fermi level $k=k_F$ corresponding to $N=100$.

Figure 8

Shell energies $\delta E\left(N\right)$ as functions of particle number $N$ for symmetric shapes ${\sigma }_1=1.0$ (a) and 2.0 (b) with ${\alpha }_2=0$. The abscissa is taken to be linear in $N^{1/3}$, which also applies in the following Figs. 9–11. Thin solid lines with dots (red) represent the quantum results, and thick solid lines (blue) represent the results of the semiclassical trace formula with the contributions of the prefragment orbit families. The individual contributions of the diameter (2,1), triangle (3,1), and square (4,1) orbits are shown by the dotted (magenta), dashed (green), and dash-dotted (cyan) lines, respectively.

Figure 9

Shell energies $\delta E\left(N\right)$ as functions of particle number $N$ for asymmetric shapes ${\alpha }_2=0.1$ (a) and 0.3 (b) with fixed elongation parameter ${\sigma }_1=2.0$. The thin solid lines with dots (red) represent quantum results and the thick solid lines (blue) represent the results of semiclassical trace formula with contributions of the prefragment orbit families. The contributions of prefragment orbits in the first (heavy) and second (light) prefragments are shown by the dashed (green) lines and dotted (magenta) lines, respectively.

Figure 10

Shell energies as functions of the particle number $N$ for symmetric deformations $({\alpha }_2=0)$ with the elongation parameter ${\sigma }_1=1.0$ (a) and 2.0 (b). The thin line with dots (red) represents the quantum result and the thick line (blue) represents the prefragment shell effect $\delta E_{\mathrm{frag}}\left(N\right)$ which is approximately evaluated in terms of the shell energy of the spherical cavity as in Eq. (33). Arrows indicate the magic numbers of the spherical prefragments.

Figure 11

Same as Fig. 10 but for asymmetric shapes. The mass asymmetry parameter is taken as ${\alpha }_2=0.1$ (a) and 0.3 (b), with the common elongation parameter ${\sigma }_1=2.0$.

Figure 12

Prefragment particle numbers $N_1$ (red triangle) and $N_2$ (blue square) for the mass asymmetry parameter ${\alpha }_2$ which minimizes the shell energy for given total particle number $N$, with the elongation parameter ${\sigma }_1=1.0$ (a) and 2.0 (b). Horizontal dotted lines indicate the spherical magic numbers $N=\cdots ,34,58,92,138,\cdots$.

Figure 13

Contour maps of the shell energy for several particle numbers on the deformation space $({\sigma }_1,{\alpha }_2)$. Solid (blue) and dotted (red) contour lines represent negative and positive shell energies, respectively. Thick solid lines represent the constant-action lines (13) of the prefragment triangle orbits.