Solving the one-dimensional penetration problem for the fission channel in the statistical Hauser-Feshbach theory

We solve the Schrödinger equation for an arbitrary one-dimensional potential energy to calculate the transmission coefficient in the fission channel of compound nucleus reactions. We incorporate the calculated transmission coefficients into the statistical Hauser-Feshbach model calculation for neutron-induced reactions on ${}^{235,238}\mathrm{U}$ and ${}^{239}\mathrm{Pu}$. The one-dimensional model reproduces the evaluated fission cross section data reasonably well considering the limited number of model parameters involved. A resonance-like structure appears in the transmission coefficient for a double-humped fission barrier shape that includes an intermediate well, which is understood to be a quantum mechanical effect in the fission channel. The calculated fission cross sections for the neutron-induced reactions on ${}^{235,238}\mathrm{U}$ and ${}^{239}\mathrm{Pu}$ all exhibit a similar structure.

Figure 1

Schematic picture of double-humped potential energy along the nuclear deformation direction, showing the double-humped fission barriers $V_A$ and $V_B$, and the class-I and class-II wells between the barriers. The initial compound nucleus state is at $E_0$ in class I, which decays through the states at $E_A$ and $E_B$ on top of each barrier.

Figure 2

Schematic picture of 1-D potential energy along the nuclear deformation direction. The initial compound nucleus state is at $E_0$, which decays through 1-D fission paths, e.g., the trajectories 1, 2, etc.

Figure 3

Calculated wave functions for the connected parabolas. The potential energy is shown by the dot-dashed curves. The solid and dotted curves are the normalized wave function (solid for the real part, and dotted for the imaginary part). (a) The system energy lies below both barrier heights, (b) the energy is between the barriers, and (c) it is above the barriers.

Figure 4

Calculated transmission coefficients for the connected parabolas as a function of the excitation energy. The top panel (a) is for the potential characterized by $V_1=6.5$, $V_2=1.0$, and $V_3=5.5$ MeV, with the curvatures of $C_1=0.6$, $C_2=0.4$, and $C_3=0.5$ MeV, and the bottom panel (b) is for the $V_1=V_3=6.0$ MeV case. The dashed and dotted curves are the WKB approximations for the inner and outer barriers.

Figure 5

Calculated fission cross section for neutron-induced reaction on ${}^{238}\mathrm{U}$. The barrier height parameters are $V_1=6.0$, $V_2=0.5$, and $V_3=5.0$ for the solid curve. The dashed curve is for $V_3=5.5$ MeV.

Figure 6

Calculated fission cross section for neutron-induced reaction on ${}^{238}\mathrm{U}$. The barrier height parameters are $V_1=6.0$, $V_2=0.5$, and $V_3=5.0$, the same as in Fig. 5, but a small perturbation $\pm \mathrm{\Delta }V$ is applied to the 1-D potential energy surface.

Figure 7

Calculated fission cross section for neutron-induced reaction on ${}^{235}\mathrm{U}$. The barrier height parameters are $V_1=5.9$, $V_2=0.5$, and $V_3=5.7$ MeV for the solid curve. The dashed curves are the case when $V_1\pm 100$ keV.

Figure 8

Calculated fission cross section for neutron-induced reaction on ${}^{239}\mathrm{Pu}$. The barrier height parameters are $V_1=5.9$, $V_2=0.5$, and $V_3=5.7$ MeV for the solid curve. The dashed curves are the case when $V_1\pm 100$ keV.