Langevin model of low-energy fission

Background: Since the earliest days of fission, stochastic models have been used to describe and model the process. For a quarter century, numerical solutions of Langevin equations have been used to model fission of highly excited nuclei, where microscopic potential-energy effects have been neglected.

Purpose: In this paper I present a Langevin model for the fission of nuclei with low to medium excitation energies, for which microscopic effects in the potential energy cannot be ignored.

Method: I solve Langevin equations in a five-dimensional space of nuclear deformations. The macroscopic-microscopic potential energy from a global nuclear structure model well benchmarked to nuclear masses is tabulated on a mesh of approximately ${10}^7$ points in this deformation space. The potential is defined continuously inside the mesh boundaries by use of a moving five-dimensional cubic spline approximation. Because of reflection symmetry, the effective mesh is nearly twice this size. For the inertia, I use a (possibly scaled) approximation to the inertia tensor defined by irrotational flow. A phenomenological dissipation tensor related to one-body dissipation is used. A normal-mode analysis of the dynamical system at the saddle point and the assumption of quasiequilibrium provide distributions of initial conditions appropriate to low excitation energies, and are extended to model spontaneous fission. A dynamical model of postscission fragment motion including dynamical deformations and separation allows the calculation of final mass and kinetic-energy distributions, along with other interesting quantities.

Results: The model makes quantitative predictions for fragment mass and kinetic-energy yields, some of which are very close to measured ones. Varying the energy of the incident neutron for induced fission allows the prediction of energy dependencies of fragment yields and average kinetic energies. With a simple approximation for spontaneous fission starting conditions, quantitative predictions are made for some observables which are close to measurements.

Conclusions: This model is able to reproduce several mass and energy yield observables with a small number of physical parameters, some of which do not need to be varied after benchmarking to ${}^{235}\mathrm{U}(n,f)$ to predict results for other fissioning isotopes.

Figure 1

A selection of shapes with different values of the neck coordinate index for shapes with $R_1=1.0,{\varepsilon }_{\mathrm{f}1}=-0.1,{\varepsilon }_{\mathrm{f}2}=0.175$, and ${\alpha }_{\mathrm{g}}=0.12$. The axis for each rotationally symmetric shape lies at the value of its neck coordinate index. The shape with neck index 14 lies near the bottom of the asymmetric fission valley for ${}^{236}\mathrm{U}$; see Sec. 2c4.

Figure 2

A schematic representation of a dynamical trajectory in the space of mesh points, with minimeshes surrounding the initial and final points on the trajectory. Along the trajectory, every time a point is reached in the outer layer of hypercells in the minimesh, a new minimesh is defined, centered about the current trajectory point.

Figure 3

The calculated energies of the second and third minima, the two outermost saddle points, the mass-symmetric and mass-asymmetric valleys, and the ridge between them as functions of the coordinate $R_1$ for ${}^{236}\mathrm{U}$.

Figure 4

Isoscalar giant quadrupole and octupole widths used in 1985 to fix the strength $k_{\mathrm{s}}$ of the surface dissipation. The width data are from Refs. [82, 83].

Figure 5

Data on symmetric or average fission-fragment kinetic energies for highly excited systems compared to the 1985 Viola two-parameter fitted systematics and to the predictions of both the wall-and-window and the surface-plus-window dissipation models with a purely macroscopic potential energy. The data sources are as follows: (1) solid circles from Ref. [89], corrected in Ref. [90]; (2) solid square from Ref. [91]; (3) downward-pointing solid triangles from Ref. [92]; (4) upward-pointing solid triangles from Ref. [93]; (5) solid diamonds from Ref. [94]; (6) open circles from Ref. [95]; (7) open squares from Refs. [96, 97, 98], tabulated in Ref. [90]; (8) open upward-pointing triangles from Ref. [99]; (9) open downward-pointing triangles from Ref. [100]; and (10) open diamonds from Ref. [101].

Figure 6

Calculated pre-evaporation fragment yields as a function of fragment mass number $A$ from the default model with a mass resolution $\mathrm{\Delta }A=4.0$ u for thermal-neutron-induced fission of ${}^{235}\mathrm{U}$, compared to evaluated experimental data [113]. The upper part has a linear yield scale, while the lower part shows the same points with a logarithmic yield scale.

Figure 7

Calculated pre-evaporation total fragment kinetic energy distribution from the default model for thermal-neutron-induced fission of ${}^{235}\mathrm{U}$, compared to evaluated experimental data [113].

Figure 8

Calculated pre-evaporation average total fragment kinetic energies as a function of fragment mass number from the default model with a mass resolution $\mathrm{\Delta }A=4.0$ u for thermal-neutron-induced fission of ${}^{235}\mathrm{U}$, compared to evaluated experimental data [113].

Figure 9

Calculated pre-evaporation fragment yields as a function of fragment mass number $A$ from the default model with a mass resolution $\mathrm{\Delta }A=3.0$ for thermal-neutron-induced fission of ${}^{235}\mathrm{U}$, compared to the data of Straede [114]. The upper part has a linear yield scale, while the lower part shows the same points with a logarithmic yield scale.

Figure 10

Calculated pre-evaporation average total fragment kinetic energies as a function of fragment mass number $A$ from the default model with a mass resolution $\mathrm{\Delta }A=3.0$ for thermal-neutron-induced fission of ${}^{235}\mathrm{U}$, compared to the data of Straede [114].

Figure 11

Calculated postevaporation fragment yields as a function of fragment mass number $A$ from the default model for thermal-neutron-induced fission of ${}^{235}\mathrm{U}$, compared to the evaluation of England and Rider [115]. The upper part of the figure has a linear yield scale, while the lower part has a logarithmic yield scale.

Figure 12

Calculated postevaporation fragment yields as a function of fragment mass number $A$ from an adjusted model with reduced dissipation, increased inertia, increased scission neck radius, and a random neck rupture, for the thermal-neutron-induced fission of ${}^{235}\mathrm{U}$, compared to the evaluation of England and Rider [115]. The upper part of the figure has a linear yield scale, while the lower part has a logarithmic yield scale.

Figure 13

Calculated pre-evaporation fragment yields as a function of fragment mass number $A$ from an adjusted model with reduced dissipation, increased inertia, increased scission neck radius, and a random neck rupture, for the thermal-neutron-induced fission of ${}^{235}\mathrm{U}$, compared to evaluated experimental data [113].

Figure 14

Calculated pre-evaporation total fragment kinetic energy distribution from the adjusted model for thermal-neutron-induced fission of ${}^{235}\mathrm{U}$, compared to evaluated experimental data [113].

Figure 15

Calculated pre-evaporation average total fragment kinetic energies as a function of fragment mass number from the adjusted model with a mass resolution $\mathrm{\Delta }A=3.5$ u for thermal-neutron-induced fission of ${}^{235}\mathrm{U}$, compared to evaluated experimental data [113].

Figure 16

Calculated pre-evaporation fragment yields as a function of fragment mass number $A$ from the adjusted model with a mass resolution $\mathrm{\Delta }A=2.50$ u for thermal-neutron-induced fission of ${}^{235}\mathrm{U}$, compared to the data of Straede [114]. The calculated trajectories are identical to those used to calculate the results in Fig. 13 except for the use of a different mass-resolution filter. The upper part of the figure shows the yields with a linear scale, while the lower part shows them with a logarithmic scale.

Figure 17

Calculated pre-evaporation average total fragment kinetic energies as a function of fragment mass number $A$ from the adjusted model with a mass resolution $\mathrm{\Delta }A=3.25$ u for thermal-neutron-induced fission of ${}^{235}\mathrm{U}$, compared to the data of Straede [114]. The calculated trajectories are identical to those used to calculate the results shown in Fig. 15 except for the use of a different mass-resolution filter.

Figure 18

Calculated pre-evaporation average total fragment kinetic energies as a function of fragment mass number $A$ from the adjusted model for various mass resolutions $\mathrm{\Delta }A$ from 0 to 5 u for thermal-neutron-induced fission of ${}^{235}\mathrm{U}$, compared to evaluated data [113]. The $498254$ calculated trajectories are identical to those used to generate the model results shown in Fig. 15, but use different mass-resolution filters.

Figure 19

Calculated average pre-evaporation total fragment kinetic energy as a function of incident neutron energy for ${}^{235}\mathrm{U}(n,f)$, using the same model as for Figs. 12, 13, 14, 15, 16, 17 (solid red circles). The model calculations have 0.2 MeV subtracted to facilitate comparison of the predicted neutron energy dependence with the several data sets. The solid black points are the data used by Madland to derive the linear evaluation shown as a red dashed line [114, 117, 118]. The dotted curve shows the 2014 evaluation of Lestone [119], while the recent measurements of Duke [120] are shown by open red squares. The calculated point at 6 MeV is for first-chance fission only, while the data include about 6% second-chance fission.

Figure 20

Logarithmic derivative with respect to neutron energy of the postevaporation fragment yields for ${}^{235}\mathrm{U}(n,f)$. The plotted quantity is $\frac{100}{3}[Y\left(E_3\right)/Y\left(E_{\mathrm{th}}\right)-1]$, where $Y\left(E_3\right)$ is the calculated postevaporation fragment yield for 3 MeV neutron-induced fission, while $Y\left(E_{\mathrm{th}}\right)$ is the calculated yield for thermal-neutron-induced fission of ${}^{235}\mathrm{U}$. The points labeled post-evap 1 (open circles) and post-evap 2 (open triangles) utilize different assumptions about the distribution of excitation energy in the heavy and light fragments for 3 MeV neutron energy. See the text for details. The data points (solid squares) are from linear fits to measured prompt yields as a function of the energy of monoenergetic neutron beams interacting with ${}^{235}\mathrm{U}$ targets from Ref. [121].

Figure 21

Calculated pre-evaporation fragment yields as a function of fragment mass number $A$ from the adjusted model with a mass resolution $\mathrm{\Delta }A=3.25$ for thermal-neutron-induced fission of ${}^{233}\mathrm{U}$ (red circles), compared to two data sets [122, 123]. The upper part of the figure shows the yields with a linear scale, while the lower part shows them with a logarithmic scale.

Figure 22

Calculated postevaporation fragment yields as a function of fragment mass number $A$ from the adjusted model for thermal-neutron-induced fission of ${}^{233}\mathrm{U}$, compared to evaluated prompt yields [115].

Figure 23

Calculated postevaporation fragment yields as a function of fragment mass number $A$ from the adjusted model for thermal-neutron-induced fission of ${}^{233}\mathrm{U}$, with 60% of symmetric trajectories removed, compared to evaluated prompt yields [115]. The upper part of the figure shows the yields with a linear scale, while the lower part shows them with a logarithmic scale.

Figure 24

Calculated pre-evaporation fragment yields as a function of fragment mass number $A$ from the adjusted model with 60% of the symmetric trajectories removed, and a mass resolution $\mathrm{\Delta }A=3.25$ for thermal-neutron-induced fission of ${}^{233}\mathrm{U}$, compared to two data sets [122, 123]. The upper part of the figure shows the yields with a linear scale, while the lower part shows them with a logarithmic scale.

Figure 25

Calculated pre-evaporation total fragment kinetic energy distribution from the adjusted model for thermal-neutron-induced fission of ${}^{233}\mathrm{U}$, for which the only data available are the mean and standard deviation of the experimental distribution [124].

Figure 26

Calculated pre-evaporation average total fragment kinetic energies as a function of fragment mass number from the adjusted model with 60% of the symmetric trajectories discarded, and a mass resolution $\mathrm{\Delta }A=3.25$ u for thermal-neutron-induced fission of ${}^{233}\mathrm{U}$, compared to two data sets [122, 123].

Figure 27

Similar to Fig. 19. The solid data points, the Madland systematics [117], the Lestone evaluation [119], and the solid circles (model) are the same as in that figure, and apply to ${}^{235}\mathrm{U}(n,f)$, while the open squares are calculated using the same adjusted model, with 60% of the symmetric trajectories discarded, for ${}^{233}\mathrm{U}(n,f)$. The data point for ${}^{233}\mathrm{U}(n,f)$ at thermal energy is from Ref. [124]. The calculated points at 6 MeV are for first-chance fission only, while the data include about 6% second-chance fission.

Figure 28

The calculated energies of the second minimum, the second (outermost) saddle point, the mass-symmetric and mass-asymmetric valleys, and the ridge between them as functions of the coordinate $R_1$ for ${}^{240}\mathrm{Pu}$.

Figure 29

Calculated pre-evaporation fragment yields for thermal-neutron-induced fission of ${}^{239}\mathrm{Pu}$ as a function of fragment mass number $A$ from the adjusted model with a mass resolution $\mathrm{\Delta }A=3.50$, compared to experimental data [125], with a linear scale.

Figure 30

Calculated postevaporation fragment yields as a function of fragment mass number $A$ from the adjusted model for thermal-neutron-induced fission of ${}^{239}\mathrm{Pu}$, compared to the evaluation of England and Rider [115]. The model uses the same parameters as for ${}^{235}\mathrm{U}(n,f)$ except for a scission neck radius of 1.8 fm instead of 1.7 fm. 85% of symmetric trajectories have been discarded to more closely approximate the symmetric yield. The upper part of the figure shows the yields with a linear scale, while the lower part shows them with a logarithmic scale.

Figure 31

Calculated pre-evaporation fragment yields as a function of fragment mass number $A$ from the adjusted model with with 85% of the symmetric yield discarded, for thermal-neutron-induced fission of ${}^{239}\mathrm{Pu}$, compared to experimental data [125]. The scission neck radius of 1.8 fm is used instead of 1.7 fm for the case of ${}^{235}\mathrm{U}(n,f)$ shown previously, while the mass resolution $\mathrm{\Delta }A=3.50$ u. The upper part of the figure shows the yields with a linear scale, while the lower part shows them with a logarithmic scale.

Figure 32

Calculated pre-evaporation total fragment kinetic energy distribution from the adjusted model for thermal-neutron-induced fission of ${}^{239}\mathrm{Pu}$, compared to the experimental data of Wagemans [125].

Figure 33

Calculated pre-evaporation average total fragment kinetic energies as a function of fragment mass number from the adjusted model with 85% of symmetric trajectories discarded, and with a mass resolution $\mathrm{\Delta }A=3.5$ u, for thermal-neutron-induced fission of ${}^{239}\mathrm{Pu}$, compared to the experimental data of Wagemans [125].

Figure 34

Calculated average pre-evaporation total fragment kinetic energy as a function of incident neutron energy for ${}^{239}\mathrm{Pu}(n,f)$, using the same model as for Figs. 30, 31, 32, 33 (solid red circles). The model calculations have 0.5 MeV subtracted to facilitate comparison of the predicted neutron energy dependence with the data set. The solid black points are the data used by Madland to derive the linear evaluation shown as a red solid line [117, 126]. The same model with all symmetric trajectories included, used to calculate the results shown in Fig. 29, predicts the energy dependence shown by the green solid squares. The calculated points at 6 MeV are for first-chance fission only.

Figure 35

Same as Fig. 20, for ${}^{239}\mathrm{Pu}(n,f)$.

Figure 36

Calculated pre-evaporation fragment yields as a function of fragment mass number $A$ from the adjusted model with 0.27% of the trajectories starting in the symmetric valley, while the remainder start in the asymmetric valley, for spontaneous fission of ${}^{240}\mathrm{Pu}$, compared to experimental data [125, 127]. The mass resolution applied to the model is $\mathrm{\Delta }A=3.0$ u. The upper part of the figure shows the yields with a linear scale, while the lower part shows them with a logarithmic scale.

Figure 37

Calculated pre-evaporation total fragment kinetic energy distribution from the adjusted model for spontaneous fission of ${}^{240}\mathrm{Pu}$, compared to the experimental data of Schillebeeckx [128].

Figure 38

Calculated pre-evaporation average total fragment kinetic energies as a function of fragment mass number from the adjusted model with 0.27% of trajectories starting in the symmetric valley, and with a mass resolution $\mathrm{\Delta }A=3.0$ u, for spontaneous fission of ${}^{240}\mathrm{Pu}$, compared to the experimental data of Wagemans [125] and Schillebeeckx [128].

Figure 39

Calculated postevaporation fragment yields as a function of fragment mass number $A$ from the adjusted model, for spontaneous fission of ${}^{252}\mathrm{Cf}$, compared to an evaluation [115]. All trajectories start in the neighborhood of the exit point in the asymmetric valley, and the scission neck radius is 2.0 fm, compared to 1.8 fm used for Figs. 36, 37, 38.

Figure 40

Calculated postevaporation fragment yields as a function of fragment mass number $A$ from the modified model, for spontaneous fission of ${}^{252}\mathrm{Cf}$, compared to an evaluation [115]. The dissipation and inertial parameters are the same as used in the adjusted model, but a scission neck radius of 2.6 fm and a neck rupture parameter of 2.1 fm are used. The upper part of the figure shows the yields with a linear scale, while the lower part shows them with a logarithmic scale.

Figure 41

Calculated pre-evaporation fragment yields as a function of fragment mass number $A$ from the modified model, for spontaneous fission of ${}^{252}\mathrm{Cf}$, compared to experimental data [113]. The dissipation and inertial parameters are the same as used in the adjusted model, but a scission neck radius of 2.6 fm and a neck rupture parameter of 2.1 fm are used. The results with two different mass resolutions are shown; $\mathrm{\Delta }A=3.0$ u (black circles) and 4.0 u (red squares). The upper part of the figure shows the yields with a linear scale, while the lower part shows them with a logarithmic scale.

Figure 42

Calculated pre-evaporation total fragment kinetic energy distribution from the modified model for spontaneous fission of ${}^{252}\mathrm{Cf}$, compared to experimental data [113].

Figure 43

Calculated pre-evaporation average total fragment kinetic energies as a function of fragment mass number from the modified model with a mass resolution $\mathrm{\Delta }A=3.0$ u (black circles) and 4.0 u (red squares), for spontaneous fission of ${}^{252}\mathrm{Cf}$, compared to experimental data [113].