²³⁶U Multi-modal Fission Paths on a Five-dimensional Deformation Surface^*

We use the macroscopic-microscopic (MM) model to calculate the PESs in a 5D deformation space. The nuclear shape is described by a 5D generalized Lawrence realistic parameterization.^[5,15] The macroscopic model is Lublin-Strasbourg Drop (LSD) model.^[16] And the microscopic model is Strutinsky correction method.^[17,18] To calculate the single particle levels, we use folded-Yukawa model^[19] as the single particle potential, and calculate the Hamiltonian on double center oscillator bases.^[20] Considering the pairing effect, we use SBCS model.^[21]

Our search technique is the watershed algorithm, which is inspired by the geographic concept of drainage basin, plus depth first algorithm.^[15] The watershed algorithm was first used in image processing for image division.^[22] To make the watershed algorithm more understandable, we illustrate it on a two-dimensional contour plot as shown in Fig. 1.

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(i) Initially, we only have the contour plot shown in Fig. 1(a)

(ii) The first step of watershed algorithm is to find the local minimum. In this step, we need to scan all the grids on the PES. Suppose that (a) there is a rainfall on a 5D surface and that (b) a rain drop flows along the steepest descent line which is recorded for each grid using linked list. The definition of a basin is the area in which the rainfall flows to the same minimum. After this step, we can divide the surface into different basins according to different minima as shown in Fig. 1(b).

(iii) Then, based on the definition of a basin, the watersheds can be defined as an area, where there is a point having a neighboring point, which does not belong to the same basin as the former point does. Thus, both points are on the watershed. After this step, watershed can be found on the surface as the dashed lines shown in Fig. 1(c).

(iv) Finally, the saddle point is defined as the point of the smallest value of the potential energy height on the watershed of the two connected basins. Red arrows on Fig. 1(d) show the location of saddle points. The steepest descent lines (stored in Step 2) connecting the basins and the saddles on the 5D surface will produce the whole path.

Compared with immersion algorithm,^[6] our algorithm shares one similarity with immersion algorithm, which is the consideration of full-degree of freedom with no minimization. There are also some differences between those two algorithms: (i) Immersion algorithm considers the surface as a whole while we divide it into different areas. (ii) Immersion algorithm raises the water (energy level) step-by-step to see at which energy the exit point gets wet, which might be time consuming, while our algorithm just searches the saddle height on the watershed, a limited area.

After applying the watershed algorithm on surface division, we can simplify the 5D surface into a graph with nodes and edges for further depth-first search algorithm. The edges mentioned here is a concept in graph theory. On the edges, we have the saddle point height of two connected basins. The root node is the ground state. To choose which minimum represents the ground state, geometric deformation should be a criterion as many experiments show that ground-state shapes of heavy elements are almost spheroids. So a minimum with a compact spheroid deformation should be chosen as the root node.

Path search algorithm should know where the starting point is and where the path ends. The starting point is the ground state, which is much easier to be determined than the final state (or the scission "line" on the PES). The grids on the PES which satisfy both conditions are realy rare. But where the scission happens is hard to be found.

In the framework of the MM model, we need to calculate the macroscopic energy by liquid drop model (LDM) as the background and add microscopic correction on it. But when the neck of the deformation nucleus becomes thinner, only few nucleons are in neck region. The hypothesis of the LDM is challenged^[23,24] because it will lose physical meaning when the radius of the neck becomes comparable with the distance between the nucleons. The PES with constrains of zero neck cannot be precisely calculated in the MM model. So an alternative criterion should be that the nucleus scissions when its neck size is smaller than a critical value or equivalently its elongation is larger than another critical value. Some of the criteria are shown in Table 1.

Table 1.  Different scission criteria used by theorists. R^crit and D^crit are the critical values of neck and elongation, respectively. R₀ is the radius of a spherical nucleus. l and r in the third row are the shape parameters, which represent for the semi-length and radius of the deformed nucleus, respectively.

Author	Criterion
Strutinsky[25–28]	${R}^{\mathrm{crit}}\approx 0.24\sim 0.27{R}_{0}$, ${D}^{\mathrm{crit}}\approx 2.3\sim 2.38{R}_{0}$
Brosa[5]	Rayleigh criterion: $2l\gt 11r$, equivalently ${R}^{\mathrm{crit}}\approx 0.3\sim 0.4{R}_{0}$
Randrup[29]	$r\lt 2.5$ fm, ${R}_{0}\approx$ 7.9 fm for 236U

Rayleigh criterion is from hydrodynamics^[30,31] to describe the rupture of cylindrical column of an incompressible liquid. With the LDM hypothesis, Rayleigh criterion can be naturally migrated into fission study with some modifications.^[5] Table 1 lists three different criteria, which can be seen as equivalent, due to the volume conservation law. Rayleigh criterion is used in our optimal path search algorithm.

So the final state is a local minimum, which satisfies the Rayleigh criterion. Many local minima can be treated as potential final states, and there are also different paths ending at those minima. But only one path has the lowest fission barrier, which is the optimal fission path, and the corresponding minimum is the final state. With Rayleigh criterion, we reproduce the fission barriers of U and Pu isotopes, and most probable fission fragment mass splitting of U, Pu, Th and Cm even-even isotopes. Those results are consistent with experiments and evaluated data.

To find other fission paths beyond the optimal one as mentioned above, we should first make clear what is the definition of another fission path. The second optimal fission path should have a different fission barrier, which is higher than the optimal one. And the second-optimal path should end at a different final state. An addition neccesary condition is that these two paths are well separated on the PES.

According to the above-mentioned conditions, the basins connecting the barrier basin and the final state basin should be removed from the candidates before the searching of the second-optimal path, because a fission path will follow the optimal path once it is intersect with the optimal one. This removement would ensure that the search algorithm leads to a total different path. A constraint, the path can only contain one final state, should be added on the search algorithm. For the third-optimal fission path, we should continue to remove the basins connecting the barrier basin and the final state on the second-optimal path, until there is no way on the PES.

The division of the PES is sensitive to the noise, which might cause the over-division. A major improvement to overcome this problem is to set the threshold of the saddle height between two connected basins. Two connected basins with lower saddle height than the threshold should be considered as one basin. In this process, we merge the basins of blur identity into one larger basin with explicit identity. The threshold used here is a quite low barrier height of 0.3 MeV, lower than Möllerʼs^[6] 0.5 MeV. After this process, the paths are well separated.