Clusterization in Ternary Fission

Abstract

This lecture notes are devoted to the new kind of ternary decay of low excited heavy nuclei called by us “collinear cluster tri-partition” (CCT) due to the features of the effect observed, namely, decay partners fly away almost collinearly and at least one of them has magic nucleon composition. At the early stage of our work the process of “true ternary fission” (fission of the nucleus into three fragments of comparable masses) was considered to be undiscovered for low excited heavy nuclei. Another possible prototype—three body cluster radioactivity—was also unknown. The most close to the CCT phenomenon, at least cinematically, stands so called “polar emission”, but only very light ions (up to isotopes of Be) were observed so far.

Manifestations of new decay channel observed in the frame of different experimental approaches are discussed. Special attention is paid to the connection between conventional binary fission and CCT.

Appendix

6.1.1 A.1 Reliability of Linear Structures in the Scatter Plot of Fragments Masses

Due to the small number of events in the linear structures discussed in Figs. 6.24(a), 6.26, 6.27, 6.29, 6.30 the question arises whether the structures have a physical reality, i.e. if they are not a random sequence of points. In order to answer this question a special simulation based on Hough transformation was performed.

The Hough transform is a feature extraction technique used in image analysis, computer vision, and digital image processing [74, 75]. The simplest case of Hough transform is the linear transform for detecting straight lines. In the image space, the straight line can be described as y=mx+b and can be graphically plotted for each pair of image points (x, y). In the Hough transform, a main idea is to consider the characteristics of the straight line not as image points x or y, but in terms of its parameters, here the slope parameter m and the intercept parameter b. For computational reasons, it is better to parametrise the lines in the Hough transform with two other parameters, commonly referred to as R and θ (Fig. 6.45).

Fig. 6.45

Parameterization of the line in Hough transform

Actually, the straight line on a plane (Fig. 6.45) can be set as follows:

$$ x\times\cos(\theta) + y\times\sin(\theta) = R, $$

(6.5)

where R—the length of the perpendicular lowered on a straight line from the beginning of coordinates, θ—angle between a perpendicular to a straight line and the axis OX, changes within the limits of 0–2π, R is limited by the size of the entrance image.

In view of step-type representation of entrance data (in the form of a matrix with elements “1”—presence of a point, “0”—it absence), the phase space (R, θ) also is represented in a discrete kind. In this space the grid to which one bin corresponds a set of straight lines with close values of R and θ is entered. For each cell of a grid (R _i ,R _i+1)×(θ _i ,θ _i+1) (in other words for each Hough transform bin) the number of points with coordinates (x, y), satisfying to the equation (6.5), where θ _i ≤θ≤θ _i+1,R _i ≤R≤R _i+1, is counted up. The size of bins is obtained empirically.

Besides the steps on R and θ (ΔR, Δθ) in the real program code realizing Hough transform, there are the additional parameters responsible for the decision—whether all the points, satisfying Eq. (6.5), are necessary to attribute an analyzed straight line. So, if the distance in pixels (cells of the matrix under analysis) between extreme points of a line segment less than set number n—it is rejected. When the code finds two line segments associated with the same Hough transform bin that are separated by less than the set distance d, it merges them into a single line segment.

A lower part of the scatter plot shown in Fig. 6.27 below M ₁=40 amu was chosen for the analysis (Fig. 6.46). The straight line (marked by the arrow) united nine points was recognized using Hough transform algorithm at appropriate choice of the principal parameters (ΔR, Δθ, n, d).

Fig. 6.46

Part of the distribution shown in Fig. 6.26(a) chosen for estimation of a reliability of the line structures. Monte-Carlo simulation were performed in the circle region marked by the dash line. See text for details

We tried to estimate the probability of a random realization of the line of such length and tilted to the abscissa axis at an arbitrary angle. A sequence of patterns within randomly distributed points inside was generated. Each circular pattern included precisely the same number of points as those in the initial distribution in Fig. 6.46. An area of circular shape was chosen in order to avoid a priori distinguished direction (for instance, diagonal in a rectangular area). Each pattern was processed with the Hough transform algorithm “tuned” earlier on revealing the line under discussion. Among one hundred patterns analyzed only two of them provided a positive answer. In other words, a probability of a random realization of the line under discussion is about 2 %.

Another approach based on the methods of morphological analysis of images [106, 107] was applied as well in order to estimate the probability of random realization of the rectangle seen in Fig. 6.26(b). This probability was estimated to be less than 1 %.