Collinear cluster tri-partition: Kinematics constraints and stability of collinearity

Background: A new mode of nuclear fission has been proposed by the FOBOS Collaboration, called collinear cluster tri-partition (CCT), and suggests that three heavy fission fragments can be emitted perfectly collinearly in low-energy fission. This claim is based on indirect observations via missing-energy events using the $2v2E$ method. This proposed CCT seems to be an extraordinary new aspect of nuclear fission. It is surprising that CCT escaped observation for so long given the relatively high reported yield of roughly $0.5\%$ relative to binary fission. These claims call for an independent verification with a different experimental technique.

Purpose: Verification experiments based on direct observation of CCT fragments with fission-fragment spectrometers require guidance with respect to the allowed kinetic-energy range, which we present in this paper. Furthermore, we discuss corresponding model calculations which, if CCT is found in such verification experiments, could indicate how the breakups proceed. Since CCT refers to collinear emission, we also study the intrinsic stability of collinearity.

Methods: Three different decay models are used that together span the timescales of three-body fission. These models are used to calculate the possible kinetic-energy ranges of CCT fragments by varying fragment mass splits, excitation energies, neutron multiplicities, and scission-point configurations. Calculations are presented for the systems ${}^{235}\mathrm{U}(n_{\text{th}},f)$ and ${}^{252}\mathrm{Cf}\left(sf\right)$, and the fission fragments previously reported for CCT; namely, isotopes of the elements Ni, Si, Ca, and Sn. In addition, we use semiclassical trajectory calculations with a Monte Carlo method to study the intrinsic stability of collinearity.

Results: CCT has a high net $Q$ value but, in a sequential decay, the intermediate steps are energetically and geometrically unfavorable or even forbidden. Moreover, perfect collinearity is extremely unstable, and broken by the slightest perturbation.

Conclusions: According to our results, the central fragment would be very difficult to detect due to its low kinetic energy, raising the question of why other $2v2E$ experiments could not detect a missing-mass signature corresponding to CCT. Considering the high kinetic energies of the outer fragments reported in our study, direct-observation experiments should be able to observe CCT. Furthermore, we find that a realization of CCT would require an unphysical fine tuning of the initial conditions. Finally, our stability calculations indicate that, due to the pronounced instability of the collinear configuration, a prolate scission configuration does not necessarily lead to collinear emission, nor does equatorial emission necessarily imply an oblate scission configuration. In conclusion, our results enable independent experimental verification and encourage further critical theoretical studies of CCT.

Figure 1

Plots (a) and (b) show the $Q$ value versus the final mass split between the lightest fragments in the decays in Eqs. (1) and (2), respectively, at zero neutron multiplicity ($\nu =0$). The $Q$ values are calculated from mass excesses taken from AME2012 [25]. No data are available for the bottom-left corner (i.e., for masses $A_{\text{Sn}}>138$). Prompt-neutron emission $\nu >0$ generally lowers the $Q$ values (see Fig. 3).

Figure 2

Schematic picture of the formation of CCT. For long timescales between two successive (sequential) breakups, there is an intermediate state (2). For sufficiently short timescales between breakups, there is no intermediate state, and the decay is a true three-body decay. Arrows indicate momentum direction. See text for explanation of acronyms.

Figure 3

The $Q$ value versus the mass split in the binary decays of (a) ${}^{235}\mathrm{U}(n_{\text{th}},f)$, (b) ${}^{252}\mathrm{Cf}\left(sf\right)$, (c) ${}^{104}\mathrm{Mo}$, and (d) ${}^{120}\mathrm{Cd}$. The $Q$ values are calculated from the mass excesses, taken from AME2012 [25]. Lines have been added to guide the eye.

Figure 4

CCT as an almost-sequential decay. In contrast to the sequential model, the Coulomb repulsion of the heavy fragment is crucial after the second scission, making the interfragment distances relevant to the kinematics. Arrows indicate momentum direction.

Figure 5

CCT as a true ternary decay. Arrows indicate momentum direction.

Figure 6

CCT as a true ternary decay, with an initial lateral momentum or spatial offset of the ternary particle from the fission axis. The arrow indicates momentum direction.

Figure 7

Final kinetic energies of the TP and LF versus the TP to LF mass split in the sequential decay of (a), (b) ${}^{235}\mathrm{U}(n_{\text{th}},f)$ and (d), (e) ${}^{252}\mathrm{Cf}\left(sf\right)$, calculated with the analytic method ($◯$, $△$) described in Sec. 3a (derivations in Appendix pp1), and with trajectory calculations (+, $\times$) described in Sec. 3b (derivations in Appendix pp1). The case when the TP is formed at the center is shown in panels (a) and (d), while the case when the LF is formed at the center is shown in panels (b) and (e) [upper and lower signs in Eqs. (A30) and (A31), respectively]. The corresponding final kinetic energy of the HF is labeled in each plot. The excitation energy of the intermediate fragment is $E_{IF}^{*}=30$ MeV. The missing trajectory calculations for $A_{\text{Si}}=28$ highlights that the intermediate steps of the decay are energetically forbidden. The corresponding $Q$ values in the fission of (c) ${}^{104}\mathrm{Mo}$ and (f) ${}^{120}\mathrm{Cd}$ are calculated from mass excesses taken from AME2012 [25]. Lines have been added to guide the eye.

Figure 8

The $Q$ value as a function of the mass split between the LF and the TP, in the decays (a) $\text{Mo}\to \text{Si}+\text{Ni}$ and (b) $\text{Cd}\to \text{Ca}+\text{Ni}$. Recall from Fig. 1 that masses under the dashed line correspond to nuclides without data for the corresponding HF ($A_{\text{Sn}}>138$), and from Fig. 3 that the most favorable mass splits are $A_{\text{Sn}}=132$ with (a) $A_{\text{Mo}}=104$ and (b) $A_{\text{Cd}}=120$. The most favorable mass split between the TP and LF are therefore found along the solid diagonal lines as $A_{\text{Ni}}=70$ with (a) $A_{\text{Si}}=34$ and (b) $A_{\text{Ca}}=50$. The $Q$ values are calculated from mass excesses taken from AME2012 [25]. Prompt neutron emission from the IF (${\nu }_2$) lowers the $Q$ values significantly (see Fig. 3).

Figure 9

Final kinetic energies of the LF and the TP versus their mass split in the sequential decay of (a) ${}^{235}\mathrm{U}(n_{\text{th}},f)$ and (b) ${}^{252}\mathrm{Cf}\left(sf\right)$, when the TP is formed at the center. The kinetic energies are calculated with the analytic method ($◯, △$) described in Sec. 3a (derivations in Appendix pp1), and with trajectory calculations (+, $\times$) described in Sec. 3b (derivations in Appendix pp1). Comparing plots in the horizontal direction, the mass split between the HF and the IF is varied and, comparing plots in the vertical direction, $E_{IF}^{*}$ is varied. The values of these parameters and the final kinetic energy of the HF are labeled in each plot. Trajectory calculations are only shown for systems that are energetically allowed and have a tip distance at the second scission of $\le 7$ fm.

Figure 10

Areas of attainable final fragment kinetic energies versus the excitation energy of the intermediate fragment in the sequential decays (a)–(c) ${}^{235}\mathrm{U}(n_{\text{th}},f)\to \text{Sn}+\text{Mo}+{\nu }_{1}n\to \text{Sn}+\text{Si}+\text{Ni}+{\nu }_{1}n$, and (d)–(f) ${}^{252}\mathrm{Cf}\left(sf\right)\to \text{Sn}+\text{Cd}+{\nu }_{1}n\to \text{Sn}+\text{Ca}+\text{Ni}+{\nu }_{1}n$. Each figure is element specific and shows results both when the TP and when the LF are formed at the center. The colors indicate formation position, as shown by the inset in panel (f). The areas are spanned between the highest and lowest kinetic energies obtained. For a given set of parameters, the final kinetic energy versus $E_{IF}^{*}$ follows a well-defined line. The bold dashed line is an example of the latter, with the most favorable set of parameters. For comparison, the horizontal lines correspond to the kinetic energy of the same fragment from compact binary fission [see text and Eqs. (10, 11, 12, 13, 14, 15)]. The masses are varied as $A_{\text{Sn}}=128–134$ and $A_{\text{Ni}}=68,70,72$, and the prompt neutron multiplicity as ${\nu }_1=0–4$. As a consequence, the other masses are in the ranges (a)–(c) $A_{\text{Mo}}=98–108, A_{\text{Si}}=28–40$, and (d)–(f) $A_{\text{Cd}}=114–124, A_{\text{Ca}}=42–56$.

Figure 11

Final kinetic energies versus the time between the first and the second scission (logarithmic scale), in the almost-sequential decays (a) ${}^{235}\mathrm{U}(n_{\text{th}},f)\to {}^{132}\mathrm{Sn}+{}^{104}\mathrm{Mo}\to {}^{132}\mathrm{Sn}+{}^{34}\mathrm{Si}+{}^{70}\mathrm{Ni}$, and (b) ${}^{252}\mathrm{Cf}\left(sf\right)\to {}^{132}\mathrm{Sn}+{}^{120}\mathrm{Cd}\to {}^{132}\mathrm{Sn}+{}^{50}\mathrm{Ca}+{}^{70}\mathrm{Ni}$. The excitation energies are $E_{IF}^{*}=40$ MeV and $\mathit{TXE}=0$ MeV, the tip distance at the second scission is $\mathrm{\Delta }{d}_0=7$ fm, and the ternary particle is set to be formed at the center. The fine solid and dashed lines represent the corresponding kinetic energies of the sequential and true ternary decay models, respectively. See text for further explanation of how these results are obtained. Less than $1\%$ of the total energy remains as potential energy in the almost-sequential and ternary model results.

Figure 12

The maximal available energy versus the corresponding scission tip distance in the fission processes (a) ${}^{235}\mathrm{U}(n_{\text{th}},f)\to {}^{132}\mathrm{Sn}+{}^{104}\mathrm{Mo}$ (solid lines) and ${}^{252}\mathrm{Cf}\left(sf\right)\to {}^{132}\mathrm{Sn}+{}^{120}\mathrm{Cd}$ (dashed lines), as well as in (b) ${}^{104}\mathrm{Mo}\to {}^{34}\mathrm{Si}+{}^{70}\mathrm{Ni}$ (solid lines) and ${}^{120}\mathrm{Cd}\to {}^{50}\mathrm{Ca}+{}^{70}\mathrm{Ni}$ (dashed lines). Results are shown both with and without an attractive nuclear interaction (bold and fine lines, respectively). Panel (c) shows a contour plot of the maximal available energy after fission of the IF [Eq. (21)] for a certain scission tip distance $\mathrm{\Delta }{d}_0$, assuming the maximum $E_{\text{IF}}^{*}$ has been provided. The latter is set by the tip distance $\mathrm{\Delta }{D}_0$ of the first fission. The first and the second fissioning systems are ${}^{235}\mathrm{U}(n_{\text{th}},f)$ and ${}^{104}\mathrm{Mo}$, respectively, and the contour lines show the available energy in MeV. In (d), the shaded regions show the $\mathrm{\Delta }{D}_0$ and $\mathrm{\Delta }{d}_0$ which lead to geometrically and energetically allowed fission of the IF, both for ${}^{235}\mathrm{U}(n_{\text{th}},f)$ (solid lines) and ${}^{252}\mathrm{Cf}\left(sf\right)$ (dashed lines). See text for further information.

Figure 13

Final fragment kinetic energies versus the relative starting position of the TP, $x_r$, in the true ternary decay model (as described in Sec. 3c, derivations in Appendix pp2) of (a) ${}^{235}\mathrm{U}(n_{\text{th}},f)$ and (b) ${}^{252}\mathrm{Cf}\left(sf\right)$. When $x_r=0$ and $x_r=1$, the TP starts touching the HF and LF, respectively. Less than $1\%$ of the total energy remains as potential energy, and $\mathit{TXE}=0$ MeV.

Figure 14

Areas of attainable final kinetic energies of the fission fragments versus the total excitation energy of all fragments in the true ternary decays (a)–(c) ${}^{235}\mathrm{U}(n_{\text{th}},f)\to \text{Sn}+\text{Si}+\text{Ni}$ and (d)–(f) ${}^{252}\mathrm{Cf}\left(sf\right)\to \text{Sn}+\text{Ca}+\text{Ni}$. Each figure is element specific, and the different areas indicate the choice of $x_r$, as indicated in the legend in panel (d). The areas are spanned between the highest and lowest kinetic energies obtained for each choice of $x_r$, by varying the neutron multiplicity as $\nu =0–4$ and the mass split as $A_{\text{Sn}}=128–134, A_{\text{Ni}}=68,70,72$, (a)–(c) $A_{\text{Si}}=34–40$, and (d)–(f) $A_{\text{Ca}}=42–56$. Less than $1\%$ of the total energy remains as potential energy.

Figure 15

Final emission angle between the light fragment and the ternary particle versus the initial lateral fission axis offset $y$ of the ternary particle when the ternary particle is initially closer to (a) the heavy fragment ($x_r<0.5$), and (b) the light fragment ($x_r\ge 0.5$), respectively. The decays are ${}^{235}\mathrm{U}(n_{\text{th}},f)\to {}^{132}\mathrm{Sn}+{}^{34}\mathrm{Si}+{}^{70}\mathrm{Ni}$ and ${}^{252}\mathrm{Cf}\left(sf\right)\to {}^{132}\mathrm{Sn}+{}^{48}\mathrm{Ca}+{}^{72}\mathrm{Ni}$, as indicated by the legend. The plots are generated according to Appendix pp3, by varying $x_r=0–1, \mathit{TXE}=0–30$ MeV, and $y=0–1$ fm. All initial momenta are set to zero.

Figure 16

Final emission angle between the light fragment and the ternary particle versus the initial lateral kinetic energy of the ternary particle when the ternary particle is initially closer to (a) the heavy fragment ($x_r<0.5$), and (b) the light fragment ($x_r\ge 0.5$), respectively. The decays are ${}^{235}\mathrm{U}(n_{\text{th}},f)\to {}^{132}\mathrm{Sn}+{}^{34}\mathrm{Si}+{}^{70}\mathrm{Ni}$ and ${}^{252}\mathrm{Cf}\left(sf\right)\to {}^{132}\mathrm{Sn}+{}^{48}\mathrm{Ca}+{}^{72}\mathrm{Ni}$, as indicated by the legend. The plots are generated according to Appendix pp3, by varying $x_r=0–1, \mathit{TXE}=0–30$ MeV, and $E_{\text{K},0}=0–0.05$ MeV. The total linear and angular momentum is set to zero, and all particles are initially set perfectly collinearly.

Figure 17

CCT as a sequential or an almost-sequential decay, as described in Secs. 3a and 3b, respectively. In contrast to the former model, the Coulomb repulsion between the fragments is relevant at all times in the latter model, making the interfragment distances relevant to the kinematics. Arrows indicate momentum direction.

Figure 18

CCT as a true ternary decay, as described in Sec. 3c. Note that $x_r$ is a relative and dimensionless coordinate, defined between 0 and 1, corresponding to the TP “touching” the HF and the LF, respectively.

Figure 19

CCT as a true ternary decay with a perturbation in the ternary particle position or momentum, as described in Sec. 3d. Note that $x_r$ is a relative and dimensionless coordinate, defined between 0 and 1, corresponding to the TP “touching” the HF and the LF, respectively.