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Description
In experimental papers [1, 2] the yields, angular and energy distributions of the pairs of light third and fourth particles, such as $\alpha$-particles pair $(\alpha_1,\alpha_2)$, were obtained for the spontaneous quaternary fission of the nuclei $^{252}$Cf, $^{248}$Cm and for the induced by thermal neutrons quaternary fission of compound nuclei $^{234}$U, $^{236}$U. Using the theoretical concepts [3-5] of ternary and quaternary fission as virtual processes [6], we consider spontaneous quaternary fission from the ground states of even-even actinides [1,2] with the sequential emission of two $\alpha$-particles from nuclei A and (A-4) with the formation of the intermediate nuclei $(A-4)$ and $(A-8)$ in the virtual states, and the subsequent binary fission of the residual fissile nucleus $(A-8)$ into light and heavy fission fragments. Induced quaternary fission occur from the excited states of compound nucleus $A$, which is formed when the neutron is captured by the target nucleus, and after that the process goes in the same way as in analogous spontaneous fission. These $\alpha$-particles, in contrast to the $\alpha$-particles that fly out in the sub-barrier $\alpha$-decay from ground states of the studied nuclei $A$ and $(A-4)$, when the energies $Q_{\alpha_1}^A$ and $Q_{\alpha_2}^{(A-4)}$ of this decays are close to $4-6$ MeV, are long-ranged, since their asymptotic kinetic energies $T_{\alpha_1} \approx 16 $ MeV and $T_{\alpha_2} \approx 13 $ MeV, are markedly larger than energy values $Q_{\alpha_1}^A$ and $Q_{\alpha_2}^{(A-4)}$.
The quaternary fission yield $N_{\alpha \alpha f}^A$ normalized to the yield of the binary fission of the nucleus $A$ for spontaneous fission using the formula [4] for the virtual quaternary fission width of nucleus $A$ can be presented as
$
N_{\alpha \alpha f}^A=\frac{1}{(2\pi)^2}\int\int\frac{(\Gamma_{\alpha_1}^A)^{(0)}(T_{\alpha_1})(\Gamma_{\alpha_2}^{(A-4)})^{(0)}(T_{\alpha_2})(\Gamma_{f}^{(A-8)})^{(0)}}
{(Q_{\alpha_1}^A-T_{\alpha_1})^2(Q_{\alpha_2}^{(A-4)}-T_{\alpha_2})^2(\Gamma_{f}^A)^{(0)}}dT_{\alpha_1}dT_{\alpha_2}, (1)
$
where index (0) denotes to the configuration of fissile nuclei, corresponding to the appearance of two deformed fission prefragments, connected by the neck; $(\Gamma_{\alpha_1}^{A})^{(0)}$ and $(\Gamma_{\alpha_2}^{(A-4)})^{(0)}$ are the width of the $\alpha$-emission from the fissile nucleus neck. In (1) the ratio of the binary fission widths $ (\Gamma_{f}^{A})^{(0)}/(\Gamma_{f}^{(A-8)})^{(0)}\approx 1$. In the case of the induced quaternary fission the energy $Q_{\alpha}^A$ should be replaced by $Q_{\alpha}^A+B_n$, where $B_n$ is neutron binding energy in compound nucleus A. Using Gamov formulae for $(\Gamma_{\alpha_1}^{A})^{(0)}$ and $(\Gamma_{\alpha_2}^{(A-4)})^{(0)}$ , taking into account the fact that the probabilities of formation of the $\alpha_1$ and $\alpha_2$ particles are close to each other and the neck radius $r_{neck}^{A}$ before the emission of $\alpha_1$-particle does not differ from the neck radius $r_{neck}^{(A-4)}$ before the emission of the second $\alpha_2$-particle, the specified estimation of the yield $N_{\alpha \alpha f}$ for spontaneousand induced quaternary can be derived.
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P. Jesinger et al., Eur. Phys. J. A. 2005. V. 24. P. 379.
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M. Mutterer et al., in Proceedings of "Dynamic. Aspects of Nuclear Fission", Slovakia, 2002, p. 191.
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S.G. Kadmensky, L.V. Titova / Physics of Atomic Nuclei. 2013. V. 76. P.16.
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S.G. Kadmensky, O.A. Bulychev / Bull. of RAS: Physics. 2016. V. 80. P. 921.
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S.G. Kadmensky, L.V. Titova, D.E. Lyubashevsky, Phys. Atom. Nucl. 83, 298 (2020).
