Theoretical Approaches Making It Possible to Describe Simultaneously ${P}$-Even ${T}$-Odd Asymmetries in Nuclear-Fission Processes Induced by Polarized Neutrons and Accompanied by the Emission of Various Light Particles

Abstract

In a special laboratory frame, the differential cross sections ${d\sigma_{nf,\nu}\left(\theta\right)}/{d\Omega}$ for the fission of nonoriented target nuclei that is induced by cold polarized neutrons $n$ and which is accompanied by the emission of light particles $\nu$, such as prescission alpha particles or prompt neutrons $n^{\prime}$, and photons can be represented as the sum of two terms. The first term is equal to the cross section for the analogous reaction induced by unpolarized neutrons, ${d\sigma_{nf,\nu}^{\left\{0\right\}}\left(\theta\right)}/{d\Omega}=\sigma_{nf,\nu}^{\left\{0\right\}}P_{\nu}^{\left\{0\right\}}\left(\theta\right)$, where $\sigma_{nf,\nu}^{\left\{0\right\}}$ is the total cross section for this reaction and $P_{\nu}^{\left\{0\right\}}\left(\theta\right)$ is the angular distribution of light particles $\nu$ emitted in this reaction. The second term in this sum, ${d\sigma_{nf,\nu}^{\left\{1\right\}}\left(\theta\right)}/{d\Omega}$, depends linearly on the neutron polarization vector $\boldsymbol{\sigma}_{n}$ and describes $P$-even $T$-odd asymmetries in the original cross section. By employing the concepts of space isotropy and parity conservation, it is possible to represent the cross section ${d\sigma_{nf,\nu}^{\left\{1\right\}}\left(\theta\right)}/{d\Omega}$ as the sum of two scalar functions, $d\sigma_{nf,\nu}^{\left\{1\right\}}\left(\theta\right)/d\Omega=\left({d\sigma_{nf,\nu}^{\left\{1\right\}}\left(\theta\right)/d\Omega}\right)_{\textrm{ev}}+\left({d\sigma_{nf,\nu}^{\left\{1\right\}}\left(\theta\right)/d\Omega}\right)_{\textrm{odd}}$, which are related to the correlation functions ($\boldsymbol{\sigma}_{n}[\mathbf{k}_{\mathrm{LF}},\mathbf{k}_{v}]$) and ($\boldsymbol{\sigma}_{n}[\mathbf{k}_{\mathrm{LF}},\mathbf{k}_{v}]$)($\mathbf{k}_{\mathrm{LF}},\mathbf{k}_{v}$) that are, correspondingly, even and odd under the transformation $\theta\to\pi-\theta$, where $\mathbf{k}_{\mathrm{LF}}$ and $\mathbf{k}_{\nu}$ are the wave vectors of the light fission fragment and the light particle, respectively. These correlation functions can be expressed in terms of the quantities $\left({\beta_{nf,\nu}\left(\theta\right)}\right)_{\textrm{ev}\left({\textrm{odd}}\right)}\equiv(d\sigma_{nf,\nu}^{\left\{1\right\}}\left(\theta\right)/d\Omega)_{\textrm{ev}\left({\textrm{odd}}\right)}/\sigma_{nf,\nu}^{\left\{0\right\}}$, whose experimental values can be found from the experimental values of the asymmetry coefficient $D_{nf,\nu}\left(\theta\right)$ introduced earlier [1] and the angular distribution $P_{\nu}^{\left\{0\right\}}\left(\theta\right)$ of light particles by employing the relation $\left({\beta_{nf,\nu}\left(\theta\right)}\right)_{\textrm{ev}\left({\textrm{odd}}\right)}=\left({D_{nf,\nu}\left(\theta\right)P_{\nu}^{\left\{0\right\}}\left(\theta\right)}\right)_{\textrm{ev}\left({\textrm{odd}}\right)}$. Within the quantum-mechanical approach, the theoretical values of $\left({\beta_{nf,\nu}\left(\theta\right)}\right)_{\textrm{ev}\left({\textrm{odd}}\right)}$ can be obtained by means of the formula $\left({\beta_{nf,\nu}\left(\theta\right)}\right)_{\textrm{ev}\left({\textrm{odd}}\right)}=\Delta_{\nu,\textrm{ev}\left({\textrm{odd}}\right)}\dfrac{d}{d\theta}\left({P_{\nu,\textrm{ev}\left({\textrm{odd}}\right)}^{\left\{0\right\}}\left(\theta\right)}\right)$, which takes into account the angle $\Delta_{\nu,\textrm{ev}\left({\textrm{odd}}\right)}$ of rotation of the light-particle wave vector $\mathbf{k}_{\nu}$ about the wave vector $\mathbf{k}_{\mathrm{LF}}$ of the light fission fragment under the effect of Coriolis interaction associated with the collective rotation of the fissioning system about the axis orthogonal to its symmetry axis. The angle of rotation is determined from a comparison of the experimental and theoretical values of $\left({\beta_{nf,\nu}\left(\theta\right)}\right)_{\textrm{ev}\left({\textrm{odd}}\right)}$ with the aid of the maximum-likelihood method. Because of taking into account quantum interference effects, $\Delta_{\nu,\textrm{ev}\left({\textrm{odd}}\right)}$ may in general take not only positive (as is the case within the semiclassical method of trajectory calculations [1]) but also negative values. The use of this result permits reaching reasonable agreement between the experimental and theoretical values of $\left({\beta_{nf,\nu}\left(\theta\right)}\right)_{\textrm{ev}\left({\textrm{odd}}\right)}$ for all $\nu$ particles simultaneously in the cases of ${}^{\mathrm{235}}$U, ${}^{\mathrm{239}}$Pu, and ${}^{\mathrm{241}}$Pu target nuclei. In the case of the ${}^{\mathrm{233}}$U target nucleus, however, the quantity$\left({\beta_{nf,\alpha}\left(\theta\right)}\right)_{\textrm{ev}}$, which is independent of the angle $\theta$, should be supplemented with $\left({\widetilde{\beta}_{nf,\alpha}}\right)_{\mathrm{ev}}$ in order to reach the same degree of agreement. The appearance of the latter may in principle be due [1] to the violation of axial symmetry of the fissioning system because of the effect of its bending and wriggling vibrations in the vicinity of the scission point.