1. On the cold polarized beam of the high-flux reactor at the Institut Laue-Langevin (ILL), the large collaboration of the institutes of Russia and Germany performed an experiment on searching the \(T\)-odd three-vector angular correlation in the ternary fission of nuclei \({}^{233}\)U:

$$W(\mathbf{P}_{\textrm{lf}},\mathbf{P}_{\alpha},\mathbf{S})=\textrm{Const.}\{1+D\mathbf{S}[\mathbf{P}_{\textrm{lf}}\times\mathbf{P}_{\alpha}]\},$$
(1)

where \(D\) is the correlation coefficient, \(\mathbf{S}\) is the unit vector in the direction of polarization of neutron beam, \(\mathbf{P}_{\textrm{lf}}\) and \(\mathbf{P}_{\alpha}\) are unit vectors in the directions of momenta of a light fragment and long-range \(\alpha\) particle of ternary fission, respectively. A great surprise, even for the authors, was the observation that \(D\) is not zero, but on the order of \(10^{-3}\) [1].

Of course, the discovered effect does not indicate the violation of the time invariance. Most probably, it can be explained by the Coriolis mechanism [2], because the fissioning nucleus rotates.

The revealed phenomenon was studied in detail, and it was of interest to measure the same correlation in ternary fission of nuclei \({}^{235}\)U. Such an experiment was posed on a beam of polarized cold neutrons of the ILL reactor in 2005. Nature presents another surprise to the authors: it appeared that, instead of the three-vector correlation detected at ternary fission of nuclei \({}^{233}\)U, at the ternary fission of nuclei \({}^{235}\)U a completely another effect was revealed, namely, the macroscopic effect of rotation of a polarized fissioning nucleus [3, 4]. This effect manifested itself as an asymmetry of the number of counts for the coincidence between signals from \(\alpha\)-particle detectors and signals from fission-fragment detectors with respect to the polarization direction of the neutron beam. The authors analyzed all possible instrumental effects and arrived at the explanation that the angular distributions (ADs) of \(\alpha\) particles mix in the direction towards or against the direction of rotation of the deformation axis of the fissioning nucleus at the time instance of scission of a neck connecting two future fragments. In the interpretation of the authors, the asymmetry of number of counts for coincidences measured in the experiment is the difference between the shifted ADs referred to their sum in the position of \(\alpha\) particle detector. This hypothesis was ‘‘confirmed’’ by the Monte Carlo calculations [5]. Goennenwein et al. [3] called this phenomenon the ROT effect (from rotation).

This discovery of a new phenomenon in the nuclear fission physics stimulated the formulation of experiments on searching a similar effect in emission of \(\gamma\) rays and neutrons by excited fragments. The sought effects of asymmetry of the number of counts for coincidences of these particles with fragments were indeed revealed. We do not explain how \(\gamma\) rays and neutrons emitted not at the instance of neck scission may demonstrate ROT effects; we refer the interested reader to papers [6–8]. The authors of a majority of publications in this topic, including the author of this paper, in their initial publications, following work [3], explained the discovered asymmetries by the shift in ADs of registered particles in the direction of rotation of the nucleus. In fact, we can provide a very simple and vivid explanation of the mechanism of appearance of asymmetry of the number of counts for coincidences between a particle and a fragment when the polarization direction of the neutron beam is reversed. Below, we describe this mechanism.

2. A nucleus with a nonzero spin rotates, and, if it is polarized, then there arises the rotation axis \(Z\). Fission nuclei are deformed, and the deformation axis must rotate about this preferred axis. In fission of a rotating nucleus, at the initial time instance of neck scission because of the orbital rotation of asymmetric ‘‘dumbbell’’, the fission fragments acquire a tangential component of the velocity, and, consequently, their trajectories, instead of straightlined ones in the direction of the deformation axis caused by Coulomb repulsion of fragments, become hyperbolic, which leads to a shift in the trajectory of fragments from the direction of the deformation axis at the time instance of neck scission by a small angle. This angle may be used to determine the rotation velocity of the fissioning nucleus; however, for this purpose, we need to know the direction of the deformation axis at the time instance of neck scission and the configuration of the asymmetric dumbbell.

Nuclear fission, as a rule, is accompanied by emission of the particle whose AD is formed with respect to the deformation axis at the time instance of the neck scission or correlates with it and, as a rule, is independent of the trajectories of fragments. An exception is the AD of \(\alpha\) particle of ternary fission, because at the initial instance a low-energy \(\alpha\) particle can be involved by the Coulomb field of fragments in rotation about the fission axis. It is the AD of the particle that is the label of direction of the deformation axis at the time instance of the neck scission. The trajectories of fragments deviate relative to exactly this direction.

3. The installations intended to measure the ROT effects are certainly diverse from the point of view of the used measurement methods. However, for the sake of simplicity of our arguments, we provide a schematic design of the installation in Fig. 1. The target containing the fission nuclei and the detectors of fission fragments are placed in the ‘‘plane’’ orthogonal to the beam of longitudinally polarized neutrons.

Fig. 1
figure 1

Schematic design of the experimental installation for detecting the ROT effect. 1, detectors of conditional particles accompanying nuclear fission; 2, detectors of nuclear fission fragments; 3, target containing fission nuclei; 4, profile of beam of longitudinally polarized neutrons; 5, working gas used in fragment detectors; and 6, housing of the fission chamber.

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The fission chamber is surrounded by the detectors of particles accompanying the act of nuclear fission. We register the coincidences between the signals from the detectors of fragments and the signals from all detectors of particles. For the sake of simplicity, we consider a situation with coincidence of the signal from a fragment with the signal from one of the detectors of particles in the upper hemisphere. A particle accompanying the nuclear fission must be emitted from the neck at the time instance of its scission (scission point) and, clearly, from the ‘‘point’’ characterizing the center of mass of the asymmetric dumbbell. Because the latter rotated at the time instance of neck scission, this rotation affects the trajectory of the particle to a less degree than the trajectory of the fragment. Let us assume that the considered detector of particles is located to the left of the maximum of AD with respect to the conventional AD of the particle (Fig. 2). We think that the fissioning nucleus rotates in the clockwise direction if we look at the installation scheme. The particle falls into the detector along the ‘‘rotating’’ trajectory also in the clockwise direction, but slightly deviating from the straightlined one, whereas the fragment deviates in the clockwise direction by a small angle \(\theta\). We can easily see in Fig. 3 that, due to the difference in the effects of shift in the trajectories of the particle and fragment, the angle between the ‘‘points’’ of registration of the particle and fragment in the corresponding detectors increased. Keeping in mind that the AD of a particle is nothing else as the probability of detecting a particle at a given angle, we arrive at the conclusion that, at rotation of the asymmetric dumbbell in the clockwise direction, the probability of registering a particle at the position of the considered detector increased. Obviously, when a fissioning nucleus rotates in the counterclockwise direction, the angle decreases (Fig. 3), and, consequently, the probability of registration reduces. Such a situation takes place for all particle detectors in the upper hemisphere of the experimental installation. In the lower hemisphere the situation is opposite. It is clear that the counting rate for coincidences in a counter corresponding to the rotation of the dumbbell in the clockwise direction must increase, because the probability of registration of a particle increased. Therefore, the counting rate for coincidences in a counter corresponding to rotation of the dumbbell in the counterclockwise direction decrease. The difference of the counting rates in these two counters of events of coincidence between the particle and fragment referred to their sum is the counting-rate asymmetry measured in the experiment for the particle detector at a given angle to the detector of fragments with respect to the polarization direction of the neutron beam.

Fig. 2
figure 2

Schematic representation of AD of registered particles in coincidence with a fragment.

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Fig. 3
figure 3

Schematic image of shifts of particle and fragment. (\(a\)) The neutron beam is not polarized. The fissioning nucleus does not rotate. (\(b\)) The nucleus rotates in the clockwise direction. The particle deviates by a small angle, whereas the fragment deviates by the angle \(\theta\). The angle between the particle and fragment increases, compared to the left panel. (\(c\)) The nucleus rotates in the counterclockwise direction. The angle between the particle and fragment decreases, compared to the left panel.

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4. Above, we have considered a special case of AD of particles. In general, the sign and magnitude of the asymmetry of the number of counts for coincidences for each particle detector depend on the sign and value of the derivative of AD of particles at the place of a given particle detector. The asymmetry coefficient as a function of angle between the detectors of particle and fragment can be written as

$${A}$$
(2)
$${}=\{N^{+}(\varphi)-N^{-}(\varphi)\}/\{N^{+}(\varphi)+N^{-}(\varphi)\}$$
$${}=2\theta\{[{\partial F}(\varphi)/{\partial}\varphi]\}/F(\varphi)].$$

Here, \(N^{+}\)(\(\varphi\)) is the number of coincidences of signals from the detector of particles located at an angle of \(\varphi\) to the axis of abscissa at a positive shift in the trajectory of fragment, \(N^{-}\)(\(\varphi\)) is that at the negative shift, and \(F(\varphi)\) is a function describing the AD of a particle.

Knowing \(F(\varphi)\), we can compute \(\theta\). It is clear that the values of derivatives of the AD at different angles depend on the anisotropy coefficient of AD. Unlike the ternary fission of nuclei, where the AD anisotropy is rather large, for \(\gamma\) rays and neutrons the anisotropy is not so large; therefore, the asymmetry measured in the experiment, as well as the angle \(\theta\), are very small.

It is worth noting that in all experiments on revealing the ROT effect, the only measured value is the asymmetry of the number of counts for coincidences of the signals from the detectors of particles with the signals from detectors of fragments of nuclear fission. It is clear from the above discussed that, to explain the asymmetry observed in the experiment, we need not employ the shifts in AD. However, if we use the data of event counters of coincidences between the particle and fragment at two opposite directions of neutron beam polarization to construct the AD corresponding to each of the directions of neutron beam polarization, then we find out that the constructed ADs of the particle are shifted with respect to each other by an angle of 2\(\theta\), as it was shown in Fig. 4. This shift in ADs appeared due to shift in the trajectory of the fragment, but in the literature it is given as a consequence of AD ‘‘rotation’’.

Fig. 4
figure 4

Angular distribution of conventional particles accompanying fission. 1 and 2 are the angular distributions at rotation of the nucleus in the clockwise and counterclockwise directions, respectively; 3, their difference in an enlarged scale.

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As an example of applying the proposed model of the ROT effect, we consider the correlation of signs of the asymmetry coefficients at ternary fission of \({}^{235}\)U nuclei published in work [3]. We borrowed Fig. 5 from this paper. In this figure the experimental installation is schematically shown that was used to reveal the ROT effect at ternary fission of \({}^{235}\)U nuclei. The signs of the asymmetry coefficients shown in the upper part of the installation were obtained with the direction of the beam of longitudinally polarized neutrons denoted by \(\boldsymbol{\sigma}_{+z}\), which means that the rotation of the fissioning nucleus is in the clockwise direction if we look at the figure. The symmetric angular distribution of \(\alpha\) particles has a maximum at 82\({}^{\circ}\); therefore, the left group of detectors of \(\alpha\) particles was placed at 68\({}^{\circ}\), whereas the right one was placed at an angle of 112\({}^{\circ}\). We considered coincidences between the signals from the detectors of \(\alpha\) particles and the signals from the detectors registering light fragments. As noted above, at the initial time instance of its production, a low-energy \(\alpha\) particle is entrained by the rotating Coulomb field of fragments and rotates by a small angle much smaller than the angle of shift in the light fragment. For simplicity of our arguments, in the first approximation we ignore this angle. For the detectors placed at an angle of 68\({}^{\circ}\), the angle between the \(\alpha\) particle and the shifted fragment increases, and, consequently, the probability of registering an \(\alpha\) particle increases too. When we reverse the direction of the neutron beam polarization, on the contrary, the angle between the \(\alpha\) particle and the shifted fragment decreases; consequently, the probability of registration decreases. The set of events of the considered phenomenon shows that the measured asymmetry is positive, as is shown in the scheme. In order not to overload the reader with redundant information, we report that, for the group of detectors of \(\alpha\) particles placed at an angle of 112\({}^{\circ}\), the situation is inverse, and the signs of the asymmetry coefficients appear to be negative, as also shown in the figure. Clearly, the pattern for the lower group of detectors of \(\alpha\) particles is asymmetric to the pattern for the upper group.

Fig. 5
figure 5

Image of correlations of signs of asymmetry coefficients at measurements of the ROT effect in ternary fission of \({}^{235}\)U nuclei with registration of coincidences of the signals from \(\alpha\) detectors with the signals from the detectors of fragments. The pattern reproduces the sign correlation in measurements in the neutron beam polarization direction \(\boldsymbol{\sigma}_{+z}\) when the nucleus rotates in the clockwise direction and in the reverse direction \(\boldsymbol{\sigma}_{-z}\) when the nucleus rotates in the counterclockwise direction.

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5. Thus, above, we showed that the qualitative explanation of the ROT effect developed in this work and applied to the ternary fission of \({}^{235}\)U nuclei rather satisfactory agrees with the experimental data obtained in work [3]. Of course, our task was not to perform calculation of the absolute values of the asymmetry coefficients. However, the proposed alternative explanation of the nature of the ROT effect will be useful for interpretation and planning of new experiments.