From Current Literature

Neutron–Electron Interaction

The forces acting between a neutron and an electron cannot be Coulomb forces, since the neutron has no electric charge. Nor can these forces be of nuclear character, since the electron is not endowed with a nucleonic charge. The interaction under consideration is, basically, explained by two causes. First, the neutron has an anomalously large magnetic moment, equal to \(\mu_n = 1.91\) nuclear magnetons. Therefore the forces of interaction of the neutron magnetic moment with the electromagnetic field of the electron must manifest themselves. Second, in accordance with meson theory, the neutron undergoes continuous transformations into a proton and a \(\pi\)-meson

\[ n \rightleftarrows p + \pi^- . \]

From an estimate of the magnetic moments of the particles one comes to the conclusion that approximately 20% of the time the neutron is in the dissociated state. Therefore, when the electric field of the electron penetrates into the meson cloud of the dissociated neutron (of size of order \(10^{-13}\) cm), attractive forces between the electron and the neutron must appear\(^1\). The second part of the interaction described is of the greatest interest.

According to Fermi’s estimate\(^1\), the cross section for the interaction of a neutron with an electron is \(\sigma_e \approx 10^{-37}\ \text{cm}^2\). With present-day laboratory techniques, processes with such a cross section cannot be detected experimentally. However, in a concrete experiment the electrons are not free, but are part of an atom, whose size is of order \(10^{-8}\) cm. If, moreover, the experiment is carried out with thermal neutrons, whose wavelength \(\lambda = \frac{h}{p}\) is also of order \(10^{-8}\) cm, then interference phenomena of the scattered neutron wave play an essential role. The amplitude of the neutron wave coherently scattered by the nucleus \((a)\) will add to the amplitude of scattering by the electrons \((b)\), so that the total interaction with the atom will be equal to

\[ \sigma_a = 4\pi(a + b)^2 = \sigma_q + 2\sqrt{\sigma_q \sigma_e} + \sigma_e, \]

where \(\sigma_e\) is negligibly small. On the other hand, the amplitude of the neutron wave scattered by the electrons of an atom (the dimensions of the scattering region are comparable with the wavelength of the incident radiation) will depend on the scattering angle and on the neutron energy.

In the first experiment, performed by Fermi and Marshall\(^2\), the scattering of thermal neutrons in xenon was studied. The authors assumed that the interaction of the neutron magnetic moment with the atom would be excluded, since the Xe atom has no magnetic moment. In addition, interference of the neutron wave on a group of atoms would not occur (the molecule consists of one atom). They associated the effect with neutron dissociation. In the above-mentioned and subsequent\(^3\) experiments, the ratio was measured of the number of neutrons scattered at angles \(45^\circ\) and \(135^\circ\) to the direction of the initial neutron beam. In the center-of-mass system of the colliding neutron and atom

nuclear scattering is isotropic, while electronic scattering (by the atom’s electrons) is substantially anisotropic. The ratio of the number of neutrons scattered at angles \(45^\circ\) and \(135^\circ\) is equal to:

\[ \frac{d\sigma(45^\circ)}{d\sigma(135^\circ)} = 1+\frac{2ab}{\sigma_{\mathrm{я}}/4\pi} \left[ \left(\int nF\,d\lambda\right)_{45^\circ} - \left(\int nF\,d\lambda\right)_{135^\circ} \right], \]

where \(\sigma_{\mathrm{я}}\) is the total cross section of nuclear scattering, \(F\) is the atomic scattering factor for monochromatic neutrons by an atom’s electron, and \(n\) takes into account the energy distribution of neutrons over wavelengths. When comparing the experimental results with the calculated ones, a correction is introduced for the motion of the center of gravity (transition to the laboratory reference system). For Xe this correction\(^3\) is \(2.1\%\), whereas the effect of scattering by electrons is an order of magnitude smaller.

The large errors of the first experiment allowed only the order of magnitude\(^2\) of the energy of interaction of the neutron with the electron to be determined at a distance between them equal to the classical electron radius \(r_0 = 2.8\cdot 10^{-13}\,\mathrm{cm}\): \(V_0 = -500 \div 5000\,\mathrm{eV}\). If the interaction is characterized by a potential well of width \(r_0\), then the depth of this well is equal to \(V_0\). Later the experiment was repeated\(^3\) under more refined conditions: the collimation was improved and the power of the thermal-neutron beam was increased, and the efficiency of neutron detection was increased. The experiments were carried out with Ar, Kr, and Xe. In the experiments with Ar, the experimental correction for the motion of the center of gravity was determined. From the experiments with Kr and Xe the interaction energy of the neutron with the electron was calculated: \(V_0 = -5020 \pm 13\%\,\mathrm{eV}\) (Kr) and \(-2860 \pm 16\%\,\mathrm{eV}\) (Xe). The average of these quantities is given in the table.

Experimental value found for the energy of interaction of the neutron with the electron
(\(V_0\) in eV)

Simultaneously with the neutron-scattering experiments, Rabi’s group determined the energy of interaction of neutrons with electrons from measurements of the transmission of monochromatic thermal neutrons with energies of the order of \(10^{-2}\,\mathrm{eV}\) through molten metals\(^4,5,6\). In this method the cross section for the interaction of thermal neutrons with the atoms of a liquid (\(\sigma_a\)) is found as a function of the neutron energy \(\lambda\)

\[ \sigma_a = \sigma_{\mathrm{я}} + 2abF(\lambda) + P(\lambda), \]

where \(\sigma_{\mathrm{я}}\) is the total cross section of interaction of the neutron with the nucleus, \(F(\lambda)\) takes into account the interference of neutrons on the atom’s electrons, and \(P(\lambda)\) the interference of neutrons in the liquid. In the energy range of neutrons \(0.1—2\,\text{\AA}\) the total cross section changes by several percent, but the numerous corrections make it difficult to determine the cross section for the interaction of neutrons with electrons, \(\sigma_e = 4\pi b^2\).

In the first experiments, performed with molten lead\(^4\) and bismuth\(^5\), only the order of magnitude of the interaction energy was determined: \(V_0 \sim 2500\,\mathrm{eV}\).

In the subsequent experiment with molten Bi, the necessary corrections were accurately taken into account: the cross section for neutron capture by the nucleus, proportional to \(\lambda\) (for \(0.026\) ev it is equal to \(0.035\cdot 10^{-24}\ \mathrm{cm}^2\)), the correction for the motion of the center of gravity, proportional to \(0.011\lambda^2\), and neutron scattering in the liquid \(P(\lambda)=0.095\lambda^2\). The resulting interaction energy is given in the second line of the table.

In a brief communication by a group working on the reflection of neutron beams\({}^{7}\), the value is given for the interaction energy of a neutron with an electron, obtained from the measurement of the critical angle of total reflection of neutrons (third line of the table). A well-collimated neutron beam is directed onto a mirror surface polished with optical precision. If the phase of the neutron wave does not change upon scattering (\(a>0\)), then there exists a critical angle between the direction of the neutron beam and the plane of the mirror, beginning with which total reflection of neutrons occurs. In this case the critical angle is determined by the scattering cross section. To compensate for nuclear scattering in the experiments described, the bismuth mirror was coated with liquid oxygen (coherent scattering of neutrons by Bi and O nuclei is equal in absolute value but opposite in phase). From the measured critical angle of reflection, the cross section for the interaction of neutrons with electrons was calculated.

As is seen from the table, the results obtained by different methods agree within the errors of measurement. Averaging the independent data, we find the mean of the experimental values \(V_0=-4530\pm 530\) ev. This energy is three orders of magnitude smaller than the energy of the nuclear interaction of a particle, and for this reason its accurate measurements are difficult.

In the first experiments\({}^{2,4}\) it was mistakenly assumed that, in the collision of a neutron with an atom having no magnetic moment, the dissociation of the neutron fully explains the interaction of the neutron with the electrons of the atom. In this, the forces acting between the magnetic moment of the neutron and the electrostatic field of the electron, which in the experiments is not compensated, were not taken into account. Foldy\({}^{8,9}\), applying the relations of electrodynamics, calculated this interaction and found it equal to \(V_0=-4080\) ev, i.e. the difference between this part of the interaction and the experimental value is less than \(450\) ev. This difference must be connected with the dissociation of the neutron.

The interaction taking into account neutron dissociation is calculated by meson theory. Applying the pseudoscalar variant of meson theory for the total energy of interaction of a neutron with an electron, a value\({}^{10}\) \(V_0=-5380\) ev was obtained. In the symmetrical pseudoscalar variant of the theory\({}^{11}\), the total interaction energy is \(1208\) ev greater than the interaction energy of the magnetic moment with the electric field of the electron. The remaining variants of meson theory predict an even larger value of the interaction. Reference 9 summarizes the experimental and theoretical works dealing with the interaction of the neutron with the electron. It is noted that meson theory encounters a new difficulty in explaining the small magnitude of the interaction of the electron with the neutron, which possesses an anomalously large magnetic moment. The difficulties noted cannot be resolved by the existing variants of meson theory.

I. V.

CITED LITERATURE

1. E. Fermi, Lectures on Atomic Physics, IL, Moscow, 1952.
2. E. Fermi and L. Marshall, Phys. Rev., 72, 1139 (1947). Nauchno-referativnyi sb., issue IV, abstract 58 (1948).
3. M. Hamermesh, G. K. Ringo and A. Wattenberg, Phys. Rev., 85, 483 (1952).

1. W. W. Havens, I. I. Rabi and L. J. Rainwater, Phys. Rev., 72, 634 (1947). Scientific Abstracts Collection, issue IV, abstract 57 (1948).
2. L. J. Rainwater, I. I. Rabi and W. W. Havens, Phys. Rev., 75, 1295 (1949).
3. W. W. Havens, L. J. Rainwater and I. I. Rabi, Phys. Rev., 82, 345 (1951).
4. J. A. Harvey, D. J. Hughes and M. D. Goldberg, Phys. Rev., 87, 220 (1952).
5. L. L. Foldy, Phys. Rev., 83, 688 (1951).
6. L. L. Foldy, Phys. Rev., 87, 693 (1952).
7. B. D. Fried, Phys. Rev., 86, 434 (1952).
8. S. Borowitz, Phys. Rev., 86, 567 (1952).