PRODUCTION OF LOW-ENERGY NEUTRONS BY FILTRATION THROUGH GRAPHITE*)

In theoretical works on neutron scattering (Wick, Pomeranchuk, and others) it has long been pointed out that crystalline substances must be almost transparent to neutrons with energies on the order of \(10^\circ\) K. Indeed, let us consider elastic (for simplicity) scattering in an ideal crystal. The angle of deflection of a neutron beam \(\theta\) must satisfy the Bragg–Wulff condition (where \(d\) is the distance between two layers of nuclei, and \(\lambda\) is the de Broglie wavelength for neutrons):

\[ 2d \sin \frac{\theta}{2} = \lambda . \]

In all other directions scattering is impossible, since waves scattered by individual nuclei mutually cancel as a result of interference. Obviously, if \(\lambda > 2d_{\max}\) (twice the distance between the most widely separated layers), then scattering is altogether impossible. Thus, in an ideal

*) Anderson, Fermi, Marshall, Phys. Rev. 70, 11 and 12, 1946.

neutrons with \(\lambda > 6.63\) Å cannot be scattered in a graphite crystal, which corresponds to \(2.3^\circ\) K. Fermi and his coworkers succeeded not only in proving the existence of this effect, but also in applying it to obtain a beam of slow neutrons.

Initially the effect was found and interpreted in the above sense in experiments with random, imperfect geometrical conditions. At the top of a small “boiler” (an apparatus in which a controlled chain fission reaction of \(U_{235}\) takes place) a graphite column was placed, and at various points in it the average neutron energy was measured. It turned out that at the top of the column it is less than the theoretical value (roughly speaking, the average thermal energy).

To observe the effect in pure form, an apparatus was constructed consisting of a cadmium tube which separated from among the thermal neutrons wandering in the graphite column a directed beam (as is known, cadmium has a large absorption cross section for neutrons which are in a state of large scattering cross section). The inside of the cadmium tube is filled with graphite. Neutrons with wavelength less than \(2d_{\max}\) (relatively fast ones) are scattered in the graphite and absorbed by the cadmium tube. The length of the tube was taken as 23 cm. Slower neutrons pass unhindered through the graphite. On their path at the end of the tube one may place various samples and, with the aid of a proportional counter, study the attenuation of the neutron beam due to scattering in the sample.

One of the samples was pre-calibrated, i.e., the attenuation of a beam of monochromatic neutrons of various energies had been studied in it. Such neutrons were obtained with the aid of a mechanical selector with rotating sectors. A comparison of the absorption coefficients gave an effective energy value of \(18^\circ\) K, corresponding to a neutron velocity of 533 m/sec and \(\lambda = 7.15\) Å, in excellent agreement with the theoretical value. A series of effective experiments is described which, one may think, do not exhaust the possibilities of this method. The scattering cross section per atom calculated for crystalline sulfur proved equal to \(2.89 \cdot 10^{-24}\,\text{cm}^2\). The value for amorphous sulfur, \(7.06 \cdot 10^{-24}\,\text{cm}^2\), is also large. The difference is explained by the fact that amorphous sulfur possesses inhomogeneities on the order of the de Broglie wavelength and is “turbid” for neutrons. On the day after preparation, the same sample showed a decrease of the cross section to \(3.31 \cdot 10^{-24}\,\text{cm}^2\), since it had partially passed into the crystalline state.

Heating crystals also increases their “turbidity.” Thus, heating graphite from \(20^\circ\) C to \(370^\circ\) C leads to an increase in the cross section by more than a factor of 2.5.

When low-energy neutrons are scattered in water, the hydrogen protons cannot recoil, since they are “fixed” by the forces of chemical bonding (they are in a potential well created by the electron cloud). The distribution of neutron momenta after collision with such a fixed proton is represented by a sphere with its center at the origin of coordinates. The distribution of neutron momenta in scattering by a free proton is also represented by a sphere, but one with half the radius. This sphere touches the first one from the inside. It can be shown that the differential cross section per unit area of the sphere is the same in both cases (for example, by considering the density of points of contact). Since the surface of the first sphere is 4 times greater, the scattering cross section of low-energy neutrons is also 4 times greater than the scattering cross section of neutrons with energies of several volts. This ratio was indeed confirmed with high accuracy in an experiment with the apparatus described above.

A. D. Sakharov