Interference Phenomena in the Scattering of Slow Neutrons

L. M. Khromchenko

I. Introduction

The work of 1932–1936 on the study of the properties of slow, so-called “thermal,” neutrons led physicists to the idea of a whole series of phenomena that ought to occur in the propagation of neutron waves.

Indeed, these works showed that neutrons of the thermal group have velocities distributed approximately according to Maxwell’s law, with a maximum corresponding to a de Broglie wavelength of the neutron \(\lambda = 1.7\ \text{Å}\). This is the same order of wavelengths as in the classical experiments on the scattering of X-rays, approximately corresponding to the order of the lattice constants of most crystals. It was natural to expect that the passage of a beam of slow neutrons through crystals would be accompanied by the same phenomena of interference, diffraction, etc., as the passage of X-rays.

These a priori considerations, expressed in 1936 by Elsasser, gave rise to the performance of several special experiments which qualitatively showed the presence of effects of this kind.

On the other hand, somewhat later the experiment itself prompted investigators to bring in the hypothesis of interference of neutron waves in order to explain a number of observed facts.

As is known, one of the principal characteristics of nuclear processes is their effective cross section, composed of the absorption cross section and the scattering cross section. This effective cross section was regarded as an essentially nuclear quantity, independent of the physical and chemical state of the substance. Hence followed the concept of additivity of the cross section, i.e., the possibility of considering the cross section of a complex molecule as equal to the sum of the cross sections of its components.

The experiments of Whitaker and Bayer in 1939 forced this point of view to be revised in many respects. It could explain the behavior of the absorption cross section, characterized mainly by the energy structure of the nucleus. For the scattering cross section, however, where one has to take into account the mechanism

scattering of neutron waves and their interaction with matter became considerably more complicated.

The concept of the additivity of cross sections and of their independence from the physical structure of matter had to be critically reconsidered and, in many cases, rejected.

Results of this kind, unexpected from the standpoint of earlier concepts and explicable from the standpoint of interference of neutron waves, were provided by three groups of experiments carried out in the period 1939–1941.

The first, and fundamentally the most important, group included experiments on passing a beam of slow neutrons through the same substances in different crystalline modifications. In a number of cases these experiments gave sharply different scattering cross sections for single crystals and polycrystals. As a rule, the cross sections of polycrystals were larger, which qualitatively corresponded to the picture expected in the presence of neutron diffraction. For a single crystal with a fixed position of the crystal planes, only narrow bands of the neutron spectrum proved to satisfy the Laue conditions, while the spectrum as a whole had to pass through without scattering, apart from the removal of neutrons from the beam due to capture and incoherent-scattering processes. Thus, in a first approximation the cross section of a single crystal could be regarded as an upper limit to the cross section for capture and incoherent scattering.

In the case of a polycrystal, coherent scattering also had to be added to these two processes, since all possible spatial orientations of the microcrystals made the fulfillment of the Laue conditions much more probable.

The second group of experiments studied the dependence of the cross sections of alloys on the percentage content of their components, the degree of order in the alloy, and other factors. Here too, in order to explain a number of phenomena, it was necessary to turn to the mechanism of propagation of neutron waves and to take into account the presence of interference.

And, finally, the third series, devoted to measuring the cross sections of complex compounds and comparing them with the cross sections of the constituent elements, showed a noticeable departure from additivity in a whole series of cases. From the standpoint under consideration, this phenomenon was entirely understandable, since cross sections measured for substances in the most diverse physical states (gas, liquid, polycrystal, etc.) were being compared, and interference phenomena that occurred in some cases were absent in others.

These experiments compelled the firm introduction into nuclear physics of the concept of the scattering cross section as a quantity characterizing not only nuclear properties, but also the physical state of the substance under study.

The theory of this question somewhat anticipated experiment. Comparatively early, from 1937 on, theorists began to study the question of the passage of slow neutrons through crystals. A whole series of works was devoted to the consideration of this problem in various aspects, as

both earlier ones (Wick, Pomeranchuk) and comparatively new ones (Halpern, Hammermesh and Johnson, Sanger and Teller, Weinstock).

Here, above all, both the similarity of the problem under consideration to the scattering of X-rays (the coherent part of scattering) and the essential differences, associated with the much lower energy of the neutrons and their large mass (the incoherent part), were emphasized. For this reason the incoherent part of the scattering was subjected to the most detailed analysis.

The authors considered the interactions of neutrons with the thermal vibrations of lattice atoms, the influence of spin and of the isotopic composition of nuclei, and other factors affecting the scattering processes. The result of this cycle of works was new criteria for the basic regularities of neutron scattering, a number of calculation formulae, quantitative estimates, and fundamental conclusions. All this has been little developed experimentally and still awaits its proof and confirmation by experiment.

It therefore seems to us quite timely to give a systematic review of all the material available on neutron interference and to draw some general conclusions about the present state of the question.

II. EXPERIMENTS

In his note\(^1\) concerning the analogy between the scattering of slow neutrons and the scattering of X-rays, which follows from the same order of magnitude of their wavelengths, Elsasser proposed using Laue’s criterion for coherent scattering of slow neutrons. As the initial formula he adopted the expression for the intensity of coherent scattering of X-rays by a polycrystalline powder, proposed by Laue in 1926\(^2\). In the subsequent calculation of the intensity of scattered neutrons, Elsasser proceeded from the assumption of a Maxwellian distribution of the velocities of thermal neutrons and considered the case of using an indicator that absorbs neutrons according to the law \(1/v\).

The angular distribution of the intensity of the scattered neutrons then took the form:

\[ \frac{I_{11}}{I_0}=N\bar{\sigma}\cdot e^{-aB_0^2}, \quad \text{where} \quad a=\frac{h^2}{8\sin^2\vartheta\cdot mkT}. \]

In this expression \(I_{11}\) is the intensity of the scattered neutrons registered by the indicator; \(B_0\) characterizes the type of crystalline lattice; \(m\) is the mass of the neutron; \(N\) is the number of scattering centers per unit volume; \(\bar{\sigma}\) is the mean cross section of such a center. The condition \(aB_0^2=1\) makes it possible to determine the critical angle (the opening angle of the cone around the direction of the primary beam, inside which the intensity falls off very rapidly). For metals of the Fe, Cu type this critical angle was found to be of the order of \(25^\circ\).

For verification of this assumption, von Halban and Preiswerk\(^{3}\) carried out experiments observing the angular dependence of scattered neutrons. A cylinder of Fe was used as the scatterer.

Fig. 1.

The observation was carried out for two temperatures of the neutron source: \(300^\circ\) and \(90^\circ\) K. The scheme of the experimental apparatus is given in Fig. 1. The source \((\mathrm{Rn}+\mathrm{Be})\) was located in a Dewar, inside and outside of which paraffin was placed. The beam aperture was equal to \(27^\circ\) (the beam was cut out by two Cd diaphragms). The indicators were Dy plates, arranged in a Cd chamber around the axis of the incident beam at intervals of \(13^\circ\).

The results obtained by the authors are shown in Fig. 2. Here it is still impossible to give any quantitative interpretation of these curves, since the geometry of the apparatus is very imperfect and the nonparallelism of the beam considerably complicates the picture. Moreover, the picture itself is a summed one, since, besides coherent scattering, there is an incoherent background. Qualitatively one can only state that the dependence of the angular distribution on temperature agrees with the theoretically expected picture: with increasing \(\lambda\) (decreasing \(T\)) the number of neutrons scattered through large angles increases.

Thus, apart from the qualitative compatibility of the obtained curves with Elsasser’s calculations, no more exact comparison with them can be made.

In the same year, 1936, Mitchell and Powers\(^{4}\), in a considerably more perfect arrangement, performed experiments on the observation of Bragg reflection of slow neutrons from the surface of single crystals. The reflectors were single crystals of MgO \((2d = 4\,\text{Å};\ \theta_1 = 22^\circ)\), mounted in the form of a ring around the beam axis. The geometry of the experiment is clear from Fig. 3.

Detection of slow neutrons was carried out by ionization chambers with \(\mathrm{BF}_3\) and \(\mathrm{B}_4\mathrm{C}\).

Measurements were made for two positions of the crystals: corresponding to the Bragg angle of \(22^\circ\) (the sum of coherent and incoherent scattering) and with the crystals alternately deviated from the Bragg position clockwise and counterclockwise (incoherent scattering). In both cases the background of fast neutrons was subtracted—the chamber front was covered with cadmium.

Fig. 2.

Since complete removal of the crystals did not affect this background, it could be considered not to be connected with scattering. On the other hand, removal of Cd from the aperture of the chamber, in the absence of crystals, did not increase the number of neutrons counted by the chamber, i.e., the entire background consisted of fast neutrons.

Fig. 3.

These three cycles of measurements made it possible to estimate the fraction of coherently and incoherently scattered neutrons. For the described case of MgO crystals, the ratio of the number of coherently scattered neutrons to the number scattered incoherently proved to be \(0.4 \pm 0.06\).

As a check, the same measurements were made with the MgO crystals replaced by blocks of polycrystalline Al of equivalent scattering power. The corresponding ratio

\[ \frac{n\ \text{coher.}}{n\ \text{incoher.}} = 0.01 \pm 0.04, \]

i.e., in the case of the polycrystal the Bragg effect disappeared completely.

Such a forty-percent increase in intensity in the case of incidence at the Bragg angle, with an accuracy of 6%, may serve as almost convincing proof of the presence of Bragg reflection of slow neutrons. For complete certainty, the presence of at least one more maximum is lacking (investigations by angles).

Preiswerk’s experiments\(^5\) on Al crystals gave results not contradicting the work of Mitchell and Powers.

After a three-year interval, the question of the propagation of neutron waves was again raised by experimenters in a somewhat different connection.

In 1939, Whitaker and Bayer\(^6\), while studying the passage of slow neutrons through paramagnetic salts, encountered an unexpectedly large difference between the cross sections of the salts and the sum of the cross sections of their components. This prompted a special series of experiments to check the observed effect. It turned out that the noted nonadditivity of the cross sections does indeed occur in a number of cases.

Almost simultaneously,\(^7\) Whitaker and Bayer noted the fact of a noticeable dependence of the cross section for slow neutrons on the crystalline modification of the substance being studied.

In 1940 these authors summarized the results of their experiments in a special article,\(^8\) devoted to interference effects in the scattering of slow neutrons.

In contrast to previous works, the authors studied the cross sections quantitatively, passing a beam of slow neutrons through a scatterer and observing the attenuation of the beam in this process. The “transmission” of cry-

became: \(P=e^{-N\sigma X}\); thus, for a known \(N\)—the number of nuclei per unit volume—and \(X\)—the thickness of the scatterer—the attenuation of the beam characterizes the cross section of the process \(\sigma\). As is clear from the description of the measurement procedure, \(\sigma\) was the total cross section, taking into account both capture and scattering.

Fig. 4.

Figure 4 gives a diagram of the Whitaker and Beyer apparatus. From it it is clear that here we are dealing with a comparatively well-collimated beam of neutrons. The thickness of the scatterers was chosen so as to give an attenuation of the beam by 30–40%, i.e., not yet to have to reckon with the complications introduced by multiple scattering. Special attention was paid by the authors to the process of dehydrating the scatterers, since water was one of the most dangerous contaminants.

The most characteristic results were given by measurements of the cross sections of substances in different crystalline states, brought together in Table 1.

Table 1

Cross sections of substances in different crystalline states \(\times 10^{24}\ \mathrm{cm}^{-2}\)

Substance	Amorphous	Polycrystal	Single crystal	\(g/\mathrm{cm}^2\)
Fe	—	\(12.0 \pm 0.2\)	\(7.0 \pm 1.0\)	1.6
Fe	—	—	\(6.1 \pm 1.0\)	8.8
Ni	—	\(19.8 \pm 0.5\)	\(14.1 \pm 1.2\)	4.4
\(\mathrm{SiO}_2\)	\(8.0 \pm 1.0\)	\(8.8 \pm 1.0\)	\(4.5 \pm 0.6\)	3.7
\(\mathrm{SiO}_2\)	fused quartz	sand	\(4.1 \pm 0.6\)	1.3

For the scatterers used, the cross sections of single crystals are, as a rule, smaller than the cross sections of polycrystals. This effect of the “transparency” of a single crystal is seen most distinctly in the case of \(\mathrm{SiO}_2\): the cross section of the single crystal is almost two times smaller than the cross section of the polycrystal. If one does not regard as possible a defect in the geometry of the experiment—the entry into the detector, together with the transmitted beam, of Laue spots—then one must in fact assume that in the single crystal only an insignificant part of the neutron spectrum satisfies the Laue conditions. The difference would probably have been still greater for a monochromatic neutron beam. No noticeable dependence on the thickness of the scatterer was observed.

The results of the second part of the work of Whitaker and Beyer—the measurement of cross sections of chemical compounds and of the elements composing them—are summarized in Table 2.

Table 2

Cross sections of chemical compounds and elements \(\times 10^{24}\ \mathrm{cm}^{-2}\)

Compound	\(\sigma\) measured	\(\sigma\) additive	Compound	\(\sigma\) measured	\(\sigma\) additive
MnO	19.9	17.2	NiO	22.3	23.9
MnS	19.1	15.1	HDO	54.0	54.5
MnO\(_2\)	25.2	21.3	D\(_2\)O	16.0	—
MnSO\(_4\)	33.6	31.5	C	\(4.8 \pm 0.1\)	graphite
Fe\(_2\)O\(_3\)	39.2	36.3	Cu	\(10.5 \pm 0.4\)	rolled
CuO	16.2	14.6	\({}^{\alpha}\)Fe	\(12.0 \pm 0.2\)	armco
uS	16.5	12.5	Mn	\(13.1 \pm 0.6\)	metal grains
Cu\(_2\)O	29.5	25.1	Ni	\(19.8 \pm 0.5\)	rolled layer
Cu\(_2\)S	27.7	23.0	O	4.1	gas
ZnO	7.7	8.6	S	\(2.0 \pm 0.6\)	powder
ZnS	9.7	6.5	Zn	\(4.5 \pm 0.5\)	metal grains

The accuracy of the determinations of the cross sections, according to the authors’ estimate, is of the order of 3%. These results are very difficult to interpret quantitatively: the cross sections of compounds (polycrystals) were compared with additive cross sections, and in calculating the latter use was made of known and newly measured cross sections of elements in the most diverse physical states (gas, liquid, polycrystal, etc.). However, qualitatively it is quite obvious that additivity is almost nowhere obeyed. Therefore, the necessity of treating the cross section for slow neutrons as a quantity depending also on the physical state of the substance emerges quite clearly even from these measurements, which are far from exhaustive.

The third section of the work was devoted to the study of cross sections of alloys. Whitaker and Beyer studied an Fe—Ni alloy, the so-called “permalloy.” The results of comparing additive cross sections with the cross sections of alloys, measured for their mono- and polycrystalline modifications, are given in Table 3.

Table 3

Cross sections of alloys \(\times 10^{24}\ \mathrm{cm}^{-2}\)

Alloy	Additive \(\sigma\)	Polycrystal	Single crystal
Permalloy (78)	18.2	\(12.5 \pm 0.7\)
Permalloy (73)	17.8		\(10.2 \pm 1.3\)
Permalloy (68)	17.4		\(10.2 \pm 1.3\)
Permalloy (45)	15.6	\(16.0 \pm 0.8\)

Additive cross sections for different percentage contents of permalloy were calculated on the basis of the Fe and Ni cross sections measured for their polycrystalline modification. In this case as well, one may note the high transparency of single crystals and the divergence of the additive cross sections from those actually measured.

It is hardly expedient to discuss this picture in greater detail, since scattering by two types of nuclei, in the general case, will have different phases and amplitudes of the scattered waves, and the assumed interference may give effects of different sign. The data of Whitaker and Beyer’s work are clearly insufficient for this kind of analysis.

The authors themselves regard the difference they observed in the transmission of alloys at different ratios of its components as the result of one of two processes: either a change in the lattice and in the sizes of the crystallites, or a change in the degree of order in the alloy.

The clarification of which of the indicated factors is responsible for the difference in transmission was the subject of the work of Nix, Beyer, and Dunning^9. In their experiments, a series of Fe—Ni alloys was subjected to a special cooling and treatment regime. As a result, for each percentage content of the alloy they obtained two specimens with exactly identical crystallite sizes, but with a different degree of order. Since the transmission for these two types proved to be different, this effect must be attributed to the change in order in the alloy. The different “neighbors” of each atom may, as indicated above, give interference effects of different sign and produce the irregular change in cross section that was obtained in the experiments of Nix, Beyer, and Dunning.

Another detail of the work of Whitaker and Beyer, the dependence of the interference effects on the energy of the neutrons, was checked in a special experiment by Whitaker, Bright, and Murphy^10. If one assumes that the interference is realized for neutron de Broglie wavelengths corresponding to the given geometry, then a change of these \(\lambda\) should have caused substantial changes in the cross section. In order to observe effects of this kind, the authors measured the cross section of a large quartz single crystal (\(10.97\ \mathrm{g/cm^2}\)) for three groups of neutrons: “cold” (\(T = 100^\circ\mathrm{K}\)), thermal at \(300^\circ\mathrm{K}\), and so-called “residual” neutrons transmitted by Cd and activating In and Rh (energy of the order of \(1.5\ \mathrm{eV}\)). The cross sections obtained for these neutron groups are given in Table 4.

Table 4

Total cross section for quartz crystals \(\times 10^{24}\ \mathrm{cm}^{-2}\)

Neutrons Detector Cross section	Cold B \(2.3 \pm 0.7\)	Residual In, Rh \(7.2 \pm 1.2\)	C-group In, Rn \(3.2 \pm 1.2\)	C-group B \(3.0 \pm 0.7\)

As we see, for In-neutrons ($\lambda$ smaller by approximately a factor of 7) the cross section increased almost twofold in comparison with the cross section for thermal neutrons; the “transparency” of the single crystal disappeared to a considerable extent. The difference in the cross section for thermal neutrons, in comparison with the measurements of Whitaker and Beyer, may be attributed to the different thicknesses of quartz.

The results of this work, from our point of view, are very difficult to interpret in the sense desired by the authors: as proof of the presence in a single crystal of interference phenomena that disappear with a change in the order of $\lambda$.

In fact, as the work of Whitaker and Beyer showed, a single crystal is much more “transparent” than a polycrystal, i.e., thanks to the sharp selectivity of the Bragg conditions, only a negligible part of the neutron spectrum is scattered coherently in it. Thus its cross section characterizes, in the main, capture and incoherent scattering. The capture cross section for the elements under consideration is small and, with decreasing $\lambda$, should have a tendency to decrease, unless one assumes the presence of a special level near $E \sim 1.5$ eV. Thus, the decrease in the “transparency” of the single crystal characterizes only the growth of incoherent scattering. On the influence of energy exchange between the neutron and the lattice on scattering, and the exposition of our point of view on these questions in connection with the coherence criterion—see Chapter III.

It is clear, however, that the experimental arrangement described did not, and in essence could not, give an answer to the question of the influence of neutrons on the coherent part of the scattering.

The second cycle of investigations of interference phenomena in crystals was reported in 1940 by Rasetti$^{11}$.

As the material for the scatterer in his experiments, Rasetti chose calcite crystal with a very perfect lattice, possessing the further advantage that for each element in it practically only one isotope is present.

Measurements with calcite gave results still more striking than the measurements of Whitaker and Beyer on quartz. For a very perfect single crystal of calcite Rasetti obtained the cross section:

\[ \sigma = 6.6 \cdot 10^{-24}\ \text{cm}^{2}. \]

The additive cross section calculated by him for $\mathrm{CaCO}_{3}$ proved to be equal to $21.2 \cdot 10^{-24}\ \text{cm}^{2}$, i.e. approximately three times larger.

In these experiments the cross section was also determined by the transmission method. The “transparency” of the crystal, according to Rasetti’s observations, depended noticeably on the degree of perfection of the crystal: for a less perfect specimen he obtained $\sigma = 7.8 \cdot 10^{-24}\ \text{cm}^{2}$. The scattering was practically exponential up to rather large thicknesses: $20\ \text{g}/\text{cm}^{2}$.

In order to elucidate the dependence of scattering on the dimensions and orientation of the scattering crystals, Rasetti carried the process of “polycrystallization” of his single crystal further.

The initial single crystal with \(\sigma = 7.8 \cdot 10^{-24}\ \text{cm}^2\) Rasetti crushed into ever smaller pieces. Then, selecting crystals of the same size with the aid of a sieve, he measured the cross sections as a function of \(n\), the mean number of crystals traversed by the beam in the scatterer. In all these cases the thickness of the scatterer was \(5\ \text{g}/\text{cm}^2\).

Table 5

Scattering in calcite as a function of crystal size

\(n\)	Mass scattering coefficient \(\mu\) in \(\text{cm}^2/\text{g}\)	\(\sigma \cdot 10^{24}\ \text{cm}^{-2}\)
1	\(0.047 \pm 0.002\)	7.8
13	\(0.070 \pm 0.003\)	11.7
39	\(0.079 \pm 0.003\)	13.2
150	\(0.098 \pm 0.003\)	16.3
\(1.5 \cdot 10^4\)	\(0.139 \pm 0.005\)	23.2

A recalculation of the results of these measurements is given in Table 5.

The last of the values indicated in the table corresponds to a powder obtained chemically and is, possibly, overestimated because of incomplete removal of water.

From these data the author was able to estimate roughly the percentage of neutrons scattered coherently, comparing \(\sigma\) of the single crystal with \(\sigma\) at \(n \ne 1\). The largest value was obtained by him for \(n = 13\); in this case approximately \(1\%\) of the incident neutrons were scattered coherently.

Rasetti regards the picture obtained as evidence that a single crystal scatters, for the most part, incoherently, and that this scattering is proportional to the thickness of the material traversed. As the number of orientations of the crystallites increases, the probability of coherent scattering increases, and the cross section of the polycrystal already gives a noticeable fraction of coherent scattering.

Further experiments by Rasetti were directed toward clarifying the factors that could influence interference effects.

First of all, he compared the cross sections of substances in the solid (polycrystalline) form with their cross sections in the liquid state. The materials used were lead and bismuth. However, Rasetti did not succeed in observing any clearly defined difference in the cross sections for these two states. A comparison of the cross sections of the same substances in the mono- and polycrystalline form gave some difference, although a considerably less clear one than in the case of calcite. Recalculations of these results are given in Table 6.

Table 6

Measurements with lead and bismuth

Substance	State	\(\mu\ \text{cm}^2/\text{g}\)	\(\sigma \cdot 10^{24}\ \text{cm}^2\)
Pb	single crystal	\(0.0213 \pm 0.001\)	7.2
Pb	polycrystal	\(0.0275 \pm 0.001\)	9.3
Bi	single crystal	\(0.019 \pm 0.01\)	6.6
Bi	liquid	\(0.024 \pm 0.02\)	8.4

Considering the reasons that make the effect in Pb so much less noticeable than in calcite, Rasetti settles on three that can give a diffuse background in addition to Laue scattering:

a) the mixture of isotopes in the case of Pb,
b) the spin effect in the interaction of a neutron with a nucleus,
c) thermal vibrations of the atoms in the lattice, since the Debye temperature of Pb is considerably below room temperature.

Rasetti immediately rejects the influence of the isotope effect in this case, since for monoisotopic Bi the difference in cross sections is of the same order as in Pb.

To check the influence of thermal vibrations, the author carried out a special experiment: the cross section of a Bi single crystal was measured at crystal temperatures of \(300^\circ\) and \(80^\circ\ \mathrm{K}\). The difference obtained was very small.

The author was unable to estimate the influence of the third possible factor—the spin. It proved impossible to grow a crystal of radiogenic lead containing about \(90\%\) of \(\mathrm{Pb}^{206}\), whose spin is zero.

The works considered from 1939–1940 led to a number of new experiments aimed at clarifying the various factors that affect the scattering of slow neutrons in crystals.

One such attempt to understand the cross section for slow neutrons as a function of their energy is found in Henry H. Hanstein’s paper, published in 1941.\(^{12}\) The purpose of the work was to compare the cross sections for slow neutrons of the thermal group with the cross sections for resonance neutrons In and I. The energy of the neutrons of both groups (respectively \(0.9\ \mathrm{eV}\) and \(25\text{--}100\ \mathrm{eV}\)) is obviously higher than thermal, so that complications connected with interference phenomena were not to be expected. For resonance neutrons \(I\), moreover, the energy is considerably higher than the binding energy in a molecule, so that the observed cross sections characterize the free nucleus.

The neutron source in Hanstein’s experiments was a cyclotron. This made it possible to obtain a well-collimated and sufficiently intense neutron beam. The geometry of the apparatus is noticeably more refined than in earlier works, which used Rn + Be sources. The statistics are also better; the accuracy of the measurements of \(\sigma\), estimated by the author at \(5\%\), can easily be improved. The collimator consisted of a cylindrical cavity filled with \(\mathrm{B}_4\mathrm{C}\) and lined on the inside with Cd—In—Cd layers (in the experiments with resonance In neutrons). The diaphragms were made of the same triple layer. The detectors were In foils, whose activity was measured with an ionization chamber with a string electrometer. Hanstein’s apparatus is shown in Fig. 5. The monitor (also an In foil) made it possible to normalize the measurements for different cyclotron outputs to a definite standard uranium.

The “transmission” of a specimen \(P\) was measured—selective filters served as follows: for In neutrons, In; for thermal neutrons, Cd.

Below, in Table 7, we give Hanstein’s results devoted to the comparison of cross sections. The cross sections determined by him for resonance

Fig. 5.

neutrons of In are compared by the author with the cross sections for thermal neutrons, partly measured by him as well, but chiefly taken from the data of Whitaker and Beyer.

Table 7

Cross sections of elements \(\times 10^{24}\ \mathrm{cm}^{-2}\)

Element	Form	\(\sigma\) for In neutrons \((E \simeq 0.9\ \mathrm{eV})\)	Thermal neutrons \((E \simeq 0.03\ \mathrm{eV})\), total \(\sigma\)	Thermal neutrons \((E \simeq 0.03\ \mathrm{eV})\), scattering \(\sigma\)
H	cetane	\(21.0 \pm 1.0\)	49.0
D	\(\mathrm{D_2O}\)	3.3	5.7
C	graphite	\(4.9 \pm 0.2\)	4.9	4.8
Al	metal	\(1.5 \pm 0.1\)	1.5	1.6
Fe	Armco	\(11.1 \pm 0.3\)	12.0	10.3
Ni	rolled	\(16.1 \pm 0.8\)	19.8	12.4
Cu	rolled	\(8.3 \pm 0.3\)	10.5	8.6
Zn	metal	\(4.2 \pm 0.2\)	4.5	5.2
Sn	metal	\(5.7 \pm 0.3\)		4.9
Pb	metal	\(9.6 \pm 0.8\)	12.5	12.9
Bi	metal	\(8.7 \pm 0.5\)	8.9	8.9

The results for D, C, Fe, Ni, Cu, Pb, and Bi can be compared directly, since they were measured for the same samples. For further comparisons the table gives scattering cross sections measured by Goldhaber and Briggs\(^{13}\).

Approximately half of the substances studied by Hanstein have cross sections for thermal neutrons higher than for In resonance neutrons. Cross sections of the same order for neutrons of both groups are found for C, Al, Fe, Zn, and Bi, and comparison with the data of Goldhaber and Briggs shows that in them the principal process is scattering.

In the remaining cases, where the cross sections for thermal neutrons are larger, it is difficult to estimate what causes this effect: the dependence of the capture cross sections on the neutron energy (the \(1/v\) law) or the presence of interference phenomena in the scattering of thermal neutrons.

The second task of Hanstein’s work (the experiments with I are not considered by us) was to check scattering in quartz and sand (the case studied for C-neutrons by Whitaker and Beyer) both for C-neutrons and for neutrons of the In resonance group. The cross sections measured for this case are given in Table 8.

Table 8

Cross sections \(\times 10^{24}\ \mathrm{cm}^{-2}\)

	In resonance neutrons	Thermal neutrons
Quartz single crystal \((4.5\ \mathrm{g/cm^2})\)	\(7.5 \pm 0.5\)	\(4.3 \pm 0.3\)
\(\mathrm{SiO_2}\) (sand) \((3.3\ \mathrm{g/cm^2})\)	\(8.8 \pm 0.8\)	\(8.8 \pm 1.0\)

The cross section for \(\mathrm{SiO_2}\) for thermal neutrons is taken from the work of Whitaker and Beyer, but is directly comparable with the author’s results, since the measurements were made with one and the same sand.

The upper line of the table—the comparison of \(\sigma\) for thermal and In-neutrons for a quartz single crystal—is a reproduction of the experiment of Whitaker, Bright, and Murphy. The agreement of the results is sufficiently good for In-neutrons; for C-neutrons Hanstein’s value is closer to the data of Whitaker and Beyer.

The comparison of the results in columns II and III is essential. Whereas for thermal neutrons the cross sections of the mono- and polycrystal differ by almost a factor of two, for In-neutrons both cross sections are of the same order.

It seems very probable that in this case we indeed have a cross section determined mainly by incoherent processes. We have already indicated above that an increase in diffuse scattering with increasing neutron energy is quite compatible with the mechanism of this process known to us. For this kind of scattering, the role of structure is less critical than for scattering strictly according to Laue. Thus, the results of this part of the work give an interesting supplement to the previous measurements of the “transparency” of single crystals and of its dependence on neutron energy.

The author also checked the transmission effect in alloys, noted for thermal neutrons in works \(^{8}\) and \(^{9}\). In the case of In-neutrons, comparison of the cross section of permalloy with the cross section of the physical mixture of its components gave no deviations from additivity. This may be regarded as a new confirmation that nonadditivity

sections is caused by interference phenomena and therefore is associated with a definite range of neutron wavelengths.

In G. Carroll’s paper,^14 which appeared in the same year, 1941, one of the tasks was likewise to check interference phenomena. The substance investigated was cetane \((n\,C_{16}H_{34})\); thus the cross sections found characterized the interaction of a neutron with a proton—a case of fundamental importance from the standpoint of studying the nature of nuclear forces.

The cross sections in Carroll’s experiments were also determined by the “transmission” method. The author determined the cross section for two modifications of cetane: liquid and solid. For experiments of this kind the choice of cetane was fortunate: its melting point is \(16.1^\circ\mathrm{C}\), which made it easy to convert it from the liquid state into the solid. On solidification, cetane formed large, well-formed crystals with a lattice constant of about \(1.5\ \text{Å}\), very suitable for interference experiments. We give the measurement results in Table 9.

Table 9

Cross section of cetane \((n\,C_{16}H_{34})\) for slow neutrons

Substance	State	\(g/\mathrm{cm}^{2}\)	% transmission	Cross section per proton \(\times 10^{24}\ \mathrm{cm}^{-2}\)
Cetane	solid	0.1721	47.7	\(44.9 \pm 0.4\)
Cetane	liquid	0.1573	47.1	\(50.1 \pm 0.4\)

As is seen from the table, Carroll succeeded in observing a difference in the cross sections of solid and liquid cetane of about \(10\%\), which, with the accuracy of the measurements indicated by him, makes it possible to regard the observed effect as not accidental.

Thus, the phenomenon which in the case of Pb and Bi Rasetti was unable to establish—for cetane may be regarded as recorded. If the observed difference in cross sections is interpreted as the result of interference—the most logical explanation—we arrive at the possibility of very interesting conclusions about the presence of measurable coherent processes even in such a hydrogen-containing substance as cetane. Such an influence of structure in the case of organic, chain molecules is very curious.

Coherent scattering in cetane may give yet another illustration of the widely discussed problem of neutron interference in ortho- and parahydrogen. Let us briefly recall the essence of the question. As early as 1936, Teller^15 expressed the idea that the study of collisions of slow neutrons with ortho- and parahydrogen should clarify the question of the dependence of nuclear forces on spin. Since the distance between the two protons in the \(H_2\) molecule is of the order of the length

neutron waves of slow neutrons must interfere with one another. If the interaction of a neutron with a proton depends on the mutual orientation of their spins, these interference effects must manifest themselves differently in the case of ortho- and parahydrogen.

Further calculations by Schwinger and Teller16 showed the pattern of behavior of \(\sigma\)-ortho and \(\sigma\)-para for different neutron energies, for cases of forces that depend and do not depend on spins, and made it possible to estimate quantitatively the expected effects.

The experiments of Brickwedde, Dunning, Hoge, and Manley17, set up to test this theory, although they did not give complete agreement with the calculation, nevertheless made it possible to answer certain fundamentally important aspects of the question.

First of all, the results of the experiment unambiguously showed that the forces between proton and neutron depend on spins: \(\sigma_p\) proved to be considerably smaller than \(\sigma_0\). Further, for spin-dependent forces two cases are possible: the presence of a stable or a virtual singlet deuteron level. The interference effects for these two cases are expected to be sharply different. In the case of a real singlet deuteron level, the scattered waves corresponding to the singlet and triplet interactions will have identical phases; with a virtual singlet deuteron level the phases will be opposite, and in parahydrogen we shall have destructive interference.

The results of the experiments described—the magnitude of \(\sigma_p\) and \(\sigma_0\) and their dependence on temperature—convincingly proved that the singlet state of the deuteron is virtual.

We shall not dwell in greater detail on the further development of this question; despite the large group of works devoted to it, some discrepancies between theory and experiment have still not been overcome.

The presence of phenomena of this kind, however, compels one to think through more carefully experiments on the scattering of slow neutrons in hydrogen-containing substances. Here, besides the already considered mechanism of interference caused by the crystalline structure of the substance, one must take into account the possibility of a purely ortho- and para-state effect.

Carroll’s work, of course, still gives no data for analyzing the role of factors of this kind in coherent scattering. However, the field of further experiments suggested by it is undoubtedly of fundamental interest and topical importance.

III. THEORY

The theoretical analysis of the problem of the scattering of slow neutrons in crystals was first carried out by Wick18 in 1937. As the author emphasizes, the analogy between the scattering of slow neutrons and X-rays is limited to the region of strictly coherent scattering. Outside this

region, owing to the much smaller energy of the neutrons, the problem becomes different and requires special consideration.

In the cited article by Wick, it is chiefly incoherent processes of this kind that are considered, in which an exchange of energy takes place between the neutron and the atoms of the crystal lattice. This exchange may occur in two ways: either the neutron gives up part of its energy—sound vibrations are excited in the crystal lattice—or the neutron absorbs energy—the vibrations are damped.

To clarify the mechanism of the first process, Wick considered a completely “cold” crystal, in which lattice vibrations are absent. Then only such processes can occur in which the neutron gives up its energy—a quantum of sound vibrations is emitted. Since the conservation laws must be satisfied in this case, in order for such a process to take place the initial momentum of the neutron must be

\[ P > mV + \frac{h}{2\lambda_s}. \tag{1} \]

In this expression \(m\) is the mass of the neutron, \(V=v_s\lambda_s\) is the velocity of sound in the crystal for the limiting case of long waves.

Thus, for \(p<mV\) no scattering is possible. This condition is easily reduced to the form:

\[ p < \frac{h}{2d} \quad \text{or} \quad \lambda > 2d . \tag{2} \]

We see that scattering in a “cold” crystal practically ceases for \(\lambda>2d\), i.e. under the same condition as Laue scattering.

Restrictions of this kind are, of course, absent in the absorption of sound quanta. These processes, however, play an ever smaller role as the temperature of the crystal is lowered. A quantitative estimate made by Wick shows that for neutrons with \(\lambda>2d\) the crystal, already at practically attainable temperatures, should become completely “transparent” (if capture is not taken into account).

This result of Wick’s leads to interesting consequences concerning the limit of neutron slowing down at low temperatures.

The next part of the problem considered by Wick is the determination of the probability of scattering. The propagation of the scattered wave is described by the Schrödinger equation. The cross section of a rigidly fixed nucleus is:

\[ \sigma_0=\frac{4\pi h}{m}\,|C|^2, \]

where \(C\) is a constant entering the expression for the amplitude of the scattered wave and characterizing the nucleus.

In a real lattice, however, the nuclei will execute vibrations about some equilibrium position. In Wick’s calculations this is characterized by introducing the coordinate \(u\), giving the displacement of the atom from the equilibrium position. The expression for the scattered wave is expanded in powers of \(u\), and only the linear part of this

decomposition. The cross section obtained consists of a coherent part (terms not containing \(u\)) and an incoherent part.

Averaging over all orientations gives the cross section characterizing scattering in a polycrystal:

\[ \sigma=\sigma_0\frac{h^3}{16\pi MP^2}\iiint \left\{n_s+\begin{array}{c}0\\[-2mm]1\end{array}\right\}\frac{s}{\nu_s}\,ds_xds_yds_z. \tag{3} \]

Here \(M\) is the mass of the nucleus (in a simple cubic lattice), \(n_s\) is the mean number of sound quanta per natural vibration of frequency \(\nu_s\). The 0 or 1 standing in braces corresponds to absorption or emission of a quantum \(s=1/\lambda_s\). The integration is carried out over all values of the variables satisfying condition (1).

Starting from (3), Wick considered two limiting cases. The first is a completely “cold” crystal \((n_s=0)\) and \(\lambda\) only slightly smaller than \(2d\). For this case:

\[ d\sigma=\sigma_0\frac{6d}{Mhv^3}\,\varepsilon\,d\varepsilon, \tag{4} \]

where \(\varepsilon\) is the loss of energy by the neutron in the collision; it can have only certain values following from the conservation laws.

The second case is absorption of sound quanta at very low temperatures and \(\lambda<2d\). An approximate estimate gives for this process:

\[ \sigma=\left(1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots\right)\sigma_0\frac{3h^3}{16\pi Mp^2Vd^3}\left(\frac{T}{\Theta}\right)^3, \tag{5} \]

where \(\Theta\) is the characteristic temperature of the crystal.

As is seen from (5), the cross section of this kind of process is very small.

In a second paper on the same subject\({}^{19}\), Wick considered the complications introduced into the picture just considered by the presence of isotopy and spin. That such effects cannot be neglected in a real estimate is shown by the substantial dependence of scattering on the mutual orientation of spins, which was found in experiments with ortho- and parahydrogen. An influence of the isotopic composition must also, undoubtedly, make itself felt if isotopes with strongly differing scattering constants are present in comparable amounts.

In the presence of several isotopes: \(a,\beta,\gamma\ldots\) \((a+\beta+\gamma+\cdots=1)\), instead of one constant \(C\) it is necessary to take into account the presence of many constants: \(C_\alpha, C_\beta, C_\gamma\), etc. In this case Wick represented the scattering cross section in the form of a sum:

\[ \sigma_{\mathrm n}+\sigma_{\mathrm{int}}=\frac{4\pi h}{m}\left\{\alpha|C_\alpha|^2+\beta|C_\beta|^2+\cdots\right\}, \tag{6} \]

where \(\sigma_{\mathrm{int}}\) is the interference cross section, analogous to \(\sigma_0\) in the first paper, and \(\sigma_{\mathrm n}\) characterizes “independent” scattering, in which the individual

waves are scattered completely incoherently. This independent scattering takes place as if each atom were an independent linear oscillator. The frequencies of these oscillators, however, must be distributed so as to give the actual vibrational spectrum of the lattice.

This type of scattering proves possible also for \(\lambda > 2d\), which, in interference scattering, had proved impossible.

To take account of the influence of spin, the author replaces the scattering constant by a scattering operator of the form:

\[ S=A+2B(I,S), \tag{7} \]

where \(A\) and \(B\) characterize the properties of the nucleus, and \(I\) and \(S\) are the spin vectors of the nucleus and, respectively, of the neutron.

Taking account of the isotopic composition and spin, the author gives the general solution in the form:

\[ \sigma_{\text{int}}+\sigma_{\text{n}}= \frac{4\pi h}{m}\left\{ \alpha\left[|A_\alpha|^2+|B_\alpha|^2 I_\alpha(I_\alpha+1)\right]+ \beta\left[|A_\beta|^2+|B_\beta|^2 I_\beta(I_\beta+1)\right]+\ldots \right\}. \tag{8} \]

For the case of a monoisotopic crystal at low temperature and \(\lambda > 2d\), these results make it possible to estimate the spin interaction, since then the scattering must be determined only by it.

In a paper by Pomeranchuk\({}^{20}\), published shortly after the works of Wick mentioned above, the same problem of the scattering of slow neutrons by a crystal lattice is treated. The consideration is conducted by the author, however, in a somewhat different aspect: he is interested in neutron-slowing processes, and therefore the processes are divided into elastic and inelastic ones.

Pomeranchuk solves the problem by the method of perturbation theory, with the interaction energy expressed as a \(\delta\)-function of the distance \(r\) between the nucleus and the neutron:

\[ V=a\cdot\delta(r), \tag{9} \]

where

\[ a=AI+B\cdot\vec K\cdot\vec S . \]

The coefficient \(a\) consists of two terms: one independent of spin and one reflecting the spin interaction of the nucleus and the neutron. It should be taken into account here that \(A\) and \(B\) may be regarded as constants only for sufficiently slow neutrons.

For the quantum-mechanical system neutron–lattice (whose atoms execute thermal vibrations), Pomeranchuk determines the wave function and the corresponding transition matrices.

The probability of elastic scattering of slow neutrons with \(\lambda>d\), i.e. with energy \(E_n<\dfrac{h^2}{2md}\), Pomeranchuk gives in the form:

\[ W_{el}= \frac{mpp}{\pi h^4} \left\{ \sum_s C_s A_s^2 - \left(\sum_s C_s A_s\right)^2 + \sum_s \frac{B_s}{4} C_s j_s(j_s+1) \right\} = \frac{mpp}{\pi h^4}a. \tag{10} \]

Each isotope is characterized by its concentration \(C_s\) and by the coefficients \(A_s\) and \(B_s\); \(\rho\) is the number of nuclei per unit volume, \(j_s\) is the quantum number determining the nuclear moment.

The mean free path for elastic scattering is

\[ \lambda_{el}=\frac{p}{mW_{el}} . \tag{11} \]

Pomeranchuk considers inelastic scattering for neutrons with an energy sufficient to excite the Debye thermal vibrations of the lattice, but insufficient to excite the nucleus.

In this case only the process of excitation of one sound quantum is taken into account, since the simultaneous emission of several quanta is much less probable. In this case the probability of inelastic scattering will be

\[ W_i=\frac{4m^2E^{7/2}}{63\pi^3\hbar^7M}\,a\left(\frac{1}{S_l^3}+\frac{1}{S_t^3}\right)\sqrt{2m}. \tag{12} \]

In this expression \(m\) and \(M\) are the masses of the neutron and the nucleus. \(S_l\) and \(S_t\) are the longitudinal and transverse velocities of the sound wave.

The author emphasizes that the probability of inelastic scattering increases rapidly with the increase of the neutron energy loss, and that the greatest role is played by processes in which the energy loss is of the order of the initial energy of the neutron.

The mean free path for inelastic scattering is

\[ \lambda_i=\lambda_{el}\,\frac{7}{8}\frac{M}{m}\left(\frac{k\Theta}{E}\right)^3 . \tag{13} \]

As is seen from this expression, for \(E<k\Theta\), \(\lambda_i\) becomes much greater than \(\lambda_{el}\), i.e. inelastic scattering becomes insignificant in comparison with elastic scattering.

The expressions obtained for \(\lambda\) allowed the author to estimate the relative probability of three processes: elastic and inelastic scattering and capture. Such an estimate is very important from the point of view of determining the boundary of the process of slowing down slow neutrons in crystals.

It is obvious that if \(\lambda_{\text{capt}}>\lambda_i\), neutrons may be slowed down to thermal equilibrium with the lattice; otherwise capture may occur before their complete slowing down.

In order to make such a comparison, it is necessary to know the constants \(A\) and \(B\). It proved possible to determine them only for hydrogen, where experiments on scattering in ortho- and parahydrogen made it possible to take into account the effect of the spin interaction.

For hydrogen:

\[ \frac{\lambda_i}{\lambda_{\text{capt}}}=\left(\frac{30}{T_0}\right)^{7/2}, \tag{14} \]

where \(T_0=\frac{E}{k}\), and \(\Theta\) has been taken equal to \(91^\circ\).

Thus, for neutron energies less than \(30^\circ\), the slowing down of neutrons in solid hydrogen ceases.

It may be assumed that for other lattices as well, when \(k\Theta \gg E \gg kT\), neutrons cannot be slowed down to thermal equilibrium with the lattice.

The next stage in the analysis of the scattering of slow neutrons by crystals was the work of Halpern, Hamermesh, and Johnson\(^{21}\), published in 1941. This work was a generalization and further development of the authors’ separate notes on the passage of neutrons through ferromagnetic bodies. Since the authors regard the polarization of neutrons in such passage as the result of interference of coherent nuclear and magnetic scattering, further calculations of polarization effects required a more detailed investigation of nuclear scattering in crystals. In contrast to the preceding works, the main attention here is devoted to the mechanism of coherent scattering. The treatment of the question is analogous to the corresponding theory in the case of X-rays. All numerical calculations were carried out for the Fe element, which was of greatest interest to the authors in view of the indicated direction of the work.

The experimental and theoretical material that had accumulated by that time enabled the authors to consider in greater detail the mechanism of interaction of neutrons with matter and, on a number of points, to compare theory with experiment.

Real scatterers are regarded by the authors as a conglomerate of microcrystals; by the latter is meant a combination of elementary cells, oriented without defects and therefore scattering strictly coherently. From this point of view, a single crystal is a mosaic structure consisting of microcrystals oriented only approximately alike. Macroscopically such a single crystal behaves as a perfect one, but for de Broglie waves of the order of the lattice constants the individual microcrystals will not scatter coherently. Therefore, in order to determine the scattering from such a single crystal, the authors calculate the amplitude of the scattered wave for each of the microcrystals and then sum the intensities.

A polycrystal is an aggregate of many small single crystals (crystallites) oriented in space. To calculate scattering from a polycrystal, the authors integrate the diffraction integral over all neutron directions and then average over all orientations of the microcrystals.

For the case of a simple cubic lattice with only one type of nucleus, the Laue condition leads to the following expression for the scattering intensity:

\[ I=\frac{N\cdot \varepsilon}{8\pi}\left(\frac{\lambda}{d}\right)^2 \sum_{1}^{1<2d/\lambda}\frac{1}{|l|}. \tag{15} \]

The cross section entering here, \(\sigma=4\pi c^2\), characterizes coherent scattering by a single nucleus. \(N\) is the number of atoms in the crystal, equal to the number of unit cells for a simple cubic lattice; \(l_i\) are Miller indices.

This expression makes it possible to estimate approximately the effect of secondary extinction—the weakening of the primary beam as it passes into the depth of the crystal, owing to the shielding of the lower-lying microcrystals by the microcrystals situated above. The estimate was made for incidence exactly at the Bragg angle, i.e. for maximum extinction.

The ratio of the intensities of the scattered and primary beams is

\[ \frac{I}{I_0} = \frac{L\cdot l\cdot \sigma\cdot \lambda}{8\pi d^5 |l|} \sim N^2\cdot \sigma\cdot \lambda^2\cdot l\cdot L, \tag{16} \]

if \(l\) is understood as the side of a microcrystal, and \(L\) as that of the crystallite.

For Fe this estimate gave the authors \(I/I_0 \sim 6\cdot 10^{-3}\), i.e. in the case of polycrystalline Fe, secondary extinction may be completely neglected in further calculations. Effects of this kind may be manifested only in the case of large single crystals with \(L\sim 1\ \mathrm{cm}\).

Expression (15) gives the scattering effect in a completely “cold” crystal. To take account of the thermal motion of the nuclei in the lattice, the authors introduce the temperature factor proposed by Debye and Waller for X-rays:

\[ \exp(-A|l|^2). \tag{17} \]

If one takes into account the case of non-monochromatic neutrons, and assumes the spectrum of thermal neutrons to coincide approximately with the Maxwell distribution, then the ratio of the cross section of coherent scattering by the crystal to the scattering cross section by an independent nucleus will be

\[ \frac{\bar{\sigma}}{\sigma} = \frac{1}{4\pi} \left(\frac{\lambda_A}{d}\right)^2 \sum_{|l|<2d/\lambda} \frac{\Phi(\lambda_A |l|/2d)\cdot \exp(-A|l|^2)}{|l|}, \tag{18} \]

where \(\Phi\) is an analytic function giving the Maxwell distribution, and

\[ \lambda_A=\frac{h}{(2M_0 kT)^{1/2}} \]

at neutron-source temperature \(T\).

For Fe, which at room temperature forms a body-centered cubic lattice, expression (18) is modified somewhat: the number of unit cells is equal to half the number of atoms, and all reflections with \(\sum_2 l_i\) even disappear, while those with odd \(\sum_2 l_i\) give quadrupled intensity.

The authors consider (18) for several limiting cases. For \(\lambda_A\to 0\) the sum can be approximated by an integral. The result of integration gives \(\bar{\sigma}/\sigma=1\), i.e., as was to be expected, for small \(\lambda\) the effect of the crystal structure disappears.

As \(\lambda\) increases, the number of terms in the sum decreases, but the intensity of each Debye ring increases as \(\lambda^2\). As \(\lambda\) grows, there will be points at which the rings disappear (at \(\lambda = \dfrac{2d}{l}\) the \(l\)-th ring disappears), and for \(\lambda > 2d\) coherent scattering vanishes completely. This behavior of \(\bar{\sigma}\) as a function of \(\lambda\) for Fe is illustrated by a series of curves (Fig. 6).

Fig. 6.

The theory set forth requires abandoning the generally accepted point of view regarding the scattering cross section of slow neutrons as a quantity independent of the neutron energy.

In view of the existence of this kind of dependence in the region of the Debye rings, the question of the agreement between the usually adopted Maxwellian distribution and the actual spectrum of thermal neutrons acquires additional importance. As the calculations of Pomeranchuk showed, complete slowing down of neutrons is not possible, so that the behavior of the spectrum on the side of large \(\lambda\) is not described by a Maxwellian curve. On the side of small \(\lambda\), the true spectrum must also fall off considerably more slowly than the Maxwell curve, since the entire spectrum extends beyond the absorption range of Cd. The authors consider the distribution obtained by Dunning and collaborators\({}^{22}\) with the aid of a mechanical velocity selector to be closest to the true one, and from it the curves of Fig. 6 have been recalculated (using graphical integration).

In addition to coherent scattering, the authors briefly consider two types of incoherent scattering that form the background between the Debye rings.

The first type—inelastic scattering due to energy exchange between the neutron and the lattice—the authors regard as having little significance for slow neutrons and \(\Theta > T\), since it is small even for X-rays.

The second type—isotopic irregularities, which give a spherically symmetric background, since the amplitudes scattered by different isotopes are, in the general case, different both in magnitude and in sign. An effect of the same kind is also produced by spin interaction.

The authors give this incoherent cross section in the form:

\[ \sum_p |b_p|^2 \frac{\left[i_p a_0^p + (i_p+1)a_1^p\right]^2}{2i_p+1} - \left( \sum_p |b_p| \frac{i_p a_0^p + (i_p+1)a_1^p}{2i_p+1} \right)^2 + \sum_p |b_p|^2 \frac{i_p(i_p+1)(a_1^p-a_0^p)^2}{(2i_p+1)^2}. \tag{19} \]

The first two terms in this expression characterize the isotopic background; the last term, the spin effect; by \(b_p\) is meant the rela-

...the relative concentration of the \(p\)-th isotope, \(i_p\) is its spin, and \(a_0^p\) and \(a_1^p\) are the amplitudes of the scattered waves for the total angular momentum respectively \((i_p - 1/2)\) and \((i_p + 1/2)\).

The authors attribute to this second mechanism a large part of the incoherent scattering, at least in the case of Fe.

On the basis of the theory set forth and the data of Whitaker and Beyer, the authors estimated the coherent cross section for Fe:

\[ \sigma = 5 \cdot 10^{-24}\ \text{cm}^2 \]

and the nuclear amplitude

\[ C = 7.05 \cdot 10^{-13}\ \text{cm}. \]

A rough estimate of possible coherent cross sections for Fe isotopes gave the following values of \(\sigma\):

\[ \sigma \mathrm{Fe}^{56} = 9.7 \cdot 10^{-24}\ \text{cm}^2, \]

\[ \sigma \mathrm{Fe}^{54} = 7.4 \cdot 10^{-24}\ \text{cm}^2, \]

the amplitudes having opposite signs. Here \(\mathrm{Fe}^{53}\) is not taken into account, because of its negligible concentration, and by \(\mathrm{Fe}^{54}\) one means \(\mathrm{Fe}^{54}\) and \(\mathrm{Fe}^{57}\).

The value of \(\sigma\) also made it possible to estimate, very approximately, the dimensions of Fe microcrystals. For the case of large extinction, the upper limit \(l\) is found to be of the order of

\[ 5 \cdot 10^{-4}\ \text{cm}. \]

In view of the substantial importance of the concept of coherence in the interpretation of interference phenomena, it seems necessary to dwell on this question and to discuss it in somewhat greater detail.

In the works considered above we encountered such an understanding of coherence in which all processes of inelastic scattering of neutrons fell into the category of completely incoherent ones. If one takes into account that the ability of scattered waves to interfere with one another is determined above all by the relation of their phases, then the fundamental inaccuracy of such a classification is obvious. In the process of inelastic scattering the wavelength changes, i.e. the period of oscillation, but there is a correlation between the initial phases of the primary and the scattered waves. Absolutely incoherent processes of an inelastic character may therefore be expected mainly in the case of capture of a neutron by a nucleus with subsequent emission of a new neutron, i.e. in passage through the compound-nucleus stage.

From this point of view, inelastic scattering both as a consequence of the rocking of the lattice—displacement of its atoms from their equilibrium positions—and as a consequence of a change in the period, i.e. of an inexact coincidence of phases, leads to phenomena that are transitional from strictly coherent elastic scattering to a completely incoherent spherically symmetric background. The result of processes of this kind is not interference of waves regularly reflected, according to Bragg, from a fixed crystalline plane and reaching the detector in strictly identical phase, but scattering that is to some extent diffuse, owing to the blurring of phases, yet on which the conditions of interference nevertheless still exert a substantial influence.

Experimentally, the existence of such diffuse scattering in the case of X-rays was established by Preston\(^{23}\).

The work of Seeger and Teller considered below gives an interesting treatment of such inelastic processes “at the boundary” of the Bragg region. As is seen from it, this kind of inelastic scattering with a small change in wavelength is the most probable for sufficiently slow neutrons.

The aim of Seeger and Teller’s work is to analyze those restrictions which the conditions for interference in crystals impose on inelastic scattering. As this analysis showed, the indicated restrictions not only cause the disappearance of scattering at very small neutron energies, but also affect the processes of inelastic scattering at neutron velocities comparable with the velocity of sound in the crystal.

The authors do not consider the influence of spin and isotopic composition. The calculation is restricted to one-quantum processes, since in the neutron-energy region under consideration,

\[ E_n \ll \frac{M E_0^2}{E_0}, \tag{20} \]

processes of this kind are more probable than multiquantum ones. In (20), \(M\) denotes the mass of the atom whose amplitude of oscillation is maximal, \(E_0\) the energy of oscillation of such an atom in the lattice. The latter energy becomes equal to \(kT\) at higher temperatures and to \(E_0\) at low ones.

In the region of neutron energies determined by (20), elastic scattering is still more probable than the one-quantum process.

The further analysis is restricted to this region of energies and is carried out in momentum space, with the aid of the so-called “reciprocal” lattice. The relation of the points of the reciprocal lattice to the ordinary one is given by the Bragg condition.

In this system, elastic scattering is characterized by the equality of the momenta of the incident and scattered neutron:

\[ |P_i|=|P_s|. \]

Graphically this means that the end of \(\vec P_s\) lies on the surface of a sphere of radius \(|P_i|\), described about the origin of \(\vec P_i\). Thus elastic scattering is possible if on the surface of this sphere there lies at least one point of the reciprocal lattice, apart from the end of \(\vec P_i\). In the absence of such a point, elastic scattering is impossible. Inelastic scattering remains possible, in which lattice vibrations with momentum \(P_L=h\cdot k_L\) are excited or damped. In the system of the reciprocal lattice, \(\vec P_L\) is constructed as a segment joining the end of \(\vec P_s\) with the nearest lattice point.

Zachariasen’s analysis\(^ {25}\) for analogous processes in the case of X-rays showed that the intensity of inelastic scattering will be maximal,

if \(P_L\) is as small as possible, i.e., the Bragg conditions are as nearly fulfilled as possible. This follows from the fact that the intensity of inelastic scattering is proportional to the square root of the amplitude of the lattice vibrations that arise or disappear in the scattering process, and that in thermal equilibrium the amplitude of the vibrations increases as the frequency decreases.

Since for neutrons, in contrast to X-rays, the change in momentum \(P_s\) upon scattering is important, it is most essential to consider the regions near the boundary of the sphere of radius \(|P_i|\). Fig. 7 gives the section of this sphere by the plane of the drawing and shows the boundary region on an enlarged scale. In this case the part of the circumference under consideration may be replaced by a straight line, to which \(\vec P_s\) is normal, as a radius. \(\Delta P_s\) gives the change of \(\vec P_s\) in inelastic scattering. In Fig. 7 only one point of the reciprocal lattice is shown, the one nearest to the end of \(\vec P_s\). Its being inside the sphere corresponds to emission, outside—to absorption of a quantum.

Fig. 7.

For the processes of greatest intensity, i.e., vibrations of low frequency:

\[ \Delta E_n = v_n \Delta P_s. \tag{21} \]

Here \(v_n\) is the initial velocity of the neutron, and \(\Delta P_s\) is determined from the conservation conditions: \(|P_s| \pm \Delta P_s = |P_i|\).

On the other hand:

\[ \Delta E_n = \hbar k_L \cdot v_L = P_L \cdot v_L, \tag{22} \]

where \(v_L\) is the velocity of sound in the crystal. Hence:

\[ \frac{\Delta P_s}{P_L} = \frac{v_L}{v_n}. \tag{23} \]

For an approximate estimate of the possible scattering pattern the authors consider the case of constant \(v_L\) and monochromatic neutrons. Then, for a given \(v_n\), the ratio \(\dfrac{\Delta P_s}{P_L}\) is constant. In this case two cases are possible:

\[ 1)\qquad \frac{|\Delta P_s|}{P_L} = \frac{v_L}{v_n} < 1. \]

In this case the ends of the possible \(\vec P_s\) lie on the surface of a hyperboloid of revolution (Fig. 8). The two sheets of the hyperboloid correspond to the processes of absorption and emission of a quantum. These processes are most probable near the Bragg region. As \(v_n\) increases, the branches of the hyperbola

approach a plane (sphere), and the theory of inelastic scattering of neutrons becomes completely analogous to the corresponding theory for X-rays.

Fig. 8.

\[ 2)\qquad \frac{|\Delta P_s|}{P_L}=\frac{v_L}{v_n}>1 . \]

The ends of the possible \(\vec P_s\) then lie on the surface of an ellipsoid of revolution surrounding the reciprocal-lattice point (Fig. 9).

Depending on whether the point lies inside the sphere or outside it, either emission or absorption of a quantum will occur.

In the case \(v_L>v_n\) and small \(|P_i|\), it may turn out that not a single reciprocal-lattice point lies inside the sphere. If, in addition, the crystal temperature is sufficiently low, then inelastic scattering, like elastic scattering, cannot occur at all—a result noted earlier by Wick.

Thus, an analysis of the possible inelastic processes shows that, for monochromatic neutrons with \(v_n<v_L\), inelastic scattering gives sharply limited areas near the Bragg maximum.

For \(v_n>v_L\), inelastic scattering gives diffuse spots around the Laue spots.

Fig. 9.

The consideration of another aspect of the process of inelastic scattering—its dependence on the temperature of the scattering crystal—was the subject of a paper by P. Weinstock\(^{26}\), the most recent work of this cycle.

Weinstock derived a somewhat more general expression for the scattering cross section than had been done in the preceding papers\(^{21}\). The temperature dependence of the scattering, which in the work of Halpern, Hamermesh, and Johnson had been taken into account by analogy with X-rays, through the introduction of an exponential factor of the Debye–Waller type, was treated in greater detail.

In Weinstock’s work a polycrystalline scatterer is considered, with such a size of the microcrystals that the effect of secondary extinction can be neglected. The isotopic composition, magnetic interaction, and spin are not taken into account by the author. As the author points out, the results obtained by him should not depend on spin, since the same spin factor enters both the expression for the cross section of the crystal and that for the cross section of a free nucleus. The neutrons are considered monochromatic.

The problem of scattering of slow neutrons by crystals is solved in this work by the method of the Born approximation, with the use of the Dirac three-dimensional \(\delta\)-function of the coordinates of the neutron and the nucleus for the interaction energy.

For elastic scattering of slow neutrons by bound nuclei, the applicability of such a method of solution was, in its time, proved by Fermi. The author justifies its application to the consideration of inelastic processes by the circumstance that the inelastic amplitude is in general small in comparison with the elastic one.

A rough estimate of the ratio of inelastic scattering to elastic scattering for \(N \gg 1\) gives the expression:

\[ L \sim \frac{\hbar^{2} k^{2}/2m}{\hbar \omega_s}\left(\frac{m}{M}\right) = \frac{E_n}{E_s}\left(\frac{m}{M}\right), \tag{24} \]

where \(E_n\) is the initial energy of the neutron, \(E_s\) is of the order of the energy of the \(s\)-th oscillator, and \(\left(\frac{m}{M}\right)\) is the ratio of the masses of the neutron and the nucleus.

For room-temperature neutrons, for many substances: \(\frac{E_n}{E_s}\sim 1\). In these cases \(L\) is determined by the mass ratio, and thus, for heavy elements, inelastic processes are much less probable than elastic ones.

For processes of first order (one-quantum inelastic processes), emission of a quantum is more probable than its absorption. These phenomena become equally probable at such crystal temperatures, \(T\), for which: \(k_0 T \gg \hbar\omega_s\) (\(k_0\) is Boltzmann’s constant).

The elastic-scattering cross section in the reciprocal-lattice system, characterized by the vector \(\vec{\tau}\) \(\left(\tau \leq \frac{k}{\pi},\right.\) where \(k\) is the wave number), is given by Weinstock in the form:

\[ \bar{\sigma}_{el}^{\tau} = \sum_{\tau}\bar{\sigma}_{el}^{\tau} = (\pi \sigma/2BK^{2})\sum_{\tau}\left(\frac{1}{\tau}\right)\exp(-2W_{0}^{\tau}). \tag{25} \]

In this expression \(\sigma\) is the elastic cross section for a free nucleus:

\[ \sigma=\frac{m}{\hbar^{4}\pi}|A|^{2}. \]

\(B\) is the volume per nucleus in the crystal (\(B=d^{3}\) for a simple cubic lattice).

\[ W_{0}^{\tau} = (6\pi^{2}\hbar^{2}\tau^{2}/Mk_{0}\Theta) \left[\left(\frac{1}{4}\right)+\left(\frac{T}{\Theta}\right)^{2}Q_{1}\left(\frac{\Theta}{T}\right)\right], \]

where \(Q_{1}\) is the Debye function:

\[ Q_{1}(z)=\int_{0}^{z}\left\{\beta/[\exp(\beta)-1]\right\}\,d\beta. \]

Taking into account only one-quantum processes, the inelastic-scattering cross section was found by Weinstock in the form:

\[ \bar{\sigma}_{in}^{\tau} = (\pi\sigma k_{0}^{2}T^{2}\tau/\hbar_{x}c^{3}k^{2}M) \exp(-2W_{0}^{\tau}) \left\{Q_{1}\left(\frac{\Theta}{T}\right) +\left(\frac{1}{4}\right)\left(\frac{\Theta}{T}\right)^{2}\right\}. \tag{26} \]

These expressions for \(\bar{\sigma}\) are mathematically rather complex and do not give a clear representation of their dependence on the temperature of the scatterer.

Therefore, the author carried out calculations for the case of polycrystalline Fe and neutrons at 300° K. The results of the calculations are represented graphically in Fig. 10, giving the dependence of \(\sigma_{el}/\sigma\) on \(T\), and in Fig. 11, describing the dependence on temperature of \(\sigma_{in}/\sigma\) and \(\sigma_{\text{total}}/\sigma\).

Fig. 10.

From the curve in Fig. 10 it is seen that, when the temperature changes from \(0^\circ\) to \(1000^\circ\) K, the inelastic cross section increases approximately to \(19\%\), and this increase is linear in the region \(150^\circ\)—\(400^\circ\) K. Fig. 11 demonstrates the comparative constancy of the total (elastic \(+\) inelastic) scattering cross section from \(0^\circ\) to \(400^\circ\) K. Its decrease for \(T = 1000^\circ\) K is of the order of \(7\%\), i.e. effects of this kind are quite amenable to measurement and experimental verification.

The dependence of the scattering cross section on the temperature of the crystal has not yet been measured by anyone; thus the theory set forth has not yet received experimental confirmation. An indirect theoretical check may, however, be considered to be the agreement of Weinstock’s calculations with the corresponding results of Zachariasen\(^{25}\) for diffuse scattering of X-rays in that limiting case in which the two theories should coincide, i.e. for neutrons of such energies that the change in their energies in the scattering process may be neglected.

Fig. 11.

The results of the calculations of Seeger and Teller, on the one hand, and of Weinstock, on the other, give a picture that fully justifies the point of view set forth above regarding inelastic processes as intermediate between strictly coherent scattering and a spherically symmetric background of incoherent scattering, since the interference conditions exert a substantial influence on them.

IV. GENERALIZATION

The works considered in Chapters II and III have shown us that, in the nine years that have passed since its emergence, the problem of the interference of slow neutrons has undergone repeated and many-sided investigation.

The interest aroused by this circle of questions is quite understandable. The overwhelming majority of the solid bodies known to us are crystals, and therefore taking into account the influence of the crystalline structure, in order to obtain the true picture of scattering in the light of the works considered above, is absolutely necessary.

On the other hand, the development of a method for obtaining diffracted neutron beams of sufficient intensity could place in the hands of experimenters an instrument so urgently needed for modern nuclear research: monochromatic neutrons of long wavelengths.

From this point of view it is important to note the obvious lag of experiment behind theory. As we saw in Chapter III, the theoretical problem of the scattering of neutrons by crystals has already been developed from many sides. The mechanism of energy exchange between the neutron and the lattice has been considered; the relative probability of one- and many-quantum processes, of the emission and absorption of quanta, of elastic and inelastic processes, and of capture has been estimated; the limit of neutron slowing-down in crystals has been evaluated; the influence of spin and isotopic disorder has been taken into account; the dependence of scattering on the temperature of the crystals has been examined; and the presence of phenomena of a more “subtle” origin—effects in regions adjacent to the Bragg region, etc.—has been shown.

Experimentally, however, we so far have only a roughly qualitative picture of the influence of crystalline structure on neutron scattering. Perhaps the most vivid analogy sought with X-rays is provided by one of the earliest works—the work of Mitchell and Powers[^4]. But even in this case, the absence of at least one more order of reflection deprives the results of the character of complete reliability.

The results of later experiments persistently indicate the presence of interference phenomena, but none of them gives a sufficiently clear picture of diffraction. If one compares this with the almost classical completeness achieved by the corresponding questions in the theory of X-rays, then the present state of the question of neutron interference may still be considered quite rudimentary.

The few experimental results obtained serve rather as an indication of those regions where a deeper study promises to yield interesting material.

Such, for example, are the phenomena of “supertransparency” of single crystals at low temperatures and for sufficiently slow neutrons, predicted by the theory; the whole complex of questions concerning the scattering of neutrons in hydrogen-containing substances, and much else.

The clarification of these questions by experiment, the “raising” of experiment to the level of theory, which in turn stimulates its further development—such is the most urgent task in the field considered.

In conclusion, I should like to express my gratitude to Corresponding Member of the Academy of Sciences of the USSR P. I. Lukirskii for fruitful discussion of all the main questions touched upon in the present article.

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