FROM CURRENT LITERATURE

APPLICATION OF NEUTRON DIFFRACTION TO THE STUDY OF THE STRUCTURE OF CRYSTALLINE SUBSTANCES

The structure of crystalline substances largely determines their physical properties. The great importance is therefore clear of methods that make it possible to study the atomic–molecular structure of crystals. Among such methods the principal role is played by X-ray structural analysis, i.e., the application to structural investigations of the diffraction of X-rays by crystals. This method makes it possible to study both individual crystals and polycrystalline or powder specimens. The apparatus and technique of X-radiography are now at a high level, but, despite this, there exist a number of fundamentally important questions that are insoluble (or soluble only with exceptional difficulty) by X-ray methods. Electron diffraction, carried out in 1926, did not help in this respect either. Naturally, after the discovery of neutrons the supposition arose that the diffraction of these particles might be applied to the study of the structure of crystalline substances, with the aim of clarifying questions that, in most cases, cannot be resolved by X-radiography. Experiments performed up to the present time have confirmed this supposition.

The present article describes work on neutronography—so one may briefly call the application of neutron diffraction to the study of the structure of crystalline substances. In the corresponding sections, various shortcomings of X-radiography are examined and the great possibilities of neutron structural analysis are substantiated.

Since the neutron is uncharged, neither an electric nor a magnetic field acts on it; it has no ionizing ability. These facts deprive us of the possibility of making use of the usual sources and accelerators and compel us to introduce a number of changes into recording instruments in order to make them suitable for work with neutrons.

1. APPARATUS FOR NEUTRON STRUCTURAL ANALYSIS

The source of neutrons is nuclei participating in nuclear reactions. The most commonly used reaction is: $\mathrm{Be}^9(\alpha,n)\mathrm{C}^{12}$. Instead of beryllium one may take B, Mg, P, Al, etc., but the effectiveness of these reactions is much less. A beryllium source is a sealed ampoule filled with emanation, Ra, and Be. Another source of neutrons is the cyclotron, in which the bombardment of deuterons by deuterons is carried out. The following reaction takes place: $\mathrm{H}^2(d,n)\mathrm{He}^3$. In general, similar reactions that can be used as sources.

neutrons, there are quite a few, but, unfortunately, they are of negligibly small intensity. Therefore, before the advent of more powerful sources, it had not been possible to determine quantitatively the diffraction of neutrons by crystals.

The most powerful source of neutrons is provided by piles for chain nuclear reactions. These sources have made it possible to study neutron diffraction effects quantitatively. As a result of the fact that the neutrons in the pile are in thermal equilibrium with the moderator substance, they have an almost Maxwellian velocity distribution. Therefore it may be necessary to monochromatize the neutron beam. Monochromatization can be accomplished in two ways: by a mechanical velocity selector and by a crystal monochromator. Mechanical selectors transmit neutrons of a definite velocity, absorbing all the rest. The method of the crystal monochromator makes use of the wave properties of neutrons. As is known, crystals consist of regularly arranged atoms. Wulff (in Moscow) and Bragg (in England) considered

Labels in the figure: Incident beam of neutrons; Diffracted beam; Cadmium slit; Neutron counter; Pile shielding.

Fig. 1. Spectrograph for neutron-structural analysis.

the scattering of X-rays (and in the present case neutron beams of a definite wavelength are also subject to these laws) as reflection from atomic planes, i.e. planes passing through the atoms of a crystal. Hence the law of reflection is obtained: \(n\lambda = 2d \sin \theta\), where \(\lambda\) is the wavelength of the incident radiation, \(d\) is the interplanar distance, \(\theta\) is the angle between the incident beam and the atomic plane, and \(n\) is the order of reflection. From this it is clear that if white radiation is directed onto a crystal, then radiation with a wavelength satisfying the Wulff–Bragg equation will be reflected from the crystal in a definite direction. This is the essence of the crystal monochromator, which is most often used in neutronography with a pile as the source.

The velocity distribution of neutrons from the pile is such that, using various monochromators, one can obtain neutrons with wavelengths from 5 to 1.02 Å at sufficient intensity.

For the registration of individual neutrons, ordinary Geiger–Müller counters filled with BF\(_3\) are used. In some cases another method of registration is applied—the method of photographing neutron beams. Neutrons do not act directly on the photographic emulsion,

therefore nuclear reactions have to be used. In practice this is accomplished by introducing boron salts into the photographic emulsion. The α-particles formed as a result of the reaction \(B^{10}(n,\alpha)Li^7\) cause blackening of the film. Another method may be used, based on the reaction \((n,\beta)\): a thin sheet of indium is placed in front of an ordinary photographic plate. The reaction products—β-particles—cause blackening of the photographic film.

Spectrographs for neutron-structure analysis are, in the main, similar to those used with X-rays. A typical spectrograph is shown in Fig. 1. The neutrons pass through a channel in the concrete lining of the boiler

Fig. 2. Apparatus for monochromatizing neutrons.

fall onto the crystal. Around the axis passing through the crystal there rotates a massive frame on which a neutron counter is mounted. The counter measures the number of neutrons scattered by the crystal at a definite angle. Since the lining of the boiler is not an entirely opaque screen for neutrons, the counter must be shielded from lateral radiation. For this reason the counter is surrounded by a layer of paraffin to slow fast neutrons and by a layer of \(B_4C\), which absorbs these neutrons. This makes the counter very bulky and requires bulky spectrographs.

In the case when a monochromatic beam of neutrons is required for analysis, the radiation from the boiler is first directed onto a crystal which, according to the Wulff–Bragg condition, reflects neutrons of one definite energy. This apparatus is shown in Fig. 2. In order to predo-

protection from neutrons scattered by the monochromator and from the direct beam, the monochromating crystal is placed in a paraffin casing and casings of \(B_4C\), lead, and cadmium. Only holes for the incident and reflected beams are made in them. Next a spectrograph similar to that described is installed.

For the study of resonance absorption, a spectrograph with a bent crystal was constructed and used.

Fig. 3. Spectrum, by velocities, of neutrons emitted by a pile.

With the aid of such apparatus, the velocity spectrum of neutrons from a pile was obtained (Fig. 3). The Maxwell curve, calculated for \(T = 548^\circ K\), is somewhat distorted as a result of the occurrence of reflections of higher orders from the crystal (with \(n = 2\) and \(n = 3\)).

II. SOME QUESTIONS OF THEORY

The present review does not aim to give a complete exposition of the theory of neutron scattering and diffraction. Only questions directly related to structural analysis are discussed.

Although the scattering of thermal neutrons and of X-rays is analogous in many respects, there are nevertheless certain substantial differences. For X-rays, the role of scatterers is played by bound electrons. The scattering power of an atom can be accurately calculated by summing the scattering amplitudes of all the electrons of the atom, taking into account their mutual positions. This gives the atomic scattering factor, which is determined by the magnitude of the relative displacement of the whole atom with respect to the scattering of an individual bound electron. Atomic factors for X-rays can be determined experimentally and calculated theoretically.

For neutrons, scattering is almost entirely nuclear, and the little that is known about the structure of nuclei and nuclear forces does not yet allow us to calculate the scattering theoretically. With the exception of this fundamental difference in the scattering mechanism, there are also several other substantial differences between neutron and X-ray diffraction, which will be considered in connection with the experimental data.

Scattering Theory

The theory of neutron scattering gives the relation between the scattering amplitude (or the scattering cross section) and the potential of nuclear forces. To find this relation it is necessary to solve the wave equation, which for the case of the interaction of a neutron with a nucleus is as follows:

\[ \Delta \psi + k'^2 \psi = 0, \quad \text{where} \quad k'=\sqrt{\frac{2m(E-V)}{\hbar^2}} \]

is the wave number inside the nucleus, whose potential function is \(V(r)\). The incident wave corresponding to the incident beam of neutrons will be represented by the same expression, but with \(V=0\), and

\[ k=k'=\frac{2\pi}{\lambda}, \quad \text{where} \quad \lambda=\frac{h}{mv} \]

is the wavelength of the incident neutrons according to the de Broglie wave theory of particles, \(h\) is Planck’s constant, \(m\) is the mass, and \(v\) the velocity of the neutron.

As a result of the fact that the scattering of thermal neutrons is spherically symmetric, the solution of the wave equation must be sought in the form of the sum of a plane and a spherical wave:

\[ \psi=e^{ik'z}+f\frac{e^{ik'r}}{r}. \]

\(f\) is the amplitude of the scattered wave, which is a function of the scattering nuclei and of the incident neutrons. It ultimately determines the scattering. It can be shown that \(f\) depends on the phase shift between the incident and scattered wave, \(\eta_0\), and that this phase shift depends on the nuclear potential \(V(r)\). As a result of solving the wave equation one can obtain this dependence:

\[ f=\frac{e^{2i\eta_0}-1}{2ik}. \tag{1} \]

Since the phase shift is very small (apart from proximity to a resonance level), this exact expression may be expanded in a series, neglecting terms \(\eta_0^2\) and higher. Then

\[ |f|=\frac{\eta_0}{k}. \]

Since the scattering cross section is related to the scattering amplitude by the relation \(\sigma=4\pi |f|^2\), then

\[ \sigma=4\pi\left(\frac{\eta_0}{k}\right)^2. \]

In the figure: \(r\sim 10^{-13}\,\mathrm{cm}\), \(\lambda\sim 10^{-8}\,\mathrm{cm}\); dashed line — undisturbed wave; solid line — resultant wave.

Fig. 4. Transformation of a neutron wave in the potential well of a nucleus.

The dependence of \(\eta_0\) on \(V(r)\) is easy to understand from the following geometrical picture, which shows the transformation of the incident wave in the potential well of the nucleus (Fig. 4). The internal wave must be smoothly joined to the external waves with the same value and slope at the boundaries, which can be done only when the length of the wave inside the well is small in comparison with its value outside the nucleus. In Fig. 4 the dashed lines show the undisturbed wave, and the solid line the resultant wave, which is smoothly joined to the internal one. Curve \(A\)

shows the case of a negative phase shift, and curve B—of a positive one. At the resonance level the phase shift assumes the value \(\pi/2\), and the amplitude of the wave inside the well becomes equal to the amplitude of the wave outside the nucleus, as is seen from curve C.

From examining Fig. 4 it is seen that negative phase shifts are more probable than positive ones, which are observed only in the vicinity of resonance levels. Moreover, a positive phase shift is more probable for light elements with a small ordinal number, since the width of resonance levels is the greater, the smaller \(Z\) is.

The phase shift can be determined from various experiments, which will be discussed further on. They can also be determined from the theory:

\[ \eta_0=-c\int_0^\infty V(r)r^3\,dr. \]

Thus the nuclear potential does not enter directly into the solution of the wave equation, but acts on the phase shift \(\eta_0\), which enters into the solution. In nuclear physics this equation is used for determining \(V(r)\) from the experimentally found \(\eta_0\).

It should be noted that the scattering amplitude has a sign opposite to that of the phase shift: if \(\eta_0\) is positive, then \(f\) is negative, and conversely. This is directly evident from equation (1).

Influence of the Binding of Nuclei, Isotopes, and Spin Orientation on Scattering

The scattering cross section for a given nucleus depends on whether the nucleus is in a free or bound state. The cross section for an immobile nucleus, in comparison with a free nucleus, will be larger by the square of the ratio of the neutron mass to the reduced mass of the nucleus and neutron, i.e. by

\[ \left(\frac{1}{\mu}\right)^2=\left(\frac{A+1}{A}\right)^2, \]

where \(A\) is the mass number of the scattering nucleus. The numerical value of this expression practically lies within the limits from 4 to 1. Coherent scattering by crystals corresponds to scattering by immobile nuclei.

In the case of neutrons, the role of different isotopes of one element as scatterers is not the same. This is a very important fact, since it enables us, by neutron-diffraction methods, to determine not only the elements being analyzed, but also their individual isotopes, which is inaccessible to X-ray and electron-diffraction methods.

The cross section of isotope scattering by bound nuclei in the case of diffuse scattering is determined by the formula: \(\sigma_{\text{diff}}=4\pi\Sigma(p_i f_i^2)\), where \(f_i\) is the scattering amplitude of a bound nucleus of an individual isotope, and \(p_i\) is its content in the sample. In the case of an isotope distribution in a sample, coherent scattering may occur, and then the scattering amplitudes of each isotope must be combined algebraically into the total amplitude before being squared: \(\sigma_{\text{coh}}=4\pi(\Sigma p_i f_i)^2\).

Since neutron scattering depends on the relative orientation of the spins of the neutron and the scattering nucleus, there will be two amplitudes: for parallel and antiparallel spins; here, too, one must reckon with the possibility of both coherent and diffuse scattering.

It must be said, however, that isotope and spin effects usually give diffuse scattering. Diffuse scattering also has scattering due to thermal motion, which causes displacements of atoms from their equilibrium positions (the nodes of the crystal lattice). The total scattering cross section of a crystal—diffuse and coherent—will be

by the sum of all cross sections: \(\sigma_{\mathrm{cr}}=\sigma_{\mathrm{coh}}+\sigma_{I_d}+\sigma_{S_d}+\sigma_{T_d}\), where \(\sigma_{I_d}\), \(\sigma_{S_d}\), and \(\sigma_{T_d}\) are the cross sections of diffuse scattering for isotopic, spin, and temperature effects.

III. APPLICATIONS OF NEUTRONOGRAPHY

For neutrons, the relation between the scattering cross section and the atomic number of an element has no systematic character. This has its disadvantages, but it is precisely this that also determines the principal advantages of neutronography over X-ray structural analysis. The possibility of distinguishing isotopes in polyatomic structures has already been mentioned. In addition, it proves possible to study crystal systems consisting of several kinds of atoms, both differing considerably in atomic number (for example PbO), and, conversely, neighboring one another in the Mendeleev system of elements (FeCo). Finally,

Fig. 5. Neutronograms of NaH and NaD crystals.

because, for neutrons, the scattering cross sections on hydrogen and deuterium are quite comparable with the scattering cross sections for other nuclei, neutronographic analysis of hydrogen-containing structures becomes feasible. Such analysis by means of X-rays is almost not carried out, because of the exceptionally weak scattering by the latter of the hydrogen atom.

As an example one may consider the study of the structures of NaH and NaD, whose neutronograms are shown in Fig. 5. From a comparison of the intensities of the diffraction maxima it is immediately seen that H and D participate in the diffraction and that the scattering amplitudes of them have different signs, since those peaks which are strong in NaH are weak or disappear altogether in NaD.

It had been established radiographically that the sodium atoms form a face-centered cubic lattice, but the position of the hydrogen atoms could not be determined. As a result of this, one had to choose between two possible structures for NaH (among the known structures of the face-centered cubic type): the NaCl structure and the ZnS structure.

Exhaustive results on the structure of NaH and on the signs of the scattering amplitude can be obtained from a consideration of the theoretically calculated and experimentally found intensities of the diffraction maxima.

Only by assigning \(f_{\mathrm{Na}}\) and \(f_{\mathrm{H}}\) different signs and assuming the NaCl structure for NaH do we obtain good agreement between the experimental and theoretically calculated intensities for all the diffraction maxima. Under all other assumptions the experiment disagrees with theory. Hence it is clear that NaH has the NaCl structure and that the scattering phases for H and D have different signs. Consequently, the scattering phases for Na and D have the same sign. The almost complete disappearance of the diffraction maxima (200), (220), (222), (400) indicates that the coherent scattering amplitudes of Na and H are approximately equal. Thus, for coherent scattering,

\[ f_{\mathrm{Na}}=-f_{\mathrm{H}}. \]

Table I

From this example two further conclusions may be drawn: 1) to all the known intensity factors for X-rays (with the exception of the polarization factor), in the case of neutron analysis there is added one more: the sign of the scattering amplitude, which plays a significant role in the diffraction pattern of polyatomic crystals; 2) from neutronograms of NaH and NaD one can determine the cross section of coherent scattering for H and D. It turned out that

\[ \sigma_{\mathrm{coh}}^{\mathrm{H}}=2.0\cdot 10^{-24}\ \mathrm{cm}^{2}, \]

\[ \sigma_{\mathrm{coh}}^{\mathrm{D}}=4.0\cdot 10^{-24}\ \mathrm{cm}^{2}. \]

The total scattering cross section for them is:

\[ \sigma^{\mathrm{H}}=80\cdot 10^{-24}\ \mathrm{cm}^{2}, \]

\[ \sigma^{\mathrm{D}}=7\cdot 10^{-24}\ \mathrm{cm}^{2}. \]

Since the isotope effect is excluded, the strong difference in the cross sections for hydrogen is explained by the considerable dependence of neutron scattering by protons on spin. The estimates given make clear

strong background of incoherent scattering in NaH, which greatly complicates the calculation of the intensity of neutronograms of compounds containing hydrogen. In contrast to this, the diffuse scattering from deuterium is many times weaker. Hence there follows a practical conclusion: a crystallographic study can be carried out much more easily with deuterium preparations than with hydrogen ones, while their structures are, of course, identical. This method was used in studying the structure of ice, and heavy ice was subjected to the analysis.

Undoubtedly, this procedure will find wide application in the study of the structures of organic compounds, where hydrogen plays the dominant role. X-ray diffraction is powerless in this field.

IV. METHOD OF CRYSTALLINE POWDERS

The method of crystalline powders requires monochromatic neutrons; it therefore provides for the use of a spectrograph with a crystal monochromator, similar to that described above. The neutronogram is obtained by counting the intensity of the neutrons at various reflection angles with the aid of a highly efficient BF₃ counter. Typical neutronograms obtained from diamond and aluminum samples are presented in Fig. 6. The diffraction maxima are located at those positions dictated by the structure of the crystals and by the wavelength of the neutron beam.

Fig. 6. Neutronograms of diamond and aluminum.

Within the limits of experimental error, it turned out that diffuse scattering is absent in diamond and aluminum. This made it possible to conclude that neutron scattering is independent of spin in these elements (aluminum and diamond are almost monoisotopic). A certain increase in diffuse scattering at large reflection angles is explained by a temperature effect, and it is greater for aluminum, which has a lower characteristic temperature.

FROM CURRENT LITERATURE

The interpretation of neutronograms can be carried out with the aid of the roentgenography equation, omitting only the polarization factor:

\[ \frac{P_{hkl}}{P_0} = \frac{\lambda^3 r}{4\pi l}\cdot \frac{h\rho'}{\rho}\cdot \frac{e^{-\mu h\sec\theta}}{\sin^2 2\theta}\, j_{hkl}F_{hkl}^{2}N^{2}, \]

where \(P_{hkl}\) is the total intensity of reflection from the planes \(\{hkl\}\), measured by a counter with slit width \(r\) and distance from the specimen \(l\). \(P_0\) is the intensity of the incident beam, \(\lambda\) is the neutron wavelength, \(\rho'\) is the density of the powder, \(\rho\) is the density of the solid crystal, \(e^{-\mu h}\) is the absorption factor, \(\theta\) is the Wulff–Bragg angle, \(j_{hkl}\) is the multiplicity factor, \(N\) is the number of elementary cells in \(1\ \mathrm{cm}^3\) of crystal, and \(F_{hkl}\) is the structure factor, which, for the case of neutron diffraction, is related to the scattering amplitude by the relation \(F = k\cdot f_T\) (for different \(\{hkl\}\), different). Since the temperature correction has not been taken into account in the equation, in order to obtain \(f_0\) the measured value \(f_T\) must be multiplied by the temperature factor \(f_0=f_T e^W\), where \(W\) is a function of the angle and of the characteristic temperature.

The calculation of neutronograms by the formula written above is carried out as follows: 1) from all the experimental data \(F\), the structure factor, is determined; 2) knowing \(F\), from the formula \(F=kf_T\), \(f_T\) is determined; 3) from the formula \(f_0=f_Te^W\), \(f_0\) is determined. From the equation \(\sigma=4\pi |f_0|^2\) one can find \(\sigma_{\mathrm{coh}}=\sigma_{\mathrm{cr}}\).

The calculation of the neutronogram of aluminum is given in Table II.

Table II

Table III

For this sample, a scattering cross section equal to \(1.46 \cdot 10^{-24}\ \mathrm{cm}^2\) was obtained, whereas the theoretically calculated value is \(1.48 \cdot 10^{-24}\ \mathrm{cm}^2\). Such agreement once again indicates the absence of a spin effect in aluminum.

In the case of monoatomic crystals, measurement of the relative intensity \(\dfrac{P_{hkl}}{P}\) for one diffraction maximum, plus knowledge of the characteristic temperature (this gives us \(W\)), is sufficient for determining the coherent scattering cross section. For diatomic crystals (NaCl, etc.) two maxima must be measured. From these data we obtain the scattering cross section for both kinds of nuclei, but we cannot determine which cross section belongs to which kind. In the case of X-ray diffraction this is possible immediately, for the X-ray scattering amplitude is proportional to the atomic number (the number of electrons in the atom).

In the case of neutron analysis it is necessary to obtain one more neutronogram from a sample containing one of the elements under investigation. Then comparison of the values of \(\sigma\) in the two experiments will allow us to associate the value of \(\sigma\) precisely with the nucleus.

Such experiments were carried out with compounds containing sodium: NaCl, NaBr, NaF, and Na (Fig. 7). The results of determining the scattering cross section for Na, Br, Cl, and F from these experiments are given in Table III.

The table gives the scattering amplitudes for all the elements analyzed and the scattering cross sections for bound nuclei.

From the table it may be seen that the coherent scattering cross section for bromine and chlorine is somewhat smaller than the total scattering cross section for these elements. Since bromine and chlorine each contain two isotopes, this must be attributed to the isotope effect. In the case of fluorine, the measured coherent scattering cross section proved larger than the total scattering cross section. This is explained by experimental errors in one of the measurements.

Fig. 7. Neutronograms of Na, NaBr, NaF, NaCl.

The coherent scattering cross sections for sodium, determined from four types of crystals, are in good agreement. The mean value is

\[ \sigma_{\mathrm{coh}}^{(\mathrm{Na})}=1.51 \cdot 10^{-24}\ \mathrm{cm}^2. \]

This value is considerably smaller than the previously determined total scattering cross section, equal to \(3.7 \cdot 10^{-24}\ \mathrm{cm}^2\). Since sodium is monoisotopic, its scattering must strongly depend on spin. One can determine \(\sigma_{S\lambda}=\sigma_{\mathrm{cr}}-\sigma_{\mathrm{coh}}\), and then the amplitudes of spin scattering for parallel and antiparallel spins of the neutron and the Na nucleus.

V. TRANSMISSION OF NEUTRONS THROUGH CRYSTALLINE POWDERS

The theory of this question has no analogue in the case of X-rays, and therefore it is of interest to clarify its application to structural analysis and nuclear physics.

The cross section of coherent scattering in the case of crystalline powders has the following expression:

\[ \sigma_{\mathrm{coh}}=\frac{\lambda^{3}N}{\delta}\sum_{hkl} j_{hkl}d_{hkl}e^{-2W}F_{hkl}\sigma_{v}, \]

where the summation is extended over all maxima of coherent scattering. In this expression \(N\) is the number of elementary cells in \(1\ \mathrm{cm}^{3}\), \(j\) is the multiplicity factor, \(e^{W}\) is the temperature factor, \(F\) is the structural factor, \(d\) is the interplanar spacing, and \(\sigma_{v}\) is the scattering cross section of the bound nucleus. In addition, the Bragg–Wulff equation \(n\lambda=2d\sin\theta\) restricts the reflected wavelengths, i.e., taking instead of \(\sin\theta\) its greatest value, 1, we obtain the condition: \(\lambda \leq 2d\) (for \(n=1\)). Consequently, if \(\lambda\) is greater than twice the largest interplanar spacing, then the condition \(\lambda \leq 2d\) is violated and there will be no coherent scattering. The scattering cross section will become equal to zero. In practice, however, this may not occur because of the spin and isotope effect.

Fig. 8. Observed and calculated scattering cross sections for Be as a function of energy.

Similar experiments, which are in full agreement with the theory of this question, were carried out with samples of Be and BeO. On the spectrograph described above the passage of thermal neutrons through finely ground powders of these materials was studied. The results of the experiments are shown in Figs. 8 and 9. For a compound containing two kinds of atoms (for example, BeO), the equation written above must be slightly modified. The calculation showed that the resulting curve must depend on the signs of the scattering amplitudes of both nuclei. Comparison of the experimental data with the theoretical curves for both cases (identical and opposite signs) clearly shows that the experiment well confirms the theory and that the scattering phases for Be and O have the same sign.

For samples containing one isotope and having no dependence of scattering on spin, all scattering will be wholly located in the maxima of coherent scattering, and the scattering cross section for \(\lambda > 2d\) will become equal to zero. Conversely, if we have a monoisotopic

sample, and for \(\lambda > 2d\) we observe diffuse scattering, we may confidently say that it is caused by the spin effect; moreover, from the scattering intensity one can judge the magnitude of the nuclear spin. By this method it was established that Be, Bi, and Al have a small dependence of scattering on spin.

Each absorption jump in the graph corresponds to \(\lambda = 2d_{hkl}\). Hence, knowing the corresponding neutron wavelength and the indices of the reflecting planes \(\{hkl\}\), one can determine the parameters of the crystal lattice from the quadratic formulae.

This method was used to obtain beams of long waves by filtering neutrons through BeO columns. The spectrum obtained is given in Fig. 10. All wavelengths satisfying the Wulff–Bragg equation were scattered; only \(\lambda > 2d\) remained in the spectrum. These beams of waves, with a maximum near \(5\,\text{\AA}\) and almost monochromatic, have found application in studies of the interaction of neutrons with gas molecules.

Fig. 9. Observed and calculated total scattering cross sections for BeO as a function of energy.

In the figure: “Calculated curves for BeO in the case of identical phases” and “opposite phases”; vertical axis: “Total scattering cross section \(\sigma\)”; horizontal axis: \(E(\mathrm{eV})\).

VI. SIGN OF THE SCATTERING PHASE

The sign of the phase shift between the incident and scattered neutron wave and, consequently, the sign of the scattering amplitude plays a significant role in neutronography. It is therefore important to determine it for all elements. The sign of the phase shift can be determined in various ways, which were indicated in the preceding sections; for example, by measuring the transmission of neutrons through powders of diatomic specimens or by considering the diffraction pattern of crystals. Usually the sign of the phase shift is determined by comparing experimental and theoretical data on the intensities of diffraction maxima. A typical comparison is given in Fig. 11, where a number of maxima and theoretical calculations of the intensity are shown for identical and oppo-

Fig. 10. Spectrum of neutrons after filtration through BeO columns.

opposite phase shifts. This comparison once again shows that Be and O scatter with the same sign.

The fact that the diffraction maximum (111) for NaCl, NaBr, and NaF is smaller than the maximum (200) (Fig. 7) shows that Na, Cl, Br, and F scatter with the same sign of phase shift. This conclusion was again made in connection with theoretical calculations.

Fig. 11. Comparison of calculated and observed diffraction patterns for BeO.

A comparison of the observed and calculated diffraction patterns for manganese and oxygen is shown in Fig. 12.

Fig. 12. Comparison of calculated and observed diffraction patterns for MnO.

The sign of the scattering amplitude can also be determined, in addition, by measuring the angle of total internal reflection for neutrons.

All these investigations showed that most elements have a positive scattering amplitude. Table IV gives a summary of the signs of the scattering amplitudes for 31 elements.

It turned out that only four of them have a negative scattering amplitude.

Table IV

In the theoretical part it was noted that just such a pattern should be expected, owing to the greater probability of a negative phase shift.

It is of interest that different isotopes may have different signs of the scattering amplitudes. Experiments have established, for example, that Li⁷ has a negative amplitude, while Li⁶ has a positive scattering amplitude. The same observation can be made for the case of different orientations of the spins of the nucleus and the neutron.

VII. PHOTOGRAPHING NEUTRON DIFFRACTION BY THE LAUE METHOD

In the Laue method nonmonochromatic beams of neutrons are used; it would seem that it should be easy to carry out. However, in this case a counter is inapplicable, and a photographic method of recording is required in the variant described above. In this case it is not individual neutrons that are recorded, but their beams, and the sensitivity of the method is therefore extremely low. Nevertheless, several Laue photographs have been obtained. The exposure in this case was more than 10 hours. From this it may be concluded that in neutronography the principal role will belong to the crystalline-powder method.

The experiments carried out up to the present time must be regarded as the beginning of the main work in this field. They have shown the enormous possibilities of neutronography and confirm the theory underlying it—the interaction of neutrons with crystals, molecules, and nuclei.

However, one cannot consider that, with the emergence of neutronography, X-ray structural analysis has completely lost its significance. On the contrary, these two methods, in their reasonable combination, are a powerful instrument in the hands of the investigator in the study of any crystalline structures.

R. P. Ozerov.

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