STRUCTURAL NEUTRONOGRAPHY

R. P. Ozerov

It is quite obvious that methods enabling the investigation of the atomic and molecular structure of substances are of enormous importance, since in the final analysis it is precisely this structure that determines many physicochemical properties. Among such methods, X-ray structural analysis occupies a fundamental place. It is difficult to enumerate all the fields in which X-ray diffraction is applied; an extraordinarily large number of problems have been solved by this method. And yet there are a number of fundamental questions that cannot be solved (or can be solved only with exceptional difficulty) by X-ray methods. The same may be said of electron diffraction.

Naturally, after the discovery of neutrons and after the experimental realization of neutron interference phenomena, the suggestion arose that diffraction of these particles might be applied to structural investigations. Experiments carried out up to the present time have confirmed these assumptions.

Because neutrons have no charge, the mechanism of their scattering is entirely different from that for X-rays and electrons. Therefore the fields of application of neutron and X-ray analysis are somewhat different; they as it were complement one another. This is also where the value of neutronography lies. It becomes still more obvious if one recalls that neutrons possess a magnetic moment and that, consequently, coherent magnetic scattering makes it possible to study the magnetic structure of substances.

To carry out neutronographic investigations an intense beam of slow neutrons is required. Therefore, before the construction of nuclear reactors, structural work with neutrons was scarcely performed; their number increased sharply in connection with the development of powerful neutron sources. It is quite understandable that these works are mainly of a somewhat preparatory character (analogous, for example, to the compilation of certain tables of intensity factors for X-ray structural analysis).

analysis, without which no serious structural investigations are possible); however, a number of fundamental questions have already been solved by neutronographic methods, and these will be discussed below.

A major contribution to theoretical neutron physics has been made by Soviet theorists, including A. I. Akhiezer and I. Ya. Pomeranchuk, who brought together a considerable number of their works concerning the interaction of neutrons with matter in a monograph¹ that is used extensively in the present article.

I. SOME QUESTIONS OF THEORY

Although the scattering of thermal neutrons and the scattering of X-rays are in many respects analogous phenomena, there are, however, essential differences that compel one to consider neutron scattering quite independently.

For X-rays the role of scatterers is played by bound electrons. The scattering power of an atom can be calculated with a high degree of accuracy by summing the scattering amplitudes of all the electrons of the atom, taking into account their relative positions. This gives the atomic scattering factor, which is determined by the magnitude of the scattering of the whole atom relative to the scattering of a single bound electron. Thus, atomic factors for X-rays can be determined experimentally and calculated theoretically.

For neutrons the scattering is almost always nuclear, and the little that is known about the structure of the nucleus and about nuclear forces does not yet permit one to calculate the scattering theoretically. From this fundamentally different mechanism of scattering there follow several other features, which will be considered in the corresponding sections.

1. Scattering of neutrons by nuclei

According to collision theory, the wave equation describing the scattering of neutrons by nuclei is as follows:

\[ \Delta\psi + k'^2\psi = 0, \tag{1.1} \]

where

\[ k'=\sqrt{\frac{2\mu\left(E-V(r)\right)}{\hbar^2}} \tag{1.2} \]

is the wave number of the neutron in the nucleus, \(V(r)\) is the potential function of the scattering nucleus, \(\mu\) is the reduced mass of the system, and \(E\) is the energy of the neutron. For \(V=0\), this equation describes the moti-

the neutron outside the nucleus, and \(k'\) in this case is equal to the wave number \(k\):

\[ k'\big|_{V=0}=\sqrt{\frac{2mE}{\hbar^2}}=\frac{2\pi}{\lambda}=k, \tag{1.3} \]

where \(m\) is the mass and \(\lambda\) is the wavelength of the neutrons. The neutron beam in this case is described by a plane wave:

\[ \psi\sim e^{ikz}. \tag{1.4} \]

The solution of equation (1.1) at large distances from the scattering nucleus (for large \(r\)) may be sought in the form of the sum of a plane wave incident along the \(z\)-axis and a wave scattered by the nucleus:

\[ \psi=e^{ik'z}+f(\theta)\frac{e^{ikr}}{r}. \tag{1.5} \]

In this expression \(f(\theta)\) is the scattering amplitude and \(\theta\) is the scattering angle.

Solving equation (1.1), we obtain a relation between the scattering amplitude \(f(\theta)\) and the nuclear potential. Thus, the scattering amplitude \(f(\theta)\) completely determines the scattering.

The solution of equation (1.1) in the general case also gives the following expression for \(f(\theta)\) (see, for example, \({}^{3}\)):

\[ f(\theta)=\frac{1}{2ik}\sum_{l=0}^{\infty}(2l+1)\left(e^{2i\eta_l}-1\right)P_l(\cos\theta), \tag{1.6} \]

where \(P_l(\cos\theta)\) is the Legendre polynomial of order \(l\), and \(\eta_l\) is the change in phase of the radial part of the wave function relative to the phase in the absence of the scattering nucleus.

Let us establish the relation between the cross section and the scattering amplitude. The probability that a scattered particle, per unit time, will pass through a surface element \(dS\), located at a distance \(r\) from the scattering center, is equal to

\[ \frac{v}{r^2}\,|f(\theta)|^2\,dS, \tag{1.7} \]

where \(v\) is the density of the incident beam. Since the element of solid angle \(d\omega=\dfrac{dS}{r^2}\), expression (1.7) can be rewritten as

\[ v\,|f(\theta)|^2\,d\omega. \tag{1.8} \]

The differential scattering cross section \(d\sigma_s\) is defined as the ratio of (1.8) to the density of the incident beam \(v\). Thus:

\[ d\sigma_s=|f(\theta)|^2\,d\omega. \tag{1.9} \]

Since \(d\omega=2\pi\sin\theta\,d\theta\), expression (1.9) is rewritten as

\[ d\sigma_S=2\pi |f(\theta)|^2\sin\theta\,d\theta . \tag{1.10} \]

To find the integral scattering cross section \(\sigma_S\), (1.10) must be integrated with respect to \(\theta\) over the limits from \(0\) to \(\pi\):

\[ \sigma=2\pi\int_0^\pi |f(\theta)|^2\sin\theta\,d\theta . \tag{1.11} \]

Substituting in (1.11) expression (1.6) for \(f(\theta)\), taking into account the orthogonality of the Legendre polynomials and the fact that

\[ \int_0^\pi P_l^2(\cos\theta)\sin\theta\,d\theta=\frac{2}{2l+1}, \tag{1.12} \]

we obtain

\[ \sigma_S=\frac{4\pi}{k^2}\sum_l (2l+1)\sin^2\eta_l . \tag{1.13} \]

Neither equation (1.6) nor equation (1.13) contains the nuclear potential \(V(r)\), although both give a relation between the quantities determining scattering and \(V\). The point is that both (1.6) and (1.13) contain \(\eta_l\)—the phase shift, which is related in a completely definite way to \(V(r)\). Thus, although \(V\) does not enter the expressions for \(f(\theta)\) and \(\sigma_S\), it is connected with these quantities through \(\eta_l\).

Up to now we have imposed no restrictions on the energy of the scattered neutrons. Let us now turn to slow neutrons. These include those neutrons whose wavelength \(\lambda\) is considerably greater than the width of the potential well of the scattering nucleus, or, what is the same thing, than the range of the nuclear forces or the radius of the nucleus \(r_0\). Thus, for slow neutrons

\[ \lambda \gg r_0 . \tag{1.14} \]

Taking into account that

\[ \lambda=\frac{h}{mv}, \tag{1.15} \]

and substituting into (1.14) and (1.15) the known value \(r_0 \approx 10^{-13}\,\text{cm}\), we obtain the criterion for slow neutrons: neutrons with \(E\) up to \(20\,\text{MeV}\) are called slow. For structural purposes neutrons with \(\lambda \simeq 1\,\text{\AA}\) are needed. Such a wavelength corresponds to neutrons with energy \(E \simeq 1\,\text{eV}\), and therefore these are slow neutrons. Such energy is possessed by neutrons that have undergone moderation, i.e. are in thermal equilibrium with the molecules of the moderator substance. Hence they have been given the name thermal. And, finally, we shall have to encounter neutrons whose energy corresponds to a temperature of \(20^\circ\text{K}\). These are “cold” neutrons with \(\lambda \simeq 5\text{–}7\,\text{\AA}\).

It is easy to show that slow neutrons (including thermal ones) are scattered by nuclei spherically symmetrically, i.e., with \(l=0\). Indeed, for a scattering event to occur, the impact parameter \(p\) must be smaller than the range of action of the nuclear forces, i.e.

\[ p<r_0 . \tag{1.16} \]

Let us estimate the magnitude of \(p\). From classical mechanics we obtain

\[ p=\frac{M}{m v}, \tag{1.17} \]

where \(M\) is the relative angular momentum and \(v\) is the relative velocity of the neutron. In the semiclassical case

\[ M \simeq l\hbar . \tag{1.18} \]

Substituting (1.18) into (1.17), we obtain

\[ p \simeq \frac{l\hbar}{m v} \simeq l\lambda . \tag{1.19} \]

Let us rewrite condition (1.16), taking into account expression (1.19). We obtain

\[ l\lambda<r_0, \tag{1.20} \]

or

\[ l\cdot 10^{-8}<10^{-13}. \tag{1.20'} \]

This inequality can be satisfied only for \(l=0\). Hence it is clear that the scattering of slow neutrons is spherically symmetric. It follows from this that all \(\eta_l\), except \(\eta_0\), may practically be taken equal to zero:

\[ \eta_l\big|_{l\ne 0}=0 . \tag{1.21} \]

Mathematically this follows from the solution of equation (1.1) for the special case of slow neutrons, as a result of which for \(\eta_l\) one obtains the expression

\[ \eta_l\sim k^{2l+1}, \tag{1.22} \]

where \(k\) is the wave number in nuclear units; for slow neutrons \(k\ll 1\). From this formula the conclusion (1.21) made above also follows.

Taking \(l=0\) in (1.6), (1.10), (1.13), we obtain:

\[ f(\theta)=f=-\frac{1}{2ik}\left(e^{2i\eta_0}-1\right). \tag{1.23} \]

As was to be expected, the scattering amplitude of slow neutrons does not depend on the scattering angle:

\[ d\sigma_s=f^2\sin\theta\,d\theta, \tag{1.24} \]

\[ \sigma_s=4\pi\frac{\sin^3\eta_0}{k^3}=4\pi f^2 . \tag{1.25} \]

Usually the phase shift \(\eta_0\) is very small, except in the immediate vicinity of a resonance level. Therefore in (1.23) the term \(e^{2i\eta_0}\) may be expanded in a series and one may retain only the first power of \(\eta_0\). Then we obtain

\[ f=\frac{\eta_0}{k} \tag{1.26} \]

and

\[ \sigma_s=4\pi \frac{\eta_0^{2}}{k^{2}} . \tag{1.27} \]

The phase shift may take both positive and negative values. However, positive phase shifts occur only near resonance levels and are therefore less probable than negative ones. Since it has become customary to regard the scattering amplitude and phase shifts as having opposite signs, positive values are consequently more probable. This has also been confirmed experimentally.

The phase shifts \(\eta_0\) can be determined from various experiments, which will be discussed below. They can also be calculated theoretically:

\[ \eta_0=C\int_{0}^{\infty} V(r)\,r^{2}dr . \tag{1.28} \]

In nuclear physics, equation (1.28) is used to find \(V(r)\) from experimentally determined \(\eta_0\).

In all these derivations, absorption of neutrons by nuclei was neglected. Such neglect is legitimate under the condition

\[ \frac{\sigma_a}{\sigma_s}\gtrsim 10^4, \tag{1.29} \]

which is satisfied for the overwhelming majority of elements of the periodic system (exceptions include, for example, cadmium, gadolinium, and a few other strongly neutron-absorbing elements). In all the works described below, condition (1.29) is satisfied, and the neglect of absorption is legitimate.

2. Factors Affecting Neutron Scattering

The magnitude of the scattering cross section depends on whether the nucleus is in a free or bound state. In the case of a rigidly bound nucleus, the scattering cross section is greater than the scattering by a free nucleus by the square of the ratio of the neutron mass to the reduced mass of the neutron and the scattering nucleus, i.e.

\[ (\sigma_s)_{\mathrm{bound}} = \left(\frac{1}{\mu}\right)^2 (\sigma_s)_{\mathrm{free}} = \left(\frac{A+1}{A}\right)^2 (\sigma_s)_{\mathrm{free}} . \tag{1.30} \]

Between the cases of rigid binding and a free nucleus there exist intermediate cases, characterized by incomplete binding. For such binding the correction assumes some average value from

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

to 1. In considering elastic scattering of neutrons by crystals, the nuclei may be assumed to be rigidly bound; in the case of gases, the nuclei are practically free.

In what follows, the symbol \(\sigma_{\mathrm{cryst}}\) is understood to mean the scattering cross section of an atom rigidly fixed in a crystal lattice, i.e., the cross section determined by formula (1.30).

Because neutrons are scattered by the atomic nucleus and not by the electron shell, the scattering cross sections for different isotopes of the same element are, generally speaking, not equal to one another. For X-rays this phenomenon does not occur. Therefore a single-atom crystal that is completely homogeneous from the X-ray point of view will, for neutrons, be an inhomogeneous, disordered “alloy” of several isotopes. Therefore in (1.25) there must be substituted an amplitude \(\overline{f}_u\) averaged over all \(i\) isotopes:

\[ \overline{f}_u=\sum_i g_i f_i . \tag{1.31} \]

Here \(g_i\) is the abundance and \(f_i\) the scattering amplitude of each isotope. In this connection, in order to take coherent scattering into account, it is first necessary to add algebraically the scattering amplitudes of each isotope multiplied by the abundance, and only after this to square the result. Thus*):

\[ \sigma_{\mathrm{coh}}=4\pi\left(\sum_i g_i f_i\right)^2 . \tag{1.32} \]

In doing this it is necessary to pay special attention to the signs of the scattering amplitudes.

The total cross section of isotopic scattering is equal to

\[ \sigma=4\pi\sum_i g_i f_i^2 . \tag{1.33} \]

The difference between (1.33) and (1.32) gives the incoherent isotopic scattering \(\sigma_{DI}\).

Thermal vibrations of atoms in the crystal lattice, in neutron scattering, increase the diffuse background. The corresponding correction, as for X-rays, can be introduced by means of the temperature factor:

\[ f_0=f_T e^w , \tag{1.34} \]

*) In what follows the sign \(S\) in \(\sigma_S\) is omitted; \(f\) is understood to mean the amplitude of coherent scattering.

where \(f_0\) and \(f_T\) are the scattering amplitudes at \(0^\circ\mathrm{K}\) and at a certain temperature \(T^\circ\mathrm{K}\), respectively, and

\[ w=\frac{6h^2}{Mk\Theta}\left(\frac{\varphi(x)}{x}+\frac{1}{4}\right) \left(\frac{\sin\theta}{\lambda}\right)^2 . \tag{1.35} \]

In (1.34), \(k\) is Boltzmann’s constant, \(M\) is the mass of an atom of the lattice, \(\Theta\) is the characteristic temperature, \(x=\frac{\Theta}{T}\), and

\[ \varphi(x)=\frac{1}{x}\int_0^x \frac{z\,dz}{e^z-1} \tag{1.36} \]

is the Debye function. Naturally, the temperature effect is most pronounced for large \(\Theta\).

If it is necessary to reduce the temperature effect to a minimum (for example, in order to increase the resolving power of crystal spectroscopic instruments), one must choose crystals with a large characteristic temperature \(\Theta\), since \(w\sim \frac{1}{\Theta}\).

So far nothing has been said about the dependence of scattering on the mutual orientation of the spins of the neutron and the scattering nucleus. As is known, the total spin of the system in the act of scattering can assume two values: \(I+\frac{1}{2}\)—for parallel orientation, and \(I-\frac{1}{2}\)—for antiparallel orientation of the spin of the neutron \(\left(\frac{1}{2}\right)\) and of the nucleus \(I\). Let us assign to each of these cases its own scattering amplitude: \(f_{I+\frac{1}{2}}\) and \(f_{I-\frac{1}{2}}\). Then, as in the case of the isotopic effect, we obtain one averaged amplitude:

\[ \bar f_c= \left(\frac{I+1}{2I+1}\right)f_{I+\frac{1}{2}} + \left(\frac{I}{2I+1}\right)f_{I-\frac{1}{2}} . \tag{1.37} \]

The expressions in parentheses are the corresponding statistical weights of one or the other orientation of the two spins; their role in (1.37) is analogous to the role of the factors \(g_i\) in (1.32) and (1.33).

Equality of the two amplitudes means the absence of spin dependence. In this case the averaged amplitude is equal to one of the amplitudes.

As a result of spin dependence, both coherent and incoherent scattering can be observed. The first type of scattering is due to those collisions that occur without a change in the total spin of the system; the second, to those in which the mutual orientation of the spins changes. This is explained simply by the fact that interference phenomena take place only in the case when,

when scattering occurs as a result of the interaction of the scattered particles with an entire group of definitely arranged atoms. In the case of a collision with a change of the total spin, it is possible to specify exactly the atom which is responsible for the scattering. Consequently, such scattering is incoherent. Therefore the values of the cross section for spin scattering will be different for these two cases. The cross section of coherent spin scattering is equal to

\[ \sigma_{\mathrm{coh}}=4\pi\left\{\frac{I+1}{2I+1}f_{I+\frac12}+\frac{I}{2I+1}f_{I-\frac12}\right\}^{2}. \tag{1.38} \]

Analogously to (1.33), the total cross section of spin scattering is

\[ \sigma=4\pi\left\{\frac{I+1}{2I+1}f^{2}_{I+\frac12}+\frac{I}{2I+1}f^{2}_{I-\frac12}\right\}. \tag{1.39} \]

The difference between these two quantities will give the cross section of incoherent spin scattering:

\[ \sigma-\sigma_{\mathrm{coh}}=\sigma_{DS}=4\pi\frac{(I+1)I}{(2I+1)^{2}}\left(f_{I+\frac12}-f_{I-\frac12}\right)^{2}. \tag{1.40} \]

If it were possible to separate \(\sigma_{DS}\) in pure form, one could find the quantity \(\left|f_{I+\frac12}-f_{I-\frac12}\right|\), which is in some sense a measure of the dependence of scattering on spin. In practice, because of the large number of factors causing incoherent scattering of neutrons, the experiments described below make it possible to determine only the upper limit of this quantity.

Summarizing what has been set forth for the case of neutron scattering by crystals, one may conclude: the total scattering cross section will be a sum consisting of the cross sections of coherent scattering and of diffuse scattering arising as a result of isotopic \(\sigma_{DI}\), temperature \(\sigma_{DT}\), and spin \(\sigma_{DS}\) effects:

\[ \sigma_{\mathrm{cryst}}=\sigma_{\mathrm{coh}}+\sigma_{DI}+\sigma_{DT}+\sigma_{DS} \tag{1.41} \]

—this is in the general case. In many particular cases, however, some terms of this sum may vanish. Thus, for example, for a monoisotopic element \(\sigma_{DI}=0\); for crystals with a large elastic constant \(\sigma_{DT}=0\); for elements whose scattering does not depend on the spin of their nuclei, \(\sigma_{DS}=0\).

Up to now we have not been concerned with interference phenomena arising as a result of the regular arrangement of scattering centers in space. Let us consider them.

3. Scattering of Neutrons in Gases, Liquids, and Solids

At present, the theory of the scattering of X-rays in gases and liquids from the structural point of view has been developed sufficiently well (see, for example, \({}^{3}\)). The difference between X-rays and neutrons in this respect lies only in the mechanism of scattering; the interference phenomena are almost completely analogous. Let us generalize the theory of the scattering of X-rays in gases, liquids, and solids to the case of slow neutrons, noting at the same time the similarity and the difference of the phenomena.

If one disregards the interaction between the incident and the scattered radiation (neglecting extinction, which plays an essential role only in scattering by crystals), the act of scattering may be regarded as the combination of two processes. Incident X-rays, for example, cause the electrons of an atom to oscillate—the first process; the electrons, oscillating with the same frequency, become a source of coherent electromagnetic oscillations—the second process. Since the scattered rays are coherent, they interfere according to laws that depend on the positions of the scattering centers.

The intensity of X-ray scattering at an angle \(\theta\) by a single bound electron, according to the classical theory, after Thomson, is expressed as follows:

\[ I_s = I_0 \cdot \frac{e^4}{m^2 c^4 r^2} P(\theta), \tag{1.42} \]

where \(I_0\) is the intensity of the incident radiation, \(e\) the charge and \(m\) the mass of the electron, \(c\) the speed of light, and \(r\) the distance from the scattering atom to the point of observation; \(P(\theta)\) is the polarization factor, depending on the orientation of the electric vector of the incident radiation. Since \(P(\theta) \ne \mathrm{const}\), but depends on \(\theta\), the scattering is not spherically symmetric; in this lies the difference from neutron scattering. For unpolarized X-rays,

\[ P(\theta)=\frac{1+\cos^2\theta}{2}. \tag{1.43} \]

The polarization factor for neutrons when they are scattered by nuclear forces is equal to unity.

If there were no interference phenomena in the scattering of X-rays by an atom, then the scattering intensity in this case would be \(Z\) times greater than the scattering by an electron. However, owing to the fact that the electrons in the atom are arranged regularly and the wavelength of X-rays is of the same order as the size of the atom, the rays scattered by the electrons of one atom will interfere. Naturally, the interference phenomena

play the greatest role at \(\theta=\frac{\pi}{2}\) and the smallest at \(\theta=0\). Thus we have arrived at the atomic amplitude of x-ray scattering \(f_p\), determined by the ratio of the scattering amplitude of an atom to the scattering amplitude of one electron. The square of this quantity is called the atomic factor. Therefore

\[ I_a=I_e\cdot f_p^2 . \tag{1.44} \]

As was already said above, \(f_p\) can be calculated theoretically with a high degree of accuracy:

\[ f_p=\int_0^\infty \rho(r)\,\frac{\sin(ksr)}{ksr}\,4\pi r^2dr, \tag{1.45} \]

where \(\rho(r)\) is the electron density in the atom and \(s=2\sin\theta\).

The main difficulty lies in finding \(\rho(r)\). For this there are, basically, two methods: the Hartree method and the Thomas–Fermi method. The first method is more laborious and cumbersome, but accurate; the second is simpler, but gives less accuracy; for heavy atoms both methods give good results, confirmed by experimental data.

Fig. 1. Comparison of the dependence of the scattering amplitudes of neutrons, x-rays, and electrons by copper atoms.

It is possible to determine \(f_p\) rather accurately and simply from experiment, by measuring the intensities of the lines of a definite powder x-ray pattern and, in the calculation, excluding all the other intensity factors.

Let us compare the atomic scattering amplitudes for neutrons \(f_n\) and for x-rays \(f_p\) from the point of view of their dependence on \(\theta\) (in Fig. 1 such a comparison is given\(^4\); the scattering amplitude for electrons \(f_e\), which in principle differs in no way from \(f_p\), is also given there). Of course, the strong dependence of \(f_p\) on \(\theta\) is harmful for structural investigations, since to a considerable extent it masks weak maxima. (A more detailed comparison of \(f_n\) and \(f_p\) will be given below.)

The pattern obtained in the scattering of x-rays by monatomic gases at low pressure differs in no way from

of the course of the atomic factor with only a change of the vertical scale. This occurs because each atom of such a gas scatters independently of the others; hence

\[ (I_1)_{\text{m.d.}}=I_aNf_p^2=I_aN, \tag{1.46} \]

where \(N\) is the number of scattering centers of the gas per unit volume.

Fig. 2. Dependence \(I(\theta)\) for a monatomic gas at high pressures.

At high pressures the picture changes somewhat as a result of the fact that there appears a certain distance of closest approach, identical for all pairs of atoms, equal to twice the radius of the atom. The resulting interatomic interference decreases the scattering intensity at small angles and (if the atomic factor is not taken into account) leads to the appearance of a diffraction maximum in the region of small \(\theta\). According to Debye’s calculations\(^5\), the scattering intensity in this case is described by the following expression:

\[ (I_1)_{\text{b.d.}}=I_aN\left\{1-\frac{\Omega}{V}\cdot \varphi(ksa)\right\}, \tag{1.47} \]

where \(a\), roughly speaking, is the radius of the atom,

\[ \Omega=\frac{4}{3}\pi a^3N \]

is the total volume of the atoms in the volume \(V\), and

\[ \varphi=3\left(\frac{\sin u-u\cos u}{u^3}\right). \]

The quantity \(\frac{\Omega}{V}\) characterizes the density of the gas; for small \(\frac{\Omega}{V}\) formula (1.47) passes into (1.46).

If the dependence of the scattering intensity on the angle is plotted without taking the atomic factor into account (i.e., \(\frac{I}{f_p^2}=\psi(\theta)\)), then a curve with one maximum is obtained (Fig. 2; \(\frac{\Omega}{V}=\frac{1}{2}\), \(\frac{a}{\lambda}=3\)).

The interference pattern becomes considerably more complicated if, instead of a monatomic gas, the X-rays are scattered by a diatomic gas. To the intermolecular interference considered above there is now added intramolecular interference.

Indeed, X-rays scattered by different atoms of the same molecule will be in constant phase relations and therefore must interfere. The calculated

taking into account intramolecular interference, the scattering intensity \(I_2\) is equal to

\[ I_2=4NI_a\left\{\frac{1}{2}\left[1+\frac{\sin ksl}{ksl}\right]-\frac{\Omega}{V}\left(\frac{\sin \frac{ksl}{2}}{\frac{ksl}{2}}\right)^2\varphi(ksa)\right\}, \tag{1.48} \]

where \(l\) is the distance between the atoms in the molecule, and \(2a\) is, in this case, the diameter of the “sphere of action” of the whole molecule.

This formula passes into (1.46) for \(l\to 0\) and \(\dfrac{\Omega}{V}\to 0\). The term in square brackets describes intramolecular interference, which gives an intensity maximum at large scattering angles (as compared with the intermolecular one).

Figure 3 presents, in general form, the interference pattern for a diatomic gas at different pressures \(\left(\dfrac{\Omega}{V}=0,\ \dfrac{1}{4},\ \dfrac{1}{2},\ \dfrac{3}{4};\ \dfrac{l}{\lambda}=\dfrac{a}{\lambda}=3\right)\). It is characteristic that the first maximum, which depends on intermolecular interference, decreases as the density of the gas decreases. The second, however, at a larger angle, remains unchanged at all densities. It may be used to determine the distance between atoms in a molecule.

Fig. 3. Dependence \(I(\theta)\) for a diatomic gas at different pressures.

In the study of X-ray scattering in liquids, a similar approximation \(\left(\dfrac{\Omega}{V}\to 1\right)\) proved to be invalid. In this case it is necessary to take into account a certain ordering in the arrangement of the atoms of the liquid. This was done\(^6\) by introducing a certain function \(\rho(r)\), such that \(4\pi\rho(r)\,dr\) is equal to the number of atomic centers in a spherical layer of thickness \(dr\) at a distance \(r\) from some specified atom. Naturally, for large \(r\) in a liquid, where there is no long-range order, \(\rho(r)=\rho_0\), the mean macroscopic atomic density. The greater the difference \([\rho(r)-\rho_0]\), the greater the ordering in the arrangement of the atoms. It was already established long ago by X-ray methods that in liquids there are certain regions in which the atoms are arranged quite regularly; the dimensions of these regions, however, do not exceed 6–8 atomic radii. We shall discuss this in greater detail below.

The expression for the scattering intensity in liquids was found with the aid of the function \(g_0(r)=\rho(r)-\rho_0\):

\[ I_{\varkappa}=\frac{1}{2}N I_a\left[1+\int_0^\infty \frac{4\pi r^2 g_0(r)}{ksr}\sin ksr\,dr\right]. \tag{1.49} \]

In this expression the integral plays the role of a kind of structural factor. If it is equal to zero, the scattering by such a liquid becomes analogous to scattering by an ensemble of atoms independent of one another.

By means of a similar equation, X-ray diagrams of liquid substances and, in particular, of a fairly large number of liquid metals have been deciphered.

Fig. 4. X-ray diagram of liquid mercury and the curve \(\rho(r)\).

Figure 4 shows an X-ray diagram of liquid mercury without allowance for the atomic and polarization factors, \(\dfrac{I_{\varkappa}}{f_p^2P(\theta)}\), and the curve \(\rho(r)\), obtained from the X-ray diagram*).

*) A detailed discussion of Fig. 4 will be given below, in the corresponding section.

In passing from scattering by a liquid to scattering by solid crystalline substances, the picture becomes considerably more complicated. The radiograph consisting of sharp diffraction maxima, located at quite definite positions determined by the well-known Bragg–Wulff equation,

\[ n\lambda = 2d\sin\theta, \tag{1.50} \]

is simpler, where \(d\) is the distance between the atomic planes of the crystal and \(n\) is the order of reflection; however, interpretation of the radiograph and determination of the structure of the crystal lattice at present present a considerable difficulty.

In crystals the role of the function \(\rho(r)\) is played by the structure amplitude \(F\), which depends, first, on the positions of the atoms in the crystal lattice and, second, on the constituent atoms themselves. The square of the structure amplitude \(|F|^{2}\) is called the structure factor.

The intensity of reflection of x rays from crystals depends on many factors of both physical and geometrical origin; it depends on whether the crystal is thin or thick (the criterion of thinness is reflection by the crystal of a negligible fraction of the radiation incident on it), mosaic or ideal, on the size of the microcrystallites of a polycrystalline substance, and on many other causes.

In this article we shall be interested only in the dependence of the reflection intensity on the structure factor. This dependence is different for mosaic and ideal crystals because of strong extinction in the latter case.

For mosaic crystals the intensity of reflection \(I_{hkl}\) from the plane \((hkl)\) is proportional to the structure factor*):

\[ \frac{I_{hkl}}{I_{0}} \sim |F|^{2}, \tag{1.51} \]

whereas for ideal crystals \(\dfrac{I_{hkl}}{I_{0}}\) is proportional to the structure amplitude:

\[ \frac{I_{hkl}}{I_{0}} \sim |F|. \tag{1.52} \]

In some cases this difference plays an essential role. In the scattering of x rays that have passed through a plate of crystalline powder of thickness \(t\), the expression

*) \(I_{hkl}\) is proportional to the area of the diffraction maximum.

for the scattering intensity is as follows:

\[ \frac{I_{hkl}}{I_0} = \frac{j_{hkl}l^4}{4\pi r\sin^2 2\theta}\, \frac{\rho'}{\rho}\, \frac{N\lambda^4 e^2}{m^2c^4}\, \left|F\right|^2 P(\theta)e^{-2w}. \tag{1.53} \]

In this formula \(I_{hkl}\) is the intensity of the X-rays reflected from a part of the ring of length \(l\) at a distance \(r\), \(j_{hkl}\) is the repetition factor, \(\rho'\) is the density of the powder, \(\rho\) is the density of the continuous crystal, \(N\) is the number of unit cells per unit volume of the crystal, and \(e^{-2w}\) is the temperature factor.

Let us now discuss the question of how far, for neutrons, the conclusions drawn in this paragraph concerning interference phenomena in X-rays upon their scattering are valid.

The mechanism of the scattering act for X-ray and neutron waves is quite different. Therefore, of course, formulas (1.42), (1.43), and (1.44) do not apply to neutrons. In deriving the subsequent formulas, however, the scattering mechanism was not touched upon; rather, the behavior of the scattered radiation was considered, taking into account the positions of the atoms and their interaction. Therefore all structural conclusions will also be valid for neutrons.

In order to make formulas (1.46), (1.47), (1.48), (1.49) valid for neutrons, it is necessary everywhere, instead of \(I_a\), to put the quantity \(\frac{\sigma}{4\pi}=f_n^2\), leaving the interference terms unchanged. Consequently, for a monatomic gas at low pressures we have:

\[ (I_1)_{\text{l.p.}}^{\text{n}} \sim \frac{\sigma}{4\pi}N, \tag{1.46'} \]

at high pressures:

\[ (I_1)_{\text{h.p.}}^{\text{n}} \sim \frac{\sigma}{4\pi}N \left\{1-\frac{\Omega}{V}\cdot\varphi(ksa)\right\}, \tag{1.47'} \]

for a diatomic gas:

\[ (I_2)^{\text{n}} \sim \frac{\sigma}{4\pi}N \left\{ \frac{1}{2}\left[1+\frac{\sin(ksl)}{ksl}\right] -\frac{\Omega}{V} \left(\frac{\sin \frac{ksl}{2}}{\frac{ksl}{2}}\right)^2 \varphi(ksa) \right\}, \tag{1.48'} \]

and for monatomic liquids:

\[ (I_{\text{liq}})^{\text{n}} \sim \frac{\sigma}{8\pi}N \left[ 1+ \int_0^\infty \frac{4\pi r^2 g_0(r)}{ksr}\sin(ksr)\,dr \right]. \tag{1.49'} \]

The X-ray diffraction patterns shown in Figs. 2, 3, and 4 in transformed form (with the atomic and polarization factors \(I\) excluded) should in principle be identical to neutron diffraction patterns

with another vertical scale. Indeed, if \((I_a)^{\mathrm p}\) depends in a complicated way on \(\theta\), then

\[ \frac{(I_a)^{\mathrm p}}{f_{\mathrm p}^{\,2}(\theta)} \]

is in this respect a “constant” and is proportional to \(k(I)^{\mathrm n}\) (\(k\) is the scale factor). Consequently, for neutrons we may expect the interference pattern shown in Figs. 2, 3, and 4 and described by equations \((1.46')\), \((1.47')\), \((1.48')\), \((1.49')\).

When considering the diffraction of neutrons in crystalline substances, it is natural to suppose that all the conclusions of X-ray crystallography made for weakly absorbing crystals will also be valid for neutrons. The work on neutron crystallography carried out at the present time \({}^{7,8}\) and partly \({}^{9}\) confirms this supposition and notes the far-reaching analogy between these two cases. We have no possibility here of dwelling in greater detail on neutron crystallography.

When considering the diffraction of neutrons in crystalline powders (in transmission), the formula (1.53), transformed in accordance with what was set out above, remains essentially valid:

\[ \left(\frac{I_{hkl}}{I_0}\right)^{\mathrm n} = \frac{j_{hkl}lt}{4\pi r\sin^2 2\theta}\, \frac{\rho'}{\rho}\, \lambda^3 e^{-\mu t\sec\theta}\, |F|^2 N^2 e^{-2w}, \tag{1.53'} \]

where \(I_{hkl}\) is the intensity of the reflection from the planes \((hkl)\), measured by a counter with slit width \(l\) at a distance \(r\) from the specimen [the corresponding \(I_{hkl}\) in (1.53)].

To what extent experiment agrees with theory will be stated below.

All the neutron formulas of this section were derived under the assumption that all atoms of the substance are identical and that the scattering does not depend on the orientations of the spins. In the contrary case the picture becomes more complicated.

4. Optical properties of neutrons

Solving the wave equation for a neutron outside and inside a crystal and then finding the difference \(k^2-k_0^2\), one can obtain an expression for the refractive index \(n=\dfrac{k}{k_0}\) (\(k\) is the wave number of the neutron inside, and \(k_0\) outside, the crystal). If condition (1.29) is satisfied, the following expression \({}^{9}\) is valid for the refractive index (see also \({}^{66}\)):

\[ n^2-1 = -\frac{h^2 N\sqrt{4\pi\sigma}}{8\pi^2 mE} = -\lambda^2\frac{Nf}{\pi}, \tag{1.54} \]

where \(N\) is the number of atoms per unit volume (\(f\) is taken with its sign). Substituting in (1.54) the values \(\lambda \simeq 10^{-8}\,\text{cm}\), \(f \simeq 10^{-12}\,\text{cm}\), and \(N \simeq 10^{23}\,\dfrac{1}{\text{cm}^3}\), we obtain for the refractive index the numerical value \(n^2 - 1 \simeq 10^{-6}\). Hence it is evident that the refractive index differs only very little from unity. Here again the analogy with X-rays appears.

Using the boundary conditions for the wave function and its normal derivative, and knowing the refractive index, one can find an expression for the reflection coefficient \(R\):

\[ R=\left|\frac{\sqrt{\,n^2-\sin^2\theta\,}-\cos\theta}{\sqrt{\,n^2-\sin^2\theta\,}+\cos\theta}\right|^2, \tag{1.55} \]

where \(\theta\) is the angle between the direction of the incident ray and the normal to the reflecting surface.

Analysis of (1.55) shows that for \(n^2>1\) (which corresponds to a negative scattering amplitude) the reflection coefficient is always less than unity. Total “internal”*) reflection in this case (for \(f<0\)) does not occur. In the opposite case, for \(f>0\), \(n^2<1\), and at certain angles \(\theta\), \(R=1\). It is easy to find the limiting angle from which \(R\) becomes equal to unity, i.e. the angle of total “internal” reflection:

\[ \sin\theta_0=n. \tag{1.56} \]

Since \(n\simeq 1\), it follows that \(\theta_0\simeq \dfrac{\pi}{2}\). Using instead of \(\theta_0\) the angle \(\varphi_0=\dfrac{\pi}{2}-\theta_0\) (the angle of limiting grazing), and considering small angles of incidence, one can obtain the value of \(\varphi_0\):

\[ \varphi_0=\lambda\sqrt{N\frac{f}{\pi}}; \tag{1.57} \]

the angle \(\varphi_0\), for the same values of \(\lambda\), \(\rho\), and \(f\), has the value \(\sim 10'\).

The normal component of the neutron wave inside the crystal under total “internal” reflection decays exponentially. This deepens still further the analogy with optics. The distance \(d'\), over which the amplitude of the transmitted wave decreases by a factor of \(e\) in comparison with its value at the surface, is determined by the formula

\[ d'=\frac{i}{k\sqrt{\,n^2-\sin^2\theta\,}}. \tag{1.58} \]

For the previous values of \(k\), \(n\), and \(\theta\), \(d'\simeq 100\,\text{\AA}\).

*) Quotation marks are used because \(\theta_0\) should more properly be called the angle of total external reflection.

The study of total “internal” reflection is considerably complicated by the small values of the angles. In addition, appreciable distortions are introduced into this effect by slight contaminations of the surface of the reflecting mirror. However, from experiments on total internal reflection of neutrons one can determine the sign of the scattering amplitude, which is very important for neutron-structural analysis.

5. Comparison of the Scattering of X-rays and Neutrons by Matter

Let us summarize the conclusions concerning the difference in scattering for neutrons and X-rays. This difference is what determines the advantages of neutronography over other methods of structural analysis.

There are three fundamental differences between the scattering amplitudes of neutrons \(f_n\) and X-rays \(f_p\) (leaving aside the dependence on \(\theta\)): 1) \(f_n\) is negative for some elements (H, Li, Ti, Mn), whereas \(f_p\) is always positive; 2) \(f_p\) is proportional to the number of electrons in the atom and, consequently, to the atomic number of the element in Mendeleev’s periodic system (\(f_p \sim Z\)); \(f_n\) is not explicitly related to the atomic number and, in going from one element to another, changes nonmonotonically (for example, the scattering amplitude of hydrogen is of the order of \(f_n\) for lead); 3) \(f_p\) is the same for all isotopes of an element, whereas \(f_n\) differs for different isotopes, and at times may even differ in sign.

Almost all the advantages of neutronography are based on these (mainly on the last two) differences, although they also introduce a number of inessential inconveniences. A comparison of the neutron and X-ray methods of structural analysis will be made at the end of the article, after the presentation of work on neutronography.

II. APPARATUS FOR NEUTRON-STRUCTURAL ANALYSIS

Before the creation of uranium reactors there were no powerful sources of neutrons. When it was necessary to use thermal neutrons, a Ra-Be source was employed, and the fast particles emitted by it were slowed down, for example, by paraffin. The intensity of a Ra-Be source is small. The necessity of monochromatizing the neutrons after their slowing down, leading to a further decrease in intensity, made it almost impossible to use this source for the purposes of neutronography. For the same reason, neutrons obtained in nuclear reactions in a cyclotron also could not be used. Therefore, naturally, neutronography could arise only after the creation of nuclear reactors.

Slowed neutrons, on emerging from such a boiler, have an almost Maxwellian velocity distribution, with its maximum at room temperature of the moderator so situated that, after monochromatization, one can obtain neutron beams of sufficient intensity with wavelengths from 5 to 0.02 Å, i.e., precisely those required for structural investigations.

Neutrons are devoid of electric charge and therefore do not possess ionizing power. Neutrons are detected by the particles that arise in the nuclear reaction of a neutron with some specially “substituted” nucleus, most often \(B^{10}\); the \(\alpha\)-particles produced in this reaction are recorded.

If individual neutrons are recorded, Geiger–Müller counters filled with \(\mathrm{BF}_3\) are used. In photographic recording either 1) boron salts are introduced into the emulsion and the \(\alpha\)-particles cause blackening of the film, or 2) a thin sheet of indium is placed in front of an ordinary X-ray film, and the electrons produced in the reaction \((n,\beta)\) are recorded by the photographic film. By the latter method several neutronograms\(^{10,11}\) have been obtained from single crystals, taken in white radiation and quite analogous to similar X-ray photographs.

For monochromatizing neutrons, generally speaking, three methods may be used: a mechanical selector (see, for example,\(^{16}\)), the crystal-diffraction method, and the method of the pulsed cyclotron\(^{17}\). Let us note at once that the simplest of these is the crystal-diffraction method, while the most complicated is the pulsed-cyclotron method, which requires complex radio-engineering circuits for its implementation. This is a substantial advantage of crystal spectrographs.

The resolving power of crystal spectrographs is considerably higher than that of mechanical selectors and no worse than in the pulsed-cyclotron method. However, the possibilities of the method in this respect are far from exhausted.

Recently a somewhat unusual mechanical selector has been constructed\(^{18,19}\), whose principle of operation resembles rather a pulsed cyclotron than a velocity selector. The resolving power of this instrument is somewhat higher than that of the earlier ones, but remains lower than that of crystal spectrometers.

In neutronography the crystal-diffraction method of monochromatization is almost always used.

Neutron spectrographs are in the main similar to spectrographs used with X-rays\(^{12,13,14,15}\). A typical spectrograph with a flat crystal is shown in Fig. 5. Neutrons pass through a channel in the concrete lining of the boiler and fall on the crystal. Cadmium diaphragms are usually mounted in this channel. Because in neutron spectrographs a high-

some degree of collimation, the system of slits has a great length (on the order of several meters). The configuration of cadmium slits and plates depends on the method of focusing.

Around an axis passing through the middle of the crystal there rotates a frame on which a neutron counter is mounted, counting the intensity of the beam diffracted at a definite angle. The counter is positioned along the diffracted ray so that the recorded neutrons traverse a considerable path in the counter, and its efficiency is thereby brought almost to

Fig. 5. Neutron spectrograph with a flat single crystal.¹²

100%. However, since the capture cross section of \( \mathrm{B}^{10} \) in the region of low energies depends on the neutron velocity as \( \frac{1}{v} \), the efficiency of the counter increases as the neutron energy decreases. Therefore the results obtained must then be corrected for the efficiency of the counter.

Because the lining of the reactor is not completely impermeable to neutrons, the counter must be shielded from side radiation. The counter is therefore surrounded by a layer of paraffin for slowing down fast neutrons and by a layer of \( \mathrm{B}_4\mathrm{C} \), which absorbs these neutrons. This makes the counter very bulky and requires spectrographs of similar bulk.

According to exactly the same principle as in X-ray spectral analysis, a spectrograph with a bent crystal, operating in transmission (after Cauchois), was constructed¹⁵ (Fig. 6). In this case the system of slits was mounted in such a way that the neutron beam was made convergent (in the absence of the crystal)

at the point \(F\). The radius of curvature of the crystal and the distance from the crystal to the counter were chosen in such a way that the neutron rays reflected from the planes (200) of the NaCl crystal, arranged perpendicular to the outer boundary, were focused on the slit in front of the counter. In connection with this, the slit size could be reduced to 1 mm. Therefore, when working with such a spectrograph, one can use small specimens under study.

Fig. 6. Diagram of a focusing neutron spectrograph with a bent crystal.

A certain shortcoming of the crystal-diffraction method is the limitation of the range of energies investigated, although this range is considerably larger than with a mechanical selector. The lower limitation is imposed by condition (1.50), from which it follows that

\[ \lambda_{\max}=\frac{1}{\sqrt{2mE_{\min}}}=2d \]

(\(d\) is the interplanar spacing of the crystal), while the upper limitation is due, first, to a decrease in the resolving power of the instrument and, second, to experimental difficulties in working with very small angles \(\theta\) (the collimating device). In addition, at low energies considerable distortions are introduced by higher orders of reflection. The resolving power of the instrument varies with the interplanar spacing \(d\) as

\[ \frac{1}{d}, \]

and the useful energy interval as

\[ \frac{1}{d^2}. \]

Therefore, for neutron spectroscopy it is advantageous to use crystals with small \(d\). For example, for the (200) plane of a LiF crystal the useful energy interval lies in the range from 0.04 to 65 eV.

Thus, when working with a crystal spectrometer, it is very important to choose the crystal in such a way as to broaden, as far as possible, the range of energies investigated without worsening the resolving power of the instrument and while reducing to a minimum the distortions introduced by higher orders of reflection.

Using a spectrograph with a flat crystal, a spectrum of neutrons from a boiler was obtained (Fig. 7). The Maxwellian curve, calculated for the temperature \(T = 548^\circ\ \mathrm{K}\), is somewhat distorted as a result of the occurrence of higher-order reflections (with \(n = 2\) and 3).

Fig. 7. Spectrum of neutrons emitted by a reactor.

Fig. 8. Apparatus for monochromatizing neutrons.

In the case where structural analysis requires a monochromatic beam of neutrons, the radiation from the pile is first directed onto a monochromator\({}^{20}\)—a fixed crystal which, according to equation (1.50), reflects neutrons of one definite wavelength. The monochromating crystal is ground in such a way that its external plane makes an angle of \(6^\circ\) with the atomic plane. This method, previously developed\({}^{21}\) for X-ray spectral analysis, makes it possible to reduce the width of the reflected beam; this apparatus is shown in Fig. 8.

To protect against neutrons scattered by the monochromator, against \(\gamma\)-radiation, and against the direct beam, the monochromating crystal is placed in a paraffin jacket and in a jacket made of \(\mathrm{B_4C}\), lead, and cadmium. In these, only openings are made for the incident and reflected beams. A spectrograph similar to that described above is then installed.

In some of the works described below, installations specially developed for solving the given problem were used. They will be described in the corresponding sections.

III. NEUTRONOGRAPHY OF POLYCRYSTALLINE SPECIMENS

1. Diffraction of neutrons in polycrystalline specimens

The neutronogram of a polycrystalline specimen is the dependence of the intensity of neutron scattering by the specimen on the scattering angle. In general form this dependence is represented by a diffuse background on which the maxima of coherent scattering are situated. A monochromatic beam of neutrons is used as the radiation. (In the works described below, a wavelength \(\lambda = 1.057 \,\text{Å}\) was used.) The position of the diffraction maxima, determined by the Wulff–Bragg equation (1.50), is governed by the structure of the specimen. The intensity of the maxima \(\dfrac{I_{hkl}}{I_0}\), determined in the case of transmission of neutrons through the specimen by equation (1.53′), depends both on the structure of the specimen and on the magnitude of the scattering amplitude of the constituent nuclei.

In the case of a monoatomic specimen the structural amplitude \(F_{hkl}\) is related to the amplitude of coherent scattering \(f_0\) in a relatively simple way:

\[ F_{hkl}=R_{hkl}f_0, \tag{3.1} \]

where

\[ R_{hkl}=\sum_i \exp 2\pi i\,(hx_i+ky_i+lz_i) \tag{3.2} \]

is the geometrical part of the structural amplitude. (The summation is extended over all atoms of the unit cell.) In

in a known structure, calculating \(R_{hkl}\) presents no difficulty.

The interpretation of a neutronogram, allowing one to determine the amplitude and cross section of coherent scattering, is carried out as follows: 1) from the experimentally known \(\dfrac{I_{hkl}}{I_0}\), the structural factor \(|F|^2\) is determined by formula \((1.53')\); 2) \(f_0\) is determined by formula (3.1); and 3) \(\sigma_{\mathrm{coh}}\) is found by formula (1.25). If it is necessary to find the phase shift upon scattering, this can be done from (1.26).

From comparison of the obtained \(\sigma_{\mathrm{coh}}\) with the cross section for total scattering, one can draw certain conclusions about the magnitudes of \(\sigma_{DS}\) and \(\sigma_{DI}\). However, in doing so one must take account of the increase in background as a result of multiple scattering. This phenomenon plays a more significant role in neutronography than in X-ray structural analysis, because of the low absorption of neutrons by matter. It can be avoided by choosing very thin specimens, but this leads to a decrease in the scattering intensity. Therefore, for multiple scattering a correction is usually calculated that agrees well with the magnitude of the diffuse background at small scattering angles (when \(\sigma_{DS}\) and \(\sigma_{DI}=0\)).

By the method described, neutronograms of diamond, aluminum, and other elements and compounds were obtained and interpreted. In Fig. 9,a a neutronogram of diamond is presented \(^{20}\). The specimen was prepared from powder passed through an 800-mesh sieve. The scattering intensity plotted along the ordinate axis is expressed as the difference between the readings of the counter with a cadmium shutter and without it.

From Fig. 9,a it is seen that even at small diffraction angles there is some background of diffuse scattering. Let us analyze the reasons for its appearance.

Natural carbon consists of a mixture of two isotopes \(C^{12}\) and \(C^{13}\), with the isotope \(C^{13}\) being present in a small amount (1.1%). Taking into account that the nucleus \(C^{12}\) has zero spin, and neglecting the admixture of \(C^{13}\), one may conclude that in this case practically \(\sigma_{DI}=0\) and \(\sigma_{DS}=0\). Diffuse scattering depending on thermal vibrations is usually very small at small angles and appears only at large angles. Therefore the appearance of a diffuse background in diamond was attributed to multiple scattering. Indeed, the corresponding correction reduced the diffuse scattering to zero.

It is practically very difficult to calculate the coefficient of proportionality \(K^2\) between \(I_{hkl}\) and \(\sigma_{\mathrm{coh}}\) in \((1.53')\). Therefore the neutronogram of diamond, expressed most clearly because of the practically absent diffuse background, was taken as a standard from which \(K^2\) was determined. This was done as follows: from measurements of neutron transmission through the same specimen, it was

Figure 9. Neutronograms of diamond (а), graphite (б), aluminum (в), and lead (г).

Panel г — Lead.
Axis labels: \(I\); \(\theta\).
Reflections indicated: \((111)\), \((200)\), \((220)\), \((311)\), \((222)\).

Panel в — Aluminum.
Axis labels: count rate in 1 min; counter angle.
Reflections indicated: \((111)\), \((200)\), \((220)\), \((311)\), \((222)\), \((331)\), \((420)\), \((422)\).

Panel б — Graphite.
Axis labels: count rate per min.
Reflections indicated: \((002)\), \((100)\), \((010)\), \((110)\), \((101)\), \((010)\), \((101)\), \((004)\).

Panel а — Diamond.
Axis labels: count rate per min; counter angle.
Reflections indicated: \((111)\), \((220)\), \((311)\), \((400)\), \((331)\), \((422)\).

the cross section of total scattering was found, amounting to

\[ \sigma_{\text{cryst}} = 5.2 \cdot 10^{-24}\ \text{cm}^{2}; \]

since \(\sigma_D = 0\), \(\sigma_{\text{coh}}\) was set equal to this value (by (1.41)), and the corresponding value of \(f_0\) was substituted into (3.1) and (1.53′); determining from the experiment \(I_{hkl}\) and knowing the position and indices of the diffraction maximum \((hkl)\), \(K^2\) was determined.

This made it possible subsequently to calculate the neutronograms of the remaining substances.

From the neutronogram of graphite (Fig. 9, б) (only one maximum, (002), was measured, since the others could not be measured with sufficient accuracy) the value \(\sigma_{\text{coh}} = 5.5 \cdot 10^{-24}\ \text{cm}^{2}\) was obtained.

This is somewhat greater than the value of the cross section of coherent scattering of carbon obtained with the diamond specimen, but the difference lies almost within the limits of measurement error. One should place somewhat more confidence in the first value.

The neutronogram of aluminum\(^{20}\) (Fig. 9, в) was obtained with a specimen consisting of a thick (\(1.3\ \text{cm}\)) plate of rolled aluminum. Before the experiment the plate was heated several times almost to the melting point and quenched. This was done to remove texture and to refine the grains. The use of a solid specimen contributed to an increase in the intensity of reflection due to the factor

\[ \frac{\rho'}{\rho} \]

in (1.53′) and accordingly reduced the measurement errors.

Diffuse scattering by aluminum is small and increases with increasing angle. Since aluminum is monoisotopic, this diffuse background must be attributed to the temperature effect, especially since aluminum has a low characteristic temperature.

Measurement of the neutronogram gave the value \(\sigma_{\text{coh}} = 1.46 \cdot 10^{-24}\ \text{cm}^{2}\). The good agreement of this quantity with the previously obtained values \(\sigma_{\text{cryst}} = 1.48 \cdot 10^{-24}\ \text{cm}^{2}\) indicates a weak dependence of the scattering on spin for Al, if such a dependence exists at all.

In the same way the neutronogram of lead\(^{22}\) (Fig. 9, г) was obtained and calculated. The scattering amplitude \(f_{\text{Pb}}\) proved equal to \(0.96 \cdot 10^{-12}\ \text{cm}\). The good agreement of the cross sections of coherent and total scattering (\(11.5\) and \(11.6 \cdot 10^{-24}\ \text{cm}^{2}\), respectively) indicates the absence in lead of a dependence of scattering on spin.

Let us note that the neutronograms of diamond, aluminum, and lead constitute the simplest cases.

In the case of monatomic substances, measurement of \(I_{hkl}\) for one diffraction maximum plus the value of the characteristic temperature (which gives \(w\)) is sufficient for determining the cross section of coherent scattering. For diatomic substances

(NaCl, etc.) two maxima must be measured. From these data one can obtain the coherent scattering cross section for both kinds of nuclei, but one cannot unambiguously assign the cross section to a definite kind. In X-ray analysis this can be done at once, since for X-rays the scattering amplitude is proportional to the ordinal number (the number of electrons of the atom). In the case of neutron analysis it is necessary to obtain one more neutronogram of a substance containing one of the components under investigation. Then the coincidence of the values of \(\sigma\) in two experiments makes it possible to relate the value of \(\sigma\) exactly to the nucleus.

Such experiments were carried out with substances containing sodium\({}^{20}\): Na, NaBr, NaF, and NaCl (Fig. 10). The results of determining \(\sigma_{\mathrm{coh}}\) and their comparison with the previously obtained \(\sigma_{\mathrm{cryst}}\) (taken from other sources) are given in Table I.

Table I

Crystal	\(\sigma_{\mathrm{coh}}\) in \(10^{-24}\ \mathrm{cm}^2\)	\(\sigma_{\mathrm{coh}}\) in \(10^{-24}\ \mathrm{cm}^2\)	\(f\) in \(10^{-12}\ \mathrm{cm}\)	\(f\) in \(10^{-12}\ \mathrm{cm}\)	\(\sigma_{\mathrm{cryst}}\) in \(10^{-24}\ \mathrm{cm}^2\)
Crystal	Na	\(X\)	Na	\(X\)	\(\sigma_{\mathrm{cryst}}\) in \(10^{-24}\ \mathrm{cm}^2\)
Na	1.50	—	0.345	—	—
NaBr	1.41	5.3	0.335	0.65	7.5
NaCl	1.64	12.8	0.358	1.01	15.0
NaF	1.52	5.2	0.348	0.65	4.5

For all these compounds the structure factor is equal to:

\[ F^2= \begin{cases} (f_{\mathrm{Na}}+f_x)^2 & \text{for even } h,\ k,\ l;\\ (f_{\mathrm{Na}}-f_x)^2 & \text{for odd } h,\ k,\ l. \end{cases} \tag{3.3} \]

Since the maxima (111) are smaller than the maxima (200), the scattering amplitudes of all the elements entering into the compounds are of the same sign.

Let us note, incidentally, that the neutronogram of MnO\({}^{74}\) (see Fig. 33, below) has a strong maximum (111) and a weak (200), although the structure of MnO is similar to the structure of NaCl and (3.3) is valid for it. This indicates that the scattering amplitudes of Mn and O are of different signs.

These experiments are less accurate than those described above. To determine \(\sigma_{\mathrm{coh}}\), the maxima (111) and the maximum (110) in the neutronogram of Na were measured. The four values of \(\sigma_{\mathrm{coh}}^{\mathrm{Na}}\) obtained for sodium are in good agreement and amount, on the average—

with \(\sigma_{\mathrm{coh}}^{\mathrm{Na}}=1.51\cdot 10^{-24}\ \mathrm{cm}^2\). This value is considerably smaller than the previously determined \(\sigma_{\mathrm{cryst}}=3.7\cdot 10^{-24}\ \mathrm{cm}^2\). Since sodium is monoisotopic, there must be a strong dependence of the scattering of sodium neutrons on spin. From the difference \((\sigma_{\mathrm{cryst}}-\sigma_{\mathrm{coh}})\), using (1.40) and (1.37), one can determine the scattering amplitudes for parallel and antiparallel spins.

It is seen from Table I that the coherent-scattering cross sections of bromine and chlorine are somewhat smaller than \(\sigma_{\mathrm{cryst}}\), since both elements have two isotopes; this must be attributed to the isotope effect.

Knowing the abundances of both isotopes and the difference \((\sigma_{\mathrm{cryst}}-\sigma_{\mathrm{coh}})\), from formulas (1.32) and (1.33) one can determine the scattering amplitudes for both isotopes. This was not done, since the results obtained do not yet possess a sufficient degree of accuracy.

In the case of fluorine, \(\sigma_{\mathrm{coh}}\) proved to be somewhat larger than \(\sigma_{\mathrm{cryst}}\). This can be explained only by experimental errors in one of the measurements. Evidently, fluorine has no spin dependence of scattering.

Fig. 10. Neutronograms of some compounds containing sodium.

To determine the scattering amplitudes of nickel and its isotopes,\(^{23}\) samples of oxides NiO, Ni\(^{58}\)O, Ni\(^{60}\)O, and Ni\(^{62}\)O were studied in the same way (Fig. 11). Interest in such compounds arose, in particular, because the difference between the total and coherent scattering cross sections of nickel reaches a considerable value: \(3.9\cdot 10^{-24}\ \mathrm{cm}^2\). Since the nuclei of nickel have no spin, this discrepancy can be attributed only to a considerable difference between the scattering amplitudes of the individual nickel isotopes.

The structure of nickel oxide is similar to the structure of NaCl. Therefore formula (3.3) is also valid in this case. From consideration of Fig. 11 one can immediately say that Ni, Ni\(^{58}\), and Ni\(^{60}\) have a positive, while Ni\(^{62}\) has a negative scattering amplitude.

Measurement of the intensities of the diffraction maxima made it possible to determine the amplitudes of all the nickel isotopes studied. The results—

Fig. 11. Neutronograms of oxides of nickel isotopes.

—of these measurements are given in Table II. The last column gives the scattering amplitudes of each isotope multiplied by their corresponding abundances. According to (1.32), the algebraic sum of these quantities should give the scattering amplitude of natural nickel. The good agreement of the results indicates the high accuracy of the experiment.

Table II

Isotope	Amplitude of coherent scattering, in \(10^{-12}\ \mathrm{cm}\)	\(\sigma_{\mathrm{coh}}\), in \(10^{-24}\ \mathrm{cm}^{2}\)	\(g_i f_i\)
\(\mathrm{Ni}^{58}\)	1.48	27.6	1.00
\(\mathrm{Ni}^{60}\)	0.28	0.97	0.073
\(\mathrm{Ni}^{63}\)	−0.85	9.1	−0.032
\(\mathrm{Ni}\)	1.03	13.3	\(\sum g_i f_i = 1.04\)

In the same way, the scattering amplitudes of silver \(^{24}\), iron \(^{25}\), and each of their isotopes were determined, and it was established \(^{26}\) that titanium has a negative scattering amplitude.

2. Study of the Structure of Sodium Hydride and Ice

The results of measuring the cross section of coherent scattering by sodium were needed subsequently in the study of the structure of sodium hydride NaH and in determining the scattering amplitude of hydrogen and deuterium.²⁷ The fact that the scattering power of an atom for X-rays is proportional to the number of electrons makes scattering by hydrogen very small, elusive in comparison with the other elements. For neutrons, the scattering amplitude of hydrogen is of the same order as for most other elements. Therefore the problem, extremely difficult for X-ray analysis, of determining the position of hydrogen atoms in its crystalline compounds becomes an ordinary problem of neutron-structure analysis.

The simplest problem in this respect was the determination of the position of hydrogen in the hydrides of the alkali elements (LiH, NaH, KH, and CsH). It had been established by X-ray analysis that the metal atoms form a cubic face-centered lattice. An answer concerning the position of the hydrogen atoms was obtained only in the first case,²⁸ since the scattering power of lithium is only three times greater than the scattering of hydrogen. It turned out that LiH forms a lattice analogous to the NaCl structure. The suggestion was made that all the other hydrides indicated crystallize in the same system.

To verify this assumption, for certain simple reasons, NaH was chosen. To study the scattering of deuterium, NaD was also investigated; however, owing to its considerable contamination with metallic Na and NaOH, the results in this case proved less accurate.

In Fig. 12 are presented neutronograms of NaD and NaH. The fact that the neutronograms are not identical indicates that H and D take part in diffraction and that the scattering amplitudes of H and D have different signs, since those maxima that are strong for NaH are weak or disappear altogether for NaD.

The X-ray data do not permit a final choice between the two structures possible for NaH: a structure of the NaCl type and a structure of the ZnS type. Other structures are excluded from consideration on the basis of the dimensions of the unit cell.

Exhaustive results on the structure of NaH and on the signs of the scattering amplitude can be extracted from a comparison of the theoretically calculated and experimentally found intensities of the diffraction maxima (Table III). Only by assigning \(f_{\mathrm{Na}}\) and \(f_{\mathrm{H}}\) opposite signs and assuming for NaH the NaCl structure did the authors obtain good agreement between the experimental and theoretical data on the relative intensities of the maxima. In

under all other assumptions the experiment is at variance with the theory. Hence it is clear that NaH has the NaCl structure and that the scattering amplitudes for Na and H have opposite signs. Consequently, Na and D scatter with the same sign.

Fig. 12. Neutronograms of NaH and NaD.

Table III

Diffraction maxima	Intensities: calculation for structures ZnS	Intensities: calculation for structures NaCl	experiment
111	$f_{\mathrm{Na}}^{2}+f_{\mathrm{H}}^{2}$	$(f_{\mathrm{Na}}-f_{\mathrm{H}})^{2}$	37
200	$(f_{\mathrm{Na}}-f_{\mathrm{H}})^{2}$	$(f_{\mathrm{Na}}+f_{\mathrm{H}})^{2}$	$<2$
220	$(f_{\mathrm{Na}}+f_{\mathrm{H}})^{3}$	$(f_{\mathrm{Na}}+f_{\mathrm{H}})^{2}$	0
311	$f_{\mathrm{Na}}^{2}+f_{\mathrm{H}}^{2}$	$(f_{\mathrm{Na}}-f_{\mathrm{H}})^{2}$	25
222	$(f_{\mathrm{Na}}-f_{\mathrm{H}})^{3}$	$(f_{\mathrm{Na}}+f_{\mathrm{H}})^{3}$	0
400	$(f_{\mathrm{Na}}+f_{\mathrm{H}})^{3}$	$(f_{\mathrm{Na}}+f_{\mathrm{H}})^{2}$	$<5$
331	$f_{\mathrm{Na}}^{2}+f_{\mathrm{H}}^{2}$	$(f_{\mathrm{Na}}-f_{\mathrm{H}})^{2}$	15
420	$(f_{\mathrm{Na}}+f_{\mathrm{H}})^{3}$	$(f_{\mathrm{Na}}+f_{\mathrm{H}})^{2}$	0

Using the previously obtained value of $f_{\mathrm{Na}}$ and measuring the neutronogram of NaH with the aid of the diamond standard, one can deter-

divide the scattering amplitude of hydrogen. The calculations gave \(f_{\mathrm H} = -0.39\cdot 10^{-12}\,\text{cm}\). This gives a coherent-scattering cross section equal to \(\sigma_{\mathrm{coh}} = 2.0\cdot 10^{-24}\,\text{cm}^2\).

The same was done with the neutronogram of NaD. However, for the reason indicated above, the results obtained were very approximate. Therefore we shall at once give more reliable results, taken from another source: \(f_D = 0.64\cdot 10^{-12}\,\text{cm}\) and \(\sigma_{\mathrm{coh}}^D = 5.2\cdot 10^{-24}\,\text{cm}^2\).

Having calculated the difference \((\sigma_{\mathrm{cryst}}-\sigma_{\mathrm{coh}})\), one can determine \(\sigma_D\) for hydrogen and deuterium. The quantities of the total cross section for H and D have values \(80\) and \(7.4\cdot 10^{-24}\,\text{cm}^2\), respectively \(^{27,29}\). Since the isotope effect is excluded, the strong diffuse scattering of hydrogen is explained by the significant dependence of neutron scattering by protons on spin, the so-called spin incoherence.

The \(f_{\mathrm H}\) determined above corresponds to the average scattering amplitude (1.37), which for hydrogen (spin \(I=\frac12\)) is equal to

\[ f_{\mathrm H}=2\left\{\frac34 f_1+\frac14 f_0\right\} \tag{3.4} \]

(the factor 2 arose because the nuclear binding was taken into account). In addition, knowing \(\sigma_{DS}^{\mathrm H}=\sigma_{\mathrm{cryst}}^{\mathrm H}-\sigma_{\mathrm{coh}}^{\mathrm H}\) and using expression (1.40), which for hydrogen is rewritten as:

\[ \sigma_{DS}^{\mathrm H}=4\pi\cdot 4\cdot \frac{3}{16}\left(f_1-f_0\right)^2, \tag{3.5} \]

one can determine the scattering amplitudes for both spin orientations. Substituting the values of \(f_{\mathrm H}\) and \(\sigma_{DS}^{\mathrm H}\) obtained above, one can obtain two pairs of solutions, one of which agrees well with the results obtained earlier \(^{30}\):

\[ \begin{aligned} f_1&=+0.520\cdot 10^{-12}\,\text{cm},\\ f_0&=-2.35\cdot 10^{-12}\,\text{cm}. \end{aligned} \tag{3.6} \]

It is interesting to note that the scattering amplitudes for different spin states not only differ greatly from one another, but also have different signs. This explains such a strong dependence of scattering on spin for hydrogen.

Similar calculations can also be performed for deuterium. Starting from the value of the coherent-scattering cross section given above, from \(\sigma_{\mathrm{cryst}}^{D}=7.4\cdot 10^{-24}\,\text{cm}^2\) and the spin of the deuterium nucleus \((I=1)\), one can determine the scattering amplitudes for both spin orientations:

\[ \begin{aligned} f_{3/2}&=0.62\cdot 10^{-12}\,\text{cm}, & f_{3/2}&=0.23\cdot 10^{-12}\,\text{cm},\\ &\text{or}\\ f_{1/2}&=0.03\cdot 10^{-12}\,\text{cm}, & f_{1/2}&=0.82\cdot 10^{-12}\,\text{cm}. \end{aligned} \tag{3.7} \]

The choice between these quantities, as in the case of sodium, has not yet been made.

The very strong dependence of neutron scattering by hydrogen on the orientation of the spins \(({}^{\mathrm H}\sigma_{DS}=78\cdot 10^{-24}\ \text{cm}^2)\) makes understandable the large background of incoherent scattering in neutronograms of hydrogen-containing substances, which considerably complicates the calculation of intensities. In contrast to this, \({}^{\mathrm D}\sigma_{DS}=2.2\cdot 10^{-24}\ \text{cm}^2\) is much smaller, and the background in this case is considerably weaker. Hence there follows a conclusion of practical character: a crystallographic study can be carried out much more easily and accurately with deuterium preparations than with hydrogen ones, while their structures, of course, are identical.

Fig. 13. Various assumptions concerning the positions of the hydrogen atoms in the structure of ice. (○ — probability of finding a hydrogen atom at this point is equal to \(\frac{1}{2}\).)

This consideration was used in studying the structure of ice. X-rayographically, ice had been studied for a long time, but it proved impossible to decipher its structure completely by this method. It was only established that ice crystallizes in the hexagonal system, with the oxygen atoms situated at the vertices and at the centers of tetrahedra.

As to the position of the hydrogen atoms, it was only possible to make assumptions. In this connection several hypotheses were advanced concerning the position of the hydrogen atoms in the crystal lattice of ice. The first of them assumed that the hydrogen atoms are located midway between the oxygen atoms (Fig. 13, a) and that the unit cell consists of four water molecules. Somewhat later, for water molecules in the solid state, the same structure was proposed as they have in water vapor. (As is known, in a free water molecule the O—H distance is \(0.96\ \text{Å}\), and the angle between the “bond lines” of the oxygen atom with the hydrogen atoms is \(105^\circ\).) Proceeding from such a molecular structure, one can construct the structure of ice shown in Fig. 13, b, in which the oxygen atom will have only two nearest hydrogen atoms. In this case the unit cell will contain 12 molecules of \(\mathrm{H_2O}\). However, these conclusions contradicted certain thermodynamic principles\(^{34}\). Therefore the latter hypothesis was somewhat supplemented in order to overcome these contradictions: it was proposed\(^{34}\) that

hydrogen atoms can, with different probabilities, be located at various distances between the oxygen atoms; however, the probability amplitude is maximal at a distance of 0.96 Å from some oxygen atom (Fig. 13, c). And finally, the last hypothesis assumed that the hydrogen atoms rotate around the oxygen atoms (Fig. 13, d).

The choice among these four hypotheses could be made with the aid of neutronography. For this purpose a neutronogram of ice\(^ {35}\) was obtained (Fig. 14), and, according to the considerations expressed, heavy ice was used. The interpretation was carried out as follows. Theoretical neutronograms were constructed for all four assumed structures. For this,

Fig. 14. Neutronogram of D\(_2\)O.

in order to calculate the neutronograms it was necessary to know the scattering amplitudes of oxygen and deuterium. From earlier experiments it had been established that the scattering of neutrons by oxygen does not depend on spin and that the scattering amplitude is positive. Direct measurements of the total scattering cross section gave for O the value

\[ \sigma_{\mathrm{cryst}}^{\mathrm{O}} = 4.2 \cdot 10^{-24}\ \mathrm{cm}^{2}. \]

This value was equated to \(\sigma_{\mathrm{coh}}^{\mathrm{O}}\), and from this the scattering amplitude of oxygen was calculated.

The scattering amplitude of deuterium could have been determined from the experiment described with NaD, but, as has already been indicated, the accuracy in this case would have been very low. Therefore, as was already mentioned above, experiments were carried out to determine the scattering amplitude of deuterium. Such experiments with ThD\(_2\) gave for \(\sigma_{\mathrm{coh}}^{\mathrm{D}}\) the value \(5.2 \cdot 10^{-24}\ \mathrm{cm}^{2}\). Starting from these quantities, the neutronograms presented in Table IV were constructed.

With the aid of the diamond standard, a comparison was made of the four calculated neutronograms with the experimental one.

Table IV

\(\dfrac{\sin\theta}{\lambda}\)	(b) \(hkl\)	(b) \(I\)	(b) \(I\)	(a) \(hkl\)	(a) \(I\)	(a) \(I\)	(g) \(I\)	(v) \(I\)	\(I_{\mathrm{exp}}\)
0.1271	110	406	899	100	363	767	304	386	830
0.1358	002	240	899	002	202	767	166	219	830
0.1442	111	253	899	101	212	767	177	230	830
0.1860	112		128	102		79	97	104	118
0.2206	300		199	110		72	131	140	130
0.2401	113		212	103		14	101	70	95
0.2542	220	57	304	200	41	201	14	6	98
0.2590	302	47	304	112	15	201	47	3	98
0.2639	221	66	304	201	67	201	9	63	98
0.2717	004	32	304	004	78	201	0	41	98
0.2745	311	102
0.2836	213	29	104
0.2885	222	35	104	202	199	253	6	77	97
0.3000	114	40	104	104	54	253	0	29	97

Fig. 15. Graphical comparison of the experimental and theoretical neutronograms of \(\mathrm{D_2O}\) for four possible ice structures.

In Fig. 15 the results are presented graphically. A comparison of the theoretical intensities with the experimental ones convincingly supports the third hypothesis (Fig. 13, v).

Consequently, ice crystallizes in the hexagonal system; its unit cell consists of 12 \(\mathrm{H_2O}\) molecules, the oxygen atoms are located at the corners and at the centers of tetrahedra with a distance between neighbors of \(2.76\ \text{Å}\); the hydrogen atoms may be arranged along the straight line between oxygen atoms, with probability maxima at a distance of \(0.96\ \text{Å}\) from the oxygen atoms.

The good agreement of the theoretical and experimental neutronograms indicates that the value \(\sigma^{\mathrm{D}}_{\mathrm{coh}}\) was determined from the experiments with \(\mathrm{ThD_2}\) with sufficient accuracy.

3. Passage of Neutrons through Crystalline Powders

The cross section for coherent scattering of neutrons as they pass through crystalline powders has the following expression:

\[ \sigma_{\mathrm{coh}}=\frac{\lambda^{2}N}{8}\sum_{(hkl)} j_{hkl}\,d_{hkl} e^{-2w/R\sigma}, \tag{3.8} \]

where \(\sigma\) is the cross section for scattering by bound nuclei [the meanings of the remaining letters were given earlier in (1.53′) and (3.2)]. The Wulff equation (1.50) restricts the reflected wavelengths, since, taking instead of \(\sin\theta\) its greatest value, 1, we obtain the reflection condition:

\[ \lambda \leq 2d_{\max}. \tag{3.9} \]

Consequently, if \(\lambda\) is greater than twice the largest interplanar spacing, this condition is violated and the scattering cross section becomes equal to zero. However, this may not be the case because of the spin and isotopic effect. Therefore, if neutrons with \(\lambda>2d_{\max}\) are passed through the mono-isotopic crystalline powders under study, all the scattering will be diffuse, and it will be possible to obtain information concerning the spin dependence of the scattering.

Similar experiments were carried out with samples of graphite, Be, and BeO \(^{36,37}\). In the spectrograph described above, the passage of thermal neutrons through finely ground powders of these materials was studied. The experimental results for BeO are presented in Fig. 16. For a compound containing two kinds of atoms, expression (3.8) must be changed only slightly. Calculation showed that the resulting curve should depend on the signs of the scattering amplitudes of both nuclei. Comparison of the experimental and theoretical data for BeO showed that the scattering amplitudes of Be and O have the same sign.

For samples containing one isotope and having no spin dependence of the scattering, all scattering will be completely located in the maxima of coherent scattering, and the scattering cross section for \(\lambda>2d\) will become zero. Conversely, if there is a mono-isotopic sample and for \(\lambda>2d\) neutron scattering is observed, this scattering can with certainty be attributed to the spin effect. From the measurement of \(\sigma_D\) one may judge the magnitude of the effect.

One may proceed in a somewhat different way. When Maxwell-distributed neutrons pass through a sufficiently long column of crystalline substance, neutrons with \(\lambda<2d\) will leave the beam. Therefore, from the Maxwellian curve there will remain only

Fig. 16. Dependence of the neutron scattering cross section on energy when passing through BeO.

Calculated curves for BeO in the case of:

— identical phases

--- opposite phases

Vertical axis: Scattering cross section, in units of \(10^{-24}\ \text{cm}^2\)

Horizontal axis: \(E\) (eV)

Fig. 17. Spectrum of filtered BeO neutrons.

Vertical axis: Intensity

Horizontal axis: \(\lambda\) (Å)

...the part corresponding practically to a monochromatic beam of cold neutrons with a temperature of about \(20^\circ K\) (Fig. 17) and with \(\lambda > 2d\) (when filtered through graphite the effective neutron wavelength is \(\simeq 7\ \text{\AA}\), through BeO—\(5\ \text{\AA}\)). If such neutrons are passed through substances with the corresponding lattice parameters \((2d_{\max} < \lambda)\), it is theoretically possible to obtain the values \(\sigma_{DS}\) in pure form. In practice, however, the matter is complicated by the isotopic composition of the substances under study and, chiefly, by imperfection of the crystal lattice.

The experiments carried out with Be, Al, and \(\mathrm{Bi}^{38}\) are, as a result, very unconvincing and give the quantity \(\left| f_{I+\frac12} - f_{I-\frac12} \right|\) [see (1.40)] within broad limits, which include also 0, i.e., the absence of a spin effect. Indeed, other works indicate the absence of spin incoherence in these elements.

Let us further note that the neutron beam obtained by filtration through BeO was used to study the interaction of neutrons with gas molecules (see below).

From experiments on the passage of neutrons through crystalline powders one can determine the scattering amplitudes of elements. According to (3.8), amplitudes were determined for Ca, Ti, Tl, and \(\mathrm{Zn}^{39}\); however, the results proved to be good only for Zn \((f_{\mathrm{Zn}} = 0.58 \cdot 10^{-12}\ \text{cm})\). The reason for this probably lies partly in the imperfection of the method and partly in a certain difficulty of the experiment. It is characteristic that the greatest discrepancy was obtained for elements with comparatively large neutron-absorption cross sections. In this case the picture changes somewhat (Fig. 18; compare with Fig. 16) and is coarsened by the absorption correction.

Measurement of neutron transmission has found application in metallophysics. Studies were carried out on the transmission of neutrons through a number of alloys and mechanical mixtures of two compounds—titanium carbide and tungsten carbide \({}^{40}\). It was established by x-ray diffraction that, when WC dissolves in TiC, a solid substitutional solution is formed.

In measuring the total neutron cross section it turned out that the cross section of a mechanical mixture of WC and TiC is equal to the sum of the cross sections of the elements entering into the mixture. With increasing WC concentration the total cross section increased in complete agreement with the increase in total absorption (Fig. 19, curve \(b\)). It is clear that each component scatters independently of the other. A different picture was obtained when measuring neutron transmission through solid solutions with different WC concentrations. After first revealing a certain decrease, the total cross section rapidly increased with increasing WC concentration

Fig. 18. Same as in Fig. 16 for TiC (the dashed curve is without subtraction of absorption).

(Fig. 19, curve a). The result indicates that the transmission of neutrons is sensitive to the structure of the solid solution and to the concentration of WC in the TiC solvent. This allows one to hope that similar experiments will make it possible to determine the solubility limits of solid solutions. In the present case one may say that the solubility limit (if it exists at all) lies in the region of high concentrations of tungsten carbide, greater than 50 mole percent WC.

Fig. 19. Dependence of the total cross section for neutrons of a mechanical mixture (b) and a solid solution (a) TiC + WC on the molar concentration of WC.

All that has been said above is valid for polycrystalline specimens and powders only in the case where their microcrystallites are oriented randomly in space. If this condition is not met, i.e., if there is some preferential orientation in the specimens, the results obtained will be different.

In practice, such an anisotropic distribution of orientations of microcrystallites—texture—is observed rather often. This occurs, for example, in rolling, drawing, and pressing of materials. Texture significantly changes the X-ray patterns of deformed specimens, and X-rays have long been used to study this phenomenon.

Experiments involving the transmission of neutrons through rolled aluminum^41 and pressed graphite^43 showed that texture can also be studied by means of neutronography. In this direction, certain advantages of neutron-structural analysis over X-ray analysis are also becoming apparent. The point is that X-ray

radiographically; first, texture is usually studied in a very small volume and, second, investigation of materials with a large coefficient of absorption of X-rays is difficult. Neutronography is free of these shortcomings. However, for the time being these theoretical advantages remain theoretical, since experiments have been carried out only with aluminum and graphite, i.e., with substances that have already been sufficiently studied radiographically.

Figure 20 presents curves for rolled aluminum, analogous to Figs. 16 and 18. Curve \(a\) is the result of measurements of the cross section for a neutron beam parallel to the rolling direction; curve \(б\) is for a beam perpendicular to the rolling direction; curve \(в\) was obtained when neutrons were passed through cast aluminum (without texture). The dashed curve represents the absorption correction. The large difference in the curves for these three cases indicates a strong texture in the rolled aluminum.

Fig. 20. Dependence of neutron transmission through rolled aluminum on energy.

Fig. 21. The same as in Fig. 20, for pressed graphite.

The neutronograms presented confirm the previously obtained results, according to which rolled aluminum possesses

of the so-called limited texture, i.e., one such that, in addition to the predominant direction (111), oriented along the rolling direction, there is also a fixed plane (100), coinciding with the rolling plane.

The poor resolving power of the instruments on which the experiment was performed (a combination of a mechanical selector with a rotating shutter and a crystal spectrograph) somewhat distorts the result.

A simpler and more reliable result was obtained with pressed graphite. In this case only one diffraction peak was investigated; this is possible because, in the crystal lattice of graphite, one of the parameters (along the \(c\)-axis) is considerably greater than the other two.

Figure 21 presents the curves obtained when measuring the transmission of neutrons parallel and perpendicular to the pressing direction.

The indicated advantage of graphite made it possible, in this case, to approach the study of texture quantitatively.

If we denote by \(\varphi\) the angle between the pressing direction and the axis of symmetry of the crystal (for graphite this is the \(c\)-axis), then the probability of finding this axis in the solid angle \(d(\cos\varphi)\,d\beta\) is equal to

\[ P(\cos\varphi)\,d(\cos\varphi)\,d\beta, \tag{3.10} \]

where \(\beta\) is the azimuthal angle.

The function \(P(\cos\varphi)\) is represented in the following form:

\[ P(\cos\varphi)=\frac{1}{4\pi} +x\left(\cos^2\varphi-\frac{1}{3}\right) +y\left(\cos^4\varphi-\frac{1}{5}\right). \tag{3.11} \]

The quantities \(x\) and \(y\) indicate the degree of correlation between microcrystals, while the factors \(\frac{1}{3}\) and \(\frac{1}{5}\) arise from the normalization condition (3.10). For \(x=y=0\), \(P=\frac{1}{4\pi}\); in this case texture is absent.

The curves presented in Fig. 21 made it possible to find the values of \(x\) and \(y\). The values \(x=-0.07\) and \(y=+0.04\) gave the best agreement with the experimental data, and consequently the distribution function of orientations is obtained as

\[ P(\cos\varphi)=\frac{1}{4\pi}\left(1.2-0.9\cos^2\varphi+0.5\cos^4\varphi\right). \tag{3.12} \]

It follows from this that \(P(1)<P(0)\) and that the \(c\)-axis of graphite has a tendency, during pressing, to be oriented perpendicular to the direction in which the pressure is applied, while the planes

basis of the hexagonal system—to be oriented along this direction.

The same conclusion can be drawn simply from consideration of Fig. 21.

4. Neutronographic Study of Ordering Phenomena in Alloys

In an ordinary solid solution, different kinds of atoms are distributed statistically at random in the sites of the crystal lattice. However, under certain conditions a situation may arise in which partial, and sometimes complete, ordering occurs in the arrangement of the atoms: one kind of atom occupies some lattice sites, and another kind occupies other sites. Such a system may be regarded as two crystal lattices inserted into one another.

The phenomenon of ordering can be studied by various physical methods, but most often this is done by X-ray diffraction. The principle of the method is as follows.

If in a disordered alloy all crystallographic planes are identical in their reflecting powers (for X-rays and neutrons), then in an ordered alloy this may no longer be the case, because some planes consist only of atoms of one kind or the other.

On the other hand, it is known that some lines of a powder X-ray diffraction pattern, taken from specimens having a crystal lattice with a basis, disappear as a result of additional interference on planes passing through the basis atoms.

In ordered alloys these lines may reappear, since the reflecting properties of planes consisting of atoms of different kinds are different (see^43). The lines of the X-ray diffraction pattern that arise as a result of ordering are called superstructure lines.

The intensity of superstructure lines depends on the difference between the scattering amplitudes of the atoms entering into the alloy. The structure amplitude of the principal lines is

\[ F \approx |f_A + f_B|, \tag{3.13} \]

and of the superstructure lines

\[ F \approx |f_A - f_B|, \tag{3.14} \]

where \(f_A\) and \(f_B\) are the scattering amplitudes of both kinds of atoms.

In the case of incomplete ordering, the superstructure lines have lower intensity; in this case

\[ F \approx |r f_A - w f_B|. \tag{3.15} \]

where \(r\) and \(w\) are quantities characterizing the degree of ordering of the alloy (\(r+w=1,\ 0\leq r,\ w\leq 1\)). From the appearance and from the intensity of the superstructure lines one judges the state of the alloys. This applies both to x-rays and to neutrons.

The sensitivity of the methods (x-ray and neutron) depends on the difference in the scattering amplitudes of the atoms entering into the alloy. It is quite obvious that an x-ray study of ordering in alloys consisting of elements located close to one another (or adjacent) in the periodic table (with close \(Z\)) is extremely difficult, and sometimes even impossible, because of the extremely weak intensity of the superstructure lines in comparison with the fundamental ones. Since the neutron scattering amplitude does not depend on \(Z\) and varies from element to element nonmonotonically (see Table VII on p. 541), such a complication does not arise for neutrons.

Fig. 22. Neutronograms of ordered and disordered FeCo alloys.

In Fig. 22 are shown neutronograms of disordered and ordered specimens of the alloy FeCo\(^{44}\). The ratio of the intensities of the superstructure (100) line and the fundamental (110) line almost reaches the theoretical value \((1:6)\), which indicates a high degree of ordering in the alloy.

Since for neutrons the scattering amplitudes may have opposite signs, the expressions for the line intensities may also interchange places \([(3.14)\) for fundamental and \((3.13)\) for superstructure lines]. An extremely interesting case is the alloy Ni\(_3^{60}\)Mn\(^{55}\), since for Ni\(^{60}\) and Mn\(^{55}\) the scattering amplitudes are close in magnitude and opposite in sign. On the x-ray photograph of such an ordered alloy the fundamental lines would be absent, while the superstructure lines would be of ordinary intensity.

In earlier works (45, see also 47), the study of the ordering of NiFe alloys was carried out in a somewhat different way: the change in transmission of polychromatic neutrons, obtained from a Ra-Be source, through alloy powder was measured as a function of the degree of its ordering (i.e., as a function of the preliminary heat treatment). In this way a regular increase in the transmissivity of the specimens was established with increasing ordering. In contrast to the data obtained earlier, ordering in Fe-Ni alloys was observed at nickel contents from 35 to 90 atomic percent. The maximum ordering, as was to be expected, occurred at the nickel concentration corresponding to the alloy $\mathrm{Ni}_3\mathrm{Fe}$.

However, the results obtained depended to a considerable extent on the grain sizes of the specimens, and the reliability of the results obtained in this way is very low. Work carried out somewhat later confirmed the strong dependence of the effect on the grain size46.

The theory of this phenomenon is very complex because of the complexity of the processes that occur in it, and therefore a theoretical correction cannot be calculated.

In contrast to this, the method described above is free from such a dependence*) (in the absence of texture and with sufficient fineness of the grains of the specimen). Therefore the results obtained are quite reliable.

It may be noted that neutronography is subject to the same drawback as X-ray structural analysis. It may turn out that the scattering amplitudes of the atoms entering the alloy (for example, the alloy $\mathrm{Cu}_3\mathrm{Au}$) are identical. Then the superstructure lines will be absent. However, for neutrons this phenomenon has a random rather than a systematic (as for X-rays) character. Therefore, in this as in many other problems of structural analysis, it is necessary to combine both methods.

IV. DIFFRACTION OF NEUTRONS IN GASES AND LIQUIDS

1. Scattering of Neutrons by Gases

It had long been noted that, in the scattering of neutrons by molecular gases, the cross section obtained was larger than the sum of the cross sections of the atoms entering into the molecule48. This discrepancy, observed with neutrons whose wavelength is of the same order as the distance between the nuclei in the molecule, decreased substantially with increasing neutron energy. Such

*) Such a dependence does exist, it is true, but it is of an entirely different character, and its influence is easily eliminated.

the phenomenon could be explained, basically, by two causes: the thermal motion of molecules (the Doppler effect) and interference phenomena in scattering. Calculation of the corrections for the thermal motion of the molecules reduces the difference, but it remains considerable.

An exact theoretical calculation of the interference phenomena in the scattering of cold neutrons by gas molecules obtained, for example, by filtration through BeO powder ($\lambda \simeq 5\,\text{\AA}$) cannot at present be carried out (except in the case of hydrogen). The problem becomes tractable if the mass of the neutron is neglected in comparison with the masses of the nuclei making up the molecule, i.e., if it is solved in the quasiclassical approximation: neutrons are treated as waves of the corresponding wavelength, and the molecules are regarded as rigid. The problem for X-rays was solved in the same way (see Section I), and, as was already indicated, the results obtained were quite satisfactory.

Table V

Molecules	$\sigma_{\text{exp}}$ in $10^{-24}\ \text{cm}^2$	$\sum \sigma_{\text{atom}}$ per molecule in $10^{-24}\ \text{cm}^2$	$\sigma$, calculated according to classical theory, in $10^{-24}\ \text{cm}^2$
CO$_2$	24.5	13.0	24.8 or 4.1 for opposite phases
NO$_2$	57.8	34.0	55 or 41 for opposite phases
O$_2$	16.2	8.2	13.2
N$_2$	47.4	30	44.4
CF$_4$	41.5	21	38 or 7.5 for opposite phases
H$_2$	170	42

Table V gives the results of measurements of the scattering cross sections of cold neutrons by various gases at room temperature$^{49}$. The rather good agreement of the experimental data with calculations obtained in the quasiclassical approximation indicates the applicability of such a theory to this phenomenon.

The cross section must depend on the relative sign of the scattering amplitudes of the atoms composing the molecule. For CO$_2$, CF$_4$, and NO$_2$, good agreement is obtained only if one assumes that oxygen, nitrogen, carbon, and fluorine scatter neutrons with phases of the same sign.

For hydrogen the quasiclassical theory is, of course, not valid, but in this case an exact calculation is possible. However, satisfactory agreement with experiment for \(H_2\) has not been obtained. This is apparently explained by the strong distortions introduced into the experiment by the thermal motion of the molecules, since in this case the velocity of the molecules exceeds the velocity of the neutrons, and it would be more correct to say that the molecules strike the neutrons, and not the reverse. This leads to a strong dependence of the cross section on temperature. The value of \(\sigma\) varies within the limits from \(170\cdot 10^{-24}\) to \(80\cdot 10^{-24}\,\mathrm{cm}^2\) when the temperature of hydrogen is changed from room temperature to that of liquid air.

Fig. 23. Neutronograms of oxygen and carbon dioxide.

In the works described, the total cross section was measured, and the scattering cross section was obtained after introducing corrections for absorption. Recently experiments have been carried out to find the total interference pattern in neutron scattering by gases. The objects studied (the gases \(CO_2\) and \(O_2\))\(^{50}\) (see also \(^{51}\)) were contained in a steel cylinder under a pressure of \(60\ \mathrm{atm}\), and a monochromatic beam of neutrons with \(\lambda = 1.06\,\mathrm{\AA}\), obtained by reflection from the plane (200) of a NaCl crystal, was directed at it. With a counter the scattering intensity of the neutrons by the gas was measured in the range from 5 to \(90^\circ\). Since the scattering of neutrons by the gas is small, periodic corrections were made for the background of fast neutrons scattered by the walls of the cylinder; the background amounted to approximately half of all the neutrons counted.

A correction was made for multiple scattering, which in this experiment reached \(10\%\). No attempts were made to reduce it.

In Fig. 23 the results of the experiment are presented for oxygen \((\mathrm{O}_2)\) and carbon dioxide \((\mathrm{CO}_2)\). The theoretical curves were obtained in the quasiclassical approximation (1.48) for small pressures, i.e. with \(\Omega/V = 0\) [for \(\mathrm{CO}_2\) this formula was slightly modified, since (1.48) is valid for molecules consisting of atoms of a single kind]. As we shall see below, \(60\ \mathrm{atm}\) for oxygen was rightly regarded as a small pressure; however, this proved erroneous for \(\mathrm{CO}_2\). In the calculation the values \(l_{\mathrm{O}_2}=1.20\cdot 10^{-8}\ \mathrm{cm}\) and \(l_{\mathrm{CO}_2}=1.16\cdot 10^{-8}\ \mathrm{cm}\) were used.

The theoretical curves were corrected for the Doppler effect (similarly to how this was done in \(^{53}\)) and reduced to the laboratory coordinate system. No correction was made for the paramagnetism of oxygen. The vertical scale was chosen so that the experimental data coincided as well as possible with the theoretical ones.

For oxygen the agreement was very good; for carbon dioxide it was good only at large angles. The discrepancy at small angles is quite natural and was to be expected. The fact that the departure from theory was found at small angles indicates significant intermolecular interference. Therefore it is clear that in this case neglecting intermolecular phenomena was incorrect. The theoretical curve for \(\mathrm{CO}_2\) must be calculated from (1.48) with \(\Omega/V \ne 0\). This, in accordance with Fig. 3, will not change the curve at large angles and will lower it somewhat at small ones. With the proper choice of \(\Omega/V\), agreement with experiment will be complete.

Similar experiments were carried out with gaseous deuterium. The gas used was in a chamber at a pressure of \(21\ \mathrm{atm}\) and at the temperature of liquid nitrogen. From the experiment it was possible to determine the quantity

\[ \frac{\left(f_{3/2}-f_{1/2}\right)^2}{\left(2f_{3/2}+f_{1/2}\right)^2}=0.173, \tag{4.1} \]

where \(f_{3/2}\) and \(f_{1/2}\) are the amplitudes of deuterium scattering for parallel and antiparallel orientations of the neutron and nuclear spins \((I=1)\), respectively.

Starting from (4.1) and from the value of the total cross section of deuterium, two pairs of solutions for \(f_{3/2}\) and \(f_{1/2}\) were obtained:

\[ \left. \begin{aligned} f_{3/2}&=0.26\cdot 10^{-12}\ \mathrm{cm}, & f_{3/2}&=0.638\cdot 10^{-12}\ \mathrm{cm},\\ f_{1/2}&=0.826\cdot 10^{-12}\ \mathrm{cm} &\text{or}\quad f_{1/2}&=0.07\cdot 10^{-12}\ \mathrm{cm}. \end{aligned} \right\} \tag{4.2} \]

These values of the amplitudes are in contradiction with the results obtained earlier from measurements of the cross sections of atomic and molecular deuterium[^52]. On the other hand, comparison of (4.2) with (3.8) shows that for one pair of values the results, if they do not coincide sufficiently well, are in any case closer to one another. Since (3.8) were obtained from experiments with crystalline specimens, they are more reliable than (4.2).

Such experiments can be used for two purposes: starting from the known structure of molecules, to determine their scattering ability, or, conversely, from the interference pattern obtained, to determine the structure of the scattering substances. Whereas the work described with deuterium pursued the first aim, the experiments with molten metals described below pursued the second aim.

2. Scattering of Neutrons by Liquids

By means of neutron diffraction the structure of liquid metals (lead and bismuth) and liquid sulfur[^54] was studied (see also [^55]). In doing so, a special high-intensity method was used, based on the fact that the interference term in equation (1.49) is a function only of \(\dfrac{\sin \theta}{\lambda}\). Indeed, for solving the structural problem it is entirely sufficient to determine this term, in order then, by means of a Fourier transformation, to find the distribution function \(\rho(r)\) (see Section I, 3).

The equation (1.49) transformed for neutrons has the form

\[ x\left(\frac{4\pi}{\sigma}\frac{d\sigma}{d\omega}-1\right) = \int_{0}^{\infty} r g_0(r)\sin(4\pi x r)\,dr, \tag{4.3} \]

where \(x=\dfrac{\sin\theta}{\lambda}\), and \(\sigma\) is the scattering cross section of one bound atom and \(\dfrac{d\sigma}{d\omega}\) is the differential scattering cross section. In view of the fact that for neutrons the quantity \(\dfrac{d\sigma}{d\omega}\) is also a function only of \(\dfrac{\sin\theta}{\lambda}\), no additional complications arise.

Let us return to the method. The dependence of the interference only on \(\dfrac{\sin\theta}{\lambda}\) makes it possible to use not monochromatic neutron beams, but beams with a certain range of wavelengths, though with a definite value of \(\dfrac{\sin\theta}{\lambda}\); this considerably increases the luminosity-

ness of the method. The diffraction apparatus shown in Fig. 24 differs from the usual one in that the incident neutron beam is not parallel, but homocentric, and in that the reflecting atomic planes of the monochromator crystal are perpendicular to the external face (the flat crystal works in transmission). The monochromator selected neutrons from the incident beam and focused them on the central slit, which made it possible, to a considerable extent, to get rid of incoherent scattering.

During operation the collimator slit, the monochromator crystal, and the central slit remained in fixed positions. The scattering specimen could be moved along guides along the central ray and rotated about a vertical axis. Thus it was possible to vary the parameters \(\varphi\) and \(\gamma\) indicated in Fig. 24.

For neutrons passing along the central ray, the Wulff condition (1.50) is valid:

\[ \lambda = 2d \sin \theta_0 \tag{4.4} \]

and, consequently,

\[ \frac{\sin \theta}{\lambda}=\frac{\sin \theta}{2d\sin\theta_0}, \tag{4.5} \]

Fig. 24. Apparatus for a neutronographic study of liquids.

where \(d\) is the corresponding interplanar spacing of the monochromator crystal, and \(\theta\) and \(\theta_0\) are indicated in the drawing.

For neutrons not passing along the central ray, the parameters \(\varphi\) and \(\gamma\) must be chosen so that, for them as well, \(\dfrac{\sin\theta}{\lambda}\) would have the same value. The choice of \(\varphi\) and \(\gamma\) can be made analytically from the conditions

\[ \left(\frac{dx}{d\alpha}\right)_{\alpha=0}=0;\qquad \left(\frac{d^2x}{d\alpha^2}\right)_{\alpha=0}=0. \tag{4.6} \]

However, the analytical form of these equations is very complicated, and therefore the solution was carried out graphically. The experimental setup is shown in Fig. 24 and requires no explanation.

In order to eliminate all factors introducing errors into the experiment and creating background, measurements were carried out with: 1) the shutter closed (the background of fast neutrons and cosmic radiation was measured); 2) the shutter open without the chamber (scattering by air was measured); 3) an empty chamber (scattering by the walls of the vessel was measured); 4) the chamber filled with the liquid under investigation

Fig. 25. Neutronogram of liquid sulfur. Also shown (dashed line) is the X-ray diffraction pattern without allowance for the atomic and polarization factors and without preservation of the vertical scale.

Fig. 26. Neutronogram of liquid lead.

(liquid scattering was measured), and 5) with a piece of Plexiglas inserted in place of the chamber (first, to monitor the intensity of the neutron beam; second, to introduce a correction for different solid angles; and third, to monitor the efficiency of the counter).

As a result of such measurements, neutronograms were obtained for liquid sulfur (Fig. 25), bismuth, and lead (Fig. 26), representing the dependence of the differential scattering cross section per atom into a unit solid angle,

\[ \frac{d\sigma}{d\omega}, \]

on

\[ \frac{\sin\theta}{\lambda}. \]

The vertical scale was obtained from the normalization conditions for \(\frac{d\sigma}{d\omega}\). Measurements were carried out twice (solid and dashed lines). As was to be expected, the curves obtained are essentially analogous to the X-ray diffraction pattern of liquid mercury presented in Fig. 4. This once again confirms the correctness of the conclusion concerning the validity, for neutrons, of the interference equations given in Section I.

The fact that incoherent scattering of neutrons is almost isotropic makes it easy to take it into account by measuring the background at \(\dfrac{\sin\theta}{\lambda}\) values smaller than \(\dfrac{1}{2r_0}\) (where \(r_0\) is the radius of the atoms of the liquid under study), since in this range coherent scattering is absent (because of the smallness of the sulfur atoms this operation could not be carried out for it). To find the distribution function \(\rho(r)\), equation (4.3) was, as already indicated above, transformed by Fourier methods:

\[ r g_0(r)=8\int_0^\infty x\left(\frac{4\pi}{\sigma}\cdot\frac{d\sigma}{d\omega}-1\right)\sin(4\pi x r)\,dx. \tag{4.7} \]

This equation is valid, of course, for a liquid consisting of identical atoms, and under the assumption that the nuclear spin does not affect the scattering.

For all three elements investigated, formulas (4.3) and (4.7) are valid, since these elements have no isotopic or spin effects (because \(\sigma_{\mathrm{cryst}}=\sigma_{\mathrm{coh}}\)).

The integration was carried out graphically over the 32 points found from the experiment.

As the results of the experiment, the functions \(\rho(r)\) were found for the three liquids studied. In Fig. 27 the curve \(\rho(r)\) (and, for comparison, \(\rho_0\)) for liquid lead is given. From the curves the coordination number and the distance of closest approach between the atoms of the liquid structures studied were determined; the results are summarized in Table VI.

Fig. 27. Distribution function of atomic density \(\rho(r)\) for lead.

In lead the nearest neighbors (12 in all) are not sharply separated from the remaining atoms. It had previously been established by X-ray diffraction\(^ {66}\) that in liquid lead the nearest neighbors are eight atoms, while four more are located at a somewhat greater distance. An exact solution of this question can be obtained with a more careful performance of the experiment and with the use of

Table VI

Element	Crystal lattice in the solid state	Coordination number in the liquid state	Coordination number in the solid state	Smallest distance between atoms in the liquid state, Å	Smallest distance between atoms in the solid state, Å	Distances with an increase in the mean atomic density, \(\rho>\rho_0\)	Distances with a decrease in the mean atomic density, \(\rho<\rho_0\)
Sulfur	Rhombic	2	\(2^*\)	2.1	\(2.12^*\)	2.1 4 5	2.6 6
Lead	Face-centered cubic	\(12 \pm 1\)	\(12^*\)	3.4	\(3.49^*\)	3.4 6.4	4.7 8
Bismuth	Rhombohedral	8	\(3^*\)	3.10	\(3.10^*\)	3.1 6.7	5.3 7.7

Note. Figures marked with an asterisk are taken from other sources.

larger values of \(\dfrac{\sin \theta}{\lambda}\). In other respects the agreement with the X-ray data is very good.

The X-ray study of liquid metals that have close packing in the solid state made it possible to put forward the supposition\(^{57}\) that the close packing is preserved when the metal melts. The results obtained with lead confirm this supposition. This supposition is also supported by the fact that the diffraction maxima on the roentgenogram of liquid lead occur at the same values of \(\dfrac{\sin \theta}{\lambda}\) as can be theoretically expected from the structure of solid lead. On this basis it is possible to index the neutronogram.

The preservation of the positions of the maxima of coherent scattering, the preservation of the minimum distance between atoms, and the preservation of the coordination number of sulfur upon melting make it possible to suppose that the sulfur atoms tend to form groups with a configuration analogous to the structure of solid sulfur. These considerations are in complete agreement with the previously obtained X-ray results of the study of liquid and solid sulfur\(^{58}\).

The results obtained for molten bismuth indicate a lesser degree of ordering in this liquid. The nearest atoms are not separated from the rest sufficiently sharply; however, it is evident that at a distance of \(3.1\ \text{Å}\) there are eight atoms. On melting, bismuth’s coordination number increases, i.e. its structure becomes denser. This may explain the decrease in the volume of bismuth upon melting. Bismuth shows no tendency to form crystal-like groups.

From the curves it is evident that, at distances exceeding six to eight interatomic distances, all \(r\) are equally probable. This is in agreement with what was said above about crystal-like groups and indicates that in liquids there is an ordered arrangement of atoms only in small regions.

It was also established that when the temperature of molten metals is lowered, the ordered regions are rearranged and the atoms form groups with a structure approaching the crystalline structure of the solid metal. In the neutronogram of liquid lead (Fig. 26) there are strong maxima at values of \(\dfrac{\sin\theta}{\lambda}\) corresponding to the (111) and (311) reflections of crystalline lead. However, on the slopes of the clearly expressed strong peaks (111) and (311), two additional small subsidiary maxima are also noticeable. Their position is difficult to determine with sufficient accuracy, but it is known that in the neutronogram of crystalline lead, between the maxima (111) and (311), two more must be present: (200) and (220) (see Fig. 9, g). Their occurrence is also observed in the neutronogram. The same picture is observed with bismuth. This once again confirms the supposition about the rearrangement of atoms when the temperature of liquid metals is lowered.

Fig. 28. Neutronogram of glass.

Liquid neutronograms and radiographs similar to those shown above have such a simple form only for monatomic liquids, such as molten metals. For complex molecular liquids the picture becomes more complicated and does not have such a regular character. The same applies to amorphous bodies of the liquid type. Figure 28 shows a neutronogram of boron-free glass, obtained by the method described above1. The curve is not similar to the neutronograms of liquid metals; nevertheless, everything said above is applicable to this case as well: by the same equations and in the same way one can determine the structure of the glass under investigation.

V. THE AMPLITUDE OF COHERENT SCATTERING OF NEUTRONS BY NUCLEI AND METHODS FOR MEASURING IT

It is quite clear that no serious structural work can be carried out without knowledge of the amplitudes with which neutrons are scattered by nuclei. The determination of their values is at present the principal task of neutron-structural analysis, and is of a somewhat preparatory character. Therefore the center of gravity of all the work carried out up to the present time lies precisely in this field of neutronography. The determination of the scattering amplitude (where the determination of its sign is of essential importance) can be carried out in many ways; some of them have already been described above.

The most direct and, in our opinion, the most accurate determination of the magnitude of the scattering amplitude can be achieved by measuring the intensity of diffraction maxima, in the manner described in Section III, 1.

In this case coherent scattering is separated from incoherent scattering most completely and, moreover, the diffraction pattern is expressed most vividly (see, for example, Fig. 9, a, b, c, d).

By this method it is possible to determine the absolute value of the amplitude; its sign, however, generally speaking, cannot be determined. This can be done only in the study of neutronograms of binary compounds by comparing the intensity of the diffraction maxima with their theoretical values, calculated on the assumption of identical and opposite signs of the amplitudes (as was done, for example, with NaH).

The relative sign of the amplitude can be determined by measuring the intensity of reflection from those planes of the atomic lattice of complex crystals (consisting of several kinds of atoms) which alternately contain atoms of first one element and then another element49,60. For example, the (111) planes of an NaCl crystal are separated from one another by equal distances and alternately contain atoms either of Na or of Cl. The intensities of the diffraction maxima (111) of different orders [(222)—second order, (333)—third order, etc.] are determined by equations (3.3), since even orders contain only even indices, and odd orders—odd ones. Thus, by measuring maxima of different orders, obtained upon reflection of monochromatic neutrons from the same planes, one can determine the relative signs of the scattering amplitudes. (In doing this it is necessary to take into account a certain decrease in intensity from order to order in connection with geometrical factors, thermal vibrations in the lattice, etc.) By this method the signs of the scattering amplitudes were found for a large number of elements, and it was established that only 2 of them

have negative amplitudes. These elements turned out to be lithium and manganese. (The negative amplitude of hydrogen had become known earlier, and that of titanium—somewhat later^26.)

The absolute value of the scattering amplitude can also be determined from formula (3.8) by measuring the transmission of neutrons through crystalline powders. By this method the amplitudes of Zn and of some other elements were determined. However,

Fig. 29. Determination of cross sections from the measurement of neutron transmission through a crystalline powder.

as has already been pointed out, such measurements are less accurate and in most cases give amplitude values that do not agree with other data.

Somewhat more accurate results are obtained from the same experiments, but by a simpler method^61. As was already stated above, the scattering cross section at \(\lambda > 2d\) is zero. The attenuation of a neutron beam of such wavelength occurs due to absorption, following the law \(\frac{1}{v}\) (segment \(AB\) in Fig. 29), and due to incoherent scattering. If the straight line \(AB\) is extended, then its intersection with the ordinate corresponding to

zero wavelength ($v \to \infty$) (point $M$) will give, obviously, the cross section of incoherent scattering. On the other hand, at $\lambda = 2d_{\max}$ there will be a jump in the cross section, caused by diffraction phenomena. With an increase in the neutron energy in the region $\lambda \ll 2d$ the cross section again decreases linearly with decreasing wavelength. The intersection of the continuation of the rectilinear segment $CD$ with the ordinate corresponding to $\lambda = 0$ (point $N$) gives, obviously, the total cross section of a free nucleus. Thus, the quantity $\sigma_N \left( \dfrac{A+1}{A} \right)^3 - \sigma_M$ gives the cross section of coherent scattering of the element under investigation. This method is attractive by virtue of its simplicity, but its accuracy, like that of other methods based on similar experiments, is low.

The cross sections of nickel, copper, and manganese determined in this way differ rather strongly from the values determined earlier and most frequently encountered in the literature.

The only method that makes it possible to determine directly the sign of the scattering amplitude is the observation of total “internal” reflection of neutrons from solid and liquid mirrors. In Section I, 4 it was shown that total “internal” reflection occurs only for a positive scattering amplitude. Therefore the presence of total “internal” reflection would quite unambiguously indicate the positivity of the amplitude.

Two difficulties arise in carrying out such an experiment: 1) from formula (1.57) it is evident that the limiting glancing angle amounts to several minutes, and its measurement is therefore extremely difficult; 2) as has already been indicated, the effect is extremely sensitive to all kinds of contamination. For these reasons the method described should be regarded, mainly, as a way of determining the absolute sign of the amplitude.

In the experiment a monochromatic beam of neutrons with wavelength $\lambda = 1.87\,\text{\AA}$ was used, obtained by reflection from a CaF$_2$ crystal. A well-collimated beam was directed onto a mirror made of the material under study, and the intensity of neutrons reflected at different angles was measured with a counter. When the glancing angle was varied from $0$ to $\varphi_0$, the intensity of the neutrons reflected from the surface of the mirror remained practically constant. At a glancing angle equal to $\varphi_0$, the intensity dropped sharply to the background value. From this drop the magnitude of $\varphi_0$ was inferred.

Such experiments showed that Be, Zn, Cu, Ni, Fe, and C have positive scattering amplitudes$^{49}$. Combining this method with those described earlier (especially with the first), one can obtain reliable data concerning the sign and magnitude of the scattering amplitude.

However, in some cases it is possible to increase the accuracy of amplitude measurements on the basis of similar experiments by applying liquid molecular “mirrors.” In that case, if the liquid is not monatomic, the critical glancing angle is determined by the previous formula (1.57), but instead of \(f\) one takes the scattering amplitude of the whole molecule, analogous to (1.31):

\[ f_{\mathrm{mol}}\sim \sum_i g_i f_i, \tag{5.1} \]

where \(g_i\) is the relative content and \(f_i\) the scattering amplitudes of the \(i\) atoms entering into the liquid molecule. It follows from elementary calculations that the errors in measurements of the critical glancing angle \(\Delta\varphi_0\) and of the scattering amplitudes \(\Delta f\) are related by

\[ \left|\frac{\Delta\varphi'_0}{\Delta\varphi_0}\right| = \sqrt{\frac{\displaystyle\sum_i g_i f_i}{f}}\cdot \frac{\Delta f'}{\Delta f}, \tag{5.2} \]

where primes denote the errors obtained in experiments with monatomic mirrors, i.e., in experiments in which the scattering amplitude of an atom is determined directly.

If we now somewhat inaccurately assume that \(\Delta\varphi_0=\Delta\varphi'_0\), we obtain the relation between \(\Delta f\) and \(\Delta f'\):

\[ \Delta f= \sqrt{\frac{\displaystyle\sum_i g_i f_i}{f}}\cdot \Delta f'. \tag{5.3} \]

If atoms having scattering amplitudes of different signs enter into the liquid molecule, it may turn out that \(\sum_i g_i f_i\) becomes considerably smaller than \(f\). For example, when determining the scattering amplitude of hydrogen \(f_{\mathrm H}\) from experiments with reflection of neutrons from triethylbenzene \((\mathrm{C}_{12}\mathrm{H}_{18})\), equation (5.3) takes the form

\[ \Delta f= \sqrt{\frac{12}{30}\cdot\frac{f_{\mathrm C}-1.5\,f_{\mathrm H}}{f_{\mathrm H}}} \simeq 0.2\,\Delta f'. \tag{5.4} \]

The gain in accuracy is approximately a factor of five.

By selecting another organic compound with an even smaller \(\sum_i g_i\), one can attain still greater accuracy. One may proceed in this way until the values of \(\varphi_0\) decrease so much that the relative error in \(\Delta\varphi_0\) will become significant and the determination of \(\varphi_0\) itself will become extremely difficult. Knowing the geometr—

ical parameters of the experimental setup, it is easy to determine the optimal values of \(f_{\mathrm{mol}}\). A similar procedure was used to determine the amplitude of hydrogen using tristilbenzene \((C_{12}H_{18})^{62}\). The critical glancing angle in this case was approximately \(6'\) for wavelengths \(\lambda \approx 8 \text{ Å}\).

A well-collimated beam of thermal polychromatic neutrons was directed (Fig. 30) onto the surface of the liquid under an angle of \(5.3'\). In this case, those neutrons whose wavelength was equal to or greater than

\[ \lambda_0=\frac{5.3'}{\sqrt{\frac{N}{\pi}\left(f_C-1.5 f_H\right)}} . \]

Fig. 30. Determination of the critical glancing angle in reflection from liquid mirrors.

were reflected from the surface of the liquid. The limiting wavelength \(\lambda_0\) was determined by secondary reflection from a beryllium mirror, which had previously been calibrated with monochromatic neutrons in order to find the function \(\varphi_0=f(\lambda)\). Then, from formula (1.57), the quantity \((f_C-1.5 f_H)\) was found. Starting from the known value of the scattering amplitude of carbon, the amplitude of hydrogen was found, amounting to \(-0.375\cdot 10^{-12}\) cm. This value agrees fairly well with those obtained earlier; however, the discrepancy exceeds the errors of the experiment.

At present the scattering amplitudes have been determined for a large number of elements and their isotopes\(^{22}\). All values of the amplitudes known at present are collected in Table VII.

Scattering amplitudes and cross sections
Table VII

Investigated substances	\(Z\)	Element	Isotope	Nuclear spin	\(f_{\mathrm{cor}}\) in \(10^{-12}\) cm	\(\sigma_{\mathrm{cor}}\) in \(10^{-24}\) cm\(^2\)	\(\sigma_{\mathrm{cryst}}\) in \(10^{-24}\) cm\(^2\)
NaH	1	H	H\(^1\)	\(1/2\)	\(-0,40\)	2,0	80,0
NaD, ThD, D\(_2\)O			H\(^2\)	1	0,64	5,2	7,4
LiF, LiCl	3	Li			\(-0,18\)	0,4
			Li\(^6\)	1	0,7	\(\sim 6\)
			Li\(^7\)	\(3/2\)	\(-0,25\)	0,8	\(\sim 2\)
BeO, Be	4	Be	Be\(^9\)	\(3/2\)?	0,78	7,7	7,5
Diamond, graphite	6	C	C\(^12\)	0	0,64	5,2	5,2
KN\(_3\)	7	N	N\(^14\)	1	0,85	9,1	10,0
Various oxides	8	O	O\(^16\)	0	0,58	4,2	4,2
NaF, CaF\(_2\)	9	F	F\(^19\)	\(1/2\)	0,55	3,8	\(\sim 3,5\)
NaCl, Na, etc.	11	Na	Na\(^23\)	\(3/2\)	0,35	1,5	3,5
MgO, Mg	12	Mg			0,44	2,4	4,2
Al	13	Al	Al\(^27\)	\(5/2\)	0,35	1,5	1,5
PbS	16	S			0,31	1,2	\(\sim 1,2\)
NaCl, KCl and CuCl	17	Cl			0,99	12,2	15
KCl	19	K			0,35	1,5	\(\sim 2\)
CaO, CaF\(_2\)	20	Ca			0,49	3,0	3,5
			Ca\(^40\)	0	0,49	3,0	3,2
			Ca\(^44\)	0	0,18	0,4
Ti, TiC	22	Ti			\(-0,38\)	1,8	\(\sim 6\)
V, VC	23	V	V\(^51\)	\(7/2\)	\(<0,09\)	\(<0,1\)	5
Cr, FeCr	24	Cr			0,37	1,7	3,8
MnO, Ni\(_3\)Mn	25	Mn	Mn\(^55\)	\(5/2\)	\(-0,33\)	1,35	2,2
FeO, Fe, Fe\(_2\)O\(_3\)	26	Fe			0,96	11,4	11,7
			Fe\(^54\)	0	0,42	0,2	2,5
			Fe\(^56\)	0	1,00	12,6	13
			Fe\(^57\)	(?)	0,23	0,64	2
Co, CoO, FeCo	27	Co	Co\(^59\)	\(7/2\)	0,28	1,0	\(\sim 5\)
Ni, NiO, Ni\(_3\)Mn	28	Ni			1,03	13,4	17,3

Table VII. Continued

Substances studied	\(Z\)	Element	Isotope	Nuclear spin	\(f_{\mathrm{cor}}\) in \(10^{-12}\) cm	\(\sigma_{\mathrm{cor}}\) in \(10^{-24}\) cm\(^2\)	\(\sigma_{\mathrm{cryst}}\) in \(10^{-24}\) cm\(^2\)
			Ni\(^{58}\)	0	1.47	27.0	27.0
			Ni\(^{60}\)	0	0.28	0.97	1.0
			Ni\(^{62}\)	0	−0.85	9.1	9.0
Cu, Cu\(_2\)O, CuCl	29	Cu			0.76	7.3	7.8
Zn, ZnO, CuZn	30	Zn			0.59	4.3	4.2
Ge, GeO\(_2\)	32	Ge			0.84	8.8	8.5
As, As\(_2\)O\(_3\)	33	As	As\(^{75}\)	\(3/2\)	0.63	5.0	\(\sim 7.0\)
Mn, Se	34	Se			0.89	10.0	10.0
NaBr, KBr	35	Br			0.67	5.7	6.0
RbCl	37	Rb			0.55	3.8	5.5
SrO	38	Sr			0.57	4.1	9.5
ZrC, ZrN	40	Zr			0.62	4.9	\(\sim 7\)
Nb	41	Nb	Nb\(^{93}\)	\(9/2\)	0.69	6.0	6.2
Mo	42	Mo			0.64	5.2	7.4
Pd	46	Pd			0.63	5.0	4.8
Ag, AgCl	47	Ag			0.61	4.6	7
			Ag\(^{107}\)	\(1/2\)	0.83	8.7	10
			Ag\(^{109}\)	\(1/2\)	0.43	2.3	6
Sn, SnO, SnO\(_2\)	50	Sn			0.61	4.6	4.9
Sb, Sb\(_2\)O\(_3\)	51	Sb			0.54	3.7	4.2
NaI, KI	53	I	I\(^{127}\)	\(5/2\)	0.52	3.4	3.8
CsC	55	Cs	Cs\(^{133}\)	\(7/2\)	0.49	3.0	\(\sim 7\)
Ta, TaC	73	Ta	Ta\(^{181}\)	\(7/2\)	0.70	6.1	7.0
W, WO\(_3\)	74	W			0.51	3.3	5.7
Pt	78	Pt			0.95	11.2	11.2
Au, Cu\(_3\)Au	79	Au	Au\(^{197}\)	\(3/2\)	0.77	7.5	\(\sim 9\)
Pb, PbS	82	Pb			0.96	11.5	11.6
Bi	83	Bi	Bi\(^{209}\)	\(9/2\)	0.89	10.1	10
Th, ThO\(_2\)	90	Th	Th\(^{232}\)	0	1.01	12.8	12.8

In the first column of the table are given those compounds or pure elements that were studied in order to determine the scattering amplitudes. In the second, third, and fourth columns are given the atomic number of the element \(Z\), its symbol, and its individual isotope, if measurements with an isotope were carried out. In the fifth column is given the nuclear spin; in the sixth, the value of the neutron-scattering amplitude for each element or its isotope. In the seventh and eighth columns are given the cross sections

Fig. 31. Values of scattering amplitudes as a function of \(A\).

of coherent and total scattering for bound nuclei. From the table it is seen that in most cases the last two quantities either coincide or are close to one another; this indicates that strong spin incoherence or strong isotopic diffuse scattering are rather rare phenomena. However, for some elements they reach a considerable magnitude: spin incoherence is greatest for hydrogen, for which \(\sigma_{Ds}^{\mathrm H}=78\cdot10^{-24}\ \text{cm}^2\), while the isotopic effect is especially large for nickel \((\sigma_{Di}^{\mathrm{Ni}}=3.9\cdot10^{-24}\ \text{cm}^2)\). Even for lead, which has a large number of isotopes, only \(0.1\cdot10^{-24}\ \text{cm}^2\) remains for the isotopic background; moreover, this value may possibly result from an error in one of the measurements.

Recently there have been attempts to find some regularity in the dependence of the scattering amplitude on the mass number of the element[^63].

In Fig. 31 the data of Table VII are presented graphically: along the abscissa axis are plotted the mass numbers of the nuclei, and along the ordinate axis—the scattering amplitudes. Cases in which the amplitude values for each isotope of a single element are not known, but only its value for the element as a whole is known, are marked with crosses. Indeed, essentially three groups of amplitude values are indicated: the first—from \(A = 0\) to \(A = 55\), the second—from \(A = 55\) to \(A = 150\), and the third—from \(A = 150\) and higher. Attempts have been made to explain such a picture theoretically by connecting the scattering amplitudes with the sizes of nuclei. However, it is unlikely that so simple a dependence obtains.

VI. MAGNETIC SCATTERING OF NEUTRONS

In addition to the nuclear interaction, about which, and only about which, we have been speaking up to now, magnetic forces may arise between a neutron and an atom, due to the presence in the neutron of a magnetic moment. These magnetic forces depend substantially on the mutual orientation of the magnetic moments of the neutron and the scattering atom. Scattering by magnetic forces, or, in other words, magnetic scattering, can be characterized by the magnetic-scattering amplitude \(f_{\mathrm{m}}\), in contrast to the nuclear-scattering amplitude \(f_{\mathrm{n}}\) mentioned up to now. The total scattering amplitude of the atom will be the algebraic sum of both amplitudes:

\[ f = f_{\mathrm{n}} \pm f_{\mathrm{m}} . \tag{6.1} \]

The indefinite sign before \(f_{\mathrm{m}}\) appears because the sign depends on the orientation of the magnetic moment of the neutrons relative to the moment of the atom.

It was shown theoretically[^64] that the amplitudes of magnetic and nuclear scattering are of the same order. This conclusion was confirmed by numerous experimental data. Therefore the case in which \(f_{\mathrm{m}} > f_{\mathrm{n}}\) is of interest. In this case the total scattering amplitude will have different signs for different orientations of the magnetic moments of the neutron and the atom.

This property can be used to obtain polarized neutrons, i.e., neutrons whose magnetic moments lie in one plane and are oriented in one direction.

To polarize neutrons it is necessary to obtain total “internal” reflection from a magnetized mirror, for which the total scattering amplitude has different signs for dif-

orientations of the magnetic moments of the neutron and the atom. Such a mirror may be magnetized cobalt, for which \(f_M>f_{\mathrm{я}}^{65}\). As was already said above, total “internal” reflection of neutrons occurs only for a positive scattering amplitude, and therefore only those neutrons for which the spin orientation corresponds to a positive scattering amplitude will be reflected. In this way one can obtain a polarized neutron beam without resorting to preliminary monochromatization, i.e., while retaining a high intensity also in the polarized beam.

For iron the amplitude of nuclear scattering is greater than the magnetic one, and obtaining a polarized beam in this case is possible only by varying the limiting glancing angle \({}^{65,67,68}\), i.e., preliminary monochromatization is necessary. Naturally, when magnetic scattering is taken into account, formulas (1.54), (1.55), and (1.57) become invalid. In this case, in solving the Schrödinger equations inside the crystal it is necessary to take into account not only the nuclear but also the magnetic interaction \({}^{66}\). This leads to the following expressions for the refractive indices [provided condition (1.29) is satisfied]:

\[ n_{\pm}^{2}-1=-\frac{\lambda^{2}Nf}{\pi}\pm\frac{\mu B}{E}, \tag{6.2} \]

where \(n_{+}\) and \(n_{-}\) are the refractive indices for parallel and antiparallel orientation of the magnetic moment of the neutron \(\mu\) and of the atomic moment, and \(B\) is the magnetic induction of the specimen. In accordance with this, equation (1.57) is rewritten as

\[ (\varphi_{0})_{\pm}\simeq \sqrt{\frac{\lambda^{2}Nf}{\pi}\mp\frac{\mu B}{E}}. \tag{6.3} \]

In connection with the addition of the new term, the optical effects for ferromagnetic materials, generally speaking, increase approximately twofold, and the technical side of the experiment is somewhat simplified.

Another method of obtaining polarized neutrons is to pass a neutron beam through a plate of magnetized iron. The neutron beam that has passed through the plate, as a result of the magnetic interaction of the neutrons with the atoms, becomes polarized, though not completely, but only to a greater or lesser degree. The degree of polarization can be measured with the aid of another magnetized plate, which can be rotated through a definite angle in a plane perpendicular to the neutron beam. Thus such a system becomes analogous to an optical polarization instrument, the first plate playing the role of polarizer and the second that of analyzer.

A similar system can be used to study regions of spontaneous magnetization (domains) of ferromagnets[^69].

Domains in an unmagnetized material are oriented at random, and therefore a polarized neutron beam, when passing through such a material, is partially depolarized. If an unmagnetized ferromagnetic plate is placed between the neutron polarizer and analyzer described above, then the magnitude of the neutron depolarization can be used to judge the sizes of the domains. In this way, experiments with iron foil of thickness of the order of 0.01 mm showed that the domain sizes lie in the range from \(10^{-3}\) to \(7\cdot10^{-5}\) cm.

In addition to depolarization, the domain structure of ferromagnets is the source of effects of another kind. When a neutron beam passes through an unmagnetized ferromagnet, the neutrons undergo multiple refraction at the boundaries between domains. This was used to explain the presence of intense scattering through small angles (of the order of \(1'\)) when a neutron beam passes through a ferromagnet[^70]. In confirmation of this, when the specimen was magnetized the scattering disappeared. Similar experiments also provide a method for studying domains—a method for studying the magnetic structure of ferromagnets.

To study the magnetic structure, samples of certain manganese and iron compounds were investigated, by the transmission method[^71],[^72]: \(\mathrm{MnF_2}\), \(\mathrm{MnSO_4}\), \(\mathrm{MnO}\), and \(\alpha\)-\(\mathrm{Fe_2O_3}\). Figure 32 presents the result of measuring the transmission of neutrons through \(\mathrm{MnF_2}\). The dotted curve was constructed theoretically (opposite to what was described in Section V), proceeding from the nuclear interaction and the following values of the total cross sections:

\[ \sigma_{\mathrm{Mn}}=\left(1.8+2.14\sqrt{E}\right)\cdot10^{-24}\ \mathrm{cm}^2, \]

\[ \sigma_{\mathrm{F}}=3.3\cdot10^{-24}\ \mathrm{cm}^2 \]

(for free atoms; in addition, absorption by fluorine was neglected). The region of diffraction peaks (at \(\lambda \simeq d\)), lying in the range from 4 to 5.5 Å, was excluded from consideration for simplicity. The solid curve represents the experimental results. The difference between experiment and theory was attributed to paramagnetic scattering.

Neutron diffraction patterns obtained with the same samples showed that these substances belong to three types of magnetic structures: 1) in the neutron diffraction patterns of \(\mathrm{MnF_2}\) and \(\mathrm{MnSO_4}\), the magnetic scattering fell off according to a law analogous to the atomic factor, which indicates a completely chaotic distribution of the magnetic moments of the atoms and the absence of interaction between them in these substances; 2) in the neutron diffraction pattern of \(\mathrm{MnO}\), the magnetic scattering had a form analogous to nuclear scattering in monatomic liquids

(on Fig. 33, bottom; a very diffuse maximum at small angles); such a picture indicates that there are some small regions in which the magnetic moments are oriented in an ordered manner, and that the magnetic and nuclear structures have different parameters, and, finally, 3) in the neutronogram of $\alpha$-$\mathrm{Fe_2O_3}$ there were

Fig. 32. Passage of neutrons through a crystalline paramagnetic powder.

strong maxima of coherent scattering at positions not allowed from the point of view of the chemical structure.

The results obtained are well explained by, and confirm, modern ideas about the magnetic properties of various substances. From the point of view of these ideas, there are essentially three types of magnetic structures with an ordered arrangement of elementary magnetic moments2: ferromagnetics, whose stable state is the parallel arrangement of the magnetic moments of atoms; antiferromagnetics with antiparallel moments; and metamagnetics with alternating planes in which all magnetic moments are directed in one direction (the moments of neighboring planes being antiparallel).

At \(0^\circ\)K each atomic moment of an antiferromagnet is surrounded by oppositely directed moments; there is, one may say, a magnetic superstructure. With increasing temperature this order begins to break down, and upon reaching a certain temperature (the Curie point) thermal motion completely destroys the regions of spontaneous antiferromagnetism, and order remains only in small regions between which there is no correlation; above the Curie temperature the substance exhibits typical paramagnetic properties. At the Curie point, in connection with magnetic transformations, certain anomalies of physical properties were observed in antiferromagnets. On the basis of the discovery of these anomalies, the presence of antiferromagnetism had earlier been inferred, since they could not be explained by anything else. This was the only, and moreover indirect, method.

Fig. 33. Neutronogram of MnO at temperatures \(300^\circ\)K and \(80^\circ\)K.

In Fig. 33 are presented neutronograms of MnO\(^{74}\) (see also \(^{75}\)), obtained at 80 and \(300^\circ\)K. On the neutronogram of MnO at room temperature there are normal diffraction maxima of coherent nuclear scattering of a face-centered cubic lattice and a background of magnetic scattering at small angles. The appearance of superstructure maxima—strong at \(\theta = 12^\circ\) and weaker at larger angles—on the neutronogram at \(80^\circ\)K can be explained only by the antiferromagnetic properties of MnO. (It was shown\(^{76}\) that in this temperature interval ...

structure of MnO undergoes no structural changes.) The Curie temperature of this specimen is \(122^\circ\mathrm{K}\), and therefore coherent magnetic scattering is absent from the neutronogram obtained at room temperature. In contrast, the Curie temperature of \(\alpha\)-\(\mathrm{Fe_2O_3}\) is \(950^\circ\mathrm{K}\), and therefore this substance has antiferromagnetic properties at room temperature.

Naturally, the parameters of the magnetic lattice of an antiferromagnet are twice as large as the parameters of the atomic lattice. (For ferromagnets they coincide, and it is more difficult for them to separate magnetic scattering from nuclear scattering.) Therefore, of course, the indices of the magnetic and nuclear maxima do not coincide with one another.

The further application of neutronography to the study of magnetic structure and magnetic transformations of the crystal lattice will undoubtedly yield a great deal. This especially concerns ferro-, antiferro-, and metamagnetics. Extremely useful in this respect would be the use of polarized neutrons, which in a magnetic sense are a kind of “monochromatic” neutrons. The diffraction of polarized neutrons in domains would depend essentially on their orientation and dimensions, just as the diffraction of monochromatic beams on the microcrystals of polycrystalline specimens does.

As has already been indicated above, the advantages of neutronography over other methods of structural analysis—X-ray and electron methods—are based mainly on the different properties of the scattering amplitudes for neutrons and for X-ray and electron rays. These differences, set forth in section I, 5, make it possible by neutronography:

1. To study the crystal structures of hydrogen-containing substances with accurate localization of the hydrogen atoms (similarly to how this was done with \(\mathrm{NaH}\) and \(\mathrm{H_2O}\)). Such a study is possible by X-ray methods*) only in certain exceptional cases or on the basis of various indirect considerations.

2. To study systems consisting of two or several elements with close \(Z\) (on the basis of which, for example, it became possible to study ordering in \(\mathrm{FeCo}\) and \(\mathrm{Ni_3Mn}\) alloys). Such experiments, generally speaking, do not give a result in X-ray

*) Everything that will be said below about X-rays applies fully also to electrons.

physical investigation because of the equality of the scattering abilities of atoms with close \(Z\).

1. To investigate systems consisting of atoms with \(Z\) greatly differing from one another (as was done, for example, with PbS, ThD\(_2\), WO\(_2\), etc.). The strong scattering of X-rays by the heavy element completely masks scattering by the light one, in consequence of which such investigations are impossible by X-ray diffraction.

2. To study systems consisting of definite isotopes of one and the same element (for example, the disappearance of the fundamental lines of the ordered alloy Ni\(_3^{60}\)Mn). The scattering of X-rays by different isotopes of one and the same element is exactly the same, owing to the identity of their electron shells.

3. To find \(\sigma_{DS}\) and determine the scattering amplitudes for different spin states, which gives certain information on the dependence of scattering on the mutual orientation of the spins of the neutron and the nucleus and is very essential for modern atomic and nuclear physics.

4. In view of the presence of a magnetic moment in the neutron, to study the magnetic structure of substances. The extraordinary importance of such investigations and the peculiar uniqueness of neutron diffraction in this respect have already been mentioned above.

In addition, in contrast to X-rays, neutrons are weakly absorbed in the overwhelming majority of elements. Therefore some experiments requiring transmission through samples and, consequently, impracticable with X-rays, are easily carried out neutronographically.

Among the shortcomings of neutron structural analysis one should first of all mention the complexity of powerful neutron sources—reactors. As for the other shortcomings of neutronography (the coincidence of scattering amplitudes of different elements, which was mentioned above; the impossibility of applying the less accurate but simpler photographic method of neutron registration, etc.), they are few in number and do not play any substantial role.

In view of the comparative complexity of the experimental technique of neutronography and in view of the wide prevalence and productivity of X-ray analysis, the most valuable and fruitful approach is a reasonable combination of these two methods of structural investigations. Such a combination unusually broadens the possibilities for studying crystalline structures.

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1. Reference number as shown in the source. ↩

2. 5* ↩