Neutron Diffraction Analysis

a method of studying the structure of molecules, crystals, and liquids by using neutron scattering. Data on the atomic and magnetic structure of crystals are obtained from neutron diffraction experiments, and information on the thermal vibrations of atoms in molecules and crystals is obtained from neutron scattering experiments. In neutron scattering, the neutrons exchange energy with the object under study; in this case, the scattering is said to be inelastic. Neutron diffraction analysis was first studied by E. Fermi from 1946 to 1948; the fundamental principles were first proposed in 1948 in a review by the American scientists E. Wollan and C. Shull.

Neutron diffraction experiments are conducted with neutron beams obtained from nuclear reactors. (Electron accelerators with special targets have been proposed for use in neutron diffraction analysis.) A typical unit for neutron diffraction studies is illustrated in Figure 1, a. The neutron diffraction apparatus, such as diffractometers and various types of neutron spectrometers, is placed in the immediate vicinity of the reactor in the path of a neutron beam. The neutron flux density in beams from the most powerful reactors is several orders of magnitude less than the quantum flux density from an X-ray tube; neutron diffraction apparatus and experiments are therefore complicated. It is for this reason that the specimens used in neutron diffraction analysis are significantly larger than the specimens used in X-ray diffraction analysis. Experiments can be carried out over a broad range of different variables, such as pressure, magnetic field, and temperature (1–1500°K and higher).

Figure 1. Diagram of neutron diffraction apparatus for studying poly-crystalline specimens: (1) collimation system to define neutron beam; (2) monochromator unit for isolating neutrons with a certain fixed energy, or wavelength, from the continuous spectrum of neutrons obtained from the nuclear reactor; (3) neutron spectrometer with neutron detector (4) for measuring neutron radiation intensity at different scattering angles Θ. The specimen is placed at the center of the spectrometer, (b) Neutron diffraction pattern of a polycrystal-line specimen of BiFeO₃

A neutron diffraction pattern, that is, the dependence of the scattered intensity I of the neutrons on the scattering angle θ, is shown in the Figure 1,b; the specimen is polycrystalline BiFeO₃. The neutron diffraction pattern is the set of maxima owing to coherent nuclear or magnetic scattering (see below) observed against a background of diffuse scattering.

The successful use of neutron diffraction analysis is made possible by the neutron’s fortunate combination of properties as an elementary particle. Contemporary neutron sources—nuclear reactors—produce thermal neutrons over a broad range of energies with a maximum around 0.06 electron volt. The de Broglie wavelength of neutrons of this energy (approximately 1 angstrom [A]) is of the same order of magnitude as the interatomic distances in molecules and crystals. This fact makes possible the diffraction of neutrons in crystals; the method of structural analysis by neutron diffraction is based on this. The commensurability of the energy of thermal neutrons and the energy of the thermal vibrations of atoms and molecular groups in crystals and liquids ensures optimal use of inelastic scattering of neutrons in neutron spectroscopy. Because the neutron has a magnetic moment that can interact with the magnetic moments of atoms in crystals, magnetic diffraction of neutrons can be realized in magnetically ordered crystals. This is the basis for magnetic neutron diffraction analysis.

Structural analysis by neutron diffraction. Neutron diffraction is one of the main modern methods of structural analysis of crystals, along with X-ray and electron diffraction. The geometric theories of diffraction of the three types of radiation—X rays, electrons, and neutrons—are identical, but the physical natures of the interaction with matter differ. It is this difference that determines the specifics and fields of application of each of these methods. X rays are scattered by the electron shells of atoms; neutrons are scattered by atomic nuclei (through short-range nuclear forces); and electrons are scattered by the electric potential of atoms. As a result of this, structural analysis by neutron diffraction has a number of distinctive features. The scattering power of atoms is characterized by the atomic scattering amplitude f. The special character of the interaction of neutrons with nuclei leads to a situation in which the atomic scattering amplitude of neutrons f_n (usually designated by the letter b) for different elements depends in an irregular way (in contrast to the f of X rays) on the atomic number Z of the element in the periodic table.

In particular, the scattering powers of light and heavy elements prove to be of the same order of magnitude. The study of the atomic structure of compounds of light and heavy elements is thus a specific branch of neutron diffraction analysis. This pertains, above all, to compounds that contain the lightest element—hydrogen. The position of hydrogen atoms in crystals of compounds containing hydrogen combined with other light atoms (atoms having Z < 30) can be determined in certain favorable cases by X-ray and electron diffraction. The determination by neutron diffraction of the position of hydrogen atoms is no more complicated than is the determination of the position of most other elements. Here, a significant methodological advantage is achieved by replacing the hydrogen atoms in the molecule under study by the hydrogen isotope deuterium.

Neutron diffraction has been used to determine the structures of a large number of organic compounds, hydrides, and crystalline hydrates; the method has also been used to refine the structure of the various modifications of ice and hydrogen-containing ferroelectrics. This has provided extensive new data for developing the crystal chemistry of hydrogen.

Another field that makes optimal use of neutron diffraction analysis is the study of compounds of elements that have similar values of Z (such elements are practically indistinguishable in X-ray analysis, since their electron shells contain nearly the same number of electrons); examples of such compounds are spinel (MnFe₂O₄) and Fe-Co-N alloys. The extreme case is the study of compounds of different isotopes of a given element, which are absolutely indistinguishable by X-ray diffraction but which can be distinguished by neutron diffraction just as readily as can different elements.

The intensities of the maxima of the coherent scattering I (hkl), where h, k, and I are the Miller crystallographic indices, which are related to the structural amplitudes F (hkl) by certain equations, are found experimentally in structural analysis by neutron diffraction. Then, the nuclear density function p (x,y,z) is constructed by using Fourier series whose coefficients are the quantities F (hkl). Summation of the series, like most other calculations in structural analysis, is carried out by high-speed computers using special programs. The maxima of the function p (x,y,z) correspond to the positions of the atomic nuclei.

As an example of this, a projection of the nuclear density of part of the unit cell of the cobalt-containing vitamin B₁₂ is illustrated in Figure 2,a. In this projection, the central atom of the core of the molecule—the cobalt atom—has the smallest value of b; that is, cobalt is the “lightest” of the atoms as compared

Figure 2. (a) Nuclear density in unit cell of cobalt-containing vitamin B₁₂ obtained by Fourier synthesis method. The central maximum, which corresponds to the cobalt atom, is faint because of the small atomic scattering amplitude of cobalt. This makes possible a more precise determination of the positions of light atoms in the cell, for example, nitrogen, oxygen, and hydrogen, (b) Nuclear density in the peripheral CH₃ group. The nuclear density for the hydrogen atoms is indicated by a broken line because the atomic scattering amplitude of hydrogen is negative.

with the other atoms—nitrogen, carbon, oxygen, and even hydrogen. Because of this, more precise localization of all the atoms is possible. The nuclear density in the terminal methyl group (CH₃) is illustrated in Figure 2,b. The hydrogen atoms clearly stand out in the figure as minima. This situation is due to the negative value of b for protons.

There are some differences in the nature of the results obtained by X-ray and neutron diffraction analysis. In the former case, the position of the centen of gravity of the atomic electron cloud is determined from the experiment, while in the latter case, the center of mass of the centrode (centroid) of the thermal vibrations of the nucleus is determined. In some precision experiments, this leads to a difference in the interatomic distances obtained by X-ray and neutron diffraction methods. On the other hand, this difference can be used to study the distribution in regions of the electron density in molecules and crystals that correspond to such features as the covalent chemical bond (Figure 3) or an unshared electron pair.

Figure 3. Distribution of the electron density in the cyanuric acid molecule constructed by the difference method from combined X-ray and neutron diffraction analyses (difference Fourier synthesis). The maxima at the center of the C–0, C–N, and N–H bonds correspond to the electron density, which is responsible for the covalent bond. Half of the symmetric picture is given.

Neutron spectroscopy. The closeness of the values of the energy of thermal neutrons and the energy of thermal vibrations of atoms in crystals makes it possible to measure the latter with high precision in experiments on the inelastic scattering of neutrons. In this case, during the interaction, part of the energy of the neutron is transferred to the molecule or crystal, exciting specific types of vibrations; the inverse process of energy transfer from the crystal to the neutron is also possible. A distinction is made between inelastic coherent and incoherent scattering of neutrons. Coherent inelastic scattering of slow neutrons is determined by the dynamics of all the particles in the crystal and may be considered to be a collision of the neutron with the collective thermal vibrations of the lattice (phonons), in which the energy and momentum (or more accurately, the quasi-momentum) of the colliding particles are conserved. Experiments on the inelastic coherent scattering of neutrons by single crystals of the compound under study therefore yield complete information on phonons in the crystal—phonon dispersion curves—that cannot be obtained by other methods. The phonon dispersion curves (acoustical and optical branches) of a germanium crystal for two crystallographic directions are presented in Figure 4. The agreement of the experimental results with calculations made on the basis of a specific theoretical model indicates the validity of the

Figure 4. Dependence of frequency ν = ω/2π of phonon vibrations on the wave number q (phonon dispersion curves) in a germanium crystal for two directions - [111] (at left) and [100] (at right). Branches of the longitudinal (L) and transverse (T) optical (O) and acoustic (A) vibrations are given.

model and also makes possible the calculation of a number of parameters of the interatomic interaction forces.

In incoherent inelastic scattering, neutrons are scattered by individual atoms of the crystal, but, because of the strong coupling of the nuclei in the lattice, the other nuclei affect the scattering of slow neutrons, so that also in this case all the particles participate collectively in the scattering. Such scattering may therefore also be considered as a neutron-phonon collision in which the energy of the colliding particles is conserved but the momentum is not. Experiments on the inelastic incoherent scattering of slow neutrons by single crystals and by poly-crystalline specimens can be used to obtain the phonon spectrum of the crystal. By neutron spectroscopy it is possible to conduct studies over a broad range of wave vectors and to extend the studies to lower frequencies (approximately 20 cm^-1) than by other methods (chiefly optical). Moreover, in this case, the scattering is not limited by selection rules—all vibration modes are active in the neutron experiment. The large cross section for incoherent scattering of neutrons by protons makes hydrogen-containing compounds good objects for such studies. Some data, such as relaxation times and mobility, can also be obtained on the dynamics of liquids and amorphous solids.

Magnetic neutron diffraction analysis. The atoms of certain elements, such as the transition metals, rare-earth elements, and actinides, have a nonzero spin magnetic moment and/or a nonzero orbital magnetic moment. Below a certain critical temperature, the magnetic moments of these atoms in pure metals or in compounds are ordered, that is, an ordered atomic magnetic structure emerges (Figure 5). This significantly affects the properties of the magnetic substance. Magnetic neutron diffraction

Figure 5. Magnetic structure of antiferromagnet MnO. The shaded circles are manganese ions, and the solid circles are oxygen ions. The arrows indicate the directions of the magnetic moments.

analysis is virtually the only method of detecting and investigating the magnetic structure of metals. The presence of magnetic ordering is usually detected by the appearance of additional maxima owing to coherent magnetic scattering in neutron diffraction patterns; the intensity of these additional maxima is temperature dependent. We can determine the type of magnetic structure of the crystal and the magnitude of the magnetic moment of the atoms from the position and intensity of these maxima. Moreover, the absolute direction of the magnetic moments of the crystal can be determined in experiments with single crystals, and the distribution of spin density (that is, the density of that fraction of the electrons whose spins are not compensated within the atom itself) in the unit cell of the crystal can be constructed.

The spin density of the 3d electrons in the unit cell of iron is shown in Figure 6,a. The slight deviation from sphericity in the spin density distribution becomes fully evident if the spherically symmetric part of the distribution is subtracted from the total distribution (Figure 6,b). The shape of the spin-density maxima allows us to draw certain conclusions as to the structure of the electron shell of the iron atom in the crystal. In particular, the elongation of the maxima along the axes of a cube shows that of the two possible d sublevels of the iron atom, e_g and t_2g, which result from the removal of the degeneracy owing to the crystal field, the e_g sublevel is preferentially filled in this case. Figure 6,c shows the distribution of magnetization in the unit cell of iron obtained from special neutron measurements; this distribution is due to the partial polarization of the 4s electrons (as neutron diffraction measurements have shown, the 4s electrons, along with the 3d electrons, make a certain contribution to the magnetic properties of iron). The inelastic coherent magnetic scattering of neutrons makes it possible to study the dynamic state of magnetically ordered crystals, that is, the elementary excitations in such crystals (spin waves, or magnons).

With the neutron diffraction method it is possible to solve a broad range of problems that pertain to various questions of the structure of matter, such as the structure of biopolymers and amorphous bodies, the microstructure of special alloys, and phase transitions.