STUDY OF ALLOY ORDERING BY NEUTRON DIFFRACTION

Ordering is observed in solid substitutional solutions. Such a solid solution is obtained when, upon alloying two metals, the atoms of the dissolved metal replace the atoms of the solvent in the crystal lattice of the latter. The distribution of the atoms of both components of the solid solution may be random; if, for example,

![Fig. 1]

Disordered  Ordered

● Fe atoms
○ Co atoms
◉ 50% Fe atoms and 50% Co atoms

Fig. 1.

an alloy of composition \(AB\) is considered, then at each site of the crystal lattice an atom \(A\) or an atom \(B\) may occur with equal probability. If the probabilities are not equal, we are dealing with a certain ordering. Complete ordering occurs in the case when atoms of one kind are located at some sites, and atoms of the other kind at other sites (see Fig. 1). The ordered state is also called a superstructure.

The study of ordering may be carried out by the following method. Reflection of X-rays from atomic planes, for example from the \((100)\) planes of a simple cubic lattice (Fig. 2), occurs according to the Wulff–Bragg law:

\[ n\lambda = 2d \sin \vartheta . \tag{1} \]

This means that first-order reflection of a definite wavelength \(\lambda\) takes place at a definite angle of incidence \(\vartheta\) and at a definite interplanar spacing \(d\). Let us pass from a simple cubic lattice to a body-centered cubic lattice. This is equivalent to the fact that

that between the planes (100) we place a new plane (200), completely identical to the former in its reflecting properties, since the plane (200) is just as densely populated with atoms as the plane (100).

In this case condition (1) is violated \(\left(d_{200}=\frac{1}{2}d_{100}\right)\), and there will be no reflection at the former angle \(\vartheta\). The factor that takes into account the relative change in the intensity of reflection in going from a simple to a complex lattice is called the structure amplitude. Consequently, for the plane (100) of a space-centered cubic lattice the structure amplitude is equal to zero. It is also equal to zero for the planes (111), (210), (300), etc. Therefore, the space-centered cubic lattice gives reflections from planes with indices (110), (200), (211), (220), (310), etc.

Fig. 2

Exactly the same picture is obtained in the case of a disordered alloy, since in this case the reflecting ability of the planes (100) and (200) is completely identical. (Atoms \(A\) and atoms \(B\) are distributed uniformly over all planes.)

A completely different picture is observed when a superstructure is present in the alloy. In this case the plane (100) consists only of atoms \(A\) (or \(B\)), while the plane (200) consists of atoms \(B\) (or \(A\)), since the plane (200) contains atoms located at the center of the cubic cell. Now the reflection from the plane (100) will not disappear entirely, but will only be weakened, because the structure amplitude for the plane (100) is equal to the difference of the scattering amplitudes of atoms \(A\) and \(B\), and this difference is not equal to zero. The lines of the X-ray pattern that arise as a result of ordering are called superstructure lines.

Thus, the structure amplitude of the principal lines is:

\[ F \sim |f_A+f_B|, \tag{2} \]

and of the superstructure lines:

\[ F \sim |f_A-f_B|, \tag{3} \]

where \(f_A\) and \(f_B\) are the scattering amplitudes for atoms of type \(A\) and \(B\), respectively.

In the case of incomplete ordering the superstructure lines have lower intensity, with

\[ F \sim |r f_A-w f_B|, \tag{4} \]

where \(r\) and \(w\) are quantities characterizing the degree of ordering of the alloy \((r+w=1;\ 0 \le r,w \le 1)\).

The sensitivity of the X-ray method for studying ordering depends on the difference of the scattering amplitudes of the atoms entering into the alloy: the smaller this difference, the less intense the superstructure lines will be.

structures, the method is nevertheless less sensitive. Since the amplitude of scattering of X-rays is proportional to the atomic number \(Z\), the study of ordering in alloys whose components have close atomic numbers is always difficult.

This shortcoming is avoided by neutron-structural analysis, for for neutrons the scattering amplitude changes sharply and nonmonotonically with change of atomic number and is different for different isotopes. Moreover, for some elements the neutron scattering amplitudes are negative, which substantially broadens the possibilities of neutron-structural analysis (see, for example, \({}^{1}\)). Therefore a neutronographic study of the phenomenon of ordering in alloys makes it possible to obtain results in cases inaccessible to X-ray diffraction. Such an investigation was carried out\({}^{2}\), for example, for the alloys FeCo and Ni\(_3\)Mn.

Table I gives the scattering amplitudes of X-rays and neutrons for certain elements.

The table shows that the difference in scattering amplitudes for X-rays in the case of the FeCo and Ni\(_3\)Mn alloys is considerably smaller than for neutrons.

Table I

Table II gives the theoretically calculated relative intensities of the superstructure line (100) and the fundamental line (110) for wavelengths: X-rays \(0.7\ \text{\AA}\) and neutrons \(1.06\ \text{\AA}\), with complete ordering in the case of the FeCo alloy. From the table the impossibility of applying X-rays to the given case without the use of any special methods is evident, since it is impossible to detect a diffraction maximum 1390 times smaller than the principal one. However, the study of ordering can be carried out without difficulty with neutrons.

Table II

Relative intensities of the superstructure (100) and fundamental (110) lines for the FeCo alloy

The experiment confirmed the theory. In Fig. 3 are shown neutronograms of ordered and disordered specimens of the FeCo alloy. The disordered specimen was obtained by quenching the alloy at \(850^\circ\text{C}\). The ordered specimen was slowly cooled at \(750^\circ\text{C}\) for 100 hours. In the neutronogram of the ordered specimen, the superstructure lines (100), (111), and (210) are clearly visible; while in the neutronogram of the disordered specimen they are absent. The ratio of the intensities of the lines (100) and (110) is equal to \(1:5\), whereas the theoretical

\(^*\) The scattering amplitudes of X-rays are given for \(\sin\vartheta/\lambda = 0.2\).

calculation gave 1:6. The fairly good agreement indicates a high degree of ordering in the alloy under study.

The alloy Ni\(_3\)Mn is especially interesting because Ni and Mn have scattering amplitudes of different signs. Therefore the intensity of the superstructure lines for Ni\(_3\)Mn is comparable with the intensity of the fundamental lines, and in some cases may even exceed the latter.

Fig. 3.
(In the figure: vertical axis—Intensity; horizontal axis—\(\theta\). Upper curve: Ordered; lower curve: Not ordered. Reflections marked: (100), (110), (111), (200), (210), (211).)

In Fig. 4 are presented neutronograms of ordered and disordered specimens of the alloy Ni\(_3\)Mn. The superstructure lines have the indices (100), (110), (210), and (211). The difference in indices from the FeCo alloy is explained by the fact that the alloy Ni\(_3\)Mn possesses a face-centered

Fig. 4.
(In the figure: vertical axis—Intensity; horizontal axis—\(\theta\). Upper curve: Ordered; lower curve: Not ordered. Reflections marked: (100), (110), (111), (200), (210), (211), (220).)

lattice, in contrast to the body-centered lattice of the FeCo alloy.

The intensity of the superstructure lines is half the theoretically expected value. This indicates that the specimen obtained by slow cooling at \(550^\circ\)C for 100 hours is not completely ordered. The ordered structure of the alloy corresponds to a face-centered lattice with Mn atoms at the corners and Ni atoms at the centers of the faces.

As was mentioned, different isotopes of one element have different scattering amplitudes (see, for example, Table I). This creates

possibility of a simpler, easier, and more accurate study of certain alloys with a given isotope or with a combination of them (which, however, increases the background of incoherent scattering). An extremely curious fact, which has no place in other methods of structural analysis, is that the alloy \(\mathrm{Ni}^{60}\mathrm{Mn}\) will not give the principal lines on the neutron pattern, since the scattering amplitudes of \(\mathrm{Ni}^{60}\) and Mn are equal in magnitude and opposite in sign. The superstructure lines of this alloy will have ordinary intensity. \(\mathrm{Ni}^{60}\) is more preferable in the study of Ni–Fe alloys, whereas \(\mathrm{Ni}^{58}\), or ordinary Ni, is suitable for the study of Ni–Co alloys.

In a number of cases neutron diffraction is subject to the same defect as X-ray structural analysis: these are cases in which different nuclei have practically identical scattering amplitudes. For neutrons such coincidences are rare and, moreover, are not systematic (as for X-rays), but accidental in character. For example, for gold and copper, alloys of which are easily studied by means of X-rays, the neutron scattering amplitudes proved to be almost equal (see Table I), and no substantial difference between the neutron patterns of disordered and ordered \(\mathrm{Cu}_3\mathrm{Au}\) alloys is observed. The converse conclusion will also be correct: since there is no difference between the neutron patterns, the neutron scattering amplitudes for Cu and Au are equal. Thus the value of the neutron scattering amplitude for gold was first determined, and its magnitude was subsequently confirmed by experiments with specimens of the pure metal.

R. Ozerov

References Cited

1. R. P. Ozerov, UFN 33, 413 (1949).
2. C. G. Shull, S. Siegel, Phys. Rev. 75, 1008 (1949).