DETERMINATION OF POSSIBLE TYPES OF ORDERED STRUCTURES IN SPINEL SUBLATTICES

L.I. Leontiev*, D.G. Soldatov**, V.B. Fetisov**, K.Yu. Shunyaev*

*Institute of metallurgy, Ural’s Division of Russian Academy of Science, 101Amundsen Str., Ekaterinburg 620016, Russia,

**Urals State Economy University, 62 8 marta Str., Ekaterinburg 620219, Russia

It is often necessary to know possible types of ordered structures, which can appear when atoms of different components of a solid solution occupy a particular lattice or sublattice. This applies, in the first place, to interpretation of diffraction experimental data for identification of ordered phases. The knowledge of possible types of ordering is also required for calculation of the energy component of the free energy in models of the quasichemical type, specifically the clusters variation model. It was shown [1] that in the clusters variation method the internal energy of the solution may be expressed not as the energy of basis figures and subfigures, whose physical meaning is not quite clear, but as the energy of some set of fully ordered structures (superstructures) having a largest possible degree of long-range order. This set should be specified in the lattice at hand, first limiting the interaction radius in accordance with the maximum distance in the chosen basis figure.

I.Kanomori [2, 3] was the first to pose and solve the problem of determining possible types of ordered structures in systems with a limited interaction radius. The proposed approach was developed further in [4, 5]. Several more solutions of the problem were advanced, for example [6-7]. A method, which does not assume a priori the presence of the translation symmetry in the ordered phase, was proposed [8]. An analysis of ordering in alloys with most frequent lattices (BCC, FCC and HCP) showed that the obtained results agreed in the main with the data of other researchers. Moreover, the proposed approach yielded not only known solutions, but also unique results. It was found in particular that some lattices could form structures with a perfect order in parallel planes, but without a long-range order in the direction perpendicular to those planes, which were referred to as unidimensionally disordered structures. A situation may arise in layered structures when ordering takes place in mutually parallel chains and is absent in the direction normal to the chains. This situation is possible, for example, with ordering of defects in a high-temperature superconductor of the Y-Ba-Cu-O system.

Hunting for possible types of ordered structures is reduced to minimization of the internal energy of the system. If we consider a binary АcВ1-c solid solution and limit ourselves to a pair approximation, part of the internal energy, which depends on the mutual location of the atoms, will be written as

, / (1)

where рi is the probability that a pair of the i-th order is formed in the solution; Vi is the product of the ordering energy in the i-th coordination sphere and the corresponding coordination number.

If we restrict ourselves to the interaction of first neighbors, then

/ (2)

and the minimum energy is determined primarily by the sign of V1. If V1 > 0, then p1 = 0, i.e. the system tends to decompose into initial components and does not form ordered phases. If V1 < 0, р1 should have maximum possible values. To determine these values, it was proposed [8] to divide the lattice into sublattices so that nearest neighbors occupy different sublattices and the number of sublattices is as small as possible. In this case, replacement of atoms of one species by atoms of another species sequentially in the sublattices leads to formation of ordered structures, which are characterized by a maximum possible probability that nearest-neighbor pairs of unlike atoms appear.

The situation is much more complicated if interactions at distances longer than the shortest-range interactions are taken into account. Restricting ourselves to the interaction of second neighbors makes it necessary to minimize the expression

/ (3)

The situation is rendered by far more complex by an obvious fact that the probabilities p1 and p2 are independent. Therefore it was proposed [8] to consider three cases.

1. . In this case, the ordering interaction of first neighbors is the main interaction and one may reasonably assume that nearest neighbors will be ordered first, as was defined only for the interaction of first neighbors, and second neighbors will be ordered next. To determine the type of possible ordered structures, the sublattices, which were obtained with respect to the interaction of first neighbors, should be divided such that they do not contain second neighbors and their number is a minimum again.

2. . In this case, the interaction of second neighbors is significant and it is reasonable to alter the sequence. First the lattice of the solution is divided into sublattices, which are free of second neighbors, and then the sublattices are divided again into sublattices not containing first neighbors. This procedure means that the second term in (3) is minimized first and the first term is minimized second.

Fig. 1. Fragment of an octahedral sublattice of the spinel ()

Fig. 2. Two methods for division of an octa-sublattice of the spinel taking into account the interaction of first and second neighbors ()


Fig. 3. Ordered structures comprising D[A0.5B1.5]O4 in an octa-sublattice of the spinel. Black: species A atoms /
Fig. 4. Ordered structures comprising D[AB]O4 in an octa-sublattice of the spinel

Preparation of nanoamorphous metals, alloys and metal oxides The spinel octa-sublattice consists of tetrahedrons of nearest neighbors, which are weakly connected one to another (just one bond). Figure 1 presents a fragment of the spinel octa-sublattice showing two neighboring planes with the coordinate along the axis z. As can be seen, the sites in each of the planes are located in chains, which are not connected by a nearest bond. The chains in the neighboring planes are mutually perpendicular. If the interaction of first neighbors is considered, each of the chains breaks up into two sublattices, while the whole lattice is divided into four sublattices (see Fig. 1). This division is ambiguous, because renumbering of sites in a chain does not change the number of pairs of nearest neighbors in the sublattices. Ordered structures may be D[A0.5B1.5]O4 andD[AB]O4. Ordering in the chains is realized in directions of the (110) type if it is absent in the perpendicular direction. Such chains go through the plane in the direction of the z axis for the D[A0.5B1.5]O4 structure and are located in each plane for the D[AB]O4 solution.If second neighbors are considered, the ambiguity is eliminated and two variants of division to four sublattices (Fig. 2) are possible. Note that the second variant (Fig. 2b) is also realized with similar-order ordering energies of first and second neighbors. Thus, if the interaction of second neighbors in the octahedral sublattice of the spinel is taken into account, two ordered structures with the composition D[A0.5B1.5]O4 (Fig. 3) and two structures with the composition D[AB]O4 (Fig. 4) may appear.

If the interaction of first neighbors is considered, the tetrahedral sublattice is divided unambiguously into two sublattices, each representing simply an FCC lattice. This division corresponds to one ordered structure having the composition A0.5B0.5[D2]O4. If the interaction of second neighbors is considered, each of the sublattices breaks up into four sublattices with a unidimensional disorder. Both one disorder direction of the (100) type and two mutually perpendicular disorder directions (an individual direction for each of the FCC lattice components) are possible. The issue of ordering in the FCC lattice was discussed in detail elsewhere [8].

So, an analysis of some types of ordered structures in spinel sublattices shows that one may expect the appearance of structures with a maximum possible nearest order provided the long-range order is not present in all directions. This applies primarily to a substitution in the octahedral sublattice. Note that all the detected types of order may also relate to magnetic ordering.

The authors are thankful to the Russian Foundation for Basic Research for financial support (grant № 02-03-32877)

References

  1. Krashaninin V.A., Shunyaev K.Yu., Men A.N. // Izv. AN SSSR. Metally. 1990. No. 2. pp. 154-164.
  2. Kanamori I. // Progr. Theor. Phys. 1966. V. 35, No. 1. pp. 16-35.
  3. Kabiragi M., Kanaromi I. // Ibid. 1966. V.54, No. 1. pp. 30-44.
  4. Dyck D. van, Ridden K. de, Amelinckx S. // Phys. Stat. Sо1. 1980. V.59A. pp.513-530.
  5. Ridder K. de, Dyck D. van, Amelinckx S. // Ibid. 1980. V. 61A. pp. 231-250.
  6. Larikov L.N., Geichenko V.V., Falchenko V.M. Diffuse Processes in Ordered Alloys. Kiev, Naukova Dumka, 1975. 112 p.
  7. Men B. A., Katsnelson M.L. // Phys. Stat. Sol. 1985 V. 87А. pp. 93-102.
  8. Shunyaev K.Yu. Varshavsky M.T., Men A.N. // Izv. AN SSSR. Metally. 1986, No. 4. pp. 96-99.
  9. Krupichka S. Physics of Ferrites and Akin Magnetic Oxides. Moscow, Mir, 1976. V. 1. 353 p.

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