Equations of state are considered that correspond to a model of static concentration waves which describe order–disorder phase transitions of the substitution type in binary alloys. A phase diagram associated with a three-dimensional order parameter that describes ordering in Cu–Au alloys is presented as an example. It is established that the equations of state of the theory of concentration waves allow additional solutions not considered in existing theories of ordering based on the hypothesized predominant role of pair interactions. These solutions correspond to states that are partially ordered and stable over a wide area of the phase diagram.

The main stages of constructing a theory of high-order elastic moduli are considered. The first stage is developing a way to determine the dependence of the irreducible energies of interaction between atoms in each cluster. The second stage is finding a way of summing the irreducible potential energies of all clusters so that the result is equal to the total potential energy of the crystal, even though some clusters contain the same atoms.

A way of forming and depositing dosed droplets on the surface of a solid substrate with or without variable composition is proposed. Contact melting closely related to phase diagram is used to obtain such substrates and droplets of different concentrations. This relationship allows studies of the concentration and temperature dependences of surface characteristics under identical conditions, increasing the accuracy of their determination. It is shown that the microstructure of a substrate of variable composition affects the spreading of droplets over its surface.

Abstract The paper describes features of calculating the magnetization of solid solution based on statistical approach on the example of a one-dimensional crystal. Calculations are made in approximation of Weiss effective field. The result shows the error of the third Gilleo hypothesis, which is still widely used to forecast the magnetic properties of solid solutions.

The paper proposes a new method of transferable calculation of elastic constants of the second, third and fourth orders. The method is based on the assumption that the potential energy of bulk crystals may be presented as the decomposition into irreducible energies of two, three, and four atomic clusters. To justify this decomposition, we proved that the irreducible energy of atomic clusters obeys additional symmetry O(3)xN(CI), which depends on the number of atoms constituting clusters (No), and not depends on cluster structure. Additional symmetry allows defining the bases invariants, which depend on interatomic distances, and irreducible energies of clusters are depended on bases invariants only. In turn, these dependences made it possible to calculate the elastic modulus of complexes that exist in different crystal modifications as functions of the same phenomenological parameters of the model of interatomic interactions. Then we presented an example of the calculation of elastic constants BCC, HCP and FCC of the metal phases of Co polymorphs. The results confirm that the proposed method is transferable and makes it possible to calculate the elastic moduli of the second, third and fourth orders with the accuracy not worse than was achieved using the quantum chemistry models.

The three-sublattice ordering observed in alloys used for special purpose hard magnets is considered. Possible phase diagrams describing the ordering of alloys in systems of intermetallides with RCo2 structure are presented. It is shown that, according to the static concentration wave (SCW) approximation, the main area of the phase diagram is occupied by incongruently melting partially ordered alloys.

In modern studies, experimental methods for estimating nonlinear elastic moduli are often replaced with calculations of these quantities in mathematical simulation. However, different reliable models and experiments give different values of nonlinear elastic moduli. This work proposes a poßsible solution to the problem of which sets of elastic moduli should be used to predict the properties of substances. Two sets of the second-({cαβ,γδ}), third-({cαβ,γδ}), and fourth-order elastic moduli ({cαβ,γδ,μη,τρ}) of hexagonal Gd crystal proposed in the literature are considered as examples. The elastic moduli are defined as partial derivatives of the nonequilibrium Landau potential ΦL {uαβ{ with respect to components of the tensor of homogeneous deformation of the crystal (Ü). Necessary information about the nonequilibrium Landau potential as a function of {uαβ{ is given in the second section. An analytical way of deriving relationships between generally independent values of nonlinear elastic moduli caused by symmetry is proposed in Sections 3 and 4. The approach is based on using the integral rational basis of invariants (IRBI), which have the form of polynomials {uaß{. Aspects of the theory of phase transitions based on IRBI are discussed in Section 3. The set of polynomials of the second, third, and fourth order included in the list of basic invariants {J i (P)}, and the form of Landau potential Φ({J i (P)}, are clearly defined. The forms of the chosen dependences {J i (P)} and (ΦL{J i (P)}) are defined in Section 4 to compare the results from different works. Once the forms of Landau potential ΦL{J i (P)} and ΦL{uαβ} are defined, they are compared. The comparison results allow derivation of the nontrivial relationships between the components of the third-C III and fourth-rank elastic moduli tensors C IV due Gd hexagonal symmetry. Two different sets of calculated elastic moduli of Gd crystals, found in two different works, are given in Section 5. The criteria for selecting the most suitable set of numerical values of elastic moduli are described in Sections 6 and 7. One criterion is a comparison of load limits calculated from isothermal Gd elastic moduli and experimentally determined numerical values of the limits of stability of a certain Gd phase. In the last section, we show how the criterion based on comparing the phase stability limits allows us to dertermine which sets of third-rank elastic moduli should be used in, e.g., predicting Raman spectra.

Born’s criterion of crystal stability with respect to small variations of homogeneous deformations is formulated in a quasi-harmonic approximation. It is shown that the third-order Landau potential with respect to a tensor’s components of Lagrangian deformation is sufficient for predicting the critical pressure at which the cubic structure becomes unstable. The accuracy of prediction is no worse than that of the available experimental data.

The total potential energy of a crystal is presented as an expansion by irreducible interactions in clusters containing pairs of atomic triplets and quadruplets. The potential energy of the clusters in the adiabatic approximation is a function of vectors \(\vec r_{ik}\) connecting the centers of atoms in the clusters. Arguments (basic functions) affecting the potential energy of clusters are found using the model with allowance for the exchange symmetry of atoms and the irreducibility of the considered energies of doublet, triplet, and quadruplet atoms and interactions. This allows us to present arguments of the potential energy in the form of summed integral numbers (latticed sums) multiplied by a fixed value of the crystal unit cell parameter, and to set the numerical values of the potential energy arguments as a function of vectors \(\vec r_{ik}\). A potential corresponding to the thermodynamic additivity concept of crystal energy is selected as the model potential of pairwise interaction. Secondand third-order moduli of the rigidity of Co crystals with A1, A2, and HCP structures are calculated using this model of multiatom interactions.

The sharply nonmonotonic dependence of the number of structurally excited states of small clusters on energy n ( e i ) 0 of excited state ei is observed for the first time in a numerical experiment. The physical nature of observed features in dependence n ( e i ) 0 is discussed.

theory of the elastic moduli of crystals and amorphous substances is constructed that considers the internal symmetry of the irreducible energy of triatomic clusters. Allowing for the internal symmetry of clusters allows us to build three invariants by which the potential energy of any free triatomic cluster depends on the atomic coordinates. Direct calculation of the elastic modulus using an analytic form of the dependence of invariants on atomic coordinates shows that the adopted model leads to the following relationship between independent components of the third-rank elasticity tensor: C_xx, yy, zz - (C_xx, yz,yz + C_yy, xz, xz + C_zz, xy, xy) + 2C_xy, yz, zx = 0 This result is independent of the global symmetry of a substance and represents a generalization of the widely known Cauñhy discrepancies, which follow from models that consider only pair interactions. Fifteen laws of conservation that determine the movement of a point representing the atomic coordinates of cluster atoms in 9-dimensional configuration spaces of triatomic clusters are identified. The symmetry group of a triatomic cluster that determines these laws of conservation is identified.

A new criterion for separating the roles of anisotropic and isotropic interactions in the formation of second-, third-, and fourth-order elastic moduli is proposed. The applicability of this criterion is illustrated by considering an example of cubic crystals.

Elastic moduli of Co with A 1-type structure are calculated using models in which the total energy is presented in the form of the energy of clusters consisting of pairs, triplets, and quadruplets of identical atoms. The irreducible energies of the clusters are assumed to contain two terms: E ∝ −τ−6, corresponding to the mutual attraction of the particles, and E ∝ −τ−6, corresponding to their mutual repulsion. Here, τ stands for the half-edge of an elementary cell. All 27 types of models expected in accordance with symmetry theory are analyzed. Of these, the 13 that yielded elastic modulus values satisfying the structural stability criteria are selected. The potentials of the selected models are used to predict the rigidity moduli of Co films with an A2-type structure.

The physical content of an equation of state in the form of σ = σ(υ) is discussed as it applies to the theory of the elastic properties of crystals. Here, σ is pressure and υ is a change in the volume of the crystal under the effect of pressure. We assume that the energy of a deformed state is determined using an elastic modulus no higher than the third order. The forms of the analytical dependences of σ = σ(υ) for the eight most popular equations of state in the mechanics of finite deformations of a continuous medium then coincide, and with the same equation of the state predicted by the Mott model. As an example, differences between the values of the elastic moduli of Cu and Al, calculated using different equations of state for the mechanics of finite deformations of continuous media in the Mott model, are discussed.

The influence of substitution of Fe ions for manganese on the structure, phase transitions, magnetoresistance, Mn-55 NMR and Fe-57 Mossbauer spectra in the ceramic La0.6Sr0.3Mn1.1-xFexO3 (x=0-0.15) samples has been studied by X-ray diffraction, electron microscopy, magnetic, Mn-55 NMR and Fe-57 Mossbauer methods. The real rhombohedral perovskite structure R (3) over barc) is established to contain the different valence manganese ions (Mn3+ and Mn4+), anion and cation vacancies as well as nanostructural clusters with Mn2+ located in the A-sites. Temperature dependences of the a lattice parameter, a(T), demonstrate the anomalies near the Curie temperature, T-c. Wide asymmetric Mn-55 NMR spectra confirm the high frequency electron double exchange between Mn3+ and Mn4+ ions and irregularity of their surrounding by other ions and defects. According to the Mossbauer spectroscopy data Fe3+ ions similar to 80%) substitute for Mn3+ and partially Mn4+ in the B-positions. The rest of Fe3+ (Fe2+) ions and clusters with Mn2+ are located in the A-positions. The temperatures of metal-semiconductor and ferromagnet-paramagnet phase transitions are reduced with increasing x, and the magnetic irregularity increases due to the weakening high frequency Mn3+ <-> Mn4+ double exchange by Fe3+ ions. The amount of ferromagnetic phase is also reduced. The anomalous hysteresis is interpreted as a result of anisotropy of exchange interaction between the ferromagnetic matrix and antiferromagnetic cluster containing Mn-A(2+) ions. The phase diagram demonstrates the strong correlated interrelation among magnetic, transport and magnetoresistance properties. (C) 2014 Elsevier B.V. All rights reserved.