content_image/418766.jpg content_image/418765.jpg
structure in first-principles calculations. 

The correctness of our results may be supported by the following experimental evidence $^{1}$: firstly, the metastable ω phase was obtained at atmospheric pressure as a result of cooling of the β phase under a pressure of 6 GPa with subsequent unloading at room temperature. Secondly, between 2 . 2 and 4 . 8 GPa on cooling of the β phase there forms a two-phase mixture of a stable α and a metastable ω phase. And lastly, it was found that at atmospheric pressure the ω phase in the TiZr alloy, when heated above 698 K, transforms into an α phase $^{1}$. In Ref.3 the tem- perature of this transformation was defined as T = 623 K for P = 0 . 0001 GPa. This value differs by only 13 K from the temperature T ω → α P =0 = 610 K we have calculated for the ω → α transition. 

<img src='content_image/418766.jpg'>

The above evidence suggests, in our opinion, that in the TiZr alloy with ideal crystal lattice the phase dia- gram should look as it is depicted in Fig.4. Of course, it must be taken into account that the Debye-Gr¨ uneisen model, used for calculating the thermodynamic poten- tials, is a rather rough approximation and, obviously, cannot ensure good accuracy, especially at high temper- atures when the anharmonic effects become of consider- able importance. It should be noted that at room tem- perature the pressure calculated for the ω → β transition is nearly half as large as the experimental value. Because of low temperatures, we do not believe this discrepancy to be connected with the choice of the Debye model for describing the thermodynamic properties. It is rather due to the deviation of real alloys from the ideal peri- odic structures used in calculating the total energy in the ground state. 

## III. CALCULATION OF THE TENDENCY TOWARD DECOMPOSITION 

To estimate the tendency toward decomposition in ω − TiZr in the ground state, the total energy was cal- culated for four structures (see Fig.5). Structure I was represented by layers of pure titanium and pure zirco- nium alternating along the z axis and separated by in- termediate mixed Ti-Zr layers. In the two-layer structure 

II two layers of pure titanium alternated along the z axis with two layers of pure zirconium with no intermediate layer. For structure III were chosen six Ti layers alternat- ing with six Zr layers without intermediate layer. And lastly, in variant IV five Ti layers were separated by an intermediate layer from five Zr layers. 

The free energy was calculated by the scalar rel- ativistic full-potential linearized augmented-plane-wave (FPLAPW) method, using the WIEN2K package $^{7}$. In the first two variants the number of atoms per unit cell was six (3 Ti atoms and 3 Zr atoms). In variants III and IV the number of atoms in the cell amounted to 18 (9 atoms of each species). For variants I and II struc- tural optimization of the c/a ratio was performed, and the equilibrium atom positions were defined with the pro- cedure of minimizing the forces acting on atoms. 

<img src='content_image/418765.jpg'>

In Fig.6 the volume dependence of the lattice relax- ation energy change Δ E = E$_{relax}$ − E 0 is displayed for calculation variants I and II. Here E 0 is the total energy of the system with atomic arrangement corresponding to the ideal ω lattice; E$_{relax}$ is the system energy after min- imization of the forces acting on atoms for a given vol- ume. As seen from the figure, with decreasing volume the atoms become displaced from the positions correspond- ing to the ideal ω lattice, the displacement magnitude depending on the volume and the structure type. For the structure of type I, corresponding to the most uni- form distribution of Ti and Zr atoms at V < 15 ˚ A $^{3}$, there occurs a sharp decrease of Δ E due to significant atomic rearrangement. For the two-layer system (II) such a re- arrangement is not observed in the considered interval of volume change. In Fig.7, for V = 13 ˚ A 3 are depicted the (110) planes, the arrows indicating the direction of atomic displacements on relaxation for the lattices of type I (a) and II (b) (the z axis is pointing upwards). It is seen that in both cases the atomic displacements are directed only along the z axis. In the two-layer system (II) the atoms of titanium and zirconium are displaced in op- posite directions, whereas in system I the atomic chain 

