total number of AI3Mg 2 microparticles was N = 1.96.108 cm -~, and the mean radius was r = 5.4"10 -~ cm. We must select a similar theoretical distribution.

Table i give values of the functionals S l and $2, which were obtained from system (9) by substituting Vk, rk, and the moments of the experimental histogram at rg = 12.25.10 -4 cm. The can simultaneously become zero at 0 < e < 0.05 and ~ m 0.02. Only in this case as ~ in- creases do they pass from negative values (at ~ = 0) through zero to positive values (at

= 0.05). Successively narrowing the range of = and e, we finally arrive at their reliable values. The functionals S l and S 2 can also be minimized by the method described in [5].

At nonzero ~ and E, the dissolution and growth of AI3Mg 2 microparticles as the alloy is heated depend to an extent on the individual properties of the microparticies at the phase boundary: the degree of equilibrium and nonequilibrium faceting and the rate of reaction with magnesium and aluminum atoms. The nature of the variation of the values of these para- meters as a function of the duration of the isothermal heating of the alloy or as a function of the temperature can give an idea of the kinetic features of the processes at the phase boundary. Such information can be obtained after analysis of the corresponding experimental histograms by the method proposed here.

LITERATURE CITED

I. V.I. Psarev, Izv. Vyssh. Uchebn. Zavad., Fiz., No. 6, 73 (1984). 2. V.I. Psarev, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 12, 53 (1990). 3. L.S. Schulman and P. E. Selden, J. Stat. Phys., 19, No. 3, 293 (1978). 4. V.I. Psarev and A. F. Kulikov, in: Physical Methods of Solid State Research [in Russian]~

No. l, 131, UPI (Ural Polytechnic Institute), Sverdlovsk (1975). 5. V.I. Psarev and A. F. Kulikov, Zavod. Lab., No. 3, 303 (1975). 6. M.I. Kendall and A. Stuart, The Advanced Theory of Statistics, Vol. i, The Distribution

Theory [Russian translation], Nauka, Moscow (1966). 7. C. Wagner, Z. Elektrochim., 65, No. 7/8, 581 (1961). 8. Yu. I. Makovichuk and V. !. Psarev, in: Some Problems of the Physical Kinetics of Solids

[in Russian], No. 2, Chuvash. Univ., Chebokarsy (1976), pp. 35-55.

CRYSTAL-CHEMICAL FEATURES OF PHASES WITH THE AI5 STRUCTURE

N. M. Kormin, N. O. Solonitsyna, and ~. V. Kozlov

UDC 548.314.736

Since Hume-Rothery factors are important in the stability of bounded and unbounded solid solutions and many ordered and intermediate phases, an attempt has been made to study the effect of the size factor and the electron density on the range of binary phases with the AI5 structure. It has been found that a • deviation of the atomic radii is predominant and the electron concentration is limited, its val- ue depending on the electron configurations of the constituent components.

Phases of the AI5 type occupy an important place in a number of intermetallic compounds with stoichiometry AB 3. These compounds are formed only on the basis of elements of iso- electronic subgroups Ti, V, and Cr. Component A is an s, p, or d metal. The unit cell of the AI5 structure, shown in Fig. i, has space group Pm3n.

Most of these compounds have narrow regions of homogeneity. For vanadium phases the the regions of homogeneity attain a considerable width (up to 10-12 at. %). In many systems the actual composition of Al5-type phases is observed to deviate from its ideal stoichiometric composition, toward the B component as a rule.

Phases with AI5 structure are formed by peritectic or peritectoid reactions or, more rarely, the phases melt congruently or are precipitated from the bcc solid solution during ordering. These compounds are usually stable over a wide temperature range. Individual

Tomsk Civil Engineering Institute. Translated from Izvestiya Vysshikh Uchebnykh Zave- denii, Fizika, No. 12, pp. 85-88, December, 1991. Original article submitted June 13, 1991.

0038-5697/91/3412-1113512.50 �9 1992 Plenum Publishing Corporation 1113

Fig. i. Unit cell of the

of species A; o) atoms of

~ / / O 6 -L~ ,, / species B.

compounds of this type are stable only at high temperatures or are metastable phases in gen- eral [i]. A high degree of long-range order is noted in most phases [2].

The formation of primary continuous and bounded solid solutions in the phase diagram is due to different crystal-chemical factors: atomic, size, electronic, and electrochemical. For solid solutions the stability analysis is associated with the names of Hume-Rothery (size factor) and Darken and Gurri (joint role of electronegativity and size factor) [3]. The empirical Hume-Rothery rules and the technique of Darken-Gurri diagrams were fruitful for predicting the regions of the stability of clearly ordered phases [4, 5]. Recently correlations have been detected between the empirical relations and the electronic structure of alloys [6-8].

Since ordered phases are formed mainly from a disordered bcc solution, the Hume-Rothery criteria (the ratio of the atomic radii of the components is no more than 1.15) also place limitations on the size factor in ordered phases. In close-packed structures, genetically derived from fcc and hcp solid solutions, the size factor of the phases formed is less than 15% in most cases. The size factor is observed to vary over a wider range (up to 40%) in structures that are not close-packed (formed on the basis of a bcc lattice). The applica- tion of the technique of Darken-Gurri diagrams to ordered phases showed that those phases within the "ellipse of favorableness" become disordered in the solid state and the phase not within that ellipse as a rule as stable up to melting. Such crystal-chemical analysis was not carried out for compounds with the AI5 structure.

THE SIZE FACTOR

The sizes of atoms in alloys differ from that of the atoms of the pure components. The rule of additivity in relation to atomic size (Vegard law) as a rule is not satisfied even in solid solutions [3]. It is natural to expect the additivity rule not to be satisfied in the AI5 structure, where a different interatomic distance is observed even in the nearest neighborhood of the atoms. Atoms of species A have 12 atoms of species B at a distance dAB = a/2s in their nearest neighborhood while atoms of species B have 14 neighbors at a dis- tance of a/2 to a/2~i\5. Nevitt [9] suggests that a necessary condition for formation of a type AI5 phase is that the atomic radii of the components differ by no more than 15% for a coordination number of 12. Palatnik and Fal'ko [i0] consider the covalent radius to be more applicable for a number of phases. Comparison of the covalent radii with the apparent atomic size gave a more satisfactory agreement. Further on we show that the concept of the size factor is justified.

The dependences of the number of phases formed on the size factor 6 = Ii - rA/rBI and on the ratio of atomic radii rA/rB, where r A and r B are the radii of the pure components (after Titumi et al.) for a coordination number of 12 [3] are shown for type AI5 compounds in Fig. 2 as histograms with a 0.05 step.

The distribution of the phases according to the size factor (Fig. 2a) is polynomial, with a smeared principal maximum in the interval 6 = 0.0-0.I0. A principal maximum is formed by 80% of the compounds and the alloying components in them are d elements of the platinum se- ries and p elements with a relatively small atomic radius (AI, Ga, Ge). At 6 > 0.i0 compounds are formed on the basis of V and Nb with p elements (Pb, Sn, Sb) and with an s element (Be). A limitation on the size factor thus clearly exists for compounds which are of the AI5 type and resemble solid solutions and ordered phases.

The distribution of the phases according to the ratio of atomic radii (Fig. 2b) is also polynomial with two principal maxima on either side of the value rA/r B = 1.00. The first maximum lies in the interval rA/r B = 0.9-1.0 and is formed by compounds in which the A com- ponent is a d metal from the end of the long periods with a filled or nearly filled d shell

1114

2~

f6

I

0,05 !I

tZ

f6

8

F-1 I 0,25 ~ 0,8 (2 zA/z~

I I

Fig. 2. Distribution of phases with the AI5 structure: a) according to the size factor; b) according to the ratio of atomic radii.

zZ b

6O

a t6 ~ +0

$ 20

$ # $ 6 e/A, electrons/atom 2 zt 6 8 8/U, electrons/atom

Fig. 3. Electron-concentration distribution of phases: a) with the AI5 structure; b) with the composition AB s. The horizontal shading denotes ordered phases and the vertical shading, AI5.

(platinum and the precious metals). The second maximum at rA/r B = 1.0-i.I and a separate maximum at rA/r B > 1.2 is formed by compounds based on V and Nb, with a p element as the A component. The value rA/r B = 1.0 thus divides the known compounds with the AI5 structure into two groups and is critical. The compound with a single s metal (Be) has the smallest value rA/r B = 0.81. The AI5 structure was not detected at rA/r B > 1.3 and rA/r B < 0.81. An increase in the ratio of radii of the pure components is accompanied by a simultaneous growth of the electron concentration, which is determined by the number of valence electrons.

THE ELECTRON CONCENTRATION

Let us consider the region of stability of the AI5 structure with respect to the electron concentration, which is taken to mean the total number of s + p + d electrons of the pure com- ponents above the filled p6 shells, per atom in the unit cell. The distribution of phases with the AI5 structure according to electron concentration is shown in Fig. 3a as a histograms with a 0.5 step. The general limitation on the electron concentration is e/a = 3-7 electrons/ atom. The distribution is bimodal in which the first maximum is in the interval e/a = 4.0-5.6 electrons/atom and is formed by compounds in which p elements are the A component. A second, smeared maximum is formed by compounds in which d metals (e/a = 5.5-7 electrons/atom) are the A component. The interval e/a = 3-4 electrons/atom is characteristic of compounds with precious and polyvalent metals.

The electron concentration thus divides all the compounds into three groups (indicated by different shading).

Primarily ordered phases and compounds with the AI5 structure form in metal systems with the composition AB3, regardless of the type of initial lattice [4]. Comparison of the elec- tron-concentration distribution of ordered phases and structures of type AI5 (Fig. 3b) shows that the region of stability of AI5 compounds lies in the region of destabilization of ordered phases. Analysis of Fig. 3b shows that ordered phases with the superstructure LI2, D022 , DOl9 et al., and phases with the AI5 structure complement each other as to electron concentration. The latter indicates that the AI5 structure can evidently be classified among ordered phases.

Conclusions. In summary, our analysis of the effect of the atomic size factor on the region of stability of compounds with the AI5 structure makes it possible to extend the

1115

Hume-Rothery criterion to them and to assign compounds with the given structure to the class of ordered phases in regard to electron structure.

LITERATURE CITED

i.

2.

3.

4.

5.

6.

7.

8.

9.

I0.

E. M. Savitskii, Yu. V. Efimov, N. D. Kozlova, et al., Superconducting Compounds of Transition Metals [in Russian], Nauka, Moscow (1976). S. V. Vonsovskii, Yu. A. Izyumov, and E. Z. Kurmaev, Superconductivity of Transition Metals and Their Alloys and Compounds [in Russian], Nauka, Moscow (1977). W. Pearson, Crystal Chemistry and Physics of Metals and Alloys, Wiley-lnterscience (1972). N M. Kormin, Author's Abstract of Candidate's Dissertation, Tomsk. Gos. Univ., Tomsk (1980). N M. Matveeva and ~. V. Kozlov, Ordered Phases in Metal Systems [in Russian], Nauka, Moscow (1989). A P. Blandin, Stability of Phases in Metals and Alloys [Russian translation], Mir, Mos- cow (1970), pp. 47-57. V Heine, M. Cohen, and D. Weir, Pseudopotential Theory [Russian translation], Mir, Moscow (1973). V E. Panin, Yu. A. Khon, I. I. Naumov, et al., Theory of Phases in Alloys [in Russian], Nauka, Novosibirsk (1984). M V. Nevitt, Intermetallic Compounds [Russian translation], Metallurgiya, Moscow (1970), pp. 162-177. L S. Palatnik and I. I. Fal'ko, Dokl. Akad. Nauk SSSR, 217, No. 3, 573 (1974).

ISING-NAKANO MODEL FOR AN AMORPHOUS ALLOY

A. I. Olemskoi and Yu. V. Sherstennikov UDC 536.42

A lattice model, taking into account the possibility of an atom being pinned at a site upon vitrification, has been developed for binary alloys. The Hamiltonian found for the problem is used in the average-field approximation to obtain equa- tions for the parameters of atom pinning and long-range compositional order. The conditions under which mixing of the components promotes vitrification are analyzed.

A consistent microscopic theory that would represent the diverse aspects of the behavior of metallic glass in a unified manner is lacking at present. This is because the metallic glass state is neither thermodynamically stable or metastable but is a set of successively changing labile liquid states frozen into a limited region of phase space [i]. Naturally, the theory representing vitrification of a glass should be kinetic in essence [2]. If the goal set is to describe the conditions for the existence of the metallic glass phase outside the dependence on processes of the structural relaxation type, various modifications of the thermodynamic approach may be expected to be useful. One such approach was developed in our earlier work [3] by analogy with the theory of spin glass. Another [4] is based on the Ising- Nakano model [5], within the framework of which a pseudospin variable describes the excited (liquid-like) and pinned state of an atom at a site.

In this work we provide further details of the Ising-Nakano model as applied to solu- tions. The practical need to develop such a model stems from the well-known circumstance that the metallic glass state is realized in alloys and not in pure metals, as a rule [i]. We note that the proposed model describes the relation between the ordering and stratification processes and melting as well as the effect of the compositional disorder in the alternation of atoms of different species on the topological disorder in a metallic glass [3].

We confine the discussion to a binary A-B solid solution with pairwise interaction. Introducing the occupation numbers

Institute of Metal Physics, Academy of Sciences of the Ukrainian SSR. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 89-93, December, 1991. Original article submitted July 2, 1991.

1116 0038-5697/91/3412-1116512.50 �9 1992 Plenum Publishing Corporation