- Email: [email protected]
The calculations of cell dimensions of stoichiometric phases and binary and ternary solid solutions with the A15 structure
Gv2z-3697/82/c6o54745$03.al/0 Pugamon Press Ltd.
THE CALCULATIONS OF CELL DIMENSIONS OF STOICHIOMETRIC PHASES AND BINARY AND TERNARY SOLID SOLUTIONS WITH THE Al5 STRUCTURE W. B. PEARSON Departments of Physics and of Chemistry, University of Waterloo, Waterloo, Ontario, Canada NZL 3GI (Received I June 1981;accepted in revised form 7 October 1981) Abstrati-Equations that successfully reproduce the unit cell dimensions of stoichiometric (i) transition metaltransition metal (T-T) and (ii) transition metal-non-transition metal (T-B) phases with the R-Wolfram or Al5 structure in terms of the diameters of the component atoms for CN 12 [J. Less-Common Metals, 81, 161(Ml)], are hear applied to binary and ternary solid solutions with the Al5 structure, to discover how well and under what conditions they reproduce the observed lattice parameter variation. That in the main, they perform satisfactorily gives further credence to the model on which they are based, which shows how the various interatomic contacts within the interpenetrating icosahedra and CN 14 polyhedra of the structure, combine together in controlling the cell dimensions of the phases. The mode1also indicates that it is the contacts between the transition metal atoms that are paramount in controlling a. This fact leads to a specific prediction for the lattice parameter variation in transition-metal-rich solid solutions of T-B phases, which is essentially substantiated for phases of niobium. 1.
lNTRoDucTlm
It has been shown[l] that the unit cell dimensions of intermetallic binary phases, A,B,, with what we shall call “ordinary” crystal structures are controlled by a single type of atomic array emanating from a given atom, e.g. A-A-A, A-B-A, A-B-B-A, etc. However, the problem is more complex in phases with tetrahedrally closepacked structures which, if regular, are composed of interpenetrating polyhedra of coordination number (CN) 12, 14, 15, or 16, which have triangulated surfaces. The /3-Wolfram or Al5 structure, AsB, is tetrahedrally close packed, being composed of interpenetrating icosahedra and CN 14 polyhedra. Owing to the interpenetration of the polyhedra it is improbable that any single array of atoms can act alone to control the cell dimension, a. For example the A atom at the centre of the CN 14 polyhedron has two close A neighbours at a/2, 4 close B neighbours at a%‘5/4, and eight more remote A neighbours at ad3/2v2. If these interatomic contacts were separately to control a the expected dependences of a on DA would be respectively 2.0 D.+,, 0.89D~ and 1.63DA, whereas the observed dependence is 1.4 DA for phases with the A15 structure. Thus the problem is not trivial. The Shoemakers[2] tackled it for tetrahedrally close-packed phases generally by recognizing the nonsphericity of the atomic sites at the centres of the polyhedra and adopting different sets of radii for atoms at the centres of the CN 12, 14, 15 and 16 polyhedra, and for ligands to atoms with surface coordination number (SCN) five and with SCN six. Geller[3] and Pauling[41 and their followers reproduced the cell dimensions of phases with the Al5 structure on the basis of the A-B aV5/4 = and ad5/4 = (RA+ Ru)[3] distances: 1/2(3R, t Re)[4]; in the latter the radii are weighted according to the stoichiometric composition. In both treatments, sets of self-consistent atomic radii are assumed which best reproduce the observed cell dimensions.
A more complex method of deriving radii and calculating the cell dimensions of phases with the Al5 structure has been devised by Tarutani and Kudo[5]. There are two distinct groups of phases with the Al5 structure: T-T phases in which transition metal atoms occupy both site-sets in the structure (Tc~,~T& and T-B phases in which non-transition metals occupy the B sites. Their dimensional behaviours differ as first noted by Johnson and Douglass 161. Recently we have been able to deal with the problem of several different interatomic contacts within the same coordination polyhedron acting together to control the cell dimension, obtaining equations that reproduce the cell dimensions of the stoichiometric T-T and T-B phases effectively from standard CN 12 atomic diameters obtained from the elemental structures[7]. Since elemental diameters are used rather than ad hoc diameters as in[2-4], the resulting model provides a real explanation of how the cell dimensions are controlled in such phases. From a base equation that expresses the differences between the calculated interatomic distances and the appropriate diameter sums with relative weighting according to the multiplicity of the contacts within the polyhedra, non-linear equations for the T-B and T-T phases were obtained for a in terms of DA and l)B, the atomic diameters for CN 12[7]. The base equation also included an effectiveness factor, 0 I a 5 1, expressing the effectiveness of the eight remote A-A contacts in controlling a. The base equation as now correctly written[8], also includes weighting for the relative site-set multiplicity, although this omission did not introduce errors in the equations for a of phases with the Al5 structure, since no contacts ‘occur between atoms on B sites at the stoichiometric composition. Furthermore, to increase its generality an effectiveness factor, 05 a 5 1, etc., is attached to each of the expressions for interatomic con547
W. B. PEARSON
548
tacts occurring in the equation. Thus the base equation for phases with the A15 structure is 54k = 12a(aq5/4-(l/2)0,-(1/2)DB)+ CN 12 Polyhedron 3x2/?(a/2-D,)+ 3 x 4y(aV5/4- (l/2)0, - W2PE) + 3 x 8S(a~3/2~2 - DA) CN 14 Polyhedron (1) where k in an adjustable parameter. Experience from the Al5 and CuzMg structures shows[8] that (i) if all contacts in the structure are closer than the appropriate radius sums, the effectiveness factors a, p,. . are independent of DA and Dg; k is a constant and the equation relating a, DA and Dn is linear, (ii) if one of the contacts in the structure is at a greater distance than the appropriate radius sum, the value of the effectiveness factor concerned, may change from 0 to 1 as a linear function of DA or Dn as appropriate, and k may also be a linear function of DA or &, in which case the equation relating a, DA and Da is non-linear, and finally (iii) if the coupling between the contacts in the polyhedra breaks down so that they no longer jointly control a, k may be a function of a. The 54 contacts involved in the weighting in eqn (I) are correct for the stoichiometric formula A,B. Thus the A-B contacts that occur in both CN 12 and CN 14 polyhedra occur twice in the equation. Through the effectiveness factors, this allows account to be taken of the fact that a particular contact may have no influence on a in one polyhedron, but the same contact may influence a in the other polyhedron, owing to its joint action with the other contacts in the polyhedron. Thus for T-B phases we find OL= 0, /3 = y = 1, whereas for T-T phases a = j3 = y = 1 and for both, S varies linearly with DA between the limits where its value changes from oto 1. In[7] the dependence of the effectiveness factor (here S in eqn 1) on DA was determined from the slopes of lines for phases with various A components on plots of a against Da, thus giving equations for a in terms of DA and I)B for the T-B and T-T phases, the adjustable parameter being determined from comparison of calculated and observed a values. The equations obtained were: for B-T phases a = IO.1429Ds- 1.7231DA+O.8971 D:-0.6034) /{OS493DA-- 1.1334) (2) valid between DA= 2.484& below which 6 = 0 and DA = 3.121 A, above which 6 = 1. for T-T phases a = {0.2222De - 1.4782DA t 0.8444 Dfg - 0.6833}/{0.5172DA - 0.%72}- S
(3)
valid between DA= 2.458 A, below which 6 = 0 and DA = 2.984 A, above which 8 = 1. In eqn (3) S represents an apparent valency effect with values of 0 for Cr, 0.025 %, for MO, 0.050 8, for Group V and 0.150 A for Group IV A components. The significance of the valency
effect is discussed in the Appendix. With elemental atomic diameters of [9], eqn (2) reproduces the observed cell dimensions of the 32 T-B phases to which it applies with a mean error of (0.012)A, or )0.008)A neglecting very divergent results for five phases. Similarly eqn (3) reproduces the observed cell dimensions of the 29 phases to which it applies with a mean error of 10.006(A. The comparison is made with cell dimensions adjusted to stoichiometry by Waterstrat[lO], and a diameter of 3.246 A for Sn” is used[7]. These equations can be applied explicitly to binary and ternary solid solutions and also to disorder. Because of the occurrence of the apparent valency effect, the application of these equations to solid solutions is not just an extension of Vegard’s law [11] applied to the site-sets on which substitution occurs. This paper therefore seeks to establish the extent to which, and under what conditions, the equations may reproduce the lattice parameter variation of solid solutions with the A15 structure, thus providing a further test of the validity of the model on which they are based. Furthermore, it investigates a significant prediction regarding the variation of the cell dimensions in transition-metal-rich solid solutions of T-B phases, that arises from the model. 2. CALCULATIONOFLATTICEP-
EBVARIATIONIN BINARY ANDTERNARYSOLIDSOLUTIONS
In applying these equations to binary or ternary solid solutions, the exact number of contacts of each type formed by the component atoms can be explicitly entered in a base equation similar to (1) and equations for a in terms of DA, Dg (DC) and concentration developed, or more simply, the average diameters for atoms on each site-set calculated from the composition can be entered directly in the final eqns (2) and (3). Since the simpler procedure appears to give the same numerical results, we adopt it here. Most of the experimental data available in the literature has not been confirmed by compositional analysis of the samples, and, for example, Fig. 10 of[l2] illustrates the divergent results that may be obtained from several different investigations on the same system, particularly for T-B alloys. In contrast, the inset in our Fig. 1 shows the agreement that may be obtained for the same system (Ga-V) for results confirmed by compositional analysis. In Figs. 1 and 2 we have plotted only data for alloys whose compositions were determined by analysis, except for some data on refractory transitionmetal alloys (T-T phases) that appear to be reliable. We use these data to examine how well eqns (2) and (3) reproduce the observed lattice spacing variation, but the case of replacement of B atoms by T atoms in T-rich T-B phases is considered separately in section 3. In A-rich T-T alloys we would expect that the size of the valency effect would decrease pro rata of substitution as the T(A) atoms replace the TcB) atoms which are the source of the electrons for the valency effect. Alternatively the size of the valency effect might remain unchanged over the first few atomic percent of substitution, although ultimately at hypothetical complete substitution (T(A)~T~A)),it must be zero. Secondly in
The calculations of ceil dimensions of stoichiometric phases and binary and ternary solid solutions 0t
20
v.
549
B
25
30 1
06)
Id,
5.22
4.85 F
5.20 Go’+ 5.18 480 5.16 5.14 5.12 I
475 P1-Cr,
470
465 I5
20
25
30
35
40
01% B
0
15
30 at
45
60
75
% Cr
F&l. Lattice parameter variation with at.% B for T-T binary solid solutions: Nb-Au[l7], Nb-Pt[l6], NMr[l8], Nt+Os[28], Mo-Ir[l9], V-Au[20], V-Pt[21], V-1$29], V-Rh[30], Cr-pt[31] and Cr-Ir[32]. Insets: Similar data for T-B binary solid solutions: V-Ga 01121, q[i3], 0[14], A[151 and Nb-SnL231.Symbols indicate measured lattice parameters. Filled symbols indicate measurements for two-phase alloys. Full lines indicate calculated lattice parameters (see text). Coarse broken lines for V-Au and V-Pt alloys indicate lattice parameters calculated without pro rata reduction in the size of the valency effect.
Fig. 2. Lattice parameter variation with at.% T for (VI-,T,)&a ternary solid solutions, where T = Ti, Cr, Mn, Fe or Co[14,22]. 0 data for compositionally analysed alloys, 13 data for alloys not analysed. Full lines represent variation of lattice parameters calculated without valency effect. Coarse broken lines represent calculations of o with introduction of a valency effect pro rata of substitution by Mn, Fe and Co.
B-rich T-T alloys as the T(s) component replaces TcI\j on the A sites, we assume in calculations that the correction for the valency effect decreases pro rata of substitution. We also note that unless the atomic diameters differ by very little, the slope of a with composition is not in general expected to be the same on the A-rich side of Aa, as on the B-rich side, since substitution occurs on different crystallographic sites, and for the substituted site, the rate of change of the average atomic diameter with composition is three times greater on the A-rich than the B-rich side of AaB. Figure 1 that gives data for binary solid solutions, shows that on the B-rich side of A,B eqn (2) well reproduces the variation of a with composition for V&a[l2,13,14,15] which is the only example of a T-B phase. With the exception of Nb#t[l6], eqn (3) also appears to reproduce the variation of a with composition satisfactorily for Tea,-rich T-T phases on the basis that the correction for the valency effect is reduced pro rata of substitution. In T(A)-rich T-T phases the rate of change of a with composition is reasonably reproduced by eqn (3) for Nb-Au[l7], NLPt[M], Nb-Ir[l8] and Mo-Ir [ 191alloys on the basis of a pro rata reduction of
the size of the valency effect with composition; however, for the V-Au alloys[20] over the 5 at.% range of solid solution, it appears that there is no decrease in the size of the valency effect. A similar situation appears to obtain for V-Pt alloys[21] and in the T-T alloys of Cr the valency effect is zero. The available data for ternary solid solutions of T-B or T-T phases of the type Ax(B,B’), (A,A’)3B are not particularly interesting, since generally there is little departure from a linear variation of the lattice spacings between those of the binary end components, and the effectiveness of eqns (2) or (3) in reproducing the observed lattice spacings depends on how well they reproduce those of the binary phases concerned. The one exception is the data of Girgis et al. [ 14,221 for the substitution of vanadium by other transition metals in VJGa, since substituents Ti and Cr are components that occupy the A sites in T-T phases, whereas Mn, Fe and Co are regarded as components occupying B sites in T-T phases. We therefore expect Mn, Fe and Co to create a valency effect pro rata of substitution in VsGa, regardless of whether they occupy the A or B sites, whereas Ti and Cr should not.
W. B. PEARSON
550
Figure 2 which shows data for homogeneous alloys whose composition was determined by analysis [ 14,221, bears out these expectations. Since neutron diffraction results of Girgis and Fischer reported in [14,22] indicate that the substituting transition metal atoms are located mainly on the A sites, this location is assumed in the calculations. It is seen (full line) that the variation of a with composition for (V, Cr)aGa and (V, I&Ga alloys is well reproduced by eqn (2), using averaged diameters for the A site and assuming stoichiometric AsB composition for the solid solutions. Generally the analysed compositions are not exactly stoichiometric, being either Aor B-rich. The a values have also been calculated for the exact analysed compositions, but generally they depart little from the lines obtained on the assumption of stoichiometric composition (A, A’)3Ga. The a values of the two most Ti-rich (V, T&Ga alloys lie below the calculated line. The data are for as-cast alloys; data for these alloys annealed at 800°C indicate that they are no longer homogeneous, and the a values decrease further[22]. Therefore the a values of these as-cast alloys appear to be too low because of partial decomposition on cooling. When the (V, MnfrtGa, (V, Fe)3Ga and (V, Co)sGa alloys are considered, it is seen that the a values calculated from eqn (2) (full lines in Fig. 2) are greater than the observed values as expected, because the substituted atoms now contribute electrons to vanadium creating a valency effect. When the size of the valency effect is calculated to increase pro rata with the increasing substitution of V by Mn, Fe or Co up to 25 at.% and the correction for it to decrease pro rata with the decreasing concentration of V in the alloys (eg. -0.050 [x/25] I(75 -x)/75] A for V7+Xo,Ga&, reasonably good agreement of observed and calculated a values is obtained, except for (V, MnbGa alloys in the range from 20 to 25 at.% Mn, as shown by the broken lines in Fig. 2.
3. LATTICE PAUAMhT BK VARIATION IN ERICH SOtID ADDS OF T-B PHASES
Equation (2) for T-B phases was obtained on the basis that the 14 contacts to the transition metal atom at the centre of the CN 14 polyhedron together control a, whereas in eqn (3) for T-T phases all contacts in both CN 12 and 14 polyhedra were involved. Both of these situations maximise the influence of contacts between the transition metal atoms in controlling u. Recognition of this domin~ce of contacts between the transition metal atoms in controlling a, leads to the prediction that eqn (2) should not correctly give the variation of LJwith composition for T-rich T-B phases. As substitution of the transition metal for the B component proceeds to the ultimate T,T, changeover must occur from the behaviour given by eqn (2) to that given by eqn (3) where all contacts in both polyhedra together control a. The difference ofl_Tr- (la-a) calculated with eqns (3) and (2) for complete substitution of A on B sites is found to be 0.071 A for A=Cr and to increase linearly with Da to 0.080 8, for A=Nb, regardless of the size of the valency effect, which is taken as zero at 100%A in eqn (3). Thus the prediction calls for the observed a to be
about 0.003 A per at.% larger in T-rich T-B solid solutions, than calculated from eqn (2). The insets in Fig. 1 for Nb75+,SnX-,1231 and V75+xGaz-x [ 12-151 show that the observed a values are indeed larger than the values calculated from eqn (2). The same is found to hold for Nb7s+XA1~--x[24], [25] and Nb75+xGe2s-xi261alloys if we Nb 7s+XGa25-X consider also these data, although the compositions of the alloys are not explicitly stated to have been determined by analysis. Extrapolating the observed lattice parameters of the Nb phases to 100% Nb gives an a value of approximately 5.250A for the Sn, Al and Ge alloys and of about 5.235 A for the Ga alloys. Thin-film Nb-Ge samples[27] extending to 954at.% Nb, with compositions determined by microprobe analysis, further confirm the extrapolation to 5.250 A. These values compare reasonably with a vahes of 5.260-5.265 8, obtained by extrapolating a for the T-T phases Nb-Au, Nb-Pt and Nb-Ir to 100% Nb. Thus the lattice parameters of both T-B and T-T phases in Nb-rich solid solutions extrapolate to about the same a value of 5.2502 0.020 A at loo% Nb. The caiculated values of a for substitution to 100% Nb obtained from eqns (2) and (3) respectively are 5.195 and 5.275 A. Thus for T-B phases of Nb, the prediction that the observed variation of a with composition on the Nb-rich side of stoichiometric composition will not be correctly given by eqn (2), but a changeover must occur to the values given by eqn (3), is substantially confirmed. The lattice parameters of V-rich T-B and T-T phases of vanadium, V75trGa25-x [12-1.51,VX+~ALIZ~-~ [ZO]and VTs+rPtz+., 1211extrapolate to 4.800 2 0.010 A at 100%V. This value is intermediate between those calculated from eqns (2) and (3) for substitution to 100%V, respectively 4.761 A and 4.834& so the situation is uncertain. However it was noted above that the size of the valency effect did not appear to decrease pro rata of substitution of V on the Au or Pt sites in the V-Au and V-Pt phases over the several atomic percent of V-rich solid solution. Therefore the a values extrapolated to 100%V must be lower than the value calculated from eqn (3) for hypothetical substitution to 100% vanadium, when the valency effect must have become zero. A CONCLUSIONS Equations developed for the cell dimensions of stoichiometric T-B and T-T phases AsB with the A15 structure in terms of the elemental diameters of the component atoms for CN 12, appear generally to be able to reproduce the variation of a with composition in binary and ternary solid solutions (insofar as the limited experimental data permit comparison) when the apparent valency effect correction that appears in the equation for T-T phases i) is reduced pro rata of substitution in B-rich solid solutions (although data for NbsPt prove an exception), ii) is reduced in size pro rata of substitution in A-rich solid solutions (though there appears to be no reduction in the case of V3Au and VsPt phases). In ternary (V, T)3Ga solid solutions, when the T substituent is a metal that occurs on the B sites of T-T phases a valency effect is introduced pro rata of substitution, but
The calculations of cell dimensions of stoichiometric phases and binary and ternary solid solutions as we expected, not when the T substituent is a metal
that occurs on the A sites of T-T phases. Finally, the model on which the equation for the T-B and T-T phases are developed, shows that it is the transition metal-transition metal contacts that are paramount in controlling the cell dimensions of phases with the A15 structure. This leads to the prediction that in A-rich T-B phases as the transition metal replaces the non-transition metal atoms on the B sites, the observed lattice parameters of the solid solution will be larger than given by the equation for T-B phases and will approach the value given by the equation for T-T phases at substitution to a hypothetical 100% of transition metal. This prediction is largely substantiated for phases of niobium. Thus the study does not appear to have revealed any very serious disagreements with the model from which the equations for the cell dimensions of the T-B and T-T phases were derived, and which shows how the cell dimensions are controlled by atomic interactions on the basis of their elemental CN 12 diameters. Indeed the observed lattice parameter variation of T-rich T-B solid solutions suggests a confirmation of the model. Acknowledgements-I am grateful to Dr. R. M. Water&at for further discussions. The work was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. REFERENCES
I. Pearson W. B., Phil. Trans. R Sot. London 298,415 (1980). 2. Shoemaker C. B. and Shoemaker D. P., Developments in the Structural Chemistry of Alloy Phases, p. 107 Plenum Press, New York (1%9). 3. Geller S., Acta Crystallogr. 9,885 (1956). 4. Pauling L., Acta Crystallogr. IO, 374 (1957). 5. Tarutani Y. and Kudo M., J. Less-Common Metals 55, 221 (1977). 6. Johnson G. R. and Douglass D. H., 1. Low Temp. whys. 14, 565 (1974). Pearson W. B., J. Less-Common Metals 81, 161(1981). :: Pearson W. B.. To be published. 9. Teatum E., Gschneidnkr K. and Waber J., Los AIamos Rep. LA-2345, available from U.S. Department of Commerce, Washington, D.C. (1960). 10. Waterscat d. M., j. Sohd SLateChem. 37,370 (1981). 11. Vegard L., Z. Phys. 5, 17, (1921). 12. Das B. N., Cox J. E., Huber R. W. and Meussner R. A., Met. Trans. SA, 541 (1977). 13. Zeglei S. T. and Downey J. W., Trans. Met. Sot. AJME 227, 1407(1%3). 14. Girgis K., Odoni W. and Ott H. R., J. Less-Common Metals 46, 175(1976). 15. Miiller A., Z. Naturforsch. A24, 1134(1969). 16. Moehlecke S., Cox D. E. and Sweedler A. R., Solid State Commun., 23,703 (1977). 17. Wire M. S. and Webb G. W., J. Phys. Chem. Solids 42, 233 (1981). 18. Giessen B. C., Koch R. and Grant N. J., Trans. Met. Sot. AJME 230,1268 (1964). 19. Michalik S. J. and Brophy J. H., Trans. Met. Sot. ACME227, 1047(1%3). 20. Fliikiger R., Susz Ch., Heiniger F. and Muller J., J. LessCommon Metals 40, 103(1975). 21. Water&at R. M., Met. Trans. 4, 455 (1973), and private communication. 22. Girgis K., J. Less-Common Metals 65, 1 (1979). 23. Junod A., Muller J., Rietschel H. and Schneider E., .f. Phys. Chem. SoIids 39,317 (1978).
551
24. Jorda J.-L., Fliikiger R. and Muller J., J. Less-Common Metals 75, 227 (1980).
25. Jorda J.-L., Fliikiger R. and Muller J., J. Less-Common Metals 55, 249 (1977). 26. Jorda J.-L., Fliikiger R. and Muller J., 1. Less-Common Metals 62, 25 (1978). 27. Stewart G. R., Newkiik L. R. and Valencia F. A., Phys. Rev. BZI, 5055(1980). 28. Water&at R. M. and Manuszewski R. C., J. Less-Common Met. 51,55 (1977). 29. Giessen B. C., Dangel P. N. and Grant N. J., .I. LessCommon Metals 13,62 (1%7). 30. Waterstrat R. M. and Manuszewski R. C., J. Less-Common Metals 52,293 (1977).
31. Waterstrat R. M., Met. Trans. 4, 1585(1973). 32. Waterstrat R. M. and Manuszewski R. C.. .f. Less-Common Metals 32, 79 (1973). 33. Pearson W. B., Acta Crystallogr. A36, 724 (1980). 34. Pauling L., J. Am. Chem. Sot. 69,542 (1947).
APPENDM
When, say, a is plotted against Da for series of phases with a given structure and the same A component, the lines for phases of different A are parallel and separated according to their DA values. Sometimes, however, the separations of the lines for phases of daerent A components can not be accounted for on the basis of the DAvalues and a “valency effect” is seen to occur. Thus the valencies of the A components dilTerfrom those in their elemental structures from which their DA values were determined, and hence the elemental DAvalues do not account for the relative positions of the lines of a vs Da. Alternatively, if a values at a given W value are plotted against DA for the phases of the various A components, phases of A atoms of valency I, 2, 3,... are found to lie severally on different, but parallel lines. The reason why plots of a vs Da and DA reveal changes of atomic valency and therefore of size in the alloys compared to the elemental structures, but are unin!luenced by differences of CN in the phases from CN 12 of the diameters used, is that the changes of valency are diierent from A atoms of valency 1, 2,3,. . whereas the differences of CN from CN 12 are the same for all phases with the given structure. When the component atoms have standard valencies, such as three or five for arsenic, antimony and bismuth, the valency effects can be interpreted quantitatively from the data obtained. With transition metal atoms showing a valency effect, quantitative interpretation may be obtained, e.g.[33], but generally it is only relative. In the case of the T-T phases of chromium, the number of electrons required by Cr to satisfy the observed interatomic distances (calculated with Pauling’s equation[34]) is close to six, so a valency effect of zero is assumed, since six electrons are also presumed to be used in the elemental structure. On the other hand phases of the Group V and Group IV A components, require considerably more electrons than the 5 and 4 assumed in their elemental structures; hence the assigned S values of 0.050 and O.l5OA, which express the required size daerences in eqn (3). Since the B components of these T-T phases come from Group VII and hiier groups and include Au and Hg, they have the electrons required without changing their atomic valency and size from that of their elemental structures. In a true valency effect the value of S must be the same for all elements of the same Group, but here a value of S = 0.025A is required for phases of MOcompared to S = 0 for phases of Cr. Thus assuming a correct elemental diameter for MO, it is seen that some additional small effect is being included in the apparent valency effect. Alternately a smaller CN 12 diameter for MO would remove the effect maLing S = 0 for both Cr and MOalloys; however, we have not previously noticed evidence in MOalloys with other crystal structures, that a CN 12 diameter of 2.800A is too large.
THE CALCULATIONS OF CELL DIMENSIONS OF STOICHIOMETRIC PHASES AND BINARY AND TERNARY SOLID SOLUTIONS WITH THE Al5 STRUCTURE W. B. PEARSON Departments of Physics and of Chemistry, University of Waterloo, Waterloo, Ontario, Canada NZL 3GI (Received I June 1981;accepted in revised form 7 October 1981) Abstrati-Equations that successfully reproduce the unit cell dimensions of stoichiometric (i) transition metaltransition metal (T-T) and (ii) transition metal-non-transition metal (T-B) phases with the R-Wolfram or Al5 structure in terms of the diameters of the component atoms for CN 12 [J. Less-Common Metals, 81, 161(Ml)], are hear applied to binary and ternary solid solutions with the Al5 structure, to discover how well and under what conditions they reproduce the observed lattice parameter variation. That in the main, they perform satisfactorily gives further credence to the model on which they are based, which shows how the various interatomic contacts within the interpenetrating icosahedra and CN 14 polyhedra of the structure, combine together in controlling the cell dimensions of the phases. The mode1also indicates that it is the contacts between the transition metal atoms that are paramount in controlling a. This fact leads to a specific prediction for the lattice parameter variation in transition-metal-rich solid solutions of T-B phases, which is essentially substantiated for phases of niobium. 1.
lNTRoDucTlm
It has been shown[l] that the unit cell dimensions of intermetallic binary phases, A,B,, with what we shall call “ordinary” crystal structures are controlled by a single type of atomic array emanating from a given atom, e.g. A-A-A, A-B-A, A-B-B-A, etc. However, the problem is more complex in phases with tetrahedrally closepacked structures which, if regular, are composed of interpenetrating polyhedra of coordination number (CN) 12, 14, 15, or 16, which have triangulated surfaces. The /3-Wolfram or Al5 structure, AsB, is tetrahedrally close packed, being composed of interpenetrating icosahedra and CN 14 polyhedra. Owing to the interpenetration of the polyhedra it is improbable that any single array of atoms can act alone to control the cell dimension, a. For example the A atom at the centre of the CN 14 polyhedron has two close A neighbours at a/2, 4 close B neighbours at a%‘5/4, and eight more remote A neighbours at ad3/2v2. If these interatomic contacts were separately to control a the expected dependences of a on DA would be respectively 2.0 D.+,, 0.89D~ and 1.63DA, whereas the observed dependence is 1.4 DA for phases with the A15 structure. Thus the problem is not trivial. The Shoemakers[2] tackled it for tetrahedrally close-packed phases generally by recognizing the nonsphericity of the atomic sites at the centres of the polyhedra and adopting different sets of radii for atoms at the centres of the CN 12, 14, 15 and 16 polyhedra, and for ligands to atoms with surface coordination number (SCN) five and with SCN six. Geller[3] and Pauling[41 and their followers reproduced the cell dimensions of phases with the Al5 structure on the basis of the A-B aV5/4 = and ad5/4 = (RA+ Ru)[3] distances: 1/2(3R, t Re)[4]; in the latter the radii are weighted according to the stoichiometric composition. In both treatments, sets of self-consistent atomic radii are assumed which best reproduce the observed cell dimensions.
A more complex method of deriving radii and calculating the cell dimensions of phases with the Al5 structure has been devised by Tarutani and Kudo[5]. There are two distinct groups of phases with the Al5 structure: T-T phases in which transition metal atoms occupy both site-sets in the structure (Tc~,~T& and T-B phases in which non-transition metals occupy the B sites. Their dimensional behaviours differ as first noted by Johnson and Douglass 161. Recently we have been able to deal with the problem of several different interatomic contacts within the same coordination polyhedron acting together to control the cell dimension, obtaining equations that reproduce the cell dimensions of the stoichiometric T-T and T-B phases effectively from standard CN 12 atomic diameters obtained from the elemental structures[7]. Since elemental diameters are used rather than ad hoc diameters as in[2-4], the resulting model provides a real explanation of how the cell dimensions are controlled in such phases. From a base equation that expresses the differences between the calculated interatomic distances and the appropriate diameter sums with relative weighting according to the multiplicity of the contacts within the polyhedra, non-linear equations for the T-B and T-T phases were obtained for a in terms of DA and l)B, the atomic diameters for CN 12[7]. The base equation also included an effectiveness factor, 0 I a 5 1, expressing the effectiveness of the eight remote A-A contacts in controlling a. The base equation as now correctly written[8], also includes weighting for the relative site-set multiplicity, although this omission did not introduce errors in the equations for a of phases with the Al5 structure, since no contacts ‘occur between atoms on B sites at the stoichiometric composition. Furthermore, to increase its generality an effectiveness factor, 05 a 5 1, etc., is attached to each of the expressions for interatomic con547
W. B. PEARSON
548
tacts occurring in the equation. Thus the base equation for phases with the A15 structure is 54k = 12a(aq5/4-(l/2)0,-(1/2)DB)+ CN 12 Polyhedron 3x2/?(a/2-D,)+ 3 x 4y(aV5/4- (l/2)0, - W2PE) + 3 x 8S(a~3/2~2 - DA) CN 14 Polyhedron (1) where k in an adjustable parameter. Experience from the Al5 and CuzMg structures shows[8] that (i) if all contacts in the structure are closer than the appropriate radius sums, the effectiveness factors a, p,. . are independent of DA and Dg; k is a constant and the equation relating a, DA and Dn is linear, (ii) if one of the contacts in the structure is at a greater distance than the appropriate radius sum, the value of the effectiveness factor concerned, may change from 0 to 1 as a linear function of DA or Dn as appropriate, and k may also be a linear function of DA or &, in which case the equation relating a, DA and Da is non-linear, and finally (iii) if the coupling between the contacts in the polyhedra breaks down so that they no longer jointly control a, k may be a function of a. The 54 contacts involved in the weighting in eqn (I) are correct for the stoichiometric formula A,B. Thus the A-B contacts that occur in both CN 12 and CN 14 polyhedra occur twice in the equation. Through the effectiveness factors, this allows account to be taken of the fact that a particular contact may have no influence on a in one polyhedron, but the same contact may influence a in the other polyhedron, owing to its joint action with the other contacts in the polyhedron. Thus for T-B phases we find OL= 0, /3 = y = 1, whereas for T-T phases a = j3 = y = 1 and for both, S varies linearly with DA between the limits where its value changes from oto 1. In[7] the dependence of the effectiveness factor (here S in eqn 1) on DA was determined from the slopes of lines for phases with various A components on plots of a against Da, thus giving equations for a in terms of DA and I)B for the T-B and T-T phases, the adjustable parameter being determined from comparison of calculated and observed a values. The equations obtained were: for B-T phases a = IO.1429Ds- 1.7231DA+O.8971 D:-0.6034) /{OS493DA-- 1.1334) (2) valid between DA= 2.484& below which 6 = 0 and DA = 3.121 A, above which 6 = 1. for T-T phases a = {0.2222De - 1.4782DA t 0.8444 Dfg - 0.6833}/{0.5172DA - 0.%72}- S
(3)
valid between DA= 2.458 A, below which 6 = 0 and DA = 2.984 A, above which 8 = 1. In eqn (3) S represents an apparent valency effect with values of 0 for Cr, 0.025 %, for MO, 0.050 8, for Group V and 0.150 A for Group IV A components. The significance of the valency
effect is discussed in the Appendix. With elemental atomic diameters of [9], eqn (2) reproduces the observed cell dimensions of the 32 T-B phases to which it applies with a mean error of (0.012)A, or )0.008)A neglecting very divergent results for five phases. Similarly eqn (3) reproduces the observed cell dimensions of the 29 phases to which it applies with a mean error of 10.006(A. The comparison is made with cell dimensions adjusted to stoichiometry by Waterstrat[lO], and a diameter of 3.246 A for Sn” is used[7]. These equations can be applied explicitly to binary and ternary solid solutions and also to disorder. Because of the occurrence of the apparent valency effect, the application of these equations to solid solutions is not just an extension of Vegard’s law [11] applied to the site-sets on which substitution occurs. This paper therefore seeks to establish the extent to which, and under what conditions, the equations may reproduce the lattice parameter variation of solid solutions with the A15 structure, thus providing a further test of the validity of the model on which they are based. Furthermore, it investigates a significant prediction regarding the variation of the cell dimensions in transition-metal-rich solid solutions of T-B phases, that arises from the model. 2. CALCULATIONOFLATTICEP-
EBVARIATIONIN BINARY ANDTERNARYSOLIDSOLUTIONS
In applying these equations to binary or ternary solid solutions, the exact number of contacts of each type formed by the component atoms can be explicitly entered in a base equation similar to (1) and equations for a in terms of DA, Dg (DC) and concentration developed, or more simply, the average diameters for atoms on each site-set calculated from the composition can be entered directly in the final eqns (2) and (3). Since the simpler procedure appears to give the same numerical results, we adopt it here. Most of the experimental data available in the literature has not been confirmed by compositional analysis of the samples, and, for example, Fig. 10 of[l2] illustrates the divergent results that may be obtained from several different investigations on the same system, particularly for T-B alloys. In contrast, the inset in our Fig. 1 shows the agreement that may be obtained for the same system (Ga-V) for results confirmed by compositional analysis. In Figs. 1 and 2 we have plotted only data for alloys whose compositions were determined by analysis, except for some data on refractory transitionmetal alloys (T-T phases) that appear to be reliable. We use these data to examine how well eqns (2) and (3) reproduce the observed lattice spacing variation, but the case of replacement of B atoms by T atoms in T-rich T-B phases is considered separately in section 3. In A-rich T-T alloys we would expect that the size of the valency effect would decrease pro rata of substitution as the T(A) atoms replace the TcB) atoms which are the source of the electrons for the valency effect. Alternatively the size of the valency effect might remain unchanged over the first few atomic percent of substitution, although ultimately at hypothetical complete substitution (T(A)~T~A)),it must be zero. Secondly in
The calculations of ceil dimensions of stoichiometric phases and binary and ternary solid solutions 0t
20
v.
549
B
25
30 1
06)
Id,
5.22
4.85 F
5.20 Go’+ 5.18 480 5.16 5.14 5.12 I
475 P1-Cr,
470
465 I5
20
25
30
35
40
01% B
0
15
30 at
45
60
75
% Cr
F&l. Lattice parameter variation with at.% B for T-T binary solid solutions: Nb-Au[l7], Nb-Pt[l6], NMr[l8], Nt+Os[28], Mo-Ir[l9], V-Au[20], V-Pt[21], V-1$29], V-Rh[30], Cr-pt[31] and Cr-Ir[32]. Insets: Similar data for T-B binary solid solutions: V-Ga 01121, q[i3], 0[14], A[151 and Nb-SnL231.Symbols indicate measured lattice parameters. Filled symbols indicate measurements for two-phase alloys. Full lines indicate calculated lattice parameters (see text). Coarse broken lines for V-Au and V-Pt alloys indicate lattice parameters calculated without pro rata reduction in the size of the valency effect.
Fig. 2. Lattice parameter variation with at.% T for (VI-,T,)&a ternary solid solutions, where T = Ti, Cr, Mn, Fe or Co[14,22]. 0 data for compositionally analysed alloys, 13 data for alloys not analysed. Full lines represent variation of lattice parameters calculated without valency effect. Coarse broken lines represent calculations of o with introduction of a valency effect pro rata of substitution by Mn, Fe and Co.
B-rich T-T alloys as the T(s) component replaces TcI\j on the A sites, we assume in calculations that the correction for the valency effect decreases pro rata of substitution. We also note that unless the atomic diameters differ by very little, the slope of a with composition is not in general expected to be the same on the A-rich side of Aa, as on the B-rich side, since substitution occurs on different crystallographic sites, and for the substituted site, the rate of change of the average atomic diameter with composition is three times greater on the A-rich than the B-rich side of AaB. Figure 1 that gives data for binary solid solutions, shows that on the B-rich side of A,B eqn (2) well reproduces the variation of a with composition for V&a[l2,13,14,15] which is the only example of a T-B phase. With the exception of Nb#t[l6], eqn (3) also appears to reproduce the variation of a with composition satisfactorily for Tea,-rich T-T phases on the basis that the correction for the valency effect is reduced pro rata of substitution. In T(A)-rich T-T phases the rate of change of a with composition is reasonably reproduced by eqn (3) for Nb-Au[l7], NLPt[M], Nb-Ir[l8] and Mo-Ir [ 191alloys on the basis of a pro rata reduction of
the size of the valency effect with composition; however, for the V-Au alloys[20] over the 5 at.% range of solid solution, it appears that there is no decrease in the size of the valency effect. A similar situation appears to obtain for V-Pt alloys[21] and in the T-T alloys of Cr the valency effect is zero. The available data for ternary solid solutions of T-B or T-T phases of the type Ax(B,B’), (A,A’)3B are not particularly interesting, since generally there is little departure from a linear variation of the lattice spacings between those of the binary end components, and the effectiveness of eqns (2) or (3) in reproducing the observed lattice spacings depends on how well they reproduce those of the binary phases concerned. The one exception is the data of Girgis et al. [ 14,221 for the substitution of vanadium by other transition metals in VJGa, since substituents Ti and Cr are components that occupy the A sites in T-T phases, whereas Mn, Fe and Co are regarded as components occupying B sites in T-T phases. We therefore expect Mn, Fe and Co to create a valency effect pro rata of substitution in VsGa, regardless of whether they occupy the A or B sites, whereas Ti and Cr should not.
W. B. PEARSON
550
Figure 2 which shows data for homogeneous alloys whose composition was determined by analysis [ 14,221, bears out these expectations. Since neutron diffraction results of Girgis and Fischer reported in [14,22] indicate that the substituting transition metal atoms are located mainly on the A sites, this location is assumed in the calculations. It is seen (full line) that the variation of a with composition for (V, Cr)aGa and (V, I&Ga alloys is well reproduced by eqn (2), using averaged diameters for the A site and assuming stoichiometric AsB composition for the solid solutions. Generally the analysed compositions are not exactly stoichiometric, being either Aor B-rich. The a values have also been calculated for the exact analysed compositions, but generally they depart little from the lines obtained on the assumption of stoichiometric composition (A, A’)3Ga. The a values of the two most Ti-rich (V, T&Ga alloys lie below the calculated line. The data are for as-cast alloys; data for these alloys annealed at 800°C indicate that they are no longer homogeneous, and the a values decrease further[22]. Therefore the a values of these as-cast alloys appear to be too low because of partial decomposition on cooling. When the (V, MnfrtGa, (V, Fe)3Ga and (V, Co)sGa alloys are considered, it is seen that the a values calculated from eqn (2) (full lines in Fig. 2) are greater than the observed values as expected, because the substituted atoms now contribute electrons to vanadium creating a valency effect. When the size of the valency effect is calculated to increase pro rata with the increasing substitution of V by Mn, Fe or Co up to 25 at.% and the correction for it to decrease pro rata with the decreasing concentration of V in the alloys (eg. -0.050 [x/25] I(75 -x)/75] A for V7+Xo,Ga&, reasonably good agreement of observed and calculated a values is obtained, except for (V, MnbGa alloys in the range from 20 to 25 at.% Mn, as shown by the broken lines in Fig. 2.
3. LATTICE PAUAMhT BK VARIATION IN ERICH SOtID ADDS OF T-B PHASES
Equation (2) for T-B phases was obtained on the basis that the 14 contacts to the transition metal atom at the centre of the CN 14 polyhedron together control a, whereas in eqn (3) for T-T phases all contacts in both CN 12 and 14 polyhedra were involved. Both of these situations maximise the influence of contacts between the transition metal atoms in controlling u. Recognition of this domin~ce of contacts between the transition metal atoms in controlling a, leads to the prediction that eqn (2) should not correctly give the variation of LJwith composition for T-rich T-B phases. As substitution of the transition metal for the B component proceeds to the ultimate T,T, changeover must occur from the behaviour given by eqn (2) to that given by eqn (3) where all contacts in both polyhedra together control a. The difference ofl_Tr- (la-a) calculated with eqns (3) and (2) for complete substitution of A on B sites is found to be 0.071 A for A=Cr and to increase linearly with Da to 0.080 8, for A=Nb, regardless of the size of the valency effect, which is taken as zero at 100%A in eqn (3). Thus the prediction calls for the observed a to be
about 0.003 A per at.% larger in T-rich T-B solid solutions, than calculated from eqn (2). The insets in Fig. 1 for Nb75+,SnX-,1231 and V75+xGaz-x [ 12-151 show that the observed a values are indeed larger than the values calculated from eqn (2). The same is found to hold for Nb7s+XA1~--x[24], [25] and Nb75+xGe2s-xi261alloys if we Nb 7s+XGa25-X consider also these data, although the compositions of the alloys are not explicitly stated to have been determined by analysis. Extrapolating the observed lattice parameters of the Nb phases to 100% Nb gives an a value of approximately 5.250A for the Sn, Al and Ge alloys and of about 5.235 A for the Ga alloys. Thin-film Nb-Ge samples[27] extending to 954at.% Nb, with compositions determined by microprobe analysis, further confirm the extrapolation to 5.250 A. These values compare reasonably with a vahes of 5.260-5.265 8, obtained by extrapolating a for the T-T phases Nb-Au, Nb-Pt and Nb-Ir to 100% Nb. Thus the lattice parameters of both T-B and T-T phases in Nb-rich solid solutions extrapolate to about the same a value of 5.2502 0.020 A at loo% Nb. The caiculated values of a for substitution to 100% Nb obtained from eqns (2) and (3) respectively are 5.195 and 5.275 A. Thus for T-B phases of Nb, the prediction that the observed variation of a with composition on the Nb-rich side of stoichiometric composition will not be correctly given by eqn (2), but a changeover must occur to the values given by eqn (3), is substantially confirmed. The lattice parameters of V-rich T-B and T-T phases of vanadium, V75trGa25-x [12-1.51,VX+~ALIZ~-~ [ZO]and VTs+rPtz+., 1211extrapolate to 4.800 2 0.010 A at 100%V. This value is intermediate between those calculated from eqns (2) and (3) for substitution to 100%V, respectively 4.761 A and 4.834& so the situation is uncertain. However it was noted above that the size of the valency effect did not appear to decrease pro rata of substitution of V on the Au or Pt sites in the V-Au and V-Pt phases over the several atomic percent of V-rich solid solution. Therefore the a values extrapolated to 100%V must be lower than the value calculated from eqn (3) for hypothetical substitution to 100% vanadium, when the valency effect must have become zero. A CONCLUSIONS Equations developed for the cell dimensions of stoichiometric T-B and T-T phases AsB with the A15 structure in terms of the elemental diameters of the component atoms for CN 12, appear generally to be able to reproduce the variation of a with composition in binary and ternary solid solutions (insofar as the limited experimental data permit comparison) when the apparent valency effect correction that appears in the equation for T-T phases i) is reduced pro rata of substitution in B-rich solid solutions (although data for NbsPt prove an exception), ii) is reduced in size pro rata of substitution in A-rich solid solutions (though there appears to be no reduction in the case of V3Au and VsPt phases). In ternary (V, T)3Ga solid solutions, when the T substituent is a metal that occurs on the B sites of T-T phases a valency effect is introduced pro rata of substitution, but
The calculations of cell dimensions of stoichiometric phases and binary and ternary solid solutions as we expected, not when the T substituent is a metal
that occurs on the A sites of T-T phases. Finally, the model on which the equation for the T-B and T-T phases are developed, shows that it is the transition metal-transition metal contacts that are paramount in controlling the cell dimensions of phases with the A15 structure. This leads to the prediction that in A-rich T-B phases as the transition metal replaces the non-transition metal atoms on the B sites, the observed lattice parameters of the solid solution will be larger than given by the equation for T-B phases and will approach the value given by the equation for T-T phases at substitution to a hypothetical 100% of transition metal. This prediction is largely substantiated for phases of niobium. Thus the study does not appear to have revealed any very serious disagreements with the model from which the equations for the cell dimensions of the T-B and T-T phases were derived, and which shows how the cell dimensions are controlled by atomic interactions on the basis of their elemental CN 12 diameters. Indeed the observed lattice parameter variation of T-rich T-B solid solutions suggests a confirmation of the model. Acknowledgements-I am grateful to Dr. R. M. Water&at for further discussions. The work was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. REFERENCES
I. Pearson W. B., Phil. Trans. R Sot. London 298,415 (1980). 2. Shoemaker C. B. and Shoemaker D. P., Developments in the Structural Chemistry of Alloy Phases, p. 107 Plenum Press, New York (1%9). 3. Geller S., Acta Crystallogr. 9,885 (1956). 4. Pauling L., Acta Crystallogr. IO, 374 (1957). 5. Tarutani Y. and Kudo M., J. Less-Common Metals 55, 221 (1977). 6. Johnson G. R. and Douglass D. H., 1. Low Temp. whys. 14, 565 (1974). Pearson W. B., J. Less-Common Metals 81, 161(1981). :: Pearson W. B.. To be published. 9. Teatum E., Gschneidnkr K. and Waber J., Los AIamos Rep. LA-2345, available from U.S. Department of Commerce, Washington, D.C. (1960). 10. Waterscat d. M., j. Sohd SLateChem. 37,370 (1981). 11. Vegard L., Z. Phys. 5, 17, (1921). 12. Das B. N., Cox J. E., Huber R. W. and Meussner R. A., Met. Trans. SA, 541 (1977). 13. Zeglei S. T. and Downey J. W., Trans. Met. Sot. AJME 227, 1407(1%3). 14. Girgis K., Odoni W. and Ott H. R., J. Less-Common Metals 46, 175(1976). 15. Miiller A., Z. Naturforsch. A24, 1134(1969). 16. Moehlecke S., Cox D. E. and Sweedler A. R., Solid State Commun., 23,703 (1977). 17. Wire M. S. and Webb G. W., J. Phys. Chem. Solids 42, 233 (1981). 18. Giessen B. C., Koch R. and Grant N. J., Trans. Met. Sot. AJME 230,1268 (1964). 19. Michalik S. J. and Brophy J. H., Trans. Met. Sot. ACME227, 1047(1%3). 20. Fliikiger R., Susz Ch., Heiniger F. and Muller J., J. LessCommon Metals 40, 103(1975). 21. Water&at R. M., Met. Trans. 4, 455 (1973), and private communication. 22. Girgis K., J. Less-Common Metals 65, 1 (1979). 23. Junod A., Muller J., Rietschel H. and Schneider E., .f. Phys. Chem. SoIids 39,317 (1978).
551
24. Jorda J.-L., Fliikiger R. and Muller J., J. Less-Common Metals 75, 227 (1980).
25. Jorda J.-L., Fliikiger R. and Muller J., J. Less-Common Metals 55, 249 (1977). 26. Jorda J.-L., Fliikiger R. and Muller J., 1. Less-Common Metals 62, 25 (1978). 27. Stewart G. R., Newkiik L. R. and Valencia F. A., Phys. Rev. BZI, 5055(1980). 28. Water&at R. M. and Manuszewski R. C., J. Less-Common Met. 51,55 (1977). 29. Giessen B. C., Dangel P. N. and Grant N. J., .I. LessCommon Metals 13,62 (1%7). 30. Waterstrat R. M. and Manuszewski R. C., J. Less-Common Metals 52,293 (1977).
31. Waterstrat R. M., Met. Trans. 4, 1585(1973). 32. Waterstrat R. M. and Manuszewski R. C.. .f. Less-Common Metals 32, 79 (1973). 33. Pearson W. B., Acta Crystallogr. A36, 724 (1980). 34. Pauling L., J. Am. Chem. Sot. 69,542 (1947).
APPENDM
When, say, a is plotted against Da for series of phases with a given structure and the same A component, the lines for phases of different A are parallel and separated according to their DA values. Sometimes, however, the separations of the lines for phases of daerent A components can not be accounted for on the basis of the DAvalues and a “valency effect” is seen to occur. Thus the valencies of the A components dilTerfrom those in their elemental structures from which their DA values were determined, and hence the elemental DAvalues do not account for the relative positions of the lines of a vs Da. Alternatively, if a values at a given W value are plotted against DA for the phases of the various A components, phases of A atoms of valency I, 2, 3,... are found to lie severally on different, but parallel lines. The reason why plots of a vs Da and DA reveal changes of atomic valency and therefore of size in the alloys compared to the elemental structures, but are unin!luenced by differences of CN in the phases from CN 12 of the diameters used, is that the changes of valency are diierent from A atoms of valency 1, 2,3,. . whereas the differences of CN from CN 12 are the same for all phases with the given structure. When the component atoms have standard valencies, such as three or five for arsenic, antimony and bismuth, the valency effects can be interpreted quantitatively from the data obtained. With transition metal atoms showing a valency effect, quantitative interpretation may be obtained, e.g.[33], but generally it is only relative. In the case of the T-T phases of chromium, the number of electrons required by Cr to satisfy the observed interatomic distances (calculated with Pauling’s equation[34]) is close to six, so a valency effect of zero is assumed, since six electrons are also presumed to be used in the elemental structure. On the other hand phases of the Group V and Group IV A components, require considerably more electrons than the 5 and 4 assumed in their elemental structures; hence the assigned S values of 0.050 and O.l5OA, which express the required size daerences in eqn (3). Since the B components of these T-T phases come from Group VII and hiier groups and include Au and Hg, they have the electrons required without changing their atomic valency and size from that of their elemental structures. In a true valency effect the value of S must be the same for all elements of the same Group, but here a value of S = 0.025A is required for phases of MOcompared to S = 0 for phases of Cr. Thus assuming a correct elemental diameter for MO, it is seen that some additional small effect is being included in the apparent valency effect. Alternately a smaller CN 12 diameter for MO would remove the effect maLing S = 0 for both Cr and MOalloys; however, we have not previously noticed evidence in MOalloys with other crystal structures, that a CN 12 diameter of 2.800A is too large.