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Dimensional analysis of phases with the A15 β-W structure
Journal of the Less-Common
DIMENSIONAL
Metak,
ANALYSIS
227
77 (1981) 227 - 240
OF PHASES WITH THE Al5 /3-W STRUCTURE
W. B. PEARSON Departments of Physics N2L 3Gl (Canada)
and of Chemistry,
University
of Waterloo,
Waterloo,
Ontario
(Received May 22,198O)
Summary It is found that the method of dimensional analysis of binary metallic phases M,N, ,recently developed by the author, can be applied successfully to a tetrahedrally close-packed structure, e.g. the Al5 or 0-W type MNs. Linear equations are developed for the cell edge a in terms of D1, and DN , the atomic diameters for coordination number (CN) 12 of the atoms, and of such valency effects as are apparent; these reproduce the observed a values of 63 phases to within an average of a fraction of a per cent. Differences are found between those phases which have transition metal M components and those which do not. In the former it is the M-N contacts in the icosahedron of N atoms surrounding M that control a, whereas in the latter it is the contacts in the CN 14 polyhedron of four M and ten N atoms surrounding the N atoms that together control a. A pseudo-valency effect occurs for the N atoms in phases with transition metal M components. This appears to result in part from valency considerations and in part from geometrical constraints related to the two close N-N contacts. In this paper we compare phases with the Al5 structure, rare earth dihydrides with the fluorite structure and rare earth nitrides with the rock salt structure, in each of which two or more sets of atomic contacts act simultaneously to control the cell dimension. A lemma is provided for such cases to indicate how the individual expected cell edge dependences on DM or DN are to be combined to give the overall expected dependence.
1. Introduction In this paper we extend the method of analysis of the unit cell dimensions of binary metallic phases M,N, [l - 71 to a tetrahedrally close-packed structure - the Al5 or 0-W type. We do this to see whether the method of analysis is applicable to a structure with a cell dimension that may be controlled by several different sets of interatomic contacts, some of which are (much) closer and some of which are further apart than the appropriate radius sums, and to discover how these sets of interatomic contacts combine 0022-5088/81/0000-0000/$02.50
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228
in giving the overall dependence of the cell edge on the diameters of the component atoms. In this pursuit we also make comparison with the behaviour of dihydrides of the rare earths having the fluorite structure and with the nitrides of the rare earths having the rock salt structure, for which previous analyses [5,7] have shown that two independent atomic arrays act together in controlling the cell dimension. The method of analysis which we have developed avoids the usual difficulties due to atomic coordination (i.e. more than one coordination number (CN) for a component atom in a structure or very disparate distances to the neighbours forming the convex coordination polyhedron) and it generally reveals a difference between the valency of an atom in an intermetallic phase and that in its elemental structure from which its atomic diameter was derived. The analysis involves examining the variation of the unit cell dimensions of phases with a given structure as a function of DM and D, , the atomic diameters of the component atoms for CN 12, regardless of the actual CN of the atoms in the structure. Linear relationships are obtained between the unit cell dimensions and DM and DN , which then can be interpreted in terms of the atomic arrays that control the structural dimensions. When the valency of a component atom in the structure differs from that in its elemental structure from which its CN 12 diameter was derived, instead of all phases with a given structure and N component lying on a single line of a versus &, they lie on a series of parallel lines for M components of valency 1, 2, 3,... provided that the appropriate atomic arrays control the cell dimensions. This behaviour is referred to as a valency effect. Generally such data allow quantitative determination of the valency state of the M components in the intermetallic phases compared with their states in their elemental structures. The reason why differences of valency may be detected, and differences of coordination from CN 12 in the phases are without effect on the dimensional equations obtained, is that differences of coordination are the same for all phases considered whereas the differences of valency vary with the of the component concerned. Thus the method appears to valency 1,2,3,... be ideal in that it avoids the coordination problem, which at best is but a nuisance, but it can generally detect changes of valency (that normally are not apparent), knowledge of which is essential for a successful dimensional analysis since valency is an intrinsic property of the atom and its size. The success of this method of analysis presumes the validity of Pauiing’s equation [ 81: R( 1) - R(n) = 0.3 log n where n equals the atomic valency divided by the atomic CN. If his equation holds it does not matter what the actual CNs of the atoms in a given structure are, since change of atomic diameter from CN 12 to some other CN would only add or subtract a constant term to or from the dimensional equations obtained. Although cases where Pauling’s equation does not hold have been recognized (see for example refs. 9 and lo), the analyses that have been carried out show that it is generally a sufficiently good approximation, particularly in relation to the scatter of measured lattice parameter data, owing to the variation of composition and the general accuracy of the measurements.
229
Normally the arrays of atomic contacts that control the cell dimensions are identified by their observed dependences on DM and DN and those expected from calculation of the interatomic distances. Thus, for example, if calculation shows that a should be proportional to && for a particular atomic array, then generally the observed dependence of (I on DM cannot be larger than 1.4 if these contacts are to control a; it may, however, be less. In the structures examined so far it has generally been found that a single array (often of many possible arrays with the appropriate radius sums equal to, or less than, the observed distances) controls a particular cell edge. However, in the dihydrides of the rare earths with the fluorite structure and in the rare earth nitrides with the rock salt structure, it appears that two atomic arrays act simultaneously to control the cell edge [ 51. A similar situation is expected to occur in phases with tetrahedrally close-packed structures, which are composed of interpenetrating CN 12,14,15 or 16 polyhedra with triangulated surfaces. The question of how the expected cell edge dependences on & or DN should be combined if more than one atomic array, or set of atomic contacts, act together to control a particular cell dimension has not yet been studied. Water&at’s [ 111 recent examination of phases with the tetrahedrally close-packed Al5 structure and his establishment of accurate cell dimensions, adjusted to stoichiometric composition when necessary, make this structure the most attractive to study in this respect. The cubic Al5 structure MN3 is composed of interpenetrating icosahedra (CN 12) and CN 14 polyhedra, which are therefore centred and have triangulated surfaces. There are three sets of interatomic distances involved in these polyhedra: 2 N-N at a/2 A, 12-4 M-N at flu/4 A and 8 N-N at &&2/2fi A. The 2 N-N contacts are invariably closer than DN , whereas the 8 N-N contacts are always at a distance greater than D, . The 12 M-N contacts are generally as close as or closer than + (DM + DN). In 1956 Geller [ 121 derived a self-consistent set of atomic radii that, taken as a = (4@)(R, + RN), reproduced the observed cell constant of 31 phases with the Al5 structure to within 0.02 A and’he discussed atomic valency in phases with that structure. This paper was followed by comments by Pauling [13] who did rather better with another set of radii, taking a = (4/fi)(iR, + $RN). Recently Water&rat [ 111 has carried out an examination of 64 phases with the structure using the near-neighbour diagram method [ 141.
2. Dependence of the cell dimension on diameters of component atoms in the Al5 structure In examining the dependence of the cell edge on the atomic diameters DILland DN for CN 12 [ 91 we find differences between phases with transition metal M components and those with other M components. The dependence of a on D, is indicated in Fig. 1 for transition metal, gold and mercury M components and in Fig. 2 for other metal M components. With transition
230
(4
lb)
Fig, 1. (a) The variation of a with DN for phases with the Al5 structure MN3 formed by transition metal, gold and mercury M components: 0, nickel;a, cobalt; 0, ruthenium; x , rhodium; 0, osmium; 0, iridium; l, technetium;., palladium; A, platinum;V, gold; +, mercury. (b) The variation of a at DN = 2.90 A with DM for Al5 phases formed by transition metal, gold and mercury M components. The different behaviour of chromium, molybdenum, group V and group IV N component alloys is illustrated.
metal M components, a depends on about 1.4& and the N components appear to exhibit a valency effect since phases of groups IV, V and VI N components lie on different but generally parallel lines. In the case of phases with other metal M components, a again on the average depends on about 1.4DN but there is no significant valency effect involving the N atoms. The dependence of a on DM at a constant DN value of 2.9 A, also shown in Figs. 1 and 2, differs for the two groups of M components. For the transition metals, gold and mercury a depends on 0.45&,, and for the other metals it depends on 0.325&. A true valency effect involving the SnNa alloys is apparent in Fig. 2, when the elemental DM value of tin for CN 12 (3.090 i$) is used. It corresponds to tin of valency four. Taking instead the CN 12 diameter (3.246 A) corresponding to Sn2+ removes the valency effect and places the point for SnNa phases on the line of slope 0.3250, for the other phases. The finding of a divalent tin state in these alloys does not affect the Matthias relationship regarding superconducting transition temperatures and the average number of outer electrons, since it is still appropriate to count four electrons for tin in that relationship. The dependence of a on &, DN and an apparent valency effect x for phases with transition metal, gold and mercury M components (shown in Fig. 1) is given by the equation
231
o,tB
04
(b)
Fig. 2. (a) The variation of a with DN for phases with the Al5 structure formed by non transition metal M components: +, silicon; a, germanium;o, arsenic; FJ, gallium; x , aluminium; A, tin;o, antimony; 0, tellurium;r, indium; A, bismuth; 0, thallium; 0, lead. (b) The variation of a at DN = 2.90 A with D, for Al5 phases formed by non transition metal M components.
U = 0.45&
+ 1.4&
-0.035X
- 0.132
(1)
where x has values of 0, 1,2 and 5 respectively for chromium alloys, molybdenum alloys, alloys of group V and alloys of group IV N components, and Dniland DN are the atomic diameters for CN 12 [ 91. This equation reproduces the observed a values for these 30 phases to within an average of 10.0101 A, as shown in Table 1; the apparent valency effect is zero for the four alloys CrsRu, CrsRh, CrsOs and CrsIr. Only in these four alloys of the 30 considered are the interatomic distances almost exactly in accord with the number of valency electrons provided by the two component atoms. In all the remaining alloys the 2 N-N, 4 N-M and 8 N-N distances taken together are considerably closer than can be accounted for by the valency electrons provided by the N atoms. The data in Fig. 2 for non tr~sition metal M component are represented by the equation a = O-325&, + 1.40~ + 0.126
(2)
232 TABLE 1 Comparison of calculated and observed a values for phases with transition metal M components Phase
aoh
(A)
(cl,&
-a,~)
x
IO3 (A)
Nd’f TiaIr Tispt TiaAu TiaHg
5.012 5.032 5.098 5.189
-13 14
Zr3 Au %Hg
5.486 5.558
-9 37
V3Ni
8 6
V3Ir V3Pd vapt V3Au
4.680 4.688 4.786 4.797 4.788 4.826 4.817 4.884
Nb3Rh Nb3 OS NbaIr Nb3Pt Nb3 Au
5.130 5.135 5.134 5.149 5.203
v3co
V3Rh v3os
Phase
=obs (A) klc
-“ohs) x
lo36%
N3M -4 3
-8 -12 0
Ta3Pt Ta3 Au
5.143 5.193
Cr3 Ru CraRu Cr3Cs Cr3Ir CraPt
4.677 4.674 4.676 4.682 4.674
Mo30s MoaIr Mo3Tc Mo3Pt
4.969
4.970 4.960
4.991
11 11 -14 -5 0 -3 32 2 4 17 10
-21 -2 -19 -11 -9 -5 7 3
For the 30 phases the average error is (0.010 1A.
which reproduces the observed a values of 33 of these phases (excluding PbZr,) to within IO.01651 A*. The result for PbZr, is ignored as it appears to be inaccurate; it seems to be inaccurate on Water&at’s near-neighbour diagram [ 111 - possibly the composition of the phase examined was nonstoichiometric to the zirconium-rich side.
3. Calculation of unit cell dimensions of phases with the Al5 structure Calculation of the lattice parameters of phases with the Al5 structure is not a prime objective of this work; it is merely a means of establishing that suitable dimensional equations have been obtained so that further information *There is actually a very small but regular valency effect due to the M components that cannot be removed by adjustments to the coefficients of DM or DN. The equation a = 0.3250, + 1.40, + 0.007~ + 0.109, where y has values 0, 1, 2, 3 and 4 for M components from groups VI, V, IV and III and Sn2+ respectively, reproduces the observed a values of the 33 phases to within 10.0151 A. However, the improvement is so small for the introduction of six more variables that it is ignored.
233
about the structure can be derived from them. Nevertheless, since we have reproduced the cell constants of 63 of the 64 phases known at present to within a mean error of IO.0131A, it is interesting to recall Geller’s [12] and Pauling’s [ 131 earlier calculations for the 32 phases known then. The compelling aspect of eqn. (1) for the transition metal M component phases is that it vindicates Pauling’s choice of a talc =
4 L (D, + 30,)
= 0.4470,
+ 1.342DN
Ji;4
rather than Geller’s choice of a talc
=
AL(DM+DN)= Jj;2
0.894DM + 0.894DN
Indeed, an empirical equation acalc= 0.447DM + 1.342DN - 0.031x + 0.03, with values of DM, DN and x as for eqn. (l), reproduces the observed lattice parameters of the 30 phases slightly better than does eqn. (1). However, it should be noted that our empirical eqn. (2) for phases with non transition metal M components does not agree exactly with Pauling’s choice. In calculating the cell dimensions of binary phases from atomic diameters, the minimum number of variables DM and DN required for n phases is 2n. If these diameters do not reproduce the cell dimensions sufficiently accurately, so that a further adjustment is required for each phase, the maximum number of variables required is then 3n and these give the cell dimension of each phase exactly. The average number of variables required per phase must therefore lie between two and three, and the effectiveness of any method of calculation can thus be judged accordingly. In calculating the cell constants of 32 phases both Geller [ 121 and Pauling [ 131 made adjustments to 20 atomic radii; in addition Pauling’s equation also has two coefficients ($ and+) so that ((2 X 32) + 20 + 2}/32 = 2.69 variables or parameters are needed per phase. Our eqns. (1) and (2) involve a total of 11 extra variables, whence {( 2 X 63) + 11)/63 = 2.17 parameters are required per phase. This lower number of variables per phase, compared with the results of Geller and of Pauling, is balanced by a somewhat lower precision in reproducing &,bs. However, our mean error of 10.0131 A should be compared with the mean error of IO.0571A in calculating a for the 63 phases from 0.447DM + 1.342DN when only two extra variables are introduced. If, more logically, CN 14 diameters are used for the N components, this mean error is increased by about 0.05 A as most deviations are positive. The significant point about the present results is that standard CN 12 atomic diameters can be used to describe the cell dimensions of a tetrahedrally close-packed phase in the same manner as has been applied to phases with other types of structures [ 1 - 71, although notably more parameters are required in the process. For example, both a and c parameters of 43 known phases with the Th2Zn1, structure can severally be described with only ((2 X 43) + 3}/43 = 2.07 variables per phase [l].
234
4. Atomic array controlling the cell edge of phases with transition metal M components That Pauling’s coefficients, 0.447& + 1.3420N, reproduce the observed cell dimensions of phases with the Al5 structure which have transition metal M components implies that the cell dimensions of these phases are indeed controlled by arrays of M-N contacts. This is because his coefficients arise from the assumption that the calculated M-N distances give a, when the atomic diameters are weighted according to the stoichiometric proportion in the MN3 formula. This observation that arrays of M-N contacts control cafor these phases is also confirmed by Waterstrat’s [ll] near-neighbour diagram if phases with transition metal M components are separated from those with other metal M components (Fig. 3). It is seen that the former lie parallel to the line for 12-4 M-N contacts, whereas the latter do not; instead they follow a line which is the weighted root mean square of the 2 N-N, 12-4 M-N and 8 N-N contacts (weighted according to their multiplicities), indicating that the interatomic distances in the interpenetrating CN 12 and CN 14 polyhedra, and their multiplicities, simultaneously control a (see Section 5). Nevertheless, the observation that the phases with transition metal M components lie along the line for 12-4 M-N contacts that is based on f (DM + D, ) weighting, rather than along a steeper line based on $ (& + 30,) weighting, is presently an enigma. The reason possibly rests m the valency effect involving the N components and its effect on J&/D,. It is significant that, in this case where the M-N contacts control the cell dimension, the atomic diameters are weighted in stoichiometric proportion, i.e. 1:3, as intuitively grasped by Pauling, since it seems we have not generally found such behaviour in analysing the cell dimensions of other phases where M-N contacts control the cell dimensions. For example, it does occur in phases with the sphalerite structure MN where a is proportional to l.ODM + l.O&, but it does not occur in phases with the AsNas structure [ 151; it certainly does not hold for Laves phases with the MgCua structure although here a may not be controlled by arrays of M-N contacts. Complete understanding of these facts must await further experience in quantitative analysis of the cell dimensions of metallic phases. At present the reason appears to be as follows: where the stoichiome~ic proportion applies in the Al5 and sphalerite structures the inverse ratio of the numbers of M-N contacts (12-4 in the former, 4-4 in the latter) is the same as in the stoichiometric formula (1:3 and l:l), whereas this is not the case for the AsNas structure (( 2 + 6) :( 1 + 3) = 2 : 1) where the stoichiometric proportion does not apply. 5. Expected dependence of cell edge on IIM or DN when more than one array of contacts s~ul~eou~y exert control One of the objectives of this paper is to discover how to combine the expected cell edge dependences on DM and/or DN for two or more arrays of
235
-04 -05 -06 -04
-04 _fie
e----q. -06
u
-07
DN
-ZJ-
____--
-05 ~
----
j
I-
i
-03 -04 -05
_
____------
_-_06 -07)
I- ;-r
1 1
-02 -03
t/ __/-@/_I
-04
_____------;2_;yJ_N -05
/
-06
F-
-09
’ 090
_____-----
095
loo
105
, IO
I15
120
%‘D~
Fig. 3. Lower part: near-neighbour diagram (see ref. 11; ref. 16, pp. 52 and 53, and references cited therein) for the Al5 structure. The broken line is the weighted root mean square for the 2 N-N, 12-4 M-N and 8 N-N contact lines. Upper two parts: phases with the Al 5 structure formed by transition metal, gold and mercury M components and 3d transition metal N components (top) or 4d and 5d transition metal N components (next), showing adherence to the line for 12-4 M-N contacts. The symbols are the same as in the caption to Fig. 1. Middle two parts: phases with the A15 structure formed by non transition metal M components and 3d transition metal N components (upper) and 4d or 5d transition metal N components (lower), showing adherence to the weighted root-mean square line. The symbols are the same as in the caption to Fig. 2. atoms or sets of interatomic contacts, when they act simultaneously in controlling the cell edges of phases with a given structure. This is a problem which has not yet been tackled. We have as examples to consider the following: (1) rare earth dihydrides MHs with the fluorite structure, where it seems that (110) M-M and (111) M-H contacts simul~eou~y control a; (2) rare earth nitrides MN with the rock salt structure where (110) M-M and (100) M-N arrays simultaneously control (x [ 5 J ; (3) tetrahedrally close-packed phases with the Al5 structure and non transition metal M components. In the carbides and nitrides with the rock salt structure and the dihydrides with the fluorite structure, transition metal M phases have an CI dependence on l.4DM and show a valency effect (Fig. 4). Since the M-M distance equals a@, the l&Z& dependence of a indicates that the M-M
236
Fig. 4. The variation of (I with DM for transition metal and rare earth M dibydrides with the fluorite structure (upper), for carbides with the rock salt structure (middle) and for nitrides with the rock salt structure (lower). The broken lines represent e values which equal fiD~. Adjusted rare earth diameters, except for lanthanum and lutetium, are used [4]. contacts control a, even though the observed distances are generally slightly greater than I&., for CN 12 and the M-N distances are closer than the radius sums. This, however, is not too surprising since it has heen shown for the true Hlgg interstitial phases (RN/R, < 0.59) with the hexagonal AICraC
structure that the very compressed M-C contacts exert no influence in controlling the cell dimensions [6] . In contrast, rare earth M nitrides with the rock salt structure and dihydrides with the fluorite structure have a dependences on 2.0&, and 2.24, respectively, indicating that both the (110) M-M and M-N contacts act together in controlling a, since the former give an expected fiEu and the latter give an expected l.O&, or 0.82&, dependence. A combination of the two effects by simple addition is perhaps to be understood because in the rock salt structure the octahedra of N atoms surrounding M and the cube-octahedra of M atoms surrounding M are quite independent and are not connected to each other as polyhedra. Similarly in the fluorite structure
237
the cube of N atoms su~o~d~g M and the cube-octahedra of M atoms surrounding M are entirely independent of each other. Thus the si~ation differs from that in the tetrahedrally close-packed A15-type structure, where the three different sets of neighbours about N, with contacts to N which combine to control a, are all part of the same CN 14 polyhedron about N. Thus perhaps there is good reason why their individual influences on a combine in some form of average to give the expected dependence of a on DN , rather than combining by simple addition as in the case of rare earth nitrides with the rock salt structure and d~y~ides with the fluorite structure. Finally, one further demonstration can be given to attest to the fact that the different electronic structure of the rare earths, compared with that of the transition metals, results in their M-N contacts also controlling the cell edge in interstitial rock salt and fluorite structures, whereas M-N contacts do not control the cell edge of transition metal compounds with these structures. The rare earth carbides with the rock salt structure are all defective having the formula MsC. With only one-third of the carbon sites occupied s~tistic~y, it is improbable that the M-C distances would exert any influence on the length of the cell edge. In contrast with the stoichiometric rare earth nitrides MN this is exactly what is observed: for the rare earth carbides a depends on 1.4&, exactly the same as for the stoichiometric transition metal carbides MC (Fig. 4), indicating that it is controlled only by the (1101 M-M contacts [5] . Thus we perceive a lemma for understanding how the effects of more than one set or array of ~~mtomic contacts may combine to control the cell dimensions. The lemma states that if two (or more) atomic arrays combine together in controlling the cell dimensions and if the two arrays give coordination polyhedra about the central atom that are completely independent of each other, then the expected cell edge dependences on each array combine additively. If, however, the atoms involved in the different sets of interatomic contacts are part of a single coordination polyhedron about the central atom, then the expected dependences of the cell edge on the different sets of contacts combine together in some average, which might also be weighted according to the multiplicities of the contacts. Indeed, it has already been shown in Section 4 that phases with non transition metal M components follow, on the near-neighbour diagram, the line which is the weighted root mean square for the 2 N-N, 12-4 M-N and 8 N-N contacts (weighted according to their multiplicities). Although this line is not quite linear, it can be represented approximately by the equation (I’M -ufiJZ)/B,
= -0.455
+ (D,/D,
- 1.1)0.63
(3)
whence a = 0.453DM + 1.407&. If, however, the weighted average is confined to the 10 N-N and the 4 N-M contacts in the CN 14 polyhedron surrounding N, the equation of the line so obtained is a = 0.323& + 1.4160, which reproduces eqn. (2) almost exactly, indicating that it is the contacts in
238
the polyhedron about N that control the cell dimensions of this group of phases. Atomic diameters are not now weighted in proportion to stoichiometric composition, as M-N contacts alone do not control a.
6, Pseudo-valency effect in phases with transition metal M components A true valency effect, as explained in Section 1, is a function of the group number or the valency of the component atoms alone, although it depends not directly on this but on the difference between the (effective) valency in the phase and that in the elemental structure from which its CN 12 diameter was determined. However, in the apparent valency effect involving phases with transition metal, gold and mercury M components (eqn. (1)), we find a difference not only in respect of group but also in respect of period for chromium alloys, where the valency effect is zero, and molybdenum alloys, and a smaller effect (which we have ignored) for vanadium alloys on the one hand and niobium and tantalum alloys on the other hand. Any geometrical constraint that is separate from the arrays of atoms which control the cell dimensions (here the arrays of M-N contacts; see Sections 3 and 4) could result in effects similar to those found for a true valency effect, but could be dependent on both group and period of the atom - as is atomic size. Indeed, here we have evidence from these data of such an effect being combined with a true valency effect. It results from the influence of DN through the 2 N-N contacts which are very compressed. These contacts may make an additional contribution to increase a, which is not included in the contribution arising from the M-N contacts. The relative size of this contribution gets smaller as the true valency effect gets larger (i.e. on going from group VI to group IV N components). This is the first occasion where such an effect (which we shall caIl a pseudo-valency effect) has been revealed, but there seems little doubt about its indirection since neither the true valency effect nor the geometrical constraint is observed in alloys of non transition metal M components, where a weighted average of the contributions of the 2 N-N, 4 M-N and 8 N-N contacts controls the cell dimensions of the phases (Section 5). Fu~hermore, the effect does not result from the geometrical constraint alone, since values of the adjustment of acalc are not linear with DN or even &/I&+
7. Discussion This dimensional analysis of phases with the A15 structure has established the prime ~po~~ce of contacts between transition metals in controlling the cell dimensions. When both components are transition metals, the unlike 12-4 M-N contacts control the cell dimensions and the N atoms have an average number of electrons per contact which is different from that of their elemental structures adjusted to CN 12, since there is a valency
239
effect (to some extent modified by geometrical constraints arising from the 2 N-N contacts). When the M component is not a transition metal, like contacts (N-N) between the transition metal atoms assume a greater role and the non transition metal-transition metal M-N contacts assume a lesser role, since now the cell dimension is controlled by a weighted average of all three types of contact in the CN 14 polyhedron. The 8 N-N distances are always larger than DN . As these come closer the free energy of the phase is lowered with a con~omi~t penalty for further compressing the 2 close N-N contacts; however, the low multiplicity of these contacts favours the process. The 4 M-N contacts remain relatively close to their appropriate radius sums. The overall balancing of these energies which involves both the interatomic distances and the multiplicity of the contacts may, and in most Al5 phases does, result in the atoms being closer than expected from valence electron considerations. The consequences of such energy balancing have already been explicitly recognized in Laves phases with the MgCu, structure [ 161, which is also ~~~e~y close packed. These findings are equivalent to a statement that in transition metaltransition metal phases it is the M-N contacts of the icosahedra of N atoms surrounding M that control a, whereas in non transition metal-transition metal phases it is the contacts in the CN 14 polyhedra of 4 M and 10 N atoms surrounding N that control a. The former, certainly, is new information!
8. Conclusions (1) It is possible to analyse the cell dimensions of a tetrahedrally closepacked structure in which several sets of interatomic contacts simultaneously may control the cell dimensions, and to obtain equations in terms of the CN 12 diameters of the component atoms (and valency effects if they exist) which reproduce the observed cell dimensions on average to within a fraction of a per cent. (2) The arrays of atoms that control the cell dimension of phases with the Al5 structure which are formed with transition metal M components are different from those that control the cell dimension of these phases formed with other M components. This difference is equivalent to the M-N distances in the icosahedron of N atoms surrounding M controlling a in the former, and the contacts in the CN 14 polyhedron of 4 M and 10 N atoms surrounding N controlling u in the latter. (3) When more than one atomic array, or set of interatomic distances, control the cell dimensions the individual expected dependence6 of a on Du or DN combine to give the overall dependence. This is additive if the coordination polyhedra formed by the two arrays are independent of each other; however, they combine by an averaging process if the sets of interatomic distances are involved in a single (i.e. the same) coordination polyhedron.
(4) If a geometrical constraint resulting from atomic contacts which are different from those that generally control the cell dimensions exists, it may appear in the analysis in a manner somewhat similar to a valency effect; valency effects occur when the valency of an atom in the phase is different from that in its elemental structure from which its CN 12 diameter was determined. It may be possible to make some distinction between the two effects as cell dimensions altered by valency changes should depend on the group number of the atom alone, whereas those resulting from geometrical constraints (size per se) should depend on both the group number and the period of the atoms concerned. Acknowledgments 1 am grateful to Dr. R. M. Wa~rs~at for providing me with a copy of his paper prior to its publication. This work was supported by a grant from the Natural Sciences and Engineering Research Council of Canada.
References 1 W. B. Pearson, Aeta Crystallogr., Sect. B, 35 (1979) 1329. 2 W. B. Pearson, 2. Kristallogr., 151 (1980) 301. 3 W. B. Pearson, 2. Krktallogr., 152 (1980) 23. 4 W. B. Pearson, J. Less-Common Met., 71 (1980) 85. 5 W. B. Pearson, J. Less-~ornrn~~ Met., 72 (1980) 107. 6 W. B. Pearson, Actu Crystal&r., Sect. A, 36 (1980) 724. 7 W. B. Pearson,Philos. Truns. R. Sot., London, 298 (1980) 415. 8 L. Pauling, J. Am. Chem. Sot., 69 (1947) 542. 9 E. Teatum, K. Gschneidner and J. Waber, Los Alamos Rep, LA-2345, 1960, available from U.S. Department of Commerce, Washington, DC. 10 W. H. Zachariasen, J. Inorg. Nucl. Chem., 35 (1972) 3437. 11 R. M. Waterstrat, J. Solid State Chem., (1980) in the press. 12 S. Geller, Acta Crystal&r., 9 (1956) 885; 10 (1957) 380. 13 L. Paufing, Acta C~sta~lo~., 10 (1957) 374,685. 14 W. B. Pearson, Acta Crystallogr., Sect. B, 24 (1968) 7,1415, 15 W. B. Pearson, in L. H. Bennett (ed.), Theory &Alloy Phase Formation, The Metallurgical Society of AIME, Warrendale, PA, 1980, p. 262. 16 W. B. Pearson, The Crystal Chemists and Physics o~~eta~ and Alloys, Wiley-Interscience, New York, 1972, pp. 59,60.
DIMENSIONAL
Metak,
ANALYSIS
227
77 (1981) 227 - 240
OF PHASES WITH THE Al5 /3-W STRUCTURE
W. B. PEARSON Departments of Physics N2L 3Gl (Canada)
and of Chemistry,
University
of Waterloo,
Waterloo,
Ontario
(Received May 22,198O)
Summary It is found that the method of dimensional analysis of binary metallic phases M,N, ,recently developed by the author, can be applied successfully to a tetrahedrally close-packed structure, e.g. the Al5 or 0-W type MNs. Linear equations are developed for the cell edge a in terms of D1, and DN , the atomic diameters for coordination number (CN) 12 of the atoms, and of such valency effects as are apparent; these reproduce the observed a values of 63 phases to within an average of a fraction of a per cent. Differences are found between those phases which have transition metal M components and those which do not. In the former it is the M-N contacts in the icosahedron of N atoms surrounding M that control a, whereas in the latter it is the contacts in the CN 14 polyhedron of four M and ten N atoms surrounding the N atoms that together control a. A pseudo-valency effect occurs for the N atoms in phases with transition metal M components. This appears to result in part from valency considerations and in part from geometrical constraints related to the two close N-N contacts. In this paper we compare phases with the Al5 structure, rare earth dihydrides with the fluorite structure and rare earth nitrides with the rock salt structure, in each of which two or more sets of atomic contacts act simultaneously to control the cell dimension. A lemma is provided for such cases to indicate how the individual expected cell edge dependences on DM or DN are to be combined to give the overall expected dependence.
1. Introduction In this paper we extend the method of analysis of the unit cell dimensions of binary metallic phases M,N, [l - 71 to a tetrahedrally close-packed structure - the Al5 or 0-W type. We do this to see whether the method of analysis is applicable to a structure with a cell dimension that may be controlled by several different sets of interatomic contacts, some of which are (much) closer and some of which are further apart than the appropriate radius sums, and to discover how these sets of interatomic contacts combine 0022-5088/81/0000-0000/$02.50
@ Elsevier Sequoia/Printed
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228
in giving the overall dependence of the cell edge on the diameters of the component atoms. In this pursuit we also make comparison with the behaviour of dihydrides of the rare earths having the fluorite structure and with the nitrides of the rare earths having the rock salt structure, for which previous analyses [5,7] have shown that two independent atomic arrays act together in controlling the cell dimension. The method of analysis which we have developed avoids the usual difficulties due to atomic coordination (i.e. more than one coordination number (CN) for a component atom in a structure or very disparate distances to the neighbours forming the convex coordination polyhedron) and it generally reveals a difference between the valency of an atom in an intermetallic phase and that in its elemental structure from which its atomic diameter was derived. The analysis involves examining the variation of the unit cell dimensions of phases with a given structure as a function of DM and D, , the atomic diameters of the component atoms for CN 12, regardless of the actual CN of the atoms in the structure. Linear relationships are obtained between the unit cell dimensions and DM and DN , which then can be interpreted in terms of the atomic arrays that control the structural dimensions. When the valency of a component atom in the structure differs from that in its elemental structure from which its CN 12 diameter was derived, instead of all phases with a given structure and N component lying on a single line of a versus &, they lie on a series of parallel lines for M components of valency 1, 2, 3,... provided that the appropriate atomic arrays control the cell dimensions. This behaviour is referred to as a valency effect. Generally such data allow quantitative determination of the valency state of the M components in the intermetallic phases compared with their states in their elemental structures. The reason why differences of valency may be detected, and differences of coordination from CN 12 in the phases are without effect on the dimensional equations obtained, is that differences of coordination are the same for all phases considered whereas the differences of valency vary with the of the component concerned. Thus the method appears to valency 1,2,3,... be ideal in that it avoids the coordination problem, which at best is but a nuisance, but it can generally detect changes of valency (that normally are not apparent), knowledge of which is essential for a successful dimensional analysis since valency is an intrinsic property of the atom and its size. The success of this method of analysis presumes the validity of Pauiing’s equation [ 81: R( 1) - R(n) = 0.3 log n where n equals the atomic valency divided by the atomic CN. If his equation holds it does not matter what the actual CNs of the atoms in a given structure are, since change of atomic diameter from CN 12 to some other CN would only add or subtract a constant term to or from the dimensional equations obtained. Although cases where Pauling’s equation does not hold have been recognized (see for example refs. 9 and lo), the analyses that have been carried out show that it is generally a sufficiently good approximation, particularly in relation to the scatter of measured lattice parameter data, owing to the variation of composition and the general accuracy of the measurements.
229
Normally the arrays of atomic contacts that control the cell dimensions are identified by their observed dependences on DM and DN and those expected from calculation of the interatomic distances. Thus, for example, if calculation shows that a should be proportional to && for a particular atomic array, then generally the observed dependence of (I on DM cannot be larger than 1.4 if these contacts are to control a; it may, however, be less. In the structures examined so far it has generally been found that a single array (often of many possible arrays with the appropriate radius sums equal to, or less than, the observed distances) controls a particular cell edge. However, in the dihydrides of the rare earths with the fluorite structure and in the rare earth nitrides with the rock salt structure, it appears that two atomic arrays act simultaneously to control the cell edge [ 51. A similar situation is expected to occur in phases with tetrahedrally close-packed structures, which are composed of interpenetrating CN 12,14,15 or 16 polyhedra with triangulated surfaces. The question of how the expected cell edge dependences on & or DN should be combined if more than one atomic array, or set of atomic contacts, act together to control a particular cell dimension has not yet been studied. Water&at’s [ 111 recent examination of phases with the tetrahedrally close-packed Al5 structure and his establishment of accurate cell dimensions, adjusted to stoichiometric composition when necessary, make this structure the most attractive to study in this respect. The cubic Al5 structure MN3 is composed of interpenetrating icosahedra (CN 12) and CN 14 polyhedra, which are therefore centred and have triangulated surfaces. There are three sets of interatomic distances involved in these polyhedra: 2 N-N at a/2 A, 12-4 M-N at flu/4 A and 8 N-N at &&2/2fi A. The 2 N-N contacts are invariably closer than DN , whereas the 8 N-N contacts are always at a distance greater than D, . The 12 M-N contacts are generally as close as or closer than + (DM + DN). In 1956 Geller [ 121 derived a self-consistent set of atomic radii that, taken as a = (4@)(R, + RN), reproduced the observed cell constant of 31 phases with the Al5 structure to within 0.02 A and’he discussed atomic valency in phases with that structure. This paper was followed by comments by Pauling [13] who did rather better with another set of radii, taking a = (4/fi)(iR, + $RN). Recently Water&rat [ 111 has carried out an examination of 64 phases with the structure using the near-neighbour diagram method [ 141.
2. Dependence of the cell dimension on diameters of component atoms in the Al5 structure In examining the dependence of the cell edge on the atomic diameters DILland DN for CN 12 [ 91 we find differences between phases with transition metal M components and those with other M components. The dependence of a on D, is indicated in Fig. 1 for transition metal, gold and mercury M components and in Fig. 2 for other metal M components. With transition
230
(4
lb)
Fig, 1. (a) The variation of a with DN for phases with the Al5 structure MN3 formed by transition metal, gold and mercury M components: 0, nickel;a, cobalt; 0, ruthenium; x , rhodium; 0, osmium; 0, iridium; l, technetium;., palladium; A, platinum;V, gold; +, mercury. (b) The variation of a at DN = 2.90 A with DM for Al5 phases formed by transition metal, gold and mercury M components. The different behaviour of chromium, molybdenum, group V and group IV N component alloys is illustrated.
metal M components, a depends on about 1.4& and the N components appear to exhibit a valency effect since phases of groups IV, V and VI N components lie on different but generally parallel lines. In the case of phases with other metal M components, a again on the average depends on about 1.4DN but there is no significant valency effect involving the N atoms. The dependence of a on DM at a constant DN value of 2.9 A, also shown in Figs. 1 and 2, differs for the two groups of M components. For the transition metals, gold and mercury a depends on 0.45&,, and for the other metals it depends on 0.325&. A true valency effect involving the SnNa alloys is apparent in Fig. 2, when the elemental DM value of tin for CN 12 (3.090 i$) is used. It corresponds to tin of valency four. Taking instead the CN 12 diameter (3.246 A) corresponding to Sn2+ removes the valency effect and places the point for SnNa phases on the line of slope 0.3250, for the other phases. The finding of a divalent tin state in these alloys does not affect the Matthias relationship regarding superconducting transition temperatures and the average number of outer electrons, since it is still appropriate to count four electrons for tin in that relationship. The dependence of a on &, DN and an apparent valency effect x for phases with transition metal, gold and mercury M components (shown in Fig. 1) is given by the equation
231
o,tB
04
(b)
Fig. 2. (a) The variation of a with DN for phases with the Al5 structure formed by non transition metal M components: +, silicon; a, germanium;o, arsenic; FJ, gallium; x , aluminium; A, tin;o, antimony; 0, tellurium;r, indium; A, bismuth; 0, thallium; 0, lead. (b) The variation of a at DN = 2.90 A with D, for Al5 phases formed by non transition metal M components.
U = 0.45&
+ 1.4&
-0.035X
- 0.132
(1)
where x has values of 0, 1,2 and 5 respectively for chromium alloys, molybdenum alloys, alloys of group V and alloys of group IV N components, and Dniland DN are the atomic diameters for CN 12 [ 91. This equation reproduces the observed a values for these 30 phases to within an average of 10.0101 A, as shown in Table 1; the apparent valency effect is zero for the four alloys CrsRu, CrsRh, CrsOs and CrsIr. Only in these four alloys of the 30 considered are the interatomic distances almost exactly in accord with the number of valency electrons provided by the two component atoms. In all the remaining alloys the 2 N-N, 4 N-M and 8 N-N distances taken together are considerably closer than can be accounted for by the valency electrons provided by the N atoms. The data in Fig. 2 for non tr~sition metal M component are represented by the equation a = O-325&, + 1.40~ + 0.126
(2)
232 TABLE 1 Comparison of calculated and observed a values for phases with transition metal M components Phase
aoh
(A)
(cl,&
-a,~)
x
IO3 (A)
Nd’f TiaIr Tispt TiaAu TiaHg
5.012 5.032 5.098 5.189
-13 14
Zr3 Au %Hg
5.486 5.558
-9 37
V3Ni
8 6
V3Ir V3Pd vapt V3Au
4.680 4.688 4.786 4.797 4.788 4.826 4.817 4.884
Nb3Rh Nb3 OS NbaIr Nb3Pt Nb3 Au
5.130 5.135 5.134 5.149 5.203
v3co
V3Rh v3os
Phase
=obs (A) klc
-“ohs) x
lo36%
N3M -4 3
-8 -12 0
Ta3Pt Ta3 Au
5.143 5.193
Cr3 Ru CraRu Cr3Cs Cr3Ir CraPt
4.677 4.674 4.676 4.682 4.674
Mo30s MoaIr Mo3Tc Mo3Pt
4.969
4.970 4.960
4.991
11 11 -14 -5 0 -3 32 2 4 17 10
-21 -2 -19 -11 -9 -5 7 3
For the 30 phases the average error is (0.010 1A.
which reproduces the observed a values of 33 of these phases (excluding PbZr,) to within IO.01651 A*. The result for PbZr, is ignored as it appears to be inaccurate; it seems to be inaccurate on Water&at’s near-neighbour diagram [ 111 - possibly the composition of the phase examined was nonstoichiometric to the zirconium-rich side.
3. Calculation of unit cell dimensions of phases with the Al5 structure Calculation of the lattice parameters of phases with the Al5 structure is not a prime objective of this work; it is merely a means of establishing that suitable dimensional equations have been obtained so that further information *There is actually a very small but regular valency effect due to the M components that cannot be removed by adjustments to the coefficients of DM or DN. The equation a = 0.3250, + 1.40, + 0.007~ + 0.109, where y has values 0, 1, 2, 3 and 4 for M components from groups VI, V, IV and III and Sn2+ respectively, reproduces the observed a values of the 33 phases to within 10.0151 A. However, the improvement is so small for the introduction of six more variables that it is ignored.
233
about the structure can be derived from them. Nevertheless, since we have reproduced the cell constants of 63 of the 64 phases known at present to within a mean error of IO.0131A, it is interesting to recall Geller’s [12] and Pauling’s [ 131 earlier calculations for the 32 phases known then. The compelling aspect of eqn. (1) for the transition metal M component phases is that it vindicates Pauling’s choice of a talc =
4 L (D, + 30,)
= 0.4470,
+ 1.342DN
Ji;4
rather than Geller’s choice of a talc
=
AL(DM+DN)= Jj;2
0.894DM + 0.894DN
Indeed, an empirical equation acalc= 0.447DM + 1.342DN - 0.031x + 0.03, with values of DM, DN and x as for eqn. (l), reproduces the observed lattice parameters of the 30 phases slightly better than does eqn. (1). However, it should be noted that our empirical eqn. (2) for phases with non transition metal M components does not agree exactly with Pauling’s choice. In calculating the cell dimensions of binary phases from atomic diameters, the minimum number of variables DM and DN required for n phases is 2n. If these diameters do not reproduce the cell dimensions sufficiently accurately, so that a further adjustment is required for each phase, the maximum number of variables required is then 3n and these give the cell dimension of each phase exactly. The average number of variables required per phase must therefore lie between two and three, and the effectiveness of any method of calculation can thus be judged accordingly. In calculating the cell constants of 32 phases both Geller [ 121 and Pauling [ 131 made adjustments to 20 atomic radii; in addition Pauling’s equation also has two coefficients ($ and+) so that ((2 X 32) + 20 + 2}/32 = 2.69 variables or parameters are needed per phase. Our eqns. (1) and (2) involve a total of 11 extra variables, whence {( 2 X 63) + 11)/63 = 2.17 parameters are required per phase. This lower number of variables per phase, compared with the results of Geller and of Pauling, is balanced by a somewhat lower precision in reproducing &,bs. However, our mean error of 10.0131 A should be compared with the mean error of IO.0571A in calculating a for the 63 phases from 0.447DM + 1.342DN when only two extra variables are introduced. If, more logically, CN 14 diameters are used for the N components, this mean error is increased by about 0.05 A as most deviations are positive. The significant point about the present results is that standard CN 12 atomic diameters can be used to describe the cell dimensions of a tetrahedrally close-packed phase in the same manner as has been applied to phases with other types of structures [ 1 - 71, although notably more parameters are required in the process. For example, both a and c parameters of 43 known phases with the Th2Zn1, structure can severally be described with only ((2 X 43) + 3}/43 = 2.07 variables per phase [l].
234
4. Atomic array controlling the cell edge of phases with transition metal M components That Pauling’s coefficients, 0.447& + 1.3420N, reproduce the observed cell dimensions of phases with the Al5 structure which have transition metal M components implies that the cell dimensions of these phases are indeed controlled by arrays of M-N contacts. This is because his coefficients arise from the assumption that the calculated M-N distances give a, when the atomic diameters are weighted according to the stoichiometric proportion in the MN3 formula. This observation that arrays of M-N contacts control cafor these phases is also confirmed by Waterstrat’s [ll] near-neighbour diagram if phases with transition metal M components are separated from those with other metal M components (Fig. 3). It is seen that the former lie parallel to the line for 12-4 M-N contacts, whereas the latter do not; instead they follow a line which is the weighted root mean square of the 2 N-N, 12-4 M-N and 8 N-N contacts (weighted according to their multiplicities), indicating that the interatomic distances in the interpenetrating CN 12 and CN 14 polyhedra, and their multiplicities, simultaneously control a (see Section 5). Nevertheless, the observation that the phases with transition metal M components lie along the line for 12-4 M-N contacts that is based on f (DM + D, ) weighting, rather than along a steeper line based on $ (& + 30,) weighting, is presently an enigma. The reason possibly rests m the valency effect involving the N components and its effect on J&/D,. It is significant that, in this case where the M-N contacts control the cell dimension, the atomic diameters are weighted in stoichiometric proportion, i.e. 1:3, as intuitively grasped by Pauling, since it seems we have not generally found such behaviour in analysing the cell dimensions of other phases where M-N contacts control the cell dimensions. For example, it does occur in phases with the sphalerite structure MN where a is proportional to l.ODM + l.O&, but it does not occur in phases with the AsNas structure [ 151; it certainly does not hold for Laves phases with the MgCua structure although here a may not be controlled by arrays of M-N contacts. Complete understanding of these facts must await further experience in quantitative analysis of the cell dimensions of metallic phases. At present the reason appears to be as follows: where the stoichiome~ic proportion applies in the Al5 and sphalerite structures the inverse ratio of the numbers of M-N contacts (12-4 in the former, 4-4 in the latter) is the same as in the stoichiometric formula (1:3 and l:l), whereas this is not the case for the AsNas structure (( 2 + 6) :( 1 + 3) = 2 : 1) where the stoichiometric proportion does not apply. 5. Expected dependence of cell edge on IIM or DN when more than one array of contacts s~ul~eou~y exert control One of the objectives of this paper is to discover how to combine the expected cell edge dependences on DM and/or DN for two or more arrays of
235
-04 -05 -06 -04
-04 _fie
e----q. -06
u
-07
DN
-ZJ-
____--
-05 ~
----
j
I-
i
-03 -04 -05
_
____------
_-_06 -07)
I- ;-r
1 1
-02 -03
t/ __/-@/_I
-04
_____------;2_;yJ_N -05
/
-06
F-
-09
’ 090
_____-----
095
loo
105
, IO
I15
120
%‘D~
Fig. 3. Lower part: near-neighbour diagram (see ref. 11; ref. 16, pp. 52 and 53, and references cited therein) for the Al5 structure. The broken line is the weighted root mean square for the 2 N-N, 12-4 M-N and 8 N-N contact lines. Upper two parts: phases with the Al 5 structure formed by transition metal, gold and mercury M components and 3d transition metal N components (top) or 4d and 5d transition metal N components (next), showing adherence to the line for 12-4 M-N contacts. The symbols are the same as in the caption to Fig. 1. Middle two parts: phases with the A15 structure formed by non transition metal M components and 3d transition metal N components (upper) and 4d or 5d transition metal N components (lower), showing adherence to the weighted root-mean square line. The symbols are the same as in the caption to Fig. 2. atoms or sets of interatomic contacts, when they act simultaneously in controlling the cell edges of phases with a given structure. This is a problem which has not yet been tackled. We have as examples to consider the following: (1) rare earth dihydrides MHs with the fluorite structure, where it seems that (110) M-M and (111) M-H contacts simul~eou~y control a; (2) rare earth nitrides MN with the rock salt structure where (110) M-M and (100) M-N arrays simultaneously control (x [ 5 J ; (3) tetrahedrally close-packed phases with the Al5 structure and non transition metal M components. In the carbides and nitrides with the rock salt structure and the dihydrides with the fluorite structure, transition metal M phases have an CI dependence on l.4DM and show a valency effect (Fig. 4). Since the M-M distance equals a@, the l&Z& dependence of a indicates that the M-M
236
Fig. 4. The variation of (I with DM for transition metal and rare earth M dibydrides with the fluorite structure (upper), for carbides with the rock salt structure (middle) and for nitrides with the rock salt structure (lower). The broken lines represent e values which equal fiD~. Adjusted rare earth diameters, except for lanthanum and lutetium, are used [4]. contacts control a, even though the observed distances are generally slightly greater than I&., for CN 12 and the M-N distances are closer than the radius sums. This, however, is not too surprising since it has heen shown for the true Hlgg interstitial phases (RN/R, < 0.59) with the hexagonal AICraC
structure that the very compressed M-C contacts exert no influence in controlling the cell dimensions [6] . In contrast, rare earth M nitrides with the rock salt structure and dihydrides with the fluorite structure have a dependences on 2.0&, and 2.24, respectively, indicating that both the (110) M-M and M-N contacts act together in controlling a, since the former give an expected fiEu and the latter give an expected l.O&, or 0.82&, dependence. A combination of the two effects by simple addition is perhaps to be understood because in the rock salt structure the octahedra of N atoms surrounding M and the cube-octahedra of M atoms surrounding M are quite independent and are not connected to each other as polyhedra. Similarly in the fluorite structure
237
the cube of N atoms su~o~d~g M and the cube-octahedra of M atoms surrounding M are entirely independent of each other. Thus the si~ation differs from that in the tetrahedrally close-packed A15-type structure, where the three different sets of neighbours about N, with contacts to N which combine to control a, are all part of the same CN 14 polyhedron about N. Thus perhaps there is good reason why their individual influences on a combine in some form of average to give the expected dependence of a on DN , rather than combining by simple addition as in the case of rare earth nitrides with the rock salt structure and d~y~ides with the fluorite structure. Finally, one further demonstration can be given to attest to the fact that the different electronic structure of the rare earths, compared with that of the transition metals, results in their M-N contacts also controlling the cell edge in interstitial rock salt and fluorite structures, whereas M-N contacts do not control the cell edge of transition metal compounds with these structures. The rare earth carbides with the rock salt structure are all defective having the formula MsC. With only one-third of the carbon sites occupied s~tistic~y, it is improbable that the M-C distances would exert any influence on the length of the cell edge. In contrast with the stoichiometric rare earth nitrides MN this is exactly what is observed: for the rare earth carbides a depends on 1.4&, exactly the same as for the stoichiometric transition metal carbides MC (Fig. 4), indicating that it is controlled only by the (1101 M-M contacts [5] . Thus we perceive a lemma for understanding how the effects of more than one set or array of ~~mtomic contacts may combine to control the cell dimensions. The lemma states that if two (or more) atomic arrays combine together in controlling the cell dimensions and if the two arrays give coordination polyhedra about the central atom that are completely independent of each other, then the expected cell edge dependences on each array combine additively. If, however, the atoms involved in the different sets of interatomic contacts are part of a single coordination polyhedron about the central atom, then the expected dependences of the cell edge on the different sets of contacts combine together in some average, which might also be weighted according to the multiplicities of the contacts. Indeed, it has already been shown in Section 4 that phases with non transition metal M components follow, on the near-neighbour diagram, the line which is the weighted root mean square for the 2 N-N, 12-4 M-N and 8 N-N contacts (weighted according to their multiplicities). Although this line is not quite linear, it can be represented approximately by the equation (I’M -ufiJZ)/B,
= -0.455
+ (D,/D,
- 1.1)0.63
(3)
whence a = 0.453DM + 1.407&. If, however, the weighted average is confined to the 10 N-N and the 4 N-M contacts in the CN 14 polyhedron surrounding N, the equation of the line so obtained is a = 0.323& + 1.4160, which reproduces eqn. (2) almost exactly, indicating that it is the contacts in
238
the polyhedron about N that control the cell dimensions of this group of phases. Atomic diameters are not now weighted in proportion to stoichiometric composition, as M-N contacts alone do not control a.
6, Pseudo-valency effect in phases with transition metal M components A true valency effect, as explained in Section 1, is a function of the group number or the valency of the component atoms alone, although it depends not directly on this but on the difference between the (effective) valency in the phase and that in the elemental structure from which its CN 12 diameter was determined. However, in the apparent valency effect involving phases with transition metal, gold and mercury M components (eqn. (1)), we find a difference not only in respect of group but also in respect of period for chromium alloys, where the valency effect is zero, and molybdenum alloys, and a smaller effect (which we have ignored) for vanadium alloys on the one hand and niobium and tantalum alloys on the other hand. Any geometrical constraint that is separate from the arrays of atoms which control the cell dimensions (here the arrays of M-N contacts; see Sections 3 and 4) could result in effects similar to those found for a true valency effect, but could be dependent on both group and period of the atom - as is atomic size. Indeed, here we have evidence from these data of such an effect being combined with a true valency effect. It results from the influence of DN through the 2 N-N contacts which are very compressed. These contacts may make an additional contribution to increase a, which is not included in the contribution arising from the M-N contacts. The relative size of this contribution gets smaller as the true valency effect gets larger (i.e. on going from group VI to group IV N components). This is the first occasion where such an effect (which we shall caIl a pseudo-valency effect) has been revealed, but there seems little doubt about its indirection since neither the true valency effect nor the geometrical constraint is observed in alloys of non transition metal M components, where a weighted average of the contributions of the 2 N-N, 4 M-N and 8 N-N contacts controls the cell dimensions of the phases (Section 5). Fu~hermore, the effect does not result from the geometrical constraint alone, since values of the adjustment of acalc are not linear with DN or even &/I&+
7. Discussion This dimensional analysis of phases with the A15 structure has established the prime ~po~~ce of contacts between transition metals in controlling the cell dimensions. When both components are transition metals, the unlike 12-4 M-N contacts control the cell dimensions and the N atoms have an average number of electrons per contact which is different from that of their elemental structures adjusted to CN 12, since there is a valency
239
effect (to some extent modified by geometrical constraints arising from the 2 N-N contacts). When the M component is not a transition metal, like contacts (N-N) between the transition metal atoms assume a greater role and the non transition metal-transition metal M-N contacts assume a lesser role, since now the cell dimension is controlled by a weighted average of all three types of contact in the CN 14 polyhedron. The 8 N-N distances are always larger than DN . As these come closer the free energy of the phase is lowered with a con~omi~t penalty for further compressing the 2 close N-N contacts; however, the low multiplicity of these contacts favours the process. The 4 M-N contacts remain relatively close to their appropriate radius sums. The overall balancing of these energies which involves both the interatomic distances and the multiplicity of the contacts may, and in most Al5 phases does, result in the atoms being closer than expected from valence electron considerations. The consequences of such energy balancing have already been explicitly recognized in Laves phases with the MgCu, structure [ 161, which is also ~~~e~y close packed. These findings are equivalent to a statement that in transition metaltransition metal phases it is the M-N contacts of the icosahedra of N atoms surrounding M that control a, whereas in non transition metal-transition metal phases it is the contacts in the CN 14 polyhedra of 4 M and 10 N atoms surrounding N that control a. The former, certainly, is new information!
8. Conclusions (1) It is possible to analyse the cell dimensions of a tetrahedrally closepacked structure in which several sets of interatomic contacts simultaneously may control the cell dimensions, and to obtain equations in terms of the CN 12 diameters of the component atoms (and valency effects if they exist) which reproduce the observed cell dimensions on average to within a fraction of a per cent. (2) The arrays of atoms that control the cell dimension of phases with the Al5 structure which are formed with transition metal M components are different from those that control the cell dimension of these phases formed with other M components. This difference is equivalent to the M-N distances in the icosahedron of N atoms surrounding M controlling a in the former, and the contacts in the CN 14 polyhedron of 4 M and 10 N atoms surrounding N controlling u in the latter. (3) When more than one atomic array, or set of interatomic distances, control the cell dimensions the individual expected dependence6 of a on Du or DN combine to give the overall dependence. This is additive if the coordination polyhedra formed by the two arrays are independent of each other; however, they combine by an averaging process if the sets of interatomic distances are involved in a single (i.e. the same) coordination polyhedron.
(4) If a geometrical constraint resulting from atomic contacts which are different from those that generally control the cell dimensions exists, it may appear in the analysis in a manner somewhat similar to a valency effect; valency effects occur when the valency of an atom in the phase is different from that in its elemental structure from which its CN 12 diameter was determined. It may be possible to make some distinction between the two effects as cell dimensions altered by valency changes should depend on the group number of the atom alone, whereas those resulting from geometrical constraints (size per se) should depend on both the group number and the period of the atoms concerned. Acknowledgments 1 am grateful to Dr. R. M. Wa~rs~at for providing me with a copy of his paper prior to its publication. This work was supported by a grant from the Natural Sciences and Engineering Research Council of Canada.
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