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Dimensional analysis of phases with the A15 structure: Further considerations
Journal of the Less-Common Metals,
81 (1981)
161
161 - 172
DIMENSIONAL ANALYSIS OF PHASES WITH THE Al5 STRUCTURE: FURTHER CONSIDERATIONS
W. B. PEARSON Departments of Physics N2L 3Gl ~Ga~ada)
and of Chemistry,
University
of Waterloo,
Waterloo,
Ontario
(Received April 22,198X)
Summary A previous study of the cell dimensions of MNs phases with the Al5 structure as a function of the atomic diameters L), and DN for CN 12 is continued. It is now shown that, when several contacts from an atom to its M and N neighbours act together to control the cell dimension Q, the dependence of Q on D, cannot be independent of D, and vice versa. This leads to new non-linear equations for the dependence of a on D, and DN that reproduce the observed a values of the stoichiometric phases to about the limit of their accuracy, from CN 12 atomic diameters alone. The observations now suggest that the cell dimensions of the transition metal-transition metal (T-T) phases are controlled by all contacts in both coordination polyhedra acting together rather than by the 12-4 M-N contacts alone. Also discussed are the size of the tin atom in phases with the Al5 structure, the valency effect of the N atoms in T-T but not in non-transition metal-transition metal (B-T) phases and the relative atomic volumes in T-T and B-T phases. Finally, the dimensional behaviours of T-T and B-T phases with the A15 and CsCl structures are compared and found to be completely analogous, so the particular properties of phases with the Al5 structure cannot be attributed to their two very close N-N contacts.
1, Introduction A recent analysis of the unit cell dimensions of phases MN3 with the Al5 structure was undertaken [l] to discover how several contacts from a single atom to its surrounding M and N neighbours combine to give the expected dependence of a on D, and DN. The study revealed a distinct difference in dimensional behaviour between those phases in which both M and N components are transition metals (T-T phases) and those in which the M component is not a transition metal (B-T phases), and it was shown that cell dimensions of the phases could generally be reproduced with sufficient accu0022-5088/81/0000-0000/$02.50
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racy from atomic diameters for coordination number (CN) 12; it is not necessary to assume special radii [ 2,3] . There remained questions for future consideration such as the extent to which the dependence of a on D, is independent of DN and vice versa when several contacts from an atom to its M and N neighbours act together to control a and questions regarding the observation of valency effects in T-T but not B-T phases. (A valency effect occurs when the average number of electrons per bond (adjusted to CN 12) of, for example, the N components in phases with a given structure differs from that of the N components in their elemental structures (adjusted to CN 12). Since these differences will not be the same for N components of valency 1,2, 3,. . . , the effect is detected in plots of Q uersus DM and a versus D, for CN 12 and can be interpreted in terms of the different CN 12 diameters of the N components in phases with the structure compared with their elemental CN 12 diameters.) In addition several details of the analysis [1] have been questioned in discussion with colleagues. In this paper we address these questions. For convenience, it will be recalled [1] that the cell dimensions of T-T phases with the A15 structure were reproduced to an average accuracy of 10.010 I W by the equation a = 0.45&,
+ 1.4ODn - 0.035s
- 0.132
(1)
where DEIIand DN are the elemental diameters of the M and N components for CN 12 and 5’ is a valency effect with values 0, 1,2 and 5 for chromium, molybdenum, group V metal N components and group IV metal N components respectively. The cell dimensions of B-T phases were reproduced less satisfactorily (average precision, 10.01651 A) by the equation a = 0.325&
+ 1.400,
+ 0.126
(2)
2. Size of the tin atom in phases with the Al5 structure In ref. 1 a diameter of 3.246 A was assumed for the tin atom, corresponding to a valency of 2. This assumption has been questioned, particularly as Geller [2] and Pauling [3] both found a size corresponding to a valency of 4 to be appropriate for tin in these phases. Perhaps the most convincing way to show that a diameter for tin corresponding to a valency of 4 is less suitable than that corresponding to a valency of 2, which is free of any assumptions, is to plot the 2 N-N distances a/2 minus DN against DM (Fig. l), the diameters being taken for CN 14 and CN 12 respectively. The major data set of the niobium and tantalum phases clearly varies linearly; placing the points for the tin phases at DM = 3.246 A, corresponding to Sn(II), puts them on this line, whereas placing them at D, = 3.090 A for Sn(IV) removes them from the line. Similar behaviour is found for the tin alloys of vanadium and molybdenum, and similar results are also obtained when the observed M-N distance a5112/4 minus (& + ON)/2 is plotted against Q,, (Fig. 1).
163
Fig. 1. B-T phases with the Al5 structure: upper lines, difference between the observed M-N distance a5r1’/4 and (0~ + DN)/2; lower lines, difference between the observed 2 N-N distances a/2 and DN plotted against DM for niobium (O), tantalum (n), molybdenum (0) and vanadium (X ) alloys. Tin alloys (M) are plotted against diameters appropriate for valencies of 2 and 4 as indicated. D, is for CN 12 and D, is for CN 14.
In contrast it will be recalled that in both ref. 2 and ref. 3 the assumption was made that the cell edge could be calculated from the M-N distances with suitably adjusted atomic radii. With this assumption, a size corresponding to Sn(IV) seemed to be suitable. However, with our eqn. (4) (Section 3), the average error in acalc for four tin phases using the CN 12 diameter of Sn(I1) is approximately 10.0081A, whereas it is approximately 10.0481A using the CN 12 diameter of Sn(IV).
3. Interpretation of the cell dimensions of B-T phases In the analysis of the cell dimensions of B-T phases [l] we adopted a procrustean bed with length tailored to fit the most complete data set (niobium and tantalum alloys with a of the order of 0.3&) in order to reproduce the observed cell dimensions as accurately as possible with a minimum of added parameters. Thus we ignored the question whether the dependence of u on D, is independent of DN and vice versa when several different contacts from an atom to its M and N neighbours combine to control the cell dimension, a condition that was first met in the analysis of phases with the Al5 structure. Whereas in ref. 1 the dependence of a on D, for molybdenum phases agrees well enough with that adopted for niobium and tantalum phases (and there are so few data points for the zirconium and chromium phases
that any slopes would be acceptable), the vanadium phases definitely show a greater dependence of a on DM, e.g. 0.42DM. Thus it is possible that the coefficient of the dependence of Q on D, increases with decreasing DN and data for the B-T phases could be interpreted in this way. Since, for example, a is expected to depend on 2Dn for the indi~d~ 2 N-N contacts and on 0.894D~ + 0.894Dx for the indi~d~ 4 M-N contacts, whereas the observed dependence of a on 1.400, is in one case larger and in the other case smaller, it is by no means obvious that the observed dependences of a on D, and DN should be independent of each other. Indeed, it can now be demonstrated that they are not. The differences in the observed interatomic distances and the appropriate radius sums in the B-T phases can be expressed with weighting for rn~~p~city as I,,=,(;
-Dn)
+4(q-+DM-+Dn)
+8”is-Dnj
(3)
which gives a = 0.3240, + 1.4250, + k with coefficients approximately equal to those of eqn. (2) if Q, the empirical factor reducing the weighting of the remote 8 N-N contacts, has a value of 0.6. In eqn. (3) the value of k must be constant, corresponding to the constant term in eqn. (2). We now suppose that the value of DM in eqn. (3) is increased by x: and a increases by 2~15~~~in order to maintain the contribution to 14k constant for the term ~5~‘~ - 2DM - 20,. This will increase the difference in the first term of eqn. (3) by 2x/5112 and the difference in the third term by 4 X 61’2,x/51’2, so that k will not satisfy its required condition of constancy. Therefore the dependence of a on D, cannot be independent of D, and vice versa under these conditions where several contacts from the same atom to its M and N neighbours combine to control u. Such findings do not change the conclusions in ref. 1 that the contacts in the polyhedron about N are responsible for controlling a in B-T phases; they merely show that the contribution of D, to this control increases as DN decreases. This situation can be expressed in terms of the weighting factor 0 d ar< 1 that has a value of 0 below D, = 2.484 A and then increases linearly with DN, achieving its value of 1 at DN = 3.121 A and giving rise to the following equation: u=
0.1429DM - 1.7231 DN + 0.8971 DN2 - 0.6034 0.5493D3, - 1.1334
(4)
Equation (4) is valid in the range of DN values from 2.484 to 3.121 A and reproduces the observed cell dimensions [4] of 27 B-T phases from CN 12 diameters [ 53 to better than 10.0081A (neglecting five phases with deviations that are greater than 3~) or to 10.0121A for all 32 phases whose DN values lie within the valid range. This significant improvement over the result of eqn. (2) is achieved at the expense of three additional parameters in the equation and confirms that the dependence of a on D, is not independent of DN and vice versa.
165
4. Interpretation of the cell dimensions of T-T phases (1) In ref. 1, Fig. 3, phases with the same M components on the nearneighbour diagram (NND) were connected by lines and these lay parallel to the line for 12-4 M-N contacts determined from ~25l’~/4 = (Dnn:+ DN)/2 rather than parallel to an M-N contact line determined from a51j2/4 = (DM + 3&)/4 which gives coefficients of the dependence of a on Q,., and DN that are similar to those of eqn. (1). (An NND [6] is a plot of a reduced strain versus &,,/DN for the interatomic contacts occurparameter (DM - d,)/D, ring in the structure and for actual phases with the structure. dEAis any arbitrarily chosen M-M distance in the structure expressed in terms of a (and/or other cell dimensions) such that dM = fu = I&. This establishes the zero for the (D, - d,)/D, scale.) It is now seen that these lines connecting the phases are not significant since they are influenced by the valency effects of the different N atoms to which the M atoms are combined. If lines are drawn through phases formed by the same N components instead, they lie nearly parallel to the line for M-N contacts, determined from a5112/4 = (D, + 3&)/4, and the enigma in ref. 1 disappears. If indeed corrections for the valency effect (eqn. (1)) are applied to the elemental CN 12 diameters of the N components in calculating the positions of the phases on the NND, they all lie on a single line (Fig. 2). The correc-
-03
-04
-05 UM- dM Do -06
-07
090
095
100
105
I IO
II5
120
DM’DN
Fig. 2. An NND (0~ -a3 1’2/2)/D~ vs. DMIDN for the Al 5 structure. T-T phases of N components: +, titanium; 0, zirconium; X, vanadium; 0, niobium; 0, tantalum; A, chromium;V, molybdenum. In plotting these positions, DN values adjusted for the valency effect have been used. The diagram also shows the average of the lines for the 2 N-N, 12-4 M-N and 8 N-N contacts (- - -) with weighting for multiplicity of the contacts (unlabelled) and the line for the 12-4 M-N contacts weighted for composition (i.e. with (DM + 6&)/4).
166
tions for the valency effect calculated from eqn. (1) are respectively 0 A, -0.025 A, -0.050 A and -0.125 A for the diameters of chromium, molybdenum, group V N components and group IV N components. (2) In ref. 1 we preferred Pauling’s equation [ 3) a51i2/4 = (D, + 30,)/4, which gives a = 0.44’7D,
+ 1.3420,
(5)
as the explanation of the dimensional behaviour of the T-T phases (eqn. (1)) rather than ref. 1, eqn. (3), which gives essentially the same result, since there was then no reason to disagree with previously accepted work. Reference 1, eqn. (3), represented the line on the NND that was obtained as the r.m.s. average of the 2 N-N, 12-4 M-N and 8 N-N contact lines, weighted for contact multiplicity. Although such an averaging process happened to be fairly satisfactory because of the positions of the lines on the diagram, it is not generally satisfactory, as the result is subject to the zero chosen for the NND. According to the new procedure in ref. 7, the averaged positions of the three contact lines on the NND with weighting for multiplicity give a line that lies parallel to the phases plotted with valency correction for DN, as shown in Fig. 2. That this line lies considerably above these points only means that the observed coefficient of the dependence of a on DN is about 0.055DN larger than 1.335DN calculated for the line (i.e. about 1.390 DN). Thus either Pauhng’s eqn. (5) based on control of a by the M-N contacts with weighting of the diameters for composition or an equation depending on the average of the 2 N-N, 12- 4 M-N and 8 N-N contacts with weighting for their multiplicity could account for the observed variation of a with D, and DN for the T-T phases. We now seek evidence to distinguish between these two possibilities. First, we have recently examined MN, phases with many crystal structures for which the cell dimensions are controlled by M-N contacts, searching for definite evidence that it is necessary to weight the atomic diameters according to the stoichiometric proportions in calculating the cell dimensions. The results are at present inconclusive but we have not been able to show that it is a necessary procedure. Secondly, the valency effect observed for the N components in these phases (discussed in Section 5) is seen to involve all three different contacts to the N atoms which favours control of a thereby rather than by the M-N contacts alone. Thirdly, if we assume that the 2 N-N, 12-4 M-N and 8 N-N contacts control a and that the dependences of a on DM and DN are not independent of each other (certainly the dependence of a on DM for the vanadium phases is rather larger than that for the titanium and niobium phases) and follow the procedure adopted for the B-T phases (Section 3), we obtain the equation a=
0.22220,
- 1.4782DN + 0.8444Dn2 - 0.6833 0.51720,
- 0.9672
- 0.0255
(6)
which is valid between DN values of 2.458 W (a! = 0) and 2.984 A (Q = 1). S is the valency effect with values of 0, 1,2 and 6 respectively for chromium, molybdenum, group V N components and group IV N components. This
167 equation reproduces the observed Q values [4] to 10.0061 A of 27 phases whose N components have diameters within this range, excluding PtCr, whose reported cell dimension is clearly too small. (We have subsequently discovered that the a value for PtCrs is wrongly listed; it should have a value of about 4.714 A, bringing the agreement of the calculated and observed values to within this range.) Such precision probably exceeds the limit to be expected in view of the uncertain composition of the phases and other experimental errors, and it is a significant improvement over that provided by eqn. (1); it is also noted that the precision is achieved with standard CN 12 diameters of the atoms [ 51. Such evidence suggests (although still without certainty) that we were probably wrong in concluding [l] that the cell dimensions of the T-T phases are controlled by the M-N contacts rather than by all interatomic contacts in both coordination polyhedra, suitably weighted for multiplicity, with the influence therein of the remote 8 N-N contacts increasing as DN increases.
5. Valency effects The valency effect involving the N components in the T-T phases is a recognition that a for the vanadium and titanium phases, for example, is smaller (relative to the chromium phases) than can be accounted for by the differences in their atomic diameters. Vanadium and titanium are therefore assumed to use more valency electrons in bonding to their neighbours than they normally possess, and hence their smaller sizes. Since the M components come from group VII and higher groups and include gold and mercury, they can provide the required electrons without altering their normal valencies and CN 12 diameters. Using Pauling’s equation [S] R(1) -R(n) = 0.3 log n, the number of valency electrons required by chromium, molybdenum, group V and group IV N components can be calculated from eqn. (1) and also from the interatomic distances for each of the individual phases and then averaged. That the results of these two calculations agree closely has only the elements of a circular argument, except that it demonstrates self-consistency since in the calculations for the individual phases it was necessary to assume normal valencies and D, values for CN 12 for the M component, so that differences between a5”‘/4 and (D, + On)/2 were attributed to the N atoms. Furthermore, it emphasizes that the two close N-N contacts result from valency bonding rather than from geometrical constraints elsewhere in the structure. No such treatment can be accorded to the B-T phases. First, there are no detectable differences in a for, say, the chromium, vanadium and titanium phases that cannot be attributed to the atomic diameters of these elements for CN 12 alone. Thus the lack of a valency effect is consistent with insufficient valency electrons of the M components, which are elements of groups II - VI. Secondly, the B-T phases show a characteristic common to many intermetallic phases that the interatomic distances are generally shorter than
168
can be accounted for by the number of available valency electrons. Indeed, B-T phases of niobium formed by M components larger than DNI= 3.28 A do not even have enough valency electrons to account for the 12-4 M-N distances alone! This leads us to consider the relative atomic volumes of the two groups of phases. 6. Comparison of cell volumes The differences of behaviour of the T-T and B-T phases result in ratios of the observed cell volumes to those calculated as 2V, + 6V, (where V is the elemental atomic volume) in the approximate ranges 1.00 - 1.047 for the T-T phases and 0.90 - 0.98 for the B-T phases. (The V, values used here are those corrected for the valency effects. This correction is applied by multiplying the elemental atomic volume by the ratio Rv3/R3, where R, is the CN 12 radius corrected for the valency effect and R is the elemental CN 12 radius.) The first range is abnormal for intermetallic phases, whereas the second range is normal since volume ratios generally lie in the range from about 0.80 to 1.0 [9] . Nevertheless, some reason must be found to explain why the structures of the B-T phases are more compressed when there are apparently less (insufficient) valency electrons available. The differences in volume ratios are not themselves the result of the valency effect, since the average for the T-T phases of chromium (whose valency effect is zero) is 1.02, whereas it is 0.917 for the B-T phases of chromium. Figure 1 shows that a51j2/4 - (&,, + DN)/2 decreases rapidly with increasing I&, which must be an unfavourable feature of the structure as the M-N contacts are becoming more compressed. The reason for this is that the observed dependence of a on 0.325& (eqn. (2)) is much smaller than the dependence of a on 0.8940, that is required to keep this difference constant as D, increases. Therefore the obvious solution for the structure is to increase the dependence of a on DNI.However, this is impossible because of the influence also of the 2 N-N and 8 N-N contacts in the averaging process with weighting for multiplicity. Thus the physical significance of the averaging process assumed in ref. 1 is apparent when several different sets of contacts from the central atom to neighbours forming its convex coordination polyhedron act together to control a. The next obvious solution for the structure to avoid the compression is for (I to increase for all B-T phases, but again there appear to be reasons why this may not be possible. The only feature apparent in Fig. 1 that limits the stability of the phases is that a51j2/4 - (DNI+ On)/2 must be approximately zero or negative. The only phase for which it is clearly positive (i.e. M-N contacts “not made”) is SiNb3 which is metastable. Possibly these two reasons taken together account for the compression in the B-T phases compared with the T-T phases, and it must therefore be accorded to the geometrical arrangement in the structure, whereas the T-T phases appear to have sufficient outer electrons for the interatomic distances to be controlled by normal valency bonding interactions.
169
7. Comparison of dimensional behaviour of T-T and B-T phases with the Al5 and C&l structures The question arises whether the valency effect of the T-T phases and the differences in volume ratios of the T-T and B-T phases are particular to the atomic arrangement in the Al5 structure and especially to the two close N-N contacts. This can be answered by examining similar phases with the CsCl structure where such contacts do not occur. In the CsCl structure the M-N contacts are expected mainly to control a but there is also the possibility that M-M or N-N contacts influence a if a < DM or u < DN . Furthermore, there are a reasonable number of T-T and B-T phases with the CsCl structure, so that some analysis is possible. Figure 3 shows that the behaviour of the T-T phases with the CsCl structure is entirely analogous to that of T-T phases with the Al5 structure (see ref. 1, Fig. 1). There is a valency effect involving the group IV and V transition metals titanium, zirconium, hafnium, vanadium and tantalum (N) and also the volume ratios of the phases with V, values corrected for a valency effect are all greater than unity. The cell dimensions are given to an accuracy of 10.0061A (except for cobalt and nickel phases of titanium, zirconium and hafnium) by the equation a = 0.66D,
+ 0.65DN - 0.075s - 0.424
(7)
where S, the valency effect, has the relative values 0 for FeCo and FeRh (i.e. alloys formed by two M metals), 1 for group V N metals and 2 for group IV N components. Thus the transition metal valency effect and the volume ratios greater than unity are nothing particular to the occurrence of the two close N-N contacts in T-T phases with the Al5 structure. Although the data for B-T phases with the CsCl structure on a uersus D, and a uersus D, plots are somewhat scattered and various, there is no evidence of a valency effect involving the transition metals (N) in agreement
I 24
2.6
28
26
DN (ii, (a)
I
I/
I
28
, 30
I
I 32
D,(i) @I
Fig. 3. T-T phases (MN) with the CsCl structure: (a) a us. DN; (b) a at D, = 2.7 A us. DM. The M components are identified as follows: 0, titanium; A, zirconium; +, hafnium; X, vanadium; 0, tantalum; 0, iron.
27
I/
26 t 2.4
27 O,(i)
30
33
3.6
D,(i)
(b)
(a)
Fig. 4. B-T phases (MN) with the CsCl structure: (a) a us. DN; (b) a at DN = 2.7 w US.DM. M components are identified as follows: 0, beryllium; 0,silicon;~, arsenic; X, gallium; 0, aluminium ; 0, magnesium ; +, scandium; A, indium.
observations for phases with the Al5 structure. Secondly, when a 2 DM (and including two indium and certain scandium (M) phases where a < &), the behaviour is seen (Fig. 4) to resemble that of B-T phases with the Al5 structure, a being given by with
u = 0.43&
+ 0.650,
+ 0.40
(8)
to an average accuracy of 10.0171A, neglecting the discordant results of five out of 29 phases. From direct plots of a uersus DM it is also confirmed that a is approximately proportional to 0.40, when a 2 DNI. The cell volume ratios of these B-T phases are all less than unity, similar to those of B-T phases with the Al5 structure. Most of the ratios lie in the range from 0.78 to 0.92. The behaviour of B-T phases with both structures is further similar in that a for the CsCl phases appears to be controlled by the combined action of the 8-8 M-N and 6 N-N contacts as has already been demonstrated for the 2 N-N, 4 M-N and 8 N-N contacts of B-T phases with the Al5 structure [1] . Phases with the CsCl structure formed by such N components as copper, silver, gold, zinc, cadmium and mercury and rare earth M components [lo] have an a dependence on D, of 1.0 or slightly less, which corresponds to M-M contacts controlling a, whereas phases of the same N components with non-rare-earth M atoms have an a dependence of about 0.577&, which corresponds to M-N contacts controlling a (u31j2/2 = (Dw + &)/2, whence a = 0.577& + 0.5770N). The dependence of a on O.43& and 0.65DN for B-T phases when a > D, is clearly different and requires the assumption that the 8 M-N and 6 N-N contacts from the N atom to its surrounding dodecahedron act together in controlling a. Averages of the lines for these contacts
171
on the NND or averages of the difference of a - DN and ~3l’~/2 - (D, + 0,)/2, both weighted for the multiplicity of the contacts [ 71, give approximately a = 0.320, + 0.76&. A less strong weighting of the large N-N distances would increase the dependence of a on D, and decrease that of a on DN, agreeing sufficiently well with the observed values of eqn. (8), and a more complex equation analogous to eqn. (4) is indicated. Thus not only is the principle of dimensional control of B-T phases with the CsCl structure shown to be similar to that of B-T phases with the Al5 structure but it is also shown to be different to other phases with the CsCl structure that do not contain transition metals. The overall similarity of the dimensional behaviour of T-T and B-T phases with the Al5 and CsCl structures indicates that the behaviour of phases with the Al5 structure cannot be attributed to the particular influence of the two very close N-N contacts. 8. Conclusions (1) It now appears most probable that the cell dimensions of T-T phases are controlled by all 2 N-N, 12-4 M-N and 8 N-N contacts in both coordination polyhedra acting together rather than by the 12-4 M-N contacts with weighting of the diameters for composition. (2) The cell dimensions of the B-T phases are controlled by the 2 N-N, 4 M-N and 8 N-N contacts in the polyhedron about N acting together. (3) When several contacts from an atom to its M and N neighbours act together to control the cell dimension a, its dependence on D, is not independent of D, and vice versa. (4) Equations for calculating a in which this is recognized, being based on a weighting of contact multiplicity and a variable weighting of the remote 8 N-N contacts that depends on D,, reproduce the observed a values from standard atomic diameters for CN 12 alone to about the limit of their experimental accuracy. (5) Valency effects involving the N atoms in T-T phases are compatible with the ability of the M atoms to provide the necessary electrons, such that interatomic distances in these phases appear to be controlled by normal valency bonding, whereas in B-T phases insufficient electrons are present to account for the interatomic distances and they appear to result from geometrical constraints within the structure. (6) All evidence provided is consistent with tin having a size corresponding to a valency of 2 in phases with the Al5 structure. Acknowledgments I wish to thank Professor E. Hellner, Drs. R. M. Water&rat and E. Koch and Mrs. Reinhardt for critical comments and discussion of ref. 1 and for providing me with various calculated data.
172
This work was supported by a grant from the Natural Sciences and Engineering Research Council of Canada.
References 1 2 3 4 5 6 7 8 9 10
W. B. Pearson, J. Less-Common Met., 77 (1981) 227. S. Geller, Acta Crystallogr., 9 (1956) 885. L. Pauling, Acta Crystallogr., 10 (1957) 374. R. M. Waterstrat, J. Solid State Chem., (1981) in the press. E. Teatum, K. Gschneidner and J. Waber, LA Rep. 2345, 1960 (Los Alamos Scientific Laboratory) (available from U.S. Department of Commerce, Washington, DC). W. B. Pearson, Acta Crystallogr., Sect. B, 24 (1968) 7,1415. W. B. Pearson, Acta Crystallogr., Sect. B, 37 (1981) in the press. L. Pauling, J. Am. Chem. Sot., 69 (1947) 542. W. B. Pearson, The Crystal Chemistry and Physics of Metals and Alloys, Wiley-Interscience, New York, 1972, pp. 570, 658. W. B. Pearson, Philos. Trans. R. Sot., London, Ser. A, 295 (1980) 415,431,432.
81 (1981)
161
161 - 172
DIMENSIONAL ANALYSIS OF PHASES WITH THE Al5 STRUCTURE: FURTHER CONSIDERATIONS
W. B. PEARSON Departments of Physics N2L 3Gl ~Ga~ada)
and of Chemistry,
University
of Waterloo,
Waterloo,
Ontario
(Received April 22,198X)
Summary A previous study of the cell dimensions of MNs phases with the Al5 structure as a function of the atomic diameters L), and DN for CN 12 is continued. It is now shown that, when several contacts from an atom to its M and N neighbours act together to control the cell dimension Q, the dependence of Q on D, cannot be independent of D, and vice versa. This leads to new non-linear equations for the dependence of a on D, and DN that reproduce the observed a values of the stoichiometric phases to about the limit of their accuracy, from CN 12 atomic diameters alone. The observations now suggest that the cell dimensions of the transition metal-transition metal (T-T) phases are controlled by all contacts in both coordination polyhedra acting together rather than by the 12-4 M-N contacts alone. Also discussed are the size of the tin atom in phases with the Al5 structure, the valency effect of the N atoms in T-T but not in non-transition metal-transition metal (B-T) phases and the relative atomic volumes in T-T and B-T phases. Finally, the dimensional behaviours of T-T and B-T phases with the A15 and CsCl structures are compared and found to be completely analogous, so the particular properties of phases with the Al5 structure cannot be attributed to their two very close N-N contacts.
1, Introduction A recent analysis of the unit cell dimensions of phases MN3 with the Al5 structure was undertaken [l] to discover how several contacts from a single atom to its surrounding M and N neighbours combine to give the expected dependence of a on D, and DN. The study revealed a distinct difference in dimensional behaviour between those phases in which both M and N components are transition metals (T-T phases) and those in which the M component is not a transition metal (B-T phases), and it was shown that cell dimensions of the phases could generally be reproduced with sufficient accu0022-5088/81/0000-0000/$02.50
0 Elsevier Sequoiai~inted
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racy from atomic diameters for coordination number (CN) 12; it is not necessary to assume special radii [ 2,3] . There remained questions for future consideration such as the extent to which the dependence of a on D, is independent of DN and vice versa when several contacts from an atom to its M and N neighbours act together to control a and questions regarding the observation of valency effects in T-T but not B-T phases. (A valency effect occurs when the average number of electrons per bond (adjusted to CN 12) of, for example, the N components in phases with a given structure differs from that of the N components in their elemental structures (adjusted to CN 12). Since these differences will not be the same for N components of valency 1,2, 3,. . . , the effect is detected in plots of Q uersus DM and a versus D, for CN 12 and can be interpreted in terms of the different CN 12 diameters of the N components in phases with the structure compared with their elemental CN 12 diameters.) In addition several details of the analysis [1] have been questioned in discussion with colleagues. In this paper we address these questions. For convenience, it will be recalled [1] that the cell dimensions of T-T phases with the A15 structure were reproduced to an average accuracy of 10.010 I W by the equation a = 0.45&,
+ 1.4ODn - 0.035s
- 0.132
(1)
where DEIIand DN are the elemental diameters of the M and N components for CN 12 and 5’ is a valency effect with values 0, 1,2 and 5 for chromium, molybdenum, group V metal N components and group IV metal N components respectively. The cell dimensions of B-T phases were reproduced less satisfactorily (average precision, 10.01651 A) by the equation a = 0.325&
+ 1.400,
+ 0.126
(2)
2. Size of the tin atom in phases with the Al5 structure In ref. 1 a diameter of 3.246 A was assumed for the tin atom, corresponding to a valency of 2. This assumption has been questioned, particularly as Geller [2] and Pauling [3] both found a size corresponding to a valency of 4 to be appropriate for tin in these phases. Perhaps the most convincing way to show that a diameter for tin corresponding to a valency of 4 is less suitable than that corresponding to a valency of 2, which is free of any assumptions, is to plot the 2 N-N distances a/2 minus DN against DM (Fig. l), the diameters being taken for CN 14 and CN 12 respectively. The major data set of the niobium and tantalum phases clearly varies linearly; placing the points for the tin phases at DM = 3.246 A, corresponding to Sn(II), puts them on this line, whereas placing them at D, = 3.090 A for Sn(IV) removes them from the line. Similar behaviour is found for the tin alloys of vanadium and molybdenum, and similar results are also obtained when the observed M-N distance a5112/4 minus (& + ON)/2 is plotted against Q,, (Fig. 1).
163
Fig. 1. B-T phases with the Al5 structure: upper lines, difference between the observed M-N distance a5r1’/4 and (0~ + DN)/2; lower lines, difference between the observed 2 N-N distances a/2 and DN plotted against DM for niobium (O), tantalum (n), molybdenum (0) and vanadium (X ) alloys. Tin alloys (M) are plotted against diameters appropriate for valencies of 2 and 4 as indicated. D, is for CN 12 and D, is for CN 14.
In contrast it will be recalled that in both ref. 2 and ref. 3 the assumption was made that the cell edge could be calculated from the M-N distances with suitably adjusted atomic radii. With this assumption, a size corresponding to Sn(IV) seemed to be suitable. However, with our eqn. (4) (Section 3), the average error in acalc for four tin phases using the CN 12 diameter of Sn(I1) is approximately 10.0081A, whereas it is approximately 10.0481A using the CN 12 diameter of Sn(IV).
3. Interpretation of the cell dimensions of B-T phases In the analysis of the cell dimensions of B-T phases [l] we adopted a procrustean bed with length tailored to fit the most complete data set (niobium and tantalum alloys with a of the order of 0.3&) in order to reproduce the observed cell dimensions as accurately as possible with a minimum of added parameters. Thus we ignored the question whether the dependence of u on D, is independent of DN and vice versa when several different contacts from an atom to its M and N neighbours combine to control the cell dimension, a condition that was first met in the analysis of phases with the Al5 structure. Whereas in ref. 1 the dependence of a on D, for molybdenum phases agrees well enough with that adopted for niobium and tantalum phases (and there are so few data points for the zirconium and chromium phases
that any slopes would be acceptable), the vanadium phases definitely show a greater dependence of a on DM, e.g. 0.42DM. Thus it is possible that the coefficient of the dependence of Q on D, increases with decreasing DN and data for the B-T phases could be interpreted in this way. Since, for example, a is expected to depend on 2Dn for the indi~d~ 2 N-N contacts and on 0.894D~ + 0.894Dx for the indi~d~ 4 M-N contacts, whereas the observed dependence of a on 1.400, is in one case larger and in the other case smaller, it is by no means obvious that the observed dependences of a on D, and DN should be independent of each other. Indeed, it can now be demonstrated that they are not. The differences in the observed interatomic distances and the appropriate radius sums in the B-T phases can be expressed with weighting for rn~~p~city as I,,=,(;
-Dn)
+4(q-+DM-+Dn)
+8”is-Dnj
(3)
which gives a = 0.3240, + 1.4250, + k with coefficients approximately equal to those of eqn. (2) if Q, the empirical factor reducing the weighting of the remote 8 N-N contacts, has a value of 0.6. In eqn. (3) the value of k must be constant, corresponding to the constant term in eqn. (2). We now suppose that the value of DM in eqn. (3) is increased by x: and a increases by 2~15~~~in order to maintain the contribution to 14k constant for the term ~5~‘~ - 2DM - 20,. This will increase the difference in the first term of eqn. (3) by 2x/5112 and the difference in the third term by 4 X 61’2,x/51’2, so that k will not satisfy its required condition of constancy. Therefore the dependence of a on D, cannot be independent of D, and vice versa under these conditions where several contacts from the same atom to its M and N neighbours combine to control u. Such findings do not change the conclusions in ref. 1 that the contacts in the polyhedron about N are responsible for controlling a in B-T phases; they merely show that the contribution of D, to this control increases as DN decreases. This situation can be expressed in terms of the weighting factor 0 d ar< 1 that has a value of 0 below D, = 2.484 A and then increases linearly with DN, achieving its value of 1 at DN = 3.121 A and giving rise to the following equation: u=
0.1429DM - 1.7231 DN + 0.8971 DN2 - 0.6034 0.5493D3, - 1.1334
(4)
Equation (4) is valid in the range of DN values from 2.484 to 3.121 A and reproduces the observed cell dimensions [4] of 27 B-T phases from CN 12 diameters [ 53 to better than 10.0081A (neglecting five phases with deviations that are greater than 3~) or to 10.0121A for all 32 phases whose DN values lie within the valid range. This significant improvement over the result of eqn. (2) is achieved at the expense of three additional parameters in the equation and confirms that the dependence of a on D, is not independent of DN and vice versa.
165
4. Interpretation of the cell dimensions of T-T phases (1) In ref. 1, Fig. 3, phases with the same M components on the nearneighbour diagram (NND) were connected by lines and these lay parallel to the line for 12-4 M-N contacts determined from ~25l’~/4 = (Dnn:+ DN)/2 rather than parallel to an M-N contact line determined from a51j2/4 = (DM + 3&)/4 which gives coefficients of the dependence of a on Q,., and DN that are similar to those of eqn. (1). (An NND [6] is a plot of a reduced strain versus &,,/DN for the interatomic contacts occurparameter (DM - d,)/D, ring in the structure and for actual phases with the structure. dEAis any arbitrarily chosen M-M distance in the structure expressed in terms of a (and/or other cell dimensions) such that dM = fu = I&. This establishes the zero for the (D, - d,)/D, scale.) It is now seen that these lines connecting the phases are not significant since they are influenced by the valency effects of the different N atoms to which the M atoms are combined. If lines are drawn through phases formed by the same N components instead, they lie nearly parallel to the line for M-N contacts, determined from a5112/4 = (D, + 3&)/4, and the enigma in ref. 1 disappears. If indeed corrections for the valency effect (eqn. (1)) are applied to the elemental CN 12 diameters of the N components in calculating the positions of the phases on the NND, they all lie on a single line (Fig. 2). The correc-
-03
-04
-05 UM- dM Do -06
-07
090
095
100
105
I IO
II5
120
DM’DN
Fig. 2. An NND (0~ -a3 1’2/2)/D~ vs. DMIDN for the Al 5 structure. T-T phases of N components: +, titanium; 0, zirconium; X, vanadium; 0, niobium; 0, tantalum; A, chromium;V, molybdenum. In plotting these positions, DN values adjusted for the valency effect have been used. The diagram also shows the average of the lines for the 2 N-N, 12-4 M-N and 8 N-N contacts (- - -) with weighting for multiplicity of the contacts (unlabelled) and the line for the 12-4 M-N contacts weighted for composition (i.e. with (DM + 6&)/4).
166
tions for the valency effect calculated from eqn. (1) are respectively 0 A, -0.025 A, -0.050 A and -0.125 A for the diameters of chromium, molybdenum, group V N components and group IV N components. (2) In ref. 1 we preferred Pauling’s equation [ 3) a51i2/4 = (D, + 30,)/4, which gives a = 0.44’7D,
+ 1.3420,
(5)
as the explanation of the dimensional behaviour of the T-T phases (eqn. (1)) rather than ref. 1, eqn. (3), which gives essentially the same result, since there was then no reason to disagree with previously accepted work. Reference 1, eqn. (3), represented the line on the NND that was obtained as the r.m.s. average of the 2 N-N, 12-4 M-N and 8 N-N contact lines, weighted for contact multiplicity. Although such an averaging process happened to be fairly satisfactory because of the positions of the lines on the diagram, it is not generally satisfactory, as the result is subject to the zero chosen for the NND. According to the new procedure in ref. 7, the averaged positions of the three contact lines on the NND with weighting for multiplicity give a line that lies parallel to the phases plotted with valency correction for DN, as shown in Fig. 2. That this line lies considerably above these points only means that the observed coefficient of the dependence of a on DN is about 0.055DN larger than 1.335DN calculated for the line (i.e. about 1.390 DN). Thus either Pauhng’s eqn. (5) based on control of a by the M-N contacts with weighting of the diameters for composition or an equation depending on the average of the 2 N-N, 12- 4 M-N and 8 N-N contacts with weighting for their multiplicity could account for the observed variation of a with D, and DN for the T-T phases. We now seek evidence to distinguish between these two possibilities. First, we have recently examined MN, phases with many crystal structures for which the cell dimensions are controlled by M-N contacts, searching for definite evidence that it is necessary to weight the atomic diameters according to the stoichiometric proportions in calculating the cell dimensions. The results are at present inconclusive but we have not been able to show that it is a necessary procedure. Secondly, the valency effect observed for the N components in these phases (discussed in Section 5) is seen to involve all three different contacts to the N atoms which favours control of a thereby rather than by the M-N contacts alone. Thirdly, if we assume that the 2 N-N, 12-4 M-N and 8 N-N contacts control a and that the dependences of a on DM and DN are not independent of each other (certainly the dependence of a on DM for the vanadium phases is rather larger than that for the titanium and niobium phases) and follow the procedure adopted for the B-T phases (Section 3), we obtain the equation a=
0.22220,
- 1.4782DN + 0.8444Dn2 - 0.6833 0.51720,
- 0.9672
- 0.0255
(6)
which is valid between DN values of 2.458 W (a! = 0) and 2.984 A (Q = 1). S is the valency effect with values of 0, 1,2 and 6 respectively for chromium, molybdenum, group V N components and group IV N components. This
167 equation reproduces the observed Q values [4] to 10.0061 A of 27 phases whose N components have diameters within this range, excluding PtCr, whose reported cell dimension is clearly too small. (We have subsequently discovered that the a value for PtCrs is wrongly listed; it should have a value of about 4.714 A, bringing the agreement of the calculated and observed values to within this range.) Such precision probably exceeds the limit to be expected in view of the uncertain composition of the phases and other experimental errors, and it is a significant improvement over that provided by eqn. (1); it is also noted that the precision is achieved with standard CN 12 diameters of the atoms [ 51. Such evidence suggests (although still without certainty) that we were probably wrong in concluding [l] that the cell dimensions of the T-T phases are controlled by the M-N contacts rather than by all interatomic contacts in both coordination polyhedra, suitably weighted for multiplicity, with the influence therein of the remote 8 N-N contacts increasing as DN increases.
5. Valency effects The valency effect involving the N components in the T-T phases is a recognition that a for the vanadium and titanium phases, for example, is smaller (relative to the chromium phases) than can be accounted for by the differences in their atomic diameters. Vanadium and titanium are therefore assumed to use more valency electrons in bonding to their neighbours than they normally possess, and hence their smaller sizes. Since the M components come from group VII and higher groups and include gold and mercury, they can provide the required electrons without altering their normal valencies and CN 12 diameters. Using Pauling’s equation [S] R(1) -R(n) = 0.3 log n, the number of valency electrons required by chromium, molybdenum, group V and group IV N components can be calculated from eqn. (1) and also from the interatomic distances for each of the individual phases and then averaged. That the results of these two calculations agree closely has only the elements of a circular argument, except that it demonstrates self-consistency since in the calculations for the individual phases it was necessary to assume normal valencies and D, values for CN 12 for the M component, so that differences between a5”‘/4 and (D, + On)/2 were attributed to the N atoms. Furthermore, it emphasizes that the two close N-N contacts result from valency bonding rather than from geometrical constraints elsewhere in the structure. No such treatment can be accorded to the B-T phases. First, there are no detectable differences in a for, say, the chromium, vanadium and titanium phases that cannot be attributed to the atomic diameters of these elements for CN 12 alone. Thus the lack of a valency effect is consistent with insufficient valency electrons of the M components, which are elements of groups II - VI. Secondly, the B-T phases show a characteristic common to many intermetallic phases that the interatomic distances are generally shorter than
168
can be accounted for by the number of available valency electrons. Indeed, B-T phases of niobium formed by M components larger than DNI= 3.28 A do not even have enough valency electrons to account for the 12-4 M-N distances alone! This leads us to consider the relative atomic volumes of the two groups of phases. 6. Comparison of cell volumes The differences of behaviour of the T-T and B-T phases result in ratios of the observed cell volumes to those calculated as 2V, + 6V, (where V is the elemental atomic volume) in the approximate ranges 1.00 - 1.047 for the T-T phases and 0.90 - 0.98 for the B-T phases. (The V, values used here are those corrected for the valency effects. This correction is applied by multiplying the elemental atomic volume by the ratio Rv3/R3, where R, is the CN 12 radius corrected for the valency effect and R is the elemental CN 12 radius.) The first range is abnormal for intermetallic phases, whereas the second range is normal since volume ratios generally lie in the range from about 0.80 to 1.0 [9] . Nevertheless, some reason must be found to explain why the structures of the B-T phases are more compressed when there are apparently less (insufficient) valency electrons available. The differences in volume ratios are not themselves the result of the valency effect, since the average for the T-T phases of chromium (whose valency effect is zero) is 1.02, whereas it is 0.917 for the B-T phases of chromium. Figure 1 shows that a51j2/4 - (&,, + DN)/2 decreases rapidly with increasing I&, which must be an unfavourable feature of the structure as the M-N contacts are becoming more compressed. The reason for this is that the observed dependence of a on 0.325& (eqn. (2)) is much smaller than the dependence of a on 0.8940, that is required to keep this difference constant as D, increases. Therefore the obvious solution for the structure is to increase the dependence of a on DNI.However, this is impossible because of the influence also of the 2 N-N and 8 N-N contacts in the averaging process with weighting for multiplicity. Thus the physical significance of the averaging process assumed in ref. 1 is apparent when several different sets of contacts from the central atom to neighbours forming its convex coordination polyhedron act together to control a. The next obvious solution for the structure to avoid the compression is for (I to increase for all B-T phases, but again there appear to be reasons why this may not be possible. The only feature apparent in Fig. 1 that limits the stability of the phases is that a51j2/4 - (DNI+ On)/2 must be approximately zero or negative. The only phase for which it is clearly positive (i.e. M-N contacts “not made”) is SiNb3 which is metastable. Possibly these two reasons taken together account for the compression in the B-T phases compared with the T-T phases, and it must therefore be accorded to the geometrical arrangement in the structure, whereas the T-T phases appear to have sufficient outer electrons for the interatomic distances to be controlled by normal valency bonding interactions.
169
7. Comparison of dimensional behaviour of T-T and B-T phases with the Al5 and C&l structures The question arises whether the valency effect of the T-T phases and the differences in volume ratios of the T-T and B-T phases are particular to the atomic arrangement in the Al5 structure and especially to the two close N-N contacts. This can be answered by examining similar phases with the CsCl structure where such contacts do not occur. In the CsCl structure the M-N contacts are expected mainly to control a but there is also the possibility that M-M or N-N contacts influence a if a < DM or u < DN . Furthermore, there are a reasonable number of T-T and B-T phases with the CsCl structure, so that some analysis is possible. Figure 3 shows that the behaviour of the T-T phases with the CsCl structure is entirely analogous to that of T-T phases with the Al5 structure (see ref. 1, Fig. 1). There is a valency effect involving the group IV and V transition metals titanium, zirconium, hafnium, vanadium and tantalum (N) and also the volume ratios of the phases with V, values corrected for a valency effect are all greater than unity. The cell dimensions are given to an accuracy of 10.0061A (except for cobalt and nickel phases of titanium, zirconium and hafnium) by the equation a = 0.66D,
+ 0.65DN - 0.075s - 0.424
(7)
where S, the valency effect, has the relative values 0 for FeCo and FeRh (i.e. alloys formed by two M metals), 1 for group V N metals and 2 for group IV N components. Thus the transition metal valency effect and the volume ratios greater than unity are nothing particular to the occurrence of the two close N-N contacts in T-T phases with the Al5 structure. Although the data for B-T phases with the CsCl structure on a uersus D, and a uersus D, plots are somewhat scattered and various, there is no evidence of a valency effect involving the transition metals (N) in agreement
I 24
2.6
28
26
DN (ii, (a)
I
I/
I
28
, 30
I
I 32
D,(i) @I
Fig. 3. T-T phases (MN) with the CsCl structure: (a) a us. DN; (b) a at D, = 2.7 A us. DM. The M components are identified as follows: 0, titanium; A, zirconium; +, hafnium; X, vanadium; 0, tantalum; 0, iron.
27
I/
26 t 2.4
27 O,(i)
30
33
3.6
D,(i)
(b)
(a)
Fig. 4. B-T phases (MN) with the CsCl structure: (a) a us. DN; (b) a at DN = 2.7 w US.DM. M components are identified as follows: 0, beryllium; 0,silicon;~, arsenic; X, gallium; 0, aluminium ; 0, magnesium ; +, scandium; A, indium.
observations for phases with the Al5 structure. Secondly, when a 2 DM (and including two indium and certain scandium (M) phases where a < &), the behaviour is seen (Fig. 4) to resemble that of B-T phases with the Al5 structure, a being given by with
u = 0.43&
+ 0.650,
+ 0.40
(8)
to an average accuracy of 10.0171A, neglecting the discordant results of five out of 29 phases. From direct plots of a uersus DM it is also confirmed that a is approximately proportional to 0.40, when a 2 DNI. The cell volume ratios of these B-T phases are all less than unity, similar to those of B-T phases with the Al5 structure. Most of the ratios lie in the range from 0.78 to 0.92. The behaviour of B-T phases with both structures is further similar in that a for the CsCl phases appears to be controlled by the combined action of the 8-8 M-N and 6 N-N contacts as has already been demonstrated for the 2 N-N, 4 M-N and 8 N-N contacts of B-T phases with the Al5 structure [1] . Phases with the CsCl structure formed by such N components as copper, silver, gold, zinc, cadmium and mercury and rare earth M components [lo] have an a dependence on D, of 1.0 or slightly less, which corresponds to M-M contacts controlling a, whereas phases of the same N components with non-rare-earth M atoms have an a dependence of about 0.577&, which corresponds to M-N contacts controlling a (u31j2/2 = (Dw + &)/2, whence a = 0.577& + 0.5770N). The dependence of a on O.43& and 0.65DN for B-T phases when a > D, is clearly different and requires the assumption that the 8 M-N and 6 N-N contacts from the N atom to its surrounding dodecahedron act together in controlling a. Averages of the lines for these contacts
171
on the NND or averages of the difference of a - DN and ~3l’~/2 - (D, + 0,)/2, both weighted for the multiplicity of the contacts [ 71, give approximately a = 0.320, + 0.76&. A less strong weighting of the large N-N distances would increase the dependence of a on D, and decrease that of a on DN, agreeing sufficiently well with the observed values of eqn. (8), and a more complex equation analogous to eqn. (4) is indicated. Thus not only is the principle of dimensional control of B-T phases with the CsCl structure shown to be similar to that of B-T phases with the Al5 structure but it is also shown to be different to other phases with the CsCl structure that do not contain transition metals. The overall similarity of the dimensional behaviour of T-T and B-T phases with the Al5 and CsCl structures indicates that the behaviour of phases with the Al5 structure cannot be attributed to the particular influence of the two very close N-N contacts. 8. Conclusions (1) It now appears most probable that the cell dimensions of T-T phases are controlled by all 2 N-N, 12-4 M-N and 8 N-N contacts in both coordination polyhedra acting together rather than by the 12-4 M-N contacts with weighting of the diameters for composition. (2) The cell dimensions of the B-T phases are controlled by the 2 N-N, 4 M-N and 8 N-N contacts in the polyhedron about N acting together. (3) When several contacts from an atom to its M and N neighbours act together to control the cell dimension a, its dependence on D, is not independent of D, and vice versa. (4) Equations for calculating a in which this is recognized, being based on a weighting of contact multiplicity and a variable weighting of the remote 8 N-N contacts that depends on D,, reproduce the observed a values from standard atomic diameters for CN 12 alone to about the limit of their experimental accuracy. (5) Valency effects involving the N atoms in T-T phases are compatible with the ability of the M atoms to provide the necessary electrons, such that interatomic distances in these phases appear to be controlled by normal valency bonding, whereas in B-T phases insufficient electrons are present to account for the interatomic distances and they appear to result from geometrical constraints within the structure. (6) All evidence provided is consistent with tin having a size corresponding to a valency of 2 in phases with the Al5 structure. Acknowledgments I wish to thank Professor E. Hellner, Drs. R. M. Water&rat and E. Koch and Mrs. Reinhardt for critical comments and discussion of ref. 1 and for providing me with various calculated data.
172
This work was supported by a grant from the Natural Sciences and Engineering Research Council of Canada.
References 1 2 3 4 5 6 7 8 9 10
W. B. Pearson, J. Less-Common Met., 77 (1981) 227. S. Geller, Acta Crystallogr., 9 (1956) 885. L. Pauling, Acta Crystallogr., 10 (1957) 374. R. M. Waterstrat, J. Solid State Chem., (1981) in the press. E. Teatum, K. Gschneidner and J. Waber, LA Rep. 2345, 1960 (Los Alamos Scientific Laboratory) (available from U.S. Department of Commerce, Washington, DC). W. B. Pearson, Acta Crystallogr., Sect. B, 24 (1968) 7,1415. W. B. Pearson, Acta Crystallogr., Sect. B, 37 (1981) in the press. L. Pauling, J. Am. Chem. Sot., 69 (1947) 542. W. B. Pearson, The Crystal Chemistry and Physics of Metals and Alloys, Wiley-Interscience, New York, 1972, pp. 570, 658. W. B. Pearson, Philos. Trans. R. Sot., London, Ser. A, 295 (1980) 415,431,432.