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Bonding and polyhedral atomic volumes for the transition metal borides
157
Journal of the Less-Common Metals,47 (1976) 157 - 163 @ Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
BONDING AND POLYHEDRAL ATOMIC VOLUMES TRANSITION METAL BORIDES*
FORREST
FOR THE
L. CARTER
Naval Research Laboratory,
Washington, D.C. 20375
(U.S.A.)
Summary The results of Pauling’s metallic radii and PAV calculations provide an estimate of effective charges, valences, atomic volumes, and non-integral coordination numbers for a wide selection of rare earth and transition metal binary borides. This uniform treatment suggests that effective charges for the transition metals are generally less than one, and that their valence is lower than the usual Pauling maximum of six. In addition, it was noted that the boron PAV volume was related to the non-isonomicity of the cell and that the use of f orbit& in the BOA technique provides a useful discussion of bonding in some of the rare earth diborides not readily treated by s, p, and d hybridization alone.
Introduction Bonding in the transition metal borides is a topic of considerable interest because of the interplay of many factors. These include the size effect, boron electron deficiency, the persistence of B-B bond formation, band filling effects, and the ability of transition metals not only to serve as a source or sink for electrons but also to change valency and bond hybridization. The problem is further complicated by erroneous data resulting from the difficulties attending the preparation of borides, the necessary chemical analyses, and the X-ray structure determinations. While molecular orbital (M.O.) treatments of the boranes and the icosahedral boron building block are well known, and while a limited group of diborides (AIBz-type) were previously treated [ 11 using approaches employed here, a systematic treatment of the transition metal borides, either by M.O. or valence bond techniques, is not available. Accordingly, it seems appropriate not only to employ Pauling’s semi-empirical metallic radii [2] to estimate bond orders, valencies, and charges from known structures, but also to utilize newer concepts in this attempt to understand the bonding in these borides. The object of this paper, then, is to consider the better known transition metal borides, from the points of view of Pauling’s metallic radii (PMR), *Paper presented at the 5th International Symposium on Boron and Borides, Bordeaux, France, September 8 - 11,1975.
158 TABLE
1
The higher borides “ORONOI Faces A’
“a
/vert.
vol.
"1
24'14 7/9
13.367 7.549
1.497* .799
24/14 7/9
13.439 7.590
1.426
B
24/14 9/13
18.718 8.655
L"B6
LU B
24/14 9/13
-4
Srn Bl B2 B3
LUB.
f&f.'
L”B12E
METALLIC Eff.
RADII
PA” Faces
ca1c.
Charqe
Valence 2.95
A’ vol.
CN"
3.04
24/14 7/9
26.17 6.48
24.00 6.49
-.01
.O6
2.55 3.00
24/14 7/9
24.68 6.65
24.00 6.49
1.590* ,793
1.71 -.28
1.47 3.30
30156 9/13
35.07 5.92
30.00 8.67
18.403 R.509
1.512* .793
1.63 -.27
1.21 3.36
30/56 9/13
32.88 6.09
29.89 8.64
23/38 10/16 9/14 ll/lS
16.690 7.595 11.223 8.429
1.611 ,781 .a29 .791
.96 -.73 .52 -.38
2.71+ 3.64 2.34 3.29
25/46 lo/16 9/14 9/14
30.53 4.69 6.70 5.22
23.86 8.90 8.91 8.60
LU Bl B2 B3
23/38 lo/16 9/14 11/18
15.325' 6.957 10.373 7.776
1.463** .774 821 :785
1.47 -.91 .50 -.55
2.11 3.87 2.41 3.50
25/46 lo/16 9/14 10116
26.15 4.71 6.84 5.25
23.94 8.87 8.64 8.57
M01_XB3 (6)
M0l Mo2 B
17/27 15/27 Y/14
10.062 10.089 9.043
1.296 1.296 .802
.04 .04 .04
4.91 4.63 2.98
17/27 15/27 9/14
16.56 16.86 6.83
16.15 14.80 7.60
c-MO>BS
MO Bl B2 B3
20,'36 U/18 11/E 14/24
10.135 7.415 6.554 6.569
1.256* .786 .784 .788
1.18 -.48 -.53 -.39
4.59 3.47 3.53 3.38
20/36 9/14 lo/l6 12/20
15.29 4.81 4.72 5.13
19.24 8.99 9.50 11.91
Snl2B5 (71
Sml Sm2 B1 B2 B3 B4 B5
20/36 22/40 12/20 10'16 12/20 9/14 S/12
19.e 21.718 8.140 10.891 8.145 7.999 14.280
1.628 1.630 .?I39 .823 .789 .786
.70 .68 -.51 14 -:52 -.62 .14
2.83 2.42 3.37 2.64 3.37 3.48 2.66
20,'36 21,'38 lo/16 9/14 Y/14 Y/14 S/12
31.41 32.78 4.78 5.99 4.77 4.68 6.72
18.11 18.81 8.73 8.96 a.72 8.73 7.75
WzBs
W Bl B2 B3 B4
20/36 9/14 II/18 11118 12'20
9.931 7.170 7.150 6.401 6.421
1.254* .781 .780 .782 ,785
1.50 -.65 -,65 -.60 -.50
4.86 3.68 3.68 3.62 3.52
20/36 9/14 9/14 lo/16 12/20
14.99 4.59 4.58 4.63 5.01
19.28 8.98 8.98 9.55 11.90
"Sl2
SnlB 6
;
Snl
%ach * indicates b Earh + indicates CThrsr
cells
a necessary
.800
.818
.04 increment
Vc > estimated
For Srn~B5 are not
69 -:C&
CELL
/Pert.
valence space
in d-character.
by 0.25.
filling,
the
error
is probably
in Sml.
the Polyhedral Atomic Volumes (PAV), and the Bidirectional Orbital Approximation (BOA), as well as the newly quantified concept of coordination number [3]. The first two techniques (PMR and PAV) are useful in detecting structures requiring either compositional-structural refinement or obviation.
Results The calculations are summarised in Tables 1,2, and 3, for selected transition metal borides. The first column lists the compounds in order of composition, structure type, and atomic number. For recent structure determinations (references in parentheses), the labelling of non-equivalent atoms is that of the crystallographer, and for earlier work, that of Pearson [4] or Wyckoff [5]. The next two columns give the number of faces and vertices as well as the volume of the Voronoi cells. These space-filling cells are calculated
159
TABLE
2
Di- to monoborides
“ORONOI
cpaI 1,Wf.i RUB2
X” B
R&3*
w Bl 92 83
PA" C LL JG--FaCeS /vert. ""1.
14/20 10/16
1.243 0.794
.37 -.19
4.94 3.21
14/20 12/20
14.97 5.97
13.74 8.11
1.282 ,792
.55 -.28
5.73 3.28
14/m lb,'25
15.40 5.91
13.89 R.15
10.315 8.306
Re
(81
RADII Eff. Charge
R;
B WB*
METALLIC
Faces3-/vert. vol.
CalC:VakUlCe
CN"
17/27 14/18 U/18 9/14
10.536 8.973 7.225 7.237
1.284 .a06 .778 .778
.59 .14 -.74 -.74
5.06 2.83 3.74 3.73
17/27 16/25 9/14 9/14
15.79 6.47 4.48 4.48
16.25 8.07 8.98 R.98
14/17 14/18 14/18
IO.119 8.132 10.322
1.244 .792 .795
.31 -.25 -.12
5.07 3.28 3.lh
14/18 16/25 lfl/32
14.68 5.83 5.85
13.55 8.09 7.52
18/2? 9/14
15.097 10.434
1.534** .795
-.7O .45
4.31 2.34
14.015 9.892
1.47e* .795
-1.20 .6O
4.411 2.26
9.010 9.736 7.656 6.795
1.189 1.189 .791 .782
.64 .il -.39 -.65
4.35 4.78 3.31 3.60
LUB2
LU B
201'36 11/18
CraBa
cr1 cr2 Bl B2
20/36 17/30 lb‘18 11/H
Mn,B4
Mnl ml2 Bl B2
20/36
8.722
1.181
;;;3; IV18
10.272 7.747 6.475
1.181 .797 .770
.80 .79 -.19 -1.01
4.67 4.06 3.11 4.00
11.276 10.526 8.825 8.340 7.664
1.263 1.263 .791 .791 .778
.42 .42 -.26 -.25 -.64
4.87 4.97 3.30 3.30 3.72
-65
IrBxC x=1.35 (9‘
IX-1 I=2 ;: B6
15/26 16/28 14,'24 16/27 xv20
CrB
CT B
17/30 11/B
9.744 7.357
1.189 .782
-.65
4.96 3.60
17/30 9/14
12.59 4.52
lii.'H, ii.
NiB
Ni B
17/30 U/18
9.089 6.953
1.130* .7?8
.73 -.73
4.60 3.73
17/30 WI.4
11.51 4.53
15.74 R.91
FeB
Fe B
17/30 ll/lE
9.261 7.241
1.173 .77?
.77 -.77
5.33 3.?7
17/30 9/14
12.07 4.43
16.16 8.82
MOB
"0 B
17/30 11/18
11.621 8.830
1.296 .792
.28 -.28
5.35 3.28
17/30 9/14
15.55 4.90
16.12 8.82
aEach %b,
* indicates
zs treated
'An ordered
a nec,xmary
as having
version
.*4 increment
an anisotropic
of this
structure
0’)
in d-character.
Rl. was
assumed.
for atoms of zero (or equal) radii in much the same way as are Wigner-Seitz cells, but they differ from those space-filling polyhedra of Gorter [II] ; these latter are often associated with other atoms at the polyhedra vertices. The Voronoi volumes can be used to indicate the intrinsic sizes associated with structure sites, independent of atom sizes. Thus, for the borides LuBrz, LuB,, LuB*, LuB,, we see that intrinsic metal volumes decrease in the order LuBe, LuB,, LuBa, and then LuBlz. The next three columns contain the results of the metallic-radii calculations carried out in a self-consistent manner, allowing charge transfer and rehybridization of both boron (p-character) and metai atoms (d-character increase) under the condition that all boron electrons are involved in bond formation. The calculated valence, V,, for an atom, i, is just equal to the sum of its bond orders, nij, for bonds with its neighborsj (see ref. 2, p. 400). The listed atomic effective charges correspond to the charge transferred, as corrected for bond polarization due to electronegativity dif-
160 TABLE
3
The sub-borides VORONOl
Cpd. (Ref.)
Faces /vert.
A' VOl.
R= -1
METALLIC RADII Eff. ca1c. Charge ValelUX?
PAV CELL Faces /vert. "",I.
CNv
I?/30 17/30 15/26 17/3c 12/20 11/18 13/22
9.341 8.727 9.096 9.291 7.061 6.793 8.217
1.157 1.150 1.150 1.157 ,771 .770 .783
.61 .?1 -61 .61 -.95 -1.00 -.57
5.45 5.80 6.06 5.24 3.96 4.00 3.57
17/30 17/30 14124 17/30 9/14 9/14 9/14
11.56 11.19 10.84 11.72 4.30 4.22 4.70
15.38 15.60 13.03 15.98 8.70 8.90 8.82
17/30 16/28 9/14 12/20
9.503 9.529 6.885 7.830
1.157 1.150 .770 .792
.57 57 -l:oo -.27
5.34 5.48 4.04 3.27
17/30 15,'26 9/14 12/20
11.70 11.41 4.18 5.09
15.25 14.07 8.78 9.79
z RU3 'ml4 RU5 Rub Bl 82 83 84
14/24 17/30 14/24 17/30 17j30 16/27 13/22 9/14 11/18 11/18
10.972 9.744 12.091 10.151 l.2.000 12.219 9.819 7.359 8.350 6.137
1.244 1.208 1.244 1.206 .244 .244 .799 .770 .778 .7?0
.35 .21 21 :30 30 :32 -.02 -.90 -.66 -.90
5.46 5.53 5.52 5.04 5.01 4.59 3.05 4.04 3.74 4.61++
141'24 17/30 14,'24 17/30 17/30 16/27 9/14 9/14 9/14 9/14
13.81 12.84 13.92 13.14 14.28 1::;~
13.58 L5.86 13.23 15.90 14.99 15.07 8.23 8.79 8.63 8.94
ClSBl
cr1 Cr2 Bl B2
14/24 16/28 10/16 13/22
9.721 10.750 9.680 8.450
.186 .186 .820 .790
.08 .06 -.08 -.15
5.50 4.48 2.483.03
14,'24 :6&
11.97 1:.;;
Y/14
5:os
13.37 14.91 9.67 8.95
Fe28
Fe B
15/x 10/16
10.020 7.687
1.171 .784
.39 -.79
5.75 3.52
15,'26 lo/l6
11.55 4.64
14.49 9.57
NizB
Ni B
15/26 lo/l6
9.537 7.352
1.156 .??6
.39 -.79
6.02 3.79
15/26 lo,'16
11.01 4.41
14.50 9.48
Ru?BI
RUl RU2 Ru3 B
15/26 15/24 15j26 9/14
12.070 13.363 12.108 9.460
1.245 1.245 1.245 .785
19 :19 .19 -.44
5.36 4.72 5.10 3.51
15/26 15,'24 15/26 Y/14
13.95 14.51 14.05 4.77
13.95 14.22 14.64 8.31
NiaB (10)
Nil Ni2 B
14/24 17/30 9/14
9.856 10.580 7.552
1.137 1.156 .770
.33 .33 -1.00
5.54 4.93 4.27+
14,'24 17/30 9.04
10.90 11.83 4.21
13.32 14.11 8.58
NisBs ortho
Nil Ni2
(loi
::: Bl BZ B3
NiaBp Monocl (10)
Nil Ni2 ",;
R"11Be
aEach Each
+ indicates - indicates
V > estimated Vz < estimated
valence valence
by an additional 0.25 by 0.25 electron.
4.18 4.59 3.84
electron.
ferences, where the quantity of charge transferred is such that the number of electrons available (estimated valence) is equal to the calculated valence, V,. The values listed of the single bond metallic radii, R r, which are a function of both charge and hybridization, are then used to calculate the polyhedral atomic volumes in much the same way as the Voronoi cells are calculated. The faces of PAV cells, however, are placed midway between the surfaces of the atomic spheres of differing radii, RI. This placement of the cell faces then divides equally (approximately) the bonding electrons between the two atoms, thus justifying the designation of “polyhedral atomic volume”. The final column in Tables 1, 2, and 3 gives the coordination number, CNv, based on the pyramidal volumes, Vi, of the i’th face according [ 31 to the equation l/CNV = ~(Vi/V,)2, where VT is the total volume (i.e., the listed PAV’s). This kind of definition for coordination number not only gives the usual result for regular polyhedra (Le., 8 for an octahedron) but gives the appropriate low weight to small, unimportant faces. For example, the PAV cell for atom Bl in WB2 (Fig. 1) has 9 small faces of a total of 16 with a CNv of 8.07 (Table 2).
161
Fig. 1. These PAV polyhedra are drawn with the x and y Cartesian axes in the xy crystallographic plane with x axes parallel. Faces parallel to the y axis are seen end-on.
In Tables 1,2, and 3, the presence of reference marks, following metallic radii calculations, indicates either d-character deviations (*) from normality or failure of the data to converge (+,-). A surplus of such marks and a high effective charge, greater than k1.0, casts suspicion upon the composition and/or structural analysis. For the transition metal borides, the effective charges differ from the transferred charges by no more than 0.1 electron; however, for the rare earth borides, the effective charge is more positive than the formal transferred charge by about 0.2 to 0.4 electrons. We note that, excluding U and the rare earths, the maximum possible number of unpaired d electrons according to these PMR calculations occurs for the ruthenium and cobalt sub-borides at 2.4 (max.).
Discussion The average PAV volume for boron undergoes a steady decrease from about 6.5 A3 for the duodecaborides to about 4.5 A3 for the subborides where only metal-boron bonds are formed. In terms of these latter contacts (M-B) one may define [3] a fractional coordination number as f(M-B) =
162
C VF(M-B)/C VF (all contacts) such #at f(M-I3) = 0 for boron and f(M-B) = 1 for NiaB. In view of the average PAV decrease noted above for increasing f(M-B), it is surprising to find that for the borides of some metals, for example Sm in SmB4 and Sm2B5, the PAV volume increases with f(M-B). As suggested in Fig. 1, part of the reason for this is related to the shape or isonomicity [ 111 of the boron PAV cell. In these polyhedra, the atom position is marked by an X. Note that the more asymmetry borons, B5 of Sm,B, and B1 of WBs, are larger than their symmetric counterparts. These are in contrast with the small symmetric cells for boron in CrB and FesB where f(B-M) approaches 1 (0.849 and 0.937, respectively). Whilst the PMR results for the duodecaborides appear reasonable, those for the hexaborides of Sm and Lu indicate that the boron octahedron is too compact, giving rise to an excessive charge transfer, too small a metal valence, and, related, too large a metal PAV volume. By expanding the octahedron and tilting its axes away from the cubic axes, the above-mentioned difficulties could be ameliorated. However, at the same time, the symmetry requirements on the borons will be relaxed, suggesting an instability in the cubic structure for the duodecaborides of the smaller metal atoms (see ref. 12). The use of the positional parameters of ThB4 gives moderately reasonable results for SmB4 but poor results for LuB4 since the B-B bonds are somewhat too strong for Bl and B3. For the vacancy-disordered compound Mo~_~B~, the PMR calculations are reasonable with 60% occupancy of the Mol site; PAV results assume full occupancy. The high charge calculated for W,B,, plus the increased W d-character, suggest that this structure should be questioned, as indicated by Lundstrom [S] . The difficulty with the related Ru,B, structure [4, 51 is even greater, and raises questions concerning its existence. The only difficulti~ occurring in Table 2 concern the rare earth diborides with the AlB, structure. Here, the metal-metal bonds are so short that the normal rare-earth radii give rise to impossible valencies (e.g., 7 to 8). By assuming an anisotropic radius corresponding to high d (or f) character for bonds in the basal plane, the results for TbB, approach reasonableness. This ~sumption of anisotropy has been used previously by the author to discuss the SmCo, structure [ 133 where the anisotropy arose from either anisotropic f electron shielding or f orbital bonding participation. Using six C-type bidirectional orbit& modelled after those for HCP metals [14] but with high basal plane f-character (-16%), one can form six metal-metal bonds and six metal-boron bonds (3 above and 3 below) the basal plane. The other six borons oan be bonded by a set of G-type bidirectional orbitals mutually orthogonal to the other six orbit&. This result is in contrast to an earlier attempt using the BOA method [ 11. Valence bond photoemission studies should be able to distinguish between the normal f electron distribution and that correspond~g to anisotropic f electron shielding and bond hybridization. LuB,, without anisotropic bonding, does not give satisfactory PMR results. Although CrsB4 and MnsB4 appear reasonable in Table 2, NbsB, and TasB4 [4, 51 are less so with too strong a B2-B2 bond (bond order aO.95).
163
The satisfactory PMR and PAV results for disordered IrBr,ss were calculated by using an ordered arrangement corresponding to the special space group Cl. The best results for MOB (Table 3) and WB (not listed) were obtained with a boron parameter of u = 0.344. Among the sub-borides, Table 3, the positions of B2 and B4 in Rurr Bs need reconsideration. For the various nickel borides we note that the boron charges are all high, suggesting some systematic problem in their preparation or the operation of a special bonding effect.
References 1 2 3 4 5 6 7 8 9 10 11 12 13 14
F. L. Carter, in R. S. Roth and S. J. Schneider (eds.), Solid State Chemistry, Nat. Bureau Stand. (U.S.) Spec. Publ. No. 364, 1972, p. 515. L. Pauling, The Nature of the Chemical Bond, Cornell Univ. Press, Ithaca, New York, 3rd Edn., 1960, p. 393 f.f. F. L. Carter, submitted for publication in Acta Crystallogr. W. B. Pearson, in G. V. Raynor (ed.), A Handbook of Lattice Spacings and Structures of Metals and Alloys-2, Vol. 8, Pergamon Press, Oxford, 1967. R. W. G. Wyckoff, Crystal Structures, Vols. 1 and 2, Interscience, 2nd. Edns., 1963, 1964. T. Lundstrom and I. Rosenberg, J. Solid State Chem., 6 (1973) 299. P. H. Schmidt, A. S. Cooper and S. J. LaPlaca, The synthesis and structure of Sm2B5, accepted for publication, Acta Crystallogr. T. Lundstrom, Ark. Kemi, 30 (1968) 115. T. Lundstrom and L. Tergenius, Acta Chem. Stand., 27 (1973) 3705. S. Rundqvist and S. Pramatus, Acta Chem. Stand., 21 (1967) 191. E. W. Gorter, J. Solid State Chem., 1 (1970) 279. M. C. Nichols, R. W. Mar and Q. Johnson, J. Less-Common Met., 33 (1973) 317. F. L. Carter, in P. E. Field (ed.), Proc. Rare Earth Conf., 9th, VP1 and State Univ., Blacksburg, Va., Oct. 1971, Vol. 2, p. 617. F. L. Carter, Proc. Rare Earth Res. Conf., 5th, Iowa State Univ., Ames, Iowa, Aug. 30 -Sept. 1,1965, Book Two, p. 103.
Discussion T. LUNDSTRGM I want to make a comment RuzBb. You can test the strength give an unacceptable result.
on your paper. of your theory
There is no boride with the composition by a calculation on RuzB,. It should
F. L. CARTER In attempting to use Pauling Metallic Radii calculations on the literature values given for Ru2B5, I have found that self-consistent, reasonable values for this compound are not obtainable. In particular, the boron distances are too close, giving rise to unreasonable boron valencies. It is unlikely that this compound exists as described.
Journal of the Less-Common Metals,47 (1976) 157 - 163 @ Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
BONDING AND POLYHEDRAL ATOMIC VOLUMES TRANSITION METAL BORIDES*
FORREST
FOR THE
L. CARTER
Naval Research Laboratory,
Washington, D.C. 20375
(U.S.A.)
Summary The results of Pauling’s metallic radii and PAV calculations provide an estimate of effective charges, valences, atomic volumes, and non-integral coordination numbers for a wide selection of rare earth and transition metal binary borides. This uniform treatment suggests that effective charges for the transition metals are generally less than one, and that their valence is lower than the usual Pauling maximum of six. In addition, it was noted that the boron PAV volume was related to the non-isonomicity of the cell and that the use of f orbit& in the BOA technique provides a useful discussion of bonding in some of the rare earth diborides not readily treated by s, p, and d hybridization alone.
Introduction Bonding in the transition metal borides is a topic of considerable interest because of the interplay of many factors. These include the size effect, boron electron deficiency, the persistence of B-B bond formation, band filling effects, and the ability of transition metals not only to serve as a source or sink for electrons but also to change valency and bond hybridization. The problem is further complicated by erroneous data resulting from the difficulties attending the preparation of borides, the necessary chemical analyses, and the X-ray structure determinations. While molecular orbital (M.O.) treatments of the boranes and the icosahedral boron building block are well known, and while a limited group of diborides (AIBz-type) were previously treated [ 11 using approaches employed here, a systematic treatment of the transition metal borides, either by M.O. or valence bond techniques, is not available. Accordingly, it seems appropriate not only to employ Pauling’s semi-empirical metallic radii [2] to estimate bond orders, valencies, and charges from known structures, but also to utilize newer concepts in this attempt to understand the bonding in these borides. The object of this paper, then, is to consider the better known transition metal borides, from the points of view of Pauling’s metallic radii (PMR), *Paper presented at the 5th International Symposium on Boron and Borides, Bordeaux, France, September 8 - 11,1975.
158 TABLE
1
The higher borides “ORONOI Faces A’
“a
/vert.
vol.
"1
24'14 7/9
13.367 7.549
1.497* .799
24/14 7/9
13.439 7.590
1.426
B
24/14 9/13
18.718 8.655
L"B6
LU B
24/14 9/13
-4
Srn Bl B2 B3
LUB.
f&f.'
L”B12E
METALLIC Eff.
RADII
PA” Faces
ca1c.
Charqe
Valence 2.95
A’ vol.
CN"
3.04
24/14 7/9
26.17 6.48
24.00 6.49
-.01
.O6
2.55 3.00
24/14 7/9
24.68 6.65
24.00 6.49
1.590* ,793
1.71 -.28
1.47 3.30
30156 9/13
35.07 5.92
30.00 8.67
18.403 R.509
1.512* .793
1.63 -.27
1.21 3.36
30/56 9/13
32.88 6.09
29.89 8.64
23/38 10/16 9/14 ll/lS
16.690 7.595 11.223 8.429
1.611 ,781 .a29 .791
.96 -.73 .52 -.38
2.71+ 3.64 2.34 3.29
25/46 lo/16 9/14 9/14
30.53 4.69 6.70 5.22
23.86 8.90 8.91 8.60
LU Bl B2 B3
23/38 lo/16 9/14 11/18
15.325' 6.957 10.373 7.776
1.463** .774 821 :785
1.47 -.91 .50 -.55
2.11 3.87 2.41 3.50
25/46 lo/16 9/14 10116
26.15 4.71 6.84 5.25
23.94 8.87 8.64 8.57
M01_XB3 (6)
M0l Mo2 B
17/27 15/27 Y/14
10.062 10.089 9.043
1.296 1.296 .802
.04 .04 .04
4.91 4.63 2.98
17/27 15/27 9/14
16.56 16.86 6.83
16.15 14.80 7.60
c-MO>BS
MO Bl B2 B3
20,'36 U/18 11/E 14/24
10.135 7.415 6.554 6.569
1.256* .786 .784 .788
1.18 -.48 -.53 -.39
4.59 3.47 3.53 3.38
20/36 9/14 lo/l6 12/20
15.29 4.81 4.72 5.13
19.24 8.99 9.50 11.91
Snl2B5 (71
Sml Sm2 B1 B2 B3 B4 B5
20/36 22/40 12/20 10'16 12/20 9/14 S/12
19.e 21.718 8.140 10.891 8.145 7.999 14.280
1.628 1.630 .?I39 .823 .789 .786
.70 .68 -.51 14 -:52 -.62 .14
2.83 2.42 3.37 2.64 3.37 3.48 2.66
20,'36 21,'38 lo/16 9/14 Y/14 Y/14 S/12
31.41 32.78 4.78 5.99 4.77 4.68 6.72
18.11 18.81 8.73 8.96 a.72 8.73 7.75
WzBs
W Bl B2 B3 B4
20/36 9/14 II/18 11118 12'20
9.931 7.170 7.150 6.401 6.421
1.254* .781 .780 .782 ,785
1.50 -.65 -,65 -.60 -.50
4.86 3.68 3.68 3.62 3.52
20/36 9/14 9/14 lo/16 12/20
14.99 4.59 4.58 4.63 5.01
19.28 8.98 8.98 9.55 11.90
"Sl2
SnlB 6
;
Snl
%ach * indicates b Earh + indicates CThrsr
cells
a necessary
.800
.818
.04 increment
Vc > estimated
For Srn~B5 are not
69 -:C&
CELL
/Pert.
valence space
in d-character.
by 0.25.
filling,
the
error
is probably
in Sml.
the Polyhedral Atomic Volumes (PAV), and the Bidirectional Orbital Approximation (BOA), as well as the newly quantified concept of coordination number [3]. The first two techniques (PMR and PAV) are useful in detecting structures requiring either compositional-structural refinement or obviation.
Results The calculations are summarised in Tables 1,2, and 3, for selected transition metal borides. The first column lists the compounds in order of composition, structure type, and atomic number. For recent structure determinations (references in parentheses), the labelling of non-equivalent atoms is that of the crystallographer, and for earlier work, that of Pearson [4] or Wyckoff [5]. The next two columns give the number of faces and vertices as well as the volume of the Voronoi cells. These space-filling cells are calculated
159
TABLE
2
Di- to monoborides
“ORONOI
cpaI 1,Wf.i RUB2
X” B
R&3*
w Bl 92 83
PA" C LL JG--FaCeS /vert. ""1.
14/20 10/16
1.243 0.794
.37 -.19
4.94 3.21
14/20 12/20
14.97 5.97
13.74 8.11
1.282 ,792
.55 -.28
5.73 3.28
14/m lb,'25
15.40 5.91
13.89 R.15
10.315 8.306
Re
(81
RADII Eff. Charge
R;
B WB*
METALLIC
Faces3-/vert. vol.
CalC:VakUlCe
CN"
17/27 14/18 U/18 9/14
10.536 8.973 7.225 7.237
1.284 .a06 .778 .778
.59 .14 -.74 -.74
5.06 2.83 3.74 3.73
17/27 16/25 9/14 9/14
15.79 6.47 4.48 4.48
16.25 8.07 8.98 R.98
14/17 14/18 14/18
IO.119 8.132 10.322
1.244 .792 .795
.31 -.25 -.12
5.07 3.28 3.lh
14/18 16/25 lfl/32
14.68 5.83 5.85
13.55 8.09 7.52
18/2? 9/14
15.097 10.434
1.534** .795
-.7O .45
4.31 2.34
14.015 9.892
1.47e* .795
-1.20 .6O
4.411 2.26
9.010 9.736 7.656 6.795
1.189 1.189 .791 .782
.64 .il -.39 -.65
4.35 4.78 3.31 3.60
LUB2
LU B
201'36 11/18
CraBa
cr1 cr2 Bl B2
20/36 17/30 lb‘18 11/H
Mn,B4
Mnl ml2 Bl B2
20/36
8.722
1.181
;;;3; IV18
10.272 7.747 6.475
1.181 .797 .770
.80 .79 -.19 -1.01
4.67 4.06 3.11 4.00
11.276 10.526 8.825 8.340 7.664
1.263 1.263 .791 .791 .778
.42 .42 -.26 -.25 -.64
4.87 4.97 3.30 3.30 3.72
-65
IrBxC x=1.35 (9‘
IX-1 I=2 ;: B6
15/26 16/28 14,'24 16/27 xv20
CrB
CT B
17/30 11/B
9.744 7.357
1.189 .782
-.65
4.96 3.60
17/30 9/14
12.59 4.52
lii.'H, ii.
NiB
Ni B
17/30 U/18
9.089 6.953
1.130* .7?8
.73 -.73
4.60 3.73
17/30 WI.4
11.51 4.53
15.74 R.91
FeB
Fe B
17/30 ll/lE
9.261 7.241
1.173 .77?
.77 -.77
5.33 3.?7
17/30 9/14
12.07 4.43
16.16 8.82
MOB
"0 B
17/30 11/18
11.621 8.830
1.296 .792
.28 -.28
5.35 3.28
17/30 9/14
15.55 4.90
16.12 8.82
aEach %b,
* indicates
zs treated
'An ordered
a nec,xmary
as having
version
.*4 increment
an anisotropic
of this
structure
0’)
in d-character.
Rl. was
assumed.
for atoms of zero (or equal) radii in much the same way as are Wigner-Seitz cells, but they differ from those space-filling polyhedra of Gorter [II] ; these latter are often associated with other atoms at the polyhedra vertices. The Voronoi volumes can be used to indicate the intrinsic sizes associated with structure sites, independent of atom sizes. Thus, for the borides LuBrz, LuB,, LuB*, LuB,, we see that intrinsic metal volumes decrease in the order LuBe, LuB,, LuBa, and then LuBlz. The next three columns contain the results of the metallic-radii calculations carried out in a self-consistent manner, allowing charge transfer and rehybridization of both boron (p-character) and metai atoms (d-character increase) under the condition that all boron electrons are involved in bond formation. The calculated valence, V,, for an atom, i, is just equal to the sum of its bond orders, nij, for bonds with its neighborsj (see ref. 2, p. 400). The listed atomic effective charges correspond to the charge transferred, as corrected for bond polarization due to electronegativity dif-
160 TABLE
3
The sub-borides VORONOl
Cpd. (Ref.)
Faces /vert.
A' VOl.
R= -1
METALLIC RADII Eff. ca1c. Charge ValelUX?
PAV CELL Faces /vert. "",I.
CNv
I?/30 17/30 15/26 17/3c 12/20 11/18 13/22
9.341 8.727 9.096 9.291 7.061 6.793 8.217
1.157 1.150 1.150 1.157 ,771 .770 .783
.61 .?1 -61 .61 -.95 -1.00 -.57
5.45 5.80 6.06 5.24 3.96 4.00 3.57
17/30 17/30 14124 17/30 9/14 9/14 9/14
11.56 11.19 10.84 11.72 4.30 4.22 4.70
15.38 15.60 13.03 15.98 8.70 8.90 8.82
17/30 16/28 9/14 12/20
9.503 9.529 6.885 7.830
1.157 1.150 .770 .792
.57 57 -l:oo -.27
5.34 5.48 4.04 3.27
17/30 15,'26 9/14 12/20
11.70 11.41 4.18 5.09
15.25 14.07 8.78 9.79
z RU3 'ml4 RU5 Rub Bl 82 83 84
14/24 17/30 14/24 17/30 17j30 16/27 13/22 9/14 11/18 11/18
10.972 9.744 12.091 10.151 l.2.000 12.219 9.819 7.359 8.350 6.137
1.244 1.208 1.244 1.206 .244 .244 .799 .770 .778 .7?0
.35 .21 21 :30 30 :32 -.02 -.90 -.66 -.90
5.46 5.53 5.52 5.04 5.01 4.59 3.05 4.04 3.74 4.61++
141'24 17/30 14,'24 17/30 17/30 16/27 9/14 9/14 9/14 9/14
13.81 12.84 13.92 13.14 14.28 1::;~
13.58 L5.86 13.23 15.90 14.99 15.07 8.23 8.79 8.63 8.94
ClSBl
cr1 Cr2 Bl B2
14/24 16/28 10/16 13/22
9.721 10.750 9.680 8.450
.186 .186 .820 .790
.08 .06 -.08 -.15
5.50 4.48 2.483.03
14,'24 :6&
11.97 1:.;;
Y/14
5:os
13.37 14.91 9.67 8.95
Fe28
Fe B
15/x 10/16
10.020 7.687
1.171 .784
.39 -.79
5.75 3.52
15,'26 lo/l6
11.55 4.64
14.49 9.57
NizB
Ni B
15/26 lo/l6
9.537 7.352
1.156 .??6
.39 -.79
6.02 3.79
15/26 lo,'16
11.01 4.41
14.50 9.48
Ru?BI
RUl RU2 Ru3 B
15/26 15/24 15j26 9/14
12.070 13.363 12.108 9.460
1.245 1.245 1.245 .785
19 :19 .19 -.44
5.36 4.72 5.10 3.51
15/26 15,'24 15/26 Y/14
13.95 14.51 14.05 4.77
13.95 14.22 14.64 8.31
NiaB (10)
Nil Ni2 B
14/24 17/30 9/14
9.856 10.580 7.552
1.137 1.156 .770
.33 .33 -1.00
5.54 4.93 4.27+
14,'24 17/30 9.04
10.90 11.83 4.21
13.32 14.11 8.58
NisBs ortho
Nil Ni2
(loi
::: Bl BZ B3
NiaBp Monocl (10)
Nil Ni2 ",;
R"11Be
aEach Each
+ indicates - indicates
V > estimated Vz < estimated
valence valence
by an additional 0.25 by 0.25 electron.
4.18 4.59 3.84
electron.
ferences, where the quantity of charge transferred is such that the number of electrons available (estimated valence) is equal to the calculated valence, V,. The values listed of the single bond metallic radii, R r, which are a function of both charge and hybridization, are then used to calculate the polyhedral atomic volumes in much the same way as the Voronoi cells are calculated. The faces of PAV cells, however, are placed midway between the surfaces of the atomic spheres of differing radii, RI. This placement of the cell faces then divides equally (approximately) the bonding electrons between the two atoms, thus justifying the designation of “polyhedral atomic volume”. The final column in Tables 1, 2, and 3 gives the coordination number, CNv, based on the pyramidal volumes, Vi, of the i’th face according [ 31 to the equation l/CNV = ~(Vi/V,)2, where VT is the total volume (i.e., the listed PAV’s). This kind of definition for coordination number not only gives the usual result for regular polyhedra (Le., 8 for an octahedron) but gives the appropriate low weight to small, unimportant faces. For example, the PAV cell for atom Bl in WB2 (Fig. 1) has 9 small faces of a total of 16 with a CNv of 8.07 (Table 2).
161
Fig. 1. These PAV polyhedra are drawn with the x and y Cartesian axes in the xy crystallographic plane with x axes parallel. Faces parallel to the y axis are seen end-on.
In Tables 1,2, and 3, the presence of reference marks, following metallic radii calculations, indicates either d-character deviations (*) from normality or failure of the data to converge (+,-). A surplus of such marks and a high effective charge, greater than k1.0, casts suspicion upon the composition and/or structural analysis. For the transition metal borides, the effective charges differ from the transferred charges by no more than 0.1 electron; however, for the rare earth borides, the effective charge is more positive than the formal transferred charge by about 0.2 to 0.4 electrons. We note that, excluding U and the rare earths, the maximum possible number of unpaired d electrons according to these PMR calculations occurs for the ruthenium and cobalt sub-borides at 2.4 (max.).
Discussion The average PAV volume for boron undergoes a steady decrease from about 6.5 A3 for the duodecaborides to about 4.5 A3 for the subborides where only metal-boron bonds are formed. In terms of these latter contacts (M-B) one may define [3] a fractional coordination number as f(M-B) =
162
C VF(M-B)/C VF (all contacts) such #at f(M-I3) = 0 for boron and f(M-B) = 1 for NiaB. In view of the average PAV decrease noted above for increasing f(M-B), it is surprising to find that for the borides of some metals, for example Sm in SmB4 and Sm2B5, the PAV volume increases with f(M-B). As suggested in Fig. 1, part of the reason for this is related to the shape or isonomicity [ 111 of the boron PAV cell. In these polyhedra, the atom position is marked by an X. Note that the more asymmetry borons, B5 of Sm,B, and B1 of WBs, are larger than their symmetric counterparts. These are in contrast with the small symmetric cells for boron in CrB and FesB where f(B-M) approaches 1 (0.849 and 0.937, respectively). Whilst the PMR results for the duodecaborides appear reasonable, those for the hexaborides of Sm and Lu indicate that the boron octahedron is too compact, giving rise to an excessive charge transfer, too small a metal valence, and, related, too large a metal PAV volume. By expanding the octahedron and tilting its axes away from the cubic axes, the above-mentioned difficulties could be ameliorated. However, at the same time, the symmetry requirements on the borons will be relaxed, suggesting an instability in the cubic structure for the duodecaborides of the smaller metal atoms (see ref. 12). The use of the positional parameters of ThB4 gives moderately reasonable results for SmB4 but poor results for LuB4 since the B-B bonds are somewhat too strong for Bl and B3. For the vacancy-disordered compound Mo~_~B~, the PMR calculations are reasonable with 60% occupancy of the Mol site; PAV results assume full occupancy. The high charge calculated for W,B,, plus the increased W d-character, suggest that this structure should be questioned, as indicated by Lundstrom [S] . The difficulty with the related Ru,B, structure [4, 51 is even greater, and raises questions concerning its existence. The only difficulti~ occurring in Table 2 concern the rare earth diborides with the AlB, structure. Here, the metal-metal bonds are so short that the normal rare-earth radii give rise to impossible valencies (e.g., 7 to 8). By assuming an anisotropic radius corresponding to high d (or f) character for bonds in the basal plane, the results for TbB, approach reasonableness. This ~sumption of anisotropy has been used previously by the author to discuss the SmCo, structure [ 133 where the anisotropy arose from either anisotropic f electron shielding or f orbital bonding participation. Using six C-type bidirectional orbit& modelled after those for HCP metals [14] but with high basal plane f-character (-16%), one can form six metal-metal bonds and six metal-boron bonds (3 above and 3 below) the basal plane. The other six borons oan be bonded by a set of G-type bidirectional orbitals mutually orthogonal to the other six orbit&. This result is in contrast to an earlier attempt using the BOA method [ 11. Valence bond photoemission studies should be able to distinguish between the normal f electron distribution and that correspond~g to anisotropic f electron shielding and bond hybridization. LuB,, without anisotropic bonding, does not give satisfactory PMR results. Although CrsB4 and MnsB4 appear reasonable in Table 2, NbsB, and TasB4 [4, 51 are less so with too strong a B2-B2 bond (bond order aO.95).
163
The satisfactory PMR and PAV results for disordered IrBr,ss were calculated by using an ordered arrangement corresponding to the special space group Cl. The best results for MOB (Table 3) and WB (not listed) were obtained with a boron parameter of u = 0.344. Among the sub-borides, Table 3, the positions of B2 and B4 in Rurr Bs need reconsideration. For the various nickel borides we note that the boron charges are all high, suggesting some systematic problem in their preparation or the operation of a special bonding effect.
References 1 2 3 4 5 6 7 8 9 10 11 12 13 14
F. L. Carter, in R. S. Roth and S. J. Schneider (eds.), Solid State Chemistry, Nat. Bureau Stand. (U.S.) Spec. Publ. No. 364, 1972, p. 515. L. Pauling, The Nature of the Chemical Bond, Cornell Univ. Press, Ithaca, New York, 3rd Edn., 1960, p. 393 f.f. F. L. Carter, submitted for publication in Acta Crystallogr. W. B. Pearson, in G. V. Raynor (ed.), A Handbook of Lattice Spacings and Structures of Metals and Alloys-2, Vol. 8, Pergamon Press, Oxford, 1967. R. W. G. Wyckoff, Crystal Structures, Vols. 1 and 2, Interscience, 2nd. Edns., 1963, 1964. T. Lundstrom and I. Rosenberg, J. Solid State Chem., 6 (1973) 299. P. H. Schmidt, A. S. Cooper and S. J. LaPlaca, The synthesis and structure of Sm2B5, accepted for publication, Acta Crystallogr. T. Lundstrom, Ark. Kemi, 30 (1968) 115. T. Lundstrom and L. Tergenius, Acta Chem. Stand., 27 (1973) 3705. S. Rundqvist and S. Pramatus, Acta Chem. Stand., 21 (1967) 191. E. W. Gorter, J. Solid State Chem., 1 (1970) 279. M. C. Nichols, R. W. Mar and Q. Johnson, J. Less-Common Met., 33 (1973) 317. F. L. Carter, in P. E. Field (ed.), Proc. Rare Earth Conf., 9th, VP1 and State Univ., Blacksburg, Va., Oct. 1971, Vol. 2, p. 617. F. L. Carter, Proc. Rare Earth Res. Conf., 5th, Iowa State Univ., Ames, Iowa, Aug. 30 -Sept. 1,1965, Book Two, p. 103.
Discussion T. LUNDSTRGM I want to make a comment RuzBb. You can test the strength give an unacceptable result.
on your paper. of your theory
There is no boride with the composition by a calculation on RuzB,. It should
F. L. CARTER In attempting to use Pauling Metallic Radii calculations on the literature values given for Ru2B5, I have found that self-consistent, reasonable values for this compound are not obtainable. In particular, the boron distances are too close, giving rise to unreasonable boron valencies. It is unlikely that this compound exists as described.