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A ternary boron-rich phase Al1.00Cu0.79B25 grown from a molten AlCuB mixture
Journal of the Less-Common
Metals,
108 (1985)
ATERNARY 3URON-R~~H~HA~E AMOLTEN~-~u-RM~XT~RE IWAMI HIGASHI
The Institute
and YASUO
ofPhysical
August
177
177 - 188
~~_~~u*.,~*~~R~~N
FROM
TAKAHASNI
and Chemical
Research,
Wake, Saitamo 352431 (Japan)
23, 1984)
Summary B 2s crystal was grown from a high temperature molten An A~I.c&uo.,~ Al-&-B mixture and the structure was determined. The crystal has tetragonal symmetry with a = 9.002(S) 8, G = 5.069(2)& Z = 2 and space group P&2. 874 unique reflections (MO Kcr; 26’ < 120”) were measured by means of a single-crystal X-ray diffractometer and were used in the block diagonal least-squares refinement to a final R value of 5.3%. The boron framework in the crystal is essentially the same as that of tetragonal a-B (P4,jnnm; u = 8.74 8; c = 5.07 A). The aluminium atom is situated at a large hole surrounded tetrahedraily by four IQ2 icosahedra. However, as in AiBe0.sB24.2, the aluminium site is split into two positions separated by distances of +0.247(2) A from the centre of the hole. The copper atoms are distributed between two holes Cu(1) and Cu(2) with occupancies of 76.0(4)% and 23(4)% respectively. An unusually long intra-icosahedral B-l3 bond length of 2.026(33 A is found between the boron atoms which are bonded to the Cu(1) atom with a similar bond length of 2.049(2) A,
1. Introduction A large number of metals react with boron to form various kinds of boride compounds. Among these the metallic elements, copper and aluminium, are unique in that their high temperature melts dissolve a large quantity of boron [1] and grow Br2 icosabedral crystals during the cooling process [Z, 3 1. The crystals obtained from the Cu-3 melt have structures of the rhombohedral P-3 type with various chemical compositions such as CUE& f41, CuLzs f51, CuBwsq 161andCuBw [7] ; the crystals obtained from the Al-B melt are a-AlBrs [S, 91 and y-A1Br2 [lo, 111 which consist of Blz icosahedra and B?, umts, Accordingly it seemed to be of interest to investigate crystal growth from high temperature ternary Al-Cu-B melts. We have obtained three phases from the ternary mixture: (1) A11.&u0,79B25 with tetragonal symmetry, (2) a phase with a structure of the rhombohedral 0-B type and a composition compatible with that of CU~AI~.,B,,~ [1!2] and (3) an unknown @ Elsevier
Sequoia/Printed
in The Netherlands
178 TABLE 1 Comparison of the crystal data with those of related materials Formula unit Crystal system a (A) c (8) Space group d, (cmP3) d, (cme3) z
A~Lo&~o.T&s* Tetragonal 9.002(3) 5.069(2) P&22* 2.8b 2.81 2
J%o
(a
t&r.
Tetragonal 8.740(15) 5.068( 10) P42/nnm 2.31 2.33 1
‘Determined by structure analysis. bFlotation in a mixture of tribromomethane
B)
BSOCZ
Tetragonal 8.753(4) 5.093(15) P42/nnm 2.43 2.426 1
B&2
AlBeo.aB24.2
Tetragonal 8.634(4) 5.128(3) PZ2m 2.46 2.463 1
Tetragonal 8.82 5.08 P421nnm 2.51 2.48 2
and acetone,
phase of monoclinic symmetry with approximately the same composition as phase (2). The ternary phase examined here is basically the same as that of tetragonal o-B [ 131 with respect to the boron structure, and is similar to AlBe,,sB24.Z [14] particularly with respect to the aluminium distribution. However, there are obvious differences between some of the structural details of this crystal and those reported previously for similar phases. The basic crystal data are compared with those of related compounds including BsOCZand BsONZ [ 151 in Table 1.
2. Experimental
details
2.1. Preparation of crystals A vertical A120s tube furnace equipped with Sic resistors and a BN crucible was used. The starting materials were boron (purity, 99.5%), copper (purity, 99.999%) and aluminium (purity, 99.99%). A mixture consisting of 0.625 g of aluminium, 1 g of boron and 29 g of copper (atomic ratio 1:4:20) was heated to 1500 “C, kept at this temperature for 1 h and then allowed to cool to room temperature. The crystals grown in the reaction mixture were separated by dissolving the excess metals in HNOs. The crystals thus obtamed were of two types. The first type was needle shaped and the second type was plate like or irregularly shaped. These crystals were examined by means of X-ray diffraction and chemical analysis. Oscillation and Weissenberg photographs indicated that the needles had a tetragonal lattice. The Laue symmetry and the extinction condition were consistent with the space groups Ph2, P4,nm and P4&nm. The lattice constants (Table 1) of the needles were measured using a single-crystal diffractometer with MO Kol radiation (X = 0.71073 A). Two different phases were found in the second type of crystal: one was of trigonal symmetry with similar crystal data to those of A12.,CuZB104 [12] and the other was an unknown monoclinic ternary boride. The average composition of the needles as determined by
179
and that of the second type of crystal chemical analysis was A11,25CU~,~~B~~, consisting of the two phases was A13.3C~2,3B,04.Further structural investigation of both the trigonal and monoclinic phases in the second type of crystal is now in progress in our laboratory. 2.2. Intensity measurement A specimen with dimensions 0.1 mm X 0.1 mm X 0.25 mm was cut from a needle and was used for the measurement of the X-ray intensities. The reflections with 0 < h Q 21,O G Ft< 21 and 0 < I < 12, for which 28 was less than 120”, were examined using a Rigaku automated four-circle diffractometer with ~aphite-monochromatized MO Ka radiation (h = 0.71073 A). The o (28 G 30”) or o - 28 (30” < 26 < 120”) scan mode was used at a rate of 2” min-’ in o. Background counts of 10 s were measured on each side of the scanning width (Ao = 1.2 + 1.5 tan 0). Three standard reflections were examined after every 100 reflections and no noticeable variations were observed. The intensities were corrected for the Lorentz and polarization effects. No absorption (II = 23 cm-l) or extinction corrections were made. Reflections with F, < 2.50,~ were excluded from the data and finally 874 unique reflections were obtained by averaging equivalents. 2.3. Determination and refinement of the structure Despite the finding that the possible space groups, as indicated by the preliminary X-ray diffraction work, were different from those of the related phases (Table l), the present crystal was assumed to have the same boron structure as that of tetragonal a-B. This assumption was made because the lattice constants were approximately equal and a structure based on the possible space groups P4,nm or P&z2 could be derived from the tetragonal o-B structure with only a slight modification. If the (001) diagonal glide planes at z = +1/4 or the { 110) mirror planes are eliminated from the tetragonal cw-Bstructure (space group, P4$znm), a structure of lower symmetry which belongs to space group P4,nm or P&42 is obtainable. Thus we examined two structures with space group P4gzm or P&22, and finally concluded that the correct space group is P&22; a least-squares refinement for the structure in the space group P4gzm was stopped at an R value of about 30%. The structure was solved by iterative Fourier methods starting with the structure of tetragonal a-B. The assignments of the three metal sites were made with reference to the composition determined by chemical analysis and the metallic distribution in A1Be0.sB24.2[14]. Thus the atom located at the same position as aluminium in A1Be0.sB24.2was assigned as aluminium and the two atoms situated at new sites were assigned as Cu(1) and Cu(2). Finally a block diagonal least-squares refinement of the structure including the occupancies of the metal sites was made. Anisotropic temperature factors were applied to all the metal sites except Cu(2) which showed a very low occupancy. The function min~ized was &.u(]F~~~- lF,/)2 with w = 1/0~~~. The values obtained for
180
and R
w
Z:w(lFoI - lFcl)2 1’2
=
1
=4Fo12
I
were 0.053 and 0.034 respectively. A difference Fourier synthesis subsequently calculated showed no significant maxima. The chemical composition of the crystal determined from the structure analysis (Table 1) was compatible with the composition Ali_,,Cu o,7sB25 obtained by chemical analysis. A slight increase in the aluminium content in the composition determined by chemical analysis can be attributed to the presence of free aluminium in the sample crystals; the same tendency was observed in LiA1Bi4 [16] and MgAlBi4 [17] which were prepared by similar techniques to those used to obtain the present crystal. The validity of the assignment of the metal sites will be discussed later. The atomic scattering factors and the dispersion correction factors were taken from ref. 18. All the calculations were performed using the UNICS-III program [19] on a FACOM M200 computer.
3. Results and discussion The final atomic coordinates and temperature factors are given in Table 2. The boron framework of the present crystal (Fig. 1) is basically the same as that of tetragonal a-B. Although the boron structure has been described extensively by Hoard et al. [13], a brief description will be given here to facilitate the discussion. The boron framework consists of four Bi2 icosahedra (B(1) - B(6) in Table 2) and two isolated boron atoms (B(7) in Table 2) per unit cell. The icosahedra are centred about the positions $, a, 1.2 3 1.3 1 A* L 3 2 of the space group P&22. They are oriented so 4’ 4’ 4’ 4’ 4’ 4’ 4’ 4’ 4’ 4 that one of the quasi-fivefold axes is approximately parallel to the c axis and one of the quasi-mirror planes is approximately parallel to the { 110) plane. The isolated boron atoms are situated at the positions 0, 0, i; f , $, 0. Each of the isolated boron atoms combines the external bonds of four different icosahedra which surround the boron atom in the arrangement of a bisphenoidally distorted tetrahedron. Two apex atoms of the icosahedron are involved in such a combination, and each of the remaining ten apex atoms is connected to a neighbouring icosahedron. Thus every icosahedron is linked to two isolated boron atoms and 10 neighbouring icosahedra by direct intericosahedral bonds. There are four infinite B12 icosahedral chains per unit square plane lattice (a a X a A) running along the c axis. In the same way, if the single boron atom is disregarded, four infinite channnels per unit square plane lattice are seen in the same direction as the icosahedral chains. Unlike the
181 TABLE 2 Final atomic coordinates and temperature factors Atom
Site
x
Y
z
B (8’)
Occupancy
B(1) B(2) B(3) B(4) B(5) B(6) B(7)
8i 8i 8i 8i 8i 8i
0.3259(3) 0.2369(3)
0.4050(5) 0.0850(5) 0.3803(4) 0.0856( 5) 0.4128(5) 0.584814)
0.41 0.45 0.49 0.45 0.48 0.44
100 100 100 100 100 100
2b
0
0.0832(3) 0.0809(3) 0.1232(3) 0.2303(3) 0.3140(3) 0.2404(3) 0
1
0.32
100
Al*
4e
0
0
:0487(5)
_b
50
Cu(l)
2c
0
1
1
F
z-
_b
76.0(4)
2d
1
0
1
3.6
Cu(2)
0.1304(3) 0.0901(3) 0.0937(3) 0.2485(3)
z
4
(%)
2.8(4)
Atom
UIl
u22
u33
Ul2
u13
u;3
B eqc (A2)
Al Wl)
86(13) 172(5)
62(12) 172(l)
263( 20) 578(7)
-20(9) 129(4)
0 0
0 0
1.1 2.4
aThe aluminium atoms fully occupy the holes centred at the 2a site (0, 0, 0; k, $, a). However, ’ 1 812?
A k 2
the aluminium site in each hole is split into two positions (O,O, 0 k 0.049; 0.049).
Therefore the occupancy of the aluminium site is 50%.
bAnisotropic temperature factors given by the expression is exp{-10P4
X 2n2( Ulrh20*2 + U22k2b* + U3312c* + 2U&ka*b*
+ 2U&la*c*
+
+ 2U23kZb*c*)) CThe equivalent isotropic temperature factors are calculated from the relation
8
B eq = - $x.x.U..a.*a.*a.a. I I Ill J 3
IJ
icosahedral chains, which are all crystallographically equivalent, the channels are divided into two types. One of the channels is centred at x = 0, y = 0 or x = $_, y = i and accommodates both the single boron atoms (Fig. 1) and the alummium atoms (Fig. Z), and the other is centred at x = 0, y = a or x = a, y = 0 and contains the Cu(1) and Cu(2) atoms (Fig. 2); in Fig. 2 the B(7), Cu(2) and Cu( 1) atoms which overlap the Al, Cu(1) and Cu(2) atoms respectively are omitted. Thus the first type of channel has an Al-B-Al-B chain at its centre and the second type has a Cu(l)-Cu(2)-Cu(l)-Cu(2) chain at its centre. It should be noted here that the aluminium site is split into two positions along the Al-B-Al-B chain by displacements of rtO.247(2) A from the midpoint of, the nearest two boron atoms. Thus the Al-B distance in the chain is either 2.288(3) A or 2.781(3) A. The Cu(l)-Cu(2) distance is 2.535(2) A, which is just half tbe c axis length. The occupancy of the ~uminium site is 50% (Table 2), i.e. the hole which accommodates the aluminium is fully occupied. The occupancies of the Cu(1) and Cu(2) sites
182
Fig. 1. The boron framework as seen along the c axis. The z coordinate of the centre of each icosahedron is either i or + as indicated in the figure.
are 76.0(4)% and 2.8(4)% respectively. The nature of the aluminium distribution is exactly the same as that found in A1Beo.sB24.2;the split in the aluminium site in this phase is +0.27 A. However, there is no correspondence regarding the copper distribution because the copper sites in the boron framework of A1Beo.sB24.2are completely vacant and the beryllium atoms are statistically distributed over the apex sites of the icosahedron. The B-B bond lengths are given in Table 3 and are compared with their equivalents in related phases. Unlike the icosahedra in the related phases which belong to a space group of higher symmetry (Table l), the icosahedron of the present crystal has no centre of symmetry and is rather distorted. This can be ascribed to the incorporation of copper atoms into Cu(1) and Cu(2) sites with very different occupancies; if the occupancy of one copper site were the same as that of the other, the crystal structure would have the (110) mirror planes belonging to the space group P4,/nnm. However, there is no significant difference in the size of the icosahedron. The average B-B bond lengths within the icosahedron are 1.818 A, 1.81 A, 1.796 A and 1.811 A for the present crystal, tetragonal a-B, B5sC2 and AlBeo.sB24.2respectively.
183
-b
Fig. 2. The distribution of metal atoms. They are projected along the c axis; B(7), Cu( 1’) and Cu( 2’) which overlap Al, Cu(2) and Cu( 1) respectively are omitted. The z coordinates of the overlapping atoms are all 3. The symmetry code for the aluminium atom in the figure is identical with that used in Table 4.
The striking feature of the present crystal is the presence of an intraicosahedral B-B bond (B(5)-B(5) in Table 3) with an unusually long bond length (2.026(3) A). The boron atoms giving such a long bond length interact strongly with the Cu(1) atom with a similar bond length (2.049(4) A), resulting in an almost regular triangle of B(5)-Cu(l)-B(5) linkage (Fig. 3). Similar bonds are also present at the opposite side of the icosahedron where Cu(2) is in contact with the two boron atoms giving a B(l)-Cu(2)-B(1) triangle with sides of 1.91 A for (%(2)-B(l) and 1.95 A for B(l)-B(1) (Fig. 3). The assignment of the Cu(2) site may be questionable because the value of 1.91 A observed for the distance from this site to the nearest boron atoms seemed to be rather short for a Cu-B bond length but acceptable for a B-B bond. From the following viewpoints, however, the site assignment is considered to be reasonable. (1) In the boron framework of AlBe 0.8B24.2 [14] no atoms have been found at the same positions as the Cu(1) and Cu(2) sites which have similar chemical environments. This means that boron and aluminium atoms do not incorporate into the copper sites. (2) The metallic radius of copper is noticeably shorter than that of aluminium: 1.28 a (12 coordination) or 1.24 A (8 coo~~ation) for copper,
184 TABLE
3
Comparison
of the B-B
bond
lengths
with their equivalents
in tetragonal
Q-B, B.&z
and
A1BeO.sB24.2
Bond
Symmetry codea
B(l)-B(1) B(l)-B(2) B(l)-B(2) B(l)-B(3) B(1)-B(5)b B(l)-B(6) B(2)-B(3) B(2)-B(4) B(2)-B(4)b B(2)-B(6) B(3)-B(4) B(3)-B( 5) B(3)-B(6) B(3)-B(7)C B(4)-B( 5) B(4)-B( 5) B(4)-B(6) B(5)-B(5) B( 5)-B(6) B(6)-B(6)b ‘Symmetry (y, -x,
-2);
B-B
1 0 1 0 2 0 0 0 3 1 0 0 0 0 0 1 1 1 0 4 codes: 4, (a -
bIntericosahedral CBond between dBond between
none
bond length (A)
A~~.ooCuo.w&s
Tetragonal
1.951(3) 1.809(3) 1.825(3) 1.801(4) 1.844(3) 1.822(3) 1.817(3) 1.885(3) 1.767(3) 1.772(3) 1.814(3) 1.756(4) 1.821(3) 1.,725(3) 1.822(3) 1.805(3) 1.764(3) 2.026(3) 1.772(3) 1.681(3)
1.85(3) 1.80(2) 1.81(l) 1.822(g) 1.86(2) 1.79(l) 1.79(2) 1.81(l) 1.66(2) 1.79(l) 1.79(2) 1.822(8) 1.84(l) 1.601(5) 1.80(2) 1.81(l) 1*79(l) 1.85(3) 1.79(l) 1.71(3)
or 0, (x, y, z);
y, $ -
X, + -
1, (-$--
y, $ -
(Y-B
X, $-
BsoC2
AlBe0.824.4
1.847(5) 1.814(6) 1.816(3) 1.806(5) 1.867(4) 1.814(6) 1.781(6) 1.800(3) 1.668(3) 1.758(6) 1.781(6) 1.806(5) 1.864(7) 1.638(4)d 1.814(6) 1.668(6) 1.758(6) 1.847(6) 1.814(6) 1.683(7)
1.918 1.797 1.794 1.799 1.818 1.811 1.799 1.861 1.715 1.779 1.799 1.799 1.833 1.661 1.797 1.794 1.779 1.918 1.811 1.660
z);
2, (y, -x,
1 -z);
3,
z).
B-B bond. an apex atom of the icosahedron an apex atom of the icosahedron
and the isolated and the isolated
boron atom (B(7)). carbon atom.
and 1.43 A (12 coordination) or 1.39 A (8 coordination) for aluminium [20]. Further, short bond lengths in the range of 2.03 - 2.08 A are found even for Al-B bonds in Blz icosahedral phases such as (r-A1B12 [ 8, 91, A11_1Beo.6B22[21], y-AIB12 [lo, 111, LiA1B14 [16] and MgAlB14 [17, 221. Therefore the interatomic distance of 1.91 A is highly probable for the Cu-B bond. The boron coordination around the metal sites is presented in Fig. 4, and the distances between the boron atoms and the metal sites are given in Table 4. As in the cases of LiA1B14 [16] and MgA1B14 [17], the metal atoms show significant anisotropic thermal motion. There is a close interrelation between the thermal ellipsoid of each metal atom and the geometry of the hole accommodating it. This is interpreted as follows. As mentioned above, the Cu(1) atom is bonded much more strongly to the four B(5) atoms than to the other eight boron atoms (Table 4) and thus the copper atom should naturally vibrate more in parallel with the (li0) plane than in the direction
[ii01
0r
L-[so11
[ii01
[ii01 or
Cu(l)-B(5)
: 2.05i
B (5)-B(5)
: 2.03i
C”(2) -B(l)
: 1.91 i
B (1) -B(l)
: 1.95 i
[llOI
Fig. 3. Strong interactions between copper atoms and B 12 icosahedra. hedral B-B bonds involved in such interactions have unusually long indicated in the figure.
TABLE
4
Distances
between
the metals and boron
first neighboursa
Symmetry code b
Bond
Al-2B(2) Al-2B(2) Al-2B(3) Al-2B(3) Al-2B(4) Al-2B(4) Al-B(7) Al-B(7) Cu(l)-4B(l) Cu(l)-4B(4) Cu(l)-4B(5) Cu(2)-4B(l) Cu(2F4B(2) Cu( 2)-4B( 5)
Distance
0, 1 4, 7 0, 1 4, 7 0, 1 4,7 0 4 5, 8, 10, 12 0, 0, 0, 0, 6,
2, 2, 3, 3, 9,
14, 14, 15, 15, 11,
16 16 16 16 13
2);
4, (-Y,
1 -
2); 9, (y, -x, +x,
x, -2);
+ -y,
5, (-Y, 1 -
-$
x, 1-s);
2);
10,
+ 2);
13,
(++y,-++x,+-z);16,(+y,~-x,+-z).
($ -x,
AlBeo.sB24.2
2.261(2) 2.353(2) 2.331(2) 2.709(2) 2.234(2) 2.327(2) 2.288(3) 2.781(3) 2.465(2) 2.692(2) 2.049(2) 1.906(2) 2.616(2) 2.537(2)
2.185 2.293 2.254 2.681 2.185 2.293 2.266 2.814
in AlBee.sB24.2. z); 2, (-x, 1 -y,
6, (1 -Y,
x, 1-s);
$ + y, -+
+ 2);
(+ +x,+-y,
(A)
AhGo.79Bzs
aAl-B distances are compared with the equivalents bSymmetry codes: 0 or none, (x, y, z); 1, (-x,-y,
(-+
The intra-icosabond lengths as
-+
+ 2);
7, (Y, -x, 11, ($ 14,
(-i
x, -+
z); 3, (1 -x,-y, --a);
8, (Y, 1 -x,
+ y, -$
+ y, + +x,
+ 2); 12,
+ -2);
15,
(cl
c
23
(d)
Fig. 4. Coordinations of the boron atoms around the copper and aluminium atoms showing thermal ellipsoids at the 50% level for Cu( 1) with symmetry codes; (b) designations of the atoms around AI with symmetry and At: (a) designations of the atoms around &(I) codes; (c) designations of the atoms around Cu(2) with symmetry codes; (d) stereoscopic pair for Cu(1); (a) stereoscopic pair for Al; (f) stereoscopic pair for Cu(2). The symmetry codes for the atoms in (a), (b) and (c) are identical with those used in Table 4.
188
perpendicular to it (Fig. 4(a)). A similar mode of thermal vibration is expected to be associated with the Cu(2) site because it is centred in a chemical environment (Fig. 4(c) and Table 4) similar to that of the Cu(1) site; owing to the low occupancy, anisotropic refinement could not be applied to this site. The coordination polyhedron surrounding the aluminium site is ellipsoidal (Fig. 4(b) and Table 4) with its major axis along the c axis. Therefore, as shown in Fig. 4(b), the aluminium atom should necessarily vibrate more strongly along the c axis than in parallel with the (001) plane.
References 1 M. Hansen, Constitution of Binary Alloys, Metallurgy and Metallurgical Engineering Series, McGraw-Hill, New York, 1958. 2 I. Higashi, Y. Takahashi and T. Atoda, J. Less-Common Met., 37 (1974) 199. 3 V. I. Matkovich, J. Economy and R. F. Giese, Jr., J. Am. Chem. Sot., 86 (1964) 2337. 4 I. Higashi, T. Sakurai and T. Atoda,J. Less-Common Met., 45 (1976) 283. 5 S. Andersson and B. Callmer, J. Solid State Chem., 10 (1974) 219. 6 J. Rexer and G. Petzov, Metallurg (Moscow), 24 (1970) 1083. 7 J. 0. Carlson and T. Lundstrom, J. Less-Common Met., 22 (1970) 317. 8 I. Higashi, T. Sakurai and T. Atoda, J. Solid State Chem., 20 (1977) 67. 9 J. S. Kasper, M. Vlasse and R. Naslain, J. Solid State Chem., 20 (1977) 281. 10 R. E. Hughes, M. E. Leonowicz, J. T. Lemley and L.-T. Tai, J. Am. Chem. Sot., 99 (1977) 5507. 11 I. Higashi, J. Solid State Chem., 47 (1983) 333. 12 R. Mattes, L. Marosi and H. Neidhard, J. Less-Common Met., 20 (1970) 223. 13 J. L. Hoard, R. E. Hughes and D. E. Sands, J. Am. Chem. Sot., 80 (1958) 4507. 14 K. Krogmann and H. J. Becher, 2. Anorg. Allg. Chem., 392 (1972) 197. 15 G. Will and K. H. Kossobutzki, J. Less-Common Met., 47 (1976) 33. 16 I. Higashi, J. Less-Common Met., 82 (1981) 317. 17 I. Higashi and T. Ito, J. Less-Common Met., 92 (1983) 239. 18 J. A. Ibers and W. C. Hamilton (eds.), International Tables for X-ray Crystallography, Vol. 4, Kynoch, Birmingham, 1974, pp. 72,74,79,149. 19 T. Sakurai and K. Kobayashi, Rep. Inst. Phys. Chem. Res., Tokyo, 55 (1979) 69. 20 L. Pauling, The Nature of the Chemical Bond, Cornell University Press, Ithaca, NY, 1940 (Japanese translation, 1950, p. 445). 21 I. Higashi, J. Solid State Chem., 32 (1980) 201. 22 V. I. Matkovich and J. Economy, Acta Crystallogr., Sect. B, 26 (1970) 616.
Metals,
108 (1985)
ATERNARY 3URON-R~~H~HA~E AMOLTEN~-~u-RM~XT~RE IWAMI HIGASHI
The Institute
and YASUO
ofPhysical
August
177
177 - 188
~~_~~u*.,~*~~R~~N
FROM
TAKAHASNI
and Chemical
Research,
Wake, Saitamo 352431 (Japan)
23, 1984)
Summary B 2s crystal was grown from a high temperature molten An A~I.c&uo.,~ Al-&-B mixture and the structure was determined. The crystal has tetragonal symmetry with a = 9.002(S) 8, G = 5.069(2)& Z = 2 and space group P&2. 874 unique reflections (MO Kcr; 26’ < 120”) were measured by means of a single-crystal X-ray diffractometer and were used in the block diagonal least-squares refinement to a final R value of 5.3%. The boron framework in the crystal is essentially the same as that of tetragonal a-B (P4,jnnm; u = 8.74 8; c = 5.07 A). The aluminium atom is situated at a large hole surrounded tetrahedraily by four IQ2 icosahedra. However, as in AiBe0.sB24.2, the aluminium site is split into two positions separated by distances of +0.247(2) A from the centre of the hole. The copper atoms are distributed between two holes Cu(1) and Cu(2) with occupancies of 76.0(4)% and 23(4)% respectively. An unusually long intra-icosahedral B-l3 bond length of 2.026(33 A is found between the boron atoms which are bonded to the Cu(1) atom with a similar bond length of 2.049(2) A,
1. Introduction A large number of metals react with boron to form various kinds of boride compounds. Among these the metallic elements, copper and aluminium, are unique in that their high temperature melts dissolve a large quantity of boron [1] and grow Br2 icosabedral crystals during the cooling process [Z, 3 1. The crystals obtained from the Cu-3 melt have structures of the rhombohedral P-3 type with various chemical compositions such as CUE& f41, CuLzs f51, CuBwsq 161andCuBw [7] ; the crystals obtained from the Al-B melt are a-AlBrs [S, 91 and y-A1Br2 [lo, 111 which consist of Blz icosahedra and B?, umts, Accordingly it seemed to be of interest to investigate crystal growth from high temperature ternary Al-Cu-B melts. We have obtained three phases from the ternary mixture: (1) A11.&u0,79B25 with tetragonal symmetry, (2) a phase with a structure of the rhombohedral 0-B type and a composition compatible with that of CU~AI~.,B,,~ [1!2] and (3) an unknown @ Elsevier
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178 TABLE 1 Comparison of the crystal data with those of related materials Formula unit Crystal system a (A) c (8) Space group d, (cmP3) d, (cme3) z
A~Lo&~o.T&s* Tetragonal 9.002(3) 5.069(2) P&22* 2.8b 2.81 2
J%o
(a
t&r.
Tetragonal 8.740(15) 5.068( 10) P42/nnm 2.31 2.33 1
‘Determined by structure analysis. bFlotation in a mixture of tribromomethane
B)
BSOCZ
Tetragonal 8.753(4) 5.093(15) P42/nnm 2.43 2.426 1
B&2
AlBeo.aB24.2
Tetragonal 8.634(4) 5.128(3) PZ2m 2.46 2.463 1
Tetragonal 8.82 5.08 P421nnm 2.51 2.48 2
and acetone,
phase of monoclinic symmetry with approximately the same composition as phase (2). The ternary phase examined here is basically the same as that of tetragonal o-B [ 131 with respect to the boron structure, and is similar to AlBe,,sB24.Z [14] particularly with respect to the aluminium distribution. However, there are obvious differences between some of the structural details of this crystal and those reported previously for similar phases. The basic crystal data are compared with those of related compounds including BsOCZand BsONZ [ 151 in Table 1.
2. Experimental
details
2.1. Preparation of crystals A vertical A120s tube furnace equipped with Sic resistors and a BN crucible was used. The starting materials were boron (purity, 99.5%), copper (purity, 99.999%) and aluminium (purity, 99.99%). A mixture consisting of 0.625 g of aluminium, 1 g of boron and 29 g of copper (atomic ratio 1:4:20) was heated to 1500 “C, kept at this temperature for 1 h and then allowed to cool to room temperature. The crystals grown in the reaction mixture were separated by dissolving the excess metals in HNOs. The crystals thus obtamed were of two types. The first type was needle shaped and the second type was plate like or irregularly shaped. These crystals were examined by means of X-ray diffraction and chemical analysis. Oscillation and Weissenberg photographs indicated that the needles had a tetragonal lattice. The Laue symmetry and the extinction condition were consistent with the space groups Ph2, P4,nm and P4&nm. The lattice constants (Table 1) of the needles were measured using a single-crystal diffractometer with MO Kol radiation (X = 0.71073 A). Two different phases were found in the second type of crystal: one was of trigonal symmetry with similar crystal data to those of A12.,CuZB104 [12] and the other was an unknown monoclinic ternary boride. The average composition of the needles as determined by
179
and that of the second type of crystal chemical analysis was A11,25CU~,~~B~~, consisting of the two phases was A13.3C~2,3B,04.Further structural investigation of both the trigonal and monoclinic phases in the second type of crystal is now in progress in our laboratory. 2.2. Intensity measurement A specimen with dimensions 0.1 mm X 0.1 mm X 0.25 mm was cut from a needle and was used for the measurement of the X-ray intensities. The reflections with 0 < h Q 21,O G Ft< 21 and 0 < I < 12, for which 28 was less than 120”, were examined using a Rigaku automated four-circle diffractometer with ~aphite-monochromatized MO Ka radiation (h = 0.71073 A). The o (28 G 30”) or o - 28 (30” < 26 < 120”) scan mode was used at a rate of 2” min-’ in o. Background counts of 10 s were measured on each side of the scanning width (Ao = 1.2 + 1.5 tan 0). Three standard reflections were examined after every 100 reflections and no noticeable variations were observed. The intensities were corrected for the Lorentz and polarization effects. No absorption (II = 23 cm-l) or extinction corrections were made. Reflections with F, < 2.50,~ were excluded from the data and finally 874 unique reflections were obtained by averaging equivalents. 2.3. Determination and refinement of the structure Despite the finding that the possible space groups, as indicated by the preliminary X-ray diffraction work, were different from those of the related phases (Table l), the present crystal was assumed to have the same boron structure as that of tetragonal a-B. This assumption was made because the lattice constants were approximately equal and a structure based on the possible space groups P4,nm or P&z2 could be derived from the tetragonal o-B structure with only a slight modification. If the (001) diagonal glide planes at z = +1/4 or the { 110) mirror planes are eliminated from the tetragonal cw-Bstructure (space group, P4$znm), a structure of lower symmetry which belongs to space group P4,nm or P&42 is obtainable. Thus we examined two structures with space group P4gzm or P&22, and finally concluded that the correct space group is P&22; a least-squares refinement for the structure in the space group P4gzm was stopped at an R value of about 30%. The structure was solved by iterative Fourier methods starting with the structure of tetragonal a-B. The assignments of the three metal sites were made with reference to the composition determined by chemical analysis and the metallic distribution in A1Be0.sB24.2[14]. Thus the atom located at the same position as aluminium in A1Be0.sB24.2was assigned as aluminium and the two atoms situated at new sites were assigned as Cu(1) and Cu(2). Finally a block diagonal least-squares refinement of the structure including the occupancies of the metal sites was made. Anisotropic temperature factors were applied to all the metal sites except Cu(2) which showed a very low occupancy. The function min~ized was &.u(]F~~~- lF,/)2 with w = 1/0~~~. The values obtained for
180
and R
w
Z:w(lFoI - lFcl)2 1’2
=
1
=4Fo12
I
were 0.053 and 0.034 respectively. A difference Fourier synthesis subsequently calculated showed no significant maxima. The chemical composition of the crystal determined from the structure analysis (Table 1) was compatible with the composition Ali_,,Cu o,7sB25 obtained by chemical analysis. A slight increase in the aluminium content in the composition determined by chemical analysis can be attributed to the presence of free aluminium in the sample crystals; the same tendency was observed in LiA1Bi4 [16] and MgAlBi4 [17] which were prepared by similar techniques to those used to obtain the present crystal. The validity of the assignment of the metal sites will be discussed later. The atomic scattering factors and the dispersion correction factors were taken from ref. 18. All the calculations were performed using the UNICS-III program [19] on a FACOM M200 computer.
3. Results and discussion The final atomic coordinates and temperature factors are given in Table 2. The boron framework of the present crystal (Fig. 1) is basically the same as that of tetragonal a-B. Although the boron structure has been described extensively by Hoard et al. [13], a brief description will be given here to facilitate the discussion. The boron framework consists of four Bi2 icosahedra (B(1) - B(6) in Table 2) and two isolated boron atoms (B(7) in Table 2) per unit cell. The icosahedra are centred about the positions $, a, 1.2 3 1.3 1 A* L 3 2 of the space group P&22. They are oriented so 4’ 4’ 4’ 4’ 4’ 4’ 4’ 4’ 4’ 4 that one of the quasi-fivefold axes is approximately parallel to the c axis and one of the quasi-mirror planes is approximately parallel to the { 110) plane. The isolated boron atoms are situated at the positions 0, 0, i; f , $, 0. Each of the isolated boron atoms combines the external bonds of four different icosahedra which surround the boron atom in the arrangement of a bisphenoidally distorted tetrahedron. Two apex atoms of the icosahedron are involved in such a combination, and each of the remaining ten apex atoms is connected to a neighbouring icosahedron. Thus every icosahedron is linked to two isolated boron atoms and 10 neighbouring icosahedra by direct intericosahedral bonds. There are four infinite B12 icosahedral chains per unit square plane lattice (a a X a A) running along the c axis. In the same way, if the single boron atom is disregarded, four infinite channnels per unit square plane lattice are seen in the same direction as the icosahedral chains. Unlike the
181 TABLE 2 Final atomic coordinates and temperature factors Atom
Site
x
Y
z
B (8’)
Occupancy
B(1) B(2) B(3) B(4) B(5) B(6) B(7)
8i 8i 8i 8i 8i 8i
0.3259(3) 0.2369(3)
0.4050(5) 0.0850(5) 0.3803(4) 0.0856( 5) 0.4128(5) 0.584814)
0.41 0.45 0.49 0.45 0.48 0.44
100 100 100 100 100 100
2b
0
0.0832(3) 0.0809(3) 0.1232(3) 0.2303(3) 0.3140(3) 0.2404(3) 0
1
0.32
100
Al*
4e
0
0
:0487(5)
_b
50
Cu(l)
2c
0
1
1
F
z-
_b
76.0(4)
2d
1
0
1
3.6
Cu(2)
0.1304(3) 0.0901(3) 0.0937(3) 0.2485(3)
z
4
(%)
2.8(4)
Atom
UIl
u22
u33
Ul2
u13
u;3
B eqc (A2)
Al Wl)
86(13) 172(5)
62(12) 172(l)
263( 20) 578(7)
-20(9) 129(4)
0 0
0 0
1.1 2.4
aThe aluminium atoms fully occupy the holes centred at the 2a site (0, 0, 0; k, $, a). However, ’ 1 812?
A k 2
the aluminium site in each hole is split into two positions (O,O, 0 k 0.049; 0.049).
Therefore the occupancy of the aluminium site is 50%.
bAnisotropic temperature factors given by the expression is exp{-10P4
X 2n2( Ulrh20*2 + U22k2b* + U3312c* + 2U&ka*b*
+ 2U&la*c*
+
+ 2U23kZb*c*)) CThe equivalent isotropic temperature factors are calculated from the relation
8
B eq = - $x.x.U..a.*a.*a.a. I I Ill J 3
IJ
icosahedral chains, which are all crystallographically equivalent, the channels are divided into two types. One of the channels is centred at x = 0, y = 0 or x = $_, y = i and accommodates both the single boron atoms (Fig. 1) and the alummium atoms (Fig. Z), and the other is centred at x = 0, y = a or x = a, y = 0 and contains the Cu(1) and Cu(2) atoms (Fig. 2); in Fig. 2 the B(7), Cu(2) and Cu( 1) atoms which overlap the Al, Cu(1) and Cu(2) atoms respectively are omitted. Thus the first type of channel has an Al-B-Al-B chain at its centre and the second type has a Cu(l)-Cu(2)-Cu(l)-Cu(2) chain at its centre. It should be noted here that the aluminium site is split into two positions along the Al-B-Al-B chain by displacements of rtO.247(2) A from the midpoint of, the nearest two boron atoms. Thus the Al-B distance in the chain is either 2.288(3) A or 2.781(3) A. The Cu(l)-Cu(2) distance is 2.535(2) A, which is just half tbe c axis length. The occupancy of the ~uminium site is 50% (Table 2), i.e. the hole which accommodates the aluminium is fully occupied. The occupancies of the Cu(1) and Cu(2) sites
182
Fig. 1. The boron framework as seen along the c axis. The z coordinate of the centre of each icosahedron is either i or + as indicated in the figure.
are 76.0(4)% and 2.8(4)% respectively. The nature of the aluminium distribution is exactly the same as that found in A1Beo.sB24.2;the split in the aluminium site in this phase is +0.27 A. However, there is no correspondence regarding the copper distribution because the copper sites in the boron framework of A1Beo.sB24.2are completely vacant and the beryllium atoms are statistically distributed over the apex sites of the icosahedron. The B-B bond lengths are given in Table 3 and are compared with their equivalents in related phases. Unlike the icosahedra in the related phases which belong to a space group of higher symmetry (Table l), the icosahedron of the present crystal has no centre of symmetry and is rather distorted. This can be ascribed to the incorporation of copper atoms into Cu(1) and Cu(2) sites with very different occupancies; if the occupancy of one copper site were the same as that of the other, the crystal structure would have the (110) mirror planes belonging to the space group P4,/nnm. However, there is no significant difference in the size of the icosahedron. The average B-B bond lengths within the icosahedron are 1.818 A, 1.81 A, 1.796 A and 1.811 A for the present crystal, tetragonal a-B, B5sC2 and AlBeo.sB24.2respectively.
183
-b
Fig. 2. The distribution of metal atoms. They are projected along the c axis; B(7), Cu( 1’) and Cu( 2’) which overlap Al, Cu(2) and Cu( 1) respectively are omitted. The z coordinates of the overlapping atoms are all 3. The symmetry code for the aluminium atom in the figure is identical with that used in Table 4.
The striking feature of the present crystal is the presence of an intraicosahedral B-B bond (B(5)-B(5) in Table 3) with an unusually long bond length (2.026(3) A). The boron atoms giving such a long bond length interact strongly with the Cu(1) atom with a similar bond length (2.049(4) A), resulting in an almost regular triangle of B(5)-Cu(l)-B(5) linkage (Fig. 3). Similar bonds are also present at the opposite side of the icosahedron where Cu(2) is in contact with the two boron atoms giving a B(l)-Cu(2)-B(1) triangle with sides of 1.91 A for (%(2)-B(l) and 1.95 A for B(l)-B(1) (Fig. 3). The assignment of the Cu(2) site may be questionable because the value of 1.91 A observed for the distance from this site to the nearest boron atoms seemed to be rather short for a Cu-B bond length but acceptable for a B-B bond. From the following viewpoints, however, the site assignment is considered to be reasonable. (1) In the boron framework of AlBe 0.8B24.2 [14] no atoms have been found at the same positions as the Cu(1) and Cu(2) sites which have similar chemical environments. This means that boron and aluminium atoms do not incorporate into the copper sites. (2) The metallic radius of copper is noticeably shorter than that of aluminium: 1.28 a (12 coordination) or 1.24 A (8 coo~~ation) for copper,
184 TABLE
3
Comparison
of the B-B
bond
lengths
with their equivalents
in tetragonal
Q-B, B.&z
and
A1BeO.sB24.2
Bond
Symmetry codea
B(l)-B(1) B(l)-B(2) B(l)-B(2) B(l)-B(3) B(1)-B(5)b B(l)-B(6) B(2)-B(3) B(2)-B(4) B(2)-B(4)b B(2)-B(6) B(3)-B(4) B(3)-B( 5) B(3)-B(6) B(3)-B(7)C B(4)-B( 5) B(4)-B( 5) B(4)-B(6) B(5)-B(5) B( 5)-B(6) B(6)-B(6)b ‘Symmetry (y, -x,
-2);
B-B
1 0 1 0 2 0 0 0 3 1 0 0 0 0 0 1 1 1 0 4 codes: 4, (a -
bIntericosahedral CBond between dBond between
none
bond length (A)
A~~.ooCuo.w&s
Tetragonal
1.951(3) 1.809(3) 1.825(3) 1.801(4) 1.844(3) 1.822(3) 1.817(3) 1.885(3) 1.767(3) 1.772(3) 1.814(3) 1.756(4) 1.821(3) 1.,725(3) 1.822(3) 1.805(3) 1.764(3) 2.026(3) 1.772(3) 1.681(3)
1.85(3) 1.80(2) 1.81(l) 1.822(g) 1.86(2) 1.79(l) 1.79(2) 1.81(l) 1.66(2) 1.79(l) 1.79(2) 1.822(8) 1.84(l) 1.601(5) 1.80(2) 1.81(l) 1*79(l) 1.85(3) 1.79(l) 1.71(3)
or 0, (x, y, z);
y, $ -
X, + -
1, (-$--
y, $ -
(Y-B
X, $-
BsoC2
AlBe0.824.4
1.847(5) 1.814(6) 1.816(3) 1.806(5) 1.867(4) 1.814(6) 1.781(6) 1.800(3) 1.668(3) 1.758(6) 1.781(6) 1.806(5) 1.864(7) 1.638(4)d 1.814(6) 1.668(6) 1.758(6) 1.847(6) 1.814(6) 1.683(7)
1.918 1.797 1.794 1.799 1.818 1.811 1.799 1.861 1.715 1.779 1.799 1.799 1.833 1.661 1.797 1.794 1.779 1.918 1.811 1.660
z);
2, (y, -x,
1 -z);
3,
z).
B-B bond. an apex atom of the icosahedron an apex atom of the icosahedron
and the isolated and the isolated
boron atom (B(7)). carbon atom.
and 1.43 A (12 coordination) or 1.39 A (8 coordination) for aluminium [20]. Further, short bond lengths in the range of 2.03 - 2.08 A are found even for Al-B bonds in Blz icosahedral phases such as (r-A1B12 [ 8, 91, A11_1Beo.6B22[21], y-AIB12 [lo, 111, LiA1B14 [16] and MgAlB14 [17, 221. Therefore the interatomic distance of 1.91 A is highly probable for the Cu-B bond. The boron coordination around the metal sites is presented in Fig. 4, and the distances between the boron atoms and the metal sites are given in Table 4. As in the cases of LiA1B14 [16] and MgA1B14 [17], the metal atoms show significant anisotropic thermal motion. There is a close interrelation between the thermal ellipsoid of each metal atom and the geometry of the hole accommodating it. This is interpreted as follows. As mentioned above, the Cu(1) atom is bonded much more strongly to the four B(5) atoms than to the other eight boron atoms (Table 4) and thus the copper atom should naturally vibrate more in parallel with the (li0) plane than in the direction
[ii01
0r
L-[so11
[ii01
[ii01 or
Cu(l)-B(5)
: 2.05i
B (5)-B(5)
: 2.03i
C”(2) -B(l)
: 1.91 i
B (1) -B(l)
: 1.95 i
[llOI
Fig. 3. Strong interactions between copper atoms and B 12 icosahedra. hedral B-B bonds involved in such interactions have unusually long indicated in the figure.
TABLE
4
Distances
between
the metals and boron
first neighboursa
Symmetry code b
Bond
Al-2B(2) Al-2B(2) Al-2B(3) Al-2B(3) Al-2B(4) Al-2B(4) Al-B(7) Al-B(7) Cu(l)-4B(l) Cu(l)-4B(4) Cu(l)-4B(5) Cu(2)-4B(l) Cu(2F4B(2) Cu( 2)-4B( 5)
Distance
0, 1 4, 7 0, 1 4, 7 0, 1 4,7 0 4 5, 8, 10, 12 0, 0, 0, 0, 6,
2, 2, 3, 3, 9,
14, 14, 15, 15, 11,
16 16 16 16 13
2);
4, (-Y,
1 -
2); 9, (y, -x, +x,
x, -2);
+ -y,
5, (-Y, 1 -
-$
x, 1-s);
2);
10,
+ 2);
13,
(++y,-++x,+-z);16,(+y,~-x,+-z).
($ -x,
AlBeo.sB24.2
2.261(2) 2.353(2) 2.331(2) 2.709(2) 2.234(2) 2.327(2) 2.288(3) 2.781(3) 2.465(2) 2.692(2) 2.049(2) 1.906(2) 2.616(2) 2.537(2)
2.185 2.293 2.254 2.681 2.185 2.293 2.266 2.814
in AlBee.sB24.2. z); 2, (-x, 1 -y,
6, (1 -Y,
x, 1-s);
$ + y, -+
+ 2);
(+ +x,+-y,
(A)
AhGo.79Bzs
aAl-B distances are compared with the equivalents bSymmetry codes: 0 or none, (x, y, z); 1, (-x,-y,
(-+
The intra-icosabond lengths as
-+
+ 2);
7, (Y, -x, 11, ($ 14,
(-i
x, -+
z); 3, (1 -x,-y, --a);
8, (Y, 1 -x,
+ y, -$
+ y, + +x,
+ 2); 12,
+ -2);
15,
(cl
c
23
(d)
Fig. 4. Coordinations of the boron atoms around the copper and aluminium atoms showing thermal ellipsoids at the 50% level for Cu( 1) with symmetry codes; (b) designations of the atoms around AI with symmetry and At: (a) designations of the atoms around &(I) codes; (c) designations of the atoms around Cu(2) with symmetry codes; (d) stereoscopic pair for Cu(1); (a) stereoscopic pair for Al; (f) stereoscopic pair for Cu(2). The symmetry codes for the atoms in (a), (b) and (c) are identical with those used in Table 4.
188
perpendicular to it (Fig. 4(a)). A similar mode of thermal vibration is expected to be associated with the Cu(2) site because it is centred in a chemical environment (Fig. 4(c) and Table 4) similar to that of the Cu(1) site; owing to the low occupancy, anisotropic refinement could not be applied to this site. The coordination polyhedron surrounding the aluminium site is ellipsoidal (Fig. 4(b) and Table 4) with its major axis along the c axis. Therefore, as shown in Fig. 4(b), the aluminium atom should necessarily vibrate more strongly along the c axis than in parallel with the (001) plane.
References 1 M. Hansen, Constitution of Binary Alloys, Metallurgy and Metallurgical Engineering Series, McGraw-Hill, New York, 1958. 2 I. Higashi, Y. Takahashi and T. Atoda, J. Less-Common Met., 37 (1974) 199. 3 V. I. Matkovich, J. Economy and R. F. Giese, Jr., J. Am. Chem. Sot., 86 (1964) 2337. 4 I. Higashi, T. Sakurai and T. Atoda,J. Less-Common Met., 45 (1976) 283. 5 S. Andersson and B. Callmer, J. Solid State Chem., 10 (1974) 219. 6 J. Rexer and G. Petzov, Metallurg (Moscow), 24 (1970) 1083. 7 J. 0. Carlson and T. Lundstrom, J. Less-Common Met., 22 (1970) 317. 8 I. Higashi, T. Sakurai and T. Atoda, J. Solid State Chem., 20 (1977) 67. 9 J. S. Kasper, M. Vlasse and R. Naslain, J. Solid State Chem., 20 (1977) 281. 10 R. E. Hughes, M. E. Leonowicz, J. T. Lemley and L.-T. Tai, J. Am. Chem. Sot., 99 (1977) 5507. 11 I. Higashi, J. Solid State Chem., 47 (1983) 333. 12 R. Mattes, L. Marosi and H. Neidhard, J. Less-Common Met., 20 (1970) 223. 13 J. L. Hoard, R. E. Hughes and D. E. Sands, J. Am. Chem. Sot., 80 (1958) 4507. 14 K. Krogmann and H. J. Becher, 2. Anorg. Allg. Chem., 392 (1972) 197. 15 G. Will and K. H. Kossobutzki, J. Less-Common Met., 47 (1976) 33. 16 I. Higashi, J. Less-Common Met., 82 (1981) 317. 17 I. Higashi and T. Ito, J. Less-Common Met., 92 (1983) 239. 18 J. A. Ibers and W. C. Hamilton (eds.), International Tables for X-ray Crystallography, Vol. 4, Kynoch, Birmingham, 1974, pp. 72,74,79,149. 19 T. Sakurai and K. Kobayashi, Rep. Inst. Phys. Chem. Res., Tokyo, 55 (1979) 69. 20 L. Pauling, The Nature of the Chemical Bond, Cornell University Press, Ithaca, NY, 1940 (Japanese translation, 1950, p. 445). 21 I. Higashi, J. Solid State Chem., 32 (1980) 201. 22 V. I. Matkovich and J. Economy, Acta Crystallogr., Sect. B, 26 (1970) 616.