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V. Adasch et al. / Journal of Solid State Chemistry 179 (2006) 2916–2926
2925
only significant exception is B521. Therefore it can be
assumed, the Wade rules, its extension by Jemmis and the
transfer to solid state compounds by Longuet-Higgins and
Roberts are also usable for MgB .
elements [5]. By use of a Cu/Mg melt we were able to
obtain single crystals of MgB . Structure solution and
1
2
refinement were done with single crystal data resulting in
the composition MgB12 which was confirmed by WDX
measurements. The crystal structure is similar to g-AlB12
and consists of B12 icosahedra and B21 units in a ratio 2:1.
These B21 units which are here observed for the first time
are built of two condensed icosahedra with one missing
apex and an additional boron atom. The content of 7.3
magnesium atoms we have found in the formula unit
corresponds nearly to the expected value of 8 which can be
explained by the required number of electrons to stabilize
1
2
This mnopq rule applied to the B21 unit (m ¼ 2, n ¼ 21,
o ¼ 1, p ¼ 1, q ¼ 0) gives 25 electron pairs. Regarding the
1
7 exohedral 2e2c bonds the B unit requires 67 electrons.
21
Because 3 ꢂ 21 ¼ 63 electrons are from the boron atoms, 4
additional electrons are needed and should be supplied by
the Mg atoms. The unit cell of MgB12 contains 16 B12
icosahedra and 8 B21 units. Therefore, the framework of
the boron polyhedra needs 64 electrons. The crystal
structure contains about 29.15 Mg atoms which is in
excellent agreement with the expected value regarding the
simplicity of the model and the complexity of the crystal
structure.
The application of the mnopq rule to a-AlB12 and g-
AlB12 leads to an electron need of 6 for the B20 unit (m ¼ 2,
n ¼ 20, o ¼ 1, p ¼ 2, q ¼ 0) as it was already stated by
Higashi [10,17]. The observed Al contents are in good
agreement. For the unit cell of g-AlB12 with 16 icosahedra
and 8 B20 units 80 electrons are required. The observed
content are 25.2 Al atoms (75.6 electrons). For a-AlB12
with a real composition AlB13.6 and half the unit cell
volume 12.8 Al atoms (38.4 electrons) are found.
If the boron polyhedra framework of g-AlB12 should be
stabilized by magnesium about 38 Mg atoms would be
required and the expected composition is MgB9.3. A careful
analysis of the Al positions in g-AlB12 shows that only 36
atoms can be placed into the crystals structure without
forcing unusual short Mg–Mg contacts. This might be the
reason for the formation of the B21 units in MgB12.
According to the mnopq rules the B21 unit demands fewer
electrons than the B20 unit. Therefore, the electrons, which
are necessary for the stabilization can be supplied by a
metal with fewer valence electrons.
2
ꢀ
4ꢀ
the boron polyhedra [(B ) ] and [(B ) ]2. MgB12 is a
21
1
2
4
convincing example for the stabilization of 3D networks of
boron polyhedra by the uptake of electrons from suitable
metal atoms. The electron need can be derived directly
from the geometry of the corresponding polyhedra.
Especially the comparison to g-AlB12 with its similar but
not identical crystal structure supports the mnopq-concept
developed by Jemmis et al. [18] for complex boron
polyhedra.
Acknowledgments
Thanks are due to the Bayerisches Geo-Institut (BGI,
¨
Universitat Bayreuth) for the access to WDX measure-
ments and to Detlev KrauXe for his support.
This work was supported by the program ‘‘Neue
Werkstoffe in Bayern’’, project B21092.
References
[
[
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several other examples. Higashi et al. [23] reported on a
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[
[
3] H. Okamoto, Phase Diagrams for Binary Alloys, ASM, Ohio, 2000.
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1
2
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1
7
[
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MgB . Here the unit cell contains 21.13 Al atoms
¨
1
2
3
corresponding to a number of 63.4 electrons. Furthermore
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[
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4
. Summary
[
[
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2
1.-26.8.2005, Hamburg, Germany.
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Adasch, Volker
