CRYSTAL SYSTEMS

It has been discussed in earlier Chapters (Chaps.3–5) that a perfect crystalline
structure should possess a long range order comprising both translational
and rotational symmetries, which should be maintained in three dimensions.
However, crystalline order can also be maintained in some ways other than
translational symmetry and they are called “aperiodic crystals.” Now, three
alternatives to translational symmetry are known: incommensurately modulated crystals, incommensurate composite crystals, and quasi crystals. The
modulated structures are obtained from the structures having translational
symmetry by giving displacements of the atoms in the periodic structure by
equal amounts. Incommensurate composite structures are formed in layered
compounds by two interpenetrating periodic structures which are mutually
incommensurate. The discovery of quasi crystals has added up one more dimension to crystallography. Influenced by the discovery of a number of quasi
crystals or quasi periodic crystals, International Union of Crystallography has
redefined the term crystals to mean “any solid having an essentially discrete
diffraction diagram.” This broader definition leads to the understanding that
microscopic periodicity are sufficient but not necessarily the only condition
for crystallinity. A distinct property of quasi crystals that has been found
from the diffraction pattern is that it shows fivefold rotation and also other
crystallographic point symmetries.
We have seen in Chap.3 (Fig.10.3) that there cannot be any crystalline
substance with fivefold of symmetry as the motifs having that symmetry cannot make any compact structure, and same is true for sevenfold, eightfold,
or tenfold rotation symmetries. It was accepted in classical crystallography.
that these symmetries are not possible to preserve both translational and
rotational symmetries in the long range in stable and metastable states of
crystalline solids till the year 1984 [1]. However, quasi crystals as mentioned
above lack translational symmetry but rotational symmetries are allowed according to any point group in three-dimensional space. The important logic
behind this classical idea was that no compact structure can be formed having
fivefold symmetry, but the building principle to form a compact structure can
be revised if the motifs are not exactly similar and tilling can be made without
overlapping or leaving any gap. This tiling will definitely be aperiodic as they
lack translational symmetry and can be taken as a model of quasi crystals.
An aperiodic tiling of the plane can be formed with two different proto tiles.
In the simplest form, the proto tiles are rhombuses with equal edges but of
different angles between the edges. The skinny one has angle 36? and the fat
one has angle 72?, that is, a multiple of (360/10)?.
Now, not following this matching rule for joining the proto tiles, an infinite
number of tilings can be formed, which can be either periodic or aperiodic.
One periodic tiling is given above. When this matching rule is followed, the
Penrose tiling can be obtained [2].
British mathematician of Oxford, Roger Penrose, devised a pattern in a
nonperiodic fashion using two different types of tiles (Fig.10.4b). The motif
of this Penrose tiles is rhombi, which may be arranged in a plane or in three
dimension (rhombohedra) so that they obey certain matching rules other than
those symmetries discussed before and yet these constitute patterns. Such 2D
or 3D tilling have several important properties and among them the most important is that they possess self similarity, which means that any part of the
tiling repeats again within a predictable area or volume. This Penrose tiling
shows crystalline properties in a number of ways. The edges occur in five
different orientations only and thus represent fivefold rotation symmetry. In
1984, when Shechtman et al. [3] published in their paper the electron diffraction pattern of Al-Mn alloy, the diffraction pattern showed tenfold symmetry
and that was the first experimental evidence of the presence of symmetry
hitherto unpredictable in crystalline matter and it was the evidence of the
existence of a new crystalline state knownas quasi crystal. If closely observed
there lays a similarity between 3D Penrose pattern and icosahedral quasi crystals. Atomic structures of quasi crystals can be constructed by placing atoms
in the Penrose tiling having same atoms at all the vertices and also the similar edges. A three-dimensional structure can then be constructed by stacking
the Penrose tilings. This leads to a crystal, which is quasi periodic in two
dimension but periodic in third direction. Figure 10.4c, which is the electron
diffraction micrograph of rapidly solidified Al-Fe-Cu alloy system distinctly,
shows the presence of fivefold of rotation symmetry.
A Fourier transform explains very well the diffraction patterns obtained
from Al-Mn quasi crystal. The symmetry that determines the type of quasi
crystal is found in its electron diffraction patterns. Figures 10.5–10.7 given
below show the diffraction pattern and the simulation of diffraction patterns,
which like others represent the eightfold and tenfold rotation symmetry
observed in the electron diffraction or the zero layer precession X-ray
photographs.
Since 1984, many stable and also metastable quasi crystals have been
found, and these are often binary or ternary intermetallic alloys with aluminum
as one of the primary component. Some of these stable quasi crystals with
aluminum as a major component are In addition to these stable state quasi crystals, there are much more binary
and ternary alloys that form metastable quasi crystalline states.
These quasi crystals are materials with perfect long range order but with
no three-dimensional translational periodicity. The first property is manifested
by the symmetric diffraction spots and the second property is manifested by
the presence of noncrystallographic rotation symmetry, that is, either fivefold
or 8-, 10-, or 12-fold rotation symmetry. Since quasi crystals do not show the
translational periodicity at least in one dimension, it is mathematically more
difficult to interpret its diffraction pattern. As for normal classical crystals,
we require three integer values known as Miller indices (hkl), to label the
observed reflections, because of it having three-dimensional periodicity, for
quasi crystals we require at least five linearly independent indices for polygonal
quasi crystals and six indices for icosahedral quasi crystals, giving rise to
generalized Miller indices. It can then be said that while in three-dimensional
space quasi crystals fail to show the required periodicity, in higher dimensional
space they exhibit periodicity.
The growth morphology of the stable decagonal quasi crystals Al-Ni-Co
or Al-Mn-Pd shows that they grow as decaprismaic (ten prism faces with the
tenfold axis as rotation axis). Al-Cu-Fe quasi crystals, which are icosahedral
quasi crystals, grow with triacontahedral shape, which exhibit 30 rhombic
faces perpendicular to the twofold rotation axes. Though the interpretation
of the patterns involves more mathematical complexicity, the experimental
techniques involved HRTEM (The high resolution transmission electron microscopy) and maximum entropy method (MEM) for the interpretation [4].
Conclusion: A crystalline material should then not be regarded as those materials which only preserve the ten number of macroscopic symmetry in threedimensional space, but it would be more appropriate to redefine the crystalline
materials as those that show a regular diffraction spot and having, in addition to the classically existing ones, also 5, 6, 8, 10 or 12 rotation axes of
symmetries.