Crystallography
It is concerned with the geometry of arrangements of atoms in crystals & the various
consequences of such arrangements.
Brief Overview
In mineralogy and crystallography, a crystal structure is a unique arrangement
of atoms in a crystal. A crystal structure is composed of a motif, a set atoms arranged in
a particular way, and a lattice. Motifs are located upon the points of a lattice, which is an
array of points repeating periodically in three dimensions. The points can be thought of
as forming identical tiny boxes, called unit cells, that fill the space of the lattice. The
lengths of the edges of a unit cell and the angles between them are called the lattice
parameters. The symmetry properties of the crystal are embodied in its space group. A
crystal's structure and symmetry play a role in determining many of its properties, such
as cleavage, electronic band structure, and optical properties.
Basic Terminologies you should know in Crystallography:
Motif:
It is a recurring thing in a pattern.
Symmetry Operation:
It is an operation that leads to superimposition of an object on itself. In 2D there are 3
single step symmetry operations, namely: Translation, Rotation & Reflection. In 3D there
are 4, ‘inversion’ being the additional operation.
In 5D there are 6 distinct
transformations and so on… (But you should only be concerned till up to 3-Dimensions)
Translation T : By a translation each point of the space is shifted in the same direction by
the same amount. Every translation has a direction and a distance. Hence, it is like a
vector.
Reflection
: Each point of space is reflected in a plane, the reflection plane or mirror
plane, such that all points of this plane, and only these points, are fixed points. In
addition, each line and each plane perpendicular to the mirror plane is left invariant as a
whole.
To reflect an object means to produce its mirror image. Every reflection has a mirror line.
A reflection of an "R" is a backwards "R".
This is an example of an object that can be divided into two parts by a plane. Since the
two parts are mirror images of one another, this symmetry element is called a mirror
plane. Operation of this element on one half of the teacup generates the other: if a half
teacup is held with its sliced edge against a mirror, the appearance of the whole is
regenerated. Teacups are rarely sliced in real life but you could try it out with an apple or
a pear. If the two similar parts produced have no symmetry remaining, as in the case of
the teacup, they are called asymmetric units.
Rotation:
A rotation is a transformation, Rdegrees , that moves every point around a
fixed point (usually the origin). To rotate an object means to turn it around. Every rotation
has a center and an angle.
Inversion:
A body has an inversion center if corresponding points of the body are
located at equal distances from the center on a line drawn through the center. A body
having an inversion center will come into coincidence with itself if every point in the body
is inverted, or "reflected," in the inversion center. A cube has such a center at the
intersection of its body diagonals. An inversion is the reflection of the whole space in a
point P, which is called the centre of inversion. The point P is the only fixed point. Each
line or plane through P is mapped onto itself as a whole because it is reflected in P.
Inversion changes the sign of the coordinate of each point with respect to a center of
symmetry or inversion point.
Mathematically speaking, if the coordinates of any point on one hand, relative to the
centre of inversion, are x, y and z, then the coordinates of the corresponding point on the
other hand will be -x, -y and -z.
Symmetry Element:
It is the locus of a point left invariant (unmoved) by the operation. In simple words, it is
the entity about which a symmetry operation acts.
Important Crystallographic theories:
1) What makes a crystal a crystal is the arrangement of atoms and molecules
which is related from one part to another by the operation of translation. So if
you don’t have a periodicity of translation you cant have a crystal. This is the
essence of Crystallization
2) An object or space possess symmetry when there is an operation or set of
operations that maps it into congruence with itself (cant tell its been moved)
3) Symmetry operation acts on everything, on all of the space and not just on
one local domain
Crystal, Lattice & Unit Cell.
A crystal or crystalline substance is one whose constituent atoms and molecules or ions
are arranged in an orderly pattern. This definition of a crystal is distinct from the popular
concept based on observation of external symmetry of crystals, in which some crystals
appear cubic, others needle-shaped and so on. The regular external shape is obtained
only when the conditions of crystallization are favorable to development of flat, geometric
faces. In most instances, particularly with metals, these conditions are absent, and the
crystals have irregular surfaces even though the internal arrangement is perfectly
geometric.
Lattice is an array of fictitious points that summarizes the translational periodicity of the
crystal. It has the property that something hung over a particular position & orientation
relative to a lattice point will be hung over all the lattice points in the same manner. This
means that every point has identical surroundings: the grouping of lattice points about
any given point is identical to the grouping about any other point in the lattice. In other
words, if we could move about in the lattice, we would not be able to distinguish one
point from another because rows and planes near each point would be identical. If we
were to wander among the atoms of a solid metal or chemical compound, we would find
the view from any lattice point exactly the same as that from any other.
The unit cell is the fundamental unit from which the entire crystal/object may be
constructed by purely translational displacements. A unit cell must posses the overall
symmetry of the crystal, i.e. it is the smallest unit containing all the information required
to generate the entire pattern. The unit cell is a tiny box containing one or more motifs, a
spatial arrangement of atoms. The unit cells stacked in three-dimensional space
describes the bulk arrangement of atoms of the crystal.
As already mentioned, the only single step symmetry operations in 2D are Translation,
Rotation & Rotation (for 2D). The following mathematical proof justifies the existence of
1-fold, 2-fold, 3-fold, 4-fold, and 6-fold symmetry in patterns and corroborates the
nonexistence of all other rotational symmetries in crystallography.
P
cos α = (1-p)/2
α
n = 360o/ α
4
-3/2
-
3
-1
180o
2-fold axis
2
-1/2
120o
3-fold axis
1
0
90o
4-fold axis
0
½
60o
6-fold axis
-1
1
360o
1-fold axis
-2
3/2
-
-
-
This is also evident from the following 2-D tiling or wallpaper patterns:
Restriction: entire plane must be able to be built up via translation alone of the unit cell,
i.e. no gaps.
Hence there are only 5 distinct 2D lattice nets: