CRYSTAL STRUCTURE
THE SPACE LATICE
The atomic arrangement in a crystal is called crystal structure. It is very convenient to imagine periodic arrangement of points in space about which these atoms are located. This leads to the concepts of space lattice
UNIT CELL
A space lattice can be defined by referring to a unit cell. The unit cell is the smallest unit which, when repeated in space indefinitely, generates the space lattice. The square obtained by joining four neighboring lattice points represents a unit cell. Since each lattice point is common to four unit cells meeting at the corner, the effective number of lattice points in the unit cell is ¼ * 4 = 1 only one. Instead, the unit cell can be visualized by taking one lattice point in the centre of the square. These two possible ways of choosing unit cell.
If we associate with each lattice point, a group of atoms or molecules identical in composition, called the basis or the pattern, crystal structure is generated.
Lattice + basis = crystal structure
LATTICE PARAMETERS OF AN UNIT CELL
The lines drawn parallel to the lines of intersection of any three faces of the unit cell which do not lie in the same plane are called crystallographic axes. Naturally the three translational vectors a, b and c lies along the crystallographic axes x, y and z. In imagination see that a unit cell is with three crystallographic axes x, y and z. The intercept a, b and c define the dimensions of a unit cell and are known as primitives.
The angles between the three crystallographic axes are known as interfacial angles. The angle between b and c is alpha and c and a, is beta, and that between a and b is gamma. The primitive a, b and c and interfacial angles between alpha, beta and gamma are basic lattice parameters because they determine the form and actual size of the unit cell. The init cell formed by the primitive a, b and c is called primitive cell. A primitive cell will have only one lattice point. If there are two or more lattice points, then it is non primitive cell. Most of the unit cells of various crystal lattices contain two or more lattice points, and hence it is not necessary that the unit cell should be primitive cells.
We know that a three dimensional space lattice is generated by repeated translation of three non coplanar a, b and c. there are only fourteen distinguishable ways of arranging points in three-dimensional space. These 14 space lattice are known as Bravais lattices. They belong to seven crystal systems. The below table1 lists the seven crystal system and the relationship between the lattice parameters.
Table1: Crystal system
Crystal system Unit vector Angles
Cubic a=b=c α=β=γ=90°
Tetragonal a=b≠c α =β= γ=90°
Orthorhombic a≠b≠c α =β= γ=90°
Monoclinic a≠b≠c α =β=90°=≠90°
Triclinic a=b=c α =β= γ≠90°
Hexagonal a=b≠c α = β=90° γ=120°
The different Bravais lattices and their names are listed in the below table2. There are fourteen Bravious lattices.
Table2-Bravious lattices and crystal types
Crystal type Bravais lattices Symbols
Cubic simple P
Body centered I
Face centered F
Tetragonal Simple P
Body centered I
Orthorhombic simple P
Base-centered C
Body centered I
Face centered F
Monoclinic simple P
Base centered C
Triclinic simple P
Trigonal simple P
Hexagonal simple P
CRYSTAL SYMMETRY
Crystal has inherent symmetry. The definite arrangement of the faces and edges of a crystal is known as crystal symmetry. It is a powerful tool for the study of the internal structure of crystals. Crystals possess different symmetries or symmetry elements. They are described by certain operations. A symmetry operation is one that leaves the crystal and its environment invariant. It is an operation performed on an object or pattern which brings it to a position which is absolutely indistinguishable from the old position. The seven crystal system are characterized by three symmetry elements
- The centre of the symmetry
- The planes of symmetry and
- The axes of symmetry
Since the cubic system is the simplest and also most common, symmetries present in a cubic system will be discussed in details.
CENTRE OF SYMMETRY
Considering a unit cell of cubic lattice the point at the body centre represent the centre of symmetry. Any line passing through it meets the surface of the crystal at equal distances in both directions. Since centre lies at equal distances from various symmetrical position it also known as centre of inversion. For every lattice point given by the position vector r vector, there will be a corresponding lattice point at the position negative r vector. Thus it is equivalent to reflection through a point.
PLANE OF SYMMETRY
An imaginary plane passing through a crystal, such that portions on the two basis of the plane are exactly alike, is known as plane of symmetry. In the case of the cube there are three planes of symmetry parallel to the faces of the cube and six diagonal planes of symmetry. These planes divide the crystal in to two halves such that they are mirror images of each other with respect to the plane. The total number of planes of symmetry in a cubic crystal is 3 +6 = 9.
AXIS OF SYMMETRY
A body is said to possess rotational symmetry about an axis if after rotation of the body about this axis through some angle, it appears as it was prior to rotation. If a cube is rotated through 90° about an axis normal to one of its faces at its midpoint, it brings the cube in to in to self coincident or a congruent position. Hence during one complete rotation about this axis, i.e. through 360°, at four positions the cube is coincident with its original position. Such an axis is called fourfold positions axis of symmetry or tetrad axis. There are three tetrad axes for every cube. In general, if a rotation through an angle 360°/n degrees about an axis brings the crystal in to congruent position, then that axis is called n-fold axis symmetry.
If n=1, then the crystal has to be rotated through an angle 360°/1=360° about an axis to achieve self-coincidence. Such an axis is called identity axis. Each crystal possesses an infinite number of such axes.
If n=2, then the crystal has to be rotated through an angle 360°/2=180° about an axis to achieve self-coincidence. Such an axis is called Diad axis. In a cub, a line joining the mid points of a pair of opposite parallel edge provides a Diad axis. In a cube, a line joining the midpoint points of a pair of opposite parallel edges provides a Diad axis. Since in a cube, there are 12 such edges, the number of Diad axes. Since in a cube, there are 12 such edges, the number of Diad axes is 6.
If n=3, for every 120° rotation congruence is achieved and the axis is termed triad axis. This axis is passing through a solid diagonal acts as triad axis and since there are four solid diagonals in a cube, it has four triad axes.
If n=4, for every 90° rotation congruence is achieved and the axis is termed tetrad axis. We have already seen that a cube has three tetrad axes.
If n=6, the corresponding angle of rotation is 60° and the axis of rotation is called a hexad axis.
CRYSTALLOGRAPHIC SYMMETRY ELEMENTS OF THE CUBE
- centre of symmetry 1
- Plane of symmetry 9 (Straight planes 3 and Diagonal planes-6)
- Tetrad axes 3
- Triad axes 4
- Diad axes 6
Total number of symmetry elements = 23
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