## 1. Unit cell and Crystal

A unit cell is defined as the basic structural part in the composition of materials. It is analogous to a brick used in the building construction.

When many unit cells repeat in a three-dimensional space, a crystal is obtained. The structure of a crystal is same as that of a repeating unit cell. Hence crystal structure may be classified as:

- mono-atomic crystal,
- diatomic crystal, and
- multi-atomic crystal.

**Molecular crystal**: Diatomic
and multi-atomic crystals are also known as molecular crystals. There exist
thousands of varieties of crystal structures. Many of them have complex nature.
Complex crystal structure has two or more types mixed together, such as in Sulphur,
Gallium, Phosphorus, Uranium etc.

# 2. Bravais Crystal System

Crystals have unit cells of various geometries. The
geometries are defined in terms of their linear dimensions **a**, **b**
and **c**, and angular dimensions **α**, **β** and **Ɣ**.

Dimensions **a**, **b**
and **c **are along the **x**, **y** and **z** axes respectively;
and angles **α**, **β** and **Ɣ** between **xy**, **yz** and **zx**
axes respectively.

Bravais has classified 14 space lattices into 7 crystal systems. Crystals have inherent symmetry of different types. Different crystals exhibit vivid varieties of symmetry.

The **cubic** crystal is most symmetric while the **triclinic**
is least symmetric. Symmetry decreases as we move from cubic crystal system
towards triclinic crystal system.

# 3. Crystal Symmetry

Crystals usually have definite shapes. Symmetry of one kind or the other exists in them. Crystal symmetry means the property of space a crystal occupies to coincide with itself as a result of certain transformations.

The regularities in position of similar faces, edges etc. due
to these transformations determine the symmetry of a crystal. The symmetries
may be of **simple form** or **mixed form.**

Cubes and octahedrons have simple form of crystal symmetry. All the phases of these shapes are identical. The faces are square in cubes and equilateral triangles in octahedron.

## 3.1 Different Kinds of Crystal Symmetry

Two or more simple forms of crystals, as one unit, make a mixed symmetry. Many minerals are the examples. Whether simple or mixed, the symmetries may further be classified into

- centre of symmetry,
- planes of symmetry,
- axes of symmetry.

**Centre of symmetry**: A centre of symmetry has such a
centre point that any line drawn through it intersects the crystal at equal
distances on either side.

A crystal can have one or more planes of symmetry, one or more axes of symmetry; but can never have more than one centre of symmetry.

Majority of crystals are not centre-symmetrical.

**Plane of symmetry**: A crystal possesses a plane of
symmetry when a plane is such that the two opposite side regions are exact
image of each other.

Any line perpendicular to the plane intersects the crystal surface at equal distances on either side. It occurs only in ideal crystals having faces exactly of the same size. A cube has nine planes of symmetry.

**Axis of symmetry**: Axis of symmetry is the symmetry
around an imaginary line through the centre of the crystal about which the
crystal on rotation presents exactly the same nature.

The axes of symmetry are of different kinds viz.

- two-fold symmetry,
- three-fold symmetry,
- four-fold symmetry, and
- six-fold symmetry.

The 2-fold, 3-fold, 4-fold and 6-fold symmetries mean axis rotation θ of 180°, 120°, 90° and 60° respectively to obtain same coincidence of properties in the crystal. In general,

360/θ = n

Where n = 2, 3, 4, … etc. indicates the number of fold of symmetry. Generally a symmetry of more than six-fold axis means an isotropic material.

## 4. Primitive Unit Cell

Primitive cells are those unit cells which contain atoms at corner lattice points only. So these cells have least number of total atoms and the least volume of atoms per unit cell.

All unit cells of Figure (i) namely simple cube (SC), simple tetragon (ST), simple orthorhombic (SO), simple rhombohedral (SR) etc. are primitive cells. All those unit cells which do not fall under this category are non-primitive cells.

## 5. Coordination Number

Each atom in a crystal is surrounded by a number of atoms. The surrounding atoms are located at different distances.

The coordination number is defined as the number of nearest and equidistant atoms with respect to any other atom in a unit cell.

Take for example atom **A** as reference atom in the unit
cell of Figure which is simple cube (SC). The atoms marked 1, 2 and 3 belonging
to the same unit cell are nearest and equidistant to atom **A**.

The atoms 4, 5, and 6 in the neighboring unit cells are also
exactly nearest and equidistant as atoms 1, 2, and 3. The atoms marked 7, 8, 9
and 10 are not the nearest with respect to atom **A**.

In this way, there are 6 atoms nearest and equidistant with respect to any other atom. Hence coordination number is 6 in case of simple cube (SC).

In a face centered cubic (FCC) structure, one atom lies at each corner of the cube in addition to one atom at the centre of each face. For any corner atom, there are four atoms at each corner that surround it. Each face centre has an atom.

There are four atoms nearest to the atom on four atomic planes below it and four above it in the atomic plane. The coordination number of face centered cubic structure is thus 4 + 4 + 4 = 12.

In a body centered cubic (BCC) structure, there is an atom at each corner of the unit cell. One atom is at the body centre of the cube. There are eight atoms lying at equal distances from each atom. Thus the coordination number of BCC structure is 8.

Coordination number can never exceed 12. In dense liquids, it is about 10.

## 6. Relation between Atomic Radius and Lattice Constant

Atoms in the unit cells are assumed to be rigid spheres in contact.
The radius of atoms **r** in the unit cells bears some relation with the
lattice constant **a**. Cubical unit cells are found in a majority of
materials. Hence relations between **r** and **a **for cubical unit cells
are discussed.

**Simple Cube (SC)**: There are eight atoms situated at
the eight corners. The corner atoms touch each other. The lattice constant **a**
i.e. side of the unit cell and the radius of the atom **r** on front face
only is shown in Figure (a). The relation between **r **and **a** in SC
is shown in Figure (a).

**Face Centered Cube (FCC)**: There are eight corner
atoms and one atom at the centre of each face. The corner atoms do not touch
each other but each corner atom touches the central atom of each face. The
front face arrangement is shown in Figure (b). The relation between **r **and
**a** in SC is shown in Figure (b).

**Body Centered Cube (BCC)**: There are eight atoms at
the corners of the unit cell and one atom at the centre. The corner atoms do
not touch each other, but each corner atom touches the central atom. The
relation between **r **and **a** in SC is shown in Figure (c).

## 7. Effective Number of Atoms

The effective number of atoms per unit cell **M** is
different from total number of atoms per unit cell.

The atom at the corner of a cubical unit cell has only 1/8 of it inside the boundary of that unit cell. The remaining 7/8 of it lies in the surrounding unit cells of the crystal.

Similarly the atom at the face in FCC is shared 1/2 by that atom and 1/2 by the neighboring atom.

In BCC, the atom at the centroid is wholly occupied by that unit cell in which it lies.

Thus, the effective number of atoms are 1, 2 and 4 in SC, BCC and FCC respectively. Figures explain these details. The effective number of atoms for different unit cells are summarized in Table 2.

## 8. Atomic Packing Fraction

The atomic packing fraction is defined as the ratio of total
volume of atoms per unit cell **v** to the total volume of unit cell **V**.
It is also known as relative density of packing or Atomic Packing Factor (APF)
or Atomic Packing Efficiency (APE). This may be expressed by

## 9. Calculation of Density or Bulk Density

The density **ρ
**of a
material is expressed as:

ρ = mass of unit cell/volume of unit cell = M/V

Where,
M = m.N_{e}, and **m** = mass of one atom. The mass of one atom **m**
may be obtained by:

The above equation shows that the density is directly
proportional to the effective number of atoms per unit cell. Hence elements
having FCC structure possesses higher densities. Gold (Au) belonging to this
kind of structure has density of 19300 kg/m^{3}.

However, only this fact cannot be taken as a rule. Aluminum
(Al) also belongs to FCC structure but has a lower density of 2700 kg/m^{3}
only. Infact, two other factors viz. atomic weight and lattice constant also
influence the density.

Iridium (Ir) and Osmium (Os) are the heaviest elements having
density of almost the same value 22500 kg/m^{3}. The structure of
iridium is FCC and that of osmium is HCP.