Crystal Structures

In a simple cubic structure, the atoms are arranged such that the lattice parameter (a) is directly equal to twice the atomic radius (r), as the atoms touch each other along the edge of the unit cell. Therefore:
a = 2r

For FCC structures, the Burgers vector is given by:

where is the lattice parameter. Substituting the given lattice parameter for Cu:

The force between two ions is calculated using Coulomb's law:

Here, C, m (sum of ionic radii), and F/m:

The angle between two directions and in a cubic crystal is given by:

For and :

A cube has the following symmetry elements:
• 1 identity element.
• 6 planes of symmetry.
• 3 axes of twofold rotation.
• 4 axes of threefold rotation.
• 3 axes of fourfold rotation.
Summing these, there are a total of 26 symmetry elements.

In an FCC unit cell, the metallic radius is given by:

For Gold ( ):

For Platinum ( ):

The packing factor is the fraction of volume occupied by atoms in the unit cell:

For diamond cubic, this value is approximately , indicating of the volume is occupied by silicon atoms.

The formula for interplanar spacing in cubic systems is:

This relationship holds for all cubic materials, where are Miller indices, and is the lattice constant.

The interplanar spacing in cubic materials is given by:

For FCC with , , , and :

For a simple cubic structure:

For the (010) plane:

For the (020) plane:

In a BCC structure, the relationship between the lattice constant and atomic radius is:

Substituting :

The density of a BCC structure is given by:

For BCC, there are 2 atoms per unit cell. Substituting the given values:

The packing factor is calculated as:

For FCC, the packing factor is:

where . This indicates that of the unit cell volume is occupied by atoms.

In an FCC unit cell, atoms are present at the 8 corners of the cube (each contributing atom) and at the centers of all 6 faces (each contributing atom). The total number of lattice points is:

From Bragg's law:

For (220) reflection, , with , , and being the lattice parameter: