How to find second nearest neighbours for different type of cubic systems ?

Question

What would be the number of second nearest neighbours in Simple Cubic, BCC, FCC, HCP and different structures like those of NaCl, ZnS etc. and what is the approach to find them ?

Answer 1

To find the nearest neighbors you should first know the following terms:

Unit Cells: The smallest amount of the solid whose properties resemble the properties of the entire solid irrespective of the amount taken is called a unit cell. It is the smallest repeating unit of the solid. Any amount of the solid can be constructed by simply putting as many unit cells as required.

Primitive Unit Cell: In a primitive unit cell the same type of atoms are present at all the corners of the unit cell and are not present anywhere else. It can be seen that each atom at the corner of the unit cell is shared by eight unit cells (four on one layer, as shown, and four on top of these). Therefore, the volume occupied by a sphere in a unit cell is just one-eighth of its total volume. Since there are eight such spheres, the total volume occupied by the spheres is one full volume of a sphere. Therefore, a primitive cubic unit cell has effectively one atom.

Each primitive cell contains 1 site.

Cubic System

Conventional cells
For a simple cubic lattice, a conventional cell = a primitive cell
NOT true for body-centered or face-centered cubic lattices 

How can we see it?
sc: one conventional cell has one site (same as a primitive cell)
bcc: one conventional cell has two sites (twice as large as a primitive cell)
fcc: one conventional cell has four cites (1 conventional cell=4 primitive cells)

Simple cubic 

Lattice point per conventional cell: 1=8×1/8
Volume (conventional cell): a³
Volume (primitive cell) :a³
Number of nearest neighbors: 6
Nearest neighbor distance: a
Number of second neighbors: 12
Second neighbor distance: √2a
Packing fraction: pi/6≈0.524

6 nearest neighbors: ±1,0,0,0,±1,0 and 0,0,±1
12 next nearest neighbors: ±1,±1,0,0,±1,±1 and (±1,0,±1)

bcc

 

Lattice point per conventional cell: 2=8×1/8+1

Number of nearest neighbors: 8

Number of second neighbors: 6
Second neighbor distance: a
Packing fraction: √3/8pi≈0.680

8 nearest neighbors: ±1/2,±1/2,±1/2
6 next nearest neighbors: ±1,0,0, 0,±1,0 and (0,0,±1)

fcc 

Sodium Chloride NaCl

Face-centered cubic lattice
Na+ ions form a face-centered cubic lattice
Cl- ions are located between each two neighboring Na+ ions
Equivalently, we can say that
Cl- ions form a face-centered cubic lattice
Na+ ions are located between each two neighboring Na+ ions