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): a^{3}

Volume (primitive cell) :a^{3}

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