Question

Lets assume, a represents the edge length (lattice constant) of a BCC unit cell and R represents the radius of the atom in the unit cell. Draw a BCC unit cell and show the atoms in the unit cell. Derive the relationship between the a and R.

Answer 1

Answer:

$4\ R=\sqrt 3\ a$

Explanation:

Given that

Lattice constant = a

Radius of unit cell cell =R

Atom is in BCC structure.

In BCC unit cell (Body centered cube)

1.Eight atoms at eight corner of cube which have 1/8 part in each cube.

2.One complete atom at the body center of the cube

So the total number of atoms in the BCC

 Z= 1/8 x 8 + 1 x 1

Z=2

In triangle ABD

$AB^2=AD^2+BD^2$

$AB^2=a^2+a^2$

$AB=\sqrt 2\ a$

In triangle ABC

$AC^2=AB^2+BC^2$

AC=4R

BC=a

$AB=\sqrt 2\ a$

So

$16R^2=2a^2+a^2$

$4\ R=\sqrt 3\ a$

So the relationship between lattice constant and radius of unit cell

$4\ R=\sqrt 3\ a$