Proving the Relationship 4r √3a in a Body-Centered Cubic Unit Cell

In this article, we will delve into the geometric and atomic structure of a body-centered cubic (BCC) unit cell to prove the relationship 4r √3a, where 4r is the total length of the body diagonal in terms of the atomic radii, and √3a is the length of the body diagonal in terms of the edge length of the cubic unit cell.

Definition of Key Terms

To begin, let's define the terms we’ll be using throughout this discussion:

Definitions:

r Radius of the atom. a Edge length of the cubic unit cell.

Structure of a Body-Centered Cubic (BCC) Unit Cell

A BCC unit cell has a unique atomic arrangement consisting of:

2 atoms per unit cell: one at each corner of the cube and one at the center of the cube. Each corner atom is shared by 8 unit cells, contributing 1/8 of an atom per unit cell. This configuration results in a total of 1 corner atom and 1 center atom per unit cell.

Geometric Analysis of the BCC Unit Cell

The atoms in a BCC structure touch along the body diagonal, which runs from one corner of the cube to the opposite corner. This geometric arrangement allows us to derive the relationship between the atomic radii and the edge length of the unit cell.

Length of the Body Diagonal

To find the length of the body diagonal, we can use the Pythagorean theorem in three dimensions:

Mathematical Derivation:

The body diagonal d of a cube with edge length a is given by:

d √(a2 a2 a2) √3a2 a√3

Atoms along the Body Diagonal

Considering the structure of the BCC unit cell:

There is an atom at the center of the cube contributing a radius r on each side, resulting in a total of 2r. There are two corner atoms each contributing a radius r. Thus, the total length of the body diagonal d can be expressed in terms of the atomic radii:

d 4r

Equating the Two Expressions

Now, we equate the two expressions for the body diagonal:

a√3 4r

Rearranging this equation yields:

4r √3a

Conclusion

This derivation demonstrates that in a BCC unit cell, the relationship 4r √3a holds true, relating the atomic radius r to the edge length a of the unit cell.