Introduction:
An FCC (face-centered cubic) crystal has a crystal structure with atoms arranged in a cubic lattice, where atoms are located at the corners and the center of each face of the unit cell. Dislocations are line defects in the crystal structure that affect the mechanical properties of materials. The strain energy per unit length of a dislocation depends on its Burgers vector, which represents the magnitude and direction of the lattice distortion caused by the dislocation.

Explanation:
To understand why the strain energy per unit length of a dislocation with a Burgers vector a/2 is three times that of a dislocation with a Burgers vector a/6, we need to consider the geometry and the energy associated with the dislocations.

1. Burgers vector:
The Burgers vector of a dislocation represents the lattice distortion caused by the dislocation. It is defined as the vector that connects equivalent lattice points on either side of the dislocation line. In an FCC crystal, the Burgers vector of an edge dislocation is given by b = a/2, where a is the lattice parameter.

2. Strain energy per unit length:
The strain energy per unit length of a dislocation represents the energy required to create and maintain the dislocation in the crystal lattice. It depends on the Burgers vector and the elastic properties of the material.

3. Calculation:
In an FCC crystal, the strain energy per unit length (U) of a dislocation is given by the equation:
U = μb^2/4π(1-ν), where μ is the shear modulus and ν is the Poisson's ratio.

For a dislocation with a Burgers vector of a/2, the strain energy per unit length (U1) is:
U1 = μ(a/2)^2/4π(1-ν)

For a dislocation with a Burgers vector of a/6, the strain energy per unit length (U2) is:
U2 = μ(a/6)^2/4π(1-ν)

4. Comparison:
To compare U1 and U2, we can take the ratio:
U1/U2 = (μ(a/2)^2/4π(1-ν)) / (μ(a/6)^2/4π(1-ν))

Simplifying the equation, we get:
U1/U2 = (a/2)^2 / (a/6)^2
U1/U2 = (a^2/4) / (a^2/36)
U1/U2 = 9

Therefore, the strain energy per unit length of a dislocation with a Burgers vector a/2 is three times that of a dislocation with a Burgers vector a/6.