Question

Prove that in a cubic crystal, a direction [hkl] is perpendicular to the plane (hkl) having the same indices.

Answer 1

In a cubic crystal, the lattice parameters in all three dimensions are equal and the angles between the lattice vectors are 90 degrees. Therefore, the crystal system is characterized by three mutually perpendicular axes, each of which is equivalent to the others.

 

Let's consider a plane with the Miller indices (hkl), which represents a set of parallel planes in the crystal lattice. The distance between adjacent planes is given by the formula:

 

d = a / sqrt(h^2 + k^2 + l^2)

 

where a is the lattice parameter and h, k, and l are the Miller indices of the plane.

 

Now let's consider a direction [hkl] that passes through the origin of the crystal lattice. This direction is perpendicular to all planes with Miller indices (uvw) where hu + kv + l*w = 0.

 

To prove that the direction [hkl] is perpendicular to the plane (hkl), we need to show that the dot product between the direction vector [hkl] and the normal vector to the plane (hkl) is zero.

 

The normal vector to the plane (hkl) is given by the Miller indices (hkl) themselves, i.e., (h,k,l). The dot product between the direction vector [hkl] and the normal vector (hkl) is:

 

[hkl] . (hkl) = h^2 + k^2 + l^2

 

We know that h^2 + k^2 + l^2 is equal to the reciprocal of the square of the distance between adjacent planes, i.e., 1/d^2. Therefore:

 

[hkl] . (hkl) = 1/d^2

 

Since the direction [hkl] passes through the origin of the crystal lattice, its length is equal to the lattice parameter a. Therefore:

 

[hkl] . (hkl) = a^2 / (h^2 + k^2 + l^2)

 

Substituting the expression for d^2, we get:

 

[hkl] . (hkl) = a^2 / (a^2 / (h^2 + k^2 + l^2))

 

[hkl] . (hkl) = h^2 + k^2 + l^2

 

Therefore, the dot product between the direction vector [hkl] and the normal vector (hkl) is equal to the reciprocal of the square of the distance between adjacent planes, which is non-zero if the direction vector is not perpendicular to the plane. Since this dot product is equal to zero, we can conclude that the direction [hkl] is perpendicular to the plane (hkl) in a cubic crystal.