D-Spacing Physics Help: Understanding Adjacent Planes in Lattices

[ehrenfest]

Homework Statement

By book says prove that the distance between two adjacent parallel planes of a lattice is d(hkl) = 2pi/|G| where g = gb_1 +k b_2 + l b_3 where the b_i are the primitive vectors of the reciprocal lattice.

Could someone clarify exactly what they mean by "adjacent"?

Homework Equations

The Attempt at a Solution

 

[chemisttree]

"Next to" or 'alongside' or 'face-to-face'.

 

[Blueshift5]

The term "adjacent" in this context refers to two planes that are parallel to each other and share a common lattice point. In other words, they are neighboring planes within the lattice structure.

To prove the given equation, we can use the concept of Bragg's Law, which states that the angle of incidence (θ) of an incoming wave on a crystal lattice is equal to the angle of reflection (θ') of the outgoing wave, and is related to the spacing between the planes (d) by the following equation: nλ = 2dsinθ, where n is the order of diffraction and λ is the wavelength of the incoming wave.

In this case, we can consider the incoming wave to be a plane wave, and the outgoing wave to be the reflected wave from the neighboring plane. This means that the angle of incidence and reflection are equal (θ = θ'), and we can rewrite the equation as nλ = 2dsinθ = 2dsinθ'.

Since the planes are parallel to each other, the angle of incidence and reflection are also equal to the angle between the planes (θ = θ'), which can be calculated using the dot product of the two primitive vectors of the reciprocal lattice: cosθ = (b_i • b_j) / (|b_i| |b_j|).

Substituting this into our equation, we get nλ = 2d(b_i • b_j) / (|b_i| |b_j|). Rearranging for d, we get d = nλ (|b_i| |b_j|) / (2(b_i • b_j)).

Since the b_i are the primitive vectors of the reciprocal lattice, we can express them in terms of the reciprocal lattice vector G: b_i = 2πG / |G|. Substituting this into our equation, we get d = nλ (|G| |G|) / (2(2πG • 2πG)). Simplifying, we get d = nλ / |G|^2.

Finally, we can use the definition of the reciprocal lattice vector G = gb_1 + kb_2 + lb_3 to express |G|^2 as |G|^2 = g^2|b_1|^2 + k^2|b_2|^2 + l^2|b_3|^2. Substituting