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Bragg planes and Brillouin zone construction

The construction of Bragg Planes in the context of Brillouin zones can be understood by considering Bragg’s Law

λ = 2dsinθ

where θ is the angle between the incident radiation and the diffracting plane, λ is the wavelength of the incident radiation and d is the interplanar spacing of the diffracting planes. (Further information on Bragg’s Law is found in the TLP on X-ray diffraction).

In reciprocal space this can be expressed in the form

k' – k = g

where k is the wave vector of the incident wave of magnitude 2π/λ, k' is the wave vector of the diffracted wave, also of magnitude 2π/λ, and g is a reciprocal lattice vector of magnitude 2π/d:

This can be shown graphically using the Ewald sphere construction:

Here 000 is the origin of the reciprocal lattice and O is the centre of the sphere of radius . If the angle subtended at O between 000 and G on the above diagram is 2θ, simple geometry shows that

which can be rearranged into the more familiar form

i.e., the Bragg equation.

The equation

k' – k = g

can be rearranged in the form

k' = k + g

Hence

k'.k' = (k + g).(k + g) = k.k + g.g + 2k.g

Since k'.k' = k.k because diffraction is an elastic scattering event, it follows that

g.g + 2k.g = 0

To construct the Bragg Plane, it is convenient to replace k by -k in this equation so that both k and g begin at the origin, 000, of the reciprocal lattice. Hence, the equation can be written in the form

Constructing the plane normal to g at the midpoint, , then means that any vector k drawn from the origin, 000, to a position on this plane satisfies the Bragg diffraction condition:

 

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