Working with Reciprocial Lattice

Question

In solid matter physics the lattice structure can equivalently be described via standard Bravais modelling but also by considering it's reciprocial lattice. Mathematically the reciprocial lattice can be indeed interpreted as Fourier trafo of given Bravais lattice.

I often read that by threating a lot of problems concerning solid matter there are advantages to "work" with the reciprocial lattice.

What are concretely theese advantages? Does anybody have some nice examples justifying the advantage of usage of reciprocial lattice? Does it have physical interpretation?

My considerations: Up to now I only know that for given Bravais lattice ${\displaystyle \mathbf {R} _{n}=n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}}$ and it's corresponding reciprocial ${\displaystyle \mathbf {G} _{m}=m_{1}\mathbf {b} _{1}+m_{2}\mathbf {b} _{2}}$ there exist the nice identity ${\displaystyle e^{i\mathbf {G} _{m}\cdot \mathbf {R} _{n}}=1}$.

But is that all?

Answer 1

Have you seen so-called tight-binding models yet, describing electrons hopping on a lattice? Here's a simple example with spinless particles in one dimension: $$H=t\sum_i c_i^\dagger c_{i+1},$$ where $c_i$ annihilates a particle at site $i$, and $c_i^\dagger$ creates one. $t$ sets the hopping energy scale. Since the electrons are moving around, using an occupation basis in real space (on the ordinary lattice) is at best tedious, at worst hopeless. But if we Fourier transform, $$H=t\sum_k c_k^\dagger c_k \cos ka,$$ where $a$ is the lattice constant. In other words, by using the repeated unit cell structure we have diagonalized the Hamiltonian and found an appropriate basis set. Usually, when there is a periodic structure, a Fourier transform is helpful - whether it is to solve this model, analyze X-ray scattering data, or to decompose a wave form into frequencies.