Reciprocal lattices

Reciprocal lattices can be informally defined as the representation of a lattice in a different space i.e, not the real space. Formally, reciprocal lattices are defined as the fourier transform of the given lattice from the real space to the space of spatial frequencies or the reciprocal space.

Mathematical representation

If the given lattice is a Bravais lattice which we know is defined as $$\bar r = n_{1}\bar a_{1} + n_{2}\bar a_{2} + n_{3}\bar a_{3}$$ where $a_{1}$, $a_{2}$, $a_{3}$ are the primitive vectors of the Bravais lattice and if the reciprocal lattice obtained is a Bravais lattice of the form $$\bar g = m_{1}\bar b_{1} + m_{2}\bar b_{2} + m_{3}\bar b_{3}$$ where $b_{1}$, $b_{2}$, $b_{3}$ are the primitive vectors of the reciprocal lattice i.e, the reciprocal vectors, then we have it that $$g.r = 2\pi N$$ for all N belonging to the set of integers. This mathematical definition is not entirely rigorous since the real proof is very complicated for the discussion at hand and hence has been shortened for better understanding. For this discussion, we will omit 2D lattices and focus on finding reciprocal vectors for 3D lattices only.

Reciprocal vectors for 3D lattices

Mathematically if $a_{1}$, $a_{2}$, $a_{3}$ are the primitive vectors of the given Bravais lattice, then the reciprocal vectors $b_{1}$, $b_{2}$, $b_{3}$ can be defined as $$b_{1} = \dfrac{2\pi}{V} a_{2} \times a_{3}$$ $$b_{2} = \dfrac{2\pi}{V} a_{3} \times a_{1}$$ $$b_{3} = \dfrac{2\pi}{V} a_{1} \times a_{2}$$ where $V = a_{1}.(a_{2} \times a_{3}) = a_{2}.(a_{3} \times a_{1}) = a_{3}.(a_{1} \times a_{2})$. Now, the experiment will give an understanding of how reciprocal vectors work.