Living in the universe of waves

1. The reciprocal lattice

In real space the lattice structure is represented as the Bravias lattice. In the frequency or momentum space the lattice generated by the real Bravias lattice is known as the reciprocal lattice.
If \(b=2\pi/a\) is the lowest reciprocal lattice vector, then physically it means a wave which travels \(2\pi\) phase in between atomic distance of \(a\). As it is the vector it should satisfy the condition \[\vec{b} \cdot \vec{a} = 2\pi n\].
For arbitrary 3D Bravias lattice the reciprocal lattice vectors are found from the following principles.
- If \(\vec{a}= \left(a_{1}, a_{2}, a_{3} \right)\) is the set of Bravias lattice vectors defining arbitrary Bravias lattice, then \(a_{1} \cdot a_{2} \times a_{3}\) defines a volume. The reciprocal lattice is defined as: \[b_{i} = 2\pi \frac{a_{j} \times a_{k}}{a_{i} \cdot \left[ a_{j} \times a_{k} \right]}\] Multiplying \(b_{i}\) by \(a_{i}\) will give the \(2\pi\).
- The lowest value of the reciprocal lattice vector \(b_{i}\) is physically represented as the \(2\pi\) rotation of the wave between two lattice points. The higher values of the reciprocal vectors represent the \(2 \pi n\) rotation of the phase.
- The reciprocal lattice of different crystal structures
  
  Reciprocal lattice Bravias Lattice
  
  Cubic Cubic
  
  FCC BCC
  
  BCC FCC
  
  Hexagonal Hexagonal
- The Wigner-Seitz cell of the Reciprocal lattice is known as the first Brillouin zone.

In a 3D reciprocal lattice one can find the planes where infinite lattice points can lie on a single plane. This plane is called the lattice plane. The planes which are parallel to these planes are called the family of the planes. In a family of the plane all the reciprocal lattice points can lie.
Miller indices are the intercept of the reciprocal plane. In other words if a reciprocal vector is represented as \(k = a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3}\), then the miller indices of the plane perpendicular to this plane is represented as \(\left(a_{1}, a_{2}, a_{3} \right)\).

Different types of the Miller indices

Form of Miller indices	Reciprocal	Direct	Meaning
\(\left( h,k,l \right)\)	Yes		Single lattice plane
\(\left\{ h,k,l \right\}\)	Yes		Family of equivalent lattice planes
\(\left[ h,k,l \right]\)		Yes	Single lattice plane
\(\left\langle h,k,l \right\rangle\)		Yes	Family of equivalent lattice planes