We want reciprocal lattice vectors such that the reciprocal vector is the inverse in magnitude of the real vector and is normal to the planes separating the original vector.

So,

$$\left| {{\bf{a}}*} \right| = {1 \over {{d_{100}}}} = {1 \over {\left| {\bf{a}} \right|\cos (\gamma - {\pi \over 2})}}$$ and $${{{\bf{a}}*} \over {\left| {{\bf{a}}*} \right|}} = {{{\bf{b}} \times {\bf{c}}} \over {\left| {{\bf{b}} \times {\bf{c}}} \right|}}$$

Therefore,

$${\bf{a}}* = {{{\bf{b}} \times {\bf{c}}} \over {{\bf{a}}.{\bf{b}} \times {\bf{c}}}}$$and similarly:

$${\bf{b}}* = {{{\bf{c}} \times {\bf{a}}} \over {{\bf{a}}.{\bf{b}} \times {\bf{c}}}}$$

$${\bf{c}}* = {{{\bf{a}} \times {\bf{b}}} \over {{\bf{a}}.{\bf{b}} \times {\bf{c}}}}$$

### Fourier Analysis of Periodic Potential

The periodic potential of a lattice is given by:

\(U({\rm{r}}) = \sum\limits_k {{U_k}} \exp (i2\pi {\rm{K}}.{\rm{r}})\), where *U _{k} *is the coefficient of the potential, and

**r**is a real position vector

However only values of

**K**are allowed which are reciprocal lattice vectors (

**S**).

** Proof:**

\(U({\rm{r}}) = \sum\limits_S {{U_S}} \exp (i2\pi {\rm{S}}{\rm{.r}})\)

since *U*_{(r)} = *U*_{(r + R)}, where **R** is a lattice vector,

\(\sum\limits_S {{U_S}} \exp (i2\pi {\rm{S}}{\rm{.r}}) = \sum\limits_S {{U_S}} \exp (i2\pi {\rm{S}}.({\rm{R}} + {\rm{r}}))\)

\(\sum\limits_S {{U_S}} = \sum\limits_S {{U_S}} \exp (i2\pi {\rm{S}}.{\rm{R}})\)

λ = exp( i 2 π* S R* )

* S R = n*, where

*n*is an integer.

Only possible values are of the form:

* G = ha* + kb* + lc** as

*and h, k, l are integers.*

**GR**= h + k + lNote: This is strictly the crystallographer’s definition of reciprocal lattice vectors. In solid-state physics, the 2π factor is included as a scalar within **S. **The 2π factor may be omitted depending on the application.