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DoITPoMS Teaching & Learning Packages Reciprocal Space Mathematical representation of reciprocal lattice

Mathematical representation of reciprocal lattice

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 Uk 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 GR = h + k + l and h, k, l are integers.

Note: 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.