PX262 - K2 - reciprocal lattice and reciprocal space

the structure of crystals can be found from x-ray diffraction measurements, which probe the reciprocal lattice
since a crystal has periodicity, the properties (like the potential, the electron density) are also expected to be periodic functions
fourier series:

f (x) = a_{0} + \sum_{n \neq 0} (a_{n} \cos (\frac{2 π n x}{a}) + b_{n} \sin (\frac{2 π n x}{a}))

periodicity:

f (x) = f (x + a) = f (x + 2 a) = \dots

complex representation:

F (x) = \sum_{n} F_{n} \exp (i \frac{2 π n x}{a}) = \sum_{n} F_{n} e^{i G_{n} x}

where, $F_{n} = a_{n} + i b_{n}$ , and $G_{n} = 2 π n / a$

F_{n} = \frac{1}{a} \int_{0}^{a} F (x) \exp (i \frac{2 π n x}{a}) d x

clearly, $\exp (i G_{n} a) = 1 = \exp (i 2 π n)$
the points sit on a lattice with a spacing of $2 π / a$
for a 3D lattice, were each point is located by three integers: $n_{1}, n_{2}, n_{3} :$

{\vec{R}}_{n_{1} n_{2} n_{3}} = n_{1} \vec{a} + n_{2} \vec{b} + n_{3} \vec{c}

looking for: $\exp (i G_{h, k, l} - {\vec{R}}_{n_{1} n_{2} n_{3}}) = 1$
choosing ${\vec{G}}_{h, k, l} = h \vec{A} + k \vec{B} + l \vec{C}$

\begin{aligned} \vec{A} & = \frac{2 π (\vec{b} \times \vec{c})}{\vec{a} \cdot (\vec{b} \times \vec{c})} \\ \vec{B} & = \frac{2 π (\vec{c} \times \vec{a})}{\vec{a} \cdot (\vec{b} \times \vec{c})} \\ \vec{C} & = \frac{2 π (\vec{a} \times \vec{b})}{\vec{a} \cdot (\vec{b} \times \vec{c})} \end{aligned}

eg: a simple tetragonal lattice with primitive vectors:

\begin{matrix} \vec{a} = a (1, 0, 0) = a \hat{x} \\ \vec{b} = a (0, 1, 0) = a \hat{y} \\ \vec{c} = c (0, 0, 1) = c \hat{z} \\ c > a \end{matrix}

the position vectors for the points on the lattice are:

{\vec{R}}_{n_{1}, n_{2}, n_{3}} = n_{1} \vec{a} + n_{2} \vec{b} + n_{3} \vec{c}

the number of nearest neighbours that any/each point has is given by the shortest distance between any two points, which is $a$ since $c > a$
each point on the lattice has four nearest neighbours, ie: ${\vec{R}}_{1, 0, 0}, {\vec{R}}_{- 1, 0, 0}, {\vec{R}}_{0, 1, 0}, {\vec{R}}_{0, - 1, 0}$
finding the position vectors of the reciprocal lattice, ${\vec{G}}_{h, k, l} :$

{\vec{G}}_{h, k, l} = h \vec{A} + k \vec{B} + l \vec{C}

\begin{matrix} \exp (i {\vec{G}}_{h, k, l} \cdot {\vec{R}}_{n_{1}, n_{2}, n_{3}}) = 1 \\ \vec{A} = \frac{2 π a c}{a^{2} c} \hat{x} = \frac{2 π}{a} (1, 0, 0) \\ \vec{B} = \frac{2 π a c}{a^{2} c} \hat{y} = \frac{2 π}{a} (0, 1, 0) \\ \vec{C} = \frac{2 π a^{2}}{a^{2} c} \hat{z} = \frac{2 π}{c} (0, 0, 1) \end{matrix}

this is another tetragonal lattice with lattice constant $2 π / a$ and $2 π / c$

{\vec{G}}_{0, 0, 1} = \frac{2 π}{c} \hat{z}

this is a vector which specifies the normal vector of a plane perpendicular to the $z$ -axis