Fourier Transformation

Fourier Transformation for whole volume

Fourier transfomation

$$\mathfrak{F}[ f ] (\bm{k}) \equiv \int \left. d^3r e^{-i\bm{k} \cdot \bm{r}} f(\bm{r}) \right.$$ (47)

Inverse transformation

$$f(\bm{r}) = \frac{1}{V} \sum_{\bm{k}}\left. e^{i\bm{k} \cdot \bm{r}} \mathfrak{F}[ f ] (\bm{k}) \right.$$ (48)

$$= \frac{1}{(2\pi)^3} \int \left. d^3 ke^{i\bm{k} \cdot \bm{r}} \mathfrak{F}[ f ] (\bm{k}) \right.$$ (49)

最初の式は、有限の体積を考え周期的境界条件を課した場合で、2番目の式は体積無限大の極限と考えれば良い。

Fourier Transformation within unit cell

When a function $f$ has periodicity for translation vectors $\bm{T}$

$$f(\bm{r} + \bm{T}) = f(\bm{r}),$$ (50)

the fourier transformation within the unit cell can be defined as

$$\mathfrak{F}_c [ f ] (\bm{G}) \equiv \frac{1}{\Omega_c} \int_{\Omega_c} d^3 r f(\bm{r}) e^{-i \bm{G} \cdot \bm{r}},$$ (51)

$$f(\bm{r}) = \sum_{\bm{G}} \mathfrak{F}_c [ f ] (\bm{G}) e^{i \bm{G} \cdot \bm{r}}.$$ (52)

Note, the position of volume factor differs from the whole volume fourier transformation.

relation

Let $f$, $g$ periodic functions in unit cell.

$$\int_{\Omega_c} d^3 f^{*} (r) g (r) = \Omega_c \sum_{\bm{G}} \mathfrak{F}_c [ f ] ^{*}(\bm{G}) \mathfrak{F}_c [ g ] (\bm{G})$$ (53)

Linear combination of shifted functions

A function $g(\bm{r})$ is defined in whole space. New periodic function is generated by linear combination:

$$g_c ( \bm{r} ) \equiv \sum_{\bm{T}} g(\bm{r} + \bm{T} ).$$ (54)

Its Fourier transformation can be calculated as

$$\mathfrak{F}_c [ g_c ] (\bm{G}) = \frac{1}{\Omega_c} \int_{\Omega_c} d^3 r g_c( \bm{r} ) e^{-i \bm{G} \cdot \bm{r}}$$ (55)

$$= \frac{1}{\Omega_c} \sum_{\bm{T}} \int_{\Omega_c} d^3 r g( \bm{r} + \bm{T} ) e^{-i \bm{G} \cdot \bm{r}}$$ (56)

$$= \frac{1}{\Omega_c} \sum_{\bm{T}} \int_{\Omega_c} d^3 r g( \bm{r} ... ... e^{-i \bm{G} \cdot (\bm{r} + \bm{T})} \quad ( e^{i \bm{G} \cdot \bm{T} } = 1 )$$ (57)

$$= \frac{1}{\Omega_c} \int d^3 r g( \bm{r} ) e^{-i \bm{G} \cdot \bm{r}}$$ (58)

$$= \frac{1}{\Omega_c} \mathfrak{F}[ g ] (\bm{G} ).$$ (59)