PHY.K02UF Molecular and Solid State Physics
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PHY.K02UF Molecular and Solid State Physics | ||||
A three-dimensional Discrete Fourier Transform can be used to find a periodic function that goes through a three-dimensional set of data points. Consider a $P\times Q \times R$ grid of equally spaced data points that take the values $f_{uvw}$ at position $\vec{r} = u\Delta x\hat{x} + v\Delta y\hat{y} + w\Delta z\hat{z}$ where $u$, $v$, and $w$ are integers and $\Delta x$, $\Delta y$, and $\Delta z$ are the spacing of the points. We seek a three-dimensional periodic function $f(x,y,z)$ that passes through all of the points. This function can be expressed as a Fourier series,
$$f(x,y,z) = \sum\limits_{h=h^*}^{h^*+P-1}\sum\limits_{k=k^*}^{k^*+Q-1}\sum\limits_{l=l^*}^{l^*+R-1}F_{hkl}\exp \left(i2\pi \left(\frac{hx}{P\Delta x}+\frac{ky}{Q\Delta y}+\frac{lz}{R\Delta z}\right)\right).$$There are $P\times Q \times R$ points, and $P\times Q \times R$ complex exponential terms are needed to fit the data points. The exponential terms are indexed by the integers $h$, $k$, and $l$. There are $P$ values of $h$, $Q$ values of $k$, and $R$ values of $l$ in the sum: $h = h^*,h^*+1,h^*+2,\cdots,\,h^*+P-1$; $k = k^*,k^*+1,k^*+2,\cdots,\,k^*+Q-1$; $l = l^*,l^*+1,l^*+2,\cdots,\,l^*+R-1$. The starting values $h^*$, $k^*$, and $l^*$ can be any integers, corresponding to the fact that there are infinitely many periodic functions that will pass through all of the data points. The smoothest function $f(x,y,z)$ is obtained for $h^*=\text{Int}(-P/2+1)$, $k^*=\text{Int}(-Q/2+1)$, $l^*=\text{Int}(-R/2+1)$, where $\text{Int}(x)$ rounds down to the nearest integer. This choice corresponds to using only reciprocal lattice vectors in the first Brillouin zone.
To determine the values of $F_{hkl}$, the data points are substituted into expression for the Fourier series. At the data points, $x=u\Delta x$, $y=k\Delta y$, and $z=l\Delta z$.
$$f_{uvw} = \sum\limits_{h=h^*}^{h^*+P-1}\sum\limits_{k=k^*}^{k^*+Q-1}\sum\limits_{l=l^*}^{l^*+R-1}F_{hkl}\exp \left(i2\pi\left(\frac{hu}{P}+\frac{kv}{Q}+\frac{lw}{R}\right)\right).$$Multiply by $\exp \left(-i2\pi\left(\frac{h'u}{P}+\frac{k'v}{Q}+\frac{l'w}{R}\right)\right)$ and sum over $u$, $v$, and $w$.
$$\sum\limits_{u=0}^{P-1}\sum\limits_{v=0}^{Q-1}\sum\limits_{w=0}^{R-1}f_{uvw}\exp \left(-i2\pi\left(\frac{h'u}{P}+\frac{k'v}{Q}+\frac{l'w}{R}\right)\right) = \sum\limits_{u=0}^{P-1}\sum\limits_{v=0}^{Q-1}\sum\limits_{w=0}^{R-1}\sum\limits_{h=h^*}^{h^*+P-1}\sum\limits_{k=k^*}^{k^*+Q-1}\sum\limits_{l=l^*}^{l^*+R-1}F_{hkl}\exp \left(-i2\pi\left(\frac{(h-h')u}{P}+\frac{(k-k')v}{Q}+\frac{(l-l')w}{R}\right)\right).$$In the sum on the right side, the sums over $u$, $v$, and $w$ will be zero unless $h=h'$, $k=k'$, and $l=l'$. When $h=h'$, $k=k'$, and $l=l'$, the right side evaluates to $PQRF_{hkl}$. This yields an expression for the Fourier coefficients,
$$F_{hkl} = \frac{1}{PQR} \sum\limits_{u=0}^{P-1}\sum\limits_{v=0}^{Q-1}\sum\limits_{w=0}^{R-1}f_{uvw}\exp\left(-i2\pi\left( hu/P+kv/Q+lw/R\right)\right).$$This is the Discrete Fourier Transform of a three-dimensional data series $f_{uvw}$. In the form below, $f_{uvw}$ can be input as a rank-3 tensor $f[u][v][w]$. When the button is pushed, the Fourier coefficients are calculated using a discrete Fourier transform. Fits with other values of $h^*$, $k^*$, and $l^*$ can be displayed by inputting integer values for $h^*$, $k^*$, and $l^*$ and pressing the button.
| $f_{uvw}$
$F_{hkl}$ |