## X-ray Crystallography and the Fourier TransformWhen a monochomatic X-ray diffracts off a crystal it performs part of a mathematical operation: the Fourier transform...
Tony Phillips
## Introduction
X-ray crystallography has been essential, since the beginning of the 20th century, to our understanding of matter; recently, as knowledge of the chemical composition of proteins has progressed, the determination of their 3-dimensional structure has become indispensable for the correct interpretation of their functions. Our main access to this information is ## Fourier series: temperature distribution in a wire
Fourier series and the Fourier transform were invented as a method of data analysis. For example, let us follow Jean-Baptiste Joseph Fourier (1768-1830) in studying the time evolution of the temperature distribution in a circular loop of circumference $a$, given an initial distribution of temperature $f(x), 0\leq x\leq a$; (we require $f(a)= f(0)$). We start by calculating what are now known as the (Taken as an initial distribution separately, each of the $\cos~ \frac{2\pi h}{a}x$ and $\sin~ \frac{2\pi h}{a}x$ functions determines a simple solution, as does a constant function; the linearity of the heat equation allows these separate solutions to be combined, with coefficients $a_h$ and $b_h$, to give the complete solution to the problem.) It is useful to simplify the formulas by using Euler's identities $\cos~ t = \frac{1}{2}(e^{it} + e^{-it})$, $\sin~ t = \frac{1}{2i}(e^{it} - e^{-it})$ and grouping terms to yield $$f(x) = \sum_{h=-\infty}^{\infty}c_h e^{i\frac{2\pi h}{a}x}$$ and $$c_h = \frac{1}{a}\int_0^a f(x)~e^{-i\frac{2\pi h}{a}x}~dx;$$ in general, the $c_h$ are complex numbers. ## Electron density distribution in a crystal
The dual operations of -
*The phase problem.*The absolute values $|c_{hkl}|$ alone, although they contain a great deal of information about the molecule in question, do not allow $\rho(x,y,z)$ to be completely reconstructed. A simple example comes from temperature distributions like those studied by Fourier. Consider (with circumference $a = 2\pi$) the temperature distributions $f(x) = \cos ~x + \cos~3x = \frac{1}{2}(e^{ix}+e^{-ix}) + \frac{1}{2}(e^{3ix} + e^{-3ix})$ and $g(x) = \cos ~x - \cos~3x = \frac{1}{2}(e^{ix}+e^{-ix}) - \frac{1}{2}(e^{3ix} - e^{-3ix})$. So the coefficents for $f$ are all $\frac{1}{2}$, while those for $g$ are $c_{-3} = c_3 =-\frac{1}{2}, c_{-1} =c_1 = \frac{1}{2}$. The absolute values are the same, but the distributions are different, as shown below. X-ray crystallographers have devised many ways to get around this limitation, called "the phase problem;" they are beyond the scope of this column. The temperature distributions $f(x) = \cos ~x + \cos~3x$ (red) and $g(x) = \cos ~x - \cos~3x$ (blue), on a circular loop of circumference $2\pi$ (plotted radially) are quite different but have Fourier coefficients with the same absolute values.
## The reciprocal lattice
X-rays interact with a crystal through interaction with parallel families of planes. Suppose as before that the unit cell in the crystal is an $a\times b\times c$ rectangular parallelipiped (when $a,b$ and $c$ are all different, this structure is called
## X-ray diffraction: how a monochromatic plane wave performs Fourier analysis on the electron density distribution.
The diffraction corresponding to a diffraction vector ${\bf s}$ and a single electron at position ${\bf r}$ multiplies the amplitude of the scattered wave by a phase factor $e^{-2\pi i {\bf r}\cdot{\bf s}}$. If $\rho({\bf r})$ is the electron density function in the crystal, the effect on ${\bf s}$ will sum to $$ F({\bf s}) = \int_{\mbox{crystal}}\rho({\bf r})e^{-2\pi i {\bf r}\cdot{\bf s}}~ d{\bf r}.$$ So the It is possible to rewrite this integral in terms of the $(h,k,l)$ vectors in reciprocal space. For each such vector ${\bf H}$ we write, using the reciprocal coordinates ${\bf X} = (X,Y,Z)$ $$ F(h,k,l) = \int_{\mbox{unit cell}}\rho({\bf X})e^{-2\pi i {\bf X}\cdot(h,k,l)}~ d{\bf x}.$$ In this case the inverse Fourier transform $$\rho({\bf X})=\int F(h,k,l)e^{2\pi i {\bf X}\cdot(h,k,l)}~ dV$$ where $dV$ is volume in reciprocal space, can be approximated by a Fourier series $$\rho({\bf X})=\sum_{h=-\infty}^\infty~\sum_{k=-\infty}^\infty~\sum_{l=-\infty}^\infty F(h,k,l)e^{2\pi i {\bf X}\cdot(h,k,l)}$$ which can be compared with the Fourier series for $\rho(x,y,z)$ given at the start of this column.
Here is a nice example, from Kevin Cowtan's Interactive Structure Factor Tutorial. The example is 2-dimensional, and shows how rapidly the structure factors,
Cowtan's simulation leads to the approximate Fourier synthesis of the target from just the seven largest structure factors: those corresponding to $(h,k) = (0,1), (1,0), (-1,2), (-2,1), (1,2), (3,-2), (3,1)$. Here is how the synthesis proceeds, step by step, each time adding in the next structure factor. These images are from his tutorial, and are used with permission. The unit cell (not orthorhombic!) is outlined in dots.
The AMS encourages your comments, and hopes you will join the discussions. We review comments before they're posted, and those that are offensive, abusive, off-topic or promoting a commercial product, person or website will not be posted. Expressing disagreement is fine, but mutual respect is required.
Tony Phillips |
Welcome to the These web essays are designed for those who have already discovered the joys of mathematics as well as for those who may be uncomfortable with mathematics. Search Feature Column Feature Column at a glance |