X-ray Fourier Transform

In mathematics, a Fourier transform is an operation that converts one real function into another. In the case of FTIR, a Fourier transform is applied to a function in the time domain to convert it into the frequency domain. One way of thinking about this is to draw the example of music by writing it down on a sheet of paper. Each note is in a so-called "sheet" domain. These same notes can also be expressed by playing them. The process of playing the notes can be thought of as converting the notes from the "sheet" domain into the "sound" domain. Each note played represents exactly what is on the paper just in a different way. This is precisely what the Fourier transform process is doing to the collected data of an x-ray diffraction. This is done in order to determine the electron density around the crystalline atoms in real space. The following equations can be used to determine the electrons' position:

where p(xyz) is the electron density function, and F(hkl) is the electron density function in real space. Equation 1 represents the Fourier expansion of the electron density function. To solve for F(hkl), the equation 1 needs to be evaluated over all values of h, k, and l. The resulting function F(hkl) is generally expressed as a complex number (as seen in equation above) with |F(q)| representing the magnitude of the function and ϕ representing the phase.