## Fourier Transforms in Crystallography

What *is* a Fourier Transform? This question is best answered
with the aid of an example.
A Fourier transform is a representation of some function in terms of a
set of sine-waves. The set of sine-waves of different frequencies is
orthogonal, and it can be shown that any
continuous function can be represented by summing enough sine-waves of
the appropriate frequency, amplitude and phase.
Let us consider an imaginary one-dimensional crystal. There are three
atoms in the unit cell; two carbons and an oxygen. The electron
density in the unit cell looks like this:

Now we will try and represent this function in terms of sine
waves. The first sine wave has a frequency of **2**, that is there
are two repeats of the wave across the unit cell. One peak represents
the oxygen, and the other the two carbons:

The second sine wave has a frequency of **3**; three repeats of the
wave across the unit cell. It has a different phase, in other words we
start at a different place on the wave. The amplitude is also
different:

Finally, we introduce a sine wave with a frequency of **5**. Two of
the peaks of this wave are lined up with the carbon atoms:

Now we add them all together:

Note that the sum of the three sine-waves is a good approximation to
the original unit cell. Thus we can see that the unit cell can be
represented quite well using only three sine-waves, given the correct
choice of frequency, amplitude and phase.

Now we will look at the Fourier Transform of the same unit cell. Note
that the result consists of a series of peaks, the largest of which
are at **2**, **3** and **5** on the x-axis. These correspond
exactly to the sine-wave frequencies which we used to reconstruct the
unit cell. If you look carefully you will also see that the heights of
the peaks correspond to the amplitudes of the three waves:

The smaller peaks in the Fourier transform correspond to additional
smaller waves which would have to be added to get a perfect fit to the
original density. Thus we can see that the Fourier Transform tells us
what mixture of sine-waves is required to make up any function.

Of course the sine-waves go on for ever, and so there will be lots more
copies of the unit cell beyond the pictures here. Also, the Fourier
Transform has some other features: it has values for both positive and
negative frequency, and the values are complex and not real. These
features combine to determine the phase of any particular sine-wave
component.

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