The electron densities of atoms in a crystal are Fourier transforms of the structure factor.
In the article on scattering factor, we have restricted the electron density ρ to a lattice point, which is too simplistic. We have also described the scattering factor (eq27) as
or in its integrated form
If we now extend the distribution of ρ through the unit cell, eq40 becomes the structure factor:
where ρ’ = ρ(xyz) is the electron density at coordinates xyz in the unit cell and , i.e.
Let dV be an infinitesimal volume of the unit cell with edges dx, dy, dz that are parallel to the unit cell axes of a, b, c (volume of unit cell is V). The ratio of dV/V must be equal to (dxdydz)/abc and we can rewrite eq41 as
A Fourier series is an expansion series used to represent a periodic function and is given by:
or by its complex form
Due to the repetitive arrangement of atoms in a crystal, the electron density in a crystal is also periodic and can be expressed as a Fourier series:
In three dimensions, eq43 becomes
Substitute eq44 in eq42
If n ≠ –h,
which makes Fhkl = 0.
Similarly, Fhkl = 0 if m ≠ -k or o ≠ -l. Therefore, the surviving term in the triple summation in eq45 corresponds to the case of n = –h, m = –k, o = –l, giving
When n = –h, m = –k, o = –l, eq44 becomes
Since
Substitute eq46 in eq47
As mentioned in the previous article, the intensity of a diffraction signal is proportional to the square of the magnitude of the three-dimensional structure factor, i.e. . If we know the value of Fhkl (which in principle is the square root of the intensity of a peak from an X-ray diffraction experiment ) and having indexed the plane contributing to this intensity peak (i.e. knowing the h, k, l values), we can determine ρ(xyz) using a mathematical software. The solution to ρ(xyz) is an electron density map that elucidates bond lengths and bond angles of the compound. However, a problem called the phase problem arises.