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 *F _{hkl}* = 0.

Similarly, *F _{hkl}* = 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 *F _{hkl}* (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.