The scattering factor describes the amplitude of scattered rays from a single atom.

The presence or absence of an intensity peak depends on the interference of scattered X-rays at the detector of the powder X-ray diffractometer. As the interference of scattered X-rays is the sum of amplitudes of X-ray waves from many atoms in a sample, we have to begin with the analysis of the resultant amplitude of scattered rays from a single atom (** scattering factor**), which is in turn the sum of amplitudes of waves scattered by the electrons in the atom.

The amplitude of a travelling wave is expressed by the equation:

We can also expressed the general wave equation in its complex form:

The complex form of the wave equation is equivalent to the travelling wave equation if we consider only the real part of it: *Re*(*y*), which is the value that is physically observed in an X-ray diffraction experiment. The reason eq25 is preferred over the travelling wave equation is that it is easier to manipulate mathematically to derive the scattering factor. Recall that *ω* = 2*π*/*T* (angular frequency), *k* = 2*π*/λ (wave number) and *Φ* is the phase difference. If we assume that the scattering of X-rays by an atom is elastic, i.e. there is no change in frequency of the X-ray between the incident and scattered rays, the term *kx*–*ωt* becomes a constant and we can simplify eq25 to:

where *A’ = Ae ^{i(kx-ωt)}*.

The amplitudes of the scattered X-rays not only differ in terms of their phase as suggested by eq26 but are also proportional to the number of electrons in the atom, *ρdV*, where *ρ *and *V* are electron density and volume of the atom respectively. If we define *A’* as the number of electrons in the atom, eq26 becomes

where the amplitude of the scattered X-rays from an atom is denoted by *f*, the scattering factor of an atom.

To determine an expression for *Φ*, we note that the ratio of path difference *δ* and wavelength is equal to the ratio of phase difference and 2*π*, i.e.

Substitute eq19 in eq28,

Since and from eq23, , eq29 becomes:

In three dimensions, we let ** a** =

**and I**

*r***I = I**

*a***I =**

*r**r*,

Substitute eq30 in eq27

where

Since *ρ* is spherically symmetrical, we can represent the volume element *dV* in eq31 in spherical coordinates where *dV* = *r ^{2}*

*sinμdrdμdΦ*(see above diagram). Eq31 becomes:

Integrating both sides,

Note that *d(cosμ) = -sinμdμ*; when *μ = π*, *cosμ = -1*; when* μ = 0*, *cosμ = 1*. So the second integral in eq32 becomes

Eq32 now becomes

Since

where . *ρ* is a function of *r* and remains within the integral.

So far, we have developed an expression (eq33) that describes the resultant amplitude of scattered rays from a single atom. To further analyse the interference of scattered X-rays from multiple atoms in a sample, we have to derive another expression called the structure factor.