Classical treatment of X-ray scattering

In the classical theory of scattering (Cohen-Tannoudji et al. 1977, James 1982), atoms are considered to scatter as dipole oscillators with definite natural frequencies. They undergo harmonic vibrations in the electromagnetic field, and emit radiation as a result of the oscillations. 

The equation of motion for a single harmonic oscillators of mass m, and force 

When the oscillation is damped, with damping factor k, and damping kx proportional to the speed of motion x, one obtains 

The corresponding time-dependent value of the damped oscillating dipole equals 

The oscillating dipole is a source of electromagnetic radiation of the same frequency, polarized in the direction of the oscillations. At large distances, the wave is spherical. According to the electromagnetic theory, the resulting electric vector at a point in the equatorial plane of the dipole is a 2/ r c2 times the moment of the dipole at time t — r /c. The amplitude of the spherically scattered wave at unit distance in the equatorial plane is therefore 

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