My textbook, Solid-State Physics, Fluidics, and Analytical Techniques in Micro- and Nanotechnology, by Madou, says the following in a section on x-ray diffraction:
X-rays are scattered by the electrons in atoms because electromagnetic radiation (including x-rays) interacts with matter through its fluctuating electric field, which accelerates charged particles. You can think of electrons oscillating in position and, through their accelerations, re-emitting electromagnetic radiation. The scattered radiation interferes both constructively and destructively, producing a diffraction pattern that can be recorded on a photographic plate.
This explanation is fine, but I was hoping to have mathematics accompanying this explanation, so that I could familiarise myself with (or, at least, have some exposure to) the mathematics of this process. Therefore, I have attempted to do this myself. At the moment, the part that I am stuck on is the mathematical description of a fluctuating electric field and accelerating charged particle.
The Wikipedia article for Maxwell's equations only has a single mention of fluctuation:
An important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at a constant speed ($c$) in a vacuum.
And the Wikipedia articles for electric field has no mentions of fluctuation. Therefore, I am left wondering how one uses Maxwell's equations to describe a fluctuating electric field?
With regards to the description of the accelerating charged particle, this question asks, "how and why do accelerating charges radiate electromagnetic radiation?", which, although different to what was written in the textbook, seems to likely be related. However the question and its answers do not include any mathematical descriptions, which is what I'm primarily interested in.
I would greatly appreciate it if people would please take the time to provide a basic mathematical description of these two phenomena, along with some accompanying explanations to assist a novice such as myself in understanding them.