X-ray scattering: Mathematical description of a *fluctuating electric field* and *accelerating charged particle*

Question

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.

Answer 1

The basics are simple.

It is easy to show that a function of the form $\vec{E} = \vec{E_0} f(\vec{k}\cdot \vec{r} - \omega t)$ is a valid solution to Maxwell's equations in vacuum, as long as $\vec{E_0} \cdot \vec{k} = 0$ and $\omega/k = c$.

The function $f$ is arbitrary, but is usually assumed to be some sort of sinusoidal oscillation. e.g. $\vec{E} = \vec{E_0} \sin(kx - \omega t)$. A fluctuating electric field.

Because $\nabla \times \vec{E} = -\partial \vec{B}/\partial t$, it is also easy to show that there must be an accompanying magnetic field that is in phase with $\vec{E}$, but at right angles to it and $\vec{k}$ and that the amplitude of $\vec{B}$ is $E_0/c$. e.g. $B = B_0 \sin(kx - \omega t)$, where $\vec{E_0}\cdot \vec{B_0} = \vec{k}\cdot \vec{B_0} = 0$ and $B_0 = E_0/c$.

When this wave encounters an electron it exerts a Lorentz force $$\vec{F} = q(\vec{E} + \vec{v} \times \vec{B}),$$ where $\vec{v}$ is thevelocity of the electron. Since the amplitude of the B-field is $c$ times smaller than the E-field amplitude, then so long as $v \ll c$, then the magnetic component of the force can be ignored. One then uses Newton's second law to derive the acceleration of the eletron. The accelerated electron acts like an accelerating electric dipole and emits "electric dipole radiation" with the appropriate 3-dimensional radiation pattern of such a system (i.e no radiation along the axis of oscillation). i.e. $$m_e \ddot{\vec{r}} = -e \vec{E_0} \sin (kx - \omega t)\ \ \ {\rm and}$$ $$ \ddot{\vec{p}} = -e \ddot{\vec{r}}$$

The above is for a "free electron" - either genuinely free electrons or electrons that are only loosely bound in atoms compated with the energies of photons that are incident upon them. This is known as "Thomson scattering" and has a frequency-independent scattering cross-section.

To generalise to more firmly bound systems one treats the atom as a weakly damped oscillator, with a restoring force set by the nuclear attraction, a damping term which is due to the oscillating system emitting radiation and a driving force given by the Lorentz force due to the incoming wave (as before).

The solutions to such a system are just the usual solutions for a driven harmonic oscillator (e.g. radiation $\propto \omega^4$ at frequencies below resonance and a resonant peak in emission at the "natural frequencies" - which are associated with allowed transitions in the atom).

This classical model breaks down when the scattering becomes inelastic (Compton scattering) and part of the photon momentum is transferred to the atom. Unfortunately this does start to set in at X-ray wavelengths and a classical treatment becomes quantitatively inappropriate.

Answer 2

Scattering of electromagnetic waves by a single atom can be treated in the model of Lorentz oscillators: a unit charge with mass $m_e$ bound to a nucleus driven by the incident electric field.

Resonances are typically at ultraviolet frequencies.

For visible light this model can explain refractive index and dispersion.

For x-rays this explains how the phase velocity is generally slightly larger than $c$ (refractive index slightly lower than unity).

X-ray diffraction then occurs because of scattering by three-dimensional arrays of such oscillators.