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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.

The Pointer
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  • I'm not quite sure I understand what you're looking for here. Something in the vein of this part of the Wiki article on EM radiation? – ACuriousMind Jan 18 '20 at 09:19
  • @ACuriousMind Yes. Basically, as I mentioned, I'm trying to understand the mathematical components of the quoted section of the textbook (this section does not have any mathematical descriptions). At the moment, I'm having trouble doing this for the concepts of fluctuating electric field and accelerating charged particle. A mathematical description of these two particular aspects of the textbook excerpt is what I'm looking for here; similar to how Maxwell's equations are laid out in the Wikipedia article you provided (although, [...] – The Pointer Jan 18 '20 at 09:23
  • [...] the Wiki article obviously does not provide a description for the particular case of the textbook excerpt, which is what I'm looking for). The additional request I made for accompanying explanations is just due to the fact that I'm a novice, so a bit of hand-holding is nice for the sake of learning. – The Pointer Jan 18 '20 at 09:29
  • @ACuriousMind Is that clear? Or should I add further clarification? I'm really just looking for the mathematical description of what I mentioned. – The Pointer Jan 18 '20 at 10:00
  • I think the main mathematical thing you're looking for is the https://en.wikipedia.org/wiki/Retarded_potential ,which is basically what @ACuriousMind linked to but with nonzero charge and current density. I dont really have the expertise to give a short & self contained answer and I imagine a full explanation would be quite long; thus the comment. Jackson, I believe , is a very good book for these kind of topics ( I'm currently reading the chapter on scattering and the retarded potential is the thing on which everything is build upon) – ctsmd Jan 18 '20 at 15:09
  • heres something about the retarded potential though it is not quite about scattering https://physics.stackexchange.com/questions/329292/why-is-dipole-the-simplest-source-in-electrodynamics – ctsmd Jan 18 '20 at 15:12
  • @ctsmd But how do retarded potentials relate to fluctuating electric fields? – The Pointer Jan 18 '20 at 15:18
  • I've only just started the chapter but I think the main Idea is that the fluctuating electric fields act on the molecule/ball/system/whatever youre looking at and induce movement/polarization/oscillations/whatever else leading to time varying charge & current densities and the solution of the maxwell equations with those charge and current densities included is described by the retarded potential. Basically describing how the electromagnetic field fluctuates away from the charges/current (so radiation). – ctsmd Jan 18 '20 at 15:29
  • @ctsmd Hmm, interesting. Although, in this case, it's an accelerating charged particle, right? – The Pointer Jan 18 '20 at 15:30
  • yeah, but generally you can talk about any kind of charge/current density by superposition – ctsmd Jan 18 '20 at 15:32
  • Emission and reception of radio and TV waves can be treated as oscillation of free charges in conductors. But X-ray lengths have the scale of an atom. I'm afraid that the description of oscillating electrons are only heuristic, and the mathematical description relates to collision and scattering of waves/particles. – Claudio Saspinski Jan 18 '20 at 21:52
  • @ClaudioSaspinski But can’t this be described mathematically? Say, as an electromagnetic wave? And what would be the relevant equation of an accelerating charged particle? – The Pointer Jan 18 '20 at 21:56
  • @The Pointer there is an introduction of the maths of X-ray scattering in the wikipedia article about X ray crystallography – Claudio Saspinski Jan 18 '20 at 22:07
  • @ClaudioSaspinski but is this indeed a description of a “fluctuating electric field”? – The Pointer Jan 18 '20 at 22:09

2 Answers2

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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.

ProfRob
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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.

  • But I'm looking for a mathematical description of a fluctuating electric field and accelerating charged particles. How does this relate to that? – The Pointer Jan 20 '20 at 10:05
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    @ThePointer The accelerating charged particle is the electron in the Lorentz oscillator. The fluctuating field is the driving force. The math is the standard differential equation for the driven oscillator: http://hyperphysics.phy-astr.gsu.edu/hbase/oscdr.html –  Jan 20 '20 at 10:55
  • Thanks for that. So the fluctuating field is the "sinusoidal driving force" term, $F_0 \cos(\omega t + \varphi_d)$, in the second order linear DE $m\dfrac{d^2 x}{dt^2} + c\dfrac{dx}{dt} + kx = F_0 \cos(\omega t + \varphi_d)$? So which term is the electron? – The Pointer Jan 20 '20 at 11:05
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    @ThePointer The electron is the charged mass of the harmonic oscillator, bound with a Hooke's law "spring constant" $k$. –  Jan 20 '20 at 11:13
  • Hmm, the image seems to suggest that $m$ is "bound" with the spring constant $k$, so I'm presuming that $m$ is the charged mass? – The Pointer Jan 20 '20 at 11:20
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    This is about visible optical properties, but it mentions what happens at high frequencies (x-rays): https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-007-electromagnetic-energy-from-motors-to-lasers-spring-2011/lecture-notes/MIT6_007S11_lec22.pdf –  Jan 20 '20 at 11:49
  • Ahh, yes, I see. On page 10, it has that $m$ is the electron mass, right? – The Pointer Jan 20 '20 at 11:54