Time-average Poynting vector

[Wox]

I'm having a problem in expressing the time-average norm of the Poynting vector of the scattered electromagnetic field from a crystal, as expressed in several textbooks.

Concider a monochromatic plane wave
$$\begin{split} \bar{E}_{\text{mono}}(t,\bar{x})&=\bar{E}_{0}e^{i(\bar{k}\cdot\bar{x}-\omega t})\\ \bar{B}_{\text{mono}}(t,\bar{x})&=\frac{1}{\omega}\bar{k}\times \bar{E}_{\text{mono}}(t,\bar{x}) \end{split}$$
Then the scattered field of this wave from a crystal is given by (in the kinematic approximation)
$$\begin{split} \bar{E}_{\text{scat}}(t, \bar{x})&=\frac{r_{e}}{\|\bar{x}\|}.\|\bar{E}_{0}\| .K.((\hat{x}\cdot\hat{E}_{0})\hat{x}-\hat{E}_{0}).e^{i(\bar{k}_{\text{scat}}\cdot \bar{x}-\omega t)}\\ \bar{B}_{\text{scat}}(t,\bar{x})&=\frac{1}{\omega}\bar{k}_{\text{scat}}\times \bar{E}_{\text{scat}}(t,\bar{x})\\ K&=\sum_{j=1}^{N}{f_{j}e^{-i\bar{Q}\cdot(\bar{\Delta x}_{j}+\bar{\delta x}_{j}(t))}}\\ \bar{Q}&=\bar{k}_{\text{scat}}-\bar{k}\\ \bar{k}_{\text{scat}}&=\|\bar{k}\|.\hat{x}(t) \end{split}$$
where $\bar{\Delta x}_{j}$ the equilibrium positions of the atoms, $\bar{\delta x}_{j}(t)$ their thermal displacement, $r_{e}$ the classical electron radius and $f_{j}$ atomic scattering factors (which are complex numbers).

The Poynting vector and the derived intensity (time-averaged norm of the Poynting vector) are defined as
$$\begin{split} \bar{P}(t,\bar{x})=&\frac{1}{\mu_{0}}\mathcal{R}e( \bar{E})\times \mathcal{R}e( \bar{B})\\ I(\bar{x})=&\left<\|P\|\right>_{t}=\frac{1}{T}\int_{0}^{T} \|P\|dt \end{split}$$
where $T$ a period of time which is much longer than the period of the atomic vibrations and the period of the monochromatic plane wave. If we write $K=M_{K}e^{i\phi_{K}(t)}$ then this is written as
$$I(\bar{x})=c\epsilon_{0}.\frac{r_{e}^{2}}{\bar{x}^{2}}.\bar{E}_{0}^{2} .M_{K}^{2}.(1-(\hat{x}\cdot\hat{E}_{0})^{2}).\frac{1}{T}\int_{0}^{T} cos^{2}(\bar{k}\cdot\bar{x}-\omega t+\phi_{K}(t))dt$$
At least, that's how I would do it. But several textbooks write something else:
$$I(\bar{x})= c\epsilon_{0}.\frac{r_{e}^{2}}{\bar{x}^{2}}.\bar{E}_{0}^{2} .\left<K\right>_{t}^{2}.(1-(\hat{x}\cdot\hat{E}_{0})^{2}).\frac{1}{T}\int_{0}^{T} cos^{2}(\bar{k}\cdot\bar{x}-\omega t)dt$$
which allows use to write the simple expression (time average of the $cos^{2}(\ldots)$ is $1/2$)
$$I(\bar{x})=c\epsilon_{0}.\frac{r_{e}^{2}}{\bar{x}^{2}}.\bar{E}_{0}^{2} .\left<K\right>_{t}^{2}.\frac{(1-(\hat{x}\cdot\hat{E}_{0})^{2})}{2}$$
The $\left<K\right>_{t}^{2}$ term leads to the structure factor, the Laue interference function and the Debye-Waller factor. But I have no idea how the time average (an integral) can be done for the factor $K$ and the $cos^{2}(\ldots)$ separately. Can someone give me a hint or maybe a reference on this?

 

[eljose79]

Thank you for bringing up this interesting problem. Your approach seems to be correct, but the textbooks may be using a simplification for the sake of clarity. Let me explain:

The time-averaged norm of the Poynting vector is defined as the average of the squared magnitude of the Poynting vector over a period of time. This is equivalent to taking the time average of the cosine squared term in your expression. However, the time average of the factor K would be more complicated to calculate, as it involves both the thermal displacement of the atoms and the complex atomic scattering factors.

To simplify the calculation, the textbooks may be assuming that the thermal displacements and atomic scattering factors are constant over time, or that they vary slowly enough that their average over a period of time is equivalent to their value at a specific time. In this case, the time average of the factor K would simply be the square of the average of the atomic scattering factors, which is what is written in the textbook expression.

If you want to consider the time dependence of the thermal displacements and atomic scattering factors, you would need to take the time average of the cosine squared term and the factor K separately, as you mentioned in your post. This would result in a more complicated expression, which may not have a simple closed form solution.

I hope this helps clarify the discrepancy between the two expressions. If you require a more detailed explanation, I recommend consulting a textbook on scattering theory or electromagnetic theory, which may have a more thorough discussion on the time average of the Poynting vector in the context of scattered electromagnetic fields.