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X-ray/neutron scattering from crystals and liquids is well-described by the scattering theory to give the (dynamic) structure factor which is a function of momentum and energy:

$$S(\mathbf{k},\omega)= \int d\omega \, e^{i \omega t}\langle \hat{\rho}^{\dagger}(\mathbf{k}, t) \hat{\rho}(\mathbf{k}, 0)\rangle$$

Often, the energy-dependence is ignored (because you can't measure it) and one says that they measure the (static) structure factor

$$S(\mathbf{k}) = \int d\omega \,S(\mathbf{k},\omega)$$

From here, people model the temperature dependence of the structure factor by (quoting wikipedia)

$$S(\mathbf{k}, T\neq0) = S(\mathbf{k}) \, \langle\textrm{exp}(i\mathbf{k}\cdot \mathbf{u}) \rangle^2 \sim S(\mathbf{k}) \textrm{exp}(-k^2 \langle u^2 \rangle)$$

Where the term $\langle\textrm{exp}(i\mathbf{q}\cdot \mathbf{u}) \rangle^2 $ is the Debye-Waller factor that is temperature-dependent and comes from atoms moving randomly at finite temperature ($u$ is the average displacement which depend on temperature).

But, because the number of atoms hasn't changed, the total amount of scattering is conserved, and we need the following relation

$$\int d\mathbf{k}\, S(\mathbf{k}) \textrm{exp}(-k^2 \langle u^2 \rangle)= \int d\mathbf{k}\, S(\mathbf{k}) = \int d\mathbf{k} \,d\omega\, S(\mathbf{k},\omega)$$

The introduction of the Debye-Waller factor however, causes the integral to decrease, and it is commonly said that the rest of the integral comes from inelastic scattering that has a $1- \textrm{exp}(-k^2 \langle u^2 \rangle)$ dependence to exactly balance things.

My question is, where is the energy going? Specifically, where exactly in $(\mathbf{k},\omega)$-space is it located?

For example, is it going to increased Compton scattering off of the atoms, into more phonons, or both? Is it going to high or low $k$,$\omega$? Is there anisotropy? I'm having trouble locating a general answer to this, but I am sure it should have been answered decades ago. Ideally I would like to see how these processes give back the missing $1- \textrm{exp}(-k^2 \langle u^2 \rangle)$ term

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  • $\begingroup$ Actually, you don't even have to involve inelastic scattering to balance things. For example, in a crystal, the maximum of each Bragg peak will decrease by this $\exp(-k^2<u^2>)$ but will also broaden. Thus the total integral can be the same at any temperature. $\endgroup$ Commented Feb 18, 2019 at 14:10
  • $\begingroup$ @E. Bellec, thanks for your comment. I am aware that one thing that happens is that the peak broadens when you take an energy integrated look at the peak, but is that broadening inelastic or elastic scattering? Ordinary xrd can't tell. Since it's occurring from dynamic defects (moving atoms) I think it can only be inelastic. And beyond this, how is thermal phonon scattering accounted for in this picture? $\endgroup$
    – KF Gauss
    Commented Feb 18, 2019 at 16:17
  • $\begingroup$ @E.Bellec, One should be careful about this. The Debye-Waller factor categorically does not give rise to a broadening of the Bragg peaks. It only results in a reduction in intensity. $\endgroup$
    – Xcheckr
    Commented Nov 13, 2020 at 0:52

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I need to put this as an answer since it's too long. You can have an elastic broadening due to Xray-phonon elastic scattering. Take the scattering of one phonon for example. The scattering amplitude due to a phonon of wavevector k at second order is (in 1D, with N atoms and lattice paramter=1 for simplicity)

$A(q) = \sum\limits_{n=-N/2}^{n=N/2} e^{iq(n+u\cos(kn))}$

Where $u$ is the phonon amplitude (related to the number of phonon and therefore to the temperature and energy cost of this phonon). Doing a Taylor expansion to second order in $u$ and taking only into account the terms for $q$ close to $0$ (or to a bragg value) you get

$\begin{align} A(q) &\approx \sum\limits_{n=-N/2}^{n=N/2} e^{iqn}[1+iqu\cos(kn)-\frac{(qu)^2}{2}\cos^2(kn)] \\ &= \sum\limits_{n=-N/2}^{n=N/2} e^{iqn}[(1-\frac{(qu)^2}{4})+iqu\cos(kn)-\frac{(qu)^2}{4}\cos(2kn)]\end{align}$

If you continue the calculation and look at the diffracted intensity $I(q)=|A(q)|^2$, you'll see that the first term correspond to a Bragg with intensity lowered by the phonon due to the $-\frac{(qu)^2}{4}$ term, this is the Debye-Waller factor. Then the first cosine term will give small peaks at $q=\text{Bragg}\pm k$ and finally the last term is even smaller peaks at $q=\text{Bragg}\pm 2k$.

So you see that each elastic scattering on a thermal phonon will create small peaks in the Bragg tail. But the total integrated intensity will still be the same. Since the phonon at low wavevector $k$ cost a small amount of energy, they will have the most important contribution. The tail of the Bragg decreases rapidly when $q$ is far from a Bragg value.

No need for dynamical effects since the phonon is so slow regarding the Xray speed, you can make the assumption that the phonon is frozen during the elastic scattering.

I Hope it helped.

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  • $\begingroup$ Thanks a lot E. Bellec this is really helpful. Sorry if it sounds like I'm a dummy, but the energy exchange to phonons can be readily be seen with high resolution inelastic x-ray scattering as well as inelastic neutron scattering, so I don't think it's valid to say the phonons are too slow to be seen. But from your answer it seems like you can just substitute $u \textrm{cos}(kn)$ with $u \textrm{cos}(kn-\omega t)$, with the phonon energy being $\omega$ and everything would be fine. $\endgroup$
    – KF Gauss
    Commented Feb 19, 2019 at 14:00
  • $\begingroup$ In that case, does this all boil down to saying that thermal phonon scattering grows as approximately $1-\mathrm{exp}(-q^2 u^2)$? $\endgroup$
    – KF Gauss
    Commented Feb 19, 2019 at 14:00
  • $\begingroup$ @ KF Gauss You are right. My point was that you don't need to involve inelastic scattering to have the same total integrated intensity. But if you take all into account, you will also have this Xray inelastic scattering by phonon. But on that point, I'm not strong enough to tell you how to calculate it. You may indeed be right on that $u\cos(kn-\omega t)$ $\endgroup$ Commented Feb 19, 2019 at 14:06
  • $\begingroup$ @ KF Gauss I guess you ask how the DW factor is derived from my formula. You simply make the approximation on the first factor $(1-\frac{(qu)^2}{4})\approx e^{-(qu)^2/4}$ (I may have a factor wrong here). Then you have to do a thermal average on all phonon to get the $<u^2>$. $\endgroup$ Commented Feb 19, 2019 at 14:18
  • $\begingroup$ I see so you are saying that even a static modulation (due to defects or whatever) can give rise to the same intrinsic idea. I was fixated on dynamics because only they can give rise to the temperature dependence. I'll have to think about this some more. $\endgroup$
    – KF Gauss
    Commented Feb 19, 2019 at 14:19

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