Physics

XRTS is describing the scattering of an X-ray photon on electrons. In the WDM community, the term summarizes Raman, Rayleigh and Thomson scattering processes [Glenzer and Redmer, 2009].

In a typical setup, a light-source provides photons carying momentum \(\hbar\vec{k_0}\) and energy \(\hbar\omega_0\). This light scatteres off a target, often a WDM or plasma state, defined by a set of plasma paramters, such as density, temperature, and ionization. A detector, located at an angle \(\theta\) collects scattered photons, with energy \(\hbar\omega_s\), and momentum \(\hbar\vec{k}_s\). The transferred quantities shall be denoted with \(\vec{k}\) (the scattering vector) and \(\omega\) (photon frequency shift).

Since the plasma is isotropic, we simplify that only the absolute value of the scattering vector \(\vec{k}\) has to be considered. This quantity can be expressed as

\[k = \sqrt{k_s^2 + k_0^2 - 2 k_s k_0 \cos \theta} \approx 2 \frac{\omega_0}{c} \sin\left(\frac{\theta}{2}\right),\]

where the approximation holds when \(k\) is small compared to \(k_0\). A sketch of the geometry of a typical XRTS experiment can be seen in the figure.

_images/XRTS_geometry.svg

An actual XRTS signal arises from several distinct mechanisms that are interpreted according to the chemical picture introduced by Chihara [Chihara, 2000]. In this model, and in jaxrts, the total electron–electron dynamic structure factor is decomposed into a sum of contributions that correspond to different physical origins—namely, elastic (el), free–free (ff), bound–free (bf), and free–bound (fb) interactions between electrons and ions.

\[S_{ee}^{\text{tot}}(k, \omega) = S_{ee}^{\text{el}}(k, \omega) + S_{ee}^{\text{ff}}(k, \omega) + S_{ee}^{\text{bf}}(k, \omega) + S_{ee}^{\text{fb}}(k, \omega)\]

Each of these four contributions have to be defined, in order to generate a spectrum. See Generating a first spectrum on how this is done. A comprehensive list of available models can be found under Models implemented.

\(S_{ee}^{\text{tot}}\) is related to the intensity \(I\) measured in an experiment via:

\[I(k, \omega) \propto \left(\frac{\omega + \omega_0}{\omega_0}\right)^\nu S_{\text{ee}}^\text{tot}(k, \omega) \circledast R\left(\omega\right)\quad, \label{eqn:signal}\]

Here, \(R\) is the combined source instrument function (jaxrts.setup.Setup.instrument), \(\circledast\) is the convolution, and the exponent \(\nu\) accounts for the frequency redistrubution correction (jaxrts.setup.Setup.frc_exponent, see [Crowley and Gregori, 2013].