Contrast function

The contrast of RSoXS is strongly dependent on the dispersion (\(\delta\)) and the absorption (\(\beta\)) components of the refractive index (n) as represented in the equations below. The refractive index [\(n(E)\)] at a photon energy \(E\) can be calculated using equation:

\[\begin{equation} \begin{aligned} n(E)=&1-\delta(E)+i\beta(E)\\ =&1-\frac{r_0}{2\pi}\lambda^2\sum_j N_j[f_{1j}(E)+if_{2j}(E)] \end{aligned} \label{eq:contrast1} \end{equation}\]

The contrast function (\(C\)) can be estimated from equation: \begin{equation} C\propto{(\Delta n)^2}\propto{(\Delta\delta)^2+(\Delta\beta)^2} \label{eq:contrast2} \end{equation} Where \(r_0\) is the classical electron radius, \(\lambda\) is the wavelength of incident X-rays, \(N_M\) the number density of an atom (\(M\)), \(f_1\) and \(f_2\) are the real and imaginary parts of the complex atomic scattering factor, and the summation is performed for all atoms (\(M\)).

Kramers–Kronig relations: \begin{equation} \chi_1(\omega)=\frac{1}{\pi} \mathcal{P} \int_{-\infty}^{\infty} \frac{\omega^{\prime} \chi_2\left(\omega^{\prime}\right)}{\omega^{\prime 2}-\omega^2} d \omega^{\prime}+\frac{\omega}{\pi} \mathcal{P} \int_{-\infty}^{\infty} \frac{\chi_2\left(\omega^{\prime}\right)}{\omega^{\prime 2}-\omega^2} d \omega^{\prime} \end{equation}

\[\begin{equation} \chi_2(\omega)=-\frac{2}{\pi} \mathcal{P} \int_0^{\infty} \frac{\omega \chi_1\left(\omega^{\prime}\right)}{\omega^{\prime 2}-\omega^2} d \omega^{\prime}=-\frac{2 \omega}{\pi} \mathcal{P} \int_0^{\infty} \frac{\chi_1\left(\omega^{\prime}\right)}{\omega^{\prime 2}-\omega^2} d \omega^{\prime} \end{equation}\]