## Introduction to SAS Till now, we have obtained all required background knowledge for scattering experiments in theory, and the problem now is extracting information that is of our interest. A quick recap, for our system, we have the [[Coherent and incoherent scattering#Decompose scattering cross-section for many particles|coherent scattering differential cross-section]] as $\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}_{\text{coh}} = \langle b \rangle^2 \left\langle \sum_{i,j} e^{i\mathbf{q}\cdot(\mathbf{R}_i - \mathbf{R}_j)} \right\rangle$ $\mathbf{q}$ can be adjusted by changing $\theta$, where incident and scattered beam has the angle $2\theta$, and $q=\frac{4\pi}{\lambda}\sin(\theta)$. >[!Note] >The scattering pattern in previous sections ([[Scattering in crystals]], [[Coherent and incoherent scattering]]) are caused by (absolute) SLD. That is, a discrete function that describing at where the photon/neutron can be scattered. And the scattering pattern is formed by coherent part, so we care about whether the scattering is coherent or incoherent. >But in SAS, we only care about the relative scattering length density. In this way, we use the modified **continues SLD** as our scattering length density. It is the relative value that contributes to the scattering pattern, and almost all scattering events are incoherent. So we don't often talk about coherence in this case. ### Know about SAS data and plot If use Bragg condition as an reference, $n\lambda=2\mathrm{d}\sin(\theta)$, the structure than could be detected by a wavevector with size $q$ would be $d=\frac{2\pi}{q}$. This helps us selecting proper $q$ for characterization. ![[Drawing 2024-04-15 22.55.03.excalidraw.svg]] Intensity is proportional to scattering cross-section. At x axis we labeled the scattering angle and real space distance $d$ to be detected. - $d>1\ nm$, $2\theta \lesssim 20^\circ$, we detect **mesoscopic structure**, and this is the SAS regime (SANS, SAXS); - $d<1\ nm$, $20^\circ \lesssim2\theta <180^\circ$, we detect microscopic structure (or say atomic), this is the WAS regime (neutron diffraction, XRD, WAXS). Now focus on the SAS signal, the following sketch shows critical regions inside a SAS plot. ![[Drawing 2024-04-15 23.24.25.excalidraw.svg]] - Guinier region, $q \sim 0.01-0.1\ nm^{-1}$. Determination of size of scatterers. - Intermediate region, $q \sim 0.05-0.5\ nm^{-1}$. Determination of shape (form factor), structure, fractal dimension. - Porod region, $q \sim 0.5-10\ nm^{-1}$. High $q$ region, determination of surface scattering, surface fractals. >[!Notice] >It is worth noting that one could always extract [[Structure factor S(q) and density autocorrelation functions]] and [[Form factor]] as long as the intensity data is obtained, regardless of the regions presented above. But the interpretation of these factors are only *valid* (i.e., corresponds to actual structures or forms in real space) in certain regions.