## Coherent and incoherent scattering After scrutinizing the Fraunhofer approximation and scattering of individual, and multiples, a natural question aroused: how scattering behaves above atomic level? How the measured differential scattering cross-section (or measured intensity) evolves with $q$ reflect the structure and geometry information of our soft mat samples, like polymers and colloids? Although we may not answer these questions in this page, we would go closer to reveal how structural information could be obtained from arranged atoms. ### Coherent scattering, incoherent scattering and absorption **Coherent scattering**: interaction with all scattering centers (particles); ![[Drawing 2024-04-14 18.41.03.excalidraw.svg]] **Incoherent scattering**: interaction with one (or specific) scattering center (particles); ![[Drawing 2024-04-14 18.54.57.excalidraw.svg]] **Absorption**: absorbed by the particles. ![[Drawing 2024-04-14 18.56.56.excalidraw.svg]] One example for incoherent scattering is [Compton scattering](https://en.wikipedia.org/wiki/Compton_scattering), an interaction between x-ray and the outer shell electrons. This scattering is unselective and "kick" that electron out, at the cost of wavelength change (momentum change of the photon). It coexist with coherent scattering. ### Decompose scattering cross-section for many particles Recall in the [[Fraunhofer scattering#Scattering cross-section and scattering length density under Fraunhofer condition|Fraunhofer scattering]], the (microscopic differential) scattering cross-section is given as: $\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega} = \sum_{i,j} b_i b_j e^{-i\mathbf{q}\cdot(\mathbf{r}_i - \mathbf{r}_j)} $ (Here we change $j,\ k$ to $i,\ j$, this makes no difference) Since we now care about more than one particles (atoms, ions, etc.), the scattering length $b_{i}$ are not necessarily the same. And for neutrons, even the same elements, we may have different values due to isotopes and neutron spin (nucleus difference). It makes more sense to find an **average value** for them (for we don't really care, in soft mat studies, how atoms arrange in polymers or colloids). $b_i = \langle b \rangle + \delta b_i$$b_i b_j = \langle b \rangle^2 + \langle b \rangle (\delta b_i + \delta b_j) + \delta b_i \delta b_j$ Here the $\delta b_{i}$ is just a random value deviate from the ensemble average. **Not Delta function**. For random variable $\delta b_{i}$, we have $\sum_{i,j} b_i b_j = \sum_{i,j} \langle b \rangle^2 + \langle b \rangle \left( \sum_i \delta b_i + \sum_j \delta b_j \right) + \sum_{i,j} \delta b_i \delta b_j $ Since $\delta$ distribution around $\langle b \rangle$, it is safe to consider $\sum_i \delta b_i=\sum_j \delta b_j=0$, this gives $\langle b^2 \rangle = \langle b \rangle^2 + \langle \delta b^2 \rangle$ Now go back to the expression of cross-section $\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega} =\sum_{i,j} b_i b_j e^{-i\mathbf{q}\cdot(\mathbf{r}_i - \mathbf{r}_j)}$. We wanna decouple $b$ and $R$ to make further simplification (for sure $b$ is only real space dependent). The decoupling requires the uncorrelation of $b_{i}$ and $R_{i}$, assume this is valid, we have $\begin{aligned} \frac{\mathrm{d}\sigma}{\mathrm{d}\Omega} &= \sum_{i,j} b_i b_j e^{-i\mathbf{q}\cdot(\mathbf{R}_i - \mathbf{R}_j)} \\ &= \sum_{i\neq j} b_i b_j e^{-i\mathbf{q}\cdot(\mathbf{R}_i - \mathbf{R}_j)} + N\langle b^2 \rangle \\ &= \langle b \rangle^2 \sum_{-i\neq j} e^{i\mathbf{q}\cdot(\mathbf{R}_i - \mathbf{R}_j)} + N\langle b^2 \rangle \\ &= \langle b \rangle^2 \sum_{i,j} e^{-i\mathbf{q}\cdot(\mathbf{R}_i - \mathbf{R}_j)} + N(\langle b^2 \rangle - \langle b \rangle^2) \\ &=\text{coherent}+\text{incoherent} \end{aligned} $ We - first separate $i=j$ and $i\neq j$, convert $i=j$ to $N\langle b^2 \rangle$, - then applied $\langle b_i b_j \rangle = \langle b_i \rangle \langle b_j \rangle = \langle b \rangle^2$ for $i\neq j$, - then use the relation obtain above $\langle b^2 \rangle = \langle b \rangle^2 + \langle \delta b^2 \rangle$, - at last add $i=j$ back, we get the final result. In this expression: $\langle b \rangle^2 \sum_{i,j}e^{i\mathbf{q}\cdot(\mathbf{R}_i - \mathbf{R}_j)}$ is the **coherent** part, $N(\langle b^2 \rangle - \langle b \rangle^2)$ is the **incoherent** part. The incoherent part does not carry structure information and acts as a flat background. But it could be useful in sometimes. We are more interested in the coherent part since it contains all the structural information. We write $\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}_{\text{coh}} = \langle b \rangle^2 \sum_{i,j} e^{-i\mathbf{q}\cdot(\mathbf{R}_i - \mathbf{R}_j)} =N \langle b \rangle^2 S(q)$ And $S(q)=\frac{1}{N} \sum_{i,j}e^{-i\mathbf{q}\cdot(\mathbf{R}_i - \mathbf{R}_j)} $ $\mathbf{R}_i$ are the positions of nucleus. This $S(q)$ is the structure factor of point scatterers. Further discussion on the structure factor is shown in [[Structure factor S(q) and density autocorrelation functions#Structure factor and density autocorrelation function]]. >[!Notice] >The decoupling has some restriction, in general, we can accept >- $b_{i}$ and $b_{j}$ are independent; > >But in crystal lattice, >- $R_{i}$ and $R_{j}$ are strongly correlated, and >- $b_{i}$ and $R_{i}$ could also be strongly correlated > >The last statement is important because we assumed $b_{i}$ and $R_{i}$ are uncorrelated. There are some specific case, even in crystal lattice, would also satisfy this condition >- The atoms are sitting in given positions in the lattice, but different elements distribute randomly, this is the case for high entropy alloys. >- Or we have the crystal consist of only one isotope. Then the $b_{i}$ is a constant and no dependence to space at all. >- Or we do not care about the exact elements distribution, only the lattice structure, then this $b$ can be considered as an average value. > >See [[Scattering in crystals]] for more information. >[!Notice] >***ALL ABOVE*** in this section is for scattering of many **particles**, i.e., a cluster of atoms or ions or molecules. And this $S(q)$ is not the structure factors of large structures (although all information are now contained in it). It is sometimes called ***Atomic structure factor***. >Although we went further, but we still do not have a direct illustrations on how large structures (like polymers and colloids) arrange. >**Pay attention to the meanings of $\mathbf{R}$ and $\mathbf{r}$!** > >Also for actual measurements, $S(\mathbf{q})$ are likely to be averaged. This ensures observables are real. But in theory parts we keep the original form.