## Scattering length density in practice In [[Fraunhofer scattering#Scattering cross-section and scattering length density under Fraunhofer condition]] we introduced the concept of scattering length and scattering length density under Fraunhofer approximation. In this page, we will show how they are converted to bulk material level and applied in actual scattering experiments. ### Scattering length for elements For each atom, the scattering length $b$ has different values, and can be searched or find in literatures. Here are websites to get these numbers: Neutron: [Neutron scattering lengths and cross sections (nist.gov)](https://www.ncnr.nist.gov/resources/n-lengths/) X-ray: [NIST X-Ray Form Factor, Atten. Scatt. Tables Form Page](https://physics.nist.gov/PhysRefData/FFast/html/form.html) #### For neutron People typically use scattering cross-section $\sigma$ for neutron scattering directly. The table below has some values for common elements and isotopes. | Nucleus | $\sigma_{coh}$ | $\sigma_{inc}$ | | ------- | -------------- | -------------- | | 1H | 1.8 | 80.2 | | 2H | 5.6 | 2.0 | | C | 5.6 | 0.001 | | O | 4.2 | 0.0008 | | Al | 1.5 | 0.0082 | | V | 0.02 | 5.08 | | Fe | 11.5 | 0.4 | | Co | 1.0 | 4.8 | | Cu | 7.5 | 0.55 | | 36Ar | 24.9 | 0.225 | The units are barn, $1\ \text{barn} = 10^{-24}\ \text{cm}^{2}$ $^{1}\text{H}$ and $^{2}\text{H}$ (aka H and D) has different coherent scattering cross-sections, and this can be used for [[Contrast variation|contrast variation]]. What's more, coherent scattering length of $\rm H$ is negative,$-3.7406 \textrm{\ fm}$. This enables us to do some complex contrast variation. $\text{Al}$ can be used for neutron windows, and $\text{V}$ can be used as sample containers for its low coherent scattering, or as a calibration standard. Scattering length and scattering cross-section has the relation $\sigma_{\text{coh}} = 4\pi b_{\text{coh}}^2$. Spin also leads to a phase change, and affect the scattering lengths and scattering length density. >[!Info] >Above table provide SLD, and to calculate scattering length of elements, as suggested at the beginning, use [Neutron scattering lengths and cross sections (nist.gov)](https://www.ncnr.nist.gov/resources/n-lengths/). Here we have some values for $\rm H,\ C,\ O$. > > > [!Note]- > >Here is the caption. > > > > | Column | Unit | Quantity | > > |--------|------|---------------------------------------------------------------------| > > | 1 | --- | Isotope | > > | 2 | --- | Natural abundance (For radioisotopes the half-life is given instead)| > > | 3 | fm | bound coherent scattering length | > > | 4 | fm | bound incoherent scattering length | > > | 5 | barn | bound coherent scattering cross section | > > | 6 | barn | bound incoherent scattering cross section | > > | 7 | barn | total bound scattering cross section | > > | 8 | barn | absorption cross section for 2200 m/s neutrons | > > > > For $\rm H$: > > > > | Isotope | conc | Coh b | Inc b | Coh xs | Inc xs | Scatt xs | Abs xs | > > |---------|-----------|---------|---------|--------|--------|----------|---------| > > | H | --- | -3.7390 | --- | 1.7568 | 80.26 | 82.02 | 0.3326 | > > | 1H | 99.985 | -3.7406 | 25.274 | 1.7583 | 80.27 | 82.03 | 0.3326 | > > | 2H | 0.015 | 6.671 | 4.04 | 5.592 | 2.05 | 7.64 | 0.000519| > > | 3H | (12.32 a) | 4.792 | -1.04 | 2.89 | 0.14 | 3.03 | 0 | > > > >For $\rm C$: > > > >| Isotope | conc | Coh b | Inc b | Coh xs | Inc xs | Scatt xs | Abs xs | > > |---------|-------|--------|-------|--------|--------|----------|---------| > > | C | --- | 6.6460 | --- | 5.551 | 0.001 | 5.551 | 0.0035 | > > | 12C | 98.9 | 6.6511 | 0 | 5.559 | 0 | 5.559 | 0.00353 | > > | 13C | 1.1 | 6.19 | -0.52 | 4.81 | 0.034 | 4.84 | 0.00137 | > > > > For $\rm O$: > > > > | Isotope | conc | Coh b | Inc b | Coh xs | Inc xs | Scatt xs | Abs xs | > > |---------|--------|-------|-------|--------|--------|----------|----------| > > | O | --- | 5.803 | --- | 4.232 | 0.0008 | 4.232 | 0.00019 | > > | 16O | 99.762 | 5.803 | 0 | 4.232 | 0 | 4.232 | 0.0001 | > > | 17O | 0.038 | 5.78 | 0.18 | 4.2 | 0.004 | 4.2 | 0.236 | > > | 18O | 0.2 | 5.84 | 0 | 4.29 | 0 | 4.29 | 0.00016 | #### For x-ray The scattering length can be calculated using $b_i = \frac{e^2}{4\pi \varepsilon_0 m_e c^2} f_1 = r_e f_1$, here $f_{1}$ is the so-called [x ray form factor](https://en.wikipedia.org/wiki/Atomic_form_factor#X-ray_form_factors), and $r_{e}$ is the [classical electron radius](https://en.wikipedia.org/wiki/Classical_electron_radius), $r_{e}=2.818\times 10^{-15}\ m$. It is still suggested to check the websites for simplicity reasons. X-ray is not isotope or spin sensitive. ### Scattering length density for materials Unlike the space dependent definition in [[Fraunhofer scattering#Scattering cross-section and scattering length density under Fraunhofer condition]], here for practical reason we use the simple form to calculate the scattering length density. The idea is taking an average over certain volumes and integrate the atomic SLD. Namely, $\rho_{mat}=\frac{\sum_{i}^{n}b_{i}}{\bar{V}}$ $\bar{V}$ is the volume containing $n$ atoms, $b_{i}$ is the SLD of the relevant atom. This can be then converted to the following form with mass density rather than the volume here. >[!Note] >A normal question is, will such an average really reflects the nature of SLD derived from atomic level? >Consider a small drop of water, molecules are randomly oriented. Integrate from a point inside this water drop. >![[Drawing 2024-04-23 15.03.08.excalidraw.svg]] >the average profile of $\rho_{mat}$ with respect to integration distance $r$ would be >![[Drawing 2024-04-23 15.09.36.excalidraw.svg]] >One may see that after the integration range gets larger, the SLD for material could naturally represent the atomic SLD. >Since we are considering the small angle scattering, the length scale we care about is $10\ nm -1\ \mu m$. Under such length scale, it is safe to write scattering length density as an position independent term by integrating over volume $V$ and take an average. For x-ray: $\rho_{x}=Z r_{e} n_{a}$ The $Z$ is total atomic number, like for $\mathrm{H_{2}O}$ is 1+1+8=10. $r_{e}$ is the classic electron radius, $r_{e}=2.818\times 10^{-15}\ m$, $n_{a}$ is the atomic number density, $n_{a}=\frac{\rho N_{A}}{M}$, $\rho$ is mass density, $M$ is molar mass. For neutron: $\rho_{n} =bn_{a}$ This $b$ is total scattering length, like for $\mathrm{H_{2}O}$ is $2b_{H}+b_{O}$, and the value can be read from the table. These $\rho$ could be spatial dependent, but only due to the **spatial uneven distributed materials**, not small displacement at atomic level. >[!Note] >To have an idea on the SLD values, for $\mathrm{H_{2}O}$, $\rho_{x-ray}=9.4\times 10^{10}\ cm^{-2}$, and $\rho_{neu} = -0.5577\times 10^{10}\ cm^{-2}$. >[!Info] >A paper showing neutron scattering length is [[Neutron scattering lengths and cross sections]]. The material SLD and atomic SLD is also stated in [[SANS_NR_Intro.pdf]] (https://www.ncnr.nist.gov/summerschool/ss10/pdf/SANS_NR_Intro.pdf).