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Tutorial: Scattering Length Density

Contributors: Jeppe Breum Jacobsen, Jacob Kirkensgaard, Arwen Tyler.


Contrast variation technique: a photosynthetic membrane in different concentrations of D$_2$O. The signal from the lipids is masked out in 5-25% D$_2$O, and protein components in 40-45%. From Jakubauskas et al., 2021, Front. Chem. 9:631370.

Before you start

  • It's a good idea to have SasView installed for this tutorial. You can download it from sasview.org.
  • You will need to make calculations by hand. A spreadsheet and possibly pen and paper are recommended.

Learning Outcomes

After completing this tutorial you will
  • Understand the difference between scattering length densities for X-rays and neutrons.
  • Be able to calculate scattering length densities for both X-rays and neutrons.
  • Be able to calculate percentages of D$_2$O and H$_2$O to match out certain parts of a sample.
  • Decide on SANS/SAXS and solvent based on your specific sample.

Introduction

Scattering length density (SLD) is a measure of a material's ability to scatter. SLD varies greatly depending on whether X-rays (SAXS) or neutrons (SANS) are used. This is because X-rays scatter from the electrons of a material, while neutrons scatter from the nuclei. In general, the SLD of a material can be calculated as: $$ \text{SLD} = \dfrac{\sum_{i=1}^n b_i}{V}, $$ where $b_i$ is the (coherent) scattering length of atom $i$, and $V$ is the molecular volume of the material. For X-rays, the scattering length of an atom is roughly proportional to the atomic number $Z$ of that atom. However, for neutrons, the scattering lengths of atoms vary irregularly throughout the periodic table. This is illustrated in the figure below.


Comparison of coherent scattering cross sectioins (square of the coherents scattering lenghts) for neutrons (red) and X-rays (blue).

For neutrons, it is common to list the scattering length of an atom in units of fm (femtometers, $10^{-15}$ m). For our applications, the X-ray scattering length of an atom is essentially given by $b=Zr_e$, where $r_e = 2.82$ fm is the classical electron radius. This approximation holds because SAXS uses hard X-rays ($E>5-10 \text{ keV}$) with wavelengths dissimilar from those often absorbed by atoms in soft matter.

Part I: Calculate SLDs

In this exercise we will calculate the scattering length densities for the two most commonly used solvents in SAS experiments, H$_2$O and D$_2$O for both X-rays and neutrons.
    Exercise: Calculate scattering length densities.
  • Calculate the volume of H$_2$O and D$_2$O from the density and molecular weight. They can be found on wikipedia: water and heavy water
  • Calculate the scattering lengths for X-rays using $b=Zr_e$.
  • For neutrons, use this database from NIST to find the coherent scattering length (under column "Coh b"). Note that deuterium is 2H.
  • Calculate the $\text{SLD}$ for both X-rays and neutrons using the formula above.
  • Fill out the table below and give your answers in $10^{-6} \text{ Å}^{-2}$.
    • SLD X-rays ($10^{-6} \text{ Å}^{-2}$) Neutrons ($10^{-6} \text{ Å}^{-2}$)
    H$_2$O

    9.45

    -0.56
    D$_2$O

    9.41

    6.38
  • Notice the difference between H$_2$O and D$_2$O in, respectively, an X-ray and a neutron scattering experiment!
  • Show answers
In practice, one should use a SLD Calculator. In Sasview, the SLD Calculator can be found under "Tools". Alternatively one can use an online calculator from NIST.

Most SAS experiments are performed in a solvent, which has a SLD of its own. If the SLD of the sample is similar to the solvent's SLD, the scattering from the sample will be weak. This difference or excess scattering length density is $$ \Delta \text{SLD} = \text{SLD}_{\text{sample}} - \text{SLD}_{\text{solvent}}. $$ The intensity of scattering from the sample is proportional to $\Delta \text{SLD}^2$, which we call contrast.

Part II: Contrast matching

Sometimes, we want to match the SLD of the sample to the solvent, to make the sample or a part of the sample invisible. This technique is called contrast variation. In the exercise below, you will learn how to match out a protein.

    Exercise: Match out a protein in SANS
  • Since there is only a difference in SLD between H$_2$O and D$_2$O for neutrons, this exercise only relates to SANS.
  • Assume that the SLD of a protein is $1.3 \times 10^{-6} \text{ Å}^{-2}$.
  • Calculate the fraction of H$_2$O and D$_2$O in your solvent in order to match out the protein ($\Delta \text{SLD}=0$). You can use the numbers you calculated in the previous exercise
  • Display hint

    To do this, you have to isolate x in the following equation \begin{equation} \Delta \text{SLD} =\text{SLD}_{\text{protein}} - x \times \text{SLD}_{\text{H}_2\text{O}} - (1-x) \times \text{SLD}_{\text{D}_2\text{O}} = 0, \end{equation} where $x$ is the fraction of H$_2$O in the solvent.

Part III: Sodium dodecyl sulfate

This section focuses on the detergent sodium dodecyl sulfate (SDS), CH$_3$(CH)$_2$OSO$_3$Na. SDS is widely used to solubilize membrane proteins in biophysical experiments, including SAXS and SANS. As a first exercise, calculate the SLD of SDS for both X-rays and neutrons with the online calculator from NIST. For this you only need the molecular formula of SDS and the density which can be found on wikipedia. For the X-ray source, assume an energy of 10 keV. You should see a table like the one below.



Under "Scattering length density" is the SLD for neutrons as "real". The SLD for X-rays is given as "X-ray SLD" under "real". Both marked in red.

Next, determine the percentage of H$_2$O and D$_2$O in your solvent in order to match out SDS in a SANS experiment. (The answer to this question is given below the table as the "Contrast match point")

In reality, the SLD of SDS varies between the hydrophilic head group and the hydrophibic tail group. In addition to this, SDS forms micelles in solution with aggregation number $N\sim66$. In the exercise below, we calculate the forward scattering ($I(q=0)$) of SDS micelles, which is given by \begin{equation} I(q=0) = n N^2 \left( \Delta \mathrm{SLD}_\mathrm{tail} V_\mathrm{tail} + \Delta \mathrm{SLD}_\mathrm{head} V_\mathrm{head}\right)^2, \end{equation} where $n$ is the number density of the micelles.

    Exercise: Calculate SLD and forward scattering of SDS
  • In the image below the head/tail structure of the SDS molecule can be seen, as well as some SAXS data with a SDS concentration of 5 mg/mL.
  • Calculate the SLD for the headgroup (SO$_4$Na, $V=60.5 \text{ Å}^3$), and tailgroup (C$_{12}$H$_{25}$, $V=355 \text{ Å}^3$) for X-rays.
  • Calculate the number density $n$ of micelles. (Start by calculating the concentration in $\text{mol/cm}^3$).
  • Finally, calculate the forward scattering
  • How does it compare to the data?


In some applications, it can be useful to deuterate the detergents. This is done by replacing the hydrogen in the detergent with deuterium. Read more about this technique in the Tutorial on invisible detergents.

Part IV: Designing a neutron scattering experiment

In this part, we will design a SANS experiment to study the size and polydispersity of SiO$_2$ nanoparticles. Additionally, a cationic surfactant (CetylTrimethylAmmonium Bromide - CTAB) is added to the solution in a concentration close to the critical micelle concentration (CMC). The added CTAB is expected to form micelles around the nanoparticles with a thickness 2-3 nm. The added CTAB prohibits aggregation of SiO$_2$ nanoparticles. For our experiment we wish to gain information on both the SiO$_2$ nanoparticles and the CTAB coating. First we would like to find all the relevant SLDs.

    Task 1: Find SLDs
  • Use your preferred tool to fill out the SLDs in the table below.
    • Material Chemical formula Density (g/cm$^3$) Neutron SLD ($10^{-6} \text{ Å}^{-2}$) X-ray SLD ($10^{-6} \text{ Å}^{-2}$)
    Silica SiO$_2$ 2.2
    CTAB C$_{19}$H$_{42}$BrN 1.0
    Water H$_2$O 1.0 -0.561 9.44
    Heavy water D$_2$O 1.1
Next we want to calculate the $\Delta \text{SLD}$ between all the different substances. We will put $\Delta \text{SLD}^2$ in the table. That means that the values above the diagonal should be the square of the neutron SLD, while the values below the diagonal should be the square of the X-ray SLDs. This is done, because the measure scattering in proportioanl to SLD squared (scattering cross section).

    Task 2: Find contrasts
  • Fill out the table below with $\Delta \text{SLD}^2$. Neutrons above diagonal, X-rays below diagonal.
    • Material SiO$_2$ CTAB Water Heavy water
    SiO$_2$ 0 16.3
    CTAB 0
    Water 0
    Heavy water 0
  • Which features will you expect to see with X-rays?
  • If you use normal/heavy water as your solvent, which features can you expect to see with neutrons?
Before we make a decision on what kind of experiment(s?) to perform, we should also consider incoherent scattering. Incoherent scattering contributes $|q|$-independent background to the scattering. This makes it difficult to see features in the data, especially at low $q$-values, where the signal is weaker. However, this is essentially only a problem for neutrons. Go back to the SLD calculator, and find the incoherent scattering cross section for H$_2$O and D$_2$O for neutrons.

    Task 3: Decide on experiment
  • Given everything we now know from task 1 and 2 which experiments should we perform? Display answer
    1. X-ray scattering with water as solvent. From the table in task 2, we see that the $\Delta \text{SLD}^2$ for silica is much higher than that of CTAB. It is even better in heavy water, but this is such a small difference that it is not worth the cost of the D$_2$O.
    2. Neutron scattering with heavy water as solvent. The $\Delta \text{SLD}^2$ for silica is ~$5.2$ times greater than that of CTAB. This isn't a huge difference, but we don't have to worry about the incoherent scattering from water. Additionally, we could match out the silica by adding some amount of H$_2$O to the solvent.
  • What should the percentages of H$_2$O and D$_2$O be in the neutron experiment to match out the silica?

Perspectives

If you want to learn how to fit a SasView model to data taken collected in different solvent contrasts, you can try the Tutorial on Simultaneous Fitting.

Challenges

  • Coming soon...

Feedback

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  • Reporting issues and bugs via our GitHub page. This could be typos, dead links etc., but also insufficient information or unclear instructions.
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  • Posting or answering questions in the SAStutorials forum.

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