Scattering Length Density¶

Hard X-rays interact with the electrons in a material. A region that is dense with electrons scatters X-rays more strongly than a region that is sparse. The quantity that captures this is the scattering length density (SLD), given the symbol \(\rho\) and measured in units of Å\(^{-2}\).

These values are typically calculated for common materials by adding up the number of atoms of each species within the molecule. You will rarely need to calculate this value from scratch as they can be computed regularly using online tools. What matters for building models is understanding the role SLD plays.

The two-phase model¶

The simplest scattering model treats the sample as two regions with different SLDs: the particle (with SLD \(\rho_1\)) surrounded by a matrix or solvent (with SLD \(\rho_2\)). Examples would include a lipid nanoparticle in in water or proteins in a buffer solution.

Typical SLD values¶

These values give a sense of the typical magnitudes. The contrast between lipid chains and water is relatively small — this is why lipid scattering experiments require careful experimental design to maximize signal. Certain experiments are designed to maximize and manipulate scattering contrast. The most common example is the use of a neutron scattering facility, where water can be easily replaced with deuterium (heavy water) to increase contrast. An alternative solution is to use resonant X-rays which can generate scattering contrast through the use of a rapidly changing complex index of refraction, more on this later.

Representing SLD in code¶

In our Python models, SLD values will appear as parameters. The convention used throughout this tutorial is:

rho_particle = 10.0e-6   # SLD of the sphere, in Å^-2
rho_matrix   =  9.47e-6  # SLD of water, in Å^-2

delta_rho = rho_particle - rho_matrix

In the normalized form factor \(P(q)\) that we will compute in the next pages, \((\Delta\rho)^2\) appears as an overall prefactor that scales the intensity without changing the shape of the curve. For now we will fold it into a proportionality constant and focus on the shape of \(P(q)\).

A question to keep in mind¶

Consider what happens when \(\rho_1 = \rho_2\): the contrast \(\Delta\rho = 0\), and the intensity goes to zero. A particle that is indistinguishable from its surroundings produces no scattering signal at all.

rho_silica     = 18.8e-6
rho_water      =  9.47e-6
rho_heavy_water = 6.40e-6

delta_rho_water       = rho_silica - rho_water        # 9.33e-6 Å^-2
delta_rho_heavy_water = rho_silica - rho_heavy_water  # 12.4e-6 Å^-2

This idea — that you cannot see what you cannot distinguish from the background — will come back in Section 3 when we build hollow sphere models which are appropriate to simulate vesicle scattering and model biological membranes.

Complex SLD and the index of refraction¶

The atomic scattering factor¶

In standard (non-resonant) SAXS at hard X-ray energies, each atom scatters X-rays in proportion to its number of electrons \(Z\). The SLD is therefore proportional to the local electron density, which is what the table above reflects.

Near an absorption edge — a photon energy at which the X-rays are resonantly absorbed by a specific element — this picture changes. The atomic scattering factor picks up two energy-dependent correction terms:

where \(f_0 \approx Z\) is the non-resonant (Thomson) term, \(f'(E)\) is the real anomalous dispersion correction, and \(f''(E)\) is the imaginary absorption correction. Both \(f'\) and \(f''\) can vary by orders of magnitude across an absorption edge, and \(f'\) can even become negative.

The SLD therefore becomes complex near an edge:

where \(\rho_{\text{real}}\) encodes scattering (refraction) and \(\rho_{\text{imag}}\) encodes absorption. The contrast \(\Delta\rho\) used in the form factor becomes a complex number as well, and the intensity is then proportional to \(|\Delta\rho|^2 = (\Delta\rho_{\text{real}})^2 + (\Delta\rho_{\text{imag}})^2\).

The complex index of refraction¶

X-ray optical techniques — including RSOXS, reflectometry, and GISAXS — often describe materials using the complex index of refraction:

where \(\delta\) (delta) is the refractive index decrement (it controls phase shifts and refraction) and \(\beta\) (beta) controls absorption and attenuation. Both \(\delta\) and \(\beta\) are dimensionless and positive for ordinary materials.

Converting between the two representations in Python¶

The following utility function performs both conversions. Add it to your notebook for reference — you will use it if you work with resonant scattering data:

import numpy as np

HC = 12398.4  # hc in eV·Å — converts photon energy to wavelength

def wavelength_from_energy(energy_eV):
    """Return X-ray wavelength in Å for a given photon energy in eV."""
    return HC / energy_eV

def sld_from_optical_constants(delta, beta, energy_eV):
    """
    Convert optical constants (delta, beta) to complex SLD.

    Parameters
    ----------
    delta : float
        Refractive index decrement (dimensionless).
    beta : float
        Absorption index (dimensionless).
    energy_eV : float
        Photon energy in eV.

    Returns
    -------
    rho_real, rho_imag : float
        Real and imaginary SLD components in Å^-2.
    """
    lam = wavelength_from_energy(energy_eV)
    prefactor = 2 * np.pi / lam**2
    return prefactor * delta, prefactor * beta

def optical_constants_from_sld(rho_real, rho_imag, energy_eV):
    """
    Convert complex SLD to optical constants (delta, beta).

    Parameters
    ----------
    rho_real : float
        Real SLD component in Å^-2.
    rho_imag : float
        Imaginary SLD component in Å^-2.
    energy_eV : float
        Photon energy in eV.

    Returns
    -------
    delta, beta : float
        Dimensionless optical constants.
    """
    lam = wavelength_from_energy(energy_eV)
    prefactor = lam**2 / (2 * np.pi)
    return prefactor * rho_real, prefactor * rho_imag

As a concrete example, here is what happens near the carbon K-edge (~285 eV), where soft X-ray scattering experiments on organic materials are commonly performed:

# Typical optical constants for a polymer near the carbon K-edge
energy = 285.0   # eV — near the carbon 1s absorption edge
delta  = 2.5e-4  # dimensionless
beta   = 1.5e-5  # dimensionless

rho_real, rho_imag = sld_from_optical_constants(delta, beta, energy)
lam = wavelength_from_energy(energy)

print(f"Photon energy:  {energy:.1f} eV")
print(f"Wavelength:     {lam:.4f} Å")
print(f"rho_real:       {rho_real:.3e} Å^-2")
print(f"rho_imag:       {rho_imag:.3e} Å^-2")

$ git add chapter_02_sphere.ipynb
$ git commit -m "add SLD conversion utility functions"

What's next: The Sphere Form Factor — the mathematical function that describes how a sphere of radius \(R\) scatters X-rays as a function of \(q\).