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Physics Background

This page describes the physical models and equations implemented in XRayLabTool.

Refractive index for X-rays

For X-rays, the complex refractive index of a material is:

\[n = 1 - \delta - i\beta\]

where δ (dispersion) and β (absorption) are both small positive numbers (typically 10⁻⁵ to 10⁻⁸).

Optical constants δ and β

The dispersion and absorption coefficients are calculated from:

\[\delta = \frac{\lambda^2}{2\pi} r_e \frac{\rho N_A}{M} \sum_i n_i f_{1,i}(E)\]

\[\beta = \frac{\lambda^2}{2\pi} r_e \frac{\rho N_A}{M} \sum_i n_i f_{2,i}(E)\]

where:

  • $\lambda$ — X-ray wavelength (m)
  • $r_e = 2.818 \times 10^{-15}$ m — classical electron radius (Thomson scattering length)
  • $\rho$ — mass density (g/cm³)
  • $N_A$ — Avogadro's number
  • $M$ — molecular weight (g/mol)
  • $n_i$ — number of atoms of element $i$ per formula unit
  • $f_{1,i}(E), f_{2,i}(E)$ — energy-dependent atomic scattering factors for element $i$

Atomic scattering factors

The scattering factors $f_1$ and $f_2$ describe how an atom scatters X-rays. They are tabulated as functions of photon energy and show sharp features at absorption edges.

At high energies far from absorption edges, $f_1 \to Z$ (the atomic number), meaning the atom scatters like $Z$ free electrons.

The data tables are from the CXRO (Center for X-ray Optics) Henke tables, covering energies from 30 eV to 30 keV for elements H through U.

Derived quantities

Energy–wavelength conversion

\[\lambda [\text{Å}] = \frac{12.39842}{E [\text{keV}]}\]

Critical angle for total external reflection

\[\theta_c = \sqrt{2\delta} \quad (\text{radians})\]

Attenuation length

The 1/e intensity attenuation length:

\[L = \frac{\lambda}{4\pi\beta}\]

Scattering length density (SLD)

\[\text{SLD} = \frac{2\pi}{\lambda^2} (\delta + i\beta)\]

SLD is energy-independent to first order (the $\lambda^2$ in δ and β cancels with the $1/\lambda^2$ prefactor), making it useful for reflectometry.

Electron density

\[\rho_e = \frac{\rho N_A Z_{mol}}{M}\]

where $Z_{mol}$ is the total number of electrons per formula unit.

Physical constants

The package uses the following constants (CODATA values):

ConstantSymbolValue
Thomson scattering length$r_e$2.8179403227 × 10⁻¹⁵ m
Speed of light$c$299792458 m/s
Planck constant$h$6.626068 × 10⁻³⁴ J·s
Elementary charge$e$1.60217646 × 10⁻¹⁹ C
Avogadro's number$N_A$6.02214199 × 10²³ mol⁻¹

References