X-ray Optics Materials
X-ray refractive index n = 1 − δ − iβ of Be, diamond, Si, Ni, Cu, Mo, Pt, Au, Kapton and Si₃N₄ from 1 to 30 keV, from the Henke scattering factors: critical angle, attenuation length, μ/ρ and transmission, with absorption edges.
Choosing a material fills its density. Thin films and membranes are often less dense than the bulk: enter a measured value.
Decrement δ
3.4077e-6
n = 1 − δ − iβ at λ = 0.12398 nm
Absorption index β
1.0279e-9
β/δ = 3.017e-4, Compton 22.4 %
Critical angle θ_c
2.6106 mrad
0.1496° grazing, √(2δ)
Attenuation length
9598 µm
intensity 1/e, λ/(4πβ)
Mass attenuation μ/ρ
0.5638 cm²/g
photoabsorption + Compton
Transmission
98.96 %
through 100.0 µm
Model
δ = (r_e λ²/2π) Σ n_q f₁,q, β = (r_e λ²/2π) Σ n_q f₂,q + (λ/4π) Σ n_q σ_inc,q
θ_c = √(2δ), μ = 4πβ/λ, T = exp(−μt)
f₁ and f₂ are the Henke scattering factors (CXRO tables, 1–30 keV) of independent atoms, so values within about 50 eV of an absorption edge depend on the chemical state and are only approximate. β adds Compton (incoherent) scattering to photoabsorption, with the NIST XCOM cross-sections σ_inc (Hubbell et al. 1975), as CXRO does: at 10 keV it is 22 % of β for beryllium and 6 % for carbon, at 30 keV 93 % for beryllium and 12 % for silicon. Rayleigh (coherent) scattering is left out, as in CXRO: in a narrow-beam measurement it would add 18 % to beryllium's attenuation and 2 % to silicon's at 10 keV, about 10 % for both at 30 keV.
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
δ, β = (r_e λ²/2π) Σ n_q f₁,₂ with the Henke f₁, f₂ of independent atoms (CXRO tables, interpolated within 0.2 %); θ_c = √(2δ), μ = 4πβ/λ. Approximate within ~50 eV of an edge. β adds Compton scattering, (λ/4π) Σ n_q σ_inc,q with the NIST XCOM σ_inc (Hubbell et al. 1975), as CXRO does (within 0.2 % of CXRO's β). Rayleigh scattering is left out, as in CXRO: it would add 18 % to Be's narrow-beam attenuation at 10 keV and about 10 % to Be's and Si's at 30 keV.
- Henke B. L., Gullikson E. M., Davis J. C. (1993). X-ray interactions: photoabsorption, scattering, transmission, and reflection at E = 50–30,000 eV, Z = 1–92. At. Data Nucl. Data Tables 54, 181–342. doi:10.1006/adnd.1993.1013
- CXRO X-ray Database (Lawrence Berkeley National Laboratory). Atomic scattering factor files (sf/<element>.nff) and index-of-refraction calculator; default densities. henke.lbl.gov
- Kaloyeros A. E., Pan Y., Goff J., Arkles B. (2020). Review—Silicon nitride and silicon nitride-rich thin film technologies: state-of-the-art processing technologies, properties, and applications. ECS J. Solid State Sci. Technol. 9, 063006. Table I: bulk Si₃N₄ 3.10–3.20 g/cm³ (α 3.184), CVD films ≈ 2.8; the Si₃N₄ preset uses 3.18, not CXRO's 3.44. doi:10.1149/2162-8777/aba447
- Hubbell J. H., Veigele W. J., Briggs E. A., Brown R. T., Cromer D. T., Howerton R. J. (1975). Atomic form factors, incoherent scattering functions, and photon scattering cross sections. J. Phys. Chem. Ref. Data 4, 471–538. doi:10.1063/1.555523
- Berger M. J., Hubbell J. H., Seltzer S. M. et al. XCOM: Photon Cross Sections Database (NIST Standard Reference Database 8). Incoherent scattering cross-sections, queried 2026-11-03. physics.nist.gov
- Als-Nielsen J., McMorrow D. (2011). Elements of Modern X-ray Physics, 2nd ed., Wiley. Ch. 3: refractive index and critical angle. doi:10.1002/9781119998365