X-Ray Refractive Index Calculator

Calculator Inputs

Common Material Preset

Presets fill chemistry values only. Enter μ/ρ for your chosen energy.

Material Name

X-Ray Energy (keV)

Density (g/cm³)

Molar Mass (g/mol)

Electrons per Molecule / Formula Unit

Mass Attenuation Coefficient μ/ρ (cm²/g)

Sample Thickness (mm)

μ/ρ Energy Exponent m

Sweep Minimum Energy (keV)

Sweep Maximum Energy (keV)

Sweep Points

Formula Used

1) X-ray wavelength
λ (Å) = 12.3984198433 / E(keV)

2) Electron density
ρ_e = ρ × N_A × Z_eff / M

3) Real decrement
δ = r_e λ² ρ_e / (2π)

4) Absorption term
μ = (μ/ρ) × ρ
β = μ λ / (4π)

5) Complex refractive index
n = 1 − δ − iβ

6) Extra outputs
θ_c ≈ √(2δ) | Attenuation length = 1/μ | Transmission = e^−μt

This page uses the common X-ray optics sign convention with a negative imaginary term. If your source uses a different sign convention, δ and β magnitudes stay the same, but the printed complex sign can differ.

How to Use This Calculator

Enter a material name so your output and downloads stay labeled.
Fill density, molar mass, and electrons per molecule or formula unit.
Enter the X-ray energy and the mass attenuation coefficient at that same energy.
Add thickness if you want transmission, attenuation length, and phase-shift context.
Set the energy sweep range and points to inspect how δ, β, and transmission trend.
Submit the form. The result appears above this form, directly below the header.

Example Data Table

Illustrative inputs only. Replace μ/ρ with tabulated values for your exact chemistry and energy.

Material	Energy (keV)	Density (g/cm³)	Molar Mass (g/mol)	Electrons / Unit	μ/ρ (cm²/g)	Thickness (mm)
Quartz (SiO₂)	8.0	2.65	60.0843	30	9.20	1.00
Water	10.0	1.00	18.0153	10	5.33	5.00
Alumina (Al₂O₃)	12.0	3.95	101.9613	50	3.80	0.75

Frequently Asked Questions

1) Why is the X-ray refractive index usually below one?

For X-rays, the real part is commonly written as 1 − δ. Since δ is positive and very small, the real part becomes slightly less than one.

2) What does delta represent?

Delta is the refractive index decrement. It controls phase behavior, critical angle, and how strongly a material changes the X-ray wavefront.

3) What does beta represent?

Beta is the absorption-related part of the complex index. Larger β generally means stronger attenuation and lower transmission through the sample.

4) Why do I need electrons per molecule?

The calculator converts composition into electron density. That value is needed to estimate δ from density, molar mass, and chemical makeup.

5) Why do I still need μ/ρ?

Beta and transmission depend on attenuation. The easiest manual route is entering a mass attenuation coefficient for your exact material and energy.

6) Is the energy sweep exact for beta?

No. The graph uses your reference μ/ρ and a power-law exponent for practical trend checking. Exact studies should use tabulated attenuation data across the sweep.

7) What is the critical angle output for?

It helps when estimating total external reflection behavior in X-ray optics, grazing-incidence setups, and mirror or surface interaction problems.

8) Which units are expected?

Use keV for energy, g/cm³ for density, g/mol for molar mass, cm²/g for μ/ρ, and millimeters for thickness. Outputs are labeled clearly.