Bragg's Law Calculator

Understanding Bragg's Law

Bragg's Law describes the fundamental relationship governing X-ray diffraction from crystalline materials. When X-rays interact with regularly spaced atomic planes, constructive interference occurs only when the path difference between waves reflected from adjacent planes equals an integer multiple of the wavelength.

When to Use This Calculator

• Phase identification: Convert peak positions (2θ) to d-spacings for database matching
• Lattice parameter determination: Calculate d-spacings to extract unit cell dimensions
• Peak prediction: Determine expected 2θ positions for known crystal structures
• Experimental planning: Check if reflections are accessible with your X-ray source

d-Spacing and Crystal Structure

The interplanar spacing (d) depends on the crystal system and Miller indices (hkl):

• Cubic: d = a / √(h² + k² + l²)
• Tetragonal: 1/d² = (h² + k²)/a² + l²/c²
• Hexagonal: 1/d² = (4/3)(h² + hk + k²)/a² + l²/c²
• Orthorhombic: 1/d² = h²/a² + k²/b² + l²/c²

Energy vs. Wavelength

X-ray wavelength and energy are related by: λ (Å) = 12.398 / E (keV). This calculator accepts either input format. Synchrotron beamlines typically specify energy in keV, while laboratory sources are characterized by wavelength.

Practical Considerations

• 2θ vs θ: Diffractometers typically display 2θ (detector angle). The Bragg angle θ is half this value. This calculator accepts both formats.
• Zero-point error: Experimental 2θ values may have systematic offset. Calibrate with a standard (Si, LaB₆) for precise d-spacing determination.
• Kα splitting: At high 2θ (>~70°), Kα₁/Kα₂ doublet splitting becomes visible. Use Kα₁ for precise work.
• Accessible range: Most laboratory diffractometers cover 2θ = 10–140°. Some reflections may be inaccessible with certain X-ray sources.

Diffraction Order (n)

While Bragg's Law includes diffraction order n, in practice n=1 is almost always used. Higher-order reflections (n=2, 3...) appear at the same position as lower-order reflections from planes with smaller d-spacings. For example, the (200) reflection is conventionally reported rather than second-order (111).

Common X-ray Sources

The choice of X-ray source affects accessible d-spacing range and resolution:

• Cu Kα (λ = 1.5406 Å): Most common for general crystallography. Good balance of penetration and resolution.
• Mo Kα (λ = 0.7107 Å): Higher penetration, useful for absorbing samples and single-crystal work.
• Co Kα (λ = 1.7890 Å): Reduces fluorescence from Fe-containing samples.
• Synchrotron: Tunable wavelength, high intensity for specialized applications.