About

Bragg's Law describes the condition for constructive interference of X-rays scattered by crystal lattice planes. The relationship nλ = 2d sin(θ) connects the integer diffraction order n, the incident wavelength λ, the interplanar spacing d, and the glancing angle θ. An error of 0.01 Å in d propagates directly into lattice parameter determination, producing incorrect unit cell dimensions and potentially misidentified crystal phases. This calculator solves for any single unknown given the remaining three parameters. It enforces physical constraints: sin(θ) ≤ 1, positive d and λ, and integer n ≥ 1. Results assume elastic scattering with negligible absorption and a perfect infinite crystal approximation. Real samples introduce peak broadening described by the Scherrer equation, which this tool does not address.

Formulas

The Bragg condition for constructive interference from a set of parallel lattice planes is:

nλ = 2d sin(θ)

Solving for each unknown individually:

λ = 2d sin(θ)n

d = nλ2 sin(θ)

θ = arcsin(nλ2d)

n = 2d sin(θ)λ

The maximum observable diffraction order is:

n_max = floor(2dλ)

Where n = diffraction order (positive integer), λ = wavelength of incident radiation in Å, d = interplanar spacing in Å, and θ = Bragg angle (glancing angle between incident beam and lattice plane) in °. Note that 2θ is the angle typically measured on a diffractometer. The conversion factor is 1 Å = 10⁻¹⁰ m = 0.1 nm.

Reference Data

Radiation Source	Target / Line	Wavelength (Å)	Energy (keV)	Common Use
X-ray tube	Cu Kα₁	1.54056	8.048	General powder XRD
X-ray tube	Cu Kα₂	1.54439	8.028	Unfiltered XRD
X-ray tube	Cu Kβ	1.39222	8.905	Secondary line
X-ray tube	Mo Kα₁	0.70930	17.479	Single crystal diffraction
X-ray tube	Mo Kα₂	0.71359	17.374	Unfiltered Mo source
X-ray tube	Co Kα₁	1.78897	6.930	Iron-rich samples
X-ray tube	Co Kα₂	1.79285	6.915	Unfiltered Co source
X-ray tube	Fe Kα₁	1.93604	6.404	Specialized studies
X-ray tube	Cr Kα₁	2.28970	5.415	Stress analysis, large d
X-ray tube	Ag Kα₁	0.55941	22.163	High-energy diffraction
X-ray tube	W Lα₁	1.47642	8.398	Heavy-element analysis
Synchrotron	Tunable	0.1 - 5.0	2.5 - 124	High-resolution, anomalous
Neutron reactor	Thermal neutrons	1.0 - 2.5	0.033 - 0.082	Magnetic structures, H detection
Electron gun	Electrons (100 kV)	0.0370	100	TEM diffraction
Electron gun	Electrons (200 kV)	0.0251	200	HRTEM / SAED
Common crystal d-spacings for calibration
Si (111)	-	d = 3.1355 Å		Standard reference
Si (220)	-	d = 1.9201 Å		Standard reference
Si (311)	-	d = 1.6375 Å		Standard reference
NaCl (200)	-	d = 2.8200 Å		Teaching / calibration
LaB₆ (110)	-	d = 2.9388 Å		NIST SRM 660
Al₂O₃ (012)	-	d = 3.4790 Å		Corundum standard
Quartz (101)	-	d = 3.3434 Å		Geological reference

Frequently Asked Questions

The Bragg angle θ is the glancing angle between the incident beam and the crystal plane. Diffractometers measure 2θ, the angle between the incident and diffracted beams. This calculator uses θ (the Bragg angle). To convert from a diffractometer reading, divide the measured 2θ value by 2 before entering it.

Bragg's Law requires sin(θ) ≤ 1. When solving for θ, if nλ2d exceeds 1, the diffraction condition cannot be satisfied. This occurs when the wavelength is too long relative to the lattice spacing for the requested order. Reduce n, use shorter wavelength radiation, or verify your d-spacing value.

Thermal expansion changes the lattice parameter a and therefore the interplanar spacing d. For metals, typical linear thermal expansion coefficients are 10 - 25 ×10⁻⁶ K⁻¹. A 100 K temperature change shifts d by approximately 0.1 - 0.25%, producing measurable peak shifts in high-resolution diffractometers. Additionally, the Debye-Waller factor reduces peak intensity at elevated temperatures due to increased atomic thermal vibration.

Yes. Bragg's Law is universal for all wave-particle scattering. For neutrons, the de Broglie wavelength of thermal neutrons (1.0 - 2.5 Å) falls conveniently in the range of crystal lattice spacings. For electrons at 100 kV, λ ≈ 0.037 Å, requiring very small Bragg angles. Enter the appropriate wavelength for your radiation source.

The order n represents a set of virtual planes with spacing dn. In practice, higher-order reflections (e.g., n = 2 from (111) planes) are equivalent to the first-order reflection from (222) planes. Most XRD analysis sets n = 1 and uses Miller indices (hkl) to represent higher orders. This calculator supports arbitrary n for educational and specialized applications.

The conversion is 1 nm = 10 Å. Cu Kα₁ radiation at 1.54056 Å equals 0.154056 nm. This calculator accepts wavelength in Å by default, which is the conventional unit in crystallography. Use the unit toggle to switch between Å and nm.