Bragg's Law Calculator
Calculate X-ray diffraction angles using Bragg's Law nλ = 2d sin θ. Free online calculator for X-ray crystallography, XRD analysis, and materials science with charts.
About This Calculator
Bragg's Law Calculator helps you compute the diffraction angle of X-rays incident on a crystalline material. This tool is essential for students, researchers, and professionals in X-ray crystallography, materials science, solid-state physics, and structural biology. The calculator uses the fundamental Bragg equation nλ = 2d sin θ to determine the angle at which constructive interference occurs between X-rays scattered from adjacent atomic planes.
The formula nλ = 2d sin θ relates four key parameters: n, the order of diffraction (a positive integer); λ, the wavelength of incident X-rays in picometers (pm); d, the interplanar spacing in the crystal lattice in pm; and θ, the Bragg angle — the angle between the incident X-ray beam and the crystal plane. The calculator computes θ from the other three inputs and also determines the maximum possible diffraction order n_max = ⌊2d/λ⌋ for the given crystal and wavelength.
How to Use
Enter the X-ray wavelength λ (pm), the interplanar spacing d (pm), and the diffraction order n (integer ≥ 1). Click "Calculate Bragg Angle" to compute the Bragg angle θ in degrees and radians, along with sin(θ) and the maximum possible order. The breakdown table shows all computed values, and the bar chart displays the Bragg angles for every valid diffraction order from n = 1 up to n_max.
Applications of Bragg's Law
Bragg's Law is fundamental to X-ray crystallography, the primary method for determining atomic structures of crystalline materials. It has been used to discover the double-helix structure of DNA, determine protein structures for drug development, characterize new materials and minerals, and analyze stress and strain in engineering materials. The same principle applies to neutron diffraction and electron microscopy techniques.
Frequently Asked Questions
What is Bragg's Law?
Bragg's Law, formulated by William Henry Bragg and William Lawrence Bragg, describes the condition for constructive interference of X-rays scattered from crystal planes. The equation nλ = 2d sin θ relates the X-ray wavelength λ, the interplanar spacing d, the diffraction order n, and the Bragg angle θ. This law is fundamental to X-ray crystallography and materials science.
How does X-ray diffraction work?
In X-ray diffraction, incident X-rays scatter off atoms in a crystalline material. When the scattered waves from different atomic layers interfere constructively, a diffraction peak occurs. Bragg's Law predicts the angles at which these peaks appear, allowing scientists to determine the crystal structure by analyzing the diffraction pattern.
What is the maximum diffraction order?
The maximum diffraction order n_max is the largest integer satisfying nλ < 2d, or equivalently n_max = floor(2d/λ). Since sin θ cannot exceed 1, the condition nλ/(2d) ≤ 1 limits the observable orders. For example, with λ = 120 pm and d = 280 pm, n_max = 4, giving Bragg angles of 12.37°, 25.38°, 40.01°, and 59.00°.
What units should I use for wavelength and distance?
You must use consistent units for both wavelength and interplanar distance. Common choices are picometers (pm), ångströms (Å), or nanometers (nm). The calculator uses picometers by default. For reference, typical X-ray wavelengths are around 100 pm and crystal lattice spacings range from 100 to 500 pm.
What is X-ray crystallography used for?
X-ray crystallography is used to determine the three-dimensional atomic structure of crystals. It has been instrumental in discovering the structure of DNA, proteins, viruses, minerals, and advanced materials. The technique works by analyzing the diffraction pattern produced when X-rays pass through a crystalline sample and applying Bragg's Law to interpret the atomic positions.
Can I use this calculator for neutron or electron diffraction?
Yes, Bragg's Law applies to any wave phenomenon that undergoes diffraction by a periodic lattice. Neutron diffraction and electron diffraction follow the same equation nλ = 2d sin θ. However, the wavelength must be calculated from the particle's momentum using the de Broglie relation λ = h/p for neutrons and electrons.
Who discovered Bragg's Law?
Bragg's Law was discovered by British physicists William Henry Bragg and his son William Lawrence Bragg in 1912. They were jointly awarded the Nobel Prize in Physics in 1915 for their pioneering work in X-ray crystallography. Lawrence Bragg, at age 25, remains the youngest Nobel laureate in science.