Learn / Diffraction

Bragg’s law explorer.

Discover how families of planes arise from a repeating lattice and why their spacing determines a Bragg diffraction angle.

Jump to the interactive activity ↓

THE CONCEPT / WHY DIFFRACTION HAPPENS

What is Bragg's law?

When X-rays encounter a repeating crystal structure, waves scattered from neighboring lattice planes can reinforce one another. Bragg's law gives the angles at which that constructive interference is possible.

nλ = 2d sin θ

λ is the X-ray wavelength; d is the distance between planes; θ is the angle between the incoming beam and those planes; n is the diffraction order. A powder diffractometer typically reports 2θ, the angle between the incident and diffracted beams.

Why this matters

Smaller d → larger 2θ, at the same wavelength.
Longer λ → larger 2θ, at the same plane spacing, while the Bragg condition remains possible.
PXRD connection: measured peak positions let researchers estimate plane spacings. A shift toward higher 2θ corresponds to smaller d, for the same indexed reflection and wavelength.

Incoming rays are parallel; diffracted rays are parallel. Scattering points are offset intentionally. The diagram is schematic and not to scale.

Explore the lattice

LIVE DIFFRACTION ANGLE · 2θ 22.21° d = 4.00 Å See the full diagrams ↓

GUIDED ACTIVITY · SQUARE LATTICE

Change the lattice spacing or select a plane family. Watch the diffraction angle respond.

1 · Lattice parameter, a (Å) a is the repeat distance between adjacent blue dots. Move the slider beside the lattice diagram to watch it expand or contract.

2 · Highlight a plane family A two-dimensional square-lattice example; the dots are lattice points, not individual atoms.

Wavelength preset

Wavelength λ (Å)Verify the wavelength and optics used for your experiment.

Advanced calculation · input d, 2θ or q

Use this to interpret a diffraction measurement or explore a known plane spacing. Enter d, 2θ, or q at a chosen X-ray wavelength.

Known quantity

Why enter a known value?

Predict where a reflection could occur if you know the plane spacing d.

Plane spacing d (Å)

Diffraction order nStart with n = 1. For higher orders, d remains the selected plane spacing.

WATCH THE RELATIONSHIP CHANGE

From lattice spacing to diffraction angle

nλ = 2d sin θ

Change a Plane spacing d Diffraction angle 2θ

Move the lattice slider to see how changing the repeat distance shifts the diffraction angle.

Plane spacing · d—Å

Bragg angle · θ—degrees to lattice plane

Diffraction angle · 2θ—degrees · possible reflection position

1 · The repeating latticeStructure / spacing

Adjust lattice parameter a4.00 Å

Blue a · lattice repeatRed d · plane spacing

2 · The diffraction angleθ — · 2θ —

Blue indicates the incident beam; red indicates a diffracted direction. The red arc labeled 2θ expands or contracts as the Bragg angle changes. Ray geometry is schematic.

What changed?

Smaller plane spacing produces a larger Bragg angle at a fixed wavelength.

WHY SOME DIRECTIONS REINFORCE AND OTHERS DO NOT

What happens away from the Bragg angle?

Scattered waves are still present away from the Bragg condition, but they generally arrive with different phases. Compare two equal-amplitude contributions from neighboring planes: they add when their extra path is a whole number of wavelengths, and cancel when that path difference is a half-integer number of wavelengths.

Bragg condition satisfied Constructive · in phase

2θ = —Extra path = nλ

Successive contributions reinforce one another in this direction. This is a possible diffraction-peak position, not a predicted peak intensity.

An off-Bragg direction Destructive · half-cycle offset

2θ = —Extra path = (n − ½)λ

At this specifically chosen off-Bragg angle, two ideal equal-amplitude contributions cancel. That does not mean real crystals scatter zero intensity at all off-Bragg angles.

Scientific scope: this is a two-wave phase illustration, not a simulated diffraction pattern. Real crystalline scattering depends on many planes, finite-size broadening, atomic structure factors, and the instrument. IUCr explanation of phase and Bragg diffraction.

Model assumptions, limitations & sources

What the models represent: the guided lattice is an ideal 2D square lattice, with d(10) = a and d(11) = a/√2. Dots are lattice points, not atoms or a specific MOF. The wave comparison adds only two equal-amplitude contributions.

What the models cannot determine: peak intensity, phase identity, structural factors, systematic absences, full diffraction patterns, or experimental uncertainty. Real diffraction arises from scattering and interference, not literal mirror reflections from atomic planes.

Measurements: wavelength presets are rounded values; detector calibration, instrumental effects and uncertainty propagation are not included. Here q = 4πsinθ/λ = 2πn/d, with q in Å⁻¹.

Reference: IUCr Online Dictionary of Crystallography — Bragg's law.