Bragg's Law Calculator

Solve Bragg's Law, nλ = 2d sin θ, for the diffraction angle given the X-ray wavelength, crystal plane spacing, and reflection order.

Quick Facts

Formula
nλ = 2d sin θ
Constructive interference occurs only when the path difference between reflected waves is a whole number of wavelengths.
Common source
Cu Kα radiation: λ ≈ 1.5406 Å
The default wavelength for most laboratory powder X-ray diffractometers.
Physical limit
sin θ ≤ 1
Since sin θ cannot exceed 1, only orders with nλ ≤ 2d produce a reflection.

Your Results

Calculated
Bragg angle (θ)
-
Angle from the crystal plane
Diffraction angle (2θ)
-
Angle read on a diffractometer
sin θ = nλ / 2d
-
Ratio solved for θ
Maximum order (n_max)
-
Highest integer n with sin θ ≤ 1

Ready

Enter the wavelength, d-spacing, and order, then press Calculate.

Understanding Bragg's Law

Bragg's Law describes how X-rays reflect off the parallel atomic planes inside a crystal. When radiation of wavelength λ strikes a set of lattice planes spaced d apart at an angle θ, each plane reflects a small fraction of the beam. Those reflected waves interfere constructively — reinforcing rather than cancelling out — only when the extra distance travelled by the wave reflecting off the deeper plane is a whole number of wavelengths. That condition is written as:

nλ = 2d sin θ

What each symbol means

  • n — the order of reflection, a positive integer (1, 2, 3…) describing how many whole wavelengths fit the path difference.
  • λ — the wavelength of the incident radiation, typically X-rays, in angstroms (Å) or nanometers.
  • d — the spacing between the parallel crystal lattice planes doing the reflecting, in the same length unit as λ.
  • θ — the Bragg angle, measured between the incident beam and the crystal plane itself (not the surface normal).

Solving for the angle

Given λ, d, and n, the Bragg angle is found by rearranging the law: θ = arcsin(nλ / 2d). This calculator computes θ directly, along with 2θ — the angle between the incoming and outgoing beam that a diffractometer detector actually measures — and the ratio sin θ = nλ/2d used to get there.

Why there is a maximum order

Because sine can never exceed 1, a reflection can only exist while nλ/2d ≤ 1, i.e. nλ ≤ 2d. For a fixed wavelength and spacing, this puts a hard ceiling on which integer orders are physically observable — the calculator reports this as the maximum order n_max = floor(2d/λ). Push n past that ceiling and no real angle satisfies the equation; the calculator flags it as impossible rather than returning a number.

Typical X-ray wavelengths

  • Cu Kα (copper): λ ≈ 1.5406 Å — the most common laboratory powder-diffraction source.
  • Mo Kα (molybdenum): λ ≈ 0.7107 Å — shorter wavelength, used for small-molecule single-crystal work.
  • Co Kα (cobalt): λ ≈ 1.7902 Å — favored for iron-containing samples to reduce fluorescence.
  • Cr Kα (chromium): λ ≈ 2.2909 Å — used for large unit cells and residual-stress measurements.

Frequently Asked Questions

What is Bragg's Law?
Bragg's Law, nλ = 2d sin θ, is the condition for constructive interference when a wave reflects off evenly spaced crystal planes. n is the integer order, λ is the wavelength, d is the spacing between planes, and θ is the angle between the incident beam and the plane. When the equation holds, reflected waves from successive planes stay in phase and produce a detectable diffraction peak.
What is the difference between θ and 2θ?
θ, the Bragg angle, is measured between the incoming beam and the crystal plane. 2θ is the angle between the incoming beam and the outgoing (diffracted) beam — the angle an X-ray diffractometer detector actually scans and records. The two are simply related: 2θ is always twice θ.
Why does Bragg's Law have a maximum diffraction order?
Because sin θ cannot be greater than 1, the equation nλ = 2d sin θ only has a solution while nλ ≤ 2d. Dividing through gives the largest usable order, n_max = floor(2d / λ). Any integer order above that would require sin θ greater than 1, which is impossible, so no reflection occurs at that order.
What wavelength should I use?
Use the wavelength of your actual radiation source. Laboratory X-ray diffractometers most commonly use copper Kα radiation at about 1.5406 Å; molybdenum (about 0.7107 Å), cobalt (about 1.7902 Å), and chromium (about 2.2909 Å) sources are also common depending on the sample. Keep the wavelength and d-spacing in the same length units.

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