Bragg Diffraction

Bragg's Law calculator for crystallography: relates wavelength, interplanar distance, and diffraction angle.

Input parameters

What do you want to calculate?

Diffraction order (n)

Integer representing the order of the reflection.

110

Grazing angle (θ)

Calculated automatically.

Interplanar distance (d)

Perpendicular distance between adjacent atomic planes.

0.520

Wavelength (λ)

Wavelength of the incident beam (e.g. X-rays).

0.110

Results

Bragg Angle (θ)

22.64

Detector angle (2θ)

45.29

Photon Energy (E)

8.051

keV

Path Diff / plane (d·sinθ)

0.77

Total Path Diff (2d·sinθ)

1.54

Crystal and Beam Visualization (Cross Section)

Fundamentals & Explanation

What is Bragg's Law?

Bragg's Law was formulated by William Lawrence Bragg and William Henry Bragg in 1913, a discovery for which they were awarded the Nobel Prize in Physics in 1915. It describes the fundamental conditions under which an X-ray, electron, or neutron beam interacts with the periodic structure of a crystal to produce constructive interference.

Unlike classical geometric optics, the Bragg angle ( $\theta$ ) is not measured from the surface normal, but from the crystallographic plane itself (it is a grazing angle).

Interference and the Main Equation

Constructive interference (and thus, the observation of a diffraction peak) occurs only when the optical path difference between rays scattered by adjacent atomic planes is equal to an integer multiple of the incident wavelength.

Bragg's Equation

n \lambda = 2d \sin(\theta)

$n$ : Diffraction order (positive integer: 1, 2, 3...).
$\lambda$ : Wavelength of the incident beam (typically in Å or nm).
$d$ : Interplanar distance (atomic spacing) between crystallographic planes.
$\theta$ : Grazing angle of incidence (Bragg angle).

Calculation Analysis and Variables

The calculator allows you to solve Bragg's equation by isolating the variable of interest:

Solving for Spacing (d)

d = \frac{n \lambda}{2 \sin(\theta)}

Solving for Angle (θ)

\theta = \arcsin\left(\frac{n \lambda}{2d}\right)

Analytical Consideration: When calculating $\theta$ , the equation reveals the non-linearity of diffraction. Small changes in $d$ at high angles produce significantly larger angular shifts in the spectrum (higher dispersion), which makes high-angle $2\theta$ peaks crucial for accurate unit cell determination. The relative error in d is proportional to cot(θ), which tends to zero as θ → 90°, maximizing precision.

Crystallography and Energy (XRD)

In X-ray diffraction, the wavelength $\lambda$ (Å) is inversely related to the photon energy $E$ (keV) through the Planck-Einstein approximation:

E = \frac{hc}{\lambda}

hc = 12398.4 \text{ eV}\cdot\text{\AA} \approx 12.398 \text{ keV}\cdot\text{\AA}

Standard example: The characteristic $K\alpha$ emission line of Copper (used in most powder diffractometers) has $\lambda \approx 1.5406$ Å, which corresponds to a photon energy of ~8.05 keV.

Geometric Diffraction Limit

Since the mathematical sine function is bounded between -1 and 1, Bragg's Law imposes an unavoidable physical constraint:

\sin(\theta) \le 1 \implies n\lambda \le 2d

If the wavelength used is greater than twice the interplanar spacing of the crystal ( $\lambda > 2d$ ), it is mathematically and physically impossible to observe diffraction for that plane. For an order n, the condition generalizes to $n\lambda \le 2d$ .

Typical Sources

Cu Kα: 1.5406 Å

Mo Kα: 0.7107 Å

Co Kα: 1.78897 Å

Ag Kα: 0.5594 Å

Conversions

1 Å = 10⁻¹⁰ m

1 nm = 10 Å

1 pm = 10⁻² Å

Academic References

[1] Bragg, W. L. (1913). The Diffraction of Short Electromagnetic Waves by a Crystal. Proceedings of the Cambridge Philosophical Society, 17, 43-57. (Original publication of Bragg's Law).
[2] Kittel, C. (2004). Introduction to Solid State Physics(8th ed.). John Wiley & Sons. (Fundamentals of reciprocal lattices and crystal diffraction).
[3] Cullity, B. D., & Stock, S. R. (2001). Elements of X-Ray Diffraction (3rd ed.). Prentice Hall. (Canonical text for practical analysis and diffraction considerations in materials).

Bragg Diffraction

Input parameters

Results

Fundamentals & Explanation

What is Bragg's Law?

Interference and the Main Equation

Calculation Analysis and Variables

Crystallography and Energy (XRD)

Geometric Diffraction Limit

Typical Sources

Conversions

Academic References

You may also like:

What is Bragg's Law?

Interference and the Main Equation

Calculation Analysis and Variables

Crystallography and Energy (XRD)

Geometric Diffraction Limit

Typical Sources

Conversions

Academic References