Bragg's Law — Physica

Equivalent forms

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

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

One path-difference equation links the angle, the wavelength, and the spacing of atoms — the foundation of all crystallography.

Symbol	Name	SI unit	Dimension	Range
$n$	Diffraction order Integer order of the Bragg reflection	dimensionless	1	1 – 5
$d$	Interplanar spacing Distance between adjacent crystal lattice planes	m	L	0.1 – 1
$theta$	Glancing angle Angle between the incident beam and the crystal planes	rad	1	0.087 – 1.4
$lambda$	Wavelengthoutput Wavelength of the diffracted radiation	m	L	0.01 – 1

Unit systems

Symbol	SI	CGS	Imperial
$n$	dimensionless	dimensionless	dimensionless
$d$	m	cm	in
$theta$	rad	rad	deg
$lambda$	m	cm	in

Where it holds

Valid for elastic (Rayleigh) scattering of X-rays, neutrons, or electrons from a periodic lattice with sharp planes. Assumes negligible refraction (index near 1 for X-rays) and a wavelength comparable to or smaller than the interplanar spacing.

Discovery

W. H. Bragg and W. L. Bragg · 1913

The father-and-son Braggs realized that the sharp X-ray reflections von Laue had seen from crystals obeyed a simple path-difference rule. Their law turned crystals into rulers for X-ray wavelengths and X-rays into probes of crystal structure, earning them the 1915 Nobel Prize.

Try this

How did scientists 'see' the double-helix shape of DNA without any microscope powerful enough to resolve atoms?

X-rays of unknown wavelength reflect strongly from crystal planes spaced 0.28 nm apart at a glancing angle of 30 degrees in first order. What is the X-ray wavelength?

Research status: stable

Real-world applications

X-ray crystallography solved the structures of DNA, proteins, and countless materials.
Powder diffraction identifies crystalline phases in geology, pharmaceuticals, and metallurgy.
Bragg mirrors and crystal monochromators select narrow X-ray bands at synchrotrons.
Neutron and electron diffraction map magnetic and surface structures using the same condition.

Common misconceptions

theta is measured from the surface normal — in Bragg's law it is the glancing angle from the plane itself.
X-rays reflect off individual atoms like a mirror — the sharp peaks come from coherent addition across many planes, not single-atom reflection.
Any wavelength can diffract from any crystal — only wavelengths shorter than 2d can satisfy the condition.

Experimental verification

Rotating a single crystal in a monochromatic X-ray beam produces sharp reflections exactly at the angles predicted by Bragg's law; powder diffraction gives rings whose radii encode the d-spacings, routinely matched to sub-percent accuracy.

Derivation

Consider two parallel rays reflecting from adjacent planes separated by d.

The lower ray travels an extra distance d*sin(theta) entering and d*sin(theta) leaving, a total path difference of 2*d*sin(theta).

Constructive interference requires this to equal an integer number of wavelengths:

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

.

Limiting cases

\lambda > 2d⟶ No diffractionsin(theta) cannot exceed 1, so wavelengths longer than twice the spacing produce no Bragg peak — why visible light cannot resolve atomic planes.

\theta \to 90^\circ⟶ n\lambda \to 2dBack-reflection (normal incidence to the planes) gives the maximum usable path difference of 2d per order.

What if…

What if you use second-order reflection

(n = 2)

?

For the same planes and wavelength, sin(theta) doubles, so the reflection appears at a larger glancing angle — successive orders fan out, letting you index the lattice.

What if you shine visible light

(\lambda \sim 500\,\mathrm{nm})

on the crystal?

Since 500 nm far exceeds $2d (\sim 0.56\,\mathrm{nm})$ , sin(theta) would need to exceed 1 — impossible. That is why atomic structure demands X-rays, not visible light.

1

Finding the X-ray wavelength

Given ·

n:: 1
d:: 0.28
theta:: 0.5236

Find · lambda

Steps

Rearrange Bragg's law: $\lambda = 2*d*\sin (\theta )/n$ .
$\sin (30\,\mathrm{deg}) = 0.5$ .
$\lambda = 2*0.28*0.5/1 = 0.28\,\mathrm{nm}$ .

Result ·

\lambda = 2*d*\sin (\theta )/n = 2*0.28*\sin (30\,\mathrm{deg})/1 = 2*0.28*0.5 = 0.28\,\mathrm{nm}

.

Diffraction Grating Equation→Single Slit Diffraction→Young's Double Slit Interference→