Bragg's Law & X-ray Diffraction

A crystal does not reflect X-rays over a range of angles. It reflects them at a handful of sharp angles and nowhere else, and those angles are a measurement of the gaps between its atoms.

Worked example — Bragg's Law & X-ray Diffraction

Spacing of the (111) planes	0.20871 nm
Bragg angle θ	21.677°
Where the detector sits, 2θ	43.354°
Extra distance the lower wave travels	0.15418 nm
Orders this spacing can show	2
Reflections from the whole crystal	8

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Lesson

The theory — Bragg's Law & X-ray Diffraction

Bragg's law, nλ = 2d sin θ, is the condition for a crystal to reflect a wave at all. Waves bouncing off successive planes of atoms travel paths differing by 2d sin θ; where that is a whole number of wavelengths they add, and everywhere else the contributions from the millions of planes below arrive at every phase and cancel. The angle is the measurement — d follows from it and from a wavelength that is known.

What each symbol means

λ: the wavelength of the incoming wave, known because the source is a particular metal target
d: the perpendicular distance between successive planes of atoms, the thing being measured
θ: the angle between the incoming beam and the PLANES, not the normal to them; the detector sits at 2θ
n: which order of reflection; sin θ ≤ 1 caps it at ⌊2d/λ⌋

Assumes: That the crystal is perfect and infinite, and that the only thing deciding whether a beam appears is geometry. A real crystal is a few thousand cells across, which broadens every peak from a spike into a line of measurable width — and that width is itself used to measure crystallite size.
Breaks when: On brightness. This page computes WHERE the peaks are, from spacings and centring alone, and says nothing about how strong each one is. That depends on which atoms sit where and how strongly each scatters, and it is the whole subject of the structure factor. Rock salt's (111) is a good example: geometry allows it and the page shows it, but sodium and chlorine scatter differently and partly cancel, so in a real pattern it is weak rather than absent.

Reading a distance of 0.2 nanometres off a dial 🖖

The planes of atoms in copper are 0.20871 nm apart. That is about one atom wide, and no microscope in 1913 or now resolves it directly.

What you can do is bounce X-rays off the crystal and turn the detector until a beam appears. For copper it appears at 2θ = 43.354°. Waves reflecting off successive planes travel different distances, the extra path is 2d sin θ, and a beam survives only where that extra path is a whole number of wavelengths. So the angle and the wavelength give you d, and the wavelength is known because the source is a copper target and always has been.

Every protein structure, every mineral, and the shape of DNA were measured this way. The quantity in the answer is a distance nobody has ever seen.

A wave cannot measure a gap much smaller than itself 🖖

The sine of an angle is capped: sin θ ≤ 1, so nλ ≤ 2d, and a set of planes spaced d shows exactly ⌊2d/λ⌋ orders. Not approximately — that many and no more.

Copper radiation at 0.154184 nm against copper's 0.20871 nm planes gives 2. Molybdenum at 0.071073 nm gives 5, and the whole pattern crowds toward smaller angles. Now press With green light: 500 nm against the same crystal gives ⌊0.00083⌋, which is zero, and the pattern panel is empty.

That is the entire reason X-ray crystallography uses X-rays. Not because they penetrate, and not because they are energetic. Because their wavelength happens to be the size of the thing being measured, and nothing longer can do the job at all.

The lattice is identified by the reflections that are missing 🖖

Geometry says every set of planes (h k l) has a spacing a/√(h²+k²+l²) and should reflect somewhere. It does not.

Press A silenced reflection. Iron's (100) planes are 0.28665 nm apart and a beam should appear at 2θ = 31.203°. The pattern has no peak there. Iron is body-centred, so there is an atom at the middle of every cell, and for (100) that atom lies exactly halfway between the planes — its reflection arrives half a wavelength out of step and cancels the one from the corners precisely.

Which families go silent depends only on where the extra atoms sit. Simple cubic shows everything; body-centred shows only h+k+l even; face-centred only h, k and l all even or all odd. So you can tell one lattice from another without resolving a single atom, by writing down the angles where nothing happened.

Problem solved in full

A powder pattern, and the crystal that made it 6 steps
An unknown cubic metal is ground to a powder and put in a diffractometer running copper Kα radiation, λ = 0.154184 nm. Beams come out at 2θ = 44.710°, 65.079°, 82.412° and 99.048°, and nowhere else below 100°. Find the type of lattice and the size of its unit cell.

Open this problem in the tool
1. Halve each angle to get θ, because 2θ is where the detector sits and θ is the angle in Bragg's law: 22.355°, 32.540°, 41.206° and 49.524°.
  
  $θ = \frac{1}{2} (2 θ) : 22.35 5^{\circ}, 32.54 0^{\circ}, 41.20 6^{\circ}, 49.52 4^{\circ}$
2. Bragg's law with n = 1 says sin θ = λ/2d, and for a cubic crystal d = a/√(h²+k²+l²). Putting those together gives sin²θ = (λ²/4a²)(h²+k²+l²). Everything in the bracket is the same for all four peaks, so the four values of sin²θ are in the same ratio as the four values of h²+k²+l² — which are whole numbers.
  
  $sin θ = \frac{λ}{2 d}, d = \frac{a}{h ^{2} + k ^{2} + l ^{2}} ⟹ sin^{2} θ = \frac{λ ^{2}}{4 a ^{2}} (h^{2} + k^{2} + l^{2})$
3. The four values of sin²θ are 0.144659, 0.289318, 0.433976 and 0.578635. Divide each by the first: 1.0000, 2.0000, 3.0000, 4.0000. So h²+k²+l² runs 1 : 2 : 3 : 4 across the peaks, or some common multiple of that.
  
  $sin^{2} θ = 0.144659, 0.289318, 0.433976, 0.578635 ⟹ 1 : 2 : 3 : 4$
4. Now use the silences. A simple cubic lattice allows every (h k l), so its sums would run 1, 2, 3, 4, 5, 6 — and a peak at ratio 5 would exist. There is none. A face-centred lattice allows only all-even or all-odd, giving sums 3, 4, 8, 11 and ratios 1 : 1.333 : 2.667 : 3.667, which is not what was measured. A body-centred lattice allows only h+k+l even, giving sums 2, 4, 6, 8 — ratios 1 : 2 : 3 : 4, exactly the observed pattern. The metal is body-centred cubic, and the first peak is (110).
  
  $P \to 1, 2, 3, 4, 5 \dots F \to 3, 4, 8, 11 (1 : 1.333 : 2.667) I \to 2, 4, 6, 8 (1 : 2 : 3 : 4)$
5. With the first peak identified as (110), h²+k²+l² = 2, so a = λ√2 / (2 sin θ₁) = 0.154184 × 1.41421 ÷ (2 × 0.380341) = 0.286650 nm.
  
  $a = \frac{λ 2}{2 sin θ _{1}} = \frac{0 . 154184 \cdot 1 . 41421}{2 \cdot 0 . 380341} = 0.286650 nm$
6. Check it against the tool rather than against the back of the book. Press Iron, leave the source on copper Kα, and the Spacing row reads 0.20269 nm for (110) with the beam at 2θ = 44.710° — the first measured angle. The Reflections row says 6, and the pattern panel shows peaks at exactly the four angles in the question and at two more above 100°.
  
  $d_{110} = \frac{a}{2} = 0.20269 nm, 2 θ = 44.71 0^{\circ}$
Answer

Body-centred cubic with a = 0.286650 nm, which is α-iron. What did the work was not the four angles by themselves but their ratios, and then the gaps: the peak that a simple cubic lattice would have shown at ratio 5, and the one a face-centred lattice would have shown at ratio 1.333, are both missing. The structure was read off the absences.

$body-centred cubic, a = 0.286650 nm$

References (3)

Father and son, the year they worked out that a crystal reflects X-rays only at particular angles and what those angles say: W. H. Bragg and W. L. Bragg, "The reflection of X-rays by crystals." Proceedings of the Royal Society of London A 88(605), 1913, pp. 428–438. DOI 10.1098/rspa.1913.0040
The paper the law is usually traced to, read to the Cambridge Philosophical Society at the age of 22: W. L. Bragg, "The Diffraction of Short Electromagnetic Waves by a Crystal." Proceedings of the Cambridge Philosophical Society 17, 1913, pp. 43–57. Reprinted in X-ray and Neutron Diffraction, Pergamon, 1966, pp. 109–125.
The standard reference for powder patterns, systematic absences and reading a lattice off a set of angles: B. D. Cullity and S. R. Stock, Elements of X-Ray Diffraction, 3rd edition, Prentice Hall, 2001.

By the 42Math team · Reviewed against published sources. Corrections welcome.