How this instrument works
Fire X-rays at crystalline matter; almost everything passes straight through — except at certain sharp tilts, where one bright beam springs out sideways. William Lawrence Bragg worked out why in 1912, within months of Laue's first diffraction photographs: rays turned back by one sheet of atoms travel 2d·sin θ further than rays turned back by its neighbour above. Make that surplus path some whole number of wavelengths and every layer in such stacks arrives in step. He and his father shared 1915's Nobel Prize in Physics for it, and at 25 he is still its youngest science laureate.
Angles only land somewhere useful when wavelength and spacing are of similar size, which is precisely why this is X-ray work rather than optics. Copper tubes emit Kα radiation at 1.5406 Å; silicon (111) sheets sit 3.135 Å apart, rock salt (200) 2.82 Å, graphite (002) 3.35 Å. Pair any of those and θ falls between roughly 10° and 45°, exactly what benchtop powder diffractometers sweep in an afternoon. Reactor neutrons at 1.8 Å and synchrotron beams obey identical arithmetic, as do fibre Bragg gratings, whose 'planes' are periodic index ripples burned into glass cores.
One caution: this relation fixes where peaks appear and says nothing whatever about how bright they are. Intensities come from structure factors, and plenty of geometrically legal reflections vanish outright because atoms inside one unit cell cancel each other. It also presumes rigid, endless, flawless stacks, with single scattering per photon. Thick perfect crystals scatter repeatedly, so dynamical theory broadens every peak into Darwin plateaux some arcseconds wide, nudging its centre, since crystalline refractive index for X-rays sits just below unity. Grains under about 100 nm smear peaks in yet another way, described by Scherrer.
- Leave Diffraction order at 1 for a first-order reflection, or raise it to 2 or 3 to find where those same sheets reinforce again.
- Put your source into X-ray wavelength — 0.154 nm for copper Kα, 0.0711 nm for molybdenum Kα — using the nanometre unit, since 1 Å is 0.1 nm.
- Enter Lattice plane spacing for the family you are chasing, again in nanometres: 0.314 nm here, 0.3135 nm for silicon (111).
- Read Bragg angle (degrees). It is measured from the atomic sheet itself, so double it before comparing with a diffractometer scale marked in 2θ.
- If the sheet reports that no reflection exists, n·λ has overtaken 2d — drop the order or move to a shorter wavelength.
Worked example — copper Kα off 3.14 Å planes
A powder sample goes into a benchtop diffractometer fitted with a copper tube, so X-ray wavelength = 1.54e-10 m, or 0.154 nm. The planes under study lie 3.14e-10 m apart, and Diffraction order stays at 1. Then sin θ = 1 × 1.54 ⁄ (2 × 3.14) = 0.245223, and Bragg angle (degrees) comes back as 14.195009.
That detector arm, though, is graduated in 2θ, so your peak turns up at 28.39° on the chart — down in the crowded low-angle region where most inorganic phases show their strongest lines. Raise Diffraction order to 2 and those same planes reinforce again at 29.369891°, which is 58.74° in 2θ. Third order lands at 47.363758°, a fourth barely survives at 78.781303°, and a fifth cannot exist at all, because 5 × 1.54 overtakes 2 × 3.14.
Questions
Is the Bragg angle measured from the plane or from the normal?
From the plane itself. This trips up almost everyone arriving from optics, where Snell's law, Brewster's angle and every Fresnel coefficient reference a perpendicular dropped onto the surface. Bragg chose the glancing convention because he was thinking about reflection off stacked sheets, and crystallography has kept it ever since. An answer of 14.195009° here means the beam grazes those planes at 14.2°, striking them 75.8° from their normal.
Why does my instrument display 2θ rather than θ?
Because a detector measures how far your beam has been deflected from its original path, and that deviation is twice the Bragg angle. Sample and detector are geared so one turns through θ while an arm swings through 2θ, keeping reflecting planes bisecting both beams. Published powder patterns are therefore plotted against 2θ, so halve any peak position from a database before entering it as a check.
What does the diffraction order actually change?
It counts how many entire wavelengths fit into the extra path between adjacent planes. Order 2 needs twice the path, so it demands a steeper tilt from the same spacing. Note that order n from spacing d is arithmetically identical to order 1 from spacing d/n, which is why working crystallographers usually pin n at 1 and absorb the order into Miller indices — reflection (222) instead of second order from (111).
When does the formula return no answer at all?
Whenever n·λ exceeds 2d, because sin θ cannot pass 1. That ceiling is called a Bragg cutoff, and it explains why visible light diffracts off nothing crystalline: 550 nm runs roughly two thousand times too long for a 0.3 nm spacing. Neutron scientists exploit this limit deliberately, filtering out short wavelengths with a cooled beryllium block whose largest spacing sets a hard 3.96 Å threshold.
Does this work for electrons and neutrons as well as X-rays?
Yes, provided you enter their de Broglie wavelength. Davisson and Germer settled the matter in 1927 by bouncing 54 eV electrons off nickel and finding peaks exactly where this relation predicted, which proved matter behaves as waves. Reactor neutrons run near 1.8 Å, comparable to X-rays; 100 kV electrons sit near 0.037 Å, so their Bragg angles are under a degree, which is why transmission electron microscopy diffracts in near-forward geometry.
Should I correct for the crystal's refractive index?
Not for routine phase identification. An X-ray refractive index differs from 1 by about one part in a hundred thousand, shifting peaks by a few arcseconds — invisible against typical powder peak widths of 0.1°. It does matter in high-resolution diffraction of epitaxial semiconductor films, where lattice mismatch is read from splittings of that same magnitude, and there the correction is applied before any strain is quoted.