Crystal structure is determined from diffraction of X-rays, with n lambda equals 2 d sin theta.
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X-ray diffraction determines the arrangement of atoms in a crystal by measuring how X-rays scatter from the regular planes of atoms. Waves scattered from successive planes interfere constructively when the path difference equals a whole number of wavelengths, which is Bragg's law: nλ = 2d sin θ. Here n is an integer, λ is the X-ray wavelength, d is the spacing between the planes and θ is the glancing angle. Measuring the angles at which strong reflections occur gives the interplanar spacings, from which the unit cell and atomic positions can be deduced. X-ray diffraction is a key technique for determining the structures of minerals, metals and biological molecules.
Key ideas: Use n lambda = 2 d sin(theta) with theta measured from the plane, not from the normal. For cubic crystals d = a / sqrt(h^2 + k^2 + l^2). Cu K-alpha X-rays have wavelength 0.1542 nm. Peak positions give cell size and peak intensities give atom arrangement. Powder diffraction is the standard method for phase identification.
Common mistakes and connections: Typical error is using the angle 2 theta reported on diffractometers directly as theta. Another is forgetting that wavelength must be comparable to spacing, about 0.1 nm. Electrons and neutrons can also diffract. Higher orders n appear at larger angles. Links to unit cell, crystal systems and structural determination of proteins and DNA.
Bragg's law n lambda = 2 d sin theta: X-rays reflecting from lattice planes give peaks at angles that reveal spacings; shown pattern is silicon. The interactive lesson shows this as a live animation or chart.
X-rays of wavelength 0.154 nm reflecting from planes spaced 0.20 nm apart give a first-order reflection at a glancing angle of about 22.6°.
Crystal structure is determined from diffraction of X-rays, with n lambda equals 2 d sin theta.
X-ray Diffraction and Bragg's Law is a advanced topic in States of Matter & Solutions. It builds on beginner and intermediate topics, so review those first if needed.
Study these earlier topics in States of Matter & Solutions first: Tetrahedral and Octahedral Voids, Crystal Systems, Bravais Lattices.
The next topics in States of Matter & Solutions are Crystal Defects and Schottky Defect.
Yes. The interactive lesson includes a self-check question on X-ray Diffraction and Bragg's Law, and the States of Matter & Solutions quiz draws on every topic in this area.
X-ray Diffraction and Bragg's Law is topic 66 of 109 in States of Matter & Solutions.