Bragg's Equation

Bragg’s equation expresses the geometric condition that must be satisfied for X-rays scattered by successive planes of atoms in a crystal to reinforce each other - that is, to produce constructive interference and therefore a measurable reflection. It is the single most important relationship in X-ray crystallography, because it converts the easily measured quantity (the diffraction angle 2θ) into the scientifically meaningful quantity (the interplanar spacing d). Without this link, diffraction patterns would be angles without meaning; with it, they become fingerprints of crystal structure.

Why Single Crystals Are Impractical for Routine Work

Diffraction of X-rays from a set of atomic planes can only occur when the crystal grain happens to be oriented correctly relative to the incoming beam. For an unknown mineral with unknown d-spacings, the probability that a single grain is in exactly the right orientation when placed in the beam is effectively negligible - roughly comparable, as the source describes it, to winning the lottery. This is made worse by the fact that every mineral possesses many different sets of atomic planes - {100}, {010}, {001}, {111}, {110}, and so on - each capable of diffracting at its own specific angle. A grain oriented to reflect from one set of planes cannot simultaneously reflect from any of the others. To systematically sample all d-spacings in a single crystal, the instrument and sample would need to be rotated through a large range of orientations in a complex, exacting procedure. For routine identification work, using a powder of the mineral - many tiny grains in all possible orientations at once - is far simpler and equally effective. ^[1]

Why X-Rays Are Diffracted by Crystals

The X-rays used in diffractometers have wavelengths of around 1 or 2 Å - very nearly the same as the spacing between atoms in most mineral structures. This is not coincidence; it is the prerequisite for diffraction. If the X-ray wavelength were much larger or much smaller than atomic spacings, the waves would not interact with the crystal lattice in the geometrically systematic way needed to produce a diffraction pattern. Because the wavelength and spacing are comparable, the regularly arranged atomic planes act as a diffraction grating, and the scattered waves combine to produce diffraction maxima - effectively, reflections - at specific angles. ^[1]

Derivation of the Bragg Equation

Consider two parallel planes of atoms separated by a distance d, with X-rays impinging at angle θ measured from the plane surface (not from the normal - note that this definition differs from the convention used in ordinary optics). Ray 1 reflects from the upper plane; Ray 2 penetrates to the lower plane and reflects from there before exiting at the same angle. The extra distance Ray 2 travels compared to Ray 1 is the path segment pqr, which equals 2 × pq. Because pq = d sinθ, the extra path length is 2d sinθ. Constructive interference - and therefore a detectable reflection - occurs when this extra distance equals an integer number of wavelengths. Writing n for that integer and λ for the X-ray wavelength, the condition is: ^[1]

This is the Bragg equation. The integer n is called the order of the reflection. A first-order reflection (n = 1) occurs when the path difference equals exactly one wavelength; second-order (n = 2) when it equals two wavelengths; and so on. Each set of atomic planes can, in principle, produce multiple orders of reflection at progressively larger angles. ^[1]

Worked Example: Halite

The {111} planes in halite (NaCl) have an interplanar spacing d₁₁₁ = 3.2555 Å. Using CuKα radiation (λ = 1.5418 Å) and solving the Bragg equation for successive values of n: ^[1]

| Order (n) | θ (°) | 2θ (°) | | ------------- | --------------- | ---------------- | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | 1 | 13.70 | 27.40 | ^[1] | | 2 | 28.27 | 56.54 | | | 3 | 45.27 | 90.54 | ^[1] | | 4 | 71.30 | 142.60 | ^[1] |

Reflections of the fifth order and beyond are not possible for this plane because θ would need to exceed 90°, which has no physical meaning. The example illustrates a key practical consequence: a single set of atomic planes generates multiple peaks in the diffraction pattern, and these higher-order peaks must not be confused with first-order peaks from planes with different d-spacings. ^[1]

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