X-Ray Crystallography

X-ray crystallography is the application of X-rays to the study of minerals. The physical, chemical, and optical properties of minerals establish that their structures consist of regular, repeating arrays of atoms, but none of those classical techniques reveal what those arrangements actually are. X-rays can - by interacting with those arrays, they produce diffraction patterns from which the precise positions of atoms and ions can be worked out. Coupled with knowledge of chemical composition, this structural information has driven an ever-expanding understanding of why minerals behave as they do in geological environments and, by extension, how the geological processes that formed them operated. ^[1]

X-ray diffractometers are widely used in routine mineral analysis and identification. The basic instrument has three components: an X-ray tube, a sample holder, and a detector. Its most important practical feature is that it can identify very small samples of an unknown mineral and can handle materials like clays and zeolites that are too fine-grained to be studied by hand sample or optical microscope. Mineral identification using the powder method requires minimal sample preparation, and a typical diffraction run takes 20 minutes or less. ^[1]

What X-Rays Are

X-rays are part of the continuous electromagnetic spectrum, occupying a wavelength range of 0.1 to 10 angstroms (Å; 1 Å = 0.1 nm = 10⁻¹⁰ m). This places them far beyond the visible spectrum at the short-wavelength, high-energy end. They are generated whenever a stream of high-energy electrons strikes a material; both the energy of those electrons and the nature of the target material determine the wavelength or wavelengths of X-rays that are produced. ^[1]

X-Ray Generation

In a conventional diffractometer, X-rays are generated in a cathode ray tube. A metal filament (the cathode) is heated to boil off free electrons, which are then accelerated toward a metal target (the anode) by a voltage of several tens of kilovolts. The energy of each electron in electron-volts equals its charge times the accelerating voltage. When these high-energy electrons strike atoms in the target, two distinct types of X-ray spectrum are produced simultaneously. ^[1]

The Continuous Spectrum

The continuous spectrum is produced when incoming electrons are slowed or stopped through one or more collisions without dislodging bound electrons from the target atoms. As each electron decelerates, the kinetic energy it loses is released as electromagnetic radiation spanning a broad range of wavelengths, along with a large amount of heat. The accelerating voltage determines the shortest wavelength (and therefore the highest-energy X-rays) that can be produced; no radiation with a wavelength shorter than this limit is generated. The continuous spectrum behaves as background noise and must be removed before the data can be used. ^[1]

The Characteristic Spectrum (Kα and Kβ)

The characteristic spectrum is produced by a different mechanism: an incoming high-energy electron dislodges a tightly held electron from the innermost K shell of a target atom, creating a vacancy. Almost immediately, an electron from a higher-energy outer shell drops in to fill the vacancy, releasing radiation whose energy equals the energy difference ΔE between that outer shell and the K shell. The wavelength of this radiation is given by λ = 12.4 ch/ΔE (in angstroms). Because electrons drop in most commonly from the L shell (producing Kα radiation) or the M shell (producing Kβ radiation), and because the energy gap from M to K is larger than from L to K, Kβ has a shorter wavelength than Kα. The combined Kα peak is typically about ten times as intense as Kβ, making Kα the practical radiation of choice for diffractometry. ^[1]

Kα radiation itself is not a single peak but actually consists of two closely spaced wavelengths, Kα1 and Kα2, because there are two slightly different energy levels in the L shell. Kα1 has a slightly shorter wavelength than Kα2 and about twice its intensity. For copper - the most commonly used target metal in diffractometry - the weighted average Kα wavelength (written Cu Kᾱ) is 1.5418 Å. The characteristic spectrum is specific to each element, meaning the pattern of wavelengths produced is an elemental fingerprint of the target. ^[1]

Producing Monochromatic X-Rays

In practice, perfect monochromaticity is never achieved. The two components Kα1 and Kα2 overlap in wavelength and are difficult to isolate even with the best solid-state detector, and a very narrow wavelength band would have too little intensity to be useful. The Kα weighted average is therefore used in all calculations. ^[1]

X-Ray Detection

Photographic film, once widely used to capture diffraction patterns, has been almost entirely replaced by electronic detectors. Three types are in common use. A scintillation counter uses a crystal scintillator that emits a brief flash of light when struck by an X-ray photon; a photomultiplier converts this flash to an electrical spike, which is recorded as a count. A gas-proportional counter is a hollow metal tube filled with inert gas, with a thin wire along its axis maintained at ~1.5 kV relative to the tube wall; incoming X-ray photons ionize the gas by dislodging electrons, the resulting electrons migrate to the wire and ions to the wall, and the current pulse is counted. Solid-state detectors, based on semiconductor technology, convert the energy of each photon directly to an electrical pulse; their operation requires very cold temperatures, supplied by liquid nitrogen or a Peltier device. All three detector types are highly sensitive and output directly to a computer for immediate analysis. ^[1]

None of these detectors produces an image showing where individual atoms sit - atoms are far too small for that to be possible. What they record instead is the distribution of diffracted X-ray intensity as a function of angle, from which atomic positions are inferred indirectly. ^[1]

Target Metals and Their Wavelengths

The table below lists the characteristic wavelengths for the metals most commonly used as X-ray tube targets. The Kᾱ value (weighted average of Kα1 and Kα2) is the wavelength used in all diffraction calculations. ^[1]

| Metal | Kβ (Å) | Kα1 (Å) | Kα2 (Å) | Kᾱ (Å) | | --------- | --------------- | ----------------- | ----------------- | ---------------------- | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | Mo | 0.63225 | 0.70926 | 0.71354 | 0.7107 | ^[1] | | Cu | 1.38217 | 1.54051 | 1.54433 | 1.5418 | ^[1] | | Co | 1.62073 | 1.78892 | 1.79279 | 1.7902 | ^[1] | | Fe | 1.75653 | 1.93597 | 1.93991 | 1.9373 | ^[1] | | Cr | 2.08479 | 2.28962 | 2.29351 | 2.2909 | ^[1] |

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