Powder diffraction

Powder diffraction is a scientific technique using X-ray, neutron, or electron diffraction on powder or microcrystalline samples for structural characterization of materials. An instrument dedicated to performing such powder measurements is called a powder diffractometer. Powder diffraction stands in contrast to single crystal diffraction techniques, which work best with a single, well-ordered crystal.

Explanation

The most common type of powder diffraction is with X-rays, the focus of this article, although some aspects of neutron powder diffraction are mentioned. (Powder electron diffraction is more complex due to dynamical diffraction and is not discussed further herein.) Typical diffractometers use electromagnetic radiation (waves) with known wavelength and frequency, which is determined by their source. The source is often X-rays, and neutrons are also common sources, with their frequency determined by their de Broglie wavelength. When these waves reach the sample, the incoming beam is either reflected off the surface, or can enter the lattice and be diffracted by the atoms present in the sample. If the atoms are arranged symmetrically with a separation distance d, these waves will interfere constructively only where the path-length difference 2d sin θ is equal to an integer multiple of the wavelength, producing a diffraction maximum in accordance with Bragg's law. These waves interfere destructively at points between the intersections where the waves are out of phase, and do not lead to bright spots in the diffraction pattern. Because the sample itself is acting as the diffraction grating, this spacing is the atomic spacing. The distinction between powder and single crystal diffraction is the degree of texturing in the sample. Single crystals have maximal texturing, and are said to be anisotropic. In contrast, in powder diffraction, every possible crystalline orientation is represented equally in a powdered sample, the isotropic case. Powder X-ray diffraction (PXRD) operates under the assumption that the sample is randomly arranged. Therefore, a statistically significant number of each plane of the crystal structure will be in the proper orientation to diffract the X-rays. Therefore, each plane will be represented in the signal. In practice, it is sometimes necessary to rotate the sample orientation to eliminate the effects of texturing and achieve true randomness. Mathematically, crystals can be described by a Bravais lattice with some regularity in the spacing between atoms. Because of this regularity, we can describe this structure in a different way using the reciprocal lattice, which is related to the original structure by a Fourier transform. This three-dimensional space can be described with reciprocal axes x*, y*, and z* or alternatively in spherical coordinates q, φ*, and χ*. In powder diffraction, intensity is homogeneous over φ* and χ*, and only q remains as an important measurable quantity. This is because orientational averaging causes the three-dimensional reciprocal space that is studied in single crystal diffraction to be projected onto a single dimension.

When the scattered radiation is collected on a flat plate detector, the rotational averaging leads to smooth diffraction rings around the beam axis, rather than the discrete Laue spots observed in single crystal diffraction. The angle between the beam axis and the ring is called the scattering angle and in X-ray crystallography always denoted as 2θ (in scattering of visible light the convention is usually to call it θ). In accordance with Bragg's law, each ring corresponds to a particular reciprocal lattice vector G in the sample crystal. This leads to the definition of the scattering vector as:

| G | = q = 2 k sin ⁡ ( θ ) = 4 π λ sin ⁡ ( θ ) . {\displaystyle |G|=q=2k\sin(\theta )={\frac {4\pi }{\lambda }}\sin(\theta ).}

In this equation, G is the reciprocal lattice vector, q is the length of the reciprocal lattice vector, k is the momentum transfer vector, θ is half of the scattering angle, and λ is the wavelength of the source. Powder diffraction data are usually presented as a diffractogram in which the diffracted intensity, I, is shown as a function either of the scattering angle 2θ or as a function of the scattering vector length q. The latter variable has the advantage that the diffractogram no longer depends on the value of the wavelength λ. The advent of synchrotron sources has widened the choice of wavelength considerably. To facilitate comparability of data obtained with different wavelengths the use of q is therefore recommended and gaining acceptability.

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