Characterization Techniques

Bragg's law and XRD peak indexing

Bragg's law describes when X-rays scattered by a family of parallel lattice planes add up constructively: n\lambda = 2d\sin\theta, where n is the order of reflection, \lambda is the X-ray wavelength, d is the interplanar spacing and \theta is the Bragg angle. The diffractometer plots the angle 2\theta between the incident and diffracted beams, so the Bragg angle is always half of the plotted angle. A first-order reflection means n = 1.

For a cubic crystal of lattice parameter a, the spacing of the (hkl) planes is d_{hkl} = a/\sqrt{h^2+k^2+l^2}. Substituting this into Bragg's law gives \sin^2\theta = \dfrac{\lambda^2}{4a^2}(h^2+k^2+l^2), so the \sin^2\theta values of the peaks in a cubic powder pattern are proportional to N = h^2+k^2+l^2. Peak indexing therefore works with ratios: dividing each \sin^2\theta by the smallest one gives integers that can be matched to allowed values of N, without needing a or \lambda.

In an FCC lattice the atoms at the cube corners and face centres cause destructive interference unless h, k and l are all odd or all even. The first allowed reflections are therefore (111), (200), (220), (311) and (222), with N = 3, 4, 8, 11, 12 and \sin^2\theta in the ratio 3:4:8:11:12. A peak that does not fit this sequence, or a gap in it, indicates a missing reflection, and the missing index can be found from the gap.

Between two peaks of a cubic pattern, \sin y/\sin x = \sqrt{N_y/N_x}. For example, the (220) peak is the third peak of an FCC powder pattern after (111), so \sin y/\sin x = \sqrt{8/3} \approx 1.6.

Once the indices of a peak are known, the lattice parameter follows from a = d_{hkl}\sqrt{h^2+k^2+l^2}. For example, if the third peak of an FCC metal is (220), \lambda = 1.54 Å and 2\theta = 60^\circ, then d = \lambda/(2\sin 30^\circ) = 1.54 Å and a = 1.54\sqrt{8} \approx 4.36 Å.

• Bragg angle — half of the measured 2\theta; always substitute \theta into \sin\theta.
• allowed reflection — a reflection that survives the structure-factor extinction rules of the lattice type.
• indexing by ratio — comparing \sin^2\theta values of peaks rather than absolute angles.

SEM signals and interaction volume

When a focused electron beam strikes a specimen, the electrons scatter inside a pear-shaped region called the interaction volume. The size of this region grows with beam energy and shrinks as the mean atomic number of the specimen rises. Each signal leaves the specimen from only part of this volume, and the lateral spread of that part sets how sharp the image can be.

Secondary electrons (SE) are low-energy electrons, usually below about 50 eV, knocked out of atoms near the surface. Because they escape only from the top few nanometres, the SE sampling volume is small, and SE images have high resolution and show surface topography. The SE yield is higher on edges and steep faces, which gives the bright edges typical of SE micrographs.

Backscattered electrons (BSE) are beam electrons that are scattered back out of the specimen with energies close to the primary beam energy. They come from greater depth and spread laterally, so their sampling volume is larger and a BSE image has poorer resolution than an SE image.

The BSE yield increases with atomic number, so heavier phases look brighter. This gives compositional, or atomic-number, contrast that distinguishes chemically distinct phases in a polished section.

• sampling volume — the region of the specimen from which a given signal can emerge; it limits the attainable lateral resolution.
• topographic contrast — brightness that depends on surface inclination, recorded by the SE detector.
• characteristic X-rays — emitted when inner-shell vacancies are filled; analysed by energy-dispersive spectroscopy (EDS) to find the elements present.
• EBSD and Kikuchi lines — backscattered electrons that diffract from crystal planes form Kikuchi lines, from which the crystallographic orientation of a grain is determined.
• fractography — examination of fracture surfaces with SE imaging to reveal failure mechanisms.

Unidentified content (stem missing)

Homogeneous nucleation of a solid from its melt requires a nucleus larger than a critical radius. Balancing the volume free-energy gain against the surface energy \gamma gives r^* = \dfrac{2\gamma T_m}{L\,\Delta T}, where L is the volumetric latent heat, T_m is the equilibrium melting temperature and \Delta T = T_m - T is the undercooling. Because r^* is inversely proportional to \Delta T, a larger undercooling gives a smaller critical nucleus and makes nucleation easier. Solidification well below T_m therefore starts from smaller nuclei than solidification just below T_m.

Magnetization M is the magnetic moment per unit volume, M = \mu/V, with units of A m^{-1}. When a sample is specified by mass and density, its volume is V = W/\rho and must be converted to SI units before use. For example, a 20 g sample of density 8 g cm^{-3} has V = 2.5 cm^3 = 2.5\times10^{-6} m^3, and a moment of 5\times10^{-7} A m^2 then gives M = 0.2 A m^{-1}. Unit conversion between cm^3 and m^3 is a common error.

A diffraction peak has a finite width for two reasons. Small crystallites broaden peaks, and the Scherrer equation \beta = K\lambda/(D\cos\theta) estimates the crystallite size D from the full width at half maximum \beta in radians, with K \approx 0.9 as a typical shape factor. Lattice microstrain also broadens peaks, with a contribution 4\varepsilon\tan\theta that grows with angle. Williamson-Hall analysis plots \beta\cos\theta against \sin\theta to separate the size term from the strain term. The precise lattice parameter comes from peak positions, not widths, and residual macrostress is found from peak shifts measured at several tilt angles, as in the \sin^2\psi method.

• critical radius — the minimum nucleus size that can grow spontaneously.
• undercooling — the difference between the equilibrium melting temperature and the actual temperature.
• microstrain — small, varying lattice distortions that broaden peaks.