Chap 8 Analytical Instruments

XRD

Measure X-Rays “Diffracted” by the specimen and obtain a diffraction pattern Interaction of X-rays with sample creates secondary “diffracted” beams (actually generated in the form of cones) of X-rays related to interplanar spacings in the crystalline powder according to a mathematical relation called “Bragg’s Law”: nλ = 2d sinθ where n is an integer λ is the wavelength of the X-rays d is the interplanar spacing generating the diffraction and θ is the diffraction angle λ and d are measured in the same units, usually angstroms. We will derive the Bragg law a bit more rigorously later but for a powder specimen in a diffractometer having a statistically infinite amount of randomly oriented crystallites, diffraction maxima (or peaks) are measured along the 2θ diffractometer circle.

The aspects of crystallography most important to the effective interpretation of XRD data are: conventions of lattice description, unit cells, lattice planes, d-spacing and Miller indices, crystal structure and symmetry elements, the reciprocal lattice (covered in a separate document)

Lattice Planes Lattice planes are defined in terms of the Miller indices, which are the reciprocals of the intercepts of the planes on the coordinate axes. In Fig. 1-5, the plane shown intercepts a at 100, b at 010 and c at 002. The Miller index of the plane is thus calculated as 1/1(a), 1/1(b), 1/2(c), and reduced to integers as 2a,2b,1c. Miller indices are by convention given in parentheses, i.e., (221). If the calculations result in indices with a common factor (i.e., (442)) the index is reduced to the simplest set of integers (221). This means that a Miller index refers to a family of parallel lattice planes defined by a fixed translation distance (defined as d) in a direction perpendicular to the plane. If directions are negative along the lattice, a bar is placed over the negative direction, i.e. (2 2 1) Families of planes related by the symmetry of the crystal system are enclosed in braces { }. Thus, in the tetragonal system {110} refers to the four planes (110), ( 1 10), ( 1 1 0) and (1 1 0). Because of the high symmetry in the cubic system, {110} refers to twelve planes. As an exercise, write the Miller indices of all of these planes.

Spacing of Lattice Planes

In a and c are in the plane of the paper, and b is perpendicular to the plane of the page. The notation shown for the d spacing and the relationship to the particular lattice plane (i.e., d001, d101, d103) with the Miller index for the particular plane shown in the subscript (but usually without parentheses) are standard notation used in crystallography and x-ray diffraction.

Scherrer’s Formula for Estimation of Crystallite Size If there is no inhomogeneous strain, the crystallite size D can be estimated from the peak width with the Scherrer’s formula: D = kλ/BcosθB Where λis the X-ray wavelength, B is the full width of height maximum of a diffraction peak, θB is the diffraction angle, and k is the Scherrer’s constant of the order unity for usual crystal.

Disadventages Compared with electron diffraction, XRD is the low intensity of diffracted X-rays, particularly for low Z materials. XRD is more sensitive to high Z materials. For low Z materials, neutron or electron diffraction is more suitable. Because of the small diffraction intensity, XRD requires large amount of specimens for measurements.

Electron Spectroscopy EDS: Energy Dispersive X-ray Spectroscopy AES: Auger Electron Spectroscopy XPS: X-ray Photoelectron Spectroscopy, similar to EDS but has a lower energy X-ray is used to eject the electrons from an atom via photoelectric effect. RBS: Rutherford Backscattering Spectrometry, use of high energy beams of low mass ions to penetrate into the sample and cause back scattering of the ions. SIMS: secondary ion mass spectrometry,

EDS AES

Each atom in the Periodic Table has a unique electronic structure with a unique set of energy levels, both X-ray and Auger spectral lines are characteristic of the element in question.