3.1 Experimental techniques
3.1.4 X- ray Diffraction
Figure 3.11: M¨ossbauer spectroscopy hyperfine interactions: Isomer Shift (blue), Quadrupole Splitting (red), and magnetic splitting (green) [25].
∆EQ = eQVzz 2
where Vzz is the principal electric field gradient tensor.
Magnetic Hyperfine Splitting
For magnetic materials a magnetic field is present at the nucleus. The interaction of the nuclear dipole moment and the field results in ‘Zeeman’ splitting, yielding a sextet pattern (see fig. 3.11, (green)). The energies of nuclear transitions are modified when the nucleus is in a magnetic field, the energy levels split into 2I+ 1 sublevels [27]. The ground state (I = 1/2) of 57Fe in a magnetic field Bz is split into two energy sublevels(M I =±1/2). And the excited state (I = 3/2) split into four sublevels (MI =±3/2,±1/2. The selection rule ∆MI = 0,±1 restricts the two transitions lines shown in grey in the green spectra [27].
Figure 3.12: The Bragg-Brentano setup consist of an X-ray source, a detector and a sample mount [28].
wheren 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, (see fig. 3.13).
Figure 3.13: Geometric derivation of Bragg’s law: Constructive interference occures when waves scattered from adjacent lattice planes are in phase, while Destructive interference occurs when they are out of phase [28].
Rotation of the sample should take place so that the diffracted X-rays can be detected at various angles. Conversion of the diffraction peaks to d-spacings al- lows identification of the compound by comparing to a standard reference patterns because each compound has a set of unique d-spacings [28]. X-ray diffractometers consist basic parts; an X-ray tube, a sample holder, and an X-ray detector [28]. X- rays are generated in a cathode ray tube by heating a filament to produce electrons, accelerating the electrons toward a target thorough a potential difference, and bom- barding the target material with electrons. When electrons have sufficient kinetic energy to dislodge inner shell electrons of the target material, characteristic X-ray spectra are produced [28]. These spectra consist of several components, the most common beingKα and Kβ, (see fig.3.14). The one componentKα consists, in part, ofKα1 and Kα2, whereKα1 has a slightly shorter wavelength and twice the inten- sity as Kα2. The specific wavelengths are characteristic of the target material (Cu, Fe, Mo, Cr). Filtering, by foils or crystal monochrometers, is required to produce monochromatic X-rays needed for diffraction. Kα1 and Kα2 are sufficiently close in wavelength such that a weighted average of the two is used. Copper is the most com- mon target material for single-crystal diffraction, with CuKα1 radiation= 1.5418A.
Figure 3.14: Lines occur because bombarding electrons knock out electrons from K shell (n = 1), which are filled by electrons in higher shells. Electrons falling from L shell (n = 2) give rise to Kα lines, whereas electrons from M shell (n = 3) give the Kβ lines [28].
In many papers, also in the last decade, the most commonly applied configura- tion for XRD uses a copper radiation source (e.g., Amstaetter et al. 2012; Bharde et al. 2005, 2006; Wu et al. 2013)[29]. By using copper radiation when analyzing iron minerals however, it is often overlooked that the choice of radiation should be compatible to the sample composition in order to obtain a diffractogram of good quality[29]. A mismatch therein can lead to a high background signal that, espe- cially in combination with a low peak intensity, makes interpretation more difficult and can even lead to incorrect phase identification[29]. The most striking exam- ple thereof is when dealing with magnetite and/or maghemite. Diffractograms of these two minerals are only distinguishable by a few low intensity peaks[29]. Both can form in similar environments and be converted into one another, in the case of magnetite to maghemite simply by exposure to ambient air[29]. These X-rays are collimated and directed onto the sample. As the sample and detector are rotated, the intensity of the reflected X-rays is recorded. When the geometry of the incident X-rays impinging the sample satisfies the Bragg law, constructive interference occurs and a peak in intensity occurs. A detector records and processes this X-ray signal and converts the signal to a count rate which is then output to a device such as a printer or computer monitor [28]. The geometry of an X-ray diffractometer is such that the sample rotates in the path of the collimated X-ray beam at an angleθwhile the X-ray detector is mounted on an arm to collect the diffracted X-rays and rotates at an angle of 2θ. The instrument used to maintain the angle and rotate the sample is termed a goniometer [28]. For typical powder patterns, data are collected at 2θ
from 5° to 70°, angles that are preset in the X-ray scan. X-ray powder diffraction is most widely used for the identification of unknown crystalline materials (e.g. min- erals, inorganic compounds). Determination of unknown solids is critical to studies in geology, environmental science, material science, engineering and biology. Other applications include characterization of crystalline materials; identification of fine- grained minerals such as clays and mixed layer clays that are difficult to determine optically; determination of unit cell dimensions; measurement of sample purity[28].
With specialized techniques, XRD can be used to: determine crystal structures by using Rietveld refinement; determine of modal amounts of minerals (quantitative analysis); characterize thin films samples; make textural measurements, such as the orientation of grains, in a polycrystalline sample [28]. There are some strengths and some limitations of X-ray Powder Diffraction (XRD), strengths include; Power- ful and rapid (less than20 min) technique for identification of an unknown mineral [28]. In most cases, it provides an unambiguous mineral determination; Minimal sample preparation is required; XRD units are widely available; Data interpretation is relatively straight forward. Limitations include; Homogeneous and single phase material is best for identification of an unknown; Must have access to a standard reference file of inorganic compounds; requires tenths of a gram of material which must be ground into a powder; for mixed materials, detection limit is approximately two percent of sample; for unit cell determinations, indexing of patterns for non- isometric crystal systems is complicated; peak overlay may occur and worsens for high angle ”reflections”[28]. X-ray Diffraction is a high-tech, non-destructive tech- nique for analyzing a wide range of materials including fluids, metals, minerals, polymers, catalysts, plastics, pharmaceuticals, thin film coatings, ceramics, solar cells and semi conductors [28]. The technique finds innumerable practical applica- tions in various industries, including microelectronics, power generation, aerospace, and many more. The XRD analysis can easily detect the existence of defects in a particular crystal, its resistance level to stress, its texture, its size and degree of crystallinity and virtually any other variable relating to the sample’s basic structure [28].
The Scherrer equation is used to calculate the average crystallite sizes. It is important to note that Scherrer’s equation can only be applied for average sizes up to about 100–200 nm (depending on the instrument, sample and signal-to-noise ratio), because diffraction-peak broadening decreases with increasing crystallite size and it becomes difficult to separate the peak broadening due to crystallite size from the broadening due to other factors.
D= Kλ
βcosθ (3.2)
whereK is the shape factor (the typical value for sphere is 0.89),λis the wavelength of the incident beam,β is the broadening of the most intense peak at full width half maximum height (FWHM), θ is the Bragg’s angle and D is the diameter of the crystallite.