# X-Ray Diffraction Background and Fundamentals

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X-Ray Diffraction Background and Fundamentals
Prof. Thomas Key School of Materials Engineering

Crystalline materials are characterized by the orderly periodic arrangements of atoms.
The (200) planes of atoms in NaCl The (220) planes of atoms in NaCl The unit cell is the basic repeating unit that defines a crystal. Parallel planes of atoms intersecting the unit cell are used to define directions and distances in the crystal. These crystallographic planes are identified by Miller indices.

Bragg’s law is a simplistic model to understand what conditions are required for diffraction.
dhkl For parallel planes of atoms, with a space dhkl between the planes, constructive interference only occurs when Bragg’s law is satisfied. In our diffractometers, the X-ray wavelength l is fixed. Consequently, a family of planes produces a diffraction peak only at a specific angle q. Additionally, the plane normal must be parallel to the diffraction vector Plane normal: the direction perpendicular to a plane of atoms Diffraction vector: the vector that bisects the angle between the incident and diffracted beam The space between diffracting planes of atoms determines peak positions. The peak intensity is determined by what atoms are in the diffracting plane. draw the diffraction vector on this slide, or make a second slide explicitly illustrating the diffraction vector

Our powder diffractometers typically use the Bragg-Brentano geometry.
Detector X-ray tube Φ w q 2q Angles The incident angle (ω) is between the X-ray source and the sample. The diffracted angle (2q) is between the incident beam and the detector. In plane rotation angle (Φ) In “Coupled 2θ” Measurements: The incident angle w is always ½ of the detector angle 2q . The x-ray source is fixed, the sample rotates at q °/min and the detector rotates at 2q °/min.

Coupled 2θ Measurements
Detector Motorized Source Slits X-ray tube Φ w q 2q In “Coupled 2θ” Measurements: The incident angle w is always ½ of the detector angle 2q . The x-ray source is fixed, the sample rotates at q °/min and the detector rotates at 2q °/min. Angles The incident angle (ω) is between the X-ray source and the sample. The diffracted angle (2q) is between the incident beam and the detector. In plane rotation angle (Φ)

The X-ray Shutter is the most important safety device on a diffractometer
X-rays exit the tube through X-ray transparent Be windows. X-Ray safety shutters contain the beam so that you may work in the diffractometer without being exposed to the X-rays. Being aware of the status of the shutters is the most important factor in working safely with X rays.

The wavelength of X rays is determined by the anode of the X-ray source.
Electrons from the filament strike the target anode, producing characteristic radiation via the photoelectric effect. The anode material determines the wavelengths of characteristic radiation. While we would prefer a monochromatic source, the X-ray beam actually consists of several characteristic wavelengths of X rays. K L M

Why does this sample second set of peaks at higher 2θ values?
Hints: It’s Alumina Cu source Detector has a single channel analyzer 006 113 Kα1 Kα2

Diffraction Pattern Collected Where A Ni Filter Is Used To Remove Kβ
Ka1 Ka2 What could this be? W La1 Due to tungsten contamination K alpha 1 and K alpha 2 overlap heavily at low angles and are easier to discriminate at high angles. Kb 9

Wavelengths for X-Radiation are Sometimes Updated
Copper Anodes Bearden (1967) Holzer et al. (1997) Cobalt Cu Ka1 Å Å Co Ka1 Å Å Cu Ka2 Å Å Co Ka2 Å Å Cu Kb Å Å Co Kb Å Å Molybdenum Chromium Mo Ka1 Å Å Cr Ka1 Å Å Mo Ka2 Å Å Cr Ka2 Å Å Mo Kb Å Å Cr Kb Å Å Often quoted values from Cullity (1956) and Bearden, Rev. Mod. Phys. 39 (1967) are incorrect. Values from Bearden (1967) are reprinted in international Tables for X-Ray Crystallography and most XRD textbooks. Most recent values are from Hölzer et al. Phys. Rev. A 56 (1997)

Calculating Peak Positions

Body Centered Cubic (BCC) a Common BCC Metals Chromium Iron (α) Molybdenum Tantalum Tungsten a

(110) Body Centered Cubic (BCC) a Common BCC Metals Chromium Iron (α) Molybdenum Tantalum Tungsten

Face Centered Cubic (FCC) a Common FCC Metals Aluminum Copper Gold Lead Nickel Platinum Silver a

(100) Face Centered Cubic (FCC) Common FCC Metals Aluminum Copper Gold Lead Nickel Platinum Silver

Planes and Family of Planes
<001> c Cubic (001) (010) (100) b (011) (110) (101) a (planes) (111) (111) (111) <directions>

Other Families of Planes

Higher Order Planes (Half Planes)

Not all Planes Produce Peaks
Peak Intensity Structure Factors Only if [Fhkl]2 ≠0 does a peak appear where Io = Intensity of the incident X-ray beam p = Multiplicity factor (a function of the crystallography of the material) C = Experimental constant (related to temperature, absorption, fluorescence, and crystal imperfection). Temperature factor=e-2M; Absorption factor = A(θ). LP = Lorentz-Polarization factor. ƒn = atomic scattering factor of atom ‘n’ is a measure of the scattering efficiency u,v,w are the atomic positions in the unit cell h,k,l are the Miller indices of the reflection. N is number of atoms in the unit cell The summation is performed over all atoms in the unit cell. These calculations are easily doable for simple structures

Structure Factors: Useful Knowledge
(1) Atomic scattering factors vary as a function of atomic number (Z) and diffraction angle (θ) Values can be looked up in tables Linear extrapolations are used for calculating the values between those listed. (2)

X-Ray Diffraction Patterns
111 200 220 311 222 BCC or FCC? Relative intensities determined by:

A random polycrystalline sample that contains thousands of crystallites should exhibit all possible diffraction peaks 200 220 111 222 311 2q 2q 2q For every set of planes, there will be a small percentage of crystallites that are properly oriented to diffract (the plane perpendicular bisects the incident and diffracted beams). Basic assumptions of powder diffraction are that for every set of planes there is an equal number of crystallites that will diffract and that there is a statistically relevant number of crystallites, not just one or two.

Why are peaks missing? The sample is a cut piece of Morton’s Salt
111 200 220 311 222 JCPDF# The sample is a cut piece of Morton’s Salt JCPDF# is supposed to fit it (Sodium Chloride Halite)

It’s a single crystal (a big piece of rock salt)
111 200 220 311 222 2q The (200) planes would diffract at °2q; however, they are not properly aligned to produce a diffraction peak The (222) planes are parallel to the (111) planes. At °2q, Bragg’s law fulfilled for the (111) planes, producing a diffraction peak.

Questions

Multiplicity (p) Matters

Is that enough information?
Example 5 Radiation from a copper source - Is that enough information? “Professor my peaks split!”

X-radiation for diffraction measurements is produced by a sealed tube or rotating anode.
Sealed X-ray tubes tend to operate at 1.8 to 3 kW. Rotating anode X-ray tubes produce much more flux because they operate at 9 to 18 kW. A rotating anode spins the anode at 6000 rpm, helping to distribute heat over a larger area and therefore allowing the tube to be run at higher power without melting the target. Both sources generate X rays by striking the anode target wth an electron beam from a tungsten filament. The target must be water cooled. The target and filament must be contained in a vacuum.

Spectral Contamination in Diffraction Patterns
Ka1 Ka1 Ka2 Ka1 Ka2 Ka2 K alpha 1 and K alpha 2 overlap heavily at low angles and are easier to discriminate at high angles. The Ka1 & Ka2 doublet will almost always be present Very expensive optics can remove the Ka2 line Ka1 & Ka2 overlap heavily at low angles and are more separated at high angles W lines form as the tube ages: the W filament contaminates the target anode and becomes a new X-ray source W and Kb lines can be removed with optics W La1 Kb

Divergence slits are used to limit the divergence of the incident X-ray beam.
The slits block X-rays that have too great a divergence. The size of the divergence slit influences peak intensity and peak shapes. Narrow divergence slits: reduce the intensity of the X-ray beam reduce the length of the X-ray beam hitting the sample produce sharper peaks the instrumental resolution is improved so that closely spaced peaks can be resolved.

Varying Irradiated area of the sample
the area of your sample that is illuminated by the X-ray beam varies as a function of: incident angle of X rays divergence angle of the X rays at low angles, the beam might be wider than your sample “beam spill-off”

The constant volume assumption
In a polycrystalline sample of ‘infinite’ thickness, the change in the irradiated area as the incident angle varies is compensated for by the change in the penetration depth These two factors result in a constant irradiated volume (as area decreases, depth increase; and vice versa) This assumption is important for many aspects of XRPD Matching intensities to those in the PDF reference database Crystal structure refinements Quantitative phase analysis This assumption is not necessarily valid for thin films or small quantities of sample on a ZBH

One by-product of the beam divergence is that the length of the beam illuminating the sample becomes smaller as the incident angle becomes larger. The length of the incident beam is determined by the divergence slit, goniometer radius, and incident angle. This should be considered when choosing a divergence slits size: if the divergence slit is too large, the beam may be significantly longer than your sample at low angles if the slit is too small, you may not get enough intensity from your sample at higher angles Appendix A in the SOP contains a guide to help you choose a slit size. The width of the beam is constant: 12mm for the Rigaku RU300.

Detectors point detectors position sensitive detectors
observe one point of space at a time slow, but compatible with most/all optics scintillation and gas proportional detectors count all photons, within an energy window, that hit them Si(Li) detectors can electronically analyze or filter wavelengths position sensitive detectors linear PSDs observe all photons scattered along a line from 2 to 10° long 2D area detectors observe all photons scattered along a conic section gas proportional (gas on wire; microgap anodes) limited resolution, issues with deadtime and saturation CCD limited in size, expensive solid state real-time multiple semiconductor strips high speed with high resolution, robust

Sources of Error in XRD Data
Sample Displacement occurs when the sample is not on the focusing circle (or in the center of the goniometer circle) The greatest source of error in most data A systematic error: S is the amount of displacement, R is the goniometer radius. at 28.4° 2theta, s=0.006” will result in a peak shift of 0.08° Can be minimized by using a zero background sample holder Can be corrected by using an internal calibration standard Can be analyzed and compensated for with many data analysis algorithms For sample ID, simply remember that your peak positions may be shifted a little bit Can be eliminated by using parallel-beam optics

Other sources of error Axial divergence Flat specimen error
Due to divergence of the X-ray beam in plane with the sample creates asymmetric broadening of the peak toward low 2theta angles Creates peak shift: negative below 90° 2theta and positive above 90° Reduced by Soller slits and/or capillary lenses Flat specimen error The entire surface of a flat specimen cannot lie on the focusing circle Creates asymmetric broadening toward low 2theta angles Reduced by small divergence slits; eliminated by parallel-beam optics Poor counting statistics The sample is not made up of thousands of randomly oriented crystallites, as assumed by most analysis techniques The sample might be textured or have preferred orientation Creates a systematic error in peak intensities Some peaks might be entirely absent The sample might have large grain sizes Produces ‘random’ peak intensities and/or spotty diffraction peaks

sample transparency error
X Rays penetrate into your sample the depth of penetration depends on: the mass absorption coefficient of your sample the incident angle of the X-ray beam This produces errors because not all X rays are diffracting from the same location Angular errors and peak asymmetry Greatest for organic and low absorbing (low atomic number) samples Can be eliminated by using parallel-beam optics or reduced by using a thin sample m is the linear mass absorption coefficient for a specific sample

Techniques in the XRD SEF
X-ray Powder Diffraction (XRPD) Single Crystal Diffraction (SCD) Back-reflection Laue Diffraction (no acronym) Grazing Incidence Angle Diffraction (GIXD) X-ray Reflectivity (XRR) Small Angle X-ray Scattering (SAXS)

Available Free Software
GSAS- Rietveld refinement of crystal structures FullProf- Rietveld refinement of crystal structures Rietan- Rietveld refinement of crystal structures PowderCell- crystal visualization and simulated diffraction patterns JCryst- stereograms