X-ray diffraction

Equipment Bruker D8 Analytical X-ray Systems

X-ray beam source Bruker D8 ADVANCE uses an x-ray tube with a Cu anode as the primary x-ray beam source. In this component x-rays are generated when a focused electron beam accelerated across a high voltage field bombards a stationary solid Cu target. As electrons collide with atoms in the target and slow down, a continuous spectrum of x-rays is emitted, which is termed Bremsstrahlung radiation. The high energy electrons also eject inner shell electrons in atoms through the ionization process. When a free electron fills the shell, an x-ray photon with energy characteristic of the target material is emitted. Common targets used in x-ray tubes include Cu and Mo, that emit 8 keV and 14 keV x-rays with corresponding wavelengths of 1.54 Å and 0.8 Å, respectively.

Wavelengths for X-Ray source Copper Anodes Bearden (1967) Holzer et al. (1997) Cobalt Cu Ka1 1.54056Å 1.540598 Å Co Ka1 1.788965Å 1.789010 Å Cu Ka2 1.54439Å 1.544426 Å Co Ka2 1.792850Å 1.792900 Å Cu Kb 1.39220Å 1.392250 Å Co Kb 1.62079Å 1.620830 Å Molybdenum Chromium Mo Ka1 0.709300Å 0.709319 Å Cr Ka1 2.28970Å 2.289760 Å Mo Ka2 0.713590Å 0.713609 Å Cr Ka2 2.293606Å 2.293663 Å Mo Kb 0.632288Å 0.632305 Å Cr Kb 2.08487Å 2.084920 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)

d dSin The path difference between ray 1 and ray 2 = 2d Sin BRAGG’s EQUATION Deviation = 2 Ray 1 Ray 2     d  dSin The path difference between ray 1 and ray 2 = 2d Sin For constructive interference: n = 2d Sin

θ - 2θ Scan The θ - 2θ scan maintains these angles with the sample, detector and X-ray source Normal to surface Only planes of atoms that share this normal will be seen in the θ - 2θ Scan NanoLab/NSF NUE/Bumm

Occurs throughout the bulk Takes place at any angle Powder diffraction data can be collected using either transmission or reflection geometry, as shown below. Because the particles in the powder sample are randomly oriented, these two methods will yield the same data Reflection Diffraction Occurs from surface Occurs throughout the bulk Takes place at any angle Takes place only at Bragg angles ~100 % of the intensity may be reflected Small fraction of intensity is diffracted

Incident X-rays Fluorescent X-rays Electrons Scattered X-rays SPECIMEN Heat Fluorescent X-rays Electrons Scattered X-rays Compton recoil Photoelectrons Coherent From bound charges Incoherent (Compton modified) From loosely bound charges Transmitted beam X-rays can also be refracted (refractive index slightly less than 1) and reflected (at very small angles) Refraction of X-rays is neglected for now.

How does it work? In powder XRD method, a sample is ground to a powder (±10µm) in order to expose all possible orientations to the X-ray beam of the crystal values of , d and  for diffraction are achieved as follows: 1.  is kept constant by using filtered X- radiation that is approximately monochromatic. 2. d may have value consistent with the crystal structure 3.  is the variable parameters, in terms of which the diffraction peaks are measured.

How does XRD Works??? Every crystalline substance produce its own XRD pattern, which because it is dependent on the internal structure, is characteristic of that substance. The XRD pattern is often spoken as the “FINGERPRINT” of a mineral or a crystalline substance, because it differs from pattern of every other mineral or crystalline substances.

Basic Component Of XRD Machine Therefore any XRD machine will consist of three basic component. Monochromatic X-ray source () Sample-holder (goniometer). Data collector- such as film, strip chart or magnetic medium/storage. By varying the angle , the Bragg’s Law conditions are satisfied by different d-spacing in polycrystalline materials. Plotting the angular positions and intensities of the resultant diffraction peaks produces a pattern which is characterised of the sample

X-ray Components A typical X-ray instrument is built by combining high performance components such as X-ray tubes, X-ray optics, X-ray detectors, sample handling device etc. to meet the analytical requirements. A consequent modular design is the key to configure the best instrumentation. .

Diffraction Pattern Collected Where A Ni Filter Is Used To Remove Kβ K alpha 1 and K alpha 2 overlap heavily at low angles and are easier to discriminate at high angles. Kb

Typical experimental data from Bruker XRD TiO2 2-theta intensitas 20 405 20.05001 357 20.10002 381 20.15002 371 20.20003 376 20.25004 356 20.30005 370 20.35006 395 20.40006 373 20.45007 335 20.50008 397 I 101 Anatase 110 Rutile 2

101 Anatase 110 Rutile

SC Lattice = SC Reciprocal Crystal = SC Examples of 3D Reciprocal Lattices weighed in with scattering power (|F|2) SC 001 011 101 111 Lattice = SC 000 010 100 110 No missing reflections Reciprocal Crystal = SC Figures NOT to Scale

BCC Lattice = BCC Reciprocal Crystal = FCC 002 022 202 222 011 101 020 000 Lattice = BCC 110 200 100 missing reflection (F = 0) 220 Reciprocal Crystal = FCC Weighing factor for each point “motif” Figures NOT to Scale

FCC Lattice = FCC Reciprocal Crystal = BCC 002 022 202 222 111 020 000 200 220 100 missing reflection (F = 0) 110 missing reflection (F = 0) Weighing factor for each point “motif” Reciprocal Crystal = BCC Figures NOT to Scale

Sample preparation

Make a mine powder

Sample holder

Side Drift Mount Designed to reduce preferred orientation – great for clay samples, (and others with peaks at low 2-theta angles)

Film, pellets, crystals mineral specimens

Sample holder

Specimen Holders for X-ray Diffraction

Match The Sample/Measurement Conditions With The Diffraction Pattern 1 2 3

Misinterpreting X-Ray Diffraction Results

Rock Salt Why are peaks missing? The sample is made from Morton’s Salt JCPDF# 01-0994 111 200 220 311 222 The sample is made from Morton’s Salt JCPDF# 01-0994 is supposed to fit it (Sodium Chloride Halite)

It’s a single crystal 111 200 220 311 222 2q The (200) planes would diffract at 31.82 °2q; however, they are not properly aligned to produce a diffraction peak The (222) planes are parallel to the (111) planes. At 27.42 °2q, Bragg’s law fulfilled for the (111) planes, producing a diffraction peak.

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.

Hint: Why are the intensities different? Which of these diffraction patterns comes from a nanocrystalline material? 66 67 68 69 70 71 72 73 74 2 q (deg.) Intensity (a.u.) Hint: Why are the intensities different? 1o 0.0015o These two diffraction patterns come from the exact same sample (silicon). The apparent difference in peak broadening is due to the instrument optics, not due to specimen broadening These diffraction patterns were produced from the exact same sample The apparent peak broadening is due solely to the instrumentation 0.0015° slits vs. 1° slits optical cofigurations Scan speed ( stepsize) http://prism.mit.edu/xray

Crystallite Size Broadening Scherrer’s Formula Peak Width B(2q) varies inversely with crystallite size The constant of proportionality, K (the Scherrer constant) depends on the how the width is determined, the shape of the crystal, and the size distribution the most common values for K are 0.94 (for FWHM of spherical crystals with cubic symmetry), 0.89 (for integral breadth of spherical crystals with cubic symmetry, and 1 (because 0.94 and 0.89 both round up to 1). K actually varies from 0.62 to 2.08 For an excellent discussion of K, refer to JI Langford and AJC Wilson, “Scherrer after sixty years: A survey and some new results in the determination of crystallite size,” J. Appl. Cryst. 11 (1978) p102-113. Remember: Instrument contributions must be subtracted

Scherrer’s Formula t = thickness of crystallite / crystallite size K = constant dependent on crystallite shape (0.89) l = x-ray wavelength B = FWHM (full width at half max) or integral breadth qB = Bragg Angle

Scherrer’s Formula What is B? B = (2θ High) – (2θ Low) B is the difference in angles at half max Peak 2θ low 2θ high Noise

When to Use Scherrer’s Formula Crystallite size <1000 Å Peak broadening by other factors Causes of broadening Size Strain Instrument If breadth consistent for each peak then assured broadening due to crystallite size K depends on definition of t and B Within 20%-30% accuracy at best Sherrer’s Formula References Corman, D. Scherrer’s Formula: Using XRD to Determine Average Diameter of Nanocrystals.

Scherrer’s Example

Scherrer’s Example = 0.89*1.54 Ǻ / ( 0.00174 * Cos (98.25/ 2 ) ) t = 0.89*λ / (B Cos θB) λ = 1.54 Ǻ = 0.89*1.54 Ǻ / ( 0.00174 * Cos (98.25/ 2 ) ) = 1200 Ǻ B = (98.3 - 98.2)*π/180 = 0.00174 Simple Right! Target Metal  Of K radiation (Å) Mo 0.71 Cu 1.54 Co 1.79 Fe 1.94 Cr 2.29

Methods used to Define Peak Width 46.7 46.8 46.9 47.0 47.1 47.2 47.3 47.4 47.5 47.6 47.7 47.8 47.9 2 q (deg.) Intensity (a.u.) Full Width at Half Maximum (FWHM) the width of the diffraction peak, in radians, at a height half-way between background and the peak maximum Integral Breadth the total area under the peak divided by the peak height the width of a rectangle having the same area and the same height as the peak requires very careful evaluation of the tails of the peak and the background FWHM 46.7 46.8 46.9 47.0 47.1 47.2 47.3 47.4 47.5 47.6 47.7 47.8 47.9 2 q (deg.) Intensity (a.u.)

Remember, Crystallite Size is Different than Particle Size A particle may be made up of several different crystallites Crystallite size often matches grain size, but there are exceptions TEM images collected by Jane Howe at Oak Ridge National Laboratory. http://prism.mit.edu/xray

Anistropic Size Broadening The broadening of a single diffraction peak is the product of the crystallite dimensions in the direction perpendicular to the planes that produced the diffraction peak. http://prism.mit.edu/xray

Instrumental Peak Profile A large crystallite size, defect-free powder specimen will still produce diffraction peaks with a finite width The peak widths from the instrument peak profile are a convolution of: X-ray Source Profile Wavelength widths of Ka1 and Ka2 lines Size of the X-ray source Superposition of Ka1 and Ka2 peaks Goniometer Optics Divergence and Receiving Slit widths Imperfect focusing Beam size Penetration into the sample 47.0 47.2 47.4 47.6 47.8 2 q (deg.) Intensity (a.u.) Patterns collected from the same sample with different instruments and configurations at MIT http://prism.mit.edu/xray

What Instrument to Use? The instrumental profile determines the upper limit of crystallite size that can be evaluated if the Instrumental peak width is much larger than the broadening due to crystallite size, then we cannot accurately determine crystallite size For analyzing larger nanocrystallites, it is important to use the instrument with the smallest instrumental peak width Very small nanocrystallites produce weak signals the specimen broadening will be significantly larger than the instrumental broadening the signal:noise ratio is more important than the instrumental profile http://prism.mit.edu/xray

Smaller Crystals Produce Broader XRD Peaks

Comparison of Peak Widths at Crystallite Sizes FWHM (deg) 100 nm 0.099 50 nm 0.182 10 nm 0.871 5 nm 1.745 Rigaku XRPD is better for very small nanocrystallites, <80 nm (upper limit 100 nm) PANalytical X’Pert Pro is better for larger nanocrystallites, <150 nm http://prism.mit.edu/xray

Decrease crystallite size A = anatase, R = rutile, B = brokite, (B)=TiO2(B) Wahyuningsih, S., 2009

Polycrystalline films on Silicon Why do the peaks broaden toward each other? Solid Solution Inhomogeneity Variation in the composition of a solid solution can create a distribution of d-spacing for a crystallographic plane CeO2 19 nm 45 46 47 48 49 50 51 52 2 q (deg.) Intensity (a.u.) ZrO2 46nm CexZr1-xO2 0<x<1

Many factors may contribute to the observed peak profile Instrumental Peak Profile Crystallite Size Microstrain Non-uniform Lattice Distortions Faulting Dislocations Solid Solution Inhomogeneity The peak profile is a convolution of the profiles from all of these contributions http://prism.mit.edu/xray

Thank you for your attending! Workshop & Analysis Informations: Dr. Sayekti Wahyuningsih, M.Si Dr. Yoventina Iriani, M.Si Laboratorium MIPA Terpadu FMIPA Universitas Sebelas Maret Phone / fax : (0271) 663375