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. A=2d),, sin9 + For parallel planes of atoms, with a space d,, between the planes, constructive interference only occurs when Bragg's law is satisfied. — In our diffractometers, the X-ray wavelength 2. is fixed. = Consequently, a family of planes produces a diffraction peak only at a specific angle 0. — 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.Our powder diffractometers typically use the Bragg-Brentano geometry. + Angles — The incident angle (w) is between the X-ray source and the sample. — The diffracted angle (26) is between the incident beam and the detector. — In plane rotation angle (®) + In “Coupled 28" Measurements: — The incident angle w is always % of the detector angle 20 . — The x-ray source is fixed, the sample rotates at 6 °/min and the detector rotates at 26 °/min. 4Coupled 26 Measurements Motorized xray PS, Source Slits In “Coupled 26” Measurements: — The incident angle w is always % of the detector angle 20 . — The x-ray source is fixed, the sample rotates at 6 */min and the detector rotates at 26 °/min. Angles — The incident angle (w) is between the X-ray source and the sample. — The diffracted angle (26) is between the incident beam and the detector. - In plane rotation angle (©) 5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. @ Outbound Inbound electron electron ha ————— MV election MIV MII MII fon MI; po ray LI photon Kai LU +e LI Atom 7 MADWhy does this sample second set of peaks at higher 20 values? * Hints: — It's Alumina — Cusource — Detector has a single channel analyzer Intensity (AU) 20 (Degrees)Diffraction Pattern Collected Where A Ni Filter Is Used To Remove K, ial Kod | Wha "| Due to tungsten contamination Ta Te Scio i, Elke) =hv= 42 = 8 A AA) 42°43 44 45 46 47 +48 49 Two-Theta (deg) Ts in) No fitWavelengths for X-Radiation are Sometimes Updated Copper Bearden Holzer et al, Cobalt Bearden Holzer et.al. Anodes (1967) (1997) Anodes (1987) (1997) Cu Kot 1.54056A 1.540598 A Co Ket 1.788965A 1.789010 A Cu Kaz 1.54439A 1.544426 A Co Kaz 1.792850A 1.792900 A cukp 1.39220A 1.392250 A CokB 1.62079A 1.620830 A Molybdenum Chromium Anodes Anodes Mo Kert 0.709300A 0.709319.A Cr Kort 2.28970A 2.289760 A Mo Ko2 0.713590A 0.713609 Cr koe 2.293606A 2.293663 A Mo KB 0.632288A 0.632305 A cr kp 2.08487 2.084920 A + Often quoted values from Cullity (1956) and Bearden, Rev. Mod. Phys. 39 (1967) are incorrect. ~ Values from Bearden (1967) are reprinted in internationat Tables for X-Ray Crystallography and most XRD textbooks. + Most recent values are from Hélzer et al. Phys. Rev. A 56 (1997) 10Calculating Peak Positions verplanar spaciog of te GA pe dedwegesde deheem ee de (Lo cone! 4 409 = = ot Jara r+ a eee eae ® tal Peas sate * BY + aarp Seal ain ale wesLattice Parameters & Atomic Radii Body Centered Cubic (BCC) 4R a= B Common BCC Metals - Chromium — Iron (a) = Molybdenum — Tantalum — Tungsten 12Lattice Parameters & Atomic Radii Body Centered Cubic (BCC) Common BCC Metals — Chromium — Iron (a) = Molybdenum — Tantalum - Tungsten 13Lattice Parameters & Atomic Radii Face Centered Cubic (FCC) a=2RV/2 Common FCC Metals — Aluminum — Copper — Gold — Lead — Nickel — Platinum — Silver 44Lattice Parameters & Atomic Radii (100) Face Centered Cubic (FCC) a=2RV2 Common FCC Metals — Aluminum _ = Copper Maggs | — Gold — Lead — Nickel — Platinum — Silver 45Planes and Family of Planes AO (001) (010) (100) = i ([S) k= —S} Vv (011) (710) (ion) a ZF M<—/ ~ al ixN Figure 1.18, The (110) family of planes. . I y ZY (planes) 14) (111) (171) 16Other Families of Planes Figure 1.17. Examples of several eubie erystal planes. 7Higher Order Planes (Half Planes) we — a RieNot all Planes Produce Peaks Peak Intensity Structure Factors 2 > Only if [F,' #0 does a peak appear Ta = 1pCLy [Fy] fexp[2zi(hu, +kv, +hw,)] where = Intensity of the incident X-ray beam = Muhiplicity factor (a function of the crystallography of the material) — C= Experimental constant (related to temperature, absorplion, fluorescence, and crystal imperfection) = J, = atomic scattering factor of atom ‘nis a measure of the scattering efficiency — U,V,WW are the atomic pasitions in the unit cell ,k,] are the Millar indices of the reflection. ‘Temperature facior=e%; Absarption factor = A(é). = Nis number of atoms in the unit cell = L, = Lorentz-Potarization factor. > The summation is performed over all atoms in the unit cell These calculations are easily doable for simple structures 19Structure Factors: Useful Knowledge + Atomic scattering factors vary as a function of afomic number (Z) and diffraction angle (8) + Values can be looked up in tables — Linear extrapolations are used for caiculating the values between those listed. (2) ‘Calculating structure factors involves complex expon relationships to determine the values of the exponenti ial functions, Use the following 20its unit cell can be reduced to two identical atoms. Atom #1 is at 0.0,0.and atom #2 is at, ¥, Yo. For this case we have . o Note: For atoms of the same type, f= f Observations: (Hf the sum (i + &-+£) = even in Equation (5), Fay = 2fand Gi) Hf the sum (h +k +1) = odd in Equation (5), Fay Oand ‘This, diffiaetions from BEC planes where b+ k + [8 odd are of zero intensity, They are forbidden reflections. These reflections are usually omitted from the reciprocal lattice. 213. PCC Structure ‘The FCC unit cell has. four atoms located at (40,0), (%.%4.0), (4.0.4), and (0, 4.4), It follows that, for the same kind of atoms, the structure fuetor the FCC structure is given by the expression, ( TEA, k, and (are all even or all odd (ie. unmixed then the sums +k, he and K+ Care all ‘even integers, and each term in Equation (6) equals 1. Therefore, Fray =4f. However, if, A, and f fare mixed integers, then Fy; ~/(1+1-2) 22Compound Structure Factors Consider the compound ZnS (sphalerite). Sulphur atoms occupy FCC sites with zine atoms displaced by "4 14 “a from these sites, The unit cell ‘can be reduced to four atoms of sulphur and 4 atoms of zinc. Consider a general unit cell for this type of structure, Many important compounds adopt this structure; examples include ZnS, GaAs, InSb, InP. and (AlGaAs. It can be reduced to 4 atoms of type A at 000, 0 14%, 4 014, % 4 O ic. in the FCC position and 4 atoms of type BB at the sites 4 4 Ya from the A sites, This can be expressed as; ‘The structure factors for this structure are: ‘if h, k, P mixed (just like FCC) fx if) if h, k, Pall odd n=Ja) Ph, k, Palleven and ht ket P= 40+ fa) ifh, k Fall even and let kt = 2n where nod (e.g. 200) 2a where n=even (c.z. 400) 23X-Ray Diffraction Patterns A= 2d, sin@ 100 o 20 40 o 0 100 * BCC or FCC? + Relative intensities determined by: Ta = 1pCL,[Fyul ; 24Arandom polycrystalline sample that contains thousands of crystallites should exhibit all possible diffraction peaks Intensity (AU) 30 35 40 45 50 55! 60 20 (Degre es) + 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 stabefcaly eevont ‘number of crystallites, not just one or two,Why are peaks missing? JCPOF# 01-0994 Intensity (AU) 25 30 35 40 45 50 55 60 20 (Degrees) *The sample is a cut piece of Morton’s Salt *JCPDF# 01-0994 is supposed to fit it (Sodium Chloride Halite) 26It’s a single crystal (a big piece of rock salt) 200 Intensity (AU) 25 30 35 40 45 50 55 60 20 (Degrees) ‘The (200) planes would diffract at 31.82 The (222) planes are parallel to the (111) “20; however, they are nat properly planes. aligned to produce a diffraction peak ‘At 27.42 °26, Brago's law fulfited for the (111) planes, producing a diffraction peak.Questions a 429Multiplicity (p) Matters Hexagonal at ft eee TE $ = oy ee € - + | (SS 3 Onturhombic ay OME ts * ; ; z : : Mati a os a = Irene ner 303132vee L < wt ££ inFour circle diffimetomerer 35 2eExample 5 Radiation from a copper source - Is that enough information? “Professor my peaks split!” 36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 i Diffraction Patterns Kot | Ka2 87 8889 80 81 82 93 S4 95 96 ‘Two-Theta (deq) * The Kat & Ko2 doublet will almost always be present WLat — Very expensive optics can remove the Ker2 line KB — Kort & Ko2 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- a2 43 44° 48 46" 47 da 49” Tay Source | | Two-Theta (deg) +W and Kf lines can be removed3&th optics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. 39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 your sample — “beam spill-off” at low angles, the beam might be wider than y& 40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 44divergence Is that tne engin of the beam illuminating the sample becomes smaller as the euawgerg * inedentbeaa|e becom 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 — ifthe slit is too small, you may not get enough intensity from your 185mm Radius Goniometer, XRPD 2 40 60 80 100 2Theta (deg)43Detectors * point 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 ggturation - CCD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 erfgrdss#ost data . 20 = -————_(in radians) — Asystematic error: * Sis 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 45 — Can he corrected hy tising an internal calibratianOther sources of error * Axial divergence — 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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 - aribers for Stee and low absorbi x (low “at Ae be. on nn by © Varallent oo optics or reduced by using a thin sample 11s the linear mass absorption coefficient for a specific sample 47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) 48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 49