Crystal Structure Analysis

1 Crystal Structure AnalysisX-ray DiffractionElectron DiffractionNeutron Diffraction Essence of diffraction: Bragg DiffractionReading: West 5A/M 5-6G/S 3218 Elements of Modern X-ray Physics, 2ndEd. by Jens Als-Nielsen and Des McMorrow, John Wiley & Sons, Ltd., 2011 (Modern x-ray physics & new developments)X-ray Diffraction, by Warren, General Publishing Company, 1969, 1990 (Classic X-ray physics book)Elements of X-ray Diffraction, 3rd Ed., by Cullity, Addison-Wesley, 2001 (Covers most techniques used in traditional materials characterization)High Resolution X-ray Diffractometry and Topography, by D. Keith Bowen and Brian K. Tanner, Taylor & Francis, Ltd.

2 , 1998 (Semiconductors and thin film Analysis )Modern Aspects of Small-Angle Scattering, by H. Brumberger, Editor, Kluwer Academic Publishers, 1993 (SAXS techniques)Principles of Protein X-ray Crystallography, 3rdEd. by Jan Drenth, Springer, 2007 (Crystallography)REFERENCES219 SCATTERINGE lastic (E = E)X-rays scatter by interaction with the electron density of a are scattered by nuclei and by any magnetic moments in a are scattered by electric/magnetic is the process in which waves or particles are forced to deviate from a straight trajectory because of scattering centersin the propagation medium. p' pqE' E h Momentum transfer:Energy change:Inelastic (E E)q2sin2p Elastic scattering geometry Rayleigh ( >> dobject) Mie ( dobject) Geometric ( << dobject) Thompson (X-rays)Epc For X-rays: Compton (photons + electrons) Brillouin (photons + quasiparticles) Raman (photons + molecular )COMPTON SCATTERINGX-ray sourceGraphiteTargetCrystal (selects wavelength)Collimator (selects angle) Compton (1923) measured intensity of scattered X-rays from solid target, as function of wavelength for different angles.

3 He won the 1927 Nobel :peak in scattered radiation shifts to longer wavelength than source. Amount depends on (but not on the target material).A. H. Compton. Phys. ,409 (1923).DetectorComptonCOMPTON SCATTERINGC ompton s explanation: billiard ball collisions between particles of light (X-ray photons) and electrons in the materialClassical picture:oscillating electromagnetic field causes oscillations in positions of charged particles, which re-radiate in all directions at same frequency and wavelengthas incident radiation (Thompson scattering).Change in wavelength of scattered light is completely unexpected classically ep pBeforeAfterElectronIncoming photon pscattered photonscattered electronOscillating electronIncident light waveEmitted light waveConservation of energyConservation of momentum 1/ 222 22 4eeehm chp cm c eh pipp 1 cos1 cos0echmc 12 Compton wavelength 10mcehmc From this Compton derived the change in wavelength: ep pBeforeAfterElectronIncoming photon pscattered photonscattered electronCOMPTON SCATTERING223 Note that there is also an unshiftedpeak at each of this is elastic scatter.

4 Some comes from a collision between the X-ray photon and the nucleus of the atom. 1 cos0 Nhmc NemmsinceCOMPTON SCATTERING224 >>COMPTON SCATTERINGC ontributes to general background noiseDiffuse background from Compton emission by gamma rays ina positron emission tomography (PET) (18F)X-RAY SCATTERING wide-angle diffraction ( > 5 ) small-angle diffraction ( close to 0 ) X-ray reflectivity (films)elastic (Thompson, E = 0)inelastic ( E 0) Compton X-ray scattering resonant inelastic X-ray scattering (RIXS) X-ray Raman scatteringX-rays: 100 eV ( soft ) 100 keV ( hard ) photons 12,400 eV X-rays have wavelengths of 1 ,somewhat smaller than interatomic distances in solidsDiffraction from crystals!

5 First X-ray: 1895 Roentgen1901 Nobel (in ) = 12400/E (in eV)226 DIFFRACTIOND iffraction refers to the apparent bending of waves around small objects and the spreading out of waves past small our context, diffraction is the scattering of a coherent wave by the atoms in a Crystal . A diffraction pattern results from interference of the scattered the change in the direction of awavedue to a change in L. BraggW. H. Braggdiffraction of plane waves von LaueCrystal space description (Bragg) (k) space description (von Laue)227 OPTICAL INTERFERENCE =n , n= 0, 1, 2, .. =n , n= 1/2, 3/2, .. : phase differencen: orderperfectly in phase:perfectly out of phase:BRAGG S LAW OF DIFFRACTIONWhen a collimated beam of X-rays strikes pair of parallel lattice planes in a Crystal , each atom acts as a scattering center and emits a secondary wave.

6 All of the secondary waves interfere with each other to produce the diffracted beam Bragg provided a simple, intuitive approach to diffraction: Regard Crystal as parallel planes of atoms separated by distance d Assume specular reflection of X-rays from any given plane Peaks in the intensity of scattered radiation will occur when rays from successive planes interfere constructively2 229 BRAGG S LAW OF DIFFRACTIONAC sind ACB 2 sind ACBn 2 sinnd Bragg s Law:When Bragg s Law is satisfied, reflected beams are in phase and interfere constructively. Specular reflections can occur only at these peak is observed unless the condition for constructive interference( =n , with nan integer)is precisely met:230 DIFFRACTION ORDERS1storder:12 sind 2ndorder:222 sind By convention, we set the diffraction order = 1 for XRD.

7 For instance, when n=2 (as above), we just halve the d-spacing to make n= sind 22( / 2) sind the 2ndorder reflection of d100occurs at same as 1storder reflection of d200 XRD TECHNIQUES AND APPLICATIONS powder diffraction single- Crystal diffraction thin film techniques small-angle diffraction phase identification Crystal Structure determination radial distribution functions thin film quality crystallographic texture percent crystalline/amorphous Crystal size residual stress/strain defect studies in situ Analysis (phase transitions, thermal expansion coefficients, etc) superlattice structureUses:POWDER X-RAY DIFFRACTION uses monochromatic radiation, scans angle sample is powder allorientations simultaneously presented to beam some crystals will always be oriented at the various Bragg angles this results in cones of diffracted radiation cones will be spotty in coarse samples (those w/ few crystallites)crystalliteno restriction on rotational orientationrelative to beam2332sinhklhkld 234 TransmissiongeometryDEBYE-SCHERRER we can use a diffractometer to intercept sections of the cones2352sinhklhkld BASIC DIFFRACTOMETER SETUP236 General Area Detector Diffraction System (GADDS)

8 DIFFRACTOMETERSTHIN FILM SCANS2384-axis goniometerTHETA-2 THETA GEOMETRY X-ray tube stationary sample moves by angle theta, detector by 2theta239 THETA-THETA GEOMETRY sample horizontal (good for loose samples) tube and detector move simultaneously through theta240 POWDER DIFFRACTOGRAMS increasing , decreasingdMinimum d?min/2d In powder XRD, a finely powdered sample is probed with monochromatic X-rays of a known wavelength in order to evaluate the d-spacings according to Bragg s K radiation: = peak positions depend on: d-spacings of {hkl} systematic absences 241 ACTUAL EXAMPLE: PYRITE THIN FILMFeS2 cubic (a = ) Random Crystal orientations On casual inspection, peaks give usd-spacings, unit cell size, Crystal symmetry, preferred orientation, Crystal size, and impurity phases (none!)

9 111200210211220311Cu K = 2 ThetaIntensity powder pattern 2 = d= [2sin( )] = = d111reference pattern from ICDD(1,004,568+ datasets)d-SPACING FORMULAS2432 Layered Cuprate Thin film, growth oriented along c axis(hkl)(001)(002)(003)c = (00l)EXAMPLE 2: textured La2 CuO4 Epitaxialfilm is textured. (It has crystallographic orientation).Many reflections are missing 244 POWDER DIFFRACTIONPeak positionsdetermined by size and shape of unit cell Peak intensitiesdetermined by the position and atomic number of the various atoms within the unit cellPeak widthsdetermined by instrument parameters, temperature, and Crystal size, strain, and imperfections245we will return to this OF X-RAYSX-rays beams are usually generated by colliding high-energy electrons with 1sSiegbahn notationX-ray emission spectrum247 Generating BremsstrahlungGenerating Characteristic X-raysGENERATION OF X-RAYSCo K 1 : Cu K 1: (~8 keV)Mo K 1.

10 /hchE Side-window Coolidge X-ray tubeX-ray energy is determined by anode material, accelerating voltage, and monochromators: 1/ 2()CZ Moseley s Law:248 ROTATING ANODES 100X higher powers possible by spinning the anodeat > 6000 rpm to prevent melting it brighter source249 SYNCHROTRON LIGHT SOURCESSOLEIL brightest X-ray sources high collimation tunable energy pulsed operationGeV electron accelerators250 Bremsstrahlung( braking radiation )Australian SynchrotronMONOCHROMATIC X-RAYSF ilters (old way)A foil of the next lightest element (Ni in the case of Cu anode) can often be used to absorb the unwanted higher-energy radiation to give a clean K beam Crystal MonochromatorsUse diffraction from a curvedcrystal (or multilayer)