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X-RAY DIFFRACTION X- Ray Sources Diffraction: Bragg’s Law Crystal Structure Determination Elements of X-Ray Diffraction B.D. Cullity & S.R. Stock Prentice Hall, Upper Saddle River (2001)

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For electromagnetic radiation to be diffracted the spacing in the grating should be of the same order as the wavelength In crystals the typical interatomic spacing ~ 2-3 Å so the suitable radiation is X-rays Hence, X-rays can be used for the study of crystal structures Beam of electrons Target X-rays A accelerating charge radiates electromagnetic radiation

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Intensity Wavelength ( ) Mo Target impacted by electrons accelerated by a 35 kV potential 0.2 0.6 1.0 1.4 White radiation Characteristic radiation → due to energy transitions in the atom KK KK

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Target Metal Of K radiation (Å) Mo0.71 Cu1.54 Co1.79 Fe1.94 Cr2.29

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Heat Incident X-rays SPECIMEN Transmitted beam Fluorescent X-rays Electrons Compton recoilPhotoelectrons Scattered X-rays Coherent From bound charges Incoherent (Compton modified) From loosely bound charges X-rays can also be refracted (refractive index slightly less than 1) and reflected (at very small angles)

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Incoherent Scattering (Compton modified) From loosely bound charges Here the particle picture of the electron & photon comes handy Electron knocked aside 22 No fixed phase relation between the incident and scattered waves Incoherent does not contribute to diffraction (Darkens the background of the diffraction patterns)

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Vacuum Energy levels Nucleus Characteristic x-rays (Fluorescent X-rays) (10 −16 s later seems like scattering!) Fluorescent X-rays Knocked out electron from inner shell

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A beam of X-rays directed at a crystal interacts with the electrons of the atoms in the crystal The electrons oscillate under the influence of the incoming X-Rays and become secondary sources of EM radiation The secondary radiation is in all directions The waves emitted by the electrons have the same frequency as the incoming X-rays coherent The emission will undergo constructive or destructive interference Incoming X-rays Secondary emission

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Sets Electron cloud into oscillation Sets nucleus into oscillation Small effect neglected

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Oscillating charge re-radiates In phase with the incoming x-rays

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

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Incident and scattered waves are in phase if Scattering from across planes is in phase In plane scattering is in phase

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Extra path traveled by incoming waves AY Extra path traveled by scattered waves XB These can be in phase if and only if incident = scattered But this is still reinforced scattering and NOT reflection

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Bragg’s equation is a negative law If Bragg’s eq. is NOT satisfied NO reflection can occur If Bragg’s eq. is satisfied reflection MAY occur Diffraction = Reinforced Coherent Scattering Reflection versus Scattering ReflectionDiffraction Occurs from surfaceOccurs throughout the bulk Takes place at any angleTakes place only at Bragg angles ~100 % of the intensity may be reflectedSmall fraction of intensity is diffracted X-rays can be reflected at very small angles of incidence

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n = 2d Sin n is an integer and is the order of the reflection For Cu K radiation ( = 1.54 Å) and d 110 = 2.22 Å n Sin 10.3420.7ºFirst order reflection from (110) 20.6943.92º Second order reflection from (110) Also written as (220)

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In XRD n th order reflection from (h k l) is considered as 1 st order reflection from (nh nk nl)

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Crystal structure determination Monochromatic X-rays Panchromatic X-rays Monochromatic X-rays Many s (orientations) Powder specimen POWDER METHOD Single LAUE TECHNIQUE Varied by rotation ROTATING CRYSTAL METHOD

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THE POWDER METHOD

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Intensity of the Scattered electrons ElectronAtomUnit cell (uc)Scattering by a crystal A B C

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Scattering by an Electron Sets electron into oscillation Emission in all directions Scattered beams Coherent (definite phase relationship) A

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x z r P Intensity of the scattered beam due to an electron (I) For a wave oscillating in z direction For an polarized wave

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For an unpolarized wave E is the measure of the amplitude of the wave E 2 = Intensityc I Py = Intensity at point P due to E y I Pz = Intensity at point P due to E z

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Scattered beam is not unpolarized Polarization factor Comes into being as we used unpolarized beam Rotational symmetry about x axis + mirror symmetry about yz plane Forward and backward scattered intensity higher than at 90 Scattered intensity minute fraction of the incident intensity Very small number

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B Scattering by an Atom Scattering by an atom [Atomic number, (path difference suffered by scattering from each e −, )] Scattering by an atom [Z, ( , )] Angle of scattering leads to path differences In the forward direction all scattered waves are in phase f → (Å −1 ) → 0.2 0.4 0.6 0.8 1.0 10 20 30 Schematic

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Coherent scatteringIncoherent (Compton) scattering Z Sin( ) /

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C Scattering by the Unit cell (uc) Coherent Scattering Unit Cell (uc) representative of the crystal structure Scattered waves from various atoms in the uc interfere to create the diffraction pattern The wave scattered from the middle plane is out of phase with the ones scattered from top and bottom planes

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d (h00) BB Ray 1 = R 1 Ray 2 = R 2 Ray 3 = R 3 Unit Cell x M C N R B S A (h00) plane a

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Extending to 3D Independent of the shape of uc Note: R 1 is from corner atoms and R 3 is from atoms in additional positions in uc

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If atom B is different from atom A the amplitudes must be weighed by the respective atomic scattering factors (f) The resultant amplitude of all the waves scattered by all the atoms in the uc gives the scattering factor for the unit cell The unit cell scattering factor is called the Structure Factor (F) Scattering by an unit cell = f(position of the atoms, atomic scattering factors) In complex notation Structure factor is independent of the shape and size of the unit cell

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Structure factor calculations A Atom at (0,0,0) and equivalent positions F is independent of the scattering plane (h k l) Simple Cubic

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B Atom at (0,0,0) & (½, ½, 0) and equivalent positions F is independent of the ‘l’ index C- centred Orthorhombic Real Both even or both odd Mixture of odd and even e.g. (001), (110), (112); (021), (022), (023) e.g. (100), (101), (102); (031), (032), (033) (h + k) even (h + k) odd

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C Atom at (0,0,0) & (½, ½, ½) and equivalent positions Body centred Orthorhombic Real (h + k + l) even (h + k + l) odd e.g. (110), (200), (211); (220), (022), (310) e.g. (100), (001), (111); (210), (032), (133)

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D Atom at (0,0,0) & (½, ½, 0) and equivalent positions Face Centred Cubic Real (h, k, l) unmixed (h, k, l) mixed e.g. (111), (200), (220), (333), (420) e.g. (100), (211); (210), (032), (033) (½, ½, 0), (½, 0, ½), (0, ½, ½) Two odd and one even (e.g. 112); two even and one odd (e.g. 122)

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E Na + at (0,0,0) + Face Centering Translations (½, ½, 0), (½, 0, ½), (0, ½, ½) Cl − at (½, 0, 0) + FCT (0, ½, 0), (0, 0, ½), (½, ½, ½) NaCl: Face Centred Cubic

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(h, k, l) unmixed (h, k, l) mixed e.g. (100), (211); (210), (032), (033) Zero for mixed indices If (h + k + l) is even If (h + k + l) is odd Presence of additional atoms/ions/molecules in the uc (as a part of the motif ) can alter the intensities of some of the reflections

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Structure Factor (F) Multiplicity factor (p) Polarization factor Lorentz factor Relative Intensity of diffraction lines in a powder pattern Absorption factor Temperature factor Scattering from uc Number of equivalent scattering planes Effect of wave polarization Combination of 3 geometric factors Specimen absorption Thermal diffuse scattering

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Multiplicity factor LatticeIndex Multiplicity Planes Cubic(100)6 [(100) (010) (001)] ( 2 for negatives) (110)12 [(110) (101) (011), ( 110) ( 101) (0 11)] ( 2 for negatives) (111)8 [(111) (11 1) (1 11) ( 111)] ( 2 for negatives) (210)24 (210) = 3! Ways, ( 210) = 3! Ways, (2 10) = 3! Ways, ( 2 10) = 3! Ways, (211)21 (321)48 Tetragonal(100)4[(100) (010)] (110)4 [(110) ( 110)] (111)8 [(111) (11 1) (1 11) ( 111)] ( 2 for negatives) (210)6 (211)21 (321)48

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Polarization factor Lorentz factor

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Intensity of powder pattern lines (ignoring Temperature & Absorption factors) Valid for Debye-Scherrer geometry I → Relative Integrated “ Intensity ” F → Structure factor p → Multiplicity factor POINTS As one is interested in relative (integrated) intensities of the lines constant factors are omitted Volume of specimen m e, e (1/dectector radius) Random orientation of crystals in a with Texture intensities are modified I is really diffracted energy (as Intensity is Energy/area/time) Ignoring Temperature & Absorption factors valid for lines close-by in pattern

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In crystals based on a particular lattice the intensities of particular reflections are modified they may even go missing Crystal = Lattice + Motif Diffraction Pattern Position of the Lattice points LATTICE Intensity of the diffraction spots MOTIF

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Reciprocal Lattice Properties are reciprocal to the crystal lattice The reciprocal lattice is created by interplanar spacings

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A reciprocal lattice vector is to the corresponding real lattice plane The length of a reciprocal lattice vector is the reciprocal of the spacing of the corresponding real lattice plane Planes in the crystal become lattice points in the reciprocal lattice ALTERNATE CONSTRUCTION OF THE REAL LATTICE Reciprocal lattice point represents the orientation and spacing of a set of planes

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Reciprocal Lattice The reciprocal lattice has an origin!

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Note perpendicularity of various vectors

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Reciprocal lattice is the reciprocal of a primitive lattice and is purely geometrical does not deal with the intensities of the points Physics comes in from the following: For non-primitive cells ( lattices with additional points) and for crystals decorated with motifs ( crystal = lattice + motif) the Reciprocal lattice points have to be weighed in with the corresponding scattering power (|F hkl | 2 ) Some of the Reciprocal lattice points go missing (or may be scaled up or down in intensity) Making of Reciprocal Crystal (Reciprocal lattice decorated with a motif of scattering power) The Ewald sphere construction further can select those points which are actually observed in a diffraction experiment

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

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

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

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The Ewald * Sphere * Paul Peter Ewald (German physicist and crystallographer; 1888-1985) The reciprocal lattice points are the values of momentum transfer for which the Bragg’s equation is satisfied For diffraction to occur the scattering vector must be equal to a reciprocal lattice vector Geometrically if the origin of reciprocal space is placed at the tip of k i then diffraction will occur only for those reciprocal lattice points that lie on the surface of the Ewald sphere See Cullity’s book: A15-4

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K = K = g = Diffraction Vector Ewald Sphere The Ewald Sphere touches the reciprocal lattice (for point 41) Bragg’s equation is satisfied for 41

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http://www.matter.org.uk/diffraction/x-ray/powder_method.htm Diffraction cones and the Debye-Scherrer geometry Film may be replaced with detector

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Powder diffraction pattern from Al 420 111 200 220 311 222 400 331 422 1 & 2 peaks resolved Radiation: Cu K , = 1.54 Å Note: Peaks or not idealized peaks broadend Increasing splitting of peaks with g Peaks are all not of same intensity

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n 22 Sin Sin 2 ratioIndex 138.5219.260.330.113111 244.7622.380.380.144200 365.1432.570.540.298220 478.2639.130.630.4011311 582.4741.2350.660.4312222 699.1149.5550.760.5816400 7112.0356.0150.830.6919331 8116.6058.30.850.7220420 9137.4768.7350.930.8724422 Determination of Crystal Structure from 2 versus Intensity Data

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Extinction Rules Structure Factor (F): The resultant wave scattered by all atoms of the unit cell The Structure Factor is independent of the shape and size of the unit cell; but is dependent on the position of the atoms within the cell

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Bravais Lattice Reflections which may be present Reflections necessarily absent SimpleallNone Body centred(h + k + l) even(h + k + l) odd Face centredh, k and l unmixedh, k and l mixed End centred h and k unmixed C centred h and k mixed C centred Bravais LatticeAllowed Reflections SCAll BCC(h + k + l) even FCCh, k and l unmixed DC h, k and l are all odd Or all are even (h + k + l) divisible by 4 Extinction Rules

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n 2 → Intensity Sin Sin 2 ratio Determination of Crystal Structure from 2 versus Intensity Data

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The ratio of (h 2 + K 2 + l 2 ) derived from extinction rules SC1234568… BCC1234567… FCC3481112… DC381116…

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2 → Intensity Sin Sin 2 ratio 121.50.3660.1343 2250.4220.1784 3370.600.3628 4450.7070.50011 5470.7310.53512 6580.8480.71916 7680.9270.85919 FCC

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h 2 + k 2 + l 2 SCFCCBCCDC 1100 2110 3111 4200 5210 6211 7 8220 9300, 221 10310 11311 12222 13320 14321 15 16400 17410, 322 18411, 330 19331

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Structure factor calculation Consider a general unit cell for this type of structure. It can be reduced to 4 atoms of type A at 000, 0 ½ ½, ½ 0 ½, ½ ½ 0 i.e. in the fcc position and 4 atoms of type B at the sites ¼ ¼ ¼ from the A sites. This can be expressed as: The structure factors for this structure are: F = 0 if h, k, l mixed (just like fcc) F = 4(f A ± if B ) if h, k, l all odd F = 4(f A - f B ) if h, k, l all even and h+ k+ l = 2n where n=odd (e.g. 200) F = 4(f A + f B ) if h, k, l all even and h+ k+ l = 2n where n=even (e.g. 400) Consider the compound ZnS (sphalerite). Sulphur atoms occupy fcc sites with zinc atoms displaced by ¼ ¼ ¼ from these sites. Click on the animation opposite to show this structure. The unit cell can be reduced to four atoms of sulphur and 4 atoms of zinc. Many important compounds adopt this structure. Examples include ZnS, GaAs, InSb, InP and (AlGa)As. Diamond also has this structure, with C atoms replacing all the Zn and S atoms. Important semiconductor materials silicon and germanium have the same structure as diamond.

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Bravais lattice determination Lattice parameter determination Determination of solvus line in phase diagrams Long range order Applications of XRD Crystallite size and Strain Temperature factor Scattering from uc Number of equivalent scattering planes Effect of wave polarization Combination of 3 geometric factors Specimen absorption Thermal diffuse scattering

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Diffraction angle (2 ) → Intensity → 90 180 0 Crystal 90 180 0 Diffraction angle (2 ) → Intensity → Liquid / Amorphous solid 90 180 0 Diffraction angle (2 ) → Intensity → Monoatomic gas 300 310 Schematic of difference between the diffraction patterns of various phases

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