Properties Of X-rays - Stanford University

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X-ray DiffractionInteraction of WavesReciprocal Lattice and DiffractionX-ray Scattering by AtomsThe Integrated Intensity

Basic Principles of Interaction of WavesPeriodic waves characteristic:Frequency : number of waves (cycles) per unit time – cycles/time.[ ] 1/sec Hz. Period T: time required for one complete cycle – T 1/ time/cycle. [T] sec. Amplitude A: maximum value of the wave during cycle. Wavelength : the length of one complete cycle. [ ] m, nm, Å. A simple wave completescycle in 360 degreesE (t , x) A exp kx t x x A exp 2 t A exp 2 t c 2 , k 2

Basic Principles of Interaction of WavesConsider two waves with the same wavelength and amplitude but displaced adistance x0.The phase shift: x 2 0 E1 (t ) A exp t E1 (t ) A exp t Waves #1 and #2 are 90o out of phasex0Waves #1 and #2 are 180o out of phaseWhen similar waves combine, the outcome can be constructive or destructive interference

Superposition of WavesResulting wave is algebraic sum of the amplitudes at each pointSmall difference in phaseLarge difference in phase

Superposition of WavesThomas Young's diagram of double slit interference (1803)Δ𝑥 dLThe angular spacing of the fringes is given by𝜃𝑓 𝜆𝑑,where 𝜃𝑓 1𝐿𝜆𝑑

The discovery of X-ray diffraction and its use as aprobe of the structure of matter The reasoning: x-rays have a wavelength similar tothe interatomic distances in crystals, and as a result,the crystal should act as a diffraction grating. 1911, von Laue suggested to one of his researchassistants, Walter Friedrich, and a doctoral student,Paul Knipping, that they try out x-rays on crystals.Max von Laue April 1912, von Laue, Friedrich and Knipping hadperformed their pioneering experiment on coppersulfate.First diffraction pattern from NaCl crystal

The discovery of X-ray diffraction and its use asa probe of the structure of matter They found that if the interatomic distances in thecrystal are known, then the wavelength of the X-rayscan be measured, and alternatively, if the wavelengthis known, then X-ray diffraction experiments can beused to determine the interplanar spacings of acrystal. The three were awarded Nobel Prizes in Physics fortheir discoveries.Friedrich & Knipping's improved set-upZnS Laue photographs alongfour-fold and three-fold axes

The Laue EquationsIf an X-ray beam impinges on a row of atoms, each atom can serve as a source ofscattered X-rays.The scattered X-rays will reinforce in certain directions to produce zero-, first-, andhigher-order diffracted beams.

The Laue EquationsConsider 1D array of scatterers spaced a apart.Let x-ray be incident with wavelength .a cos cos 0 h In 2D and 3D:a cos cos 0 h 2Db cos cos 0 k 3Dc cos cos 0 l The equations must be satisfied simultaneously, it is in general difficult to produce a diffractedbeam with a fixed wavelength and a fixed crystal.

Lattice Planes

Bragg’s LawAB BC 2d sin

Bragg’s LawIf the path AB CD is a multiple of the x-ray wavelength λ, then two waves willgive a constructive interference:n AB CD 2d sin 2d sin n The diffracted waves will interfere destructively if equation is not satisfied.Equation is called the Bragg equation and the angle θ is the Bragg angle.n

Bragg’s LawThe incident beam and diffracted beam are always coplanar.The angle between the diffracted beam and the transmitted beam is always 2 .Since sin 1:n sin 12dFor most crystals d 3 ÅFor n 1: 2d 6ÅCu K 1 1.5406 ÅRewrite Bragg’s law:dn 2 sin 2d sin 1 h2 k 2 l 2 2da2ad100 a21 0 0a1d 200 a22 0 0 2UV radiation 500 Å 2d sin

Reciprocal lattice and Diffractiond*hkls s0 d*hkl hb1 kb 2 lb3s s0 s s0 It is equivalent to the Bragg law since s s 0 2 sin s s0 2 sin d*hkl 2d hkl sin 1d hkl

Reciprocal lattice and Diffractions s0 2 sin d*hkl 1d hkl

Sphere of Reflection – Ewald SphereOC OB d*hkl1 sin 1 *1d hkl 2d hkl sin 22d hkl

Sphere of Reflection – Ewald Sphere

Kinematical x-ray diffractionSilicon lattice constant:aSi 5.43 ÅX-ray wavelength: 1.5406 ÅFor cubic crystal:1 h2 k 2 l 2 2da2Bragg’s law:2d sin n

Two perovskites: SrTiO3 and CaTiO3Differences: Peak position – d-spacing.Peak intensity – atom type: Ca vs Sr.OTiSr/CaaSTO 3.905 ÅaCTO 3.795 Å

Scattering by an ElectronElementary scattering unit in an atomis electronClassical scattering by a single freeelectron – Thomson scatteringequation:e 4 1 cos 2 2 I I 0 2 4 2 mc R 2 2 The polarization factor of anunpolarized primary beame4 262 7.94 10cmm 2c 4If R few cm:I 10 26I01 mg of matter has 1020 electronsAnother way for electron to scatter ismanifested in Compton effect.Cullity p.127

Scattering by an AtomAtomicScatteringFactorf amplitude of thewave scatteredby an atomamplitude of thewave scatteredby one electron

Scattering by an AtomScattering by a group of electrons at positions rn:Scattering factor per electron:f e exp 2 i / s s 0 r dVAssuming spherical symmetry for the chargedistribution (r ) and taking origin at the center ofthe atom: f e 4 r 2 r 0sin krdrkrFor an atom containing several electrons: f f en 4 r 2 n r nn0sin krdrkrf – atomic scattering factor4 sin k f sin Calling Z the number of electrons per atom we get: n 04 r 2 n r dr Z

Scattering by an Atomf amplitude of the wave scattered by an atomamplitude of the wave scattered by one electronThe atomic scattering factor f Z for anyatom in the forward direction (2 0):I(2 0) Z2As increases f decreases functionaldependence of the decrease depends onthe details of the distribution of electronsaround an atom (sometimes called theform factor)f is calculated using quantum mechanics

Electron vs nuclear densityPowder diffraction patterns collected using Mo K radiation and neutron diffraction

Scattering by an Atom

Scattering by a Unit CellFor atoms A & C 2 1 MCN 2d h 00 sin d h 00 AC ahFor atoms A & B 3 1 RBS ABABxMCN ACACa/h 2 hxphase 2 3 1 2 3 1 aIf atom B position: u x / a 3 1 For 3D:2 hx 2 hua 2 hu kv lw Fhkl amplitude scattered by atoms in unit cellamplitude scattered by single electron

Scattering by a Unit CellWe can write:Ae i fe 2 i hu kv lw F f1ei 1 f 2 ei 2 f 3ei 3 .NF f ne1i nN f n e 2 i hun kvn lwn 1 I Fhkl Fhkl Fhkl2

Scattering by a Unit CellExamplesUnit cell has one atom at the originF fe 2 i 0 fIn this case the structure factor is independent of h, k and l ; it will decrease with fas sin / increases (higher-order reflections)

Scattering by a Unit CellExamplesUnit cell is base-centered F fe 2 i 0 fe 2 i h / 2 k / 2 f 1 e i h k F 2fh and k unmixedF 0h and k mixed(200), (400), (220) Fhkl 4 f 2(100), (121), (300) Fhkl 0 22“forbidden”reflections

Body-Centered Unit CellExamplesFor body-centered cell F fe 2 i h0 k 0 l 0 fe 2 i h / 2 k / 2 l / 2 f 1 e i h k l F 2fwhen (h k l ) is evenF 0when (h k l ) is odd(200), (400), (220) Fhkl 4 f 2(100), (111), (300) Fhkl 0 22“forbidden”reflections

Body-Centered Unit CellExamplesFor body centered cell with different atoms:F f Cl e 2 i h 0 k 0 l 0 f Cse 2 i h / 2 k / 2 l / 2 f Cl f Cse i h k l F f Cl f Cswhen (h k l) is evenF f Cl f Cswhen (h k l) is oddCs (200), (400), (220) Fhkl f Cs f Cl (100), (111), (300) Fhkl f Cs f Cl 2222Cl

Face Centered Unit cellThe fcc crystal structure has atoms at 000, ½½0, ½0½ and 0½½: NFhkl f n e 2 i hun kvn lwn f 1 e i h k e i h l e i k l 1If h, k and l are all even or all odd numbers (“unmixed”), then the exponentialterms all equal to 1 F 4fIf h, k and l are mixed even and odd, then two of the exponential terms willequal -1 while one will equal 1 F 0Fhkl 216f 2,0,h, k and l unmixed even and oddh, k and l mixed even and odd

The Structure FactorNFhkl f n e 2 i hun kvn lwn 1Fhkl amplitude scattered by all atoms in a unit cellamplitude scattered by a single electronThe structure factor contains the information regarding the types (f ) and locations(u, v, w ) of atoms within a unit cellA comparison of the observed and calculated structure factors is a common goal ofX-ray structural analysesThe observed intensities must be corrected for experimental and geometric effectsbefore these analyses can be performed

Integrated IntensityPeak intensity depends onStructural factors: determined by crystal structureSpecimen factors: shape, size, grain size and distribution, microstructureInstrumental factors: radiation properties, focusing geometry, type of detectorWe can say that:𝐼 𝑞 𝐹 𝑞2

Integrated IntensityI hkl (q) K phkl L P A T Ehkl F (q)2K – scale factor, required to normalize calculated and measured intensities.phkl – multiplicity factor. Accounts for the presence of symmetrically equivalent points in reciprocal lattice.L – Lorentz multiplier, defined by diffraction geometry.P – polarization factor. Account for partial polarization of electromagnetic wave.A – absorption multiplier. Accounts for incident and diffracted beam absorption.Thkl – preferred orientation factor. Accounts for deviation from complete random grain distribution.Ehkl – extinction multiplier. Accounts for deviation from kinematical diffraction model.Fhkl – the structure factor. Defined by crystal structure of the material

The Multiplicity FactorThe multiplicity factor arises from the fact that in general there will be several setsof hkl -planes having different orientations in a crystal but with the same d and F 2valuesEvaluated by finding the number of variations in position and sign in h, k and land have planes with the same d and F 2The value depends on hkl and crystal symmetryFor the highest cubic symmetry we have:110, 1 10, 1 1 0, 1 1 0, 101, 10 1, 1 0 1, 1 01, 011, 0 1 1, 01 1, 0 1 1p100 6p110 12111, 11 1, 1 1 1, 1 11, 1 1 1, 1 1 1, 1 1 1, 1 1 1p111 8100, 1 00, 010, 0 1 0, 001, 00 1

The Polarization FactorThe polarization factor p arises from the fact that an electron does not scatteralong its direction of vibrationIn other directions electrons radiate with an intensity proportional to (sin )2:The polarization factor (assuming that the incident beam is unpolarized):1 cos 2 2 P 2

The Lorentz-Polarization FactorThe Lorenz factor L depends on the measurement technique used and, for thediffractometer data obtained by the usual θ-2θ or ω-2θ scans, it can be written asL 11 sin sin 2 cos sin 2 The combination of geometric corrections are lumped together into a singleLorentz-polarization (LP ) factor:1 cos 2 2 LP cos sin 2 The effect of the LP factor is to decrease theintensity at intermediate angles and increase theintensity in the forward and backwardsdirections

The Absorption FactorAngle-dependent absorption within the sample itself will modify the observedintensity Absorption factor for infinitethickness specimen is:A 2 Absorption factor for thinspecimens is given by: 2 t A 1 exp sin where μ is the absorption coefficient, t is the total thickness of the film

The Extinction FactorExtinction lowers the observed intensity of very strong reflections from perfectcrystalsprimary extinctionsecondary extinctionIn powder diffraction usually this factor is smaller than experimental errors and therefore neglected

The Temperature FactorThe temperature factor is givenby:sin 2 B 2 ewhere the thermal factor B isrelated to the mean squaredisplacement of the atomicvibration:B 8 2 u 2This is incorporated into theatomic scattering factor:f f 0e M f 2 e 2 MScattering by C atom expressed in electronsAs atoms vibrate about their equilibrium positions in a crystal, the electrondensity is spread out over a larger volumeThis causes the atomic scattering factor to decrease with sin / (or S 4 sin / )more rapidly than it would normally:

Diffracted Beam Intensity hklI Fhkl F Fhkl2I C (q) KAp ( LP) F (q) I b2where K is the scaling factor, Ib is the background intensity, q 4sinθ/λ is thescattering vector for x-rays of wavelength λFor thin films:2 2 2 t 1 cos 2 I C (q) K 1 exp F(q) Ib 2 sin cos sin

Face Centered Unit cell The fcc crystal structure has atoms at 000, ½½0, ½0½ and 0½½: i h k i h l i k l N ihu kv lw Fhkl fne n n n f e e e 1 1

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