Diffraction: Powder Method - Stanford University

1 diffraction : Powder MethodDiffraction MethodsDiffraction can occur whenever Bragg s law is monochromatic x-rays and arbitrary setting of a single crystal in a beam generally will not produce any diffracted of satisfying Bragg s law: Continuously vary Continuously vary during the main diffraction methods: Method LauevariablefixedPowderfixedvariable = Principal diffraction MethodsLaue Method : single crystal sample, fixed , variable used for orienting single crystalsPowder Method : polycrystalline sample, variable , fixed used in the determination of crystalline structure of materials in Powder formSingle crystal diffractometer Method : single crystal sample, thin film sample, rotating , w, and ,fixed used for determining complex crystal structures from single crystal and thin film materialsPowder DiffractionUnderstanding Powder diffraction patternsPattern componentCrystal structureSpecimen propertyInstrumental parameterPeakPositionUnit cell parameters.

2 A,b,c,a,b,gAbsorptionStrain -StressRadiation (wavelength)InstrumentSample alignmentBeam axial divergencePeakIntensityAtomic parameters:x,y,z,B,..Preferred orientationAbsorptionPorosityGeometry/co nfigurationRadiation (LP)PeakshapeCrystallinityDisorderDefect sGrain sizeStrain -StressRadiation (spectral purity)GeometryBeam conditioningBold key parametersItalic significant influencePowder DiffractionMultiple single crystallites are irradiated simultaneously by a monochromatic beamFor a single dhkl: Powder DiffractionMultiple single crystallites are irradiated simultaneously by a monochromatic beamFor many dhkl:SamplePowder DiffractionIncident beam 1/ Diffracted beamAccording to Euclid.

3 The angles in the same segment of a circle are equal to one another and the angle at the center of a circle is double that of the angle at the circumference on the same base, that is, on the same arc .X-ray diffraction from Polycrystalline Materials2 a= 180o 2 aPaPFor any two points S and D on the circumference of a circle, the angle ais constant irrespective of the position of point Powder DiffractometerPowder diffractometers working in the Bragg-Brentano ( /2 ) geometry utilize a parafocusing geometry to increase intensity and angular resolutionBragg-Brentano geometryPowder DiffractionSample lengthSample irradiated area 2sin 2= sin 2 1sin 2= sin + 2 = 1+ 2 Powder DiffractionSample lengthSample irradiated area = 2o = 1o = 1/2o = 1/4oR = 320 mmPowder DiffractionFixed SlitsDivergence Slit.

4 Match the diffraction geometry and sample size At any angle beam does not exceed sample sizeRL asin (rad)Receiving Slit: As small as possible to improve the resolution Very small slit size reduces diffracted beam intensityPowder DiffractionVariable Slits Vary aperture continuously during the scan Length of the sample is kept constantPowder DiffractionSample Displacement = cos Powder DiffractionSample Displacements = mms = mms = mms = mms = mms = mmR = 320 mmPowder DiffractionIntensity scale: Linear Logarithmic Square RootPowder DiffractionHorizontal scale: 2 1/d 1/d2 Powder DiffractionPeak shape sin2d 2211sin2sin2 a a KKd 3221avea a a KKK Powder diffraction of Powder DiffractometerBasic Principles of Crystal Structure AnalysisThe angular positions of diffracted peaks gives information on the properties (size and type) of the unit cellThe intensities of diffracted peaks gives information on the positions and types of atoms within the unit cellGeneral procedure.

5 Index the pattern assign hklvalues to each peak determine the number of atoms per unit cell (chemical composition, density, and size/shape of unit cell determine the positions of atoms in the unit cell from the measured intensitiesIndexing the pattern an assumption is made as to which of the 7 crystal systems the unknown structure belongs and then, on the basis of the assumption, the correct Miller indices are assigned to each of Cubic PatternsFor cubic unit cell:222ohkladhkl so Bragg s law becomes: 2222222244 sinsinoadhkl so:constant for a given crystalalways equal to an integerbecause of restrictionson h,k,l different cubiccrystal structures willhave characteristicsequences of diffractedpeak positions 202222224sinsinaslkh Indexing of Cubic PatternsCharacteristic line sequences in the cubic system: Simple cubic:1, 2, 3, 4, 5, 6, 8, 9, 10, 11.)

6 Body-centered cubic:2, 4, 6, 8, 10, 12, 14, 16, .. Face-centered cubic:3, 4, 8, 11, 12, 16, 19, 20, .. Diamond cubic:3, 8, 11, 16, 19, ..Indexing of Cubic PatternsIndexing of Cubic PatternsSteps in indexing a cubic pattern:measure sample & list anglescalculate (sin2 )/s for three Bravais latticesIf the observed lines are from a particular lattice type, the (sin2 )/s values should be of non Cubic PatternsDeviations from cubicIndexing of non Cubic PatternsNon-cubic structures --much more complex! 2222222222222222222222222222222232222222 22111143sin2coscos11 3cos2cos11sin2 cossinhkldahkldachkldabchhkkldachklhkklh ldahklhldabcac a a a a a bb b Cubic:Tetragonal:Orthorhombic:Rhombohedr al:Hexagonal:Monoclinic:Indexing of non Cubic PatternsTetragonal System: The sin2 must obey relation: 2222sinClkhA where:222244cCaA andare constants for any patternAcan be found from hk0 indices: 222sinkhA (h 2+ k 2) are 1, 2, 4, 5, 8.

7 Then C can be found from other lines: 2222sinClkhA Indexing of non Cubic PatternsHexagonal System:Orthorhombic System: 2222sinClkhkhA where:222243cCaA and2222sinClBkAh where:2222222,2,2cCbBaA The trick is to find values of the coefficients A, Band Cthat account for all the observed sin2 s when h, k and l assume various integral valuesNumber of Atoms in the Unit CellAfter establishing shape and size we find the number of atoms in that unit cell. Note that the X-ray density is almost always larger than the measured bulk density We need to know the unit cell volume We need to index the Powder pattern in order to obtain the unit cell parametersCoVNnM unit cell volumedensityAvogadro snumbermolecular weightNumber of Atoms in the Unit volume of the unit cell in 3 sum of the atomic weights of the atoms in the unit cell density g/cm3V A For simple elements:AnA1 where n1is number of atoms per unit cellAis atomic weight of an elementFor compounds.

8 MnA2 where n2is number of molecules per unit cellMis molecular weightDetermination of Atomic PositionsRelative intensities determine atomic procedure is trial and error. There is no known general Method of directly calculating atom positions from observed Intensity is given by:where: NilwkvhuinnnnefF 2 Phase problemExample CdTeWe did chemical analysis which revealed: atomic percent as Cd atomic percent as TeMake Powder diffraction and list sin2 Pattern can be indexed as cubic and calculated lattice parameter is: a= The density is determined as g/cm3thenMolecular weight of CdTe is , number of molecules per unit cell is 945 = , or 4.

9 AExample CdTePowder pattern is consistent with FCCTwo structures that would be consistent with 4 molecules per unit cell are NaCl and ZnSNaCl --Cd at 000 & Te at + fcc translationsZnS --Cd at 000 & Te at + fcc translations 22 CdTe22 CdTe1616 FffFff 222 CdTe22 CdTe22 CdTe161616 FffFffFff if (h+k+l) is evenif (h+k+l) is oddif (h+k+l) is oddif (h+k+l) is odd multiple of 2if (h+k+l) is even multiple of 2 Atomic numbersCd 48Te 52 Example CdTeThe Powder diffraction FileExperience has shown that the ensemble of d-spacings ( d s) and intensities ( I s) is sufficiently distinctive in order to identify phasesPhase determination can be performed by a comparison of a set of experimental d s and I s with a database of d -Ifilesd -spacings are independent of wavelengthIntensities are relative (most intense = 100 or 1000 or 1) Powder diffraction File (PDF) DatabaseJ.

10 D. Hanawalt (1902 1987) Powder diffraction File (PDF) Database1919 --Hull pointed out that Powder diffraction could be used for routine chemical analyses Powder pattern is characteristic of the substance the patterns of a combination of phases will superimpose only a small amount is needed1938 --Hanawalt, Rinnand Frevelat Dow Chemical compiled diffraction data on about 1000 compounds as 3 x5 file cards and devised a simple means of classifying the data1942 --the American Society for Testing Materials (ASTM) published the first edition of diffraction data cards (1300 entries)Card image for Ce2(SO4)