Rietveld-Method part I Dr. Peter G. Weidler Institute of Functional Interfaces IFG 1 KIT 10/31/17 The Research University in the Helmholtz Association Name of Institute, Faculty, Department www.kit.edu
Overview introduction, overview data collection background contribution, peak-shape function, refinement of profile parameters, refinement of structural parameters, use of geometric restraints, calculation of e.s.d.'s, interpretation of R values some common problems and possible solution QA with Rietveld -2-
History Hugo M. Rietveld (1932) The Rietveld refinement --> least squares approach --> refinement of theoretical line profile (calculated from a known or postulated crystal structure) --> match with measured profile "Line Profiles of Neutron Powder-diffraction Peaks for Structure Refinement." (Rietveld,H.M.(1967). Acta Crystallogr.,22,151-2.) H. M. Rietveld (1969). "A profile refinement method for nuclear and magnetic structures". Journal of Applied Crystallography 2 (2): 65 71. doi:10.1107/s0021889869006558. -3-
History Gregori Aminoff Prize 1995 --> Ewald, Guinier, Kratky -4-
History -5- relevance of RM
Rietveld-Method what we want! THIS!!! good match of your model with the observed data ==> amount/composition crystallographic info... -6-
Rietveld Software and Utility Software or Features DBWS (Free Dos Structure Refinement Software) FullProf (Free Dos and Mac Structure Refinement Software) RIQAS (Commercial Dos Quantitative Phase Analysis Software) GSAS (Free Dos Structure Refinement Software) SiroQuant (Commercial MS-Windows Quantitative Phase Analysis Software) Quasar (Commercial MS-Windows Quantitative Phase Analysis Software) FAT-Rietan (Free Dos and Mac Structure Refinement Software) ARITVE (Free Dos Glass Modelling Software) Riet7/SR5 (Dos Structure Refinement Software - No Quantitative Analysis) LHPM from ANSTO (Structure Refinement for DOS, WIN) XND (Structure Refinement for DOS) SIMREF and SIMPRO (Structure Refinement for DOS) Koalariet (Developmental Structure Refinement for Win95) BGMN fundamental parameters Rietveld (Structure Ref. WIN OS/2, Linux) XRS-82 Rietveld (Structure Refinement for DOS) ANSTO GUI LHPM-Rietica for Win32 Rietveld and Related Software BRASS - Bremen Rietveld Analysis and Structure Suite TOPAS - Alan Coelho more info under http://www.ccp14.ac.uk/ -7-
Rietveld-Method Parameters refinable (simultaneously) For each phase j present: Global parameters: xj, yj, zj, Bj, Nj xj, yj, zj position coordinates, Bj an isotropic thermal parameter, Nj site-occupancy multiplier for all the jth atom in the unit cell 2θ-Zero Instrumental profile Profile asymmetry Background Wavelength Specimen displacement Specimen transparency Absorption Scale factor (--> quantitative phase analysis) Specimen-profile breadth parameters Lattice parameters Overall temperature factor Individual anisotropic thermal parameters Preferred orientation Crystallite size and micorstrain (--> profile parameters) Extinction -8-
Rietveld-Method some formulas The DREAM is Sy = 0!!! -9-
Rietveld-Method - 10 - some formulas
Rietveld-Method - 11 - finding a solution...
Rietveld-Method - 12 -...what a solution?
Rietveld-Method - 13 -...some details
closer look on some details - 14 - data collection background contribution, peak-shape function, refinement of profile parameters, refinement of structural parameters, use of geometric restraints, calculation of e.s.d.'s, interpretation of R values some common problems and possible solution
data collection For Rietveld refinement, it is essential that the powder diffraction data be collected appropriately incorrect relative intensities and/or 2θ values --> no amount of time spent on refinement will yield sensible results For reflection geometry sample has to be `infinitely thick' i.e. X-ray beam is totally absorbed by the sample However, for highly absorbing materials, a potential source of error is surface roughness. --> can reduce the intensity of low-angle reflections --> leads to anomalously low thermal parameters in refinement. - 15 -
data collection: variable vs. fixed slits: For Bragg-Brentano geometries incident beam to be kept on sample at all angles to ensure a constant-volume condition However, varying slit leads to a progressive angular-dependent defocussing quality of the data deteriorates. slit opening needs to have a precision of at least 1% (reproducible to a few microns over the entire 2θ range) Recommendation: do not use variable slits for a Rietveld refinement - 16 -
data collection: step width and time Step width: at least five steps (not more than ten) across the top of each peak (i.e. step size = FWHM/5) Step time: time per step should approximately compensate for the gradual decline in intensity with 2θ maximum 2θ value should be chosen to give the maximum useful data (.e. as high as possible). different counting times per ranges - 17 -
data collection: preferred orientation PO If intensities show strong dependence e.g. all 00l reflections are strong and all hk0 weak preferred orientation of the crystallites should be suspected. Although many Rietveld refinement programs allow refinement of a preferred-orientation parameter with respect to a specific crystallographic vector based on the March model (Dollase, 1986), this is usually only a crude approximation to reality, so elimination (or minimization) of the problem experimentally is to be preferred. - 18 -
data collection: particle size ideal particle size approx. 1±5 µm If crystallites larger --> non-randomness may become a problem i.e. not all crystallite orientations are equally represented example: quartz α-sio2 10x10x0.2mm³...some calculations yielding the number of particles in diffraction conditions: - 19 -
data collection: diffractometer Calibration of diffractometer careful calibration of 2θ-values with standard material e.g. NIST Si SRM 640b and/or fluorophlogopite mica SRM 675 Any diffractometer can be adjusted so that the deviations of the measured peak positions from the correct ones are less than 0.01(2). Set-up: diffractometer should give a low background and maximum peak resolution (small peak widths) monochromatic radiation e.g. Cu Kα1 rather than Cu Kα1,2 if possible Synchrotron radiation Although longer data-acquisition times are required with monochromatic radiation, its use is particularly advantageous: number of lines in pattern is halved - 20 -
data collection: data pretreatment Any temptation to smooth the diffraction data before doing a Rietveld refinement must be resisted. Smoothing introduces point-to-point correlations conclusion : only best data yields best refinement results - 21 -
Background basically two approaches estimation by linear interpolation btw. selected points btw. peaks modelled by an empirical or semi-empirical function containing several refinable parameters. Both have advantages and disadvantages For simple patterns where most peaks are resolved to the baseline, both methods tend to work well and the fit is easily verified with a plot. This means if background-subtraction approach is used, the background usually has to be re-estimated and re-subtracted several times during a refinement - 22 -
Background Refining the background appears to be the preferred method because background and structural parameters can be refined simultaneously (and std. deviations estimated in the usual way). However, polynomial functions are largely or entirely empirical. If the polynomial happens to describe the background well, then, as might be expected, this procedure also works well; but if it does not, no amount of refining coefficients of polynomial (or increasing the order of the polynomial) can correct the problem and refinement will not proceed satisfactorily. In such a case, background subtraction is the better approach - 23 -
Peak Shape Functions The accurate description of the shapes of the peaks in a powder pattern is critical to the success of a Rietveld refinement. If the peaks are poorly described, the refinement will not be satisfactory The peak shapes are a function of both sample e.g. domain size, stress/strain, defects and instrument e.g. radiation source, geometry, slit sizes and vary as a function of 2θ. In certain cases, they can also vary as a function of indices (hkl). Accommodating all of these aspects in a single peak-shape description is nontrivial and compromises are often made - 24 -
Peak Shape Functions: pseudo-voigt pseudo-voigt function: linear combination of Lorentzian and Gaussian with η/(1- η) the pseudo-voigt mixing parameter Diffraction lines dominated by instrumental broadening, usually vary in a linear manner, from a dominant Gaussian component at low angles to a Lorentzian trend at high angles. - 25 -
Peak Shape Functions: Pearson type VII - 26 -
Peak Shape Functions: TCHZ Thompson-Cox-Hasting pseudo-voigt - 27 -
Profile Parameter structure-free approach If only a partial structural model is available, it is probably best to use a structure-free approach, in which the intensities of the reflections are simply adjusted to fit the observed ones. --> Le Bail- or Pawley-Method --> whole pattern methods Pawley method enables the e.s.d.'s of the reflection intensities to be estimated more correctly and calculates the covariances between overlapping reflections. --> Pawley method of choice --> fast check/estimation of model, amount, additional phases, background-function --> input: space group, lattice parameters, peak shape function - 28 -
Space Groups - 29 -
Space Groups - 30 -
Profile Parameter structure-free approach Pawley-Method CeO: a (Å) 5.41301 GOF 1.63 TCHZ bgr: polynomial 2 order - 31 - +/- 0.00002
closer look on some details...... to be continued - 32 - refinement of profile parameters, refinement of structural parameters, use of geometric restraints, calculation of e.s.d.'s, interpretation of R values some common problems and possible solution
Literature R.A. Young The Rietveld Method IUCr, Oxford University Press, 1993, pp.299 40 95BRL D.L. Bish & J.E. Post (Eds) Modern Powder Diffraction Reviews in Mineralogy Vol 20 Mineralogical Society of America, 1989, pp.369 30 70BRL W.I.F. David, K. Shankland, L.B. McCusker, Ch. Baerlocher Structure Determination from Powder Diffraction Data IUCr, Oxford Science Publications, 2002 (2011 reprint), pp. 337 68 160BRL International Union of Crystallography www.iucr.org - 33 -
Literature - 34 -
Acknowledgment Bruker AXS do Brasil and Bruker AXS Germany, Knielingen CEFET UMFG INCT-Acqua - 35 -