Introduction to Rietveld refinements. Luca Lutterotti Department of Materials Engineering and Industrial Technologies University of Trento - Italy
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1 Introduction to Rietveld refinements Luca Lutterotti Department of Materials Engineering and Industrial Technologies University of Trento - Italy
2 The Rietveld method Need to refine crystal structures from powder. Peaks too much overlapped: Groups of overlapping peaks introduced. Not sufficient. Peak separation by least squares fitting (gaussian profiles). Not for severe overlapping First refinement program by H. M. Rietveld, single reflections + overlapped, no other parameters than the atomic parameters. Rietveld, Acta Cryst., 151, First complete program with structures and profile parameters. Distributed 7 copies (ALGOL) Fortran version. Distributed worldwide Wide acceptance. Extended to X-ray data. Today: the Rietveld method is widely used for different kind of analyses, not only structural refinements. If the fit of the assumed model is not adequate, the precision and accuracy of the parameters cannot be validly assessed by statistical methods. Prince.
3 Principles of the Rietveld method 500 To minimize the residual function: i Iiexp where: Intensity1/ [a.u.] 1 WSS = # w i ( Iiexp " Iicalc ),w i = ! Iicalc = SF Lk Fk S (" i # " k ) Pk A + bkgi k Pk = preferred orientation function S (" i # " k ) = profile shape function (PV : %,HWHM) HWHM = U tan " + V tan " + W #3 / ' * sin & Pk = ) r cos & +, r + (! ! [degrees]
4 Non classical Rietveld applications Quantitative analysis of crystalline phases (Hill & Howard, J. Appl. Cryst. 0, 467, 1987) Nphases I calc i = S n L k F k;n S ( "i # " k;n )P k;n A + bkg i W p = n=1 k S p (ZMV ) p Nphases n=1 S n (ZMV ) n Z = number of formula units M = mass of the formula unit V = cell volume Non crystalline phases (Lutterotti et al, 1997) Using Le Bail model for amorphous (need a pseudo crystal structure)
5 Non classical Rietveld applications Microstructure: Le Bail, Profile shape parameters computed from the crystallite size and microstrain values (<M> and <!>1/) More stable than Caglioti formula Instrumental function needed Popa, 1998 (J. Appl. Cryst. 31, 176). General treatment for anisotropic crystallite and microstrain broadening using harmonic expansion. Lutterotti & Gialanella, 1998 (Acta Mater. 46(1), 101). Stacking, deformation and twin faults (Warren model) introduced. Long Range Order LRO Deformation faulting (intrinsic) Deformation faulting (extrinsic) Twin faulting Antiphase domain Milling time [h] Deformation faulting probability Crystallite size [nm] Crystallite size Microstrain Milling time [h] Microstrain
6 Rietveld Stress and Texture Analysis (RiTA) Characteristics of Texture Analysis: Powder Diffraction Quantitative Texture Analysis needs single peaks for pole figure meas. Less symmetries -> too much overlapped peaks Solutions: Groups of peaks (WIMV, done), peak separation (done) What else we can do? -> Rietveld like analysis? 199. Popa -> harmonic method to correct preferred orientation in one spectrum Ferrari & Lutterotti -> harmonic method to analyze texture and residual stresses. Multispectra measurement and refinement Wenk, Matthies & Lutterotti -> Rietveld+WIMV for Rietveld Texture analysis GSAS got the harmonic method (wide acceptance?).
7 Non classical applications: Texture Orientation Distribution Function (ODF) From pole figures From spectra
8 Non classical applications: strains & stresses Macro elastic strain tensor (I kind) Crystal anisotropic strains (II kind) Fe Cu C Macro and micro stresses Applied macro stresses
9 The classical Rietveld method The function to minimize by a least squares method (non linear): WSS = w i I exp calc # i " I i,w i = 1 i ( ) I i exp the spectrum is calculated by the classical intensity equation: I i calc = S F Nphases j=1 f j V j Npeaks k=1 L k F k, j S j ( " i # " k, j )P k, j A j + bkg i The spectrum depends on phases: crystal structure, microstructure, quantity, cell volume, texture, stress, chemistry etc. instrument geometry characteristics: beam intensity, Lorentz-Polarization, background, resolution, aberrations, radiation etc. sample: position, shape and dimensions, orientation. Each of the quantity can be written in term of parameters that can be refined (optimized).
10 The classical Rietveld method I i calc = S F Nphases j=1 f j V j Npeaks k=1 L k F k, j S ( j " i # " k, j )P k, j A j + bkg i The spectrum (at a " point i) is determined by: a background value some reflection peaks that can be described by different terms: Diffraction intensity (determines the height of the peaks) Line broadening (determines the shape of the peaks) Number and positions of the peaks
11 The classical Rietveld method I i calc = S F Nphases j=1 f j V j Npeaks k=1 L k F k, j S ( j " i # " k, j )P k, j A j + bkg i The more used background in Rietveld refinements is a polynomial function in " : N b is the polynomial degree bkg " i a n the polynomial coefficients N # b n= 0 ( ) = a n ( " i ) n For more complex backgrounds specific formulas are availables It is possible to incorporate also the TDS in the background
12 The classical Rietveld method I i calc = S F Nphases j=1 f j V j Npeaks k=1 L k F k, j S ( j " i # " k, j )P k, j A j + bkg i Starting with the Diffraction Intensities, the factors are: A scale factor for each phase A Lorentz-Polarization factor The multiplicity The structure factor The temperature factor The absorption The texture Problems: extinctions, absorption contrast, graininess, sample volume and beam size, inhomogeneity, etc.
13 The classical Rietveld method I i calc = S F Nphases j=1 f j V j Npeaks k=1 L k F k, j S ( j " i # " k, j )P k, j A j + bkg i The scale factor (for each phase) is written in classical Rietveld programs as: S j = S F f j V j S = phase scale factor (the overall Rietveld generic scale factor) j S = beam intensity (it depends on the measurement) F f = phase volume fraction j V = phase cell volume (in some programs it goes in the F factor) j In Maud the last three terms are kept separated.
14 The classical Rietveld method I i calc = S F Nphases j=1 f j V j Npeaks k=1 L k F k, j S ( j " i # " k, j )P k, j A j + bkg i The Lorentz-Polarization factor: it depends on the instrument geometry monochromator (angle #) detector beam size/sample volume sample positioning (angular) For a Bragg-Brentano instrument: L p = ( ) 1+ P h cos " ( 1+ P h )sin " cos" P h = cos ( " )
15 The classical Rietveld method I i calc = S F Nphases j=1 f j V j Npeaks k=1 L k F k, j S ( j " i # " k, j )P k, j A j + bkg i Under a generalized structure factor we include: The multiplicity of the k reflection (with h, k, l Miller indices): m k The structure factor The temperature factor: B n F k, j = mk N = number of atoms N & n=1 f n e "B n x n, y n, z n coordinates of the n th atom sin # ( ( )) e %i hx n +ky n +lz n f n, atomic scattering factor
16 Atomic scattering factor and Debye-Waller The atomic scattering factor for X-ray decreases with the diffraction angle and is proportional to the number of electrons. For neutron is not correlated to the atomic number. The temperature factor (Debye-Waller) accelerate the decreases.
17 Neutron scattering factors For light atoms neutron scattering has some advantages For atoms very close in the periodic table, neutron scattering may help distinguish them.
18 The classical Rietveld method I i calc = S F Nphases j=1 f j V j Npeaks k=1 L k F k, j S ( j " i # " k, j )P k, j A j + bkg i The absorption factor: in the Bragg-Brentano case (thick sample): A j = 1, µ is the linear absorption coefficient of the sample µ For the thin sample or films the absorption depends on " For Debye-Scherrer geometry the absorption is also not constant There could be problems for microabsorption (absorption contrast)
19 The classical Rietveld method I i calc = S F Nphases j=1 f j V j Npeaks k=1 L k F k, j S ( j " i # " k, j )P k, j A j + bkg i The texture (or preferred orientations): The March-Dollase formula is used: P k, j = 1 m k m k ) n=1 # % P MD cos " n + sin " n P MD & ( ' * 3 P MD is the March-Dollase parameter summation is done over all equivalent hkl reflections (m k ) # n is the angle between the preferred orientation vector and the crystallographic plane hkl (in the crystallographic cell coordinate system) The formula is intended for a cylindrical texture symmetry (observable in B-B geometry or spinning the sample)
20 The classical Rietveld method I i calc = S F Nphases j=1 f j V j Npeaks k=1 L k F k, j S ( j " i # " k, j )P k, j A j + bkg i The profile shape function: different profile shape function are available: Gaussian (the original Rietveld function for neutrons) Cauchy Voigt and Pseudo-Voigt (PV) Pearson VII, etc. For example the PV: 1 ( ) = I n - k ' PV " i # " k +, % ( * + ( 1# k )e #S i,k & 1+ S i,k ) ln. 0 S i,k = " i # " k / 1 k the shape parameters are: Caglioti formula : " = W + V tan# + U tan # N g ( ) n Gaussianity : " = c n # n= 0
21 The classical Rietveld method I i calc = S F Nphases j=1 f j V j Npeaks k=1 L k F k, j S ( j " i # " k, j )P k, j A j + bkg i The number of peaks is determined by the symmetry and space group of the phase. One peak is composed by all equivalent reflections m k The position is computed from the d-spacing of the hkl reflection (using the reciprocal lattice matrix): d hkl = V C s 11 h + s k + s 33 l + s 1 hk + s 13 hl + s 3 kl S=
22 Quality of the refinement Weighted Sum of Squares: N [ ( )] WSS = w i I exp calc # i " I i, w i = i=1 R indices (N=number of points, P=number of parameters): R wp = R exp = N w i I exp calc [ ( i " I i )] i=1 N # # i=1 N exp [ w i I i ], w i = ( N " P), w i = 1 exp [ w i I i ] exp I i # i=1 1 1 I i exp I i exp The goodness of fit: GofF = R wp R exp
23 The R indices The R wp factor is the more valuable. Its absolute value does not depend on the absolute value of the intensities. But it depends on the background. With a high background is more easy to reach very low values. Increasing the number of peaks (sharp peaks) is more difficult to get a good value. R wp < 0.1 correspond to an acceptable refinement with a medium complex phase For a complex phase (monoclinic to triclinic) a value < 0.15 is good For a highly symmetric compound (cubic) with few peaks a value < 0.08 start to be acceptable With high background better to look at the R wp background subtracted. The R exp is the minimum R wp value reachable using a certain number of refineable parameters. It needs a valid weighting scheme to be reliable.
24 WSS and GofF (or sigma) The weighted sum of squares is only used for the minimization routines. Its absolute value depends on the intensities and number of points. The goodness of fit is the ratio between the R wp and R exp and cannot be lower then 1 (unless the weighting scheme is not correctly valuable: for example in the case of detectors not recording exactly the number of photons or neutrons). A good refinement gives GofF values lower than. The goodness of fit is not a very good index to look at as with a noisy pattern is quite easy to reach a value near 1. With very high intensities and low noise patterns is difficult to reach a value of. The GofF is sensible to model inaccuracies.
25 Why the Rietveld refinement is widely used? Pro It uses directly the measured intensities points It uses the entire spectrum (as wide as possible) Less sensible to model errors Less sensible to experimental errors Cons It requires a model It needs a wide spectrum Rietveld programs are not easy to use Rietveld refinements require some experience (1- years?) Can be enhanced by: More automatic/expert mode of operation Better easy to use programs
26 Expert tricks/suggestion First get a good experiment/spectrum Know your sample as much as possible Do not refine too many parameters Always try first to manually fit the spectrum as much as possible Never stop at the first result Look carefully and constantly to the visual fit/plot and residuals during refinement process (no blind refinement) Zoom in the plot and look at the residuals. Try to understand what is causing a bad fit. Do not plot absolute intensities; plot at iso-statistical errors. Small peaks are important like big peaks. Use all the indices and check parameter errors. First get a good experiment/spectrum
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