Rietveld Method_Revised

nullAn introduction to Rietveld MethodAn introduction to Rietveld MethodL. Wu School of Physics, Nankai University Nov, 2009 What is a Rietveld Refinement?What is a Rietveld Refinement? -a standard treatment of powder diffraction data to make the final structural model achieve the accepted criterion; -a best known method that fully makes use of the step-mode scanned data to dig out a lot of structural and other information; - a procedure for structural solution in nature. History ReviewHistory ReviewRietveld originally introduced the Profile Refinement method (Using step-scanned data rather than integrated Powder peak intensity) (1966,1967) Rietveld developed first computer Program for the analysis of neutron data for Fixed-wavelength diffractometers (1969) Malmos & Thomas first applied the Rietveld for analysis of method (RR) x-ray powder data collected on a Ginier Hagg focusing Camera (1977) Khattack & Cox first applied the RR to x-ray powder data collected on a diffractometer (1977) Conference on Diffraction Profile Anlysis Sponsored by IUCr in Poland, suggested the term “Rietveld Method”(1978) Wiles and Yang developed a general computer program (D.B.W) for both x-ray & neutron diffraction data (fixed wavelength)(1981) Von Dreele, Jorgensen and Windsor extended to the program to the neutron diffraction data (1982) Fitch et al, 193 refined parameters,UO2 DAs.4D2O (1982) A. Le Bail, Whole-pattern decompostion, Le Bail Method(1988) Morris et al, Ga2(HPO3)3.4H2O, Extracting 551 reflections, 10 O atoms (1992)What can we get to perform a Rietveld refinement? What can we get to perform a Rietveld refinement? Lattice Parameters Quantitative phase Analysis Atomic Positions Grain size Atomic Occupancy Incommensurate Structure Debye Temperatures Structure factors Crystallinity Phase transitions Magnetic structures ……How RM works?How RM works?The RM refines a structure by minimizing a quantity through the Newton-Raphson algorithmwhere, yi is the observed intensity at a certain 2, yc,i is the calculated intensity at the same angle, wi is a weight, we usually take wi=1/yi i=1,2,…n =( 1 2 …p), the parameters to be refined.nullStructural model Raw dataRietveld RefinementRefined modelshiftGiven a solution =opt(1, 2… p) that approximately satisfy the above equation. To find a better solution, we begin an iterative process by expanding into a Taylor series.Given a solution =opt(1, 2… p) that approximately satisfy the above equation. To find a better solution, we begin an iterative process by expanding into a Taylor series.S is the scale factor of the phase  Lh contains the Lorentz, polarisation and multiplicity factors. Fh is the structure factor Ah is the absorption correction Ph is the preferred orientation function Ω is the reflection profile function that models both instrumental and sample effects S is the scale factor of the phase  Lh contains the Lorentz, polarisation and multiplicity factors. Fh is the structure factor Ah is the absorption correction Ph is the preferred orientation function Ω is the reflection profile function that models both instrumental and sample effects nullwhere, fi atomic scattering factor for ith atom xi, yi and zi the fractional coordinates for ith atomThe mean square displacement of the atom in a direction normal to the reflecting planesnullGoodness of neuron diffractionnullWhere RELAX is relaxtion factors that are used to control the shifts to avoid divergence; and CC is a multiplier.i is a shift. null What we need to perform a RR? A set of step-mode scanned data, usually 2=10-120˚ or more, step  2=0.02˚ collecting time is instrument dependent from 1-20s for laboratory diffractometer; An initial structural model having roughly accurate lattice constants, correct space group and approximate atomic positions nullHow we obtain an initial structural model? - solid solutions usually adopt same structure types of their parent compounds; NaSr4-xBaxB3O9 (0≤x≤4) Compounds with same chemical formula YBa2Cu3O7 and NdBa2Cu3O7 but always alert that exceptions are not uncommon La2CuO4 and Nd2CuO4 Try and error Ab inito structure determination nullRaw dataIndexing |F| extracting Atom assignmentDifference Fourier Rietveld refinement Crystal structureSteps of structure determination from powder dataIs the compound known? Crystallographic Structure DatabasesIs the compound known? Crystallographic Structure DatabasesICSD (Minerals and Inorganics) http://www.fiz-karlsruhe.de/ Minerals and Inorganic Over 60000 entries Cambridge Structure Data Bank） http://www.ccdc.cam.ac.uk Organics & Organometallics Over 250000 entriesICDD diffraction data http:http://www.icdd.com/ Inorganic & Organic Over 140000 entries NIST Crystal Data http://www.nist.gov/srd/nist3.htm Inorganic & Organic Over 230000 entriesnullA new structural database(2003): aimed at freely retrieving data18000 Patterns already!nullXRD patterns of Solid solution NaSr4-xBaxB3O9 (0≤x≤4) :First cubic borate only with BO3 Cubic borates are estimated below 1%.L. Wu, X.L. Chen, et. al. 2004Isotructural compounds: LnBa3B9O18 (Ln=Y, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb) Isotructural compounds: LnBa3B9O18 (Ln=Y, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb) X.Z. Li, X.L. Chen, et. al. 2004NaCaBO3: A structure with partial soluble sites determined from SDPD NaCaBO3: A structure with partial soluble sites determined from SDPD Wu, Chen et al(2005)Parameters in PCR fileParameters in PCR fileThe parameters in PCR file can be divided into three categories -relating only to samples, refinable such as atomic positions, temperature factors -relating both to samples and instruments such as scale factors, FWHM (Full width at half maximum) -user-specified parameters such as BKPOS, Nba Modeling backgroundsModeling backgroundsThe background intensity bi at the ith step may be obtained by any of the following three method. a specified background function, usually a polynomial; linear interpolation between user-selected points in the pattern A user-supplied function Control flagsControl flagsThe choice of background type is indicated by a control flag Comment line(4) Job Npr Nph Nba Nex… 0 5 1 0 2 Nba: =0 Refine background with a polynomial =1 Read background from file COFHIL.bac =2,3,…,N linear interpolation between N given points … nullnullWhere Bm are parameter to be refined BKPOS is a user-specified parameter, origin of polynomial function, non-refinable. If 2=BKPOS, we see bi=B0 Users can look into their data files to set the values of BKPOSProfile functions (I)Profile functions (I)Gaussian (G) Npr=0 Lorentzian (L) Npr=1Parameter to be refined: Hk, Full Width at Half Maximum (FWHM)Hk=0.2Hk=0.2Profile functions (II)Profile functions (II)Mod.I Lorentzian Npr=2 Parameter to be refined: Hk, Mod.I Lorentzian Npr=3 Profile functions (III)Profile functions (III)Psudo-Voigt Npr=5Parameters to be refined: Hk, η0, Xη0 = shapePseudo-Voigt functions Hk=0.2Pseudo-Voigt functions Hk=0.2Profile functions (III)Profile functions (III)Pearson VIIParameters to be refined: Hk, m0, X,YProfile functions (IV)Profile functions (IV) (Mod-TCHZ pV) L(x) and G(x) have different FWHMs HL and HGParameters to be refined: HG and HLFull width at half Maximum (FWHM) Full width at half Maximum (FWHM) For Npr=0…6, Hk=HG For Npr=7, HL is required apart from HG Typical variations of FWHM vs 2 Typical variations of FWHM vs 2 Summary for the parameters to be refined with different profilesSummary for the parameters to be refined with different profilesNpr=0, Gaussian: U, V, W, Ig 3 Npr=5, pv: U,V,W,Ig, η0(Shape), X 5 NPr=6, Pearson VII: U,V,W,Ig, η0(Shape), X,Y 6 NPr=7, TCHZpv: U,V,W,Ig, X,Y,Sz 6nullPreferred orientations (I)Preferred orientations (I)Nor=0, Rietveld-Toraya ModelG1 and G2 are refinable parameters H is the acute angle between d*H and the normal to the crystallites (platy habit)Note: preferred orientation vector Pr1,Pr2 and Pr3 is needed to specify a priori by usersPreferred orientations (II)Preferred orientations (II)Nor=1, modified March’s ModelG1 and G2 are refinable parametersG1<1, platy habit, G1=1, no preferred orientation G1>1 Needle-like habitnullWDT·FWHMWDT·FWHMWDT>5, preferably 10Monochromator polarization correctMonochromator polarization correct Incident angle to a monochromator CTHM=cos22=0.8009 for a graphite monochromator, CuK Asymmetry correction for profilesAsymmetry correction for profilesP1, P2, P3 , and P4 are parameters to be refinedAsyLim: peaks below this 2 angle limit are corrected for asymmetry nullOccupancyOccupancym is the site multiplicity, M is the multiplicity of the general site for a given space group. For example, KCaFCO3, P-6m2(187) K+1 occupies 1(a) site; O-2 3(k) site; general site 12(o), Occ(K+)=1/12=0.08333, Occ(O-2)=0.25, both atoms’ chemical occupancy=1.0 EPS and Relax factors EPS and Relax factors Forced termination when shifts10000 Too less sample. Sample should fully cover the sample holder window; Overflow in low angle region; Too less angle region 2>120˚; Too large EPS that leads to false minimum; Improper profile function; Too less WDT values .Error messages (I)Error messages (I)‘Hole in Matrix’ : the number of parameters to be refined NPR larger than the number of codewords For example, you set NPR= 12, while one codeword 80.5 is missing or 101.0 is mistyped as 11.0 - ‘Negative FWHM’: HG2 <0, meaningless! Increasing the negative U,V, or W while set smaller Relax values in the ensuing the refinements Error messages (II)Error messages (II)- ‘No scattering factor’ : atom identifier ‘TYP’ is not recognizable by Fullprof. For example, Ca+2 is accepted while Ca2+ is not accepted. ‘Too many reflections’: For a given point, there are too many reflections contributing to the intensity that are beyond the software’s capacity. Usually this results from the false FWHM ‘Invalid integer’ or ‘Invalid real’: Examine the format of parameters Note: The software does not always give the correct lines where errors occur. Look into the nearby lines!Calculation of bond length, angle and bond valence sumCalculation of bond length, angle and bond valence sumESDsESDsnullBond valence sum is a good indicator of the structural validity. For details, see I.D.Brown, Acta Crystallogr. B48, 141(1992)Structure factors extracting by Le Bail Method Structure factors extracting by Le Bail Method Linear extrapolated backgroundnullIndexing:one of most important steps in SDPD Organizers: 18.881 18.881 18.881 90 90 90 (vol 6734 Å3) P1: 13.349 13.349 9.439 90 90 90 (Tetragonal - 1638Å3) P2: 18.878 18.878 18.878 90 90 90 P3: 13.354 13.354 9.442 90 90 90 (Tetragonal - 1638Å3) P4: no solution P5: 18.878 18.878 18.878 90 90 90 P6: 18.88 18.88 18.88 90 90 9050% successful only ! Possibly because of too small default maximum volume limit. P1 and P3 provide the same (correct) subcell.Data quality and indexing resultsData quality and indexing resultsC7H17ClN2O2 a=8.875 b=16.408 c=7.137 =93.84 P21/n ICDD-PDF 43-1748 8 impurity peaks, ZP error= 0.1 ICDD-PDF 46-1964(1995) 3 impurity peaks, ZP error=0.1 New data of diffractormeter New data of synchrotron ITO 13 < DICVOL91= TREOR< Crysfire = McMaille Impurity is the No. 1 killer ; Decreasing zero point error is important; High instrumental resolution is desired. Dicvol: a useful indexing software Dicvol: a useful indexing software A dichotomy procedure is applied to find the solution. Qi=1/di2=hi2a*2+ki2b*2+li2c*2 [a-=a0+np, a+=a-+p] [b-=b0+np, b+=b-+p] [c-=c0+np, c+=c-+p] Calculate Qi –(hkl), Qi +(hkl), and see if the observed Qoi lies within the [Qi –, Qi +].FullProf_SuiteFullProf_SuitePeak search Peak search nullnullCeO2 a=0.5411nm Fm-3m(225) Z=4nullYou can set a larger value if you data are of a quite poor qualityAlways try cubic first nullFigure of Merit M(16)=125 F(16)=32.2(0.0098,51)Lattice constant and its estimated standard deviationX-ray diffraction pattern of KCaFCO3X-ray diffraction pattern of KCaFCO3Delete the small peak!nullBy deleting 2 weak peaks, now you have totally remaining 24 peaks. Save the peak file in Dicvol format For indexing, you can further delete the peaks with 2>80˚. Now you only have 17 peaks left. For indexing, you can further delete the peaks with 2>80˚. Now you only have 17 peaks left. 13nullnullnullFigure of MeritFigure of MeritWhere Q20is the value of Q (1/d2) for 20th observed line N20 is the number of different calculated Q values up to Q20  the average error in Q valueDe Wolff FOMSmith and Snyde FOMWhere Nobs the number of observed lines, Ncal the number of calculated line up to Nth line For reliable indexing results M,F>10!nullMulti-solutions? Hexagonal Lattice a=5.1030(2) Å, c=4.4563(3) Å FOM: M(17)=247.9, F(17)=130.9 Orthorhombic Lattice a= 4.4555(5)Å, b= 4.4188(4)Å, c= 2.5512(5)Å, V= 50.23Å3 FOM: M(17)=180.6, F(17)=97.95.14.422.55ao≈ ch, bo≈ ahsin60o co ≈ 1/2ah, Hints for indexingHints for indexing Select well-defined peaks always; Omit the high-angle peaks Be alert with the weak peaks After getting a solution, calculate the all possible peaks to identify the remaining observed peaks Then re-indexing again by adding the identified peaks to the original data. Put space group Pmmm herePut space group Pmmm hereUsing Powder Cell to calculate all possible diffraction linesnullReflection conditions, 0kl: k+l=2n, h0l:h=2n Extinction symbol, Pna-, Space group: Pna21(33) or Pnam(62) nullSpace group conversiona=8.5Å, b=7.0Å,c=5.4Å Pnam8.57.05.45.48.57.0a=8.5Å, b=5.4Å,c=7.0Å PnmaStandard space groupNon-standard space groupSpace group conversion (II)Space group conversion (II)Non-standard space group for No.62 Pnma Pmnb, Pbnm, Pcmn, Pmcn, Pnammbnnmaa’=b, b’=a, c’=c Pmnb Pnmanull8.5cnmaPnmaab5.47.0abcnamPnam8.55.47.0Space group transformationsnull7.0ccmPcmnab5.48.5n6 equivalent space groupsStandard space groupPrepare your pcr file Prepare your pcr file Using FullProf PCR Editor; Correct on a readable old pcr nullPrepare your pcr file(I)Pcr=1 CODFIL.pcr is re-written with updated parameters.Prepare your pcr file (II)Prepare your pcr file (II)nullchenhongMAC MXP18A-HFFormCon2.0nullhuming理学 DMAX 2000nullRefine scale factor S nullRwp=962 138, S=0.5E-2 0.64E-3nullRefine zero point along with SnullRwp=62.8 ZP≈0.05nullRefine background along with S and ZPnullRwp=32.9%Refine lattice parameters along with othersRefine lattice parameters along with othersnullRwp=28.9%nullRefine peak profile along with other parametersnull Rwp=18.9%Refine asymmetry Refine asymmetry null Rwp=16.8%nullRefine atomic coordinates: first two atoms Pb and S the number of parameters to be refined:22nullRwp=14.0%Further refine atomic coordinates of 3 O atoms Rwp=13.1%nullRefine temperature factors along other parametersnullRwp=12.5%nullRp=8.92%, Rwp=10.8%, Rexp=6.57%nullCalculate the bond lengths and bond valences And the results are stored in *.disnullababcPbSO4Rp=7.3%-16.6% 5.82% 8.91% Rwp=8.2-20.0% 7.83% 10.8% Rexp=1.5%-7.0% 4.83% 6.71% GodF=1.3-7.4 1.6 1.6 Rp=7.3%-16.6% 5.82% 8.91% Rwp=8.2-20.0% 7.83% 10.8% Rexp=1.5%-7.0% 4.83% 6.71% GodF=1.3-7.4 1.6 1.6 Compare with the Rietveld Refinement Round RobinTotal 23 respondentsBackground excludednull