1、陈小龙 中国科学院物理研究所2006.10 杭州,Lecture notes,Rietveld 方法原理,Isostructural Compounds,Nd2CuO4 Gd2CuO4 a=0.39419nm a=3.8938nm c=1.21627nm c=11.8810nm ZNd=0.353 ZGd=0.349,Nd(Gd),XRD patterns of Solid solution NaSr4-xBaxB3O9 (0x4) :First cubic borate only with BO3,Cubic borates are estimated below 1%.,L. Wu, X.
2、L. Chen, et. al. 2004,Phase transition of BaTiO3 from tetragonal to cubic at about 132C,YBa3B3O9: Phase transition and structure determination,S. G.: P63cm (No. 185) a=9.4235(4), c=17.602(1) 1100C,S. G.: R-3 (No. 148) a=13.0441(1), c=9.5291(1) 1140C,X.Z. Li, X.L. Chen, et. al. 2004,LiAlB2O5: Search
3、for new SHG materials,B3O75- & AlB2O75-,Structural Data for LiAlB2O5,Final Refinement of New compound of LiAlB2O5,Structure vs Temperature:KCaCO3F,0.8120(3) 0.1880(3) 0.5 by x-ray data,LiSr4B3O9 : A comparison between structure determination from single-crystal and powder X-ray diffraction,a= 14.947
4、0 S.G: Ia-3d (ZB/ZSr)2=(3/38)2 0.6%,Single-crystal: Rint=0.0745R1(all data)=0.0695wR2(all data)=0.1887 with weighting scheme: W=1/2(Fo2)+(0.0000P)2+359.71P where P=(Fo2+2Fc2)/3 SDPD: RB=0.07Rp=0.0609Rwp=0.0811Rexp=0.0314,What is a Rietveld Refinement?,-a standard treatment of powder diffraction data
5、 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.,What can we get to perform a Rietveld refinement?,Lattice P
6、arameters Quantitative phase Analysis Atomic Positions Grain size Atomic Occupancy Incommensurate Structure Debye Temperatures Structure factors Crystallinity Phase transitions Magnetic structures,History Review,Rietveld originally introduced the Profile Refinement method (Using step-scanned data ra
7、ther 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 refinement method (RR) for analysis of x-ray powder data collected on a Ginier Hag
8、g 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 bot
9、h 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),Aminoff Prize, Stockholm,1995,H.M. Rietveld Acta crystallogr., 22, 151 (1967). H.M.Rive
10、ted, J. Appl. Crystallogr., 2, 65 (1969).,Structural model,Raw data,Rietveld Refinement,Refined model,shift,How RM works?,The RM refines a structure by minimizing a quantity through the Newton-Raphson algorithm,where, yi is the observed intensity at a certain 2, yc,i is the calculated intensity at t
11、he same angle,wi is a weight, we usually take wi=1/yii=1,2,n =( 1 2 p), the parameters to be refined.,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.,A1=b,1= 0+ 1,1 is a shift.,S
12、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 is the reflection profile function that models both instrumental and sample effects,where, fi atomic sc
13、attering factor for ith atomxi, yi and zi the fractional coordinates for ith atom,The mean square displacement of the atom in a direction normal to the reflecting planes,Where RELAX is relaxtion factors that are used to control the shifts to avoid divergence; and CC is a multiplier.,What we need to
14、perform a RR?A set of step-mode scanned data,usually 2=10-120 or more, step 2=0.02collecting time is instrument dependent from 1-20sfor laboratory diffractometer;An initial structural model having roughly accurate latticeconstants, correct space group and approximate atomic positions,How we obtain a
15、n initial structural model?- solid solutions usually adopt same structure types of their parent compounds;NaSr4-xBaxB3O9 (0x4)Compounds with same chemical formulaYBa2Cu3O7 and NdBa2Cu3O7but always alert that exceptions are not uncommonLa2CuO4 and Nd2CuO4Try and errorAb inito structure determination,
16、Is the compound known? Crystallographic Structure Databases,ICSD (Minerals and Inorganics) http:/www.fiz-karlsruhe.de/ Minerals and Inorganic Over 60000 entriesCambridge Structure Data Bank） http:/www.ccdc.cam.ac.uk Organics & Organometallics Over 250000 entries,ICDD diffraction data http:http:/ Ino
17、rganic & Organic Over 140000 entriesNIST Crystal Data http:/www.nist.gov/srd/nist3.htm Inorganic & Organic Over 230000 entries,A new structural database(2003): aimed at freely retrieving data,18000 Patterns already!,Parameters in PCR file,The parameters in PCR file can be divided into three categori
18、es-relating only to samples, refinablesuch as atomic positions, temperature factors -relating both to samples and instrumentssuch as scale factors, FWHM (Full width at half maximum) -user-specified parameterssuch as BKPOS, Nba,Codewords(I),codewords are used to control parameters when to be refined,
19、 when to be fixed and when to be constrained and etc. A codeword is formed asC=S(10P+CC) Where S stands for the sign mark, P is an ordinal number set by users from 1 to p, the maximum number of parameters,Codeword(II),For example, an atom Ca position is (0,0,z) with z to be refined from its initial
20、value 0.1. The codeword in your PCR file looks like the followingCa1 Ca+2 0.0 0.0 0.1 0.0 0.0 120.5 Here, S=1, P=12, and CC=0.5. That means that z of Ca1 is the 12nd parameter to be refined in the iterative process, and x and y of Ca1 occupy special positions not needed to be refined.,Codeword(III),
21、Another example: the lattice constants of a tetragonal compound are to be refined. The codeword in your PCR file looks as follows3.891 3.891 11.73251.0 51.0 61.0In this case, constraint is put on a and b by using the same codeword since a=b always holds in tetragonal compounds,Codeword(IV),One more
22、example: the occupancies of two kinds of atoms at one site are to be refined. Solid solutions are the most common among this kind of refinements. The codewords in your PCR file are set asY Y+3 0.8 10.3Yb Yb+3 0.2 -10.3 Only in this way are the occupancies guaranteed to satisfy Occ(Y)+Occ(Yb)=1,Codew
23、ord(V),each parameter usually controlled by one codeword. Be alert that one codeword should be given to two or more parameters that are irrelevant;- there is no limit to choose ordinal numbers.But we usually set the first ordinal numbers to global parameters such as zero point, background parameters
24、 and the etc.,Modeling backgrounds,The 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 flags,The choi
25、ce of background type is indicated by a control flag Comment line(4) Job Npr Nph Nba Nex0 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,Where Bm are parameter to be refined BKPOS is a user-specified parame
26、ter, 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 BKPOS,Profile functions (I),Gaussian (G)Npr=0,Lorentzian (L) Npr=1,Parameter to be refined: Hk, Full Width at Half Maximum (FWHM),Hk=0.2,Profile functions (II),Mod.I L
27、orentzian Npr=2,Parameter to be refined: Hk,Mod.I Lorentzian Npr=3,Profile functions (III),Psudo-VoigtNpr=5,Parameters to be refined: Hk, 0, X,0 = shape,Pseudo-Voigt functions Hk=0.2,Profile functions (III),Pearson VII,Parameters to be refined: Hk, m0, X,Y,Profile functions (IV),(Mod-TCHZ pV),L(x) a
28、nd G(x) have different FWHMs HL and HG,Parameters to be refined: HG and HL,Full width at half Maximum (FWHM),For Npr=06, Hk=HG,For Npr=7, HL is required apart from HG,Typical variations of FWHM vs 2,Summary for the parameters to be refined with different profiles,Npr=0, Gaussian: U, V, W, Ig 3 Npr=5
29、, pv: U,V,W,Ig, 0(Shape), X 5 NPr=6, Pearson VII: U,V,W,Ig, 0(Shape), X,Y 6 NPr=7, TCHZpv: U,V,W,Ig, X,Y,Sz 6,Preferred orientations (I),Nor=0, Rietveld-Toraya Model,G1 and G2 are refinable parameters H is the acute angle between d*H and the normal to the crystallites (platy habit),Note: preferred o
30、rientation vector Pr1,Pr2 and Pr3 is needed to specify a priori by users,Preferred orientations (II),Nor=1, modified Marchs Model,G1 and G2 are refinable parameters,G11 Needle-like habit,Systematic line-shift,Bragg-Brentano Geometry Specimen displacementSpecimen Transparency,SYCOS,SYSIN,: the linear
31、 absorption coefficient of the sample,WDTFWHM,WDT5, preferably 10,Monochromator polarization correct, Incident angle to a monochromator,CTHM=cos22=0.8009 for a graphite monochromator, CuK,Asymmetry correction for profiles,P1, P2, P3 , and P4 are parameters to be refined,AsymLim: peaks below this 2 a
32、ngle limit are corrected for asymmetry,Occupancy,m 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 occupan
33、cy=1.0,Agreement Factors (I),Profile Factor,Weighted Profile Factor,Expected Weighted Profile Factor,Agreement Factors (II),Goodness of fit indicator,Bragg Factor,Crystallographic RF factor,Variations of agreement factors and esd.,Hill & Madsen, Powder Diffraction(1987),An estimation of S for an ide
34、al refinement,take,S1 since NP,Gaussian,Lorentzian,R.J. Hill and H.D. Flack, J.Appl. Cryst. 20 (1987) 356-361,Durbin-Watson statistic parameters,d 4-QD: negative serial correlation,Variations of d, eds, Rwp and RB vs cycles,R.J. Hill and H.D. Flack, J.Appl. Cryst. 20 (1987) 356-361,Comments on agree
35、ment factors,RF and RB are more indicative of structural model fits Rp and Rexp are more indicative of overall profile fits RF ,RB , Rp and Rexp are not good indices for the refinements of different patterns S should be as close as 1 d is a more sensitive index over RF ,RB , Rp and Rexp At least Rp,
36、 Rwp and Rexp should be given when submitting a paper to a journal,Quantitative Phase Analysis,where, Wj is the weight fraction for the jth phase; Sj is scale factor for the jth the phase;Zj is the number formula units per cell for the jth phase;Mj is the mass of the formula unit; Vj is the unit cel
37、l volume;tj Brindley coefficient that comes into effect when thelinear absorption coefficients of phases in powderdiffer a lot to each other.,KCaFCO3,CaCO3,Multiphase Rietveld Analysis,Result in *.out file,No absorption correction is applied,To obtain a satisfactory quantitative phase analysis based
38、 on the Rietveld method, we should be cautioned: - Sample should be carefully prepared: powder is homogeneous in compositions and have a sufficient number of grains with random orientations; Structures of phases are well known; Absorption correction is applied whenever the phases differ a lot in the
39、ir linear absorption coefficients. The Brindley coefficients can be consulted in the Fullprof Manual.,Calculation of bond length, angle and bond valence sum,ESDs,Bond valence sum is a good indicator of the structural validity. For details, see I.D.Brown, Acta Crystallogr. B48, 141(1992),EPS and Rela
40、x factors,Forced termination when shiftsEPS*esd Recommended EPS=0.1,R_at: Atomic parameters, including atomic coordinates and etc. R_an: anisotropic parameters R_pr: profile parameters, lattice parameters, preferred orientations and the etc; R_gl: global parameters, such as zero shift, backgrounds a
41、nd theetc.,2 Excluded 2 regions,Suggested turn-on sequences for the parameters,Scale factors Zero shift Background FWHM Shape1, X, Y, 6. Lattice parameters (if accurate to some extent otherwise do it before refining FWHM) 7. Atomic coordinates 8. Temperature factors, occupancies 9. Preferred orienta
42、tion, GauSiz, LorSiz,Some factors affecting refinement results,Low instruments resolution. RS=0.1-0.2mm Too low counts. Strongest counts 10000 Too less sample. Sample should fully cover the sample holder window; Overflow in low angle region; Too less angle region 2120; Too large EPS that leads to fa
43、lse minimum; Improper profile function; Too less WDT values .,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, meani
44、ngless! Increasing the negative U,V, or W while set smaller Relax values in the ensuing the refinements,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,
45、 there are too many reflections contributing to the intensity that are beyond the softwares capacity. Usually this results from the false FWHMInvalid 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
46、nearby lines!,chenhong,MAC MXP18A-HF,FormCon2.0,huming,理学 DMAX 2000,Refine scale factor S,Rwp=962 138, S=0.5E-2 0.64E-3,Refine zero point along with S,Rwp=62.8 ZP0.05,Refine background along with S and ZP,Rwp=32.9%,Refine lattice parameters along with others,Rwp=28.9%,Refine peak profile along with
47、other parameters,Rwp=18.9%,Refine asymmetry,Rwp=16.8%,Refine atomic coordinates: first two atoms Pb and Sthe number of parameters to be refined:22,Rwp=14.0%,Further refine atomic coordinates of 3 O atoms Rwp=13.1%,Refine temperature factors along other parameters,Rwp=12.5%,Rp=8.92%, Rwp=10.8%, Rexp=
48、6.57%,Calculate the bond lengths and bond valences And the results are stored in *.dis,a,b,a,b,c,PbSO4,Fourier synthesis,Set Fou=4 in your Pcr file,PbSO4,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 Robin,
