A Voigtian as Profile Shape Function in Rietveld Refinement

Transcription

1 352 J. Appl. Cryst. (1984). 17, A Voigtian as Profile Shape Function in Rietveld Refinement By M. AHTEE, L. UNONIUS, M. NURMELA* AND P. SUORTTI Department of Physics, University of Helsinki, Siltavuorenpenger 20 D, SF Helsinki 17, Finland (Received 5 March 1984; accepted 30 May 1984) Abstract A modification of the Rietveld refinement program is introduced by replacing the Gaussian profile function with a Voigtian. The performance of the program is tested by an application to simulated and measured diffraction patterns of NaTaO3, and excellent results are obtained when the tails of the reflections are included at distances 10 to 20 times the half-width of the reflections. Comparison between Gaussian and Voigtian refinements show large differences in the thermal parameters. This is due to the high level of background that is assumed in the Gaussian refinement to compensate for the lack of tail overlap. Application of the Voigtian refinement to the measured pattern of Ni yields a thermal parameter that is close to the literature values, while the Gaussian analysis gives a value which is 35% too large. The isotropic temperature factors of the room-temperature structure of NaTaO 3, respectively, drop down to the values B(Ta) = 0.09(2), B(Na) = 1.21 (3), B(O 1) = 0.59(3) and B(O2)=0.61(3)A 2, when the Voigtian analysis with a proper background is carried out. The role of background in powder pattern refinement is discussed, and it is suggested that a calculation of the average TDS should be included in the refinement. Introduction The use of Rietveld's (1969) structure refinement method by profile fitting has lately increased considerably. Originally, this method was applied to fixedwavelength neutron data (Hewat, 1973a, b), and later adapted to the X-ray case (Malmros & Thomas, 1977). Subsequent applications include synchrotron radiation data (Glaer, Hidaka & Bordas, 1978) and timeof-flight neutron powder patterns (Von Dreele, Jorgensen & Windsor, 1982). A review of the Rietveld method including all these applications has been given by Albinati & Willis (1982). Some problems of the Rietveld method arise from the overlap of reflections and non-ideal powder samples. The latter include primary extinction, prefer- *Present address: Bureau of Physical Computation, University of Helsinki, Finland. red orientation and granularity effects, and these have been only marginally included in the present methods. The actual profile and the true background of a Bragg reflection are unknown due to overlap, and studies of various model functions are numerous. Much of this work is summaried by Young & Wiles (1982), who compare six analytical functions in Rietveld refinements of X-ray data. In their conclusion, they emphasie the need for a better profile function that properly incorporates the sample and instrument parameters. We have earlier (Suortti, Ahtee & Unonius, 1979) studied the performance of the Voigt function in profile fitting both in the neutron and X-ray (fixedwavelength) powder pattern of nickel. The Voigt function, which is a convolution of a Gaussian function giving the shape of the main peak and a Lorent function giving the correct asymptotic behaviour of the reflection, approximates well the observed profile so that the measured profiles of the powder reflections could be fitted over a much larger angular region than using the pure Gaussian. In the neutron case, the overall fit was very good, the improvement being one order of magnitude over the Gaussian fit. The intensity in the tails of the reflections, which was recovered by the Voigtian profile, was found to affect the thermal parameters substantially, and the analysis suggested that the thermal parameters from the Gaussian refinement are systematically too large. This early work also demonstrated the importance of a correct determination of the background. In this paper, we report implementation of the Voigt function in the Rietveld refinement program for fixedwavelength neutron data (Hewat, 1973a, b). It has been demonstrated that the Voigt function could be approximated closely by the so-called pseudo-voigt, which is the sum of Gaussian and Lorentian functions with an adjustable mixing parameter (Young & Wiles, 1982). However, the actual convolution of the two functions is to be favoured if independent factors of Gaussian or Lorentian form can be assumed to contribute to the reflection profile. Computation time is not increased substantially, and convolution also gives an option of an easy Fourier analysis of the profiles. In order to test the program without ambiguities of overlapping reflections and unknown /84/ L" 1984 International Union of Crystallography

2 M. AHTEE, L. UNONIUS, M. NURMELA AND P. SUORTTI 353 background of a measured pattern, we have constructed simulated powder patterns based on the roomtemperature structures of NaTaO3 and nickel. In this way, we have also been able to isolate certain problems of the whole-pattern-fitting structure refinement method. The revised program has been published (Ahtee, Unonius, Nurmela & Suortti, 1984) and is available on request. 2. Calculation of the Voigt function When the profile of a powder reflection is given in a functional form, appropriate parameters are the fullwidth at half-maximum intensity (FWHM), 2w, and the integral breadth/3, which is the area A under the reflection divided by the maximum intensity I(0). For a single-parameter function, such as Gaussian or Lorentian, the ratio 2w//3 is a characteristic constant, but in the following we will study a two-parameter function, namely Voigtian, for which 2w//3 may be selected within certain limits. The Voigt function is the convolution of the Lorent and Gauss functions and can be written as (Langford, 1978) I(q~)=/3LIL(O)I~(O)Re )o9~--~-c, q~+iy, (1) where ~o = A(20) is the angular distance from the centre of the reflection,/3l and/3g are the integral breadths and 1L(0) and IG(0) the maximum intensities of the Lorent and Gauss functions, y=/3l/g 1/2 /3G, and oj() is the complex error function og() = exp( - Z)erfc( - i). (2) The relative weights of the Lorentian and Gaussian functions are given by y; the value y = 0 corresponds to a pure Gaussian and the value y=~ to a pure Lorentian function. The area under the Voigtian is A= /3LIL(O) /3j(O) (3) and by normaliing this to unity the Voigtian reduces to I(~0) = fl~ Re ]~o~,--~- G q9 + iy. (4) The integral breadth is thus /3=/3~exp(- yz)/erfc y. (5) The ratio of the half-width to the integral breadth, 2w//3, is related to y through ( ft ~/2 2w//3 + iy = 5 exp(y2)erfc y. Re ~,o~-~ e xp(y2)erfc y The actual relation between 2w//3 and y, which is (6) cumbersome, was approximated by 2w (l~_)l/z l +ay+by2 (7) =2 1 +cy+dy 2' where the values of the parameters a, b, c and d are determined by a least-squares fit. This approximation is more precise than that given by de Keijser, Langford, Mittemeijer & Vogels (1982), the maximum difference from the exact value being less than 0.16% (at y=0.15). The values of the parameters are: a =0" , b = , c = and d= Fast algorithms to calculate the complex error function have been developed (Gautschi, 1969; Sundius, 1973; K61big, 1978). In the Rietveld method, the calculated intensity Ycal,i is compared with the observed intensity Yobs.~ at each value 20i. The least-squares parameters are adjusted to minimie the quantity Z2 : ~~wi Yobs,i-- c Ycal,i, (8) where c is the scale factor and the weighting scheme _ (9) Wi- 0"/2 -- Ycalc,i Yobs,i is calculated from the raw data before subtracting the background. Following Rietveld's treatment, the variation of the FWHM 2w of the Voigtian is described by the expression, originally from Caglioti, Paoletti & Ricci (1958), (2w)2 = U tan/o+ V tano+ W. (10) In the examples included in this study the behaviour of the ratio 2w//3 as a function of the scattering vector S = 2sinO/2 turns out to be characteristic of the instrument and can be approximated with the following expression (Suortti, 1980). 2_ww /3 =2 exp --~ps 2, (11) where p is a constant; here it will be called the Voigt parameter. 3. Simulated diffraction patterns of NaTaO3 and Ni The revised computer program was tested on a simulated diffraction pattern that corresponds to the roomtemperature structure of NaTaO3, as determined by Ahtee & Darlington (1981). In this way the complicating factors of measured patterns, such as unknown background and variations of thermal diffuse scattering (TDS), were eliminated, and the factors affecting the refinement could be studied separately. The integrated intensity of a Bragg reflection hkl

3 354 A VOIGTIAN AS PROFILE SHAPE FUNCTION can be calculated from (see e.g. Warren, 1969) Phkt F2kl I hkl = C Aabs, (12) sin2ohu sinohu where Phkl is the multiplicity of the reflection, Fhk I the structure factor including the isotropic temperature factors for different atoms, Ohk~ the diffraction angle and C a normaliation factor. The absorption factor Aabs depends on sample geometry and may be a function of 0. The profile shape of the reflections was Ta B assumed to be Voigtian given by (4). The half-width I'qa x parameters U, V and W in (10) were given typical B (A ~) values of U=3500, V=-6500 and W=5200. The O(1) x value of the Voigt parameter p in (11) was taken from B (A 2) the graph given by Suortti (1980, p. 15) as (2) x The simulated diffraction pattern was then gen- y crated by adding up the distributions of all the reflections contributing at the particular Bragg angle. B (A 2) The tail of each reflection was taken to extend 40w from the centre of the reflection. A linear background was also added but the asymmetry at low angles, which is caused by the finite volume of the diffracting sample and the finite height of the receiving slit, was Voigt parameter p (M) not included. The actual asymmetry is visible in Fig. 1, where the measured pattern is compared with the simulated pattern. The simulated pattern was made to resemble the measured pattern as closely as possible with an iterative procedure. The measured diffraction pattern of NaTaO3 was originally analysed by the Rietveld method, which uses the Gaussian profile function (Ahtee & Darlington, 1980), and therefore the simu- Cut-off value lated pattern was analysed in the same way. In particular, the background was estimated from the minima between the reflections. The input parameters of the simulated pattern were changed until the refinement yielded the model parameters of the measured pattern. Table 1 demonstrates that there is a 2000 [ 1000 NaTaO= measured-simulated 1 measured -- simulated deg Fig. 1. Measured and simulated diffraction patterns for NaTaO3 at room temperature. Table 1. Structural data for orthorhombic NaTaO 3, space group Pcmn, from different methods A: As determined by Ahtee & Darlington (1980) using the conventional Gaussian version of the Rietveld program. B: The corresponding input data used in construction of the simulated NaTaO3 powder pattern shown in Fig. 1. C: The resulting data from the analysis of the simulated NaTaO3 pattern with the Voigt version with the cut-off value of 20w. 4w 6w 10w 20w 40w Input values (M) A B C 0.384(14) (3) (12) (3) (9) (2) 1.507(34) (8) (4) (1) ) "0082(7) 0.842(29) 0"590 0"566(7) (3) (1) - 0"0295(2) "0294(1) (3) "284441) 0-897(18) 0" (4) Lattice parameters a (A) (2) (0) b (A) (2) "7948(I) c (A) (2) (0) R factor for the integrated intensities RI o "235(2) 0"012 Table 2. The effect of the cut-off value on the lattice parameter a, temperature factor B and the Voigt parameter p in the case of simulated nickel data a (A) B (M) p (M) 3"5239 0"3277 0"2323 3"5239 0"3240 0" "2335 3"5239 0"3380 0"2283 3"5239 0" "3400 0"2250 substantial difference between the temperature factors of the Voigtian model and those recovered by a Gaussian analysis. The remarkable feature of the above analysis is that the overlapping of the tails of the Voigtian profiles very closely reproduced the observed apparent background. This means that if the true profiles are Voigtians, as suggested by studies where detailed analysis was possible (Suortti et al., 1979; Suortti, 1980), a substantial part of the diffracted intensity is lost in the background when conventional Rietveld analysis with a Gaussian profile function is applied. This affects the thermal parameters, particularly that of Ta in the present example, as Ta gives the dominant contribution to scattering at high angles where many reflections overlap. This situation does not manifest itself in the conventional refinement, as the background level is a hidden variable (or actually many variables), and observed and calculated patterns may

4 M. AHTEE, L. UNONIUS, M. NURMELA AND P. SUORTTI 355 Table 3. Structural parameters of nickel based on the Rietveld refinement of the experimental neutron powder diffraction data In column 2 the profile shape function is Gaussian, in columns 3 and 4 Voigtian. In column 4 the visually estimated background is systematically lowered. The actual value of B is about 0"36 A 2 (Suortti, 1980). The different R factors are defined as 1 Rt = Y'll(obs) - - l(calc)l/y'/(obs), c 1 Rv = ~ly(obs) - - y(calc)l/y'jy(obs)l, C Gaussian Cut-off value 3w Temperature factor B (,~2) 0-470(25) Voigt parameter p (ilk 2) 0 Scale factor (8) Lattice parameter a (A) (1) Half-width parameters U 1230(40) V (130) W 2940(80) Counter ero point - 23"50(2) R~ 0"018 R v 0"072 Rwp 0"101 Number of observations 236 Number of least-squares parameters 7 Iteration cycles 5 CPU time (s), 5-36 Burroughs 7800 show better agreement than that of Fig. 1. A simulated powder neutron diffraction pattern was also calculated from the structural data of nickel. This was used for systematic studies of the effects of cut-off. In the Rietveld program, where the Gaussian profile function is used, the reflections are included only within 3w about the point of calculation, as at this limit the intensity is only 0-2% of the peak value. The tail of the Voigt function falls off much slower, as demonstrated in Fig. 2. The results collected in 1.00 t/t(o) 0"75 0'50 0"25 O.OC 0 " ' ' 2w ' ' 1. ~- = 0"65 Lorentian \ 2Z, i=o,o a. i ~ J i ~olw Fig. 2. Behaviour of the Voigt function with different values of the ratio 2w/fl. Voigtian 20w Voigtian with lowered background 20w 0.393(9) 0.367(8) 0-110(4) 0" ] 54(3) (3) (2) (0) (0) 1128(15) 1121(13) (50) (40) 2870(30) 2910(30) (8) (7) Table 2 show that the lattice parameter is not affected by the cut-off value, whereas the value of the thermal parameter B increases and the value of the Voigt parameter decreases towards the input values when the cut-off value is increased. The above results were utilied in the Voigtian analysis of the simulated pattern from NaTaO3. The maximum cut-off value of 20w yielded the structural and thermal parameters shown in Table 1. These are very close to the input parameters, which substantiates the correctness of the refinement procedure. 4. Results and conclusions The preceding results show that the revised version of the Rietveld program performs well and also gives valuable guidelines for application to measured patterns. The experimental neutron diffraction data on nickel and NaTaO3, which were measured with the D1A high-resolution powder diffractometer at the ILL (Grenoble), were analysed using the Gaussian and Voigtian versions of the Rietveld program. The results are collected in Tables 3 and 4. Initially, the same visually estimated background was used in both analyses. Even with this choice, the

5 356 A VOIGTIAN AS PROFILE SHAPE FUNCTION Table 4. Structural parameters of NaTaO3 based on the Rietveld refinement of the experimental neutron powder diffraction data In column 2 the profile shape function is Gaussian, in columns 3 and 4 Voigtian. In column 4 the background is deduced from the simulated pattern. Voigtian with Gaussian Voigtian lowered background Cut-off value 3w 14w 20w Ta B (A 2) 0"385(15) 0-332(15) 0.093(13) Na x - 0"0042(14) - 0"0036(14) - 0"0025(13) B (A 2) -0"0116(11) 1"47(4) -0"0105(11) 1"42(4) -0"0121(9) 1"21(3) O( 1 ) x 0"4405(5) 0"4407(5) 0"4399(4) 0"0080(6) 0"0069(6) (6) B (A ) 0"85(3) 0"78(3) 0"59(3) 0(2) x 0"2852(3) 0"2853(3) 0"2860(3) y - 0"0298(2) - 0"0299(2) - 0"0297(2) 0"2845(3) 0"2843(3) 0"2840(3) B (A 2) 0"90(2) 0"92(2) 0"611(17) Voigt parameter p (A,2) (10) 0"303(8) Scale factor 0.738(4) 0.734(3) 0.741(3) Lattice parameters a (A) (2) 5"4841(2) 5"4855(2) b (/~,) (3) 7"7935(3) 7"7948(3) c (/~) (2) 5"5201(2) 5"5213(2) Half-width parameters U 3470(120) 3190(110) 3438(96) V (200) (200) (165) W Counter ero point 5170(80) 5-40(20) 5090(70) 5.50(20) 5272(63) 5.70(20) RI 7" 16 7"05 4"69 Rp 13"17 13"41 10"62 Rwp Number of observations 14" " " Number of least-squares parameters Iteration cycles CPU time (s), 325" "7 Burroughs 7800 structural parameters (lattice constants, atomic positions and scale factor) remain the same well inside the standard deviations in both the Gaussian and the Voigtian refinements. The same is true with the parameters describing the characteristics of the diffractometer (the counter ero point and the three half-width parameters). In the case of nickel, the actual value of the thermal parameter is known from the previous measurements and calculations, and the use of a Voigtian improves the fit considerably. A further improvement is obtained when the visually estimated background is sytematically lowered to a level that better agrees with the probable level. This improves the R factors, and several refinement programs include optimiation of the background, often with a manyparameter function (Young & Wiles, 1982). However, this may be another dubious way of improving the fit, and alternative approaches should be studied, if finding the non-bragg background is the objective. The problem of background becomes far more obvious in the case of NaTaO3. Again, the tail contribution missed by the Gaussian shows up in the B factors, which are somewhat smaller in the Voigtian refinement (Table 4). However, comparison with the final parameters of the simulated pattern, shows that the correct values may still be much smaller. Therefore, another refinement in the analysis of the experimental data was performed by adapting the background level deduced from the simulated pattern. The thermal parameters are the only ones noticeably affected, and they come close to the values of the simulated pattern (Table 1). The improvement of the R factors is even more remarkable than in the case of nickel. The results of the above calculations can be summaried as follows. Firstly, a modification of the Rietveld refinement is developed by replacing the Gaussian profile function with the Voigt function, and application to a simulated pattern demonstrates that the input values are retrieved to the level allowed by the statistical fluctuations introduced into the data. Secondly, the analysis of the data from nickel substantiates the improvement provided by the Voigtian profile function, when the thermal parameters are

6 M. AHTEE, L. UNONIUS, M. NURMELA AND P. SUORTTI 357 concerned. Thirdly, the problem of background is fully exposed by the analysis of the data on NaTaO3. In effect, the model can be fitted to the observed pattern using Gaussian functions and the apparent background between the reflections as well as by Voigtian functions of long overlapping tails and a nearly negligible background. The structural parameters are not affected, only the B values, and so the approach is to be decided by the objective of the study. If the thermal parameters, and particularly their changes, say, over a structural transition, are of interest, a careful study of background is due. An ideal solution would be an analytical approach in cases of some simple structures, but in a case of a sufficiently complicated structure the calculation of TDS must be reduced to that of the average TDS, as suggested by Sabine & Clarke (1977). It has been demonstrated that, when the tails overlap, the use of the average background nearly eliminates the need for a correction for TDS included in the Bragg reflections (Suortti, 1980). The average TDS can be calculated from the thermal parameters, and so this can be included in the refinement, at least in principle. The above findings may be stated also as follows: any meaningful refinement of the background must be strongly constrained. The constraints come from the profile function and to some degree from the relation of the TDS to the integrated intensities of the reflections. The profile function is eventually derived from the shape of sufficiently isolated reflections, and the possibility of incorporating this feature in the refinement procedure is studied. One of the authors (MA) gratefully acknowledges the financial support of the National Research Council for Science, Finland. References AHrEE, M. & DARLINGTON, C. N. W. (1980). Acta Cryst. B36, AHTEE, M., UNONIUS, L., NURMELA, M. & SUORTTI, P. (1984). Report Series in Physics, Univ. of Helsinki, HU-P-230. ALBINATI, A. & WILLIS, B. T. M. (1982). J. Appl. Cryst. 15, CAGLIOTI, G., PAOLETTI, A. & RICCI, F. P. (1958). Nucl. Instrum. 3, GAUTSCHI, W. (1969). Commun. ACM, 12, 635. GLAZER, A. M., HIDAKA, M. & BORDAS, J. (1978). J. Appl. Cryst. 11, HEWAT, A. W. (1973a). J. Phys. C, 6, HEWAT, A. W. (1973b). UK At. Energy Auth. Res. Group, Report R-7350 (unpublished). KEIJSER, TH. H. DE, LANGEORD, J. 1., MITTEMEIJER, E. J. & VOGELS, A. I. P. (1982). J. Appl. Cryst. 15, KOLBIG, K. S. (1978). CERN Computer Centre Program Library No. C335. LANGFORD, J. I. (1978). J. Appl. Cryst. 11, MALMROS, G. & THOMAS, J. O. (1977). J. Appl. Cryst. 10, RIETVELD, H. M. (1969). J. Appl. Cryst. 2, SABINE, T. M. & CLARKE, P. J. (1977). J. Appl. Cryst. 10, SUNDIUS, Z. (1973). J. Raman Spectrosc. l, SUORTTI, P. (1980). Accuracy in Powder Diffraction. Natl Bur. Stand. Spec. Publ. No. 567, edited by S. BLOCK & C. HUBBARD, pp SUORTTI, P., AHTEE, M. & UNONIUS, L. (1979). J. Appl. Cryst. 12, VON DREELE, R. B., JORGENSEN, J. D. & WINDSOR, C. G. (1982). J. Appl. Cryst. 15, WARREN, I. E. (1969). X-ray Diffraction. Reading, MA: Addison-Wesley. YOUNG, R. A. & WILES, D. I. (1982). J. Appl. Cryst. 15,