Model Calculations of the Accuracy of Structure Factor Determination From Textured Powder Samples

Crystal structure analysis is based on absolute values of structure factors |Fhkl|. These values can be measured by powder diffraction. In random powder diffraction many superpositions occur in the 
diffraction diagram over the Bragg angle θ. If non-random (textured) samples are used and intensity measurements are done in the three-dimensional reciprocal space, i.e. over θ and the sample rotation angles αβ then these superpositions can be deconvoluted to a good deal. In order to perform this deconvolution a two-step procedure was developed and was implemented in a computer code. In the first step the texture is determined from a low number of reflections having no or only two-fold overlap. In the second step the deconvolution is done for all reflections. This includes the deconvolution of “systematically” overlapped reflections, e.g. those with the same h2


INTRODUCTION
Crystal structure analysis is the determination of atom positions in the unit cell of the crystal lattice. The atom positions are obtained from the electron density distribution which can be represented by a Fourier expansion e-2rithx+kY+''J p (xyz) -f E Fhkl" (1) hkl where Fhk are the structure factors. The values of the Fhk! can be represented as a function of hkl on the reciprocal lattice, Fig. (a). If the Fhk! are known then p (x,yz) can be calculated by Eq. (1). Structure factors are usually complex quantities. If they are to be measured by X-ray diffraction, only their absolute values can be obtained. The phases are lost.
The electron density function p (xyz) is of limited variance given by the size of an atom. Hence, the Fhkz must obey certain internal relationships which can be used to obtain some information about the missing phases. This is the reason why crystal structure analysis is possible at all, even with missing phases.
The absolute values of the structure factors IFhkll are mostly determined from single crystal diffraction experiments (Klug and Alexander, 1974;Luger, 1980), measuring the reflected intensities, which are proportional to [Fhkz [2,in all reciprocal lattice points (up to a reasonable upper limit).

Figure
Intensity distribution in the reciprocal space (two-dimensional): (a) reciprocal lattice of a single crystal; (b) the reciprocal lattice points in a polycrystal lie on spheres (or circles); (c) intensity distribution on two such spheres (circles) in a powder with non-random orientation distribution (texture).
consists of (infinitely) many structurally identical single crystals which have, however, different orientations g of their crystal axes. Hence, also their respective reciprocal spaces are rotated through g. Then infinitely many reciprocal lattices are superposed. Hence, the reciprocal space of a polycrystalline sample consists of spheres on which the intensity [Fhk[ is continuously distributed, Fig. l(b).
If the orientation distribution of the crystallites is random then the intensity distribution on each sphere is constant, i.e. it is independent of the polar angles specifying a point on a sphere. In this case there is a simple relationship between the intensities [Fhkt [2 of any hkl of a single crystal and the powder diffraction intensities on the spheres.
In crystals with big unit cells there are, however, many "superpositions" of different hkl, the spheres of which have the same (or nearly the same) radius. In this case the unique relationship between single crystal intensities and polycrystal intensities cannot be inverted uniquely. This is the essential restricting condition which renders crystal structure analysis from powder diffraction much more difficult than from single crystal diffraction.
If the orientation distribution of the crystallites is not random then the spheres carry intensity distributions which vary with the angles a on the sphere as is shown schematically in Fig. l This is the principle of crystal structure analysis from textured powder samples.

EXPERIMENTAL TECHNIQUE
Single crystal X-ray diffraction is mostly carried out with small single crystals which may be considered as having no extension (compared to the dimensions of the experimental equipment). This situation can practically not be achieved with polycrystalline samples. In X-ray powder diffraction, focusing methods are often preferred which require flat (extended) samples (see e.g. Klug and Alexander, 1974;Bish and Post, 1984;Jenkins and Snyder, 1996).
Powder diffraction with random samples requires the diffraction vector s to be oriented only in one direction with respect to the sample. It is then usually chosen perpendicular to the fiat sample surface. This allows to use the Bragg-Brentano technique with the sample surface tangent to the focusing circle (or more correctly to the focusing cylinder), Fig. 2

(a).
Powder diffraction with non-random (textured) samples requires the diffraction vector s to be brought into many orientations a/ with respect to the sample surface. This requires a sample orientation device which is most often chosen in the form of an Eulerian cradle fixed on the 8--to axis of a 8-28 powder diffractometer. The tilt angle a can then be achieved either by the goniometer angle to or by the Eulerian cradle angle X, i.e. by to-tilt or x-tilt. The most common method in texture analysis is x-tilt as is illustrated in Fig. 2(b). This leads to peal broadening due to a violation of the focusing conditions, and this, in turn, leads to abundant peak overlap.
As an example, Fig. 3 shows peak profiles measured with a texture goniometer (Wcislak and Bunge, 1996) at different sample tilt angles a. One sees the strong peak broadening with increasing sample tilt.
Powder diffraction diagrams as they have to be evaluated for crystal structure analysis are shown in Fig. 4. They correspond to a polyoxadiazole derivate used later in the model calculations.
The sample orientation angle fl corresponds to a rotation of the sample about its surface normal direction. This rotation does not change the focusing conditions Thereby I0 is the primary intensity, C (0, a) is an instrumental factor containing, for instance, divergencies and absorption, B is the peak profile function centred at the ideal position 0 hkl and having a width described by the parameter b which depends on 0 and a, m hkt is the multiplicity (according to crystal symmetry and the form of the hkl) of the reflection hkl, and finally Phkt(a,,) is the pole density distribution function (i.e. the pole figure). The pole distribution functions of the different reflections hkl are not independent of each other. Rather they are integrals over the orientation distribution function f(g) of the crystallites of the sample. It is Phkl(OZ, fl) l+/-/3f(g) d, g= {<Pl,,q_}- Thereby is the rotation angle about the normal direction to hkl, i.e. about the sample direction c/3 and about the diffraction vector s (for details see e.g. Bunge, 1993). Equations (2) and (3) must be considered for two different situations: At the beginning of the investigation IFhkzl and f(g) are unknown.
Later on f(g) has first been determined, then only the IF,kl remain as unknowns which have to be determined from measured intensity data.
In order to develop an effective deconvolution procedure it is helpful to estimate the expected number of unknowns as well as the number of available experimental input data from which the unknowns are to be calculated.
The number of unknown values [Fhkl[ is given by the volume of the reciprocal unit cell depending on the reciprocal lattice vectors a*, b*, and c*, the crystal symmetry and the considered range 0max of the powder diffraction diagram. The influence of crystal symmetry can be approximated by the multiplicity mgen of the general form.
In the example considered later with the lattice parameters of Eq. (22) and 0max 66 (CuI( radiation) this leads to Nhkt= 756.
The texture f(g) can be represented by its series expansion coefficients C ", Eq.
(9), the number of which depends on the "sharpness" of the texture. The texture can be varied deliberately by the experimenter in certain (rather wide) limits. Hence, it is in the hands of the experimenter how many texture coefficients have to be determined. Reasonable ranges may be estimated by 0 < Nc < Nmax 500 (5) depending on crystal symmetry (the lower the crystal symmetry the higher is Nmax).
Thereby it must be taken into consideration that the resolution for [Fhkll is the better the sharper the texture, i.e. the higher the number Nc.
However, since the experimenter is able to choose the texture himself, it is possible to combine different measurements with different degrees of texture sharpness, for instance, starting at first with a sample with a flat texture and then, when some resolution has already been reached, proceeding to another sample prepared from the same powder but now with a sharper texture. Hence, in the example considered later on, the number of unknowns is in the order of Nhkl + Nc < 756 + 500 1256.
The number NI of measurable intensity data (input data) I(0, a, ) depends on the resolving power of the used texture goniometer in all three variables 0 a/3: thereby Orange is the available 0 range, A0 is the resolving step in 0, x is the measurable part of the pole sphere (i.e. of the hemisphere) and As, A/3 are the resolved steps in the two pole figure angles As an example we may assume the following values: A0 0.05 , 0min 6 , 0max 66 , As A3 5 , x 0.7; (8) then we obtain N1 1200-1080 1.296.106 (9) different intensity values. This shows that the number of experimental data points is by far higher than the number of unknowns, i.e. the number of [Fhkl[ and C together. This may be compared with the situation in "random" powder diffraction. In this case there are no unknown texture coefficients C but the number of input data is drastically smaller by the factor Na. In the example given above there would remain only NI--1200 intensity values (assuming the same angular resolution in 20) compared with 756 unknowns.
In random powder diffraction, i.e without the necessity of sample tilt, usually higher angular resolution in 20 can be achieved, e.g. 0.005 Then the number of available intensity data would be Nr= 12000 in this case.
It must be mentioned that very small angular steps As and A/ may lead to "redundant" data which do not carry independent information (depending on the sharpness on the texture).
A DECONVOLUTION PROCEDURE Equation (2) contains four types of quantities: The three-dimensional intensity distribution function I(O, o,fl) as the primary experimental input data.
The instrumental quantities C, B and b. The texture of the sample described by Phkl(OZ, 1) or f(g).
The unknown structure factors IFhktl.
The instrumental quantities are, in principle, independent of the particular studied sample. Hence, they can be assumed to have been measured beforehand. So, they may be assumed to be known in the actual deconvolution.
The texture of the sample is unknown. It is described by a not-too-high number of parameters C'(Eq. (10)). Hence, it can be determined from parts of the diffraction diagram with no or only small overlap in 0. In these parts of the diffraction diagram the to types of unknowns, i.e. the texture factors and the structure factors, can easily be separated by employing well-established methods of quantitative texture analysis (see e.g. Dahlem-Klein et al., 1993).
Finally, deconvolution for all IFhkt[ can be done also in the strongly overlapped parts of the diffraction diagrams but then with already known texture.

TEXTURE DETERMINATION
The texture is expressed in terms of a series expansion (see e.g. Bunge, 1993)  ) sin a da dfl 47r.
Also the peak profile functions B are normalized over the peak width B{O Ohgt, b} dO 1.
Since the instrumental function C is known, the factor I0. [Fhkt[ 2" m m,t follows by integrating Eq. (13) over a/3 according to Eq. (11). Then Eq. (13) gives the normalized functions Phkl(O, ) from which the coefficients C can be determined by solving Eq. (10). In this latter equation, M(A) unknowns C are contained in each of several systems of linear equations (one such system for every combination of indices A and u.). Hence, the solution can be obtained if at least M(A) functions Pha(a,13) are known. The numbers M(A) depend on A and on crystal symmetry. (This procedure corresponds to well-established methods of texture analysis, see e.g. Bunge, 1993.) It must be mentioned that the solution of Eq. (10) for C is also possible if Phkt(a, ) are not known in the full range of cq3 (incomplete pole figures) so that the normalization condition (11) cannot be used explicitly. Also, the solution is still possible for low-order superpositions of a few hkl. The solution procedures for these cases are, however, not described in detail here (for details see e.g. Dahlem-Klein et al., 1993).

DETERMINATION OF IF,kll
Once the texture is known, the Phkt(O, l) in Eq. (2)  For any given hkl these coefficients Ahkl(O,a,) become zero for 10-Ohktl > 1 / 2 A 0max as is illustrated in Fig. 5(b). This fact can be used for an approximative, successive least-squares solution (Bronstein and Semendjajew, 1991) as is illustrated in Fig. 5(d) (Hedel et al., 1994).
The matrix Am,, is illustrated schematically in Fig. 5(d).
Then a partial system of equations can be selected out of the full system containing only 0 values out of an interval A0ma,, wide enough to contain the full width of the central peak hkleentr. This system contains only the unknowns IFhkt 12 the diffraction peaks of which contribute intensity to the chosen 0 interval. They are indicated in Fig. 5(b) (16).) This partial system is then solved for the IFhgtlel. including the positivity condition of Eq. (16). If xn turns out to be negative it is put to zero. For the final solution only those hkl are kept which lie inside the central interval A0entr, Fig. 5(b) (which has the size of one experimental step in 0). This procedure is repeated for all intervals A0 between 0min and 0ma taken as A0entr. After that, solutions for all IF,zl have been obtained as is shown schematically in Fig. 5(b).

EXACT SUPERPOSITIONS
The described deconvolution procedure is essentially based on the resolving power of texture measurement in the sample rotation angles a. It is able to resolve superpositions of hkl in the Bragg angle 0 which are closer than the experimental step A0 and, even more so, with respect to the width b of the profile function B. In fact, the width b of the profile functions in texture analysis is often higher than that used in high-resolution powder diffraction. The loss of resolving power in O is, however, by far overcompensated by the gain of information (and resolving power) according to the sample angles aft. Equation (14) shows very clearly the increase of the number of equations due to the variables aft compared with the situation in random powders where only one intensity value is available for each value 0 (see e.g. the example given in Eq. (9)).
From the above-said it follows that the deconvolution of superposed hkl does not depend on their distance in O. In fact, even hkl having exactly the same 0 value may be deconvoluted in the same way. Systematic superpositions of this type occur, for instance, in cubic crystal symmetry where all reflections with the same sum h2+k2+lhave exactly the same 0hkt. (As an example, this is the case for 621,540, 443 in a cubic primitive lattice.) The deconvolution pr.ocedure is thus able to deconvolute also such exact coincidences in the same way as nearly coincident reflections.

TEST OF THE DECONVOLUTION PROCEDURE BY MODEL CALCULATIONS
In order to test the deconvolution procedure, it was applied to theoretically constructed intensity distributions Eq. (2) assuming [Fhkt 12 according to a known crystal structure. Also the texture of the sample was assumed to be known. Particularly we used textures consisting of several preferred orientations gi with Gaussian spread about them together with a random component f(g) Vo + yvi'e-lA#l<'l:, g= Ag'gi.
(17) Thereby vi are the volume fractions of the components, Ag is the distance from the preferred orientation and wi characterizes the spread width. The coefficients C of one of these components can be easily written in the form C,,, exp{ -(l/4)Azw} -exp{ -(l/4)(A + 1)2w7} , ....
The deconvolution procedure was tested by comparing input and output texture coefficients c" ' in terms of an accuracy measure Rc: Rc E,x,,. Ic'"(input) C'"(output)l.
E,. c '"(input) As the final result of the deconvolution procedure, the input and output structure factors were compared by the accuracy measure Re: The intensity function (2) contains the peak profile function B{O--Ohkl, b} which fully enters the coefficients Ahkt in Eq. (14). Hence, the type of this function and the width b (0, a) must be considered with the highest possible accuracy. The function depends on the particular features of the used diffractometer. In the present case, it was found that peak profiles for all 0 and a could be best represented by a pseudo-Voigt function (see e.g. Wcislak and Bunge, 1996) n(o) 7" nGauss(0) t_ (1 7)" nCauchY(0), where is a "shape parameter" and the width a depends on 0 and a. For modelling purposes it is sufficient to assume r/= 1. For real measurements, of course, the experimental function must be used (see e.g. Wcislak and Bunge, 1996).

RESULTS
At first the accuracy of texture determination was tested using a texture consisting of the components listed in Table 1. In Fig. 6 input and output pole figures are compared. Those of Fig. 6(a) were used for texture determination case (a) and those of Fig. 6(b) were not used. It is seen that the agreement between input and output pole figures for both groups is reasonably good at least considered at a qualitative level. Rc. The starting texture was that according to no. 5 in Table 1. In Fig. 7(a) the texture sharpness was varied by varying the spread width of the components. Curve a in this figure corresponds to only non-overlapped pole figures whereas curve b includes two-fold overlapped ones. It is seen that in case (a) the accuracy increases with increasing spread width whereas in case (b) it decreases. This latter fact can be understood since with decreasing texture sharpness (i.e. increasing  Next, the total deconvolution procedure was tested by comparing input and output IFhktl values. The used deconvolution method is an approximation depending on the size of the interval AOmax according Fig. 5. Hence, the size of this interval was varied. The result is shown in Fig. 8. It is seen that the accuracy converges to a constant value above A0max 0.45. Hence, in the following, lower values of A0max were no more used. The quality of the approximation, illustrated in Fig. 5, can also be judged by Fig. 9. Here the solutions of the partial systems (for the same unknown xn IFht I) are plotted as a function of 0 of the central interval of the partial system. The central value of Fig. 9 was then kept for the total solution. If the partial solutions were exact, then they should be identical, so that Fig. 9 were a straight horizontal line. Hence, the deviation from the straight line illustrates the degree of accuracy obtained by the approximative nature of the solution. Also the absolute deviation of the solution from the theoretical (input) value can be seen. The accuracy of the deconvolution procedure also depends on the type of texture, i.e. how many texture components it contains. This is shown in Fig. 10 the deconvolution for the IFhkl] values was done using, in the second step, either the theoretical input texture or the texture obtained from the first step of the procedure. It is seen that the deconvolution works extremely well when the texture is exactly known. The essential part of error is due to the errors introduced through the uncertainty of texture determination. (In Fig. 10 the "sharpness" of the texture is expressed in terms of the "texture index", see e.g. Bunge, 1982.) The same trend can be seen when the texture sharpness is varied either by the spread width of the texture components, Fig. l(a), or by the random component, Fig. l(b). In both figures the use of the ideal input texture allows a very accurate deconvolution of the overlapped peaks.   from isolated peaks alone are much better than with included overlapped ones. Both figures also show that deconvolution is no more possible (even with the ideal texture) when the texture approaches the random distribution (as is evident from the general considerations given above). It is interesting to know, how experimental errors in the measurement of diffraction spectra influence the results. Figure 12(a) shows the influence of a shift of the whole diffraction diagram, i.e. the peak profile function of each peak is shifted out of its centre. In Fig. 12(b) the influence of a wrong peak width b is illustrated. From Fig. 7, curves b, it was evident that a great deal of uncertainty is introduced into texture analysis by the uncertainty of overlapping factors obtained by the used algorithm of texture analysis, i.e. the iterative series expansion method (Dahms and Bunge, 1989).
In order tO quantify the "degree of overlap", illustrated in Fig. 4, a correlation factor Corr hkt, h'k'r of the two reflections hkl and h'k'l' was introduced: This factor considers overlap in all three angles 0 a .H ence, it depends on the actual texture. The correlation factor can vary between zero and one.
In Fig. 13 the distribution of the reflections hkl according to their correlation factors with respect to neighbouring reflections h'k'l' is shown. The different symbols correspond to different assumed values of the lattice parameter 'a' (compared to that of Eq. (22), with b and c fixed).
It is seen that the number of non-overlapped or only moderately overlapped pole figures decreases drastically with increasing lattice constants.
Finally Fig. 14 shows the accuracy parameter Re as a function of the assumed lattice parameter 'a'. It is seen that the accuracy decreases with increasing lattice parameter due to the uncertainty in texture determination as a consequence of pole figure overlap. If the texture is exactly known (i.e. the input texture is used) then deconvolution according to Eq. (16) reaches a very high degree of accuracy (curve c). The accuracy parameters (19) and (20) compare input and output parameters IF and C in a global way, i.e. they are averages over all hl, respectively all #,. For the final structure analysis, however, the individual and particularly the maximal deviations are even more important.
In Fig. 1  This representation also shows the IFhktl values which obtained negative solutions.
They were put to zero. In the logarithmic scale of Fig. 15 they are plotted at the lower limit of the diagram. If the deconvolution is done with the theoretical input texture, Fig. 15(c), then the retrieved IFhkt values have a very high accuracy in the order of 0.3% of the maximum value. In this case no negative values were obtained in the solution. This illustrates again that the by far largest part of the errors comes through texture determination in the first step of the procedure.
The distribution of the residues between input and output IFhktl is shown in Fig. 16. The absolute values follow approximately a normal distribution. In the distribution of the relative residues, the ones put to zero are seen as a separate "population" in this curve.

CONCLUSIONS
Crystal structure analysis is based on absolute values of structure factors [Fhkl[. least only low-order superposition. Model calculations have shown that the deconvolution of reflections hkl in the second step can be done with very good accuracy when the texture is correctly known. The essential part of uncertainty comes in through the uncertainty of texture determination from a limited number of reflections and/or a limited number of sample orientations c. The uncertainty is higher the higher the degree of.superposition of reflections in this first step. From these results it can be concluded that structure factor determination with an error value of, say, 5% should be possible in structures with lattice parameters at least up to the ones used in the present example.
Texture determination was modelled assuming conditions as they are usual in conventional texture analysis. The model calculations show that this is not yet optimal for the purpose of crystal structure analysis. They show, however, also that the assumed conditions of texture analysis are still far from their theoretically possible limits ofaccuracy. Hence, it can be estimated in which way the experimental conditions must be changed in order to improve the accuracy of structure factor determination.
Provided such experimental conditions can be established, it should be possible to determine IFhktl values from textured powder samples with accuracies better than 5% for a reasonably wide class of structures. Strictly speaking, the determination of structure factors from textured powders should bear the capability of reaching the highest accuracy compared to both other methods, i.e. single crystals and random powders.
On the one hand, a powder represents a much better statistical average than a small single crystalline individuum which may contain unknown lattice imperfections.
On the other hand, the orientation distribution function of the crystallites in the powder is fully taken into consideration. Hence, the uncertainties due to nonrandom orientation distribution, as they may enter "random powder diffraction", are avoided in this method.