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Volume 68 
Part 6 
Pages 736-749  
November 2012  

Received 1 May 2012
Accepted 7 August 2012
Online 18 October 2012

Analysing Friedel averages and differences

aSchool of Chemistry and Centre for Science at Extreme Conditions, University of Edinburgh, Kings Buildings, West Mains Road, Edinburgh EH9 3JJ, Scotland,bLaboratoire de Cristallographie, Ecole Polytechnique Fédérale de Lausanne, BSP, Lausanne, CH-1015, Switzerland,cSwiss-Norwegian Beamline, ESRF, BP 220, Grenoble, F-38043 Cedex 9, France, and dChimie minérale, analytique et appliquée, University of Geneva, Geneva, Switzerland
Correspondence e-mail: s.parsons@ed.ac.uk

A version of this paper will be published in the next edition of Volume A of International Tables for Crystallography.

Various practical applications of the average (A) and difference (D) of Friedel opposites are described. Techniques based on the resonant-scattering contribution to Friedel differences are applied to see whether a crystal is centrosymmetric or not, and to determine the point group of the crystal. For the validation of a structural study, plots of Aobs against Amodel, and Dobs against Dmodel are used extensively. Moreover, it is useful to display both plots on the same graph. Intensity measurements on a crystal of NaClO3 were made at three different speeds, with two different radiations and two different diffractometers, and treated with two different software packages and four different absorption corrections. The evaluation of these numerous data sets reveals underlying deficiencies. For comparison, plots of Aobs against Amodel, and Dobs against Dmodel are presented for two centrosymmetric crystals.

1. Introduction

In small-molecule crystallography it has been customary in crystal structure analysis to make no use of the contribution of resonant scattering other than in the specific area of absolute-structure and absolute-configuration determination. One may trace the causes of this situation to the weakness of the resonant-scattering contribution, to the high cost in time and labour of collecting intensity data sets containing measurements of all Friedel opposites and to the lack of any perceived or real need for the additional information that might be obtained from the effects of resonant scattering. A natural consequence of this situation is that reputable and authoritative texts on crystallography, introductory or advanced, contain no or very fragmentary information on this topic.

On the experimental side the turning point came with the widespread distribution of area detectors for small-molecule crystallography, giving the potential to measure, at no extra cost, the full-sphere data sets leading to the intensity differences between Friedel opposites, hkl and [\overline{hkl}]. Further impetuses for development were both the realization that standard uncertainties on the Flack parameter (Flack, 1983[Flack, H. D. (1983). Acta Cryst. A39, 876-881.]) were unreliable and the ever-increasing need for improvements in structure validation.

The validation of a structure analysis of a non-centrosymmetric crystal structure by way of plots of observed against model values of the average (A) and difference (D) intensities of Friedel opposites has been introduced by Flack et al. (2011[Flack, H. D., Sadki, M., Thompson, A. L. & Watkin, D. J. (2011). Acta Cryst. A67, 21-34.]). The average and difference of Friedel opposites are defined by

[A ({ h k l}) = \textstyle{1\over 2} [{{{| {F({ h k l})} |}^2} + {{| {F({ \overline h \overline k \overline l})} |}^2}}], \eqno (1a)]

[D ({ h k l}) = {| {F({ h k l})} |^2} - {| {F({\overline h \overline k \overline l})} |^2}. \eqno (1b)]

In general D(hkl) is small compared to A(hkl). A compound with an appreciable resonant-scattering contribution has D(hkl) approximately 0.01A(hkl), whereas a compound with a small resonant-scattering contribution has D(hkl) approximately 0.0001A(hkl). Flack et al. (2011[Flack, H. D., Sadki, M., Thompson, A. L. & Watkin, D. J. (2011). Acta Cryst. A67, 21-34.]) made a study of 29 crystal structures published in 2007. It was found that these crystal structure determinations could be separated into three categories based on the appearance of the Dobs against Dmodel plot of the acentric reflections. In the first category, these plots had the data points arranged around a straight line of slope 1 passing through the origin, and the conventional R values calculated on D were the lowest, ranging from 40 to 70%. It was deduced for structure determinations in this first category that the resonant-scattering contribution to the observed Friedel differences was significant, and that random uncertainties and systematic errors were minor in the intensity data. In the third category, the plots of Dobs against Dmodel had the data points arranged about the Dobs axis (where Dmodel = 0), the range of values of |Dobs| was larger, even much larger, than the range of values of |Dmodel|, and the R values on D were high with values between 90 and 100%. Flack et al. (2011[Flack, H. D., Sadki, M., Thompson, A. L. & Watkin, D. J. (2011). Acta Cryst. A67, 21-34.]) interpreted the results of the third category as indicating that there was no problem with the Dmodel values but that the Dobs were entirely dominated by random uncertainties and systematic errors which combined to obscure, almost entirely, the resonant-scattering contribution to the difference intensity of Friedel opposites. In the second category, the results were intermediate.

Various procedures are now described, arranged in the sequence in which they would be of use in a structure analysis. §2[link] deals with a novel method for the determination of the status of centrosymmetry of a crystal in which the contribution of resonant scattering is significant. §3[link] gives a detailed example of the extension of the Rmerge technique to distinguish between the possible point groups in a chosen Laue class. In §4[link] we deal with the validation of the intensity data after structure refinement. This is carried out on our own carefully designed measurements, data correction and refinements on a single crystal of a model compound, NaClO3. It is shown that these new data confirm and amplify the conclusions of Flack et al. (2011[Flack, H. D., Sadki, M., Thompson, A. L. & Watkin, D. J. (2011). Acta Cryst. A67, 21-34.]) and bring the analysis full circle. The problem in absolute-structure determination is shown to lie in the data measurements and corrections. The paper concludes in §5[link] with the presentation of the analyses of the intensity data from two centrosymmetric crystals.

2. Status of centrosymmetry and resonant scattering

We make use of the average and difference intensities of Friedel opposites given in equations (1a)[link] and (1b)[link]. It is necessary to recall a few basic facts concerning these quantities. A data set of intensities needs to contain both reflections hkl and [\overline{hkl}] in order to obtain the observed values of Aobs(hkl) and Dobs(hkl). A(hkl) is a centrosymmetric function as A(hkl) = A([\overline{hkl}]). On the other hand, D(hkl) is antisymmetric as D(hkl) = -D([\overline{hkl}]). In the model of a centrosymmetric crystal structure, |Fmodel(hkl)|2 = |Fmodel([\overline{hkl}])|2 and consequently Dmodel(hkl) = 0 in this case. The values of Dobs(hkl) of a centric reflection are entirely due to random uncertainties and systematic errors in the intensity measurements. On the other hand, the set of Dobs(hkl) of acentric reflections contains contributions both from the random uncertainties and the systematic errors of the data measurements, and from the differences between |F(hkl)|2 and |F([\overline{hkl}])|2 which arise through the effect of resonant scattering. Consequently, an appropriate study of the set of Dobs(hkl) of potentially acentric reflections of a crystal structure of unknown space group may supply useful indications on whether the structure is centrosymmetric or non-centrosymmetric.

The Bijvoet ratio, defined by [chi] = <D2>1/2/<A>, is the ratio of the root-mean-square value of D to the mean value of A. At the outset of a structure analysis, two independent estimates of the Bijvoet ratio are available and their comparison leads to useful information as to whether the crystal structure is centrosymmetric or not.

The first estimate arises from considerations of intensity statistics through the analysis of an ensemble of random structures leading to the definition of the Bijvoet ratio as a value called Friedifstat, whose functional form was derived by Flack & Shmueli (2007[Flack, H. D. & Shmueli, U. (2007). Acta Cryst. A63, 257-265.]) and Shmueli & Flack (2009[Shmueli, U. & Flack, H. D. (2009). Acta Cryst. A65, 322-325.]). One needs only to know the chemical composition of the compound and the wavelength of the X-radiation used in order to calculate Friedifstat. Flack & Bernardinelli (2008[Flack, H. D. & Bernardinelli, G. (2008). Acta Cryst. A64, 484-493.]) provide a spreadsheet application for its evaluation.

The second estimate of the Bijvoet ratio, Friedifobs, is obtained from the observed diffraction intensities. One tricky point in the evaluation of Friedifobs is the different variation of A and D with sin([theta])/[lambda]. One must thus work with suitably normalized values. The normalization of the Aobs values proceeds in the normal way by determining the values of the overall scale factor and isotropic displacement parameter from a Wilson plot of log(Aobs/<A>) against [sin([theta])/[lambda]]2 and applying these to obtain normalized values of Aobs and <Aobs>. Dobs values are normalized in the same way using the same values of the overall scale factor and isotropic atomic displacement parameter. However, for the Dobs a further adjustment is made to allow for the sin([theta])/[lambda] variation of Friedifstat. To aid in this procedure, a further spreadsheet application, available as supplementary material to Flack et al. (2011[Flack, H. D., Sadki, M., Thompson, A. L. & Watkin, D. J. (2011). Acta Cryst. A67, 21-34.]), calculates Friedifstat at various values of sin([theta])/[lambda] in the range 0.0 to 0.7 Å-1. Expressing this variation as a polynomial, Friedifstat = c0 + c1sin([theta])/[lambda] + c2[sin([theta])/[lambda]]2, the preliminary normalized Dobs are modified to become Dobs/{1 + (c1/c0)sin([theta])/[lambda] + (c2/c0)[sin([theta])/[lambda]]2}. From these further-normalized Dobs values, <Dobs2> can be obtained to give Friedifobs. A second tricky point in the calculation is to make sure that only acentric reflections of any of the non-centrosymmetric point groups in the chosen Laue class are selected for the calculation of Friedifobs. In this way one is sure that if the point group of the crystal is centrosymmetric, all of the chosen reflections are centric, and if the point group of the crystal is non-centrosymmetric, all of the chosen reflections are acentric. The necessary selection is achieved by taking only those reflections that are general in the Laue group and these are indicated in Table 1[link]. It is easy to prove that these will necessarily be acentric in any index-2 non-centrosymmetric subgroup of the Laue group. At the present time, the calculation of Friedifobs is not available in distributed software. On comparison of Friedifstat with Friedifobs one is able to state with some confidence that:

Table 1
Classification of reflections in centrosymmetric point groups

All reflections are centric and all have a value of zero for the root-mean-square D. H.A. signifies hexagonal axes and R.A. rhombohedral axes.

Point group h [epsilon], order of stabilizer General (g) or special (s)
[\bar 1] hkl 1 g
       
2/m hkl 1 g
  h0l, 0k0 2 s
       
mmm hkl 1 g
  0kl, h0l, hk0 2 s
  h00, 0k0, 00l 4 s
       
4/m hkl 1 g
  hk0 2 s
  00l 4 s
       
4/mmm hkl 1 g
  hk0, h0l, hhl 2 s
  hh0, h00 4 s
  00l 8 s
       
[\overline{3}] (H.A) hkl 1 g
  00l 3 s
       
[\overline{3}] (R.A) hkl 1 g
  hhh 3 s
       
[\overline{3}m]1 (H.A.) hkl 1 g
  hh0, h0l 2 s
  00l 6 s
       
[\overline{3}1m] (H.A.) hkl 1 g
  h00, hhl 2 s
  00l 6 s
       
[\overline{3}m] (R.A.) hkl 1 g
  [\bar h]h0, hhl 2 s
  hhh 6 s
       
6/m hkl 1 g
  hk0 2 s
  00l 6 s
       
6/mmm hkl 1 g
  hk0, h0l, hhl 2 s
  h00, hh0 4 s
  00l 12 s
       
[m\overline{3}] hkl 1 g
  0kl, hh0 2 s
  hhh 3 s
  00l 4 s
       
[m\overline{3}m] hkl 1 g
  hhl, 0kl 2 s
  hh0 4 s
  hhh 6 s
  h00 8 s

(i) If Friedifobs is much lower than Friedifstat, then the crystal structure is either centrosymmetric, or non-centrosymmetric with the crystal twinned by inversion in a proportion close to 50:50, and random uncertainties and systematic errors in the intensity data set are minor.

(ii) If Friedifobs is close in value to Friedifstat, then the crystal is non-centrosymmetric, not twinned by inversion, and random uncertainties and systematic errors in the intensity data set are minor. However, data from a centrosymmetric crystal with large random uncertainties and systematic errors may also produce this result.

(iii) If Friedifobs is much larger than Friedifstat, then either the data set is dominated by random uncertainties and systematic errors or the chemical formula is erroneous.

Example 1. Measurements were made on the Swiss-Norwegian Beamline (BM01A) at the European Synchrotron Radiation Facility in Grenoble, France, on the compound 1-methyl-4-oxotetrahydro-2H-imidazol-2-iminium tetrachlorocopper(II) (Udupa & Krebs, 1979[Udupa, M. R. & Krebs, B. (1979). Inorg. Chim. Acta, 33, 241-244.]) [Cambridge Structural Database (CSD; Allen, 2002[Allen, F. H. (2002). Acta Cryst. B58, 380-388.]) refcode: CRINCC] at 100 K with a radiation of wavelength 0.7000 Å. The crystal is known to be centrosymmetric (space group P21/c) and has a significant resonant-scattering contribution, Friedifstat = 498. The intensity data were merged and averaged in point group 1 giving 5372 measurements. The general reflections were separated from the special reflections (h0l and 0k0) for Laue group 2/m and collected into sets of reflections equivalent in 2/m. This led to 724 sets (2896 reflections) for which all four 2/m-symmetry-equivalent measurements were available. Normalized values of Aobs and Dobs were calculated for these 724 sets, from which <Aobs> and <Dobs2>1/2 could be obtained to give Friedifobs = 164. The comparison of 498 for Friedifstat with 164 for Friedifobs indicates that the crystal structure is centrosymmetric. Other selected statistics on CRINCC are given in §3[link].

Example 2. Measurements and analysis of the compound potassium hydrogen (2R,3R) tartrate (CSD refcode: ZZZRZW) are described in §3[link]. One finds there the values of <Anormalized> and <Dnormalized2>1/2 from the measurements of those sets of general reflections in the Laue group mmm which contain all eight mmm-symmetry-equivalent reflections. The value of Friedifobs is 217 compared to a Friedifstat value of 174. The agreement is good and allows the deduction that the crystal is neither centrosymmetric nor twinned by inversion in a proportion near to 50:50, nor that the data set is unsatisfactorily dominated by random uncertainty and systematic error. Clearly the crystal is non-centrosymmetric as justified by the results of the structure analysis and the excellent agreement between Dobs and Dmodel after refinement, as shown in Fig. 1[link]. 2AD plots are described in detail in §4.2.2[link].

[Figure 1]
Figure 1
Dobs against Dmodel of all Friedel pairs with 2Aobs against 2Amodel for weak Friedel pairs for ZZZRZW. On the left of the plot, Dobs - Dmodel and 2Aobs - 2Amodel of all Friedel pairs are displayed at constant abscissa.

Example 3. Details of the relevant intensity measurements, structure refinement and data analysis for 1,3,4,6-tetra-O-acetyl-2-(trifluoromethylsulfonyl)-D-mannopyranose (Zhu & Jiang, 2007[Zhu, H.-Y. & Jiang, S.-D. (2007). Acta Cryst. E63, o2833.]) (CSD refcode: UNEVAK01) are given in Flack & Bernardinelli (2008[Flack, H. D. & Bernardinelli, G. (2008). Acta Cryst. A64, 484-493.]). The Laue group is [\bar 1] so all reflections are general. One finds Friedifstat = 70 and Friedifobs = 499. The huge discrepancy between the two shows that the observed values of D are dominated by random uncertainty and systematic error. This result is confirmed by the Dobs against Dmodel plot obtained after least-squares refinement of the crystal structure. The plots resemble those of SEZPUJ presented in Fig. 2 of Flack et al. (2011[Flack, H. D., Sadki, M., Thompson, A. L. & Watkin, D. J. (2011). Acta Cryst. A67, 21-34.]).

3. Resolution of non-centrosymmetric ambiguities

It was shown in §2[link] that under certain circumstances it is possible to determine whether or not the space group of the crystal investigated is centrosymmetric. Suppose that the space group was found to be non-centrosymmetric. In each Laue class, there is one centrosymmetric point group and one or more non-centrosymmetric point groups. For example, in Laue class 2/m, such point groups are 2 and m, and in Laue class mmm we need to distinguish between the point groups 222, 2mm, m2m and mm2, and of course between the space groups based on them. We shall show that it is possible in practice to distinguish between these non-centrosymmetric point groups using intensity differences between Friedel opposites caused by resonant scattering. The first technique, which works well in our test example, is a generalization of the use of conventional merging R factors applied to all point groups within a chosen Laue class. The second technique relies on intensity enhancement within specific zones or lines of reflections, and turns out not to be entirely satisfactory. The techniques are demonstrated by a practical example.

3.1. Practical example on ZZZRZW

Intensity measurements on a crystal of ZZZRZW were made on the Swiss-Norwegian Beamline (BM01A) at the European Synchrotron Radiation Facility in Grenoble, France. A wavelength of 0.7469 Å was used at 100 K. The crystal structure is well established and occurs in space group P212121. Least-squares refinement on the data, merged and averaged in point group 222, displays conventional R factors RA = 3.1%, RD = 51.1% and RAweak = 10.4% (i.e. for those reflections with Aobs < |Dobs|max) (see Flack et al., 2011[Flack, H. D., Sadki, M., Thompson, A. L. & Watkin, D. J. (2011). Acta Cryst. A67, 21-34.]), and the Dobs against Dmodel plot in Fig. 1[link] shows the data to be distributed about the straight line Dobs = Dmodel. The data are thus of very good quality with a clear signal from the resonant scattering in the Friedel opposites, Friedifstat = 174. There were 20 679 intensity measurements in the raw data with one reflection having been measured as many as nine times. The experimental data are available as supplementary material (ktar.hkl).1 The Laue group was assumed to be mmm.

3.1.1. Rmerge on ZZZRZW

The steps in the data treatment are as follows:

(i) All reflections with identical reflection indices were collected together and averaged. This is a sorting and merging of reflections under point group 1 and makes no assumption at all as to the real point group of the intensity measurements. [For a crystal in Laue class mmm, the reflections were sorted and averaged in point group 1. In fact, the suitable merging group for reflections in any given Laue class is the highest subgroup common to all point groups in the Laue class. This merging group may be found by study of the figure of maximal subgroups of the three-dimensional point groups (Hahn & Klapper, 2002[Hahn, Th. & Klapper, H. (2002). Point groups and crystal classes, Figure 10.1.3.2. International Tables for Crystallography, Vol. A, Space-Group Symmetry, edited by Th. Hahn, 5th ed. Dordrecht: Kluwer Academic Publishers.]) and these are indicated in Table 2[link].]

Table 2
Laue classes with corresponding merging groups for the calculation of Aobs and Dobs

Laue class Point groups Merging group Index (Laue/merging)
[\bar 1] 1, [\bar 1] 1 2
2/m 2, m, 2/m 1 4
mmm 222, 2mm, m2m, mm2, mmm 1 8
4/m 4, [\overline{4}], 4/m 2 4
4/mmm 422, [\overline{4}]2m, [\overline{4}]m2, 4mm, 4/mmm 2 8
[\overline{3}] 3, [\overline{3}] 3 2
[\overline{3}m] 32, 3m, [\overline{3}m] 3 4
6/m 6, [\overline{6}], 6/m 3 4
6/mmm 622, [\overline{6}]2m, [\overline{6}]m2, 6mm, 6/mmm 3 8
[m\overline{3}] 23, [m\overline{3}] 23 2
[m\overline{3}m] 432, [\overline{4}3m], [m\overline{3}m] 23 4

(ii) The reflections were separated into seven classes corresponding to the general and special reflections of the Laue group mmm as indicated in Table 1[link]. The special reflections h00, 0k0 and 00l have a multiplicity of 2, the special reflections 0kl, h0l and hk0 have a multiplicity of 4, and the general reflections hkl have a multiplicity of 8. Reflections that are general reflections in point group mmm will necessarily be general reflections in all of its non-centrosymmetric subgroups (i.e. 2mm, m2m, mm2, 222) within the same Laue class (mmm). The same technique was used for selecting reflections in §2[link].

(iii) Intensity measurements were gathered together into sets of reflections equivalent under the Laue group mmm. The count of reflections in each set and the corresponding number of sets are shown in Table 3[link] for the general reflections hkl in mmm. It is somewhat surprising that, despite the large redundancy of the raw data, there is nevertheless a considerable proportion of data for which the full complement of eight reflections had not been measured.

Table 3
For the general reflections hkl of mmm, the number of measurements on ZZZRZW in each set and the corresponding number of sets of reflections symmetry equivalent under point group mmm

No. of measurements in set No. of sets
8 589
7 27
6 55
5 62
4 41
3 29
2 30
1 59
0 undetermined

(iv) Restricting the analysis to the 589 sets containing eight general reflections (hkl, [{\overline h}{\overline k}{\overline l}], [h{\overline k}{\overline l}], [{\overline h}kl], [{\overline h}k{\overline l}], [h{\overline k}l], [{\overline h}{\overline k}l], [hk{\overline l}]), merging R factors were calculated on the |Fobs|2 in the usual way. The values, given throughout in percentage units, are reported in the first line of Table 4[link]. One sees that the true point group of the crystal 222 has a significantly lower value of R|F|2 than the other point groups. Next, the |Fobs|2 of inversion-related reflections (i.e. hkl and [{\overline h}{\overline k}{\overline l}]; [h{\overline k}{\overline l}] and [{\overline h}kl]; [{\overline h}k{\overline l}] and [h{\overline k}l]; [{\overline h}{\overline k}l] and [hk{\overline l}]) were converted into their average (A) and difference (D) values. The A and D values were then averaged separately under the five point groups of Table 4[link] using the relationships A(hkl) = A([{\overline h}{\overline k}{\overline l}]) and D(hkl) = -D([{\overline h}{\overline k}{\overline l}]). The merging RA and RD values are reported in lines 2 and 3 of Table 4[link]. It should be noted that the merging RD value in a centrosymmetric point group is 100%, not by coincidence, but by definition. Once again RD merge of the true point group has the lowest value, which is now noticeably different from the other choices of point group. The R merging on D appears as a more sensitive value than the merging R|F|2.

Table 4
R merging values (%) for the 589 sets of general reflections of mmm which have all eight measurements on ZZZRZW in the set

Rmerge mmm 2mm m2m mm2 222
R|F|2 2.42 2.31 2.29 2.31 1.80
RA 1.30 1.30 1.30 1.30 1.30
RD 100.0 254.4 235.7 258.1 82.9

(v) Similar calculations to those described in (iv) were performed on sets of general reflections of mmm which had less than the full complement of eight measurements. In this case it is not possible to undertake the calculation of the Rmerge on A and D as these necessarily require all eight mmm-symmetry-equivalent reflections to be present to obtain the values of A and D. However it is possible to calculate the Rmerge on |Fobs|2. The results are given in Table 5[link]. From a comparison of Tables 4[link] and 5[link] it is clear that the R merging values are more reliable with sets containing a full complement of eight reflections. Moreover, the calculation using A and D is preferable to that on |Fobs|2.

Table 5
R merging values (%) for the 27 sets of general reflections of mmm which have seven measurements on ZZZRZW in the set, and the 55 and 27 sets that have six and seven measurements, respectively, in the set

Rmerge mmm 2mm m2m mm2 222 No. of measurements
R|F|2 3.67 3.78 3.16 3.44 2.96 7 measured in set
R|F|2 3.90 3.73 3.39 3.36 3.19 6 and 7 measured in set

(vi) Similar calculations to (iv) were performed on the mmm special reflections 0kl, h0l and hk0 where a full complement of four reflections had been measured in each set. There were 75 0kl, 52 h0l and 129 hk0 reflections. The results are presented in Table 6[link]. It is very clear from Table 6[link] that merging R factors either on |F|2 or D from these special reflections do not provide reliable indications of the point group of the crystal.

Table 6
R merging values (%) on |F|2, A and D from ZZZRZW for special reflections 0kl, h0l and hk0 of mmm containing a full complement of four reflections

  Rmerge mmm 2mm m2m mm2 222
0kl R|F|2 1.85 1.85 1.25 1.55 1.85
h0l R|F|2 1.20 1.01 1.20 0.98 1.20
hk0 R|F|2 1.52 1.29 1.18 1.52 1.52
0kl RA 0.68 0.68 0.68 0.68 0.68
h0l RA 0.72 0.72 0.72 0.72 0.72
hk0 RA 0.75 0.75 0.75 0.75 0.75
0kl RD 100.0 100.0 72.8 137.3 100.0
h0l RD 100.0 107.4 100.0 93.1 100.0
hk0 RD 100.0 115.8 86.4 100.0 100.0
3.1.2. Intensity enhancement analysis on ZZZRZW

This method is based on analytical expressions for <D2>, derived for an ensemble of random structures, and the classification of reflections for all non-centrosymmetric point groups given in Table 1[link] of Shmueli & Flack (2009[Shmueli, U. & Flack, H. D. (2009). Acta Cryst. A65, 322-325.]). Special zones and lines of reflections are there shown to have an intensity enhancement which is characteristic of the point group of the crystal. In preparation for the applications to real data, the classification presented in Table 1 of Shmueli & Flack (2009[Shmueli, U. & Flack, H. D. (2009). Acta Cryst. A65, 322-325.]) has been supplemented to include the centrosymmetric point groups that are presented in Table 1[link].

To apply this technique one calculates <A> and <D2>1/2 over various sets of data. Now, as the values of A and D vary with sin([theta])/[lambda], it is the normalized values of A and D that have to be prepared and averaged. The normalization technique that we used is exactly the same as that described in §2[link]. Also, as the calculation requires values of A and D, one only uses those sets of reflections that have a full complement of measurements, i.e. eight for general reflections hkl, four for 0kl, h0l and hk0, two for h00, 0k0 and 00l. The results are given in Table 7[link].

Table 7
<Anormalized> and <D2normalized>1/2 for all sets of reflections of ZZZRZW with a full complement of measurements

  No. of sets of reflections <Aobs normalized> <D2obs normalized>1/2
hkl 589 1.73 0.038
0kl 75 1.59 0.027
h0l 52 1.43 0.019
hk0 129 1.70 0.021
h00 5 1.40 0.008
0k0 7 3.87 0.039
00l 1 0.15 0.009

The contents of Table 7[link] need to be consulted in conjunction with Table 1 of Shmueli & Flack (2009[Shmueli, U. & Flack, H. D. (2009). Acta Cryst. A65, 322-325.]) and Table 1[link]. The general reflections hkl give a baseline value of <A>. The values of <A> for the reflection classes 0kl, h0l and hk0 clearly indicate that there is no intensity enhancement in these zones and preclude the point group of the crystal being 2mm, m2m and mm2 for which one zone has an enhancement factor of 2. For the classes h00, 0k0 and 00l there are very few reflections indeed, making the results unreliable at the very best. It may just be a happy coincidence that <A> for 0k0 is about twice the value for general reflections hkl as required for point group 222. Concerning the values of root-mean-square D, these do not give any clear indication. According to Table 1[link] of Shmueli & Flack (2009[Shmueli, U. & Flack, H. D. (2009). Acta Cryst. A65, 322-325.]), r.m.s. D should have a value of zero for point group 222 but the 0kl, h0l and hk0 zones have approximately the value of the general hkl reflections.

3.2. ZZZRZW and CRINCC

From all of the above information we note that it is the merging R factors on D, derived using only general reflections of mmm for which a full complement of eight reflections have been measured, that provide the clearest indication of the point group of the crystal. For applications to other Laue classes, Table 1 of Shmueli & Flack (2009[Shmueli, U. & Flack, H. D. (2009). Acta Cryst. A65, 322-325.]) and Table 1[link] provide a complete list of general and special reflections.

For completeness, we report the principal statistics for the centrosymmetric structure CRINCC, Laue group 2/m, already described as example 1 in §2[link]. Table 8[link], similar to Table 4[link], gives the R merging values for the 724 sets of general reflections of 2/m which have all four measurements in the set. RD values show no preference between the three point groups, whereas from R|F|2 one might be tempted to believe that the point group is 2. Table 9[link], similar to Table 7[link], gives normalized <Aobs> and <Dobs2>1/2 for sets of reflections with a full complement of measurements. Space-group-absent reflections have been omitted. No pairs of special non-space-group-absent reflections 0k0 were available. The normalized <Aobs> is compatible with a symmetry-enhancement value of 2 for the h0l reflections in agreement with Table 1[link].

Table 8
R merging values (%) for CRINCC for the 724 sets of general reflections of 2/m which have all four measurements in the set

Rmerge 2/m m 2
R|F|2 2.23 2.05 1.85
RA 1.29 1.29 1.29
RD 100.0 98.3 101.7

Table 9
<Anormalized> and <D2normalized>1/2 for all sets of reflections of CRINCC with a full complement of measurements

  No. of sets of reflections <Aobs normalized> <D2obs normalized>1/2
hkl 724 3.16 0.05
h0l 80 8.84 0.12

4. Data validation demonstrated on NaClO3

We here present the use of A and D in data validation following structure solution and least-squares refinement. A simple model compound, NaClO3, has been used in this enterprise.

4.1. Intensity measurements and corrections on NaClO3

Diffraction intensity measurements were made on one single crystal (dimensions 0.01 × 0.16 × 0.18 mm) of NaClO3 crystallized from aqueous solution. NaClO3 crystallizes in space group P213 (No. 198) (a [\simeq] 6.54 Å), Z = 4 with both the Na and the Cl atoms in special positions 4a (x,x,x) on the threefold axis and an O atom in a general position 12b. Measurements were made at 150 K. The value of Friedifstat (see, for example, Flack et al., 2011[Flack, H. D., Sadki, M., Thompson, A. L. & Watkin, D. J. (2011). Acta Cryst. A67, 21-34.]) is 114 for Mo K[alpha] and 492 for Cu K[alpha] radiation. The goniometers used allowed full orientational freedom of movement of the crystal and the intensities were recorded on area detectors. A full sphere of data was measured giving between 3260 and 3412 reflections with Mo K[alpha] radiation, and between 3581 and 4622 reflections with Cu K[alpha] radiation.

Using Mo K[alpha] radiation, three distinct data sets were measured on one instrument with different exposure times per frame: i.e. 60 s (labelled MoK[alpha]-slow), 10 s (labelled MoK[alpha]-medium) and 1 s (labelled MoK[alpha]-fast). Two different data-reduction software packages were used, labelled SftW1 and SftW2. These produce a list of integrated intensities from the raw frames and apply corrections for standard systematic effects such as Lorentz-polarization.

Using Cu K[alpha] radiation, two distinct data sets were measured on an instrument different from the one used for the Mo K[alpha] data sets. The set labelled CuK[alpha]-slow was measured over a period of 24 h with times per frame of 4 and 20 s for low- and high-angle detector positions. The set labelled CuK[alpha]-medium used 2 and 5 s per frame. These two data sets were processed only with the SftW1 data-reduction software package.

The five data sets of intensity measurements (MoK[alpha]-fast, MoK[alpha]-medium, MoK[alpha]-slow, CuK[alpha]-medium and CuK[alpha]-slow) were corrected for absorption following four different procedures. These are now described.

AbsNo: No absorption correction was applied.

AbsSe: A semi-empirical absorption correction was applied. A semi-empirical absorption correction uses redundancy in the intensity data to undertake its absorption correction. Symmetry-equivalent reflections are used to produce a correction which makes the corrected intensities as nearly equal as possible in a least-squares sense. The correction function is the sum of a series of spherical harmonic functions each with an adjustable coefficient, as described for example in Blessing (1995[Blessing, R. H. (1995). Acta Cryst. A51, 33-38.]). The maximum order of spherical harmonics is generally a parameter chosen by the user. As the criterion of a semi-empirical correction is the near-equality of symmetry-equivalent reflections, the correction produced is not a pure absorption correction but rather a general-purpose correction. The origin of the disparity in intensity between symmetry-equivalent reflections is not identified. The point group used to identify equivalent reflections when applying this procedure to NaClO3 was 23 for all data sets.

AbsGr: For SftW1, an absorption correction was applied using the method of Clark & Reid (1995[Clark, R. C. & Reid, J. S. (1995). Acta Cryst. A51, 887-897.]), whereas for SftW2 the correction was obtained by numerical integration based on a Gaussian grid over the volume of the crystal. For these procedures it is necessary to index the crystal faces and measure their distance from the centre of the crystal.

AbsGrSe: An absorption correction as carried out for AbsGr was followed by a low-order semi-empirical absorption correction.

After each of these absorption corrections, the intensity data were merged and averaged in point group 23. Refinement of the 32 absorption-corrected data sets was carried out by linear(ized) least squares working on |F|2. 18 parameters were refined: scale factor, Flack parameter (Flack, 1983[Flack, H. D. (1983). Acta Cryst. A39, 876-881.]), isotropic extinction parameter, x(Na), x(Cl), x(O), y(O), z(O), U11(Na), U12(Na), U11(Cl), U12(Cl), U11(O), U22(O), U33(O), U12(O), U23(O), U13(O). There were 247 or 248 (95 acentric pairs, 1 unpaired acentric and 56 or 57 centric) reflections in the Mo K[alpha], and 197 (74 acentric pairs, 1 or 0 unpaired acentric and 48 or 49 centric) reflections in the Cu K[alpha] least-squares refinements.

Table 10[link] contains the R values and Flack parameter of the various refinements. File AFAD.NaClO3.allplots.pdf in the supplementary material 1 contains the plots of Aobs against Amodel, Dobs against Dmodel with 2Aobs against 2Amodel in the same range, and (D/A)obs against (D/A)model for Aobs > 0.01Amax of all of the refinements. The Aobs against Amodel plots are on logarithmic axes whilst the others are on linear axes. In all cases the x and y axes are arranged to span the same domain of values. In this way, a plot with a satisfactory agreement between the observed and model values follows a straight line of slope 1 passing through the origin. Some of the plots have been reproduced in the body of the paper by way of examples.

Table 10
Residuals for structure refinements on NaClO3 for the five data sets (MoK[alpha]-fast, MoK[alpha]-medium, MoK[alpha]-slow, CuK[alpha]-medium and CuK[alpha]-slow) each treated with four different absorption-correction procedures (AbsNo, AbsSe, AbsGr and AbsGrSe) and two data-reduction software options (SftW1 and SftW2)

RA = [\textstyle\sum _{hkl}^{\rm paired\, acentric} | A_{\rm obs}({hkl} ) - A_{\rm model}(hkl)|/\sum_{hkl}^{\rm paired\, acentric} |A_{\rm obs}(hkl)|], RD = [ \textstyle \sum _{hkl}^{\rm paired\, acentric} | D_{\rm obs}({hkl}) - D_{\rm model}(hkl) |/ \sum _{hkl}^{\rm paired\, acentric}| D_{\rm obs}(hkl)|]. RAweak is an R value on A limited to reflections with Aobs < |Dobs|max. Rint is an internal R factor with respect to point symmetry 23. R|F|2 is a conventional R factor on all data.

Data Absorption correction Software RA (%) RD (%) RAweak (%) RD/A (%) Rint (%) R|F|2 all (%) Flack x
MoK[alpha]-fast AbsNo SftW1 2.3 90.9 10.5 90.0 7.79 2.03 0.09 (13)
MoK[alpha]-fast AbsSe SftW1 2.3 89.0 16.7 92.7 6.30 2.04 0.11 (12)
MoK[alpha]-fast AbsGr SftW1 2.3 88.5 9.1 89.5 7.63 2.00 0.11 (13)
MoK[alpha]-fast AbsGrSe SftW1 2.3 89.1 16.7 92.9 6.26 2.04 0.11 (12)
MoK[alpha]-med AbsNo SftW1 2.0 65.7 6.0 62.2 3.32 1.32 0.02 (8)
MoK[alpha]-med AbsSe SftW1 2.0 68.1 5.7 67.6 2.98 1.30 0.02 (8)
MoK[alpha]-med AbsGr SftW1 2.0 62.3 6.9 61.5 3.30 1.30 0.01 (8)
MoK[alpha]-med AbsGrSe SftW1 1.9 70.3 5.6 69.8 2.98 1.30 0.02 (8)
MoK[alpha]-slow AbsNo SftW1 2.3 59.6 7.8 52.9 2.91 1.31 0.05 (8)
MoK[alpha]-slow AbsSe SftW1 2.3 80.3 6.1 75.1 4.64 1.55 0.04 (9)
MoK[alpha]-slow AbsGr SftW1 2.3 63.4 6.7 54.1 2.97 1.31 0.05 (8)
MoK[alpha]-slow AbsGrSe SftW1 2.3 79.2 6.3 74.9 4.64 1.55 0.04 (9)
MoK[alpha]-fast AbsNo SftW2 2.9 96.4 11.4 96.0 16.26 2.52 0.13 (14)
MoK[alpha]-fast AbsSe SftW2 2.2 86.9 21.6 94.3 4.41 1.85 0.09 (12)
MoK[alpha]-fast AbsGr SftW2 2.9 96.5 6.2 96.2 15.95 2.51 0.15 (14)
MoK[alpha]-fast AbsGrSe SftW2 2.2 86.9 21.6 94.3 4.41 1.85 0.09 (12)
MoK[alpha]-med AbsNo SftW2 2.7 94.3 5.1 88.0 13.86 2.03 0.02 (11)
MoK[alpha]-med AbsSe SftW2 1.8 67.9 5.5 61.4 2.59 1.21 0.02 (8)
MoK[alpha]-med AbsGr SftW2 2.7 94.2 4.8 88.9 13.53 1.98 0.01 (11)
MoK[alpha]-med AbsGrSe SftW2 1.8 67.8 5.7 61.1 2.60 1.21 0.02 (8)
MoK[alpha]-slow AbsNo SftW2 3.6 95.5 4.6 92.1 14.76 2.28 0.01 (11)
MoK[alpha]-slow AbsSe SftW2 2.1 52.0 6.6 46.9 2.16 1.20 0.03 (8)
MoK[alpha]-slow AbsGr SftW2 3.1 94.9 4.1 91.5 14.41 2.12 0.01 (11)
MoK[alpha]-slow AbsGrSe SftW2 2.1 51.3 6.6 45.8 2.17 1.21 0.03 (8)
CuK[alpha]-med AbsNo SftW1 3.4 65.5 5.0 49.6 8.99 2.23 0.03 (3)
CuK[alpha]-med AbsSe SftW1 3.0 39.1 4.2 35.8 5.11 2.07 0.02 (3)
CuK[alpha]-med AbsGr SftW1 2.6 59.2 3.8 45.6 7.31 1.96 0.02 (3)
CuK[alpha]-med AbsGrSe SftW1 2.8 41.0 4.5 31.7 4.95 1.90 0.02 (3)
CuK[alpha]-slow AbsNo SftW1 3.7 82.0 3.9 49.0 10.76 2.60 0.02 (3)
CuK[alpha]-slow AbsSe SftW1 4.1 36.9 3.4 25.4 5.23 2.28 0.01 (3)
CuK[alpha]-slow AbsGr SftW1 3.2 69.3 4.1 39.5 7.89 2.30 0.01 (3)
CuK[alpha]-slow AbsGrSe SftW1 3.7 32.2 3.0 23.0 5.18 2.05 0.00 (2)

4.2. Interpretation of NaClO3 measurements

4.2.1. Aobs against Amodel plots

These plots are on logarithmic axes, which entail both advantages and disadvantages. A distinct advantage is that the data may be presented compactly in one plot. However, the logarithmic axes have a tendency to exaggerate the spread of the data points at low A values and to compress the spread at high A values. One of the poorest plots is that of MoK[alpha]-fast-AbsNo-SftW2 shown in Fig. 2[link](a) and one of the best is that of CuK[alpha]-slow-AbsGr-SftW1 shown in Fig. 2[link](b). For MoK[alpha]-fast-AbsNo-SftW2 there is a slight spread of data points around the ideal line of slope 1 passing through the origin whereas CuK[alpha]-slow-AbsGr-SftW1 is almost perfect. In none of the plots is there any `outlier' or `rogue' reflection. Table 10[link] shows that the range of RA values is between 1.8 and 4.1%. One may say that these plots reveal decently refined crystal structures. In accordance with expectation, the noisiest plots are those of data measured at high speed and the best ones have been measured at low speed. A notable feature is that data-reduction software SftW1 consistently produces a positive intercept on the Aobs axis (indicating that the intensity of weak reflections is overestimated), largest for the fast data collection and smallest for the slow data collection. The plots for SftW2 do not show this effect and seem to pass very close to the origin of the plot. These effects are very clearly visible in the 2AD plots presented in §4.2.2[link]. However, the RA values produced by SftW1 are consistently lower than those of SftW2. The absorption corrections have worked well on the A data for which the software has most probably received the most tests and validation.

[Figure 2]
Figure 2
Aobs against Amodel on logarithmic axes for MoK[alpha]-fast-AbsNo-SftW2 in (a) and CuK[alpha]-slow-AbsGr-SftW1 in (b).
4.2.2. The 2AD plots

These plots contain all of the Dobs, Dmodel data points plotted on linear axes of identical length. The plots also contain 2Aobs, 2Amodel data points lying within the domain of values of the Dobs, Dmodel data. On the left of each plot, Dobs - Dmodel and 2Aobs - 2Amodel of all Friedel pairs are displayed at constant abscissa. These show the spread of residual 2A and D obtained after refinement, and hence the overall uncertainties and errors in the data. For each individual Friedel pair of reflections, 2A and D are, respectively, the sum and difference of |F(hkl)|2 and |F([\overline{hkl}])|2 and, consequently, their standard uncertainties u(2A) and u(D) are identical. In the 2AD plots, the (weak) Dobs, Dmodel data are being compared with the weak 2Aobs, 2Amodel data. If the 2Aobs, 2Amodel plot has a different appearance to the Dobs, Dmodel plot, one is led to suspect that systematic errors are affecting the data. Table 10[link] shows RD, RA and RAweak values, the latter being RA limited to those reflections with Aobs < |Dobs|max which appear in the 2AD plots.

One of the poorest cases is that of MoK[alpha]-fast-AbsNo-SftW2 shown in Fig. 3[link](a). The range of values in Dobs (|Dobs|max = 117) is much larger than in Dmodel (|Dmodel|max = 16). The arrangement of the Dobs, Dmodel data points is very different from that of the 2Aobs, 2Amodel points. It would appear that the random uncertainties and systematic errors in the Dobs values are masking the resonant-scattering signal. Since this data set has been measured with a very short exposure time per frame, it is natural that the data are noisy and RD is high at 96.4% (RA = 2.9%).

[Figure 3]
Figure 3
Dobs against Dmodel of all Friedel pairs with 2Aobs against 2Amodel for weak Friedel pairs for MoK[alpha]-fast-AbsNo-SftW2 in (a) and CuK[alpha]-slow-AbsGrSe-SftW1 in (b). On the left of the plot, Dobs - Dmodel and 2Aobs - 2Amodel of all Friedel pairs are displayed at constant abscissa.

One of the best cases is CuK[alpha]-slow-AbsGrSe-SftW1 shown in Fig. 3[link](b). The range of values in Dobs (-58 to 115) is similar to that in Dmodel (-40 to 79). Moreover, most of the Dobs, Dmodel data points are distributed about the ideal line of slope 1 passing through the origin. With RD at 32.2% (RA = 3.7%), the result is most satisfactory. It is apparent that the Dobs values are dominated by the resonant-scattering contribution with random uncertainties and systematic errors making only a small contribution.

Let us examine the effect of the speed of data collection by comparing plots in which the same method of absorption correction has been applied. We have chosen the series MoK[alpha]-fast-AbsGr-SftW1, MoK[alpha]-medium-AbsGr-SftW1 and MoK[alpha]-slow-AbsGr-SftW1 shown in Fig. 4[link]. For MoK[alpha]-fast-AbsGr-SftW1, the Dobs, Dmodel data points show a bunch of values around the origin with a vague tendency to follow a line of slope greater than 1 passing through the origin, RD is 88.5%. In Fig. 4[link](a) one Dobs, Dmodel data point at (-120.5, -7.6) has been omitted. The range of |Dobs| is approximately seven times that of |Dmodel|. For MoK[alpha]-medium-AbsGr-SftW1, the Dobs, Dmodel data points show a more extended bunch around the origin but with a clear tendency to follow a line of slope 1 with RD at 62.3%. The range of |Dobs| is twice that of |Dmodel|. For MoK[alpha]-slow-AbsGr-SftW1, the arrangement of the Dobs, Dmodel data points is very clearly a straight line of slope 1 passing through the origin. There are a few outliers. Comparison with the other sets of three plots (fast, medium and slow) with identical absorption correction shows the same behaviour as the AbsGr set. This experiment makes it very clear that random uncertainties can very easily produce Dobs against Dmodel plots in which the data points are arranged around the Dobs axis and that the plot becomes more satisfactory if the intensities are measured with more care and more slowly.

[Figure 4]
Figure 4
Dobs against Dmodel of all Friedel pairs with 2Aobs against 2Amodel for weak Friedel pairs for MoK[alpha]-XXX-AbsGr-SftW1. (a) XXX = fast, (b) XXX = medium and (c) XXX = slow. On the left of the plot, Dobs - Dmodel and 2Aobs - 2Amodel of all Friedel pairs are displayed at constant abscissa.

The effect of an absorption correction can be seen in the series of four plots of the data collection at slow speed: MoK[alpha]-slow-AbsNo-SftW2, MoK[alpha]-slow-AbsSe-SftW2, MoK[alpha]-slow-AbsGr-SftW2 and MoK[alpha]-slow-AbsGrSe-SftW2 seen in Fig. 5[link]. In AbsNo, even excluding some outliers, the data points are arranged around the Dobs axis and the range of |Dobs| is much larger than that of |Dmodel|. An absorption correction by numerical integration (AbsGr) improves things very little. In the plot of the semi-empirical correction AbsSe, the Dobs and Dmodel values are now about on the same scale. AbsSe and AbsGrSe show straight lines of slope 1. For most data sets, one sees that the Se and GrSe corrections produce very similar Dobs, Dmodel plots, suggesting that the Gr part of the correction is not helping very much. Two explanations spring to mind. Maybe the indexing of the crystal faces and the distance measurements are inadequate or maybe the systematic error present in the Gr plots is not due to absorption.

[Figure 5]
Figure 5
Dobs against Dmodel of all Friedel pairs with 2Aobs against 2Amodel for weak Friedel pairs for MoK[alpha]-slow-YYY-SftW2. (a) YYY = AbsNo, (b) YYY = AbsSe, (c) YYY = AbsGr and (d) YYY = AbsGrSe. On the left of the plot, Dobs - Dmodel and 2Aobs - 2Amodel of all Friedel pairs are displayed at constant abscissa.

There is also an effect of the speed of data collection on the quality of a semi-empirical absorption correction. Fig. 6[link] shows plots for MoK[alpha]-fast-AbsNo-SftW2, MoK[alpha]-fast-AbsSe-SftW2 and MoK[alpha]-medium-AbsSe-SftW2. For the fast data set there is a slight improvement on applying a semi-empirical correction, RD changing from 96.4 to 86.9%, though the apparent positive gradient is actually the result of just a few data points. By contrast, with the exception of one or two outliers, the Dobs, Dmodel data for the medium data set follow the 2Aobs, 2Amodel data tolerably well, with RD = 67.9%. For the slow data set RD = 52.0%. The ability of the semi-empirical method to correct for systematic errors depends on the quality of the individual intensity measurements. If the latter are subject to high random uncertainties because they have been collected rapidly, the resulting semi-empirical correction is inadequate. This shows that a data-collection strategy aimed at obtaining rapidly a data set with a very high redundancy is not appropriate for absolute-structure determination when using a semi-empirical absorption correction.

[Figure 6]
Figure 6
Dobs against Dmodel of all Friedel pairs with 2Aobs against 2Amodel for weak Friedel pairs for MoK[alpha]-fast-AbsNo-SftW2 in (a), MoK[alpha]-fast-AbsSe-SftW2 in (b) and MoK[alpha]-medium-AbsSe-SftW2 in (c). On the left of the plot, Dobs - Dmodel and 2Aobs - 2Amodel of all Friedel pairs are displayed at constant abscissa.

The effect of a change of data-reduction software can be seen by comparing MoK[alpha]-slow-AbsNo-SftW1 and MoK[alpha]-slow-AbsNo-SftW2 shown in Fig. 7[link]. In both Dobs against Dmodel plots there are outliers, but those of SftW2 have considerably larger values of |Dobs| than SftW1. Moreover, the Dobs, Dmodel data points of SftW1 follow reasonably closely a straight line of slope 1 passing through the origin, whereas those of SftW2 are bunched around the Dobs axis. Simultaneously the 2Aobs, 2Amodel data points of SftW1 appear to be more noisy than those of SftW2.

[Figure 7]
Figure 7
Dobs against Dmodel of all Friedel pairs with 2Aobs against 2Amodel for weak Friedel pairs for MoK[alpha]-slow-AbsNo-ZZZ. (a) ZZZ = SftW1 and (b) ZZZ = SftW2. On the left of the plot, Dobs - Dmodel and 2Aobs - 2Amodel of all Friedel pairs are displayed at constant abscissa.
4.2.3. (D/A)obs against (D/A)model plots

In an attempt to improve the quality of absolute-structure determination, Parsons & Flack (2004[Parsons, S. & Flack, H. (2004). Acta Cryst. A60, s61.]) and Parsons (2011[Parsons, S. (2011). Acta Cryst. A67, C191.]) have developed and investigated the use of the ratio D/A as observables in a least-squares technique. The hypotheses underlying this approach are that: (i) there are significant systematic errors on all intensities under the restriction that the difference in these errors between reflections hkl and [\overline{hkl}] is small, and (ii) random uncertainties in the intensity measurements are assumed to be small.

Indeed, in the past, using the now obsolete four-circle serial diffractometer, it was possible under certain conditions, see e.g. Le Page et al. (1990[Le Page, Y., Gabe, E. J. & Gainsford, G. J. (1990). J. Appl. Cryst. 23, 406-411.]), to produce intensity measurements obeying the above conditions very closely. We may make a simple model encapsulating these hypotheses by writing

[\let\normalbaselines\relax\openup2pt\matrix { \left| {F\left({ h k l} \right)} \right|_{\rm obs}^2 = \left [{s\left({hkl} \right) + {\Delta}s\left({hkl} \right)} \right] \left| {F\left({ h k l} \right)} \right|_{\rm model}^2, \cr \left| {F\left({ \overline h \overline k \overline l} \right)} \right|_{\rm obs}^2 = \left [{s\left({hkl} \right) - {\Delta}s\left({hkl} \right)} \right] \left| {F\left({ \overline h \overline k \overline l} \right)} \right|_{\rm model}^2.}\eqno(2)]

In these, s(hkl) and [Delta]s(hkl) are, respectively, the average and half-difference of the systematic errors to reflections hkl and [\overline{hkl}]. This leads to

[\eqalignno{(D/ A)_{\rm obs} &\simeq (D/A)_{\rm model} + {{2{\Delta}s\left({hkl} \right)} \over {s\left({hkl} \right)}} - {\left [{{{{\Delta}s\left({hkl} \right)} \over {s\left({hkl} \right)}}} \right]^2} (D/A)_{\rm model}&\cr &\quad - {{{\Delta}s\left({hkl} \right)} \over {2s\left({hkl} \right)}} (D/A)_{\rm model}^2, &(3)\cr}]

from which it may be seen that for small relative systematic errors, [Delta]s/s, between reflections hkl and [\overline{hkl}], (D/A)obs [\simeq] (D/A)model. The major part of the systematic errors, s, cancels out in this approach. In the supplementary material we also present plots of (D/A)obs against (D/A)model in order to be able to judge how well this approach applies to modern-day measurement techniques. So that the plots conform to the premises of this procedure, we eliminate all intensity measurements which have Aobs < 0.01Amax. These few eliminated reflections are thus weak. In Table 10[link] one can see that for all data sets other than MoK[alpha]-fast, the R values on (D/A) are smaller than the corresponding ones on D. Indeed, the slow data sets are superior to the medium ones, as are the Cu K[alpha] ones to those using Mo K[alpha]. Moreover, comparison of the plots of (D/A) to those on D shows that the former are cleaner and closer to the ideal plot of slope 1 passing through the origin. The ratio procedure is clearly providing some degree of correction of the systematic errors not undertaken by the various absorption corrections we applied. The plots of the MoK[alpha]-fast refinements show that the ratio procedure does very little for data which are dominated by random uncertainties. In all, the D/A ratio is performing according to expectation, with the best performance being obtained for data sets with a small random uncertainty and appreciable resonant-scattering contribution.

5. 2AD plots from centrosymmetric crystal structures

In the course of the investigations described in §§2[link][link] to 4[link], it became clear that we had neglected to produce 2AD plots of any centrosymmetric crystal structure to serve for the purposes of comparison and reference. As a precaution, only reflections which are general in the point group and for which both the hkl and [\overline{hkl}] reflections have been measured were used to produce these plots. Technically, these are more difficult to produce, as the file of observed and model intensities from the final least-squares cycle contains reflections merged and averaged in the centrosymmetric point group of the model. Now, although the available |Fobs|2 and |Fmodel|2 values are identical to Aobs and Amodel, respectively, and one knows that all Dmodel are zero, the Dobs values cannot be obtained from this file. They may, however, be obtained from a file of observed intensity data which have been merged and averaged either in point group 1 or in the merging group indicated in Table 2[link]. In general, the latter intensity data are not on an absolute scale and a suitable scale factor is determined by comparison with the data in the file obtained at the completion of least-squares refinement. As the averaging procedures for producing the Aobs and the Dobs values are different, one needs to plot (2N)1/2A values to compare with D values. We have nevertheless continued to call this a 2AD plot. N is equal to the order of the point group of the crystal if the merging and averaging have been carried out in point group 1, and is the value of index given in Table 2[link] if the merging group of Table 2[link] has been used.

Intensity measurements (14 997 Bragg reflections) on a crystal of 2-(4-(di-p-tolylmethylene)cyclohexa-2,5-dien-1-ylidene)malononitrile (local code: tk3d) were made on the Swiss-Norwegian Beamline (BM01A) at the European Synchrotron Radiation Facility in Grenoble, France. A wavelength of 0.69830 Å was used at 100 K. The crystal structure occurs in space group P21/c, the compound has composition C24H18N2 giving Friedifstat = 2. Least-squares refinement on the data, merged and averaged in point group 2/m, displays conventional R factors R|F|all = 3.9%, R|F| = 3.7%, RA = 5.6% and RAweak = 6.3% (i.e. for those reflections with Aobs < |Dobs|max) (see Flack et al., 2011[Flack, H. D., Sadki, M., Thompson, A. L. & Watkin, D. J. (2011). Acta Cryst. A67, 21-34.]). The data are thus of very good quality. The 2AD plot is in Fig. 8[link]. Some systematic error is affecting the D data. (Three outliers have not been reproduced in Fig. 8[link].) Moreover, as with the NaClO3 data treated with data-reduction software SftW1, the mean line of the 2Aobs, 2Amodel data has a positive intercept at Amodel = 0. The 2AD plot of CRINCC, described in §§2[link] and 3[link], shows similar features, as seen in Fig. 9[link].

[Figure 8]
Figure 8
Dobs against Dmodel of all Friedel pairs with 2Aobs against 2Amodel for weak Friedel pairs for tk3d. On the left of the plot, Dobs - Dmodel and 2Aobs - 2Amodel of all Friedel pairs are displayed at constant abscissa.
[Figure 9]
Figure 9
Dobs against Dmodel of all Friedel pairs with 2Aobs against 2Amodel for weak Friedel pairs for CRINCC. On the left of the plot, Dobs - Dmodel and 2Aobs - 2Amodel of all Friedel pairs are displayed at constant abscissa.

6. Concluding remarks

The technique demonstrated in §2[link] shows that information on the status of centrosymmetry of a crystal may be obtained from the observed average and differences of the Friedel opposites without the need for a model of the crystal structure. The necessary condition for success is that the resonant-scattering contribution to the Friedel differences be clearly expressed in the observed diffraction data. Unfortunately this contribution is often masked by the random uncertainties and systematic errors in the diffraction data. Nevertheless, the technique of §2[link] has its place in the toolkit of the structure analyst.

The measurements on ZZZRZW described in §3[link] reveal a weakness in the data-collection strategy as required for the type of analysis described therein. One recalls that one reflection was measured as many as nine times. On the other hand, Table 3[link] shows that for a significant number of sets of reflections, symmetry equivalent under mmm, not all of the eight reflections in each set had been measured. This lack of measurements is to the detriment of the use of Rmerge in determining the point group of the crystal. The measurements were performed with the crystal in one single orientation on its mount. This is the method that seems to be used universally. Although it should be possible to collect a complete sphere of data on a diffractometer equipped with a four-circle goniometer, this can add substantially to data-collection times, and so is rarely achieved in practice. A more even coverage of the reflections in reciprocal space could doubtless be obtained by undertaking measurements with the crystal in more than one orientation on its mount. One would require to know what are optimal values for the number of different orientations of the crystal and their mutual angular offsets. We know of no publications dealing with such matters.

Following the introduction of the Wilson intensity statistics (Wilson, 1949[Wilson, A. J. C. (1949). Acta Cryst. 2, 318-321.]), Rogers (1950[Rogers, D. (1950). Acta Cryst. 3, 455-464.]) presented a detailed analysis of new methods for determining crystal classes and space groups. Rogers' method made use both of the established techniques of merging R and space-group absences, together with the symmetry enhancement of the intensity of zones and lines of special reflections. In practice, the latter proved to be of limited reliability and the method is not used in practice nowadays. Our analysis of symmetry enhancement of intensity in §3.1.2[link] confirms the unreliability of this method.

We have shown that the use of Rmerge for determining the point group of a crystal is viable with a good data set for a crystal in point group 222. In the future we shall investigate the use of this technique to point groups with fewer symmetry-equivalent reflections in the Laue group. In particular, we think that it will be worthwhile to investigate crystals in the following point groups: 2, m and mm2.

Concerning the contents of §4[link], we have paid the greatest attention not to identify the producers of the instrumentation and software that have been used in the study on NaClO3. The intention of the current paper is to reveal general problems of data collection and correction arising in absolute-structure determination and not to undertake a witch-hunt of instrumentation and software in their design, manufacture, implementation and documentation. We believe that the effects that we describe in this paper are general phenomena and not specific shortcomings of a particular instrument or software. Consequently, the instrumentation and software have not been identified.

It is our intention to investigate the behaviour of the normal probability plots of A and D as a means of providing further information for the validation of absolute-structure determinations.

The results presented in §5[link] on centrosymmetric crystals show that the 2Aobs against 2Amodel plots show similar features to those of non-centrosymmetric crystals. Watkin (2011[Watkin, D. J. (2011). Personal communication.]) has found similar results with other centrosymmetric crystal structures. Somewhat to our surprise, there appear to be no results on this matter in the extensive literature devoted to the experimental study of deformation electron densities, which has been carried out in the main on centrosymmetric crystals.

The techniques described in §4[link] provide an objective method for the validation of the observed contribution of resonant scattering to the measured diffraction intensities. This leads us to suggest the following protocol of alternative choices for the publication of a structure analysis on a non-centrosymmetric crystal structure. Choice (a): Average all Friedel opposites. A 2AD plot, RA, RD and RAweak values should not be reported. Make no report or comment on the Flack paramter (Flack, 1983[Flack, H. D. (1983). Acta Cryst. A39, 876-881.]), absolute structure or absolute configuration. Authors should not be required to justify this choice of procedure. Currently there are implementation problems with this procedure. Most single-crystal structure-factor least-squares software neither directly allows the calculation and use of A nor provides a way of simultaneously using data sets containing both A and |F|2 values, the latter arising from reflections for which only one member of a Friedel pair was measured. Also, up to version 2.4.2, the CIF core dictionary (http://www.iucr.org/resources/cif/dictionaries/cif_core ) has no data items for recording A. Choice (b): Do not average Friedel opposites. Provide a 2AD plot, RA, RD and RAweak values which should attest to a satisfactory agreement between Dobs and Dmodel. Provide a value of the Flack parameter (Flack, 1983[Flack, H. D. (1983). Acta Cryst. A39, 876-881.]), and comment on absolute structure and absolute configuration as appropriate.

These choices provide the structure analyst with a free hand in dealing with the compound under study and justifying the results in accordance with the objectives of the study. A disturbing aspect of the analysis of Friedel averages and differences is that one may readily obtain an excellent fit on the averages whilst having a poor or nonexistent fit on the differences. One must not assume that a good fit of the averages implies a good fit of the differences.

For the determination of the absolute structure of a non-centrosymmetric crystal structure, our study shows clearly that one obtains the best results by measuring slowly and to a high redundancy. Systematic effects have to be corrected for with care and may easily hide the resonant-scattering effect in the Friedel opposites. The 2AD plots provide a very powerful method for the validation of the absolute-structure determination.

Some words of a more general nature are in order. Over the years, procedures have appeared in the literature which attempt to improve the quality of absolute-structure determination. These procedures rely on advanced statistical methodology resulting in the reduction of the uncertainty of the absolute-structure determination. Little or no account is made of intensity-measurement and data-correction methodology in the description of these improved techniques. The results on the NaClO3 crystal point in exactly the opposite direction. It is rather the intensity measurement and correction which need to be studied and undertaken with improved precision to come to a more satisfactory result. For the purposes of absolute-structure determination, the model of a crystal twinned by inversion is physically acceptable and realizable, and has stood the test of time. Our own plots of the weighted sum of squares against the Flack parameter (Flack, 1983[Flack, H. D. (1983). Acta Cryst. A39, 876-881.]) have always shown that full-matrix refinement in an iterative linearized least-squares procedure leads to a Flack parameter at the minimum of the weighted sum of squares. There are nevertheless indications that the weak point in the iterative linearized least-squares procedure is the calculation of the standard uncertainty of the Flack parameter, which apparently is sometimes too large and sometimes too small. Smaller standard uncertainties are not necessarily better or more realistic.

We suggest some lines of action which might lead to improvement in absolute-structure determination. With regard to semi-empirical absorption correction, it would appear that the algorithms used need more study and improvement in order to obtain more realistic corrections to the Dobs values. Moreover, it is expected that the weakness in absorption correction by numerical or analytical integration arises from the indexing of the crystal faces and the measurement of the crystal dimensions. More powerful and appropriate optical instrumentation on the diffractometer may well provide a solution to this problem. Moreover, the techniques used, as implemented in software, for the transformation of frames of diffraction images into lists of integrated intensities need more study and evaluation.

Acknowledgements

This paper is dedicated to Dr David J. Watkin of the Chemical Crystallography Laboratory of the University of Oxford, England, upon his retirement. It was DJW who had the inspiration, reported in Flack et al. (2011[Flack, H. D., Sadki, M., Thompson, A. L. & Watkin, D. J. (2011). Acta Cryst. A67, 21-34.]), to produce the very first plots of Aobs against Amodel and Dobs against Dmodel. To say the least, the results on his model compound measured with both Mo K[alpha] and Cu K[alpha] radiations left him perplexed, even depressed. The authors would like to thank the Swiss-Norwegian Beamline Consortium for providing access to synchrotron radiation, and we are grateful to Professor Jacqui Cole of the Cavendish Laboratory, University of Cambridge, for providing the crystal of tk3d used in the synchrotron measurements.

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Acta Cryst (2012). A68, 736-749   [ doi:10.1107/S0108767312034885 ]