Version 4cth February 2008


Applications and properties of the Bijvoet intensity ratio
  by
H.D. Flack and G. Bernardinelli


Acta Cryst. (2008) A64, xxx-xxx? Submit it to Dieter.



Synopsis


Abstract


1. Introduction

Flack & Shmueli (2007) presented the theoretical derivation of the mean-square Friedel intensity difference <D**2> in triclinic space group P1 with a centrosymmetric substructure. <D**2> was then used to define a Bijvoet intensity ratio CHI as that of the root-mean-square Friedel intensity difference to the average reflection intensity.  This value may be calculated from a knowledge of the chemical composition of the compound and the X-radiation used.  If it is required to make an allowance for a centrosymmetric substructure, the chemical composition of the latter must also be known.  Shmueli & Flack (2008) have extended the analysis to all monoclinic and orthorhombic non-centrosymmetric space groups and have evaluated the symmetry-enhancing effects on the intensity for special rows and zones of reflections.
  
Previously Flack et al. (2006) made a preliminary presentation for some crystal-structure determinations of the relationship between the Bijvoet ratio and the standard uncertainty (u) of the Flack (1983) parameter obtained by least-squares refinement.  However the work presented in Flack et al. (2006) was limited in certain respects.  In the first place only an approximate form of the Bijvoet ratio due to Girard et al. (2003) was used and  secondly the analysis was based on a small set of pseudo-centrosymmetric structures.  Consequently the current work presents the full relationship between the Bijvoet ratio and the standard uncertainty of the Flack parameter and moreover probes more deeply the practical properties of the Bijvoet ratio.


2. Relating u to CHI

The various data sources that have been used are given in Table 1.  Data on crystal-structure analyses have been assigned an unique code SFA to aid in identification.  The code S takes a value of N to indicate a non-centrosymmetric structure WITHOUT any centrosymmetric substructure, a value of P to indicate a non-centrosymmetric structure WITH a centrosymmetric substructure, and a value of C for a centrosymmetric structure which was (incorrectly) refined as being non-centrosymmetric.  F is either 0, 1, i or a to indicate the Friedel coverage** of the intensity data of 0%, 100%, 0% < F < 100%, or 0% =< F =< 100%.  The single-character A is a code indicating the source of the data.  Structures in data set N1K, which were obtained over a long period of time, using various equipment and measurement procedures have been marked as non-centrosymmetric since no centrosymmetric substructures were noticed and remarked upon at the time of the structure analysis when automated procedures like checkcif/PLATON were not available and graphic display was less professional.

FOOTNOTE [**Friedel coverage is a measure of the completeness of the diffraction-intensity data with regard to inversion in the origin of reciprocal space.  If for each value of h k l the intensity of the Friedel opposite -h -k -l (or one symmetry-equivalent to it) has NOT been measured, then the Friedel coverage is 0%.  However, if for each value of h k l, both the reflection h k l and its Friedel opposite -h -k -l (or one symmetry-equivalent to it) have been measured and used separately in the least-squares refinement, then the Friedel coverage is 100%.  Centrosymmetric reflections do not count in this evaluation.]
  
For much of the above data the values obtained for the Flack parameter itself (i.e. the x in x(u)) have been extensively analysed and discussed in Flack & Bernardinelli (2006), Flack et al. (2006) and Djukic et al. (2008).  In the current section we are almost exclusively concerned with the value of the standard uncertainty of the Flack parameter (i.e. the u in x(u)) and in particular its relation to CHI, the Bijvoet intensity ratio as defined by Flack & Shmueli (2007).  For convenience we use 10**4.CHI which is named Friedif for an arrangement without any centrosymmetric substructure and Friedif-centro if there is a centrosymmetric substructure.  As shown in Flack & Shmueli (2007) Friedif >= Friedif-centro >= 0.0 . As examples we may quote Friedif = 1526 for C35H36FeO2P (CuKa); Friedif = 232 for C49H27ClF20O4P2Ru (MoKa); Friedif = 342 for C20H16CrO6 (MoKa); Friedif = 92 for C14H17NO2S (MoKa);  Friedif = 7 for C6H12O6 (MoKa);  Friedif = 36 for C6H12O6 (CuKa).

Figure 1 shows the scatter diagram of u vs Friedif on logarithmic axes for the data sets of non-centrosymmetric and psuedo-centrosymmetric crystal structures detailed in Table 1.  The justification for incorporating the latter with those not containing a centrosymmetric substructure is presented in section 2.1 following a consideration of the results on centrosymmetric structures.  Although there is a fair amount of scatter of the data points in Figure 1, several aspects are clear in it.  Firstly there is no grouping of data points based on Friedel coverage.  Thus for the purposes of the analysis of the standard uncertainty of the Flack parameter, data sets with large and small Friedel coverage may be treated together.  Nevertheless one should remember from Flack et al. (2006) that the value of x itself is sensitive to Friedel coverage.  Secondly, it is clear that the data show an empirical linear relationship log(u) = m.log(Friedif) + c.  Now although the logarithmic axes provide the best graphical display for presenting all data simultaneously, they cover up the important salient feature that the slope m is essentially equal to -1, giving u = c'/Friedif showing that u is inversely proportional to Friedif with the mean value of u.Friedif (i.e. <u.Friedif>) taken over a data set being an empirical constant.  As an illustration, Figure 2 shows the nice scatter diagram of u vs 1/Friedif for the non-centrosymmetric structures lacking a centrosymmetric substructure and Table 1. shows values of <u.Friedif> of 13.9, 8.4, 13.6 and 9.6 for these data.

Using <u.Friedif> = 8.0 one finds that u = 0.04 corresponds to a value of Friedif of 200 and a value of u = 0.10 corresponds to a value of Friedif of 80.  These two values of u are the limiting values chosen by Flack & Bernardinelli (2000) as upper limits in the general case (u = 0.04) and with a compound known to be enantiopure (u = 0.10) for absolute-structure determination.  Consequently the corresponding values of Friedif (200, general and 80 enantiopure) are lower limits for absolute-structure determination calculable from a knowledge of the elemental composition of the compound and the wavelength of the X-rays.  These two values are thus of practical use in the choice of compound and radiation wavelength for absolute-configuration determination prior to experimentation and in the evaluation of the value of u obtained.  The present Figure 1 with its derived values supercedes Figure 1 of Flack et al. (2006).

   
2.1 Centrosymmetric structures

Figure 3 shows data for centrosymmetric structures supplementing the non-centrosymmetric structures already shown in Figure 2.  Once again the Friedel coverage has no clear effect on the value of u as all data sets of centrosymmetric structures occupy the same part of the scatter diagram.  It is however very clear that the data points of the centrosymmetric structures lie in an area of the scatter diagram above that of the non-centrosymmetric ones.  Indeed Table 1 shows values of <u.Friedif> for the centrosymmetric structures of 25.9(C1B), 23.1(C0B), 32.7(CiB) and 2900(C1X).  The first three data sets imply that on average the values of Friedif for these centrosymmetric structures should be divided by a factor of about 3 to bring them into line with the non-centrosymmetric structures.  The analysis of Flack & Shmueli (2007) shows that the value of Friedif is decreased in the presence of a centrosymmetric substructure for a structure in space group P1.  Thus although it is encouraging to see that these data sets indicate that reduced values of Friedif are in operation, it may appear odd that for a centrosymmetric structure the appropriate values of Friedif (i.e. zero) and u (i.e. infinitely large) have not been indicated.  However one has to recall that although the real crystals of these compounds are centrosymmetric, the models used to represent them are non-centrosymmetric.  In this respect the well-established instability of least-squares refinement of pseudo-centrosymmetric structures is of importance (Ermer & Dunitz, 1970; Marsh, 1981, 1986; Watkin, 1994).  The freedom offered to a centrosymmetric crystal structure of a non-centrosymmetric least-squares refinement results in a better statistical fit to the intensity data at the expense of distorted molecular geometry and a physically-unrealistic non-centrosymmetric distortion to the whole structure.  Certain other factors also give rise to underestimated values of u.  These are non full-matrix least-squares refinement (Flack & Bernardinelli, 2000, 2006), the use of shift-limiting constraints (Watkin, 1994) and the Levenberg-Marquardt stabilizing or damping procedure (Levenberg, 1944; Marquardt, 1963; Flack & Bernardinelli, 2000).

The data set C1X of five crystal structures discussed in section 5 of Flack et al. (2006) show large standard uncertainties u on the Flack parameter.  All of these structures have a checkCIF/PLATON misfit parameter (Spek, 2003) to a centrosymmetric structure of 100%.  The values of u.Friedif for these structures are indeed so high: 5000(ARAQUF), 5110(BACZAH), 226(EWOJAB), 42(SUWZIT) and 4120(WUWNAD) that they have not been included in Figure 3.

As promised in Section 2 we now return to the examination of non-centrosymmetric crystal structures with a centrosymmetric substructure.  In Figure 1 it was seen that the corresponding data sets followed the same general trend as the data sets of non-centrosymmetric crystal structures without a centrosymmetric substructure. Moreover for the data sets of pseudo-centrosymmetric crystal structures <u.Friedel> takes values of 10.2(P1B), 14.7 (P0B) and 8.7(PaD) very similar to those of the data sets of non-centrosymmetric crystal structures without a centrosymmetric substructure.  All in all both the form of the Flack & Shmueli's CHI and the freedom of a non-centrosymmetric refinement for the centrosymmetric subtructure combine to make the presence of the centrosymmetric substructure minimal for the analysis of u in terms of Friedif.


3. Bijvoet intensity ratio values

3.1  sinq/l dependence
  
Flack & Shmueli (2007) have discussed theoretically the effects on the Bijvoet intensity ratio of the dependence of scattering factors on sinq/l and of isotropic atomic displacement parameters.  Here we present results from an experimental study of K (2R,3R)H-tartrate at room temperature (compound 9BER01 in Table 2) measured with Mo Ka radiation.  Following a least-squares refinement on the whole set of intensity data, values of 10**4 CHI were calculated both from the model (calculated) and observed intensities progressively increasing the limit on sinq/l of the reflections included in the sums to take in an extra 100 reflections at a time giving approximately 10 bins.  Obviously centrosymmetric and unpaired general reflections were not used in this evaluation.  The result is shown in Figure 4.  We concentrate our attention on 10**4 CHI model as this contains no random and systematic measurement uncertainties and should be comparable to the value obtained from the Flack & Shmueli (2007) formula.  10**4 CHI model is essentially linearly dependent on sinq/l and has a value calculated over all the Friedel pairs in the intensity data (sinq/l =< 0.71 A**-1) of 537 which is much higher than that given by the Flack & Shmueli (2007) formula of 174 based on elemental composition alone.  However the latter formula, in the form of equation (8) of Flack & Shmueli (2007) is strictly valid only at sinq/l = 0 (although its range of applicability could be extended as detailed in section 4 of that paper).  Consequently the value to be compared with the Flack & Shmueli (2007) formula is 10**4 CHI model extrapolated back to sinq/l = 0.0 .  From Figure 4 an extrapolated value of 144 is obtained.  The agreement is good in view of the inevitable lack of intensity data at small values of sinq/l.
    
The increase of CHI with respect to sinq/l is due to SQRT(<D**2>) and <A> having different dependencies on sinq/l. SQRT(<D**2>) decreases less rapidly than <A> as sinq/l increases. The increase of CHI with sinq/l is hence an artefact and definitely NOT an indication that the r.m.s Friedel intensity difference increases with sinq/l.  Figure 5 shows, for the same data set of K (2R,3R)Htartrate, individual values of 10**4 |D| / <A0> where <A0> is the Wilson statistic average intensity at sinq/l = 0.00.  The large differences are at low sinq/l and it is these which are important in structure refinement especially for absolute-structure determination.  The fact that the variation with sinq/l of SQRT(<D**2>) and <A> are different is an indication that <A> is not the most appropriate normalizing factor to use with  SQRT(<D**2>).  It is not known whether a practical normalizing factor leading to a much diminished dependence of the Bijvoet intensity ratio on sinq/l can be found.  From the form of the expression for <D**2>, it might be that a normalizing factor such as SQRT[SUM(f**2i.f"**2i)] could be useful but this would depend on it being observable in some way or another.


3.2 Data sources and formulae

Data were taken from our own structure determinations and also from selected analyses which had been published in 2007 in Acta Crystallographica B, C or E up to mid August 2007.  All had refined values of the Flack parameter x(u) and intensity data (model and observed) available.  For each compound several values of the Bijvoet ratio were calculated and the numerical values are presented in Table 2.  We now detail the formulae used for these calculations.  
The standard uncertainties u(A) and u(D) of A(hkl) and D(hkl) and their mutual-uncertainty coefficient g(AD) (akin to a correlation coefficient) are calculated by applying standard propagation of uncertainty formulae to u[I(hkl)]=u+ and u[I(-h-k-l)]=u- assuming the mutual-uncertainty coefficient of I(hkl) and I(-h-k-l) to be zero.  This gives 4*u(A)**2 = u(D)**2 = (u+**2 + u-**2) and g(AD) = (u+**2 - u-**2)/(u+**2 + u-**2).  CHI(CompF&S) is calculated using Equation 8 of Flack & Shmueli (2007) relying only on a knowledge of the chemical composition and wavelength, and assuming there to be no centrosymmetric substructure.  CHI(Imodel) and CHI(Iobs) are calculated directly from the model and observed intensities of Friedel pairs by using the equation above for CHI.  The extrapolated values at sinq/l are obtained as described in section 3.1 .  As <D**2>obs contains contributions both from the real difference in intensity between Friedel opposities (i.e. DXi for the ith Friedel pair) and random and systematic uncertainties (i.e. ei for the ith Friedel pair) we report two ways to correct CHI(Iobs) for the random and systematic uncertainties.  In the first procedure, which leads to CHI(Iobscor), we use the values of the standard uncertainties of Dobsi as an expression of the random and systematic uncertainties.  Writing Dobsi = DXi + ei with ei = N(0,sigma**2i), where N(mu,v) indicates a normal distribution of mean mu and variance v, we obtain <D**2> = 1/N SumitoN (DXi + ei)**2 =  1/N SumitoN(DXi**2) + 1/N SumitoN(ei)**2 + 2/N SumitoN(DXi*ei).  Taking expectations, E[<D**2>] =  <DX**2> + <sigma**2>.  For practical purposes one writes <Dobscor**2> = <Dobs**2> - <u**2> and uses <Dobscor**2> to obtain CHI(Iobscor).  In the second procedure, which leads to CHI(Iobcecr) we make use of the intensity data of centrosymmetric reflections.  These are ones for which Imodel(hkl) = Imodel(-h-k-l) for specific classes of reflection depending on the geometric crystal class of the crystal (e.g. for orthorhombic space group P212121, 0kl, h0l and hk0 are centrosymmetric reflections).  The D(hkl) of centrosymmetric reflections contain only the contribution of random and systematic uncertainties in the intensity data.  Letting M be the number of symmetry-equivalent reflections for general values of hkl (e.g. in oP212121, M = 4), one obtains  <Dobcecr**2> = <Dobs**2> - (1/M)<Dcentro**2>.  The factor of 1/M arises in the following way.  If eps is a random variable distributed like N(0,sigma**2), and kappa = 1/M Sumi1toM (epsi), then kappa is distributed like N(0,sigma**2/M).  The intensities used in Icentro are unaveraged whereas those in Iobs are obtained by averaging M intensities where M is the order of the point group of the crystal.  Having thus defined the content of Table 2 we can now discuss its content.


3.3 Comparing CHI CompF&S with CHI Imodel (avres and sinq/l=0)

Whether or not the model is a good representation of the real crystal, the two values CHI CompF&S and CHI Imodel should be approximately equal.  However we note for most structures that CHI Imodel(avres) is larger or much larger than CHI compF&S, although there are a few cases (viz 9LHI01 and PEFXII) where CHI Imodel(avres) is smaller.  As we have demonstrated in section 3.1, the sinq/l dependence of CHI can be considerable and it is thought preferable to compare CHI CompF&S with CHI Imodel (sinq/l=0).  The latter values were calculated in a manner similar to that described in section 3.1 arranging the limits of sinq/l so as to always have ~10 bins.  In general the graphs of 10**4 CHI versus sinq/l present the same characteristics as seen in Figure 4.  In all cases CHI Imodel (sinq/l=0) is smaller than CHI CompF&S and in the cases of METWIS, SEZPUJ, PEFXII, EZEQAB and YIFZAP considerably smaller.  For these latter cases it may be that CHI F&Scomp is too large and has to be reduced to take account of the presence of a centrosymmetric substructure. They are further discussed in Section 3.7.  For the other structures there is no indication of a centrosymmetric subtructure by checkCIF/PLATON.  In view of the inevitable uncertainty in performing the extrapolation of CHI Imodel to sinq/l = 0, we judge the agreement between CHI CompF&S and CHI Imodel (sinq/l=0) as satisfactory.  Clearly the comparison of CHI CompF&S and CHI Imodel (avres) is unsatisfactory but understandable in view of the dependence of the Bijvoet intensity ratio on sinq/l.


3.4 Symmetry-enhancement factors

The analysis of Flack & Shmueli (2007) is applicable to the triclinic space group P1 with a centrosymmetric substructure.  A-priori it was not obvious to us that this analysis could be used per se for other non-centrosymmetric space groups and as a consequence we undertook the work presented by Shmueli & Flack (2008) reporting relevant theoretical intensity statistics for low non-centrosymmetric space groups.  On the practical side we used the data sources described in section 3.2 to study 10 structure analyses in the triclinic space group P1 [i.e. all those in Acta Cryst. (2007) B, C and E to mid-August 2007] and a comparable number chosen arbitrarily from the same source in the common orthorhombic space group P212121 together with selected examples in monoclinic space groups P21 and C2.  Results are given in Table 2a.  These show, as seen in section 3.3, that the Bijvoet ratio Imodel sinq/l=0 is in reasonable agreement with the value CompF&S, baring out Shmueli & Flack's (2008) analysis that there is no symmetry-enhancement factor for general reflections in these low-symmetry non-centrosymmetric space groups.

Further a few tests were made for symmetry-enhancement factors on selected lines of reflections.  We chose to study the 0k0 reflections in structures in the monoclinic space groups P21 and C2. From the 2007 Acta Cryst. sources we selected those structures in these two space groups which had the largest values of the cell parameter b (to maximize the number of 0k0 reflections) and which had good values of the Flack parameter and its standard uncertainty.  The results are given in Table 3 where the 0k0 reflections can be compared to general hkl reflections.  With the proviso that the number of 0k0 reflections in these structure determinations is very small and does not allow for satisfactory statistics, the main conclusions of Shmueli & Flack (2008) are upheld.  The <A> of the 0k0 reflections is considerably larger than that of the hkl reflections and the corresponding  rmsD is also increased.  It is not possible to say more from such a small set of data.


3.5 Classes of reflections with large |D|

For the planning of experiments, it is useful to know a-priori for an unknown crystal structure whether there are regions of reciprocal space where larger than average values of |D| may be found.  It has already been seen in section 3.4 and Shmueli & Flack (2008) that certain zones and lines of reflections have increased values of the rmsD.  The appropriate zones or lines of reflections may be found in Table 1 of Shmueli & Flack (2008) by seeking the largest values of SQRT(<D**2>/rho) for the space group under consideration.  Unfortunately the number of enhanced reflections tends to be rather small.  In passing it should be noted that reflections showing no intensity difference between Friedel opposites have a value of  SQRT(<D**2>/rho) = 0 in this Table.  It has already been seen in section 3.1 and Figure 5 that reflections with large |D| tend to lie at low sinq/l.  Examination of Figure 6 for the same compound shows that an empirical linear relationship between |D| and A exists, so roughly the reflections of strongest intensity will on average have the largest absolute value of the Friedel intensity difference.  Consequently and regrettably those reflections which tend to have a large absolute value of the Friedel intensity difference are exactly those likely to suffer from the effects of extinction.
 

3.6 CHI obs - quality of intensity data

Comparing CHI(Iobs) with CHI(Iobscor) there are several cases where <u**2> is greater than <Dobs**2> making a square root of a negative quantity appear in the calculation of CHI(Iobscor).  In another case (i.e. CIKCUV) the corrected value of CHI is rather small.  For all these cases it seems that the standard uncertainties of the intensity data are largely overestimated.  One notices also that the low values given in Table 2 of the rms delA/u and delD/u confirm this overestimation.  An opposing case is YIFZAP for which the difference between CHI(Iobs) and CHI(Iobscor) is very small.  The standard uncertainties of the intensity data seem to have been underestimated in this case.

For 9BER01 the unaveraged centrosymmetric reflections seem to have provided a satisfactory correction for random and systematic effects in the intensity data.  However for PEFXII and EZEQAB the correction is unsatisfactory.  More tests are required but the intensity data available with a structure determination in Acta Crystallographica B, C and E have already been averaged in the crystal point group resulting in the loss of the D(h) for the centrosymmetric reflections necessary for this calculation.  A further limitation of this correction is that centrosymmetric reflections are in short supply in low-symmetry space groups (i.e. none in geometric class 1, only 0k0 in m, only h0l in 2, only hk0 in mm2 but 0kl, h0l and hk0 in 222), see Shmueli & Flack (2008) Table 1.  Ideally CHI obs would be preferable to CHI model as it does not in any way depend on a structural model being available.  However if it is not possible to make a reasonable estimate of the effect of random and systematic uncertainties, its use can not be relied upon.

Turning to Table 2c it comes as somewhat as a surprise that there are often a large number of general reflections for which the intensity of the Friedel opposite is not available.  It is not at all clear why this should be or why a data-measurement strategy leading to a large number of unpaired Friedel opposites should have been chosen.  We hence suggest that a good way of judging the quality of an intensity data set of a non-centrosymmetric crystal structure is to count and report the number of unpaired general reflections as a complement to the number of measured pairs.  Moreover we have also examined the values of the A to D mutual-uncertainty coefficient g(AD) as defined in section 3.2 .  Small values of |g(AD)| mean that the standard uncertainties of Iobs(hkl) and Iobs(-h-k-l) are approximately equal, whereas values of |g(AD)| close to unity mean that one of Iobs(hkl) and Iobs(-h-k-l) has been measured with a far greater uncertainty than the other.  Consequently the largest absolute value of the AD mutual-uncertainty coefficient, |g(AD)|max, is a good guide to the homogeneity of the data collection.  So although for routine structure analysis it might be considered satisfactory merely to have intensity measurements of a sufficiently large number of Friedel opposites, we consider that for accurate electron-density measurements and absolute-configuration determination of low-Friedif compounds, a small number of unpaired Friedel opposites and a value of |g(AD)|max which is close to zero should be a requirement.  For compound 9BER01 we examined scattergrams of |g(AD)| vs sinq/l,  |g(AD)| vs A and  |g(AD)| vs 10**4|D|/<A0>model but these do not show any apparent relationships between the variable considered.

Concerning the measures of the least-squares fit of the data to the model, examination of the Friedel R factors and rms deviates is instructive.  The A(h) are being better fit than the D(h).  Although the R values clearly show that A is being better fit than D, the rms delD/u and delA/u seem to indicate that the GOF of D and A are about the same although in general that of D is smaller than that of A.  However on closer inspection on sees that the u(A) and u(D) are essentially identical where |D| is on average 1000 times smaller than A.  Thus |delD| should be much smaller than |delA|, and the rms value of |delD|/u should be much smaller than that of A.  Hence the observed rms of delD/u although it looks small and acceptable is in fact much larger than it should be and is unacceptable in the same way as the R factors of A and D.


4. Software

Values of 10**4 CHI(CompF&S) were calculated with the spreadsheet application for Microsoft ExcelTM 2003/2007 described in Flack & Shmueli (2007).  The software has been extended to allow the calculation with CrKa radiation as well as CuKa and MoKa.  The calculation of the value Rescat of Girard et al. (2003) has been removed.  The current version of the software is available at http://crystal.flack.ch/pubclns.html .  The values obtained from the spreadsheet have been verified by extensive hand calculations.


5. Concluding remarks

 - Standard presentations of point-group / space-group determination as in ITA, or Hauser or Palatinus all start by talking about Friedel's Law as a justification of methodology.  Better to look at it as an interpretation of the properties of the A - always has the point symmetry of the crystal plus the inversion - hence the Laue class without any reliance on Friedel's Law.  The analysis of the D, which has the 'anti'point group symmetry uncovers extra information.
 
 - In the context of using broken reciprocal-space symmetry (i.e. where symmetry-related reflections are no longer equivalent, as occurs when a sample suffers from site-specific radiation damage or when using plane-polarized X-radiation on resonant scatterers located in an anisotropic local atomic environment) for phase determination, Schiltz (2007) writes " ... This requires a paradigm shift in the data processing strategy, since the usual separation of the data merging and phasing steps is abandoned."
       
 
6. Acknowledgements

Dr. M. Hoyland of the IUCr Research and Development group is thanked for providing a list of structures published in 2007 in Acta Crystallographica B, C and E containing numerical values of the Flack parameter.


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Guo, H.-M., Jian, F.-F., Zhao, P.-S., Huang, B.-Y. & Lin, C.-H. (2007). Acta Cryst. E63, o2622.

Hahn, Th. & Looijenga-Vos, A. (2002). International Tables for Crystallography, Vol. A, Space-group symmetry, edited by Th. Hahn, Table 2.1.2.1. Dordrecht: Kluwer Academic Publishers.

King, G., Bergin, E., Müller-Bunz, H. & Gilheany, D. G. (2007). Acta Cryst. E63, o3278.

Kündig, E. P., Datta Chaudhuri, P., House, D. & Bernardinelli, G. (2006). Angew. Chem. Int. Ed., 45, 1092-1095.

Levenberg, K. (1944). Quart. Appl. Math. 2, 164-168.

Li, H.-Y., Huang, F.-P., Jiang, Y.-M. & Ng, S. W. (2007). Acta Cryst. E63, m219-m220.

Li, X.-J., Sun, S.-J., Wang, J. & Wang, Y.-F. (2007). Acta Cryst. E63, o619-o620.

Ma, A.-Q. (2007). Acta Cryst. E63, m1073-m1075.

Marquardt, D. W. (1963). SIAM J. Appl. Math. 11, 431-441.

Marsh, R. E. (1981). Acta Cryst. B37, 1985-1988.

Marsh, R. E. (1986). Acta Cryst. B42, 193-198.

Moskalev, N. V., Gribble, G. W. & Jasinski J. P. (2007). Acta Cryst. E63, o1279-o1281.

Rassat, A. & Fowler, P. W. (2004). Chem. Eur. J. 10, 6575-6580.

Schiltz, M. (2007). Swiss Society for Crystallography Newsletter No. 72, 14-15.

Scharwitz, M., Schäfer, S., van Almsick, T. & Sheldrick, W. S. (2007). Acta Cryst. E63, m1111-m1113.

Shen, Q., Wang, J. & Ealick, S. E. (2003). Acta Cryst. A59, 371-373.

Shmueli U. & Flack, H. D. (2008). Acta Cryst. A64, In preparation.

Spek, A. L. (2003). J. Appl. Cryst. 36, 7-13.

Suresh, J., Suresh Kumar, R., Perumal, S. & Natarajan, S. (2007). Acta Cryst. C63, o315-o318.

Tamura, M., Yoshinari, N., Igashira-Kamiyama, A. & Konno, T. (2007). Acta Cryst. E63, m1641-m1642.

Tooke, D. M., Zijp, E. J., van der Vlugt, J. I., Vogt, D. & Spek, A. L. (2007). Acta Cryst. E63, m86-m88.

Wang, B.-T., Luo, S.-P., Yue, H.-D., Wang, L.-P. & Xu, D.-Q. (2007). Acta Cryst. E63, o2786.

Wardell, S. M. S. V., Souza, M. V. N. de, Wardell, J. L., Low, J. N. & Glidewell, C. (2007). Acta Cryst. B63, 101-110.

Watkin, D. (1994). Acta Cryst. A50, 411-437.

Wolff, P. M. de, Belov, N. V., Bertaut, E. F., Buerger, M. J., Donnay, J. D. H., Fischer, W., Hahn, Th., Koptsik, V. A., Mackay, A. L., Wondratschek, H., Wilson, A. J. C. & Abrahams, S. C. (1985). Acta Cryst. A41, 278-280.

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10. Tables


10.1 Table 1
      Caption: Data sources for the study of Flack standard uncertainty versus Bijvoet intensity ratio.  Structure BU/17/611 (Friedif = 6, u = 2.0, u.Friedif = 12.0) in data set N1K has not been included in Figures 2 & 3 as although its behaviour is normal it makes the scatter diagram less informative. Likewise data with u > 0.2 have been excluded from Figure 3.  Structure ATOTOS (Friedif = 1171, u = 0.33, u.Friedif = 387) in data set NiD clearly has something wrong in its reporting and has not been counted in the averages.  Structure DAFPAC  (Friedif = 242, u = 0.08, u.Friedif = 19.4) has been confirmed by the authors to be centrosymmetric but has been kept in data set PaD as originally published.
      Location:  Sheet2 of uvsFriedif.xlsx
                 Copy in Bijvoetratioapplications.Table1.pdf

* See notes.



10.2a Table 2a
      Caption: Bijvoet ratio tests. Numerical values. "u>O" in the column Iobscor means that <u**2> > <Dobs**2> and the evaluation of CHI(Iobscor) leads to the square root of a negative quantity.
      

  - CompF&S       from chemical composition assuming no centrosymmetric
                    substructure
  - Imodel        from Friedel pairs of model intensities
       avres          all pairs in data set
       sinq/l=0       extrapolated to sinq/l=0
  - Iobs          from Friedel pairs of observed intensities
       avres          all pairs in data set
       sinq/l=0       extrapolated to sinq/l=0
  - Iobscor       from Friedel pairs of observed intensities with a
                    correction for random and systematic experimental
                    uncertainties derived from the standard uncertainties
                    of the intensity observations,
  - Icentro       from Friedel pairs of unaveraged centrosymmetric
                    observed intensities
  - Iobcecr       from Friedel pairs of observed intensities with a
                    correction for random and systematic experimental
                    uncertainties derived from Friedel pairs of unaveraged
                    centrosymmetric observed intensities.


Code                        Bijvoet ratios 10**4 CHI             
         CompF&S  Imodel  Imodel   Iobs    Iobs  Iobscor
                  avres  sinq/l=0 avres  sinq/l=0
CICYIX      306     418     217     668     512    u>O  
METWIS      575     346      85     417     115    u>O  
SEZPUJ       81     128      16    1249    -115    740  
XICNED      415     758     281     874     342    394  
UNEVAK01     70     175      49    1171     409    623  
YIDJIF      693    1084     375    1422     669    712  
GIHDAD      792    1099     296    1123     301    522
XIFSIP      104     278      87    1067     411    313
WIGWUF      926    1675     615    1890     594   1548
UDUSIW       74     199      64     907     253    u>0
9BER01      174     537     144     917     301    773  
PEFXII      857     625      27    1709     331   1134  
CICXES      389     980     379    1135     331    715
EZEQAB       87     468      20     521      30    u>0  
TIBCAJ     1220    1768    1005    2054     918   1706
RIHMUR      191     323      61     554     113    468
9YAN01      618     718     183    1442    -628    u>0
PIFDOY      538     531     165    1125     408    440
SIHDET       76     198      72    1032    -145    u>0
CIJWUO      786*    222     115    1329      77   1000
RIGMAW       89     277      75     1605  -1178    u>0
CIKCUV      120     332      76     492     128    159    
KEXYOC      110     299     104     874     173    u>O  
YIFZAP      100     474     -48    2787    -298   2762
RIGHEV      248     445     167    1079     -52    687
TIBFIU      486       -       -     551       -    u>0
EDUZOT      109     363      50    1135     109    697
METSIO     1306    1477     768    1804     666   1529
GIHKEO      939    2009     410    2465     543   2323
TICFIV      365     643     205     816     222    736

      

10.2b Table 2b
      Caption:   The value "PLATON Misfit%" (IUCr, 2007; Spek, 2003 ) measures the approximate misfit of the published structure from that of the corresponding centrosymmetric structure.  "Flack parameter" is the value of the Flack(1983) parameter.


FOOTNOTE: [* In the symbol for the space group, in order to avoid having to write a preceding " Triclinic, Monoclinic, Orthorhombic, etc," the lattice centring code P, A, B, C, I or F has, in each case, been replaced by the approriate IUCr approved symbol for the Bravais-lattice type: aP, mP, mS (mC, mA, mI), oP, oS (oC,oA,oB), oI, oF, tP, tI, hP, hR, cP, cI, cF (Wolff de et al., 1985; Hahn & Looijenga-Vos, 2002).  This slight change in nomenclature not only saves space and avoids the never-ending confusion as to whether the "Triclinic, Monoclinic, Orthorhombic, etc," refers to the lattice system, the crystal system or the crystal family, but also allows the importance of the Bravais-lattice type to be emphasised by appearing naturally as part of the space-group symbol.]


Code      Space     PLATON    Flack        Diffract-       Heaviest    T(K)   sinq/l
          group*    Misfit%   parameter    ometer            atom              max  
CICYIX    aP1            -    -0.002(17)     KAPCCD           Cd       120    0.65  
METWIS    aP1           88     0.010(10)     RIGCCD           Cu       293    0.65  
SEZPUJ    aP1            -     0.12(11)      RIGAFC6S  4C      S       296    0.65  
XICNED    aP1            -     0.012(7)      SMART             I       100    0.81  
UNEVAK01  aP1            -     0.02(10)      SMART             S       293    0.59  
YIDJIF    aP1            -     0.041(5)      SMART            Ru       100    0.60  
GIHDAD    aP1            -     0.0620(10)    SMART            Br       298    0.59  
XIFSIP    aP1            -     0.00(5)       SMART            Cl       296    0.62  
WIGWUF    aP1            -    -0.016(10)     KAPCCD           Br       293    0.56  
UDUSIW    aP1            -    -0.03(7)       SMART             P       100    0.68  
9BER01    oP212121       -     0.00(3)       IPDS1             K       298    0.71  
PEFXII    oP212121       *    -0.01(2)       IPDS1            Br       200    0.64  
CICXES    oP212121       -     0.015(8)      SMART            Co       100    0.79  
EZEQAB    oP212121       -    -0.03(11)      IPDS1             S       200    0.62  
TIBCAJ    oP212121       -    -0.039(14)     SIEMP4   4C      Ir       292    0.59  
RIHMUR    oP212121       -    -0.02(3)       IPDS2            Rh       290    0.63  
9YAN01    oP212121      87    -0.01(3)       SMART            As       273    0.76  
PIFDOY    oP212121       -     0.011(19)     SMART            Cu       296    0.62  
SIHDET    oP212121       -    -0.02(8)       SMART             P       100    0.70  
CIJWUO    oP212121       -     0.41(3)       STADI-4  4C      Se       150    0.59  
RIGMAW    oP212121       -    -0.02(6)       KAPCCD            S       120    0.65  
CIKCUV    oPna21         -    -0.05(4)       SMART            Cl       100    0.62  
KEXYOC    mCc            -    -0.05(5)       KAPCCD           Cl       120    0.65  
YIFZAP    mPc            -    -0.1 (3)       XCALIBR           S       295    0.61  
RIGHEV    mP21           -     0.05(3)       SMART            Rh       158    0.63  
TIBFIU    mP21           -     0.010(10)     SMART            Co       298    0.60  
EDUZOT    mP21           -     0.04(6)       SMART            Cl       298    0.60  
METSIO    mP21           -    -0.021(3)      KAPCCD           Pt       150    0.65  
TEVMEN    mC2            -    -0.05(3)       KAPCCD           Ru       150    -     
PIFGER    mC2            -     0.001(18)     RIGAFC75 4C      Ag       296    -     
YIDHID    mC2            -     0.11(3)       SMART            Co       296    -     
SEZZAZ    mC2            -     0.011(9)      SMART            Br       294    -     
GIHKEO    mC2            -    -0.018(12)     XCALIBR          Br       293    0.62  
TICFIV    mC2            -    -0.01(2)       RIGRAXIS  ?       I       109    0.81     
CICSOX    mC2            -    -0.03(10)      SMART            Cl       294    -     
YEYLOE    mC2            -     0.05(10)      SMART             P       294    -     
CIDFIF    mC2            -    -0.04(9)       CAD4     4C      Cl       293    -     



10.2c Table 2c
      Caption: Bijvoet ratio tests. Reflection counts and Friedel R factors.  RA = SUM (|Aobs - Amodel|) / SUM (|Aobs|), RD = SUM (|Dobs - Dmodel|) / SUM (|Dobs|), wRA2 = SQRT[ SUM {(Aobs - Amodel)/ u(Aobs)}**2 / SUM {Aobs / u(Aobs)}**2] and wRD2 = SQRT[ SUM {(Dobs - Dmodel)/ u(Dobs)}**2 / SUM {Dobs / u(Dobs)}**2]   

Code       Reflection count - standard     |   Friedel R factors     |  DelA/u  DelD/u  |g(AD)| 
          Total   Pairs  Centro  Unpaired     RA    wRA2     RD   wRD2    rms     rms     max
CICYIX     2108    1007       0      94    0.032   0.037  0.710  0.619   1.42    0.75    0.89
METWIS     5011    1170       0    2671    0.041   0.053  0.652  0.563   3.03    0.70    0.92
SEZPUJ     7146     335       0    7146    0.063   0.051  1.005  1.006   2.80    1.42    0.67
XICNED     7236    3381       0     474    0.028   0.031  0.527  0.446   1.48    0.71    0.89
UNEVAK01   4322     678       0    2966    0.049   0.059  0.977  0.978   1.73    1.00    0.83
YIDJIF     7003    3134       0     735    0.044   0.053  0.458  0.349   1.58    0.64    0.81
GIHDAD     2423    1078       0     267    0.037   0.045  0.706  0.659   1.13    0.70    0.61
XIFSIP     4860    1046       0    2768    0.043   0.059  0.940  0.937   1.97    0.92    0.92
WIGWUF     1461     658       0     145    0.052   0.065  0.478  0.421   1.98    1.41    0.96
UDUSIW     2791    1328       0     135    0.047   0.061  0.966  0.970   1.68    0.96    0.73
9BER01     1771     738     293       2    0.032   0.037  0.662  0.634   2.10    0.85    0.95
PEFXII     4711    2329       0*   7+23*   0.018   0.021  0.848  0.929   0.59    1.31    0.99
CICXES     9372    3880    1013     599    0.041   0.053  0.560  0.526   1.46    0.72    0.86
EZEQAB     4281    2134       0*    7+3*   0.012   0.015  0.402  0.492   0.27    0.14    1.00
TIBCAJ     2996     547     625    1277    0.041   0.046  0.628  0.552   1.04    0.98    0.85
RIHMUR     3037    1283     463       8    0.020   0.025  0.725  0.720   2.83    1.59    0.98
9YAN01     1344     488     278      90    0.068   0.077  0.870  0.766   1.22    0.90    0.91
PIFDOY     1884     746     391       1    0.049   0.061  0.815  0.633   1.63    0.95    0.89
SIHDET     6693    2859     953      22    0.076   0.084  0.971  0.973   2.17    0.92    0.93
CIJWUO     3511     414     709    1974    0.055   0.049  0.983  0.993   1.67    1.50    0.12
RIGMAW     6032    2609     735      79    0.071   0.055  0.982  0.977   1.16    0.84    0.98
CIKCUV     2442    1149     142       2    0.037   0.046  0.782  0.744   4.86    1.53    0.92
KEXYOC     2394    1086      12     210    0.048   0.054  0.922  0.894   1.86    0.94    0.93
YIFZAP     1053     378       6     291    0.082   0.072  1.014  0.963   8.64    4.92    0.76
RIGHEV    15832 13+7524*    289     469    0.085   0.096  0.946  0.940   3.27    1.23    0.91
TIBFIU     3353    1529      98     197    0.074   0.104                 4.34            0.94
EDUZOT     1839     854      53      78    0.031   0.045  0.972  0.909   2.48    1.33    1.00
METSIO    28340 10+13637*  1020      26    0.036   0.034  0.571  0.407   1.28    0.97    0.95
GIHKEO     1879   4+784*    173     130    0.043   0.044  0.527  0.414   3.11    1.76    0.99
TICFIV     4678  4+2138*    368      26    0.030   0.041  0.541  0.552   2.80    1.26    0.88



10.2d Table 2d
      Caption: Bijvoet ratio tests. Compound code and reference to data source. Code is a local code if beginning with a 9 otherwise it is a CSD refcode.  Numerical values are available in spreadsheet files CODE.xlsx or CODE.xls .

Refcode   OldCode   Reference
CICSOX    9GUO01    Guo et al. (2007)
CICXES    9CIS01    Cisnetti et al. (2007)
CICYIX    9YAS01    V in Yasodha et al. (2007)
CIDFIF    9SUR01    Suresh et al. (2007)
CIJWUO 9BLA01/9NoP6 Blake et al. (2007)
CIKCUV    9GUZ01    Guzei et al. (2007)
EDUZOT    9ZHA02    Zhang et Zheng (2007)
EZEQAB              (S)-B4S in Chauvin et al. (2004)
GIHDAD    9WAN01    Wang, Luo et al. (2007)
GIHKEO    9GOW02    Gowda, Usha et al. (2007)
KEXYOC              Ia in Wardell et al. (2007)
METSIO    9TOO01    Tooke et al. (2007)
METWIS    9LIH01    Li, Hang et al. (2007)
PEFXII              Kündig et al. (2006)
PIFDOY    9NoP4     Fu et al. (2007)
PIFGER    9TAM01    Tamura et al. (2007)
RIGHEV    9GAI01    Gainsford et al. (2007)
RIGMAW              Cunico et al. (2007)
RIHMUR    9NoP2     Abbasi et al. (2007)
SEZPUJ    9MOS01    Moskalev et al. (2007)
SEZZAZ    9WAN02    Wang, Wang et al. (2007)
SIHDET 9KIN01/9NoP5 King et al. (2007)
TEVMEN    9ABD01    Abdur-Rashid et al. (2007)
TIBCAJ    9NoP1     Scharwitz et al. (2007)
TIBFIU    9MAA01    Ma (2007)
TICFIV    9CYM01    Cymborowski et al. (2007)
UDUSIW    9GHA01    Ghadimi et al. (2007)
UNEVAK01  9ZHU01    Zhu et al. (2007)
WIGWUF    9BEK01    Bekaert et al. (2007)
XICNED    9CHA01    Chantrapromma et al (2007)     
XIFSIP              Xia et al. (2007)      
YEYLOE    9LIS01    Li, Sun et al. (2007)
YIDHID    9YAN02    Yang, Li, et al. (2007)
YIDJIF    9CHA02    Chartrand et al. (2007)
YIFZAP    9GOW01    Gowda, Nayak et al. (2007)
9BER01              K H (2R,3R)-tartrate in Bernardinelli (2006)
9YAN01    9NoP3     Yang, Costin et al. (2007)




10.2e Table 2e
      Caption: Bijvoet ratio tests including 'centrosymmetric' data. Numerical values. "u>O" means that <u**2> > <Dobs**2> and the evaluation of CHI(Iobscor) leads to the square root of a negative quantity.

Code                        Bijvoet ratios 10**4 CHI             
         CompF&S  Imodel  Imodel   Iobs    Iobs  Iobscor  Icentro Iobcecr  Unaveraged
                  avres  sinq/l=0 avres  sinq/l=0                          centro count

9BER01      174     537     144     917     301    773     1118     727        575
PEFXII      857     625      27    1709     331   1134     1555    1521       1186
EZEQAB       87     468      20     521      30    u>0      ???     u>0          ?



10.2f  Notes to Tables 1.
        See the footnote on page XXX concerning the space group symbols.
CIJWUO: Crystal twinned by inversion.  (1-2x)(CompF&S) = 126.
EZEQAB: (1) Data files contained |F| without centrosymmetric reflections.  
        (2) There were 7 unpaired reflections and 3 pairs for which at least one of the
            |Fobs| = 0.00. The latter 3 pairs were not used.
GIHKEO: There are 4 0k0 Friedel pairs and 784 general Friedel pairs 
PEFXII: (1) Data files contained |F| without centrosymmetric reflections.
        (2) There were 7 unpaired reflections and 23 pairs for which at least one of the
            |Fobs| = 0.00. The latter 23 pairs were not used.
        (3) Neither PLATON nor MISSYM find a pseudo centre of symmetry.  However Z'=2 and these two
            molecules may be superposed almost exactly by rotation and translation.  Moreover, APART 
            from the Br atom, the inversion image of molecule 1 may be superposed on that of molecule
            2.  So apart from the Br atom, the structure is close to being centrosymmetric.
RIGHEV: There are 13 0k0 Friedel pairs and 7524 general Friedel pairs
RIGMAW: Published structure-factor cif contains no Friedel pairs.
        Unmerged fcf file provided by C. Glidewell.  Parabolic variation of CHI obs with sinq/l
SEZPUJ: (1) Paper says 6811 Friedel pairs.  Cif say "How many Friedel pairs are there?" PLATON gives
            the correct number of 335.  However all Friedel pairs are h0l.  Unacceptable data.
        (2) Paper says temperature 29K.  CIF says 29 and 296.
SIHDET: Curve of 10**4 Chi vs sinq/l looks parabolic.  Intercept of linear fit is -145,
            intercept of parabolic fit is 927 but is obviously complete rubbish.
TIBFIU: The published calculated structure factors are wrong. For ALL reflections for which both h k l and h -k l are present,
        |Fcalc(h k l)|**2  =  |Fcalc(h -k l)|**2  (i.e. Friedel opposites).
TICFIV: Published intensity data file has no Friedel opposites. Correct fcf file obtained from W. Minor.
        There are 4 0k0 Friedel pairs and 2138 general Friedel pairs.
XICNED: For general reflections without Friedel opposites, Imodelmax = 2330 and Iobsmax = 1699.
YIDHID: The "structure factor" file has:
         (a) multiple entries for some reflections, 
         (b) entries for space-group absent reflections h+k odd, and
         (c) no calculated structure amplitudes.

      

10.3 Table 3
      Caption:  Intensity statistics on axial and general reflections
      
Refcode   Space  limit                     0k0 reflections                General reflections       
          Group  sinq/l                model        observed                  model      observed
                           count    <A>    rmsD   <A>     rmsD      count   <A>   rmsD   <A>   rmsD
RIGHEV     mP21  0.53         13   11206    173   11314    608       4793   2981   121   2988   285
METSIO     mP21  0.63         10  240543  11343  216981  29680      12384  13131  1907  13161  2250
TIBFIU     mP21  0.37          8    3406      -    3485     78        369    819     -    840    30
GIHKEO     mC2   0.59          4    6842    268    6986    420       1917   1517   291   1529   359
TICFIV     mC2   0.68          4   12436    203   13119    403       1253   3635   200   3641   248
EDUZOT     mP21                0                                   
TEVMEN     mC2                 0                                  
PIFGER     mC2                 0                                  
YIDHID     mC2                 *                                  
SEZZAZ     mC2                 0                                   
CICSOX     mC2                 0                                   
YEYLOE     mC2                 1                                   
CIDFIF     mC2                 0                                

 

            
10.4 Table 4
      Caption: 
      Location:
            
10.5 Table 5
      Caption: 
      Location:


10. Figures

 10.1 Figure 1
      Caption: Plot of u vs Friedif on logarithmic axes for
                 non-centrosymmetric structures.  These are measured with
                 high and low Friedel coverage and may have a centrosymmetric
                 substructure.
      Location: Chart 5 of uvsFriedif.xlsx
                Copy in Bijvoetratioapplications.Figure1.pdf
       
 10.2 Figure 2
      Caption: Plot of u vs 1/Friedif for non-centrosymmetric structures
                 lacking a centrosymmetric substructure.
      Location: Chart 3 of uvsFriedif.xlsx
                Copy in Bijvoetratioapplications.Figure2.pdf
                 
 10.3 Figure 3
      Caption: Plot of u vs 1/Friedif for centrosymmetric structures and
                 non-centrosymmetric structures lacking a centrosymmetric substructure.
      Location: Chart 4 of uvsFriedif.xlsx 
                Copy in Bijvoetratioapplications.Figure3.pdf

 10.4 Figure 4
      Caption:  Plot of 10**4 CHI vs sinq/lmax for K (2R,3R)Htartrate at 298K with Mo Ka radiation.
      Location: Chart 2 of 9BER01A.xlsx
                Copy in Bijvoetratioapplications.Figure4.pdf

 
 10.5 Figure 5
      Caption:  Plot of 10**4 |D| / <A0> vs sinq/l for K (2R,3R)Htartrate at 298K with Mo Ka radiation.
                <A0> is the Wilson statistic average intensity at sinq/l = 0.00. 
      Location: Chart 1 of 9BER01A.xlsx
                Copy in Bijvoetratioapplications.Figure5.pdf

 
 10.6 Figure 6
      Caption:  Plot of 10**4 |D| / <A0> vs A for K (2R,3R)Htartrate at 298K with Mo Ka radiation.
                <A0> is the Wilson statistic average intensity at sinq/l = 0.00. 
      Location: Chart 3 of 9BER01A.xlsx
                Copy in Bijvoetratioapplications.Figure6.pdf


9. Appendix

Structure determinations in Acta Cryst B, C & E (2007) reporting a value of the Flack parameter supplied by M. Hoyland in mid-August 2007.  Ones marked with a REFCODE have already been analysed.


  Compd   DOI                                            Flack         Space           
                                                                       Group
aP 1 (ITA No.1)
  CICYIX  http://dx.doi.org/10.1107/S010827010701459X   	-0.002(17)   aP 1
  METWIS  http://dx.doi.org/10.1107/S1600536806053529   	0.010(10)    aP 1
  SEZPUJ  http://dx.doi.org/10.1107/S1600536807005764	    0.12(11)	   aP 1
  XICNED  http://dx.doi.org/10.1107/S1600536807016078	    0.012(7)	   aP 1
  UNEVAK01http://dx.doi.org/10.1107/S1600536807019770	    0.02(10)	   aP 1
  YIDJIF http://dx.doi.org/10.1107/S1600536807020259	    0.041(5)	   aP 1
  GIHDAD  http://dx.doi.org/10.1107/S1600536807020594	    0.0620(10)	 aP 1
  XIFSIP  http://dx.doi.org/10.1107/S1600536807026803	    0.00(5)	     aP 1
  WIGWUF  http://dx.doi.org/10.1107/S1600536807027158	    -0.016(10)	 aP 1
  UDUSIW  http://dx.doi.org/10.1107/S1600536807028838	    -0.03(7)	   aP 1                                                        
                                                                                 
mP 21 (ITA No.4)                                        
  RIGHEV  http://dx.doi.org/10.1107/S0108270107026170   	0.05(3)      mP 21   26.61
  TIBFIU  http://dx.doi.org/10.1107/S1600536807011865   	0.010(10)    mP 21   21.75
  EDUZOT  http://dx.doi.org/10.1107/S1600536807030255   	0.04(6)      mP 21   23.28
  METSIO  http://dx.doi.org/10.1107/S1600536806052044   	-0.021(3)    mP 21   15.92
          http://dx.doi.org/10.1107/S0108768106041358   	0.09(16)     mP 21    4.79
          http://dx.doi.org/10.1107/S010827010701952X   	0.11(16)     mP 21    8.94
          http://dx.doi.org/10.1107/S0108768107014838   	0.00(7)      mP 21    7.96
          http://dx.doi.org/10.1107/S0108768107010579   	0.00(13)     mP 21   28.00
          http://dx.doi.org/10.1107/S0108270106049468   	0.48(5)      mP 21    3.80
          http://dx.doi.org/10.1107/S0108270106049468     0.51(4)      mP 21    3.77
          http://dx.doi.org/10.1107/S0108270106055156   	0.13(8)      mP 21    8.77
          http://dx.doi.org/10.1107/S0108270106055053   	0.1(2)       mP 21    6.24
          http://dx.doi.org/10.1107/S0108270107011626   	-0.08(10)    mP 21   12.95
          http://dx.doi.org/10.1107/S1600536806051130   	-0.04(5)     mP 21   13.21
          http://dx.doi.org/10.1107/S1600536806050069   	0.16(6)      mP 21    3.89
          http://dx.doi.org/10.1107/S160053680605272X   	0.02(5)      mP 21   15.64
          http://dx.doi.org/10.1107/S1600536806050768   	0.01(4)      mP 21    8.78
          http://dx.doi.org/10.1107/S1600536807001882   	0.03(2)      mP 21    6.39
          http://dx.doi.org/10.1107/S1600536806056145   	-0.01(7)     mP 21    6.91  
          http://dx.doi.org/10.1107/S1600536807000931   	-0.04(8)     mP 21    7.13
          http://dx.doi.org/10.1107/S1600536807003170   	0.00(9)      mP 21    7.13
          http://dx.doi.org/10.1107/S1600536807002395   	0.013(8)     mP 21   14.58
          http://dx.doi.org/10.1107/S1600536806055073   	-0.03(8)     mP 21    7.15
          http://dx.doi.org/10.1107/S1600536806054729   	0.00(6)      mP 21    6.45
          http://dx.doi.org/10.1107/S1600536807000700   	0.00(7)      mP 21    7.40
          http://dx.doi.org/10.1107/S1600536807003637   	0.00(4)      mP 21   11.48
          http://dx.doi.org/10.1107/S1600536807007891   	0.000(10)    mP 21    6.91
          http://dx.doi.org/10.1107/S1600536807008203   	0.019(7)     mP 21   11.07
          http://dx.doi.org/10.1107/S1600536807005417   	0.3(16)      mP 21   11.14
          http://dx.doi.org/10.1107/S1600536807012780   	0.508(12)    mP 21    8.01
          http://dx.doi.org/10.1107/S1600536807013803   	0.08(8)      mP 21   16.66
          http://dx.doi.org/10.1107/S1600536807006186   	0.134(19)    mP 21    5.49
          http://dx.doi.org/10.1107/S1600536807025767   	0.015(7)     mP 21    9.41
          http://dx.doi.org/10.1107/S1600536807026086   	-0.05(7)     mP 21    5.85
          http://dx.doi.org/10.1107/S1600536807026487   	-0.01(6)     mP 21    6.74
          http://dx.doi.org/10.1107/S1600536807028978   	-0.02(8)     mP 21    5.86
          http://dx.doi.org/10.1107/S1600536807030929   	0.05(6)      mP 21    3.79
          http://dx.doi.org/10.1107/S1600536807024506   	0.07(5)      mP 21    9.21
          http://dx.doi.org/10.1107/S1600536807028607   	0.087(16)    mP 21   14.71
          http://dx.doi.org/10.1107/S1600536807029613   	-0.002(11)   mP 21    7.14
          http://dx.doi.org/10.1107/S160053680702572X   	-0.02(16)    mP 21   15.83
          http://dx.doi.org/10.1107/S1600536807021435   	0.026(15)    mP 21   11.32
          http://dx.doi.org/10.1107/S1600536807022210   	-0.01(9)     mP 21    5.74
          http://dx.doi.org/10.1107/S1600536807024786   	0(10)        mP 21    8.99
          http://dx.doi.org/10.1107/S1600536807023379   	0.08(7)      mP 21    8.92
          http://dx.doi.org/10.1107/S1600536807016558   	0.01(6)      mP 21    6.82
          http://dx.doi.org/10.1107/S1600536807019964   	0.10(3)      mP 21    7.28
          http://dx.doi.org/10.1107/S1600536807024828   	0.12(7)      mP 21    7.44
          http://dx.doi.org/10.1107/S160053680702329X   	-0.06(4)     mP 21    7.61
          http://dx.doi.org/10.1107/S1600536807014237   	-0.019(8)    mP 21   12.14
          http://dx.doi.org/10.1107/S1600536807014341   	0.00(3)      mP 21   18.22
          http://dx.doi.org/10.1107/S1600536807018223   	0(10)        mP 21    8.97
          http://dx.doi.org/10.1107/S1600536807017278   	0.43(3)      mP 21    9.61
          http://dx.doi.org/10.1107/S1600536807014006   	-0.067(2)    mP 21    5.81
          http://dx.doi.org/10.1107/S1600536807018818   	-0.08(6)     mP 21    7.42
          http://dx.doi.org/10.1107/S160053680701447X   	0.006(16)    mP 21    9.34
          http://dx.doi.org/10.1107/S1600536807019666   	0.1(2)       mP 21    6.02
          http://dx.doi.org/10.1107/S160053680701478X   	0.08(12)     mP 21   11.93
          http://dx.doi.org/10.1107/S1600536807019733   	0.39(2)      mP 21    6.94
          http://dx.doi.org/10.1107/S1600536807014067   	-0.06(9)     mP 21   15.60
          http://dx.doi.org/10.1107/S1600536807017394   	0.00(6)      mP 21   10.12
          http://dx.doi.org/10.1107/S1600536807009257   	0.03(3)      mP 21    6.00
          http://dx.doi.org/10.1107/S1600536807003133   	0.03(9)      mP 21   13.92
          http://dx.doi.org/10.1107/S1600536807010021   	0.04(6)      mP 21    9.04
          http://dx.doi.org/10.1107/S1600536807011038   	0.514(19)    mP 21   16.58
          http://dx.doi.org/10.1107/S1600536807012238   	0.13(8)      mP 21    5.65
          http://dx.doi.org/10.1107/S0108270107022469   	10(10)       mP 21   28.79
                                                        
mC 2 (ITA No.5)                                          
  TEVMEN  http://dx.doi.org/10.1107/S1600536806053815   	-0.05(3)     mC 2
  PIFGER  http://dx.doi.org/10.1107/S1600536807020442   	0.001(18)    mC 2
  YIDHID  http://dx.doi.org/10.1107/S1600536807020016   	0.11(3)      mC 2
  SEZZAZ  http://dx.doi.org/10.1107/S1600536807005077   	0.011(9)     mC 2
  GIHKEO  http://dx.doi.org/10.1107/S1600536807025184   	-0.018(12)   mC 2
  TICFIV  http://dx.doi.org/10.1107/S1600536807009129   	-0.01(2)     mC 2
  CICSOX  http://dx.doi.org/10.1107/S1600536807018867   	-0.03(10)    mC 2
  YEYLOE  http://dx.doi.org/10.1107/S1600536806055425   	0.05(10)     mC 2
  CIDFIF  http://dx.doi.org/10.1107/S0108270107011626   	-0.04(9)     mC 2
          http://dx.doi.org/10.1107/S0108270107004830   	0.12(15)     mC 2
          http://dx.doi.org/10.1107/S0108270107008967   	0.48(5)      mI 2
          http://dx.doi.org/10.1107/S1600536807021319   	0.28(17)     mC 2
                                                        
mP c (ITA No.7)                                         
  YIFZAP  http://dx.doi.org/10.1107/S1600536807024221   	-0.1(3)      mP c
          http://dx.doi.org/10.1107/S0108270106048530   	-0.02(2)     mP n
          http://dx.doi.org/10.1107/S0108270107003241   	0.467(16)    mP c
          http://dx.doi.org/10.1107/S0108270107003241   	0.015(9)     mP c
          http://dx.doi.org/10.1107/S1600536806052500   	0.000(10)    mP n
          http://dx.doi.org/10.1107/S1600536806053050   	0.000(2)     mP c
          http://dx.doi.org/10.1107/S1600536806045648   	0.48(7)      mP c
          http://dx.doi.org/10.1107/S160053680700414X   	-0.04(8)     mP n
          http://dx.doi.org/10.1107/S1600536807008859   	0.11(10)     mP n
          http://dx.doi.org/10.1107/S1600536807007234   	0.00(3)      mP c
          http://dx.doi.org/10.1107/S160053680702541X   	0.011(17)    mP c
          http://dx.doi.org/10.1107/S1600536807029273   	0.0(4)       mP a
          http://dx.doi.org/10.1107/S1600536807029698   	0.029(17)    mP c
          http://dx.doi.org/10.1107/S1600536807025858   	0.024(17)    mP n
          http://dx.doi.org/10.1107/S1600536806043546   	0.5(2)       mP c
          http://dx.doi.org/10.1107/S1600536807016261   	0.53(3)      mP c
          http://dx.doi.org/10.1107/S1600536807014213   	0.000(7)     mP c
          http://dx.doi.org/10.1107/S1600536807016145   	0.049(9)     mP c
          http://dx.doi.org/10.1107/S1600536807013025   	-0.04(6)     mP c
          http://dx.doi.org/10.1107/S0108270107034786   	0.17(7)      mP c
                                                        
mC c (ITA No.9)                                         
  KEXYOC  http://dx.doi.org/10.1107/S0108768106041358   	-0.05(5)     mC c
          http://dx.doi.org/10.1107/S0108768106041358   	-0.01(6)     mC c
          http://dx.doi.org/10.1107/S0108768107018034   	0.036(10)    mC c
          http://dx.doi.org/10.1107/S0108768107018034   	0.030(9)     mC c
          http://dx.doi.org/10.1107/S0108270107006166   	0.006(4)     mC c
          http://dx.doi.org/10.1107/S0108270106045550   	0.507(9)     mC c
          http://dx.doi.org/10.1107/S0108270107024936   	0.42(12)     mC c
          http://dx.doi.org/10.1107/S1600536806052329   	0.016(7)     mC c
          http://dx.doi.org/10.1107/S1600536807000025   	0.15(3)      mC c
          http://dx.doi.org/10.1107/S1600536807001730   	0.352(6)     mC c
          http://dx.doi.org/10.1107/S1600536807006903   	0.02(15)     mC c
          http://dx.doi.org/10.1107/S1600536807006915   	-0.01(3)     mC c
          http://dx.doi.org/10.1107/S1600536807030619   	-0.10(10)    mC c
          http://dx.doi.org/10.1107/S1600536807026116   	0.0(13)      mC c
          http://dx.doi.org/10.1107/S1600536807027237   	0.06(10)     mC c
          http://dx.doi.org/10.1107/S1600536807027973   	0.36(3)      mC c
          http://dx.doi.org/10.1107/S1600536807023641   	0.08(9)      mC c
          http://dx.doi.org/10.1107/S1600536807022350   	0.016(10)    mC c
          http://dx.doi.org/10.1107/S1600536807022842   	0.10(5)      mC c
          http://dx.doi.org/10.1107/S1600536807021861   	-0.01(2)     mC c
          http://dx.doi.org/10.1107/S1600536807023628   	0.011(10)    mC c
          http://dx.doi.org/10.1107/S1600536807024348   	0.018(17)    mC c
          http://dx.doi.org/10.1107/S1600536807017618   	0.014(7)     mC c
          http://dx.doi.org/10.1107/S1600536807014766   	0.190(17)    mC c
          http://dx.doi.org/10.1107/S160053680701238X   	0.09(3)      mC c
          http://dx.doi.org/10.1107/S1600536807010124   	-0.008(6)    mC c
          http://dx.doi.org/10.1107/S1600536807012664   	0.09(5)      mC c
                                                        
oP 21 21 2 (ITA No.18)                                   
          http://dx.doi.org/10.1107/S0108270107025358   	-0.015(5)    oP 21 21 2
          http://dx.doi.org/10.1107/S1600536807006769   	-0.179(9)    oP 21 21 2
          http://dx.doi.org/10.1107/S1600536807025731   	0.07(5)      oP 21 21 2
          http://dx.doi.org/10.1107/S1600536807027183   	-0.013(9)    oP 21 21 2
          http://dx.doi.org/10.1107/S1600536807010744   	0.000(17)    oP 21 21 2
                                                        
oP 21 21 21 (ITA No.19)
  TIBCAJ  http://dx.doi.org/10.1107/S1600536807011750   	-0.039(14)   oP 21 21 21
  RIHMUR  http://dx.doi.org/10.1107/S1600536807027912   	-0.02(3)     oP 21 21 21
  9YAN01  http://dx.doi.org/10.1107/S1600536807003029   	-0.01(3)     oP 21 21 21
  PIFDOY  http://dx.doi.org/10.1107/S1600536807021654   	0.011(19)    oP 21 21 21
  SIHDET  http://dx.doi.org/10.1107/S1600536807029479   	-0.02(8)     oP 21 21 21
  CIJWUO  http://dx.doi.org/10.1107/S0108270107026753   	0.41(3)      oP 21 21 21
  CICXES  http://dx.doi.org/10.1107/S0108270107013522   	0.015(8)     oP 21 21 21
  RIGMAW  http://dx.doi.org/10.1107/S0108270107022913   	-0.02(6)     oP 21 21 21
          http://dx.doi.org/10.1107/S0108768106044831   	-0.03(4)     oP 21 21 21
          http://dx.doi.org/10.1107/S0108768106044831   	-0.03(5)     oP 21 21 21
          http://dx.doi.org/10.1107/S0108270106050979   	-0.11(10)    oP 21 21 21
          http://dx.doi.org/10.1107/S0108270106054412   	0.02(3)      oP 21 21 21
          http://dx.doi.org/10.1107/S0108270106055338   	-0.01(6)     oP 21 21 21
          http://dx.doi.org/10.1107/S0108270106053108   	0.025(14)    oP 21 21 21
          http://dx.doi.org/10.1107/S0108270107011821   	0.0(2)       oP 21 21 21
          http://dx.doi.org/10.1107/S0108270107015387   	0.061(12)    oP 21 21 21
          http://dx.doi.org/10.1107/S0108270107027862   	-0.05(2)     oP 21 21 21
          http://dx.doi.org/10.1107/S0108270107027497   	-0.01(7)     oP 21 21 21
          http://dx.doi.org/10.1107/S0108270107026753   	0.00(3)      oP 21 21 21
          http://dx.doi.org/10.1107/S1600536806051464   	-0.001(18)   oP 21 21 21
          http://dx.doi.org/10.1107/S1600536806051488   	0.029(15)    oP 21 21 21
          http://dx.doi.org/10.1107/S160053680605286X   	-0.01(6)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536806054225   	0.462(18)    oP 21 21 21
          http://dx.doi.org/10.1107/S1600536806050963   	0.03(12)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536806054249   	0.031(13)    oP 21 21 21
          http://dx.doi.org/10.1107/S1600536806053360   	-0.007(16)   oP 21 21 21
          http://dx.doi.org/10.1107/S1600536806053207   	0.453(18)    oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807003753   	0.01(9)      oP 21 21 21
          http://dx.doi.org/10.1107/S160053680605536X   	-0.01(2)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807001456   	0.01(2)      oP 21 21 21
          http://dx.doi.org/10.1107/S1600536806055632   	-0.009(15)   oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807002292   	0.49(4)      oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807002693   	0.2(5)       oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807000657   	0.31(3)      oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807000347   	0.05(9)      oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807000141   	0.020(10)    oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807001717   	0.07(16)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807003704   	-0.09(7)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807000256   	0.010(10)    oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807002036   	0.006(13)    oP 21 21 21
          http://dx.doi.org/10.1107/S160053680605344X   	1.4(9)       oP 21 21 21
          http://dx.doi.org/10.1107/S160053680700102X   	0.000(10)    oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807006496   	-0.03(8)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807005740   	-0.14(7)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807006526   	-0.02(3)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807007829   	-0.14(5)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807007477   	0.46(2)      oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807006277   	-0.08(8)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807004606   	0.017(6)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807008409   	0.01(5)      oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807007258   	0.002(18)    oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807007398   	0.10(10)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807007982   	0.12(18)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807006393   	-0.04(8)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807004436   	-0.04(7)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807004023   	0.08(15)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807006009   	-0.003(15)   oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807012214   	0.021(10)    oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807010045   	0.05(5)      oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807029418   	0(10)        oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807029601   	-0.09(4)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807022015   	0.02(3)      oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807030176   	0.12(10)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807030875   	-0.011(5)    oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807030905   	-0.013(9)    oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807030528   	0.49(2)      oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807028280   	-0.01(4)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807029005   	-0.07(13)    oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807028127   	-0.04(15)    oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807029327   	0.04(5)      oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807027961   	0.452(15)    oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807025706   	0.02(2)      oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807023574   	0(10)        oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807015565   	0.04(6)      oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807024245   	0.00(7)      oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807022003   	0.02(6)      oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807023926   	0.50(2)      oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807025202   	0.22(16)     oP 21 21 21
          http://dx.doi.org/10.1107/S160053680702079X   	0.018(17)    oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807015978   	0.20(15)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807024531   	0.025(12)    oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807023033   	0.22(12)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807020739   	0.03(8)      oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807019071   	0.012(12)    oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807007453   	-0.02(9)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807018922   	-0.011(7)    oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807018600   	-0.030(10)   oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807014912   	-0.02(3)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807013608   	0.022(11)    oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807015309   	0.03(17)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807014687   	0.007(11)    oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807020508   	0.001(4)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807015607   	0.03(5)      oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807015619   	0.06(8)      oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807018624   	0.00(2)      oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807018879   	-0.19(14)    oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807014602   	0.020(10)    oP 21 21 21
          http://dx.doi.org/10.1107/S160053680701906X   	0.50(7)      oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807014122   	0.03(8)      oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807016868   	-0.04(7)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807014973   	0.52(6)      oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807013943   	0.41(9)      oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807016194   	0.031(13)    oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807012536   	0.001(19)    oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807009749   	0.007(6)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807009427   	0.001(6)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807009701   	-0.02(14)    oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807009853   	10(10)       oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807008380   	-0.01(4)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807010173   	0.20(8)      oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807007076   	0.60(10)     oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807014092   	0.0(2)       oP 21 21 21
          http://dx.doi.org/10.1107/S1600536807012081   	-0.03(6)     oP 21 21 21

oC 2 2 21 (ITA No.20)
          http://dx.doi.org/10.1107/S1600536806049762   	0.17(3)      oC 2 2 21
          http://dx.doi.org/10.1107/S1600536807002383   	0.106(13)    oC 2 2 21
          http://dx.doi.org/10.1107/S1600536807002127   	-0.001(13)   oC 2 2 21
          http://dx.doi.org/10.1107/S1600536807008665   	0.007(6)     oC 2 2 21
          http://dx.doi.org/10.1107/S1600536807020703   	-0.06(3)     oC 2 2 21
                                                        
oP c a 21 (ITA No.29)                                   
          http://dx.doi.org/10.1107/S0108270107015764   	-0.02(5)     oP c a 21
          http://dx.doi.org/10.1107/S1600536807003807   	0.027(11)    oP c a 21
          http://dx.doi.org/10.1107/S1600536807030723   	0.013(12)    oP c a 21
          http://dx.doi.org/10.1107/S1600536807027845   	0.002(8)     oP c a 21
          http://dx.doi.org/10.1107/S1600536807026232   	0.1(3)       oP c 21 b
          http://dx.doi.org/10.1107/S160053680702394X   	0.04(5)      oP c a 21
          http://dx.doi.org/10.1107/S1600536807023859   	-0.06(17)    oP c a 21
          http://dx.doi.org/10.1107/S1600536807021691   	0.01(9)      oP c a 21
          http://dx.doi.org/10.1107/S1600536807017667   	0.02(8)      oP c a 21
          http://dx.doi.org/10.1107/S1600536807007787   	0.531(12)    oP c a 21
          http://dx.doi.org/10.1107/S1600536807015462   	0.01(7)      oP c a 21
          http://dx.doi.org/10.1107/S1600536807009518   	0.01(5)      oP c a 21
                                                        
oP m n 21 (ITA No.31)                                     
          http://dx.doi.org/10.1107/S1600536806053554   	0.18(12)     oP m n 21
          http://dx.doi.org/10.1107/S1600536807022064   	0.012(10)    oP m n 21

oP b a 2 (ITA No.32)
          http://dx.doi.org/10.1107/S1600536807004825   	0.16(12)     oP b a 2
          http://dx.doi.org/10.1107/S1600536807009130   	0.13(9)      oP b a 2
                                                        
oP n a 21 (ITA No.33)                                     
  CIKCUV  http://dx.doi.org/10.1107/S0108270107031952   	-0.05(4)     oP n a 21   
          http://dx.doi.org/10.1107/S0108270107009663   	0.24(2)      oP n a 21
          http://dx.doi.org/10.1107/S0108270107010554   	0.56(2)      oP n a 21
          http://dx.doi.org/10.1107/S0108270107023414   	0.003(11)    oP n a 21
          http://dx.doi.org/10.1107/S0108270107021993   	0.486(11)    oP 21 n b
          http://dx.doi.org/10.1107/S160053680605361X   	0.27(3)      oP n a 21
          http://dx.doi.org/10.1107/S1600536806053190   	0.025(9)     oP n a 21
          http://dx.doi.org/10.1107/S1600536806050641   	0.092(7)     oP n a 21
          http://dx.doi.org/10.1107/S1600536807001626   	0.58(5)      oP n a 21
          http://dx.doi.org/10.1107/S160053680700342X   	0.54(2)      oP n a 21
          http://dx.doi.org/10.1107/S1600536807000645   	-0.005(11)   oP n a 21
          http://dx.doi.org/10.1107/S1600536807007866   	0.05(5)      oP n a 21
          http://dx.doi.org/10.1107/S1600536807028073   	0.028(9)     oP n a 21
          http://dx.doi.org/10.1107/S1600536807029947   	0.15(13)     oP n a 21
          http://dx.doi.org/10.1107/S1600536807029686   	0.51(3)      oP 21 n b
          http://dx.doi.org/10.1107/S1600536807027365   	0.057(6)     oP n a 21
          http://dx.doi.org/10.1107/S1600536807025068   	0.39(7)      oP n a 21
          http://dx.doi.org/10.1107/S1600536806043807   	0.13(13)     oP n a 21
          http://dx.doi.org/10.1107/S1600536807016017   	0.41(4)      oP n a 21
          http://dx.doi.org/10.1107/S1600536807019137   	0.017(12)    oP n a 21
          http://dx.doi.org/10.1107/S1600536807015280   	0.03(7)      oP n a 21
          http://dx.doi.org/10.1107/S1600536807014961   	0.003(7)     oP n a 21
          http://dx.doi.org/10.1107/S1600536807013335   	-0.01(3)     oP n a 21
          http://dx.doi.org/10.1107/S1600536807010008   	0.0(4)       oP n a 21
          http://dx.doi.org/10.1107/S1600536807011440   	0.62(4)      oP n a 21
                                                            
oC m c 21 (ITA No.36)                                     
          http://dx.doi.org/10.1107/S0108768107029709   	0.53(2)      oC m c 21
          http://dx.doi.org/10.1107/S0108270107023463   	0.030(14)    oC m c 21
          http://dx.doi.org/10.1107/S1600536807014791   	0.11(8)      oC m c 21

oA m a 2 (ITA No.40)
          http://dx.doi.org/10.1107/S1600536806048914   	0.004(14)    oA m a 2

oA b a 2 (ITA No.41)
          http://dx.doi.org/10.1107/S1600536807014055   	0.4(17)      oA b a 2
                                                        
oF d d 2 (ITA No.43)                                      
          http://dx.doi.org/10.1107/S0108270106056022   	0.007(10)    oF d d 2
          http://dx.doi.org/10.1107/S0108270107020689   	-0.003(7)    oF d d 2
          http://dx.doi.org/10.1107/S1600536806052524   	0.08(10)     oF d d 2
          http://dx.doi.org/10.1107/S1600536807007970   	0.22(2)      oF d d 2
          http://dx.doi.org/10.1107/S1600536807030693   	-0.02(2)     oF d d 2
          http://dx.doi.org/10.1107/S1600536807025299   	0.017(8)     oF d d 2
          http://dx.doi.org/10.1107/S1600536807029832   	0.009(12)    oF d d 2
          http://dx.doi.org/10.1107/S1600536807030954   	0.05(7)      oF d d 2
          http://dx.doi.org/10.1107/S1600536807025895   	0.05(7)      oF d d 2
          http://dx.doi.org/10.1107/S160053680702065X   	-0.021(18)   oF d d 2
          http://dx.doi.org/10.1107/S1600536807020260   	0.022(14)    oF d d 2
          http://dx.doi.org/10.1107/S1600536807019332   	-0.001(11)   oF d d 2
          http://dx.doi.org/10.1107/S1600536807013050   	0.010(4)     oF d d 2

oI b a 2 (ITA No.45)
          http://dx.doi.org/10.1107/S1600536807022295   	-0.03(2)     oI b a 2

oI m a 2 (ITA No.46)
          http://dx.doi.org/10.1107/S1600536807030978   	0.412(17)    oI m a 2

tP 41 (ITA No.76)                                         
  CIKDEG  http://dx.doi.org/10.1107/S0108270107033008   	0.04(6)      tP 41 not studied
          http://dx.doi.org/10.1107/S1600536807023173   	0.02(4)      tP 41

tP 43 (ITA No.78)                                         
          http://dx.doi.org/10.1107/S1600536806054201   	-0.01(12)    tP 43

tI -4 (ITA No. 82)                                                        
          http://dx.doi.org/10.1107/S1600536807001808   	0.03(2)      tI -4
          http://dx.doi.org/10.1107/S160053680702627X   	0.43(3)      tI -4
          http://dx.doi.org/10.1107/S1600536807024051   	-0.01(3)     tI -4
          http://dx.doi.org/10.1107/S1600536807009397   	0.007(13)    tI -4
          http://dx.doi.org/10.1107/S1600536807009403   	0.003(15)    tI -4
                                                        
tP 41 21 2 (ITA No.92)                                    
          http://dx.doi.org/10.1107/S0108768106047306   	0.021(12)    tP 41 21 2
          http://dx.doi.org/10.1107/S0108768106047306   	0.012(9)     tP 41 21 2
          http://dx.doi.org/10.1107/S1600536806052676   	-0.08(5)     tP 41 21 2
          http://dx.doi.org/10.1107/S1600536807030115   	0.007(3)     tP 41 21 2
          http://dx.doi.org/10.1107/S1600536807019769   	0.02(2)      tP 41 21 2

tP 42 21 2 (ITA No.94)
          http://dx.doi.org/10.1107/S1600536807010471   	0.000(10)    tP 42 21 2

tP 43 21 2 (ITA No.96)
          http://dx.doi.org/10.1107/S1600536807012305   	0.003(16)    tP 43 21 2

tP -4 21 c (ITA No.114)
          http://dx.doi.org/10.1107/S1600536807027092   	-0.02(6)     tP -4 21 c

tP -4 b 2 (ITA No.117)                                     
          http://dx.doi.org/10.1107/S010827010702269X   	0.03(3)      tP -4 b 2

tI -4 c 2 (ITA No.120)
          http://dx.doi.org/10.1107/S1600536807007611   	0.009(7)     tI -4 c 2

tI -4 2 d (ITA No.122)
          http://dx.doi.org/10.1107/S1600536807008069   	0.008(13)    tI -4 2 d

hP 32 (ITA No.145)
          http://dx.doi.org/10.1107/S160053680700757X   	0.10(2)      hP 32
                                                        
hP 31 2 1  (ITA No.152)                                    
          http://dx.doi.org/10.1107/S0108270107007214   	0.02(2)      hP 31 2 1
          http://dx.doi.org/10.1107/S0108270107007214   	0.01(3)      hP 31 2 1
          http://dx.doi.org/10.1107/S1600536807028401   	-0.02(3)     hP 31 2 1

hP 32 2 1  (ITA No.154)
          http://dx.doi.org/10.1107/S1600536806052858   	0.01(3)      hP 32 2 1
          http://dx.doi.org/10.1107/S1600536807020004   	0.02(6)      hP 32 2 1

hR 3 m (ITA No.160)
          http://dx.doi.org/10.1107/S1600536807000803   	0.02(4)      hR 3 m

hR 3 c (ITA No.161)
         http://dx.doi.org/10.1107/S1600536807009762   	0.007(10)      hR 3 c 

hP 61 (ITA No.169)
          http://dx.doi.org/10.1107/S1600536807008793   	0.02(5)      hP 61

hP -6 (ITA No.174)
          http://dx.doi.org/10.1107/S1600536807004734   	0.070(18)    hP -6

hP 63 m c (ITA No.186)
          http://dx.doi.org/10.1107/S1600536807028164   	0.557(16)    hP 63 m c
          http://dx.doi.org/10.1107/S1600536807026827   	0.02(5)      hP 63 m c

hP -6 2 c (ITA No.190)
          http://dx.doi.org/10.1107/S1600536807007052   	0.00(3)      hP -6 2 c

cP 21 3 (ITA No.198)
          http://dx.doi.org/10.1107/S1600536807027043   	0.00(4)      cP 21 3
          http://dx.doi.org/10.1107/S1600536807024026   	-0.050(15)   cP 21 3
          http://dx.doi.org/10.1107/S1600536807016595   	-0.028(9)    cP 21 3

cI -4 3 d (ITA No.220)
          http://dx.doi.org/10.1107/S1600536807027018   	-0.07(7)     cI -4 3 d

DOI NOT FOUND                                 
          
SOMETHING WRONG
          http://dx.doi.org/10.1107/S1600536807021393   	0.003(3)
            No x in paper - centro - Non-centro paper rejected after acceptance and
              DOI used for another publication
          http://dx.doi.org/10.1107/S1600536807022969   	0.064(2)
            centro with a value of x.  A Alert of PLATON not clear enough.
          http://dx.doi.org/10.1107/S1600536807016789   	0.002(7)
            centro with a value of x.  No A Alert from PLATON. 


-------------------------------------------------------------------------------        

mP 21 (ITA No.4)  Sorted for tests on 0k0 reflections

  Compd   DOI                                            Flack         Space   cell b            
                                                                       group

Good
  RIGHEV  http://dx.doi.org/10.1107/S0108270107026170   	0.05(3)      mP 21   26.61
  EDUZOT  http://dx.doi.org/10.1107/S1600536807030255   	0.04(6)      mP 21   23.28
  TIBFIU  http://dx.doi.org/10.1107/S1600536807011865   	0.010(10)    mP 21   21.75
  METSIO  http://dx.doi.org/10.1107/S1600536806052044   	-0.021(3)    mP 21   15.92
          http://dx.doi.org/10.1107/S1600536807013803   	0.08(8)      mP 21   16.66
          http://dx.doi.org/10.1107/S160053680605272X   	0.02(5)      mP 21   15.64
          http://dx.doi.org/10.1107/S1600536807028607   	0.087(16)    mP 21   14.71
          http://dx.doi.org/10.1107/S1600536807002395   	0.013(8)     mP 21   14.58
          http://dx.doi.org/10.1107/S1600536806051130   	-0.04(5)     mP 21   13.21
          http://dx.doi.org/10.1107/S1600536807014237   	-0.019(8)    mP 21   12.14
          http://dx.doi.org/10.1107/S1600536807003637   	0.00(4)      mP 21   11.48
          http://dx.doi.org/10.1107/S1600536807021435   	0.026(15)    mP 21   11.32
          http://dx.doi.org/10.1107/S1600536807008203   	0.019(7)     mP 21   11.07


Average
          http://dx.doi.org/10.1107/S1600536807017394   	0.00(6)      mP 21   10.12
          http://dx.doi.org/10.1107/S1600536807025767   	0.015(7)     mP 21    9.41
          http://dx.doi.org/10.1107/S160053680701447X   	0.006(16)    mP 21    9.34
          http://dx.doi.org/10.1107/S1600536807024506   	0.07(5)      mP 21    9.21
          http://dx.doi.org/10.1107/S1600536807023379   	0.08(7)      mP 21    8.92
          http://dx.doi.org/10.1107/S1600536806050768   	0.01(4)      mP 21    8.78
          http://dx.doi.org/10.1107/S0108768107014838   	0.00(7)      mP 21    7.96
          http://dx.doi.org/10.1107/S160053680702329X   	-0.06(4)     mP 21    7.61
          http://dx.doi.org/10.1107/S1600536807024828   	0.12(7)      mP 21    7.44
          http://dx.doi.org/10.1107/S1600536807018818   	-0.08(6)     mP 21    7.42
          http://dx.doi.org/10.1107/S1600536807000700   	0.00(7)      mP 21    7.40
          http://dx.doi.org/10.1107/S1600536807019964   	0.10(3)      mP 21    7.28
          http://dx.doi.org/10.1107/S1600536806055073   	-0.03(8)     mP 21    7.15
          http://dx.doi.org/10.1107/S1600536807029613   	-0.002(11)   mP 21    7.14
          http://dx.doi.org/10.1107/S1600536807000931   	-0.04(8)     mP 21    7.13
          http://dx.doi.org/10.1107/S1600536807003170   	0.00(9)      mP 21    7.13
          http://dx.doi.org/10.1107/S1600536807007891   	0.000(10)    mP 21    6.91
          http://dx.doi.org/10.1107/S1600536806056145   	-0.01(7)     mP 21    6.91  
          http://dx.doi.org/10.1107/S1600536807016558   	0.01(6)      mP 21    6.82
          http://dx.doi.org/10.1107/S1600536807026487   	-0.01(6)     mP 21    6.74
          http://dx.doi.org/10.1107/S1600536806054729   	0.00(6)      mP 21    6.45
          http://dx.doi.org/10.1107/S1600536807001882   	0.03(2)      mP 21    6.39
          http://dx.doi.org/10.1107/S1600536807009257   	0.03(3)      mP 21    6.00
          http://dx.doi.org/10.1107/S1600536807028978   	-0.02(8)     mP 21    5.86
          http://dx.doi.org/10.1107/S1600536807026086   	-0.05(7)     mP 21    5.85
          http://dx.doi.org/10.1107/S1600536807014006   	-0.067(2)    mP 21    5.81
          http://dx.doi.org/10.1107/S1600536807022210   	-0.01(9)     mP 21    5.74
          http://dx.doi.org/10.1107/S1600536807006186   	0.134(19)    mP 21    5.49


Poor
          http://dx.doi.org/10.1107/S0108270107022469   	10(10)       mP 21   28.79
          http://dx.doi.org/10.1107/S0108768107010579   	0.00(13)     mP 21   28.00
          http://dx.doi.org/10.1107/S1600536807014341   	0.00(3)      mP 21   18.22
          http://dx.doi.org/10.1107/S1600536807011038   	0.514(19)    mP 21   16.58
          http://dx.doi.org/10.1107/S160053680702572X   	-0.02(16)    mP 21   15.83
          http://dx.doi.org/10.1107/S1600536807014067   	-0.06(9)     mP 21   15.60
          http://dx.doi.org/10.1107/S1600536807003133   	0.03(9)      mP 21   13.92
          http://dx.doi.org/10.1107/S0108270107011626   	-0.08(10)    mP 21   12.95
          http://dx.doi.org/10.1107/S160053680701478X   	0.08(12)     mP 21   11.93
          http://dx.doi.org/10.1107/S1600536807005417   	0.3(16)      mP 21   11.14
          http://dx.doi.org/10.1107/S1600536807017278   	0.43(3)      mP 21    9.61
          http://dx.doi.org/10.1107/S1600536807010021   	0.04(6)      mP 21    9.04
          http://dx.doi.org/10.1107/S1600536807024786   	0(10)        mP 21    8.99
          http://dx.doi.org/10.1107/S1600536807018223   	0(10)        mP 21    8.97
          http://dx.doi.org/10.1107/S010827010701952X   	0.11(16)     mP 21    8.94
          http://dx.doi.org/10.1107/S0108270106055156   	0.13(8)      mP 21    8.77
          http://dx.doi.org/10.1107/S1600536807012780   	0.508(12)    mP 21    8.01
          http://dx.doi.org/10.1107/S1600536807019733   	0.39(2)      mP 21    6.94
          http://dx.doi.org/10.1107/S0108270106055053   	0.1(2)       mP 21    6.24
          http://dx.doi.org/10.1107/S1600536807019666   	0.1(2)       mP 21    6.02
          http://dx.doi.org/10.1107/S1600536807012238   	0.13(8)      mP 21    5.65
          http://dx.doi.org/10.1107/S0108768106041358   	0.09(16)     mP 21    4.79
          http://dx.doi.org/10.1107/S1600536806050069   	0.16(6)      mP 21    3.89
          http://dx.doi.org/10.1107/S0108270106049468   	0.48(5)      mP 21    3.80
          http://dx.doi.org/10.1107/S1600536807030929   	0.05(6)      mP 21    3.79
          http://dx.doi.org/10.1107/S0108270106049468     0.51(4)      mP 21    3.77

                                                        
mC 2 (ITA No.5)  Sorted for tests on 0k0 reflections

  Compd   DOI                                            Flack         Space   cell b            
                                                                       group
Good
  TEVMEN  http://dx.doi.org/10.1107/S1600536806053815   	-0.05(3)     mC 2    10.74
  PIFGER  http://dx.doi.org/10.1107/S1600536807020442   	0.001(18)    mC 2     9.05

Average
  YIDHID  http://dx.doi.org/10.1107/S1600536807020016   	0.11(3)      mC 2     7.56
  SEZZAZ  http://dx.doi.org/10.1107/S1600536807005077   	0.011(9)     mC 2     7.27
  GIHKEO  http://dx.doi.org/10.1107/S1600536807025184   	-0.018(12)   mC 2     6.76
  TICFIV  http://dx.doi.org/10.1107/S1600536807009129   	-0.01(2)     mC 2     5.91
  CICSOX  http://dx.doi.org/10.1107/S1600536807018867   	-0.03(10)    mC 2     5.70
  YEYLOE  http://dx.doi.org/10.1107/S1600536806055425   	0.05(10)     mC 2     6.37
  CIDFIF  http://dx.doi.org/10.1107/S0108270107011626   	-0.04(9)     mC 2     6.13

Poor
          http://dx.doi.org/10.1107/S1600536807021319   	0.28(17)     mC 2     8.17
          http://dx.doi.org/10.1107/S0108270107008967   	0.48(5)      mI 2     5.60
          http://dx.doi.org/10.1107/S0108270107004830   	0.12(15)     mC 2     5.39



================= REMOVE



3.8 KHtartrate at 100K and 298K

DO NOT HAVE THE RESULTS AS YET. INCOMPLETE.
Dauter (2006) and Shen et al. (2003) [See also Flack & Shmueli, 2007] from studies in macromolecular crystallography has noticed an adp-enhancing factor on CHI (adp means atomic displacement parameter).  Heavy atoms tend both to have a large resonant-scattering contribution and small adps.  Consequently their effect to CHI increases as a function of sint/l.  To what extent this depends on the anisotropy of the adps needs to be discovered.
DO WE NEED TO MAKE A TEST OF CALCULATED INTENSITIES FOR KHtartrate AND SAME WITH K REPLACED BY Li and U for:
  (1) OVERALL ISOTROPIC ADP (OR ALL ISOTROPIC ADP IDENTICAL)
  (2) INDIVIDUAL ISOTROPIC ADP - EACH ATOM TYPE HAVING AN INDIVIDUAL ISOTROPIC ADP
  (3) INDIVIDUAL ANISOTROPIC ADP] - ALSO CHANGING THE ANISOTROPY AND SIZE OF THE METAL ATOMS.
THERE MAY BE AN INFLUENCE OF A SINGLE STRONG RESONANT ATOM
=======================
