Crystal structure solution via precession electron diffraction data: The BEA algorithm

Ultramicroscopy 111 (2010) 56–61

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Crystal structure solution via precession electron diffraction data: The BEA algorithm Giovanni Luca Cascarano a, Carmelo Giacovazzo a,b,n, Benedetta Carrozzini a a b

Istituto di Cristallografia, CNR, Via Amendola 122/o, Bari, Italy Dipartimento Geomineralogico, Universita di Bari, Campus Universitario, Bari, Italy

a r t i c l e i n f o

a b s t r a c t

Article history: Received 31 March 2010 Received in revised form 14 September 2010 Accepted 29 September 2010

The statistical features of the amplitudes obtained via precession electron diffraction have been studied, with particular concern with their effects on direct phasing procedures. A new algorithm, denoted by BEA, is described: according to it, the average amplitude of the symmetry equivalent reflections is used in the Direct Methods step. Once an even imperfect structural model is available, the best amplitude among the equivalent reflections is used to improve the model. It is shown that BEA is able to provide more complete structural models, to make the phasing process more straightforward and to end with crystallographic residual much better than those usually obtained by electron diffraction. & 2010 Elsevier B.V. All rights reserved.

Keywords: Electron diffraction Precession technique Electron crystallography Direct Methods

1. Introduction Using ED for crystal structure solution has several advantages over XD, e.g.:

(i) the stronger interaction of electrons with the matter (103 stronger) allows the study of individual crystallites of nanometre size via electron beams of the same size. This property overcomes the potential of powder techniques, particularly when, as in case of synthetic compounds, a single-phase sample is not available; (ii) the weaker dependence of electron structure amplitudes on Z allows a larger detectability of light atoms; (iii) the larger sensitivity to the redistribution of valence electrons allows the study of the electrostatic potential distribution.

Accurate crystal structure solutions require accurate kinematical structure factor amplitudes. Unfortunately the stronger interaction of the electrons with the matter is responsible of many beam dynamical diffraction and secondary scattering effects, which cause strong deviations of the diffraction intensities from the kinematical values [1]: the overall consequence is a limited capacity of precisely determining the values of the structural parameters. However, if a structure model is available, dynamical effects can be taken into consideration and suitably corrected [2]. n Correspondence author at: Dipartimento Geomineralogico, Universita di Bari, Campus Universitario, Bari, Italy. Tel.: + 39 080 5929140; fax: +39 080 5929170. E-mail address: [email protected] (C. Giacovazzo).

0304-3991/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2010.09.013

Recently introduced, PED techniques [3] (the crystal is oriented along a zone axis, the beam is tilted and precessed on a conical surface) reduce the number of reflections, which are simultaneously excited and therefore allow to describe the scattering by few beam approximations. PED data may be improved by applying two-beam approximation [4–9] or taking into account the Bethe effective potential. Two-beam correction may be analytically applied to PED data if the sample is well characterized (e.g. thickness and form: otherwise the crystal structure solution may be better achieved without any correction [9]). It is commonly recognized that the presence of dynamical effects in ED data (provided they are not dominant) does not hamper the crystal structure solution, but crystal structure refinement may be difficult: that occurs because least squares are more sensitive to dynamical deviations in the amplitudes than phasing procedures. As a result, the final RESID value may be larger than that usually attained by X-ray data: in the most unfavourable conditions, up to one order of magnitude larger. Consequently a poor structural model (atomic coordinates and thermal factors only approximately estimated) is obtained. This paper is devoted to the study of some statistical features of PED amplitudes, with particular attention to the effects they produce on the efficiency of the phasing procedures, specifically on Direct Methods approaches, which so far are the most popular phasing techniques. We will consider a relatively large number of test cases to provide a sufficient statistical basis for our conclusions. A specific attention will also be dedicated to diffraction data collected by combining PED and ADT techniques. If this last method is used, the reciprocal space is sequentially sampled with a fine step: that allows collecting most reflections available in the covered space.

G. Luca Cascarano et al. / Ultramicroscopy 111 (2010) 56–61

Nomenclature XD ED PED ADT s RES

Rint

X-ray diffraction electron diffraction precession electron diffraction automated diffraction tomography (sinW/l)2 ˚ observed data resolution (A)

/FobsS w RESID

P

57

9Fobs /Fobs S9

P

9Fobs 9

; the sum is over symmetry equivalent

reflections P w9Fobs 9 P ; the sum is over symmetry equivalent w reflections 1/s2(Fobs) P 9Fobs Fcalc 9 P ; the sum is over measured reflections 9F 9 obs

Furthermore a better integration of the diffraction intensities may be performed, particularly for spots placed closer to the zero beam position: several cuts through the reflection body can be collected and the true reflection intensity may be reconstructed. In our tests we will use the PED and ADT amplitudes recently collected by M. Gemmi, E. Mugnaioli and U. Kolb, as quoted in Table 1: no attempt will be made for improving them by techniques correcting for dynamical effects. As overall result, we will obtain a number of recipes (among which the BEA algorithm), which will make the crystal structure solution via PED data more straightforward. Such recipes may be useful also for standard selected area ED data, but very likely some caution is necessary because in these cases the dynamical diffraction effects may be dominant. The test structures are listed in Table 1, where we quote also their main crystal chemical data and some basic information about the diffraction experiment.

than for XD: therefore the atomic scattering curve for ED more rapidly decreases with s than the corresponding X-ray curves. In Fig. 1 we show, as an example, carbon and sulphur atomic scattering curves for electron and X-ray scattering: ED curves have been rescaled so that their maximum values are equal to Z. The more rapid decay of the electron scattering may suggest that ED data should be detectable only at resolution smaller than for X-rays. This effect however is contrasted by the stronger electron scattering (1000 times about) than X-rays, but is strengthened by the smaller size of the crystal samples submitted to electron diffraction. As a first conclusion, PED data can really attain quite high resolution as effect of three sample parameters (electrostatic potentials, strength of the scattering, size of the crystal) and of the shorter wavelength (much shorter than for X-rays): their data quality versus the resolution, however, has still to be assessed. In the practice

2. Statistical analysis of reflection intensities It is often claimed that PED data can be measured up to better than 0.5 A˚ resolution. Even if this is true because of geometrical reasons (PED techniques are really able to explore larger volumes of reciprocal space) the quality of these data needs to be explored. We will use test cases whose data are obtained by merging of different zones and we will take into considerations parameters which can affect the phasing efficiency: in particular data resolution, completeness, data accuracy. Data resolution: ED reveals the electrostatic potential, which is the sum of nuclear and of electronic potentials: viceversa, the electrostatic potential around each atom influences the atomic scattering. The region substantially determining ED scattering amplitudes is wider

˚ for XD and, Fig. 1. Carbon and sulphur atomic scattering curves versus RES (in A) after rescaling, for ED.

Table 1 For each test structure we give: the structure code (Code), the space group (SG), the chemical content in the asymmetric unit (ASU), the data collection strategy (COLL: P stays for PED and A for ADT technique), the data resolution (RES), the number of symmetry independent reflections at 0.9 A˚ resolution (NR0.9) and at the nominal experimental resolution (NRRES), the data completeness (%) using only reflections at 0.9 A˚ resolution (COMP0.9) and using all observed reflections (COMPRES), the percentage of reflections for which symmetry equivalent reflections are present in the experimental data (%ReflBEA), calculated up to 0.9 A˚ resolution and up to RES. Code

SG

ASU

COLL

RES

NR0.9/NRRES

COMP (0.9/RES)

%ReflBEA (0.9/RES)

srtio3_s [12]n anatase [13]n aker [14]n sno2 [15]n gann [16]n srtio3 [17]n mayenite [18]n zn8sb7 [19]nn barite [20]nn mullite [21]nn charo [22]nn natrolite [23]nn

Pbnm I41/amd P421m P42/mnm P63mc Pm3m I43d P1 Pnma Pbam P21/m Fdd2

O12Ca0.48Sr3.52Ti4 O8Ti4 O14Mg2Al6Si4Ca4 O4Sn2 Ga2N2 O3Ti1Sr1 O88Al28Ca24 Zn32Sb28 O16S4Ba4 Al4.56Si1.44O9.72 Ca24K14Na10Si72O186 Na2(Al2Si3O10) (H2O)2

P P P P P P P P–A P–A P–A P–A P–A

0.26 0.25 0.40 0.29 0.25 0.26 0.54 0.77 0.77 0.76 1.18 0.75

47/464 25/267 83/339 26/212 18/170 16/180 100/228 2679/3943 225/355 129/213 2878/2878 447/743

25/6 81/22 53/23 72/24 86/32 84/43 83/44 59/54 83/82 84/86 97/97 100/99

45/41 72/60 88/75 0/0 67/60 56/63 100/85 78/70 98/98 92/90 97/97 100/99

n

Single crystal ED data collected by M. Gemmi. Single crystal ED data collected by E. Mugnaioli and U. Kolb.

nn

58

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the situation is complicated by two supplementary sources: the presence of dynamical effects, already recalled in the introduction, and the geometrical corrections (to take into account the time in which the reflections are in Bragg condition). The formulas proposed in literature for such corrections are only approximated and may be critical for high resolution reflections. For six of our test structures (see Table 1) the nominal value of ˚ but, as we will see below, the quality RES is smaller or equal to 0.4 A, of the highest resolution data is rather low. In conclusion, the shorter electron wavelength combined with PED techniques allows collecting data at very high resolution, but their measurement errors may be particularly large. However we will show below that such reflections may be useful in the phasing process, and contribute to make the structural model more complete. Data completeness: This index is quite important both for phase determination and for crystal structure refinement. Although PED techniques curtail the problem of dynamical diffraction, it is difficult to collect fully 3-dimensional data by them. Usually (due to TEM optical tilt limitations) 2-dimensional reflections from few well oriented zone axes are collected. More recently [10,11] the ADT technique has been developed which, in combination with PED, allows much larger completeness values. Five of our twelve test data were collected by combining PED with ADT (see Table 1). In the same table we show the data completeness (i.e., the number of measured/number of measurable reflections) for each structure at RES and at 0.9 A˚ resolution, and the corresponding number of symmetry independent reflections. We selected this second resolution limit because 0.9 A˚ is the typical resolution for small molecule X-ray data: data completeness and quality are usually high also for the most external resolution shell. It is easy seen from Table 1 that data completeness is quite unsatisfactory for those structures for which the nominal RES ˚ in particular data collected by value is smaller than 0.4 A: combining PED and ADT show lower resolution but better data completeness, so providing a better basis for accurate crystal structure refinement. Data quality: For XD the value of Rint is usually employed as an indicator of the data quality. It is frequently in the range (0.02–0.06). To check ED data quality we calculated Rint for all the test structures and ˚ and the interval for two resolution ranges: the intervals (N—0.9 A) ˚ (0.9 A—RES). The results are shown in Table 2: Rint(0.9—RES) is generally worse than Rint(N—0.9). The only exceptions are srtio3 and srtio3_s, for which a good statistics is not allowed owing to the very small number of measured reflections in the external range (9 and 21, respectively). Symmetry equivalent reflections are not present in our sno2 data.

Table 2 Columns 2 and 3: Rint (%) values for the observed amplitudes in the intervals ˚ and (0.9 A—RES). ˚ (N—0.9 A) Columns 4 and 5: RESID (%) values using all reflections up to the observed resolution (RESIDRES) and selecting reflections up to 0.9 A˚ (RESID0.9). Code

Rint(N—0.9)

Rint(0.9—RES)

RESIDRES

RESID0.9

srtio3_s anatase aker sno2 gann srtio3 mayenite zn8sb7 barite mullite charo natrolite

15.8 7.6 6.3 – 1.9 18.5 20.5 16.1 14.6 29.4 13.3 19.2

13.1 18.8 14.5 – 5.7 18.9 27.6 24.4 18.2 40.5 – 26.8

58.4 48.9 35.7 36.9 28.5 40.7 40.8 – 33.4 36.3 – 25.8

46.3 30.2 25.3 26.4 20.7 19.2 34.2 – 32.3 34.4 – 23.7

To check further on the quality of measurements at high and at low data resolution we used the atomic positions as determined by X-rays, the electron scattering factors and our experimental Wilson average temperature factors for calculating the RESID values in two s ranges: up to 0.9 A˚ and up to RES. They are shown in columns 4 and 5 of Table 2 (except for charo and zn8sb7, for which X-ray models are not available), and clearly indicate a degradation of the average measurement quality when the nominal resolution is very high. Indeed, the largest differences in the RESID values are obtained for the structures with RES r0.41, the smallest differences are obtained for the structures for which PED and ADT techniques were simultaneously used: for them RES is close to 0.9 A˚ The above results deserve some additional comments. If PED technique is used, intensities coming from different zone axes are merged via common reflections: such a process is very critical and often generates serious errors in the amplitude estimates. If ADT and PED techniques are combined, the illumination conditions are kept constant and therefore no merging process is needed. Moreover, since data are collected far from zone axis orientation, the use of PED+ADT further increases the likelihood of being in a kinematical condition [24]: such a combination is therefore (as we will see in the following sections) the best condition for the crystal structure solution.

3. Phasing via Direct Methods Direct Methods procedures are unconditional (one can try to find the correct solution in an arbitrary number of trials). Increasing the number of trials makes easier to obtain a good solution but makes more difficult to recognize it. To use an objective criterion for checking direct phasing efficiency 200 trials were run for each test structure using Sir2008 (Version 3.0), included in the package Il Milione [25]. Direct Methods section is automatically followed by a direct space procedure which, via electron density modification techniques and diagonal least squares, tries to improve Direct Methods models. Such approach allowed using RESID as a figure of merit for recognizing the correct solution. All the following results are referred to the best solution found within the five trials with the lowest values of RESID. We used three different protocols, characterized by three different cut-off values for the resolution (see below). RES is dictated by the experimental resolution, 0.9 A˚ represents the usual experimental resolution limit for small molecules and X-ray radiation, 0.7 A˚ is the resolution attained when high quality crystals are submitted to X-rays. We did not introduce a fourth protocol with a cut-off of, say, 1.1 A˚ resolution, because in this case we are at the superior limit of the atomic resolution range, where the crystal structure solution starts to become difficult even for small molecules and X-ray data. Below we shortly describe the results of our applications. To hold an objective criterion, each model obtained at the end of the above described procedure arises from the E-map available at the end of Direct Methods section: the peak search routines of Sir2008 interpret the map in terms of a structural model and automatically submit it to the full direct space procedure. Protocol 1: All the measured data up to RES are used in the phasing process. Sir2008 was able to find a complete solution in all the cases except for anatase (one O and one Ti atom in the asymmetric: Ti was missed. Probable reason: unsatisfactory data completeness), barite (4 atoms over 5 well located: one O was missed), mullite (one O missed, with chemical occupancy equal to 0.14), srtio3_s (Ti and Ca–Sr atoms were missed. Probable reason: unsatisfactory data completeness). Protocol 2: Only data up to 0.7 A˚ resolution were used in the phasing process. Incomplete solutions were found for aker (one O missed), barite (one O was missed), mullite (one O missed, with chemical occupancy equal to 0.14), srtio3 (one O missed), srtio3_s (two O missed).

G. Luca Cascarano et al. / Ultramicroscopy 111 (2010) 56–61

59

45 40 35

RESID

30 25

Protocol 1

20

Protocol 2

15

Protocol 4

Protocol 3 Protocol 5

10

Protocol 6

5

lit e

o

tro na

ar

lit e

ch

ul

m

rit e

b7

ba

ite en

zn

ay

8s

3 tio sr

m

an

Sr ti

o3

_s at as e ak er sn o2 ga nn

0

Test structures Fig. 2. For each test structure the RESID (%) values, automatically obtained by Sir2008 at the end of the phasing process, are given, according to Protocols 1–6.

Protocol 3: Only data up to 0.9 A˚ resolution were used in the phasing process. Sir2008 found an incomplete solution for aker (5 of 6 positions well found, one O missed), anatase (the O was missed), barite (one O was missed), mullite (one O and one Al missed, both with chemical occupancy equal to 0.14), srtio3_s (one O was missed). The comparison of the results obtained via the three protocols suggests the following: (a) all the atomic positions of the structures with the largest number of atoms in the asymmetric unit [charo (89 atoms), natrolite (10) and zn8sb7 (30)], with data collected by combining PED and ADT, are found by Sir2008, no matter the protocol. (b) The very high resolution reflections provide useful information for the success of the phasing process even if the accuracy of their measured amplitudes is lower. This behaviour may be related to the recent use of the extrapolation techniques [26,27] in protein structure determination, where phases and amplitudes of nonmeasured (because out of the measured reciprocal space) reflections are estimated via probabilistic approaches and used during the phase determination process. Such extrapolated reflections increase the power of the phasing methods even if the estimated magnitudes are very far from the correct values. In our case PED techniques are able to provide measurements that are better than the extrapolated values obtained by probabilistic techniques and therefore may provide additional power to Direct Methods. (c) The RESID values available at the end of the automatic phasing process (where peaks are labelled and thermal factors are still isotropic) are shown in Fig. 2. Most of them are far from the values obtained by Sir2008 for X-ray data (usually between 0.08 and 0.12). Usually RESID decreases when the protocol number increases. However, owing to the experimental resolution limits: (i) the RESID values approximately coincide for barite, mullite, natrolite and zn8sb7, when calculated via the protocols 1 and 2; (ii) for charo RESID cannot vary with the protocol number.

4. The BEA algorithm In Section 2 we stated that a general approach for correcting electron diffraction data for dynamic effects is still not available: it requires a deep characterization of the samples (e.g., thickness

and its variation, bending, etc.), which is not usually performed. If PED techniques are used, the dynamical effects are no more dominant but still present (see the high Rint values in Table 2, calculated for the equivalent reflections). A question arises: in absence of a theoretical formulation establishing which of the equivalent reflections is less affected by dynamical scattering can one use a practical criterion for selecting the best unique reflection? What best means may be object of a big debate. We suggest choosing as unique reflection that one which better agrees with the current structural model. The effect is certainly cosmetic (the final RESID value may be much smaller than that obtained using the weighted average of the equivalent amplitudes). But it may also be substantial: i.e., the crystal structure solution becomes more straightforward and the final structural model may be more complete. We call this algorithm best equivalent amplitude (BEA). It has some similarity with a criterion used in powder crystallography during the phasing process. When a structural model is not available the experimental diffraction profile is decomposed according to Le Bail et al. [28] or to Pawley [29] algorithms: they take into account positions and full widths at half maximum of the overlapping reflections and assign to them intensity values optimizing the fitting with the overall experimental profile (the sum of the diffraction contributions arising from the reflections). On the contrary, if a structural model is available, the experimental diffraction profile is partitioned in a way proportional to the calculated structure factors of the overlapping reflections. This practice helps the crystal structure solution and reduces the profile- and the F-residual. In the case of electron diffraction we recognize that uncorrected dynamical effects still affect the experimental data (even in the PED case), that such effects are not corrected by a posteriori techniques and that merging symmetry equivalent reflections may lead to averaged values, which are not good representatives of the true intensities. To better understand the role of BEA we explicitly state that, when Direct Methods are run, the average amplitude of the symmetry equivalent reflections must be used (no structural model is available at this stage and therefore no indication may be given on which of the equivalents is the best). As soon as a model is available, the best equivalent reflection may be recognized and the corresponding amplitude is used as coefficient of the observed Fourier maps and as observed amplitude in the diagonal least-squares procedure, cyclically and automatically performed by Sir2008. Obviously the best reflection changes with the structural model.

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Some comments are useful before describing the BEA applications:

(i) BEA may only be applied if the experimental data contains symmetry equivalent reflections. Luckily in our test data there is always a good percentage of reflections for which at least two symmetry equivalent amplitudes were measured: the unique exception is sno2 (probably, the original measurements were already merged). We show in the last column of Table 1, for each test structure, the percentage of reflections for which at least two symmetry equivalent reflections (Friedel opposite included) are present in the experimental data (%ReflBEA), calculated up to 0.9 A˚ resolution and up to RES. The best reflection among the equivalents is re-defined as soon as a new structure model is available. In accordance with the algorithm, each of the above reflections is checked several times during the phasing process for a new definition of the best among the equivalents. Particularly useful for the algorithm are the data for which all the equivalent amplitudes were measured. (ii) It is well known that phases are more important than amplitudes for describing a crystal structure. However, phases are hidden in the moduli and their extraction from wrong moduli is therefore hard. That explains why phasing is more difficult for powder than for single crystal data. It is not rare that for some simple centric structures, for which only powder data are available, Direct Methods lead to a fully correct set of phases, but in the subsequent electron density map the correct structure cannot be recognized. That is due to map distortions caused by the errors in the amplitudes. It is common practice, in powder crystallography, to modify Le Bail or Pawley amplitudes (independent of any structural model) according to the model. That introduces a bias in the phasing process: the hope is that the model available at that moment is capable of modifying the amplitudes in a virtuous way, making them closer to the true ones. If that is accomplished, a next electron density map may reveal unknown fragments of the structure and therefore make the model more complete. The cyclic iteration of such process may reduce the bias and lead to the correct structure. Obviously, if the original model is mostly wrong (in our case that may easily occur when dynamical effects are dominant), the model bias will lead to failure. BEA therefore is based on the assumption that original PED data may lead to useful initial structure models. (iii) BEA modifies the intensity of the symmetry equivalent reflections, Friedel opposite included. In particular, it is not rare that Laue opposites are both in the data set and show different intensity. We preserve their original intensity up to when BEA is applied: in that case the reflection h or –h may become the representative of the set of equivalent reflections according to which of them is the best. On the other side BEA is almost useless if the symmetry equivalent reflections have about the same measured intensity: that may correspond to the ideal case in which dynamical effects are absent. (iv) The intensity changes for each set of symmetry equivalent reflections depend on the Rint value. The larger Rint the larger the changes: accordingly, the higher resolution reflections will be more affected by the application of BEA. In our set of test structures the average value of Rint significantly varies with the structure and with the resolution: thus the expected BEA improvements will be structure dependent. (v) BEA is applied to all the measured reflections, for which equivalent amplitudes are in the data set, no matter their resolution (see column 8 in Table 1).

(vi) Using BEA when Rint is very small cannot lead to improved structural models: accordingly, for such structures, RESID will not vary if BEA is used. If the average Rint values are sufficiently large RESID is expected to significantly drop, more in the first BEA cycles, less when the structural model is stable. The final models, obtained when BEA is used or not used, are expected to differ in several aspects: completeness, labelling and accuracy of the atomic positions. (vii) The full direct space procedure is performed by actively using the data changed by BEA: in particular they are employed as coefficients of the electron density maps and as observed values in the least-squares procedures. That equally occurs in powder crystallography, when the overall intensity of a reflection cluster is decomposed according to the model. There is however a subtle distinction: in the powder case the experimental diffraction profile is not modified and is used for Rietveld refinement. In the BEA case, in absence of a criterion to correct the dynamical effects, one of the symmetry equivalent reflections is assumed as representative of the set and its observed amplitude actively used in the next step of the phasing process. The representative reflection changes during the structure refinement process: at the end its amplitude may really represent a good estimate of the true one if BEA is able to recover a good model. The BEA convergence to the correct model, however, cannot be expected to be high because of the model bias. Finally, we have to consider BEA as an additional tool for providing structural models better than those achievable by traditional procedures. To check the BEA usefulness we used the following three protocols. Protocol 4. All the measured data up to RES are used in the phasing process (as in Protocol 1) and BEA is applied. The results may be described as follows: all the atomic positions of the test structures were found, but for mullite, for which one O, with chemical occupancy equal to 0.14, was missed. Protocol 5. Only data up to 0.7 A˚ resolution were used in the phasing process (as in Protocol 2) and BEA is applied. Incomplete structures were found for aker (one O missed), mullite (one O missed, with chemical occupancy equal to 0.14), srtio3_s (Ti and Ca–Sr atoms missed). Protocol 6. Only data up to 0.9 A˚ resolution were used (as in Protocol 3) and BEA is applied. Sir2008 found an incomplete solution for aker (one O missed), mullite (one O missed, with chemical occupancy equal to 0.14), srtio3_s (Ti and Ca–Sr atoms missed). The corresponding RESID values are quoted in Fig. 2: as for the Protocols 1–3, they improve when the protocol number increases. Of course, no improvement may be expected for charo and marginal improvements for barite, mullite and natrolite because of their limited data resolution. If we compare the results attained via Protocols 4, 5 and 6 with those obtained via Protocols 1, 2 and 3, respectively, (compare protocol i with protocol i+3, because they imply the same data resolution), we observe that the use of BEA: (A) usually leads to more complete structural models. The best results are obtained using the Protocol 4, where BEA is applied to diffraction data up to RES. Indeed, compared with Protocol 1, where incomplete models were obtained for anatase, barite, mullite and srtio3_s, BEA really leads to complete structural models for all the structures (mullite is the only exception, but the missed atom has a quite low chemical occupancy). While Protocol 2 provides incomplete models for four structures (aker, barite, mullite, srtio3, srtio3_s), Protocol 5 provides complete

G. Luca Cascarano et al. / Ultramicroscopy 111 (2010) 56–61

models for all the structures except aker, mullite, srtio3_s (in the last case, however, the two cation positions are missed). Comparison between Protocols 3 and 6 shows that, in the first case, incomplete models are obtained for five structures, in the second case, only three incomplete models are provided. (B) BEA may strongly improve the RESID values, particularly when the nominal data resolution is high and makes them very close to the values usually obtained for X-ray data at this stage of refinement. Obviously, no improvement is obtained for sno2 (only unique reflections are present in our data) and small improvements, if any, are obtained for gann for which Rint is quite small. Comparing results quoted in Fig. 2 suggests that RESID values quite similar are obtained for protocols i and i+ 3, for anatase and srtio3_s, but they correspond to models with different structural completeness. A last comment on the accuracy of the models may be useful. The accuracy of the atomic positions in models with the same completeness, obtained via different protocols, does not show a large variation. For example: (i) for zn8sb7 all the protocols are able to correctly locate the 30 atoms in the asymmetric unit. The average distance /dS from the correct positions varies from 0.125 to ˚ (ii) For natrolite, 10 atoms are located with /dS varying 0.135 A. from 0.11 A˚ for the Protocol 1 to 0.08 A˚ for the Protocol 5. (iii) For ˚ charo, 89 atoms located, /dS remains in the range 0.10–0.11 A. In accordance with the above results, BEA may be considered an additional tool for a more straightforward crystal structure solution from ED data. It may be foreseen that BEA may be useful also for phasing methods based on the properties of the Fourier transforms, like charge flipping [30,31] or the VLD (vive la difference) algorithm [32,33]. Indeed the use of the ‘‘best’’ reflections as coefficients of the Fourier syntheses may help such methods to reach more easily the convergence.

5. Conclusions The statistical features of the PED data have been studied, with particular attention to their consequences on direct phasing procedures. A new algorithm, with acronym BEA, has been proposed: Direct Methods applications, performed via a modified version of Sir2008 including BEA, show that such algorithm is able to provide more complete structural models and better crystallographic residuals. More extended applications are needed to extrapolate the usefulness of BEA for the final refinement stages.

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