International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 1.6, pp. 107-131
doi: 10.1107/97809553602060000924

Chapter 1.6. Methods of space-group determination

U. Shmueli,a* H. D. Flackb and J. C. H. Spencec

aSchool of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel,bChimie minérale, analytique et appliquée, University of Geneva, Geneva, Switzerland, and cDepartment of Physics, Arizona State University, Rural Rd, Tempe, AZ 85287, USA
Correspondence e-mail:  ushmueli@tau.ac.il

This chapter describes several methods of symmetry determination of single-domain crystals. A detailed presentation of symmetry determination from diffraction data is followed by a brief discussion of intensity statistics, ideal as well as non-ideal, with an application of the latter to real intensity data from a [P\overline{1}] crystal structure. Several methods of retrieving symmetry information from a solved crystal structure are then described. This is followed by a discussion of chemical and physical restrictions on space-group symmetry, including some aids in symmetry determination, and by a brief section on pitfalls in space-group determination. The theoretical background of conditions for possible general reflections and their corresponding derivation are given along with a brief discussion of special reflection conditions. An extensive tabulation of general reflection conditions and possible space groups is presented. Other methods of space-group determination, including methods based on resonant scattering, are described, including approaches to space-group determination in macromolecular crystallography and in powder diffraction. The chapter concludes with a description and illustration of symmetry determination based on electron-diffraction methods, principally using convergent-beam electron diffraction.

1.6.1. Overview

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This chapter describes and discusses several methods of symmetry determination of single-domain crystals. A detailed presentation of symmetry determination from diffraction data is given in Section 1.6.2.1[link], followed by a brief discussion of intensity statistics, ideal as well as non-ideal, with an application of the latter to real intensity data from a [P\overline{1}] crystal structure in Section 1.6.2.2[link]. Several methods of retrieving symmetry information from a solved crystal structure are then discussed (Section 1.6.2.3[link]). This is followed by a discussion of chemical and physical restrictions on space-group symmetry (Section 1.6.2.4[link]), including some aids in symmetry determination, and by a brief section on pitfalls in space-group determination (Section 1.6.2.5[link]).

The following two sections deal with reflection conditions. Section 1.6.3[link] presents the theoretical background of conditions for possible general reflections and their corresponding derivation. A brief discussion of special reflection conditions is included. Section 1.6.4[link] presents an extensive tabulation of general reflection conditions and possible space groups.

Other methods of space-group determination are presented in Section 1.6.5[link]. Section 1.6.5.1[link] deals with an account of methods of space-group determination based on resonant (also termed `anomalous') scattering. Section 1.6.5.2[link] is a brief description of approaches to space-group determination in macromolecular crystallography. Section 1.6.5.3[link] deals with corresponding approaches in powder-diffraction methods.

The chapter concludes with a description and illustration of symmetry determination based on electron-diffraction methods (Section 1.6.6[link]), and principally focuses on convergent-beam electron diffraction.

This chapter deals only with single crystals. A supplement (Flack, 2015[link]) deals with twinned crystals and those displaying a specialized metric.

1.6.2. Symmetry determination from single-crystal studies

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U. Shmuelia and H. D. Flackb

1.6.2.1. Symmetry information from the diffraction pattern

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The extraction of symmetry information from the diffraction pattern takes place in three stages.

In the first stage, the unit-cell dimensions are determined and analyzed in order to establish to which Bravais lattice the crystal belongs. A conventional choice of lattice basis (coordinate system) may then be chosen. The determination of the Bravais lattice1 of the crystal is achieved by the process of cell reduction, in which the lattice is first described by a basis leading to a primitive unit cell, and then linear combinations of the unit-cell vectors are taken to reduce the metric tensor (and the cell dimensions) to a standard form. From the relationships amongst the elements of the metric tensor, one obtains the Bravais lattice, together with a conventional choice of the unit cell, with the aid of standard tables. A detailed description of cell reduction is given in Chapter 3.1[link] of this volume and in Part 9[link] of earlier editions (e.g. Burzlaff et al., 2002[link]). An alternative approach (Le Page, 1982[link]) seeks the Bravais lattice directly from the cell dimensions by searching for all the twofold axes present. All these operations are automated in software. Regardless of the technique employed, at the end of the process one obtains an indication of the Bravais lattice and a unit cell in a conventional setting for the crystal system, primitive or centred as appropriate. These are usually good indications which, however, must be confirmed by an examination of the distribution of diffracted intensities as outlined below.

In the second stage, it is the point-group symmetry of the intensities of the Bragg reflections which is determined. We recall that the average reduced intensity of a pair of Friedel opposites (hkl and [\overline{h} \overline{k} \overline{l}]) is given by[\eqalignno{|F_{\rm av}({\bf h})|^{2}&= \textstyle{{1}\over{2}}[|F({\bf h})|^{2}+|F(\overline{\bf h})|^{2}]&\cr &=\textstyle\sum\limits_{i,j}[(f_{i}+f_{i}^{\prime})(f_{j}+f_{j}^{\prime})+f_{i}^{\prime\prime}f_{j}^{\prime\prime}] \cos[2\pi{\bf h}({\bi r}_{i}-{\bi r}_{j})] \equiv A({\bf h}),&\cr&&(1.6.2.1)}]where the atomic scattering factor of atom j, taking into account resonant scattering, is given by[{\bf f}_{j}=f_{j}+f_{j}^{\prime}+if_{j}^{\prime\prime},]the wavelength-dependent components [f_{j}^{\prime}] and [f_{j}^{\prime\prime}] being the real and imaginary parts, respectively, of the contribution of atom j to the resonant scattering, h contains in the (row) matrix (1 × 3) the diffraction orders (hkl) and [{\bi r}_{j}] contains in the (column) matrix (3 × 1) the coordinates [(x_{j},y_{j},z_{j})] of atom j. The components of the [{\bf f}_{j}] are assumed to contain implicitly the displacement parameters. Equation (1.6.2.1)[link] can be found e.g. in Okaya & Pepinsky (1955[link]), Rossmann & Arnold[link] (2001[link]) and Flack & Shmueli (2007[link]). It follows from (1.6.2.1)[link] that[|F_{\rm av}({\bf h})|^{2}=|F_{\rm av}(\overline{\bf h})|^{2} \ {\rm or}\ A({\bf h})=A(\overline{\bf h}),]regardless of the contribution of resonant scattering. Hence the averaging introduces a centre of symmetry in the (averaged) diffraction pattern.2 In fact, working with the average of Friedel opposites, one may determine the Laue group of the diffraction pattern by comparing the intensities of reflections which should be symmetry equivalent under each of the Laue groups. These are the 11 centrosymmetric point groups: [\overline{1}], 2/m, mmm, 4/m, 4/mmm, [\overline{3}], [\overline{3}m], 6/m, 6/mmm, [m\overline{3}] and [m\overline{3}m]. For example, the reflections of which the intensities are to be compared for the Laue group [\overline{3}] are: hkl, kil, ihl, [\overline{hkl}], [\overline{kil}] and [\overline{ihl}], where [i=-h-k]. An extensive listing of the indices of symmetry-related reflections in all the point groups, including of course the Laue groups, is given in Appendix 1.4.4 of International Tables for Crystallography Volume B (Shmueli, 2008[link]).3 In the past, one used to inspect the diffraction images to see which classes of reflections are symmetry equivalent within experimental and other uncertainty. Nowadays, the whole intensity data set is analyzed by software. The intensities are merged and averaged under each of the 11 Laue groups in various settings (e.g. 2/m unique axis b and unique axis c) and orientations (e.g. [\overline{3}m1] and [\overline{3}1m]). For each choice of Laue group and its variant, an [R_{\rm merge}] factor is calculated as follows:[R_{{\rm merge},i}={{\textstyle\sum_{\bf h}\textstyle\sum_{s=1}^{|G|_{i}}|\langle|F_{\rm av}({\bf h})|^{2}\rangle_{i}-|F_{\rm av}({\bf h} {\bi{W}}_{si})|^{2}|}\over{|G|_{i}\textstyle\sum_{\bf h}\langle|F_{\rm av}({\bf h})|^{2}\rangle_{i}}},\eqno(1.6.2.2)]where [{\bi{W}}_{si}] is the matrix of the [s]th symmetry operation of the ith Laue group, [|G_{i}|] is the number of symmetry operations in that group, the average in the first term in the numerator and in the denominator ranges over the intensities of the trial Laue group and the outer summations [\sum_{\bf h}] range over the hkl reflections. Choices with low [R_{{\rm merge},i}] display the chosen symmetry, whereas for those with high [R_{{\rm merge},i}] the symmetry is inappropriate. The Laue group of highest symmetry with a low [R_{{\rm merge},i}] is considered the best indication of the Laue group. Several variants of the above procedure exist in the available software. Whichever of them is used, it is important for the discrimination of the averaging process to choose a strategy of data collection such that the intensities of the greatest possible number of Bragg reflections are measured. In practice, validation of symmetry can often be carried out with a few initial images and the data-collection strategy may be based on this assignment.

In the third stage, the intensities of the Bragg reflections are studied to identify the conditions for systematic absences. Some space groups give rise to zero intensity for certain classes of reflections. These `zeros' occur in a systematic manner and are commonly called systematic absences (e.g. in the h0l class of reflections, if all rows with l odd are absent, then the corresponding reflection condition is h0l: l = 2n). In practice, as implemented in software, statistics are produced on the intensity observations of all possible sets of `reflections conditions' as given in Chapter 2.3[link] (e.g. in the example above, h0l reflections are separated into sets with l = 2n and those with l = 2n + 1). In one approach, the number of observations in each set having an intensity (I) greater than n standard uncertainties [u(I)] [i.e. [I/u(I)\,\gt\,n]] is displayed for various values of n. Clearly, if a trial condition for systematic absence has observations with strong or medium intensity [i.e. [I/u(I)\,\gt\, 3]], the systematic-absence condition is not fulfilled (i.e. the reflections are not systematically absent). If there are no such observations, the condition for systematic absence may be valid and the statistics for smaller values of n need then to be examined. These are more problematic to evaluate, as the set of reflections under examination may have many weak reflections due to structural effects of the crystal or to perturbations of the measurements by other systematic effects. An alternative approach to examining numbers of observations is to compare the mean value, [\langle I/u(I) \rangle], taken over reflections obeying or not a trial reflection condition. For a valid reflection condition, one expects the former value to be considerably larger than the latter. In Section 3.1 of Palatinus & van der Lee (2008[link]), real examples of marginal cases are described.

The third stage continues by noting that the systematic absences are characteristic of the space group of the crystal, although some sets of space groups have identical reflection conditions. In Chapter 2.3[link] one finds all the reflection conditions listed individually for the 230 space groups. For practical use in space-group determination, tables have been set up that present a list of all those space groups that are characterized by a given set of reflection conditions. The tables for all the Bravais lattices and Laue groups are given in Section 1.6.4[link] of this chapter. So, once the reflection conditions have been determined, all compatible space groups can be identified from the tables. Table 1.6.2.1[link] shows that 85 space groups may be unequivocally determined by the procedures defined in this section based on the identification of the Laue group. For other sets of reflection conditions, there are a larger number of compatible space groups, attaining the value of 6 in one case. It is appropriate at this point to anticipate the results presented in Section 1.6.5.1[link], which exploit the resonant-scattering contribution to the diffracted intensities and under appropriate conditions allow not only the Laue group but also the point group of the crystal to be identified. If such is the case, the last line of Table 1.6.2.1[link] shows that almost all space groups can be unequivocally determined. In the remaining 13 pairs of space groups, constituting 26 space groups in all, there are the 11 enantiomorphic pairs of space groups [([P4_{1}][P4_{3}]), ([P4_{1}22][P4_{3}22]), ([P4_{1}2_{1}2][P4_{3}2_{1}2]), ([P3_{1}][P3_{2}]), ([P3_{1}21][P3_{2}21]), ([P3_{1}12][P3_{2}12]), ([P6_{1}][P6_{5}]), ([P6_{2}][P6_{4}]), ([P6_{1}22][P6_{5}22]), ([P6_{2}22][P6_{4}22]) and ([P4_{1}32][P4_{3}32])] and the two exceptional pairs of [I222] & [I2_{1}2_{1}2_{1}] and [I23] & [I2_{1}3], characterized by having the same symmetry elements in a different arrangement in space. These 13 pairs of space groups cannot be distinguished by the methods described in Sections 1.6.2[link][link] and 1.6.5.1[link], but may be distinguished when a reliable atomic structural model of the crystal has been obtained. On the other hand, all these 13 pairs of space groups can be distinguished by the methods described in Section 1.6.6[link] and in detail in Saitoh et al. (2001[link]). It should be pointed out in connection with this third stage that a possible weakness of the analysis of systematic absences for crystals with small unit-cell dimensions is that there may be a small number of axial reflections capable of being systematically absent.

Table 1.6.2.1| top | pdf |
The ability of the procedures described in Sections 1.6.2.1[link] and 1.6.5.1[link] to distinguish between space groups

The columns of the table show the number of sets of space groups that are indistinguishable by the chosen technique, according to the number of space groups in the set, e.g. for Laue-class discrimination, 85 space groups may be uniquely identified, whereas there are 8 sets containing 5 space groups indistinguishable by this technique. The tables in Section 1.6.4[link] contain 416 different settings of space groups generated from the 230 space-group types.

 No. of space groups in set that are indistinguishable by procedure used
123456
No. of sets for Laue-class discrimination 85 78 43 0 8 1
No. of sets for point-group discrimination 390 13 0 0 0 0

It goes without saying that the selected space groups must be compatible with the Bravais lattice determined in stage 1, with the Laue class determined in stage 2 and with the set of space-group absences determined in stage 3.

We thank L. Palatinus (2011[link]) for having drawn our attention to the unexploited potential of the Patterson function for the determination of the space group of the crystal. The discovery of this method is due to Buerger (1946[link]) and later obtained only a one-sentence reference by Rogers (1950[link]) and by Rossmann & Arnold (2001[link]). The method is based on the observation that interatomic vectors between symmetry-related (other than by inversion in a point) atoms cause peaks to accumulate in the corresponding Harker sections and lines of the Patterson function. It is thus only necessary to find the location of those Harker sections and lines that have a high concentration of peaks to identify the corresponding symmetry operations of the space group. At the time of its discovery, it was not considered an economic method of space-group determination due to the labour involved in calculating the Patterson function. Subsequently it was completely neglected and there are no recent reports of its use. It is thus not possible to report on its strengths and weaknesses in practical modern-day applications.

1.6.2.2. Structure-factor statistics and crystal symmetry

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Most structure-solving software packages contain a section dedicated to several probabilistic methods based on the Wilson (1949[link]) paper on the probability distribution of structure-factor magnitudes. These statistics sometimes correctly indicate whether the intensity data set was collected from a centrosymmetric or noncentrosymmetric crystal. However, not infrequently these indications are erroneous. The reasons for this may be many, but outstandingly important are (i) the presence of a few very heavy atoms amongst a host of lighter ones, and (ii) a very small number of nearly equal atoms. Omission of weak reflections from the data set also contributes to failures of Wilson (1949[link]) statistics. These erroneous indications are also rather strongly space-group dependent.

The well known probability density functions (hereafter p.d.f.'s) of the magnitude of the normalized structure factor [E], also known as ideal p.d.f.'s, are[p(|E|)=\left\{\matrix{\sqrt{2/\pi}\exp\left(-|E|^{2}/2\right)\hfill &{\rm for}\ P\overline{1}\cr 2|E|\exp(-|E|^{2})\hfill &{\rm for}\ P1}\right.,\eqno(1.6.2.3)]where it is assumed that all the atoms are of the same chemical element. Let us see their graphical representations.

It is seen from Fig. 1.6.2.1[link] that the two p.d.f.'s are significantly different, but usually they are not presented as such by the software. What is usually shown are the cumulative distributions of [|E|^{2}], the moments: [\langle |E|^{n} \rangle] for n = 1, 2, 3, 4, 5, 6, and the averages of low powers of [|E^{2}-1|] for ideal centric and acentric distributions, based on equation (1.6.2.3)[link]. Table 1.6.2.2[link] shows the numerical values of several low-order moments of [|E|] and that of the lowest power of [|E^{2}-1|]. The higher the value of n the greater is the difference between their values for centric and acentric cases. However, it is most important to remember that the influence of measurement uncertainties also increases with n and therefore the higher the moment the less reliable it tends to be.

Table 1.6.2.2| top | pdf |
The numerical values of several low-order moments of [|E|], based on equation (1.6.2.3)[link]

Moment[P\overline{1}]P1
[\langle |E| \rangle] 0.798 0.886
[\langle |E|^{2} \rangle] 1.000 1.000
[\langle |E|^{3} \rangle] 1.596 1.329
[\langle |E|^{4} \rangle] 3.000 2.000
[\langle |E|^{5} \rangle] 6.383 3.323
[\langle |E|^{6} \rangle] 15.000 6.000
[\langle |E^{2}-1| \rangle] 0.968 0.736
[Figure 1.6.2.1]

Figure 1.6.2.1 | top | pdf |

Ideal p.d.f.'s for the equal-atom case. The dashed line is the centric, and the solid line the acentric ideal p.d.f.

There are several ideal indicators of the status of centrosymmetry of a crystal structure. The most frequently used are: (i) the N(z) test (Howells et al., 1950[link]), a cumulative distribution of [z=|E|^{2}], based on equation (1.6.2.3)[link], and (ii) the low-order moments of [|E|], also based on equation (1.6.2.3)[link]. Equation (1.6.2.3)[link], however, is very seldom used as an indicator of the status of centrosymmetry of a crystal stucture.

Let us now briefly consider p.d.f.'s that are valid for any atomic composition as well as any space-group symmetry, and exemplify their performance by comparing a histogram derived from observed intensities from a [P\overline{1}] structure with theoretical p.d.f.'s for the space groups P1 and [P\overline{1}]. The p.d.f.'s considered presume that all the atoms are in general positions and that the reflections considered are general (see, e.g., Section 1.6.3[link]). A general treatment of the problem is given in the literature and summarized in the book Introduction to Crystallographic Statistics (Shmueli & Weiss, 1995[link]).

The basics of the exact p.d.f.'s are conveniently illustrated in the following. The normalized structure factor for the space group [P\overline{1}], assuming that all the atoms occupy general positions and resonant scattering is neglected, is given by[E({\bf h})=2\textstyle\sum\limits_{j=1}^{N/2}n_{j}\cos(2\pi{\bf h}{\bi r}_{j}),]where [n_{j}] is the normalized scattering factor. The maximum possible value of [E] is [E_{\rm max}=\textstyle\sum_{j=1}^{N}n_{j}] and the minimum possible value of [E] is [-E_{\rm max}]. Therefore, [E({\bf h})] must be confined to the [(-E_{\rm max},E_{\rm max})] range. The probability of finding E outside this range is of course zero. Such a probability density function can be expanded in a Fourier series within this range (cf. Shmueli et al., 1984[link]). This is the basis of the derivation, the details of which are well documented (e.g. Shmueli et al., 1984[link]; Shmueli & Weiss, 1995[link]; Shmueli, 2007[link]). Exact p.d.f.'s for any centrosymmetric space group have the form[p(|E|)=\alpha\left\{1+2\textstyle\sum\limits_{m=1}^{\infty}C_{m}\cos(\pi m|E|\alpha)\right\},\eqno(1.6.2.4)]where [\alpha=1/E_{\rm max}], and exact p.d.f.'s for any noncentrosymmetric space group can be computed as the double Fourier series[p(|E|)=\textstyle{{1}\over{2}}\pi\alpha^{2}|E|\textstyle\sum\limits_{m=1}^{\infty}\textstyle\sum\limits_{n=1}^{\infty}C_{mn} J_{0}[\pi\alpha |E|(m^{2}+n^{2})^{1/2}],\eqno(1.6.2.5)]where [J_{0}(X)] is a Bessel function of the first kind and of order zero. Expressions for the coefficients [C_{m}] and [C_{mn}] are given by Rabinovich et al. (1991[link]) and by Shmueli & Wilson (2008[link]) for all the space groups up to and including [Fd\overline{3}].

The following example deals with a very high sensitivity to atomic heterogeneity. Consider the crystal structure of [(Z)-ethyl N-iso­propyl­thio­carbamato-κS]­(tricyclo­hexyl­phos­phine-κP)­gold(I), published as [P\overline{1}] with Z = 2, the content of its asymmetric unit being AuSPONC24H45 (Tadbuppa & Tiekink, 2010[link]). Let us construct a histogram from the [|E|] data computed from all the observed reflections with non-negative reduced intensities and compare the histogram with the p.d.f.'s for the space groups P1 and [P\overline{1}], computed from equations (1.6.2.5)[link] and (1.6.2.4)[link], respectively. The histogram and the p.d.f.'s were put on the same scale. The result is shown in Fig. 1.6.2.2[link].

[Figure 1.6.2.2]

Figure 1.6.2.2 | top | pdf |

Exact p.d.f.'s. for a crystal of [(Z)-ethyl N-isopropylthiocarbamato-κS]­(tricyclohexylphosphine-κP)gold(I) in the triclinic system. Solid curve: [P\overline{1}], computed from (1.6.2.4)[link]; dashed curve: P1, computed from (1.6.2.5)[link]; histogram based on the data computed from all the reflections with non-negative reduced intensities. The height of each bin corresponds to the number of reflections (NREF) in its range of [|E|] values. The p.d.f.'s are scaled up to the histogram.

A visual comparison strongly indicates that the space-group assignment as [P\overline{1}] was correct, since the recalculated histogram agrees rather well with the p.d.f. (1.6.2.4)[link] and much less with (1.6.2.5)[link]. The ideal Wilson-type statistics incorrectly indicated that this crystal is noncentrosymmetric. It is seen that the ideal p.d.f. breaks down in the presence of strong atomic heterogeneity (gold among many lighter atoms) in the space group [P\overline{1}]. Other space groups behave differently, as shown in the literature (e.g. Rabinovich et al., 1991[link]; Shmueli & Weiss, 1995[link]).

Additional examples of applications of structure-factor statistics and some relevant computing considerations and software can be found in Shmueli (2012[link]) and Shmueli (2013[link]).

1.6.2.3. Symmetry information from the structure solution

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It is also possible to obtain information on the symmetry of the crystal after structure solution. The latter is obtained either in space group P1 (i.e. no symmetry assumed) or in some other candidate space group. The analysis may take place either on the electron-density map, or on its interpretation in terms of atomic coordinates and atomic types (i.e. chemical elements). The analysis of the electron-density map has become increasingly popular with the advent of dual-space methods, first proposed in the charge-flipping algorithm by Oszlányi & Sütő (2004[link]), which solve structures in P1 by default. The analysis of the atomic coordinates and atomic types obtained from least-squares refinement in a candidate space group is used extensively in structure validation. Symmetry operations present in the structure solution but not in the candidate space group are sought.

An exhaustive search for symmetry operations is undertaken. However, those to be investigated may be very efficently limited by making use of knowledge of the highest point-group symmetry of the lattice compatible with the known cell dimensions of the crystal. It is well established that the point-group symmetry of any lattice is one of the following seven centrosymmetric point groups: [\overline{1}], 2/m, mmm, 4/mmm, [\overline{3}m], 6/mmm, [m\overline{3}m]. This point group is known as the holohedry of the lattice. The relationship between the symmetry operations of the space group and its holohedry is rather simple. A rotation or screw axis of symmetry in the crystal has as its counterpart a corresponding rotation axis of symmetry of the lattice and a mirror or glide plane in the crystal has as its counterpart a corresponding mirror plane in the lattice. The holohedry may be equal to or higher than the point group of the crystal. Hence, at least the rotational part of any space-group operation should have its counterpart in the symmetry of the lattice. If and when this rotational part is found by a systematic comparison either of the electron density or of the positions of the independent atoms of the solved structure, the location and intrinsic parts of the translation parts of the space-group operation can be easily completed.

Palatinus and van der Lee (2008[link]) describe their procedure in detail with useful examples. It uses the structure solution both in the form of an electron-density map and a set of phased structure factors obtained by Fourier transformation. No interpretation of the electron-density map in the form of atomic coordinates and chemical-element type is required. The algorithm of the procedure proceeds in the following steps:

  • (1) The lattice centring is determined by a search for strong peaks in the autocorrelation (self-convolution, Patterson) function of the electron density and the potential centring vectors are evaluated through a reciprocal-space R value.

  • (2) A complete list of possible symmetry operations compatible with the lattice is generated by searching for the invariance of the direct-space metric under potential symmetry operations.

  • (3) A figure of merit is then assigned to each symmetry operation evaluated from the convolution of the symmetry-transformed electron density with that of the structure solution. Those symmetry operations that have a good figure of merit are selected as belonging to the space group of the crystal structure.

  • (4) The space group is completed by group multiplication of the selected operations and then validated.

  • (5) The positions of the symmetry elements are shifted to those of a conventional setting for the space group.

Palatinus & van der Lee (2008[link]) report a very high success rate in the use of this algorithm. It is also a powerful technique to apply in structure validation.

Le Page's (1987[link]) pioneering software MISSYM for the detection of `missed' symmetry operations uses refined atomic coordinates, unit-cell dimensions and space group assigned from the crystal-structure solution. The algorithm follows all the principles described above in this section. In MISSYM, the metric symmetry is established as described in the first stage of Section 1.6.2.1[link]. The `missed' symmetry operations are those that are present in the arrangement of the atoms but are not part of the space group used for the structure refinement. Indeed, this procedure has its main applications in structure validation. The algorithm used in Le Page's software is also implemented in ADDSYM (Spek, 2003[link]). There are numerous reports of successful applications of this software in the literature.

1.6.2.4. Restrictions on space groups

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The values of certain chemical and physical properties of a bulk compound, or its crystals, have implications for the assignment of the space group of a crystal structure. In the chemical domain, notably in proteins and small-molecule natural products, information concerning the enantiomeric purity of the bulk compound or of its individual crystals is most useful. Further, all physical properties of a crystal are limited by the point group of the crystal structure in ways that depend on the individual nature of the physical property.

It is very well established that the crystal structure of an enantiomerically pure compound will be chiral (see Flack, 2003[link]). By an enantiomerically pure compound one means a compound whose molecules are all chiral and all these molecules possess the same chirality. The space group of a chiral crystal structure will only contain the following types of symmetry operation: translations, pure rotations and screw rotations. Inversion in a point, mirror reflection or rotoinversion do not occur in the space group of a chiral crystal structure. Taking all this together means that the crystal structure of an enantiomerically pure compound will show one of 65 space groups (known as the Sohncke space groups), all noncentrosymmetric, containing only translations, rotations and screw rotations. As a consequence, the point group of a chiral crystal structure is limited to the 11 point groups containing only pure rotations (i.e. 1, 2, 222, 4, 422, 3, 32, 6, 622, 23 and 432). Particular attention must be paid as to whether a measurement of enantiomeric purity of a compound applies to the bulk material or to the single crystal used for the diffraction experiment. Clearly, a compound whose bulk is enantiomerically pure will produce crystals which are enantiomerically pure. The converse is not necessarily true (i.e. enantiomerically pure crystals do not necessarily come from an enantiomerically pure bulk). For example, a bulk compound which is a racemate (i.e. an enantiomeric mixture containing 50% each of the opposite enantiomers) may produce either (a) crystals of the racemic compound (i.e. crystals containing 50% each of the opposite enantiomers) or (b) a racemic conglomerate (i.e. a mixture of enantiomerically pure crystals in a proportion of 50% of each pure enantiomer) or (c) some other rarer crystallization modes. Consequently, as part of a single-crystal structure analysis, it is highly recommended to make a measurement of the enantiomeric purity of the single crystal used for the diffraction experiment.

Much information on methods of establishing the enantiomeric purity of a compound can be found in a special issue of Chirality devoted to the determination of absolute configuration (Allenmark et al., 2007[link]). Measurements in the fluid state of optical activity, optical rotatory dispersion (ORD), circular dichroism (CD) and enantioselective chromatography are of prime importance. Many of these are sufficiently sensitive to be applicable not only to the bulk compound but also to the single crystal used for the diffraction experiment taken into solution. CD may also be applied in the solid state.

Many physical properties of a crystalline solid are anisotropic and the symmetry of a physical property of a crystal is limited both by the point-group symmetry of the crystal and by symmetries inherent to the physical property under study. For further information on this topic see Part 1[link] of Volume D (Authier et al., 2014[link]). Unfortunately, many of these physical properties are intrinsically centrosymmetric, so few of them are of use in distinguishing between the subgroups of a Laue group, a common problem in space-group determination. In Chapter 3.2 of the present volume, Hahn & Klapper show to which point groups a crystal must belong to be capable of displaying some of the principal physical properties of crystals (Table 3.2.2.1[link] ). Measurement of morphology, pyroelectricity, piezoelectricity, second harmonic generation and optical activity of a crystalline sample can be of use.

1.6.2.5. Pitfalls in space-group determination

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The methods described in Sections 1.6.2[link] and 1.6.5.1[link] rely on the crystal measured being a single-domain crystal, i.e. it should not be twinned. Nevertheless, some types of twin are easily identified at the measurement stage as they give rise to split reflections. Powerful data-reduction techniques may be applied to data from such crystals to produce a reasonably complete single-domain intensity data set. Consequently, the multi-domain twinned crystals that give rise to difficulties in space-group determination are those for which the reciprocal lattices of the individual domains overlap exactly without generating any splitting of the Bragg reflections. A study of the intensity data from such a crystal may display two anomalies. Firstly, the intensity distribution, as described and analysed in Section 1.6.2.2[link], will be broader than that of the monodomain crystal. Secondly, one may obtain a set of conditions for reflections that does not correspond to any entry in Section 1.6.4[link]. In this chapter we give no further information on the determination of the space group for such twinned crystals. For further information on this topic see Part 3[link] of Volume D (Boček et al., 2006[link]) and Chapter 1.3[link] on twinning in Volume C (Koch, 2006[link]). A supplement (Flack, 2015[link]) to the current section deals with the determination of the space group from twinned crystals and those displaying a specialized metric. However, it is apposite to note that the existence of twins with overlapping reciprocal lattices can be identified by recording atomic resolution transmission electron-microscope images.

In order to obtain reliable results from space-group determination, the coverage of the reciprocal space by the intensity measurements should be as complete as possible. One should attempt to attain full-sphere data coverage, i.e. a complete set of intensity measurements in the point group 1. All Friedel opposites should be measured. The validity and reliability of the intensity statistics described in Section 1.6.2.2[link] rest on a full coverage of reciprocal lattice. Any systematic omission by resolution, azimuth and declination, intensity etc. of part of the asymmetric region of the reciprocal lattice has an adverse effect. In particular, reflections of weak intensity should not be omitted or deleted.

There are a few other common difficulties in space-group determination due either to the nature of the crystal or the experimental setup:

  • (a) The crystal may display a pseudo-periodicity leading to systematic series of weak or very weak reflections that can be mistaken for systematic absences.

  • (b) The physical effect of multiple reflections can lead to diffraction intensity appearing at the place of systematic absences. However, the shape of these multiple-reflection intensities is usually much sharper than a normal Bragg reflection.

  • (c) Contamination of the incident radiation by a [\lambda/2] component may also cause intensity due to the 2h 2k 2l reflection to appear at the place of the hkl one. Kirschbaum et al. (1997)[link] and Macchi et al. (1998)[link] have studied this probem and describe ways of circumventing it.

1.6.3. Theoretical background of reflection conditions

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U. Shmuelia

We shall now examine the effect of the space-group symmetry on the structure-factor function. These effects are of importance in the determination of crystal symmetry. If [({\bi{W}},{\bi w})] is the matrix–column pair of a representative symmetry operation of the space group of the crystal, then, by definition[\rho({\bi x})=\rho({\bi{W}}{\bi x}+{\bi w}),\eqno(1.6.3.1)]where [\rho({\bi x})] is the value of the electron-density function at the point with coordinates [{\bi x}], [{\bi{W}}] is a matrix of proper or improper rotation and [{\bi w}] is a translation part (cf. Section 1.2.2.1[link] ). It is known that the electron-density function at the point [{\bi x}] is given by[\rho({\bi x})={{1}\over{V}}\sum_{{\bf h}}F({\bf h})\exp(-2\pi i{\bf h}{\bi x}),\eqno(1.6.3.2)]where, in this and the following equations, [{\bf h}] is the row matrix [(h\, k\, l)] and [{\bi x}] is a column matrix containing x, y and z in the first, second and third rows, respectively. Of course, [{\bf h}{\bi x}] is simply equivalent to [hx+ky+lz]. If we substitute (1.6.3.2)[link], with [{\bi x}] replaced by [({\bi{W}}{\bi x}+{\bi w})] in (1.6.3.1)[link] we obtain, after some calculation,[F({\bf h}{\bi{W}})=F({\bf h})\exp(-2\pi i{\bf h}{\bi w}).\eqno(1.6.3.3)]Equation (1.6.3.3)[link] is the fundamental relation between symmetry-related reflections (e.g. Waser, 1955[link]; Wells, 1965[link]; and Chapter 1.4[link] in Volume B). If we write [F({\bf h})=|F({\bf h})|\exp[i\varphi({\bf h})]], equation (1.6.3.3)[link] leads to the following relationships:[|F({\bf h}{\bi{W}})|=|F({\bf h})|\eqno(1.6.3.4)]and[\varphi({\bf h}{\bi{W}})=\varphi({\bf h})-2\pi{\bf h}{\bi w}.\eqno(1.6.3.5)]Equation (1.6.3.4)[link] indicates the equality of the intensities of truly symmetry-related reflections, while equation (1.6.3.5)[link] relates the phases of the corresponding structure factors. The latter equation is of major importance in direct methods of phase determination [e.g. Chapter 2.2[link] in Volume B (Giacovazzo, 2008[link])].

We can now approach the problem of systematically absent reflections, which are alternatively called the conditions for possible reflections.

The reflection h is general if its indices remain unchanged only under the identity operation of the point group of the diffraction pattern. I.e., if [{\bi{W}}] is the matrix of the identity operation of the point group, the relation [{\bf h}{\bi{W}}={\bf h}] holds true. So, if the reflection h is general, we must have [{\bi{W}}\equiv{\bi{I}}], where [{\bi{I}}] is the identity matrix and, obviously, [{\bf h}{\bi{I}}={\bf h}]. The operation [({\bi{I}},{\bi w})] can be a space-group symmetry operation only if w is a lattice vector. Let us denote it by [{\bi w}_{L}]. Equation (1.6.3.3)[link] then reduces to[F({\bf h})=F({\bf h})\exp(-2\pi i{\bf h}{\bi w}_{L})\eqno(1.6.3.6)]and [F({\bf h})] can be nonzero only if [\exp(-2\pi i{\bf h}{\bi w}_{L})=1]. This, in turn, is possible only if [{\bf h}{\bi w}_{L}] is an integer and leads to conditions depending on the lattice type. For example, if the components of [{\bi w}_{L}] are all integers, which is the case for a P-type lattice, the above condition is fulfilled for all h – the lattice type does not impose any restrictions. If the lattice is of type I, there are two lattice points in the unit cell, at say 0, 0, 0 and 1/2, 1/2, 1/2. The first of these does not lead to any restrictions on possible reflections. The second, however, requires that [\exp[-\pi i(h + k + l)]] be equal to unity. Since [\exp(\pi in)=(-1)^{n}], where n is an integer, the possible reflections from a crystal with an I-type lattice must have indices such that their sum is an even integer; if the sum of the indices is an odd integer, the reflection is systematically absent. In this way, we examine all lattice types for conditions of possible reflections (or systematic absences) and present the results in Table 1.6.3.1[link].

Table 1.6.3.1| top | pdf |
Effect of lattice type on conditions for possible reflections

Lattice type[{\bi w}^{T}_{L}][{\bf h}{\bi w}_{L}]Conditions for possible reflections
P (0, 0, 0) Integer None
A [(0, \textstyle{{1}\over{2}} ,\textstyle{{1}\over{2}})] [(k+l)/2] hkl: [k+l=2n]
B [(\textstyle{{1}\over{2}}, 0 ,\textstyle{{1}\over{2}})] [(h+l)/2] hkl: [h+l=2n]
C [(\textstyle{{1}\over{2}}, \textstyle{{1}\over{2}}, 0)] [(h+k)/2] hkl: [h+k=2n]
I [(\textstyle{{1}\over{2}}, \textstyle{{1}\over{2}}, \textstyle{{1}\over{2}})] [(h+k+l)/2] hkl: [h+k+l=2n]
F [(0, \textstyle{{1}\over{2}}, \textstyle{{1}\over{2}})] [(k+l)/2] h, k and l are all even or all odd (simultaneous fulfillment of the conditions for types A, B and C).
  [(\textstyle{{1}\over{2}}, 0, \textstyle{{1}\over{2}})] [(h+l)/2]
  [(\textstyle{{1}\over{2}}, \textstyle{{1}\over{2}}, 0)] [(h+k)/2]
Robv [(\textstyle{{2}\over{3}}, \textstyle{{1}\over{3}}, \textstyle{{1}\over{3}})] [(2h+k+l)/3] hkl: [-h+k+l=3n]
  [(\textstyle{{1}\over{3}}, \textstyle{{2}\over{3}}, \textstyle{{2}\over{3}})] [(h+2k+2l)/3] (triple hexagonal cell in obverse orientation)
Rrev [(\textstyle{{1}\over{3}}, \textstyle{{2}\over{3}}, \textstyle{{1}\over{3}})] [(h+2k+l)/3] hkl: [h-k+l=3n]
  [(\textstyle{{2}\over{3}}, \textstyle{{1}\over{3}}, \textstyle{{2}\over{3}})] [(2h+k+2l)/3] (triple hexagonal cell in reverse orientation)

The reflection h is special if it remains unchanged under at least one operation of the point group of the diffraction pattern in addition to its identity operation. I.e., the relation [{\bf h}{\bi{W}}={\bf h}] holds true for more than one operation of the point group. We shall now assume that the reflection h is special. By definition, this reflection remains invariant under more than one operation of the point group of the diffraction pattern. These operations form a subgroup of the point group of the diffraction pattern, known as the stabilizer (formerly called the isotropy subgroup) of the reflection h, and we denote it by the symbol [\cal{S}_{{\bf h}}]. For each space-group symmetry operation ([{\bi{W}},{\bi w})] where [{\bi{W}}] is the matrix of an element of [\cal{S}_{{\bf h}}] we must therefore have [{\bf h}{\bi{W}}={\bf h}]. Equation (1.6.3.3)[link] now reduces to[F({\bf h})=F({\bf h})\exp(-2\pi i{\bf h}{\bi w}).\eqno(1.6.3.7)]Of course, if [{\bi{W}}] represents the identity operation, w must be a lattice vector and the discussion summarized in Table 1.6.3.1[link] applies. We therefore require that [{\bi{W}}] is the matrix of an element of [\cal{S}_{{\bf h}}] other than the identity. [F({\bf h})] can be nonzero only if the exponential factor in (1.6.3.7)[link] equals unity. This, in turn, is possible only if hw is an integer.

Let us consider a monoclinic crystal with P-type lattice (i.e. with an mP-type Bravais lattice) and a c-glide reflection as an example. Assuming b perpendicular to the ac plane, the [({\bi{W}},{\bi w}]) representation of c is given by[c{:}\ \left[\pmatrix{1 & 0 & 0\cr 0 & \overline{1} & 0\cr 0 & 0 & 1}, \pmatrix{ 0\cr y\cr 1/2}\right].]

The indices of reflections that remain unchanged under the application of the mirror component of the glide-reflection operation must be h0l. The translation part of the c-glide-reflection operation has the form (0, y, 1/2), where y = 0 corresponds to the plane passing through the origin. Hence, for any value of y, the scalar product hw is l/2 and the necessary condition for a nonzero value of an h0l reflection is l = 2n, where n is an integer. Intensities of h0l reflections with odd l will be systematically absent.

Table 1.6.3.2[link] shows the effect of some glide reflections on reflection conditions.4

Table 1.6.3.2| top | pdf |
Effect of some glide reflections on conditions for possible reflections

Glide reflection[{\bi w}^{T}]hConditions for possible reflections
[a\perp[001]] (1/2, 0, z) (hk0) hk0: [h=2n]
[b\perp[001]] (0, 1/2, z) (hk0) hk0: [k=2n]
[n\perp[001]] (1/2, 1/2, z) (hk0) hk0: [h+k=2n]
[d\perp[001]] (1/4, ±1/4, z) (hk0) hk0: [h+k=4n] [(h,k=2n)]

Let us now assume a crystal with an mP-type Bravais lattice and a twofold screw axis taken as being parallel to b. The ([{\bi{W}},{\bi w})] representation of the corresponding screw rotation is given by[2_{1}{:}\ \left[\pmatrix{\overline{1} & 0 & 0\cr 0 & 1 & 0\cr 0 & 0 & \overline{1}}, \pmatrix{ x\cr 1/2\cr z }\right].]The diffraction indices that remain unchanged upon the application of the rotation part of [2_{1}] must be of the form (0k0). The translation part of the screw operation is of the form (x, 1/2, z), where the values of x and z depend on the location of the origin. Hence, for any values of x and z the scalar product hw is k/2 and the necessary condition for a nonzero value of a 0k0 reflection is k = 2n. 0k0 reflections with odd k will be systematically absent. A brief summary of the effects of various screw rotations on the conditions for possible reflections from the corresponding special subsets of hkl is given in Table 1.6.3.3[link]. Note, however, that while the presence of a twofold screw axis parallel to b ensures the condition 0k0: k = 2n, the actual observation of such a condition can be taken as an indication but not as absolute proof of the presence of a screw axis in the crystal.

Table 1.6.3.3| top | pdf |
Effect of some screw rotations on conditions for possible reflections

Screw rotation[{\bi w}^{T}]hConditions for possible reflections
[2_{1}\parallel [100]] (1/2, y, z) (h00) h00: [h=2n]
[2_{1}\parallel [010]] (x, 1/2, z) (0k0) 0k0: [k=2n]
[2_{1}\parallel [001]] (x, y, 1/2) (00l) 00l: [l=2n]
[2_{1}\parallel [110]] (1/2, 1/2, z) (hh0) None
[3_{1}\parallel [001]] (x, y, 1/3) (00l) 00l: [l=3n]
[3_{1}\parallel [111]] (1/3, 1/3, 1/3) (hhh) None
[4_{1}\parallel [001]] (x, y, 1/4) (00l) 00l: [l=4n]
[6_{1}\parallel [001]] (x, y, 1/6) (00l) 00l: [l=6n]

It is interesting to note that some diagonal screw axes do not give rise to conditions for possible reflections. For example, let [{\bi{W}}] be the matrix of a threefold rotation operation parallel to [111] and [{\bi w}^{T}] be given by (1/3, 1/3, 1/3). It is easy to show that the diffraction vector that remains unchanged when postmultiplied by [{\bi{W}}] has the form [{\bf h}=(h h h)] and, obviously, for such h and w, [{\bf h}{\bi w}=h]. Since this scalar product is an integer there are, according to equation (1.6.3.7)[link], no values of the index h for which the structure factor [F(hhh)] must be absent.

A short discussion of special reflection conditions

The conditions for possible reflections arising from lattice types, glide reflections and screw rotations are related to general equivalent positions and are known as general reflection conditions. There are also special or `extra' reflection conditions that arise from the presence of atoms in special positions. These conditions are observable if the atoms located in special positions are much heavier than the rest. The minimal special conditions are listed in the space-group tables in Chapter 2.3[link] . They can sometimes be understood if the geometry of a given specific site is examined. For example, Wyckoff position 4i in space group [P4_{2}22] (93) can host four atoms, at coordinates[4i{:}\  0,\textstyle{{1}\over{2}},z;\, \ \textstyle{{1}\over{2}},0,z+\textstyle{{1}\over{2}};\,\ 0,\textstyle{{1}\over{2}},\overline{z};\, \ \textstyle{{1}\over{2}},0,\overline{z}+\textstyle{{1}\over{2}}.]It is seen that the second and fourth coordinates are obtained from the first and third coordinates, respectively, upon the addition of the vector [t(\textstyle{{1}\over{2}}, \textstyle{{1}\over{2}}, \textstyle{{1}\over{2}})]. An additional I-centring is therefore present in this set of special positions. Hence, the special reflection condition for this set is hkl: [h+k+l=2n].

It should be pointed out, however, that only the general reflection conditions are used for a complete or partial determination of the space group and that the special reflection conditions only apply to spherical atoms. By the latter assumption we understand not only the assumption of spherical distribution of the atomic electron density but also isotropic displacement parameters of the equivalent atoms that belong to the set of corresponding special positions.

One method of finding the minimal special reflection conditions for a given set of special positions is the evaluation of the trigonometric structure factor for the set in question. For example, consider the Wyckoff position 4c of the space group Pbcm (57). The coordinates of the special equivalent positions are[4c{:}\ \ x,\textstyle{{1}\over{4}},0; \quad\overline{x},\textstyle{{3}\over{4}},\textstyle{{1}\over{2}}; \quad\overline{x},\textstyle{{3}\over{4}},0;\quad x,\textstyle{{1}\over{4}},\textstyle{{1}\over{2}}]and the corresponding trigonometric structure factor is[\eqalign{S({\bf h})&=\exp\left[2\pi i\left(hx+{{k}\over{4}}\right)\right]+\exp\left[2\pi i\left(-hx+{{3k}\over{4}}+{{l}\over{2}}\right)\right] \cr &\quad+\exp\left[2\pi i\left(-hx+{{3k}\over{4}}\right)\right]+\exp\left[2\pi i\left(hx+{{k}\over{4}}+{{l}\over{2}}\right)\right]. }]It can be easily shown that[S({\bf h})=2\cos\left[2\pi\left(hx+{{k}\over{4}}\right)\right][1+\exp(\pi il)]]and the last factor equals 2 for l even and equals zero for l odd. The special reflection condition is therefore: [hkl{:}\, l=2n].

Another approach is provided by considerations of the eigensymmetry group and the extraordinary orbits of the space group (see Section 1.4.4.4[link] ). We recall that the eigensymmetry group is a group of all the operations that leave the orbit of a point under the space group considered invariant, and the extraordinary orbit is associated with the eigensymmetry group that contains translations not present in the space group (see Chapter 1.4[link] ). In the above example the orbit is extraordinary, since its eigensymmetry group contains a translation corresponding to [\textstyle{{1}\over{2}}{\bf c}]. If this is taken as a basis vector, we have the Laue equation [\textstyle{{1}\over{2}} {\bf c}\cdot {\bf h}=l^{\prime}], where h is represented as a reciprocal-lattice vector and [l^{\prime}] is an integer which also equals l/2. But for l/2 to be an integer we must have even l. We again obtain the condition [hkl{:}\, l=2n].

These reflection conditions that are not related to space-group operations are given in Chapter 2.3[link] only for special positions. They may arise, however, also for different reasons. For example, a heavy atom at the origin of the space group [P2_{1}2_{1}2_{1}] would generate F-centring with corresponding apparent absences (cf. the special position 4a of the space group [Pbca] and the absences it generates).

We wish to point out that the most common `special-position absence' in molecular structures is due to a heavy atom at the origin of the space group [P2_{1}/c].

1.6.4. Tables of reflection conditions and possible space groups

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H. D. Flackb and U. Shmuelia

1.6.4.1. Introduction

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The primary order of presentation of these tables of reflection conditions of space groups is the Bravais lattice. This order has been chosen because cell reduction on unit-cell dimensions leads to the Bravais lattice as described as stage 1 in Section 1.6.2.1[link]. Within the space groups of a given Bravais lattice, the entries are arranged by Laue class, which may be obtained as described as stage 2 in Section 1.6.2.1[link]. As a consequence of these decisions about the way the tables are structured, in the hexagonal family one finds for the Bravais lattice hP that the Laue classes [\overline{3}], [\overline{3}m1], [\overline{3}1m], 6/m and 6/mmm are grouped together.

As an aid in the study of naturally occurring macromolecules and compounds made by enantioselective synthesis, the space groups of enantiomerically pure compounds (Sohncke space groups) are typeset in bold.

The tables show, on the left, sets of reflection conditions and, on the right, those space groups that are compatible with the given set of reflection conditions. The reflection conditions, e.g. h or k + l, are to be understood as [h=2n] or [k+l=2n], respectively. All of the space groups in each table correspond to the same Patterson symmetry, which is indicated in the table header. This makes for easy comparison with the entries for the individual space groups in Chapter 2.3[link] of this volume, in which the Patterson symmetry is also very clearly shown. All space groups with a conventional choice of unit cell are included in Tables 1.6.4.2–1.6.4.30. All alternative settings displayed in Chapter 2.3[link] are thus included. The following further alternative settings, not displayed in Chapter 2.3[link] , are also included: space group [Pb\overline{3}] (205) and all the space groups with an hR Bravais lattice in the reverse setting with hexagonal axes.

Table 1.6.2.1[link] gives some relevant statistics drawn from Tables 1.6.4.2–1.6.4.30. The total number of space-group settings mentioned in these tables is 416. This number is considerably larger than the 230 space-group types described in Part 2 of this volume. The following example shows why the tables include data for several descriptions of the space-group types. At the stage of space-group determination for a crystal in the crystal class mm2, it is not yet known whether the twofold rotation axis lies along a, b or c. Consequently, space groups based on the three point groups 2mm, m2m and mm2 need to be considered.

In some texts dealing with space-group determination, a `diffraction symbol' (sometimes also called an `extinction symbol') in the form of a Hermann–Mauguin space-group symbol is used as a shorthand code for the reflection conditions and Laue class. These symbols were introduced by Buerger (1935[link], 1942[link], 1969[link]) and a concise description is to be found in Looijenga-Vos & Buerger (2002)[link]. Nespolo et al. (2014)[link] use them.

1.6.4.2. Examples of the use of the tables

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  • (1) If the Bravais lattice is oI and the Laue class is mmm, Table 1.6.4.1[link] directs us to Table 1.6.4.11[link]. Given the observed reflection conditions[\eqalign{&hkl{:}\ h+k+l=2n,\quad\!\! 0kl{:}\ k=2n, l=2n,\quad\!\! h0l{:}\ h+l=2n,\cr &hk0{:}\ h+k=2n,\quad\!\! h00{:}\ h=2n,\quad\!\! 0k0{:}\ k=2n,\hfill \quad\!\! 00l{:}\ l=2n,}]it is seen from Table 1.6.4.11[link] that the possible settings of the space groups are: Ibm2 (46), Ic2m (46), Ibmm (74) and Icmm (74).

    Table 1.6.4.1| top | pdf |
    Summary of Tables 1.6.4.2–1.6.4.30

    Table No.Bravais latticeLaue classPatterson symmetryComment
    1.6.4.2[link] aP [\overline{1}] [P\overline{1}]  
    1.6.4.3[link] mP 2/m [P12/m1] Unique b
    1.6.4.4[link] mS (mC, mA, mI) 2/m [C12/m1], [A12/m1], [I12/m1] Unique b
    1.6.4.5[link] mP 2/m [P112/m] Unique c
    1.6.4.6[link] mS (mA, mB, mI) 2/m [A112/m], [B112/m], [I112/m] Unique c
    1.6.4.7[link] oP mmm Pmmm  
    1.6.4.8[link] oS (oC) mmm Cmmm  
    1.6.4.9[link] oS (oB) mmm Bmmm  
    1.6.4.10[link] oS (oA) mmm Ammm  
    1.6.4.11[link] oI mmm Immm  
    1.6.4.12[link] oF mmm Fmmm  
    1.6.4.13[link] tP [4/m] [P4/m]  
    1.6.4.14[link] tP [4/mmm] [P4/mmm]  
    1.6.4.15[link] tI [4/m] [I4/m]  
    1.6.4.16[link] tI [4/mmm] [I4/mmm]  
    1.6.4.17[link] hP [\overline{3}] [P\overline{3}]  
    1.6.4.18[link] hP [\overline{3}1m] and [\overline{3}m1] [P\overline{3}1m] and [P\overline{3}m1]  
    1.6.4.19[link] hP [6/m] [P6/m]  
    1.6.4.20[link] hP [6/mmm] [P6/mmm]  
    1.6.4.21[link] hR [\overline{3}] [R\overline{3}] Hexagonal axes
    1.6.4.22[link] hR [\overline{3}m] [R\overline{3}m] Hexagonal axes
    1.6.4.23[link] hR [\overline{3}] [R\overline{3}] Rhombohedral axes
    1.6.4.24[link] hR [\overline{3}m] [R\overline{3}m] Rhombohedral axes
    1.6.4.25[link] cP [m\overline{3}] [Pm\overline{3}]  
    1.6.4.26[link] cP [m\overline{3}m] [Pm\overline{3}m]  
    1.6.4.27[link] cI [m\overline{3}] [Im\overline{3}]  
    1.6.4.28[link] cI [m\overline{3}m] [Im\overline{3}m]  
    1.6.4.29[link] cF [m\overline{3}] [Fm\overline{3}]  
    1.6.4.30[link] cF [m\overline{3}m] [Fm\overline{3}m]  
  • (2) If the Bravais lattice is oP and the Laue class is mmm, Table 1.6.4.1[link] directs us to Table 1.6.4.7[link]. If there are no conditions on 0kl, the space groups P222 to Pmnn should be searched. If the condition is [0kl{:}\ k=2n] or [l=2n], the space groups Pbm2 to Pcnn should be searched. If the condition is [0kl{:}\ k+l= 2n], the space groups [Pnm2_{1}] to Pnnn should be searched.

  • (3) If the Bravais lattice is cP and the Laue class is [m\overline{3}], Table 1.6.4.1[link] directs us to Table 1.6.4.25[link]. If the conditions are [0kl{:}\ k=2n] and [h00{:}\ h=2n], it is readily seen that the space group is [Pa\overline{3}].

  • (4) If only the Bravais lattice is known or assumed, which is the case in powder-diffraction work (see Section 1.6.5.3[link]), all tables of this section corresponding to this Bravais lattice need to be consulted. For example, if it is known that the Bravais lattice is of type cP, Table 1.6.4.1[link] tells us that the possible Laue classes are [m\overline{3}] and [m\overline{3}m], and the possible space groups can be found in Tables 1.6.4.25[link] and 1.6.4.26[link], respectively. The appropriate reflection conditions are of course given in these tables. All relevant tables can thus be located with the aid of Table 1.6.4.1[link] if the Bravais lattice is known.[link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link]

    Table 1.6.4.2| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice aP and Laue class [\overline{1}]; Patterson symmetry [P\overline{1}]

    Reflection conditionsSpace groupNo.Space groupNo.
      P1 1 [P\overline{1}] 2

    Table 1.6.4.3| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice [mP] and Laue class 2/m; (monoclinic, unique axis b); Patterson symmetry [P12/m1]

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    h0l0klhk00k0h0000l
                P2 3 Pm 6 [P2/m] 10
                           
          k     P21 4 [P2_{1}/m] 11    
                           
    h       h   Pa 7 [P2/a] 13    
                           
    h     k h   [P2_{1}/a] 14        
                           
    l         l Pc 7 [P2/c] 13    
                           
    l     k   l [P2_{1}/c] 14        
                           
    h + l       h l Pn 7 [P2/n] 13    
                           
    h + l     k h l [P2_{1}/n] 14        

    Table 1.6.4.4| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice mS (mC, mA, mI) and Laue class 2/m (monoclinic, unique axis b); Patterson symmetry [C12/m1], [A12/m1], [I12/m1]

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    hklh0l0klhk00k0h0000l
    h + k h k h + k k h   C2 5 Cm 8 [C2/m] 12
                             
    h + k h, l k h + k k h l Cc 9 [C2/c] 15    
                             
    k + l l k + l k k   l A2 5 Am 8 [A2/m] 12
                             
    k + l h, l k + l k k h l An 9 [A2/n] 15    
                             
    h + k + l h + l k + l h + k k h l I2 5 Im 8 [I2/m] 12
                             
    h + k + l h, l k + l h + k k h l Ia 9 [I2/a] 15    

    Table 1.6.4.5| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice mP and Laue class 2/m (monoclinic, unique axis c); Patterson symmetry [P112/m]

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    h0l0klhk00k0h0000l
                P2 3 Pm 6 [P2/m] 10
                           
              l P21 4 [P2_{1}/m] 11    
                           
        h   h   Pa 7 [P2/a] 13    
                           
        h   h l [P2_{1}/a] 14        
                           
        k k     Pb 7 [P2/b] 13    
                           
        k k   l [P2_{1}/b] 14        
                           
        h + k k h   Pn 7 [P2/n] 13    
                           
        h + k k h l [P2_{1}/n] 14        

    Table 1.6.4.6| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice mS (mA, mB, mI) and Laue class 2/m (monoclinic, unique axis c); Patterson symmetry [A112/m], [B112/m1], [I112/m]

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    hklh0l0klhk00k0h0000l
    k + l l k + l k k   l A2 5 Am 8 [A2/m] 12
                             
    k + l l k + l h, k k h l Aa 9 [A2/a] 15    
                             
    h + l h + l l h   h l B2 5 Bm 8 [B2/m] 12
                             
    h + l h + l l h, k k h l Bn 9 [B2/n] 15    
                             
    h + k + l h + l k + l h + k k h l I2 5 Im 8 [I2/m] 12
                             
    h + k + l h + l k + l h, k k h l Ib 9 [I2/b] 15    

    Table 1.6.4.7| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice oP and Laue class mmm; Patterson symmetry [Pmmm]

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    0klh0lhk0h000k000l
                P222 16 [Pmm2] 25 [Pm2m] 25
                [P2mm] 25 Pmmm 47    
                           
              l P2221 17        
                           
            k   P2212 17        
                           
            k l P22121 18        
                           
          h     P2122 17        
                           
          h   l P21221 18        
                           
          h k   P21212 18        
                           
          h k l P212121 19        
                           
        h h     [P2_{1}ma] 26 [Pm2a] 28 Pmma 51
                           
        k   k   [Pm2_{1}b] 26 [P2mb] 28 Pmmb 51
                           
        h + k h k   [Pm2_{1}n] 31 [P2_{1}mn] 31 Pmmn 59
                           
      h   h     [P2_{1}am] 26 [Pma2] 28 Pmam 51
                           
      h h h     [P2aa] 27 Pmaa 49    
                           
      h k h k   [P2_{1}ab] 29 Pmab 57    
                           
      h h + k h k   [P2an] 30 Pman 53    
                           
      l       l [Pmc2_{1}] 26 [P2cm] 28 Pmcm 51
                           
      l h h   l [P2_{1}ca] 29 Pmca 57    
                           
      l k   k l [P2cb] 32 Pmcb 55    
                           
      l h + k h k l [P2_{1}cn] 33 Pmcn 62    
                           
      h + l   h   l [Pmn2_{1}] 31 [P2_{1}nm] 31 Pmnm 59
                           
      h + l h h   l [P2na] 30 Pmna 53    
                           
      h + l k h k l [P2_{1}nb] 33 Pmnb 62    
                           
      h + l h + k h k l [P2nn] 34 Pmnn 58    
                           
    k       k   [Pb2_{1}m] 26 [Pbm2] 28 Pbmm 51
                           
    k   h h k   [Pb2_{1}a] 29 Pbma 57    
                           
    k   k   k   [Pb2b] 27 Pbmb 49    
                           
    k   h + k h k   [Pb2n] 30 Pbmn 53    
                           
    k h   h k   [Pba2] 32 Pbam 55    
                           
    k h h h k   Pbaa 54        
                           
    k h k h k   Pbab 54        
                           
    k h h + k h k   Pban 50        
                           
    k l     k l [Pbc2_{1}] 29 Pbcm 57    
                           
    k l h h k l Pbca 61        
                           
    k l k   k l Pbcb 54        
                           
    k l h + k h k l Pbcn 60        
                           
    k h + l   h k l [Pbn2_{1}] 33 Pbnm 62    
                           
    k h + l h h k l Pbna 60        
                           
    k h + l k h k l Pbnb 56        
                           
    k h + l h + k h k l Pbnn 52        
                           
    l         l [Pcm2_{1}] 26 [Pc2m] 28 Pcmm 51
                           
    l   h h   l [Pc2a] 32 Pcma 55    
                           
    l   k   k l [Pc2_{1}b] 29 Pcmb 57    
                           
    l   h + k h k l [Pc2_{1}n] 33 Pcmn 62    
                           
    l h   h   l [Pca2_{1}] 29 Pcam 57    
                           
    l h h h   l Pcaa 54        
                           
    l h k h k l Pcab 61        
                           
    l h h + k h k l Pcan 60        
                           
    l l       l [Pcc2] 27 Pccm 49    
                           
    l l h h   l Pcca 54        
                           
    l l k   k l Pccb 54        
                           
    l l h + k h k l Pccn 56        
                           
    l h + l   h   l [Pcn2] 30 Pcnm 53    
                           
    l h + l h h   l Pcna 50        
                           
    l h + l k h k l [Pcnb] 60        
                           
    l h + l h + k h k l Pcnn 52        
                           
    k + l       k l [Pnm2_{1}] 31 [Pn2_{1}m] 31 Pnmm 59
                           
    k + l   h h k l [Pn2_{1}a] 33 Pnma 62    
                           
    k + l   k   k l [Pn2b] 30 Pnmb 53    
                           
    k + l   h + k h k l [Pn2n] 34 Pnmn 58    
                           
    k + l h   h k l [Pna2_{1}] 33 Pnam 62    
                           
    k + l h h h k l Pnaa 56        
                           
    k + l h k h k l Pnab 60        
                           
    k + l h h + k h k l Pnan 52        
                           
    k + l l     k l [Pnc2] 30 Pncm 53    
                           
    k + l l h h k l Pnca 60        
                           
    k + l l k   k l Pncb 50        
                           
    k + l l h + k h k l Pncn 52        
                           
    k + l h + l   h k l [Pnn2] 34 Pnnm 58    
                           
    k + l h + l h h k l Pnna 52        
                           
    k + l h + l k h k l Pnnb 52        
                           
    k + l h + l h + k h k l Pnnn 48        

    Table 1.6.4.8| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice oS (oC setting) and Laue class mmm; Patterson symmetry Cmmm

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    hkl0klh0lhk0h000k000l
    h + k k h h + k h k   C222 21 [Cmm2] 35 [Cm2m] 38
                  [C2mm] 38 Cmmm 65    
                             
    h + k k h h + k h k l C2221 20        
                             
    h + k k h h, k h k   [Cm2e] 39 [C2me] 39 Cmme 67
                             
    h + k k h, l h + k h k l [Cmc2_{1}] 36 [C2cm] 40 Cmcm 63
                             
    h + k k h, l h, k h k l [C2ce] 41 Cmce 64    
                             
    h + k k, l h h + k h k l [Ccm2_{1}] 36 [Cc2m] 40 Ccmm 63
                             
    h + k k, l h h, k h k l [Cc2e] 41 Ccme 64    
                             
    h + k k, l h, l h + k h k l [Ccc2] 37 Cccm 66    
                             
    h + k k, l h, l h, k h k l Ccce 68        

    Table 1.6.4.9| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice oS (oB setting) and Laue class mmm; Patterson symmetry Bmmm

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    hkl0klh0lhk0h000k000l
    h + l l h + l h h   l B222 21 [Bm2m] 35 [Bmm2] 38
                  [B2mm] 38 Bmmm 65    
                             
    h + l l h + l h h k l B2212 20        
                             
    h + l l h + l h, k h k l [Bm2_{1}b] 36 [B2mb] 40 Bmmb 63
                             
    h + l l h, l h h   l [Bme2] 39 [B2em] 39 Bmem 67
                             
    h + l l h, l h, k h k l [B2eb] 41 Bmeb 64    
                             
    h + l k, l h + l h h k l [Bb2_{1}m] 36 [Bbm2] 40 Bbmm 63
                             
    h + l k, l h + l h, k h k l [Bb2b] 37 Bbmb 66    
                             
    h + l k, l h, l h h k l [Bbe2] 41 Bbem 64    
                             
    h + l k, l h, l h, k h k l Bbeb 68        

    Table 1.6.4.10| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice oS (oA setting) and Laue class mmm; Patterson symmetry Ammm

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    hkl0klh0lhk0h000k000l
    k + l k + l l k   k l A222 21 [A2mm] 35 [Am2m] 38
                  [Amm2] 38 Ammm 65    
                             
    k + l k + l l k h k l A2122 20        
                             
    k + l k + l l h, k h k l [A2_{1}ma] 36 [Am2a] 40 Amma 63
                             
    k + l k + l h, l k h k l [A2_{1}am] 36 [Ama2] 40 Amam 63
                             
    k + l k + l h, l h, k h k l [A2aa] 37 Amaa 66    
                             
    k + l k, l l k   k l [Aem2] 39 [Ae2m] 39 Aemm 67
                             
    k + l k, l l h, k h k l [Ae2a] 41 Aema 64    
                             
    k + l k, l h, l k h k l [Aea2] 41 Aeam 64    
                             
    k + l k, l h, l h, k h k l Aeaa 68        

    Table 1.6.4.11| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice oI and Laue class mmm; Patterson symmetry Immm

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    hkl0klh0lhk0h000k000l
    h + k + l k + l h + l h + k h k l I222 23 I212121 24 [Imm2] 44
                  [Im2m] 44 [I2mm] 44 Immm 71
                             
    h + k + l k + l h + l h, k h k l [Im2a] 46 [I2mb] 46 Imma 74
                  [Immb] 74        
                             
    h + k + l k + l h, l h + k h k l [Ima2] 46 [I2cm] 46 Imam 74
                  Imcm 74        
                             
    h + k + l k + l h, l h, k h k l [I2cb] 45 [Imcb] 72    
                             
    h + k + l k, l h + l h + k h k l [Ibm2] 46 [Ic2m] 46 Ibmm 74
                  Icmm 74        
                             
    h + k + l k, l h + l h, k h k l [Ic2a] 45 Icma 72    
                             
    h + k + l k, l h, l h + k h k l [Iba2] 45 Ibam 72    
                             
    h + k + l k, l h, l h, k h k l Ibca 73 Icab 73    

    Table 1.6.4.12| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice oF and Laue class mmm; Patterson symmetry Fmmm

    Reflection conditionsSpace groupNo.
    hkl0klh0lhk0h000k000l
    h + k, h + l, k + l k, l h, l h, k h k l F222 22
                  [Fmm2] 42
                  [Fm2m] 42
                  [F2mm] 42
                  Fmmm 69
                     
    h + k, h + l, k + l k, l [h+l=4n;h,l] [h+k=4n;h,k] [h=4n] [k=4n] [l=4n] [F2dd] 43
                     
    h + k, h + l, k + l [k+l=4n;k,l] h, l [h+k=4n;h,k] [h=4n] [k=4n] [l=4n] [Fd2d] 43
                     
    h + k, h + l, k + l [k+l=4n;k,l] [h+l=4n;h,l] h, k [h=4n] [k=4n] [l=4n] [Fdd2] 43
                     
    h + k, h + l, k + l [k+l=4n;k,l] [h+l=4n;h,l] [h+k=4n;h,k] [h=4n] [k=4n] [l=4n] Fddd 70

    Table 1.6.4.13| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice tP and Laue class 4/m; hk are permutable; Patterson symmetry P4/m

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    hk00kl[h\,{\pm} h\, l]00lh00
              P4 75 [P\overline{4}] 81 [P4/m] 83
                         
          l   P42 77 [P4_{2}/m] 84    
                         
          [l=4n]   P41 76 P43 78    
                         
    h + k       h [P4/n] 85        
                         
    h + k     l h [P4_{2}/n] 86        

    Table 1.6.4.14| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice tP and Laue class 4/mmm; hk are permutable; Patterson symmetry P4/mmm

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    hk00kl[h\,{\pm} h\, l]00lh00
              P422 89 [P4mm] 99 [P\overline{4}2m] 111
              [P\overline{4}m2] 115 [P4/mmm] 123    
                         
            h P4212 90 [P\overline{4}2_{1}m] 113    
                         
          l   P4222 93        
                         
          l h P42212 94        
                         
          [l=4n]   P4122 91 P4322 95    
                         
          [l=4n] h P41212 92 P43212 96    
                         
        l l   [P4_{2}mc] 105 [P\overline{4}2c] 112 [P4_{2}/mmc] 131
                         
        l l h [P\overline{4}2_{1}c] 114        
                         
      k     h [P4bm] 100 [P\overline{4}b2] 117 [P4/mbm] 127
                         
      k l l h [P4_{2}bc] 106 [P4_{2}/mbc] 135    
                         
      l   l   [P4_{2}cm] 101 [P\overline{4}c2] 116 [P4_{2}/mcm] 132
                         
      l l l   [P4cc] 103 [P4/mcc] 124    
                         
      k + l   l h [P4_{2}nm] 102 [P\overline{4}n2] 118 [P4_{2}/mnm] 136
                         
      k + l l l h [P4nc] 104 [P4/mnc] 128    
                         
    h + k       h [P4/nmm] 129        
                         
    h + k   l l h [P4_{2}/nmc] 137        
                         
    h + k k     h [P4/nbm] 125        
                         
    h + k k l l h [P4_{2}/nbc] 133        
                         
    h + k l   l h [P4_{2}/ncm] 138        
                         
    h + k l l l h [P4/ncc] 130        
                         
    h + k k + l   l h [P4_{2}/nnm] 134        
                         
    h + k k + l l l h [P4/nnc] 126        

    Table 1.6.4.15| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice tI and Laue class 4/m; hk are permutable; Patterson symmetry I4/m

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    hklhk00kl[h\,{\pm} h\, l]00lh00[h\,{\pm} h\, 0]
    h + k + l h + k k + l l l h   I4 79 [I\overline{4}] 82 [I4/m] 87
                             
    h + k + l h + k k + l l [l=4n] h   I41 80        
                             
    h + k + l h, k k + l l [l=4n] h h [I4_{1}/a] 88        

    Table 1.6.4.16| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice tI and Laue class 4/mmm; hk are permutable; Patterson symmetry I4/mmm

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    hklhk00kl[h\,{\pm} h\, l]00lh00[h\,{\pm} h\, 0]
    h + k + l h + k k + l l l h   I422 97 [I4mm] 107 [I\overline{4}m2] 119
                  [I\overline{4}2m] 121 [I4/mmm] 139    
                             
    h + k + l h + k k + l l [l=4n] h   I4122 98        
                             
    h + k + l h + k k + l [2h+l=4n] [l=4n] h h [I4_{1}md] 109 [I\overline{4}2d] 122    
                             
    h + k + l h + k k, l l l h   [I4cm] 108 [I\overline{4}c2] 120 [I4/mcm] 140
                             
    h + k + l h + k k, l [2h+l=4n] [l=4n] h h [I4_{1}cd] 110        
                             
    h + k + l h, k k + l [2h+l=4n] [l=4n] h h [I4_{1}/amd] 141        
                             
    h + k + l h, k k, l [2h+l=4n] [l=4n] h h [I4_{1}/acd] 142        

    Table 1.6.4.17| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice hP and Laue class [\overline{3}]; hki are permutable; Patterson symmetry [P\overline{3}]

    Reflection conditionsSpace groupNo.Space groupNo.
    [hh\overline{2h}l][h\overline{h}0l][000l]
          P3 143 [P\overline{3}] 147
                 
        [l=3n] P31 144 P32 145

    Table 1.6.4.18| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice hP and Laue classes [\overline{3}1m] and [\overline{3}m1]; hki are permutable; Patterson symmetry [P\overline{3}1m] and [P\overline{3}m1]

    Reflection conditionsClass [\overline{3}1m]Class [\overline{3}m1]
    [hh\overline{2h}l][h\overline{h}0l][000l]Space groupNo.Space groupNo.
          P312 149 P321 150
          [P31m] 157 [P3m1] 156
          [P\overline{3}1m] 162 [P\overline{3}m1] 164
                 
        [l=3n] P3112 151 P3121 152
          P3212 153 P3221 154
                 
    l   l [P31c] 159    
          [P\overline{3}1c] 163    
                 
      l l     [P3c1] 158
              [P\overline{3}c1] 165

    Table 1.6.4.19| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice hP and Laue class 6/m; hki are permutable; Patterson symmetry P6/m

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    [hh\overline{2h}l][h\overline{h}0l][000l]
          P6 168 [P\overline{6}] 174 [P6/m] 175
                     
        l P63 173 [P6_{3}/m] 176    
                     
        [l=3n] P62 171 P64 172    
                     
        [l=6n] P61 169 P65 170    

    Table 1.6.4.20| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice hP and Laue class 6/mmm; hki are permutable; Patterson symmetry P6/mmm

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    [hh\overline{2h}l][h\overline{h}0l][000l]
          P622 177 [P6mm] 183 [P\overline{6}m2] 187
          [P\overline{6}2m] 189 [P6/mmm] 191    
                     
        l P6322 182        
                     
        [l=3n] P6222 180 P6422 181    
                     
        [l=6n] P6122 178 P6522 179    
                     
    l   l [P6_{3}mc] 186 [P\overline{6}2c] 190 [P6_{3}/mmc] 194
                     
      l l [P6_{3}cm] 185 [P\overline{6}c2] 188 [P6_{3}/mcm] 193
                     
    l l l [P6cc] 184 [P6 /mcc] 192    

    Table 1.6.4.21| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice hR and Laue class [\overline{3}] (hexagonal axes); hki are permutable; Patterson symmetry [R\overline{3}]; Ov = obverse setting; Rv = reverse setting

    Reflection conditionsSpace groupNo.Space groupNo. 
    [hkil][hki0][hh\overline{2h}l][h\overline{h}0l][000l][h\overline{h}00]
    [-h+k+l=3n] [-h+k=3n] [l=3n] [h+l=3n] [l=3n] [h=3n] R3 146 [R\overline{3}] 148 Ov
                         
    [h-k+l=3n] [h-k=3n] [l=3n] [-h+l=3n] [l=3n] [h=3n] R3 146 [R\overline{3}] 148 Rv

    Table 1.6.4.22| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice hR and Laue class [\overline{3}m] (hexagonal axes); hki are permutable; Patterson symmetry [R\overline{3}m]; Ov = obverse setting; Rv = reverse setting

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo. 
    [hkil][hki0][hh\overline{2h}l][h\overline{h}0l][000l][h\overline{h}00]
    [-h+k+l=3n] [-h+k=3n] [l=3n] [h+l=3n] [l=3n] [h=3n] R32 155 [R3m] 160 [R\overline{3}m] 166 Ov
                             
    [-h+k+l=3n] [-h+k=3n] [l=3n] [h+l=3n], [l=2m] [l=6n] [h=3n] [R3c] 161 [R\overline{3}c] 167     Ov
                             
    [h-k+l=3n] [h-k=3n] [l=3n] [-h+l=3n] [l=3n] [h=3n] R32 155 [R3m] 160 [R\overline{3}m] 166 Rv
                             
    [h-k+l=3n] [h-k=3n] [l=3n] [-h+l=3n,l=2m] [l=6n] [h=3n] [R3c] 161 [R\overline{3}c] 167     Rv

    Table 1.6.4.23| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice hR and Laue class [\overline{3}] (rhombohedral axes); hkl are permutable; Patterson symmetry [R\overline{3}]

    Reflection conditionsSpace groupNo.Space groupNo.
    [hhl][hhh]
        R3 146 [R\overline{3}] 148

    Table 1.6.4.24| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice hR and Laue class [\overline{3}m] (rhombohedral axes); hkl are permutable; Patterson symmetry [R\overline{3}m]

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    [hhl][hhh]
        R32 155 [R3m] 160 [R\overline{3}m] 166
                   
    l h [R3c] 161 [R\overline{3}c] 167    

    Table 1.6.4.25| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice cP and Laue class [m\overline{3}]; hkl are cyclically permutable; Patterson symmetry [Pm\overline{3}]

    Reflection conditionsSpace groupNo.Space groupNo.
    0kl[h\,{\pm} h\, l]h00
          P23 195 [Pm\overline{3}] 200
                 
        h P213 198    
                 
    k   h [Pa\overline{3}] 205    
                 
    l   h [Pb\overline{3}] 205    
                 
    k + l   h [Pn\overline{3}] 201    

    Table 1.6.4.26| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice cP and Laue class [m\overline{3}m]; hkl are permutable; Patterson symmetry [Pm\overline{3}m]

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    0kl[h\,{\pm} h\, l]h00
          P432 207 [P\overline{4}3m] 215 [Pm\overline{3}m] 221
                     
        h P4232 208        
                     
        [h=4n] P4332 212 P4132 213    
                     
      l h [P\overline{4}3n] 218 [Pm\overline{3}n] 223    
                     
    k + l   h [Pn\overline{3}m] 224        
                     
    k + l l h [Pn\overline{3}n] 222        

    Table 1.6.4.27| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice cI and Laue class [m\overline{3}]; hkl are cyclically permutable; Patterson symmetry [Im\overline{3}]

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    hkl0kl[h\,{\pm} h\, l]h00
    h + k + l k + l l h I23 197 I213 199 [Im\overline{3}] 204
                       
    h + k + l k, l l h [Ia\overline{3}] 206        

    Table 1.6.4.28| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice cI and Laue class [m\overline{3}m]; hkl are permutable; Patterson symmetry [Im\overline{3}m]

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    hkl0kl[h\,{\pm} h\, l]h00
    h + k + l k + l l h I432 211 [I\overline{4}3m] 217 [Im\overline{3}m] 229
                       
    h + k + l k + l l [h=4n] I4132 214        
                       
    h + k + l k + l [2h+l=4n] [h=4n] [I\overline{4}3d] 220        
                       
    h + k + l k, l [2h+l=4n] [h=4n] [Ia\overline{3}d] 230        

    Table 1.6.4.29| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice cF and Laue class [m\overline{3}]; hkl are cyclically permutable; Patterson symmetry [Fm\overline{3}]

    Reflection conditionsSpace groupNo.Space groupNo.
    hkl0kl[h\,{\pm} h\, l]h00
    [h+k,h+l,k+l] k, l h + l h F23 196 [Fm\overline{3}] 202
                   
    [h+k,h+l,k+l] [k+l=4n;k,l] h + l [h=4n] [Fd\overline{3}] 203    

    Table 1.6.4.30| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice cF and Laue class [m\overline{3}m]; hkl are permutable; Patterson symmetry [Fm\overline{3}m]

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    hkl0kl[h\pm h l]h00
    [h+k,h+l,k+l] k, l h + l h F432 [209] [F\overline{4}3m] 216 [Fm\overline{3}m] 225
                       
    [h+k,h+l,k+l] k, l h + l [h=4n] F4132 210        
                       
    [h+k,h+l,k+l] k, l h, l h [F\overline{4}3c] 219 [Fm\overline{3}c] 226    
                       
    [h+k,h+l,k+l] [k+l=4n;k,l] h + l [h=4n] [Fd\overline{3}m] 227        
                       
    [h+k,h+l,k+l] [k+l=4n;k,l] h, l [h=4n] [Fd\overline{3}c] 228        

1.6.5. Specialized methods of space-group determination

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H. D. Flackb

1.6.5.1. Applications of resonant scattering to symmetry determination

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1.6.5.1.1. Introduction

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In small-molecule crystallography, it has been customary in crystal-structure analysis to make no use of the contribution of resonant scattering (otherwise called anomalous scattering and in older literature anomalous dispersion) other than in the specific area of absolute-structure and absolute-configuration determination. One may trace the causes of this situation to the weakness of the resonant-scattering contribution, to the high cost in time and labour of collecting intensity data sets containing measurements of all Friedel opposites and for a lack of any perceived or real need for the additional information that might be obtained from the effects of resonant scattering.

On the experimental side, the turning point came with the widespread distribution of area detectors for small-molecule crystallography, giving the potential to measure, at no extra cost, full-sphere data sets leading to the intensity differences between Friedel opposites hkl and [\overline{h}\overline{k}\overline{l}]. In 2015, the new methods of data analysis briefly presented here are in the stage of development and have not yet enjoyed widespread distribution, use and acceptance by the community. Flack et al. (2011[link]) and Parsons et al. (2012[link]) give detailed information on these calculations.

1.6.5.1.2. Status of centrosymmetry and resonant scattering

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The basic starting point in this analysis is the following linear transformation of [|F(hkl)|^{2}] and [|F(\overline{h}\overline{k}\overline{l})|^{2}], applicable to both observed and model values, to give the average (A) and difference (D) intensities:[\eqalign{A(hkl) &=\textstyle{{1}\over{2}}[|F(hkl)|^{2} + |F(\overline{h}\overline{k}\overline{l})|^{2}] ,\cr D(hkl) &=|F(hkl)|^{2}-|F(\overline{h}\overline{k}\overline{l})|^{2}.}]In equation (1.6.2.1)[link], [A(hkl)] was denoted by [|F_{\rm av}(hkl)|^{2}]. The expression for [D(hkl)] corresponding to that for [A(hkl)] given in equation (1.6.2.1)[link] and using the same nomenclature is[D({\bf h})=\textstyle\sum\limits_{i,j}[(f_{i}+f_{i}^{\prime})f_{j}^{\prime\prime}-(f_{j}+f_{j}^{\prime}) f_{i}^{\prime\prime}]\sin[2\pi{\bf h}({\bi r}_{i}-{\bi r}_{j})].]In general [|D(hkl)|] is small compared to [A(hkl)]. A compound with an appreciable resonant-scattering contribution has [|D(hkl)|] [\approx 0.01A(hkl)], whereas a compound with a small resonant-scattering contribution has [|D(hkl)| \approx 0.0001A(hkl)]. For centric reflections, [D_{\rm model} = 0], and so the values of [D_{\rm obs}(hkl)] of these are entirely due to random uncertainties and systematic errors in the intensity measurements. [D_{\rm obs}(hkl)] of acentric reflections contains contributions both from the random uncertainties and the systematic errors of the data measurements, and from the differences between [|F(hkl)|^{2}] and [|F(\overline{h}\overline{k}\overline{l})|^{2}] which arise through the effect of resonant scattering. A slight experimental limitation is that a data set of intensities needs to contain both reflections hkl and [\overline{h}\overline{k}\overline{l}] in order to obtain [A_{\rm obs}(hkl)] and [D_{\rm obs}(hkl)].

The Bijvoet ratio, defined by[\chi = {{\langle D^{2} \rangle^{1/2}}\over{\langle A \rangle}},]is the ratio of the root-mean-square value of D to the mean value of A. In a structure analysis, two independent estimates of the Bijvoet ratio are available and their comparison leads to useful information as to whether the crystal structure is centrosymmetric or not.

The first estimate arises from considerations of intensity statistics leading to the definition of the Bijvoet ratio as a value called Friedifstat, whose functional form was derived by Flack & Shmueli (2007[link]) and Shmueli & Flack (2009[link]). One needs only to know the chemical composition of the compound and the wavelength of the X-radiation to calculate Friedifstat using various available software.

The second estimate of the Bijvoet ratio, Friedifobs, is obtained from the observed diffraction intensities. One problematic point in the evaluation of Friedifobs arises because A and D do not have the same dependence on [\sin\theta/\lambda] and it is necessary to eliminate this difference as far as possible. A second problematic point in the calculation is to make sure that only acentric reflections of any of the noncentrosymmetric point groups in the chosen Laue class are selected for the calculation of Friedifobs. In this way one is sure that if the point group of the crystal is centrosymmetric, all of the chosen reflections are centric, and if the point group of the crystal is noncentrosymmetric, all of the chosen reflections are acentric. The necessary selection is achieved by taking only those reflections that are general in the Laue group. To date (2015), the calculation of Friedifobs is not available in distributed software. On comparison of Friedifstat with Friedifobs, one is able to state with some confidence that:

  • (1) if Friedifobs is much lower than Friedifstat, then the crystal structure is either centrosymmetric, and random uncertainties and systematic errors in the data set are minor, or noncentrosymmetric with the crystal twinned by inversion in a proportion close to 50:50;

  • (2) if Friedifobs is close in value to Friedifstat, then the crystal is probably noncentrosymmetric and random uncertainties and systematic errors in the data set are minor. However, data from a centrosymmetric crystal with large random uncertainties and systematic errors may also produce this result; and

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

Example 1

The crystal of compound Ex1 (Udupa & Krebs, 1979[link]) is known to be centrosymmetric (space group [P2_{1}/c]) and has a significant resonant-scattering contribution, Friedifstat = 498 and Friedifobs = 164. The comparison of Friedifstat and Friedifobs indicates that the crystal structure is centrosymmetric.

Example 2

The crystal of compound Ex2, potassium hydrogen (2R,3R) tartrate, is known to be enantiomerically pure and appears in space group [P2_{1}2_{1}2_{1}]. The value of Friedifobs is 217 compared to a Friedifstat value of 174. The agreement is good and allows the deduction that the crystal is neither centrosymmetric, nor twinned by inversion in a proportion near to 50:50, nor that the data set is unsatisfactorily dominated by random uncertainty and systematic error.

Example 3

The crystals of compound Ex3 (Zhu & Jiang, 2007[link]) occur in Laue group [\overline{1}]. One finds Friedifstat = 70 and Friedifobs = 499. The huge discrepency between the two shows that the observed values of D are dominated by random uncertainty and systematic error.

1.6.5.1.3. Resolution of noncentrosymmetric ambiguities

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It was shown in Section 1.6.5.1.2[link] that under certain circumstances it is possible to determine whether or not the space group of the crystal investigated is centrosymmetric. Suppose that the space group was found to be noncentrosymmetric. In each Laue class, there is one centrosymmetric point group and one or more noncentrosymmetric point groups. For example, in the Laue class mmm we need to distinguish between the point groups 222, 2mm, m2m and mm2, and of course between the space groups based on them. We shall show that it is possible in practice to distinguish between these noncentrosymmetric point groups using intensity differences between Friedel opposites caused by resonant scattering.

An excellent intensity data set from a crystal (Ex2 above) of potassium hydrogen (2R, 3R) tartrate, measured with a wavelength of 0.7469 Å at 100 K, was used. The Laue group was assumed to be mmm. The raw data set was initially merged and averaged in point group 1 and all special reflections of the Laue group mmm (i.e. 0kl, h0l, hk0, h00, 0k0, 00l) were set aside. The remaining data were organized into sets of reflections symmetry-equivalent under the Laue group mmm, and only those sets (589 in all) containing all 8 of the mmm-symmetry-equivalent reflections were retained. Each of these sets provides 4 [A_{\rm obs}] and 4 [D_{\rm obs}] values which can be used to calculate Rmerge values appropriate to the five point groups in the Laue class mmm. The results are given in Table 1.6.5.1[link]. The value of 100% for Rmerge in a centrosymmetric point group, such as mmm or 2/m, arises by definition and not by coincidence. The [R_{D}] of the true point group has the lowest value, which is noticeably different from the other choices of point group.

Table 1.6.5.1| top | pdf |
Rmerge values for Ex2 for the 589 sets of general reflections of mmm which have all eight measurements in the set

Rmerge (%)mmm2mmm2mmm2222
[R_{A}] 1.30 1.30 1.30 1.30 1.30
[R_{D}] 100.0 254.4 235.7 258.1 82.9

The cryst