feature articles
Application of Pattersonfunction
to materials characterization^{a}Institut de Ciència de Materials de Barcelona, CSIC, Campus de la UAB, Bellaterra, Catalonia 08193, Spain
^{*}Correspondence email: jordi.rius@icmab.es
The aim of this article is a general description of the socalled Pattersonfunction Acta Crystallographica Section A. The common feature of these variants of is the introduction of the experimental intensities in the form of the Fourier coefficients of originfree Pattersontype functions, which allows the active use of both strong and weak reflections. The different optimization algorithms are discussed and their performances compared. This review focuses not only on those PFDM applications related to powder diffraction data but also on some recent results obtained with electron diffraction tomography data.
(PFDM), from their origin to their present state. It covers a 20year period of methodological contributions to solution, most of them published inKeywords: direct methods; PFDM; δ recycling; SFFT; STF; clusterbased DM; powder diffraction; ab initio structure solution; precession electron diffraction; electron diffraction tomography.
1. Introduction
Pattersonfunction ), the fact that PFDM lie halfway between traditional DM and Patterson deconvolution methods is surely one of the reasons why they are not as popular as other structure solution methods. The aim of the present article is to provide a comprehensive description of the advances in PFDM during the last 20 years and, at the same time, to introduce a rational classification and consistent nomenclature for their different variants. This clarification should help to increase their dissemination and to promote their wider use. PFDM are extremely simple both theoretically and computationally, and are especially well suited to such problems where not only the strong but also the weak intensities are accessible from the experiment. This comprises most applications to materials science dealing with crystalline matter. Although the present contribution focuses on the phasing of powder Xray diffraction (PD) and electron diffraction (ED) data of inorganic materials, most of the results can be applied to any kind of material.
(PFDM) are those (DM) extracting the phase information directly from the nonorigin part of the experimental Pattersontype function. Although the first method of this category was described quite early by Rius (19932. Pattersonfunction based on ρ^{2}
Before starting with the description of PFDM, a short introduction to the quantities involved in their definition is in order. In this review, for simplicity, an equalatom P1 with N atoms in the is assumed. In addition, bold letters denote complex or vector quantities, while standard text indicates the corresponding moduli (amplitudes). The observed quantities are the normalized E amplitudes, i.e. the F amplitudes corrected for falloff in sin θ/λ (mainly due to the atomic form factor evolution and to the atomic thermal vibration; θ is half the diffraction angle and λ is the incident wavelength). For an arbitrary H reflection, the corresponding E amplitude is given by
belonging toand can be derived from the measured intensity I and the average intensity in the corresponding reciprocalspace shell, 〈I〉_{shell}. A Fourier synthesis with the E values as Fourier coefficients yields the sharpened electron density distribution (ρ) of the crystal. The E values are complex quantities with their amplitudes known but with their associated phases, φ, lost during the diffraction experiment. Especially relevant for the development of DM was the derivation of the probability distribution of the (s.f.) amplitudes by Wilson (1949), in which he assumed that the atomic positions were random variables with uniform distribution throughout the Written in terms of E, the probability distribution is (for P1 symmetry)
with the moments of P_{1}(E) being 〈E^{2}〉 = 1 and 〈E〉 = 0.9, and with associated variance
The basic assumption made, i.e. that all points in the have the same probability of hosting an atom, constitutes the `randomness' condition. An important property of equation (2) is that P_{1}(E) is independent of the number of atoms N in the Physically, the E values represent the amplitudes of a hypothetical consisting of point atoms with a scattering power equal to 1/(N)^{1/2}. Consequently, the amplitude of the s.f. of ρ^{2}, G = Gexp(iψ), is given by the simple relationship
In view of equation (4), G can also be considered experimentally accessible, so that both E and G can be used interchangeably.
2.1. The calculated amplitudes
If Φ represents the subset of refined phases belonging to the h reflections with large E values, then the G(Φ) amplitudes of the squared structure can be expressed in terms of the Fourier coefficients Eexp(iφ) by Fourier transforming ρ^{2}(Φ) = ρ(Φ)·ρ(Φ) and posterior multiplication with exp(iψ_{−H}), i.e. by means of the summation
with
The equalpeak condition is implicit in the squaring operation, whereas positivity is forced if ψ_{h} and φ_{h} are equated (only for strong reflections), i.e. by assuming that ρ^{2}(Φ) is directly proportional to ρ(Φ) (Sayre, 1952). However, the randomness condition is not included in the squaring operation and hence will depend on each particular phasing method. Traditional DM procedures were not especially robust regarding this condition, as proved by the frequently occurring uraniumatom solution. For a long time, this solution represented a serious DM limitation and it is characterized by the appearance of an outstanding strong peak in the Fourier map. Although the equalpeak and positivity conditions are not violated, this solution is clearly wrong.
2.2. The originfree modulus sum function (S_{M})
A Fourier synthesis with the G amplitudes as coefficients yields the modulus synthesis (M) of ρ^{2}, which is a Pattersonlike synthesis with a dominant origin peak. Similarly, a synthesis with coefficients G^{2} yields the true P, of ρ^{2}. If the strong origin peak is removed from M, the nonorigin peaks become dominant. If M′ denotes M with no origin peak, the phasing residual
will measure, as a function of Φ, the discrepancy between observed and calculated M′ over the whole The Fourier coefficients of M′(Φ) are G_{H}(Φ) − 〈G(Φ)〉, which can be derived from equation (5). By applying the Fourier theory, R_{M}(Φ) can be worked out to (Appendix A)
where G_{H} − 〈G〉 are the Fourier coefficients of the observed M′. The first term, K_{M}, is a phaseindependent quantity. The second term, , is the variance of the probability distribution of the G(Φ) amplitudes that can also be assumed to be phaseindependent for Φ sets satisfying the equalatom and randomness conditions (irrespective of the correctness of Φ). In general, these two assumptions are valid because the equalatom condition is implicit in the squaring operation and because, by only considering the nonorigin peaks of the modulus function (which correspond to interatomic vectors ranging over the whole unit cell), randomness is favoured. Consequently, minimizing R is essentially equivalent to maximizing the third term of equation (8), the socalled originfree modulus sum function
which, after replacing G_{−H}(Φ) by equation (5), becomes
The S_{M} sum function [originally called Z_{R} in Rius (1993)] represents one of the last advances in reciprocalspace DM. Since the true Φ corresponds to a maximum in S_{M}(Φ), a simple method for its maximization is needed. For this purpose, the order of the double summation in equation (10) is changed, so that it becomes
with Q_{h}(Φ) being
If denotes the Fourier synthesis with coefficients (G_{H} − 〈G〉)exp(iψ_{H}), equation (12) can also be expressed as the Fourier coefficient of the product function, i.e.
By following Debaerdemaeker et al. (1985), the maximum of a functional like S_{M} can be found by solving the condition for an extremum, i.e. by making
If this condition is applied to equation (11), a tangent formula (TF) is obtained which provides the new phase estimates
Depending on whether Q_{h} is expressed in terms of the ψ and φ phases [equation (12)] or as a function of [equation (13)], two different optimization algorithms result: (i) the sequential S_{M} tangent formula (STF algorithm) based on phase invariants, and (ii) the parallel S_{M} tangent formula (SFFT algorithm) based on Fourier transforms. For simplicity, the subscripts of S (i.e. M or P) have been omitted from the general algorithm designation.
In equation (9), the experimental quantities are the amplitudes. However, for certain applications it can be desirable to work directly with intensities. That this is feasible can be easily shown by considering the physical meaning of S_{M}(Φ) in equation (9), which corresponds to the integral
It is known that the principal differences between Patterson and modulus functions are the relative heights between origin and nonorigin peaks. If the origin peaks are suppressed, the resulting P′ and M′ functions may be regarded as proportional by a factor close to two (Rius, 2012b). Consequently, maximizing equation (16) is equivalent to maximizing the integral
or in terms of the respective Fourier coefficients
since . Notice that the Q_{h}(Φ) expressions, equations (12) and (13), are also valid for S_{P} simply by replacing (G_{H} − G) by and by , respectively. S_{P} is particularly useful for powder diffraction (PD), because working with experimental intensities simplifies the manipulation of overlapped intensities.
2.3. Phase algorithms based on S
2.3.1. Sequential application of the tangent formula with phase invariants (STF algorithm)
One possibility of maximizing the S_{M} sum function is by means of the iterative application of the tangent formula of equation (15) with Q_{h} given in terms of the phases of equation (12). In practice, the H summation (involving all reflections) is split into two separate sums: the K one (strong reflections) and the L one (weak reflections). Only for the K sum is no distinction made between the ψ and φ phases. This causes the three summands having phase invariants φ_{−h} + φ_{K} + φ_{h−K}, φ_{K} + φ_{−h} + φ_{h−K} and φ_{−h} + φ_{h−K} + φ_{K} to have the same Φ3_{hK} phase sum. Consequently, they can be replaced in the sum by
where
and Q_{h} becomes
According to equation (14), phases refined with the tangent formula of equation (15) will lead to an extremum in S_{M}, i.e. to a large maximum or a large minimum. However, since for strong reflections STF makes ψ and φ equal, only positive solutions are possible. In the STF algorithm the TF is applied in sequential mode. This means that, once a new φ_{h} estimate is calculated with equation (15), its value is immediately replaced in Φ. The updated Φ is then used to compute the phase estimate of the next reflection. This process is repeated until all h reflections in Φ have been treated and no significant phase variations are observed. Before starting a new iteration cycle, the ψ phases of the weak reflections are updated using equation (6) (for strong reflections this is done automatically, because ψ and φ are considered equal). This means that the STF algorithm is essentially a twostage process in which the estimates of the φ and ψ phases are updated alternately.
STF is very effective, easy to apply, and makes no use of any Fourier synthesis. It is ideal for solving smallmolecule crystal structures. However, for crystal structures with a large number of atoms in the (>500 atoms) the total number of terms in the L summation becomes prohibitive. The introduction of a higher cutoff value, E_{min}, for large E values reduces the number of invariant terms at the cost of lowering the accuracy of the calculated G(Φ). To check the efficiency of the STF algorithm, the phasing power of STF was compared with the power of the traditional TF (Karle & Hauptman, 1956), strengthened with information on the most reliable negative quartets. For crystal structures with no fixed origin, the success rate of STF was one order of magnitude higher (Table 1) (Rius et al., 1995; Sheldrick, 1990).
In retrospect, one possible explanation for the late discovery of PFDM can be found in the leading role played by the integral
in the development of DM (Cochran, 1955). It can easily be shown that this integral is closely related to the sum function
since both give similar values (in both expressions the weak E values play no role). However, conceptually, both are completely different. While further progress from Z is not evident, the sum function of equation (23) can evolve to the S_{M} function of equation (11). The reason why the latter represents an improvement is explained intuitively in Fig. 1.
At the beginning of this article it was stated that the first PFDM was published in 1993. This is only partially true, because the most reliable negative quartets (Schenk, 1973; Giacovazzo, 1976) can also be derived from Pattersonfunction arguments, i.e. by expressing the integral
as a function of the Φ phases. Since the nonorigin parts of the modulus and Patterson functions of ρ^{2} can be considered proportional, maximizing the integral of equation (24) is equivalent to maximizing equation (16), so that in this case Q_{h} takes the form (Rius, 1997)
where
and h, k, l and −h, −k, −l belong to the subset of strong reflections and X_{hkl} involves both strong and weak reflections. Notice that X_{hkl} becomes clearly negative only when the three squared amplitudes in equation (26) correspond to weak reflections. One immediate difference between equations (21) and (25) is that the estimation of a new phase with equation (25) requires the lengthy calculation of a double summation. In addition, manipulation of the mixed terms in X_{hkl} is far from trivial.
One year after the introduction of STF, the first paper on the ShakeandBake phasing method was published (DeTitta et al., 1994). It represented a radical change in DM philosophy, since it combined phase in with Fourier filtering, thus exploiting the considerable computing power already available at that moment. In this way the trend towards the uraniumatom solution of traditional reciprocal DM was compensated by the periodic reintroduction of randomness during the directspace stage by picking up the N largest Fourier peaks. This new way of preserving randomness did not rely on the weak reflections. This circumstance proved particularly useful in the solution of e.g. anomalous scatterer substructures in macromolecules. However, when weak reflections are available (as in PD or ED applications), PFDM are highly competitive. As will be shown in the next section, PFDM can also be optimized entirely in (SFFT algorithm), so that if necessary they can also be strengthened by Fourier filtering.
Between the publications of the STF and SFFT algorithms, 14 years elapsed. During this period some new PD applications of STF were explored. One example is the solution of the layered zeolitelike silicate RUB15 of the formula TMA_{8}[Si_{24}O_{52}(OH)_{4}]·20H_{2}O from laboratory PD data (TMA = tetramethylammonium cation; Oberhagemann et al., 1996). At that time it was still a common belief that DM needed intensity data at atomic resolution to be successful. However, the determination of RUB15 demonstrated that, if the electron density of the main building elements can be roughly approximated at moderate resolution to broad spherical peaks (d_{min} ≃ 2 Å), DM will work. In RUB15, both the SiO_{4} and TMA tetrahedra were handled in this way (Fig. 2). This approach allowed the solution of a series of layered zeolitelike compounds at moderate resolution in collaboration with the Institute for Mineralogy of the Ruhr University Bochum (Gies et al., 1998).
Another important result was the demonstration that STF can be applied to PD data of hemihedral compounds (Rius et al., 1999). This was confirmed by solving the crystal structures of: (i) the CAH10 binding phase in highalumina cement (space group P6_{3}/m; Guirado et al., 1998), and (ii) aerinite, a natural blue pigment employed in some Catalan romanesque mural paintings (space group P3c1; Rius et al., 2004). Both crystal structures had resisted multiple attempts at solution worldwide.
In the literature there are various methods of combining information from multiple PD patterns, e.g. by making use of the anisotropic of a material (Shankland et al., 1997). In the particular case of zeolites, the information contained in the powder patterns of the assynthesized and calcined forms can easily be exploited in a twostage procedure (RiusPalleiro et al., 2005). In the first stage, the template molecule is located by combining with STF at very low resolution (d_{min} ≃ 3.2 Å), whereas in the second stage, the framework atoms are found by again applying the STF algorithm but now strengthened with the information coming from the located template molecules (d_{min} ≃ 2.21 Å). This procedure was applied to the solution of the ITQ32 zeolite (Cantín et al., 2005).
All the STF applications described so far use the resolved reflections exclusively (except for hemihedral symmetries, where the intensities of systematically overlapping reflections were equidistributed and treated as resolved), so that strictly speaking these may be regarded as singlecrystal applications. However, the solution of the triclinic of tinticite, a partially disordered phosphate mineral, required a more sophisticated STF procedure in which not only the phases were refined but also the estimated intensities of the severely overlapped peaks (d_{min} ≃ 2.3 Å) (Rius, Torrelles et al., 2000). In the best E map, the broad spherical peaks corresponding to the [Fe^{III}O_{6}] octahedra and to the phosphate tetrahedra (the latter with partial occupancies) showed up clearly (Rius, Loüer et al., 2000). In spite of this success, the often had stability problems, undoubtedly due to the inaccurate intensity estimation of overlapping reflections from a limited number of invariant terms.
2.3.2. Parallel application of the tangent formula via Fourier transforms (SFFT algorithm)
Historically, the development of the SFFT algorithm is related to the 23rd European Crystallographic Meeting in Leuven (2006). On the occasion of that meeting, Professor Baerlocher (ETH, Zurich) showed to the author the potential of chargeflipping when applied to PD (Baerlocher et al., 2007; Palatinus, 2013). Spurred on by this result, the rationale behind chargeflipping was sought. During this search it was found that S_{M} can also be maximized by Fourier methods (Rius et al., 2007). In contrast with the STF algorithm, where the new φ_{h} are estimated sequentially, the SFFT algorithm determines the new φ_{h} (the Fourier transforms of ) in parallel, i.e. from a unique Φ_{old}. A second important difference between the two algorithms is that in SFFT the alternating update of the ψ and φ phases is done in completely separate stages (no explicit use is made of the equality between ψ and φ). The two stages of one iteration cycle are (Fig. 3)
Since the TF S_{M} and the condition ψ_{h} = φ_{h} is not applied during the it can produce either ρ or −ρ as valid solutions when starting from random phases.
leads to an extremum inThe algorithm works quite well with singlecrystal data of small and mediumsized structures at atomic resolution. The stability of the algorithm is reflected in the fact that no electrondensity modification is required after each e.g. there is no need to suppress negative values or for periodic reintroduction of randomness in Φ by selecting the N highest peaks in the Fourier map (and posterior recalculation of Φ from these peaks). It is clear that, for small crystal structures, the phase efficiencies of STF and SFFT must be similar. Table 2 compares the respective efficiencies for a selection of representative compounds.
cycle,

2.3.3. The SFFT algorithm extended to nonpositive definite ρ
Some first applications of the STF algorithm to nonpositive definite density functions in difference structures and in reconstructed surfaces (by using inplane Xray diffraction data) can be found in Rius et al. (1996) and Pedio et al. (2000), respectively. However since, for these particular applications, the SFFT algorithm is much simpler and more accurate (all phase invariants are implicitly taken into account), only SFFT is considered here. In all situations so far discussed, it has been assumed that ρ is positive definite, so that G, as given by equation (4), corresponds to the amplitude of the squared structure ρ^{2}. However, there are certain situations where positivity of ρ is violated. Neutron diffraction data from compounds with negative scatterers are typical cases (Table 3). In such cases, the corresponding nuclear density function (still designated by ρ) consists of positive and negative scatterers, so that the G values derived from equation (4) are no longer the s.f. amplitudes of ρ^{2} but of the socalled `squaredshape structure', in which the atomic peaks have the shape they have in ρ^{2} but preserve the signs they have in ρ. As was shown by Rius & Frontera (2009), the SFFT algorithm can cope with nonpositive definite ρ by simply introducing in equation (13) an m mask calculated according to the following scheme
where t ≃ 2.5, and a is a random value between −1 and 1. The introduction of m into equation (13) yields the extended Q values, i.e.
The viability of the algorithm was checked thoroughly with calculated data sets from organic compounds. Fig. 4 reproduces the Fourier map obtained by processing the intensity data of TVAL (triclinic modification of valinomycin) with the extended SFFT (Karle, 1975; Smith et al., 1975).

From the tests performed on a variety of organic compounds it was concluded that the extended SFFT algorithm has a lower convergence rate than the unextended SFFT (approximately two to three times for the studied test cases), so that the number of cycles has to be increased. This is the price that the extended form has to pay for not including the positivity (or, better, the equalsign) constraint.
For inorganic compounds no significant difference in convergence speed was detected. A perovskiterelated compound containing the strong negative neutron scatterer Mn illustrates how the extended SFFT works (Table 3). The intensities used in the calculations were extracted from the observed powder diffraction pattern by redistributing the global intensities of the overlapping peaks according to the calculated individual intensities (Frontera et al., 2004). The success rate is three from a total of 25 trials (Rius & Frontera, 2008).
Another important situation where negative peaks appear in the Fourier map is in the solution of difference structures. An example of this type of application can be found in Rius & Frontera (2008).
3. Clusterbased Pattersonfunction for powder data
3.1. Definition of atomic, experimental and effective resolutions
In contrast with other θ value beyond which no more diffraction peaks appear. It is normally expressed in terms of the corresponding dspacing value (d_{min}). The experimental resolution depends mainly on the crystallinity of the material, e.g. materials with small domain sizes have broad diffraction peaks, so that peaks and background are difficult to separate at high 2θ. Also important for the application of DM to PD is the effective resolution concept (Fig. 5). Since traditional DM use only the intensities from resolved reflections (which are highly dependent on the amount of peak overlap), the pattern region with useful information is reduced. The dspacing corresponding to the upper 2θ limit of this region gives the effective resolution (d_{eff}). Very often the effective resolution is much less than the experimental one, which hampers the successful application of DM. However, if DM are modified in such a way that clusters of intensities can be treated, the effective resolution of the pattern increases and d_{eff} and d_{min} become more similar. The introduction of `modelfree pattern matching' greatly facilitated the partition of powder patterns into sequences of cluster intensities (Pawley, 1981; Le Bail et al., 1988). The inclusion of the cluster information in the process yields better resolved peaks in the intermediate Fourier syntheses. Summarizing, in the same way that the allows one to take advantage of the whole experimental resolution of the powder pattern during the clusterbased DM allow one to increase the effective resolution during the solution process, so that it comes much closer to the experimental one.
determination methods, the experimental information used by DM is generally limited to the set of measured intensities. This is why it is very important that the data set is almost complete and atomic resolution is reached (only then will the atomic peaks show up clearly separated in the Fourier map). The experimental resolution of a powder pattern is defined by the 23.2. The clusterbased S_{P} function for PD
When PFDM are applied to powder data, the smallest unit of intensity information is the total intensity of each group of unresolved reflections (cluster). The two quantities that specify an arbitrary j cluster are:
where m are the multiplicities of all symmetryindependent reflections. In view of equations (29) and (30), if H is an arbitrary reflection of this cluster, the equidistributed intensity for H is
so that 〈I〉, its average taken over all reflections, is equal to 〈E^{2}〉 = 1.
In the clusterbased S_{P} of equation (18), the observed intensities for overlapping reflections are simply their equidistributed values (Rius, 2011). This is the best approximation to the experimental Notice also that the origin peak can be removed exactly. The of the Φ subset of phases (strong reflections) is achieved by maximizing S_{P} with the SFFT algorithm. Φ is updated from cycle to cycle and at the end of each trial the clusterbased figure of merit
is computed. The solution with the smallest RV value is taken as the correct one. To handle PD data, the following modifications in the S FFT phase algorithm are necessary (Fig. 6):
(i) The coefficients (G − 〈G〉) in stage 2 must be replaced by the (G^{2} − 〈G^{2}〉) ones, so that stage 2 in §2.3.2 becomes ψ + observed (G^{2} − 〈G^{2}〉) → → → φ_{new}.
(ii) Those ρ values below tσ(ρ) are made zero.
(iii) The calculation of the Fourier coefficients (Q) of the product function is performed either by direct Fourier transformation (FT) or by calculation (SFC) from the N highest peaks in (Fig. 6). The periodic calculation of the structure factors from the N peaks is carried out to ensure the fulfilment of the randomness condition. (With singlecrystal data this step is normally not necessary).
(iv) The intensities of overlapping reflections are redistributed according to
The physical meaning of the clusterbased S_{P} function can be best understood by writing it in terms of the cluster intensities. In view of the proportionality between the integrals of equations (17) and (24), S_{P} in equation (18) may be assumed to be proportional to
so that by equation (31) it follows
and S_{P} essentially corresponds to the sum of the products of the observed and calculated cluster intensities, divided by the number of reflections contributing to each cluster. As long as Φ fulfills the general properties of the electrondensity distribution (positivity, randomness, atomicity), the second sum in equation (34b) can be regarded as constant during the phase refinement.
3.3. Examples of application of the clusterbased SFFT algorithm
Retrospectively, the development of the clusterbased SFFT algorithm was greatly facilitated by the release of some highquality PD patterns of organic compounds collected by Dr Gozzo for the Summer School on `Structure Determination from PD Data' organized at the Swiss Light Source in 2008. These patterns had been measured with the novel MythenII microstrip onedimensional detector (Schmitt et al., 2004). For a detailed study of the S_{P} function with powder data, the pattern of (S)(+)ibuprofen was selected (Freer et al., 1993). The monoclinic contains two symmetryindependent molecules, giving rise to a cyclic hydrogenbonded dimer with the formula C_{26}H_{16}O_{4} (Fig. 7). The intensities were extracted by pattern matching using DAJUST (d_{min} = 1.10 Å for λ = 1.0 Å) (Vallcorba et al., 2012). Details of the peak profiles are given in Fig. 8. The extracted cluster intensities (total number of reflections is 1009) were processed by the XLENS_PD6 program, which has the clusterbased SFFT implemented (downloadable from http://departments.icmab.es/crystallography/software ). During the phase chemical constraints were applied every second cycle. Seven trials out of 25 were successful (50 cycles per trial). All correct solutions developed the complete structural model. Some relevant details of the model extracted from the Fourier map are listed in Table 4 (Rius, 2011).

In spite of being relatively new, the clusterbased SFFT algorithm has already solved some rather difficult unknown crystal structures from conventional laboratory PD data, e.g. those of the highly hydrated minerals sanjuanite, Al_{2}(PO_{4})(SO_{4})(OH)·9H_{2}O, Z = 4, P2_{1}/n (Colombo et al., 2011), and sarmientite, Fe_{2}^{3+}(AsO_{4})(SO_{4})(OH)·5H_{2}O, Z = 4, P2_{1}/n, V = 1156 Å^{3} (Colombo et al., 2014), or the triclinic of a new partially deprotonated mixedvalence manganese(II,III) hydroxide arsenate related to sarkinite (de Pedro et al., 2012). Clusterbased SFFT has also determined the frameworks of hybrid materials like calcium hydroxyphosphonoacetates (Colodrero et al., 2011), magnesium tetraphosphonate (Colodrero et al., 2012) or calcium glyceroxide, an active phase for biodiesel production under (LeónReina et al., 2013). Due to the presence of the organic part, synchrotron radiation is preferred for hybrid materials. This normally gives higher experimental resolution (compared with laboratory data), which helps to develop the complete model at the end of the phase stage.
4. The δ recycling method
4.1. The calculated ρ (based on the δ function)
The δ recycling method is an extremely simple phasing method. It is based on a function δ_{M}, which is the convolution of P′ (of the true structure) with a phase synthesis. Experimentally, δ_{M} is computed with the Fourier syntheses
and consists of maxima at the atomic positions and noise in between. According to Rius (2012a), the strength of δ_{M} at the r_{k} atomic positions can be approximated, for an equal atom structure, by
with
Independently, the variance of δ_{M} only depends on the amplitudes and is given by
The fact that is independent of the phase estimates allows one to fix a threshold value before the structure is solved (Rius, 2012a). In practice, the threshold value Δ = tδ_{M} with t ≃ 2.5 works well for eliminating noise. In this way an m mask can be created, which will be 0 or 1 depending on whether the corresponding δ_{M} value is below or above Δ. By multiplying δ_{M} by this mask and considering equations (36) and (37), the desired approximation to ρ is obtained
which must be always positive and uses the known E magnitudes (Rius, 2012b).
4.2. The phasing residual and the algorithm
If ρ(r) represents a positive definite density function of the crystal, e.g. the electron density or the electrostatic potential (in this second case only for structure solution purposes), it will be assumed that the condition
is only fulfilled for the true Φ values. The discrepancy between ρ(r,Φ) and ρ_{C}(r,Φ) can be measured through the residual
extended over the whole V, where for clarity the r and Φ symbols have been omitted in the integrand. By working out the squared binomial, and since the integral of ρ^{2} over the is phaseindependent (it corresponds to the value of the at the origin and is equal to 1/V ∑_{H}E_{H}^{2}), minimizing R_{δ} is equivalent to maximizing the integral
of volumewhich in view of equations (37) and (39), and because m = m^{2}, reduces, after some algebraic manipulation, to
wherein ρ_{X} corresponds to
Here, ρ_{φ} denotes the phase synthesis, i.e. a Fourier synthesis with the same phases as ρ but with constant amplitudes (in this case unity). In view of this, it follows from equation (44) that the Fourier coefficients of ρ_{X} are
The dependence of the modulus of X on the amplitude E is of a linear type (Fig. 9). By expressing ρ_{X} in equation (44) as a Fourier synthesis, equation (43) transforms into
with
Equation (46) is formally equivalent to equation (11), except for the fact that the summation extends over all H reflections, not just the strongest ones (Rius, 2012b). Consequently, the new phase estimates can also be derived by applying a tangent formula, namely
The general scheme of the δ recycling phasing procedure is described in Fig. 10. As indicated by the calculation (SFC), the structure factors are computed from the N largest peaks found in ρ_{C} [equation (39)]. The new Φ set is then used to update δ_{M}. This procedure is applied cyclically until convergence is reached. Convergence is controlled by measuring the correlation Corr between the experimental E and the updated E_{new} with the expression
4.3. Application to ED tomography data
Frequently, natural and synthetic phases only appear as submicrometric crystals, too small for collecting singlecrystal Xray data even with synchrotron radiation. Normally, structural information from these phases is obtained from PD, which combines easy sample preparation (also under nonambient conditions) with fast acquisition systems and sophisticated analytical methods. Nevertheless, PD suffers from various limitations which may be caused by the sample [(i) sufficient sample must be available; (ii) the sample must be an almost pure phase; (iii) for nanocrystals, peak broadening due to the particle size reduces the effective data resolution range] and/or by the et al. (2010) and Rozhdestvenskaya et al. (2010)]. The main advantage of electron diffraction (ED) is the ability to collect singlecrystal data from nanometric volumes. This is possible because electrons can be deflected and focused in quasiparallel probes with a diameter of 10–30 nm and because the interaction with matter for electrons is much stronger than for Xrays, allowing a good signaltonoise ratio even for diffraction from nanovolumes of crystalline material. Two of the principal problems of ED, i.e. dynamic effects and incomplete data sets, are minimized by measuring offzone. This is the basis of the automated diffraction tomography (ADT) data collection strategy (Kolb et al., 2007, 2008). In ADT, the ED patterns are acquired by rotating around an arbitrary tilt axis (not corresponding to a specific crystallographic orientation) in sequential steps of 1° within the full tilt range of the microscope. The physical limit affecting the sample rotation gives rise to incomplete data sets, i.e. to a missing wedge. The precession ED technique (PED) is used to integrate the intensity between steps. Recently, an alternative technique called rotation ED (RED) has been introduced for this purpose (Zhang et al., 2010). Of course, there are also disadvantages with ED. For certain compounds, radiation damage is still a limiting problem. In general, organic and hybrid materials are more beamsensitive than inorganic materials. The application of δ recycling to PED/ADT intensities from inorganic materials was recently analyzed by Rius et al. (2013) with some interesting results: (i) scaling with the Wilson plot procedure is accurate; (ii) δ recycling is particularly robust against missing data; (iii) unlike Xrays, where Corr [equation (49)] clearly discriminates the correct solution, the final Corr values with PED/ADT data tend to be similar for correct and wrong solutions. To circumvent this difficulty, the δ recycling phasing stage always terminates when a preset number of cycles is reached, and continues with conventional Fourier recycling methods. Convergence during Fourier recycling is controlled by the R_{CC} residual,
itself [(i) indexing of unit cells with long axes is not always trivial; (ii) systematic overlap is present in highsymmetry space groups, especially in cubic ones; (iii) accidental overlap may be severe for lowsymmetry space groups]. In addition, identification of the for crystalline phases affected by pseudosymmetry can be problematic even for good PD data [see, for example, Birkelwhich is free from scaling factors. R_{CC} is always a very reliable figure of merit and for Xrays values between 5–30 indicate correct solutions; for PED/ADT data, essentially correct solutions are found between 15 and 60, although R_{CC} values up to 80 can be reached, especially if the data are affected by large thickness variations and/or by residual dynamic scattering, if missing organic parts of the structure are not included in the calculation of the intensities, or if the measured data fail to produce well shaped peaks in the Fourier map.
It goes without saying that the possibility of solving crystal structures from phases only detected by TEM is very important in many research fields. For example, it is expected that many new mineralogical species can be found. In a recent collaboration with Kolb's group at the University of Mainz, δ recycling has solved, from PED/ATD data, the of a new porous Bi sulfate mineral appearing only as a tiny crystalline fragment (∼0.15 × 0.15 × 0.2 µm) displaying no net cleavage planes (Capitani et al., 2014). The is hexagonal and the unitcell content is [Bi_{8.18}Te_{0.82}(OH)_{6}O_{8}(SO_{4})_{2}]^{0.91+}·0.91S_{2}^{−} with Z = 2 (Fig. 11). Some relevant experimental details are: d_{min} = 1.0 Å, number of measured (unique) reflections = 2748 (452), R_{equiv} = 23.57, data completeness = 100%; λ = 0.0197 Å, T = 93 K. The structure model obtained from δ recycling was complete and can be described as a selfassemblage of Bi clusters giving rise to a onedimensional porous material with the disulfide anions inside the channels. Figures of merit for the last cycle are R_{1} = 0.2173 for 332 F_{obs} > 4σ(F_{obs}) and 0.2373 for all 452 data, i.e. of the same order as the R value between symmetryequivalent reflections (SHELX97; Sheldrick, 2008). Finally, it is worth mentioning that the O atoms could be located in the presence of the extremely heavy Bi atoms (Z = 83), a consequence of the slower scatteringpower increase with compared with Xrays.
5. Conclusions
Currently, the application of DM has reached maturity. This means that, for an ideal singlecrystal intensity data set, phasing is a rather straightforward process. However, the situation changes for inaccurate or incomplete data sets, a circumstance which occurs with increasing frequency in materials science, especially when small crystalline volumes are being analyzed. To deal with these situations, not only robust but also simple DM procedures are required which can process, in a unified manner, partial information coming from different sources, e.g. transmission microdiffraction, electron diffraction, powder diffraction, grazingincidence diffraction. Due to their simplicity, PFDM are ideal candidates for such types of applications which also benefit from rapidly evolving instrumental capabilities.
APPENDIX A
Simplification of the R_{M}(Φ) expression
Written in terms of the Fourier coefficients of M′, the integral of equation (7) is equivalent to
If K_{M} = Σ_{H}(G_{H} − G)^{2}, then
By transforming the sums into averages (N_{H} = number of reflections), the last two summations reduce to
Finally, replacement of the last two last sums in equation (52) by equation (53) leads to the simplified R_{M}(Φ) expression
Acknowledgements
The author wishes to express his gratitude to Professor Carles Miravitlles for his personal and scientific support over the years. The author also thanks the Spanish Ministerio de Economia y Competividad (Projects MAT201235247 and Consolider NANOSELECT CSD200700041) and the Generalitat of Catalonia (SGR2009) for financial support. The support and advice of Dr Nicolopoulus of NanoMEGAS SPRL in the field of electron diffraction is greatly appreciated.
References
Antel, J., Sheldrick, G. M., Bats, J. W., Kessler, H. & Müller, A. (1990). Unpublished. Google Scholar
Baerlocher, Ch., McCusker, L. B. & Palatinus, L. (2007). Z. Kristallogr. 222, 47–53. Web of Science CrossRef CAS Google Scholar
Bhat, T. N. & Ammon, H. L. (1990). Acta Cryst. C46, 112–116. CSD CrossRef CAS Web of Science IUCr Journals Google Scholar
Birkel, C. S., Mugnaioli, E., Gorelik, T., Kolb, U., Panthöfer, M. & Tremel, W. (2010). J. Am. Chem. Soc. 132, 9881–9889. Web of Science CrossRef CAS PubMed Google Scholar
Braekman, J. C., Daloze, D., Dupont, A., Tursch, B., Declercq, J. P., Germain, G. & Van Meersche, M. (1981). Tetrahedron, 37, 179–186. CSD CrossRef CAS Web of Science Google Scholar
Butters, T., Hütter, P., Jung, G., Pauls, N., Schmitt, H., Sheldrick, G. M. & Winter, W. (1981). Angew. Chem. Int. Ed. 93, 904–905. CrossRef CAS Google Scholar
Cantín, A., Corma, A., Leiva, S., Rey, F., Rius, J. & Valencia, S. (2005). J. Am. Chem. Soc. 127, 11580–11581. Web of Science PubMed Google Scholar
Capitani, G. C., Mugnaioli, E., Rius, J., Gentile, P., Catelani, T., Lucotti, A. & Kolb, U. (2014). Am. Mineral. 99, 500–510. Web of Science CrossRef Google Scholar
Cochran, W. (1955). Acta Cryst. 8, 473–478. CrossRef CAS IUCr Journals Web of Science Google Scholar
Colodrero, R. M. P., Cabeza, A., OliveraPastor, P., Papadaki, M., Rius, J., ChoquesilloLazarte, D., GarcíaRuiz, J. M., Demadis, K. D. & Aranda, M. A. G. (2011). Cryst. Growth Des. 11, 1713–1722. CSD CrossRef CAS Google Scholar
Colodrero, R. M., OliveraPastor, P., Losilla, E. R., HernándezAlonso, D., Aranda, M. A. G., LeónReina, L., Rius, J., Demadis, K. D., Moreau, B., Villemin, D., Palomino, M., Rey, F. & Cabeza, A. (2012). Inorg. Chem. 51, 7689–7698. Web of Science CSD CrossRef CAS PubMed Google Scholar
Colombo, F., Rius, J., PannunzioMiner, E. V., Pedregosa, J. C., Cami, G. E. & Carbonio, R. E. (2011). Can. Mineral. 49, 835–847. CrossRef CAS Google Scholar
Colombo, F., Rius, J., Vallcorba, O. & Miner, E. V. P. (2014). Mineral. Mag. 78, 347–360. CrossRef CAS Google Scholar
Debaerdemaeker, T., Tate, C. & Woolfson, M. M. (1985). Acta Cryst. A41, 286–290. CrossRef CAS Web of Science IUCr Journals Google Scholar
Declercq, J. P., Germain, G. & Van Meersche, M. (1972). Cryst. Struct. Commun. 1, 13–15. CAS Google Scholar
DeTitta, G. T., Langs, D. A., Edmonds, J. W. & Duax, W. L. (1980). Acta Cryst. B36, 638–645. CSD CrossRef CAS IUCr Journals Web of Science Google Scholar
DeTitta, G. T., Weeks, C. M., Thuman, P., Miller, R. & Hauptman, H. A. (1994). Acta Cryst. A50, 203–210. CrossRef CAS Web of Science IUCr Journals Google Scholar
Freer, A. A., Bunyan, J. M., Shankland, N. & Sheen, D. B. (1993). Acta Cryst. C49, 1378–1380. CSD CrossRef CAS Web of Science IUCr Journals Google Scholar
Frontera, C., GarcíaMuñoz, J. L., Aranda, M. A. G., Hervieu, M., Ritter, C., Mañosa, L., Capdevila, X. G. & Calleja, A. (2004). Phys. Rev. B, 68, 134408. Web of Science CrossRef Google Scholar
Giacovazzo, C. (1976). Acta Cryst. A32, 91–99. CrossRef IUCr Journals Web of Science Google Scholar
Gies, H., Marler, B., Vortmann, S., Oberhagemann, U., Bayat, P., Krink, K., Rius, J., Wolf, I. & Fyfe, C. (1998). Microporous Mesoporous Mater. 21, 183–197. Web of Science CrossRef CAS Google Scholar
Grigg, R., Kemp, J., Sheldrick, G. M. & Trotter, J. (1978). J. Chem. Soc. Chem. Commun. pp. 109–111. CrossRef Web of Science Google Scholar
Guirado, F., Galí, S., Chinchón, S. & Rius, J. (1998). Angew. Chem. Int. Ed. 37, 72–75. CrossRef CAS Google Scholar
Hovestreydt, E., Klepp, K. & Parthé, E. (1983). Acta Cryst. C39, 422–425. CrossRef CAS Web of Science IUCr Journals Google Scholar
Irngartinger, H., Reibel, W. R. K. & Sheldrick, G. M. (1981). Acta Cryst. B37, 1768–1771. CSD CrossRef CAS Web of Science IUCr Journals Google Scholar
Jones, P. G., Schomburg, D., Hopf, H. & Lehne, V. (1992). Acta Cryst. C48, 2203–2207. CSD CrossRef CAS Web of Science IUCr Journals Google Scholar
Jones, P. G., Sheldrick, G. M., Glüsenkamp, K.H. & Tietze, L. F. (1980). Acta Cryst. B36, 481–483. CSD CrossRef CAS IUCr Journals Web of Science Google Scholar
Karle, I. L. (1975). J. Am. Chem. Soc. 97, 4379–4386. CSD CrossRef PubMed CAS Web of Science Google Scholar
Karle, J. & Hauptman, H. (1956). Acta Cryst. 9, 635–651. CrossRef CAS IUCr Journals Web of Science Google Scholar
Kolb, U., Gorelik, T., Kübel, C., Otten, M. T. & Hubert, D. (2007). Ultramicroscopy, 107, 507–513. Web of Science CrossRef PubMed CAS Google Scholar
Kolb, U., Gorelik, T. & Otten, M. T. (2008). Ultramicroscopy, 108, 763–772. Web of Science CrossRef PubMed CAS Google Scholar
Le Bail, A., Duroy, H. & Fourquet, J. (1988). Mater. Res. Bull. 23, 447–452. CrossRef CAS Web of Science Google Scholar
LeónReina, L., Cabeza, A., Rius, J., MairelesTorres, P., AlbaRubio, A. C. & López Granados, M. (2013). J. Catal. 300, 30–36. Google Scholar
Macrae, C. F., Bruno, I. J., Chisholm, J. A., Edgington, P. R., McCabe, P., Pidcock, E., RodriguezMonge, L., Taylor, R., van de Streek, J. & Wood, P. A. (2008). J. Appl. Cryst. 41, 466–470. Web of Science CSD CrossRef CAS IUCr Journals Google Scholar
Oberhagemann, U., Bayat, P., Marler, B., Gies, H. & Rius, J. (1996). Angew. Chem. Int. Ed. 35, 2869–2872. CrossRef CAS Web of Science Google Scholar
Oliver, J. D. & Strickland, L. C. (1984). Acta Cryst. C40, 820–824. CSD CrossRef CAS Web of Science IUCr Journals Google Scholar
Palatinus, L. (2013). Acta Cryst. B69, 1–16. CrossRef CAS IUCr Journals Google Scholar
Pawley, G. S. (1981). J. Appl. Cryst. 14, 357–361. CrossRef CAS Web of Science IUCr Journals Google Scholar
Pedio, M., Felici, R., Torrelles, X., Rudolf, P., Capozi, M., Rius, J. & Ferrer, S. (2000). Phys. Rev. Lett. 85, 1040–1043. Web of Science CrossRef PubMed CAS Google Scholar
Pedro, I. de, Rojo, J. M., Rius, J., Vallcorba, O., Ruiz de Larramendi, I., Rodríguez Fernández, J., Lezama, L. & Rojo, T. (2012). Inorg. Chem. 51, 5246–5256. Web of Science PubMed Google Scholar
Poyser, J. P., Edwards, R. L., Anderson, J. R., Hursthouse, M. B., Walker, N. P., Sheldrick, G. M. & Whalley, A. J. (1986). J. Antibiot. 39, 167–169. CrossRef CAS PubMed Google Scholar
Privé, G. G., Anderson, D. H., Wesson, L., Cascio, D. & Eisenberg, D. (1999). Protein Sci. 8, 1–9. Google Scholar
Rius, J. (1993). Acta Cryst. A49, 406–409. CrossRef CAS Web of Science IUCr Journals Google Scholar
Rius, J. (1997). Acta Cryst. D53, 535–539. CrossRef CAS Web of Science IUCr Journals Google Scholar
Rius, J. (2011). Acta Cryst. A67, 63–67. Web of Science CrossRef CAS IUCr Journals Google Scholar
Rius, J. (2012a). Acta Cryst. A68, 77–81. Web of Science CrossRef CAS IUCr Journals Google Scholar
Rius, J. (2012b). Acta Cryst. A68, 399–400. Web of Science CrossRef CAS IUCr Journals Google Scholar
Rius, J., Crespi, A. & Torrelles, X. (2007). Acta Cryst. A63, 131–134. Web of Science CrossRef CAS IUCr Journals Google Scholar
Rius, J., Elkaim, E. & Torrelles, X. (2004). Eur. J. Mineral. 16, 127–134. Web of Science CrossRef CAS Google Scholar
Rius, J. & Frontera, C. (2008). Acta Cryst. A64, 670–674. Web of Science CrossRef CAS IUCr Journals Google Scholar
Rius, J. & Frontera, C. (2009). Acta Cryst. A65, 528–531. CrossRef CAS IUCr Journals Google Scholar
Rius, J., Loüer, D., Loüer, M., Galí, S. & Melgarejo, J. C. (2000). Eur. J. Mineral. 12, 581–588. CAS Google Scholar
Rius, J., Miravitlles, C. & Allmann, R. (1996). Acta Cryst. A52, 634–639. CrossRef CAS Web of Science IUCr Journals Google Scholar
Rius, J., Miravitlles, C., Gies, H. & Amigó, J. M. (1999). J. Appl. Cryst. 32, 89–97. Web of Science CrossRef CAS IUCr Journals Google Scholar
Rius, J., Mugnaioli, E., Vallcorba, O. & Kolb, U. (2013). Acta Cryst. A69, 396–407. Web of Science CrossRef CAS IUCr Journals Google Scholar
Rius, J., Sañé, J., Miravitlles, C., Amigó, J. M. & Reventós, M. M. (1995). Acta Cryst. A51, 268–270. CrossRef CAS Web of Science IUCr Journals Google Scholar
Rius, J., Torrelles, X., Miravitlles, C., Ochando, L. E., Reventós, M. M. & Amigó, J. M. (2000). J. Appl. Cryst. 33, 1208–1211. Web of Science CrossRef CAS IUCr Journals Google Scholar
RiusPalleiro, J., Peral, I., Margiolaki, I. & Torrelles, X. (2005). J. Appl. Cryst. 38, 906–911. Web of Science CrossRef CAS IUCr Journals Google Scholar
Roisnel, T. & RodríquezCarvajal, J. (2001). Mater. Sci. Forum, 378–381, 118–123. CrossRef CAS Google Scholar
Rozhdestvenskaya, I., Mugnaioli, E., Czank, M., Depmeier, W., Kolb, U., Reinholdt, A. & Weirich, T. (2010). Mineral. Mag. 74, 159–177. Web of Science CrossRef CAS Google Scholar
Sayre, D. (1952). Acta Cryst. 5, 60–65. CrossRef CAS IUCr Journals Web of Science Google Scholar
Schenk, H. (1973). Acta Cryst. A29, 77–82. CrossRef IUCr Journals Web of Science Google Scholar
Schmitt, B., Broenimann, Ch., Eikenberry, E. F., Huelsen, G., Toyokawa, H., Horisberger, R., Gozzo, F., Patterson, B., SchulzeBriese, C. & Tomikazi, T. (2004). Nucl. Instrum. Methods Phys. Res. A, 518, 436–439. CrossRef CAS Google Scholar
Shankland, K., David, W. I. F. & Sivia, D. S. (1997). J. Mater. Chem. 7, 569–572. CSD CrossRef CAS Google Scholar
Sheldrick, G. M. (1990). Acta Cryst. A46, 467–473. CrossRef CAS Web of Science IUCr Journals Google Scholar
Sheldrick, G. M. (2008). Acta Cryst. A64, 112–122. Web of Science CrossRef CAS IUCr Journals Google Scholar
Sheldrick, G. M., Davison, B. E. & Trotter, J. (1978). Acta Cryst. B34, 1387–1389. CSD CrossRef CAS IUCr Journals Web of Science Google Scholar
Smith, G. D., Duax, W. L., Langs, D. A., DeTitta, G. T., Edmonds, J. W., Rohrer, D. C. & Weeks, C. M. (1975). J. Am. Chem. Soc. 97, 7242–7247. CSD CrossRef PubMed CAS Web of Science Google Scholar
SzeimiesSeebach, U., Harnisch, J., Szeimies, G., Van Meersche, M., Germain, G. & Declerq, J. P. (1978). Angew. Chem. Int. Ed. 17, 848–850. Google Scholar
Teixidor, F., Rius, J., Romerosa, A., Miravitlles, C., Escriche, L. L., Sanchez, E., Viñas, C. & Casabó, J. (1991). Inorg. Chim. Acta, 176, 61–65. CSD CrossRef Web of Science Google Scholar
Teixidor, F., Viñas, C., Rius, J., Miravitlles, C. & Casabó, J. (1990). Inorg. Chem. 29, 149–152. CSD CrossRef CAS Web of Science Google Scholar
Vallcorba, O., Rius, J., Frontera, C., Peral, I. & Miravitlles, C. (2012). J. Appl. Cryst. 45, 844–848. Web of Science CrossRef CAS IUCr Journals Google Scholar
Williams, D. J. & Lawton, D. (1975). Tetrahedron Lett. 16, 111–114. CSD CrossRef CAS Google Scholar
Wilson, A. J. C. (1949). Acta Cryst. 2, 318–321. CrossRef IUCr Journals Web of Science Google Scholar
Zhang, D. L., Oleynikov, P., Hovmöller, S. & Zou, X. D. (2010). Z. Kristallogr. 225, 94–102. Web of Science CrossRef CAS Google Scholar
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