ALBERT

Notes: When in a powder diffraction pattern two or more peaks heavily overlap, the integrated intensities of the overlapping reflections are correlated. This paper describes a method, based on the study of the peak overlap, which estimates the number of statistically independent intensities associated with a powder diagram. Such a number can be useful for forecasting the rate of success of direct methods procedures for ab initio crystal structure solution and for evaluating a priori the efficiency of least-squares programs devoted to crystal structure refinement.

Notes: A recent statistical method for deriving information about the presence of a preferred orientation plane [Altomare, Cascarano, Giacovazzo & Guagliardi (1994). J. Appl. Cryst. 27, 1045–1050] has been revisited. Some implications of the method are theoretically discussed. The method is applied to various practical cases in order to give insight into its potential and its limits and into the correlation between its indications and full-pattern decomposition programs, peak overlapping and data resolution.

Notes: A probabilistic approach is described that is able to estimate triplet phase invariants when prior information on interatomic vectors and interatomic triangles is available. The conclusive formula is compared with the vector-interaction formula derived by Hauptman & Karle [Acta Cryst. (1962), 15, 547–550] and with a probabilistic formula obtained via maximum-entropy methods.

Notes: The second representation of a triplet invariant [Giacovazzo (1977). Acta Cryst. A33, 933–944] is a collection of special quintets. In the present paper, the triplet is embedded in many more additional quintets obtained in a special way by symmetry operations on the indices of the structure factors. The method of joint probability distribution functions has been used to derive a formula for estimating triplets via the information contained in the basis and in the cross terms of the quintet invariants. The P10 formula [Cascarano, Giacovazzo, Camalli, Spagna, Burla, Nunzi & Polidori (1984). Acta Cryst. A40, 278–283] is a special case of the new formula, here called P13. The new expression has been applied to practical cases.

Notes: Negative quartet invariants play an important role in direct procedures: they can be actively used in the phasing process or, in a passive way, for selecting the correct solution in multisolution approaches. Unfortunately their average reliability is small. The use of the second representation of the quartets may lead to improved estimates but calculations are expected to be too extensive even for modem computers. The simple and fast process described in this paper provides improved quartet estimates by embedding triplet and quintet estimates. The first applications of the method are satisfactory.

Notes: Typesetting errors, not in the proofs, were introduced in the paper by Burla, Giacovazzo, Moliterni & Gonzalez Platas [Acta Cryst. (1994). A50, 771–778]. The correct equations are given below. Page 772: formulas (3) should be changed to\cases{\varphi_{{\bf h}_1} + \varphi_{{\bf h}_2} + \varphi_{{\bf h}_3}\cr -\varphi_{{\bf h}_1} - \varphi_{{\bf kR}_i} + \varphi_{{\bf h}_{1} + {\bf kR}_i} \cr -\varphi_{{\bf h}_2} + \varphi_{{\bf kR}_i} + \varphi_{{\bf h}_2 - {\bf kR}_i}\cr -\varphi_{{\bf h}_2} - \varphi_{{\bf h}_1 + {\bf kR}_i} - \varphi_{{\bf h}_{2} + {{\bf kR}_i}}}\cases{\varphi_{{\bf h}_1} + \varphi_{{\bf h}_2} + \varphi_{{\bf h}_3}\cr -\varphi_{{\bf h}_1} - \varphi_{{\bf kR}_i} + \varphi_{{\bf h}_{1} + {\bf kR}_i} \cr -\varphi_{{\bf h}_2} - \varphi_{{\bf h}_{1} + {\bf kR}_i} - \varphi_{{\bf h}_3 - {{\bf kR}_i}}\cr -\varphi_{{\bf h}_3} + \varphi_{{\bf kR}_i} + \varphi_{{\bf h}_{3} - {{\bf kR}_i}}}\cases{\varphi_{{\bf h}_1} + \varphi_{{\bf h}_2} + \varphi_{{\bf h}_3}\cr -\varphi_{{\bf h}_1} + \varphi_{{\bf kR}_i} + \varphi_{{\bf h}_{1} - {\bf kR}_i} \cr -\varphi_{{\bf h}_2} - \varphi_{{\bf kR}_i} + \varphi_{{\bf h}_{2} + {{\bf kR}_i}}\cr -\varphi_{{\bf h}_3} - \varphi_{{\bf h}_{1} - {\bf kR}_i} - \varphi_{{\bf h}_2 + {{\bf kR}_i}}} \eqno(3)\cases{\varphi_{{\bf h}_1} + \varphi_{{\bf h}_2} + \varphi_{{\bf h}_3}\cr -\varphi_{{\bf h}_1} - \varphi_{{\bf h}_{2} + {\bf kR}_i} - \varphi_{{\bf h}_3 - {\bf kR}_i}\cr -\varphi_{{\bf h}_2} -\varphi_{{\bf kR}_i} + \varphi_{{\bf h}_{2} + {\bf kR}_i}\cr -\varphi_{{\bf h}_3} + \varphi_{{\bf kR}_i} + \varphi_{{\bf h}_3 - {\bf kR}_i}}\cases{\varphi_{{\bf h}_1} + \varphi_{{\bf h}_2} + \varphi_{{\bf h}_3}\cr -\varphi_{{\bf h}_1} + \varphi_{{\bf kR}_i} + \varphi_{{\bf h}_1 - {\bf kR}_i}\cr -\varphi_{{\bf h}_2} - \varphi_{{\bf h}_1 - {\bf kR}_i} - \varphi_{{\bf h}_{3} + {\bf kR}_i}\cr -\varphi_{{\bf h}_3} - \varphi_{{\bf kR}_i} + \varphi_{{\bf h}_{3} + {\bf kR}_i}}\cases{\varphi_{{\bf h}_1} + \varphi_{{\bf h}_2} + \varphi_{{\bf h}_3}\cr -\varphi_{{\bf h}_1} -\varphi_{{\bf h}_2 - {\bf kR}_i} -\varphi_{{\bf h}_{3} + {\bf kR}_i}\cr -\varphi_{{\bf h}_2} + \varphi_{{\bf kR}_i} + \varphi_{{\bf h}_2 - {\bf kR}_i}\cr -\varphi_{{\bf h}_3} -\varphi_{{\bf kR}_i} + \varphi_{{\bf h}_{3} + {\bf kR}_i}.}Page 772: formula (4) should be changed to\cases{\varphi_{{\bf h}_1} + \varphi_{{\bf h}_2} + \varphi_{{\bf h}_3}\cr -\varphi_{{\bf h}_2} - \varphi_{{\bf kR}_i} + \varphi_{{\bf h}_{2} + {\bf kR}_i}\cr -\varphi_{{\bf h}_3} + \varphi_{{\bf kR}_j} + \varphi_{{\bf h}_3 - {\bf kR}_j}\cr -\varphi_{{\bf h}_1} - \varphi_{({\bf h}_{2} + {\bf kR}_{i}){\bf R}_p} - \varphi_{({\bf h}_3 - {\bf kR}_{j}){\bf R}_{s}.}} Pages 773–774: formulas (15) should be changed into(a)\ \ \cases{\varphi_{{\bf h}_1} + \varphi_{{\bf h}_2} + \varphi_{{\bf h}_3}\cr -\varphi_{{\bf h}_1} + \varphi_{{\bf kR}_i} + \varphi_{{\bf h}_1 - {\bf kR}_i}\cr -\varphi_{{\bf h}_2} - \varphi_{{\bf kR}_i} + \varphi_{{\bf h}_{2} + {\bf kR}_i}\cr -\varphi_{{\bf h}_3} - \varphi_{{\bf h}_1 - {\bf kR}_i} - \varphi_{{\bf h}_{2} + {\bf kR}_i}}(b)\ \ \cases{\varphi_{{\bf h}_1} + \varphi_{{\bf h}_2} + \varphi_{{\bf h}_3}\cr -\varphi_{{\bf h}_1} + \varphi_{{\bf kR}_i} + \varphi_{{\bf h}_1 - {\bf kR}_i}\cr -\varphi_{{\bf h}_2} -\varphi_{{\bf h}_1 - {\bf kR}_i} - \varphi_{{\bf h}_{3} + {\bf kR}_i}\cr -\varphi_{{\bf h}_3} -\varphi_{{\bf kR}_i} + \varphi_{{\bf h}_{3} + {\bf kR}_i}}(c)\ \ \cases{\varphi_{{\bf h}_1} + \varphi_{{\bf h}_2} + \varphi_{{\bf h}_3}\cr -\varphi_{{\bf h}_1} + \varphi_{{\bf h}_{1}{\bf R}_p} - \varphi_{{\bf h}_{1}({\bf R}_{p}-{\bf I})}\cr -\varphi_{{\bf h}_2} -\varphi_{{\bf h}_{1}{\bf R}_p} + \varphi_{{\bf h}_{1}{\bf R}_{p} + {\bf h}_2}\cr -\varphi_{{\bf h}_3} - \varphi_{{\bf h}_{1}{\bf R}_{p} + {\bf h}_2}+ \varphi_{{\bf h}_{1}({\bf R}_{p}-{\bf I})}\cr}(d)\ \ \cases{\varphi_{{\bf h}_1} + \varphi_{{\bf h}_2} + \varphi_{{\bf h}_3}\cr -\varphi_{{\bf h}_1} + \varphi_{{\bf h}_{1}{\bf R}_p} - \varphi_{{\bf h}_{1}({\bf R}_{p}-{\bf I})}\cr -\varphi_{{\bf h}_2} - \varphi_{{\bf h}_{1}{\bf R}_{p} + {\bf h}_3} + \varphi_{{\bf h}_{1}({\bf R}_{p}-{\bf I})}\cr -\varphi_{{\bf h}_3} - \varphi_{{\bf h}_{1}{\bf R}_p} + \varphi_{{\bf h}_{1}{\bf R}_{p} + {\bf h}_3}}(e)\ \ \cases{\varphi_{{\bf h}_1} + \varphi_{{\bf h}_2} + \varphi_{{\bf h}_3}\cr -\varphi_{{\bf h}_{1}{\bf R}_p} -\varphi_{{\bf h}_3} + \varphi_{{\bf h}_{1}{\bf R}_{p} + {\bf h}_3}\cr -\varphi_{{\bf h}_2} + \varphi_{{\bf kR}_{i}{\bf R}_{p}} + \varphi_{{\bf h}_{2} - {\bf kR}_{i}{\bf R}_{p}}\cr -\varphi_{{\bf kR}_i} - \varphi_{{\bf h}_{2} - {\bf kR}_{i}{\bf R}_{p}} - \varphi_{{\bf h}_{1}{\bf R}_{p} + {\bf h}_3}}(f)\ \ \cases{\varphi_{{\bf h}_1} + \varphi_{{\bf h}_2} + \varphi_{{\bf h}_3}\cr -\varphi_{{\bf h}_{1}{\bf R}_p} -\varphi_{{\bf h}_3} + \varphi_{{\bf h}_{1}{\bf R}_{p} + {\bf h}_3}\cr -\varphi_{{\bf h}_2} -\varphi_{{\bf kR}_i} + \varphi_{{\bf h}_{2} + {\bf kR}_i}\cr +\varphi_{{\bf kR}_{i}{\bf R}_p} -\varphi_{{\bf h}_{2} + {\bf kR}_i} - \varphi_{{\bf h}_{1}{\bf R}_{p} + {\bf h}_3}}(g)\ \ \cases{\varphi_{{\bf h}_1} + \varphi_{{\bf h}_2} + \varphi_{{\bf h}_3}\cr -\varphi_{{\bf h}_{1}{\bf R}_p} - \varphi_{{\bf kR}_i} + \varphi_{{\bf h}_{1}{\bf R}_{p} + {\bf kR}_i}\cr - \varphi_{{\bf h}_2} + \varphi_{{\bf kR}_{i}{\bf R}_p} + \varphi_{{\bf h}_{2} - {\bf kR}_{i}{\bf R}_p}\cr -\varphi_{{\bf h}_3} - \varphi_{{\bf h}_{1}{\bf R}_{p} + {\bf kR}_{i}} - \varphi_{{\bf h}_{2} - {\bf kR}_{i}{\bf R}_p}}(h)\ \ \cases{\varphi_{{\bf h}_1} + \varphi_{{\bf h}_2} + \varphi_{{\bf h}_3}\cr -\varphi_{{\bf h}_{1}{\bf R}_p} - \varphi_{{\bf kR}_i} + \varphi_{{\bf h}_{1}{\bf R}_{p} + {\bf kR}_i}\cr -\varphi_{{\bf h}_2} - \varphi_{{\bf h}_{1}{\bf R}_{p} + {\bf kR}_{i}} - \varphi_{{\bf h}_{3} - {\bf kR}_{i}{\bf R}_p}\cr -\varphi_{{\bf h}_3} + \varphi_{{\bf kR}_{i}{\bf R}_p} + \varphi_{{\bf h}_{3} - {\bf kR}_{i}{\bf R}_p}}(i)\ \ \cases{\varphi_{{\bf h}_1} + \varphi_{{\bf h}_2} + \varphi_{{\bf h}_3}\cr -\varphi_{{\bf h}_{1}{\bf R}_p} -\varphi_{{\bf h}_2} + \varphi_{{\bf h}_{1}{\bf R}_{p} + {\bf h}_2}\cr +\varphi_{{\bf kR}_{i}{\bf R}_p} - \varphi_{{\bf h}_{3} + {\bf kR}_i} - \varphi_{{\bf h}_{1}{\bf R}_{p} + {\bf h}_2}\cr -\varphi_{{\bf h}_3} - \varphi_{{\bf kR}_i} + \varphi_{{\bf h}_{3} + {\bf kR}_i}}(j)\ \ \cases{\varphi_{{\bf h}_1} + \varphi_{{\bf h}_2} + \varphi_{{\bf h}_3}\cr - \varphi_{{\bf h}_{1}{\bf R}_p}- \varphi_{{\bf h}_2} + \varphi_{{\bf h}_{1}{\bf R}_{p} + {\bf h}_2}\cr -\varphi_{{\bf kR}_i} - \varphi_{{\bf h}_{3} - {\bf kR}_{i}{\bf R}_p} - \varphi_{{\bf h}_{1}{\bf R}_{p} + {\bf h}_2}\cr -\varphi_{{\bf h}_3} + \varphi_{{\bf kR}_{i}{\bf R}_p} + \varphi_{{\bf h}_{3} - {\bf kR}_{i}{\bf R}_p}}(k)\ \ \cases{\varphi_{{\bf h}_1} + \varphi_{{\bf h}_2} + \varphi_{{\bf h}_3}\cr -\varphi_{{\bf h}_{1}{\bf R}_p} -\varphi_{{\bf h}_{2} + {\bf kR}_i} - \varphi_{{\bf h}_{3} - {\bf kR}_{i}{\bf R}_p}\cr -\varphi_{{\bf h}_2} - \varphi_{{\bf kR}_i} + \varphi_{{\bf h}_{2} + {\bf kR}_i}\cr - \varphi_{{\bf h}_3} + \varphi_{{\bf kR}_{i}{\bf R}_p} + \varphi_{{\bf h}_{3} - {\bf kR}_{i}{\bf R}_p}}(l)\ \ \cases{\varphi_{{\bf h}_1} + \varphi_{{\bf h}_2} + \varphi_{{\bf h}_3}\cr - \varphi_{{\bf h}_{1}{\bf R}_p} - \varphi_{{\bf h}_{2} - {\bf kR}_{i}{\bf R}_p} - \varphi_{{\bf h}_{3} + {\bf kR}_i}\cr -\varphi_{{\bf h}_2} + \varphi_{{\bf kR}_{i}{\bf R}_p} + \varphi_{{\bf h}_{2} - {\bf kR}_{i}{\bf R}_p}\cr -\varphi_{{\bf h}_3} -\varphi_{{\bf kR}_i} + \varphi_{{\bf h}_{3} + {\bf kR}_i}}(m)\ \ \cases{\varphi_{{\bf h}_1} + \varphi_{{\bf h}_2} + \varphi_{{\bf h}_3}\cr -\varphi_{{\bf kR}_i} + \varphi_{{\bf kR}_{i}{\bf R}_p} - \varphi_{{\bf kR}_{i}({\bf R}_{p}-{\bf I})}\cr -\varphi_{{\bf h}_{1}{\bf R}_p} - \varphi_{{\bf h}_2} + \varphi_{{\bf h}_{1}{\bf R}_{p} + {\bf h}_2}\cr -\varphi_{{\bf h}_3} - \varphi_{{\bf h}_{1}{\bf R}_{p} + {\bf h}_2} + \varphi_{{\bf h}_{1}({\bf R}_{p}-{\bf I})}}(n)\ \ \cases{\varphi_{{\bf h}_1} + \varphi_{{\bf h}_2} + \varphi_{{\bf h}_3}\cr -\varphi_{{\bf kR}_i} + \varphi_{{\bf kR}_{i}{\bf R}_{p}} - \varphi_{{\bf kR}_{i}({\bf R}_{p}-{\bf I})}\cr -\varphi_{{\bf h}_{1}{\b

Notes: The joint probability distribution method described in paper I [Giacovazzo, Burla & Cascarano (1992). Acta Cryst. A48, 901–906] and paper II [Burla, Cascarano & Giacovazzo (1992). Acta Cryst. A48, 906–912] of this series has been considered in order to obtain a function that is maximized by the true crystal structure. The phasing process is carried out by maximizing such a function via a modified tangent refinement: this implies the active use of negative triplet and quartet relationships. The major effects provoked in a direct procedure by the use of numerous phase relationships with expected negative cosines are analysed. Practical applications are also described.

Notes: Extraction of structure-factor amplitudes from a powder diffraction pattern is not a straightforward procedure. Peak overlapping and background estimation are the main obstacles to the process: they may introduce strong correlations among reflection intensities and heavy errors in their estimates. The program EXTRA is described, which, on the basis of the Le Bail algorithm, is able reliably to estimate the structure-factor amplitudes in a fully automatic way.

Notes: The results of the structural determination of Li2MnO3 from X-ray powder diffraction data and the refinement by the Rietveld technique are presented. The Li2MnO3 structure has a monoclinic cell with space group C2/m(Z = 4) and cell parameters a = 4.9246 (1), b = 8.5216 (1), c = 5.0245 (1) Å, β = 109.398 (1)°; the refinement with 14 structural parameters for 165 reflections in the pattern leads to Rwp = 17.61%, Rp = 13.25%, RB = 7.07% and S = 3.52. Such a solution agrees with a single-crystal structure determination previously reported in the literature and allows other hypotheses to be rejected.

Notes: The principal limitation of the diffraction methods for crystal structure analysis from powder data is originated by the collapse of the three-dimensional reciprocal space into the one dimension of the powder diffraction pattern. The degradation of the information can make difficult even the solution of small crystal structures and can generate inefficiencies in the least-squares methods devoted to crystal structure refinement. In this paper, the current two-stage procedures, the first stage dedicated to powder-pattern decomposition and the second to direct phasing of powder data, are analysed. It is shown that in the first stage such procedures disregard a large amount of information that can become available during the process of crystal structure solution and analysis. The use of such information is essential for making direct-methods procedures more robust and for improving the accuracy of the least-squares techniques. The performances of EXTRA [Altomare, Burla, Cascarano, Giacovazzo, Guagliardi, Moliterni & Polidori (1995). J. Appl. Cryst. 28, 842–846], a program for full-pattern decomposition based on the Le Bail algorithm, and of SIRPOW.92 [Altomare, Burla, Cascarano, Giacovazzo, Guagliardi, Polidori & Camalli (1994). J. Appl. Cryst. 27, 435–436], a direct-methods program optimized for powder data, are discussed in order to offer to the reader a logical pathway for the analysis of the traditional techniques and for the proposition of a new approach. It is shown that pattern-decomposition programs based on the Le Bail algorithm are able to exploit the prior information in a more effective way than Pawley-method-based decomposition programs.