In a last step, when all integer variables have been assigned, the values of the real lattice parameters can be numerically fitted. The accuracy of the result can be assessed by the factors and , where 10 is the number of reflections, and q xyz , i , q z , i are the measured and g xyz , i , g z , i the calculated peak positions of the i th reflection.
As an example, the influences of the refraction correction on the lattice constants are demonstrated by using uncorrected values for q z. Importantly, however, as the underlying equations do not allow a unique mathematical solution, it must still be checked if the lattice obtained corresponds to the reduced unit cell. For this purpose, three reciprocal lattice vectors g 1 , g 2 and g 3 , e. Listing the lengths of these vectors in ascending order yields 7.
But as the first five vectors are coplanar, our solution matches the Buerger and reduced cell. On the basis of this solution, peak positions are calculated and plotted in Fig. In addition to the ten peaks we initially selected, all other observed peaks can now be indexed according to this unit cell. The reciprocal-space map reveals, besides the diffraction peaks of HOPG see discussion above , a highly regular sequence of Bragg peaks located at constant q z values.
In this example, the indexing procedure was performed on 16 selected reflections. X-ray diffraction of o -F 2 -6P, as grown on a plane of HOPG; the inset gives the chemical structure of the molecule. Selected peaks for the indexing procedure are marked by triangles around the maximum of the diffracted intensity; the crosses give the calculated peak positions based on indexing together with the assigned Laue indices. In marked contrast to non-substituted 6P growing in a monoclinic crystal structure Baker et al.
Note that as a result of equation 24 the Laue indices h and l can be either positive or negative. It can be proven that a , b and c are the shortest non-coplanar unit-cell vectors.
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As an illustration of the comparison between calculated peaks and experimental data, the peak positions are plotted in Fig. The transition from inclined to parallel molecular planes in these structures has been ascribed to the impact of intramolecular polar bonds by the authors.
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For o -F 2 -6P, no bulk crystal structures have been published yet, and structural characterization was limited to sub-monolayers on Ag by low-temperature scanning tunnelling microscopy Niederhausen et al. There, in contrast to non-fluorinated 6P molecules which individually adsorb in the sub-monolayer regime on the metal surface without packing, the authors find a flat lying stack arrangement of the o -F 2 -6P molecules with small lateral shifts along the row direction. The net dipole moment of o -F 2 -6P is derived as 1.
Building upon previous work Salzmann et al. In the case of powder diffraction only the lengths of the reciprocal-space vectors are used for indexing. The search of indexing solutions typically starts from the cubic end of the symmetry sequence. Each crystal system is explored independently up to a maximum input volume, unless a solution has been found with a higher symmetry. Indexing of GIXD patterns is based on the knowledge of two components of the reciprocal space vector, the in-plane component q xy and the out-of-plane component q z [Fig.
Since the lattice type cannot be assigned a priori , we suggest following an iterative approach. Moreover, boundary conditions and experimental constraints can be included in the indexing procedure with less numerical effort. If no satisfactory solution can be found with a specific lattice type, the next lower symmetry system will then be used.
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In single-crystal diffraction, where reciprocal lattice vectors are used for indexing, in a first step the model parameters are refined in a triclinic setting. If possible symmetry elements are detected, cell refinement with symmetry-bases restraints is performed Sauter et al. Determination of the symmetry profile in crystallographic structures is a persistent challenge Hicks et al.
The final goal in crystallographic analysis is the determination of the most fundamental property of the structure — the correct space group. Higher metric symmetry is usually identified by computer programs Hicks et al. We emphasize that in the present work we focus only on the first point.
For determining the lattice parameters and indexing of the diffraction pattern it is appropriate to choose the crystallographic system of the highest order which can be rationally fitted to the measured reflections. In GIXD the diffraction intensities are influenced by various parameters. This may impede the determination of the correct Laue group. For the reliable identification of systematic absences it is further necessary to obtain a reasonable number of reflections. Indexing of GIXD data of fibre-textured films is important for phase analysis as well as for the identification of new polymorphs.
In the present work, we provide a unifying framework for indexing reciprocal-space maps obtained by GIXD for monoclinic lattices and lattices of higher symmetry. Our approach of including the Bragg peak from a specular X-ray diffraction experiment into the mathematical formalism is of considerable help for indexing of GIXD patterns, where the spatial orientation of the unit cell must be considered.
Mathematical expressions with a significantly reduced number of unit-cell parameters are derived, which facilitates the computational efforts. For crystallographic lattices of higher symmetry, where the set of unit-cell parameters is reduced, the specular diffraction peak is still important for determining the orientation of the crystallographic unit cell relative to the sample surface. Procedures are described in detail for how to use the derived mathematical expressions.
We demonstrate the high value of our approach by successfully applying our formalism for indexing diffraction patterns of two organic semiconductors grown as crystalline thin films on graphite surfaces.
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Then for the monoclinic system the following expression can be derived:. From equation 11 , the following formulae for in tetragonal and hexagonal systems, respectively, can be derived:.
With and. If g xy ,1 , g z ,1 and g xy ,2 , g z ,2 are the components of two Bragg peaks with the corresponding Laue indices h 1 , k 1 and h 2 , k 2 , respectively, as a result of equation 14 the following relations arise:. Expressing in equation 28 as a function of and substituting this term in equation 29 leads to a quadratic equation for , which has the two solutions.
An optimal solution for a and b can be obtained by minimizing the following function F :. This can be achieved if and are fulfilled. Then the following solution can be obtained:. Using these substitutions, equation 17 can be rewritten as. This work was funded by Austrian Science Fund grant P Deutsche Forschungsgemeinschaft grant National Center for Biotechnology Information , U.
J Appl Crystallogr.
Published online Apr 1. Author information Article notes Copyright and License information Disclaimer. Correspondence e-mail: ta. Received Nov 12; Accepted Feb This is an open-access article distributed under the terms of the Creative Commons Attribution CC-BY Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original authors and source are cited.
Abstract Grazing-incidence X-ray diffraction studies on organic thin films are often performed on systems showing fibre-textured growth. Keywords: grazing incidence X-ray diffraction, indexing, fibre texture, uni-planar texture, mathematical crystallography. Open in a separate window.
Figure 1. Table 2 Relations for the total length g xyz , the out-of-plane component g z and the in-plane component g xy of the reciprocal-space vectors with indices hkl and of the vector uvw g spec by using direct and reciprocal lattice parameters and the volume V in triclinic, monoclinic and orthorhombic systems. Figure 2. Heinrich et al. Figure 3. Acknowledgments We thank W. Appendix A. Appendix B. Table 4 Relations for the total length g xyz , the out-of-plane component g z and the in-plane component g xy of the reciprocal-space vectors with indices hkl and of the vector uvw g spec and the volume V in trigonal and hexagonal crystallographic systems.
Appendix C. Calculating a and b from two pairs of g xy , g z in a monoclinic lattice If g xy ,1 , g z ,1 and g xy ,2 , g z ,2 are the components of two Bragg peaks with the corresponding Laue indices h 1 , k 1 and h 2 , k 2 , respectively, as a result of equation 14 the following relations arise:.
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Appendix E. References Baker, K. Polymer , 34 , — Boultif, A. Bronshtein, I. Handbook of Mathematics , 6th ed. Berlin: Springer.