research papers
Indexing grazingincidence Xray diffraction patterns of thin films: lattices of higher symmetry
^{a}Department of Neuroradiology, Vascular and Interventional Radiology, Medical University Graz, Auenbruggerplatz 9, Graz, 8036, Austria, ^{b}Institute of Solid State Physics, Technical University Graz, Petersgasse 16, Graz, 8010, Austria, ^{c}Department of Chemistry and IRIS Adlershof, HumboldtUniversität zu Berlin, BrookTaylorStrasse 2, Berlin, 12489, Germany, and ^{d}Department of Physics, Department of Chemistry and Biochemistry, Centre for Research in Molecular Modeling (CERMM), Centre for Nanoscience Research (CeNSR), Concordia University, 7141 Sherbrooke Street W., SP 26520, Montreal, Quebec, Canada H4B 1R6
^{*}Correspondence email: josef.simbrunner@medunigraz.at
Grazingincidence Xray diffraction studies on organic thin films are often performed on systems showing fibretextured growth. However, indexing their experimental diffraction patterns is generally challenging, especially if lowsymmetry lattices are involved. Recently, analytical mathematical expressions for indexing experimental diffraction patterns of triclinic lattices were provided. In the present work, the corresponding formalism for crystal lattices of higher symmetry is given and procedures for applying these equations for indexing experimental data are described. Two examples are presented to demonstrate the feasibility of the indexing method. For layered crystals of the prototypical organic semiconductors diindenoperylene and (orthodifluoro)sexiphenyl, as grown on their yet unknown unitcell parameters are determined and their crystallographic lattices are identified as monoclinic and orthorhombic, respectively.
Keywords: grazing incidence Xray diffraction; indexing; fibre texture; uniplanar texture; mathematical crystallography.
1. Introduction
Crystalline thin films are often characterized by grazingincidence Xray diffraction (GIXD) owing to the high surface sensitivity of the technique. A schematic drawing of the experimental method is shown in Fig. 1(a). To achieve this surface sensitivity, the angle of incidence of the primary beam relative to the sample surface (α_{i}) is chosen close to the critical angle of total external reflection, and the scattered Xray beam encloses an inplane angle θ_{f}, as well as an outofplane angle α_{f} relative to the surface. The primary Xray beam, described by the wavevector k_{0}, and the scattered beam k determine the scattering vector q by q = k − k_{0}. In this geometry the inplane component q_{xy} and the outofplane component q_{z} of q are given by
and
A key direction in z direction at q_{xy} = 0, which reveals the orientation of crystals relative to the substrate surface. In the important case of crystalline fibretextured films, it defines the crystallographic plane of the thinfilm crystallites which is parallel to the substrate surface (i.e. the contact plane). However, in GIXD geometry the z direction is inaccessible in To experimentally assess this direction, one needs to resort to specular Xray diffraction [cf. Fig. 1(b)], where the incidence angle of the primary beam (α_{i}) and the exit angle of the scattered Xray beam (α_{f}) are equal. This results in a scattering vector q always perpendicular to the substrate surface and the scattering vector then exclusively has a q_{z} component (with q_{xy} = 0). In the following, a diffraction peak observed via specular Xray diffraction is denoted as . The correlation between the scattering vector q and the corresponding periodicities is obtained via the Laue equation: for the appearance of a diffraction peak the scattering vector q has to be equal to a vector g.
is theIn general, indexing of GIXD patterns means assigning q_{z} and q_{xy} – are taken from experimental reciprocalspace maps and available for indexing (Smilgies & Blasini, 2007; Hailey et al., 2014). As the fibretextured crystals have a well defined crystallographic contact plane, the rotation matrix of the crystallites relative to the sample surface must be considered (Shmueli, 2006), i.e. the contact plane of the investigated crystals must be determined. In many cases, it can be deduced from specular Xray diffraction data; however, the correct assignment to a crystallographic plane with u, v and w cannot be obtained from specular Xray diffraction alone, if the is unknown (Salzmann & Resel, 2004; Smilgies & Blasini, 2007; Hailey et al., 2014; Jiang, 2015). By combining the peak positions of the GIXD pattern (q_{xy}, q_{z}) with the specular peak position (q_{spec}), mathematical expressions can be derived where the required number of unknown parameters for indexing the pattern is significantly reduced. In recent work, we demonstrated this approach for the general case of the triclinic crystallographic system (Simbrunner et al., 2018).
to the observed Bragg peaks. In the specific case of GIXD data recorded on fibretextured films, which is covered in the present manuscript, two components of the vectors –Here, we now turn our focus to crystallographic lattices of higher symmetry. Obviously, the equations for the inplane and outofplane components of the reciprocalspace vectors become less complex if the number of unitcell parameters is reduced. This can facilitate the indexing procedure; however, the parameters of the rotation matrix do not reduce. In the following, we first concentrate on monoclinic systems and subsequently present mathematical expressions for all other systems. Via two practical examples, we then explicitly demonstrate the application of these equations. We stress that our indexing procedure does not rely on knowing either the chemical structure of the material or the experimental intensities of the diffraction peaks. We show that the presented approach can succeed with a limited number of reflections for experimental data of high quality.
2. Indexing method
For the mathematical treatise a laboratory coordinate system is assumed, where the xy plane runs parallel to the substrate surface; a, b, c, α, β and γ are the parameters of the (direct) and a*, b*, c*, α*, β* and γ* are the reciprocal cell parameters (Giacovazzo, 2011), which are summarized in Table 1. If the (001) plane is parallel to the substrate surface in a GIXD experiment, the vector g with its h, k and l can be represented by the equation
where the matrix is given as
When the Laue condition q = g is fulfilled, diffraction can be observed.

If the (001) A_{001}^{*} must be transformed, i.e. the rotation matrix of the thinfilm crystallites relative to the substrate surface has to be considered (Shmueli, 2006). In particular, it has to be rotated around the which is defined by the (001) plane and the new contact plane (uvw). Then the reciprocalspace vector with the Laue/Miller indices u, v and w, g_{uvw}, is perpendicular to the contact plane. Its magnitude is purely out of plane and corresponds to the specular scan, g_{spec}. Furthermore, the outofplane component g_{z} and inplane component g_{xy} of any reciprocalspace vector g_{hkl} can be determined by their scalar product and vector (cross) product with g_{uvw}, respectively. As the magnitude of a vector and the scalar product of two vectors are both rotation invariant, the unrotated expressions, i.e. equation (3), for g_{hkl} and g_{uvw} are sufficient. Therefore, the following expressions result:
plane is not parallel to the substrate surface, the matrixwith , ,
and
where the volume is
Furthermore, from equation (3) the length of the vector , g_{xyz}, can be determined as
Using equations (5) and (6), equation (9) can be rewritten as
and by algebraic transformations the following expression can be derived:
As a result of symmetry relationships, further analogous expressions for the cell parameter triples (a, c, β) and (b, c, α) and their corresponding Miller and can be derived.
From equations (5) and (6) the following expression can be obtained:
From equation (12), the residual unitcell parameters c, α, β and the l can be determined (Truger et al., 2016). Analogously, the following equation is valid:
In a monoclinic system (α = γ = 90°), equation (11) reduces to
Equation (14) comprises – in addition to the rotation parameters u and v – only the parameters a and b, as well as the h and k. This facilitates the mathematical analysis, where the integer variables can be varied and only two real unknowns must be calculated.
In rare cases, if the net planes oriented parallel to the substrate surface have a weak et al., 2012), g_{spec} must be treated as an additional real unknown parameter, which necessitates a more extensive numerical analysis. In the most general case, if no contact plane exists, u, v and w are nonintegers. In this case it is preferable to use an alternative notation of the rotation matrix (see Appendix A).
which inhibits the acquisition of specular diffraction data (DjuricFor calculating the remaining unitcell parameters c, β and the l, the following expression results from equation (12):
Analogously to equation (14), by choosing related substitutions on the basis of the symmetric properties of the relations for g_{xyz}, g_{xy}, g_{z} and g_{spec} for the general (triclinic) case, the following expressions hold:
Furthermore, equation (13) reduces to
For orthorhombic systems, the formulae are valid with β = 90°. In Table 2 we summarize the mathematical expressions for g_{xyz}, g_{z}, g_{xy} and the volume V in triclinic, monoclinic and orthorhombic systems. In tetragonal and cubic lattices, in addition to β = 90°, a = b and a = b = c, respectively. In Appendix B useful formulae for tetragonal, trigonal and hexagonal systems are listed.
As every linear combination of the unitcell vectors obeys the Laue condition, the mathematical solutions are not unique and include superlattices (Santoro et al., 1980). The definite crystallographic solution, the is defined as the cell that satisfies the conditions derived from the reduction theory of quadratic forms (Niggli, 1928). The main conditions for reduction require that the is based on the three shortest vectors of the such a is then called a Buerger cell (Buerger, 1957). However, this cell may not be unique. An unambiguous is the `reduced cell' defined by Niggli; the criteria are listed in detail in International Tables for Crystallography (De Wolff, 2016). In general, the criteria for reduced cells demand that a^{2} ≤ b^{2} ≤ c^{2} and that the angles are either acute (type I) or obtuse (type II). Geometrical ambiguities, called metric singularities, which generate subcells of lower symmetry, may occur in some indexing cases (Santoro & Mighell, 1970; Mighell, 2000). Such singularities can often be easily detected from simple relationships between the parameters of the two cells (Boultif & Louër, 2004).
In the following we turn to employing the formalism derived above to determine the unitcell parameters of two prototypical organic semiconductors, which show as yet unknown polymorphs if grown in thin films on substrates of
(HOPG).3. Examples
3.1. Experiments
Thin films of diindenoperylene (DIP, C_{32}H_{16}) and of orthodifluorosexiphenyl (oF_{2}6P, C_{36}F_{2}H_{24}) were grown on freshly cleaved HOPG (ZYA quality) by physical vapour deposition in a high vacuum (final nominal film thickness 30 nm; base pressure < 5 × 10^{−6} Pa). The films were characterized at beamline W1 at the synchrotron radiation facility DORIS (HASYLAB, Hamburg, Germany). GIXD experiments were performed together with specular Xray diffraction using a goniometer in pseudo 2 + 2 geometry and a onedimensional detector (MYTHEN, Dectris). The wavelengths of the primary radiation were 1.1796 and 1.1801 Å for DIP and oF_{2}6P, respectively. GIXD experiments were performed using incident angles of the primary beam relative to the HOPG substrate of α_{i} = 0.13 and 0.15° for DIP and oF_{2}6P, respectively. Reciprocalspace maps were recorded by keeping the sample fixed and by performing a series of detector scans along the inplane scattering angle θ_{f} at differently fixed outofplane scattering angles α_{f}. The vertical mounting of the detector allows the simultaneous measurement of Δα_{f} = 3.5°. The diffraction pattern was transformed to using the custommade software PyGID (Moser, 2012). The resulting reciprocalspace maps give the measured intensities on a logarithmic scale by a colour code; the exact positions of the Bragg peaks in terms of q_{xy} and q_{z} were determined by Gaussian fits. The q_{z} values of the peak positions were corrected for maximum variations of 0.011 and 0.014 Å^{−1} are obtained for DIP and oF_{2}6P, respectively (Resel et al., 2016).
3.2. Diindenoperylene
A specular scan of DIP crystals grown on a HOPG substrate is shown in Fig. 2(a). The dominant diffraction peak at q_{z} = 1.872 Å^{−1} (d = 3.356 Å) can be identified as the 002 peak of the HOPG substrate. Forming the difference with the reference data recorded for the plain substrate (orange curve) reveals a clear diffraction peak at q_{spec} = 1.776 Å^{−1}, which is assigned to the DIP adsorbate and corresponds to a spacing of 3.54 Å. Note that a specular reflection from DIP at the same position has been observed before for gold substrates by Dürr et al. (2003). The authors concluded on a lying molecular orientation from their data and denoted this polymorph as the λ phase; however, no direct determination of the by indexing GIXD reciprocalspacemap data has been performed before. Our reciprocalspace map of DIP on HOPG is shown in Fig. 2(b). Strong diffraction features of the HOPG substrate are located along q_{xy} = 2.94 Å^{−1}. Additionally, weak diffraction features at q_{xy} = 2.24 Å^{−1}/q_{z} = 0.98 Å^{−1} and at q_{xy} = 2.34 Å^{−1}/q_{z} = 0.98 Å^{−1} are also due to the HOPG substrate. Bragg peaks at low q values (q^{2} = q_{xy}^{2} + q_{z}^{2}) were selected for the indexing procedure and lie at centres of the triangles in Fig. 2(b). In total, we only used ten Bragg peaks (based on their q_{xy} and q_{z} positions) together with the Bragg peak of the specular diffraction pattern (q_{spec}) for our indexing procedure. Note that q_{spec}, q_{xy} and q_{z} are experimental results which describe specific components of the scattering vector, while g_{spec}, g_{xy} and g_{z} refer to these components within the According to the Laue condition, the individual components of the scattering vector q and of the vectors g must be equal.
As known crystal structures of DIP exhibit a monoclinic et al., 2007; Kowarik et al., 2009), in a first step of indexing, equation (14) may be chosen. The combination of Bragg peaks from the reciprocalspace map (q_{xy} and q_{z}) with the peak from the specular scan (q_{spec}) reduces the number of parameters considerably to two integer numbers (instead of three) for the of the contact plane, to two integer numbers (instead of three) for the of each Bragg peak, and to two real numbers (instead of four) for the constants. In theory, by using a quadratic equation, a and b can be calculated from two independent pairs of (g_{xy}, g_{z}) (see Appendix C). Because of experimental inaccuracies, however, we prefer treating equation (14) as linear with the unknowns , and × . The first step of indexing is then based on a systematic variation of integer variables: (i) a pair of for the contact plane (the crystallographic plane of DIP crystals parallel to the substrate surface), and (ii) two for each of the selected Bragg peaks from the reciprocalspace map [Fig. 2(b)]. By using several pairs of (g_{xy}, g_{z}), sets of overdetermined linear equations are then obtained, from which the solutions with the least errors e concerning the system of equations (i.e. norm of the residuals) and the factor are selected. Furthermore, the volume of the resulting should be as small as possible. From λ_{1}, λ_{2} and λ_{3} the solution for a and b can be optimized (see Appendix D). In our case, for u = 1 and v = 2, e and f of the optimal solution are 4.8 × 10^{−3} and 1.1 × 10^{−3}, respectively. The next best solutions to the optimal choice of the h and k have almost the same error e (plus 1% and minus 1%, respectively) but a larger factor f of about 40 and 56%, respectively.
(HeinrichAs equation (16) is analogous to equation (14), two solutions (for a and c, respectively) with consistent unitcell parameter b, Miller index v and k must exist. If the are chosen to be v = 2 and w = 1, an appropriate solution can be found. Thus, a monoclinic with the contact plane (121) results and the cell parameters a, b and c and the h, k and l can be assigned to the experimental pattern. In a last step, using equation (15), the last cell parameter β can be obtained.
Alternatively, in a first step of indexing, equation (17), which is valid for any type, can be used. For this procedure we use the following specific algorithm: appropriate mathematical substitutions transform equation (17) into linear equations with three real unknowns, containing the constants a, c and β (see Appendix E). Assigning two low integers as the u and w, and choosing h and l in a specific range, is a starting point to find solutions for the constants a, c and β. In theory, a set of constants can be obtained from three equations with independent pairs of g_{xy} and g_{z}. Being linearized, these equations can be solved analytically by determining the inverse matrix of their coefficients. To account for experimental inaccuracies, however, all ten peaks can be included simultaneously, again for creating an overdetermined system of linear equations. This system can be solved either by multiplying both sides of the equations with the transpose of the coefficient matrix or by QR decomposition to solve the leastsquares problem (Bronshtein et al., 2015). In addition to an optimized approximated solution of the unknown variables, the associated error can be calculated. By varying the integer variables, the final solution to the overdetermined system is found. This is done by choosing the optimal set of equations regarding a small error and a volume of the resulting that is as small as possible. To overcome the computational complexity with an increasing number of equations, this procedure is performed stepwise. In a first step, four equations with small q values are chosen, for which the possible range of is restricted. Only systems with errors not exceeding a certain limit are included further. Thus the h and l for all reflections can be assigned and the unitcell parameters a, c and β are calculated. A modification of this algorithm relies on comparing the systematically varied coefficients of the ten linear equations and seeking clusters of related coefficients.
Thus, for u = 1 and w = 1 a solution for a first set of constants (a, c, β) and the h and k can be obtained. Then, in a second step, it can be checked if equation (18) results for all peak positions (q_{xy}, q_{z}) in integer multiples of 2π/b. If – as is the case here – this condition is fulfilled, the assumption of a monoclinic is justified. Furthermore, using equation (13) for a triclinic system and calculating α and γ does not result in a more accurate final solution. Consequently, the constant b is obtained and the k for all reflections can be assigned.
In a last step, when all integer variables have been assigned, the values of the real g_{xyz}, g_{xy} and g_{z} in Table 2 can be used. For our example of DIP on HOPG, we then obtain the following solution, which obeys the scalar product criteria for typeII cells: u = 1, v = 2, w = 1; a = 7.149, b = 8.465, c = 16.62 Å, α = 90, β = 93.14, γ = 90°, V = 1004.5 Å^{3}. The accuracy of the result can be assessed by the factors and d_{10,z} = , 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 ith reflection. As an example, the influences of the refraction correction on the constants are demonstrated by using uncorrected values for q_{z}. In this case, we would then obtain significantly different parameters instead: a = 7.194, b = 8.440, c = 16.60 Å, α = 90, β = 93.54, γ = 90°, V = 1005.7 Å^{3} (d_{10,xyz} = 0.006, d_{10,z} = 0.025).
parameters can be numerically fitted. For this procedure, the expressions forImportantly, however, as the underlying equations do not allow a unique mathematical solution, it must still be checked if the g_{1}, g_{2} and g_{3}, e.g. of the three independent Laue triples (0, 0, 1), (1, 1, 0) and (−1, 1, 0), are calculated and the matrix
obtained corresponds to the reduced For this purpose, three vectorsis formed. By multiplying its inverse matrix G^{−1} with vectors 2π(m_{1}, m_{2}, m_{3})^{T}, where m_{i} are systematically varied integers in a reasonable range (e.g. between −3 and 3), vectors of the and its superlattices can be obtained (Simbrunner et al., 2018). Listing the lengths of these vectors in ascending order yields 7.149, 8.465, 11.08, 14.30, 16.616 and 16.62 for the six shortest vectors. 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. 2(b) as crosses. In addition to the ten peaks we initially selected, all other observed peaks can now be indexed according to this unit cell.
Our result is significantly different from the known crystal structures of DIP, that is, an enantiotropic polymorph that is stable at temperatures above 423 K (Heinrich et al., 2007) showing constants of a = 7.171, b = 8.55, c = 16.80 Å, α = 90, β = 92.42, γ = 90°, V = 1029.0 Å^{3}, as well as a thinfilm phase grown on sapphire (Kowarik et al., 2009) with a = 7.09 ± 0.04, b = 8.67 ± 0.04, c = 16.9 ± 0.5 Å, α = 90, β = 92.2 ± 0.2, γ = 90°, V = 1037 ± 30 Å^{3}. In Table 3 we summarize the various results. Comparing these cells highlights that DIP grows on HOPG in a previously unknown polymorph, most likely in the same polymorph which has been referred to as λ phase in the literature before (Dürr et al., 2003; Kowarik et al., 2006; Casu et al., 2008).

3.3. orthoDifluorosexiphenyl
The experimental GIXD data for the oF_{2}6P film [the synthesis of oF_{2}6P is reported by Niederhausen et al. (2018)] as grown on HOPG are shown in Fig. 3. We now observe a well pronounced specular diffraction peak at q_{spec} = 1.636 Å^{−1}, corresponding to a spacing of 3.84 Å in the outofplane direction. Like the case of DIP above, this points to lying πstacked growth of the molecules on the HOPG substrate, a growth behaviour that has been reported before for related systems (Salzmann et al., 2012). The reciprocalspace 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.
In marked contrast to nonsubstituted 6P growing in a monoclinic et al., 1993; Resel, 2003), from the highly symmetric diffraction pattern, an orthorhombic with its contact plane parallel to the (010) plane (u = w = 0) can be assumed for oF_{2}6P on HOPG. As for the specular scan the relation (v = 2) holds, the following expressions (see Table 2) follow:
(BakerOn that basis, we obtained for the oF_{2}6P on HOPG: u = 0, v = 2, w = 0; a = 5.724, b = 7.659, c = 27.424 Å, α = β = γ = 90°, V = 1202.3 Å^{3}. The accuracy of the result can be assessed by and , as for d_{N,z} only the 10 reflections with k > 0 are included. Note that as a result of equation (24) the h and l can be either positive or negative. Again, if uncorrected values for q_{z} were used, b = 7.614 Å would follow. It can be proven that a, b and c are the shortest noncoplanar unitcell vectors.
ofAs an illustration of the comparison between calculated peaks and experimental data, the peak positions are plotted in Fig. 3(b) as crosses; there are no crystal structures known so far for molecular crystals of oF_{2}6P. The low outofplane spacing that we observe via specular diffraction strongly points to πstacked growth of this fluorinated 6P derivative, if grown on HOPG. This is remarkable insofar as nonsubstituted 6P grows in a herringbone arrangement of the molecules in all known structures and on numerous surfaces including HOPG (Resel, 2003) and, therefore, it is obviously the (onesided) fluorination of the molecule that changes its growth behaviour dramatically. A similar behaviour has been observed for pentacene (herringbone) and perfluoropentacene on HOPG (πstacking) (Salzmann et al., 2012), as well as for 6,13bis(triisopropylsilylethynyl)pentacene and pentacenequinone (Swartz et al., 2005). The transition from inclined to parallel molecular planes in these structures has been ascribed to the impact of intramolecular polar bonds by the authors. For oF_{2}6P, no bulk crystal structures have been published yet, and structural characterization was limited to submonolayers on Ag(111) by lowtemperature scanning tunnelling microscopy (Niederhausen et al., 2018). There, in contrast to nonfluorinated 6P molecules which individually adsorb in the submonolayer regime on the metal surface without packing, the authors find a flat lying stack arrangement of the oF_{2}6P molecules with small lateral shifts along the row direction. The net of oF_{2}6P is derived as 1.1 Debye owing to the polar C—F bonds at the ortho position of the outer phenyl ring, which appear to maximize their distance in neighbouring molecules.
Building upon previous work (Salzmann et al., 2011; Truger et al., 2016), the determined in the present work for oF_{2}6P (and, likewise, for DIP) can now be used to model the molecular arrangement therein, which will be the subject of a forthcoming study.
4. Discussion – determining the crystallographic system
In the case of powder diffraction only the lengths of the reciprocalspace vectors are used for indexing. In the dichotomy method the cell constants are varied in increasingly smaller intervals and the hkl indices are subsequently refined using the leastsquares method (Boultif & Louër, 1991, 2004). The search of indexing solutions typically starts from the cubic end of the symmetry sequence. Each 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 q_{xy} and the outofplane component q_{z} [Fig. 1(a)]. Since the type cannot be assigned a priori, we suggest following an iterative approach. As a consequence of imperfect data due to experimental inaccuracies, it seems that starting with a of higher symmetry is favourable, since incoherency becomes evident quickly and the tendency to find only a `local minimum' when finally optimizing the cell parameters increases with the number of variables in the equations. 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 type, the next lower symmetry system will then be used.
vector, the inplane componentIn singlecrystal diffraction, where et al., 2004). In the matrix approach to symmetry (Himes & Mighell, 1987) this is accomplished by 64 symmetry matrices to check if the transformation leads to identity. Furthermore, systems of higher symmetry imply a high impact of symmetry considerations such as diffraction intensities and (Hahn, 2006). Determination of the symmetry profile in crystallographic structures is a persistent challenge (Hicks et al., 2018) and certainly lies beyond the scope of the present work.
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 with symmetrybases restraints is performed (SauterThe final goal in crystallographic analysis is the determination of the most fundamental property of the structure – the correct ): (i) the derivation of the correct space i.e. the smallest primitive (reduced) (ii) the assignment of the correct Laue group on the basis of the symmetry of the diffraction intensities and an initial decision if a structure is centrosymmetric; (iii) the identification of any characteristic of translational symmetry elements (glide planes or screw axes); and (iv) the final decision as to whether or not the structure is centrosymmetric. Higher metric symmetry is usually identified by computer programs (Hicks et al., 2018).
Four steps are necessary to achieve this endeavour (Marsh, 1995We emphasize that in the present work we focus only on the first point. For determining the ).
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. However, it is not the shape of the that determines the type but the symmetry of the diffraction intensities (Marsh, 1995In GIXD the diffraction intensities are influenced by various parameters. This may impede the determination of the correct Laue group. Furthermore, as the inplane component of the reflections is measured, the i.e. they cannot be assigned with positive or negative sign, as is the case in our example of oF_{2}6P. For the reliable identification of it is further necessary to obtain a reasonable number of reflections.
in lattices of higher symmetry can be `degenerate',5. Conclusion
Indexing of GIXD data of fibretextured 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 reciprocalspace maps obtained by GIXD for monoclinic lattices and lattices of higher symmetry. Our approach of including the Bragg peak from a specular Xray diffraction experiment into the mathematical formalism is of considerable help for indexing of GIXD patterns, where the spatial orientation of the orthodifluorosexiphenyl, the unitcell parameters of which were successfully determined following our approach.
must be considered. Mathematical expressions with a significantly reduced number of unitcell parameters are derived, which facilitates the computational efforts. For crystallographic lattices of higher symmetry, where the set of unitcell parameters is reduced, the specular diffraction peak is still important for determining the orientation of the crystallographic 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. We find a monoclinic for diindenoperylene and an orthorhombic forAPPENDIX A
Alternative rotation parameters if no contact plane exists
If u, v and w have to be assumed to be nonintegers, it may be preferable to replace the vector g_{uvw} by a sample surface normal vector n = (sinψsinΦ, −cosψsinΦ, cosΦ)^{T} with the rotation angles Φ and ψ. Then for the monoclinic system the following expression can be derived:
The same equation is valid in the orthorhombic case.
For the tetragonal and cubic systems the following expression is valid:
Therefore, especially in the cubic case, it is much easier to solve the equation
first, as in the case of powder diffraction, and then to determine the rotation parameters.
Note that for DIP Φ = 76.11° and ψ = 149.37°. In the case of oF_{2}6P Φ = 90° and ψ = 180°.
APPENDIX B
Mathematical expressions for tetragonal, trigonal and hexagonal systems
In Table 4 we summarize the mathematical expressions for g_{xyz}, g_{z}, g_{xy} and the volume V in trigonal and hexagonal systems.
From equation (11), the following formulae for in tetragonal and hexagonal systems, respectively, can be derived:
Tetragonal:
With p = hv  ku and P = (h^{2} + k^{2})g_{spec}^{2}  2 (hu + kv ) × g_{spec}g_{z}+ (u^{2} + v^{2})g_{xyz}^{2}
Hexagonal:
With p = hv  ku and P = (h^{2} + hk +k^{2})g_{spec}^{2}  (2hu + 2kv + hv + ku )g_{spec}g_{z} + (u^{2} + uv + v^{2})g_{xyz}^{2}
In both cases, by systematically varying the integer variables u, v, h and k, the factor 2π/a must be constantly observed for every reflection, allowing the assessment of the appropriate Miller and Laue indices.
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 (h_{1}, k_{1}) and (h_{2}, k_{2}), respectively, as a result of equation (14) the following relations arise:
where g_{h,1} = (1 / g_{xy,1}) (h_{1}  ug_{z,1} /g_{spec} ), g_{h,2} = (1 / g_{xy,2}) × (h_{2}  ug_{z,2} /g_{spec} ), g_{k,1} = (1 / g_{xy,1}) (k_{1}  vg_{z,1} / g_{spec} ) and g_{k,2} = (1 / g_{xy,2}) (k_{2}  vg_{z,2} / g_{spec} ). 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
where
Consequently,
APPENDIX D
Determining a and b from λ_{1}, λ_{2} and λ_{3}
From λ_{1}, λ_{2} and λ_{3} the tentative cell parameters a_{t} = 2πλ_{1}^{−1/2}, b_{t} = 2πλ_{2}^{−1/2} and their product (ab)_{t} = (2π)^{2}λ_{3}^{−1/2} can be determined. 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:
where q = a_{t} (ab )_{t}^{2} / (2b_{t}^{2}) and D = q + 2 {[ (ab )_{t}  a_{t}b_{t}] / (3b_{t}) }^{3}.
APPENDIX E
Mathematical procedure for analytically determining the cell parameters a, c and β
For analytically determining the unitcell parameters a, c and β from equation (17) it is convenient to introduce the parameters X_{a}, X_{c} and X_{β} with the substitutional relations
Using these substitutions, equation (17) can be rewritten as
Then from three independent Bragg peak series, the parameters X_{a}, X_{c} and X_{β} can be determined using a system of linear equations. For obtaining a, c and β from equations (36)–(38) the following identity is helpful:
Acknowledgements
We thank W. Caliebe (HASYLAB, DESY) for experimental support.
Funding information
IS acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) [funding reference number RGPIN201805092] and Concordia University. Generous support by the German Research Foundation (DFG via SFB 951) is gratefully acknowledged. Funding from the Austrian Science Fund (grant No. P30222) is also acknowledged.
References
Baker, K. N., Fratini, A. V., Resch, T., Knachel, H. C., Adams, W. W., Socci, E. P. & Farmer, B. L. (1993). Polymer, 34, 1571–1587. CrossRef CAS Web of Science Google Scholar
Boultif, A. & Louër, D. (1991). J. Appl. Cryst. 24, 987–993. CrossRef CAS Web of Science IUCr Journals Google Scholar
Boultif, A. & Louër, D. (2004). J. Appl. Cryst. 37, 724–731. Web of Science CrossRef CAS IUCr Journals Google Scholar
Bronshtein, I. N., Semendyayev, K. A., Musiol, G. & Mühlig, H. (2015). Handbook of Mathematics, 6th ed. Berlin: Springer. Google Scholar
Buerger, M. J. (1957). Z. Kristallogr. 109, 42–60. CrossRef CAS Google Scholar
Casu, M. B., Biswas, I., Nagel, M., Nagel, P., Schuppler, S. & Chassé, T. (2008). Phys. Rev. B, 78, 075310. CrossRef Google Scholar
De Wolff, P. M. (2016). International Tables for Crystallography, Vol. A, SpaceGroup Symmetry, 6th ed., pp. 709–713. Chichester: Wiley. Google Scholar
Djuric, T., Ules, T., Gusenleitner, S., Kayunkid, N., Plank, H., Hlawacek, G., Teichert, C., Brinkmann, M., Ramsey, M. & Resel, R. (2012). Phys. Chem. Chem. Phys. 14, 262–272. CrossRef CAS Google Scholar
Dürr, A. C., Koch, N., Kelsch, M., Rühm, A., Ghijsen, J., Johnson, R. L., Pireaux, J. J., Schwartz, J., Schreiber, F., Dosch, H. & Kahn, A. (2003). Phys. Rev. B, 68, 115428. Google Scholar
Giacovazzo, C. (2011). Editor. Fundamentals of Crystallography, 3rd ed. Oxford University Press. Google Scholar
Hahn, Th. (2006). Editor. International Tables for Crystallography, Vol. A, SpaceGroup Symmetry, 1st online ed. Chester: International Union of Crystallography. Google Scholar
Hailey, A. K., Hiszpanski, A. M., Smilgies, D.M. & Loo, Y.L. (2014). J. Appl. Cryst. 47, 2090–2099. Web of Science CrossRef CAS IUCr Journals Google Scholar
Heinrich, M. A., Pflaum, J., Tripathi, A. K., Frey, W., Steigerwald, M. L. & Siegrist, T. (2007). J. Phys. Chem. C, 111, 18878–18881. CrossRef CAS Google Scholar
Hicks, D., Oses, C., Gossett, E., Gomez, G., Taylor, R. H., Toher, C., Mehl, M. J., Levy, O. & Curtarolo, S. (2018). Acta Cryst. A74, 184–203. CrossRef IUCr Journals Google Scholar
Himes, V. L. & Mighell, A. D. (1987). Acta Cryst. A43, 375–384. CrossRef CAS Web of Science IUCr Journals Google Scholar
Jiang, Z. (2015). J. Appl. Cryst. 48, 917–926. Web of Science CrossRef CAS IUCr Journals Google Scholar
Kowarik, S., Gerlach, A., Sellner, S., Cavalcanti, L., Konovalov, O. & Schreiber, F. (2009). Appl. Phys. A, 95, 233–239. CrossRef CAS Google Scholar
Kowarik, S., Gerlach, A., Sellner, S., Schreiber, F., Cavalcanti, L. & Konovalov, O. (2006). Phys. Rev. Lett. 96, 125504. CrossRef PubMed Google Scholar
Marsh, R. E. (1995). Acta Cryst. B51, 897–907. CrossRef CAS Web of Science IUCr Journals Google Scholar
Mighell, A. D. (2000). Powder Diffr. 15, 82–85. CrossRef CAS Google Scholar
Moser, A. (2012). PhD thesis, University of Technology, Graz, Austria. Google Scholar
Niederhausen, J., Zhang, Y., Cheenicode Kabeer, F., Garmshausen, Y., Schmidt, B. M., Li, Y., Braun, K. F., Hecht, S., Tkatchenko, A., Koch, N. & Hla, S. W. (2018). J. Phys. Chem. C, 122, 18902–18911. CrossRef CAS Google Scholar
Niggli, P. (1928). Handbuch der Experimentalphysik, Vol. 7, Part 1. Leipzig: Akademische Verlagsgesellschaft. Google Scholar
Resel, R. (2003). Thin Solid Films, 433, 1–11. CrossRef CAS Google Scholar
Resel, R., Bainschab, M., Pichler, A., Dingemans, T., Simbrunner, C., Stangl, J. & Salzmann, I. (2016). J. Synchrotron Rad. 23, 729–734. CrossRef CAS IUCr Journals Google Scholar
Salzmann, I., Moser, A., Oehzelt, M., Breuer, T., Feng, X., Juang, Z. Y., Nabok, D., Della Valle, R. G., Duhm, S., Heimel, G., Brillante, A., Venuti, E., Bilotti, I., Christodoulou, C., Frisch, J., Puschnig, P., Draxl, C., Witte, G., Müllen, K. & Koch, N. (2012). ACS Nano, 6, 10874–10883. CrossRef CAS Google Scholar
Salzmann, I., Nabok, D., Oehzelt, M., Duhm, S., Moser, A., Heimel, G., Puschnig, P., AmbroschDraxl, C., Rabe, J. P. & Koch, N. (2011). Cryst. Growth Des. 11, 600–606. Web of Science CrossRef CAS Google Scholar
Salzmann, I. & Resel, R. (2004). J. Appl. Cryst. 37, 1029–1033. Web of Science CrossRef CAS IUCr Journals Google Scholar
Santoro, A. & Mighell, A. D. (1970). Acta Cryst. A26, 124–127. CrossRef IUCr Journals Web of Science Google Scholar
Santoro, A., Mighell, A. D. & Rodgers, J. R. (1980). Acta Cryst. A36, 796–800. CrossRef CAS IUCr Journals Web of Science Google Scholar
Sauter, N. K., GrosseKunstleve, R. W. & Adams, P. D. (2004). J. Appl. Cryst. 37, 399–409. Web of Science CrossRef CAS IUCr Journals Google Scholar
Shmueli, U. (2006). Editor. International Tables for Crystallography, Vol. B, Reciprocal Space, 1st online ed. Chester: International Union of Crystallography. Google Scholar
Simbrunner, J., Simbrunner, C., Schrode, B., Röthel, C., BedoyaMartinez, N., Salzmann, I. & Resel, R. (2018). Acta Cryst. A74, 373–387. CrossRef IUCr Journals Google Scholar
Smilgies, D.M. & Blasini, D. R. (2007). J. Appl. Cryst. 40, 716–718. Web of Science CrossRef CAS IUCr Journals Google Scholar
Swartz, C. R., Parkin, S. R., Bullock, J. E., Anthony, J. E., Mayer, A. C. & Malliaras, G. G. (2005). Org. Lett. 7, 3163–3166. Web of Science CrossRef PubMed CAS Google Scholar
Truger, M., Roscioni, O. M., Röthel, C., Kriegner, D., Simbrunner, C., Ahmed, R., Głowacki, E. D., Simbrunner, J., Salzmann, I., Coclite, A. M., Jones, A. O. F. & Resel, R. (2016). Cryst. Growth Des. 16, 3647–3655. CrossRef CAS Google Scholar
This is an openaccess article distributed under the terms of the Creative Commons Attribution (CCBY) Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original authors and source are cited.