research papers
An efficient method for indexing grazingincidence Xray diffraction data of epitaxially grown thin films
^{a}Department of Neuroradiology, Vascular and Interventional Radiology, Medical University Graz, Auenbruggerplatz 9, Graz, 8036, Austria, ^{b}Institute of Solid State Physics, Graz University of Technology, Petersgasse 16, Graz, 8010, Austria, ^{c}Institute of Solid State Physics, Friedrich Schiller University Jena, Helmholtzweg 5, Jena, 07743, 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, Québec H4B 1R6, Canada
^{*}Correspondence email: josef.simbrunner@medunigraz.at
102)]. The asdetermined is identical to that reported in a previous study determined for a fibretextured PQ film. Preliminary results further indicate that the algorithm is especially effective in analysing epitaxially grown crystallites not only for various orientations, but also if different polymorphs are present in the film.
identification of thin organic films entails a number of technical and methodological challenges. In particular, if molecular crystals are epitaxially grown on singlecrystalline substrates a complex scenario of multiple preferred orientations of the adsorbate, several symmetryrelated inplane alignments and the occurrence of unknown polymorphs is frequently observed. In theory, the parameters of the reduced and its orientation can simply be obtained from the matrix of three linearly independent reciprocalspace vectors. However, if the sample exhibits unit cells in various orientations and/or with different parameters, it is necessary to assign all experimentally obtained reflections to their associated individual origin. In the present work, an effective algorithm is described to accomplish this task in order to determine the unitcell parameters of complex systems comprising different orientations and polymorphs. This method is applied to a polycrystalline thin film of the conjugated organic material 6,13pentacenequinone (PQ) epitaxially grown on an Ag(111) surface. All reciprocal vectors can be allocated to unit cells of the same constants but grown in various orientations [sixfold rotational symmetry for the contact planes (102) and (Keywords: epitaxy; indexing; mathematical crystallography.
1. Introduction
). The situation becomes even more complex in the case of thin films formed by conjugated organic molecules. Their typical growth in crystal systems of low symmetry (in most cases monoclinic and triclinic), their tendency to and thus the presence of several phases make determination a difficult task (Tolan, 1999). Additionally, unknown polymorphs of organic materials are frequently observed within thin films only and cannot be determined independently via traditional methods like singlecrystal diffraction (Jones et al., 2016). On isotropic substrates, the crystallization of molecular materials typically results in fibretextured films comprising crystallites that share a common fibre axis perpendicular to the substrate surface but are azimuthally randomly oriented (Witte & Wöll, 2004). The use of anisotropic substrates (like rubbed polymer surfaces or singlecrystalline surfaces) or anisotropic preparation methods (like offcentre spin coating, dip coating or offaxis evaporation) can result in even more distinguished textures of the crystallites (Müller et al., 1999; Brinkmann et al., 2003; Qu et al., 2016). In this context, particularly complicated cases include epitaxially grown molecular crystals on singlecrystalline surfaces (Simbrunner et al., 2011). There, multiple preferred orientations of the crystals relative to the substrate surface can occur together with several symmetryrelated inplane alignments of the crystallites. For example, for the epitaxial order of the conjugated organic molecule paraquaterphenyl on Au(111) surfaces two different preferred orientations have been found, which show 24 different inplane alignments each (Müllegger et al., 2003). For such films formed by epitaxially grown molecular crystals the identification and characterization of unknown polymorphs is a particularly challenging task (Dienel et al., 2008).
identification of thin films entails a number of technical and methodological challenges: (i) low scattering volumes translate into only a small number of observable diffraction peaks and (ii) under the presence of a substrate the crystallites grow in a (texture) (Birkholz, 2006Grazingincidence Xray diffraction (GIXD) is a wellestablished method for thinfilm characterization (Schreiber, 2004). For nonfibretextured films, in order to cover large volumes of the the sample has to be rotated around its surface normal during the GIXD experiment and at each rotation angle a reciprocalspace map has to be recorded (Fumagalli et al., 2012; Schrode et al., 2019). Fig. 1(a) gives the geometry of a GIXD experiment illustrating such a sample rotation. The primary Xray beam defining the wavevector k_{0} and the scattered Xray beam characterized by the wavevector k define the scattering vector q as q = k_{0} − k. The three components of the scattering vector q_{x}, q_{y} and q_{z} can be calculated based on the geometries of primary and scattered Xray beams by
with α_{i} and α_{f} being the incident and exit angles of the primary and scattered beam (relative to the substrate surface), respectively, and θ_{i}, θ_{f} are the inplane scattering angles. Note that the orthogonal directions of q_{x}, q_{y} and q_{z} are defined in the sample coordinate system. Fig. 1(b) illustrates (i) the path of reciprocallattice points during the sample rotation by φ_{sample}, and (ii) cuts through which correspond to reciprocalspace maps recorded at two defined φ_{sample} angles.
i.e. the assignment of to the observed Bragg peaks. In the monochromatic approach of peak indexing as it is commonly employed for singlecrystal diffraction patterns, all three components of the scattering vectors are recorded and, therefore, all three components of the reciprocallattice vectors can be determined. Three linearly independent reciprocallattice vectors are sufficient to span the Any further experimentally determined reciprocallattice vector then must fit into this specific Since complete threedimensional vectors are used, even the indexing of configurations with multiple lattices can be successfully achieved (Jacobson, 1976; Higashi, 1990; Powell, 1999; Breiby et al., 2008; Gildea et al., 2014; Dejoie et al., 2015). Plenty of indexing methods have already been described over the last decades. Typically, a set of positions of recorded reflections is converted into reciprocalspace vectors. These vectors are then analysed for periodicity to determine the basis vectors. Differences between reciprocalspace vectors are calculated and accumulated in a histogram (Kabsch, 1988, 2010), or a fast Fourier transform (FFT) is used to search for periodicities (Steller et al., 1997; Campbell, 1998; Leslie, 2006; Otwinowski et al., 2012; Sauter et al., 2004). For example, the autoindexing method incorporated into the software MOSFLM (Leslie, 1992) employs FFT autoindexing routines written by the Rossmann group at Purdue University (Rossmann & van Beek, 1999) and relies on the calculation of many difference vectors between diffraction maxima in (Kabsch, 1993). The Fourier analysis is systematically performed for about 7300 separate, roughly equally spaced directions. For each direction, the distribution of the corresponding Fourier coefficients is searched to locate the largest local maximum, and by a local search procedure increases the accuracy. From these directions, a linearly independent set of three basis vectors of a realspace is then chosen.
solutions require the indexing of the diffraction pattern,In the approach by Duisenberg, periodicities are sought by projecting all observed reciprocallattice vectors onto the normal to the plane given by three randomly selected points (Duisenberg, 1992). This method was developed for difficult cases such as twin lattices, fragmented crystals and unreliable data. For such cases, Morawiec has developed another algorithm in which systematic combinations of three reciprocallattice vectors each are formed to search for periodicities of the calculated unitcell volumes (Morawiec, 2017).
In the case of GIXD, as usually performed on fibretextured films, however, only two components (of the total three) of the reciprocallattice vectors – namely q_{z} and q_{xy} – are available for the indexing procedure. In previous work, we have presented an algorithm for indexing such diffraction patterns, where the additional presence of a specular diffraction peak is being explicitly taken into account (Simbrunner et al., 2018, 2019). Furthermore, we have described an algorithm to find the and derived mathematical expressions which can be applied when reciprocalspace vectors are obtained.
In the present work, we now aim to formulate this indexing method for GIXD patterns obtained for rotated samples of nonfibretextured films, which then provides all three components of the scattering vector. Also, in this case, the combination of the diffraction peaks obtained from GIXD with the specular diffraction peak(s) simplifies the indexing procedure considerably, so that different phases, different preferred orientations and different crystal alignments can be identified. Finally, our algorithm is applied to a film of 6,13pentacenequinone (C_{22}H_{12}O_{2}, CAS No. 3029321) epitaxially grown on a singlecrystalline Ag(111) surface readily providing the unitcell parameters of the film from GIXD data.
2. Method
2.1. Fundamentals
For the following mathematical treatise a crystalfixed Cartesian 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) unit cell.
If the (001) g with its h, k and l can be represented by the equation
plane is parallel to the substrate surface, the reciprocallattice vectorThe matrix is given as
where a^{*} = , b^{*} = , c^{*} = , = and = are the reciprocal cell parameters and V is the unitcell volume, which can be explicitly written as
When the Laue condition q = g is fulfilled, diffraction can be observed. In real space, A_{001} characterizes the matrix of vectors a_{0}, b_{0} and c_{0}, which is in the nonrotated system given by
Equations (5) and (7) are connected via
The volume V of the can be calculated by V = det(A_{001}). If, however, the (001) plane is not parallel to the substrate surface, the reciprocal vector g can be expressed as
where R(φ)R() describes a general rotation. As explained previously (Simbrunner et al., 2018), we prefer this notation. R(φ) performs a rotation in the xy plane counterclockwise by an angle and is explicitly written as
The rotation matrix R () is explicitly written as
where and are the rotation angles. This rotation matrix was previously used for the mathematical formalism in GIXD experiments, where the composite component q_{xy} is measured and, therefore, rotation in the xy plane can be neglected. It has been shown that, if the specular diffraction scan q_{spec} of the sample provides useful information on the orientation, it is preferable to write the rotation parameters as follows:
where u, v and w are the of the contact plane (uvw) and thus the of the specular scan. In the general case, u, v and w can be irrational numbers.
From equation (9), using equation (8), it follows that
With
equation (15) can be rewritten as
where a, b and c are the rotated unitcell vectors with the relations , , , , and . The explicit forms of these vectors are given in Table 1. Note that the z components are only a function of the respective and the specular scan g_{spec}.

Thus, if three reciprocal vectors g_{1}, g_{2} and g_{3} are given, the following relation holds:
where
and (h_{i}, k_{i}, l_{i}) are the corresponding triples of with
Equation (18) can be equivalently expressed as
Furthermore, as , the following relation for the determinants of G and H is valid:
The unitcell vectors must be solutions to all reciprocal vectors g_{i} which can be written as
where
From equation (21) it can be deduced that 2πG^{−1}m, the product of the inverse matrix of three reciprocal vectors with a vector m, consisting of a triple of arbitrary integers (m_{1}, m_{2}, m_{3}), leads to a vector of the (Niggli, 1928) if m matches (h_{1}, h_{2}, h_{3})^{T}, (k_{1}, k_{2}, k_{3})^{T} or (l_{1}, l_{2}, l_{3})^{T}. If a transformation matrix N exists so that m equals N(h_{1}, h_{2}, h_{3})^{T}, N(k_{1}, k_{2}, k_{3})^{T} or N(l_{1}, l_{2}, l_{3})^{T} a vector of a is obtained. According to equation (22) it is favourable to select three reciprocal vectors whose matrix results in a determinant which is as small as possible but unequal to zero. The Buerger cell (Buerger, 1957) and subsequently the is obtained by choosing the three shortest vectors which are not coplanar and whose scalar products with all reciprocal vectors yield integers.
2.2. Indexing algorithm
We now suggest the following procedure for indexing an unknown crystalline system:
(i) Forming triplets of reciprocal vectors in all possible combinations, i.e. if n vectors are given these are
triplets (g_{1}, g_{2}, g_{3}), where g_{1}, g_{2} and g_{3} are any three reciprocal vectors. According to equation (22), if the triplet corresponds to a the determinant of its matrix is indirectly proportional to the volume of this or if the matrix of the corresponding has a determinant >1, to some integer fraction of it. As many reciprocal vectors belong to one they accumulate to discernible clusters. Importantly, a vanishing determinant strongly indicates that the reciprocallattice vectors (and the corresponding Laue indices) are linearly dependent and belong to the same crystalline system. If several unit cells are contained in the sample, an overlap of smaller volumes with fractions of larger volumes occurs. A feasible strategy results in gathering the reciprocal vectors of the largest volume and repeating the procedure for the remaining reciprocal vectors.
If the crystallites are characterized by the same xy plane, this algorithm can only be used if the unitcell volume is a priori known. Otherwise, three reciprocal vectors are chosen from different subgroups with identical pairs of q_{xy} = (q_{x}^{2} + q_{y}^{2})^{1/2} and q_{z}.
and differ only in their rotational arrangement in the(ii) According to equation (19), the selected triplets of reciprocal vectors are combined into matrices. If they belong to the same system, their inverse matrices multiplied with the vectors of the corresponding will result in the vectors of the [cf. equation (21)]. This can be achieved by multiplying the inverse matrices G^{−1} with vectors 2π(m_{1}, m_{2}, m_{3})^{T}, where the m_{i} are systematically varied integers in a reasonable range (e.g. between −3 and 3). Then, vectors of the and of its superlattices are obtained (Simbrunner et al., 2018). The three shortest vectors which are not coplanar are chosen to obtain the Buerger cell. The asobtained matrices contain the vectors a, b and c of the reduced unit cells which may have various orientations. If the reciprocalspace vectors are correctly combined, i.e. if they belong to the same system, the scalar product criteria (Niggli, 1928) are intrinsically fulfilled; otherwise they are useful to eliminate falsely combined vector triplets. The associated integers are the corresponding triples of [see equation (18)]. If a contact plane exists and the specular scan q_{spec} can be measured, the triplets whose z components are (almost) integer multiples of 2π/q_{spec} can be assigned as possible solutions.
(iii) The tentative unitcell matrices are multiplied with all reciprocal vectors. If the scalar products yield integers [i.e. the corresponding according to equation (23)], the matrices and reciprocal vectors belong to the same system. Due to experimental imperfections, errors must be considered. For a system of reciprocal vectors, the with the smallest deviations from perfect integers will be chosen.
(iv) From the unitcell matrix, the cell parameters and rotation parameters can be obtained. From equation (17) it follows that the sides of the are the magnitudes and its angles are the scalar products of the matrix vectors. The Niggli 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) (Niggli, 1928). Therefore, the vectors have to be designated
accordingly. Furthermore, the angles must be adopted. By multiplying one vector with −1 two angles change to their complementary ones, e.g. for a → −a we obtain β → π − β and γ → π − γ. The rotation angles φ, ψ and ϕ can be obtained by using the equations provided in Appendix A.
(v) According to equation (21) the unitcell vectors can be calculated from every linearly independent triple of reciprocallattice vectors of the same system. This redundancy can be used to determine mean values and standard deviations of the unitcell parameters. Alternatively, the matrix of the unitcell vectors may be optimized in real space (see Appendix B) and in (see Appendix C). In a last step, the parameters may be optimized with respect to the lengths q_{xyz} and the components q_{z} of the reciprocalspace vectors, which are independent of the rotation angle φ. This can be accomplished analytically (see Appendix D).
3. Example: 6,13pentacenequinone (PQ) on Ag(111)
We now apply the above methodology to an epitaxially grown film of the conjugated organic molecule 6,13pentacenequinone (PQ, C_{22}H_{12}O_{2}, CAS No. 3029321) on an Ag(111) surface. PQ was purchased from Sigma Aldrich (purity 99%) and purified via vacuum before usage. Substrate preparation and film preparation were conducted in an ultra high vacuum (UHV) chamber with a base pressure of 1 × 10^{−8} Pa. Before thinfilm deposition, the substrate surface was cleaned by repeated cycles of Ar^{+} sputtering at an energy of 700 eV and angles of ±45° to the sample normal, followed by thermal annealing at 770 K for 30 min. Surface quality was confirmed by (LEED). PQ was deposited by thermal evaporation from a quartz crucible at a constant temperature of 480 K for 60 min with the deposition time controlled by a shutter, resulting in an approximate film thickness of 10 nm. During deposition, the substrate was kept at room temperature and the chamber pressure increased to 4 × 10^{−7} Pa. A top layer of aluminium tris(8hydroxyquinoline) (Alq3, C_{27}H_{18}AlN_{3}O_{3}, CAS No. 2085338), known to grow amorphously, was deposited from a quartz crucible at a constant temperature of 550 K for 55 min to keep the PQ crystals free from environmental influences and to reduce beam damage during the Xray diffraction experiments. Alq3 was obtained from Sigma Aldrich at sublimed grade with a purity of 99.995% trace metal basis and was used without further purification.
The sample was first investigated by specular Xray diffraction using inhouse equipment and then by GIXD using synchrotron radiation. Specular Xray diffraction was performed on a PANalytical Empyrean system using a sealed copper tube together with an Xray mirror for monochromatization and a PIXcel3D detector operating in scanning line mode (255 channels). Measured data were converted to θ being half of the scattering angle 2θ and λ = 1.5406 Å. GIXD measurements were performed at the XRD1 beamline at the Elettra Synchrotron, Trieste (Italy), using a wavelength of 1.4000 Å. Diffracted intensity was recorded with a stationary Pilatus 2M detector (Dectris) with a sampletodetector distance of about 200 mm. The primary Xray beam was slightly offset from the centre of the detector to allow for simultaneously recording both the right and the lefthand side of the reciprocalspace map, but to avoid missing peaks due to detector gaps typical for the Pilatus system. The calibration of the setup (to obtain exact values for sample–detector distance, position of the primary beam on the detector, detector inclinations etc.) was performed by measuring polycrystalline lanthanum hexaboride (LaB_{6}) (Black et al., 2010) in a capillary. The incident angle α_{i} for the thinfilm measurement was set to 0.7° to reduce the footprint of the beam on the sample and, thus, enable data evaluation with higher accuracy. The sample was rotated around its surface normal [cf. Fig. 1(b), φ_{sample}] during the GIXD measurement recording 180 images, with therefore each exposure integrating 2° in φ_{sample} of the azimuthal rotation. Note that the angle φ_{sample} is the experimentally used sample azimuth, while the angle φ is used to describe a rotation of the in the xy plane in the counterclockwise direction [cf. equations (9)–(10)].
using , withTo determine the peak positions from the diffraction images, the following process was performed: in a first step, all diffraction images recorded during the 360° sample rotation were summed up pixel by pixel resulting in a single image showing the overall integrated diffraction information. The peak positions were then fitted using a twodimensional Gaussian function with a background plane, giving the pixel position of each peak [cf. Fig. 2(a)].
To determine the peak positions as a function of the sample rotation, the intensity at the obtained peak positions was monitored throughout the 180 separate data files which translates into a curve representing intensity versus φ_{sample}. Three curves are given in Fig. 2(b), where due to both the symmetry of the and that of the substrate, several peaks are then observed for a given set of Most peaks show a full width at halfmaximum (as a measure of inplane mosaicity) of about 2°. Their positions were determined by fitting the corresponding part of the curve with a onedimensional Gaussian function with linear background. In the next step, a small area around the peak position of the summed image was fitted with a twodimensional Gaussian function in the specific data file corresponding to the rounded φ_{sample} value where the was observed. These pixel positions together with the corresponding (unrounded) φ_{sample} values were used to convert the data into using source code provided by the software package GIDVis (Schrode et al., 2019). No refraction corrections were included during the conversion. As mentioned above, the diffraction pattern was recorded simultaneously at the righthand side (RHS) as well as at the lefthand side (LHS) of the detector. Therefore, the abovedescribed data evaluation can be applied separately to the data from the RHS and the LHS, since each detector side (apart from the detector gaps) contains the same information for the data analysis, resulting in individual solutions for RHS and LHS data. Large differences between peak positions obtained from the LHS and RHS would indicate a sample misalignment, but were not observed here.
The specular Xray diffraction pattern shows dominant features of the Ag(111) substrate and a clear diffraction peak which is assigned to PQ crystals (compare Fig. 3). According to our notation the position of this peak is q_{spec} = q_{z} = 1.942 Å^{−1}. The GIXD experiments gave 227 and 279 reciprocallattice vectors with the three components q_{x}, q_{y} and q_{z} on the RHS and LHS, respectively. The different number is due to some peaks falling into the detector gaps at one detector side only. The reciprocallattice vectors could be split into 31 groups with up to 12 related pairs of q_{xy} = (q_{x}^{2} + q_{y}^{2})^{1/2} and q_{z}. Therefore, it could be concluded that there are 12 different inplane alignments of the crystallites which are oriented with the same contact plane (uvw or −u−v−w). Three reciprocal vectors from different groups were then systematically combined, and the matrix was formed according to equation (16). Note that the determinant of the matrix is indirectly proportional to the volume of the [see equation (22)], and a determinant of zero expresses that linearly dependent vectors have been chosen. In a next step the inverse matrix was multiplied with vectors of systematically varied integers, and the three shortest vectors were chosen (Buerger cell) for a guess of a If the z components of these tentative vectors were integer multiples of 2π/q_{spec} and the Niggli criteria were fulfilled, they were assigned as possible solutions. The obtained integers could be assigned as the uvw = (102) or (102) of the contact plane which finally gives the of the PQ crystals relative to the substrate surface. In a next step the tentative matrix was multiplied with all reciprocal vectors of the other groups. If the matrix was indeed a solution, triples of could be assigned to the associated reciprocal vectors, according to equation (23). In a further step, the matrix of the unitcell vectors was optimized (see Appendices B and C). On request, the used code can be provided. Fig. 4(a) shows the result of the indexing procedure for a single type of epitaxially aligned crystallites with the contact plane uvw = (102) for the RHS data; the assigned peaks are marked with blue circles.
For each solution the a, b and c were used to determine the respective constants, the contact plane [uvw = (102) or (102)] and the rotation angle φ (see Appendix A). As φ can be independently calculated from each of the three vectors, the accuracy can be checked. In our case, the mean error was about 0.15%. Rotation angles between the different solutions were obtained in steps of Δφ = 60.00° (±0.26°) for the (102) contact plane and Δφ = 60.00° (±0.94°) for the (102) contact plane. This clearly reflects the symmetry of the Ag(111) surface. By repeating the procedure, all reciprocal vectors could be allocated to unit cells with the same constants but various orientations.
vectorsIn the case of Fig. 4(b) two types of epitaxially aligned crystallites are indexed with contact planes (102) and (102), denoted by blue and red circles, respectively. Fig. 4(c) gives indexing of all diffraction peaks by the two contact planes (102) and (102), each showing six different types of inplane alignment of the crystallites. We found the following epitaxial relationships: (111)_{Ag}  ± (102)_{PQ}; the b axis [010] of PQ in (102) orientation is rotated by −7° (i.e. clockwise) with respect to the directions and the b axis [010] of PQ in (102) orientation is rotated by +7° (i.e. counterclockwise) with respect to the directions. An evaluation of the data from the LHS and RHS of the detector was then performed. As expected, no significant differences in the unitcell vectors (including contact planes, epitaxial relationships and constants) were found. The constants were determined by averaging over all 24 sets of constants. We then obtained a = 5.059 ± 0.012, b = 8.097 ± 0.026, c = 8.916 ± 0.032 Å, α = 91.64 ± 0.24°, β = 92.95 ± 0.56°, γ = 94.17 ± 0.23°, V = 363.5 Å^{3}.
In a last step, we optimized the q_{xyz} and the components q_{z} (see Appendix D) and obtained the following values: a = 5.063, b = 8.091, c = 8.916 Å, α = 91.61°, β = 92.92°, γ = 94.13°, V = 363.6 Å^{3}. These parameters differ only slightly from the values given above and the error function E_{xyz,z} decreases only minimally from 0.0113 to 0.0112 Å^{−1}.
parameters with respect to the vector lengthsThe expected peak positions of this solution are plotted together with the twodimensional reciprocalspace map in Fig. 2(c). A threedimensional representation of the experimental data and the expected peak positions is given in Fig. 4. In both, a good agreement is observed. The constants we obtain by following this protocol are in excellent agreement with the of PQ reported for thin films grown on a (HOPG) surface (Simbrunner et al., 2018). Furthermore, comparison of experimental with expected peak intensities shows good agreement. Therefore, the known solution can be used to develop a model of the arrangement of PQ molecules with respect to the Ag(111) surface. Fig. 5(a) shows the situation of the (102) contact plane and a baxis rotation angle of +7° with respect to the highsymmetry direction of the silver surface. Nearly completely flatlying molecules are found which are rotated with respect to the highsymmetry silver directions. Adjacent molecules are slightly slipped to form short contacts between neighbouring oxygen and hydrogen atoms, which highlights the role of hydrogen bonding in the formation of the PQ In turn, for the (102) contact plane a baxis rotation angle of −7° with respect to the silver highsymmetry direction [cf. Fig. 5(b)] is found, i.e. a rotation of the molecules in the opposite direction.
4. Discussion
We regard our algorithm for the analysis of Xray diffraction patterns of thin films via rotating the sample to determine reciprocallattice vectors advantageous for the following reasons: (i) the
vectors of the involved unit cells and their orientation can be determined simultaneously. The method is suitable for implementation of (semi)automatic processing. (ii) Only a few reciprocal vectors are required. Theoretically, only three vectors are sufficient to determine the parameters and orientation of the if the determinant of the matrix of the corresponding is ±1. (iii) Indexing is possible even if crystals with different crystallographic unit cells and orientations are present. Depending on measurement accuracy and available boundary conditions, about six to eight related reciprocal vectors may be sufficient for a correct assignment to the corresponding The knowledge of a contact plane as determined by available specular diffraction data can be of considerable help for selecting the proper unitcell vectors. (iv) No previous knowledge of the structure is necessary, and the intensities of the various reflections are not required. For symmetry considerations, however, the diffraction intensities must be included.We note that our method of combination of three reciprocallattice vectors has been suggested before (Duisenberg, 1992; Morawiec, 2017). In both cases this approach is used for solving difficult cases in singlecrystal diffractometry such as twin lattices, fragmented crystals and unreliable data. Duisenberg takes the end points of three observed reflection vectors to build the normal to the plane formed by any three of these points. Then all observed points are projected onto this normal. This helps in eliminating spurious vectors which do not belong to the direct cell. Morawiec combines three reciprocallattice vectors to calculate the corresponding unitcell volumes and searches for periodicities to select the vectors that support the definite parallelepiped. This procedure is related to the algorithm we suggest for selecting the proper reciprocallattice vectors. In our case, however, the direct is determined first, and then further reciprocallattice vectors are selected.
Immediately switching into real space for finding the unitcell vectors is advantageous as several possible criteria and boundary conditions may exist for reducing possible solutions: (i) unitcell volume and parameters, (ii) scalar product criteria (Niggli conditions) and (iii) including information from the specular scan if a contact plane is experimentally determined. To reduce the errors due to experimental imperfections, the
vectors of one are fitted before calculating the cell parameters and orientation angles. This is accomplished by analytically minimizing error functions, which can be defined both for reciprocal and real space.Finally, our methodology allows the direct analytical determination of the orientation parameters, i.e. the angles ϕ, ψ (or the u, v, w of the contact plane) and φ of the rotation matrices. These angles refer to the chosen reference system.
If a very large number of reciprocal vectors are obtained, alternative algorithms may be advantageous. The existing autoindexing method incorporated in MOSFLM (Leslie, 1992) – FFT autoindexing routines written by the Rossmann group at Purdue University (Rossmann & van Beek, 1999) – relies on the calculation of many difference vectors between diffraction maxima in (Kabsch, 1993). Preliminary results, however, show that our method is especially effective in analysing epitaxially grown crystallites not only with various orientations but also with various polymorphs.
5. Conclusion
In this work, we present an algorithm for indexing GIXD diffraction patterns obtained with monochromatic radiation, where threedimensional reciprocallattice vectors are determined as is done in singlecrystal diffraction experiments. Our method is particularly advantageous if the number of reflections is relatively small or the sample is inhomogeneous and consists of various crystal lattices or orientations, as is commonly found for thin films grown on singlecrystalline substrates. For easy access to epitaxial relationships the
constants of the involved unit cells and the parameters of the orientation matrix can be determined simultaneously.APPENDIX A
Determining the rotation angles ψ, ϕ and φ
The rotation parameters ψ and ϕ and the parameters Ω_{a}, Ω_{b} and Ω_{c} can be preferably obtained by using the following equations, which can easily be derived from the expressions in Table 1 and equation (14):
Note that two special cases must be considered:
(i) If a_{z} < 0 (i.e. u < 0) then → 2π − ψ [as is in the interval [π, 2π], see equation (13)].
(ii) If a_{z} = 0 and b_{z} = 0 (i.e. u = v = 0) then = 0, = 0, Ω_{a} = 0, Ω_{b} = and cos Ω_{c} = cos(β)/r_{c}.
If the ratios a_{z}/b_{z} = u/v, a_{z}/c_{z} = u/w and b_{z}/c_{z} = v/w are rational numbers, one can conclude that a contact plane (with the uvw) exists. If the specular diffraction peak is known, the indices can be directly calculated from u = (a_{z}/a)(g_{spec}/2π), v = (b_{z}/b)(g_{spec}/2π) and w = (c_{z}/c)(g_{spec}/2π).
From Table 1 the following expressions for the phase φ can be deduced:
As φ covers the range between 0 and 2π, equations (29) and (30) must be considered. This can be expressed in the following way:
Analogously, the following relations are valid:
As φ can be independently calculated from the vectors a, b and c, the accuracy of the result can be checked and a mean value of φ can be determined.
APPENDIX B
Optimizing the matrix of the unitcell vectors in real space
From equation (23) the error regarding the unitcell vector a with respect to N associated reciprocallattice vectors g_{i} and the corresponding h_{i} can be expressed as
E_{a,N} becomes minimal if
is fulfilled. From this condition the following relations can be derived:
where
Analogously, the error functions for the cell vectors b and c with respect to the measured reciprocallattice vectors g_{i} and the corresponding k_{i} and l_{i}, respectively, can be expressed as
Then, E_{b,N} and E_{c,N} become minimal for
APPENDIX C
Optimizing the matrix of the unitcell vectors in reciprocal space
In analogy to equations (4) and (9) the reciprocallattice vectors g_{i} can be written as
where a_{x}*, a_{y}*, a_{z}*, b_{x}*, b_{y}*, b_{z}*, c_{x}*, c_{y}*, c_{z}* are the components of the reciprocalspace vectors, and h_{i}, k_{i}, l_{i} are the corresponding Then, for N reciprocalspace vectors the error function E_{x,N} can be expressed as
E_{x} becomes minimal if
is fulfilled. From this condition the following relations can be derived:
where
Analogously, the error functions for the cell vectors b and c with respect to the measured reciprocallattice vectors g_{i} and the corresponding h_{i}, k_{i} and l_{i} can be expressed as
Then, E_{y,N} and E_{z,N} become minimal for
The components of the
in real space can be calculated from the following expression:APPENDIX D
Analytical optimization of the unitcell parameters with respect to q_{xyz} and q_{z}
If the q_{xyz} and the component q_{z}, which are phaseinvariant. This can be accomplished analytically by considering small errors and firstorder correction. It is convenient to use the reciprocal cell parameters and minimize the quadratic error function E_{xyz,z}:
are determined and the unitcell parameters are calculated by evaluating the reciprocal vectors as described above, in a last step they may be optimized with respect to the measured vector lengthwhere
(q_{xyz,i}, q_{z,i}) are the measured and (g_{xyz,i}, g_{z,i}) are the calculated peak positions of the ith reflection, n is the number of reflections and are the correction terms. E_{xyz,z}^{n} becomes minimal if
is fulfilled. From this condition the following relation can be derived:
where the matrix contains the inner products of the vectors
Furthermore, for the length g_{xyz,i}, the outofplane component g_{z,i} of the ith reciprocal vector and the specular scan g_{spec} the following expressions are valid:
From equation (54) the following derivatives of g_{xyz,i} can be obtained:
From equation (55), using equation (56), the following derivatives of g_{z,i} result:
From equation (52) the correction terms can be determined and added to the corresponding reciprocal cell parameters to obtain the new values. For larger errors it is advantageous to repeat this procedure. The cell parameters in real space are finally obtained by using their relations with the reciprocallattice parameters [c.f. equation (5)].
Acknowledgements
The authors acknowledge the Elettra Synchrotron Trieste for allocation of beamtime and thank Luisa Barba for assistance in using beamline XRD1.
Funding information
Financial support was given by the Austrian Science Foundation FWF (grant No. P30222). IS acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) (funding reference number RGPIN201805092), the Fonds de recherche du Québec – Nature et technologies (FRQNT) (funding reference 2020NC271447) and Concordia University. The Jena group acknowledges funding from the German BMBF, grant No. 03VNE1052C.
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