research papers
Indexing of grazingincidence Xray diffraction patterns: the case of fibretextured thin films
^{a}Department of Neuroradiology, Vascular and Interventional Radiology, Medical University Graz, Auenbruggerplatz 9, Graz, 8036, Austria, ^{b}E + E Elektronik Ges.m.b.H., Langwiesen 7, Engerwitzdorf, 4209, Austria, ^{c}Institute of Solid State Physics, Technical University Graz, Petersgasse 16, Graz, 8010, Austria, and ^{d}Department of Physics, Department of Chemistry and Biochemistry, Concordia University, 7141 Sherbrooke Street W., SP 26520, Montreal, Quebec H4B 1R6, Canada
^{*}Correspondence email: josef.simbrunner@medunigraz.at
a, b, c, α, β and γ of the crystallographic are thereby determined, as well as the rotation parameters due to the unknown of the crystals with respect to the substrate surface. The mathematical analysis exploits a combination of GIXD data and information acquired by the specular Xray diffraction. The presence of a sole specular diffraction peak series reveals fibretextured growth with a crystallographic plane parallel to the substrate, which allows establishment of the u, v and w as the rotation parameters. Mathematical expressions are derived which reduce the system of unknown parameters from the three to the twodimensional space. Thus, in the first part of the indexing routine, the integers u and v as well as the h and k of the experimentally observed diffraction peaks are assigned by systematically varying the integer variables, and by calculating the three lattice parameters a, b and γ. Because of the symmetry of the derived equations, determining the missing parameters then becomes feasible: (i) w of the surface parallel plane, (ii) the l of the diffraction peak and (iii) analogously the lattice constants c, α and ß. In a subsequent step, the reduced unitcell geometry can be identified. Finally, the methodology is demonstrated by application to an example, indexing the diffraction pattern of a thin film of the organic semiconductor pentacenequinone grown on the (0001) surface of The of the crystallites, the lattice constants of the triclinic and finally, by molecular modelling, the full solution of the asyetunknown polymorph of pentacenequinone are determined.
solutions from thin films are often performed by grazingincidence Xray diffraction (GIXD) experiments. In particular, on isotropic substrates the thin film crystallites grow in a fibre texture showing a well defined crystallographic plane oriented parallel to the substrate surface with random inplane order of the microcrystallites forming the film. In the present work, analytical mathematical expressions are derived for indexing experimental diffraction patterns, a highly challenging task which hitherto mainly relied on trialanderror approaches. The six lattice constantsKeywords: grazingincidence Xray diffraction; thin films; indexing; specular scan; mathematical crystallography.
1. Introduction
The appearance of unknown polymorphs within organic thin films is a well known phenomenon which attracts considerable interest in organic electronics and pharmaceutical science (Jones et al., 2016). Frequently used terms for this type of polymorph include substrateinduced phases, substratemediated phases or thin film phases (Bouchoms et al., 1999; Schiefer et al., 2007; Ehmann & Werzer, 2014). The presence of an isotropic substrate surface during the crystallization process can induce new types of molecular packing, because the substrate acts as a template for the crystallization process. Substrates on which such new polymorphs tend to grow typically exhibit a highly flat surface like oxidized silicon wafers, glass plates or polymer surfaces. There, the deposited organic material crystallizes with a strong showing a well defined crystallographic plane (the socalled contact, or texture plane) parallel to the substrate surface. However, no azimuthal (i.e. inplane) order between the microcrystallites forming such films is observed due to the isotropic nature of the substrate surfaces. This type of crystalline orientation is called uniplanar texture (Heffelfinger & Burton, 1960) or fibre texture (Roe & Krigbaum, 1964).
Crystal structure solutions for such thin films are typically performed by grazingincidence Xray diffraction (GIXD); the experimental geometry is schematically shown in Fig. 1(a). The primary Xray beam with the wavevector k_{0} and the scattered Xray beam with the wavevector k determine the scattering vector q by q = k − k_{0}. According to the Laue equation, diffraction occurs if the scattering vector q is equal to a reciprocallattice vector g. For organic crystallites in fibretextured films, the reciprocallattice points lie on concentric circles, as illustrated by red circles in Fig. 1(b). Keeping the sample fixed in space, a GIXD experiment then equals a cut through the threedimensional roughly perpendicular to the rings of reciprocallattice points, and a corresponding twodimensional reciprocalspace map is obtained [compare Fig. 1(b)]. Note that for thin films with defined inplane alignment of the crystallites [e.g. if grown on anisotropic substrates like graphene (Salzmann et al., 2012)] or for samples with weak statistics, the system can be artificially reduced to a fibre texture simply by a 360° rotation around the substrate normal (Röthel, 2017).
q_{z} and q_{xy} – are available for the indexing process (Smilgies & Blasini, 2007; Hailey et al., 2014). This is considerably different to the indexing procedure employed for singlecrystal diffraction patterns, where all three components of reciprocallattice vectors are recorded, as well as for powder diffraction of polycrystalline materials, where only the lengths of the scattering vectors are detected. In the case of singlecrystal diffraction, three linearly independent reciprocallattice vectors are required to span the Any other experimentally determined reciprocallattice vector has then to fit into this specific Since complete threedimensional vectors are used, even indexing of configurations with multiple lattices can be successfully achieved (Jacobson, 1976; Powell, 1999; Breiby et al., 2008; Gildea et al., 2014; Dejoie et al., 2015; Morawiec, 2017). In the case of powder diffraction, only the lengths of the reciprocalspace vectors are used and the unknown variables are then up to six unitcell parameters (in the case of a triclinic system) and a set of with a triple of three integer values each. This problem cannot be solved algebraically. One possibility, however, is the dichotomy method where the cell constants are varied in increasingly smaller intervals and the hkl indices are subsequently refined using the leastsquare method (Boultif & Louër, 1991, 2004). For simplification of the indexing process, boundary conditions can be imposed.
solutions from GIXD require the indexing of the diffraction pattern, that is, the assignment of to the observed Bragg peaks. In our specific case of GIXD on fibretextured films, two components of the reciprocallattice vectors – namelyFew programs have yet been developed for the indexing of twodimensional reciprocalspace maps (Smilgies & Blasini, 2007; Breiby et al., 2008; Hailey et al., 2014; Jiang, 2015). Certainly, the situation is relatively trivial if all lattice parameters are known. However, for a successful indexing it is still necessary to determine the contact plane of the investigated crystals in fibretextured films. For this reason, the rotation matrix of the thin film crystallites relative to the substrate surface has to be considered (Shmueli, 2006). If the lattice parameters are, however, unknown, both the lattice constants and the rotation matrix need to be determined, which represents a significantly more challenging task. Present approaches for the indexing of such systems are mainly based on trial and error, which is clearly unsatisfactory for obvious reasons.
Here, we demonstrate the analytical derivation of mathematical expressions to be employed in the indexing of twodimensional reciprocalspace maps. To this end, we use two components of the reciprocalspace vectors, the inplane part q_{xy} and the outofplane part q_{z}. A further input parameter for the indexing arises from specular Xray diffraction experiments, as in essentially all cases of crystalline organic thin films grown in a fibre texture one defined Bragg peak (or one Bragg peak series) is observed at q_{spec}, originating from the plane normal to the fibre axis of the film. This peak (series) is due to diffraction from the contact plane of the fibretextured film, which is assigned to a crystallographic plane of u, v and w (Salzmann & Resel, 2004; Smilgies & Blasini, 2007; Hailey et al., 2014; Jiang, 2015). By combining the peak positions in the GIXD pattern (q_{xy}, q_{z}) with the specular peak (q_{spec}), the required number of unknown parameters for indexing significantly reduces.
If all three components of the scattering vectors are measured, the orientation of the crystal has to be considered by including the rotation parameters. Though the number of equations is smaller than the number of unknowns, the analytical treatment is much more straightforward since it is purely based on linear equations (see Appendix E).
2. Methods
For the following mathematical treatise a laboratory coordinate system with the xy plane being parallel to the substrate surface is assumed.
2.1. Nonrotated case – contact plane (001)
In the following analysis, 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) lattice plane is parallel to the substrate surface in a GIXD experiment, the reciprocallattice 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.
In the real space, A_{001} characterizes the matrix of the lattice vectors a_{0}, b_{0} and c_{0}, which is in the nonrotated system given by
Equations (2) and (3) are connected via
The volume V of the can be calculated by
Using equation (1) and the relations given in Table 1, the in and outofplane components of the reciprocal vector g can be explicitly written as
with and . From equation (6) the unitcell parameters which are oriented inplane, namely a, b and γ, can be determined; in further consequence equation (7) leads to parameters c, α and β. The integer variables of the have to be varied and the values of q_{xy} and q_{z} from three independent Bragg peak series are required to obtain a solution for the corresponding unitcell parameters, which have to be checked if proper can be obtained for all measured diffraction peaks (Truger et al., 2016).
2.2. Rotated case – contact plane (uvw)
Obviously, the situation becomes more complex if the (001) lattice plane is not parallel to the substrate surface as the matrix has now to be transformed. In particular, it has to be rotated around the which is defined by the (001) plane (characterized by its normal vector σ_{1}) and the new contact plane (uvw), as characterized by its normal vector σ_{2}. A graphical sketch of the discussed geometry is presented in Fig. 2.
It can easily be proven that an arbitrary rotation of the lattice vectors in the real space corresponds to an identical rotation of the reciprocalspace vectors. If R is an arbitrarily chosen rotation matrix acting on the lattice vectors and R^{−1} = R^{T} is its inverse, the following relation can be deduced from equation (4):
Therefore, equation (4) can be generalized and written in the form
where a = Ra_{0}, b = Rb_{0} and c = Rc_{0} are the rotated lattice vectors and .
Based on the graphical representation it can be shown that the unit vector n of the is calculated by the vector product of σ_{1} and σ_{2}:
The angle of rotation is obtained by the scalar product
The matrix R, which describes a rotation by around the axis n, is given by (Shmueli, 2006)
Combining equations (10) to (12) yields the components of the unit vector n:
n_{3} = 0, which results in the condition n_{1}^{2} + n_{2}^{2} = 1. In a next step the angle of rotation Φ can be obtained by combining equations (10), (11) and (13) as
Finally, the reciprocallattice vector g can be written as
From equation (18), the following expressions for the radius g_{xyz} = (g_{x}^{2} + g_{y}^{2} + g_{z}^{2})^{1/2} and the outofplane part g_{z} of the reciprocallattice vector can be derived:
If the condition h = u, k = v and l = w is fulfilled, equations (19) and (20) are identical, which means that there is only a contribution from the outofplane part g_{z}, whereas the inplane part g_{xy} is zero. This is valid for the specular scan g_{spec}, which is exactly sensitive to the lattice plane parallel to the surface, and therefore can be explicitly written as
From equations (19) to (21) and by including equation (5), the following expression for the inplane part g_{xy} can be derived:
Furthermore, using equations (20) and (21), equation (19) can be rewritten as
and by algebraic transformations the following expression can be derived:
Equation (24) can be regarded as a generalization of equation (6), additionally including the two rotational integer parameters u and v, the specular scan g_{spec} and the outofplane part g_{z}. For u = v = 0 it reduces to equation (6) in the nonrotated case.
Equation (24) comprises – in addition to the rotation parameters u and v – only the lattice parameters a, b, γ and the h and k. This facilitates the mathematical analysis, where the integer variables can be varied and only three real unknowns have to be calculated. Therefore, we note that when indexing GIXD patterns, the acquisition of a specular scan is of considerable help.
In rare cases, net planes oriented parallel to the substrate surface are characterized by a weak et al., 2012). In such cases u and v must be assumed to be real (instead of integer) numbers which makes the mathematical analysis more exhaustive and an alternative notation of the rotation matrix may be chosen (see Appendix A).
which inhibits the acquisition of a specular scan (DjuricIn Table 2 we provide a summary of the derived equations and further analogous expressions due to symmetry relationships.
As the a, b and γ can be calculated from the q_{xy} and q_{z} values of three independent Bragg peak series. This can be achieved analytically by employing proper mathematical substitutions to obtain linear equations (see Appendix B).
are integers, they can be systematically varied, whereas the real unknown parametersFor calculating the remaining cell parameters from equations (20) and (21) the following expression can be derived:
Equation (25) can be regarded as a generalization of equation (7) to which it reduces for u = v = 0 in the nonrotated case.
Alternatively, by using the symmetry expressions in Table 2 the parameter sets {a, c, β, u, w} and {b, c, α, v, w} can be determined in an analogous manner as {a, b, γ, u, v}.
If one component of the uvw) and (hkl), is zero, compact expressions for the reciprocal cell parameters a*, b*and c* can be derived (see Appendix C).
which is the intersection of the planes (3. Discussion – determining the reduced cell
As discussed by Niggli, the ). Such a cell provides a unique description of the lattice and is characterized independently of lattice symmetry. The main conditions for reduction require that the is based on the three shortest vectors of the lattice; such a is then called a Buerger cell (Buerger, 1957). However, this cell may not be unique. An unambiguous is the socalled defined by Niggli (Niggli, 1928; Santoro & Mighell, 1970). The general criteria for the reduced cells are summarized in Table 3; the complete criteria, which include special conditions, are listed in the International Tables of Crystallography (De Wolff, 2016).
is defined by the cell that satisfies the conditions derived from the reduction theory of quadratic forms (Niggli, 1928

If a, b and c are the lattice vectors of the then every linear combination a′, b′ and c′ of these vectors
where n_{ij} are integers and components of the transformation matrix
can be regarded as a et al., 1980). Therefore, in general, any solution that is found when indexing a diffraction pattern must be analysed if it satisfies the conditions of the reduced cell.
which obeys the Laue condition (SantoroIn the matrix approach to symmetry (Himes & Mighell, 1987) N is represented by one of the 64 symmetry matrices to check if the transformation leads to identity (, α′ = α, β′ = β, γ′ = γ).
Equations (26) to (28) can be equivalently written as A′ = (a′, b′, c′)^{T} = NA. Considering equation (9) the following relations are valid:
where h′, k′ and l′ are the in the transformed system. Thus the transformation N which converts the lattice vectors is the same as that which converts the in the This is summarized in Table 4.
Therefore, reduction of the cell parameters to the ; Mighell, 1976; Křivý & Gruber, 1976) is equivalent to converting the as in the common reciprocal approach (Kroll et al., 2011). If there are two solutions to a diffraction pattern with the h, k, l of the unitary cell and of a the transformation matrix N can be easily obtained by linearly independent of three reflections:
(Santoro & Mighell, 1970The u, v and w can equally be used. If the determinant of the transformation matrix equals ±1, the cell volume does not change. Thus, systematically combining three linearly independent triples of respectively, and calculating their determinants can give an estimate of whether a found solution may match the Buerger cell. Furthermore, in GIXD, after finding a set of cell parameters, by calculating three linearly independent reciprocal vectors and evaluating their inverse matrix the three shortest lattice vectors can be determined (see Appendix E).
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). For this, the expressions in Table 5, which directly result from the symmetric properties of the equations in Table 2, are helpful.
4. Example: pentacenequinone on highly oriented pyrolytic graphite
We now employ our novel formalism in the indexing of a thin film of 6,13pentacenequinone (PQ, C_{22}H_{12}O_{2}), which was grown on a freshly cleaved, (HOPG) substrate by physical vapour deposition under high vacuum conditions (base pressure <5 × 10^{−6} Pa; deposition rate 0.5 nm min^{−1}; final nominal film thickness 30 nm, as determined by a quartz crystal microbalance). The film was then investigated at the beamline W1 at the synchrotron radiation source DORIS (DESY, HASYLAB, Germany). GIXD experiments together with specular Xray diffraction were performed using a goniometer in pseudo 2+2 geometry by a onedimensional detector (MYTHEN, Dectris) and a wavelength of 1.1796 Å for the primary Xray beam. The specular scan was performed in the 2θ range of 2° (q_{z} = 0.185 Å^{−1}) to 26° (2.395 Å^{−1}). For the GIXD experiments, the incident angle of the primary beam was set to α_{i} = 0.13°. The inplane scattering angle θ_{f} was varied between 3° and 40° in steps of 0.05° where for every step an outofplane scattering range of Δα_{f} = 3.5° was recorded. In total, seven scans along θ_{f} were performed so that the complete covered angular range of α_{f} was 0° to 24.5°. The diffraction pattern was transformed from real to using the custommade software PyGID (Moser, 2012). The resulting reciprocalspace map illustrates 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 integration of the intensities along q_{xy} and q_{z}, respectively, and fitted by Gaussian curves. The q_{z} values of the peak positions were corrected in terms of a maximum variation of 0.011 Å^{−1} was obtained (Resel et al., 2016).
Fig. 3 shows the specular diffraction pattern where only the region around the two dominant diffraction peaks is depicted. The peak at q_{z} = 1.873 Å^{−1} (d = 3.355 Å) agrees well with the expected peak position of the 002 reflection of graphite (d = 3.354 Å) based on the lattice constants of a = 2.459 and c = 6.708 Å (Baskin & Meyer, 1955). The second peak located at q_{spec} = 1.946 Å^{−1} (d = 3.229 Å) is assigned to the PQ crystals. Fig. 4(a) shows the diffraction pattern of the GIXD experiment. Bragg peaks at q_{xy} = 2.946 Å^{−1} and q_{z} = 0.002 Å^{−1}, q_{z} = 0.941 Å^{−1} and q_{z} = 1.880 Å^{−1} are identified as the 10−10, 10−11 and 10−12 reflections of the HOPG singlecrystal substrate. Additionally, a Debye–Scherrer ring appears at q = 1.87 Å^{−1} which is assigned to disordered 0002 planes of graphite. The diffraction features of the HOPG substrate are marked by arrows in Fig. 4(a).
The remaining Bragg peaks are assigned to PQ crystals; they are distributed within the whole reciprocalspace map. The most intense peaks with their q_{xy} and q_{z} positions were used together with q_{spec} = 1.946 Å^{−1} for the indexing routine. A total of 74 reflections of the GIXD map were included in the analysis. In a first step of indexing the u and v of the contact plane (the crystallographic plane which is parallel to the substrate surface) are varied by integer variables together with a systematic change of the of three reflections so that a first set of lattice constants a, b and γ are obtained [see equation (43) in Appendix B]. Note that with the restriction of linear independency due to linear transformation [see equation (32)] three pairs of Laue/Miller indices are, in principle, freely eligible to get a mathematically valid solution which may represent a This first set of lattice constants is used to determine the h and k of all other peak positions (q_{xy}, q_{z}) until a suitable assignment of the h and k to all 74 reflections is obtained. For this procedure, mathematical expressions in Appendix D are helpful. In total 150 integer variables and three real numbers have to be determined.
In a subsequent step, the assignment of the remaining w as well as the l and the evaluation of the lattice constants c, α and β have to be accomplished. There are two possible ways. The first possibility relies on symmetry considerations of equation (24) (see Table 2). A systematic exchange of the two the pairs of and the three lattice constants leads to a set of three equivalent equations where, finally, all parameters of the indexing are determined. The second possibility is simply using equations (21) and (25), where the remaining integer w of the contact plane, the lattice constants c, α and β as well as the l of the 74 reflections are obtained. Expressions in Appendix C can be helpful in determining the parameters c*, w and l of specific reflections. In a last step, when all integer variables have been assigned, the values of the real lattice parameters can be fitted. For this procedure expressions in Table 2 can be used.
As the underlying equations do not allow a unique mathematical solution, a manifold of possible results exist. But crystallographic restrictions constrain these mathematical solutions. The cell parameters must obey the scalar product (Niggli) criteria (see Table 3). Furthermore, one has to check if a solution has the shortest possible edges and thus is a Buerger cell.
For illustration, we depict the following two mathematical solutions, both of which obey the scalar product criteria for typeII cells:
Solution 1: u_{1} = 1, v_{1} = 0, w_{1} = 2; a_{1} = 5.067 Å, b_{1} = 8.064 Å, c_{1} = 8.882 Å, α_{1} = 91.64°, β_{1} = 93.34°, γ_{1} = 94.01°, V_{1} = 361.2 Å^{3}.
Solution 2: u_{2} = 1, v_{2} = 2, w_{2} = −2; a_{2} = 5.067 Å, b_{2} = 11.824 Å, c_{2} = 12.166 Å, α_{2} = 95.53°, β_{2} = 90.22°, γ_{2} = 95.25°, V_{2} = 722.4 Å^{3}.
In Table 6, corresponding Laue triples of some lower reflections are given. The determinants of three linearly independent triples of indices give mostly ±1 for solution 1 and ±2 for solution 2. The transformation matrix N which leads from solution 2 to solution 1 can be determined according to equation (32):
with det(N) = ½, establishing the relations between the lattice parameters, the of the contact plane and the of the 74 Bragg peaks (see Table 3).

By applying equation (18), three reciprocallattice vectors g, e.g. of the three Laue triples (1,0,1), (0,1,1) and (0,1,0) for solution 1, and (1,1,−1), (0,2,0) and (0,1,1) for solution 2, can be calculated. The determinants of the vector matrices should be as small as possible but not equal to zero. By determining their inverse matrices and by multiplying these with vectors m = 2π(m_{1}, m_{2}, m_{3})^{T}, where m_{i} are systematically varied integers between −2 and 2, lattice vectors can be obtained [see Appendix E with emphasis on equation (71)]. In both cases, listing the lengths of these vectors in ascending order yields 5.067, 8.064, 8.882, 9.219, 9.819, 9.966, 10.134, 10.479, 11.824 and 12.166 for the ten shortest vectors. The z components of these vectors are all integer multiples of (in absolute values 1, 0, 2, 1, 1, 3, 2, 1, 2, 2, respectively, and thus representing the – see the equations in Table 7). Therefore, solution 1 matches the whereas solution 2 represents a The thus obtained vectors of both solutions, though they do not coincide, but are equally rotated, span identical parallelepipeds and result in the same cell parameters a, b, c, α, β and γ. Therefore, solution 2 can be reduced very effectively by the described method.
For evaluating the reliability of powder pattern indexing, a factor F_{N} has been introduced (Smith & Snyder, 1979). For GIXD we suggest the following factors for assessing the accuracy of the obtained result:
where N is the number of reflections, (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. In our case d_{74,xyz} = 0.0022 and d_{74,z} = 0.0032. However, it should be emphasized that it is additionally necessary to prove that the obtained corresponds to the reduced cell.
Since the unitcell dimensions are considerably different to the three reported phases of PQ (Dzyabchenko et al., 1979; Nam et al., 2010; Salzmann et al., 2011), we can conclude that a new polymorph is found. Based on the the peak positions are calculated and plotted in Fig. 4(b). A total of 80 positions of Bragg peaks could be assigned to PQ crystals by their Laue indices.
If the specular scan is not known, it is then an additional unknown parameter in equation (24), which has to be solved numerically by using four pairs of input parameters q_{xy} and q_{z}. An alternative way would be to exclude the specular diffraction peak from the indexing procedure: an alternative notation of the rotation matrix may then be used [see equation (35) in Appendix A]. Even in that case a twostep separation of the indexing can be obtained. Input parameters are the total length of the scattering vectors q_{xyz} and q_{z} and the estimated parameters are the lattice constants a, b, γ and the two angles ψ and φ which express the orientation of the crystal at the substrate surface [equation (38)]. Note that q_{xyz} can be easily determined by q_{xyz}^{2} = q_{xy}^{2} + q_{z}^{2} and due to the Laue condition q_{xyz} = g_{xyz} and q_{z} = g_{z}. In our case the rotation angles ψ = 94.01° and φ = 39.78° are obtained. The other lattice constants c, α and β can be obtained from equation (37).
There are different possibilities to determine the molecular packing based on the knowledge of the crystallographic et al., 2006). In the case of organic thin films rigidbody procedures based on experimental structure factors were used (Krauss et al., 2008; Mannsfeld et al., 2011) or theoretical modelling was applied (Schiefer et al., 2007; Jones et al., 2017). Here, the molecular packing relative to the experimentally determined has been determined by theoretical modelling, where a combination of (MD) simulations and density functional theory (DFT) was used. MD simulations were carried out using the LAMMPS code in combination with the CHARMM General Force Field v.2b7 (Plimpton, 1995; Vanommeslaeghe et al., 2010). In a first step, several hundred trial structures were created by placing one molecule randomly into a slightly expanded During the subsequent MD run, the system was allowed to relax energetically while the was continuously shrinking to the experimental size. The most promising structures were further redefined using DFT geometry optimizations as implemented in the VASP package (version 5.4.1) (Kresse & Hafner, 1993, 1994; Kresse & Furthmüller, 1996a,b). The Perdew–Burke–Ernzerhof functional for the exchange and correlation (Perdew et al., 1996) and projectoraugmented wave potentials for all the elements (Blöchl, 1994; Kresse & Joubert, 1999) were used. Van der Waals corrections were included following the manybody dispersion approach of Tkatchenko et al. (2012). A planewave cutoff energy of 800 eV and a converged Monkhorst–Pack grid (Monkhorst & Pack, 1976) of 7 × 4 × 4 were used. The total energy during the selfconsistency loop of each DFT step was converged to 10^{−8} eV. Calculations were performed using the experimental volume, relaxing the atomic positions down to a threshold of 10^{−3} eV Å^{−1} on forces. Based on the molecular packing a diffraction pattern was calculated. The result is depicted in Fig. 4(c), where the intensity as well as the position of the Bragg peaks are illustrated by circles. The centre of the circles gives the peak position, while the area of the circles gives the square of the corresponding structure factors. An excellent agreement between experimentally and calculated peak intensities is found; hence, the resulting molecular packing describes the surfaceinduced phase of PQ on HOPG. The for the solved can be found in the supporting information.
(DavidThe packing of the PQ molecules within the a). Short contacts appear between O atoms and neighbouring H atoms and between terminal H atoms of neighbouring PQ molecules (Fig. 5b). In a subsequent step the orientation of the molecules relative to the substrate surface can be determined. The described indexing routine reveals the assignment of the 102 to the specular diffraction. Plotting the crystallographic plane with 102 towards the molecular packing of our solution directly reveals the orientation of the molecules relative to the substrate surface (Fig. 5a). It is found that the long molecular axes are aligned parallel to the substrate surface. The molecular plane encloses an angle of 8° to a `flaton' orientation. It seems that the enhanced intermolecular interactions via the short oxygen–hydrogen bonds (discussed above) establish a stabilization of a layer formed by tilted molecules.
can be described by a parallel stacking of the planar molecules. The stacking distance between the planar molecules is about 3.45 Å (Fig. 55. Conclusion
In the present work, we provide a unifying framework for the indexing of reciprocalspace maps obtained by GIXD on fibretextured thin films, which we successfully apply in deriving the full structure solution of an asyetunknown substratemediated polymorph of PQ.
Including the specular peak in the mathematical formalism of diffraction experiments can be of considerable help, especially in the case of GIXD where the spatial orientation of the u, v and w can be employed, and mathematical expressions can be derived in which the unknown cell parameters are considerably reduced. This significantly reduces computational efforts, as the integer variables can be systematically varied and only three real unknown parameters remain, which can be analytically calculated using q_{xy} and q_{z} of three independent diffraction peaks. In subsequent steps the remaining parameters can then be conveniently determined.
has to be considered. For the rotation parameters the integer variablesAs any linear combination of the unitcell vectors satisfies the imposed mathematical conditions no unique solution exists. Based on the well known criteria originally imposed by Niggli, the reduced
has therefore to be determined. The main conditions for reduction require that the cell is based on the three shortest vectors of the lattice. These can be obtained from any mathematical solution by the proper threedimensional linear transformation. It may be helpful to use the obtained cell parameters and to calculate three linearly independent reciprocal vectors and evaluate their inverse matrix to determine the lattice vectors of the reduced unit cell.Though our analysis primarily considers the general case of a triclinic system, it also applies to crystal systems of higher symmetries which then imply a higher impact of symmetry considerations such as that of reflection conditions.
APPENDIX A
Alternative notation of the rotation matrix
If u and v have to be assumed to be real numbers (and if not u = v = 0) it may be more convenient to rewrite equation (14) as
where and . Then equations (19) and (20) are rewritten as
with and . From equations (36) and (37) the following expression can be derived:
which comprises the h and k, the three unitcell parameters a, b, γ and the rotation angles and .
APPENDIX B
Mathematical procedure for analytically determining the cell parameters a, b and γ
For analytically determining the unitcell parameters a, b and γ, it is convenient to introduce the parameters Z_{a}^{2}, Z_{b}^{2} and with the substitutional relations
Note that Z_{a}^{2} and Z_{b}^{2} are always positive. Using these substitutions, equation (24) can be rewritten as
From three independent Bragg peak series, the parameters Z_{a}^{2}, Z_{b}^{2} and can be determined by solving the following set of equations:
where
and i = 1, 2 and 3. For obtaining a, b and from equations (39) to (41) the following identity is helpful:
APPENDIX C
Compact expressions for the reciprocal cell parameters a*, b*and c*
If hv − ku = 0 and u ≠ 0 equations (22) and (24) can be reduced to
and
respectively. Combining equations (48) and (49) the following expression can be obtained:
If hv = ku and v ≠ 0 the following relation can be deduced in an analogous way:
For hw = lu and kw = lv analogous formulas for b* and a*, which contain only one real unknown besides the integer variables, are valid.
APPENDIX D
Useful expressions for determining the Laue indices
From equation (42) the following expression for the k can be derived:
The included components are given in equations (6), (40) and (41). As a term under a root sign must not be negative the following conditions need to be satisfied:
Note that analogous conditions are valid for k and l with the proper parameters v, b and w, c, respectively. Furthermore, rough lower limits for the lattice parameters a, b and c with , and result.
APPENDIX E
General rotation and analysis for threecomponent scattering vectors
In the general case, when the individual components of the reciprocal vector g_{x}, g_{y} and g_{z} have to be considered, equation (18) has to be expanded by including an additional rotation around the [0, 0, 1] axis applying the matrix R(φ):
which performs a rotation counterclockwise by an angle . Then the reciprocal vector
can be expressed as
From equation (55), using equation (4), it follows that
With
equation (56) can be equivalently expressed as
where a, b and c are the rotated unitcell vectors with the relations a = a, b = b, c = c, a·b/(ab) = cosγ, a·c/(ac) = cosβ and b·c/(bc) = cosα. These vectors can be written explicitly as
with
Thus, if three reciprocal vectors g_{1}, g_{2} and g_{3} are given, the following relation is valid:
where
and (h_{i}, k_{i}, l_{i}) are the corresponding triples of with
Equation (68) can be equivalently expressed as
Furthermore, as
the following relation for the determinants of G and H is valid:
If the specular scan g_{spec} is known it is convenient to apply equations (15)–(17) in the rotation matrix [equation (35)]. Then the unitcell vectors a, b and c can be expressed as given in Table 7. Note that the z components are only a function of the and g_{spec}.
The unitcell vectors must be solutions to all reciprocal vectors g_{i}, which, according to equation (58) and comprising equation (71), can be written as
where
and
Therefore, if a′, b′ and c′ are vectors of a with
where N is the transformation matrix [see equations (26)–(29)], the following relation with can be derived from equation (74):
From equation (71) 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 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 (73) it is favourable to select three reciprocal vectors whose matrix results in a determinant, which is as small as possible but unequal to zero. Otherwise a which is too small and does not satisfy equation (74) for all reciprocal vectors may result. The Buerger 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.
Supporting information
C_{22}H_{12}O_{2}  α = 91.64° 
M_{r} = 308.32  β = 93.3° 
Triclinic, P1  γ = 94.01° 
a = 5.067 Å  V = 361.32 Å^{3} 
b = 8.064 Å  Z = 1 
c = 8.884 Å  T = 300 K 
x  y  z  B_{iso}*/B_{eq}  
C0  1.12031  0.04031  −0.22136  
C1  1.24568  −0.10181  −0.26824  
C2  1.17551  −0.25843  −0.20220  
C3  0.98812  −0.26410  −0.09031  
C4  1.29726  −0.40225  −0.25038  
C5  1.48021  −0.39186  −0.35938  
C6  1.54823  −0.23824  −0.42524  
C7  1.43490  −0.09590  −0.37996  
C8  0.04764  0.14505  0.38299  
C9  0.23690  0.15101  0.27128  
C10  0.30672  0.30759  0.20509  
C11  0.18452  0.45133  0.25307  
C12  0.49415  0.31333  0.09323  
C13  0.61226  0.17213  0.04748  
C14  0.54852  0.01736  0.11640  
C15  0.36249  0.00895  0.22451  
C16  0.93407  0.03193  −0.11335  
C17  0.80211  0.18396  −0.07301  
C18  0.68031  −0.13470  0.07595  
C19  0.87019  −0.12286  −0.04452  
C20  0.00155  0.44090  0.36206  
C21  −0.06605  0.28732  0.42811  
H22  1.16801  0.15982  −0.27169  
H23  1.48771  0.02293  −0.43014  
H24  1.24030  −0.52067  −0.20076  
H25  1.57405  −0.50288  −0.39471  
H26  1.68995  −0.23267  −0.51322  
H27  0.24119  0.56974  0.20334  
H28  0.54876  0.43180  0.04293  
H29  0.31504  −0.11054  0.27493  
H30  0.93330  −0.38258  −0.04010  
H31  −0.00486  0.02626  0.43331  
H32  −0.09258  0.55186  0.39726  
H33  −0.20772  0.28171  0.51612  
O34  0.84557  0.31551  −0.14001  
O35  0.63669  −0.26631  0.14284 
Acknowledgements
The experiments were performed at beamline W1, HASYLAB at DESY, Hamburg, Germany.
Funding information
Financial support was given by the Austrian Science Foundation FWF: [P30222].
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