A. Effect of the uncertainty on peak positions

The absolute uncertainty Δ(2θ) _i on the peak position is an important parameter in powder pattern indexing. Too large selected errors Δ(2θ) _i would introduce instabilities arising from more possible theoretical lines within the error interval, they would lengthen computing time and they would increase the risk of missing the correct solution. Clearly, the higher the precision, the narrower the absolute error can be set, and easier is the pattern indexing. Generally, for data collected with laboratory X-rays and CuKα radiation a standard experimental error is ±0.03° (2θ) (default value in DICVOL14 for monoclinic and higher symmetry); however, it is desirable to reduce it for the triclinic symmetry, e.g. ±0.025° (2θ) (default value in DICVOL14) and, also, when large cell axes are expected or if the density of lines is high (e.g. large-cell volume, presence of spurious lines, short wavelength). For a shorter wavelength λ, as frequently used with synchrotron X-rays, the default value in DICVOL14 is set proportional to the ratio λ/λ _{Cu Kα} . The default values must, however, be critically considered according to the diffraction line broadening. It is worth considering the absolute error on the individual lines with respect to their width and resolution in the diffraction pattern. The instrumental resolution function (IRF) of the diffractometer used to collect the data (see, for example, Louër and Langford, Reference Louër and Langford1988) is the reference for minimum line broadenings. In practice, the measured full-width at half-maximum (FWHMs) can be used for estimating the error intervals ±Δ(2θ) on peak positions. As a rule of thumb an error interval in the range 1/2–1/3 FWHM is a reasonable (starting) estimation for separated lines. [It can be noted that for non-uniform FWHMs, e.g. in case of anisotropic line broadening, line overlap or low-counting statistics, the absolute error Δ(2θ) _i can be set for each individual line i.]

With the advent of high resolution with synchrotron X-rays and the use of short wavelengths, data collection with step sizes as small as 0.001° (2θ) are reported. This is illustrated in Figure 1 with data collected at the 11-BM synchrotron beam line (Argonne Advanced Photon Source) (Papoular et al., Reference Papoular, Toby and Agafonov2011). It shows a line profile in the pattern of triclinic tetracaine hydrochloride (λ = 0.458 82 Å), collected with steps of 0.001°(2θ), with a FWHM = 0.0106° (2θ). The angular interval shown at the top of the line profile corresponds to 0.003° (2θ), i.e. ±1.5 steps or ±0.0015° (2θ), close to 1/3 FWHM. When using this small uncertainty on input peak positions the solution is obtained in 0.1 s with DICVOL14.

Figure 1. Example of diffraction line profile in the powder diffraction pattern of tetracaine hydrochloride collected with synchrotron X-rays, λ = 0.458 82 Å (11-BM synchrotron beam line, Argonne Advanced Photon Source), step size 0.001° (2θ) (Papoular et al., Reference Papoular, Toby and Agafonov2011). The FWHM is 0.0106° (2θ) and the 0.003° interval corresponds to an absolute error Δ(2θ) of 0.0015° (2θ).

B. Error magnitude and computing time

The magnitude of the absolute error |Δ(2θ) _i | has an effect on the search of indexing solutions with the dichotomy algorithm and, consequently, on computing times (CPU times). This is also dependent on the lattice symmetry since the density of lines increases according to the number of unit cell parameters. It has already been shown that DICVOL91 is tolerant of imprecision (Louër, Reference Louër, Prince and Stalick1992). Simulated powder data of orthorhombic and monoclinic materials with a random error were used in order to assess the behavior of the program (see Figure 2 in Louër, Reference Louër, Prince and Stalick1992). Successful indexing was obtained with random errors as large as 0.10° (2θ) and 0.07° (2θ), respectively. In the same review, the effect of zero-point errors (see Figure 3 in Louër, Reference Louër, Prince and Stalick1992) has also been studied. For the two examples the correct solution was found up to 0.30° (2θ) and 0.15° (2θ) zero-angle offset, respectively. This tolerance to random and systematic errors was also noted by Shirley (Reference Shirley, Block and Hubbard1980). Some powder data collected with laboratory and synchrotron X-rays are used to further illustrate the effect of the magnitude of the error parameter.

Figure 2. Search of the zero-point offset with the scanning approach based on the pairs of harmonic lines detected in the powder data set. S is the residual minimized with a least-squares analysis.

C. Zero-point offset evaluation from a scanning approach

An optional analysis of the zero-angle offset in the input data was proposed in DICVOL04. Its interest was essentially for uncalibrated data with significant zero shift. Indeed, for small shifts [e.g. ≲ 0.05° (2θ)], the extra degree of freedom (for the mixed error parameter, including the true zero-point shift and specimen-surface displacement in B–B optics) in the refinement process of the lattice parameters is generally efficient enough to solve the problem. A benefit of refining the “zero-shift” is an increase of the FoMs (see, Louër and Boultif, Reference Louër and Boultif2006).

For greater zero-point shifts the method used in DICVOL04 was based on the reflection-pair method reported by Dong et al. (Reference Dong, Wu and Chen1999). In the context of pattern indexing, some ambiguous results were often found with DICVOL04, leading to long CPU times. In the new version DICVOL14, a scanning approach has been preferred. The procedure is also based on the use of higher orders of reflections, if they can be identified unambiguously. In the present approach, the input 2θ _i positions are artificially shifted by a zero-point varying in an interval [−0.30°, +0.30° (2θ)] with a scan step width of 0.005°. For each step, the effect on the different line orders is calculated through the quantity Δm (=m–d _i /d _j , where d _i and d _j are the d-spacings of two line orders for a particular step, and m is the integer corresponding to the ratio of the d-spacings). All identified pairs of lines with an integer ratio are considered. The zero-point is then estimated from a least-squares analysis, by minimizing the residual S [=Σ(Δm/m)²]. The sum is taken over pairs of harmonics. As shown in Figure 2, for a particular example, the minimum of the residual is located between steps 37 and 38, i.e. the zero-shift [=(37 + 38)/2 × “step width”] is 0.1875° (2θ).

The scanning approach for zero-point shift estimation is stable and a unique solution is obtained, provided unambiguous higher orders of lines at low angles are identified. The method has been tested successfully on many experimental powder data sets. In particular, all 30 triclinic powder data reported in the whole NBS monograph 25 (1963–1985) were artificially degraded by a zero shift in the range 0.15°–0.30° (2θ). These data were subsequently analyzed with DICVOL14. Twenty shifted patterns were correctly indexed.

The powder data of acetylcysteine reported in the PDF entry 46-1988 (ICDD, Reference Soorya2014), with the quality “i”, are characterized by a low FoM F ₃₀ = 20(0.031, 49). They can be used to illustrate the usefulness of the scanning method implemented in DICVOL14. The unit cell is triclinic (a = 5.897 Å, b = 6.548 Å, c = 5.108 Å, α = 103.93°, β = 102.19°, γ = 96.34°, V = 184 Å³). With 0.03° (2θ) as absolute error on peak positions, an unsatisfactory solution (V = 477 Å³), with a poor FoM [F ₂₀ = 17(0.018, 65)], is found in the second volume domain (352 s). If the parameter for searching a zero-point offset is activated, a zero-point error −0.125° (2θ) is suggested and the correct solution [M ₂₀ = 33, F ₂₀ = 51(0.016, 25)] is found in 0.3 s.

A. General tests

DICVOL14 has been tested from powder data reported in the NBS Monograph 25 (Natl Bur. Stand., 1963–1985), namely 70 experimental and one calculated patterns reported in section 20 (Natl Bur. Stand., 1984) [absolute uncertainty Δ(2θ) = 0.03°], as well as the 30 patterns of triclinic compounds (including seven calculated patterns) reported in the whole NBS Monograph 25 [absolute uncertainty Δ(2θ) = 0.025°]. [For the powder data of Li₄Th₇(MoO₄)₁₆ and Li₄Th₇(WO₄)₁₆ the parameter “VOLMAX” was set to 3500 Å³.] The patterns were all correctly indexed, except for the data of triclinic KIO₃ (Section 15, a = 7.708 Å, b = 7.222 Å, c = 7.689 Å, α = 109.25°, β = 108.96°, γ = 109.37°, V = 355.9 Å³). This phase is described as a “pseudo-monoclinic perovskite-type” and only a triclinic cell with half the correct volume was obtained with DICVOL14. For the 100 indexed patterns, the average computing times were less than 1 s for orthorhombic and higher-symmetry data sets and 4 s for monoclinic cases, except for one data set (Eu₂O₃) indexed in 80 s. Thirteen powder data sets of triclinic compounds were indexed in less than 5 s, eight in the range 5–30 s, six in the range 30–160 s, and the last two data sets in 360 and 690 s. The program has also been successfully tested with experimental powder data using the density and formula weight of the material as input information. As shown by Louër and Vargas (Reference Louër and Vargas1982), substantial reduction of computing times are obtained when the density information is used, since only limited volume domains of the direct parameter-space are explored. The powder data of pharmaceutical materials listed in Boultif and Louër (Reference Boultif and Louër2004; Table 2) were also indexed.

B. Triclinic examples

To improve further the triclinic section of DICVOL14, additional powder diffraction data of materials with a triclinic cell from the PDF, with quality “C”, “*”, and “i”, have been used to check the behavior of the program. Among them, 40 examples with peculiarities were selected according to the presence of a dominant zone (several successive first lines with a common zero Miller index, No. “0” in Table I), to long edges with several higher orders of the first line as first lines (N _harm in Table I) of the data set, or having close linear parameters. They were successfully indexed; some representative examples are listed in Table I. From the input data the presence of harmonic lines as first lines in the pattern (N _harm) can be easily detected, and it is the indication of a long axis with respect to the two others. In practice, with such experimental powder data, it is necessary to act on some input control parameters to have a chance to identify the correct solution, i.e. the number of lines N (>20) used for searching solutions, the absolute error Δ(2θ) and the tolerance for spurious lines N _imp (which include impurity lines and measured line positions with an error exceeding the input absolute error).

Table I. Examples of triclinic powder data with a dominant zone or long axis indexed with DICVOL14. Data quality (PDF, ICDD, Reference Soorya2014): “C” for calculated, “*” for precise data, and “i” for indexed data. The linear cell parameters are given, No. “0” is the number of successive first lines with a common zero index, N _harm is the number of successive harmonic lines at lowest angles, N is the number of lines used for the indexing search, N _imp is the number of allowed impurity lines, N _calc is the number of calculated lines up to the 20th observed line, Δ(2θ) the input absolute error in DICVOL14 (i.e. 0.002° for calculated patterns and 0.025° for experimental patterns), and CPU is the computing time.

The powder data of n-docosane (PDF entry 53–1533 “*”) can be used for illustration. The parameters of the reduced cell are a = 4.28 Å, b = 4.82 Å, c = 28.26 Å, α = 86.41°, β = 85.95°, γ = 73.61°, and V = 557 Å³. The first five lines are 00l lines (l = 1–5), consequently the value for N was increased to 25. With the default value for the absolute error and no spurious lines [Δ(2θ) = 0.025°, N _imp = 0], a first calculation (81 s) with DICVOL14 is unsuccessful. A second run with a tolerance for one spurious line [Δ(2θ) = 0.025°, N _imp = 1] gives the correct reduced cell [M ₂₀ = 14 and F ₂₀ = 22(0.014, 67)] in 510 s. The value of N _calc = 67 indicates that many possible lines are missing from the input data and the indexing listing shows that one line among the 25 lines used for indexing has a discrepancy (2θ _calc–2θ _obs) = 0.042°2θ.

C. Impurity lines, binary patterns

Powder pattern indexing is a demanding application of the powder method. This is a consequence of the ill posed mathematical problem as noted in this statement by Vand and Johnson (Reference Vand and Johnson1968) “we cannot solve the indexing problem by the method of ordinary algebra, as no matter how many (quadratic) equations we take, we have always more unknowns than knowns”. The dichotomy algorithm is a means to circumvent this problem. Generally, data quality must be as high as possible and it is desirable to avoid impurity lines, because they introduce additional unknown parameters and methodological instabilities. However, for a few two-phase patterns, some successful results have been reported with DICVOL04 (Louër and Boultif, Reference Louër and Boultif2006). A sample of 15 powder data sets of binary-mixture patterns have been used to test the efficiency of the parameter “number of spurious lines” in DICVOL14. Data were taken from the NBS Monograph 25 (1963–1985) and from the PDF (ICDD, Reference Soorya2014) for the piracetam polymorphs. The individual data sets were added to simulate mixing and DICVOL14 was used for searching an indexing solution. Clearly, only the solution of the phase with higher density of lines could be expected. A small error Δ(2θ) (e.g., 0.01°2θ) was used to reduce instabilities and to limit the number of mathematical solutions. For 12 data sets, a correct indexing solution was obtained (Table II). For two binary patterns of triclinic/monoclinic and triclinic/triclinic phases, many solutions were proposed; however, the correct solution was ranked at the fourth position over 32 and first over 37 solutions, respectively. For one mixture of patterns of triclinic phases the program failed because the first line belonged to one phase and the second line to the second phase (which contradicts the strategy used for the triclinic symmetry).

Table II. Binary powder data sets. V is the cell volume given for both compounds, Δ(2θ) is the absolute error, N is the number of observed lines used for indexing the phase with the highest number of lines, and N _imp is the number of allowed unindexed lines. Lattices are denoted as follows: (T) tetragonal, (H) hexagonal, (O) orthorhombic, (M) monoclinic, and (Tr) triclinic. The lattice in bold characters corresponds to the solution found by DICVOL14.

^aNatl Bur. Stand. (1984).

^bNatl Bur. Stand. (1963–1985).

^cICDD (Reference Soorya2014).

The use of DICVOL14 with binary mixture data sets must be considered with caution. Only in some particular circumstances, for example in the course of solid-state syntheses or mixture of polymorphs, such application could be helpful to indentify the content of the pattern. This is illustrated with the pharmaceutical compound piracetam. In the course of the structure determination study of form I from powder diffraction data (Louër et al., Reference Louër, Louër, Dzyabchenko, Agafonov and Céolin1995) and stability study of piracetam polymorphs (Céolin et al., Reference Céolin, Agafonov, Louër, Dzyabchenko, Toscani and Cense1996), mixtures of polymorphs II and III, on one hand, and I and II, on the other hand, were observed. Powder data of these two binary mixtures were simulated from the powder data of the pure phases collected with a D–S setup, namely form I (monoclinic, PDF entry 54-2432), form II (triclinic, entry 54-1783) and form III (monoclinic, entry 54-1784). The monoclinic cells of form I and form III in mixtures I/II and III/II, respectively, were found with DICVOL14 (Table II). As already noted, this application of DICVOL14 remains marginal. Obviously, the main objective of DICVOL14 is to index single-phase data. It is always desirable to identify spurious lines, for example with the help of search/match methods for phase identification (for a review, see Langford and Louër, Reference Langford and Louër1996) or other appropriate technique, e.g. temperature- or time-dependent X-ray powder diffraction in the cases of Advil tablet (Boultif and Louër, Reference Boultif and Louër2004) and piracetam (Céolin et al., Reference Céolin, Agafonov, Louër, Dzyabchenko, Toscani and Cense1996).