Lattice Parameter Calculation For Monoclinic Crystal System

[Ben Hollinbeck]

Thetemplates are a set of Excel spreadsheets that calculate unit cell parameters (axis lengths and angles) from X-ray powder diffraction data using a least-squares best-fit fit procedure. The programs require 2Θ peak positions and hkl indexes for each peak. They also require that you know what mineral you are working on. There is one template for each of the six crystal systems (hexagonal and trigonal use the same template).

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

X-ray diffraction was used to identify the crystalline phase of the NbTe2 samples. From the obtained diffractogram (Fig. 1) and the data in ref.6 it was determined that the crystalline structure is monoclinic, described by the symmetry space group 12 (C2/m). The corresponding unit cell is shown in Fig. 2a. Given the layered nature of NbTe2 and a preferred growth orientation, the most intense reflections in the diffraction pattern correspond to the [00c] direction, Fig. 1. The stacking sequence of the single layers for this material is shown in Fig. 2b. As a consequence of the structural distortion in NbTe2, the buckled surface of the Te planes and the non-uniform Nb-Nb distances can be observed.

X-ray diffraction pattern of the NbTe2 crystal and whole pattern fitting carried out by the Rietveld method. Circles correspond to the experimental data, while the fitted pattern is indicated by the solid line. The difference between the experimental data and the fit is described by the bottom graph.

The upper surface and lateral cross-section morphologies were analyzed by SEM on freshly exfoliated surfaces, Fig. 3. The quasi-2D nature of the stacked layers is evident in the upper views, Fig. 3a,b, as well as in the cross-sectional micrograph, Fig. 3c. The images show homogeneous and flat surfaces at the scales indicated in Fig. 3a,b except when the edges of several layers are exposed, so that a terrace-like structure is apparent.

To further investigate the structural properties of niobium ditelluride, cross-sectional HRTEM images were obtained at different magnifications, Fig. 4. The atomic positions in the layered structure are readily observable as well as the high crystalline quality and homogeneity of the sample. The van der Waals gaps appear as slightly darker stripes in all images of Fig. 4. It was possible to obtain selected area diffraction patterns (SADP), as the one shown in Fig. 5a. The pattern exhibits well defined bright diffraction spots, immersed in a dark background. The good contrast between white spots and dark background is indicative of high crystalline quality. The lattice parameters and unit cell angles determined from the Rietveld analysis of the X-ray parameters were employed to obtain the indexation and simulation of the SADP (Fig. 5b,c, respectively). This was accomplished with the help of the software STEM_CELL9,10. From this analysis it was found that the zone axis of the SADP was the [010] direction. It is worth pointing out the good agreement between the calculated angles and lattice parameters obtained from the Rietveld method with those measured directly in the HRTEM images. Indeed, a good match in the atomic positions is obtained when the simulated crystalline structure and an HRTEM image are overlapped, as shown in Fig. 6.

HRTEM cross section images of NbTe2 at different magnifications. Atomic resolution may be observed in the first two images. The darker lines correspond to the van der Waals regions separating the Te-Nb-Te layers.

Superposition of a HRTEM image and the a-c plane view of the NbTe2 unit cell obtained from the Rietveld refining of the X-ray diffraction patterns. A good match in the atomic positions was obtained.

Electronic band structure and density of states of NbTe2. The arrows indicate examples of band crossings around EF that produce partial fillings of valence and conduction bands. The high symmetry directions and points of the Brillouin zone are shown in Fig. 2c. As a guide, the colors of the high symmetry points of the Brillouin zone in Fig. 2c and the color of the corresponding vertical lines in Fig. 12 are in correspondence.

A.H.B.A. and S.J.J.S. wrote the manuscript and directed the research. J.C.I. grew the NbTe2 crystals, A.H.B.A. and A.M.G.T. carried out the DFT calculations; T.S. performed and analyzed the APT measurements; F.P.D. obtained the HRTEM images; the UPS data were obtained by P.M.B.; R.P. and N.K. participated and analyzed the electrical characterization data. A.H.B.A. analyzed and model the HRTEM, X-ray diffraction, Hall effect, UPS and Raman data. The Raman spectra and Hall effect measurements were obtained by A.H.B.A.

In crystallography, the monoclinic crystal system is one of the seven crystal systems. A crystal system is described by three vectors. In the monoclinic system, the crystal is described by vectors of unequal lengths, as in the orthorhombic system. They form a parallelogram prism. Hence two pairs of vectors are perpendicular (meet at right angles), while the third pair makes an angle other than 90.

For the base-centered monoclinic lattice, the primitive cell has the shape of an oblique rhombic prism;[1] it can be constructed because the two-dimensional centered rectangular base layer can also be described with primitive rhombic axes. The length a \displaystyle a of the primitive cell below equals 1 2 a 2 + b 2 \displaystyle \frac 12\sqrt a^2+b^2 of the conventional cell above.

All that is required to run an indexing calculation are either experimental peaks or raw diffraction data.If only raw (profile) but no peak data are present when the indexing command is run, Match! willautomatically execute the raw data processingbefore the actual indexing calculation is started.

Before running the "Indexing" command you should mark the peaksto be taken into account in the indexing calculation.If you do not mark any peaks, Match! will automatically use the 20 strongest peaks (if present) thatare not yet covered by selected phases and whose relative intensity is larger than thecorresponding minimum value (which can e.g. be adjusted using the red bar on the y-axis of thediffraction pattern graphics) for indexing. You can also adjust the corresponding parameter minimumrelative intensity for automatic peak usage on the Indexing tabor the parameter "minimum relative intensity for peak correlations" on theSearch-Match tab of the Options dialog.

You can run indexing either usingthe corresponding command from the "Tools" menu, or simply bypressing the corresponding button in the main toolbar. Depending on the current settings and situation, this will either bring up adialog asking which indexing method (Treor or Dicvol) you would like to use, run the defaultindexing program, or display the table of indexing results that are already present (and from whichalso new calculations can be run).

You can define the minimum figure-of-merit as well as the maximum number of unindexedpeaks (in the Dicvol manual also called "impurity tolerance" N_IMP) for a solution to be accepted. Increasing the maximum number of unindexed peakswill increase the probability to get a solution (unit cell), so this may be a wayout if you do not get a reasonable result with the standard settings. You should keep in mind thoughthat you have to check afterwards why the unindexed peaks are present in your pattern (maybethey belong to an impurity phase or are just artifacts). Note that by ignoring unindexed peaks itis quite easy to get artificial unit cells that have nothing to do with reality!

In addition, you can define the max. peak position deviation (in degrees 2theta), i.e. themaximum 2theta difference between an experimental and its corresponding calculated peak. If the 2thetadifference is larger than this value, peaks are not regarded as being correlated (i.e. indexed).

If you know the arbitrary formula weight and density of your compound, and if the expectednumber ofmolecules or formula units in the unit cell is integer, you can enter the corresponding valuesas well as the max. density deviation as additional criteria to restrict the cell volume andhence the search space of the program. The value for the max. density deviation should be the maximumexpected density deviation plus about 5-10%. The choice of the value should also take the qualityof your diffraction data into account.

When the calculation has finished, a table of the unit cells (solutions) found by Dicvol will be displayed.Please mark one or more solutions you would like to keep, then press .You will then be taken tothe indexing solutions dialog where you canevaluate the solutions (i.e. unit cells) that you have found up to now, inspect peak data, selectcrystal system and space group, and finally export the solution or add it as a newmanual entry to thematch list (e.g. in order to proceed withstructure solution).

The crystal system and space group suggested by Dicvol are also copied to the individualsolution(s) in Match!. They can be seen in the corresponding dialog elements on the right-hand sideof the indexing results dialog if a correspondingline is marked in the solution list at the top. You can of course modify these suggestions usingthe corresponding dialog elements.

If you would like to take a look at the original Dicvol output file, you can do soby marking the corresponding solution in the solution list and clicking the View output buttonon the upper right-hand side.

The following hints on indexing have been taken from the Dicvol documentation:Be careful in using the impurity tolerance: spurious lines increases the riskto miss the correct solution!It is recommended to use a two- or three-stages procedure (i.e. triclinic lattices shouldpreferably be studied separately), for example:search in high symmetries down to orthorhombic: Line 2: n,itype,1,1,1,1,0,0search in monoclinic symmetry: Line 2: n,itype,0,0,0,0,1,0if necessary, search in triclinic symmetry: Line 2 : n,itype,0,0,0,0,0,1Note that for solutions with Monoclinic and Triclinic symmetries the program provides thereduced cell. If various equivalent solutions are found, only one of them is listed in theoutput file.Trigonal symmetry case with rhombohedral lattice: the pattern is indexed with an hexagonallattice, having a unit cell volume three times greater.Please, spend time to ensure the quality of your collected data. With accurate data, thesuccess rate ofDicvol06 is very high. Peak positions should be extracted with a profile fitting software.An interactive program should be preferred, since automatic extractions can miss lines (lowintensity, shoulder, ...).With bad data, the chance to obtain the correct solution is small and the calculationcan be time-consuming.With modern X-ray powder diffractometers (the use of monochromatic radiation is recommended),absolute errors on peak positions lower than 0.02 degrees 2theta can be routinely obtained. Forindexing purposes, errors should not (ideally) exceed 0.03 degrees in 2theta.With high resolution powder diffraction data (conventional or, particularly, synchrotron X-raysources), the absolute error is usually less than 0.02 degrees (or even 0.01 degrees with ultra-highresolution) in 2theta; consequently, a maximum peak position deviation of 0.02 (or even0.01) is recommended; the convergence of the dichotomy procedure will be improved. However, besure that this condition is true for all lines used as input data. (Remember that all mathematicalsolutions within the input limits and error bounds are found, the greater they are the greateris the number of mathematical solutions).The maximum number of unindexed lines (also called "number of impurity lines", N_IMP) canbe used in case of expected spurious lines (i.e. impurity lines, as well as observed lines outof the input error). N_IMP acts at all successive levels of the dichotomy algorithm. As soon asan indexing solution is retained, a least-squares refinement of lattice parameters is carried out.For this refinement a larger error on observed lines is considered. Then, a line rejected at thelast dichotomy level can, by chance, be accepted with the refined lattice parameters.Note that the program Dicvol06 is executable from 7 lines- 8 lines if the 'zero-shift' isrefined - (though it is not recommendable since LS refinement unstabilities can be expected).Long and short axis cases (dominant zone cases): if such cases are expected, the number Nof lines used for searching the solution should, generally, be greater than 20.The minimum value for a linear lattice parameter has been fixed to 2.5 angstroms.Reliability of indexing solutions: read paragraph 8 of ref. 5 and refs 7 and 8.Note that with the option Dicvol04 (option =0), as soon as a solution is found, onlysolutions with smallest volumes will be subsequently retained. If (for some reasons!) you arenot satisfied by the solution, you can run again the program with an input lower volume limitslightly greater than that of the found solution (the exhaustive search is then extended to ahigher volume).Note that the search is exhaustive within the limits on the input data. In particular, thesearch is constrained by the higher and smaller bounds on parameters, volumes, selected FoM andabsolute errors on peak positions. Please act on these parameters when using Dicvol06.A lattice metric singularity occurs when unit cells defining two lattices have an identicalset of calculated d-spacings. This can be observed with high symmetry lattices, simple relationsexist between the parameters of the two cells, as well as particular cell-volume ratios. A typicalcase is: an hexagonal cell [a, c, volume v] can be indexed with an orthorhombic cell [parameters: a/2,a sqrt(3), c, volume v/2]. Due to the strategy used in Dicvol, based on an analysis through decreasingsymmetry, all cells should be, in principle, displayed in the output file (except if a solution isrejected by the input maximum volume).Possible space groups: look at the hkl conditions in the output list of the reviewing of thecomplete input data provided after a solution is found from the first N lines.

3a8082e126