One of the more challenging aspects of powder diffraction is finding a unit cell that indexes the pattern. By indexing, I mean that the unit cell generates reflection positions that match most, or better yet all, of the diffraction peaks. Autoindexing is where computer software is used to search for a unit cell that will index the pattern. In contrast, indexing of single crystal diffraction patterns is a routine task that has been automated since the 1960’s. Even with multiple twins present, modern software can facilely index single crystal data. Not so, alas for autoindexing powder diffraction. This, in theory, should not be a complex task on modern computers. Noting that h, k and l values have can only have limited values if one places reasonable limits on the lengths of unit cell constants, then there are a large number of possible indices that can be assigned, so it should not be that complex to search through all possibilities. However, in reality there may be mismatches between the peak positions we measure and the actual reflection positions: two reflections may be close enough that observed position of a peak matches a location between the two reflections. There may be impurities present, so some peaks may belong to a second phase. Sample displacement, particularly in a Bragg-Brentano diffractometer, can offset all peaks. The advent of high-resolution synchrotron diffraction has certainly aided the process of indexing powder data by providing data with very minimal sample displacement and very sharp peaks.
Additional phases are not the only reason why your cell may not index all lines of the powder pattern. It is also possible that the cell you have found is a subcell of the actual lattice, so looking at conventional crystallographic subgroups (discussed in Chapter 28) might provide a related lattice that does index the pattern. Another type of supercell would be a superspace expansion to the lattice in a 4th dimension. This additional dimension is where there is lattice repetition in a direction that is an irrational1 combination of the three axes of your cell. This is known as an incommensurate or modulated lattice. This extra direction is labeled as a 4th dimension, but note that this direction is actually occurring in three dimensional space, so this is labeled as 3+1 dimensional superspace. Such structures are not common, but are not unheard of either. Theory allows for two or even three irrational lattice vectors, so 3+2 and 3+3 dimensional superspace structures are possible, and there exist examples of these from single-crystal data, but it is unlikely that they would ever be fit with powder diffaction data. GSAS-II does handle superspace groups of the 3+1 type, but not the 3+2 or 3+3 types. Note that the 230 conventional space groups, expand to 775, 3338 and 12,584 space groups, respectively for 3+1, 3+2 and 3+3 superspace. Should you need higher symmetry than 3+1, the JANA software suite should help. Indexing of superspace groups, as well as indexing of magnetic lattices, requires finding an expansion vector, often referred to as a k-vector, which will not be covered in this book. Work is being done to add k-vector search tools to GSAS-II, so perhaps this will appear in a future edition.
GSAS-II provides several tools for indexing powder diffraction patterns. These are accessed using a histogram’s “Unit Cells List” data tree entry. It should be noted that there are a number of computational methods that have been implemented for autoindexing. Sometimes one method will succeed with a list of peak positions where others fail. Should the algorithm in GSAS-II fail (which is an implementation of a published algorithm from Topas), you are recommended to also try versions of the classic autoindexing programs, TREOR, ITO and DICVOL . My rather aged CMPR program provides access to these programs, but there are probably now much better options, depending on your operating system.