Single Crystal Structure Refinement (SREF)
The electron density map generated by solution of the phase problem can be seen as a basic structure map. However, the assignment of atoms to different intensity centers is key to understanding the structure. The step of atom assignment is referred to as solving the crystal structure. The procedure can be mystifying to the beginner, but is reasonably straightforward. There are two basic procedures for structure solutions, which will be discussed below.
Known MineralIf the sample is a known mineral species, a template of a solved structure of that species may be used for initial atom assignment. This speeds solution, as the major sites can be assigned quickly. Once these have been assigned, the solution can be further manipulated to extract a better agreement between observed (F(obs)) and calculated (F(calc)) data. Typical steps include changing the site occupancy, splitting the site occupancy between two elements, allowing the sites to become anisotropic, and locating hydrogen. As hydrogen is the lightest element, it can be difficult to locate in minerals with significant "heavy" element content. Small changes are made at each step and then run through least-squares cycles, which recalculate the structure using Fourier transformations. Practiced refiners tend to follow the same steps each time.
Unknown MineralMore challenging is when the mineral is unknown or a new species. Here, a template may not be able to be used and elements must instead be assigned based on the geometry of the intensity centers and the chemistry of the material. This procedure can be much more difficult, as it is often through "trial and error" that the correct assignment is found. Once it has, the structure will be further refined as described above.
ResultsThe results of the structure refinement yield a list of atom X-Y-Z assignments in the unit cell, shape of the anisotropic intensity center for each atom (thermal parameters), and the distance of the nearest atomic neighbors. Additionally, the angle between nearest neighbors is also given. Quality of a solution is assessed by the values of R1, wR2, and GooF.
- R1, often called the R-value, is the agreement between the calculated and observed models. The formula for this value is R1= |S |Fo|-|Fc| | / |S |Fo| |. Ideal solutions would have R-values of 0, however, due to random errors, this is never achieved. R-values (listed as percents) of less than 5% are considered good solutions; high quality samples will often result in R-values lower than 2.5%.
- wR2 is similar to R1, but refers to squared F-values. This results in wR2 always having a higher value than R1.
- The final value, GooF, refers to the "goodness of fit" of the solution. In addition to the difference in F values, the GooF also takes into account the number of observed reflections and the parameters used. At the end of refinement, the GooF should approach 1.