As was explored in §3.6, if we know the structure factors, including their phases, we can use the Fourier inversion of the structure factor equation, \[\rho (x,y,z) = \frac {1}{V} \sum _{hkl} F_{hkl} \exp [-2\pi i(hx + ky + lz)] \] and obtain the scattering strength at any site in the unit cell. When this is computed for a three-dimensional grid of points across the unit cell (or better, just the unique portion of the unit cell) we get what is known as a Fourier map. The locations that have the most density are where there must be scattering matter (electrons or nuclei) in the crystal. If we use the \(F_{calc, hkl}\) values in this equation, we get back a rather uninteresting result – a map that shows the positions of the atoms that were used to compute the \(F_{calc, hkl}\) values. This is known as a Fcalc map.
Note that the above sum needs to be to infinity to obtain precise values of \(\rho \), but, of course, in practice we can only measure to \(180^\circ \) degrees, and rarely will we even get close to that. The lack of the unmeasured very high Q data will produce “ripples” in the \(\rho (x,y,z)\) values and this is known as truncation error.
We cannot directly use the \(|F_{obs, hkl}|\) values that we discussed tabulating in the previous section because those values are unphased magnitudes, but we can use the quantity \(\frac {F_{calc, hkl}}{|F_{calc, hkl}|}|F_{obs, hkl}|\) to compute the map. This is known as a Fobs map. What this does is use the phases we computed from our structure, but the approximate observed intensities. Although this map is only approximate, for several reasons, including that the \(|F_{obs, hkl}|\) values are only approximately correct and that the phases we compute for \(F_{calc, hkl}\) will not always be the correct phases for \(|F_{obs, hkl}|\) in addition to the truncation errors previously mentioned. Nevertheless, map would be expected to show any atoms that are missing from our model, as well as the atoms included in the model. The most valuable map that is commonly computed is obtained from \(\frac {F_{calc, hkl}}{|F_{calc, hkl}|}(|F_{obs, hkl}| - |F_{calc, hkl}|)\) and is known as a \(\Delta \)F map, delt-F map or an obs-calc map. This requires a bit of thought to understand, but it gives us positive peaks any place in the cell where there are atoms present that are not included in our model. Should there be scattering in our model where there is less or no actual scattering present in the material, then the \(\Delta \)F map will have negative peaks. Thus, a negative peak can be diagnostic for the misassignment of an atom type or the presence of vacancies. When an atom is slightly shifted from its actual position, a negative peak will be seen next to the actual atom location.
Note that for neutron scattering, a small number of isotopes scatters with a negative scattering length. For those atoms the sign of the Fourier peak is inverted. In a \(\Delta \)F map, a negative peak means that more of that atom is needed and a positive peak indicates too much density. It is possible with site sharing for the atomic scattering to sum to zero and then nothing will be seen on a Fourier map for that site.
There is are other types of maps that GSAS-II can produce. If the quantity \(\frac {F_{calc, hkl}}{|F_{calc, hkl}|}(2|F_{obs, hkl}| - |F_{calc, hkl}|)\) is used, known as a “2Fo-Fc” map, then one sees the missing atoms superimposed, but at double intensity, on the map with the atoms from the model. This is a preferred mode for Fourier map analysis in macromolecular crystallography, but I have not found it particularly useful for my powder diffraction analysis work. In macromolecular crystallography, one sometimes removes an amino acid from a model to see where the is then found in the Fourier map, to better determine the position and orientation. This is called an “omit map.” This can be done in GSAS-II, but will not be discussed here.
Finally, GSAS-II can compute a Patterson map, where the quantity \(|F_{obs, hkl}|^2\) is used for \(F_{hkl}\) in the summation. This gives a map where peaks correspond to vectors between pairs of atoms and the strength of the map peaks goes as the product between the scattering power from each type of atom in the pair. While not used very much at present, Patterson maps were commonly used for solving structures of “heavy atom” materials from single-crystal data, but as direct methods have improved for these types of materials, Patterson maps do not see much use. It is possible that it might be a useful tool for certain powder diffraction problems, but the one time I did try to use it for that, it did not give me what I needed. Analysis of a Patterson map requires several “tricks” that I will not try to resurrect my memories from when I did solve structures with this in the 1970’s, but one can consult the older scientific literature, which is rich with Patterson technique information.
In my experience, powder diffraction Fourier maps are not as accurate as those from single-crystal diffraction. I assume this is due to peak overlap. What I find is that only the first few peaks in a map are real and the rest tend to represent artifacts. I will examine the most intensely positive and negative regions of a \(\Delta \)F map, assigning atoms where possible, and then perform a refinement. I will then recompute the Fourier map and again look at the most and least intense regions, until I have nothing more to change.
Computing a Fourier map in GSAS-II is quite easy.
Set the Fourier map settings. These are on a phase’s General tab.
The Fourier map can be viewed directly, superimposed on the desired representation of the structure, by selecting the “Draw Options” or “Draw Atoms”. Before viewing the Fourier map, you should prepare the structure as you wish to view it. For details on that, please see Chapter 18.
Note that the “view point” is important for selecting what section of the map is displayed. The view point is placed in the center of the graphics window. If the “Show view point” option is selected in the “Draw Options” tab, it is drawn with 1 \(\mathring {\text {A}}\) lines along the a, b and c, directions, with the lines colored red, green and blue, respectively. The location of the view point can be set in a number of ways:
There are two modes that can be used to view the map, as discussed below:
GSAS-II can display the Fourier map in three dimensions for a spherical region surrounding the view point. Positive regions are shown as green dots and negative regions are shown as orange dots. The higher the level, the more dense the dots. See Fig. 20.2 for an example. To display the map in this fashion, on the “Draw Options” tab select the “Show 3D density map” option. There are then two controls that dictate how the map shown. There is a “Fraction of rho max” slider that can have values from 1.0 to 0.01. The lower the value, the more of the map that will be shown. Regions where \(|\rho |\) is below this threshold are not displayed. One can also select the radius of the sphere using the “Radius of displayed map” slider/input box.
The map can also be displayed as a 2-D slice. The slice will always viewed in the plane of the screen and will always be centered on the view point, so rotating the structure with the left mouse button changes the orientation of the plane. See Fig. 20.3 for an example. The plane can be enabled by selecting the display option for the “Show map slice as...” on the “Draw Atoms” tab between lines, colors and lines+colors. Pressing the “k” keyboard key also toggles between these modes. One can also select the slice size and the contouring levels.
I find that when I am using Fourier maps, my primary goal is to identify where the most and least intense Fourier densities are located in the structure. To do this it is useful to identify the regions with the most positive and negative Fourier densities. GSAS-II can search the map and assemble a list of Fourier peak locations as well as the density at that location. To do this, on the “General” tab use the Compute/“Search map” menu command. The number of peaks located will be determined by the “Peak cutoff %” option. If rho-max is 0.8 and this threshold is set to 50%, then only peaks with \(\rho > 0.4\) or \(\rho < -0.4\) will be located. The peaks will be listed on the “Map peaks” tab and will be displayed superimposed on the structure as small cross marks. Lines are drawn between peaks, which may help one recognize molecular fragments, but mostly that is of use for charge flipping structure solution, and not so much for Fourier maps.
When the peak list is viewed in the “Map peaks” tab, the structure/map display can be updated as peaks are selected, according to options on the “Draw Options” tab:
Peaks can be selected by clicking on them in the “Map peaks” data tab. Pressing the “n” and “p” keyboard keys in the graphics window selects the next or previous Fourier peak.
Note that if the view point is moved around the cell, it is possible to locate atoms and Fourier peaks, as well as label distances, with a set of options that work similarly to those immediately above.
My experience with Fourier peaks from powder diffraction is that they do not always correspond to actual atomic positions. Peaks that are close to an atom that is already in the structure may send a hint that there is something lacking in the fit for that atom, but could also be an artifact due to experimental error in the reflection intensities or due to truncation errors. Thus, our “hint” could be nothing significant, but could be an indication of some disorder or perhaps that the atom has significantly anisotropic thermal motion. There may not be anything that one can change. When I encounter a peak that is sufficiently remote from other atoms in the structure that it could be bonded to atoms in the structure, or if even further than that, I will wonder if there is an additional atomic fragment or a solvent, etc. Note that other than intentionally porous materials, it is fairly rare for structures to have open voids large enough to accommodate atoms or molecular fragments. Faced with a “mystery” Fourier peak that is in a location where it is not overlapping with any already included atoms, I will often test if placing an atom at that location actually improves the fit. This is a fairly easy task: Place an atom at that site and vary the occupancy, but not Uiso or coordinates. If the model is not improved by scattering from the site, the occupancy will refine to a value close to zero. If it refines to a small positive number that say 3 to 10 times the uncertainty for the occupancy value (which can be checked in the .lst file), I will withhold judgment on if I believe there really is an atom at that location until later. Note that GSAS-II makes it easy to create an atom at a map peak location: Use the option to move the view point to a selected Fourier map peak. Then select the Atoms tab and use the “Edit Atoms”/“Append view point” menu command. That places a H atom at the view point location. One can then click on the H in the Type column for the new atom, and a periodic table will be presented where the atom type can be changed.
There is also a tool in the GSAS-II visualization graphics that will find voids in the structure. If a phase is selected and then the “Draw Atoms“ tab is pressed, use the “Compute”/“Create void map” menu command. This sweeps through the unit cell with the specified grid spacing and locates points where there are no atoms (as determined by their space-filling radius) within a specified distance. Dots are then placed at those locations where there are no atoms.
