A quick guide to Use of EDM in anger to solve a structure

Solving a structure using direct methods, if you have never done it before, can be very confusing. What you will end up doing is trying various different things until either you give up or you solve it. If you do solve it, normally you will wonder why it took you so long and you will look back on some of the earlier results are recognize that they were telling you what the structure was, but you did not realise. The intention of these notes is to provide some very incomplete notes, and things to check.

What am I doing?

A key thing to understand is that direct methods are used to reduce the number of possible structures which may explain your results from infinitely large to something that can be managed – see Figure. Your mission is to first find which structures are plausible from a direct methods analysis, then refine these and finally use chemical and other tests to determine if they are correct. You often will have more than one plausible structure map, i.e. ones which contain features that are atom-like, this should not worry you. It is only in the next steps that it will (hopefully) become clear which of these, if any, is viable.

At the core of the analysis are algorithms which find, for some particular set of phases, how consistent these phases are with the existence of non-overlapping atomic scatterers in the material in a two-dimensional projection. For any particular set of phases a Figure of Merit (FOM) is returned which is an approximate (not an exact) measure of this consistency. Some of the codes (fs2D, peaks2D) do this analysis for a single set of phases, which another (fs2D_pop) will do this for a larger population of phases.

At a second level a search is done over different values of the phases controlled by a genetic algorithm. The later is somewhat complicated since one needs to carefully include safeguards to ensure that all (or as close to all) of the plausible sets of phases (those with small FOMs) are found, not just one single favorable FOM and phases very similar to this. This is done by running in parallel three subpopulations as well as using what is called sharing.

What makes the analysis complicated is that there may be more than one solution as a function of the phases, as illustrated below. In this illustrative case there are three possible phases where the FOM is small, labelled A, B and C. In a perfect case with very accurate and kinematical data the most probable solution, that with the smallest FOM A, would correspond to the true solution. You may be lucky and only have one local minima, i.e. the local minima of B and C may not exist. Without going into details, the overall problem is what is called non-convex and as a consequence one cannot be certain that there are not local minima. Worse, with imperfect data, experimental errors and dynamical artifacts it may be that it is instead one of the other local minima B or C. What the first stage of the analysis does is isolate (in this particular case) the three plausible solutions for further analysis. If you know exactly how many peaks (atoms) there are in the structure this can be done automatically, which is what is done for direct methods using x-ray diffraction data from bulk crystals. However, with electron diffraction data in projection and for surfaces we do not necessarily know how many peaks there should be so this cannot currently be done.

What should I do?

We will assume that you have measured your diffraction data, and if you have multiple exposures, have combined the data using the code mergehkl -- use mergehkl --help from the command line to get more information. We will also assume that you know the Patterson symmetry. What you should probably do is follow a branching strategy where you setup a number of different directories and in each one run for different possibilities, primarily the symmetry (see below), the resolution and the number of atoms being used in the first stage of the analysis. You can limit the reflections you use either by editing your .hkl file by hand, or by running trim2D case.hkl where case.hkl should be the relevant file. This program will ask for a little information, namely the 2D lattice parameters and angle then will output to a file trim.hkl. You can setup in trim.ins the appropriate commands (or use the edm interface). A good value for the resolution to use is 1 Angstrom down to 0.5 Angstroms; you will rarely do better by going beyond this. Particularly for surfaces somewhere between 1 Angstrom and 0.75 Angstroms resolution is reasonable.

You can either vary the effective number of atoms by changing the contents, or use the two estimated modes which either set the sum of F**2 to 1 or the sum of U**2 to 1. There is no magical reason why these later two should be right, but they are often remarkably close. If you set the number of atoms yourself you must examine the output of the Convergence Analysis to ensure that you have reasonable values, in particular no U is > 1.0 which is unphysical. The values of the U's change as the square root of the number of atoms, so vary the later by factors of 2.0 to get a broad range relatively quickly.

For each case you want to think a little about how many reflections you allow to vary. If you make this too small you are not exploring the possible phase combinations in sufficient detail so might miss the solution. If you make it too large then there may be many redundant, equivalent solution and the number which you have to search over may become too large. Unfortunately different problems give different numbers of plausible solutions, so generalities are difficult. A reasonable number is about 20% of the reflections, rarely less than 5 total and more than 25 total can be too many. Changing the Beam Definition parameter allows you to change this yourself. Experiment.

You then run the different possibilities and look at the solutions. Questions you should ask include

There is no magic answer to these questions. If for a given symmetry you get no reasonable solutions, it is probably not correct. If for no symmetry do you get a reasonable solution you need to look carefully at your data. Do you have evidence for strong dynamical effects? Did something go wrong with the measurements? Did you use the wrong lattice? Is it possible that you have a lower symmetry with domains giving an apparent higher symmetry in the diffraction pattern? In such a case image data, for instance dark field, is needed to work out what is going on.

What is the Symmetry?

Whenever you take a diffraction pattern, at least with kinematical data the symmetry is higher than what you have in reality. When you lose the phase information this adds an effective center of symmetry to the diffraction data. Hence a structure which has p3m1 will give a diffraction pattern with p6mm symmetry. This means that in general you have to test allpossible lower symmetries. The table below gives all the possible symmetries for a given diffraction (Patterson) symmetry.

Diffraction Symmetry

Possible Symmetry

p1

None for kinematical data

p2

p1 or p2

p2mm

p11m, p1m1, p11g, p1g1, p2mm

p4

p4

p4mm

p4mm, p4mg, p4gm

p3

p3

p6mm

p6mm, p3m1, p31m

How many atoms do I have?

The most important thing to realise is that the direct methods analysis is relatively insensitive to what you think is the number of atoms in your structure. The issue is more important when it comes to refining a structure since if you add more atoms you are adding variables and in many respects you will always get a better fit to the experimental data although it may be an overfit. There is no magic answer for how many to use in a refinement, you have to work this through using chemical judgement.

What you need to do for the first part, when you are creating the direct methods maps, is run for a number of different approximations for the number of atoms. What this does is change the magnitude of the structure factors (strictly speaking, the U's) and then can change which solution the algorithm considers to be most probable. When you do this change by a relatively large factor, e.g. 2, since the U's scale as the square root of the number of atoms. An alternative is (in 3.0 and later) to use a line in the case.ins file “SHKL H K Value” where for “H” and “K” put a particular reflection and for “Value” substitute a number, e.g. 0.8. This will force the U of the particular reflection to that value.

What temperature factors should I use?

This is another one of those subtle issues. If the sample is too thick for kinematical theory to hold, one way that this sometimes manifests itself is by apparently very small temperature factors. In effect they are becoming anomalously small to minic multiple scattering to higher angles. Alternatively, if you have a highly disorder sample you may have very large "real" temperature factors.

One indicator that you have an issue is whether or not the U's are rapidly increasing at higher angles (factors too large) or rapidly decreasing (too small). You can, and probably should look at these using something like Excel. While there are methods for doing the quantitatively, in general they require rather large datasets and do not work for electron diffraction data.