R-Factors

Before venturing into the processing step details, some explanations are necessary about the various R-factors used in the IPLT diffraction processing: RFriedel, RSym, RMerge and RFit. All of these R-factors have a weighted version which takes the sigma of the observed intensity into account, but error estimations for z* are ignored. Most calculations will give a breakdown into resolution shells as well as overall R-factors. The basic formula is always the same, with a sum over an intensity difference divided by a sum over intensities (with appropriate absolute values):

Note that in contrast to x-ray crystallography, the intensities in the nominator are not used as their absolute values, but rather carry their sign with them; this is a consequence of the background correction which allows intensities to go into the negative regime for the purpose of proper error distribution around the zero intensity line. In essence, the R-Factors differ in the definition and calculation of their Iref component, which are now explained.

RFriedel

This is the easiest and most readily available R-factor for single diffraction patterns; since I(h,k,z*)=I(-h,-k,-z*), the majority of reflections has a Friedel mate in the pattern (except when obstructed by the beamstop) - the reference intensity in the above formulas then simply becomes the average between the Friedel mates. The RFriedel is an excellent measure for the quality of a single pattern, as it is independent of the tilt geometry or any other datasets. It is influenced by the parameters for intensity extraction and integration, which allows their fine-tuning.

RSym

For a non-tilted image, most reflections have symmetry mates, which can be used to calculate the reference intensities. For a tilted image, however, there is no overall symmetry apart from the Friedel mates. Nevertheless, there may be zones of symmetry equivalence, depending on the spacegroup. The below figure shows these zones for the case of P4 symmetry. As a consequence, there are some symmetrized reflections that do not differ greatly in z* - those can form the reference intensities.

The RSym is dependent on the tilt-geometry. As such, this R-factor can be utilized in monitoring the tilt-geometry refinement improvements. Also, if patterns are rejected based on RSym, a possible improvement during tilt-geometry refinement should be considered; e.g. an initial rejection applied only to drastic outliers, and a more stringent limit after the refinement.

RMerge

In x-ray crystallography, a single reflection defined by an (h,k,l) triplet may be (and usually is) present multiple times; in the case of 2D electron crystallography, the unevenly distributed nature of the lattice lines along z* requires a different approach to estimating the quality of a merge. Without giving a proper explanation at this point, it can be stated that the theoretical lattice line cannot vary arbitrarily fast - hence, points in close vicinity should more or less be of the same intensity. This approximation breaks down as the window size is increased; on the other hand, for sparse regions, when the window is too small, no reference values can be obtained - a tradeoff is required, and a suitable window size must be determined heuristically.

The above figure explains the details of the RMerge calculation, valid for both an individual pattern as well as the total dataset: For each reflection (red), a local window average (yellow) from closely neighboring reflections on the same lattice line (blue) is calculated - this average is then used as the reference value in the R-factor calculation (as indicated in the formulas). The RMerge can always be used as soon as data from different images is pooled together; it can monitor the quality of scaling, it can be used as a target for tilt geometry refinement, it can serve as a filter to throw out the patterns that clearly do not go well with the overall dataset, and it allows the determination of the high-resolution cutoff for the final reflection list.

RFit

The RFit can be extracted after running a lattice line discretization; since the curve covers the continuous range of z* values, at each reflection the corresponding calculated intensity can be obtained (as illustrated in the below figure). Moreover, an estimation of the error at each calculated position is available (e.g. through bootstrapping) - allowing observed and calculated weights to be combined. The quality of each lattice line fit can be obtained by simply forming an RFit with all intensities from that lattice line; e.g. to compare two different lattice line fitting runs. The quality of each diffraction pattern can be equally obtained by forming the RFit with all intensities from the pattern.

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