The \(R\) factor

The reliability factor (\(R\) factor) is a measure for the deviation between two I(V) curves or two sets of I(V) curves. Structure optimization (search or Full-dynamic optimization sections) minimizes the \(R\) factor between the calculated and experimental I(V) curves.

As the comparison of two curves is not an unambiguous task, multiple \(R\) factor implementations exist. The R_FACTOR_TYPE parameter can be set in the PARAMETERS file to choose between Pendry’s \(R\) factor \(R_\mathrm{P}\) [21] and \(R_2\) [23]:

  • \(R_\mathrm{P}\) compares the logarithmic derivatives of the I(V) data, using a fix for the divergence of the logarithm when the intensity approaches zero.

  • \(R_2\) is based on the mean square difference of the I(V) curves (after appropriate scaling).

Note

Using \(R_\mathrm{P}\) is the default setting and highly encouraged since tests have shown that it leads to better results than \(R_2\) [23].

The Pendry \(R\) factor

The Pendry \(R\) factor, \(R_\mathrm{P}\), can have values between 0 and 2 and is defined as [5, 21]

(1)\[R_{\mathrm{P}} = \frac{\displaystyle\sum\nolimits_\mathbf{g}{\int{\mkern-5mu\left(Y^\mathrm{theo}_\mathbf{g}(E) - Y^\mathrm{exp}_\mathbf{g}(E) \right)^2 dE }}}{\displaystyle\sum\nolimits_\mathbf{g}{\int{\mkern-5mu\left(Y^\mathrm{theo}_\mathbf{g}(E)^2 + Y^\mathrm{exp}_\mathbf{g}(E)^2\right) dE}}},\]

where \(\mathbf{g}\) indexes the beams for which the \(R\) factor is calculated, and \(Y(E)\) is the Pendry \(Y\) function. The \(R\) factor can be calculated for all beams together or for each beam individually.[1] The \(Y\) functions in Eq. (1) are computed from the beam intensities \(I(E)\), their derivatives \(I'(E)=\frac{dI}{dE}\), and the imaginary part of the inner potential \(V_{0\mathrm{i}}\) (see parameter V0_IMAG) as

(2)\[Y(E) = \frac{I(E)/I'(E)}{[I(E)/I'(E)]^2 + V_{0\mathrm{i}}^2}.\]

The beam intensities enter Eq. (2) via their logarithmic derivative \(\frac{d}{dE}\big(\ln{I(E)}\big) = \frac{I'(E)}{I(E)}\). This makes the Pendry \(R\) factor insensitive to differences in the absolute intensities of the I(V) curves. The largest contributions to \(R_\mathrm{P}\) come from differences in the positions of extrema, especially minima.

An \(R_\mathrm{P}\) value of zero corresponds to perfect agreement between curves. \(R_\mathrm{P}\) equals one for statistically uncorrelated data, while values larger than one indicate anticorrelation. For close-packed surfaces, \(R_\mathrm{P}\) values larger than 0.2 indicate a problem, such as an incorrect structural model. \(R\) factors for more open, corrugated surfaces, such as missing-row-reconstructed Pt(110), may be around 0.2. \(R_\mathrm{P}\) values larger than 0.25–0.30 should be taken as an indication of poor correspondence between calculated and experimental beams. The best values of \(R_\mathrm{P}\) obtained by the Erlangen group are below 0.05.[2]

Note that some smoothing algorithms applied to both experimental and calculated beams, such as the one suggested by Pendry [21], artificially reduce the \(R\) factor, because they effectively raise the minima of the I(V) curves. At minima, where the intensities approach zero, \(R_\mathrm{P}\) is especially sensitive to small differences; artificially increasing the intensity at minima thus gives smaller \(R_\mathrm{P}\) values. Some LEED programs apply such a smoothing; in those cases smaller \(R\) factors than those obtained with ViPErLEED will be reported, but this does not indicate a better agreement between calculated and experimental I(V) curves.

By default, ViPErLEED applies no additional smoothing when calculating the \(R\) factor (e.g., during structure optimization). Thus, the EXPBEAMS file should already contain smoothed data.

Tip

We highly recommend to smooth experimental data beforehand using the I(V)-curve editor of the ViPErLEED ImageJ plugins. Using the R_FACTOR_SMOOTH parameter for smoothing the experimental I(V) curves is discouraged, as the smoothing algorithm applied there is inferior to that used by the I(V)-curve editor.