An improved reliability factor for quantitative low-energy electron diffraction

TL;DR

The paper identifies critical shortcomings of Pendry's $R$-factor for LEED $I(E)$ analysis—noninvertibility, sensitivity to tiny intensity offsets, and noise-driven instability. It introduces $R_\mathrm{S}$, derived from a smooth, offset-aware $Y_\mathrm{S}$ function that leverages $I$, $I'$, and $I''$ to stabilize minima regions while preserving the favorable local comparison property of $R_\mathrm{P}$. Through systematic comparisons, $R_\mathrm{S}$ demonstrates similar numerical values to $R_\mathrm{P}$ but with markedly reduced noise and improved robustness to imperfect data, leading to more reliable and faster convergence in high-dimensional structural optimizations. The work argues that $R_\mathrm{S}$ can replace $R_\mathrm{P}$ for LEED $I(E)$ analyses without sacrificing error estimation, while performing better under realistic experimental conditions.

Abstract

Quantitative low-energy electron diffraction [LEED $I(V)$ or LEED $I(E)$, the evaluation of diffraction intensities $I$ as a function of the electron energy] is a versatile technique for the study of surface structures. The technique is based on optimizing the agreement between experimental and calculated intensities. Today, the most commonly used measure of agreement is Pendry's $R$ factor $R_\mathrm{P}$. While $R_\mathrm{P}$ has many advantages, it also has severe shortcomings, as it is a noisy target function for optimization and very sensitive to small offsets of the intensity. Furthermore, $R_\mathrm{P} = 0$, which is meant to imply perfect agreement between two $I(E)$ curves can also be achieved by qualitatively very different curves. We present a modified $R$ factor $R_\mathrm{S}$, which can be used as a direct replacement for $R_\mathrm{P}$, but avoids these shortcomings. We also demonstrate that $R_\mathrm{S}$ is as good as $R_\mathrm{P}$ or better in steering the optimization to the correct result in the case of imperfections of the experimental data, while another common $R$ factor, $R_\mathrm{ZJ}$ (suggested by Zanazzi and Jona) is worse in this respect.

Figures (7)

• Figure 1: (a) The $Y(L)$ function of Pendry's $R$ factor (blue). The gray curve shows an invertible function, $Y_\mathrm{M}$, which forms the basis for the $R$ factor $R_\mathrm{S}$ presented in this work. The horizontal axis is the scaled logarithmic derivative $L$ of the intensity [equation (\ref{['eq:L']})]. (b) The $I(E)$ plot shows two curves (black, red) that share the same $Y_\mathrm{P}(E)$ function, shown in the bottom. This implies that Pendry's $R$ factor between the two $I(E)$ curves is $R_\mathrm{P}=0$, a value that should occur only if two curves are identical except for a constant scale factor. The red bars indicate where the two $I(E)$ curves map onto different branches of the $Y_\mathrm{P}$ function; in these regions the two $I(E)$ curves are dissimilar (not proportional to each other). The horizontal axis is the scaled energy, centered at the position of a minimum of the $I(E)$ curves. For $|V_\mathrm{0i}|=4$ eV, the plots in (b) would span 12 eV on the energy axis.
• Figure 2: Intensities and various $Y$ functions at a minimum with different intensity offsets. Note that all $Y$ functions are sensitive to relative changes only, but invariant to a scale factor of the intensity. Thus, the red intensity curve in (a) would be equivalent to a curve with a peak-valley height of 1.0 (like the other curves), but an intensity of 2.6 at the minimum. (b) For small, but non-zero intensity offset, $Y_\mathrm{P}$ of Pendry's $R$ factor reaches sharp minima and maxima next to the minimum ("cusps"). The dots indicate the sampling interval in the case of 0.5 eV energy steps and $|V_\mathrm{0i}|=4$ eV. (c) The modified $Y_\mathrm{M}$ function of equation (\ref{['eq:Ymod']}) shows a hard step at an intensity minimum with a low or negligible offset (black curve), which is also undesirable. The $Y_\mathrm{S}$ function in (d) avoids these issues and depends smoothly on the intensity offset at the minimum.
• Figure 3: (a) $R$ factors and (b) their derivatives near the minimum. The horizontal axis is the displacement $\Delta z$ of the two symmetry-equivalent uppermost Fe atoms of the $\alpha$-Fe$_2$O$_3(1{\bar{1}02)}$-$(1\times 1)$ structure kraushofer_viperleed_2025. The vertical axis for the Zanazzi--Jona $R$ factor has been scaled by a factor of 7 since the curvature of $R_\mathrm{ZJ}$ vs. $\Delta z$ is lower than that of the two other $R$ factors by roughly this factor. The dashed lines indicate the error bar of the parameter $\Delta z$ derived from $\mathrm{var}(R)$, equation (\ref{['eq:varR']}). The plot does not include $R_2$; it is smooth, but its minimum would lie close to the right border of the plot. This is consistent with a previous study finding less accurate results when minimizing $R_2$sporn_accuracy_1998.
• Figure 4: Contour plots of (a) $R_\mathrm{P}$ and (b) $R_\mathrm{S}$ as a function of the $x$ and $y$ displacement of the upper two symmetry-equivalent Fe atoms of $\alpha$-Fe$_2$O$_3(1{\bar{1}02)}$-$(1\times 1)$, based on tensor-LEED calculations. The red circles mark a few points where the gradients of $R_\mathrm{P}$ are grossly misleading, even far from the minimum. Minima are marked by small, red crosses.
• Figure 5: Experimental $I(E)$ curve and its $Y$ functions. The black curve at the bottom is $I(E)$ of the (1,2) beam of $\alpha$-Fe$_2$O$_3(1{\bar{1}02)}$-$(1\times 1)$. The upper curves are the $Y$ functions of Pendry's $R$ factor $R_\mathrm{P}$ (blue) and of the $R$ factor $R_\mathrm{S}$ introduced in this work (orange). Blue arrows mark cusps of $Y_\mathrm{P}$, where the $Y_\mathrm{P}(L)$ function reaches an extremum and folds back (cf. figure \ref{['fig:YPe']}a). The inset around $E=350$ eV shows the $Y$ functions affected by experimental noise due to insufficiently smoothed data (smoothing parameter 1.0 eV, far below the recommended range schmid_viperleed_2025).