Resolution of interatomic distances in the study of local atomic structure distortions by energy-restricted x-ray absorption spectra

Applicability of the existing criteria for signal frequencies resolution to the problem of close interatomic distances determination by energy-restricted x-ray absorption spectra is studied within the approach based on the Fourier transformation and fitting procedures. Without losing generality, theoretical signals $\chi \left(k\right)$ of different $k$ dependencies are used and among them—the signals $\chi \left(k\right)$ of x-ray absorption spectroscopy ($k$ is the photoelectron’s wave number). The last ones are calculated at different values of two interatomic distances $R_1$ and $R_2$ in a model of radial distribution of coordinating atoms in relation to the absorbing center, including the values of distances, which can’t be resolved a priori according to the criteria. It is revealed that the boundary value of $\Delta R=\left|R_2-R_1\right|$, at which the model of local structure distortions can be distinguished among other alternative ones by the used approach depends strongly upon the coincidence of the functional form of $k$ dependence used for the fitting function with that of the studied signal $\chi \left(k\right)$, well known in x-ray absorption spectroscopy. In the case of coincidence, the boundary value of $\Delta R$ obtained by the restricted intervals $\Delta k\sim 3$ or $4{\text{Å}}^{-1}$ is approximately ten times smaller than that predicted by the existing criteria. The effect of statistical noise in spectrum intensity on the established $\Delta R$ value is analyzed.

Figure 1

${\chi }_1\left(k\right)$ calculated by Eq. (10) with $R_1=2.0\text{Å}$—(a); the total signal $\chi \left(k\right)$, calculated by Eqs. (9, 10) for $\left(3+1\right)$ model with $\Delta R=0.2\text{Å}$—(b); $\left|F\left(R\right)\right|$—the results of FT of $\chi \left(k\right)$, obtained for $\Delta k=10{\text{Å}}^{-1}$—solid curve in (c) and for $\Delta k=3{\text{Å}}^{-1}$—solid curve in (d), are compared with the results of corresponding fits by $\left(3+1\right)$ model—dotted curves in (c) and (d).

Figure 2

The goodness of the fits $\left({\chi }_{\nu }^2\right)$, obtained by different fitting models for $\chi \left(k\right)$, as a function of $\Delta R=\left|R_2-R_1\right|$ value used in the direct calculations of $\chi \left(k\right)$ by Eqs. (9, 10) with ${\sigma }^2=0.005{\text{Å}}^2$. Solid curve—fit by $\left(3+1\right)$ or the true model; dotted curve—$\left(2+2\right)$; dashed-dotted curve—$\left(1+3\right)$; dashed curve—single term fit.

Figure 3

${\chi }_1\left(k\right)$ calculated by Eq. (11) with $R_1=2.0\text{Å}$—(a); the total signal $\chi \left(k\right)$, calculated by Eqs. (9, 11) for $\left(3+1\right)$ model with $\Delta R=0.2\text{Å}$—(b); $\left|F\left(R\right)\right|$—the results of FT of $\chi \left(k\right)$, obtained for $\Delta k=10{\text{Å}}^{-1}$—solid curve in (c) and for $\Delta k=3{\text{Å}}^{-1}$—solid curve in (d), are compared with the results of corresponding fits by $\left(3+1\right)$ model—dotted curves in (c) and (d).

Figure 4

The goodness of the fits $\left({\chi }_{\nu }^2\right)$, obtained by different fitting models for $\chi \left(k\right)$, as a function of $\Delta R=\left|R_2-R_1\right|$ value used in the direct calculations of $\chi \left(k\right)$ by Eq. (9) with ${\chi }_1\left(k\right)=4\text{sin}\left(2k{R}_1\right)$, ${\chi }_2\left(k\right)=2\text{sin}\left(2k{R}_2\right)$, ${\sigma }^2=0.005{\text{Å}}^2$. Solid curve—$\left(4+2\right)$ or true model; dotted curve—$\left(3+3\right)$; dashed-dotted curve—$\left(2+4\right)$; dashed curve—single term fit.

Figure 5

Boundary values of $\Delta R$ in the true $\left(3+1\right)$ model of As coordination by four In atoms, at which this model becomes preferable among others in FT analysis over $\Delta k=3{\text{Å}}^{-1}$, as a function of noise amplitude ${\epsilon }_{\text{max}}$.