International
Tables for
Crystallography
Volume I
X-ray absorption spectroscopy and related techniques
Edited by C. T. Chantler, F. Boscherini and B. Bunker

International Tables for Crystallography (2022). Vol. I. Early view chapter
https://doi.org/10.1107/S1574870720016511

Significance and tables of key physico-chemical parameters

C. T. Chantlera* and J. D. Bourkea

aSchool of Physics, University of Melbourne, Melbourne, Australia
Correspondence e-mail:  chantler@unimelb.edu.au

For general X-ray absorption spectroscopy (XAS) or X-ray absorption fine-structure (XAFS) analysis, whether focusing on the pre-edge, the X-ray absorption near-edge structure (XANES), the standard full XAFS region or the higher-energy extended XAFS region (EXAFS), a series of important parameters come into the theory across the whole International Tables for Crystallography. For near-edge and pre-edge structure, the definition of the Fermi level EF is critical. The ionization hole width ΓH and inner-shell hole excited-state lifetime τH are another pair of critical parameters that limit the information content of the near-edge spectrum, revealing key physics. The Debye temperature ΘD represents the complex dynamical behaviour of electrons in the quantum system. Much less understood is the damping of the signal due to the photoelectron wave inelastic mean free path λIMFP, the electron energy-loss function, the complex and anisotropic refractive index n and the complex and anisotropic permittivity or dielectric function ɛ. Here, an attempt is made to give a brief summary of the key issues and typical values, and also a discussion of scales and units for representation of key data and critical parameters.

Keywords: X-ray absorption; XAFS; physical parameters; Fermi energy; Debye–Waller; IMF; excited-state lifetimes.

1. Introduction

For general X-ray absorption spectroscopy (XAS; Chantler & Creagh, 2022[link]; Paolasini & Di Matteo, 2022[link]; Natoli et al., 2022[link]) or X-ray absorption fine-structure (XAFS) analysis, whether focusing on the pre-edge (de Groot, 2022[link]; Yamamoto, 2022[link]), the X-ray absorption near-edge structure (XANES; Fujikawa, 2022[link]; Joly et al., 2022[link]), the standard full XAFS region or the higher-energy extended XAFS region (EXAFS; Rehr et al., 2022[link]; Kas et al., 2022[link]), a series of important parameters come into the theory across the whole International Tables for Crystallography. Several of these are obvious but are often fitted empirically with limited theoretical or experimental provenance. Others are well-known, well-understood and regularly tabulated. This chapter presents and discusses these.

For near-edge and pre-edge structure, the definition of the Fermi level EF is critical. This also impacts upon the definition of χ, especially for low-k regions of experiment and analysis. The ionization hole width ΓH and the inner-shell hole excited-state lifetime τH are another pair of critical parameters that limit the information content of the near-edge spectrum and also reveal key physics. The complex dynamical behaviour of electrons in the quantum system, and especially the static and dynamic behaviour of anisotropic thermal ellipsoids, whether representable by correlated Debye theory and a Debye temperature ΘD or not, is a huge area which particularly impacts upon the realistic (anisotropic) thermal parameters, contrasted with crystal lattice thermal anisotropic parameters usually from crystallography (B, Bij, Uij, σij), for dynamic bonding and the broadening of theoretical scattering paths i and path lengths σi [or the equivalent mean-square relative displacement (MSRD) parameters [\sigma^{2}_{i}] and the much less significant transverse broadening parameters σi,⊥] (Fornasini, 2022[link]; Castellanoi, 2022[link]). This is particularly important for high-k, high-n-leg paths and long path lengths, where this broadening can dominate.

Perhaps much less understood is the damping of the signal due to the photoelectron wave inelastic mean free path (IMFP) λIMFP (Tanuma & Powell, 2022[link]), which in turn arises from the electron-loss function (ELF), which is closely related to the complex and anisotropic refractive index n and the complex and anisotropic permittivity or dielectric function ɛ. This can be discussed in relation to the meaning and origin of the `electron self energy' Σ(E). Less well-defined again are the multiple-electron excitations, the satellite or shake excitations, and the net consequence of these, including the impact on the `amplitude-reduction factor' [S_{0}^{2}]. For fluorescence measurements, but also as a correction for attenuation measurements, the fluorescence yield ωi or f is critical, and the current status of this parameter is also discussed.

Additional to these are typical or example structures of any and all atoms, metals, molecules, lattices, surfaces etc. Indeed, several of the above parameters are an intrinsic part of the material structure, and can be structure-, molecule- and orientation-dependent. In this chapter, an attempt is made to give a brief summary of the key issues and typical values, and also initially a discussion of scales and units for the representation of key data and critical parameters. However, the main discussion of the theory behind these parameters, or of the typical software implementation, is given in the relevant chapters elsewhere.

This volume of International Tables for Crystallography also discusses other experimental modalities, including electron energy-loss spectroscopy (EELS; Joly et al., 2022[link]; Shirley et al., 2021[link]), X-ray emission spectroscopy (XES), X-ray magnetic circular dichroism (XMCD), nuclear resonant inelastic X-ray scattering (NRIXS) and resonant inelastic X-ray scattering (RIXS) (Glatzel et al., 2022[link]), where other parameters can be very important. We do not extend the discussion of critical parameters in the space available, but refer to other work on these subjects. A summary of the contents of this chapter follows.

Section 2[link] contains a discussion of amplitude options, μ, [μ/ρ], χ, k2χ and k3χ, and the scale of absorption and XAFS, and Section 3[link] contains a discussion of the corresponding x-axis options, monochromator setting, E, k and r, and the determination of structure and its comparison with theory, both clarifying and defining terms in relation to tabulations and data formats. This both enables portability of the discussions in different chapters of this volume and in the wider literature and also engenders a discussion of data-output scales.

Section 4[link] contains a discussion of edge energies EK, ELI, ELII, ELIII, EM … and E0, the energy offset for XAFS fitting or calibration, especially explaining the current status of different approaches in the literature and different approaches from a theoretical standpoint. This briefly addresses the calibration of the x axis in any format, and the variability of the terms used currently in the literature.

Section 5[link] contains a tabulation and a discussion of the Fermi energy EF and of the significance of pre-edge structure and bound–bound transitions. This section discusses the first compilation of data for this article, primarily from a theoretical perspective. The table is particularly relevant as a key set of parameters for theoretical computations as in Section 10[link].

Section 6[link] contains a discussion, tabulation and plots of hole widths and excited-state lifetimes across the periodic table, new and reviewed, as required in theoretical predictions of edge widths, XANES and XAFS resolution. This addresses both the agreement and disagreement between theory and experiment and the information content of XAS. Section 7[link] contains a discussion, tabulation and plots of fluorescence (radiative) yields ω and Coster–Kronig probabilities f, particularly for fluorescence spectroscopy or XES, and more generally linked to Section 6[link].

Section 8[link] contains a tabulation of selected exemplar elemental crystal structures, particularly providing the values used for the theory and computation of inelastic mean free paths in Section 10[link]. The selection is partly made on the basis of provenance and material stability permitting use as reference or calibration materials.

Section 9[link] contains a brief discussion of multiple scattering, shake processes and the amplitude-reduction factor [S^{2}_{0}] and its current status.

Section 10[link] contains an extensive presentation of theory, discussion, equations, tabulation and plots of inelastic mean free paths of the (photo)electron and links to low electron energy diffraction (LEED), EELS and electron diffraction, with new tabulations and references. Section 11[link] contains a brief discussion of analysis and applications of IMFP and ELF data and Section 12[link] presents a brief outlook and conclusion.

2. Amplitude options: μ, [μ/ρ], χ, k2χ and k3χ

Many older XAFS spectra present the linear attenuation coefficient μ (cm−1) versus E (keV), and many published spectra present χ versus k, the effective photoelectron momentum (Å−1). In the interests of data portability and cross-platform analysis and intercomparison, we note that for any solid material there will be a well-defined thickness t (µm or cm) and thickness profile which can be measured, and there will be a well-defined mass M and cross-sectional area A, defining a mean M/A = (ρt) or integrated column density. This may be temperature-dependent but is to be contrasted with a usually poorly defined density ρ (g cm−3), even for ideal samples and for flat plates with no voids, cracks etc. Hence, whilst the Beer–Lambert formula can be expressed as I/I0 = exp(−μt), it is better and more readily characterizable as I/I0 = exp[−(μ/π)(ρt)], where [μ/ρ] is the mass absorption coefficient, and this applies for transmission measurements and also for XAFS. It is common and important to separate the linear attenuation coefficient μ or μtot from the linear absorption coefficient μ or μpe and the mass attenuation coefficient [μ/ρ]tot from the mass absorption coefficient [μ/ρ] or [μ/ρ]pe in different literature: the Beer–Lambert formula only applies to the absorption coefficients. For solutions or frozen solutions, the density may be fairly uniform and the thickness can be well defined by the cell depth, so once again [μ/ρ] is preferred as a more repeatable and transferable measure. Definitions of concentration, solvent and matrix are important. Conversely, for complex mixed-phase, heterogeneous grains or powders, `earth science' samples or very dilute `biomedical nanosamples' it is unlikely that a direct calibration can be made on either axis; we recommend the careful use of reference materials to enable the quantification of experimentally measured axes.

In principle, the raw data signals for upstream monitor and downstream detector or, for example, each of 100 fluorescence detector pixels, each with an energy spectrum, for sample and for blank and for dark currents etc. can be defined as a source set of spectra; however, for portability it is hoped that these sets of information are processed or preprocessed to yield, for example, [μ/ρ] as a function of the energy E.

Non-uniformity issues can be dealt with by normalization of the edge jump to unity, although this means that the resulting measurements will be relative. Theory can still compare above-edge with below-edge behaviour and experiment can compare this with standards, although note that this does differ if attenuation or photoabsorption coefficients are used.

As commented in many software and theoretical analyses in this volume, the primary information for structural fitting etc. is often k2χ or k3χ, whether versus k or r (see below). This particularly makes fits and discrepancies easier to observe in a plot or publication. If a pointwise uncertainty is propagated from [μ/ρ] versus E, then the y-axis measure should be equivalent and should fit equivalently. That is, the value and the uncertainty should scale and the fit parameters and uncertainties should be unaffected by the choice of y axis. For deposition in this context, χ is to be preferred over scaled versions, and [μ/ρ] is to be preferred over this as avoiding certain processing approximations and systematics. Conversely, for plotting fits, residuals and discrepancies it is often very convenient to plot k2χ or some similar scaled measure to highlight the weaker high-k oscillations.

3. x-axis options: monochromator setting, E, k and r

All raw data streams have a monochromator angle setting for the primary crystal, usually with a secondary monochromator crystal tuned, nontuned or detuned with respect to the primary crystal. The primary crystal usually has significant heat load and strain so, in principle, there will be an energy offset even if the Bragg angle and energy are carefully calibrated in the absence of heat load. Often there are other optics such as a harmonic rejection mirror at a synchrotron or, for example, a monochromator filter for X-ray units; see Sutter (2021[link]), Hulbert (2022[link]) and Arndt et al. (1999)[link]. On a number of beamlines an energy offset is added to the spectral nominal energy to match the determined edge energy for a metal reference foil to the relevant edge. This depends upon the bandwidth etc. and generally the correction will change with energy, and hence there may be a scale or correcting functional from the reference position across the range of XAFS. Irrespective of these details, which should be recorded in as much detail as possible for cross-platform portability of XAFS spectra, the basic x axis presented is that of energy, typically in eV, and preferably with some defined uncertainty.

As explained across these Tables and elsewhere, the standard transform to an effective photoelectron wavevector magnitude k is given by k = (2π/λ) = {[2me(EE0)]/ℏ2}1/2, which only depends upon a unique definition of both E and E0, an offset representing the edge energy for XAFS data analysis or the onset of a propagating photoelectron wave inside the material. This does not make sense for the analysis or fingerprinting of pre-edge spectral regions, and is not usually performed for similar XANES analysis for the same reasons. There are very good reasons for transforming to the k axis for XAFS analysis, but there are challenges and systematics in a free fit of E0 or in the application of a purely theoretical or empirical value; see Chantler (2023[link]) and Bunker (2023a[link]). For data deposition, we currently strongly recommend that the deposited data be in [μ/ρ] versus E, so that any beamline-dependent variation of corrections can be addressed separately. Many analysis software packages transform from k-space (the scaled space of the experimental data) to Fourier r-space. In this transform, it is common that errors or uncertainties are not propagated or are aliased, and that information content is lost. Some analysis goes further and filters the r-space spectrum with a high-pass/low-pass filter and then back-transforms to k-space. For reference spectra we recommend depositing [μ/ρ] versus E, which is appropriate for any spectral range or analysis; additional transformed spectra can be deposited as might be useful.

4. Edge energies EK, ELI, ELII, ELIII, EM … and the energy offset E0

Edge energies are problematic in that several distinct definitions are correctly presented, as summarized in Authier & Chapuis (2017)[link] and in Section 11[link] below. The absorption threshold should indicate the first allowed transition in an absorption spectrum. Many definitions are used in common parlance. Practically, these yield very different numbers in common analysis. The most commonly used are the following.

  • (i) The energy at which the open continuum channel for photoelectric absorption becomes available, producing a continuum photoelectron (wave). Subject to convergence issues, this has an exact eigenvalue from theory (cf. Fermi energy). This more commonly refers to the crystal or material zero, whilst, for example, in a free atom the threshold is the vacuum level.

  • (ii) At a higher energy, a secondary (two-step) photoionization channel becomes energetically possible (n.b. shake-up, shake-off and multi-electron excitations). In general, this is more challenging to compute theoretically and less easily separable in conventional XAS, but can be investigated incisively in RIXS, XFS and related spectroscopies. In photoemission, zero kinetic energy begins at the vacuum level, whereas in XAFS any allowed transitions begin at the Fermi level, below the edge energy. Such multi-electron processes lie at higher energies than definition (i); but also note that multi-electron processes (excitations) can also occur in bound–bound transitions.

  • (iii) Experimentally, the absorption threshold is very often defined as the inflection point in the first derivative of the experimental edge spectrum, i.e. the point of maximum slope on the rising edge for a particular subshell; this is a convenient marker for experimentalists but (1) it is source- (beamline-), monochromation- and bandwidth-dependent, (2) it is affected by pre-edge structure and the Fermi level due to potential contributions from bound–bound channels and (3) the experimental edge may, and often does, contain two or more such inflection points, and the determination of even which peak is defined as the edge, or which effective energy is used, depends upon the instrumental resolution.

  • (iv) Experimentally, the absorption threshold is sometimes defined as the point exactly 50% of the jump ratio from the background absorption (from other shells, including scattering) to the peak absorption coefficient of the XANES spectrum, defined either by the clear maximum or by the smooth line representing the background to be subtracted in the determination of χ(k) (q.v. EXAFS); this is a problematic measure, since it depends upon beamline-dependent effects [as in (iii)] and a wide variety of different predictions of the `true background level' μ0 above the edge (q.v. EXAFS).

  • (v) Computationally, an `absorption threshold' is defined for XAFS fitting (and occasionally XANES fitting) as E0, which is considered either as an arbitrary fitting coefficient or the starting point of the k transform, which in turn generates the Fourier transform for the XAFS structure χ(k); as the latter, it should be defined as per (i) above; as the former, this will often yield a function of r and errors in E0 of the order of 10 eV or more, which can result in bond-length errors of the order of 0.02 Å or more.

Both computationally and experimentally, the energy axis is often not defined except in a relative sense, so that inconsistencies between the implementations of these definitions are at this point relatively common. As a general guide, the Fermi level energy is less than the first available bound–bound transition (which will have the first inflection point), which is often below the marker for the first inflection point (usually on the main edge region and often a bound–bound transition) [definition (iii)], which is often below the continuum `edge' energy [definition (i)], which is lower or higher than the 50% marker [definition (iv)], which is lower than the several two-step channels [definition (ii)]. As it depends upon many experimental details and systematic effects of pre-analysis, definition (v) can vary quite significantly from all of the other definitions.

For this chapter, we will refer to an ideal theoretical absorption threshold given by definition (i) as [E_{K_{0}}], [E_{L{\rm I}_{0}}], [E_{L{\rm II}_{0}}], [E_{L{\rm III}_{0}}], [E_{Mi,i={\rm I}-{\rm V}_{0}}], an experimentally characterized edge energy EK, ELI, ELII, ELIII, EMi, i=I–V et seq. given by definition (iii), and an energy offset E0 usually empirically determined for an individual XAFS transform from E to k, usually referencing definition (v) (for XAFS analysis).

The best current definition with uncertainties for the experimentally characterized edge energies, and especially EK, ELI, ELII and ELIII, is given by Chantler et al. (2022)[link]. This is directly applicable for metals and elemental solids, especially noting the chemical shifts of the edge due to binding, geometry and pre-edge features. Further, the number of well-calibrated lines is sparse [see Table 4 in Chantler et al. (2022)[link] for K-edge energies] and indeed is dominated by a single publication from 1996 (Kraft et al., 1996[link]). Much more research and measurement on this is needed for most of the periodic table and for many applications.

It is common to cite Arndt et al. (1999)[link] and a sequel publication Deslattes et al. (2003)[link] or the corresponding NIST database. A useful summary is given in Bunker (2010)[link]. There are two problems: in the experimental tabulation, the measured values are equally sparse. In Table 4.2.2.4 in Arndt et al. (1999)[link], wavelengths are given with conversion factors. Whilst there are two numbers for each absorption-edge energy from Z = 10 through Z = 92, most edge energies have poor provenance or accuracy or are a rough approximation to [E_{K_{0}}] rather than EK. Similarly, for Table 4.2.2.5 therein the entries for L-edge energies are almost entirely rough approximations to [E_{L{\rm I}_{0}}], [E_{L{\rm II}_{0}}] and [E_{L{\rm III}_{0}}] rather than ELI, ELII and ELIII, with the possible stated exceptions of Z = 72, 78, 79 and 82.

For the sequel publication Deslattes et al. (2003)[link] or the corresponding NIST database, Table IV therein illustrates the potential systematic errors of ±0–16 eV or even −100 to +254 eV between theory, vapour and a set of solid samples. Table V therein provides energies [E_{K_{0}}], [E_{L{\rm I}_{0}}], [E_{L{\rm II}_{0}}], [E_{L{\rm III}_{0}}] and EK, ELI, ELII, ELIII up to Z = 100. The error between these is usually a limitation of the theory, yet the experimental edge energies are not calculated for a known experimental resolution nor follow the precise definitions above. This is a valuable work and is worthy of use in the absence of a new tabulation or more detailed analysis of the absorption-edge profiles. For X-ray fluorescence microscopy (XFM) there is a need for additional energies for M-shell edges and characteristic spectra. Hence, we can cite any generic tabulation, such as, for example, the very popular and well used X-ray Data Booklet (Williams, 2001)[link] with sources Bearden & Burr (1967)[link], Cardona & Ley (1978)[link] and Fuggle & Martensson (1980)[link].

Regarding an (the?) empirical offset energy E0 in definition (v) that is almost always fitted in XAFS analysis, it is challenging to have a reliable estimation for a given edge and material but, in principle, it should lie within the uncertainty of the calibrated experimental edge energy, with some allowance for chemical shifts, the adjustment of the Fermi level and ergo pre-edge bound–bound transitions. Equally, given a known edge hole width (see below), the `edge energy' E0 should lie (well) within a half-width (see below) of the reference edge energy, depending a little upon bound–bound structural features and the instrumental resolution.

A related set of energies correspond to the fluorescence or characteristic X-ray energies (Kα1, Kα2, Kβ …), whether for the identification of fluorescence yields or windowing as a region of interest (ROI) for a pixel-based or other detector in fluorescence. Some of the fluorescence energies and spectra are very well-defined for atomic or metallic systems, but of course can vary significantly according to bonding, molecular and local structure etc. One of the best current references on these is Chantler et al. (2022)[link]. Note that the energies must include satellite and shake processes to represent the spectra. See also Yamamoto (2022[link]), Chantler (2022[link]) and Bunker (2023a[link]).

5. Fermi energy EF

This is closely linked to the previous section. The Fermi level is also known as the Fermi energy. A first definition concentrates on a general formal definition, while a second one focuses on theoretical convergence for use by XAS users and programs. A third definition is given that is conceptually equivalent but is numerically geared towards applications in electron scattering that are needed for XANES and XAFS parameters.

  • (i) In an independent-particle approach to the description of a fermionic system (i.e. particles obeying Pauli's exclusion principle), the Fermi level is the energy value lying between the highest occupied level and the lowest unoccupied level, usually defined as their average. If the energy level spectrum is in the continuum (or almost continuum) the three levels coincide. In a many-body approach the Fermi level coincides with the chemical potential, which is the energy necessary to add or subtract a particle from the system. This definition encompasses the non-interacting case. Like all energy states, the Fermi level is measured from the vacuum level. In a molecular system or band-structure analysis we label the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) (Natoli et al., 2022[link]; de Groot, 2022[link]; Fujikawa, 2022[link]; Yamamoto, 2022[link]). In XAS spectra the Fermi level is below or at the first allowed transition, depending on the system and the absorption edge.

  • (ii) In XAS, the Fermi energy dictates possible pre-edge features and explains the possibility or impossibility of open scattering channels adding to the near-edge structure. When theoretical formalisms compute the reference Fermi energy, which is crucial for the XANES region, the convergence of the complete quantum-mechanical system is an absolute requirement, whether atomic, cluster or periodic boundary conditions are used. The lack of convergence for theoretical formalisms can at this time lead to systematic errors in the determination of the Fermi energy and corresponding pre-edge structure of the order of 1–10 eV in the X-ray regime and should be considered carefully.

    In other words, the Fermi level is the level below the ionization (edge energy) where the energy levels become un­occupied and hence available for discrete bound–bound transitions; that is, where the pre-edge structure appears. This assumes that the values of EK, ELI, ELII and ELIII are well-defined and accurate.

    The experimental definition of the edge can already include some of the pre-edge bound–bound transitions, and hence can be lower than the corresponding true ionization edge energy. It is also affected by changes in the Fermi level due to bonding and charge transfer. The parameter remains critical for theoretical predictions of the pre-edge and XANES structure and also affects the definition of E0 and k for XAFS analysis (Yamamoto, 2022[link]; Chantler, 2022[link]; Bunker, 2023b[link]).

  • (iii) While the physical concept of the Fermi level is well-defined and consistently understood in accordance with the above description, its numerical value can alternatively be defined with respect to the bottom of the highest occupied band in a condensed-matter environment. This definition is pervasive in works focused on electron scattering parameters and is particularly useful in the determination of electron inelastic mean free paths (IMFPs) for XAFS because it provides an upper bound on the permitted energy loss of scattered photoelectrons. Accordingly, we quote explicit Fermi levels in this chapter according to this definition because of their direct utility in Section 10[link]. The Fermi levels used in this chapter follow this third definition and derive from Shinotsuka et al. (2015)[link], Rumble (2018)[link] and Tanuma et al. (2011)[link] as in Table 1[link].

    Table 1
    Table of the values for Fermi energies, as defined in definition (iii) in Section 5[link], needed for theoretical computations of electron scattering parameters in this chapter

    These energies, in combination with the associated chemical potentials, are also determinative for bound–bound transitions and observable pre-edge structure, and for structure at the edge and equation (1)[link].

    ZElementFormEF (eV)EF source
    3 Li Crystal 4.74 Shinotsuka et al. (2015[link])
    4 Be Crystal 14.3 Shinotsuka et al. (2015[link])
    6 C Graphite/diamond 20.4 Tanuma et al. (2011[link])
    11 Na Crystal 3.24 Shinotsuka et al. (2015[link])
    12 Mg Crystal 7.1 Shinotsuka et al. (2015[link])
    13 Al Crystal 11.2 Shinotsuka et al. (2015[link])
    14 Si Crystal 12.5 Shinotsuka et al. (2015[link])
    19 K Crystal 2.12 Shinotsuka et al. (2015[link])
    20 Ca Crystal 4.69 Rumble (2018[link])
    21 Sc Crystal 5.8 Shinotsuka et al. (2015[link])
    22 Ti Crystal 6.0 Shinotsuka et al. (2015[link])
    23 V Crystal 6.4 Shinotsuka et al. (2015[link])
    24 Cr Crystal 7.8 Shinotsuka et al. (2015[link])
    26 Fe Crystal 8.9 Shinotsuka et al. (2015[link])
    27 Co Crystal 10.0 Shinotsuka et al. (2015[link])
    28 Ni Crystal 9.1 Shinotsuka et al. (2015[link])
    29 Cu Crystal 8.7 Shinotsuka et al. (2015[link])
    30 Zn Crystal 9.47 Rumble (2018[link])
    31 Ga Crystal 10.4 Rumble (2018[link])
    32 Ge Crystal 12.6 Tanuma et al. (2011[link])
    38 Sr Crystal 3.93 Rumble (2018[link])
    39 Y Crystal 4.4 Shinotsuka et al. (2015[link])
    41 Nb Crystal 5.3 Shinotsuka et al. (2015[link])
    42 Mo Crystal 6.5 Shinotsuka et al. (2015[link])
    44 Ru Crystal 6.9 Shinotsuka et al. (2015[link])
    45 Rh Crystal 6.9 Shinotsuka et al. (2015[link])
    46 Pd Crystal 6.2 Shinotsuka et al. (2015[link])
    47 Ag Crystal 7.2 Shinotsuka et al. (2015[link])
    48 Cd Crystal 7.47 Rumble (2018[link])
    49 In Crystal 4.82 Tanuma et al. (2011[link])
    50 Sn Crystal 5.51 Shinotsuka et al. (2015[link])
    55 Cs Crystal 1.73 Shinotsuka et al. (2015[link])
    56 Ba Crystal 3.84 Rumble (2018[link])
    64 Gd Crystal 3.5 Shinotsuka et al. (2015[link])
    65 Tb Crystal 4.0 Shinotsuka et al. (2015[link])
    66 Dy Crystal 3.5 Shinotsuka et al. (2015[link])
    72 Hf Crystal 7.9 Shinotsuka et al. (2015[link])
    73 Ta Crystal 8.4 Shinotsuka et al. (2015[link])
    74 W Crystal 10.1 Shinotsuka et al. (2015[link])
    75 Re Crystal 10.7 Shinotsuka et al. (2015[link])
    76 Os Crystal 11.4 Shinotsuka et al. (2015[link])
    77 Ir Crystal 11.2 Shinotsuka et al. (2015[link])
    78 Pt Crystal 10.6 Shinotsuka et al. (2015[link])
    79 Au Crystal 9.0 Shinotsuka et al. (2015[link])
    81 Tl Crystal 8.15 Rumble (2018[link])
    82 Pb Crystal 9.47 Rumble (2018[link])
    83 Bi Crystal 12.6 Tanuma et al. (2011[link])

Some models and theoretical predictions require the use or determination of the plasmon energy Ep and the gap energy Eg for semiconductors and insulators. In the case of the theory presented later in this chapter, however, the plasmon energy is not required or considered. The gap energy is incorporated in the treatment of Section 10[link] via the definition of the complex dielectric function of the material and thus is not required as a separate explicit parameter. These Fermi levels may also be compared with (contrasted with) ionization energies, as for example represented in Table 10.3 in Martin & Wiese (1996[link]).

6. Hole widths and excited-state lifetimes

The ionization hole width ΓH and inner-shell hole excited-state lifetime τH = ℏ/ΓH are another pair of critical parameters that both limit the information content of the near-edge spectrum and also reveal key physics.

The basic equations are as follows: a level width in eV (or rate) Γi = ΓR,i + ΓA,i + ΓCK,i, which is the sum of the widths (or rates) from fluorescence (radiative) processes, Auger (non­radiative) processes and Coster–Kronig (nonradiative) processes as and when they are allowed channels, with i = K, LI, LII, LIII, MI–V, NI–VII et seq. Similarly, the diagram X-ray line width can be estimated as the sum of the widths of levels involved in the transitions [\Gamma(K\alpha_{1})=\Gamma_{K}+\Gamma_{L_{\rm II}}], [\Gamma(K\alpha_{2})] = [  \Gamma_{K}+\Gamma_{L_{\rm III}}] et seq. and an Auger line width can be estimated as [\Gamma(KL_{k}L_{l})=\Gamma_{K}+\Gamma_{L_{k}L_{l}}\simeq\Gamma_{K}+\Gamma_{L_{ k}}+\Gamma_{L_{l}}], with the latter approximation implying invariance of the decay width regardless of the number of holes present.

A key issue relates to the energy offsets for satellite and shake processes which broaden standard XAS beyond an intrinsic or lifetime width, and similarly for most modalities of detection, but which can be separated in, for example, the RIXS plane. Hence, it is important to know what the experimental data or theory are measuring or assuming. Krause & Oliver (1979)[link] provide K and L subshell widths, extrapolated up to atomic number Z = 110. The K-shell widths (Fig. 1[link]) with claimed uncertainties of 10% for Z = 10–20 and 3–5% for Z = 20–110, are largely consistent with the later work of Campbell & Papp (2001)[link], with claimed uncertainties of 5–25% for Z = 10–30 and 5–10% for Z = 30–92, and with earlier compilations compiled from Bambynek et al. (1972[link]), McGuire (1969[link], 1970[link]), Scofield (1969[link]) and Kostroun et al. (1971[link]), with an estimated uncertainty of 10% for Z = 10–100. Nonetheless, discrepancies of 10% are seen between Campbell & Papp (2001[link]) and the earlier compilation by magnitudes and of up to 14% between Campbell & Papp (2001)[link] and Krause & Oliver (1979[link]). In Tables 2[link] and 3[link] we present total level widths following Campbell & Papp (2001[link]) to Z = 92.

Table 2
Table of edge hole widths in eV following Campbell & Papp (2001[link])

ZK shellLI totalLII totalLIII totalMI shellMIIMIIIMIVMV
3 0.03                
6 0.08                
7 0.13                
8 0.14                
9                  
10 0.24   0.01 0.01          
11 0.28 0.28 0.02 0.02          
12 0.33 0.46 0.03 0.03          
13 0.37 0.78 0.04 0.04          
14 0.43 0.90 0.05 0.05          
15 0.47 1.10 0.07 0.07          
16 0.52 1.30 0.09 0.09          
17 0.57 1.50 0.11 0.11          
18 0.66 1.80 0.13 0.13 0.14        
19 0.71 2.10 0.18 0.18 0.70        
20 0.77 2.50 0.21 0.21 1.10 1.20 1.20    
21 0.83 3.30 0.36 0.23 1.70 1.20 1.20    
22 0.89 3.90 0.52 0.25 2.10 1.20 1.20    
23 0.96 4.60 0.78 0.28 2.20 1.20 1.20    
24 1.02 5.20 0.76 0.32 2.30 1.20 1.20    
25 1.11 6.20 0.97 0.36 2.40 1.20 1.20    
26 1.19 7.00 1.14 0.41 2.40 1.23 1.23    
27 1.28 7.20 1.13 0.47 2.40 1.25 1.27    
28 1.39 6.40 0.98 0.53 2.30 1.30 1.30    
29 1.49 5.50 1.04 0.61 2.20 1.90 1.80    
30 1.62 4.80 1.06 0.68 2.10 2.10 2.15    
31 1.76 4.10 0.77 0.77 2.00 2.25 2.30 0.01  
32 1.92 3.80 0.86 0.86 2.10 2.30 2.30 0.05 0.04
33 2.09 3.80 0.95 0.94 2.40 2.25 2.25 0.06 0.06
34 2.28 3.80 1.05 1.02 2.80 2.20 2.20 0.07 0.07
35 2.49 3.80 1.14 1.11 3.20 2.10 2.15 0.07 0.07
36 2.71 3.75 1.25 1.19 3.50 1.60 1.10 0.07 0.07
37 2.96 3.75 1.34 1.27 4.00 1.90 1.95 0.07 0.07
38 3.23 3.75 1.43 1.35 4.40 1.90 1.90 0.06 0.06
39 3.51 3.75 1.53 1.43 4.90 1.95 1.95 0.06 0.07
40 3.83 3.75 1.63 1.51 5.40 2.00 2.00 0.07 0.07
41 4.16 3.80 1.73 1.60 5.80 2.05 2.05 0.09 0.10
42 4.52 3.80 1.83 1.69 6.30 2.10 2.10 0.22 0.12
43 4.91 3.80 1.93 1.78 6.70 2.15 2.15 0.50 0.14
44 5.33 3.90 2.03 1.87 7.20 2.20 2.20 0.59 0.17
45 5.77 4.00 2.13 1.96 7.60 2.25 2.25 0.61 0.21
46 6.25 3.90 2.23 2.05 8.00 2.35 2.35 0.26 0.26
47 6.76 3.80 2.32 2.15 8.40 2.45 2.55 0.30 0.31
48 7.32 3.50 2.42 2.24 8.80 2.55 2.80 0.34 0.35
49 7.90 3.00 2.53 2.34 9.20 2.70 3.05 0.38 0.39
50 8.53 2.40 2.64 2.43 9.60 2.85 3.30 0.43 0.44
51 9.20 2.30 2.74 2.53 10.00 3.00 3.60 0.47 0.48
52 9.91 2.20 2.84 2.62 10.20 3.20 3.90 0.52 0.52
53 10.70 2.10 2.95 2.72 10.40 3.35 4.30 0.56 0.56
54 11.50 2.00 3.05 2.82 10.60 3.50 4.70 0.60 0.60
55 12.30 2.00 3.15 2.92 10.80 3.70 5.00 0.63 0.63
56 13.20 2.10 3.25 3.02 11.10 3.90 5.40 0.67 0.67
57 14.20 2.20 3.35 3.12 11.40 4.10 5.80 0.70 0.70
58 15.20 2.50 3.41 3.19 11.60 4.30 6.20 0.72 0.72
59 16.20 2.70 3.48 3.27 11.80 4.50 6.70 0.75 0.75
60 17.40 2.90 3.55 3.36 12.00 4.70 7.30 0.78 0.78
61 18.50 3.10 3.63 3.44 12.20 5.00 7.80 0.82 0.82
62 19.80 3.30 3.70 3.53 12.40 5.20 8.20 0.86 0.86
63 21.10 3.60 3.77 3.62 12.60 5.40 8.20 0.90 0.90
64 22.40 3.80 3.87 3.72 12.80 5.60 8.30 0.95 0.95
65 23.80 4.00 3.93 3.80 13.00 5.80 8.20 1.01 1.01
66 25.30 4.30 4.01 3.90 13.20 6.00 8.00 1.07 1.07
67 26.90 4.50 4.09 4.00 13.40 6.30 7.80 1.13 1.13
68 28.50 4.70 4.18 4.10 13.60 6.60 7.50 1.20 1.20
69 30.20 4.90 4.26 4.20 13.80 6.85 7.10 1.27 1.27
70 32.00 5.20 4.36 4.31 13.90 7.10 6.70 1.35 1.35
71 33.90 5.40 4.46 4.43 14.10 7.30 6.00 1.43 1.43
72 35.90 5.70 4.57 4.55 14.20 7.50 5.60 1.52 1.52
73 37.90 6.00 4.69 4.68 14.30 7.80 5.70 1.61 1.61
74 40.10 6.30 4.82 4.81 14.50 8.10 6.40 1.70 1.70
75 42.20 6.70 4.95 4.95 14.60 8.40 6.90 1.79 1.79
76 44.60 7.20 5.09 5.09 14.70 8.60 7.50 1.89 1.89
77 47.00 7.90 5.23 5.24 14.80 8.90 8.00 1.99 1.99
78 49.50 8.80 5.38 5.39 14.90 9.20 8.30 2.08 2.08
79 52.10 9.80 5.53 5.54 15.00 9.50 8.50 2.18 2.18
80 54.80 10.50 5.69 5.71 15.10 9.80 8.60 2.28 2.28
81 57.60 11.10 5.87 5.89 15.10 10.10 8.70 2.38 2.38
82 60.60 11.80 5.04 6.07 15.20 10.40 8.70 2.48 2.48
83 63.60 12.30 6.22 6.27 15.20 10.70 8.60 2.58 2.58
84 66.80 12.70 6.41 6.46 15.30 11.10 8.50 2.68 2.68
85 70.00 13.00 6.60 6.66 15.30 11.40 8.40 2.78 2.78
86 73.40 13.20 6.81 6.87 15.30 11.70 8.30 2.88 2.88
87 76.90 13.50 7.02 7.08 15.40 12.10 8.20 2.98 2.98
88 80.60 13.70 7.50 7.29 15.40 12.50 8.20 3.08 3.08
89 84.40 14.00 8.00 7.51 15.40 12.90 8.00 3.18 3.18
90 88.20 14.30 8.50 7.74 15.50 13.20 8.00 3.28 3.28
91 92.10 14.70 9.10 7.97 15.50 13.60 7.90 3.39 3.39
92 96.30 16.00 10.00 8.20 15.50 14.10 7.90 3.50 3.50

Table 3
Table of edge hole widths in eV following Campbell & Papp (2001[link])

ZNINIINIIINIVNVNVINVII
36 0.40            
37 1.20            
38 1.60 0.40          
39 2.00 0.80 0.30        
40 2.40 1.20 0.60        
41 2.80 1.50 1.00        
42 3.20 2.20 1.60        
43 3.50 2.70 2.20        
44 3.90 3.20 2.80        
45 4.20 4.20 3.80        
46 4.35 6.40 5.60        
47 4.40 8.40 8.00        
48 4.40 10.80 10.50        
49 4.20 13.20 14.00        
50 3.40 17.00 17.00 0.08 0.08    
51 2.60     0.14 0.14    
52 2.40     0.17 0.20    
53 2.40     0.11 0.12    
54 2.60     0.10 0.08    
55 2.80     0.08 0.08    
56 3.10 5.00 1.30 0.08 0.08    
57 3.30 5.03 1.45 0.09 0.10    
58 3.50 5.06 1.60 0.61 0.32    
59 3.70 5.08 1.75 0.78 0.53    
60 4.00 5.10 1.90 1.05 0.80    
61 4.20 5.13 2.05 1.38 1.11    
62 4.40 5.16 2.20 1.78 1.48    
63 4.60 5.20 2.35 2.20 1.90    
64 4.90 5.23 2.50 2.45 2.20    
65 5.10 5.26 2.65 2.70 2.40    
66 5.40 5.30 2.80 2.95 2.60    
67 5.60 5.33 2.95 3.15 2.80    
68 5.80 5.36 3.15 3.35 2.95    
69 6.10 5.40 3.30 3.55 3.10    
70 6.30 5.50 3.50 3.70 3.20 0.03 0.03
71 6.60 5.50 3.65 3.80 3.30 0.03 0.03
72 6.80 5.60 3.85 3.90 3.50 0.07 0.03
73 7.00 5.70 4.00 4.00 3.65 0.08 0.04
74 7.30 5.80 4.20 4.10 3.80 0.10 0.06
75 7.50 5.90 4.40 4.10 3.90 0.15 0.11
76 7.70 6.00 4.60 4.10 3.90 0.22 0.18
77 8.00 6.10 4.75 4.10 4.00 0.31 0.27
78 8.25 6.25 4.90 4.10 3.95 0.35 0.31
79 8.50 6.40 5.05 4.10 3.90 0.37 0.33
80 8.80 6.55 5.30 4.00 3.85 0.33 0.31
81 9.10 6.70 5.60 3.90 3.80 0.29 0.27
82 9.35 6.90 5.80 3.80 3.80 0.26 0.23
83 9.60 7.20 5.95 3.80 3.80 0.22 0.20
84 9.90 7.35 6.20 3.90 3.80 0.19 0.18
85 10.10 7.60 6.40 3.90 3.85 0.17 0.16
86 10.40 7.80 6.60 4.00 3.90 0.16 0.16
87 10.70 8.10 6.80 4.10 3.95 0.15 0.15
88 10.95 8.30 7.00 4.15 4.00 0.15 0.16
89 11.20 8.50 7.25 4.20 4.05 0.15 0.17
90 11.50 8.75 7.50 4.30 4.10 0.15 0.18
91 11.60 9.20 7.75 4.40 4.20 0.29 0.25
92 12.20 9.60 8.00 4.50 4.25 0.37 0.31
[Figure 1]

Figure 1

K-shell widths versus atomic number are generally consistent within uncertainties above Z = 10 (Krause & Oliver, 1979[link]; Campbell & Papp, 2001[link]; Bambynek et al., 1972[link]; McGuire, 1969[link], 1970[link]; Scofield, 1969[link]; Kostroun et al., 1971[link]).

The LIII subshell widths have no available Coster–Kronig processes, so are relatively stable in the tabulations (Fig. 2[link]), with stated uncertainties of 25–20% for Z = 10–40 and 15–8% for Z = 41–110 (Krause & Oliver, 1979[link]), 30% for Z < 20 and 30–10% for Z = 20–40 and 5–10% for Z = 40–92 (Campbell & Papp, 2001[link]), and estimated uncertainties of 10% from the earlier compilation (McGuire, 1971[link]; Chen et al., 1981[link]). They seem consistent within uncertainty across 18 < Z < 40, with large discrepancies above 50% below Z = 18 and systematic discrepancies of around 10–11% towards higher Z. The LII subshell widths are more structured (Fig. 3[link]), with stated uncertainties of 25–20% for Z = 10–50 and 15–10% for Z = 51–110 (Krause & Oliver, 1979[link]), 30% for Z < 20 and 30–10% for Z = 20–40 and 5–10% for Z = 40–92 (Campbell & Papp, 2001[link]), and estimated uncertainties of 10% or more from the earlier compilation (McGuire, 1971[link]; Chen et al., 1981[link]). They display discrepancies above 50% below Z = 19 and systematic discrepancies of up to 70% up to Z = 31 and a variation of 20% across higher Z.

[Figure 2]

Figure 2

LIII subshell widths versus atomic number are consistent within uncertainties up to Z = 40 but diverge strongly for higher Z (Krause & Oliver, 1979[link]; Campbell & Papp, 2001[link]; Bambynek et al., 1972[link]; McGuire, 1970[link]; Scofield, 1969[link]; Kostroun et al., 1971[link]).

[Figure 3]

Figure 3

Reported LII subshell widths show strong structural anomalies across Z.

The L subshell total widths show significant structures revealing discrepancies of a factor of two or even an order of magnitude, possibly due to the inclusion or exclusion of key processes. This is exemplified in the LI subshell widths (Fig. 4[link]), with stated uncertainties of 30–25% for Z = 10–50 and 20–15% for Z = 51–110 (Krause & Oliver, 1979[link]), 10% for 10 < Z < 30 and 25–10% for Z = 30–56 and 1.5–2 eV for Z = 57–92 (Campbell & Papp, 2001[link]), and estimated uncertainties of 10% or more from the earlier compilation (McGuire, 1971[link]; Chen et al., 1981[link]). They regularly display discrepancies above 60% in a structured manner, which are certainly linked to the complexity of Coster–Kronig amplitudes.

[Figure 4]

Figure 4

LI subshell widths show large anomalies with Z. There are irregularities when a new decay channel is turned on. Coster–Kronig and super Coster–Kronig transitions, in the rare earths, for example, can wipe out the single-particle core state entirely, as has been observed in XPS.

Comparisons are more limited for the other subshell widths, which remain important for fluorescence spectroscopy, XES and RIXS. In Table 2[link] we present total level widths following Campbell & Papp (2001[link]) to Z = 92, which we also present versus atomic number in Fig. 5[link]. In summary, the uncertainty in the outer shells is generally much greater than in the inner shells. As a guiding rule, for a given energy the K-shell width is narrower than the given LII- and LIII-edge widths, which in turn are narrower than the given LI-edge widths for similar energies. Conversely, for investigating local structure around a particular element in a molecule or system the LII- and LIII-edge widths are narrower than the K-shell width for Z < 47, which in turn is narrower than the LI-edge widths, whilst for Z > 47 the LII- and LIII-edge widths are narrower than the LI-shell width, which in turn is narrower than the K-edge widths, so that there may be more information content in L-edge spectra.

[Figure 5]

Figure 5

Subshell widths versus atomic number Z → 92 following Campbell & Papp (2001[link]).

7. Fluorescence (radiative) yields ω and Coster–Kronig probabilities f

For fluorescence detection of XAFS and many related phenomena, the probability or rate of fluorescence energy is critical to the signal strength, the estimation of saturation and dead-time corrections, and the contributions of absorption and self-absorption. It is, for example, given by fσ(I) or the fluorescence yield multiplied by the probability of ionization. The symbol f is used generically for the fluorescence yield in this context in many publications. For greater clarity (as presented here and in XRF theory and experiments) the fluorescence yields can be given as ωi = ΓR,i/Γi, where ΓR,i is the radiative width, for the i = K-shell fluorescence yield, the LI-subshell fluorescence yield, the LII-subshell fluorescence yield et seq.

The symbol f is then reserved for Coster–Kronig probabilities for transfer between subshells, as in f12 for a Coster–Kronig transfer from the LI to the LII subshell. Similarly, the Auger probability for nonradiative transfer from a higher shell is given by ai. The sum of all probabilities of filling a subshell is then [\omega_{i}+a_{i}+\textstyle\sum_{j=i+1}^{k}f_{ij}=1]. Hence, [a_{i}=1-\omega_{i}-\textstyle\sum_{j=i+1}^{k}f_{ij}], aK = 1 − ωK, [a_{L_{\rm III}}=1-\omega_{L_{\rm III}}], [a_{L_{\rm II}}=1-\omega_{L_{\rm II}}-f_{23}], [a_{L_{\rm I}}] = [1-\omega_{L_{\rm I}}-f_{12}-f_{13}]. The total number of radiative photons per vacancy in the LI subshell is then [\nu_{L_{\rm I}} = \omega_{L_{\rm I}}+f_{12}\omega_{L_{\rm II}}] + [(f_{13}+f_{12}f_{23})\omega_{L_{\rm III}}]; [\nu_{L_{\rm II}}=\omega_{L_{\rm II}}+f_{23}\omega_{L_{III}}]; [\nu_{L_{\rm III}}=\omega_{L_{\rm III}}] et seq.

If component widths are provided then the yields can be trivially determined, but to determine the component widths from the yields one requires a total width or one of the partial widths. Whilst the yields are primarily used to quantify the dominance and the contributions of the different relaxation processes, the widths indicate the broadening of the edge and XANES features, and XES or fluorescence intrinsic width etc. Thus, both are important for different purposes.

Some compilations report discrete subshell fluorescence yields for the K shell and L subshells following Krause (1979),[link] as in Figs. 6[link] and 7[link] (Bunker, 2010[link]). Other compilations (Bambynek et al., 1972[link]; Bambynek, 1984[link]; Hubbell et al., 1994[link]) present higher shell yields as an effective average (Fig. 8[link]) and provide consistent K-shell yields within 20% above Z = 15 and within 10% above Z = 20. Even now, the L, M and N subshell yields remain an open question and an area of significant uncertainty and investigation. The previous section tabulated and surveyed total level widths; this section particularly presents fluorescence yields and provides a review of Coster–Kronig processes for completeness (Fig. 9[link]).

[Figure 6]

Figure 6

K-shell fluorescence (radiative) yields versus atomic number diverge by large factors at low Z but are in close agreement for higher Z (McGuire, 1969[link], 1970[link]; Walters & Bhalla, 1971b[link]; Bambynek et al., 1972[link]; Krause, 1979[link]; Bambynek, 1984[link]; Hubbell et al., 1994[link]).

[Figure 7]

Figure 7

Fluorescence (radiative) yields versus atomic number for the K shell and L subshells show significant complexity for the subshells (Krause, 1979[link]).

[Figure 8]

Figure 8

Fluorescence (radiative) yields following Hubbell present effective average fluorescence yields versus atomic number. For the K shell, these are the same fluorescence yields ωK (Bambynek et al., 1972[link]; Bambynek, 1984[link]; Hubbell et al., 1994[link]).

[Figure 9]

Figure 9

Coster–Kronig probabilities versus atomic number: this work but edited from Krause (1979[link]). There are irregularities when a new decay channel is turned on. Coster–Kronig and super Coster–Kronig transitions, in the rare earths, for example, can wipe out the single-particle core state entirely, as has been observed in XPS.

We also derive a plot of the Auger probabilities for completeness (Fig. 10[link]). Uncertainties in the Coster–Kronig and Auger probabilities are quite variable. For the purposes of fluorescence XAS detection, we tabulate K and L shells in Table 4[link]; other valuable sources for L-subshell fluorescence yields include McGuire (1971[link]) and Walters & Bhalla (1971a[link],c[link]). f1 = f12 + f13 denotes all Coster–Kronig processes from the upper original LI vacancy. We have represented the Coster–Kronig probabilities with necessary additional significant figures for ease of comparison, and here derive the consequent Auger probabilities. We do not make particular recommendations as to tabulations of preference here, but simply seek to present a useful reference on these key parameters. Information on outer subshells is presented graphically for cognate techniques. For detailed summaries for M subshells and N subshells, see McGuire (1972a[link],b[link], 1974[link], 1975[link]) and Chen et al. (1981[link]), noting that Ohno & Wendin (1985[link]) and Fuggle & Alvarado (1980[link]) revealed large differences between theoretical and experimental Coster–Kronig and hence Auger transition probabilities. See also Chen (1985[link]) and Melhorn (1985[link]) for further detailed discussion.

Table 4
Table of fluorescence yields (radiative probabilities), Coster–Kronig and Auger probabilities for K shells and L subshells edited but following Krause (1979)[link]

ZωKωLIωLIIωLIIIf1f12f13f23aKaLIaLIIaLIII
3 9.00E-04               1.00E+00      
4 3.30E-04               1.00E+00      
5 1.70E-03               9.98E-01      
6 2.80E-03               9.97E-01      
7 5.20E-03               9.95E-01      
8 8.30E-03               9.92E-01      
9 1.30E-02               9.87E-01      
10 1.80E-02               9.82E-01      
11 2.30E-02               9.77E-01      
12 3.00E-02 2.90E-05 1.20E-03 1.20E-03 9.62E-01 3.22E-01 6.40E-01   9.70E-01 3.80E-02 9.99E-01 9.99E-01
13 3.90E-02 2.60E-05 7.50E-04 7.50E-04 9.65E-01 3.25E-01 6.40E-01   9.61E-01 3.50E-02 9.99E-01 9.99E-01
14 5.00E-02 3.00E-05 3.70E-04 3.80E-04 9.59E-01 3.19E-01 6.40E-01   9.50E-01 4.10E-02 1.00E+00 1.00E+00
15 6.30E-02 3.90E-05 3.10E-04 3.10E-04 9.51E-01 3.21E-01 6.30E-01   9.37E-01 4.90E-02 1.00E+00 1.00E+00
16 7.80E-02 7.40E-05 2.60E-04 2.60E-04 9.44E-01 3.24E-01 6.20E-01   9.22E-01 5.59E-02 1.00E+00 1.00E+00
17 9.70E-02 1.20E-04 2.40E-04 2.40E-04 9.39E-01 3.19E-01 6.20E-01   9.03E-01 6.09E-02 1.00E+00 1.00E+00
18 1.18E-01 1.80E-04 2.20E-04 2.20E-04 9.34E-01 3.14E-01 6.20E-01   8.82E-01 6.58E-02 1.00E+00 1.00E+00
19 1.40E-01 2.40E-04 2.70E-04 2.70E-04 9.29E-01 3.09E-01 6.20E-01   8.60E-01 7.08E-02 1.00E+00 1.00E+00
20 1.63E-01 3.10E-04 3.30E-04 3.30E-04 9.20E-01 3.10E-01 6.10E-01   8.37E-01 7.97E-02 1.00E+00 1.00E+00
21 1.88E-01 3.90E-04 8.40E-04 8.40E-04 9.12E-01 3.12E-01 6.00E-01   8.12E-01 8.76E-02 9.99E-01 9.99E-01
22 2.14E-01 4.70E-04 1.50E-03 1.50E-03 9.02E-01 3.12E-01 5.90E-01   7.86E-01 9.75E-02 9.99E-01 9.99E-01
23 2.43E-01 5.80E-04 2.60E-03 2.60E-03 8.94E-01 3.14E-01 5.80E-01   7.57E-01 1.05E-01 9.97E-01 9.97E-01
24 2.75E-01 7.10E-04 3.70E-03 3.70E-03 8.85E-01 3.15E-01 5.70E-01   7.25E-01 1.14E-01 9.96E-01 9.96E-01
25 3.08E-01 8.40E-04 5.00E-03 5.00E-03 8.77E-01 3.17E-01 5.60E-01   6.92E-01 1.22E-01 9.95E-01 9.95E-01
26 3.40E-01 1.00E-03 6.30E-03 6.30E-03 8.68E-01 3.18E-01 5.50E-01   6.60E-01 1.31E-01 9.94E-01 9.94E-01
27 3.73E-01 1.20E-03 7.70E-03 7.70E-03 8.56E-01 3.16E-01 5.40E-01   6.27E-01 1.43E-01 9.92E-01 9.92E-01
28 4.06E-01 1.40E-03 8.60E-03 9.30E-03 8.47E-01 3.07E-01 5.40E-01 2.80E-02 5.94E-01 1.52E-01 9.63E-01 9.91E-01
29 4.40E-01 1.60E-03 1.00E-02 1.10E-02 8.39E-01 3.09E-01 5.30E-01 2.80E-02 5.60E-01 1.59E-01 9.62E-01 9.89E-01
30 4.74E-01 1.80E-03 1.10E-02 1.20E-02 8.31E-01 3.01E-01 5.30E-01 2.60E-02 5.26E-01 1.67E-01 9.63E-01 9.88E-01
31 5.07E-01 2.10E-03 1.20E-02 1.30E-02 8.22E-01 2.92E-01 5.30E-01 3.20E-02 4.93E-01 1.76E-01 9.56E-01 9.87E-01
32 5.35E-01 2.40E-03 1.30E-02 1.50E-02 8.15E-01 2.95E-01 5.20E-01 5.00E-02 4.65E-01 1.83E-01 9.37E-01 9.85E-01
33 5.62E-01 2.80E-03 1.40E-02 1.60E-02 8.09E-01 2.89E-01 5.20E-01 6.30E-02 4.38E-01 1.88E-01 9.23E-01 9.84E-01
34 5.89E-01 3.20E-03 1.60E-02 1.80E-02 8.04E-01 2.84E-01 5.20E-01 7.60E-02 4.11E-01 1.93E-01 9.08E-01 9.82E-01
35 6.18E-01 3.60E-03 1.80E-02 2.00E-02 8.00E-01 2.80E-01 5.20E-01 8.80E-02 3.82E-01 1.96E-01 8.94E-01 9.80E-01
36 6.43E-01 4.10E-03 2.00E-02 2.20E-02 7.97E-01 2.77E-01 5.20E-01 1.00E-01 3.57E-01 1.99E-01 8.80E-01 9.78E-01
37 6.67E-01 4.60E-03 2.20E-02 2.40E-02 7.94E-01 2.74E-01 5.20E-01 1.09E-01 3.33E-01 2.01E-01 8.69E-01 9.76E-01
38 6.90E-01 5.10E-03 2.40E-02 2.60E-02 7.90E-01 2.70E-01 5.20E-01 1.17E-01 3.10E-01 2.05E-01 8.59E-01 9.74E-01
39 7.10E-01 5.90E-03 2.60E-02 2.80E-02 7.85E-01 2.65E-01 5.20E-01 1.26E-01 2.90E-01 2.09E-01 8.48E-01 9.72E-01
40 7.30E-01 6.80E-03 2.80E-02 3.10E-02 7.79E-01 2.59E-01 5.20E-01 1.32E-01 2.70E-01 2.14E-01 8.40E-01 9.69E-01
41 7.47E-01 9.40E-03 3.10E-02 3.40E-02 7.13E-01 1.03E-01 6.10E-01 1.37E-01 2.53E-01 2.78E-01 8.32E-01 9.66E-01
42 7.65E-01 1.00E-02 3.40E-02 3.70E-02 7.12E-01 1.02E-01 6.10E-01 1.41E-01 2.35E-01 2.78E-01 8.25E-01 9.63E-01
43 7.80E-01 1.10E-02 3.70E-02 4.00E-02 7.11E-01 1.01E-01 6.10E-01 1.44E-01 2.20E-01 2.78E-01 8.19E-01 9.60E-01
44 7.94E-01 1.20E-02 4.00E-02 4.30E-02 7.09E-01 9.90E-02 6.10E-01 1.48E-01 2.06E-01 2.79E-01 8.12E-01 9.57E-01
45 8.08E-01 1.30E-02 4.30E-02 4.60E-02 7.05E-01 1.05E-01 6.00E-01 1.50E-01 1.92E-01 2.82E-01 8.07E-01 9.54E-01
46 8.20E-01 1.40E-02 4.70E-02 4.90E-02 7.00E-01 1.00E-01 6.00E-01 1.51E-01 1.80E-01 2.86E-01 8.02E-01 9.51E-01
47 8.31E-01 1.60E-02 5.10E-02 5.20E-02 6.94E-01 1.04E-01 5.90E-01 1.53E-01 1.69E-01 2.90E-01 7.96E-01 9.48E-01
48 8.43E-01 1.80E-02 5.60E-02 5.60E-02 6.88E-01 9.80E-02 5.90E-01 1.55E-01 1.57E-01 2.94E-01 7.89E-01 9.44E-01
49 8.53E-01 2.00E-02 6.10E-02 6.00E-02 6.81E-01 9.10E-02 5.90E-01 1.57E-01 1.47E-01 2.99E-01 7.82E-01 9.40E-01
50 8.62E-01 3.70E-02 6.50E-02 6.40E-02 4.39E-01 1.69E-01 2.70E-01 1.57E-01 1.38E-01 5.24E-01 7.78E-01 9.36E-01
51 8.70E-01 3.90E-02 6.90E-02 6.90E-02 4.48E-01 1.68E-01 2.80E-01 1.56E-01 1.30E-01 5.13E-01 7.75E-01 9.31E-01
52 8.77E-01 4.10E-02 7.40E-02 7.40E-02 4.55E-01 1.75E-01 2.80E-01 1.55E-01 1.23E-01 5.04E-01 7.71E-01 9.26E-01
53 8.84E-01 4.40E-02 7.90E-02 7.90E-02 4.61E-01 1.81E-01 2.80E-01 1.54E-01 1.16E-01 4.95E-01 7.67E-01 9.21E-01
54 8.91E-01 4.60E-02 8.30E-02 8.50E-02 4.66E-01 1.86E-01 2.80E-01 1.54E-01 1.09E-01 4.88E-01 7.63E-01 9.15E-01
55 8.97E-01 4.90E-02 9.00E-02 9.10E-02 4.70E-01 1.90E-01 2.80E-01 1.54E-01 1.03E-01 4.81E-01 7.56E-01 9.09E-01
56 9.02E-01 5.20E-02 9.60E-02 9.70E-02 4.74E-01 1.94E-01 2.80E-01 1.53E-01 9.80E-02 4.74E-01 7.51E-01 9.03E-01
57 9.07E-01 5.50E-02 1.03E-01 1.04E-01 4.78E-01 1.88E-01 2.90E-01 1.53E-01 9.30E-02 4.67E-01 7.44E-01 8.96E-01
58 9.12E-01 5.80E-02 1.10E-01 1.11E-01 4.82E-01 1.92E-01 2.90E-01 1.53E-01 8.80E-02 4.60E-01 7.37E-01 8.89E-01
59 9.17E-01 6.10E-02 1.17E-01 1.18E-01 4.85E-01 1.95E-01 2.90E-01 1.53E-01 8.30E-02 4.54E-01 7.30E-01 8.82E-01
60 9.21E-01 6.40E-02 1.24E-01 1.25E-01 4.88E-01 1.88E-01 3.00E-01 1.52E-01 7.90E-02 4.48E-01 7.24E-01 8.75E-01
61 9.25E-01 6.60E-02 1.32E-01 1.32E-01 4.90E-01 1.90E-01 3.00E-01 1.51E-01 7.50E-02 4.44E-01 7.17E-01 8.68E-01
62 9.29E-01 7.10E-02 1.40E-01 1.39E-01 4.92E-01 1.92E-01 3.00E-01 1.50E-01 7.10E-02 4.37E-01 7.10E-01 8.61E-01
63 9.32E-01 7.50E-02 1.49E-01 1.47E-01 4.93E-01 1.93E-01 3.00E-01 1.49E-01 6.80E-02 4.32E-01 7.02E-01 8.53E-01
64 9.35E-01 7.90E-02 1.58E-01 1.55E-01 4.93E-01 1.93E-01 3.00E-01 1.47E-01 6.50E-02 4.28E-01 6.95E-01 8.45E-01
65 9.38E-01 8.30E-02 1.67E-01 1.64E-01 4.93E-01 1.93E-01 3.00E-01 1.45E-01 6.20E-02 4.24E-01 6.88E-01 8.36E-01
66 9.41E-01 8.90E-02 1.78E-01 1.74E-01 4.92E-01 1.92E-01 3.00E-01 1.43E-01 5.90E-02 4.19E-01 6.79E-01 8.26E-01
67 9.44E-01 9.40E-02 1.89E-01 1.82E-01 4.90E-01 1.90E-01 3.00E-01 1.42E-01 5.60E-02 4.16E-01 6.69E-01 8.18E-01
68 9.47E-01 1.00E-01 2.00E-01 1.92E-01 4.87E-01 1.87E-01 3.00E-01 1.40E-01 5.30E-02 4.13E-01 6.60E-01 8.08E-01
69 9.49E-01 1.06E-01 2.11E-01 2.01E-01 4.83E-01 1.93E-01 2.90E-01 1.39E-01 5.10E-02 4.11E-01 6.50E-01 7.99E-01
70 9.51E-01 1.12E-01 2.22E-01 2.10E-01 4.78E-01 1.88E-01 2.90E-01 1.38E-01 4.90E-02 4.10E-01 6.40E-01 7.90E-01
71 9.53E-01 1.20E-01 2.34E-01 2.20E-01 4.72E-01 1.92E-01 2.80E-01 1.36E-01 4.70E-02 4.08E-01 6.30E-01 7.80E-01
72 9.55E-01 1.28E-01 2.46E-01 2.31E-01 4.65E-01 1.85E-01 2.80E-01 1.35E-01 4.50E-02 4.07E-01 6.19E-01 7.69E-01
73 9.57E-01 1.37E-01 2.58E-01 2.43E-01 4.57E-01 1.77E-01 2.80E-01 1.34E-01 4.30E-02 4.06E-01 6.08E-01 7.57E-01
74 9.58E-01 1.47E-01 2.70E-01 2.55E-01 4.47E-01 1.67E-01 2.80E-01 1.33E-01 4.20E-02 4.06E-01 5.97E-01 7.45E-01
75 9.59E-01 1.44E-01 2.83E-01 2.68E-01 4.85E-01 1.55E-01 3.30E-01 1.30E-01 4.10E-02 3.71E-01 5.87E-01 7.32E-01
76 9.61E-01 1.30E-01 2.95E-01 2.81E-01 5.52E-01 1.62E-01 3.90E-01 1.28E-01 3.90E-02 3.18E-01 5.77E-01 7.19E-01
77 9.62E-01 1.20E-01 3.08E-01 2.94E-01 6.03E-01 1.53E-01 4.50E-01 1.26E-01 3.80E-02 2.77E-01 5.66E-01 7.06E-01
78 9.63E-01 1.14E-01 3.21E-01 3.06E-01 6.40E-01 1.40E-01 5.00E-01 1.24E-01 3.70E-02 2.46E-01 5.55E-01 6.94E-01
79 9.64E-01 1.07E-01 3.34E-01 3.20E-01 6.72E-01 1.42E-01 5.30E-01 1.22E-01 3.60E-02 2.21E-01 5.44E-01 6.80E-01
80 9.65E-01 1.07E-01 3.47E-01 3.33E-01 6.90E-01 1.30E-01 5.60E-01 1.20E-01 3.50E-02 2.03E-01 5.33E-01 6.67E-01
81 9.66E-01 1.07E-01 3.60E-01 3.47E-01 6.96E-01 1.26E-01 5.70E-01 1.18E-01 3.40E-02 1.97E-01 5.22E-01 6.53E-01
82 9.67E-01 1.12E-01 3.73E-01 3.60E-01 6.96E-01 1.16E-01 5.80E-01 1.16E-01 3.30E-02 1.92E-01 5.11E-01 6.40E-01
83 9.68E-01 1.17E-01 3.87E-01 3.73E-01 6.94E-01 1.14E-01 5.80E-01 1.13E-01 3.20E-02 1.89E-01 5.00E-01 6.27E-01
84 9.68E-01 1.22E-01 4.01E-01 3.86E-01 6.89E-01 1.09E-01 5.80E-01 1.11E-01 3.20E-02 1.89E-01 4.88E-01 6.14E-01
85 9.69E-01 1.28E-01 4.15E-01 3.99E-01 6.85E-01 9.50E-02 5.90E-01 1.11E-01 3.10E-02 1.87E-01 4.74E-01 6.01E-01
86 9.69E-01 1.34E-01 4.29E-01 4.11E-01 6.82E-01 1.02E-01 5.80E-01 1.10E-01 3.10E-02 1.84E-01 4.61E-01 5.89E-01
87 9.70E-01 1.39E-01 4.43E-01 4.24E-01 6.77E-01 9.70E-02 5.80E-01 1.09E-01 3.00E-02 1.84E-01 4.48E-01 5.76E-01
88 9.70E-01 1.46E-01 4.56E-01 4.37E-01 6.72E-01 9.20E-02 5.80E-01 1.08E-01 3.00E-02 1.82E-01 4.36E-01 5.63E-01
89 9.71E-01 1.53E-01 4.68E-01 4.50E-01 6.64E-01 8.40E-02 5.80E-01 1.08E-01 2.90E-02 1.83E-01 4.24E-01 5.50E-01
90 9.71E-01 1.61E-01 4.79E-01 4.63E-01 6.60E-01 9.00E-02 5.70E-01 1.08E-01 2.90E-02 1.79E-01 4.13E-01 5.37E-01
91 9.72E-01 1.62E-01 4.72E-01 4.76E-01 6.64E-01 8.40E-02 5.80E-01 1.39E-01 2.80E-02 1.74E-01 3.89E-01 5.24E-01
92 9.72E-01 1.76E-01 4.67E-01 4.89E-01 6.52E-01 8.20E-02 5.70E-01 1.67E-01 2.80E-02 1.72E-01 3.66E-01 5.11E-01
93 9.73E-01 1.87E-01 4.66E-01 5.02E-01 6.42E-01 7.20E-02 5.70E-01 1.92E-01 2.70E-02 1.71E-01 3.42E-01 4.98E-01
94 9.73E-01 2.05E-01 4.64E-01 5.14E-01 6.05E-01 4.50E-02 5.60E-01 1.98E-01 2.70E-02 1.90E-01 3.38E-01 4.86E-01
95 9.74E-01 2.18E-01 4.71E-01 5.26E-01 5.95E-01 4.50E-02 5.50E-01 2.03E-01 2.60E-02 1.87E-01 3.26E-01 4.74E-01
96 9.74E-01 2.28E-01 4.79E-01 5.39E-01 5.87E-01 3.70E-02 5.50E-01 2.00E-01 2.60E-02 1.85E-01 3.21E-01 4.61E-01
97 9.75E-01 2.36E-01 4.85E-01 5.50E-01 5.80E-01 4.00E-02 5.40E-01 1.98E-01 2.50E-02 1.84E-01 3.17E-01 4.50E-01
98 9.75E-01 2.44E-01 4.90E-01 5.60E-01 5.73E-01 3.30E-02 5.40E-01 1.97E-01 2.50E-02 1.83E-01 3.13E-01 4.40E-01
99 9.75E-01 2.53E-01 4.97E-01 5.70E-01 5.65E-01 2.50E-02 5.40E-01 1.96E-01 2.50E-02 1.82E-01 3.07E-01 4.30E-01
100 9.76E-01 2.63E-01 5.06E-01 5.79E-01 5.56E-01 2.60E-02 5.30E-01 1.94E-01 2.40E-02 1.81E-01 3.00E-01 4.21E-01
101 9.76E-01 2.72E-01 5.15E-01 5.88E-01 5.48E-01 1.80E-02 5.30E-01 1.91E-01 2.40E-02 1.80E-01 2.94E-01 4.12E-01
102 9.76E-01 2.80E-01 5.24E-01 5.96E-01 5.40E-01 2.00E-02 5.20E-01 1.89E-01 2.40E-02 1.80E-01 2.87E-01 4.04E-01
103 9.77E-01 2.82E-01 5.33E-01 6.04E-01 5.38E-01 8.00E-03 5.30E-01 1.85E-01 2.30E-02 1.80E-01 2.82E-01 3.96E-01
104 9.77E-01 2.91E-01 5.44E-01 6.11E-01 5.31E-01 1.10E-02 5.20E-01 1.81E-01 2.30E-02 1.78E-01 2.75E-01 3.89E-01
105 9.77E-01 3.00E-01 5.53E-01 6.18E-01 5.22E-01 1.20E-02 5.10E-01 1.78E-01 2.30E-02 1.78E-01 2.69E-01 3.82E-01
106 9.78E-01 3.10E-01 5.62E-01 6.24E-01 5.13E-01 3.00E-03 5.10E-01 1.74E-01 2.20E-02 1.77E-01 2.64E-01 3.76E-01
107 9.78E-01 3.20E-01 5.73E-01 6.30E-01 5.05E-01 0.00E+00 5.05E-01 1.71E-01 2.20E-02 1.75E-01 2.56E-01 3.70E-01
108 9.78E-01 3.31E-01 5.84E-01 6.35E-01 4.97E-01 0.00E+00 4.97E-01 1.65E-01 2.20E-02 1.72E-01 2.51E-01 3.65E-01
109 9.78E-01 3.43E-01 5.90E-01 6.40E-01 4.88E-01 0.00E+00 4.88E-01 1.63E-01 2.20E-02 1.69E-01 2.47E-01 3.60E-01
110 9.79E-01 3.54E-01 5.98E-01 6.44E-01 4.77E-01 0.00E+00 4.77E-01 1.58E-01 2.10E-02 1.69E-01 2.44E-01 3.56E-01
[Figure 10]

Figure 10

Auger probabilities versus atomic number: this work, based upon Krause (1979[link]).

8. Exemplar elemental crystal structures

Limited crystallographic data for elemental materials for FEFF, FDMX and for DFT computations etc. is provided in Table 5[link]. As many elemental solids display complex polymorphism with numerous allotropes (boron, carbon, phosphorus, sulfur, arsenic, selenium, tin, antimony …), it is important to ensure that the reference sample or calculation matches the correct structural determination of the lattice. Those elements which are liquid or gas at room temperature are generally unsuitable as reference standards, although they will become solid and crystalline at suitably low temperatures. As reference samples, users may use oxides, well-defined or well-prepared simple binary or ternary compounds, or indeed something as complex as ferrocene. For fingerprinting, many more complex and ill-defined systems are used a benchmarks; however, we list these elemental compounds as leading to potential primary references without being exclusive. We commend some regularly updated online resources as reference materials here, such as those at https://periodictable.com/Properties/A/CrystalStructure.html and at http://wwwhomes.uni-bielefeld.de/achim/ele_structures.html and even more strongly commend the original structural determinations or the corresponding CIF files from structural databases. As an illustration for zinc metal, Nuss et al. (2010[link]) include temperature variation from XRD.

Table 5
Crystal structures used for the calculation of optical losses and electron IMFP data for various elements

In all cases structural data are sourced from Wyckoff (1963[link]).

ZElementSpace groupa (Å)b (Å)c (Å)Notes
3 Li Im-3m 3.509 3.509 3.509  
4 Be P63/mmc 2.287 2.287 3.583  
11 Na Im-3m 4.291 4.291 4.291  
12 Mg P63/mmc 3.209 3.209 5.210  
13 Al Fm-3m 4.050 4.050 4.050  
14 Si Fd-3m 5.431 5.431 5.431  
19 K Im-3m 5.247 5.247 5.247 T = 78 K
20 Ca Fm-3m 5.576 5.576 5.576  
21 Sc P63/mmc 3.309 3.309 5.273  
22 Ti P63/mmc 2.950 2.950 4.686  
23 V Im-3m 3.024 3.024 3.024  
24 Cr Im-3m 2.884 2.884 2.884  
26 Fe Im-3m 2.867 2.867 2.867 α-Iron
27 Co Fm-3m 3.548 3.548 3.548  
28 Ni Fm-3m 3.524 3.524 3.524  
29 Cu Fm-3m 3.615 3.615 3.615  
30 Zn P63/mmc 2.665 2.665 4.947  
31 Ga Bmab 4.511 4.517 7.645  
38 Sr Fm-3m 6.085 6.085 6.085  
39 Y P63/mmc 3.647 3.647 5.731  
41 Nb Im-3m 3.300 3.300 3.300  
42 Mo Im-3m 3.147 3.147 3.147  
44 Ru P63/mmc 2.704 2.704 4.282  
45 Rh Fm-3m 3.803 3.803 3.803  
46 Pd Fm-3m 3.890 3.890 3.890  
47 Ag Fm-3m 4.086 4.086 4.086  
48 Cd P63/mmc 2.979 2.979 5.618  
50 Sn I41/amd 5.820 5.820 3.175 β-Tin (white)
55 Cs Im-3m 6.067 6.067 6.067 T = 78 K
56 Ba Im-3m 5.025 5.025 5.025  
64 Gd P63/mmc 3.636 3.636 5.783  
65 Tb P63/mmc 3.601 3.601 5.694  
66 Dy P63/mmc 3.590 3.590 5.647  
72 Hf P63/mmc 3.197 3.197 5.058  
73 Ta Im-3m 3.306 3.306 3.306  
74 W Im-3m 3.165 3.165 3.165 α-Tungsten
75 Re P63/mmc 2.761 2.761 4.458  
76 Os P63/mmc 2.735 2.735 4.319  
77 Ir Fm-3m 3.839 3.839 3.839  
78 Pt Fm-3m 3.923 3.923 3.923  
79 Au Fm-3m 4.078 4.078 4.078  
81 Tl P63/mmc 3.456 3.456 5.525  
82 Pb Fm-3m 4.950 4.950 4.950  

9. Multiple scattering, shake processes and the amplitude-reduction factor S20

We comment that in XAFS analysis the fitting parameter [S_{0}^{2}] causes great pain and confusion in publications. If this is, as claimed, an amplitude-reduction factor, it must always be less than unity. However, the parameter is highly correlated with

  • (i) errors in the fitted edge offset parameter E0 discussed earlier,

  • (ii) errors in spline removal and background subtraction, as a function of k,

  • (iii) scaling of the data signal amplitude or normalization errors from fluorescence spectra and

  • (iv) the range of k fit in, for example, the Hanning window.

Hence, it is common to find [S_{0}^{2}] reported as above unity or below, for example, 0.5 as a consequence of other errors. The uncertainty in this as a physical parameter may be very large; yet the fit may still provide robust and reliable structural and local dynamical parameters of valid significance with small uncertainty.

An early table of suggested [S_{0}^{2}] values can be found as Table 1.1 of Stern (1988[link]), for example, but we note that most other parameters could give good and meaningful physical insight in different experiments and analyses with [S_{0}^{2}] values differing by for example 0.2 or more; therefore, we currently do not make any explicit recommendation.

The shake process is likely to become worse (more significant and hence impacting upon the interpretation of [S_{0}^{2}]) for high angular momentum core states. Fortunately for EXAFS, most structural determination are obtained at the K edge or LIII edge, where the behaviour of multi-electronic process, and hence amplitude reduction, at energy significantly above the threshold is likely to be monotonic and can be addressed by the fitting process to a better degree. M and N shells are more likely to suffer from Coster–Kronig or super Coster–Kronig processes, which can wipe out the single-particle feature of the core state entirely. The complexity particularly lies around the XANES region, where the adiabatic to sudden transition takes place and shake processes also commence, and indeed also where multiple scattering prevails.

However, it is hoped that if uncertainties are provided with a fitted [S_{0}^{2}] parameter then it would be consistent with a sensible value within error. Much more work is required to tie down this parameter in context with tying down other correlated parameters in conventional analysis.

10. Inelastic mean free paths of the (photo)electron and links to LEED, EELS and electron diffraction

The inelastic mean free path (IMFP) of the photoelectron is a necessary parameter for the computation and quantitative analysis of any ionizing X-ray spectroscopy. In this section, we tabulate IMFPs for various elemental solids over the range 1–2000 eV relative to the Fermi energy.

The values are calculated using the coupled plasmon model (CPM) presented by Bourke & Chantler (2015)[link]. This model is based on the optical data model presented by Penn (1987[link]) and can be considered as a generalization of the full Penn algorithm (FPA) widely used in alternative tabulations (Shino­tsuka et al., 2015[link]; Tanuma et al. 1991[link], 2011[link]). A key advantage of the CPM is its self-consistent account of excitation lifetimes, which demonstrably improve the accuracy of the IMFP for energies below 200 eV (Chantler & Bourke, 2014b[link]).

The CPM is best described in terms of its defining equation, given by [\eqalignno {\lambda(E,k)^{-1}_{N} & ={{\hbar} \over {a_{o}\pi E}} {\textstyle \int \limits_{0}^{{{E-E _{\rm F}} \over {(h/2\pi)}}} \int \limits_{q_{-}}^{q_{+}}\int\limits_{0}^{\infty}} {{2} \over {\pi\omega^{\prime} q}} {\rm Im}\left[{{-1} \over {\varepsilon_{\rm data}(0,\omega^{\prime})}} \right]&(1a) \cr &\ \quad {\times}\ {\rm Im}\left\{ {{-1} \over {\varepsilon_{M}[q,\omega,\gamma_{i}(q)_{N-1}\semi \omega_{p}=\omega_{i}]}}\right\} \,{\rm d} \omega^{\prime}\, {\rm d}q\,{\rm d}\omega, \cr \gamma_{i}(q)_{N} & = \hbar\left.{{{\rm d}\omega_{q}} \over {{\rm d}q}}\right|_{\omega_{q},q}\lambda(\omega_{q},q)^{-1}_{N}\Theta(N-\delta). & (1b)}]

In this model the IMFP, λ(E, k), is determined via integration over ℏω and ℏq, which are the energy and momentum, respectively, that may be transferred by an incident electron to the scattering material during a collision event. The energy transfer may take any value up to EEF, where E is the incident energy and EF is the Fermi energy of the material, while the momentum limits are kinematically constrained and are given by [q^{*}\pm = k \pm k\left( {1-{{\hbar\omega} \over {E}}} \right)^{1/2}, \eqno (2)]where k is the momentum of the scattered particle. This is conventionally related as k = (2mE)1/2/ℏ for the incident electron but potentially has other values for states excited within the absorbing material. The inner integral of equation (1)[link] is a summation over Mermin-type free-electron gas dielectric functions (Mermin, 1970[link]), which form a basis set for the potential electronic excitations. These terms are broadened by their γi(q) parameters, which are inverse lifetimes calculated self-consistently by equation (2)[link]. The magnitudes of the excitation channels are provided by the Im{−1/[ɛdata(0, ω′)]} term, which corresponds to an optical energy-loss function and must be provided by external data. Finally, the ωq parameter dictates the effective dispersion relation quantified by Lindhard's theory, and in turn determines the group velocity of excitations vg = dωq/dq.

The optical energy-loss function (optical ELF) provides the excitation magnitudes also commonly known as optical oscillator strengths. For this work they are derived theoretically across the entire spectrum of interest. At low energies, typically meaning for losses of less than a few tens of eV up to 100 eV, depending on the material, they are calculated using density-functional theory via the WIEN2k package (Blaha et al., 2001[link]). This package uses a linearized augmented plane-wave (LAPW) representation for the periodic band structure of the solid calculated within a self-consistent Kohn–Sham algorithm. Details of the calculation of optical spectra are given by Ambrosch-Draxl & Sofo (2006)[link], while the connection to generalized ELF data and electron IMFPs has previously been described by Chantler & Bourke (2014a[link]). Limited crystallo­graphic data used for each element in the WIEN2k calculation are provided in Table 5[link].

At higher energies the optical ELF is calculated using the FFAST package (Chantler, 1995[link], 2000[link]). FFAST is an atomic code that uses a multi-configurational Dirac–Hartree–Fock technique for the self-consistent evaluation of relativistic atomic wavefunctions and a local density approximation for the treatment of the Coulomb and exchange–correlation potentials. It is known to be highly accurate at keV energies, but is not capable of obtaining the detailed loss spectrum for energies below a few hundred eV due to its lack of account of the band structure of the solid. Further, the atomistic valence states lead to a low-energy breakdown of the real part of the form factor. To reduce this problem, we apply a transform of the imaginary part of the form factor calculated by FFAST to determine the complex refractive index n = 1 − δ + iβ in such a way that δ ≃ 0. This is then converted to a dielectric function ɛ = 1 − β2 − 2iβ and in turn to an optical ELF that can be matched with the data from WIEN2k.

For most elements studied it is possible to transition smoothly from an ELF calculated using WIEN2k to one inferred from FFAST, with no significant discontinuity in either the value or the derivate of Im{−1/[ɛ(0, ω)]}. The energies at which the transition is made vary based on element, and are given in Table 6[link]. Also provided in this table are the Fermi energies used for each element in equation (1)[link] discussed in Section 10[link].

Table 6
Table of values for energy transition between WIEN2k optical data and FFAST optical data, plus the Fermi energies needed for equation (1)[link] discussed in Section 10[link]

ZElementTransition (eV)EF (eV)EF source
3 Li 54 4.74 Shinotsuka et al. (2015[link])
4 Be 34 14.3 Shinotsuka et al. (2015[link])
11 Na 48 3.24 Shinotsuka et al. (2015[link])
12 Mg 54 7.1 Shinotsuka et al. (2015[link])
13 Al 39 11.2 Shinotsuka et al. (2015[link])
14 Si 100 12.5 Shinotsuka et al. (2015[link])
19 K 39 2.12 Shinotsuka et al. (2015[link])
20 Ca 55 4.69 Rumble (2018[link])
21 Sc 68 5.8 Shinotsuka et al. (2015[link])
22 Ti 82 6.0 Shinotsuka et al. (2015[link])
23 V 78 6.4 Shinotsuka et al. (2015[link])
24 Cr 100 7.8 Shinotsuka et al. (2015[link])
26 Fe 84 8.9 Shinotsuka et al. (2015[link])
27 Co 92 10 Shinotsuka et al. (2015[link])
28 Ni 95 9.1 Shinotsuka et al. (2015[link])
29 Cu 103 8.7 Shinotsuka et al. (2015[link])
30 Zn 62 9.47 Rumble (2018[link])
31 Ga 55 10.4 Rumble (2018[link])
38 Sr 90 3.93 Rumble (2018[link])
39 Y 83 4.4 Shinotsuka et al., 2015[link])
41 Nb 88 5.3 Shinotsuka et al. (2015[link])
42 Mo 75 6.5 Shinotsuka et al. (2015[link])
44 Ru 74 6.9 Shinotsuka et al. (2015[link])
45 Rh 77 6.9 Shinotsuka et al. (2015[link])
46 Pd 114 6.2 Shinotsuka et al. (2015[link])
47 Ag 88 7.2 Shinotsuka et al. (2015[link])
48 Cd 109 7.47 Rumble (2018[link])
50 Sn 102 5.51 Shinotsuka et al. (2015[link])
55 Cs 93 1.73 Shinotsuka et al. (2015[link])
56 Ba 87 3.84 Rumble (2018[link])
64 Gd 60 3.5 Shinotsuka et al. (2015[link])
65 Tb 60 4.0 Shinotsuka et al. (2015[link])
66 Dy 59 3.5 Shinotsuka et al. (2015[link])
72 Hf 70 7.9 Shinotsuka et al. (2015[link])
73 Ta 83 8.4 Shinotsuka et al. (2015[link])
74 W 95 10.1 Shinotsuka et al. (2015[link])
75 Re 100 10.7 Shinotsuka et al. (2015[link])
76 Os 101 11.4 Shinotsuka et al. (2015[link])
77 Ir 110 11.2 Shinotsuka et al. (2015[link])
78 Pt 112 10.6 Shinotsuka et al. (2015[link])
79 Au 104 9.0 Shinotsuka et al. (2015[link])
81 Tl 74 8.15 Rumble (2018[link])
82 Pb 72 9.47 Rumble (2018[link])

Tables 7–14[link][link][link][link][link][link][link][link] provide the optical ELF data used for each element and the resulting inelastic mean free paths. Figs. 11–21[link][link][link][link][link][link][link][link][link][link][link] illustrate these functionals.

Table 7
ELF and IMFP data calculated for Li, Be, Na, Mg, Al and Si

 LiBeNaMgAlSi
Energy (eV)ELFIMFP (Å)ELFIMFP (Å)ELFIMFP (Å)ELFIMFP (Å)ELFIMFP (Å)ELFIMFP (Å)
1.0 4.939 × 10−3 144.972 5.962 × 10−2 330.932 6.283 × 10−3 89.964 7.382 × 10−3 274.847 2.547 × 10−3 614.056 2.000 × 10−2 558.041
2.0 1.224 × 10−2 46.344 4.983 × 10−2 124.060 1.555 × 10−2 30.538 1.138 × 10−2 87.013 1.027 × 10−2 178.023 2.774 × 10−2 183.951
3.0 3.125 × 10−2 24.802 5.501 × 10−2 72.739 3.806 × 10−2 17.030 1.850 × 10−2 46.254 1.133 × 10−2 90.198 4.108 × 10−2 98.824
4.0 3.697 × 10−1 15.792 5.659 × 10−2 50.918 1.097 × 10−1 11.344 2.891 × 10−2 30.123 1.529 × 10−2 57.198 3.896 × 10−2 65.284
5.0 9.001 × 10−1 10.936 5.117 × 10−2 38.879 6.784 × 10−1 8.230 4.619 × 10−2 21.793 2.002 × 10−2 40.680 4.329 × 10−2 48.055
6.0 1.767 × 10+0 8.170 7.506 × 10−2 31.295 3.600 × 10+0 6.331 8.186 × 10−2 16.794 2.628 × 10−2 31.029 7.790 × 10−2 37.660
7.0 3.025 × 10+0 6.550 1.146 × 10−1 26.019 2.416 × 10−1 5.238 1.647 × 10−1 13.471 3.461 × 10−2 24.797 8.447 × 10−2 30.675
8.0 1.822 × 10+0 5.573 1.256 × 10−1 22.098 8.168 × 10−2 4.600 3.951 × 10−1 11.100 4.641 × 10−2 20.468 8.553 × 10−2 25.761
9.0 1.055 × 10+0 4.965 1.517 × 10−1 19.111 4.469 × 10−2 4.214 1.541 × 10+0 9.312 6.639 × 10−2 17.300 1.239 × 10−1 22.162
10.0 6.763 × 10−1 4.561 2.070 × 10−1 16.793 2.708 × 10−2 3.964 1.232 × 10+1 7.916 1.017 × 10−1 14.887 1.290 × 10−1 19.432
11.0 4.343 × 10−1 4.292 2.737 × 10−1 14.934 1.817 × 10−2 3.801 1.760 × 10+0 6.843 1.765 × 10−1 12.981 1.636 × 10−1 17.280
12.0 3.298 × 10−1 4.101 3.787 × 10−1 13.407 1.405 × 10−2 3.701 5.007 × 10−1 6.045 3.584 × 10−1 11.440 2.127 × 10−1 15.546
13.0 2.438 × 10−1 3.963 6.222 × 10−1 12.123 1.045 × 10−2 3.647 2.383 × 10−1 5.464 1.052 × 10+0 10.158 3.239 × 10−1 14.118
14.0 1.631 × 10−1 3.868 1.005 × 10+0 11.037 8.188 × 10−3 3.618 1.411 × 10−1 5.042 8.236 × 10+0 9.069 4.682 × 10−1 12.921
15.0 1.043 × 10−1 3.805 1.264 × 10+0 10.098 8.513 × 10−3 3.610 9.476 × 10−2 4.725 5.016 × 10+0 8.139 5.644 × 10−1 11.900
16.0 6.232 × 10−2 3.761 1.883 × 10+0 9.281 9.440 × 10−3 3.621 7.200 × 10−2 4.483 9.234 × 10−1 7.347 6.492 × 10−1 11.018
17.0 3.918 × 10−2 3.737 3.080 × 10+0 8.571 5.671 × 10−3 3.640 5.420 × 10−2 4.295 3.748 × 10−1 6.687 8.702 × 10−1 10.251
18.0 2.621 × 10−2 3.722 3.527 × 10+0 7.952 5.506 × 10−3 3.670 4.229 × 10−2 4.151 2.055 × 10−1 6.152 1.333 × 10+0 9.581
19.0 2.709 × 10−2 3.723 3.545 × 10+0 7.416 4.778 × 10−3 3.707 3.564 × 10−2 4.038 1.370 × 10−1 5.722 1.727 × 10+0 8.987
20.0 1.182 × 10−2 3.726 2.850 × 10+0 6.951 3.344 × 10−3 3.752 2.954 × 10−2 3.951 8.912 × 10−2 5.375 2.603 × 10+0 8.463
21.0 1.929 × 10−2 3.742 1.922 × 10+0 6.555 2.504 × 10−3 3.799 2.674 × 10−2 3.883 6.839 × 10−2 5.093 3.699 × 10+0 8.001
22.0 8.111 × 10−3 3.762 1.328 × 10+0 6.212 2.577 × 10−3 3.850 2.371 × 10−2 3.835 5.654 × 10−2 4.859 3.715 × 10+0 7.590
23.0 8.045 × 10−3 3.785 9.172 × 10−1 5.919 2.800 × 10−3 3.906 2.120 × 10−2 3.797 4.575 × 10−2 4.666 4.199 × 10+0 7.223
24.0 8.492 × 10−3 3.814 6.140 × 10−1 5.664 2.971 × 10−3 3.960 1.835 × 10−2 3.772 3.690 × 10−2 4.504 2.883 × 10+0 6.897
25.0 6.361 × 10−3 3.847 4.302 × 10−1 5.447 2.642 × 10−2 4.018 1.418 × 10−2 3.755 2.946 × 10−2 4.369 1.912 × 10+0 6.607
26.0 6.720 × 10−3 3.881 3.113 × 10−1 5.258 2.788 × 10−2 4.078 1.218 × 10−2 3.746 2.565 × 10−2 4.253 1.088 × 10+0 6.347
27.0 5.538 × 10−3 3.918 2.281 × 10−1 5.094 4.793 × 10−2 4.140 1.067 × 10−2 3.744 2.213 × 10−2 4.158 7.167 × 10−1 6.112
28.0 4.568 × 10−3 3.961 1.854 × 10−1 4.950 4.648 × 10−2 4.200 1.060 × 10−2 3.748 2.028 × 10−2 4.076 5.254 × 10−1 5.903
29.0 3.787 × 10−3 4.000 1.625 × 10−1 4.825 4.914 × 10−2 4.266 9.893 × 10−3 3.754 1.771 × 10−2 4.010 4.324 × 10−1 5.714
30.0 4.190 × 10−3 4.043 1.105 × 10−1 4.714 4.896 × 10−2 4.328 9.373 × 10−3 3.765 1.646 × 10−2 3.956 3.192 × 10−1 5.544
31.0 3.580 × 10−3 4.087 1.642 × 10−1 4.618 5.153 × 10−2 4.392 7.900 × 10−3 3.781 1.531 × 10−2 3.907 3.086 × 10−1 5.391
32.0 3.739 × 10−3 4.133 6.645 × 10−2 4.533 5.599 × 10−2 4.457 7.199 × 10−3 3.800 1.317 × 10−2 3.870 2.391 × 10−1 5.251
33.0 2.517 × 10−3 4.179 5.082 × 10−2 4.459 5.701 × 10−2 4.522 7.257 × 10−3 3.819 1.182 × 10−2 3.840 1.910 × 10−1 5.123
34.0 1.907 × 10−3 4.225 4.420 × 10−2 4.393 5.297 × 10−2 4.590 6.112 × 10−3 3.841 1.252 × 10−2 3.816 1.595 × 10−1 5.008
35.0 2.270 × 10−3 4.273 4.159 × 10−2 4.336 6.002 × 10−2 4.653 5.925 × 10−3 3.864 1.272 × 10−2 3.797 1.502 × 10−1 4.902
36.0 2.394 × 10−3 4.324 3.756 × 10−2 4.286 7.529 × 10−2 4.721 5.896 × 10−3 3.890 1.092 × 10−2 3.783 1.217 × 10−1 4.806
37.0 1.747 × 10−3 4.372 3.490 × 10−2 4.242 6.266 × 10−2 4.786 4.972 × 10−3 3.917 1.097 × 10−2 3.774 1.415 × 10−1 4.718
38.0 1.369 × 10−3 4.423 3.251 × 10−2 4.203 7.402 × 10−2 4.852 4.839 × 10−3 3.946 1.101 × 10−2 3.767 1.337 × 10−1 4.637
39.0 1.392 × 10−3 4.473 3.033 × 10−2 4.170 7.435 × 10−2 4.918 4.523 × 10−3 3.975 9.328 × 10−3 3.765 1.297 × 10−1 4.565
40.0 1.481 × 10−3 4.523 2.832 × 10−2 4.141 9.165 × 10−2 4.984 4.353 × 10−3 4.006 7.226 × 10−3 3.764 1.134 × 10−1 4.499
41.0 1.315 × 10−3 4.576 2.646 × 10−2 4.117 9.743 × 10−2 5.047 4.146 × 10−3 4.036 8.139 × 10−3 3.767 1.011 × 10−1 4.439
42.0 1.412 × 10−3 4.629 2.479 × 10−2 4.098 6.652 × 10−2 5.115 4.200 × 10−3 4.068 7.861 × 10−3 3.772 8.412 × 10−2 4.384
43.0 1.583 × 10−3 4.680 2.322 × 10−2 4.080 7.881 × 10−2 5.183 2.234 × 10−2 4.101 7.705 × 10−3 3.778 7.723 × 10−2 4.334
44.0 1.733 × 10−3 4.732 2.179 × 10−2 4.066 7.691 × 10−2 5.248 3.469 × 10−2 4.134 7.530 × 10−3 3.786 7.556 × 10−2 4.290
45.0 2.166 × 10−3 4.786 2.049 × 10−2 4.055 8.293 × 10−2 5.313 3.846 × 10−2 4.168 7.303 × 10−3 3.795 7.990 × 10−2 4.249
46.0 4.010 × 10−3 4.839 1.927 × 10−2 4.046 7.272 × 10−2 5.377 3.792 × 10−2 4.202 7.074 × 10−3 3.806 7.211 × 10−2 4.212
47.0 4.267 × 10−2 4.892 1.814 × 10−2 4.040 8.138 × 10−2 5.443 3.725 × 10−2 4.236 6.849 × 10−3 3.819 7.601 × 10−2 4.179
48.0 2.962 × 10−2 4.946 1.712 × 10−2 4.036 8.163 × 10−2 5.507 3.893 × 10−2 4.271 6.626 × 10−3 3.833 7.958 × 10−2 4.149
49.0 1.397 × 10−2 4.998 1.615 × 10−2 4.034 8.040 × 10−2 5.574 4.689 × 10−2 4.307 6.405 × 10−3 3.847 7.613 × 10−2 4.124
50.0 3.669 × 10−2 5.050 1.525 × 10−2 4.033 7.133 × 10−2 5.636 4.701 × 10−2 4.344 6.188 × 10−3 3.864 7.443 × 10−2 4.100
52.0 4.236 × 10−2 5.157 1.367 × 10−2 4.037 6.876 × 10−2 5.766 4.902 × 10−2 4.415 5.766 × 10−3 3.899 7.448 × 10−2 4.062
54.0 4.613 × 10−2 5.264 1.227 × 10−2 4.047 6.796 × 10−2 5.902 5.058 × 10−2 4.486 5.362 × 10−3 3.935 8.248 × 10−2 4.032
56.0 5.280 × 10−2 5.372 1.107 × 10−2 4.061 6.547 × 10−2 6.027 4.645 × 10−2 4.563 4.984 × 10−3 3.974 7.322 × 10−2 4.011
58.0 4.975 × 10−2 5.480 1.001 × 10−2 4.078 6.285 × 10−2 6.154 5.327 × 10−2 4.636 4.631 × 10−3 4.017 3.994 × 10−2 3.997
60.0 4.693 × 10−2 5.587 9.080 × 10−3 4.100 6.015 × 10−2 6.277 5.859 × 10−2 4.710 4.302 × 10−3 4.060 3.100 × 10−2 3.989
62.0 4.424 × 10−2 5.695 8.262 × 10−3 4.124 5.742 × 10−2 6.404 6.298 × 10−2 4.786 3.999 × 10−3 4.105 2.439 × 10−2 3.985
64.0 4.165 × 10−2 5.800 7.537 × 10−3 4.151 5.825 × 10−2 6.528 6.614 × 10−2 4.861 3.717 × 10−3 4.151 2.168 × 10−2 3.987
66.0 3.915 × 10−2 5.909 6.890 × 10−3 4.180 5.441 × 10−2 6.648 6.850 × 10−2 4.936 3.457 × 10−3 4.197 1.533 × 10−2 3.993
68.0 3.673 × 10−2 6.016 6.319 × 10−3 4.212 5.192 × 10−2 6.767 7.006 × ;10−2 5.009 3.219 × 10−3 4.245 1.628 × 10−2 4.003
70.0 3.439 × 10−2 6.122 5.799 × 10−3 4.245 4.950 × 10−2 6.885 7.107 × 10−2 5.085 2.996 × 10−3 4.293 1.478 × 10−2 4.015
72.0 3.221 × 10−2 6.226 5.346 × 10−3 4.280 4.718 × 10−2 7.007 7.129 × 10−2 5.160 2.793 × 10−3 4.341 1.456 × 10−2 4.030
74.0 3.009 × 10−2 6.331 4.922 × 10−3 4.317 4.490 × 10−2 7.128 7.118 × 10−2 5.235 6.247 × 10−2 4.393 1.448 × 10−2 4.047
76.0 2.816 × 10−2 6.439 4.556 × 10−3 4.353 4.276 × 10−2 7.250 7.043 × 10−2 5.311 6.216 × 10−2 4.444 1.003 × 10−2 4.066
78.0 2.632 × 10−2 6.542 4.219 × 10−3 4.390 4.069 × 10−2 7.366 6.937 × 10−2 5.384 6.371 × 10−2 4.495 1.258 × 10−2 4.087
80.0 2.459 × 10−2 6.648 3.909 × 10−3 4.429 3.869 × 10−2 7.480 6.805 × 10−2 5.459 6.516 × 10−2 4.546 1.096 × 10−2 4.110
82.0 2.300 × 10−2 6.752 3.636 × 10−3 4.470 3.681 × 10−2 7.596 6.641 × 10−2 5.534 6.643 × 10−2 4.598 9.842 × 10−3 4.134
84.0 2.149 × 10−2 6.857 3.380 × 10−3 4.510 3.499 × 10−2 7.713 6.462 × 10−2 5.609 6.763 × 10−2 4.650 9.589 × 10−3 4.159
86.0 2.010 × 10−2 6.955 3.151 × 10−3 4.551 3.328 × 10−2 7.824 6.268 × 10−2 5.682 6.831 × 10−2 4.702 8.469 × 10−3 4.186
88.0 1.882 × 10−2 7.058 2.943 × 10−3 4.594 3.166 × 10−2 7.942 6.063 × 10−2 5.757 6.854 × 10−2 4.755 7.175 × 10−3 4.213
90.0 1.759 × 10−2 7.162 2.747 × 10−3 4.634 3.009 × 10−2 8.048 6.349 × 10−2 5.834 6.859 × 10−2 4.807 6.884 × 10−3 4.241
92.0 1.648 × 10−2 7.264 2.572 × 10−3 4.676 2.863 × 10−2 8.162 6.072 × 10−2 5.905 6.825 × 10−2 4.860 6.546 × 10−3 4.270
94.0 1.545 × 10−2 7.365 2.412 × 10−3 4.719 2.725 × 10−2 8.270 5.852 × 10−2 5.977 6.763 × 10−2 4.913 6.764 × 10−3 4.299
96.0 1.446 × 10−2 7.466 2.260 × 10−3 4.762 2.592 × 10−2 8.378 5.632 × 10−2 6.051 6.687 × 10−2 4.965 5.493 × 10−3 4.330
98.0 1.356 × 10−2 7.567 2.123 × 10−3 4.804 2.467 × 10−2 8.485 5.417 × 10−2 6.124 6.590 × 10−2 5.018 5.351 × 10−3 4.362
100.0 1.274 × 10−2 7.666 1.998 × 10−3 4.849 2.350 × 10−2 8.597 5.206 × 10−2 6.200 6.472 × 10−2 5.070 4.814 × 10−3 4.393
106.2 1.051 × 10−2 7.977 1.662 × 10−3 4.985 2.023 × 10−2 8.925 4.582 × 10−2 6.425 6.048 × 10−2 5.229 4.767 × 10−2 4.492
112.7 8.628 × 10−3 8.294 6.616 × 10−2 5.132 1.732 × 10−2 9.260 3.983 × 10−2 6.661 5.529 × 10−2 5.398 4.563 × 10−2 4.602
119.7 7.050 × 10−3 8.634 4.472 × 10−2 5.289 1.474 × 10−2 9.614 3.427 × 10−2 6.905 5.454 × 10−2 5.579 4.290 × 10−2 4.723
127.1 5.738 × 10−3 8.992 3.708 × 10−2 5.457 1.249 × 10−2 9.978 2.924 × 10−2 7.161 4.839 × 10−2 5.770 3.947 × 10−2 4.854
134.9 4.655 × 10−3 9.371 3.066 × 10−2 5.637 1.053 × 10−2 10.374 2.477 × 10−2 7.427 4.238 × 10−2 5.973 3.565 × 10−2 4.995
143.3 3.774 × 10−3 9.775 2.532 × 10−2 5.830 8.846 × 10−3 10.792 2.089 × 10−2 7.755 3.673 × 10−2 6.189 3.170 × 10−2 5.143
152.1 3.052 × 10−3 10.188 2.084 × 10−2 6.035 7.400 × 10−3 11.235 1.753 × 10−2 8.036 3.153 × 10−2 6.422 3.074 × 10−2 5.298
161.5 2.461 × 10−3 10.629 1.710 × 10−2 6.256 6.166 × 10−3 11.700 1.465 × 10−2 8.363 2.684 × 10−2 6.673 2.681 × 10−2 5.465
171.5 1.980 × 10−3 11.091 1.399 × 10−2 6.498 5.120 × 10−3 12.187 1.220 × 10−2 8.703 2.268 × 10−2 6.937 2.314 × 10−2 5.647
182.1 1.590 × 10−3 11.579 1.142 × 10−2 6.756 4.237 × 10−3 12.701 1.012 × 10−2 9.061 1.906 × 10−2 7.214 1.980 × 10−2 5.844
193.3 1.275 × 10−3 12.080 9.294 × 10−3 7.025 3.497 × 10−3 13.237 8.378 × 10−3 9.434 1.593 × 10−2 7.506 1.681 × 10−2 6.058
205.2 1.021 × 10−3 12.609 7.549 × 10−3 7.309 2.879 × 10−3 13.811 6.919 × 10−3 9.819 1.326 × 10−2 7.813 1.418 × 10−2 6.293
217.9 8.169 × 10−4 13.173 6.120 × 10−3 7.609 2.365 × 10−3 14.408 5.703 × 10−3 10.222 1.099 × 10−2 8.130 1.190 × 10−2 6.539
231.4 6.527 × 10−4 13.767 4.953 × 10−3 7.925 1.939 × 10−3 15.044 4.690 × 10−3 10.645 9.089 × 10−3 8.462 9.928 × 10−3 6.800
245.6 5.212 × 10−4 14.389 4.003 × 10−3 8.257 1.588 × 10−3 15.713 3.853 × 10−3 11.092 7.500 × 10−3 8.809 8.256 × 10−3 7.073
260.8 4.167 × 10−4 15.048 3.237 × 10−3 8.608 1.301 × 10−3 16.418 3.165 × 10−3 11.556 6.185 × 10−3 9.174 6.852 × 10−3 7.359
276.9 3.326 × 10−4 15.739 2.612 × 10−3 8.976 1.063 × 10−3 17.169 2.595 × 10−3 12.047 5.089 × 10−3 9.552 5.667 × 10−3 7.660
294.0 2.651 × 10−4 16.466 2.103 × 10−3 9.364 8.679 × 10−4 17.958 2.123 × 10−3 12.561 4.178 × 10−3 9.950 4.674 × 10−3 7.975
312.2 2.110 × 10−4 17.236 1.691 × 10−3 9.771 7.074 × 10−4 18.787 1.735 × 10−3 13.102 3.424 × 10−3 10.367 3.844 × 10−3 8.432
331.4 1.678 × 10−4 18.044 1.358 × 10−3 10.194 5.758 × 10−4 19.678 1.415 × 10−3 13.673 2.801 × 10−3 10.808 3.155 × 10−3 8.844
351.9 1.333 × 10−4 18.893 1.088 × 10−3 10.636 4.682 × 10−4 20.613 1.153 × 10−3 14.271 2.287 × 10−3 11.270 2.584 × 10−3 9.249
373.6 1.058 × 10−4 19.791 8.712 × 10−4 11.101 3.804 × 10−4 21.585 9.383 × 10−4 14.905 1.865 × 10−3 11.753 2.113 × 10−3 9.626
396.7 8.388 × 10−5 20.856 6.967 × 10−4 11.590 3.088 × 10−4 22.619 7.626 × 10−4 15.562 1.518 × 10−3 12.265 1.725 × 10−3 10.042
421.2 6.647 × 10−5 21.953 5.565 × 10−4 12.104 2.505 × 10−4 23.710 6.190 × 10−4 16.269 1.235 × 10−3 12.799 1.406 × 10−3 10.479
447.2 5.271 × 10−5 23.112 4.447 × 10−4 12.645 2.033 × 10−4 24.869 5.025 × 10−4 16.992 1.005 × 10−3 13.370 1.146 × 10−3 10.937
474.8 4.182 × 10−5 24.263 3.553 × 10−4 13.214 1.651 × 10−4 26.091 4.080 × 10−4 17.762 8.176 × 10−4 13.963 9.337 × 10−4 11.419
504.1 3.314 × 10−5 25.443 2.835 × 10−4 13.811 1.340 × 10−4 27.356 3.308 × 10−4 18.571 6.642 × 10−4 14.587 7.594 × 10−4 11.926
535.3 2.623 × 10−5 26.677 2.258 × 10−4 14.439 1.087 × 10−4 28.701 2.679 × 10−4 19.423 5.389 × 10−4 15.250 6.168 × 10−4 12.460
568.3 2.073 × 10−5 27.961 1.796 × 10−4 15.134 8.802 × 10−5 30.135 2.167 × 10−4 20.308 4.367 × 10−4 15.942 5.002 × 10−4 13.021
603.4 1.637 × 10−5 29.308 1.427 × 10−4 15.954 7.112 × 10−5 31.641 1.751 × 10−4 21.243 3.535 × 10−4 16.676 4.053 × 10−4 13.609
640.7 1.292 × 10−5 30.724 1.132 × 10−4 16.789 5.721 × 10−5 33.452 1.414 × 10−4 22.241 2.859 × 10−4 17.444 3.280 × 10−4 14.230
680.2 1.019 × 10−5 32.253 8.977 × 10−5 17.634 4.603 × 10−5 35.393 1.141 × 10−4 23.281 2.310 × 10−4 18.257 2.652 × 10−4 14.881
722.2 8.029 × 10−6 33.852 7.112 × 10−5 18.498 3.702 × 10−5 37.183 9.203 × 10−5 24.372 1.866 × 10−4 19.110 2.143 × 10−4 15.564
766.8 6.324 × 10−6 35.527 5.632 × 10−5 19.353 2.978 × 10−5 39.120 7.421 × 10−5 25.522 1.507 × 10−4 20.207 1.731 × 10−4 16.285
814.2 4.989 × 10−6 37.303 4.466 × 10−5 20.279 2.400 × 10−5 41.182 5.994 × 10−5 26.730 1.218 × 10−4 21.232 1.400 × 10−4 17.043
864.5 3.936 × 10−6 39.179 3.542 × 10−5 21.247 1.935 × 10−5 43.354 4.844 × 10−5 27.996 9.855 × 10−5 22.288 1.132 × 10−4 17.839
917.8 3.102 × 10−6 41.142 2.805 × 10−5 22.265 1.560 × 10−5 45.629 3.912 × 10−5 29.339 7.966 × 10−5 23.424 9.151 × 10−5 18.678
974.5 2.442 × 10−6 43.208 2.220 × 10−5 23.336 1.257 × 10−5 47.966 3.155 × 10−5 30.747 6.436 × 10−5 24.641 7.393 × 10−5 19.560
1034.7 1.915 × 10−6 45.375 1.751 × 10−5 24.463 1.007 × 10−5 50.439 2.527 × 10−5 32.202 5.173 × 10−5 25.918 5.936 × 10−5 20.489
1098.6 1.494 × 10−6 47.686 1.374 × 10−5 25.647 1.066 × 10−4 53.001 2.009 × 10−5 33.756 4.126 × 10−5 27.238 4.724 × 10−5 21.464
1166.4 1.163 × 10−6 50.100 1.079 × 10−5 26.897 1.103 × 10−4 55.721 1.600 × 10−5 35.410 3.297 × 10−5 28.617 3.767 × 10−5 22.493
1238.4 9.054 × 10−7 52.649 8.474 × 10−6 28.210 9.129 × 10−5 58.605 1.275 × 10−5 37.147 2.636 × 10−5 30.049 3.006 × 10−5 23.575
1314.9 7.049 × 10−7 55.308 6.654 × 10−6 29.592 7.006 × 10−5 61.663 1.579 × 10−4 38.951 2.108 × 10−5 31.550 2.400 × 10−5 24.713
1396.1 5.488 × 10−7 58.144 5.224 × 10−6 31.045 5.222 × 10−5 64.867 1.215 × 10−4 40.784 1.686 × 10−5 33.148 1.918 × 10−5 25.911
1482.3 4.285 × 10−7 61.129 4.113 × 10−6 32.577 4.102 × 10−5 68.141 1.001 × 10−4 42.795 1.347 × 10−5 34.770 1.537 × 10−5 27.171
1573.8 3.346 × 10−7 64.274 3.237 × 10−6 34.188 3.268 × 10−5 71.660 8.004 × 10−5 44.951 1.631 × 10−4 36.458 1.233 × 10−5 28.496
1671.0 2.610 × 10−7 67.582 2.546 × 10−6 35.883 2.629 × 10−5 75.378 6.319 × 10−5 47.214 1.156 × 10−4 38.270 9.890 × 10−6 29.894
1774.1 2.034 × 10−7 71.063 2.001 × 10−6 37.677 2.132 × 10−5 79.340 5.015 × 10−5 49.533 9.485 × 10−5 40.223 7.924 × 10−6 31.362
1883.7 1.585 × 10−7 74.722 1.573 × 10−6 39.551 1.736 × 10−5 83.580 4.023 × 10−5 51.855 7.597 × 10−5 42.255 7.763 × 10−5 32.907
2000.0 1.235 × 10−7 78.556 1.235 × 10−6 41.525 1.415 × 10−5 88.127 3.234 × 10−5 54.396 6.051 × 10−5 44.362 6.455 × 10−5 34.534

Table 8
ELF and IMFP data calculated for K, Ca, Sc, Ti, V and Cr

 KCaScTiVCr
Energy (eV)ELFIMFP (Å)ELFIMFP (Å)ELFIMFP (Å)ELFIMFP (Å)ELFIMFP (Å)ELFIMFP (Å)
1.0 1.333 × 10−2 52.230 3.586 × 10−2 121.023 3.008 × 10−2 182.280 2.321 × 10−2 251.400 1.099 × 10−2 325.281 2.567 × 10−2 366.090
2.0 5.940 × 10−2 18.725 6.360 × 10−2 42.437 5.021 × 10−2 66.875 3.325 × 10−2 92.366 4.155 × 10−2 105.531 2.434 × 10−2 135.303
3.0 4.122 × 10−1 10.247 2.469 × 10−1 23.035 1.061 × 10−1 38.409 6.051 × 10−2 53.387 3.329 × 10−2 60.339 2.443 × 10−2 79.010
4.0 2.149 × 10+0 6.834 4.888 × 10−1 14.899 1.209 × 10−1 26.328 5.260 × 10−2 36.942 3.230 × 10−2 42.641 2.860 × 10−2 54.791
5.0 3.010 × 10−1 5.412 2.243 × 10−1 11.136 3.928 × 10−1 19.840 8.742 × 10−2 28.233 5.667 × 10−2 33.014 4.439 × 10−2 41.527
6.0 2.477 × 10−1 4.773 4.464 × 10−1 9.126 1.072 × 10−1 15.940 3.085 × 10−1 22.715 1.458 × 10−1 26.834 1.281 × 10−1 33.106
7.0 2.348 × 10−1 4.441 1.631 × 10+0 7.824 3.170 × 10−1 13.470 1.498 × 10−1 18.849 1.068 × 10−1 22.418 8.870 × 10−2 27.253
8.0 1.189 × 10−1 4.272 3.196 × 10+0 6.886 2.767 × 10−1 11.767 2.277 × 10−1 16.107 1.450 × 10−1 19.216 1.245 × 10−1 23.064
9.0 4.090 × 10−2 4.188 9.839 × 10−1 6.193 3.713 × 10−1 10.512 5.394 × 10−1 14.130 2.646 × 10−1 16.831 2.468 × 10−1 20.004
10.0 1.590 × 10−2 4.158 3.354 × 10−1 5.695 7.004 × 10−1 9.527 3.471 × 10−1 12.592 9.161 × 10−1 14.992 7.193 × 10−1 17.640
11.0 9.885 × 10−3 4.162 1.895 × 10−1 5.335 1.359 × 10+0 8.743 3.729 × 10−1 11.381 5.913 × 10−1 13.501 4.456 × 10−1 15.732
12.0 7.673 × 10−3 4.193 7.751 × 10−2 5.070 2.975 × 10+0 8.096 5.024 × 10−1 10.403 4.295 × 10−1 12.273 5.730 × 10−1 14.147
13.0 6.003 × 10−3 4.236 3.746 × 10−2 4.874 2.451 × 10+0 7.556 8.013 × 10−1 9.614 4.703 × 10−1 11.238 4.974 × 10−1 12.837
14.0 6.281 × 10−3 4.291 2.531 × 10−2 4.732 1.144 × 10+0 7.101 9.837 × 10−1 8.969 5.764 × 10−1 10.382 5.197 × 10−1 11.767
15.0 9.225 × 10−3 4.352 2.278 × 10−2 4.627 5.230 × 10−1 6.716 1.378 × 10+0 8.431 6.818 × 10−1 9.668 5.816 × 10−1 10.885
16.0 8.371 × 10−2 4.422 1.992 × 10−2 4.550 2.913 × 10−1 6.390 1.838 × 10+0 7.973 7.066 × 10−1 9.069 6.814 × 10−1 10.155
17.0 1.234 × 10−1 4.495 1.787 × 10−2 4.499 1.906 × 10−1 6.110 2.248 × 10+0 7.585 6.061 × 10−1 8.565 7.533 × 10−1 9.546
18.0 1.599 × 10−1 4.568 1.613 × 10−2 4.465 1.316 × 10−1 5.870 1.772 × 10+0 7.251 1.049 × 10+0 8.136 6.801 × 10−1 9.029
19.0 2.520 × 10−1 4.639 1.558 × 10−2 4.445 9.350 × 10−2 5.665 1.193 × 10+0 6.960 1.578 × 10+0 7.768 5.867 × 10−1 8.585
20.0 2.183 × 10−1 4.716 1.851 × 10−2 4.439 6.816 × 10−2 5.488 7.088 × 10−1 6.705 1.957 × 10+0 7.451 8.478 × 10−1 8.203
21.0 3.433 × 10−1 4.790 1.310 × 10−2 4.437 5.635 × 10−2 5.335 5.006 × 10−1 6.477 1.680 × 10+0 7.173 9.529 × 10−1 7.869
22.0 3.193 × 10−1 4.864 1.539 × 10−2 4.444 4.794 × 10−2 5.204 3.402 × 10−1 6.277 1.858 × 10+0 6.932 1.118 × 10+0 7.579
23.0 1.358 × 10+0 4.939 2.632 × 10−2 4.458 3.610 × 10−2 5.091 2.210 × 10−1 6.097 1.253 × 10+0 6.717 1.263 × 10+0 7.322
24.0 1.043 × 10+0 5.012 2.220 × 10−1 4.476 3.239 × 10−2 4.997 1.570 × 10−1 5.934 1.071 × 10+0 6.524 1.716 × 10+0 7.094
25.0 4.599 × 10−1 5.084 2.595 × 10−1 4.497 2.759 × 10−2 4.917 1.349 × 10−1 5.789 8.772 × 10−1 6.354 2.828 × 10+0 6.888
26.0 5.822 × 10−1 5.156 4.183 × 10−1 4.520 2.795 × 10−2 4.850 1.141 × 10−1 5.655 5.846 × 10−1 6.199 1.494 × 10+0 6.706
27.0 4.134 × 10−1 5.229 3.552 × 10−1 4.549 2.664 × 10−2 4.793 9.350 × 10−2 5.535 3.869 × 10−1 6.060 1.241 × 10+0 6.540
28.0 2.714 × 10−1 5.289 1.258 × 10+0 4.578 7.445 × 10−2 4.747 7.583 × 10−2 5.426 2.244 × 10−1 5.933 8.137 × 10−1 6.389
29.0 2.301 × 10−1 5.359 2.382 × 10+0 4.608 1.601 × 10−1 4.709 6.240 × 10−2 5.327 1.911 × 10−1 5.816 8.440 × 10−1 6.253
30.0 1.071 × 10−1 5.425 1.755 × 10+0 4.640 4.363 × 10−1 4.677 4.484 × 10−2 5.236 1.648 × 10−1 5.710 6.392 × 10−1 6.128
31.0 8.583 × 10−2 5.489 1.146 × 10+0 4.672 4.199 × 10−1 4.652 3.611 × 10−2 5.155 1.389 × 10−1 5.612 4.796 × 10−1 6.013
32.0 1.107 × 10−1 5.553 5.026 × 10−1 4.706 3.033 × 10−1 4.633 3.208 × 10−2 5.082 1.408 × 10−1 5.522 3.619 × 10−1 5.908
33.0 6.389 × 10−2 5.611 2.302 × 10−1 4.739 1.068 × 10+0 4.618 7.541 × 10−2 5.014 9.951 × 10−2 5.438 3.357 × 10−1 5.813
34.0 4.226 × 10−2 5.672 2.406 × 10−1 4.773 4.864 × 10+0 4.607 2.973 × 10−1 4.956 1.162 × 10−1 5.361 2.226 × 10−1 5.722
35.0 2.390 × 10−2 5.729 9.174 × 10−2 4.806 7.251 × 10−1 4.602 4.992 × 10−1 4.902 9.823 × 10−2 5.289 2.197 × 10−1 5.639
36.0 2.061 × 10−2 5.785 8.086 × 10−2 4.841 2.125 × 10+0 4.597 3.548 × 10−1 4.853 6.626 × 10−2 5.221 2.181 × 10−1 5.563
37.0 1.727 × 10−2 5.837 7.985 × 10−2 4.876 1.488 × 10+0 4.597 4.771 × 10−1 4.809 6.097 × 10−2 5.160 1.823 × 10−1 5.491
38.0 2.159 × 10−2 5.891 8.610 × 10−2 4.910 6.949 × 10−1 4.598 1.441 × 10+0 4.773 1.873 × 10−1 5.102 1.997 × 10−1 5.423
39.0 1.927 × 10−2 5.943 5.861 × 10−2 4.944 6.673 × 10−1 4.602 3.183 × 10+0 4.738 1.901 × 10−1 5.048 1.527 × 10−1 5.362
40.0 2.173 × 10−2 5.995 7.732 × 10−2 4.977 3.553 × 10−1 4.606 3.165 × 10+0 4.708 2.794 × 10−1 4.998 1.131 × 10−1 5.303
41.0 1.686 × 10−2 6.034 5.976 × 10−2 5.010 2.132 × 10−1 4.613 7.239 × 10−1 4.683 4.725 × 10−1 4.952 8.187 × 10−2 5.248
42.0 1.582 × 10−2 6.079 4.379 × 10−2 5.046 1.369 × 10−1 4.621 1.179 × 10+0 4.660 1.109 × 10+0 4.908 6.176 × 10−2 5.196
43.0 1.487 × 10−2 6.123 3.480 × 10−2 5.077 1.556 × 10−1 4.630 1.334 × 10+0 4.641 2.642 × 10+0 4.869 8.152 × 10−2 5.148
44.0 1.402 × 10−2 6.165 2.046 × 10−2 5.110 9.319 × 10−2 4.641 1.000 × 10+0 4.624 3.003 × 10+0 4.833 2.471 × 10−1 5.103
45.0 1.330 × 10−2 6.203 1.846 × 10−2 5.141 7.666 × 10−2 4.653 5.622 × 10−1 4.610 1.876 × 10+0 4.799 4.023 × 10−1 5.060
46.0 1.263 × 10−2 6.244 1.880 × 10−2 5.172 6.937 × 10−2 4.664 5.511 × 10−1 4.598 9.972 × 10−1 4.769 7.388 × 10−1 5.020
47.0 1.204 × 10−2 6.278 1.801 × 10−2 5.203 5.727 × 10−2 4.676 4.094 × 10−1 4.588 4.850 × 10−1 4.740 2.269 × 10+0 4.983
48.0 1.151 × 10−2 6.314 2.462 × 10−2 5.233 6.516 × 10−2 4.690 2.983 × 10−1 4.581 1.082 × 10+0 4.714 2.722 × 10+0 4.948
49.0 1.102 × 10−2 6.348 2.129 × 10−2 5.262 4.561 × 10−2 4.703 2.608 × 10−1 4.575 9.812 × 10−1 4.691 1.873 × 10+0 4.914
50.0 1.056 × 10−2 6.383 1.545 × 10−2 5.291 4.309 × 10−2 4.718 1.723 × 10−1 4.571 7.306 × 10−1 4.670 1.287 × 10+0 4.884
52.0 9.779 × 10−3 6.450 1.565 × 10−2 5.347 5.068 × 10−2 4.747 1.605 × 10−1 4.569 4.527 × 10−1 4.634 4.896 × 10−1 4.828
54.0 9.098 × 10−3 6.507 1.078 × 10−2 5.403 4.152 × 10−2 4.776 1.034 × 10−1 4.569 2.774 × 10−1 4.607 7.046 × 10−1 4.780
56.0 8.509 × 10−3 6.571 1.012 × 10−2 5.451 3.172 × 10−2 4.806 9.465 × 10−2 4.574 2.664 × 10−1 4.584 5.252 × 10−1 4.739
58.0 7.989 × 10−3 6.632 1.114 × 10−2 5.499 2.912 × 10−2 4.835 7.776 × 10−2 4.582 1.622 × 10−1 4.569 3.675 × 10−1 4.704
60.0 7.525 × 10−3 6.698 1.089 × 10−2 5.546 2.352 × 10−2 4.865 5.636 × 10−2 4.592 1.735 × 10−1 4.558 2.891 × 10−1 4.676
62.0 7.109 × 10−3 6.762 1.066 × 10−2 5.590 2.295 × 10−2 4.894 5.642 × 10−2 4.605 1.773 × 10−1 4.551 2.858 × 10−1 4.652
64.0 6.732 × 10−3 6.828 1.047 × 10−2 5.632 2.241 × 10−2 4.921 5.493 × 10−2 4.619 1.578 × 10−1 4.548 2.067 × 10−1 4.632
66.0 6.388 × 10−3 6.900 1.029 × 10−2 5.675 1.880 × 10−2 4.948 5.781 × 10−2 4.634 1.161 × 10−1 4.548 1.946 × 10−1 4.617
68.0 6.075 × 10−3 6.976 1.011 × 10−2 5.717 1.528 × 10−2 4.974 5.273 × 10−2 4.650 8.877 × 10−2 4.551 2.036 × 10−1 4.606
70.0 5.783 × 10−3 7.046 9.940 × 10−3 5.763 1.431 × 10−2 4.999 3.796 × 10−2 4.666 7.689 × 10−2 4.556 2.048 × 10−1 4.599
72.0 5.517 × 10−3 7.126 9.767 × 10−3 5.804 1.399 × 10−2 5.024 3.380 × 10−2 4.683 8.385 × 10−2 4.563 1.677 × 10−1 4.594
74.0 5.265 × 10−3 7.207 9.594 × 10−3 5.846 1.402 × 10−2 5.047 2.739 × 10−2 4.701 6.943 × 10−2 4.571 1.159 × 10−1 4.592
76.0 5.036 × 10−3 7.289 9.420 × 10−3 5.893 1.382 × 10−2 5.071 2.711 × 10−2 4.717 5.646 × 10−2 4.581 1.141 × 10−1 4.593
78.0 4.819 × 10−3 7.375 9.245 × 10−3 5.939 1.361 × 10−2 5.094 2.054 × 10−2 4.734 5.468 × 10−2 4.592 9.061 × 10−2 4.594
80.0 4.614 × 10−3 7.458 9.069 × 10−3 5.988 1.341 × 10−2 5.116 2.342 × 10−2 4.752 4.949 × 10−2 4.603 1.026 × 10−1 4.599
82.0 4.426 × 10−3 7.544 8.891 × 10−3 6.037 1.322 × 10−2 5.139 2.000 × 10−2 4.768 4.672 × 10−2 4.616 8.250 × 10−2 4.604
84.0 4.244 × 10−3 7.634 8.713 × 10−3 6.089 1.302 × 10−2 5.162 1.845 × 10−2 4.785 4.522 × 10−2 4.629 7.556 × 10−2 4.613
86.0 4.075 × 10−3 7.725 8.534 × 10−3 6.140 1.283 × 10−2 5.186 1.760 × 10−2 4.801 4.354 × 10−2 4.643 6.897 × 10−2 4.622
88.0 3.916 × 10−3 7.817 8.355 × 10−3 6.192 1.263 × 10−2 5.210 1.730 × 10−2 4.818 4.169 × 10−2 4.658 5.929 × 10−2 4.632
90.0 3.763 × 10−3 7.906 8.175 × 10−3 6.248 1.244 × 10−2 5.236 1.731 × 10−2 4.835 3.993 × 10−2 4.673 5.487 × 10−2 4.643
92.0 3.620 × 10−3 7.999 7.995 × 10−3 6.305 1.224 × 10−2 5.260 1.712 × 10−2 4.854 3.834 × 10−2 4.687 6.525 × 10−2 4.655
94.0 3.485 × 10−3 8.090 7.815 × 10−3 6.362 1.204 × 10−2 5.288 1.687 × 10−2 4.871 3.686 × 10−2 4.702 4.474 × 10−2 4.667
96.0 3.354 × 10−3 8.184 7.635 × 10−3 6.420 1.185 × 10−2 5.316 1.661 × 10−2 4.888 3.545 × 10−2 4.716 4.114 × 10−2 4.680
98.0 3.232 × 10−3 8.278 7.457 × 10−3 6.480 1.165 × 10−2 5.347 1.635 × 10−2 4.907 3.415 × 10−2 4.731 4.121 × 10−2 4.692
100.0 3.116 × 10−3 8.375 7.280 × 10−3 6.538 1.145 × 10−2 5.378 1.609 × 10−2 4.925 3.295 × 10−2 4.746 4.275 × 10−2 4.706
106.2 2.789 × 10−3 8.668 6.739 × 10−3 6.726 1.082 × 10−2 5.482 1.529 × 10−2 4.984 2.962 × 10−2 4.791 3.950 × 10−2 4.748
112.7 2.490 × 10−3 8.979 6.181 × 10−3 6.931 1.016 × 10−2 5.601 1.443 × 10−2 5.054 2.665 × 10−2 4.842 3.483 × 10−2 4.794
119.7 2.216 × 10−3 9.311 5.616 × 10−3 7.154 9.456 × 10−3 5.736 1.352 × 10−2 5.135 2.398 × 10−2 4.901 3.103 × 10−2 4.845
127.1 1.966 × 10−3 9.658 5.056 × 10−3 7.395