Excitation spectrum of core electrons

Colliex, C.

doi:10.1107/97809553602060000593

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 4.3, pp. 404-411

Section 4.3.4.4. Excitation spectrum of core electrons

C. Colliex^a

4.3.4.4. Excitation spectrum of core electrons

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4.3.4.4.1. Definition and classification of core edges

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As for any core-electron spectroscopy, EELS spectroscopy at higher energy losses mostly deals with the excitation of well defined atomic electrons. When considering solid specimens, both initial and final states in the transition are actually eigenstates in the solid state. However, the initial wavefunction can be considered as purely atomic for core excitations. As a first consequence, one can classify these transitions as a function of the parameters of atomic physics: Z is the atomic number of the element; n, l, and j = l + s are the quantum numbers describing the subshells from which the electron has been excited. The spectroscopy notation used is shown in Fig. 4.3.4.21 . The list of major transitions is displayed as a function of Z and [E_c] in Fig. 4.3.4.22 .

Figure 4.3.4.21| top | pdf |

Definition of electron shells and transitions involved in core-loss spectroscopy [from Ahn & Krivanek (1982)].

Figure 4.3.4.22| top | pdf |

Chart of edges encountered in the 50 eV up to 3 keV energy-loss range with symbols identifying the types of shapes [see Ahn & Krivanek (1982) for further comments].

Core excitations appear as edges superimposed, from the threshold energy [E_c] upwards, above a regularly decreasing background. As explained below, the basic matrix element governing the probability of transition is similar for optical absorption spectroscopy and for small-angle-scattering EELS spectroscopy . Consequently, selection rules for dipole transitions define the dominant transitions to be observed, i.e. $[l'-l=\Delta l=\pm 1\quad {\rm and}\quad j'-j=\Delta j=0,\pm1. \eqno (4.3.4.37)]$ This major rule has important consequences for the edge shapes to be observed: approximate behaviours are also shown in Fig. 4.3.4.22. A very useful library of core edges can be found in the EELS atlas (Ahn & Krivanek, 1982), from which we have selected the family of edges gathered in Fig. 4.3.4.23 . They display the following typical profiles:

(i) K edges for low-Z elements $[(3\le Z\le 14)]$ . The carbon K edge occurring at 284 eV is a nice example with a clear hydrogenic or saw-tooth profile and fine structures on threshold depending on the local environment (amorphous, graphite, diamond, organic molecules, $[\ldots]$ ); see Isaacson (1972a,b).

Figure 4.3.4.23| top | pdf |

A selection of typical profiles (K, L_2,3, M_4,5, and N_2,3) illustrating the most important behaviours encountered on major edges through the Periodic Table. A few edges are displayed prior to and others after background stripping. [Data extracted from Ahn & Krivanek (1982).]

(ii) $[L_{\it 2,3}]$ edges for medium-Z elements $[(11\,\lesssim\, Z\,\lesssim\,45)]$ . The $[L_{2,3}]$ edges exhibit different shapes when the outer occupied shell changes in nature: a delayed profile is observed as long as the first vacant d states are located, along the energy scale, rather above the Fermi level (sulfur case). When these d states coincide with the first accessible levels, sharp peaks, generally known as `white lines', appear at threshold (this is the case for transition elements with the Fermi level inside the d band). These lines are generally split by the spin-orbit term on the initial level into $[2p^{3/2}]$ and $[2p^{1/2}]$ (or and ) terms. For higher-Z elements, the bound d levels are fully occupied, and no longer contribute as host orbitals for the excited 2p electrons. One finds again a more traditional hydrogenic profile (such as for the germanium case ).
(iii) $[M_{\it 4,5}]$ edges for heavier-Z elements $[(37\,\lesssim\, Z\, \lesssim\,83)]$ . A sequence of $[M_{4,5}]$ edge profiles, rather similar to $[L_{2,3}]$ edges, is observed, the difference being that one then investigates the density of the final f states. White lines can also be detected when the f levels lie in the neighbourhood of the Fermi level, e.g. for rare-earth elements.

The deeper accessible signals, for incident electrons in the range of 100–400 kV primary voltage, lie between 2500 and 3000 eV, which corresponds roughly to the middle of the second row of transition elements (Mo–Ru) for the $[L_{2,3}]$ edge and to the very heavy metals (Pb–Bi) for the $[M_{4,5}]$ edge.
(iv) A final example in Fig. 4.3.4.23 concerns one of these resonant peaks associated with the excitation of levels just below the conduction band. These are features with high intensity of the same order or even superior to that of plasmons of conduction band electrons previously described in Subsection 4.3.4.3. It occurs with the $[M_{2,3}]$ level for the first transition series, with the $[N_{2,3}]$ level for the second series (for example, strontium in Fig. 4.3.4.23) or with the $[O_{2,3}]$ level for the third series, including the rare-earth elements. The shape varies gradually from a plasmon-like peak with a short lifetime to an asymmetric Fano-type profile, a consequence of the coupling between discrete and continuum final states of the same energy (Fano, 1961).

4.3.4.4.2. Bethe theory for inelastic scattering by an isolated atom (Bethe, 1930; Inokuti, 1971; Inokuti, Itikawa & Turner, 1978, 1979)

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As a consequence of the atomic nature of the excited wavefunction in core-loss spectroscopy, the first step involves deriving a useful theoretical expression for inelastic scattering by an isolated atom. The differential cross section for an electron of wavevector k to be scattered into a final plane wave of vector k′, while promoting one atomic electron from $[\psi_0]$ to $[\psi_n]$ , is given in a one-electron excitation description by $[{{\rm d} \sigma_n \over {\rm d}\Omega\,{\rm d}(\Delta E)}=\left({m_0\over 2\pi\hbar^2}\right)^2\ {k'\over k}|\langle\psi_n{\bf k}'|V({\bf r})|\psi_0{\bf k}\rangle|^2; \eqno (4.3.4.38)]$ see, for instance, Landau & Lifchitz (1966) and Mott & Massey (1952). The potential V(r) corresponds to the Coulomb interaction with all charges (both in the nucleus and in the electron cloud) of the atom. The momentum change in the scattering event is $[\hbar{\bf q}=\hbar({\bf k}-{\bf k}')]$ . The final-state wavefunction is normalized per unit energy range. The orthogonality between initial- and final-state wavefunctions restricts the inelastic scattering to the only interactions with atomic electrons: $[{{\rm d}\sigma_n\over {\rm d}\Omega\,{\rm d}(\Delta E)} = {4\gamma^2 \over a^2_0q^4}\,{k'\over k}\,| {\scr E}_n({\bf q},\Delta E)|^2. \eqno (4.3.4.39)]$

The first part of the above expression has the form of Rutherford scattering. γ is introduced to deal, to a first approximation, with relativistic effects. The ratio k′/k is generally assumed to be equal to unity. This kinematic scattering factor is modified by the second term, or matrix element, which describes the response of the atomic electrons: $[{\scr E}_n({\bf q}, \Delta E)=\bigg\langle\psi_n\bigg|\textstyle\sum\limits_j\exp (i{\bf q}\cdot{\bf r}_j)\bigg|\psi_0\bigg\rangle, \eqno (4.3.4.40)]$ where the sum extends over all atomic electrons at positions $[{\bf r}_j]$ . The dimensionless quantity is known as the inelastic form factor.

For a more direct comparison with photoabsorption measurements, one introduces the generalized oscillator strength (GOS) as $[{{\rm d} f({\bf q},\Delta E)\over{\rm d}(\Delta E)}={\Delta E\over R}\ {|{\scr E}_n({\bf q}, \Delta E)|^2\over (qa_0)^2} \eqno (4.3.4.41)]$ for transitions towards final states $[\psi_\varepsilon]$ in the continuum [ΔE is then the energy difference between the core level and the final state of kinetic energy $[\varepsilon]$ above the Fermi level, scaled in energy to the Rydberg energy (R)]. Also, $[f_n({\bf q}) = {E_n\over R}\ {|{\scr E}_n({\bf q})|^2 \over (qa_0)^2} \eqno (4.3.4.42)]$ for transition towards bound states. In this case, [E_n] is the energy difference between the two states involved.

The generalized oscillator strength is a function of both the energy ΔE and the momentum $[\hbar{\bf q}]$ transferred to the atom. It is displayed as a three-dimensional surface known as the Bethe surface (Fig. 4.3.4.24 ), which embodies all information concerning the inelastic scattering of charged particles by atoms. The angular dependence of the cross section is proportional to $[{1\over q^2}\ {{\rm d} f({\bf q}, \Delta E)\over {\rm d}(\Delta E)}]$ at a given energy loss ΔE.

Figure 4.3.4.24| top | pdf |

Bethe surface for K-shell ionization, calculated using a hydrogenic model. The generalized oscillator strength is zero for energy loss E below the threshold E_K. The horizontal coordinate is related to scattering angle through q [from Egerton (1979)].

In the small-angle limit $[(qr_c\ll1]$ , where [r_c] is the average radius of the initial orbital), the GOS reduces to the optical oscillator strength $[{{\rm d} f({\bf q},\Delta E)\over {\rm d}(\Delta E)} \,\rightarrow{{\rm d} f(0,\Delta E)\over {\rm d}(\Delta E)}]$ and $[{\scr E}_n({\bf q}, \Delta E) \rightarrow {\scr E}_n(0,\Delta E) = q^2\bigg|\bigg\langle\psi_n\bigg|\textstyle\sum\limits_j{\bf u}\cdot {\bf r}_j\bigg|\psi_0\bigg\rangle\bigg|^2, \eqno (4.3.4.43)]$ where u is the unit vector in the q direction. When one is concerned with a given orbital excitation, the sum over $[{\bf r}_j]$ reduces to a single term r for this electron. With some elementary calculations, the resulting cross section is $[{{\rm d}^2\sigma\over {\rm d}\Omega\,{\rm d}(\Delta E)} = {4\gamma^2 R \over \Delta E\, k^2}\, {1\over \theta^2+\theta^2_E}\, {{\rm d} f(0,\Delta E) \over {\rm d}(\Delta E)}. \eqno (4.3.4.44)]$

The major angular dependence is contained, as in the low-loss domain, in the Lorentzian factor $[(\theta^2+\theta^2_E)^{-1}]$ , with the characteristic inelastic angle $[\theta_E]$ being again equal to $[\Delta E/\gamma m_0v^2]$ . Over this reduced scattering-angle domain, known as the dipole region, the GOS is approximately constant and the inner-shell EELS spectrum is directly proportional to the photoabsorption cross section $[\sigma_{\rm opt}]$ , whose data can be used to test the results of single-atom calculations. For larger scattering angles, Fig. 4.3.4.24 exhibits two distinct behaviours for energy losses just above the edge (df/dΔE drops regularly to zero), and for energy losses much greater than the core-edge threshold. In the latter case, the oscillator strength is mostly concentrated in the Bethe ridge, the maximum of which occurs for: $[\left. {\eqalign{ (qa_0)^2 &= {\Delta E\over R}\quad\hbox{(non-relativistic formula),} \cr(qa_0)^2 &= {\Delta E\over R\,}{(\Delta E)^2 \over 2m_0c^2R}\quad \hbox{(relativistic formula)}.}} \right\} \eqno (4.3.4.45)]$

This contribution at large scattering angles is equivalent to direct knock-on collisions of free electrons, i.e. to the curve $[\Delta E=\hbar^2q^2/2m_0]$ lying in the middle of the valence-electron–hole excitations continuum (see Fig. 4.3.4.13). The non-zero width of the Bethe ridge can be used as an electron Compton profile to analyse the momentum distribution of the atomic electrons [see also §4.3.4.4.4(c)].

The energy dependence of the cross section, responsible for the various edge shapes discussed in §4.3.4.4.1, is governed by $[{1\over \Delta E}\ {{\rm d} f({\bf q}, \Delta E)\over {\rm d}(\Delta E)},]$ i.e. it corresponds to sections through the Bethe surface at constant q. Within the general theory described above, various models have been developed for practical calculations of energy differential cross sections.

The hydrogenic model due to Egerton (1979) is an extension of the quantum-mechanical calculations for a hydrogen atom to inner-shell electron excitations in an atom Z by introduction of some useful parametrization (effective nuclear charge, effective threshold energy). It is applied in practice for K and $[L_{2,3}]$ shells.

In the Hartree–Slater (or Dirac–Slater) description, one calculates the final continuum-state wavefunction in a self-consistent central field atomic potential (Leapman, Rez & Mayers, 1980; Rez, 1989). The radial dependence of these wavefunctions is given by the solution of a Schrödinger equation with an effective potential: $[V_{\rm eff}(r)=V(r)+{l'(l'+1)\,\hbar^2\over 2m_0r^2}, \eqno (4.3.4.46)]$ where $[[l'(l'+1)\hbar^2]/2m_0r^2]$ is the centrifugal potential, which is important for explaining the occurrence of delayed maxima in spectra involving final states of higher [l'] . This approach is now useful for any major $[K, L_{2,3}, M_{4,5}, \ldots]$ edge, as illustrated by Ahn & Rez (1985) and more specifically in rare-earth elements by Manoubi, Rez & Colliex (1989).

These differential cross sections can be integrated over the relevant angular and energy domains to provide data comparable with experimental measurements. In practice, one records the energy spectral distribution of electrons scattered into all angles up to the acceptance value β of the collection aperture. The integration has therefore to be made from $[q_{\rm min}\simeq k\theta_E]$ for the zero scattering-angle limit, up to $[q_{\rm max}\simeq k\beta]$ . Fig. 4.3.4.25 shows how such calculated profiles can be used for fitting experimental data.

Figure 4.3.4.25| top | pdf |

A novel technique for simulating an energy-loss spectrum with two distinct edges as a superposition of theoretical contributions (hydrogenic saw-tooth for O K, Lorentzian white lines and delayed continuum for Fe L_2,3 calculated with the Hartree–Slater description). The best fit between the experimental and the simulated spectra is shown; it can be used to evaluate the relative concentration of the two elements [see Manoubi et al. (1990)].

Setting β = π [or equal to an effective upper limit $[\theta_{\rm max}\simeq(\Delta E/E_0)^{1/2}]$ corresponding to the criterion $[q_{\rm max} r\simeq1]]$ , the integral cross section is the total cross section for the excitation of a given core level. These ionization cross sections are required for quantification in all analytical techniques using core-level excitations and de-excitations, such as EELS, Auger electron spectroscopy, and X-ray microanalysis (see Powell, 1976, 1984). A convenient way of comparing total cross sections is to rewrite the Bethe asymptotic cross section as $[\sigma_{nl} E^2_{nl}=6.51\times10^{-14}\,Z_{nl} b_{nl}{\log(C_{nl}U_{nl}) \over U_{nl}}, \eqno (4.3.4.47)]$ when the result is given in cm², $[\sigma_{nl}]$ is the total cross section per atom or molecule or ionization of the nl subshell with edge energy $[E_{nl}]$ , $[Z_{nl}]$ is the number of electrons on the nl level, and $[U_{nl}]$ is the overvoltage defined as $[E_0/E_{nl}]$ . $[b_{nl}]$ and $[c_{nl}]$ are two parameters representing phenomenologically the average number of electrons involved in the excitation and their average energy loss (one finds for the major K and $[L_{2,3}]$ edges $[b_{nl}\simeq ]$ 0.6–0.9 and $[c_{nl}\simeq]$ 0.5–0.7). These values are in practice estimated from plots of curves $[\sigma_{nl}E^2_{nl}U_{nl}]$ as a function of $[\log U_{nl}]$ , known as Fano plots. From least-squares fits to linear regions, one can evaluate the values of $[b_{nl}]$ (slope of the curves) and of $[\log c_{nl}]$ (coordinate at the origin) for various elements and shells. However, it has been shown more recently (Powell, 1989) that the interpretation of Fano plots is not always simple, since they typically display two linear regions. It is only in the linear region for the higher incident energies that the plots show the asymptotic Bethe dependence with the slope directly related to the optical data. At lower incident energies, another linear region is found with a slope typically 10–20% greater. Despite great progress over the last two decades, more cross-section data, either theoretical or experimental, are still required to improve to the 1% level the accuracy in all techniques using these signals.

4.3.4.4.3. Solid-state effects

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The characteristic core edges recorded from solid specimens display complex structures different from those described in atomic terms. Moreover, their detailed spectral distributions depend on the type of compound in which the element is present (Leapman, Grunes & Fejes, 1982; Grunes, Leapman, Wilker, Hoffmann & Kunz, 1982; Colliex, Manoubi, Gasgnier & Brown, 1985). Modifications induced by the local solid-state environment concern (see Fig. 4.3.4.26 ) the following:

(a) The threshold (or edge itself), which may vary in position, slope, and associated fine structures. From photoelectron spectroscopies (UPS, XPS), an edge displacement along the energy scale is known as a `chemical shift': it is due to a shift in the energy of the initial level as a consequence of the atomic potential modifications induced by valence-electron charge transfer (e.g. from metal to oxide). EELS is actually a two-level spectroscopy and the observed changes at edge onset concern both initial and final states. Consequently, measured shifts are due to a combination of core-level energy shift with bandgap and exciton creation. Some important shifts have been measured in EELS such as:

– carbon K: 284 to 288 eV from graphite to diamond;

Figure 4.3.4.26| top | pdf |
Definition of the different fine structures visible on a core-loss edge.
– aluminium $[L_{2,3}]$ : 73 to 77 eV from metal to Al₂O₃;
– silicon $[L_{2,3}]$ : 99.5 to 106 eV from Si to SiO₂.

However, `chemical shift' constitutes a simplified description of the more complex changes that may occur at a given threshold in various compounds. It assumes a rigid translation of the edge, but in most cases the onset changes in shape and there are no simple features to correlate through the different spectra. This remark is more relevant with the increased energy resolution that is now available. With a sub-eV value, extra peaks or splittings can frequently be detected on edges that exhibit simple shapes when recorded at lower resolution. Among others, the $[L_{32}]$ white lines in transition metals show different behaviours when involved in various environments:

– crystal-field-induced splitting for each line in the oxides Sc₂O₃, TiO₂ when compared with the metal (see Fig. 4.3.4.27 ).

Figure 4.3.4.27| top | pdf |

High-energy resolution spectra on the L_2,3 titanium edge from two phases (rutile and anatase) of TiO₂. Each atomic line L₃ and L₂ is split into two components A and B by crystal-field effects. The new level of splitting B₁B₂ that distinguishes the two spectra is not yet understood. In Ti metal, the L₃ and L₂ lines are not split by structural effects [courtesy of Brydson et al. (1989)].

– relative change in intensity ratio between different ionic species [most important when the occupancy degree n for the d band is of the order of 5, i.e. around the middle of the transition series, e.g. Mn and Fe oxides; see for instance, Rask, Miner & Buseck (1987) and Rao, Thomas, Williams & Sparrow (1984)].

– presence of a narrow white line instead of a hydrogenic profile when the electron transfer from the metal to its ligand induces the existence of vacant d states at the Fermi level (CuO compared with Cu, see Fig. 4.3.4.28 ).

Figure 4.3.4.28| top | pdf |

The dramatic change in near-edge fine structures on the L₃ and L₂ lines of Cu, from Cu metal to CuO. The appearance of the intense narrow white lines is due to the existence of vacant d states close to the Fermi level [courtesy of Leapman et al. (1982)].

(b) The near-edge fine structures (ELNES ), which extend over the first 20 or 30 eV above threshold (Taftø & Zhu, 1982; Colliex et al., 1985). These are very similar to XANES structures in X-ray photoabsorption spectroscopy: they mostly reflect the spectral distribution of vacant accessible levels and are consequently very sensitive to site symmetry and charge transfer. Several approaches have been proposed to interpret them. A molecular-orbital description [e.g. Fischer (1970) or Tossell, Vaughan & Johnson (1974)] classifies the energy levels, both occupied and unoccupied, for clusters comprising the central excited ion and its first shell of neighbours. Its major success lies in the interpretation of level splitting on edges.

A one-electron band calculation constitutes a second step with noticeable successes in the case of metals (Müller, Jepsen & Wilkins, 1982). Core-loss spectroscopy, however, imposes specific conditions on the accessible final state: the overlap with the initial core wavefunction involves a projection in space on the site of the core hole, and the dominant dipole selection rules are responsible for angular symmetry selection. When extending the band-structure calculations to energy states rather high above the Fermi level, more elaborate methods, combining the conceptual advantage of the tight-binding method with the accuracy of ab initio pseudopotential calculations, have been developed (Janssen & Sankey, 1987). This self-consistent pseudo-atomic orbital band calculation has been used to describe ELNES structures on different covalent solids (Weng, Rez & Ma, 1989; Weng, Rez & Sankey, 1989).

The most promising description at present is the multiple scattering method developed for X-ray absorption spectra by Durham, Pendry & Hodges (1981) and Vvedensky, Saldin & Pendry (1985). It interprets the spectral modulations, in the energy range 10 to 30 eV above the edge, as due to interference effects, on the excited site, between all waves back-scattered by the neighbouring atoms (see Fig. 4.3.4.29 ). This multiple scattering description in real space should in principle converge towards the local point of view in the solid-state band model, calculated in reciprocal space (Heine, 1980). As an example investigated by EELS, the oxygen and magnesium K edges in MgO have been calculated by Lindner, Sauer, Engel & Kambe (1986) and by Weng & Rez (1989) for increased numbers of coordination shells and different potential models (representing variable ionicities). Fig. 4.3.4.30 shows the comparison of an experimental spectrum with such a calculation. Another useful idea emerging from this model is the simple relation, expressed by Bianconi, Fritsch, Calas & Petiau (1985): $[(E_r-E_b)\,d\,^2=C, \eqno (4.3.4.48)]$ where [E_r] is the energy position of a given resonance peak attributed to multiple scattering from a given shell of neighbours (d is the distance to this shell), and [E_b] is a reference energy close to the threshold energy. This simple law, advertised as the way of measuring `bond lengths with a ruler' (Stohr, Sette & Johnson, 1984), seems to be quite useful when comparing similar structures (Lytle, Greegor & Panson, 1988).

Figure 4.3.4.29| top | pdf |

Illustration of the single and multiple scattering effects used to describe the final wavefunction on the excited site. This theory is very fruitful for understanding and interpreting EXELFS and ELNES features, respectively equivalent to EXAFS and XANES encountered in X-ray absorption spectra.

Figure 4.3.4.30| top | pdf |

Comparison of the experimental O K edge (solid line) with calculated profiles in the multiple scattering approach [courtesy of Weng & Rez (1989)].

Other effects, generally described as multi-electron contributions, cannot be systematically omitted. They all deal with the presence of a core hole on the excited atom and with its influence on the distribution of accessible electron states. Of particular importance are the intra-atomic configuration interactions for white lines, as explained by Zaanen, Sawatzky, Fink, Speier & Fuggle (1985) for [L_3] and [L_2] lines in transition metals and by Thole, van der Laan, Fuggle, Sawatzky, Karnatak & Esteva (1985) for $[M_{4,5}]$ lines in rare-earth elements.

(c) The extended fine structures (EXELFS) are equivalent to the well known EXAFS oscillations in X-ray absorption spectroscopy (Sayers, Stern & Lytle, 1971; Teo & Joy, 1981). Within the previously described multiscattering theory, it corresponds to the first step, the single scattering regime (see Fig. 4.3.4.29a). These extended oscillations are due to the interference on the excited atom between the outgoing excited electron wavefunction and its components reflected on the nearest-neighbour atoms. This interference is destructive or constructive depending on the ratio between the return path length (where is the radial distance with the ith shell of backscattering atoms) and the wavelength of the excited electron. Fourier analysis of EXELFS structures, from 50 eV above the ionization threshold, gives the radial distribution function around this specific site. This is mostly a technique for measuring the local short-range order. Its accuracy has been established to be better than 0.1 Å on nearest-neighbour distances with test specimens, but such performance requires correction procedures for phase shifts. The method therefore seems more promising for measuring changes in interatomic distances in specimens of the same chemical composition. The major advantage of EXELFS is its applicability for small specimen volumes that can moreover be characterized by other high-resolution electron-microscopy modes. It is also possible to investigate bond lengths in different directions by selecting the scattering angle of the transmitted electron and the specimen orientation (Disko, Krivanek & Rez, 1982). On the other hand, the major limitations of EXELFS are due to the dose requirements for sufficient SNR and to the fact that the accessible excitation range is limited to edges below ∼2–3 keV and to oscillation domains ∼200 or 300 eV at the maximum.

4.3.4.4.4. Applications for core-loss spectroscopy

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(a) Quantitative microanalysis. The main field of application of core-loss EELS spectroscopy has been its use for local chemical analysis (Maher, 1979; Colliex, 1984; Egerton, 1978, 1986). The occurrence of an edge superimposed on the regularly decreasing background of an EELS spectrum is an indication of the presence of the associated element within the analysed volume.

Methods have been developed to extract quantitative composition information from these spectra. The basic idea lies in the linear relationship between the measured signal (S) and the number (N) of atoms responsible for it (this is valid in the single core-loss domain for specimen thickness, i.e. up to several micrometres): $[S=I_0N\sigma, \eqno (4.3.4.49)]$ where [I_0] is the incident-beam intensity and σ the relevant excitation cross section in the experimental conditions used, and N is the number of atoms per unit area of specimen. As a satisfactory approximation for taking into account multiple scattering events (either elastic or inelastic in the low-loss region), Egerton (1978) has proposed that equation (4.3.4.49) be rewritten: $[S(\beta,\Delta)=I_0(\beta,\Delta)N\sigma(\beta,\Delta),\eqno (4.3.4.50)]$ where all quantities correspond to a limited angle of collection β and to a limited integration window Δ (eV) above threshold for signal measurement.

A major problem is the evaluation of the signal itself after background subtraction. The method generally used, demonstrated in Fig. 4.3.4.31 , involves extrapolating a modelized background profile below the core loss of interest. Following Egerton (1978), the choice of a power law $[B(\Delta E)=A\Delta E^{-R}]$ is satisfactory in many cases, and the signal is then defined as $[S(\Delta)=\textstyle\int\limits^{E_c+\Delta}_{E_c}[I(\Delta E)-B(\Delta E)]\,{\rm d}(\Delta E). \eqno (4.3.4.51)]$ Numerical methods have been developed to perform this process with a well controlled analysis of statistical errors (Trebbia, 1988).

Figure 4.3.4.31| top | pdf |

The conventional method of background subtraction for the evaluation of the characteristic signals S_{O K} and S_{Fe L_2,3} used for quantitative elemental analysis (to be compared with the approach described in Fig. 4.3.4.25).

In many cases, one is interested in elemental ratios; consequently, the useful formula becomes $[{N_A\over N_B}= {S_A(\beta,\Delta) \over S_B(\beta,\Delta)} \, {\sigma_B(\beta,\Delta) \over \sigma_A(\beta,\Delta)}. \eqno (4.3.4.52)]$ This can be used to determine the [N_A/N_B] ratio without standards, if the cross-section ratio $[\sigma_B/\sigma_A]$ (also called the $[k_{AB}]$ factor) is previously known: accuracy at present is limited to ±5% for most edges. But it is also possible to extract from this formula the cross-section (or k factor) experimental values for comparison with the calculated ones, if the local stoichiometry of the specimen is satisfactorily known [Hofer, Golob & Brunegger (1988) and Manoubi et al. (1989) for the $[M_{4,5}]$ edges].

Improvements have recently been made in order to reduce the different sources of errors. For medium-thickness specimens (i.e. for $[t\simeq\lambda_p]$ where $[\lambda_P]$ is the mean free path for plasmon excitation), deconvolution techniques are introduced for a safer determination of the signal. When the background extrapolation method cannot be used, i.e. when edges overlap noticeably, new approaches (such as illustrated in Fig. 4.3.4.25) try to determine the best simulated profile over the whole energy-loss range of interest. It requires several contributions, either deduced from previous measurements on standard (Shuman & Somlyo, 1987; Leapman & Swyt, 1988), or from reasonable mathematical models with different contributions for dealing with transitions towards bound states or continuum states (Manoubi, Tence, Walls & Colliex, 1990).

(b) Detection limits. This method has been shown to be the most successful of all EM techniques in terms of ultimate mass sensitivity and associated spatial resolution. This is due to the strong probability of excitation for the signals of interest (primary ionization event) and to the good localization of the characteristic even within the irradiated volume of material. Variations in composition have been recorded at a subnanometre level (Scheinfein & Isaacson, 1986; Colliex, 1985; Colliex, Maurice & Ugarte, 1989). In terms of ultimate sensitivity (minimum number of identified atoms), the range of a few tens of atoms (∼10⁻²¹ g) has been reached as early as about 15 years ago in the pioneering work of Isaacson & Johnson (1975). Very recently, a level close to the single-atom identification has been demonstrated (Mory & Colliex, 1989). A major obstacle is then often radiation damage, and consequent specimen modification induced by the very intense primary dose required for obtaining sufficient SNR values.

On the other hand, the EELS technique has long been less fruitful for investigating low concentrations of impurities within a matrix. This is a consequence of the very high intrinsic background under the edges of interest: in most applications, the atomic concentration detection limit was in the range 10⁻³ to 10⁻². The introduction of satisfactory methods for processing the systematic sources of noise in spectra acquired with parallel detection devices (Shuman & Kruit, 1985) has greatly modified this situation. One can now take full benefit from the very high number of counts thus recorded within a reasonable time (10⁶ to 10⁷ counts per channel) and detection of calcium of the order of 10⁻⁵ atomic concentration in an organic matrix has been demonstrated by Shuman & Somlyo (1987).
(c) Crystallographic information in EELS. Although not particularly suited to solving crystal-structure problems, EELS carries structural information at different levels:

In a crystalline specimen, one detects orientation effects on the intensity of core-loss edges. This is a consequence of the channelling of the Bloch standing waves as a function of the crystal orientation This observation requires well collimated angular conditions and inelastic localization better than the lattice spacing responsible for elastic diffraction. When these criteria apply, the changes in core-loss excitations with crystallographic orientation can be used to determine the crystallographic site of specific atoms (Tafto & Krivanek, 1982). An equivalent method, known as ALCHEMI (atom location by channelling enhanced microanalysis), which involves measuring the change of X-ray production as a function of crystal orientation, has been applied to the determination of the preferential site for substitutional impurities in many crystals (Spence & Tafto, 1983).

Energy-filtered electron-diffraction patterns of core-loss edges could reveal the symmetry of the local coordination of selected atomic species rather than the symmetry of the crystal as a whole. This type of information should be compared with ELNES data (Spence, 1981).

At large scattering angles, and for energy losses far beyond the excitation threshold, the Bethe ridge [or electron Compton profile (see §§4.3.4.3.3 and 4.3.4.4.2)] constitutes a major feature easily observable in energy-filtered diffraction patterns (Reimer & Rennekamp, 1989). The width of this feature is associated with the momentum distribution of the excited electrons (Williams & Bourdillon, 1982). Quantitative analysis of the data is similar to the Fourier method for EXELFS oscillations. After subtracting the background contribution, the spectrum is converted into momentum space and Fourier transformed to obtain the reciprocal form factor B(r): it is the autocorrelation of the ground-state wavefunction in a direction specified by the scattering vector q. This technique of data analysis to study electron momentum densities is directly developed from high-energy photon-scattering experiments (Williams, Sparrow & Egerton, 1984).

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