Definitions

Colliex, C.

doi:10.1107/97809553602060000593

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

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. C. ch. 4.3, pp. 391-394

Section 4.3.4.1. Definitions

C. Colliex^a

4.3.4.1. Definitions

| top | pdf |

4.3.4.1.1. Use of electron beams

| top | pdf |

Among the different spectroscopies available for investigating the electronic excitation spectrum of solids, inelastic electron scattering experiments are very useful because the range of accessible energy and momentum transfer is very large, as illustrated in Fig. 4.3.4.1 taken from Schnatterly (1979). Absorption measurements with photon beams follow the photon dispersion curve, because it is impossible to vary independently the energy and the momentum of a photon. In a scattering experiment, a quasi-parallel beam of monochromatic particles is incident on the specimen and one measures the changes in energy and momentum that can be attributed to the creation of a given excitation in the target. Inelastic neutron scattering is the most powerful technique for the low-energy range $[(\lesssim]$ 0.1 eV). On the other hand, inelastic X-ray scattering is well suited for the study of high momentum and large energy transfers because the energy resolution is limited to ∼1 eV and the cross section increases with momentum transfer. In the intermediate range, inelastic electron scattering [or electron energy-loss spectroscopy (EELS )] is the most useful technique. For recent reviews on different aspects of the subject, the reader may consult the texts by Schnatterly (1979), Raether (1980), Colliex (1984), Egerton (1986), and Spence (1988a).

Figure 4.3.4.1| top | pdf |

Definition of the regions in (E, q) space that can be investigated with the different primary sources of particles available at present [courtesy of Schnatterly (1979)].

4.3.4.1.2. Parameters involved in the description of a single inelastic scattering event

| top | pdf |

The importance of inelastic scattering as a function of energy and momentum transfer is governed by a double differential cross section: $[{{\rm d}^2\sigma\over {\rm d}\Omega\,{\rm d}(\Delta E)}, \eqno (4.3.4.1)]$ where d $[\Omega]$ corresponds to the solid angle of acceptance of the detector and d(ΔE) to the energy window transmitted by the spectrometer. The experimental conditions must therefore be defined before any interpretation of the data is possible. Integrations of the cross section over the relevant angular and energy-loss domains correspond to partial or total cross sections, depending on the feature measured. For instance, the total inelastic cross section $[(\sigma_i)]$ corresponds to the probability of suffering any energy loss while being scattered into all solid angles. The discrimination between elastic and inelastic signal is generally defined by the energy resolution of the spectrometer, that is the minimum energy loss that can be unambiguously distinguished from the zero-loss peak; it is therefore very dependent on the instrumentation used.

The kinematics of a single inelastic event can be described as shown in Fig. 4.3.4.2 . In contrast to the elastic case, there is no simple relation between the scattering angle $[\theta]$ and the transfer of momentum $[\hbar{\bf q}]$ . One has also to take into account the energy loss ΔE. Combining both equations of conservation of momentum and energy, $[{\hbar^2k'^2\over 2m_0}+\Delta E= {\hbar^2k^2\over 2m_0}, \eqno (4.3.4.2)]$ and $[q^2=k^2+k'^2 - 2kk'\cos\theta, \eqno (4.3.4.3)]$ one obtains $[(qa_0)^2={2E_0\over R}\left[1-\left(1-{\Delta E\over E_0}\right)^{1/2}\cos\theta\right]-{\Delta E\over R}, \eqno (4.3.4.4)]$ where the fundamental units $[a_0=\hbar^2/m_0e^2]$ = Bohr radius and $[R=m_0e^4/2\hbar^2]$ = Rydberg energy are used to introduce dimensionless quantities. In this kinematical description, one deals only with factors concerning the primary or the scattered particle, without considering specifically the information on the ejected electron. For a core-electron excitation of an atom, one distinguishes q (the momentum exchanged by the incident particle) and χ (the momentum gained by the excited electron), the difference being absorbed by the recoil of the target nucleus (Maslen & Rossouw, 1983).

Figure 4.3.4.2| top | pdf |

A primary electron of energy E₀ and wavevector k is inelastically scattered into a state of energy E₀ − ΔE and wavevector k′. The energy loss is ΔE and the momentum change is ħq. The scattering angle is θ and the scattered electron is collected within an aperture of solid angle dΩ.

4.3.4.1.3. Problems associated with multiple scattering

| top | pdf |

The strong coupling potential between the primary electron and the solid target is responsible for the occurrence of multiple inelastic events (and of mixed inelastic–elastic events) for thick specimens. To describe the interaction of a primary particle with an assembly of randomly distributed scattering centres (with a density N per unit volume), a useful concept is the mean free path defined as $[\Lambda=1/N\sigma \eqno (4.3.4.5)]$ for the cross section σ. The ratio t/Λ measures the probability of occurrence of the event associated with the cross section σ when the incident particle travels a given length t through the material. This is true in the single scattering case, that is when $[t/\Lambda\ll1]$ .

For increased thicknesses, one must take into account all multiple scattering events and this contribution begins to be non-negligible for $[t\,\gtrsim]$ a few tens of nanometres. Multiple scattering is responsible for a broadening of the angular distribution of the scattering electrons – mostly due to single or multiple elastic events – and of an important fraction of inelastic electrons suffering more than one energy loss. The probability of having n inelastic processes of mean free path Λ is governed by the Poisson distribution: $[P_n(t)={1\over n!}\left({t\over \Lambda}\right)^{-n}\exp\left(-{t\over\Lambda}\right).\eqno (4.3.4.6)]$ Multiple losses introduce additional peaks in the energy-loss spectrum; they are also responsible for an increased background that complicates the detection of single characteristic core-loss signals. Consequently, when the specimen thickness is not very small (i.e. for $[t\,\gtrsim 50\hbox{ nm}]$ for 100 keV primary electrons), it is necessary to retrieve the single scattering profile that is truly representative of the electronic and chemical properties of the specimen.

Deconvolution techniques have therefore been developed to remove the effects of plural scattering from the low-loss spectrum (up to 100 eV) or from ionization edges but very few treatments deal with both angle and energy-loss distributions. Batson & Silcox (1983) have made a detailed study of the inelastic scattering properties of incident 75 keV electrons through a ~100 nm thick polycrystalline aluminium film , over the full range of transferred wavevectors $[(0\rightarrow3\,{\rm \AA}^{-1})]$ and energy losses $[(0\rightarrow100\,{\rm eV})]$ . Schattschneider (1983) has proposed a matrix approach that is sufficiently elaborate to handle angle-resolved energy-loss data. Simpler deconvolution schemes have been proposed and used for processing multiple losses without specific consideration of angular truncation effects. They are valid when the data have been recorded over sufficiently large angles of collection so that all appreciable inelastic scattering is included. They are generally based on Fourier transform techniques , except for the iterative approach of Daniels, Festenberg, Raether & Zeppenfeld (1970), which has been used for energy losses up to about 60 eV (Colliex, Gasgnier & Trebbia, 1976). The most accurate current methods are the Fourier-log method of Johnson & Spence (1974) and Spence (1979), and the Fourier-ratio method of Swyt & Leapman (1982), which only applies to the core-loss region. The first is far more complete and accurate and applies to any spectral range when one has recorded a full spectrum in unsaturated conditions.

4.3.4.1.4. Classification of the different types of excitations contained in an electron energy-loss spectrum

| top | pdf |

Figs. 4.3.4.3 and 4.3.4.4 display examples of electron energy-loss spectra over large energy domains, typically from 1 to about 2000 eV. With one instrument, all elementary excitations from the near infrared to the X-ray domain can be investigated. Apart from the main beam or zero-loss peak, two major families of electronic transitions can be distinguished in such spectra:

(a) The excitations in the low or moderate energy-loss region $[(1\lt\Delta E\lt50\,{\rm eV})]$ concern the quasifree (valence and conduction) electron gas. In a pure metal, such as Al, the dominant feature is the intense narrow peak at 15 eV with its multiple satellites at about 30, 45, and 60 eV. One also detects an interband transition at 1.5 eV and a surface plasmon peak at $[\sim7]$ eV. For the more complex mineral specimen, rhodizite, this contribution lies in the intense and broad, but not very specific, peak around 25 eV. All these features are discussed in detail in Subsection 4.3.4.3.

Figure 4.3.4.3| top | pdf |

Excitation spectrum of aluminium from 1 to 250 eV, investigated by EELS on 300 keV primary electrons [courtesy of Schnatterly (1979)].

Figure 4.3.4.4| top | pdf |

Complete electron energy-loss spectrum of a thin rhodizite crystal (thickness ~60 nm). Separate spectra from eight significantly overlapping energy ranges were measured and matched. Primary energy 60 keV. Semi-angle of collection 5 mrad. Recording time 300 s (parallel acquisition). Scanned area 30 × 40 nm. Analysed mass 2 × 10⁻¹⁵ g [courtesy of Engel, Sauer, Zeitler, Brydson, Williams & Thomas (1988)].

(b) The excitations in the high-energy-loss domain $[(50\lt\Delta E\lt2000\,{\rm eV})]$ concern excitation and ionization processes from core atomic orbitals: in Al, the $[L_{2,3}]$ edge is associated with the creation of holes on the 2p level, [L_1] is due to the excitation of 2s, and K of 1s electrons. These contributions appear as edges superposed on a regularly decreasing background. In the complex specimen, the succession of these different edges on top of the monotonously decaying background is a signature of the elemental composition, the intensity of the signals being roughly proportional to the relative concentration in the associated element. Core-level EELS spectroscopy therefore investigates transitions from one well defined atomic orbital to a vacant state above the Fermi level: it is a probe of the energy distribution of vacant states in a solid, see Fig. 4.3.4.5 . As the excited electron is promoted on a given atomic site, the information involved has two specific characters: it provides the local atomic point of view and it reflects the existence of the hole created, which can be more or less screened by the surrounding population of electrons in the solid. The properties of this family of excitations are the subject of Subsection 4.3.4.4.

Figure 4.3.4.5| top | pdf |

Schematic energy-level representation of the origin of a core-loss excitation (from a core level C at energy E_c to an unoccupied state U above the Fermi level E_f) and its general shape in EELS , as superimposed on a continuously decreasing background.

The non-characteristic background is due to the superposition of several contributions: the high-energy tail of valence-electron scattering, the tails of core losses with lower binding energy, Bremsstrahlung energy losses, plural scattering, etc. It is therefore rather difficult to model its behaviour, although some efforts have been made along this direction using Monte Carlo simulation of multiple scattering (Jouffrey, Sevely, Zanchi & Kihn, 1985).

When one monochromatizes the natural energy width of the primary beam to much smaller values (about a few meV) than its natural width, one has access to the infrared part of the electromagnetic spectrum. An example is provided in Fig. 4.3.4.6 for a specimen of germanium in the energy-loss range 0 up to 500 meV. In this case, one can investigate phonon modes, or the bonding states of impurities on surfaces. This field has been much less extensively studied than the higher-energy-loss range [for references see Ibach & Mills (1982)].

Figure 4.3.4.6| top | pdf |

Energy-loss spectrum, in the meV region, of an evaporated germanium film (thickness $[\simeq]$ 25 nm). Primary electron energy 25 keV. Scattering angle < 10⁻⁴. One detects the contributions of the phonon excitation, of the Ge—O bonding, and of intraband transitions [courtesy of Schröder & Geiger (1972)].

Generally, EELS techniques can be applied to a large variety of specimens. However, for the following review to remain of limited size, it is restricted to electron energy-loss spectroscopy on solids and surfaces in transmission and reflection. It omits some important aspects such as electron energy-loss spectroscopy in gases with its associated information on atomic and molecular states. In this domain, a bibliography of inner-shell excitation studies of atoms and molecules by electrons, photons or theory is available from Hitchcock (1982).

References

Batson, P. E. & Silcox, J. (1983). Experimental energy loss function, $[Im[-1/\varepsilon(q,\omega)]]$ , for aluminium. Phys. Rev. B, 27, 5224–5239.Google Scholar

Colliex, C. (1984). Electron energy loss spectroscopy in the electron microscope. Advances in optical and electron microscopy, Vol. 9, edited by V. E. Cosslett & R. Barer, pp. 65–177. London: Academic Press.Google Scholar

Colliex, C., Gasgnier, M. & Trebbia, P. (1976). Analysis of the electron excitation spectra in heavy rare earch metals, hydrides and oxides. J. Phys. (Paris), 27, 397–406.Google Scholar

Daniels, J., Festenberg, C. V., Raether, H. & Zeppenfeld, K. (1970). Optical constants of solids by electron spectroscopy. Springer tracts in modern physics, Vol. 54, pp. 78–135. New York: Springer-Verlag.Google Scholar

Egerton, R. F. (1986). Electron energy loss spectroscopy in the electron microscope. New York/London: Plenum.Google Scholar

Engel, W., Sauer, H., Zeitler, E., Brydson, R., Williams, B. G. & Thomas, J. M. (1988). Electron energy loss spectroscopy and the crystal chemistry of rhodizite. J. Chem. Soc. Faraday Trans. 1, 84, 617–629.Google Scholar

Hitchcock, A. P. (1982). Bibliography of atomic and molecular inner-shell excitation studies. J. Electron Spectrosc. Relat. Phenom. 25, 245–275. [Updated copies of this bibliography are available from the author on request.]Google Scholar

Ibach, H. & Mills, D. L. (1982). Electron energy-loss spectroscopy and surface vibrations. New York: Academic Press.Google Scholar

Johnson, D. W. & Spence, J. C. H. (1974). Determination of the single scattering probability distribution from plural scattering data. J. Phys. D, 7, 771–780.Google Scholar

Jouffrey, B., Sevely, J., Zanchi, G. & Kihn, Y. (1985). Characteristic energy losses with high energy electrons up to 2.5 MeV. Scanning Electron Microsc. 3, 1063–1070.Google Scholar

Maslen, V. M. & Rossouw, C. J. (1983). The inelastic scattering matrix element and its application to electron energy loss spectroscopy. Philos. Mag. A47, 119–130.Google Scholar

Raether, H. (1980). Excitation of plasmons and interband transitions by electrons. Spring Tracts Mod. Phys. Vol. 88. Berlin: Springer.Google Scholar

Schattschneider, P. (1983). A performance test of the recovery of single energy loss profiles via matrix analysis. Ultramicroscopy, 11, 321–322.Google Scholar

Schnatterly, S. E. (1979). Inelastic electron scattering spectroscopy. Solid State Phys. 14, 275–358.Google Scholar

Schröder, B. & Geiger, J. (1972). Electron spectrometric study of amorphous germanium and silicon in the two phonon region. Phys. Rev. Lett. 28, 301–303.Google Scholar

Spence, J. C. H. (1979). Uniqueness and the inversion problem of incoherent multiple scattering. Ultramicroscopy, 4, 9–12.Google Scholar

Spence, J. C. H. (1988a). Inelastic electron scattering: Parts I and II. High resolution transmission electron microscopy and associated techniques, edited by P. R. Buseck, J. M. Cowley & L. Eyring, pp. 129–189. Oxford University Press.Google Scholar

Swyt, C. R. & Leapman, R. D. (1982). Plural scattering in electron energy loss analysis. Scanning Electron Microsc. 1, 73–82.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 4.3, pp. 391-394