2 experimental methods.

Journal of the ICRU Vol 7 No 1 (2007) Report 77 Oxford University Press

doi:10.1093/jicru/ndm009

2 EXPERIMENTAL METHODS 2.1

GASES (ATOMS AND MOLECULES)

(1) swarm techniques, where the derived quantities are extracted from observation of the collective motion of a large number of charged particles (‘electron swarms’), and (2) beam measurements, where single collisions between individual scattering partners are examined. 2.1.1

Electron-swarm experiments

In electron-swarm experiments, electrons (or positrons) undergo multiple elastic and inelastic collisions and diffusion under a (static) uniform weak electric field. The energy gained by the electrons from the electric field is balanced by small energy losses in elastic and inelastic collisions with the gas, and an equilibrium energy distribution on electrons is established. The so-called relaxation time is the time needed for a swarm of electrons to reach a steady state in the gas under a uniform electric field, in the absence of inelastic collisions. The relaxation time must be shorter than the drift time, since otherwise the steady state cannot be established. Swarm experiments can be performed for different electric fields Ef and gas pressures P, i.e., different gas densities; they are more effective at low energies (near and below the first excitation threshold) for which elastic scattering dominates. The quantities that are commonly measured in swarm experiments are the drift velocity w (which is the velocity of the centroid of the swarm), the electron mobility m, the transverse diffusion coefficient DT , and the longitudinal diffusion coefficient DL. The ratio DT/m is a measure of the average electron kinetic energy and is sensitive to the crosssections for elastic and inelastic scattering. A plot of DT/m versus the reduced electric field Ef /P provides information on the cross-sections for the dominant mechanisms. The parameters that can be derived from swarm experiments are electron mean free paths, mean kinetic energy of the electron swarm, mean energy loss per collision, mean momentum-transfer cross-section, smt, and

# International Commission on Radiation Units and Measurements 2007

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Experimental techniques for measuring interaction cross-sections of electrons with different targets (neutral, excited, or ionized atoms and molecules in gaseous, liquid, and solid phases) have been a subject of intense research and continuous development since the beginning of the 20th century (Franck and Hertz, 1914; Ramsauer, 1921). The literature on experimental methods for measuring electron-collision cross-sections covers various targets and different collision channels. A large part of the available experimental data pertains to atomic gases, particularly to a limited group of atoms such as hydrogen, the rare gases, the alkalis, the alkaline earths, and some metals. Only a small set of cross-section data is available for more than half of the atoms in the periodic table, and data for molecules are even more limited. The demand for accurate cross-sections of electron interactions with different atomic, molecular and ionic systems has been growing rapidly (Mason et al., 1999) in the applied-science community. Experimental studies of positron collisions have also been performed for many years (Kauppila and Stein, 1990; Kimura et al., 2000; Stein and Kauppila, 1982; Sullivan et al., 2002), but the available data are limited to a small number of collision processes and are fragmentary. Although many applications based on positron impact on materials have been developed and employed in recent years, very little basic knowledge and information for positron-scattering dynamics from gaseous targets has been gained. In the last decade of the 20th century, total crosssections for positron scattering on many large molecules have been measured, and positron DCSs also became available (Kimura et al., 2000). The present discussion is limited to experimental techniques for the measurement of elastic cross-sections of electrons (and/or positrons) impinging on gaseous atomic or molecular targets. Some experimental results will be presented in Section 5. The experimental techniques used to study the interaction of electrons (and other low-energy charged particles) with atoms and molecules in the gaseous phase and subsequently to derive

scattering cross-sections can be broadly classified into two groups:

ELASTIC SCATTERING OF ELECTRONS AND POSITRONS

Advances in technology and experimental technique have made possible the measurement of absolute grand total cross-sections in attenuation experiments to an accuracy of 3 % to 5 % (Brunger and Buckman, 1997). Three methods are currently employed. (1) Transmission method. Cross-sections are obtained by measuring the attenuation of the electron beam passing though the target gas; a state-of-the-art instrument is described by Brunger and Buckman (1997). Common problems with this technique are the determination of the gas pressure and thickness of the gas cell, the beam current, the effect of forward scattering of electrons into the acceptance solid angle of the detector, and the effect of electrons that undergo multiple collisions and are scattered back into the acceptance solid angle of the detector. (2) Recoil method. The attenuation of an atomic or molecular beam is measured, rather than the electron beam. It is extensively applied for atomic species, but of limited applicability for molecular systems. (3) Transmission method with time-of-flight discrimination. The time distribution of electrons arriving at the detector is determined both with the scattering cell empty and with the gas under study. It is still limited by the small electron fluxes that can be used at low beam energies (E , 0.1 eV), time-resolution capabilities at higher energies (E . 50 eV), and both elastic and inelastic scattering in the forward direction that cannot easily be discriminated in time-of-flight measurements.

2.1.2 Attenuation and crossed-beam experiments Two kinds of beam experiments can be mentioned: direct attenuation experiments yielding grand total (elastic plus inelastic) cross-sections and crossedbeam experiments yielding DCSs. The experiments are performed under single-collision conditions and are applicable to a wide range of electron kinetic energies. The crossed-beam method yields results that are conceptually clear and straightforward to interpret, but suffers from considerable technical difficulties in the preparation of the electron beam with finely resolved momentum and in the momentum analysis of scattered electrons. It is particularly difficult to use crossed-beam experiments for projectiles with very low energies. It is also difficult to determine the absolute value of cross-sections. In principle, the momentum-transfer cross-section can be derived from absolute measurements of the DCS, but most measurements of ds/dV cannot cover the entire range of scattering angles between 08 and 1808 owing to the presence of the primary beam and other geometrical constraints, and some extrapolation procedure is required to extend these measured data to 08 and 1808.

In a typical crossed-beam apparatus for DCS measurements (Figure 2.1), a beam of atoms under study (effusing from a capillary tube or a capillary array) is crossed with a beam of monoenergetic electrons ( produced by a tungsten filament, an electrostatic hemispherical monochromator, and an electrostatic electron optics). Electrons scattered through a particular scattering angle are energy analyzed and detected by a similar combination of

Figure 2.1. Schematic diagram of a crossed-beam experiment.

18


cross-sections for electron inelastic scattering and electron attachment. Measurements through the swarm method give average values over the electron energy distribution; interaction properties are inferred either from solutions of the Boltzmann transport equation or from Monte Carlo simulations. If the energy dependence of the momentum-transfer crosssection smt is known or assumed, the absolute value of smt can be determined from the measured transport coefficients using a solution of the Boltzmann transport equation and an iterative procedure to match the calculated values of the transport coefficients w and DL to the experimental data (Elford and Buckman, 2000). The methods usually adopted to solve the Boltzmann equation are based on an expansion of the electron velocity distribution in spherical harmonics. In the so-called ‘two-term’ solution, the scattering is assumed to be isotropic and only the first two terms in the spherical-harmonic expansion are retained. ‘Multi-term’ solutions, with more flexible scattering models, have also been used. The energy range over which momentum-transfer cross-sections can be derived from swarm experiments is determined by the range of values of the electric field strength for which transport coefficients are measurable, which is limited by the onset of electric discharge.

EXPERIMENTAL METHODS

In principle, the momentum-transfer crosssection can be derived from absolute crossed-beam measurements of DCSs, but one of the most significant problems until recently has been that most measurements of ds/dV cannot cover the entire range of scattering angles between 08 and 1808 owing to the presence of the primary beam and other geometrical constraints, and some extrapolation procedure is required to extend these measurements to 08 and 1808. Low-energy positron beams for experimental studies of cross-sections were developed in 1972. Since then, the low-energy positrons have been obtained from either radioactive (22Na) or accelerator-based sources (e.g., high-energy proton beams). Positrons are then passed through, or reflected from, a moderating material (boron, gold, or MgO). Tungsten moderators are convenient to use and effective for positron-gas and positron-solid experiments. Moderated beams of positrons are spatially spread and have a wide energy distribution, with a range of energies typically higher than about 0.5 eV. The intensities of the beams are quite low compared to typical electron sources. It is thus difficult to measure positron DCSs with the conventional crossed-beam techniques for electron scattering. The Detroit group (Dou et al., 1992a, 1992b; Hyder et al., 1986) was the first to measure DCSs for elastic scattering of positrons on rare gases with a crossed-beam geometry. For these measurements it is of utmost importance to use a positron beam of high intensity (brightness), of high quality (i.e., small component of the momentum perpendicular to the flight path), and with a narrow energy spread. Recently, buffer-gas traps have been developed to provide a thermal source of positrons that produces high-resolution energy-tunable positron beams with energies from 0.05 eV to more than 50 eV (Gilbert et al., 1999; Sullivan et al., 2002).

2.1.3

Comparison of the two techniques

The swarm and crossed-beam techniques, although often viewed as competitive, are in fact complementary. In principle, the beam method is applicable to a wide range of kinetic energies and the measurements are conceptually clearer and more straightforward to interpret. However, beam experiments suffer from practical difficulties in the preparation of an electron beam with finely resolved momentum (which make measurements difficult at low energies, of the order of a few electron volts) and in the momentum analysis of scattered electrons. Intrinsic limitations of beam experiments are the limited angular range accessible to measurement 19


electron-optical elements, a hemispherical analyzer, and a channel electron multiplier. The electron analyzer can rotate about the atomic-beam axis to access as wide a range of scattering angles as possible. The range of scattering angles is generally between about 158 and 1358. The absolute electron energy scale is usually calibrated by observing the position of a well-defined line (e.g., the 2Pg reson2 ance peak of N2 1s2s2 2S 2 at 2.198 eV or the He resonance at 19.367 eV). Recently, Read and coworkers (Cho et al., 2000; Read and Channing, 1996; Zubek et al., 1996, 1999) have developed a magnetic device for changing the scattering angle that allows direct measurements to be performed for scattering angles between 1308 and 1808. The details of experimental set-ups from different laboratories can vary appreciably; descriptions of various instruments can be found in the references (Cho et al., 2000; Gibson et al., 1996, 1998). These papers also address important issues for ensuring proper performance of the instruments. In reality, crossed-beam experiments provide relative DCS data. The conversion of measured data into absolute cross-section values is a delicate process. The currently accepted practice is to use a set of accurately known elastic DCSs as ‘standards’ to remove the effect of the instrumental response function and other factors characteristic of the instrument. This is normally performed by using the relative-flow technique, which relates the DCS for a particular target gas (at specific energy and scattering angle) to that of a reference or standard gas under identical experimental conditions. For low-energy electron scattering, the ‘standard’ DCS is that for electron-helium scattering, which is considered to be known to within a few percent for energies below the first inelastic threshold at 19.8 eV and to within 7 % for electron kinetic energies up to 50 eV. Besides helium, which is commonly used as primary standard, neon is sometimes used as a secondary standard in relative-flow experiments (Gibson et al., 1996); recent experimental and theoretical studies for this element resulted in electron elastic DCSs that agreed to within 5 % over a wide range of incident energies from below 1 eV to 3000 eV, and scattering angles from 58 to 1308 (Gibson et al., 1996). Many laboratories worldwide use the relative-flow technique for DCS measurements. Where results of various groups overlap in energy and angle, the level of agreement is quite good. When DCS values lie within the combined experimental errors, the differences are typically between 5 % and 10 %. Somewhat greater differences (up to 25 %) are observed around pronounced structures of the DCS.


2.2

crystal structure. As the degree of crystalline order decreases (e.g., for an amorphous solid, a liquid, or a polycrystalline solid consisting of many randomly oriented grains or assemblies of molecules), the diffraction effects become very weak and can be neglected for many practical applications. While the neglect of diffraction effects is a useful simplifying approximation, two further issues need to be addressed. First, the interaction potential for an electron moving in condensed matter is very different from the potential calculated for isolated atoms of the constituent elements. Nevertheless, the use of atomic potentials for computing certain characteristics of the interactions of 200 eV to 2000 eV electrons with condensed matter gives results that are in satisfactory agreement with a wide class of experimental measurements. Examples of the latter are the angular distributions of electrons elastically backscattered from surfaces (Jablonski, 1991; Werner et al., 1994), the energy distributions of the backscattered intensity for different solid angles (Jablonski, 1991; Jablonski et al., 1993; Jablonski and Jiricek, 1996; Jablonski and Zemek, 1996), the conditions for maximum probability of elastic backscattering (Jablonski and Zemek, 1996; Zommer et al., 1993), the angular distributions for photoemission (Hucek et al., 1997a; Jablonski and Zemek, 1993, 1997), and the probabilities for emission of photoelectrons and Auger electrons in a particular direction from a surface as a function of depth of electron creation from the surface (Hucek et al., 1997b; Tilinin et al., 1997; Zemek et al., 1998). The observed agreement between calculations and measurements is associated with electron-transport characteristics for which small-angle elastic-electron scattering is not important (Jablonski and Powell, 2000a). It is, of course, the region of small scattering angles for which differences between the scattering potentials for atoms in condensed matter and the corresponding isolated atoms are expected to be most important. We also point out that Cumpson and Seah (1997) evaluated the influence of two potentials (the atomic relativistic Hartree–Fock–Slater potential and a ‘muffin-tin’ potential) on calculated effective attenuation lengths for 18 solid elements. There were minor differences in the results for these two potentials for electron energies between 200 eV and 2000 eV; again, small-angle elastic scattering is not significant for this application (Jablonski and Powell, 2000a). The second issue concerns the role of multiple scattering. For isolated atoms in the gas phase, the pressure can generally be reduced to sufficiently low values that multiple elastic and multiple inelastic scattering are negligible. In condensed matter, however, it is often necessary to consider

CONDENSED PHASES

Electron elastic-scattering phenomena in solids and liquids can be very different from the corresponding phenomena for the constituent atoms or molecules in the gas phase at pressures where multiple elastic or inelastic scattering can be ignored. For crystalline solids, coherent scattering by the ordered atoms leads to electron diffraction, a powerful tool for obtaining information on the 20


and the need for measurements with standard gases to derive absolute DCS data. The swarm method is experimentally easier, at least in principle, but requires a careful numerical analysis, using either a Boltzmann transport equation or a Monte Carlo simulation, to derive absolute values of the desired cross-sections from the measured transport parameters. The swarm measurements are particularly reliable at low energies (,1 eV) and can be performed for various electric fields and gas pressures. The main limitations arise from the difficulty of computing the energy distribution of the swarm electrons and from the fact that this distribution is broad. As indicated above, swarm experiments yield absolute values of the mean momentum-transfer cross-section smt, while beam experiments with the direct-attenuation method give the grand total cross section (attenuation) and, when inelastic scattering is sufficiently well known, the total elastic cross section s; crossed-beam experiments provide the elastic DCS ds/dV and, with the aid of suitable extrapolation, the total elastic cross section. Comparison of total elastic cross sections from attenuation and crossed-beam experiments with momentum-transfer cross sections from swarm experiments is not straightforward, because smt and s are sensitive to different angular regions. Buckman and Brunger (1997) describe various techniques used for comparing these two quantities for electron scattering by atoms. For molecules, the analysis of the consistency of different experiments is more difficult, because partial-wave expansion methods are not generally applicable and the level of sophistication and accuracy of calculations for electron scattering by molecules is much less than that for electron-atom scattering. More importantly, the opening of inelastic (rotational, vibrational, and electronic excitation) channels at low incident energies makes the analysis of swarm data difficult. Also, rotational excitation cannot easily be resolved in crossed-beam experiments, and most results for vibrational excitation involve a sum over rotationally excited states.


2.2.1

energy typically about four times the threshold energy. This expectation is confirmed by measurements of the attenuation of selected electron signals (e.g., Auger-electron signals or x-ray photoelectron signals) from a substrate as a thin-film overlayer of increasing thickness is deposited. In early work of this type, it was assumed that elastic scattering of the signal electrons was negligibly small and that the observed attenuation was due entirely to inelastic scattering of the signal electrons. The derived attenuation parameter was then identified as an IMFP (the inverse of the product of the total inelastic scattering cross section and the density of atoms or molecules). It is now well established (Jablonski and Powell, 1999, 2002a; Nefedov, 1999; Powell and Jablonski, 2002; Werner, 2001) that the early assumption of negligible elastic scattering in these experiments was incorrect. As a result, the derived attenuation parameter is now termed an effective attenuation length (Jablonski and Powell, 2002a; Powell and Jablonski, 2002). For common conditions of measurement in surface analysis by Auger-electron spectroscopy and x-ray photoelectron spectroscopy (XPS), the effective attenuation length is relatively smaller than the corresponding IMFP by up to about 35 % (Powell and Jablonski, 2002). Figure 2.2 shows the dependence of the effective attenuation length on electron energy from the work of Seah and Dench (1979). Although the experimental data presented here have some unknown sources of systematic error [owing to unknown deviations of the film morphology from the ideal slab assumed in the data analysis (Powell and Jablonski, 1999)], the plots can be used as a semi-quantitative guide. Figure 2.2 displays the

Electron-energy regions of interest

The most probable energy loss of electrons traveling in condensed phases with energies greater than about 100 eV, is typically between 5 eV and 30 eV; these energy losses are due to valence-electron excitations (Powell, 1994). The cross-sections for such excitations are zero below some threshold energy corresponding to each particular transition and are then expected to rise to a maximum at an

Figure 2.2. Plots of lma 21/2 versus electron energy for elements and inorganic compounds, where lm is the effective attenuation length in monolayers [cf. Eq. (2.1)] and a is the average interatomic spacing [cf. Eq. (2.2)] (from Seah and Dench, 1979). The points were obtained from experimental measurements and the solid lines are empirical fits given by Eqs. (2.3) and (2.4).

21


multiple elastic scattering and, depending on the phenomenon of interest, single or multiple inelastic scattering (Schilling and Webb, 1970). We will therefore provide information on mechanisms and cross-sections for inelastic scattering of electrons. For convenience, in surface-analytical applications, the mean free path for inelastic scattering, usually referred to as the inelastic mean free path (IMFP), is often preferred over inelastic-scattering crosssections. As will be discussed in further detail in Section 2.2.3, Monte Carlo simulations provide a useful means of investigating multiple scattering of electrons in condensed matter (e.g., Jablonski, 1991). Simulations of this type are generally performed with differential elastic-scattering crosssections obtained from calculations for the relevant free atoms; in a solid, for example, the atoms are assumed to be arranged randomly and with a density corresponding to the bulk density. Because it can be time-consuming to obtain the desired statistical precision in Monte Carlo simulations, faster but more approximate analytical procedures have been developed for investigations of multiple scattering in condensed matter (e.g., Tilinin et al., 1997).


energy dependence of lma 21/2, where lm is an attenuation parameter in monolayers defined as

lm ¼

ln ; a

energy. These two energy regions will be considered separately in the following two Sections.

ð2:1Þ

2.2.2

a ¼ 10 ðnN Þ

1=3

nNA rm 1=3 ¼ 10 : Am 7

ð2:2Þ

Here N is the number of atoms or molecules per unit volume [in cm23, Eq. (1.7)] and n is the number of atoms in the molecule (if a compound); for a pure element n ¼ 1. Calculated values of a for elemental solids range from 0.20 nm to 0.49 nm. The plot of lma 21/2 versus electron energy, E in eV, in Fig. 2.2 was chosen by Seah and Dench because they found the following empirical relations:

lm ¼ 538E2 þ 0:41ðaEÞ1=2 monolayers;

ð2:3Þ

lm ¼ 2170E2 þ 0:72ðaEÞ1=2 monolayers;

ð2:4Þ

for elements and inorganic compounds, respectively. The solid lines in Fig. 2.2 are plots of Eqs. (2.3) and (2.4) where an average value of a was used to determine a 1/2 as a multiplying factor on the E 22 terms of Eqs. (2.3) and (2.4); these average values were a ¼ 0.27 nm and a ¼ 0.30 nm for elements and inorganic compounds, respectively. Although there is a physical basis for the E 22 term in Eqs. (2.3) and (2.4), there is no such basis for the other term or for the different values of the constants for the two types of materials. Later calculations of IMFPs as functions of electron energy (Tanuma et al., 1990) showed that the minimum IMFP value occurred at different energies in different materials (e.g., at 40 eV for Al and at 120 eV for Au). Although this result differs from the almost material-independent energy (approximately 40 eV) for the minimum effective attenuation length from Eqs. (2.3) and (2.4), it is easily explainable in terms of the different inelastic-scattering mechanisms in different materials (Tanuma et al., 1993). Nevertheless, Eqs. (2.3) and (2.4) are useful as a rough guide to values of lm and thus to the effective attenuation length, ln, from Eqs. (2.1) and (2.2). Figure 2.2 and Eqs. (2.3) and (2.4) show two energy regions. For electron energies less than about 30 eV, the effective attenuation length decreases rapidly with increasing energy, while for energies higher than about 100 eV the effective attenuation length increases slowly with increasing 22


The scattering of low-energy (, 30 eV) electrons within and near surfaces of dielectric atomic and molecular solids can be investigated by allowing monoenergetic electrons to impinge on a thin multilayer film grown in an ultra-high vacuum system by the condensation of gases or organic vapors onto a clean metal substrate held at cryogenic temperatures (15–100 K). Depending on the type of experiment, it is possible to measure the dependence on primaryelectron energy of the current transmitted through (Sanche, 1979; Sanche et al., 1982a), trapped in (Marsolais et al., 1989), or reflected from (Firment and Somorjai, 1976; Pireaux et al., 1992; Sanche and Michaud, 1984) the film. Similar measurements can be done for the positive-ion (Tolk et al., 1983), negative-ion (Sanche, 1988), and neutral-species fluxes (Alvey et al., 1986; Leclerc et al., 1990) leaving the surface. As a general rule, the film thickness must be larger than the total electron mean free path if we want to minimize effects of the metal substrate. Small amounts of molecules can be added to the film surface or mixed within the dielectric film to study the effects of dopants. Electron interactions at the interface between two dielectric materials can be studied by covering a given film with an overlayer of another substance. Elastic and quasi-elastic scattering of low-energy (up to 30 eV) electrons by solid films of rare gases or molecules has been investigated by low-energy electron transmission (LEET) spectroscopy (Bader et al., 1982, 1984; Chang and Berry, 1974; Cheng and Funabashi, 1973; Jay-Gerin et al., 1985; Grechov, 1983; Goulet and Jay-Gerin, 1986; Goulet et al., 1986, 1987, 1988; Harrigan and Lee, 1974; Hino et al., 1977; Hiraoka and Nara, 1983, 1984; Huang and Magee, 1974; Keszei et al., 1986, 1988; Michaud et al., 1988; Perluzzo et al., 1982, 1984, 1985; Plenkiewicz et al., 1985, 1986a, 1986b, 1988; Sanche, 1979; Sanche et al., 1982a, 1982b; Ueno et al., 1986), photoinjection (Bernasconi et al., 1988; Kadyshevitch et al., 1997; Marsolais et al., 1991; Pfluger et al., 1984), and elastic-reflection (Michaud and Sanche, 1987a, 1987b; Michaud et al., 1988, 1991a, 1991b) experiments. Both LEET and photoinjection experiments measure the electron current transmitted through a multilayer film deposited on a metal substrate. In the photoinjection experiment, electrons are injected into the film from the metal substrate with ill-defined energies and momenta, but the outgoing electrons which escape into the vacuum at a given energy and momentum can be

where ln is the effective attenuation length (in nm), a is the average interatomic spacing (in nm) given by 7

Electron energies lower than 30 eV


2.2.2.1 Low-energy electron transmission spectroscopy A drawing of the type of apparatus used to record LEET spectra (Sanche, 1979; Sanche et al., 1982a) and electron-stimulated-desorption yields (Azria et al., 1987; Sanche, 1984) is shown in Figure 2.3. It consists of a high-current (1029 – 1027 A) electron gun (D), a quadrupole mass spectrometer (E), a gas-introduction doser (C), a cooled target (B), and a high-resolution trochoidal electron monochromator (A) (Stamatovic and Schulz, 1970). Depending on the instrument, the target (B) can be rotated and may be cooled down to 15 K by the cryostat (F). All components shown in Figure 2.3 are housed in an ultra-high-vacuum system reaching pressures 10210 mbar. The magnetically collimated electron beam leaving the monochromator (A) with an energy spread of about 40 meV full

Figure 2.3. Diagram of typical LEET spectrometer with mass spectrometer (from Leclerc et al., 1987c). (A) trochoidal electron monochromator, (B) rotatable target, (C) gas doser, (D) high current electron gun, (E) quadrupole mass spectrometer, (F) cryostat, and (G) electrical leads.

23


width at half-maximum impinges on the film condensed onto a metal substrate (B) (i.e., the electron collector). The latter is electrically isolated from the cryostat by a sapphire disk and connected to electrical leads (G). Low-energy electron transmission spectroscopy spectra are obtained by measuring the current It (E) arriving at the metal substrate as a function of incident electron energy. In these experiments, It (E) is of the order of 1 nA and the absolute electron energy scale is calibrated to within + 0.15 eV by measuring the onset of electron transmission through the films. The metal substrate is usually a polycrystalline metal sheet which can be cleaned by resistive heating via G. The condensed films are grown using a gas-volume expansion dosing procedure (Sanche, 1979), which can be calibrated by monitoring the quantum-size effect features observed for ultra-thin films (Perluzzo et al., 1984, 1985). With this calibration, film thicknesses of 1–50 nm can usually be estimated with an accuracy better than 30 % assuming a layer-by-layer growth. Because in LEET spectroscopy, one measures the total current passing through a thin-film sample, the technique can provide absolute values of electron mean free paths. Unfortunately, the interpretation is complicated because electrons suffer multiple collisions prior to their collection at the metal substrate. While in HREEL spectroscopy some information concerning the number and nature of these collisions can be inferred by measurement of the final energy of reflected electrons, no such measurement is possible in LEET experiments. Thus, considerable effort has been extended to untangle the effects of multiple scattering from LEET spectra and, to this end, several theoretical approaches have been developed. The earliest attempts (Bader et al., 1982) employed a one-dimensional or ‘two-stream’ model similar to that described in Section 2.2.2.3. This model was used to obtain an expression for the transmitted current as a function of film thickness, elastic mean free path, and the reflection coefficients of the vacuum –film and film –metal interfaces. The elastic mean free path was obtained by fitting the behavior of the transmitted current as a function of film thickness, for numerous energies in the 0–10 eV range. Subsequently, a threedimensional electron-transport model was developed that combined a semi-analytical simulation of the transport of an excess electron in the conduction band of a film with a random sampling of the temporal succession of the various elastic- and inelastic-scattering events (Goulet and Jay-Gerin, 1985, 1986; Leclerc et al., 1987a, 1987b). The injected electron was assumed to scatter

selected with an electron analyzer. Such a measurement (Marsolais et al., 1991) is more sensitive to electron interactions near the substrate – film interface. We describe in the rest of Section 2.2.2 two types of experiments that can provide elastic and quasi-elastic (i.e., including energy losses to intra- and inter-molecular vibrational excitation) electron-scattering cross-sections, namely, LEET experiments and high-resolution electron reflectivity measurements performed with high-resolution electron-energy-loss (HREEL) spectrometers.


2.2.2.2

the cryostat. In the apparatus shown in Figure 2.4, the angle of incidence u0 may be varied from 148 to 708 from the film normal. Electrons reflected at ur ¼ 458 from the film are energy analyzed by another hemispherical deflector. Low-energy electron transmission sepctroscopy spectra can also be recorded with this type of apparatus, which allows the absolute electron energy scale to be calibrated to within 0.15 eV. Energy-loss and quasi-elastic spectra are recorded by sweeping the energy of either the monochromator or the analyzer. The energy dependence of the magnitude of a given energy-loss event (i.e., the excitation function) is obtained by sweeping the energy of both deflectors with a potential difference between them, corresponding to the probed energy loss. This measurement requires that both the monochromator and analyzer lenses maintain a constant focus on the target as a function of electron energy. Double zoom-lens systems can achieve such conditions. High-resolution electron-energy-loss spectra are usually recorded with overall resolutions ranging from 6 to 20 meV full width at half-maximum and corresponding incident currents are in the 10210 to 1029 A range. In elastic-reflectivity experiments, the spectrometer shown in Figure 2.4 is adjusted to measure electrons elastically scattered from the film at particular incident and scattered angles u0 and ur. Then, the amplitude of the elastic peak is measured as a function of incident electron energy.

High-resolution reflectivity measurements

Elastic and inelastic scattering of electrons near the surfaces of thin films and their energy dependences can be studied with a HREEL spectrometer (Pireaux et al., 1992; Sanche and Michaud, 1984) of the type shown schematically in Figure 2.4. This apparatus consists essentially of two concentric hemispherical deflectors with appropriate electron optics and a closed-cycle refrigerated cryostat. Electrons leaving the monochromator (shown on top) are focused onto the film condensed on a metal substrate secured by a press fit to the cold end of

2.2.2.3 Method for analysis of high-resolution reflectivity measurements What is needed to quantify elastic and quasi-elastic scattering from electron energy-loss spectra of multilayer films is an expression that can provide, for incident electrons at a fixed energy, the energy distribution of electrons scattered out of the film in terms of mean free paths and scattering crosssections, or probabilities. With such an expression, one hopes to obtain an estimate of these mean free paths and a suitable description of the scattering mechanisms responsible for the spectral features. An analytical solution of the general problem of electron scattering by multiple targets held between two infinite interfaces is not available. Nevertheless, this type of multiple-scattering problem has been approached by several investigators (e.g., Bader et al., 1982; Chantry et al., 1966; Tougard and Sigmund, 1982), mainly in connection with electron transport in mobility, photoelectron, and Auger-electron experiments. In order to simplify the analysis, most investigators have formulated approximate solutions valid for the specific

Figure 2.4. Schematic diagram of hemispherical electron energy loss spectrometer (adapted from Bass and Sanche, 2004). The monoenergetic electron beam produced by the monochromator strikes the target cooled by a cryostat. Electrons scattered at 458 from the plane of the target are dispersed in energy by the analyzer and counted by a channeltron.

24


isotropically from imperfections within the film and to transfer its energy via elastic and inelastic collisions. Contemporaneously with the above-described work, another more quantum-mechanical approach was developed (Plenkiewicz et al., 1985). The transmission of electrons through a dielectric film was considered to be a two-step process. Electrons were envisaged as being initially scattered at the film– vacuum interface and penetrating the film-surface potential barrier to an extent proportional to the three-dimensional electronic density of states of the conduction band. Subsequently, electrons propagated in all possible directions in a manner characterized by a single mean free path. An analytical formula was obtained for the transmitted current as a function of film thickness. By fitting the parameters of this formula to the experimental data, both the density of states of the conduction band and the mean free path could be determined in the quasi-elastic scattering region.


series (Michaud and Sanche 1984, 1987a, 1987b),

problem they were considering. A formulation of this problem can be found in the work of Tougard and Sigmund (1982) on Auger and photoelectron emission from solids. These authors included the possibility of large-angle elastic scattering, but inelastically scattered particles were still considered to move in the direction of the primary beam. Because these theories are not applicable to measurements at low energies where only backscattered electrons are detected, multiplescattering theories were developed to take into account large-angle scattering (Michaud and Sanche, 1984, 1987a, 1987b). It is possible to employ a one-dimensional or ‘two-stream’ approximation (Bader et al., 1982) to account for multiple scattering and to relate the cross-sections per scatterer within a disordered film to the energy distribution of backscattered electrons. In essence, the two-stream model assumes that electron scattering within the film is composed of two components: a forward (anisotropic) component and an isotropic component. In the one-dimensional model of Michaud and Sanche (1984, 1987a, 1987b), isotropic scattering corresponds to an equal probability of electron scattering in both the forward and backward directions (or ‘streams’). Physically, forward scattering can be associated with the long-range interaction of an electron with the film, while isotropic scattering derives from short-range interactions of an electron with a molecular target, e.g., the formation of a short-lived anionic state or resonance (Schulz, 1973a, 1973b). The total scattering probability per unit length, a, for an electron of energy Ei is defined in terms of a collision probability per unit length and unit energy range Qc(Ei, E) ¼ 2Qcr(Ei, E) þ Qcf(Ei, E) such that ð1

½2Qcr ðEi ; EÞ þ Qcf ðEi ; EÞ dE:

where Qc(E), Qc(E2T), and Qc(E2T2T0 ) are the scattering probabilities per unit length for collisions in the forward or backward direction with a loss of energy equal to E, E2T, and E2T2T0, respectively. This solution of the two-stream approximation is obtained under the following assumptions: (1) the film is a semi-infinite solid (i.e., its thickness is infinite), (2) all electrons move at an average angle with respect to a line normal to the film, (3) the scattering probability per unit length for any given energy loss is constant (i.e., an average electron-energy dependence is assumed in the range where the transition is energetically possible; outside this range, the inelastic cross-section is zero), and (4) the elastic-scattering probability is constant (i.e., unchanged after an inelastic event). The first assumption is justified experimentally for films having thicknesses greater than about 2 nm. The second one also is justified if the number of collisions, including quasi-elastic events, is large enough or if the film constituents are sufficiently disordered to produce a nearly isotropic distribution. In the two-stream approximation, such a distribution is approximated by an average direction of the electrons (Bader et al., 1982). The third and fourth assumptions are not expected to be too stringent as long as the energy of inelastically scattered electrons lies within a range where the mean free paths do not change too rapidly. This situation is usually the case for small energy losses (e.g., quasi-elastic scattering). Series (2.6) can easily be evaluated for a large number of terms and fitted to the experimental data. In the analysis of the results presented in Section 5.2.2, the series was limited to one hundred terms (i.e., a single electron could collide up to 100 times at different sites within the film). The inverse total mean free path or scattering probability per unit length a, defined by Eq. (2.5), can be derived from the thickness dependence of the

ð2:5Þ

0

The variable Qcf is the scattering probability per unit length per unit energy range, for an electron of energy Ei to lose energy E and be deflected in the forward direction. The variable Qcr is the corresponding quantity for isotropic scattering, while the factor 2 indicates that this component contributes to both the forward and backward streams. a is equal to the inverse total mean free path (i.e., l21 T ). For an electron beam of current intensity I0 incident on a vacuum-solid boundary, the backscattered current distribution can be described by the 25


a¼

ð I0 Qc ðEÞ I0 1 Qc ðEÞ Qc ðE TÞ IðEÞ ¼ þ dT 2 a 2 a a ð1 ð1 0 5I0 Qc ðEÞ þ 8 0 0 a Qc ðE TÞ Qc ðE T T 0 Þ 0 dT dT þ ; a a ð2:6Þ


2.2.3 Electron energies between 100 eV and 100 keV For electron energies between 100 eV and 100 keV, the combined effects of elastic and inelastic scattering need to be considered in the analysis of particular types of experimental measurements. It is generally necessary to model the experiment in terms describing the elastic and inelastic scattering and then to compare a measured quantity with a value determined from a calculation. Satisfactory agreement between the measured and calculated quantities then gives confidence (at some level) in the model, the parameter values describing elastic and inelastic scattering, the calculation of the measured quantity, and the experimental measurement. Liljequist (1977, 1978) and Liljequist and Ismail (1987) pointed out that the main features of electron transport can be understood in terms of the transport mean free path ltr (defined in Section 1.2.1) and the electron range R. The inverse transport mean free path, l21 tr , describes the average angular deflection per unit path length and, therefore, it is a convenient measure of the strength of elastic scattering. Liljequist introduced the parameter

at ¼

R ltr

2.2.3.1 Measurement of electron IMFP by elastic-peak electron spectroscopy Schilling and Webb (1970) measured the elastic scattering of 100–500 eV electrons from liquid mercury and found that differences between their measurements and similar data for mercury vapor were due to multiple-elastic scattering and to inelastic processes. Gergely (1981) observed that the elastically backscattered intensity from several solid elements was proportional to the electron IMFP. This approach, known as elastic-peak electron spectroscopy (EPES), is now used together with a model to describe the effects of multiple-elastic scattering to determine IMFPs of electrons near solid surfaces (Powell and Jablonski, 1999). Instruments designed for Auger-electron spectroscopy or low-energy electron diffraction (LEED) can be used to obtain IMFPs by EPES. Briefly, electrons of a desired energy from the electron gun are incident on the sample of interest and a measurement is made of the current of elastically scattered electrons, IR, with the electron energy analyzer. If it were assumed (incorrectly) that there was only one large-angle elastic-scattering event sufficient for backscattering into the acceptance solid angle of the analyzer, the elastic backscattering coefficient, Re, would be given by

ð2:7Þ

and showed that it determines the ‘shape’ of electron trajectories, apart from an overall scale factor that is determined by R. When at is much less than unity, the electron transport is rectilinear, while when at is much greater than unity, the transport is diffusion-like. From the fact that the backscattering coefficient is approximately constant for electrons with energies between about 10 keV and 50 keV (see Fig. 5.25), Liljequist concluded that at these energies at should be nearly a constant, independent of the electron energy. The proportionality between ltr and R holds approximately only for nonrelativistic electrons. For positrons, the relationship between these two parameters is more involved (see Liljequist, 1998). In a similar analysis, Werner (2001) pointed out that two competing processes, momentum relaxation and absorption, play a role in energy dissipation for electron transport in condensed phases. To quantify them, Werner considered the two parameters at and

li x¼ ; ltr

Re ¼

ð2:8Þ

IR ¼ li N seff ; I0

ð2:9Þ

where I0 is the incident electron current, li is the 26


where li is the IMFP. For applications in which only a few inelastic-scattering events occur (such as the transport of signal electrons in Auger-electron spectroscopy and XPS) the electron transport is quasi-elastic; that is, the electron energy is not changing significantly and thus the elastic and inelastic parameters are also not changing appreciably. On the other hand, if many inelastic-scattering events occur for the transport of interest, the electron energy will therefore change along a given trajectory (the slowing down regime), and the parameter at indicates the extent to which R exceeds ltr. For electron energies between 100 eV and 10 keV, Werner (2001) has shown that x is typically between 0.1 and 1 for many elemental solids (although some smaller and larger values occur), while at is typically between 1 and 10 (although larger values can occur). We now consider different types of experiments that give information on elastic- as well as inelastic-scattering parameters for condensed phases; in almost all experiments performed to date, measurements have been made with solids.

elastic reflectivity of a molecular film as described by Michaud and Sanche (1987a).


Re ¼ li N seff þ F l2i ;

ð2:10Þ

where F is a fitting parameter. Satisfactory agreement was then obtained with calculated IMFPs (Powell and Jablonski, 1999). The EPES experiments can be carried out in two ways. First, as indicated by Eq. (2.9), separate measurements are made of IR and I0, and a calculation is made of Re from Monte Carlo simulations with the IMFP considered as a parameter. Comparison of measured and calculated values of Re allows determination of the IMFP. While measurements of IR and I0 are possible, it may be difficult or inconvenient to make them with the desired accuracy. Second, relative measurements are made of the elastically reflected intensities, IR1 and IR2, for two materials under the same experimental conditions, and a calculation is made of the ratios of Re for the two materials. If the IMFP is known for one material (considered a reference material), the IMFP for the sample material can be found (again with the IMFP for the sample material being considered as a parameter in the Monte Carlo simulations). It is important in the EPES experiments for diffraction effects to be negligible in the sample material (and the reference material, if used). For elemental materials, this requirement can be easily satisfied by ion bombardment. It is convenient in many such cases to clean the sample surface by bombardment with rare-gas ions (often Ar þ ). Some argon will be implanted into the sample, but usually at an insignificant concentration. It is well known that ion bombardment disorders initially crystalline solids but the net effect of any remaining sample crystallinity in EPES experiments depends on the relative magnitudes of the depth of the ion-induced disordered region [which depends on the sample material, the ion species, and the ion energy (Dake et al., 1998)] and the information or sampling depth for the EPES experiments [which depends on the material, the electron energy, and

2.2.3.2 Analysis of elastic-scattering data from single-crystal surfaces and adsorbates Low-energy electron diffraction is a widely used technique for determining the atomic structures of single-crystal surfaces and of adsorbates on such surfaces (Clarke, 1985; Duke, 1994; Heinz, 1995; Pendry, 1994; Van Hove et al., 1986; Van Hove, 1999 ). Coherent elastic scattering of 5–500 eV electrons is considered for an assumed array of atoms, often using the muffin-tin approximation for describing the potential between neighboring ion cores and variants of the Hartree –Fock method for describing the atomic potential (Clarke, 1985). Calculations are made of the intensities of various diffracted beams as a function of electron energy, and these are compared with experimental measurements. Parameters in the calculation are then varied (e.g., the spacing of atoms normal to the surface for the outermost atomic layers) in order to optimize the agreement with the experimental results. It is essential in the LEED calculations to consider multiple scattering. The LEED experiments do not provide a direct means for evaluating differential elastic-scattering cross-sections calculated for free atoms. Nevertheless, LEED data for three low-index surfaces of copper have been analyzed to assess the values of the imaginary parts of the potentials that were used to describe electron inelastic scattering in the LEED calculations (Rundgren, 1999). Values 27


the experimental configuration (Jablonski and Powell, 2004)]; Monte Carlo simulations can yield values for the EPES information depth (Gergely et al., 1999; Jablonski and Powell, 2004). For inorganic and organic compounds, any ion bombardment for sample cleaning or for inducing atomic disorder must be performed with care because the ion bombardment will generally alter the surface composition in the bombarded region (Dake et al., 1998). Other techniques may have to be employed for surface cleaning and other experiments may need to be performed to demonstrate that diffraction effects are negligible. Further details on the requirements for these experiments and on sources of uncertainty have been published by Powell and Jablonski (1999). Most EPES experiments conducted to date have been performed for electron energies between 50 eV and 5000 eV (Powell and Jablonski, 1999, 2000). Dubus et al. (2000) evaluated theoretical models, such as semi-analytical models and solutions of the Boltzmann transport equation, that have been used to describe elastic backscattering of electrons from surfaces.

IMFP in the sample at the particular electron energy, N is the number of atoms or molecules per unit volume, and seff is the effective total elastic-scattering cross section (here for singleelastic scattering) into the analyzer. Although this model is over-simplified, it gave reasonable IMFP values for numerous materials (Jablonski et al., 1984). In later work, Jablonski (1985) showed that more accurate values of seff could be obtained from Monte Carlo simulations in which proper account was taken of multiple-elastic scattering. Equation (2.9) could then be empirically replaced by


of IMFPs could be derived from the imaginary potentials, and these could be compared with values from IMFP calculations. In addition, effective attenuation lengths could be obtained as a function of electron energy from the attenuation of the diffracted beams in the solid, and these could be compared with IMFP data and with the results of calculated effective attenuation lengths (Rundgren, 1999). These comparisons provide independent evaluations of IMFP data (Powell and Jablonski, 1999) and of electron-attenuation effects in crystalline solids; most studies of electron attenuation have been based on consideration of electron transport in amorphous solids (Jablonski and Powell, 2002a). In addition, Rundgren (1999) has shown that the inner potential of the solid (the muffin-tin zero) depends on electron energy. This energy-dependence must be considered to avoid a systematic error in determination of the lateral (in-plane) lattice parameter (Walter et al., 2000). LEED theory and related experiments have been extended in several important ways. First, improved experimental and theoretical techniques have been developed to measure and analyze the distributions of diffuse intensity in LEED experiments to determine the local structure of disordered adsorbates, the substitution of surface atoms by different chemical species, and surface vibrations (Heinz, 1995). Second, experiments similar to LEED have been performed with positron beams, i.e., low-energy positron diffraction (LEPD) (Coleman, 2002; Duke, 1995; Duke and Lessor, 1990). As noted by Coleman, LEED and LEPD differ in that the phase shifts for positron scattering are less sensitive to the atomic number than those for electron scattering, the IMFP is smaller for positrons than for electrons, uncertainties in the positron-electron correlation term for LEPD calculations are less important than the corresponding electron – electron correlation term for LEED, relativistic effects for positron scattering by atoms of high atomic numbers are also smaller than for electron scattering, and the muffin-tin model appears to work better for LEPD than for LEED for covalently bonded semiconductors. Finally, multiple-scattering methods have been developed to analyze the angular distributions of photoelectrons and Auger electrons emitted from clusters of atoms (Fadley, 1993; Kaduwela et al., 1991; Smekal et al., 2004). These methods, primarily used to obtain information on the local atomic structure of the cluster, depend on the validity of the models used to describe single and multiple scattering as well as on data for needed parameters such as the IMFP and the inner potential [although Kaduwela et al. (1991) conclude that the

latter parameters appear to have a minor effect on the results]. 2.2.3.3 Measurement of electron attenuation lengths by Auger-electron spectroscopy and XPS

2.2.3.4 Measurement of electron backscattering yields and the energy distributions and angular distributions of backscattered electrons Many experiments have been conducted to investigate the yields of electrons backscattered from surfaces and the energy and angular distributions of these backscattered electrons (e.g., Reimer, 1998). These experiments have been conducted with incident electron energies typically between 0.5 keV and 100 keV to investigate contrast mechanisms in scanning electron microscopy, electron-probe microanalysis, analytical electron microscopy, and Auger-electron spectroscopy. By convention, emitted electrons with energies greater than 50 eV are considered as backscattered electrons. Monte Carlo simulations have been used to evaluate electron backscattering yields and the energy and angular distributions of backscattered electrons in scanning electron microscopy, electron-probe microanalysis, analytical electron microscopy, and Auger-electron spectroscopy (Acosta et al., 1996; Reimer, 1998; Shimizu and 28


As indicated in Section 2.2.1, the electron effective attenuation length in Auger-electron spectroscopy and XPS may be less than the corresponding electron inelastic mean free path by up to about 35 % for common measurement conditions in surface analysis by these techniques (Powell and Jablonski, 2002) because of elastic scattering of the signal electrons along their trajectories. In principle, a careful comparison of effective attenuation lengths and inelastic mean free paths for the same material could be used to evaluate calculations of effective attenuation lengths in which atomic differential elastic-scattering cross sections were used in Monte Carlo simulations or transport crosssections in analytical expressions (Jablonski and Powell, 2002b). Unfortunately, the uncertainties in measurements of effective attenuation lengths are now typically larger than the difference between the effective attenuation lengths and the inelastic mean free path due to experimental difficulties in characterizing the morphology of thin films of thicknesses less than about 5 nm (Powell and Jablonski, 1999). It is hoped that improved techniques may in the future be developed that would permit useful comparisons of effective attenuation lengths and inelastic mean free paths for the same material.


2.2.3.5 Measurement of the angular distribution of photoelectrons Quantitative applications of XPS require knowledge of reliable theoretical models describing the relation between the photoelectron signal intensity and the concentration of a given atomic species. The photoelectron emission is anisotropic; the angular distribution of photoelectrons emitted from randomly oriented atoms and molecules is described by the following DCS (for unpolarized x rays) ds x 1 b ¼ sx 1 a ð3 cos2 u 1Þ ; ð2:11Þ dV 4p 4 where sx is the total photoelectric cross-section, u is the angle between the direction of the x rays and the direction of the emitted photoelectron, and ba is the asymmetry parameter. The latter parameter describes the anisotropy of photoemission; the value ba ¼ 0 corresponds to the case of isotropic photoelectron emission. The values of the asymmetry parameter range from 21 to 2 (Band et al., 1979), and the largest values are found for photoemission from the K shells. We note that Eq. (2.11) is based on the simplifying assumptions that the dipole approximation is valid and that relativistic effects can be neglected in the electron motion; Trzhaskavskaya et al. (2001, 2002) have reported quadrupole corrections to the differential cross section for photoelectron energies between 100 eV and 5000 eV. The values of the asymmetry parameter can be, in principle, determined from measurements of the angular distribution of photoemission from solids. Vulli (1981) performed the first experiments of this kind using a movable electron analyzer. However, the ba values determined experimentally were in poor agreement with the theoretical values. Baschenko et al. (1984) proposed an ingenious experiment in which the x-ray source and the analyzer were located on opposite sides of a thin metal foil (with thickness of several micrometers). In this way, the analyzer can rotate without restriction. Baschenko et al. (1984) have found that there are systematic deviations between the experimental and theoretical angular distributions of photoemission, in particular for the configuration where the analyzer axis and the direction of the x rays were

Figure 2.5. Outline of the XPS configuration for measurements of the angular distribution of photoemission from an Au overlayer deposited on the Al thin foil substrate (from Jablonski and Zemek, 1993).

29


parallel. This effect has been ascribed to elastic collisions of photoelectrons with atoms in the solid. For K shells, theory predicts no signal in such a geometry, while, in reality, a finite signal intensity has been observed. Similar results were reported later by Jablonski and Zemek (1988). Such experiments revealed the importance of elastic scattering in solids. These authors attempted to determine the value of the asymmetry parameter that describes the experimental angular distribution of photoemission correctly. It was found that the experimental asymmetry parameter is smaller than the theoretical value. It is advisable to measure the ratio of photoelectron signal intensities rather than a single photoelectron signal intensity. The reason is that the analyzed area, A, depends on the emission angle, a. This relation is approximately described by A ¼ A0 /cos a where A0 is the area analyzed when the analyzer axis is along the surface normal. When taking the ratio of intensities, the contribution of varying surface area is canceled. The experiments of Baschenko et al. (1984) were made with a thin aluminum foil, and the ratios of the Al 2s to the Al 2p signal intensities were measured. Jablonski and Zemek (1993) modified the experiment of Baschenko et al. (1984) by using an aluminum foil with thickness of 2 mm, which is practically transparent to x rays (Al Ka with energy of 1486 eV), as the support for an overlayer of gold. The overlayer was several-tensof-nanometers thick, this was sufficient to ensure that no signal from the support was detected (Figure 2.5) without attenuating the x rays appreciably. It was found that not only aluminum but other elemental solids can be studied with the experimental geometry of Figure 2.5 (Jablonski and Zemek, 1997).

Ding, 1992). These simulations have involved use of atomic differential elastic-scattering cross sections for describing the transport of incident electrons in the selected target solid. Comparisons of measured and simulated quantities have generally been found to be satisfactory.


Jablonski and Zemek (1993) studied the effect of elastic scattering on the angular distribution of emitted photoelectrons. They performed Monte Carlo simulations of photoelectron emission taking into account elastic collisions of photoelectrons within the sample. The simulated angular distributions were found to agree with experimental data better than the distribution obtained directly from Eq. (2.11) (i.e., by disregarding the effect of elastic scattering). Measurement of x-ray spectra

Many measurements have been made of x-ray emission spectra for many materials following excitation by an incident electron beam. These measurements have been made for electron-beam energies generally between 1 keV and 30 keV for applications in electron-probe microanalysis although some measurements have also been made for higher energies. Monte Carlo simulations have been made of electron transport in such experiments to give x-ray spectra; the calculated spectra were in satisfactory agreement with measured spectra (Acosta et al., 1998, 1999, 2002; Ding et al., 1994a; Fujii et al., 2000). 2.2.3.7 Measurement of Auger-electron spectra, x-ray photoelectron spectra, and reflection electron energy-loss spectra Many measurements have been made of Augerelectron spectra, x-ray photoelectron spectra, and reflection electron energy-loss spectra (REELS)

30


2.2.3.6

for a wide variety of materials. Such spectra have been reported in numerous publications and, for Auger-electron and x-ray photoelectron spectra, in handbooks from the manufacturers of instruments for Auger-electron spectroscopy and XPS. Seah et al. (2001) have reported measurements of Auger-electron spectra, x-ray photoelectron spectra, and reflection electron energy-loss spectra for about 60 elemental solids that were made with a fully calibrated electron spectrometer. The Auger-electron spectroscopy measurements were made with 5 keV and 10 keV incident energies, the XPS measurements were made with Al and Mg x rays, and the REELS measurements were made for a 1 keV incident beam. Ding et al. (1994b, 1995, 2000), Kwei et al. (1997), and Werner (1995, 2001) have developed Monte Carlo and other algorithms for simulating Auger-electron spectra, x-ray photoelectron spectra, and reflection electron energy-loss spectra. These simulations have also involved use of atomic differential elastic-scattering crosssections for describing the transport of incident electrons in the selected target solid. The limited number of comparisons of measured and simulated spectra made to date have been generally satisfactory (Ding et al., 1994b, 1995, 2000; Werner 1995, 2001) but future similar comparisons with the more extensive data set of Seah et al. (2001) should be useful in assessing the differential elastic-scattering cross-sections used in the simulations (as well as data for other relevant parameters).

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