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Inelastic Scattering of Electrons in Solids

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Inelastic Scattering of Electrons in Solids Scanning Electron Microscopy Volume 1982 Number 1 1982 Article 2 1982 Inelastic Scattering of Electrons in Solids C. J. Powell National Bureau of Standards Follow this and additional works at: https://digitalcommons.usu.edu/electron Part of the Biology Commons Recommended Citation Powell, C. J. (1982) "Inelastic Scattering of Electrons in Solids," Scanning Electron Microscopy: Vol. 1982 : No. 1 , Article 2. Available at: https://digitalcommons.usu.edu/electron/vol1982/iss1/2 This Article is brought to you for free and open access by the Western Dairy Center at DigitalCommons@USU. It has been accepted for inclusion in Scanning Electron Microscopy by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. Electron Beam Interactions With Solids (Pp. 19-3 I) SEM, Inc., AMF O'Hare (Chicago), IL 60666, U.S.A. INELASTIC SCATTERING OF ELECTRONS IN SOLIDS* C.J. POWELL Surface Science Division, National Bureau of Standards Washington, DC 20234 Tel. 301-921-2188 *Contribution of the National Bureau of Standards, not sub­ ject to copyright. ABSTRACT 1. INTRODUCTION The principal mechanisms and available data for the in­ Electrons incident on a solid or generated internally within elastic scatter ing of electrons in solids are reviewed. The pro­ a solid can be scattered elastically (with a change of momen­ cesses relevant for electron-probe microanalysis, electron tum but without change of energy) or inelastically (with a energy-loss spectroscopy, Auger-electron spectroscopy, and change of energy). This article reviews the principal mechan­ x-ray photoelectron spectroscopy are described and examples isms and available data for the inelastic scattering of elec­ of relevant electron energy-loss data are shown . The discu s­ trons by solids. Emphasis is placed on the processes relevant sion is based on the dielectric description of inelastic scatter­ in microanalysis by electron-probe x-ray analysis and elec­ ing and treats processes important in the excitation of both tron energy-loss spectroscopy (EELS), surface analysis by core electrons and valence electrons. Information is given on Auger-electron spectroscopy (AES) and x-ray photoelectron the cross section s for excitations of valence electrons, cross spectroscopy (XPS) and, to a lesser extent, radiation section s for ionization of core levels, inelastic mean free damage . paths of Auger electrons and photoelectrons in solids, and A summary of the relevant theory is given in Section 2. radiation damage. Sources of experimental dielectric data are identified and dis­ cussed in Section 3. The theory and the dielectric data are applied in Section 4 to several types of mea surements of cur­ rent interest. Attention is given to the physical processes affecting spectral shapes and cross sections in measurements of core-electron energy-loss spectra. The Bethe formula for inner-shell ionization is compared with experimental cross­ section data and values of the "effective" Bethe parameters are given. A brief discussion is also given of various formulas that have been utilized for the variation of inner-shell ion­ ization cross sections as a function of electron energy. The status of measurements and calculations of the inelastic mean free path for low-energy (50-2000 eV) electrons in Keywords : Inelastic electron scattering, inner-shell ioni za­ solids is described and some results of new calculations of the tion cross section , inelastic mean free path, Auger-electron dependence of mean free path on energy are presented . spectroscopy, electron energy-loss spectroscopy, electron­ Finally, some brief remarks are given on radiation damage probe microanaly sis, x-ray photoelectron spectroscopy, and with particular reference to molecular dissociation caused by radiation damage. Auger transitions in compounds. 2. THEORY The theory of inelastic scattering of electrons by solids has been summarized recently by Schnatterly (1979) and Raether ( 1980). A summary is given here of those aspects of the theory that are useful for the applications to be discussed in Section 4. Inelastic electron scattering in solids can be described in terms of a complex dielectric constant t(w,q) dependent on frequency w and momentum-transfer q. For q =0, the di­ electric constant is related to the familiar optica l constants, the refractive index n and the extinction coefficient k, by 19 C.J . Powell LIST OF SYMBOLS d 2a 2e2 - I I = --- Im [ ] 2 2 d wdq 1rNnv E ( w,q) ~ A Parameter in Eq. 18 B Parameter in Eq. 18 where N is the density of atoms, e is the electronic charge, 11 bnP Parameter in Bethe theory (Eq. 15) is Planck's constant divided by 21r, and vis the velocity of the C Velocity of light incident electrons. For small scattering angles , q "" P(02 + en Parameter in Bethe theory (Eq. 13) 0E2 ) '1' , P is the momentum of the incident electrons, 0 is the cnP Parameter in Bethe theory (Eq. 15) scattering angle, 0E = E / 2E , and E is the incident electron c(E) Parameter in Bethe theory (Eq. 14) 0 0 energy. The relation given between q, P, 0, and 0Ei s approxi­ e Electron charge E Energy loss mate although the approximation is very good in the co ntext of the discussion in thi s paper. For Eq. 2 to be valid, it ha s En Particular energy loss been assumed that E is much greater than E. It ha s also been Eo Incident electron energy 0 assumed that E ~ 50 keV so that relativistic corrections [lno ­ EnP Binding energy of electrons in nf-shell f Oscillator strength kuti (1971)] are not needed. Equation 2 can be extend ed to fn(q) Generalized oscillator strength material s with crystalline anisotropies [Raether (1980)] but thi s complication will be ignored here. Finally, we do not n Planck's constant divided by 21r consider modifications to Eq. 2 associated with the excitation k Extinction coefficient of surface plasmons as these are not significant for mo st of Electron mass Refractive index the applications to be discussed. "Effective" number of electrons per atom con­ The term tributing to energy losses from O to E Density of atoms - I N 3 p Momentum of incident electrons Im( -) q Momentum transfer E E/ + E/ Minimum momentum transfer qmin in Eq. 2 is known as the energy loss function . The denomin­ qm ax Maximum momentum transfer ator in Eq. 3 is respon sible for the differen ces that ca n occur UnP Eo/EnP between electron energy-loss spectra and optical absorption V Electron velocity spectra (which are proportional to E2) . For electron energy z Atomic number losses less than about 100 eV (i.e., losses predominantly due Znl' Number of electrons in shell with principal quan­ to valence-electron excitations), E,2 + E/ is usuall y appre­ tum number n and angular momentum quantum ciably different from unity and there is a large difference be­ number e tween energy-loss and optical-absorption spectra . For ene rgy Complex dielectric constant losses greater than about 100 eV (i.e., losses predominantl y Scattering angle due to core-electron excitations), E "" I, E ~ I, so that E/2E 1 2 0 Im ( - I IE ) "" E • Thus , electron energy-loss spectra are sim­ Inela stic mean free path 2 X-ray ma ss absorption coefficient ilar to x-ray absorption spectra. Maxima occur in the energy loss function when 3.14159 Density 4 Inela stic scattering cross section Ionization cross section for nf-shell and d 2[lm( - I/ E)] / dw2 is negativ e. Maxima in Im( - l / E) aT Total inelastic scattering cross section then occur near frequencie s for which : w Frequency E (a) there are maxima in 2 (excitation of interband tran si­ flP Plasma frequency tions); or (b) E 0, E "" 0 (excitation of volume plasmons). 1 = 2 E(w,0) = (n + ik) 2 la Model calculations show how small changes in the position and strength of interband transitions affect the position and 2 2 E1 = n - k lb strength of maxima in the energy-loss function (Powell (1969) ). le It is convenient to define the differential oscillator strength In Eq . le, Q is the density of the solid, c is the velocity of df 2wim[ - I I E(w,q)] light, and 1'-mis the x-ray mass absorption coefficient. 5 The differential inelastic scattering cross section, per atom dw 1rfl/ or molecule, for energy loss E = nw and momentum transfer 2 q in an infinite medium is where flp = (41rNe / m) v, and mis the electron ma ss. 6 The Thomas-Reiche-Kuhn sum rule is 20 Inelastic Scattering of Electrons in Solids da ,., __21re4 _____df _ J;(df / d w) dw = Z 7 14 dE mv 2 E dE where Z is the atomic number. The "effective" number of electrons per atom contributing to energy losses from O to E where df / dE is the differential oscillator strength for q =0 and c(E) is a function of E. = nw is Bethe (1930) has expressed the total cross section per atom or molecule for ionization of the nf shell (containing Zn 1elec- f E . nerr(E) = J (df / dw') dw ' 8 trons with binding energy E 111) in the form O a 2,re 4 • Zn1bn1 In [ cn1Eo ] Equations 2, and can be combined to give = 5 6 nl 15 mv 2 Enl Enl 41re4 df 9 Comparison of Eqs. 14 and I 5 indicates that dwdq mv 2 q nw d w For a relatively narrow energy-loss peak I df I df 10 111w • dw • dw "" nw J dw . dw En 1 f E 1 df 11 ""-- J max - -- dE 16 Z En, E dE where an integration has been performed over a region of the 111 differential oscillator strengt h appropriate for an energy los s where Emax is large co mpared to Enl and assumed for the centered at En = nwn.
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