Inelastic Scattering of Electrons in Solids

Scanning Electron Microscopy

Volume 1982 Number 1 1982 Article 2

1982

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. moment to be less than the binding energy of the next mo st Eq uation 9 becomes tightly bound shell. It is convenient to rewrite Eq. 15 in the form da II 17 dq mv 2 q where U E / E . Equation 17 provides a conven ient 111 = 0 111 whe re f n(q) is the generalized oscillator st rength for the mean s for comparing va lue s of a 111Ei for different elements energy loss E • Equation 11 is identical to the differential in terms of the dimensionles s variable U nl· Furthermore, the 11 4 cross section derived by Bethe (1930) to describe the inelastic linearity of a so-ca lled Fano plot of a 111E ,i U nl/ 1re Znl scattering of electrons by atoms. versus In U 111for a give n element indic ates the range of U 111 The determination of either a total or a differential cross that the Bethe equation (Eq. 15) is valid and provides a con sect ion for a loss En requires knowledge of the function venient means of determining the parameter s bnl and c 111. f n(q) , either from theory or experiment. Calculations of The majo r questions concerning the applicability of the f n(q) for inner-shell excitations in atoms have been reported preceding theory to the experimental situa tion s of interest recently by Leapman el al. (1980) and by lnokuti and Man are:

son (1983). We will assume for the moment that fn(q) is (a) By how much should the incident energy E 0 exceed the slowl y va ryin g in the region of sma ll q where th e differential threshold energy En i for Eqs. 13-15 to give accurate cross cross section is largest. Thus, sections? (b) What are the optimum values of the parameter s bn e 12 and cn 1in Eq. 15 and can they be obtained from integrations of differential oscillator strength, either measured or cal where qmin is the minimum momentum transfer correspond culated, over energy transfer (Eq. 16) and momentum trans ing to 0=0 and fn(O) is the optical oscillator strength. Equa fer (Eqs. 11-13), respectively? tion 11 can now be integrated from qmin = En/v to an "ef (c) Are there other cross-section formulations (e .g., quan fective" upper limit qmax = (mc~n / 2f /' where C11is a con tum-mechanical calculations for specific atoms or semi stant, expected [Bethe (1930)] to be approximately 4, to yield empirical formulas) that provide a better mean s than the above Bethe equations for predicting inelastic sca ttering cross sect ion s? 13 These questions will be addressed later in this article.

3. EXPERIMENT AL DIELECTRIC DAT A Similarly, the differential cross section for energy loss E is As cross sect ion s for inelastic electron scattering in solids are directly proportional to the energy-loss function (Eq. 2),

21 C.J. Powell

it is usefu l to indicate sources of experimental dielectric data strength distribution for one shell often overlaps with that and to examine certain trends. Measurements of Im ( - I / for another shell. In cases such as these, it is not possible to t(w,q)] have been reviewed recently by Schnatterly (1979) derive a total oscillator strength for a given shell. and by Raet her (1980). The majority of the measurements Information on the q-dependence of Im [ - I / dw,q)] is are of energy loss spectra, obtained by transmission through rather sparse. Measurements have been reported of the q thin specimen films for 0=0, from which values of Im dependence of energy-loss spectra associated with valence [ - 1/ dw, qmin)l"" Im ( - 1/ dw)] have been obtained. electron excitations [Raether (1980), Schnatterly (1979)] for a These measurements have been compared with values of number of materials; most measurements pertain to the dis Im[ - 1/ t( w)] determined by optical methods at visible, persion of volume plasmons with varying momentum trans ultraviolet, and x-ray frequencies with generally adequate fer. A few measurements have been made of inner-shell elec agreement. tron energy-loss spectra [Schnatterly (1979)] as a function of momentum tra nsfer although the emphasis of this work has The growing availability of sources of synchroton radia been on the spectrum shape close to the core-level threshold. tion has led to increasing amounts of optical absorption data Measurements of the q-dependence of Im [ - I / £(w,q) ] are, in the ultraviolet and x-ray spectral regions. Sources of data in general, difficult as the signal of interest can be over are Hagemann et al. (1974), Haelbich et al. (1977), Winick whelmed by electrons which have been inelastically scattered and Doniach (1980), and Weaver et al. (1981). The book through small angles (where the differential cross section is edited by Winick and Doniach (1980) also contains reviews large) and elastica lly scatiered to the ang le of observation . of the various physical processes important in photoabsorp tion. 4. APPLICATIONS Figures 1-3 are plots of Im [ - 1 / t (w,0)] for Al, Cu, and Au from the compi lation of Hagemann et al. (1974). The 4.1 Electron Energy-Loss Spectroscopy plot for Al in Fig. I shows a strong maximum at E = 15 eV We cons ider now measurements of the energy- loss spectra that has been identified as being due to volume plasmon ex of high-energy ("" 50 keV) electron s transmitted through thin citation and other structure for E > 72 eV associated with ("" 100 nm) specimen films with a zero or small scattering excitations of Lwshell electrons (see also Section 4.1). By angle. The features of interest here are those associated with contrast, the plots of Im [ - I / £ (w,0)] for Cu and Au show a broader distribution of the energy loss function than for Al the excitation of inner- shell electrons (50 s E s 2000 eV) in the region of valence-electron excitations with E < 50 eV, which are useful for microanalysis. and the structures associated with the onset of inner-shell ex Figures 7 and 8 illustrate EELS data for Al and Al2O3 [Swanson and Powell ( 1968)]. Figure 7 shows the energy-los s citations (expected at the M3 threshold at 75 eV in Cu and at the N, , N,, and N3 thresholds in Au at 84 eV, 335 eV, and spectrum of 20 keV electrons transmitted through a 300 A Al 546 eV, respectively) are not prominent. The lack of promin film. The strong loss peak at E "" 15 eV is due to volume plas ence of the inner-shell thresholds in Cu and Au is due in large mon excitation (cf. Fig. I) and the feature s with dimini shing part to the delayed onset of oscillator strength associated intensity at 30, 45, and 60 eV energy loss are due to multiple with the "centrifugal barrier" in the potentia l [Fano and plasmon excitations . The plasmon features appear on an ex Cooper (I 968), Leapman et al. (I 980)] . Comparison of the ponentially decreasing background that has intensity contri Im I - I / t (w,0)] data for Cu and Au with the corre sponding butions from additional excitation processes of large energy loss in Al (Fig. I) and to inelastic scattering by the oxide plots of £1 (w) and £2(w) [Hageman , et al. (1974)] indicates that the structure in the energy loss function for these two layers on the specimen surface . The shape of thi s background metals is largely associated with the excitation of interband is also influen ced by variat ion of the effective angular resolu transitions and a damped volume plasmon (cf. Eq. 4). tion with E; the spectrometer had a fixed angular acceptance It is clear from Figs. 1-3 (and similar data for other materi and therefore accepted a decreasing fraction of the total als) that the most probable energy loss is typically 5-40 eV. angular distribution of inelastically scatte red electrons with The cross section for inelastic scattering (Eq. 2) is therefore increasing E. Also seen in Fig. 7 is an increase in energy-loss large st for excitations of valence electrons. intensity at E "" 72 eV associated with the excitation of L Figures 4-6 are plots of nerr(E) (Eq. 8) for Al, Cu, and Au shell electrons. This region of the energy-loss spectrum is shown in greater detail in Fig. 8(a) together with similar spec from Hagemann el al. (1974). The plot of nerr(E) for Al in tra for -y - Al2OJ and ano dized Al2OJ in Figs. 8(b) and 8(c). Fig. 4 indicates that nerr(E) "satura tes" at about 3, the num The spectra in Fig. 8 indicate that there is a small chemica l ber of valence electrons, for E = 70 eV, just below the shift in the Al L-shell excitat ion threshold from the metal to threshold for L-shell excitation . The integration from 70 eV the oxide, that the near-edge struct ur es in the different forms to 1500 eV indicates an effective number of about 9 elec of Al,O3 are different, that a background (the long-d ashed trons, slightly more than the number of electrons in the L line) can be found by extrapo lation from the region of shell; note the nerr(E) does not "sa turate" again until E ex valence-electron excitation, and that a correction can be ceeds 1000 eV. Shiles et al. (1980) have made a further anal made (the short-dashed line) for combined L-shell and plas ysis of optical data for Al and have shown the existence of mon excitations. some inaccuracies in the optical data as well as some inconsis We will assume for the moment that Eo is sufficient ly high, tencies in the analysis of Hagemann el al. (I 974). The trends 50 ~ Eo ~ 2000 eV, and that the electron scattering angles are shown in Fig. 4 are preserved in the new analysis but nerr(E) sufficientl y sma ll so that qmin is "small" and Im [ - I / £(w,0)] now saturates at the expected value of 13 for E> 104 eV. "" £2 (Eq. 3). Electron energy-loss spectra measured under The plots of nerr(E) for Cu and Au in Figs. 5 and 6 are these conditions should therefore correspond closely to x-ray qualitatively different from that for Al. For Cu and Au, absorption data . We consider now a number of phenomena there is not an obvious saturation of the oscillator strength that can modify structure or give rise to new structure in for the excitations for each shell. Instead , the oscillator- EELS data, particularly in the vicinity of the core-electron 22 Inelastic Scattering of Electrons in Solids

0 1 - Im c 1

0.05 0.5

1000 6 10 20 40 200 400 1000 (eVl Fig. 1. Plot of the energy loss function for Al from optical Fig. 2. Plot of the energy loss function for Cu from optical --- data; the solid points are measurements from elec data (solid line): the dashed line gives results from tron energy-loss spectra [Hagemann et al. (1974)). electron energy-loss spectra [Hagemann et al. (1974)). ,-- :' r ----,--, I ----,--

10 I

08 I 10- -j 06 :

-, I 02

--o-;[ E wdw ---- =I µdw 0 L..=c :::L_ _J_ __Jc__j_l__c__j __ .l__[__[_j_===== "- W - - - 0<]- Jm 1/tw dw 1 10 20 ( eV) 1.0 100 200 1.00 1000

Fig. 3. Plot of the energy loss function for Au from optical 10-1 10• 101 102 103 10' data (solid line); the dashed line gives results from (eVJ electron energy-loss measurement s [Hagemann et al. Fig. 4. Plot of nerr(E) for Al (Eq. 8), dot-dashed line; the (1974). -- - other curves are results of integrations of other op tical data [Hagemann et al. (1974)). Revised values of JO nerr

nett et al. (1980) (see text).

nett

10 I I I I I I I -- -. /t:1 wdw - --- -..J µ dw -- - o lffa Fig. 5. Plot of nerr(E) for Cu, dot-dashed line [Hagemann (eV) --- et al. (1974); see also caption to Fig. 4. Fig. 6. Plot of nerr

23 C.J. Powell

Fig. 7. Energy-loss spectrum of 8.2 µg/c m 2 (300 A.)Al films for Eo = 20 keV and 0 ,., 0. The angular acceptance CD of the analyzer was a cone of half-angle ,., 0.25 mrad and the angular distribution of the incident beam was approximately Gaussian with a full width at half maximum intensity of about 1.6 mrad. The relative > t gain setting is indicated for each intensity change of (/) scale [Swanson and Powell (1968)]. z w t z thresholds that are usefu l for microanalysis. These pheno ...--20 X w mena need to be considered for both qualitative and quanti > tative analyses . ~ (I) Chemica l Shifts of Thresholds . The same element in _J w different chemical environments can be expected to show dif 0:: ferent core-electron threshold energies in EELS data. These 0 10 20 YJ 40 50 60 70 80 90 shifts, often of about 1-2 eV, correspond to the chemical ENERGY LOSS , e V shifts observed in x-ray photoelectron spectroscopy [Sieg bahn et al. (1974), Wagner et al. (1979)] and can provide useful informat ion about the chemical state of a particular element. (2) Edge Singu lar ities. The detailed shape of an x-ray ab I E1 (a) A! sorption "edge" within - I eV of the thre shold energy is in - ' I t=8.2,ug/cm 2 fluenced by the excitation of electron-ho le pairs in respon se \ to the creation of the core hole in a particular shell [Schnat \ E2 \ I terly (1979), Citrin et al. ( 1979)]. The specific edge shapes \ that can occur are of little significance in EELS experiments \ with current energy resolutions.

' '--"'., (3) Near-Edge Structure . Structure within about 50 eV of -<::, ---::,, a core threshold is often observed (Fig. 8). This structure is ----..::::..-...:::..-...::.-- due in part to atomic contributions to df / dE for a particular 0 ~ ;c;------;c;-:,:-----;-;::-;c---c-c=--c~ - -=-=-=,=-::..:-=c-=-=-~~~ ---1 70 E 90 100 110 120 130 140 150 160 element [Fano and Cooper (1968), Leapman et al. (1980)] >- 1 and in part to the details of the solid-state band structure . f:: I I Ez CJ) The absorption spectra of an element in different chemical z w \ I 2 environments can be expected to have structure at different t- \ t = 15.5 ,ug/cm z \ energies (Fig . 8). This structure can also be useful for giving w \ chemical-state information. > ', E4 (4) EXAFS. The extended x-ray absorption fine structure, ~ __J ', I broad maxima extending several hundred electronvolts above w Cl'...... __ ""~-:...:::..-- an absorption edge, ca n be analyzed to give structura l infor --.::::...-~ ~ ::..=:-==-=--==-= 0 mation of atoms about an absorption site [Teo and Joy 70 80 90 100 110 120 130 140 150 160 (1981)]. E1 (5) Delayed Onsets. For some core levels, the photo 111E2 (c) Anodized A12o3 t= 13.6,ug /cm 2 absorption will increase sharply near the core threshold \ energy and then decrease while for others the increase in \ \ photoabsorption at the threshold will be extremely small. In \ the latter cases, the photoabsorption will increase and reach a ', E4 maximum about 20-200 eV above the threshold energy due to ....__,"'-_ I a strong centr ifugal barrier in the absorption potential [Fano ---::::...~~-- --:...:::.:::_-...:::...--=-~==--- and Cooper (1968), Leapman et al. (1980)]. 0 ' ...::::::...~=--"=-= (6) Fano Profiles. The shape of an absorption "edge" may 70 80 90 100 110 120 130 140 150 160 be modified appreciably by interfering excitati on channels ENERGY LOSS ( ev) which ha ve discrete and continuum intermediate states [Fano Fig. 8. Energy-loss spectra in the region corresponding to L and Cooper (1968)]. Asymmetric absorption profiles have been observed in photoabsorption and EELS studies of the shell excitation for (a) Al, (b) -y- Ali0 3 prepared by Ml) excitations in the transition and noble metals [Dietz et oxidizing an Al film in air, and (c) anodized Al 20 3 • The backgrounds determined by extrapolation of the al. (I 980)] . A striking manifestation of the Fano interference energy-loss intensities below the absorption edges are effect is the contrast in the thre shold shapes of the Ni L1 ab shown as long-dashed lines and the additional con sorp tion spect rum , which shows a peak due to unfilled 3d tributions from electrons exciting both a volume stat es [Leap man and Grunes (1980)] and the Ni Mll absorp plasmon and an L-shell electron are shown as short tion spec trum , which shows an asymmetrical step [Jach and dashed lines [Swanson and Powell (1968)]. Powell (1981)].

24 Inelastic Scattering of Electrons in Solids

(7) Multiple Scattering. Additional intensity (and struc puted from integration of the generalized oscillator strength ture) can appear in EELS data due to a combination of (Eqs . 9 and I I). Inokuti (1971) has shown that total cross sec inner-shell excitation and valence-electron excitation (Fig. 8). tions for inelastic scattering for atoms can be calculated from Unless the specimen thickness is much less than the total in atomic properties. elastic mean free path for the particular incident energy, cor Experimental inner-shell ionization cross-section data have rection of measured EELS intensities (e.g., by deconvolution been analyzed to test for consistency with the Bethe equation of the loss spectrum measured for small energy losses) will be (Eq . 15) and to derive "effective" values of the parameters needed. bnl and cnl (Powell 1976a , 1976b). Figure 9 is a plot of experi (8) Satellites. The sudden creation of a core-electron mental values of aKE/ versus UK, as suggested by Eq. 17, vacancy will generally lead to excitation of the valence elec for a number of elements. Most of the data appears to be trons, either of a sing le valence electron (thus creating a two close to a common curve and it therefore appears that bK hole final state) or a larg e number of valence electrons (an in trinsic plasmon). Details of different "final-state" effects does not depend on Z to a significant extent. A similar plot have been described by Gadzuk (1978) and Shirley (1978). of al ,, El,,2 as a function of U l,, is shown in Fig. 10. These The measured energy-loss intensity distribution will de data indicate an increase of bl ,, with Z. pend on an integration of the differential scattering cross sec Figures 11 and 12 are Fano plot s for the experimental K tion (Eq. 9) appropriate to the angular acceptance of the par shell and L23-shell cross-section data. The range of U nl for ticular spectrometer. Leap man et al. ( 1980) have pointed out which these plots are linear indicates directly where the Bethe that the measured intensity distributions for a collection cross-section equation can adequately de scr ibe the mea sured angle of 0.1 rad can depart significantly (e.g., by up to about data. Linear regions are found typically in the range 4 ':: U nl 30% for carbon K-shell excitation with Eo = 80 keV) from the photoabsorption spectrum . The total intensity within, ':: 25. say, 50 eV of the core threshold will be given by an additional "Effective" values of the Bethe parameters bnl and cnl can integration of Eq. 9 over energy transfer for the limits of par be easily found by linear least-square s fits to the appropriate ticular interest. This total intensity, which is needed for linear regions in Figs. 11 and 12. The results of these fits quantitative microanalysis, will depend on the physical pro [Powell (1976a)] are summarized in Table I. The values of bK cesses discussed above and the chemical environment of the derived in this way were larger and the values of cK smaller particular atom. The extent to which these phenomena influ than those expected from calculated K-shell ionization cross ence observed inten sities has not been explored in sufficient sections. Values of bK and also of b l ,, were therefore cal detail to predict "sensitivity factors" or "correction factors" culated from x-ray absorption cross sections with the use of for a wide range of elements and material s. Eq. 16. The results of this calculation are shown in Table 2 It is clear now that the simple model of EELS data near a from which it can be seen that the "optical" values of bnl are co re-electron thre shold consisting of a sharp jump in the loss lower than those found from ionization cross-section mea intensity followed by an exponential decay is of limited valid surement s. The reason for these differences can be under ity . There is a need for more experimental EELS data and stoo d from the fact that the differential oscillator strength is calc ulation s of the generalized oscillator stre ngth (Eqs. 5, 9 not concentrated in a narrow range of excitation energies and I 1) so that the significant processes in a range of mate near the thre shold energy Enl. It is clear from Figs. 1-6 that rials can be identified . the differential oscillator strength is extended over a consi 4.2 Total Cross Sections for Inner-Shell Ionization derable range of excitation energies, often up to about 10 Enl (particularly for materials that have delayed onsets in the ab Consideration is given now to the total cross sections for sorption away from their thre shold En ). It might then be ioni zat ion of inner-shell electrons by electron impact. Inner 1 expected that the incident energy might have to be large as, shell vacancies can decay either by the emission of character say, 30 Enl for E to be large compared to all significant ex istic x-rays or of Auger electrons. The yield of x-rays or 0 Auger electrons is not neces sarily proportional to the vacan citation energies and for Eq . 15 to be valid with a value of cies produced by electron impact; internal redistribution of b nl consistent with x-ray absorption data (Eq. 16). the inner- shell vacancies often occurs by Auger or Coster To test the validity of the above ideas, it was decided to set Kronig processes [Bambynek et al. (I 972)] prior to the x-ray c 111= 2.42 [a value originally recommended by Mott and or Auger-electron emission of interest. In practical electron Massey (1949)] and to compute bnl as a function U 111from the probe microanalysis (EPMA) and Auger-electron spectros experimental cross-section data . Value s of b 111derived in this copy (AES), ionizations by back-scattered electrons have to way are plotted in Fig. 13 and show the expected trends . For be considered in addition to those that may be produced by low U nl, the value of b is small becau se only a relatively the incident electron beam. The backscattered electrons (due 111 small fraction of the differential oscillator strength is avail to incident and secondary electrons that have been multiply able for excitation. As U is increased, more oscillator scattered by a chain of elastic and inelastic events) also give 111 rise to a background in AES on which the Auger-electron strength becomes available until, at a sufficiently large value signal of interest appears . Total cross sections for both inner of U 111, all of the oscillator strength for the shell is available shell ionization and for all types of inelastic scatte ring are for excitation. The derived va lues of b 111in Fig. 13 appear to needed to predict overall yields in EPMA and AES and the saturate as expected for increasing U 111• Also, the "satura ratio of signal and background intensities in AES. tion" va lues of bK are now close to those expected from Total cross sections for inner-shell ionization can be com- photoabsorption data (Table 2).

25 C.J. Powell

4 ~- -~--~--~~---~--~- ~ X 10-14 (a) ( b)

a □

3 .<;u•□ ,Mn C\J _,□ - □ 2 >(1) □ C\J , . 2 □ E r<) u I

C\J :,:: 0 W I X :.:: b N 0 0 00~-- ,~o--- 2~0--~ 30-~ o~-- ,~o-- -2-'-0-- --'30 r<) 0 UK=E0 /EK

Fig. 9. Experimental value of aK E/ as a function of UK. (a)

data for C (open circles), Al (open squares), and Ni 0 (solid circles); the solid squares represent data for Al, Mn, and Cu by Fischer and Hoffman (1967). (b) data w 0 for C (triangles), Ne (cros ses) , N (squares), and 0 :::.:: (circles) [Powell (1976a)). b o

Fig. 11. Plot of experimental values of aKE/UK / l.302 x 13 12 ,------,------~ ---~ 10- versus In UK (a Fano plot based on Eq. 17.) 1 1 Successive plots have been displaced vertically for xio-14 clarity. The solid lines represent linear least-square fits for the range of UK indicated in Table 1; the

10 1- - dashed lines are extrapolations. The derived Bethe parameters are also shown in Table 1 [Powell (1976a)).

- Table l. Effective values of the Bethe Parameter s found C\J- from linear least-square s fits lo the experimental cross > sect ion data in Figs. 11 and 12 [Powell (1976a)]. Q) Shell Range of Uni Bethe Parameter s C\J

E o x)(r K-shell 4 :5 UK :5 25 bK ==0.9 (.) 4 - X • " X . . 0 CK ==0 .65 X ---- 0 I') L -shell 4 :5 U :5 20 b L,, ==0.6-0.9 C\J "' 23 L B _J • • • w - CL,, ==0.6

I')

"' Table. 2 Values of the Bethe parameter bnl found from b 0 ~- ----'~- ----~ I---- ~ x-ray absorption data with the use of Eq. 16 for the 0 10 20 30 elements indicated (Powell 1976a) U L23= Eo/EL23 Shell Bethe Paramater b" 1 Fig. IO. Experimental value of aL,, EL,,2 as a function of UL . Data shown is for P (squares) , S (circles), Cl K-shell b K = 0.55 ± 0.05 (Be, C, 0, Ne) (tri~ngles), and Ar (crosses and open circles) [Pow L -shell bL ,, = 0.55 ± 0.06 (Al, Si, S, Ar) ell (1976a)). 23

26 Inelastic Scattering of Electrons in Solids

3,------~--- - -.-----,-----~ The value of c111may not, of course, be correct so the num erical values in Fig. 13 are of limited significance . It is also possible that c 111could be a function of excitation energy (Eq. 16) and thus also of U 111• Neverthele ss, the trends in Fig. 13 appear qualitatively reasonable from the known distribu tions of differential oscillator strength. The derived values of X bL,, in Fig. 13, however, are rather lower than tho se expected from photoab sor ption data and there is thu s an apparent in consistency between the cross-section and photoabsorption Ar data. o' The preceding discussion indicates that the effective values of b 111and c111in Table I should be regarded as empirical parameters useful only in the range of U indicated. The fact that 0 111 Fano plots of the typ~ shown in Figs. 11 and 12 are linear

■ establishes the utility of the Bethe equation as an empirical 0 ,., ,:<~:~ formula but does not guarantee accuracy over a larger range Cl ('J "' of Unland consistency with optical data. Further, Fano plots __J 0 w for other shells cou ld be qualitatively different than those in s__ ~ Figs. 11 and 12 if the distributions of oscilla tor strengt h dif ,., 0 ·---p fer signific ant ly (e.g., if there are lar ge del ayed onset s of __J"' absorp tion) from those for the elements considered here. 0 A large number of theoretical and empirical formulas have b 0 2 3 4 been proposed to describe inner-shell ionization cross sec In UL23 tion s (Powell 1976b) . Figure 14 is a comparison of experi mental cross-section data for K-shell ionization and results Fig. 12. Plot of experimental values of a1."EL,,2U 1.,/ 3.906 expected from a numb er of formula s. Fig. 14(a) is a plot of x 10 - 13 versus In UL" (a Fano plot based on Eq. aKE/ for C, Ne, N, and O (also given in Fig. 9) as a func 17). See also caption to Fig. 11 [Powell (1976a)I. tion of U K. The solid line is a smooth curve drawn through the experimental point s; the same curve ha s been drawn in the other panel s to serve for comparison with other results . The da shed curv e in Fig. 14(a) is the Bethe equation, Eq. 15, 0.7 (a) ( b) with bK = 0.9 and cK = 0.65, the empirical values for the se K-shell L - shell 23 parameters found from the Fano plots (Fig. 11) and shown in 0 Table I. The other panels (Figs. 14(b), (c), (d) show plot s of 0 .6 Ar aKE/ from the formulas or calcu lation s of Worthington and Tomlin (1956), Green and Cosslett (1961), Drawin (1963), Gryzinski ( 1965), Lotz ( 1970), McGuire (1971 ), Rudge and 0.5 Schwartz (1966), and Kolbenstvedt (1967); details of the for mulas and parameters are given in Powell (1976b). It can be seen that few of the formulas fit the experimental data par 0 .4 c Cl ticularly well. The formulas of Drawin (1963) and Lotz .0 (1970), how ever, cou ld fit the experimental curve if the .. s amplitudes of the calculated cross sect ion were increa sed by 0.3 about 10% and 25%, respectively. These formulas would p appear to be particularly useful for the threshold region (1 < UK < 4). 0.2 It is clear that more measurement s are needed of inner shell ionization cross sections. We do not now have an ade quate description of the variation of the cross sections as a 0.1 function U over a large range of U • It would, of course, 0 10 20 0 10 20 30 111 111 be desirable to have measurements of the generalized oscil LJnl lator strength. Further measurements are needed of L-shell Fig. 13. Plot of the effective value of the Bethe parameter ionization cross sections to remove a discrepancy in current bnl as a function of U nl found by assuming cnl = data . There are few measurements of M-shell and N-shell 2.42 in Eq. 15. The parameter bnl was calculated ionization cross sections. The known delayed onsets in photoabsorption are expected to modify the shape of the from the experimental (a) K-shell and (b) L21-shell cross-section curve as a function of U near threshold. cross-section data plotted in Figs. 9 and 10 [Powell 111 (1976a)]. Smith and Gallon (1974) have found delayed onsets in the cross-section for N 6 1-shell ionization in Au, Pb, and Bi; little

27 C.J. Powell

4 ionization was observed until UN., reached 1.5-2.0 . For such

X 10-14 (a) (b) cases, formulas of the type plotted in Fig. 14 would have to be modified significantly. 3 4.3 Inelastic Mean Free Paths of Low-Energy Electrons in Solids 2 The inelastic mean free path of low-energy (50-2000 eV) electrons in solids is needed for quantitative surface analysis by AES and XPS. Absolute values of mean free paths are N ~ required in comparisons of inten sities for an element in dif N WT --- 8 ferent chemical environments (i.e., a matrix correction) while relative values of mean free paths are needed in comparisons

N 0 >< of the intensities from two or more elements in a single speci

>< men material (Powell (1978)]. b Seah and Dench ( 1979) have published a compilation of 3 inelastic-mean-free-path data for a variety of material s. Ac curate measurement of inelastic mean free paths is difficult , [Powell (1974)] and it is not surprising that there is consider 2 able scatter in measurements from different laboratorie s. The fact that inelastic mean free paths can differ consider ably from material to material (for a fixed electron energy) is clear from calculations such as those of Penn (1976). Figure 15 is a plot of the inelastic mean free path s for Eo = 1000 eV as a function of Z [Powell ( 1978)]. Although there are a num ber of approximations in the calculated values, particularly 0 □~--~10--~2 □-,---~□~-~10---2~□ --~30 for non-free-electron solids , it is clear from Fig . 15 that sys tematic variations of the inelastic mean free path in different material s are to be expected . Other calculation s of the inela s Fig. 14. Plot of a KE/ against UK" (a) Experimental values tic mean free path for a number of solids have been pub for C (triangles), neon (crosses), nitrogen (squares), lished recently by Szajman and Leckey (1981) and by Ashley and oxygen (circles) have been plotted from Fig. and Tung (1982). 9(b) . The solid line in Figures 14(a), (b), (c), (d) is a 60 ~---r-----,-- ---,------,--- --, smooth curve through the experimental points. The ELECTRONENERGY • FreP.. electron Soli d dashed curve is Eq. 17, the Bethe equation, with bK E= lOOOeV o Non-free-electron Solid = 0.9 and cK = 0.65. The other panels are the results of specific cross-section calculations or sug 50 gested formulas. (b) The short-dashed curve (WT) is the Worthington and Tomlin (1956) equation and the long-dashed line (GC) is the Green and Cosslett (1961) equation. (c) the short-dashed line (L) is the o<( Lotz (1970) equation, the long-dashed curve (G) is ::c 40 ,- the result of Gryzinski (1965), and the dot-dashed (!J z curve (D) is the result of Drawin (1963). (d) the LU _J short-dashed curve (M) represents calculations of z McGuire (1971) for Be, C, and 0, the long-dashed CJ

line (RS) is the result of Rudge and Schwartz (1966), ~:::, 30 and the dot-dashed curve (K) is the result of Kol z ,LU benstvedt (1967) [Powell (1976b)). , <(

D 0 LU 0 Fig. 15. Calculation of the inelastic mean free path in ele , 0 <( 0 _J 20 .. mental solids as a function of Z according to the :::, 0 u . 0 _J .. results of Penn (1976). The values shown here are <( 0 0 u 0 . 0 for E0 = 1000 eV. The solid circles denote results 0 0 0 <\ '<,po"' ~ for free-electron-like solids which are expected to be ,o . more accurate than the estimates shown as open cir 10 cles for non-free-electron-like solids. For the former type of solids, volume plasmon excitation is the pre dominant inelastic mechanism while for the other solids interband transitions become important o----~ --- ~---~ --- ~ --- - (Figs. 1-3) [Powell (1968)). 20 40 60 so 100 z

28 Inelastic Scattering of Electrons in Solids

Seah and Dench (1979) propo sed the following general 4.4 Radiation Damage relationship between the inelastic free path >..in a given mate The interaction of electrons with so lids can lead to various rial and the electron energy Eo: types of damage. Isaacson ( 1977) has reviewed damage mechanisms that have been observed in the electron micro >.. (A/E/) + BE /' 18 = scope and Pantano and Madey (1981) have summarized dam age mechanisms that are manife sted in AES. The details of where A and Bare material-dependent parameters. Wagner the damage mechanism (dissociation, desorption, reduction, et al. (1980) have analyzed the dependence of>.. on E for a 0 polymerization, oxidation, carburization, diffu sion, etc.) are number of materials for which at least severa l measurements not sufficiently documented to predict damage rates in a at different energies have been made in the same laboratory; variety of materials. It seems intuitivel y clear, however, that it was believed that the variation of>.. with E could be estab 0 damage is related to energy tran sfer between the incident lished with rea sonable accuracy even if the absolute values of electron beam and the solid and thus to the magnitude of the >..could be in error. Wagner et al. (1980) suggested the more inelastic sca ttering cross section. While not every inelastic general relation: scatte ring event will necessarily lead to specim en damage , the potential exists, in general, that the interactions which give 19 rise to desired signals or properties may also lead to damage. The problem then is to identify the signi ficant damage me for E 0 ~ 300 eV and derived values of the parameter m from chanis ms in different types of material and to optimize the exper imental data. They found that m could be as high as experime nt al con dition s wherever possible to optimize the 0.81 for Si and as low as 0.54 for Au and AlzOJ. Szajman desired-signal/ damage-rate ratio. et al. (1981) have proposed that m = 0. 75 while Ashley and One important form of specimen damage is molecular dis Tung ( 1982) have found, in calculations of>.. for a number of socia tion . Dissociation of gas-phase molecules can be in materials, that the exponent m could vary between 0. 77 (for duced by direct electron bombardment. For so lids, dissocia Al) and 0.69 (for Au). tion ca n be ca used to a greater extent by the large number of The inel astic mean free path is related to the total inelastic low-energy secon dary electrons which result from the decay scattering cross section aT by of electronic excitations produced by primary electrons and ot her electrons in the co llision cascade. The electron ic excita 20 tion s considered import ant are tho se of th e valence electrons . For these excitations, the energy-loss function is lar ge (Fig s. The total inela stic cross section can be calcula ted from an 1-3) for E ~ 50 eV and the seco ndary electrons resulting from integration of Eq. 2 over all energy and momentum transfer s the decay of the exc itation have energies for which the disso appropriate for a particular "incident" electron energy Eo. ciat ion cross section is near its maximum value. Equation s 15 and 16 can be genera lized to give It ha s been recent ly discove red that molecular dissoc ia tion in so me materials can be induced by another proce ss. Inner

3.327 E0 shell ionization of the cation of an ionic compound can decay ------A 2 1 most probably by inter-atomic Auger proce sses [Knotek and / j f maxim [ - I / E(E)] fn (c E0 E) dE Feibe lman (1978)] that can lead to a po sitively cha rged an ion with a resulting "Co ulomb exp los ion" of th e molecule. The where the energy E 0 has been expressed in electron volts and damage rate in suc h cases would then depend on the cross section for ionization of the particular cation inner- shell Emax• the upper limit for the integration, is less than E 0. The term c in Eq. 2 1 is ana logo us to en\ in Eq. 15 and thus shou ld level. Whether damage in a given material was primarily as sociated with the excitat ion of valence electro ns or the ion depend, in general, on the momentum dependence of the iza tion of inner-shell leve ls cou ld be determined from energy- loss function. For the reasons discussed in Sect ion the dependence of the damage rate on electron (or photon) 4.2, however, c ca n be regarded as an emp irical parameter. energy. Equat ion 21 has been eva luat ed using the optical data of The concept of Auger-induced damage has been extended Hagemann et al. (1974) for a number of materials. Ca lcula further by Ramaker et al. (1981) and Jennison et al. (1981). tions have been made for electron energies between 300 and The ionization of inner- shell electrons in either ionic or co 2000 eV and these values have been fitted to Eq. 19 to deri ve valent compounds can lead to decay by a core-valence-val a value for th e exponent m . If c is set equal to 2.0 [a reason ence Auger process . If the two hole s of the final state are able value based on Penn 's (1976) calculations], the exponent localized in the vicinity of a bond sufficiently long , bond rup m is found to range between 0. 75 for Al and 0.69 for Au in ture may occur by a Coulomb explosion. If, on the other close agreement with the results of Ashley and Tung (1982) . hand, the two holes are delo ca lized, damage will not occur. If c is set equal to 0.65 , the empirical value of en\ found in The extent of hole localization can be deduced from analysis analy ses of K- and L23-shell cross section s, the exponent m is of the resulting Auger-electron spect rum and damage rates in found to range between 0.65 for Al and 0 .55 for Au. While the "true" value of c is not known, it is apparent that the ex different co mpounds can be estimated [Ramaker et al. (1981)]. For materials in which Auger-induced damage is im ponent m can depend on the nature of the distribution of portant, the damage rate should be proportional to a parti oscillator strength for the energy-loss function (Fig s. 1-3) and can thu s be rea sonably expected to vary in the range of cular inner-shell ionization cross section. '= 0.55 - 0. 75 for different material s.

29 C.J. Powell

5. SUMMARY Drawin H-W . (1963). Zur spektroskopischen Temperatur und Dichtemessung von Plasmen bei Abwesenheit Thermo The basic theory for the inelastic scattering of electrons by dynamischen Gleichgewichtes. (Spectroscopic temperature atoms that was developed by Bethe and extended to solids and density measurement of plasmas in the absence of ther appears generally satisfactory. Inelastic electron scattering by modynamic equilibrium.) Z. Phys. 172, 429-452. solids can be described by the so-called energy loss function Fano U and Cooper JW. (I 968). Spectral distribution of derived from the complex frequency-dependent and momen tum-dependent dielectric constant. There is still uncertainty, atomic oscillator strengths. Rev. Mod . Phy s. 40, 441-507. however, about the values of the parameters to use in the Fischer B and Hoffman K-W. (1967). Die intensitat der cross-section formulas, particularly in the parameter related bremsstrahlung und der charakteristischen K-Rontgenstrah to the momentum dependence of the energy-loss function. lung dunner anoden. (The intensities of bremsstrahlung and The extent to which the parameters may vary with atomic characteristic K x-rays of thin anodes.) Z. Physik. 204, number and with electronic shell or subshell has not been 122-128. adequately explored. Finally, there is uncertainty about the minimum electron energy for which the cross-section equa Gadzuk JW . (1978). Many-body effects in Photemis sion. In: tions can be used with confidence. Photoemi ssion and the Electron Properties of Surfaces, B. Dielectric data (optical constants, the energy loss function) Feuerbacher, B. Fitton, and R.F. Willis (eds.), Wiley, New is available for a limited number of materials although often York, 111-136. for a restricted range of frequencies or excitation energies. Green Mand Cosslett VE. (1961). The efficiency of produc More dielectric data is required for use in cross-section for tion of characteristic x-radiation in thick targets of a pure mulas. Data is also needed to define significant physical pro element. Proc. Phys. Soc . (London) 78, 1206-1214. cesses in more detail, particularly the shape of the energy-loss function in the vicinity of inner-shell thresholds and the Gryzinski M. (1965). Classical theory of atomic collision s. I. dependence of the energy-loss function on momentum trans Theory of inelastic collisions. Phys . Rev. 138, A336-A358. fer. Comparisons are needed of calculated generalized oscil Haelbich R-P, !wan Mand Koch EE. (1977). Optical proper lator strengths and experimental data. ties of some insulators in the vacuum ultraviolet region. Pre sent ly available theory and experimental dielectric data Phy sik Oaten-Phy sics Data (Zentralstelle fur Atomkernener provide a qualitative and in some cases quantitative guide to gie-Dokum entation, Eggenstein-Leopoldshafen), 1-186. inela stic scat tering cross sections appropriate for microanal ysis (EELS), inner-shell ionization (EPMA, AES), surface Hagemann H-J, Gudat W and Kun z C. (1974) . Optical con analy sis (AES, XPS), and radiation damage by electron stan ts from the far infrared to the x-ray region: Mg, Al, Cu, bombardment. Improved under standing of inelastic-electron Ag, Au, Bi, C, and A'2O 3 • Deut sches Elektronen-Synchro intera ctions and a larger data base should lead in the future tron Report SR-74/7 (Hamburg). to more accurate mea sureme nts or improved experimental designs. lnokuti M. (1971). Inela stic collisions of fast charged parti cles with atoms and molecul es- the Bethe theory revisited. Acknowledgement Rev. Mod. Phys. 43, 297-347. The author wishes to thank Dr. C. Kun z for supplying Inokuti Mand Manson ST. (1983). Cross-sect ion s for inelas Figs. 1-6. This work was supported in part by the Offic e of tic sca ttering of electrons by atoms-selected topic s related Environment, U.S. Department of Energy. to electron micro sco py, In : Electron Beam Interaction s, SEM, Inc ., AMF O'Hare, IL (this volume), 1-17. REFERENCES Isaacso n M. (1977). Specimen damage in the electron micro scope, In: Principle s and Techniques of Electron Micros Ashley JC and Tung CJ. (1982). Electron inel astic mean free copy, M.A. Hayat (ed .), Van Nostrand-Reinhold, New paths in severa l solids for 200 eV :S E :S 10 keV. Surface and York, Vol. 7, 1-78. Interface Anal. 4, 52-55. Jach TJ and Powell CJ. (1981). Incident-energy dependence Bambynek W, Crasemann B, Fink RW, Freund H-V, Mark, of 3p electron energy-loss spectra of nickel. Sol. State H, Swift CD, Price RE and Rao PY. (1972). X-ray Ouore s Comm. 40, 967-969 . cence yields, Auger, and Coster-Kronig transition probabili ties. Rev. Mod. Phys. 44, 716-813. Jennison DR, Kelber JA and Rye RR . (1981). Localized Auger final states in covalent systems. J. Vac . Sci Tech. 18, Bethe H. (1930). Theory of pa ssage of swift corpuscular rays 466-467. through matter. Ann. der Physik. 5, 325-400. Knotek ML and Feibelman PJ . (1978). Ion desorption by Citrin PH, Wertheim GK and Schluter M. (1979). One-elec core-hole Auger decay. Phys . Rev. Lett. 40, 964-967. tron and many-body effects in x-ray absorption and emission edges of Li, Na, Mg, and Al metals. Phy s. Rev . B 20, 3067- Kolbenstvedt H. ( 1967). Simple theory for K-ionization by 3114. relativistic electrons. J. Appl. Phys. 38, 4785-4787. Leapman RD and Grunes LA . (1980). Anomalous L / L Dietz RE, McRae EG and Weaver JH. (1980). Core-electron 3 2 white-line ratios in the 3d transition metals. Phys. Rev . Lett. excitation edges in metallic Ni, Cu, Pt, and Au. Phys . Rev. B 45, 397-401. 21, 2229-2247 .

30 Inelastic Scattering of Electrons in Solids

Leapman RD, Rez P and Mayers OF . (1980) . K, L, and M Shirley DA . (1978). Many-electron and final- state effects: shell generalized oscillator streng ths and ionization cross sec Beyond the one-electron picture, In: Photoemi ssion in Solids tions for fast electro n collisions. J. Chem. Phys. 72, 1232- I, M . Cardona and L. Ley (eds.), Springer Topics in Applied 1243. Phy sics 26, Springer-Verlag, New York, 165-195 .

Lotz W. (1970) . Electron impact ionization cross section s for Siegbahn K, Allison DA and Allison JH . (1974). ESCA atoms. Z. Phys. 232, 101-107. photoelectron spectro sco py, In : Handbo ok of Spectro scopy, CRC Pre ss, Cleveland , Vol. 1, 257-752 . McGuire EJ. (1971). Inela stic scatteri ng of electrons and pro tons by the elements He to Na. Phys. Rev. A 3, 267-279 . Smith OM and Gallon TE. (1974). Auger emission from solids: The estimation of backscattering effects and ioni za Mott NF and Massey HSW . (1949) . The theory of atomic tion cross section s. J . Phys . D 7, 151-16 1. co llision s, Oxford University Press, London (second edition), 243-244 . Swanson N and Powell CJ. (1968). Excitation of L-shell elec trons in Al and Al 2O by 20 keV electrons. Phys. Rev. 167, Pantano CG and Madey TE (1981). Electron beam damage 3 592-600 . in Auger electron spectroscopy. Appl. Surf. Sci. 7, 115-141. Szajman J and Leckey RCG. (1981). An ana lytical expres Penn DR . (1976). Quantitative chemica l analysis by ESCA . sion for the calcu lation of electron mean free paths in solids. J. Elect. Spect. 9, 29-40. J. Elect. Spect. 23, 83-96. Powell CJ. ( 1969). Analysis of optical- and inelastic-electron Szajman J, Liesegang J , Jenkin JG and Leckey RCG. (198 1). scattering data . Parametric calculations. J . Opt. Soc. Am. ls there a univer sal mean-free-path curve for electron inelas 59, 738-743. tic scattering in solids? J . Elect. Spect. 23 , 97- I 02. Powe ll CJ. (1974). Attenuation lengths of low-energy elec Teo BK and Joy DC. (eds .) (1981). EXAFS spectroscopy, tron s in solids. Surf. Sci. 44, 29-46. technique s and applications, Plenum Press, New York , Powe ll CJ. (1976a). Cross sections for ionizat ion of inner 1-272. she ll electrons by electrons. Rev . Mod. Phys . 48, 33-47. Wagner CD, Riggs WM, Davi s LE, Moulder JF and Muilen Powell CJ. (1976b). Evaluation of form ulas for inner-shell berg GE. (1979). Handbook of x-ray photoelectron spectros ioni za tion cross sections, In: Proceedings of a Work shop on copy, Perkin Elmer-Physical Electronics Division, Eden the Use of Monte Carlo Ca lcu lations in Electron Probe Prairie , Minn. Microanalysis and Scanning Electron Microscopy, K.F .J . Wagner CD, Davi s LE and Riggs WM. (1980). The energy Heinrich, D.E . Newbury, and H . Yakowitz (eds.), U.S. Na dependence of the electron mean free path. Surf. Int erface tional Bureau of Standards Special Publi cation 460, Wash Anal. 2, 53-55. ington , 97-104. Weaver JH, Krafka C, Lynch OW and Koch EE. (1981) . Op Powell CJ. (1978). The phy sical basis for quantitative surface tical propertie s of met als. Physik Oat en-P hysics Data (Fach analysis by Auger electron spectro sco py and x-ray photoel ec informations-zentrum, Karlsru he , W. Germany), 1-302. tron spectro scopy, In: Quantitative Surface Analysis of Material s, N .S. McIntyre (ed.), American Society for Testing Winick H and Doniach S. (eds.) (1980). Synchroton Radia and Material s Special Technical Publication 634, Philadel tion Research, Plenum Pre ss, New York, 1-754. phia, 5-29. Worthington CR and Tomlin SG . (1956). The intensity of Raether H. (I 980) . Excitat ion of plasmon s and interband emission of characteristic x-radiation. Proc. Ph ys. Soc. transitions in solids . Springer Tracts in Modern Physics 88, (London) A 69, 401-412 . Springer-Verlag, New York, 1-196. Ramaker DE, White CT and Murday JS. (1981). Auger induced desorption of covalent and ionic systems. J . Vac . Sci. Tech . 18, 748-749. Rudge MRH and Schwartz SB. (1966) . The ionization of hydrogen and of hydrogenic positive ion s by electron impa ct. Proc. Phys. Soc. (London) 88, 563-578. Schnatte rly SE. (1979). Inelastic scattering spectroscopy, In : So lid State Physics, H. Ehrenreich, F. Seitz and D. Turnbull (eds.), Academic Press, New York, Vol. 34, 275-358. Seah MP and Dench WA. (1979). Quantitative electron spec troscopy of surface s: A stan dard data base for electron mean free paths in solids. Surf. Interface Anal. 1, 2-11. Shiles E, Sasaki T, lnokuti Mand Smith DY. (1980). Self consistency and sum-rule tests in the Kramers-Kronig analys is of optical data: Applications to aluminum . Phys. Rev . B 22, 1612-1628.