Comparison of Electron Elastic-Scattering Cross Sections ...nist.gov/data/PDFfiles/jpcrd648.pdf · C. J. Powellc – Surface and ... Dirac–Hartree–Fock potential, elastic electron

370

Comparison of Electron Elastic-Scattering Cross Sections Calculatedfrom Two Commonly Used Atomic Potentials

A. Jablonski a…

Institute of Physical Chemistry, Polish Academy of Sciences, ul. Kasprzaka 44/52, 01-224 Warsaw, Poland

F. Salvat b…

Facultat de Fisica (ECM), Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain

C. J. Powell c…

Surface and Microanalysis Science Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8

~Received 22 January 2003; revised manuscript received 29 May 2003; accepted 3 June 2003; published online 26 March 2004!

We have analyzed differential cross sections~DCSs! for the elastic scattering of elec-trons by neutral atoms that have been derived from two commonly used atomic poten-tials: the Thomas–Fermi–Dirac~TFD! potential and the Dirac–Hartree–Fock~DHF!potential. DCSs from the latter potential are believed to be more accurate. We comparedDCSs for six atoms~H, Al, Ni, Ag, Au, and Cm! at four energies~100, 500, 1000, and10 000 eV! from two databases issued by the National Institute of Standards and Tech-nology in which DCSs had been obtained from the TFD and DHF potentials. While theDCSs from the two potentials had similar shapes and magnitudes, there can be pro-nounced deviations~up to 70%! for small scattering angles for Al, Ag, Au, and Cm. Inaddition, there were differences of up to 400% at scattering angles for which there weredeep minima in the DCSs; at other angles, the differences were typically less than 20%.The DCS differences decreased with increasing electron energy. DCSs calculated fromthe two potentials were compared with measured DCSs for six atoms~He, Ne, Ar, Kr, Xe,and Hg! at energies between 50 eV and 3 keV. For Ar, the atom for which experimentaldata are available over the largest energy range there is good agreement between themeasured DCSs and those calculated from the TFD and DHF potentials at 2 and 3 keV,but the experimental DCSs agree better with the DCSs from the DHF potential at lowerenergies. A similar trend is found for the other atoms. At energies less than about 1 keV,there are increasing differences between the measured DCSs and the DCSs calculatedfrom the DHF potential. These differences were attributed to the neglect of absorptionand polarizability effects in the calculations. We compare transport cross sections for H,Al, Ni, Ag, Au, and Cm obtained from the DCSs for each potential. For energies between200 eV and 1 keV, the largest differences are about 20%~for H, Au, and Cm!; at higherenergies, the differences are smaller. We also examine the extent to which three quantitiesderived from DCSs vary depending on whether the DCSs were obtained from the TFD orDHF potential. First, we compare calculated and measured elastic-backscattered intensi-ties for thin films of Au on a Ni substrate with different measurement conditions, but it isnot clear whether DCSs from the TFD or DHF potential should be preferred. Second, wecompare electron inelastic mean free paths~IMFPs! derived from relative and absolutemeasurements by elastic-peak electron spectroscopy and from analyses with DCSs ob-tained from the TFD and DHF potentials. In four examples, for a variety of materials andmeasurement conditions, we find differences between the IMFPs from the TFD and DHFpotentials ranging from 1.3% to 17.1%. Third, we compare mean escape depths for twophotoelectron lines and two Auger-electron lines in solid Au obtained using DCSs fromthe TFD and DHF potentials. The relative differences between these mean escape depths

a!Electronic mail: [email protected]!Electronic mail: [email protected]!Electronic mail: [email protected]© 2004 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.

0047-2689Õ2004Õ33„2…Õ409Õ43Õ$39.00 J. Phys. Chem. Ref. Data, Vol. 33, No. 2, 2004409

410410 JABLONSKI, SALVAT, AND POWELL

vary from 4.3% at 70 eV to0.5% at 2016 eV at normal electron emission, and becomesmaller with increasing emission angle. Although measured DCSs for atoms can differfrom DCSs calculated from the DHF potential by up to a factor of 2, we find that theatomic DCSs are empirically useful for simulations of electron transport in solids forelectron energies above about 300 eV. The atomic DCSs can also be useful for energiesdown to at least 200 eV if relative measurements are made. ©2004 by the U.S. Secre-tary of Commerce on behalf of the United States. All rights reserved.@DOI: 10.1063/1.1595653#

Key words: Auger electron spectroscopy, differential cross section, Dirac–Hartree–Fock potential, elasticelectron scattering, elastic-peak electron spectroscopy, electron transport, solid surfaces, surface analysis,Thomas–Fermi–Dirac potential, x-ray photoelectron spectroscopy

2

3

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29

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16

Contents

1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412. Calculation of Differential Elastic-Scattering

Cross Sections by the Partial-Wave ExpansionMethod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3. Phase Shifts Calculated from the Thomas–Fermi–Dirac Potential.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4143.1. Differential Equation. . . . . . . . . . . . . . . . . . . . . 4143.2. Interaction Potential. . . . . . . . . . . . . . . . . . . . . 4163.3. Initial Condition for Integration. . . . . . . . . . . . 4173.4. Upper Limit of Integration. . . . . . . . . . . . . . . . 4173.5. Phase Shifts. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4173.6. Special Functions. . . . . . . . . . . . . . . . . . . . . . . 418

4. Phase Shifts Calculated from the Dirac–Hartree–Fock Potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4184.1. Differential Equation. . . . . . . . . . . . . . . . . . . . . 4184.2. Interaction Potential. . . . . . . . . . . . . . . . . . . . . 4184.3. Initial Condition for Integration. . . . . . . . . . . . 4184.4. Upper Limit of Integration. . . . . . . . . . . . . . . . 4184.5. Phase Shifts. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4194.6. Special Functions. . . . . . . . . . . . . . . . . . . . . . . 419

5. Transport Cross Section. . . . . . . . . . . . . . . . . . . . . . 4196. Comparison of TFD and DHF Potentials. . . . . . . . 4207. Comparisons of Differential Elastic-Scattering

Cross Sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4207.1. Comparisons of Differential Cross Sections

Obtained from the TFD and DHFPotentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

7.2. Differential Cross Sections for Hydrogen. . . . 47.3. Comparisons of Differential Cross Sections

Obtained from Different Evaluations withthe DHF Potential. . . . . . . . . . . . . . . . . . . . . . . 421

7.4. Comparisons of Differential Cross SectionsObtained from the TFD and DHF Potentialswith Measured Differential Cross Sections. . . 4

8. Comparisons of Transport Cross Sections. . . . . . . . 4318.1. Comparisons of Transport Cross Sections

Obtained from the TFD and DHFPotentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431

8.2. Comparisons of Transport Cross SectionsObtained from the TFD and DHF Potentialswith Measured Transport Cross Sections. . . . . 433

J. Phys. Chem. Ref. Data, Vol. 33, No. 2, 2004

9. Comparisons of Quantities Derived fromDifferential Cross Sections that were Obtainedfrom the TFD and DHF Potentials. . . . . . . . . . . . . 4339.1. Elastic-Backscattered Intensities for an

Overlayer Film on a Substrate. . . . . . . . . . . . . 4359.2. Electron Inelastic Mean Free Paths

Determined by Elastic-Peak ElectronSpectroscopy~EPES!. . . . . . . . . . . . . . . . . . . . 4379.2.1. EPES Experiments with a Standard

Material. . . . . . . . . . . . . . . . . . . . . . . . . . 4399.2.2. EPES Experiments without a

Standard Material. . . . . . .. . . . . . . . . . . 4419.3. Mean Escape Depths for AES and XPS.. . . . 442

10. Validity of Calculated DCS Data for Atoms inSimulations of Electron Transport in Solids. . . . . . 444

11. Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4412. Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . 44913. References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

List of Tables1. Main characteristics of published databases

containing differential and/or totalelastic-scattering cross sections. . . . . . . . . . . . . . . . 415

2. Theoretical models used for calculations of theatomic electron density or the atomic potential. . . 4

3. Values of transport cross sections,s trm , derived

from measured differential cross sections for Heand Hg at energies of 50 and 100 eV and thecorresponding calculated transport cross sections,s tr

TFD ands trDHF , obtained from Refs. 10 and

12 with use of the TFD and DHF potentials,respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

4. IMFP values for the alloy Au50Cu50 derivedfrom calibration curves for the EPESexperiments of Krawczyket al.118 that werecalculated using DCSs obtained from the TFDpotential (l in

TFD) and the DHF potential (l inDHF) at

the indicated energies from the measuredratios, I AuCu/I Au , of elastic-backscatteredintensities.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

5

7

7

8

8

8

411411ELECTRON ELASTIC-SCATTERING CROSS SECTIONS

5. IMFP values for copper derived from theabsolute EPES experiments of Dolinskiet al.120

that were calculated using DCSs obtainedfrom the TFD potential (l in

TFD) and the DHFpotential (l in

DHF) at the indicated energies,E, fromthe measured elastic-backscattering probabilities,hCu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

List of Figures1. Comparison of screening functions corresponding

to the potentials used in calculations of thedifferential elastic-scattering cross sections as afunction of radial distance. . . . . . . . . . . . . . . . . . . 422

2. Differential elastic-scattering cross sections,ds/dV, calculated for hydrogen for twopotentials as a function of scattering angle. . . . . . . 423

3. Differential elastic-scattering cross sections,ds/dV, calculated for aluminum for twopotentials as a function of scattering angle:~solid line! DHF potential.. . . . . . . . . . . . . . . . . . . 423

4. Differential elastic-scattering cross sections,ds/dV, calculated for nickel for two potentialsas a function of scattering angle. .. . . . . . . . . . . . . 424

5. Differential elastic-scattering cross sections,ds/dV, calculated for silver for two potentialsas a function of scattering angle. . . . . . . . . . . . . . 424

6. Differential elastic-scattering cross sections,ds/dV, calculated for gold for two potentialsas a function of scattering angle. .. . . . . . . . . . . . . 425

7. Differential elastic-scattering cross sections,ds/dV, calculated for curium for twopotentials as a function of scattering angle. . . . . . . 425

8. Percentage differences between differentialelastic-scattering cross sections,DDCS,calculated for hydrogen from Eq.~18! for theTFD and DHF potentials as a function ofscattering angle,u. . . . . . . . . . . . . . . . . . . . . . . . . . 426

9. Percentage differences between differentialelastic-scattering cross sections,DDCS,calculated for aluminum from Eq.~18! for theTFD and DHF potentials as a function ofscattering angle,u. . . . . . . . . . . . . . . . . . . . . . . . . . 426

10. Percentage differences between differentialelastic-scattering cross sections,DDCS,calculated for nickel from Eq.~18! for the TFDand DHF potentials as a function of scatteringangle,u. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

11. Percentage differences between differentialelastic-scattering cross sections,DDCS,calculated for silver from Eq.~18! for the TFDand DHF potentials as a function of scatteringangle,u. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

12. Percentage differences between differentialelastic-scattering cross sections,DDCS,calculated for gold from Eq.~18! for the TFDand DHF potentials as a function of scatteringangle,u. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

13. Percentage differences between differentialelastic-scattering cross sections,DDCS,calculated for curium from Eq.~18! for the TFDand DHF potentials as a function of scatteringangle,u. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

14. Comparison of phase shifts,d l1, for hydrogen

corresponding to the potentials used in thecalculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

15. Percentage differences between differentialelastic-scattering cross sections,DDCS,obtained from the Berger and Seltzer~Ref. 11!database and the Jablonskiet al. ~Ref. 12!database at energies of 1000 and 10 000 eV fromEq. ~23! as a function of scattering angle,u:~a! hydrogen;~b! aluminum;~c! nickel; ~d! silver;~e! gold; and~f! curium. . . . . . . . . . . . . . . . . . . . . . 430

16. Percentage differences between differentialelastic-scattering cross sections,DDCS, withand without exchange correction calculated atenergies of 1000 and 10 000 eV from Eq.~24! as a function of scattering angle,u: ~a!hydrogen;~b! aluminum;~c! nickel; ~d! silver;~e! gold; and~f! curium. . . . . . . . . . . . . . . . . . . . . . 431

17. Comparison of differential elastic-scatteringcross sections, ds/dV, calculated for heliumfor two potentials as a function of scattering anglewith measured differential cross sections.. . . . . . . 432

18. Comparison of differential elastic-scatteringcross sections, ds/dV, calculated for neonfor two potentials as a function of scattering anglewith measured differential cross sections.. . . . . . . 433

19. Comparison of differential elastic-scatteringcross sections, ds/dV, calculated for argonfor two potentials as a function of scattering anglewith measured differential cross sections.. . . . . . . 434

20. Comparison of differential elastic-scatteringcross sections, ds/dV, calculated for kryptonfor two potentials as a function of scattering anglewith measured differential cross sections.. . . . . . . 435

21. Comparison of differential elastic-scatteringcross sections, ds/dV, calculated for xenonfor two potentials as a function of scattering anglewith measured differential cross sections.. . . . . . . 436

22. Comparison of differential elastic-scatteringcross sections, ds/dV, calculated for mercuryfor two potentials as a function of scattering anglewith measured differential cross sections.. . . . . . . 437

23. Energy dependence of the transport crosssections calculated from the TFD and DHFpotentials:~solid line! the DHF potential;~dashedline! the TFD potential:~a! hydrogen;~b!aluminum;~c! nickel; ~d! silver; ~e! gold; and~f!curium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

24. Energy dependence of percentage differences,Ds tr , from Eq. ~25! between transport crosssections obtained from the TFD and DHFpotentials:~a! hydrogen;~b! aluminum;~c! nickel;


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~d! silver; ~e! gold; and~f! curium. . . . . . . . . . . . . 43925. Relative intensity of electrons elastically

backscattered from a gold overlayer on Ni,R(t)defined by Eq.~26!, as a function of overlayerthickness at energies of:~a! 500 eV and~b!1000eV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

26. Relative intensity of electrons elasticallybackscattered from a gold overlayer on Ni,R(t)defined by Eq.~26!, as a function of overlayerthickness at energies of:~a! 500 eV and~b! 1000eV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

27. Differential elastic-scattering cross sections, ds/dV, calculated for gold from the TFD and DHFpotentials at scattering angles exceeding 90°.. . . . 441

28. Ratio of elastic-backscattering probabilitiesRAu

5I Au /I Ni defined by Eq.~27! as a function ofassumed values of the electron inelastic mean freepath for gold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442

29. Ratio of elastic-backscattering probabilitiesRAuCu5I AuCu/I Au defined by Eq.~27! as afunction of assumed values of the electroninelastic mean free path for the alloy Au50Cu50... 443

30. Percentage differences,DRAuCu, calculated fromEq. ~29! between the calibration curves of Fig.29 for the TFD and DHF potentials as afunction of assumed values of the electroninelastic mean free path for the alloyAu50Cu50.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

31. Plot of the elastic-backscattering probability,hAu , for gold as a function of assumed valuesof the electron inelastic mean free path for normalincidence of the electron beam and aretarding-field analyzer accepting emissionangles between 5° and 55°. . . . . . . . . . . . . . . . . . . 444

32. Ratio of the mean escape depth,D, from Eqs.~31!–~38! to the mean escape depth,D,calculated with neglect of elastic scattering fortwo photoelectron lines and two Auger-electronlines in gold as a function of the emissionangle,a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

33. Percentage differences,DD, between the meanescape depths for gold calculated using the TFDand DHF potentials@from Eq. ~41!# as afunction of emission angle for the twophotoelectron lines and two Auger-electron linesin gold shown in Fig. 32. . . . . . . . . . . . . . . . . . . . . 447

1. Introduction

The elastic scattering of electrons by atoms, molecuand solids is of widespread importance in many applicatiranging from radiation dosimetry, radiation therapy, radiatprocessing, radiation sensors, and radiation protectionelectron-beam lithography, plasma physics, and mateanalysis by techniques such as electron-probe microana~EPMA!, analytical electron microscopy~AEM!, Auger-electron spectroscopy~AES!, and x-ray photoelectron spec


s,s

ntolssis

troscopy~XPS!. For these applications, knowledge of elastscattering cross sections can be needed to obtain informaon the spatial distribution of energy deposition, to enamore accurate analyses, or to define the analytical voluMore specifically, data for differential elastic-scattering crosections, total elastic-scattering cross sections, transcross sections, or phase shifts may be required.

Elastic-electron-scattering phenomena in solids anduids can be very different from the corresponding phenoena for the constituent atoms or molecules in the gas phapressures for which multiple elastic or inelastic scattercan be ignored. For crystalline solids, coherent scatteringthe ordered atoms leads to electron diffraction, a powetool for obtaining information on the atomic structure. As tdegree of atomic order decreases~e.g., for an amorphoussolid, a liquid, or a polycrystalline solid consisting of manrandomly oriented grains or assemblies of molecules!, thediffraction effects become weaker and can be neglectedmany practical purposes. Atomic elastic-scattering crosstions can then be used for describing electron transporsolids. Two other differences between elastic scatteringelectrons by neutral atoms and by solids~differences in in-teraction potential and the occurrence of multiple scatteri!will be discussed below.

We consider here data describing elastic scattering of etrons by neutral atoms for energies between 50 eV andkeV, although particular attention will be given to data fenergies between 50 eV and 10 keV. Differential elasscattering cross sections, total elastic-scattering crosstions, and transport cross sections have been calculatedtabulated elsewhere for many elements, as indicated in T1.1–12 There are also a limited number of measurementsthese cross sections for certain elements and energiesthere are measurements of other parameters for certain sand energies~to be discussed below! that depend on elasticscattering cross sections. The latter measurements, inticular, provide indirect validation of the cross-section daThe present calculations of cross sections are valuable, hever, because they can be made for all elements and fwide range of energies, and are thus of general applicab

There can be noticeable differences among cross-secdata from the sources listed in Table 1. These differenmainly arise from the different theoretical models used incalculations~e.g., relativistic or nonrelativistic theory usefor description of the scattering event! or from differences inelectrostatic potentials describing interaction with the taratom. The main characteristics of the available databasedifferential elastic-scattering cross sections and/or toelastic-scattering cross sections available in the literatureshown in Table 1.

The theoretical models used for the calculation of atomelectron densities and/or atomic potentials are shownTable 2. All of these models disregard electron correlatioi.e., the fact that a single determinant is not the most genwave function for the set of atomic electrons. Correlationpartially accounted for in the so-called multiconfigurationmethods in which the trial function is approximated by

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linear combination of several Slater determinants cosponding to a set of configurations with energies close toof the ground state.

The potentials that are used in published calculationselastic-scattering cross sections can be roughly classifiedtwo groups:

~1! potentials resulting from the statistical model of tatom, i.e. the Thomas–Fermi~TF! potential or theThomas–Fermi–Dirac~TFD! potential; and

~2! potentials derived for atoms from relativistic or nonretivistic Hartree–Fock self-consistent methods, e.g.,relativistic Dirac–Hartree–Fock~DHF! potential ~Table2! considered in the present work.

Elastic-scattering cross sections derived from the TFDtential have been frequently used in theoretical descripti~e.g., by Monte Carlo simulations! of electron transport insolids.13–35Similar calculations have been made with elastscattering cross sections from the DHF potential.36–45Theseand related calculations have been performed for appltions in surface analysis by AES~incident electron energietypically between 5 and 25 keV, and detected electron egies generally from 50 to 2500 eV! and XPS~detected elec-tron energies typically from 200 to 1500 eV!, and in bulk andthin-film analysis by EPMA~incident electron energies typcally between 5 and 30 keV! and AEM ~incident energiestypically between 50 and 300 keV!. Because self-consistencalculations provide a more accurate description of atostructure, we expect that elastic-scattering cross sectbased on the DHF potential will be more accurate than crsections calculated from the TFD potential. As a reselectron-transport calculations are expected to be moreable if the elastic-scattering cross sections were obtafrom the DHF potential rather than from the TFD potentiThe question then arises as to the magnitude of the differebetween these cross sections, particularly in view of thequent use of the latter cross sections.

We make systematic comparisons here of differenelastic-scattering cross sections and of transport crosstions ~derived from the differential cross sections as incated below! from the TFD and DHF potentials in order tdetermine the magnitudes of differences in these crosstions for a range of electron energies. These comparisonsmade for six elements spanning a large range of atomic nber, Z, i.e., H (Z51), Al (Z513), Ni (Z528), Ag (Z547), Au (Z579), and Cm (Z596). Most of the compari-sons are made with cross sections obtained from twotional Institute of Standards and Technology~NIST! data-bases, one which provided cross sections from the Tpotential10 and the other which provided cross sections frothe DHF potential.12 These comparisons are made for eletron energies between 100 and 10 000 eV, the energy racommon to both databases. Additional comparisons hbeen made between cross sections from the latter dataand those from another NIST database11 for which the DHFpotential was also used but for which a different exchancorrection was applied. Comparisons were also made

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tween differential cross sections from the TFD10 and DHF12

potentials and measurements of these cross sections forare gases and mercury.

We begin with a description of calculations of differentielastic-scattering cross sections by the partial-wave expsion method~PWEM! in Sec. 2. Information on calculationof differential cross sections from the TFD and DHF potetials is presented in Secs. 3 and 4, respectively. The transcross section, a useful parameter in descriptions of sigelectron transport in AES and XPS, is defined in Sec. 5.present comparisons of the TFD and DHF potentials forAl, Ni, Ag, Au, and Cm in Sec. 6. Differential cross sectionfor these elements from the TFD10 and DHF12 potentials arepresented and compared in Sec. 7 for electron energie100, 500, 1000, and 10 000 eV. Similar comparisons aremade of differential cross sections from the DHpotential11,12 at energies of 1000 and 10 000 eV to examithe effects of different exchange corrections in the calcutions. In addition, we compare calculated differential crosections from the TFD and DHF potentials with measucross sections for He, Ne, Ar, Kr, Xe, and Hg at selecenergies between 50 and 3000 eV. The latter comparisonmade at energies for which there were two or more indepdent experimental sets of data available. We present compsons of transport cross sections obtained from the TFDDHF potentials in Sec. 8. In Sec. 9, we present comparisof three quantities derived from differential cross sectiothat were obtained from the TFD and DHF potentiaelastic-backscattered intensities for an overlayer film onsurface; electron inelastic mean free paths determined byelastic-peak electron spectroscopy method; and mean esdepths for signal electrons in AES and XPS. Finally, wcomment in Sec. 10 on the validity of differential cross setions for neutral atoms in the simulations of electron traport in solids. Our conclusions are presented in Sec. 11.

2. Calculation of DifferentialElastic-Scattering Cross Sections

by the Partial-Wave Expansion Method

Elastic collisions of electrons with atoms can be describby means of the so-called ‘‘static-field’’ approximation, i.eas scattering of the projectile by the electrostatic field oftarget atom. It is assumed that the atomic electron denr(r ) as a function of radius,r, is spherically symmetric; foratoms with open electron shells, this implies an average oorientations. The potential energy of the projectile,V(r ), isthen also spherical. Exchange between the projectile andatomic electrons can be accounted for approximately by aing a local correction to the electrostatic interaction. In norelativistic ~Schrodinger! theory, the wave function describing the scattering event is a distorted plane wave withasymptotic form

c~r ! →r→`

exp~ iK "r !1exp~ iKr !

rf ~u!, ~1!


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where r is the position vector,p5\K is the momentum ofthe projectile, and\ is the reduced Planck’s constant. Thfirst term in Eq.~1! represents the incident parallel monoeergetic electron beam, and the second term describestrons deflected by the interaction with the scattering cenThe differential cross section~DCS! for the scattering iscompletely determined by the scattering amplitudef (u).

In nonrelativistic theory, the distorted plane waves@Eq.~1!# are solutions of the time-independent Schro¨dinger equa-tion with the potentialV(r ). In the relativistic formulationadopted here,c~r ! is a solution of the time-independenDirac equation with the same potential. To calculateDCS, the distorted wave@Eq. ~1!# is expressed as a superpsition of spherical waves~i.e., eigenfunctions of the totaangular momentum! that are computed numerically by solving the radial wave equation. In fact, only a series ofcalled phase shifts,d l , is required to determine the scatterinamplitude and the DCS.

In the relativistic~Dirac! theory, the elastic-scattering differential cross section is expressed as46

dse /dV5u f ~u!u21ug~u!u2, ~2!

where f (u) and g(u) are the direct and spin-flip scatterinamplitudes. These amplitudes are defined in terms ofphase shiftsd l

1 andd l2 for spin-up and spin-down scattering

respectively

f ~u!51

2iK (l

$~ l 11!@exp~2id l1!21#

1 l @exp~2id l2!21#%Pl~cosu!, ~3!

g~u!51

2iK (l

@exp~2id l2!2exp~2id l

1!#Pl1~cosu!.

~4!

In Eqs. ~3! and ~4!, Pl(u) are Legendre polynomials, anPl

1(u) are the associated Legendre polynomials

Pl1~z!5~12z2!1/2

dPl~z!

dz.

In the nonrelativistic limit, i.e., for slow electrons,f (u) re-duces to the Schro¨dinger scattering amplitude andg(u) van-ishes. Equations~2!–~4! for central fields were first derivedby Mott,47 and the resulting DCS is sometimes referred tothe Mott cross section.

We provide information in the following two sections ocalculations of the phase shifts for the TFD and DHF pottials. These phase shifts fully characterize the elasscattering event. Calogero48 proposed a formalism that provides absolute values of the phase shifts. Fornonrelativistic case, he introduced the phase functiond l(r )that approaches asymptotically the value of the phase shlarge distances from the nucleus

d l5 limr→`

d l~r !,


-ec-r.

e

-

e

s

--

e

at

and that is equal to zero at the nucleus. The relativistic phfunctiond l

6(r ) behaves in a similar manner. The phase funtion is described by a first-order differential equation wthe right-hand side expressed in terms of Riccati–Besfunctions. Integration of this differential equation, howevis very slowly convergent, and the method cannot be reaused for practical calculations.49

The reliability of the calculated DCSs is determinemainly by the accuracy of the adopted interaction potentAtomic potentials derived from statistical models, mostly tThomas–Fermi–Dirac model,50,51 are well described bysimple analytical expressions. Due to this fact, they habeen frequently used in calculations of elastic-scattercross sections. Relativistic self-consistent calculations pvide a more accurate description of the atomic structure;relevant algorithms, however, can be very involved. Fordatabases of Refs. 9 and 12, atomic electron densities calated with the multiconfiguration Dirac–Hartree–Fock coof Desclaux52 were used. This code provides advancatomic-structure calculations that account for exchangerelativistic effects in a consistent way. The program isimplementation of the multiconfigurational model mentionabove. In fact, one should refer to it as the multiconfiguratDirac–Fock model. We also note here that coupling betwelastic and available inelastic channels has generally bignored in DCS calculations. Such coupling could occur,example, at energies near thresholds for inner-shell exction and lead to negative-ion inner-shell resonances inelastic channel. The calculated DCSs~e.g., in Refs. 10 and12! may then have larger uncertainties at energies whsuch coupling can occur.

3. Phase Shifts Calculatedfrom the Thomas–Fermi–Dirac Potential

3.1. Differential Equation

Lin et al.53 and Bunyan and Schonfelder54 developed aneffective algorithm for calculations of relativistic differentiaelastic-scattering cross sections. Details of the corresponcalculations, with the TFD potential to describe the electroatom interaction, were published by Jablonski22 and byJablonski and Tougaard.55 Elastic-scattering cross sectionderived from this algorithm have been successfully usednumerous studies of elastic interactions of electrons wsolids;22,27–33cross sections obtained in a similar way haalso been applied to problems of the same type.13–21,23–26

Elastic-scattering cross sections derived from the TFD potial were made available in two versions of a NIST elastscattering cross-section database.8,10The main features in thecalculations of phase shifts for determination of these crsections are briefly described below.

The Dirac equation for an electron interacting with tatomic potentialV(r ) can be reduced to a first-order diffeential equation53,54

ev.


TABLE 1. Main characteristics of published databases containing differential and/or total elastic-scattering cross sections

Authors Ref. Theoretical model PotentialEnergies

~eV! Elements

M. Fink and A. C. Yates 1 Relativistic PWEM Relativistic Hartree–Fock–Slatera 100–1500 H, He, C, Ne, Ar,Kr, Rb, Xe, Cs, Au,Hg, Pb, Bi

M. Fink and J. Ingram 2 Relativistic PWEM Relativistic Hartree–Fock–Slatera 100–1500 Be, N, O, Al, Cl, V,Co, Cu, As, Nb, Ag,Sn, Sb, I, Ta

D. Gregory and M. Fink 3 Relativistic PWEM Relativistic Hartree–Fock–Slatera

NonrelativisticHartee–Fockb

100–1500 Li, Na, Mg, P, K,Ca, Sc, Mn, Ga, Br,Sr, Mo, Rh, Cd, Ba,W, Os

M. E. Riley,C. J. MacCallum,and F. Biggs

4 Relativistic PWEM Relativistic Harteree–Fockc

NonrelativisticHartree–Fockd

1000–256 000 He, Be, C, N, O, F,Ne, Al, Si, Ar, TiFe, Ni, Cu, Kr, Mo,Ag, Sn, Xe, Ta, Au,Hg, Pb, UFits made for 80elements

L. Reimer and B. Lo¨dding 5 Relativistic PWEMWKB methodj

Hartree–Fockf 1000–100 000 C, Al, Si, Ti, Fe,Cu, Ge, Mo, Ag,Sb, Au, Pb, U

Thomas–Fermi–Dirace

Z. Czyzewski,D. O. MacCallum,A. Romig, and D. C. Joy

6 Relativistic PWEM Thomas–Fermi–Dirace

Hartree–Fockf

RelativisticHartree–Fockg

20–20 000 All elements~frominternet site!

M. J. Berger,S. M. Seltzer, R. Wang,R. Wang, and A. Schechter

7 Relativistic PWEM Dirac–Hartree–Fockh 1000–1 000 000 All elements~usingenclosed software!.Tables forH, He, Be, C, N, O,Ne, Al, Si, Ar, Ti,Fe, Cu, Ge, Kr, Mo,Ag, Xe, Ta, W, Au,Pb, U

A. Jablonskiand S. Tougaard

8 Nonrelativistic PWEMRelativistic PWEM

Thomas–Fermi–Dirace,i 50–9999 All elements

R. Mayol and F. Salvat 9 Relativistic PWEM Dirac–HartreeFockh

100–100 000 000 All elements

A. Jablonskiand C. J. Powell

10 Relativistic PWEM Thomas–FermiDirace,i

50–20 000 All elements

M. J. Bergerand S. M. Seltzer

11 Relativistic PWEMWKB methodj

Dirac–Hartree–Fockh 1000–100 000 000 All elements~usingenclosed software!

A. Jablonski, F. Salvat,and C. J. Powell

12 Relativistic PWEM Dirac–Hartree–Fockh 50–300 000 All elements

aD. Liberman, J. T. Waber, and D. T. Cromer, Phys. Rev.137, A27 ~1965!; R. D. Cowan, A. C. Larson, D. Liberman, J. B. Mann, and J. Waber, Phys. R144, 5 ~1966!.

bE. Clementi,Tables of Atomic Functions~International Business Machines Corp., San Jose, CA., 1965!.cJ. B. Mann and J. T. Waber, At. Data5, 201 ~1973!.dJ. B. Mann, At. Data Nucl. Data Tables12, 1 ~1973!.eR. A. Bonham and T. G. Strand, J. Chem. Phys.39, 2200~1963!.fH. L. Cox and R. A. Bonham, J. Chem. Phys.47, 2599~1967!.gA. W. Ross and M. Fink, J. Chem. Phys.85, 6810~1986!.hJ. P. Desclaux, Comput. Phys. Commun.9, 31 ~1975!.iL. H. Thomas, J. Chem. Phys.22, 1758~1954!; A. Jablonski, Physica A183, 361 ~1992!.jWentzel–Kramers–Brillouin.


tial


J. Phys. Chem. Ref

TABLE 2. Theoretical models used for calculations of the atomic electron density or the atomic poten

Name~s! and abbreviation Short description

Thomas–Fermi~TF! Statistical model. The electron cloud isconsidered as a locally homogeneous electron gasconfined by the screened field of the nucleus.

Thomas–Fermi–Dirac~TFD! Modification of the TF model that includes theeffect of electron exchange using the exchangeenergy density derived by Dirac.

Hartree~H! The simplest self-consistent method.Nonrelativistic. The Hartree equations areobtained from the variational method with a trialfunction equal to the~nonantisymmetrized!product of orbitals.

Hartree–Fock~HF! Nonrelativistic. The HF equations are obtained byapplying the variational method with a trialfunction expressed as a single Slater determinant,i.e., as the antisymmetrized product of orbitals.The antisymmetry of the wave function accountsfor electron exchange effects in a systematic way.

Hartree–Fock–Slater~HFS!or Hartree–Slater~HS!

Nonrelativistic. The exchange potential in the HFequations is replaced by the local approximationof the Slater~proportional tor1/3, similar to theDirac correction in the TFD model!. The resultingequations are much simpler to solve than theH and HF equations.

Dirac–Hartree–Fock~DHF!or Dirac–Fock~DF!

Relativistic version of the HF method. Includesexchange in the natural way, i.e., the wavefunction is a Slater determinant with orbitals thatare solutions of the Dirac equation with anonlocal exchange operator.

Dirac–Hartree–Fock–Slater~DHFS!or Dirac–Fock–Slater~DFS!

Relativistic version of the HFS or HS method;usually the nonrelativistic local exchangepotential of Slater is adopted

ffi

nits

,

dF l6~r !

dr5

k6

rsin@2F l

6~r !#1@W2V~r !#

2cos@2F l6~r !#, l 50,1,2,..., ~5!

whereW is the total energy of the electron, and the coecientsk6 are defined as follows

k152~ l 11! for the ‘‘spin-up’’ case, j 5 l 1 12,

ln

. Data, Vol. 33, No. 2, 2004

-

k25 l for the ‘‘spin-down’’ case, j 5 l 2 12.

The system of units is such that energy is expressed in uof mc2 and lengthr is expressed in units of\/(mc), wherem is the electron mass andc is the velocity of light invacuum. For each quantum numberl, there are two solutionsF1(r ) andF2(r ), which are related to thelth order spin-upand spin-down phase shifts,d l

6

d l65tan21

K j l 11~Kr !2 j l~Kr !@~W11!tanF l61~11 l 1k6!/r #

Knl 11~Kr !2nl~Kr !@~W11!tanF l61~11 l 1k6!/r #

, ~6!

of

where K25W221, j l(x) and nl(x) are spherical Bessefunctions, andF l

6 is the asymptotic value of the solutioF l

6(r )

F l65 lim

r→`

F l6~r !.

The expression on the right-hand side of Eq.~6! is calculatedfor values ofr for which V(r )50.53 The Thomas–Fermi–Dirac potential has a finite maximum radiusr 0 , and Eq.~6!

is evaluated for this radius~see Sec. 3.4!. The Sarafyan–Butcher embedding formula56 was found to be very effectivefor integrating Eq.~5!.

3.2. Interaction Potential

The interaction potential,V(r ), was approximated by theThomas–Fermi–Dirac potential. The simple analytical fitBonham and Strand57 was used in the calculations

nt

umehi


V~r !52Ze2

rC~r !, ~7!

wheree is the electronic charge and

C~r !5(i 51

3

pi exp~2qir ! ~8!

is the screening function, i.e., the function that takes iaccount the screening by the atomic electrons. In Eq.~8!, pi

andqi are fitted parameters that depend on the atomic nber. Jablonski58 has shown that this expression well describthe TFD potential for all elements except hydrogen. For telement, the fitted parameterspi and qi published byJablonski58 are recommended.

3.3. Initial Condition for Integration

The initial value of the solutionF l6(r ) for small r has

been calculated from the series expansion

s

tia

a

o

-ss

F l6~r !5F01F1r 1F2r 21F3r 31¯ .

The coefficientsF0 , F1 , F2 , andF3 were derived by Bu-nyan and Schonfelder54

sin~2F0!52Z0 /k6,

F15W1Z12cos~2F0!

122k6 cos~2F0!,

F252F1 sin~2F0!~12k6F1!1Z2

222k6 cos~2F0!,

F352F2 sin~2F0!~122k6F1!12F1

2 cos~2F0!~12 32k

6F1!1Z3

322k6 cos~2F0!,

10y to

V,han

where the parametersZ0 , Z1 , Z2 , andZ3 are the coefficientsof the series expansion of the potentialV(r ) for small valuesof r

V~r !521

r~Z01Z1r 1Z2r 21Z3r 31¯ !.

3.4. Upper Limit of Integration

The TFD potential has a finite maximum radius,r 0 . Forthis reason, the integration of Eq.~5! was performed up tothe radiusr 5r 0 . Values ofr 0 were taken from compilationspublished by Thomas59 and Jablonski.58

The r 0 values are independent of electron energy. Forlected elements, these values~in units of the Bohr radius,a0)are as follows58,60

for Z51 r 052.975,

for Z513 r 054.156,

for Z528 r 054.442,

for Z547 r 054.614,

for Z579 r 054.771,

for Z596 r 054.826.

3.5. Phase Shifts

A considerable number of phase shiftsd l6 was found to be

necessary to reach sufficient accuracy of the differenelastic-scattering cross section calculated from Eqs.~2!–~4!.For energies up to 10 keV, the partial-wave series w

e-

l

s

summed with an accuracy of eight decimal places. AtkeV, the following numbers of phase shifts were necessarreach this accuracy

for Z51 0< l<95,

for Z513 0< l<126,

for Z528 0< l<133,

for Z547 0< l<137,

for Z579 0< l<141,

for Z596 0< l<142.

In the energy range 10 keV,E<20 keV, the summation wasperformed with an accuracy of six decimal places. At 20 kethe number of needed phase shifts was distinctly larger tfor 10 keV

for Z51 0< l<131,

for Z513 0< l<175,

for Z528 0< l<183,

for Z547 0< l<191,

for Z579 0< l<196,

for Z596 0< l<199.


inn

la

at

nc

-s-n

ed

a

p

-y

sx.

lec-x-

elf-eeness

-

by

an-,theythe


3.6. Special Functions

The algorithm used for calculating the elastic-scattercross sections requires routines to provide the special futions j l(x), nl(x), Pl(x), andPl

1(x) @Eqs.~3!, ~4!, and~6!#.Care was taken to ensure that these functions were calcuwith an accuracy of ten decimal places for 0< l<200. De-tails of the analysis and the resulting algorithm for calculing the function j l(x) have been published separately.60 Asimilar analysis was made for the remaining special futions ~see Jablonski and Tougaard9!.

4. Phase Shifts Calculatedfrom the Dirac–Hartree–Fock Potential

The calculation of phase shifts~and thus the elasticscattering DCSs! from the Dirac–Hartree–Fock potential esentially follows the scheme described by Salvat aMayol.61

4.1. Differential Equation

The phase shiftsd l6 are obtained from the larger behavior

of the radial wave functions,Pl6(r ) and Ql

6(r ), which arecalculated by integrating the radial Dirac equations

dPl6

dr52

k6

rPl

6~r !1E2V12mc2

c\Ql

6~r !, ~9a!

dQl6

dr52

E2V

c\Pl

6~r !1k6

rQl

6~r !, ~9b!

whereE is the kinetic energy of the projectile that is relatto its total energyW by E5W2mc2. The solution algorithmimplements Bu¨hring’s power series method62 and is based ona cubic-spline interpolation of the potential functionrV(r )that is tabulated on a dense grid ofr values

r 150,r 2¯,r N21,r N . ~10!

That is, between each pair of consecutive grid points,r n andr n11 , the potential function,rV(r ), is represented aspiecewise cubic polynomial inr

rV~r !5an1bnr 1cnr 21dnr 3. ~11!

Then, in the interval (r n ,r n11), the radial functions can beformally expressed by the power series

Pl6~r !5r a(

i 50

`

piri

and

Ql6~r !5r b(

i 50

`

qiri , ~12!

with coefficients determined by the values ofPl6(r n) and

Ql6(r n) at the end point of the interval~see Salvat and

Mayol63!. As these series expansions can be summed u


gc-

ted

-

-

d

to

the required accuracy~nine significant digits in the corresponding PWEM code!, truncation errors are practicallavoided.

4.2. Interaction Potential

The interaction potential is defined as

V~r !52ew~r !1Vexc~r !, ~13!

wherew(r ) is the electrostatic potential of the target atom

w~r !5Ze

r2eS 1

r E0

r

r~r 8!4pr 82dr 81Er

`

r~r 8!4pr 8dr 8D ,

~14!

and wherer(r ) is the atomic electron density which wacalculated using the self-consistent DHF code of Desclau52

The termVexc(r ) in Eq. ~13! is a local approximation tothe exchange interaction between the projectile and the etrons in the target; it should not be confused with the echange interaction considered in the TFD model and in sconsistent calculations that accounts for exchange betwatomic electrons. We use the exchange potential of Furnand McCarthy64

Vexc~r !51

2@E1ew~r !#

21

2 H @E1ew~r !#214p\2e2

mr~r !J 1/2

.

~15!

The interaction potential@Eq. ~13!# is thus completely determined by the atomic densityr(r ).

4.3. Initial Condition for Integration

The integration of the radial equations is started fromr50, with boundary values

Pl6~0!50,

and

Ql6~0!50. ~16!

In the first r interval from r 150 to r 2 @Eq. ~10!#, the con-stantsa andb are different from zero and are determinedthe angular-momentum quantum numbers~see Salvat andMayol63!. For the outer intervals (r .r 2), a5b50.

4.4. Upper Limit of Integration

After determining the values of the radial functions atr 2 ,the solution is extended outwards by using the series expsions Eq.~12! ~with a5b50! up to a certain radial distancer max, large enough to ensure that the potential energy ofelectronV(r ) is negligible as compared to its kinetic energE. For selected elements and at an energy of 10 keV,maximum ranges were the following~in a0 units!

ct

nte

s

beve

henc-ecur-3 or

thein

rs

al-

-nteicintom.eann

er

ring


for Z51 r max512.27,

for Z513 r max520.15,

for Z528 r max517.85,

for Z547 r max519.37,

for Z579 r max517.11,

for Z596 r max521.54.

At 20 keV

for Z51 r max511.96,

for Z513 r max519.75,

for Z528 r max517.48,

for Z547 r max518.80,

for Z579 r max516.74,

for Z596 r max521.13.

We see that these ranges are considerably larger thanmaximum radius,r 0 , of the TFD potential in Sec. 3.4.

4.5. Phase Shifts

At sufficiently large distancesr, the radial functionPl6(r )

adopts the asymptotic form

Pl6~r !> sinS Kr 2 l

p

21d l

6D , ~17!

where

\K51

cAE~E12mc2! ~18!

is the momentum of the projectile. Equation~17!, which con-fers a geometrical meaning to the phase shift, is not direusable to computed l

6 becausePl6(r ) reaches the form given

by Eq. ~17! only at distancesr that may be much larger thar max. Instead, the phase shift is obtained from the calculavalues of the radial functions atr max by matching the nu-merical solution to the exact solution forr .r max(V50) thatcan be expressed as a linear combination of spherical Befunctions j k(Kr ) andnk(Kr ) with indicesk5 l , l 61.59

The DHF potential requires a considerably larger numof phase shifts than the TFD potential. At 10 keV, we ha

for Z51 0< l<221,

for Z513 0< l<386,

for Z528 0< l<343,

for Z547 0< l<369,

for Z579 0< l<325,

for Z596 0< l<422.

At 20 keV

the

ly

d

sel

r

for Z51 0< l<307,

for Z513 0< l<538,

for Z528 0< l<478,

for Z547 0< l<514,

for Z579 0< l<452,

for Z596 0< l<589.

4.6. Special Functions

The only special functions used in calculations of tDCSs for the DHF potential are the spherical Bessel futions j l(x) andnl(x). The algorithm used to compute thesfunctions combines several analytical expressions and rerence relations, and yields results that are accurate to 1more significant digits for ranges ofl and x values used inthe present calculations.65

5. Transport Cross Section

Values of the transport cross section are needed inanalytical formalism describing signal-electron transportAES and XPS.66,67 Determination of numerous parameterelated to this application~e.g., depth distribution functionfor the signal electrons,68–70effective attenuation length,70–72

signal-electron mean escape depth,31 information depth,73

etc.! requires knowledge of the transport cross section vues.

The transport cross section~TCS! is defined as the quotient of the fractional momentum loss of a particle incideon the sample arising from elastic scattering by the ardensity of the sample atoms, for an infinitesimally thsample.74 This cross section is expressed as an area per aIn other words, the transport cross section describes the mfractional momentum loss with respect to initial directiodue to the elastic scattering alone. Let us denote byk theelectron momentum before elastic scattering, and byk8 theprojection on the initial direction of the momentum aftelastic scattering. Obviously,k85k cosu. Let us further de-note the fractional momentum loss, due to elastic scattealone, by

Dk5uk2k8u

uku.

We may consider the transport cross section,s tr , as theproduct of the total elastic-scattering cross section,sel , andthe mean fractional momentum loss,^Dk&

s tr5sel Dk&5sel

*4pDk~ds/dV!dV

*4p~ds/dV!dV. ~19!

Equation~19! can be transformed to

s tr52pE0

p

~12cosu!~ds/dV!sinu du. ~20!


erleependnc

leheiotacdn

--

t

.t.th

shadiathee

ie

nasi

ticHe

rissix

ngD

re

entonsgiesdfor

pe

for

thethever,tor aer-om

nc-red

ly,ilarthe

re

w-tialized

es.ski

no-es

y-aren-

de-has


As indicated frequently in the literature,6,22,33,58a change ofthe atomic potential~e.g., from the TFD potential to the DHFpotential! can produce a pronounced variations of the diffential cross section in the range of small scattering angDifferences in the remaining angular range, with the exction of deep minima in the DCS, are rather small. Howevthe value of the DCS in the region of minima is small, athus the DCS in this region does not appreciably influethe TCS. In the integrand of Eq.~20!, we have two functionsapproaching zero in the region of small scattering angsinu and ~12cosu!. Consequently, we may expect that tsensitivity of the transport cross section to the interactpotential will be much weaker than the sensitivity of the toelastic-scattering cross section. It has been shown in a rework that, indeed, values of the transport cross sectionpend only weakly on the potential for different elements afor a wide range of electron energies.33

6. Comparison of TFD and DHF Potentials

Figures 1~a!–1~f! show the TFD screening function calculated from Eq.~8! and the electrostatic part of the DHF interaction potential,rw(r )/Ze, calculated from Eq.~14! forH, Al, Ni, Ag, Au, and Cm. The TFD potential is plotted outo the atomic radiusr 0 ~Sec. 2.4!. The DHF potential isshown only out to a distance of 6 Bohr radii~6 a.u.! which isslightly larger than the radiir 0 for the considered elements

We see that both potentials agree well for small radii, up1 a.u., where the screening function has its largest valueslarger distances, there is a growing difference betweentwo potentials. There seems to be no systematic relationbetween the potentials. There are different ranges of rwhere the TFD potential is larger than the DHF potentwhile in other ranges the DHF potential is larger thanTFD potential. The lack of clear trends in the differencbetween the TFD and DHF potentials can be partially undstood by recalling that the TFD electron density varmonotonically with the atomic numberZ, whereas the DHFdensity is affected by ‘‘shell’’ effects, i.e., the contributiofrom each shell has a characteristic shape with maximaminima. As a result, the shape of the atomic electron denvaries from one element to the next.

7. Comparisons of DifferentialElastic-Scattering Cross Sections

We present a series of comparisons of differential elasscattering cross sections calculated from the TFD and Dpotentials and of these computed cross sections with msured cross sections. First, we make a systematic compaof DCSs calculated from the TFD and DHF potentials forelements~H, Al, Ni, Ag, Au, and Cm! at four representativeelectron energies~100, 500, 1000, and 10 000 eV!. Thesecross sections were obtained from Version 2.010 and Version3.0,12 respectively, of the NIST Electron Elastic-ScatteriCross-Section Database. Second, we comment on the


-s.-

r,

e

s:

nlente-d

oAteipii

l,esr-s

ndty

-Fa-on

CS

values from the two potentials for H. Third, we compaDCSs from two NIST databases11,12for which the same DHFpotential was used in the calculations but where differnumerical procedures were employed. These compariswere made for the same six elements at two electron ener~1000 and 10 000 eV!. Finally, we compare DCSs calculatefrom the TFD and DHF potentials with measured DCSsseven elements~He, Ne, Ar, Kr, Xe, Au, and Hg!.

7.1. Comparisons of Differential Cross SectionsObtained from the TFD and DHF Potentials

A change of the potential leads to variations in the shaof the calculated differential cross sections~DCSs!.6,8,22,33,58

However, comparisons of DCSs have been made onlyselected cases. For example, Jablonski and Powell33 showedthat the largest differences between DCSs resulting fromDirac–Hartree–Fock–Slater and TFD potentials occur inrange of small scattering angles; this comparison, howewas limited to a single energy of 1000 eV. We intend heresystematically analyze the differences between DCSs fowide range of atomic numbers and for a wide range of engies. Let us consider now the cross sections resulting frthe two described algorithms.

Figures 2–7 show the differential cross sections, as futions of scattering angle, calculated for the six consideelements at electron energies,E, of 100, 500, 1000, and10 000 eV using the TFD and DHF potentials. Qualitativethe cross sections from the two potentials have simshapes and have similar magnitudes. At low energies,DCSs exhibit one or more minima~with the exception ofhydrogen!. The energy range over which the minima apresent depends on the atomic number (E,690 eV for Al,E,2600 eV for Ni, E,6600 eV for Ag,E,24 000 eV forAu, andE,48 000 eV for Cm!. Both potentials lead to thesame number of minima~for a given element and energy!,and their positions depend only slightly on potential. Hoever, there are pronounced differences in the differencross sections for the two potentials that can be summaras follows:

~1! In most cases~Al, Ag, Au, and Cm!, pronounced devia-tions are visible in the range of small scattering anglThis result is consistent with observation of Jablonand Powell33 at 1000 eV.

~2! Considerable differences between DCS values areticeable in the vicinity of deep minima. Such differencmay exceed more than 1 order of magnitude@see theDCS values for Ni at 100 eV in Fig. 4~a!#.

~3! We see slight oscillations in the DCSs calculated for hdrogen using the TFD potential. These oscillationsnot visible in the DCSs calculated from the DHF potetial.

~4! The deviations between the cross sections tend tocrease with increase of electron energy. This resultalso been reported by Jablonski.22

ioLe

-Fndviveo-en

iuhia

eytioortrod

mie

ileehensn

als

wren

thallo

th

a-icl

e

eiser-

re

FD

torys

CSs

lcu-

ti-

tofer-ifts,forif-ing

thmaseinmuchcesnsns

xi-s

thes athande-

ns

therded


A useful measure of the difference between cross sectfor the two potentials is the relative percentage deviation.us define this deviation,DDCS, by

DDCS5100~DCSrelTFD2DCSrel

DHF!/DCSrelDHF, ~21!

where DCSrelTFD and DCSrel

DHF denote the relativistic differential elastic-scattering cross sections calculated from the Tand DHF potentials, respectively. The relative deviatioDDCS for H, Al, Ni, Ag, Au, and Cm at 100, 500, 1000, an10 000 eV are shown in Figs. 8–13. We see that the detions in the region of small scattering angles may reach e70% ~e.g., Al or Cm at 100 eV!. These deviations are assciated with the ‘‘tail’’ of the scattering potential for largradii. For example, the DHF potential for Al is larger thathe TFD potential for radii greater than about 1 Bohr rad@cf. Fig. 1~b!#, and consequently the DCS calculated from tDHF potential is larger than the DCS from the TFD potentfor small scattering angles~see Figs. 3 and 9!. The largestpercentage deviations, however, occur in the region of dminima in the DCSs~Figs. 3–7!, where the deviations mareach even 400%. For other scattering angles, the deviadepend on energy, and usually do not exceed 20%. Fgiven atom, the deviations decrease with increasing elecenergy due to the increasing number of partial waves ctributing to the DCS. As each partial wave corresponroughly to a particular impact parameter, the whole voluof the atom is explored in great detail at the higher energby the projectile wave function. As a result, the projectexperiences an average potential that is nearly the samthe DHF and TFD potentials. For H, the oscillations in tTFD DCSs are amplified when plotting the deviatioDDCS. As one can see in Fig. 8, the number of oscillatiodepends on energy, reaching 26 at 10 keV.

DCSs from the TFD and DHF potentials for additionelements are presented in Sec. 7.4 where comparisonmade with measured DCSs.

7.2. Differential Cross Sections for Hydrogen

Let us compare the phase shiftsd l1 calculated for hydro-

gen from the TFD and DHF potentials. The results are shoin Fig. 14. At all considered energies, the phase shifts agreasonably well up to a certain value of the angular momtum quantum numberl 5 l 0 . For largerl values, values ofd l

1

calculated for the TFD potential decrease sharply whilephase shifts for the DHF potential decrease exponentiover a wide range ofl. The oscillations in the DCS potentiaarise from the fact that there is not a sufficient numberLegendre functions in the series expressed by Eqs.~3! and~4!.

The most conspicuous difference between the TFD andDHF potentials is that the former drops to zero atr 5r 0 ,whereas the DHF potential approaches zero asymptotic~i.e., for r→`, cf. Fig. 1!. The finite range of the TFD potential has a direct influence on the phase shifts. On classgrounds, we expect the phase shifts for the TFD potentia

nst

Ds

a-n

sel

ep

nsa

onn-ses

for

s

are

nee-

ely

f

e

lly

alto

vanish forl . l 0 , wherel 0\ is the angular momentum of thprojectile corresponding to the impact parameterr 0 , i.e.,

l 05pr0

\>A 2E

27.21

r 0

a0, ~22!

where p5\K is the momentum of the projectile, and thkinetic energyE is expressed in units of eV. This situationillustrated in Fig. 14 which displays phase shifts for scatting by hydrogen atoms obtained from the TFD (r 0

52.97a0) and DHF potentials. The TFD phase shifts aseen to fall rapidly forl values of aboutl 0 , as predicted bythe classical picture. As the partial-wave series for the Tpotential effectively converges with aboutl 0 terms, the dif-ference between the TFD and DHF DCSs has an oscillastructure~see Figs. 2 and 8!, and the number of oscillationincreases withE.

7.3. Comparisons of Differential Cross SectionsObtained from Different Evaluations

with the DHF Potential

As shown in Secs. 7.1 and 7.2, differences between D~for the same element and energy! can be mainly ascribed totwo sources:

~1! differences between atomic potentials used in the calations, and

~2! differences in the maximum radius of the potential ulized in the calculations.

We expect, however, that other factors could contributethe observed differences. One possible factor could be difences among algorithms used for calculating the phase shin particular, differences in numerical procedures usedintegrating the differential equation. The resulting DCS dferences can to a large extent be controlled, e.g., by varythe integration step. In the present analysis, the algoridescribed in Sec. 4 was used in calculations of the phshifts for the TFD potential. The results obtained wereagreement to within 3–4 decimal places with results frothe algorithm described in Sec. 3. Thus, we expect that serrors are not significant here. A second factor for differenin DCSs could be associated with different approximatioused for the exchange interaction, which in the calculatiowith the DHF potential is accounted for by the local appromation @Eq. ~15!# and is neglected in the DCS calculationusing the TFD potential@Eq. ~7!#. A final factor could berelated to the interpolation procedure that is needed inprogram running the database, i.e., to provide DCS valuefiner intervals of electron energy and scattering angle tthose used for the main DCS calculations. Considerableviations ~particularly relative deviations! could occur in theregions of deep minima in the DCSs as a result of limitatioof the interpolation procedure.

To examine changes in DCSs that arise from reasons othan the potentials, we decided to compare DCSs proviby two NIST databases listed in Table 1:


-sla


FIG. 1. Comparison of screening functions corresponding to the potentialused in calculations of the differentiaelastic-scattering cross sections asfunction of radial distance~in units ofthe Bohr radius,a0): ~solid line! theDHF potential;~dashed line! the TFDpotential:~a! hydrogen;~b! aluminum;~c! nickel; ~d! silver; ~e! gold; and~f!curium. Note the finite range of theTFD potential.

a,on

i-

of

t-al-ofs–

ase

aseies:

~1! the Berger and Seltzer database,11 and~2! the Jablonski, Salvat and Powell database.12

The DCSs in each database were calculated using the sDHF potential~for each atom!. Different numerical methodshowever, were used for calculating the DCSs. In additithe exchange corrections adopted in the two databasesdifferent. Berger and Seltzer11 used a high-energy approxmation from Riley and Truhlar75 @Eq. ~9a! in Ref. 11#,whereas Jablonskiet al.12 used the exchange correction


me

,are

Furness and McCarthy64 @Eq. ~15!#. As shown by Riley andTruhlar,75 their high-energy approximation is the lowesorder term in the expansion of the Furness–McCarthy locexchange potential in powers of the inverse kinetic energythe projectile. Therefore, at low energies, the FurnesMcCarthy potential is expected to yield more accurate phshifts and DCSs.

Since the lowest energy in the Berger and Seltzer databis 1 keV, our DCS comparisons were made for two energ


FIG. 2. Differential elastic-scattering cross sections, ds/dV, calculated for hydrogen for two potentials as a function of scattering angle:~solid line! DHFpotential;~dashed line! TFD potential:~a! energy of 100 eV;~b! energy of 500 eV;~c! energy of 1000 eV; and~d! energy of 10 000 eV.

FIG. 3. Differential elastic-scattering cross sections, ds/dV, calculated for aluminum for two potentials as a function of scattering angle:~solid line! DHFpotential;~dashed line! TFD potential:~a! energy of 100 eV;~b! energy of 500 eV;~c! energy of 1000 eV; and~d! energy of 10 000 eV.



FIG. 4. Differential elastic-scattering cross sections, ds/dV, calculated for nickel for two potentials as a function of scattering angle:~solid line! DHF potential;~dashed line! TFD potential:~a! energy of 100 eV;~b! energy of 500 eV;~c! energy of 1000 eV; and~d! energy of 10 000 eV.

FIG. 5. Differential elastic-scattering cross sections, ds/dV, calculated for silver for two potentials as a function of scattering angle:~solid line! DHF potential;~dashed line! TFD potential:~a! energy of 100 eV;~b! energy of 500 eV;~c! energy of 1000 eV; and~d! energy of 10 000 eV.



FIG. 6. Differential elastic-scattering cross sections, ds/dV, calculated for gold for two potentials as a function of scattering angle:~solid line! DHF potential;~dashed line! TFD potential:~a! energy of 100 eV;~b! energy of 500 eV;~c! energy of 1000 eV; and~d! energy of 10 000 eV.

FIG. 7. Differential elastic-scattering cross sections, ds/dV, calculated for curium for two potentials as a function of scattering angle:~solid line! DHFpotential;~dashed line! TFD potential:~a! energy of 100 eV;~b! energy of 500 eV;~c! energy of 1000 eV; and~d! energy of 10 000 eV.


n-

g


FIG. 8. Percentage differences betweedifferential elastic-scattering cross sections, DDCS, calculated for hydrogenfrom Eq. ~18! for the TFD and DHFpotentials as a function of scatterinangle,u: ~a! energy of 100 eV;~b! en-ergy of 500 eV;~c! energy of 1000 eV;and ~d! energy of 10 000 eV.

ucth.ts

ssw

c

om

g,forced

heles.les

ntal-the

1 and 10 keV. Generally, the observed deviations are msmaller than those found in comparisons of DCSs fromTFD and DHF potentials that were considered in Sec. 7The differences are not visible in semilogarithmic ploshowing the DCSs as functions of scattering angle~such asFigs. 2–7!; the plots seem to be identical within the thickneof the lines. Thus, as a measure of relative deviations,used an expression similar to Eq.~22!

DDCS5100~DCSBSDHF2DCSJSP

DHF!/DCSJSPDHF, ~23!

where DCSBSDHF is the differential elastic-scattering cross se

tion taken from the Berger and Seltzer database, and DCSJSPDHF

is the differential elastic-scattering cross section taken fr


he1.

e

-

the Jablonski, Salvat, and Powell database. Plots ofDDCSare shown as a function of scattering angle for H, Al, Ni, AAu, and Cm at the two energies in Fig. 15. We see thatelements and energies where there are no pronounminima in the DCSs~Figs. 2–7!, the differences betweencross sections are very small. For H, Al, Ni, and Ag, tdifferences do not exceed 0.02% for most scattering angSlightly larger deviations are observed for scattering angnear 180° where the deviation reaches20.28% for hydrogenat 10 000 eV. A common feature of Figs. 15~a!–15~d! is afine oscillatory structure. This is probably due to the differenumber ofl terms in the partial-wave series used in the cculations of DCSs. For high-atomic-number elements,

n-

g

FIG. 9. Percentage differences betweedifferential elastic-scattering cross sections,DDCS, calculated for aluminumfrom Eq. ~18! for the TFD and DHFpotentials as a function of scatterinangle,u: ~a! energy of 100 eV;~b! en-ergy of 500 eV;~c! energy of 1000 eV;and ~d! energy of 10 000 eV.

-

-


FIG. 10. Percentage differences between differential elastic-scatteringcross sections,DDCS, calculated fornickel from Eq.~18! for the TFD andDHF potentials as a function of scattering angle,u: ~a! energy of 100 eV;~b! energy of 500 eV;~c! energy of1000 eV; and~d! energy of 10 000 eV.

vie

eth

reic

mbted

rsdo

theTo. 4or

tion

oscillations are much less pronounced. However, the detions are larger than for low- and medium-atomic-numbelements. For gold, the largest deviation is20.3%, and forcurium 61%. Furthermore, the largest deviations observfor Au and Cm at 1000 eV occur in the same positions asdeep minima in the DCSs.

We find that the DCSs from the two databases consideare practically equivalent. For the low- and medium-atomnumber elements, the agreement is within 3–4 deciplaces. Larger deviations, observed for high-atomic-numelements, should be ascribed to the different exchange potials used in the respective algorithms. Some part of the

a-r

de

d-alern-

e-

viations might possibly be due to numerical round-off erroand interpolation errors. These contributions, however,not seem to be significant.

An obvious question arises as to the magnitude ofcontribution to the DCS due to the exchange correction.evaluate this contribution, the algorithm described in Sechas been used in calculations of the DCSs that includeneglect the exchange correction@cf. Eq.~13!#. The deviationsbetween calculated DCSs were determined from an equasimilar to Eq.~23!

DDCS5100~DCSnexcDHF2DCSexc

DHF!/DCSexcDHF, ~24!

-

-

FIG. 11. Percentage differences between differential elastic-scatteringcross sections,DDCS, calculated forsilver from Eq.~18! for the TFD andDHF potentials as a function of scattering angle,u: ~a! energy of 100 eV;~b! energy of 500 eV;~c! energy of1000 eV; and~d! energy of 10 000 eV.


-

-


FIG. 12. Percentage differences between differential elastic-scatteringcross sections,DDCS, calculated forgold from Eq. ~18! for the TFD andDHF potentials as a function of scattering angle,u: ~a! energy of 100 eV;~b! energy of 500 eV;~c! energy of1000 eV; and~d! energy of 10 000 eV.

rf

gseCs f

nt 1a

all3rom

Ri-e

iffer-

where DCSexcDHF is the DCS calculated with the exchange co

rection and DCSnexcDHF is the DCS calculated with neglect o

the termVexc(r ) in Eq. ~13!. Values ofDDCS from Eq.~24!are shown as a function of scattering angle for H, Al, Ni, AAu, and Cm in Fig. 16 at the two electron energies. Wethat the contribution of the exchange potential to the Dcan be substantial, in particular, for elements and energiewhich there are deep minima in the DCSs~Ag, Au, and Cmat 1 keV!. In the vicinity of minima, the deviations betweeDCSs may exceed 35%. The DCSs for H, Al, and Ni akeV differ by up to 5%. Smaller deviations are observed


-

.e

Sor

t

10 keV where the deviations usually do not exceed 2%. Incases, the values ofDDCS are much larger, by up to 2–orders of magnitude, than the deviations between DCSs fthe two databases considered here@cf. Figs. 15~a! and 16~a!,or Figs. 15~c! and 16~c!#.

We therefore conclude that the exchange correction ofley and Truhlar75 leads to practically identical DCSs as thcorrection of Furness and McCarthy.64 The slight differencesbetween these DCSs are considerably smaller than the dences due to neglect of the exchange correction.

-

-

FIG. 13. Percentage differences between differential elastic-scatteringcross sections,DDCS, calculated forcurium from Eq.~18! for the TFD andDHF potentials as a function of scattering angle,u: ~a! energy of 100 eV;~b! energy of 500 eV;~c! energy of1000 eV; and~d! energy of 10 000 eV.

,

s:


FIG. 14. Comparison of phase shiftsd l

1 , for hydrogen corresponding tothe potentials used in the calculation~squares! the DHF potential;~circles!the TFD potential:~a! energy of 100eV; ~b! energy of 500 eV;~c! energy of1000 eV; and~d! energy of 10 000 eV.Vertical lines indicate the values ofl 0

calculated from Eq.~22!.

lcuS

dmo

Kr

lear

ropa

rtictoS

ta

teadvo

,

hetheeenat

re

CSto

entsthe1.5.

or

ig.

nda-

-

ure-ig.

Ssronoreto 1

e-botha-

7.4. Comparisons of Differential Cross SectionsObtained from the TFD and DHF Potentialswith Measured Differential Cross Sections

We present illustrative comparisons here of DCSs calated from the TFD and DHF potentials with measured DCfor six elements~He, Ne, Ar, Kr, Xe, and Hg! at energiesbetween 50 and 3000 eV. These elements were selectecause in most cases there were at least two independentsurements available for the chosen electron energies. In scases~Ne at 800 eV; Ar at 800 eV, 2000 eV, and 3000 eV;at 750 eV; Xe at 750 eV; Hg at 500 eV!, measured DCSsfrom a single source were used to extend the range of etron energy or atomic number in the comparisons. Forsuch sources, DCS measurements were available at otheergies and were in satisfactory agreement with results fother groups. The calculated cross sections in these comsons were obtained from Version 2.010 ~for the TFD poten-tial! and Version 3.012 ~for the DHF potential! of the NISTElectron Elastic-Scattering Cross-Section Database.

Several review articles have identified and discussedquirements for reliable measurements of differential elasscattering and other cross sections in electron–ainteractions.76–80 The requirements relevant to elastic-DCmeasurements include knowledge of the uncertaintiesmeasurements of the incident electron-beam current,beam energy, the effective product of path length of the bein the scattering gas and solid angle of the detector systhe scattered-electron current, and the gas pressure. Intion, the gas pressure clearly has to be low enough to aunwanted multiple-scattering effects.

Figures 17–22 show measured81–106 and calculated DCSvalues for He~at 50, 100, 300, 500, and 700 eV!, Ne ~at 100,300, 500, and 800 eV!, Ar ~at 50, 100, 300, 500, 800, 10002000, and 3000 eV!, Kr ~at 100, 200, 500, and 750 eV!, Xe

-s

be-ea-me

c-llen-mri-

e--

m

inhemm,di-id

~at 50, 100, 200, 400, and 750 eV!, and Hg~at 100, 150, 300,and 500 eV!, respectively. The standard uncertainty in tmeasured DCS values has been typically estimated byauthors of the reported DCS measurements to be betw5% and 25%. Examination of the measured DCSs for Ar100 eV in Fig. 19~b!, the element and energy for which theis the largest number of independent measurements~9!,shows that the ratio of the largest DCS to the smallest Dat a scattering angle away from the minimum can be upabout 1.8. The measurements of Vuskovic and Kurepa95 ap-pear to be systematically larger than the other measuremfor most scattering angles; if their results are excluded,ratio of the largest DCS to the smallest DCS is less thanSimilar inspections of other figures where there are threemore DCS measurements at various scattering angles~e.g.,He at 50 eV and 100 eV in Figs. 17; Ne at 100 eV in F18~a!; Ar at 500 eV in Fig. 19~b!; Kr at 100 and 200 eV inFig. 20; Xe at 100 and 200 eV in Fig. 21; and Hg at 100 a300 eV in Fig. 22! indicate that the consistency of the mesured DCSs is typically within625% although there aresome larger [email protected]., for He at 100 eV in Fig. 17~b! andfor Kr at 200 eV in Fig. 20~b!#. We also note that the measurements of Kurepa and Vuskovic84 for He at 100 eV in Fig.17~b! are also systematically larger than the other measments shown, a result consistent with the Ar data in F19~b!.

We now consider the extent to which the measured DCagree with the calculated DCSs as a function of electenergy. Argon is the element for which there are two or mDCS measurements over the energy range from 50 eVkeV ~with additional data from one group at 2 and 3 keV!.Figures 19~g! and 19~h! show that there is excellent agrement between the measured DCSs and the DCSs fromthe TFD and DHF potentials. At lower energies, the me


H

thrmti

.

thdt

enreep

-

eda1han

ain

ffele

tetirrfo

teTn

fonetiaoheene

mn.by

ringove

d ater-hermsedres

ross

00


sured DCS agree generally better with DCSs from the Dpotential than the TFD [email protected]., in the positions andmagnitudes of the deep minima in Figs. 19~a! and 19~b!#,although it is clear that the measured DCSs are smallerthe computed DCSs at other scattering angles. For otheements, there is generally close agreement between thesured DCSs and the DCSs calculated from the DHF potenat the highest available [email protected]., He at 700 eV in Fig17~e!; Ne at 800 eV in Fig. 18~d!; Kr at 750 eV in Fig. 20~d!;Xe at 750 eV in Fig. 21~e!; and Hg at 500 eV in Fig. 22~d!#.At lower energies, the measured DCSs are often smallerthe calculated DCSs for most scattering angles, and theviations between the measured DCSs and the DCSs fromDHF potential generally become larger with decreasingergy. In most cases, the DCSs from the DHF potential agbetter with the measured DCSs in the vicinity of deminima than DCSs from the TFD [email protected]., for Ne at100 eV in Fig. 18~a!; Kr at 100 eV in Fig. 20~a!; Xe in Fig.21; and Hg at 300 eV in Fig. 22~c!#. There are some situations, however, where the reverse [email protected]., for Kr at 200eV in Fig. 20~b! and Hg at 100 and 150 eV in Fig. 22#,although the result for Kr might be associated with limitangular resolution in the experiments for the particularly nrow minimum expected at a scattering angle of about 8Finally, we point out that the measured DCSs are larger tthe DCSs from the DHF potential, particularly at lower eergies, for scattering angles less than about 25°~particularlyHe in Fig. 17; Ne at 100, 300, and 500 eV in Fig. 18; Ar50, 100, 300, and 500 eV in Fig. 19; Kr at 100 and 200 eVFig. 20; Xe at 50 and 100 eV in Fig. 21; and Hg in Fig. 22!.The range of scattering angles where these positive diences occur nevertheless gets smaller with increasing etron energy.

For electron energies below 1 keV, the DCSs calculafrom the DHF potential have the same shapes as a funcof scattering angle as the measured DCSs and the conumber of minima. The absolute values of these DCSsscattering angles larger than about 25°, however, are sysatically larger than the measured DCSs, as just noted.cause for this systematic difference is believed to be theglect of absorption~or inelastic-scattering! effects that can-not be accounted for in the calculational algorithm.107 Inprinciple, these absorption effects can be described byimaginary potential;107,108 an imaginary potential in theSchroedinger equation is equivalent to a loss of particles,109 aresult to be expected due to open inelastic channels. Untunately, this approach is difficult to implement. On the ohand, it is hard to calculate the local absorption potenfrom first principles,107 because it depends on the detailsthe wave functions of excited atomic states. On the othand, the solution of the radial equations for complex pottials requires numerical algorithms different from those usfor real potentials.

A second intrinsic limitation of the DCSs calculated frothe DHF potential is the atomic polarizability correctioThis correction, which can be described empiricallymeans of a long-range polarization potential (}r 24) for the


F

anel-ea-al

ane-he-e

r-°.n

-

t

r-c-

donectr

m-hee-

an

r-

lfr-d

gas phase, increases the DCS at relatively small scatteangles and is responsible for the deviations mentioned abin this angular range.

The arguments that lead to an imaginary potential anlong-range polarization potential in elastic-electron scating are the following. In a perturbation expansion of telastic-transition rate to second order, the second-order teinvolve virtual transitions to excited states. As mentionabove, the formalism is very involved because it requi

FIG. 15. Percentage differences between differential elastic-scattering csections,DDCS, obtained from the Berger and Seltzer~Ref. 11! databaseand the Jablonskiet al. ~Ref. 12! database at energies of 1000 and 10 0eV from Eq. ~23! as a function of scattering angle,u: ~a! hydrogen;~b!aluminum;~c! nickel; ~d! silver; ~e! gold; and~f! curium.

torec.ai-

onanthagxiins

alrteroTe

7–rgm

suFobtsto

cthed

-tii

il

icisapn

lidil o

C-acc

gr

ro

t theof

o bethege

rosser-,


knowing the wave functions of excited states. A methodbypass this difficulty is based on the so-called eikonal thethat allows the calculation of the scattering amplitude to sond order in terms of only the ground-state wave function107

From the eikonal scattering amplitude, one can obtain‘‘equivalent’’ approximate local potential that yields approxmately the eikonal scattering amplitude. This potential csists of the electrostatic potential used in our calculationsa second-order term that is complex. The real part ofterm corresponds to the polarization potential, and the imnary part is the absorption potential. Alternatively, appromate absorption and polarization potentials can be obtafrom semiclassical arguments~see Ref. 108 and referencetherein!. Recently, Salvat110 has proposed a semiempiricoptical-model potential in which the absorption and shorange polarization potentials are obtained from the local dsity approximation, i.e., by assuming that the atomic electcloud behaves locally as an homogeneous electron gas.model parameters, determined by fitting a large set of expmental DCS data~essentially those presented in Figs. 122!, were found to be essentially independent of the eneof the projectile and the atomic number of the target atoAlthough this type of approach yields more accurate DCSlow energies, calculations are substantially more difficthan for the TFD and DHF potentials considered here.condensed matter, there is no empirical information availaon the magnitude of the polarizability and absorption effecIn principle, the absorption correction should be similarthat for the gas phase. The polarization potential is expeto be much weaker than for the gas phase because botpolarizing field of the projectile and the field of the inducatomic dipole will be screened by the medium.111 It shouldalso be noted that it is customary to construct a muffinpotential for solids to describe elastic-electron scatteringthe medium. This change from an atomic potential wmodify the DCSs at small scattering angles6,11 but the result-ing changes are in the opposite direction to the atompolarizability correction. The long-range polarization fieldattractive. This field therefore increases the DCS at smangles where, classically, trajectories have large impactrameters and the projectile sees only the ‘‘tail’’ of the potetial. On the other hand, the muffin-tin potential for a sovanishes outside the muffin-tin sphere. As a result, the tathe atomic potential is suppressed and the small-angle Dfor an atom in a solid is smaller~in the absence of polarization! than the corresponding DCS for a free atom. In prtice, the degree of partial cancellation of these two effewill depend on atomic number and electron energy.

8. Comparisons of TransportCross Sections

8.1. Comparisons of Transport Cross SectionsObtained from the TFD and DHF Potentials

The energy dependences of the TCSs for H, Al, Ni, AAu, and Cm obtained from the TFD and DHF potentials acompared in Fig. 23; these TCS values were obtained f

oy-

n

-d

isi--ed

-n-nhe

ri-

y.

atltr

le.

edthe

nnl

-

lla--

fS

-ts

,em

Versions 2.010 and 3.0,12 respectively, of the NIST ElectronElastic-Scattering Cross-Section Database. We see thaagreement is generally rather good. With the exceptionenergies below about 200 eV, the dependences seem tnearly identical. To evaluate the deviations betweenTCSs quantitatively, let us consider the following percentadifference,Ds tr :

Ds tr5100~s trTFD2s tr

DHF!/s trDHF , ~25!

FIG. 16. Percentage differences between differential elastic-scattering csections,DDCS, with and without exchange correction calculated at engies of 1000 and 10 000 eV from Eq.~24! as a function of scattering angleu: ~a! hydrogen; ~b! aluminum; ~c! nickel; ~d! silver; ~e! gold; and~f! curium.


ith


FIG. 17. Comparison of differential elastic-scattering cross sections, ds/dV, calculated for helium for two potentials as a function of scattering angle wmeasured differential cross sections:~solid line! DHF potential;~dashed line! TFD potential:~a! energy of 50 eV;~b! energy of 100 eV;~c! energy of 300 eV;~d! energy of 500 eV;~e! energy of 700 eV. The symbols show the measurements of Bromberg,81 McConkey and Preston,82 Gupta and Rees,83 Kurepa andVuskovic,84 Jansenet al.,85 Registeret al.,86 Wagenaaret al.,87 and Brungeret al.88

ltesn

fe

rge

00rger

ed

wasr-was

wheres trTFD ands tr

DHF are the transport cross sections resuing from the TFD and DHF potentials, respectively. Valuof Ds tr for the six elements are shown as a function of eergy in Fig. 24. Above 200 eV, the largest percentage difence is observed for hydrogen. The value ofDs tr is close to20% for H at 200 eV, and decreases with increasing eneto 10% at 20 000 eV. For low and medium atomic numbelements~Al,Ni,Ag !, the value ofDs tr is less than 5% forenergies above 200 eV. At lower energies, however,Ds tr

may exceed 40%. For high atomic number elements~Au and


-

-r-

yr

Cm!, Ds tr varies considerably in the energy range up to 10eV, and exceeds 20% at some energies. For energies lathan 1000 eV,Ds tr is less than 5%.

A similar analysis, for the same potentials, was publishby Jablonski and Powell33 for Be, C, Al, Cu, Ag, and Au. Theenergy range considered by these authors, however,smaller~100 eV–10 keV! than that considered here. Furthemore, the percentage difference between DCS valuesdefined by a slightly different relation than Eq.~25!. None-

ith


FIG. 18. Comparison of differential elastic-scattering cross sections, ds/dV, calculated for neon for two potentials as a function of scattering angle wmeasured differential cross sections:~solid line! DHF potential;~dashed line! TFD potential:~a! energy of 100 eV;~b! energy of 300 eV;~c! energy of 500eV; and ~d! energy of 800 eV. The symbols show the measurements of DuBois and Rudd,89 Gupta and Rees,90 Jansenet al.,85 Williams and Crowe,91

Wagenaaret al.,87 and Bromberg.81

in

e-

Cogu-onsao

qrelefts

l-ure-rareeon

100

ileire

byual

r athe

ay

theless, the energy dependences ofDs tr found by Jablonskiand Powell33 are very similar to the dependences shownFig. 24.

8.2 Comparisons of Transport Cross SectionsObtained from the TFD and DHF Potentialswith Measured Transport Cross Sections

Elford and Buckman112 have discussed experimental dterminations of transport cross sections~often also referred toas momentum-transfer cross sections!. For electron energiesof interest here, TCSs are determined from measured Dand Eq.~20!. Unfortunately, it is not generally possible tmeasure DCSs over the complete range of scattering andue to the difficulty of defining the interaction volume accrately for small scattering angles and to geometrical cstraints for large scattering angles. As a result, it is neceseither to extrapolate measured DCSs to scattering anglesand p, perhaps using calculated DCSs as a guide, or tomeasured DCSs as a function of scattering angle with E~2! and ~3! using a selected number of phase shifts as fparameters. The latter approach can be useful for low etron energies (E<200 eV) where the number of phase shiis sufficiently small.113,114

Ss

les

-ryf 0fits.ec-

Elford and Buckman have published ‘‘preferred’’ TCS vaues for a number of atoms based on an analysis of measments and calculations.112 Most of the preferred TCSs are foelectron energies of less than 10 eV although some datagiven for energies of up to 100 eV. We show illustrativTCSs in Table 3 for two atoms, He and Hg, that are basedmeasured DCSs for electron energies of 50 andeV.86,88,104Registeret al.86 and Brungeret al.88 derived TCSsafter performing a phase-shift analysis of their DCSs whPanajotovicet al.104 obtained TCSs after extrapolating theDCSs to 0 andp. For Hg and an energy of 100 eV, thexperimental TCSs are smaller than the calculated TCSsabout a factor of two, as would be expected from a casinspection of Fig. 22~a!.

9. Comparisons of Quantities Derivedfrom Differential Cross Sectionsthat were Obtained from the TFD

and DHF Potentials

Differential elastic-scattering cross sections are used fovariety of purposes, as noted in Sec. 1. We discuss hereextent to which certain quantities derived from DCSs m


ith

ts of


FIG. 19. Comparison of differential elastic-scattering cross sections, ds/dV, calculated for argon for two potentials as a function of scattering angle wmeasured differential cross sections:~solid line! DHF potential;~dashed line! TFD potential:~a! energy of 50 eV;~b! energy of 100 eV;~c! energy of 300 eV;~d! energy of 500 eV;~e! energy of 800 eV;~f! energy of 1000 eV; and~g! energy of 2000 eV; energy of 3000 eV. The symbols show the measuremenPanajotovicet al.,92 Wagenaaret al.,87 DuBois and Rudd,89 Cvejanovic and Crowe,93 Gupta and Rees,83 Jansenet al.,85 Srivastavaet al.,94 Vuskovic andKurepa,95 Williams and Willis,96 Bromberg,81 Dou et al.,97 and Igaet al.98


ith


FIG. 20. Comparison of differential elastic-scattering cross sections, ds/dV, calculated for krypton for two potentials as a function of scattering angle wmeasured differential cross sections:~solid line! DHF potential;~dashed line! TFD potential:~a! energy of 100 eV;~b! energy of 200 eV;~c! energy of 500eV; and~d! energy of 750 eV. The symbols show the measurements of Cvejanovic and Crowe,93 Danjo,99 Srivastavaet al.,94 Williams and Crowe,91 Wagenaaret al.,87 Bromberg,81 and Jansen and de Heer.100

inistees

Pobifi

-

cre

tho

n1

roopf

inat-ug-be

ise,ve

tter-os-dszer

eith

otalhanic-k-aterly

e of

vary depending on whether the cross sections were obtafrom the TFD or DHF potentials. Specific considerationgiven here to three quantities relevant to materials characization: ~1! elastic-backscattered intensities for an overlayfilm on a substrate;~2! electron inelastic mean free pathdetermined by elastic-peak electron spectroscopy; and~3!mean escape depths for signal electrons in AES and XComparisons will be made of these quantities for DCSstained, as before, from the TFD and DHF potentials, speccally Versions 2.010 and 3.0,12 respectively, of the NIST Electron Elastic-Scattering Cross-Section Database.

9.1 Elastic-Backscattered Intensitiesfor an Overlayer Film on a Substrate

Elastic-scattering cross sections are needed for a destion of elastic backscattering of electrons from solid surfacAn electron impinging on the solid surface may leavesolid as a result of multiple elastic collisions. The processmultiple scattering, however, is usually dominated by olarge-angle elastic collision. We have seen from Figs. 8–that the largest differences between DCSs calculated fthe DHF and TFD potentials occur at scattering angles cresponding to deep minima in the DCSs. The scatteringtential for the solid will, in general, be different from that o

ed

r-r-

S.--

ip-s.ef

e3mr-o-

the corresponding neutral atom~see also Sec. 9.4!. It is pos-sible that the deep minima in the DCSs for atoms shownFigs. 2–7 may occur for the DCSs in solids at different sctering angles and with different strengths. We therefore sgest that, where possible, experimental configurationsavoided for which the DCS has a deep minimum. Otherwsimulations describing elastic backscattering might halarge uncertainties.

Problems associated with the theory of elastic backscaing were demonstrated in studies of gold overlayers depited on nickel.115,116 In the experimental configuration useby Jablonskiet al.,115 the direction of the electron beam waat 25° with respect to the surface normal while the analyaxis was located along the surface normal~with an accep-tance angle of66°!. At energies of 300 and 500 eV, thelastic-backscattering probability was found to decrease wthickness of the gold overlayer despite the fact that the telastic-scattering cross section for gold is much larger tthat for nickel. At energies of 1000 eV or more, the elastbackscattering probability increased with overlayer thicness. These experimental results were confirmed in a lwork116 in which the experiment was performed in a slightdifferent configuration~normal incidence of the primarybeam, emission angle of 25°, and an acceptance angl


ith


FIG. 21. Comparison of differential elastic-scattering cross sections, ds/dV, calculated for xenon for two potentials as a function of scattering angle wmeasured differential cross sections:~solid line! DHF potential;~dashed line! TFD potential:~a! energy of 50 eV;~b! energy of 100 eV;~c! energy of 200 eV;~d! energy of 400 eV; and~e! energy of 750 eV. The symbols show the measurements of Wagenaaret al.,87 Nishimuraet al.,101 Ester and Kessler,102 Jansenand de Heer,100 Williams and Crowe,91 and Bromberg.81

nteth

thnde5at

nts,tds

eseen-

64°!. These results were in good agreement with MoCarlo simulations of elastic backscattering that made usdifferential elastic-scattering cross sections obtained fromTFD potential. These simulations were performed forsame configurations~incidence angle, emission angle, aacceptance angle! as in the respective experiments. The dcrease of the backscattering probability at energies up toeV was ascribed to the presence of a deep minimum loc


eofe

e

-00ed

in the DCS for Au at a scattering angle close to 155°@cf. Fig.6~b!#. For the angular range relevant to these experimethe DCS for nickel is much larger than that for gold. Ahigher energies, the minimum in the Au DCS shifts towarsmaller scattering angles@cf. Fig. 6~c!#, and the DCS for goldis larger than that for nickel.

We have repeated the Monte Carlo calculations for thexperiments at energies of 500 and 1000 eV using differ

ith


FIG. 22. Comparison of differential elastic-scattering cross sections, ds/dV, calculated for mercury for two potentials as a function of scattering angle wmeasured differential cross sections:~solid line! DHF potential;~dashed line! TFD potential:~a! energy of 100 eV;~b! energy of 150 eV;~c! energy of 300eV; and~d! energy of 500 eV. The symbols show the measurements of Holtkampet al.,103 Panajotovicet al.,104 Bromberg,105 and Peitzmann and Kessler.106

thti

at

he

ace.

rinseeths

tiataon.efra

ice-eboth-eenm-

le-s. Inzersad

ayy of

, aopy

tial cross sections for gold and nickel calculated fromDHF potentials for these atoms. We determined the elasbackscattering probability for an uncovered nickel substrh~0!, and for a gold overlayer with thicknesst deposited onnickel, h(t). It was convenient to compare values of tratio, R(t), of elastic-backscattering probabilities

R~ t !5h~ t !

h~0!, ~26!

that were obtained from either measured or calculated bscattering data. Figures 25 and 26 show comparisons of msured values ofR as a function of Au thickness from Refs115 and 116, respectively, with the corresponding valuesR determined from Monte Carlo simulations for each expemental configuration and with differential cross sectiofrom each of the two potentials. We see that both crosstions lead to reasonably good agreement with the experimtal data. There is a distinct difference, however, betweenrelative intensities calculated for each potential. In one cause of differential cross sections from the DHF poten(DCSrel

DHF) leads to ratios that are closer to the experimenresults while, in other cases, the differential cross sectifrom the TFD potential (DCSrel

TFD) lead to better agreementAs mentioned above, the experiments reported in R

115 and 116 were performed in slightly different configu

ec-e,

k-a-

of-sc-n-ee,lls

s.-

tions; nevertheless, the experimental ratios differed notably ~see Figs. 25 and 26!. Although the angle between thelectron beam and the analyzer axis was the same inexperiments~155°!, the location of these directions with respect to the surface was different. The differences betwthe experimental ratios might also be partially due to systeatic errors associated with the different experiments~e.g.,accuracy of determining the angles, different samppreparation procedure, etc.!. Finally, the observed differencemay result from the different analyzer-acceptance anglesboth experiments, hemispherical electron-energy analy~HSAs! were used of different constructions, and these hnominal acceptance angles of64° ~Ref. 116! or 66° ~Ref.115!. In general, we expect that the measured intensities mdepend on the acceptance angle, especially in the vicinitdeep minima in the DCS.

9.2. Electron Inelastic Mean Free Paths Determinedby Elastic-Peak Electron Spectroscopy „EPES…

Measurements of elastic-backscattered probabilitiestechnique known as elastic-peak electron spectrosc~EPES!, can be used to obtain inelastic mean free paths~IM-



FIG. 23. Energy dependence of the transport cross sections calculated from the TFD and DHF potentials:~solid line! the DHF potential;~dashed line! the TFDpotential:~a! hydrogen;~b! aluminum;~c! nickel; ~d! silver; ~e! gold; and~f! curium.

c

yi

oriall

ident, andex-f therentsed..2.1

FPs! at various electron energies.117 The IMFP is an impor-tant parameter that is needed for quantification of surfaanalytical techniques such as AES and XPS.117

The EPES experiments can be performed in two waMost commonly, relative measurements are made of thetensities of electrons backscattered from two surfaces,the specimen of interest and the other a ‘‘standard’’ matefor which the IMFP is considered to be sufficiently we


e-

s.n-nel

known. These measurements are made for the same incbeam energy and current, the same scattering conditionsthe same analyzer conditions. In the other type of EPESperiment, absolute measurements are made of the ratio oelastically backscattered current to the incident beam curfor a selected sample material. No standard material is uWe consider these two types of experiments in Secs. 9and 9.2.2, respectively.

r-

s-


FIG. 24. Energy dependence of pecentage differences,Ds tr , from Eq.~25! between transport cross sectionobtained from the TFD and DHF potentials: ~a! hydrogen;~b! aluminum;~c! nickel; ~d! silver; ~e! gold; and~f!curium.

iati

inanetn

allthsrohe

isat

c-sueast

ndlesWeanen-

a

of

and

ls,

9.2.1. EPES Experiments with a Standard Material

In typical EPES experiments with a standard matermeasurements are made of the ratio of the elasbackscattering probabilities for a sample of interest,hsample,and for a selected standard material,hstandard, with a‘‘known’’ IMFP. Four elemental solids were recommendeda previous analysis as suitable standards: Ni, Cu, Ag,Au.117 The same ratio is calculated from a suitable theorcal model describing the backscattering phenomenon. MoCarlo simulations of electron backscattering are usumade for this purpose, and DCS values are required forsample and standard materials at the electron energieinterest. Simulations are performed at a particular electenergy for different assumed values of the IMFP for tsample, and a so-called calibration curve is producedwhich the ratio of elastic-backscattering probabilitiesgiven as a function of the sample IMFP. The measured rcan then be used to determine the sample IMFP.

To determine experimentally the ratio of elastibackscattering probabilities, one does not need to meatheir absolute values. It is sufficient to measure elastic-pintensities for the sample and standard in the energy dibutions of backscattered electrons,I sample and I standard, re-spectively. We then have the ratio of interest,Rsample

l,c-

di-teyeofn

in

io

rek

ri-

Rsample5hsample

hstandard5

I sample

I standard. ~27!

We showed in Sec. 7.2 that DCS values from the TFD aDHF potentials can differ considerably at scattering angclose to those where deep minima occur in the DCSs.therefore now consider a ‘‘worst-case’’ situation in whichEPES experiment is performed with a material, electronergy, and experimental configuration for which there isdeep minimum in the DCS~for a single elastic-scatteringevent!. Figure 27 shows the large-scattering-angle region

TABLE 3. Values of transport cross sections,s trm , derived from measured

differential cross sections for He and Hg at energies of 50 and 100 eVthe corresponding calculated transport cross sections,s tr

TFD and s trDHF , ob-

tained from Refs. 10 and 12 with use of the TFD and DHF potentiarespectively

AtomEnergy~eV!

s trm

(a02) Ref.

s trTFD

(a02)

s trDHF

(a02)

He 50 2.51 86 2.45 2.7750 3.32 88 2.45 2.77

100 0.754 86 0.834 0.928Hg 50 12.3 104 20.4 18.4

100 4.29 104 8.70 7.70


ea

f°ro

he, tfo

eth

sFD00l-5to

at

to00dif-

nd

,aed

te-

-eu

hemD

oold


the DCS for gold at energies of 500 and 1000 eV. The vtical line with the acronym HSA indicates the average sctering angle in the experiments of Jablonskiet al.115,116Asdiscussed in Sec. 9.1, a deep minimum in the DCS occursgold at 500 eV in the vicinity of a scattering angle of 155and there are appreciable differences in the DCS values fthe TFD and DHF potentials@cf. Fig. 12~b!#; smaller differ-ences near this scattering angle occur at 1000 eV.

For an EPES experiment with normal incidence of tprimary electron beam and a mean emission angle of 25°average scattering angle will be 155°. Calibration curvesa gold sample and a nickel standard~i.e., RAu5I Au /I Ni as afunction of the IMFP assumed for Au! calculated for thisconfiguration and for electron energies of 500 and 1000are shown in Fig. 28. In these calculations, we assumedthe acceptance angle was equal to64° which is typical forhemispherical analyzers.116We see that the calibration curvederived from DCS values that were obtained from the Tand DHF potentials differ considerably, particularly at 5eV. Such differences will clearly lead to different IMFP vaues. Suppose, for example, thatRAu is measured to be 0.232at 500 eV and 1.226 at 1000 eV~these values correspond

FIG. 25. Relative intensity of electrons elastically backscattered from a goverlayer on Ni,R(t) defined by Eq.~26!, as a function of overlayer thick-ness at energies of:~a! 500 eV and~b! 1000 eV:~triangles and dashed line!experimental data taken from Jablonskiet al.115; results of Monte Carlosimulations for the same configuration as used in Jablonskiet al.115 withelastic-scattering cross sections calculated from the TFD potential~opencircles! and from the DHF potential~filled circles!.


r-t-

or,m

her

Vat

the recommended IMFPs for gold of 8.36 and 13.78 Åthese energies, respectively117!. The calibration curve in Fig.28 calculated with DCSs from the TDF potential leadssmaller IMFPs for Au, 6.93 and 12.13 Å at energies of 5and 1000 eV, respectively. We calculate the percentageferenceDl

Dl5100~l inTFD2l in

DHF!/l inDHF ~28!

from the IMFPs,l inTFD and l in

DHF , derived from calibrationcurves calculated using DCSs obtained from the TFD aDHF potentials, respectively. The resulting values ofDlfrom Eq.~28! are217.1% and212.0% at 500 and 1000 eVrespectively. This ‘‘worst-case’’ example illustrates thatsignificant systematic error can arise in IMFPs determinby EPES with DCS values from the TFD potential for marials ~in this case, Au!, electron energies~in this case, 500eV!, and measurement configuration~in this case, a scattering angle of 155°! such that there is a deep minimum in thDCS. Although a measurement with this configuration for Aat 1000 eV would not occur near a deep minimum in tDCS, there is an error of 12.0% in IMFPs determined frothe calibration curve derived with use of DCSs from the TF

ldFIG. 26. Relative intensity of electrons elastically backscattered from a goverlayer on Ni,R(t) defined by Eq.~26!, as a function of overlayer thick-ness at energies of:~a! 500 eV and~b! 1000 eV:~triangles and dashed line!experimental data from Jablonskiet al.116; results of Monte Carlo simula-tions for the same configuration as used in Jablonskiet al.116 with elastic-scattering cross sections calculated from the TFD potential~open circles!and from the DHF potential~filled circles!.

g

le

e


FIG. 27. Differential elastic-scatteringcross sections, ds/dV, calculated forgold from the TFD and DHF poten-tials at scattering angles exceedin90°: ~solid line! DHF potential;~dashed line! TFD potential: ~a! en-ergy of 500 eV;~b! energy of 1000 eV.Vertical line with the acronym HSAindicates the average scattering angfor the experiments of Jablonskiet al.~Refs. 115 and 116! and the verticalline with the acronym CMA shows theaverage scattering angle for thexperiments of Krawczyket al.118

iv

ESascen-mhiwea-uatPn

beykteeli-Fin

ve

W0

. Toof

ios,-edthes ofner-26%m-

ardio,

f

ure-ents,rlodi-

anlue

forc-da

andu-

PES

potential. This systematic error arises in part from a positerror in the DCS for Au@cf. Fig. 12~c!# and in part from anegative error in the DCS for Ni@cf. Fig. 10~c!# at the scat-tering angle of 155°.

We consider now the determination of IMFPs by EPwith a cylindrical-mirror analyzer, the instrument that hbeen most frequently used in the work discussed in a rereview.117 In this configuration, the primary electrons are icident on the surface along the normal, and the average esion angle is equal to 42.3°. For the case of Au with tconfiguration, EPES measurements at 500 eV would beremoved from deep minima in the DCS for Au while mesurements at 1000 eV would be closer to a deep minim~cf. Fig. 27!. No minima are observed in the DCS for Cuthe considered energies in the relevant angular range. Emeasurements of AuCu alloys with a CMA and an Au stadard have been reported recently by Krawczyket al.118As anexample, we consider the case of the Au50Cu50 alloy. Figure29 shows calibration curves for four electron energiestween 200 and 2000 eV for the conditions of the Krawczet al. experiments. These calibration curves were calculaassuming the same acceptance angle as for the analyzthe experiments, i.e.66°. The differences between the cabration curves found using DCSs with the TFD and DHpotentials in Fig. 29 are much less than the corresponddifferences in Fig. 28.

We quantify the differences between the calibration curin Fig. 29 by defining the percentage differenceDRAuCu

DRAuCu5100~RAuCuTFD 2RAuCu

DHF !/RAuCuDHF , ~29!

whereRAuCuTFD andRAuCu

DHF are the values ofRAuCu for the TFDand DHF potentials, respectively. Values ofDRAuCu are plot-ted in Fig. 30 as a function of the assumed IMFP values.see thatDRAuCu is practically always less than 10%. At 200

e

nt

is-sll

m

ES-

-

dr in

g

s

e

eV, the percentage differences decrease to less than 5%determine an IMFP in the alloy, we need only a fragmentthe calibration curve corresponding to measured ratRAuCu, for particular energies.118 These ratios and the corresponding IMFPs are listed in Table 4 for DCSs obtainfrom the TFD and DHF potentials. As one can see,change of potential leads to changes in the derived IMFPless than 6% at energies of 200 and 500 eV. At higher egies, the changes in the derived IMFPs decrease from 3.at 1000 eV to 1.27% at 2000 eV. The latter change is coparable to the precision of our Monte Carlo calculations.

9.2.2. EPES Experiments without a Standard Material

We consider now EPES experiments in which a standmaterial is not used.117 Measurements are made of the rathsample, of the elastically backscattered current,I sample, in aparticular solid angle~for the material and electron energy ointerest! to the current of the incident beam,I beam. Suchexperiments are more demanding than the relative measments described in Sec. 9.2.1 because each of the currI sampleandI beam, has to be measured absolutely. Monte Casimulations are made for the particular experimental contions to produce calibration curves showinghsampleas a func-tion of assumed values for the sample IMFP. The IMFP cthen be determined from the calibration curve as the vathat gives the same value ofhsampleas that measured.

As an example, Fig. 31 shows EPES calibration curvesgold in the form of plots of the calculated elastibackscattering probabilities,hAu , as a function of assumeIMFP for normal incidence of an electron beam inretarding-field analyzer for emission angles between 5°55° at electron energies of 500 eV and 1 keV. This configration was selected to correspond to some absolute E


ein

-a

mo

e

e-

-aglene

beb

e5

–

pi

nacea

alnsn

l,nor-ys-o

b-

eanownand;

the


measurements.118,119The calibration curves of Fig. 31 werobtained, as for Fig. 28, from Monte Carlo simulationswhich DCSs from the TFD potential~dashed line! and theDHF potential~solid line! were utilized. The marked ordinate values, 0.036 01 at 500 eV and 0.053 23 at 1 keV,values ofhAu obtained from the simulations with DCSs frothe DHF potential and the recommended IMFPs for Au8.36 and 13.78 Å, respectively.117 The IMFPs derived fromthe calibration curves for the TFD potential for these valuof hAu are 7.52 and 13.60 Å, respectively. Values ofDl fromEq. ~28! are210.0% and21.31%, respectively.

As another example, we consider the absolute EPESperiments of Dolinskiet al. made for Cu in which the surfaces were cleaned by concurrent K1 bombardment andheating to remove K from the surface.120 These measurements were made with normal incidence of the primary beand a RFA for which the range of accepted emission anwas between 5° and 44°. Table 5 shows IMFPs obtaifrom the experiments of Dolinskiet al. and from DCSs cal-culated from the TFD and DHF potentials for energiestween 250 and 1500 eV and the percentage differencestween these IMFPs from Eq.~28!. These percentagdifferences range between 2.36% at 250 eV to 6.42% ateV. Table 4 also shows IMFP values,l in

HFS, obtained by Do-linski et al. using the DCSs from a relativistic HartreeFock–Slater potential of Fink and Ingram.2

9.3. Mean Escape Depths for AES and XPS

The mean escape depth~MED! is a useful parameter inAES and XPS. This parameter specifies the average denormal to the surface, from which signal electrons are emted. The MED,D, is defined by121

D5*0

`zf~z,a!dz

*0`f~z,a!dz

, ~30!

where f(z,a) is the emission depth distribution functiodefined as the probability that the particle leaving the surfin a given direction originated from a specified depth msured normally from the surface into the material.121

The following analytical expression for the MED of signelectrons in AES and XPS has been derived by Jabloet al.31 from a solution of the kinetic Boltzmann equatiowithin the so-called transport approximation

D5l inl tr

l in1l trS cosa1

S1

S2D , ~31!

where

S15~12v!21/2x1C1 , ~32!

S25~12v!21/22~b/4!~3 cos2 c21!

H~cosa,v!1C2 , ~33!

x5~v/2!~12v!21/2E0

1

cosaH~cosa,v!d~cosa!,

~34!


re

f

s

x-

msd

-e-

00

th,t-

e-

ki

C152vb

16~3 cos2 u21!

3E0

1~x21x cosa!~3x221!H~x,v!

x1cosadx, ~35!

C25vb

16~3 cos2 u21!E

0

1 x~3x221!H~x,v!

x1cosadx.

~36!

In Eqs.~31!–~36!, l tr is the transport mean free path,a is theelectron emission angle with respect to the surface normauis the angle between the x-ray direction and the surfacemal in XPS,c is the angle between the direction of x raand the analyzer axis in XPS,b is the photoionization asymmetry parameter in XPS,v is the single-scattering albedgiven by v5l in /(l in1l tr), and H(x,v) is the Chan-drasekhar function.122 The transport mean free path is otained from the transport cross section

FIG. 28. Ratio of elastic-backscattering probabilitiesRAu5I Au /I Ni definedby Eq. ~27! as a function of assumed values of the electron inelastic mfree path for gold. Two calibration curves for EPES experiments are shbased on Monte Carlo simulations with DCS values obtained from TFDDHF potentials: ~solid line! DCS calculated from the DHF potential~dashed line! DCS calculated from the TFD potential:~a! energy of 500 eVand ~b! energy of 1000 eV. The calibration curves were calculated forexperimental configuration of Jablonskiet al.115,116who used a hemispheri-cal analyzer for their measurements.

er

e-

-

e-


FIG. 29. Ratio of elastic-backscattering probabilities RAuCu

5I AuCu /I Au defined by Eq.~27! as afunction of assumed values of thelectron inelastic mean free path fothe alloy Au50Cu50 . Two calibrationcurves for EPES experiments arshown based on Monte Carlo simulations with DCS values obtained fromTFD and DHF potentials:~solid line!DCS calculated from the DHF potential; ~dashed line! DCS calculatedfrom the TFD potential:~a! energy of200 eV; ~b! energy of 500 eV;~c! en-ergy of 1000 eV; and~d! energy of2000 eV. The calibration curves wercalculated for the experimental configuration of Krawczyket al.118 whoused a cylindrical-mirror analyzer fortheir measurements.

em

lin

inp-

x-sent

q.

nsing

l tr51

Ns tr, ~37!

whereN is the atomic density, i.e., the number of atoms punit volume. For AES, the emission of Auger electrons froatoms is close to isotropic for amorphous and polycrystalsolids, andb50. The MED from Eqs.~31!–~36! then be-comes

D5l inl tr~x1cosa!/~l in1l tr!. ~38!

It has been shown that the parametersC1 andC2 in Eqs.~32! and~33! are small compared toS1 andS2 , respectively,

r

e

and can be neglected for typical photoelectron linesXPS.31 The change in the MED values, due to the assumtion that C1 and C2 are equal to zero, usually does not eceed 5%. This assumption has been made in the preMED calculations.

We consider now changes in MED values from Eqs.~31!–~38! arising from the use of transport cross sections in E~37! that were computed from Eq.~20! using DCS valuesobtained from the TFD and DHF potentials. Calculatiowere made for four electron energies in gold correspondto two Auger-electron lines (AuN7VV and AuM5N67N67)

,

9

er

FIG. 30. Percentage differencesDRAuCu , calculated from Eq.~29! be-tween the calibration curves of Fig. 2for the TFD and DHF potentials as afunction of assumed values of thelectron inelastic mean free path fothe alloy Au50Cu50 .


re

ara

qctin

thchw

olia

easith

o

s-re-io

eVin

ialat

d in

rgieseenen

ized

ureder-

inionasid-r-e-

forcept-PESlues

riv


and two photoelectron lines (Au 4s and Au 4f 7/2) for XPSwith Mg Ka x rays. These energies cover a wide range~from70 to 2016 eV!. In these calculations, we selected a repsentative value123 of the anglec554° and determinedD forvariations of the electron emission angle,a, from 0° to 80°.The variation ofa corresponds to rotation of the sample inchamber with fixed positions of the analyzer and the x-source~for XPS!.

The transport mean free path is the only parameter in E~31!–~38! describing the strength of elastic-scattering effein a given solid. If the transport mean free path reachesfinity, the MED approaches the value determined fromso-called common formalism of AES and XPS in whielastic–electron collisions in the solid are neglected. Ifdenote the MED for this case byD, then Eq.~31! becomes

D5l in cosa. ~39!

To examine the effects of elastic scattering onD, it is con-venient to determine the ratio,RMED

RMED5D/D. ~40!

Figure 32 shows plots ofRMED calculated from values ofDfor the TFD and DHF potentials as a function ofa for thefour selected photoelectron and Auger-electron lines in gWe see that the MED values derived from the two potentare very similar. The differences between theD values seemto decrease with increasing emission angle and with incring energy. The latter effect can be ascribed to the decreadifferences between transport cross sections in Au fortwo potentials with increasing energy@cf. Fig. 24~e!#. Toquantify the difference between the MEDs from the two ptentials, we calculate the percentage difference,DD

DD5100~DTFD2DDHF!/DDHF, ~41!

whereDTFD andDDHF are the MEDs determined using tranport cross sections from the TFD and DHF potentials,spectively. Values ofDD for the Auger-electron and photoelectron lines in Fig. 32 are plotted as a function of emissangle in Fig. 33. As one can see, the values ofDD at normalemission vary from 4.3% at 70 eV to about 0.5% at 2016and become smaller with increasing emission angle. Selastic-scattering effects~as judged by the value ofv! are

TABLE 4. IMFP values for the alloy Au50Cu50 derived from calibrationcurves for the EPES experiments of Krawczyket al.118 that were calculatedusing DCSs obtained from the TFD potential (l in

TFD) and the DHF potential(l in

DHF) at the indicated energies from the measured ratios,I AuCu /I Au , ofelastic-backscattered intensities. The percentage differences in the deIMFPs,Dl, from Eq. ~28! are shown in the final column.

E~eV! I AuCu /I Au

l inTFD

~Å!l in

DHF

~Å!Dl~%!

200 2.144 6.91 6.58 5.02500 0.881 7.86 7.43 5.79

1000 1.081 16.17 15.66 3.261500 1.078 22.10 21.48 2.892000 0.859 24.78 24.47 1.27


-

y

s.s-

e

e

d.ls

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-

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n

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strong in gold,124 we conclude that the change of potentwill not significantly affect MED values of other elementsenergies typical for AES and XPS.

10. Validity of Calculated DCS Datafor Atoms in Simulations of Electron

Transport in Solids

Calculated DCSs for atoms have been extensively usesimulations of electron transport in solids13–45 as well as inthe three examples discussed in Sec. 9. The electron enein these particular applications have typically been betw100 eV and 25 keV although similar simulations have bemade for higher energies. The atomic data have been utilbecause calculated DCSs are available1–12 for all elementsand a wide range of electron energies, whereas measDCSs exist for a very limited number of elements and engies ~such as those presented in Sec. 7.4!.

We discuss here the validity of the atomic DCS datadescriptions of electron transport in solids. The interactpotential in a solid will clearly be different from that forfree atom. These differences are expected to have a conerable effect on the DCSs for ‘‘small-angle’’ scattering. Fotunately, for many problems of practical interest, the ‘‘larg

FIG. 31. Plot of the elastic-backscattering probability,hAu , for gold as afunction of assumed values of the electron inelastic mean free pathnormal incidence of the electron beam and a retarding-field analyzer acing emission angles between 5° and 55°. Two calibration curves for Eexperiments are shown based on Monte Carlo simulations with DCS vaobtained from TFD and DHF potentials:~solid line! DCS calculated fromthe DHF potential;~dashed line! DCS calculated from the TFD potential:~a!energy of 500 eV and~b! energy of 1000 eV.

ed

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angle’’ scattering is generally more relevant than the smangle scattering. The atomic potential can be matchedmuffin-tin potential for the corresponding solid or canempirically truncated in the solid.125 While the resulting po-tentials lead to substantial differences in the DCSs fromatomic DCSs for small-angle scattering~with the differencesdepending on Z!,11 there are often only small effects on prameters such as transport cross sections, effective attetion lengths~EALs!, and backscattering factors.6,11,38For ex-ample, Berger and Seltzer11 report that transport crossections for solid Au~calculated with the Raith125 method ofpotential truncation! differ from those for atomic Au by lessthan 0.1% for energies between 1 and 500 keV, whilesimilar differences in transport cross sections for Be arethan 3.3% for the same energy range. Cumpson and Se38

show that the differences between EALs obtained fromTF/muffin-tin potential for 18 solid elements and those frothe corresponding atomic relativistic HFS potentials couldcharacterized by a standard deviation of 2.5% at 200 eVof 1.5% at 1 keV. Similarly, as shown in Sec. 9 aelsewhere,33 elastic-backscattering probabilities, IMFPMEDs, and XPS signal intensities calculated for solids froTFD and DHF or similar atomic potentials generally diffby less than 10% and often less than 5%. These uncertaiare generally small compared to other sources of uncertain many applications.

We come now to an apparently more serious problem.have shown in Sec. 7.4 that measured DCSs for Ar agwell with DCSs calculated from the DHF potential for eletron energies of 1 keV and above. Similar results have bfound for He, Ne, Kr, and Xe.11 At lower energies, howeverthere can be differences~away from deep minima! of oftenup to a factor of 2 and occasionally by a larger factor,shown in Figs. 17–22. As noted in Sec. 7.4, these differencan be explained by the neglect of absorption and polaability corrections in most DCS calculations. The eikonmethod takes these corrections into account, and improagreement is found between the resulting calculated Dfor He and Ar and those measured.107

Figures 17–22 suggest that while atomic DCSs calculafrom the DHF potential should be useful for applications

TABLE 5. IMFP values for copper derived from the absolute EPES expments of Dolinskiet al.120 that were calculated using DCSs obtained frothe TFD potential (l in

TFD) and the DHF potential (l inDHF) at the indicated

energies,E, from the measured elastic-backscattering probabilities,hCu .The percentage differences in the derived IMFPs,Dl, from Eq. ~28! areshown in the final column. The third column shows the IMFPs reportedDolinski et al., l in

HFS, from calibration curves obtained with DCSs computed from a relativistic Hartree–Fock–Slater potential by Fink aIngram.2

E~eV! hCu

l inHFS

~Å!l in

TFD

~Å!l in

DHF

~Å!Dl~%!

250 0.0434 4.2 3.47 3.39 2.36500 0.0395 5.9 6.10 5.45 6.42

1000 0.0229 11.2 11.77 11.14 5.661500 0.0149 17.0 17.23 16.74 2.93

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electron energies of 1 keV and above, there could be systatic errors at lower energies. Nevertheless, a large bodliterature13–45,117indicates that atomic DCSs have been sucessfully used at these low energies for solids.

As an example, we now examine the use of calculaatomic DCSs for the determination of IMFPs by the EPmethod. As discussed in Sec. 9.2, IMFPs can be determfrom relative or absolute measurements of elasbackscattered intensities. The IMFPs obtained from relameasurements should not be affected appreciably by sysatic error in the calculated DCSs because, in effect, ratioDCSs for two solids are employed~unless, of course, thesystematic error in the DCS for one material was mugreater than the systematic error for the other!. In contrast,IMFPs derived from absolute measurements will have stematic errors comparable to the systematic errors in theresponding DCSs. Comparisons of IMFPs from absolEPES measurements with IMFPs calculated from expmental optical data for six elemental solids~Al, Si, Ni, Cu,Ag, and Au! as a function of electron energy between 150and 2 keV do not show clear trends.117 For two elements~Cuand Au!, the IMFPs from the EPES measurements are lowthan the calculated IMFPs for energies less than 700 eVup to about 40%; for Ag, however, the EPES IMFPs alarger than the calculated IMFPs by up to about 17% insame energy range. These results could possibly be dusystematic errors associated with the use of atomic DCSseach solid. It is also possible that the results could be durandom errors associated with varying strengths of surfelectronic excitations for different experimental configurtions and with changes of EPES intensities with differesurface roughnesses.117 The latter effects would be expecteto become larger with decreasing electron energy, and wobe more significant for more grazing angles of electron indence or emission. Further experiments are clearly neededefine the magnitudes of these effects and to clarify the mnitude of any differences between IMFPs obtained fromsolute EPES experiments and calculated IMFPs.

We point out that IMFPs calculated from experimental otical data117 have been tested experimentally in at least thindependent ways. First, Seahet al.126 analyzed the signaintensities of photoelectron and Auger-electron lines of so60 elemental solids and found that these were consistentpredicted intensities within a standard uncertainty of ab28% for lines with energies greater than 180 eV. Their coparison, based on data not only for IMFPs but also for extation cross sections and backscattering factors~for AES!,did not show energy-dependent systematic deviations forergies less than 1 keV that could be associated with possenergy-dependent errors of the IMFP predictive equatTPP-2M127 that was used in the analysis. Second, EALsrived from IMFPs have been extensively used for determition of overlayer-film thicknesses by AES and XPS.72 Com-puted EALs for SiO2 , for example, agree within 10% withthe average of those determined experimentally.128 Finally,Rundgren129 compared current–voltage curves from lowenergy electron diffraction experiments for the Cu~111! sur-

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FIG. 32. Ratio of the mean escapdepth,D, from Eqs.~31!–~38! to themean escape depth,D, calculated withneglect of elastic scattering for twophotoelectron lines and two Augerelectron lines in gold as a function othe emission angle,a: ~solid line! Dobtained from transport cross sectioncalculated with the DHF potential;~dashed line! D obtained from trans-port cross sections calculated with thTFD potential: ~a! Au N7VV Augerelectrons; ~b! Au 4s photoelectronsexcited by MgKa radiation; ~c!Au 4f 7/2 photoelectrons excited byMg Ka radiation; and ~d!Au M5N67N67 Auger electrons.

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face with calculated curves in which an energy-dependimaginary potential was used to represent the inelastic stering. This imaginary potential was derived from Cu IMFcomputed from experimental optical data.130 There was ex-cellent agreement in the widths of the diffraction peaksenergies between 40 and 190 eV.

We therefore conclude that the IMFPs determined frabsolute EPES experiments are unlikely to have a systemerror of as much as a factor of 2 for electron energiestween 150 eV and 1 keV. While measured DCSs in thisergy range for the rare gases in Figs. 17–21 can deviatup to factor of 2 from DCSs calculated from the DHF potetial, these computed DCSs seem to provide reliable IMFfrom absolute EPES experiments. It is possible that theviations found in Figs. 17–21 for the rare gases mightmuch larger than those for other atoms, although we doknow of any reason to support this speculation. Neverthelwe point out that the deviations for Hg in Fig. 22 appearbe less than those for the rare gases at similar energiesalso suggest that the absorption correction to the DCSsculated from the DHF potential might be less for a solid ththe corresponding atom. Tanumaet al.131 showed that thetotal inelastic-scattering cross section in a solid can be msmaller, by a factor of up to about 4, than these cross sectfor the corresponding free atoms. As a result, the acDCSs in a solid at energies less than 1 keV might be closethe atomic DCSs obtained from the DHF potential than sgested by the comparisons of Figs. 17–22.

We comment that IMFPs determined from both relatand absolute EPES measurements agree with IMFPs clated from experimental optical data for solid Al, Si, Ni, CGe, Ag, and Au with an average mean deviation of 17.4%electron energies between 50 eV and 5 keV.117 Most of theIMFPs from the EPES measurements were larger than


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calculated IMFPs for Al, Si, and Ge, while the reverse wtrue for Ni, Ag, and Au. There were no consistent systemadeviations as a function of electron energy, although itpossible that any such deviations could have been maskevarying effects of surface excitations and surface roughnin individual experiments. A more recent analysis of IMFfrom relative EPES experiments for 24 elemental solidsshown that the description of elastic-backscattered intensusing atomic DCSs is satisfactory for energies above ab200 eV.132

Finally, we note that calculated angular distributionselastically backscattered electrons from Au surfaces~ob-tained with atomic DCSs calculated from the TFD potenti!deviated considerably from measured distributions for engies below 200 eV.49 The calculated and measured enerdependencies of the elastic-backscattered intensity fromwithin the solid angle of a retarding-field analyzer are diffeent for energies below 300 eV.49 Similar effects have beenfound with other elements.24,133We therefore conclude fromthese observations and the discussion of IMFPs obtafrom EPES measurements that DCSs calculated from atopotentials appear to be at least empirically useful for simlations of electron transport in solids for electron energabove about 300 eV. At lower energies, the simulations mprovide useful but more approximate guides. In additiomeasurements that depend on ratios of atomic DCSs~such asrelative EPES measurements! can give reliable results~suchas IMFPs! down to at least 200 eV.

11. Summary

We have analyzed calculations of DCSs by the relativispartial-wave expansion method from two commonly uspotentials for elastic scattering of electrons with energies

for

-


FIG. 33. Percentage differences,DD,between the mean escape depthsgold calculated using the TFD andDHF potentials@from Eq. ~41!# as afunction of emission angle for the twophotoelectron lines and two Augerelectron lines in gold shown inFig. 32.

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tween 50 eV and 10 keV by neutral atoms. Calculated crsections have been used for many applications becauseare available for all elements and for a wide range of electenergies, as indicated in Table 1; in contrast, DCSs hbeen measured for a limited number of atoms and elecenergies. Different methods have been used for the crsection calculations~Tables 1 and 2! and, as a result, therare numerical differences in the cross-section data availin different tabulations and databases.1–12 We note here thaan additional database is available on a web site,134 but fewdetails of the calculations are available.135

We compared DCSs that were calculated from the Tand the DHF potentials; cross sections from the latter potials are considered more accurate because these potewere obtained from DHF electron densities compuself-consistently.52 These cross sections were obtained frotwo NIST databases for which the TFD10 and DHF12 poten-tials had been employed. Our comparisons were made foelements spanning a wide range of atomic numbers~H, Al,Ni, Ag, Au, and Cm! and for electron energies of 100, 501000, and 10 000 eV because these energies were commboth databases. While the DCSs from the two potentialssimilar shapes and magnitudes, pronounced deviations~thatcould be as large as 70%! occurred for small scatteringangles for Al, Ag, Au, and Cm. In addition, there were cosiderable differences in the DCSs in the vicinity of scatterangles for which there were deep minima in the DCSs.these angles, the differences could reach 400%, althougother angles the differences were typically less than 20The deviations between the DCSs from the two potentdecrease with increasing electron energy. A slight oscillatstructure in the DCSs for H from the TFD potential wassociated with truncation of the TFD potential at a relativsmall radius.

Another factor that could lead to differences in DCSs fro

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different sources is the choice of approximation used forexchange interaction. We compared DCSs from tdatabases11,12 in which the same DHF potential had beeused~for H, Al, Ni, Ag, Au, and Cm! in the DCS calculationsbut different procedures had been adopted for the exchacorrection. Berger and Seltzer11 used the Riley and Truhlar75

approximation while Jablonskiet al.12 used the Furness anMcCarthy64 correction. These comparisons were madeelectron energies of 1 and 10 keV. For elements and enerwhere there are no deep minima in the DCSs, the differenbetween DCSs from the two databases are very small. FoAl, Ni, and Ag, the differences generally do not exce0.02% although larger differences occurred for scatterangles near 180°; for example, a deviation of20.28% wasfound for H at 10 keV. For Au and Cm, differences of up20.3% and 61%, respectively, were found at scatterinangles where there were deep minima in the DCSs. We cclude that the exchange corrections of Riley and Truhlarof Furness and McCarthy gave essentially equivalent resWe also note that the magnitude of the exchange correcon the DCS can be substantial. The differences betwDCSs for Ag, Au, and Cm at 1000 eV can exceed 35% whdifferences between DCSs for H, Al, and Ni can be up to 5The larger deviations occur at scattering angles where thare deep minima in the DCSs.

We compared calculated DCSs from the TFD10 and DHF12

potentials with measured DCSs for six elements~He, Ne, Ar,Kr, Xe, and Hg! at energies between 50 and 3000 eV. Fthese elements, the measurements were relatively simplethere were generally two or more independent measuremavailable for the chosen electron energies. The consistencthe measured DCSs is typically within625%, althoughlarger deviations occur in some cases. For argon, the elemwith the largest number of measurements, there is exceagreement between the measured DCSs at 2 and 3 keV


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the corresponding calculated DCSs from both the TFDDHF potentials. At lower energies, the measured DCSs gerally agree better with DCSs from the DHF potential ththose from the TFD potential, particularly in the positioand the magnitudes of the deep minima, although the msured DCSs are smaller than the calculated DCSs at oscattering angles. For other elements, the measured Dagree better with the DCSs from the DHF potential thanTFD potential at the highest available energy; deviationstween the measured DCSs and the DCSs from the DHFtential typically become larger with decreasing electronergy, and can be as much as a factor of 2. There is also obetter agreement between the measured DCSs and the Dfrom the DHF potential in the vicinity of deep minima. Thincreasing differences between measured DCSs andDCSs from the DHF potential generally found with decreing energy below 1 keV are believed to be due to the negof absorption effects in the calculational algorithm. For sctering angles less than about 25°, the measured DCSs aergies less than about 500 eV are larger than the DCSsthe DHF potential, but the range of scattering angles whpositive deviations occur gets smaller with increasing eneThese deviations are associated with the neglect of an atopolarizability correction in the calculations. Considerationthe absorption and polarizibility corrections~an appreciablymore complex calculation! leads to improved agreement btween measured and calculated DCSs for He and Ar.107,110

The transport cross section, obtained from an integrathe DCS that emphasizes large-angle elastic scattering,useful parameter in simulations of electron transport in sids. We have compared TCSs derived from DCSs calculafrom the TFD and DHF potentials for H, Al, Ni, Ag, Au, anCm. For energies above 200 eV, the largest differencestween the TCSs from the two potentials is for H wheredeviation is about 20%. For Al, Ni, and Ag, the differencesTCSs are less than 5% for energies above 200 eV. Forand Cm, the differences can exceed 20% for energiesthan 1000 eV, but for higher energies the differences arethan 5%. For energies between 50 and 200 eV, the difences are generally larger and can be up to 43%~for Ni at 50eV!.

We have also examined the extent to which three quaties derived from DCSs varied depending on whetherDCSs were obtained from the TFD or DHF potential. Firwe considered elastic-backscattered intensities from an olayer film on a surface. As an example, calculations wmade of the dependence of these intensities on film thicknfor an Au film on a Ni substrate. Comparisons with two sof experimental data for slightly different configurationstwo electron energies~500 and 1000 eV! were inconclusive.For one configuration, the measured intensities at both egies agreed better with intensities calculated using DCfrom the TFD potential. For the other configuration, a simiresult was found at 1000 eV while better agreement wfound at 500 eV with intensities computed using DCSs frthe DHF potential.

Second, we considered the determination of electron


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FPs by EPES. IMFPs can be obtained from either relativeabsolute EPES measurements. For relative EPES exments, we examined a ‘‘worst-case’’ example of a materconfiguration, and energy for which there was a deep mmum in the DCS~for a single elastic-scattering event!. Weselected Au, a scattering angle of 155°, and electron enerof 500 and 1000 eV for this example, as there is a deminimum in the DCS for Au at 500 eV close to this scatteing angle and there are appreciable differences in the Dfrom the TFD and DHF potentials; there were smaller diffeences at 1000 eV. We found that Au IMFPs derived fromuse of DCSs from the TFD potential~and use of a Ni refer-ence material and a hemispherical electron energy analwith small angular acceptance! were 17.1% and 12.1% lesthan the IMFPs obtained using DCSs from the DHF potenat 500 and 1000 eV, respectively. In another example, invoing EPES measurements from an AuCu alloy, an Au reence material, an analyzer with a large angular acceptaand four energies between 200 and 2000 eV, IMFPs wDCSs from the TFD potential were about 6% larger ththose found using DCSs from the DHF potential at 200 a500 eV; the differences at 1000 and 2000 eV were 3.26%1.27%, respectively. For absolute EPES experiments,considered first the case of Au, measurements with an alyzer of large angular acceptance, and energies of 5001000 eV. IMFPs with DCSs from the TFD potential we10.0% and 1.31% less than those found with the DHF pottial at energies of 500 and 1000 eV, respectively. In anotexample involving Ni, an analyzer of large acceptance anand energies between 250 and 1500 eV, differences betwIMFPs found with the TFD potential and those from thDHF potential were 2.36%, 6.42%, 5.66%, and 2.93%energies of 250, 500, 1000, and 1500 eV, respectivelythese four examples, the differences between IMFPs fousing DCSs from the TFD and DHF potentials varied btween 1.27% and 17.1%. We conclude that, for some mrials, configurations, and electron energies, IMFPs derifrom EPES experiments and simulations with DCSs fromDHF potential could differ significantly from those obtaineusing DCSs from the TFD potential; in other cases, the dferences would be comparable to the precision of the MoCarlo simulation.

Third, we considered MEDs for the signal electronsAES and XPS. As an example, we determined MEDs for tphotoelectron lines~with excitation by MgKa x rays! andtwo Auger-electron lines of Au where the electron energrange from 70 to 2016 eV. MEDs were calculated fromformalism in which the TCS is the only parameter describthe strength of elastic-electron scattering. Differencestween MEDs obtained from TCSs derived from the TFD aDHF potentials for normal electron emission varied betwe4.3% at 70 eV and 0.5% at 2016 eV, and become smawith increasing emission angle. Since Au is an elementwhich elastic-scattering effects are relatively strong, we cclude that MEDs for signal electrons in AES and XPS wnot change significantly if TCSs from the DHF potenti

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are used in the calculation instead of those from the Tpotential.

Calculated DCSs for neutral atoms have been extensiused in simulations of electron transport in solids for a lavariety of applications. Although the interaction potentiala solid will be clearly different from that for a free atommany calculated parameters of interest~e.g., TCSs, electronattenuation lengths, electron backscattered intensities,signal intensities in AES and XPS! depend on ‘‘large-angle’’scattering in the solid for which changes of potential aoften of lesser significance than other sources of uncertaOur comparisons of measured and calculated DCSs foroms have shown, however, that measured DCSs can be mless~sometimes by a factor of 2! than the calculated DCSfor energies less than 1 keV. Nevertheless, we point outIMFPs from absolute EPES experiments agree satisfactowith IMFPs computed from experimental optical data. Tlatter IMFPs have been independently tested by other expments. We therefore believe that IMFPs from absolute EPexperiments are unlikely to have a systematic error ofmuch as a factor of two for electron energies between 150and 1 keV. We speculate that the absorption correction toDCSs for a solid may be less than that for the correspondatom. While further work is needed to understand watomic DCSs are useful for solids, we conclude thatatomic DCSs are empirically useful for simulations of eletron transport in solids for energies above about 300These DCSs can also be useful for energies down to at200 eV where relative measurements are made~such as rela-tive EPES measurements!.

A new NIST database of electron elastic-scattering crsections12 implements the most accurate physical modelthat can be utilized for all atoms and that can be handleda systematic basis~i.e., it has a robust computational schemthat can be reproduced by anyone wishing to repeat theculations!. This approach necessarily prevents the considation of second-order effects whose proper theoreticalscription would require detailed knowledge of the atomstructure beyond the electron-charge distribution ofground state. Even the latter had to be slightly ‘‘maniplated’’ ~through a spherical average! for open-shell configu-rations to obtain a central interaction potential. As a resthe required robustness and generality of the numericalculational algorithm mean a certain sacrifice in the accurof DCS values from the database. Nevertheless, the availity of partial-wave numerical DCSs from a well-definephysical model is a necessary step towards the systemation of more elaborate calculations. For instance, the effeof any improvements in the basic physics will be easierquantify when expressed as corrections to the static-fiDCSs available now.12

12. Acknowledgments

The authors wish to thank Dr. A. Dubus, Dr. M. InokuDr. D. Joy, Dr. D. Liljequist, Dr. S. Manson, Dr. S. Seltze

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Dr. R. Shimizu, and Dr. S. Tougaard for their many usecomments and suggestions on a previous draft of this pa

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