A Universal Predictive Equation for the Inelastic Mean Free Pathlengths of X-ray Photoelectrons and Auger Electrons

SURFACE AND INTERFACE ANALYSIS, VOL. 24, 38-50 (1996)

A Universal Predictive Equation for the Inelastic Mean Free Pathlengths of X-ray Photoelectrons and Auger Electrons

Werner H. Gries Research Center FTZ, Deutsche Telekom, PO Box 10 00 03, 64276 Darmstadt, Germany

A new universal inelastic mean free pathlength (1MFP)-predictive equation (the G1 equation) has been formulated for use in the analytical electron spectroscopies XPS and AES which essentially states that the IMFP of an electron traversing matter is inversely proportional to the atomic density. The equation is based on a concept of matter in which the latter consists of clusters of interaction-prone regions (identified with the orbitals of an atom) in an otherwise interaction-free space. The energy dependence and best values for two fitting parameters were obtained from a large set of IMFPs, derived by Tanuma, Powell and Penn (TPP) from published optical data. The G1 equation has been developed to allow the IMFPs to be predicted not only for regular phases of elements and compounds (a common restriction on existing IMFP-predictive equations), but also for any sub- and non- stoichiometric arrangement of atoms found at technological surfaces. Evidence is presented which substantiates the claim of universal applicability of the G1 equation.

INTRODUCTION ~~ ~ ~~~ ~~~ ~ ~

In the vicinity of a surface or other interface, the struc- ture and stoichiometry of a bulk phase tend to become significantly modified. Native oxide layers are simple examples of the chemical complexity to be expected at such material discontinuities. For instance, the 2 nm thick native oxide layer on a silicon wafer has been found' to consist of an SiO, phase only over the first nanometre at the surface, while the second nanometre towards the oxide/silicon interface is made up of a suc- cession of various suboxidic states. In addition, one may find the SiO, phase to be overlaid and/or partly invaded by carbonaceous contamination.' The composition of surfaces treated by technological processes (e.g. by ion bombardment) is bound to be much more complex.2

Quantitative depth information over the upper several nanometres of a technological surface can be obtained by angle-resolved x-ray photoelectron spectrometry (ARXPS) and by angle-resolved Auger electron spectrometry (ARAES), where signal intensities are measured as a function of take-off angle from the surface, The reliability of this depth information depends largely on the interpretation of the experimental data in the mathematical procedure used and is still under scrutiny. The problems encountered have recently been summarized, as has been the procedure preferred by this author (i.e. the angle-resolved self-ratio procedure).

Regardless of differences of data treatment, a common requirement for all evaluation procedures is the input of reliable so-called effective attenuation lengths (EAL) of the 'analytical' electrons (as x-ray photoelectrons and Auger electrons are jointly being referred to in the following) in the material constituting the surface. As is apparent from the above, such data

CCC 0142-242 1/96/010038-13 0 1996 by John Wiley & Sons, Ltd

are required not only for elements and their common compounds, but also for sub-stoichiometric compounds and, indeed, any assembly of atoms that are likely to occur at surfaces and interfaces.

As it is obviously impossible to obtain all these EAL by direct measurement of signal attenuation in all materials of interest, the approach has been to generalize from EAL measured in a representative set of materials. One of the best-known sets of generalized EAL-predictive equations is that of Seah and D e n ~ h . ~ From a large collection of experimental data based on measurement of the signal attenuation in overlayers of given thickness, they derived fitting equations for three classes of solids : elements, inorganic compounds and organic compounds. Although these equations are widely used, it is realized that the large uncertainties associated with overlayer experiments4 call for a fresh and different attempt at deriving EAL-predictive equations.

An alternative to direct determination of EAL by overlayer experiment is the derivation from inelastic mean free pathlengths (IMFPs), since the former is a projection of the latter on a specified direction vector. Monte Carlo codes e ~ i s t ~ - ~ for derivation of the EAL from a given IMFP. So the problem shifts to the devel- opment of a means for general determination of IMFPs in whatever material may be found in and on a technological surface. Again, the IMFPs have to be known for all materials that are likely to occur, and the only approach feasible is to generalize from IMFPs obtained for a representative set of materials.

Tanuma, Powell and Penn (to be referred to as TPP in the following)* have pioneered this route by publi- shing tables of IMFPs for electrons of energies 50-2000 eV in 27 elements and 29 inorganic and organic compounds. These IMFPs were derived from optical data (refractive indices and extinction coefficients in dependence on photon energy) by using an algorithm of

Receiued 1 7 July I995 Accepted I8 September 1995

39 A UNIVERSAL IMFP-PREDICTIVE EQUATION FOR XPS AND AES

Penn.' [TPP calculate values of the energy loss function Im( - 1 / ~ ) = E ~ / ( E ~ + c2 2), where E is the complex dielec- tric function E = E~ + ie2, and the real and imaginary terms are given by c1 = n2 - k2 and e2 = 2nk, where n is the index of refraction and k is the extinction coeffi- cient (obtained by measurement). The calculation procedure of T P P is explained in some detail in Ref. 8(c).] In the following, these IMFPs are referred to as 'optical' IMFPs. Based on these optical IMFPs, TPP have proposed a generalized IMFP-predictive equation (designated as TPP-1, since superseded by TPP-2 and now by TPP-2M) for both elements and compounds. [In Ref. 8(e) the TPP-2M equation is given as A = E/{EP2[& ln(yE) - (C /E) + ( D / E 2 ) ] } , where A is the IMFP (in A), E is the electron energy (in eV), E , = 28.8 J ( N , p / M ) is the free-electron plasmon energy (in eV), p is the density (in g cmP3), N , is the number of valence electrons per atom (for elements) or molecules (for compounds) and M is the atomic or molecular mass number; PM, y, C and D are parameters. The parameters are given by &, = -0.10 + 0.069pO.' + 0.944/J(Ep2 + E,'), y = O.l9l/Jp, C = 1.97 - 0.91U, D = 53.4 - 20.8U, where U = EP2/829.4 and E, is the bandgap energy (in eV) for non-conductors.]

The TPP equation (by which is meant the latest version TPP-2M) is an adaptation of the well-known Bethe equation for the stopping of charged particles in matter." The adaptation consists of terms which require two macroscopic material properties as input quantities, and which contain fitting parameters for an average best fit of the TPP IMFP to the optical IMFP. The input quantities required for prediction of the IMFP in a given material are the free-electron plasmon energy and the bandgap energy. It is obvious that these can be determined only for stoichiometric materials, and not for non-standard materials as are often found in and on solid surfaces. This places a severe limitation on the applicability of the TPP equation to technological surfaces.

One notices (and this is emphasized by TPP) that the IMFPs predicted by the TPP equation can deviate significantly (up to 50%) from the corresponding optical IMFPs for individual elements or compounds. In the absence of a simple physical model, these deviations cannot be readily explained other than (as TPP do) by questioning the general applicability of the algorithm of Penn and/or the reliability of some of the optical data. Thus, there is a need for an alternative approach which can assist in the decision on which data to trust.

It is also noted that the IMFPs calculated by use of the TPP equation exhibit a pronounced non- proportionality to the inverse of the atomic density (discussed below). This fact clashes with a concept of matter (held by this author) in which the latter presents itself to a traversing analytical electron as individual clusters of interaction-prone regions, to be identified with electron orbitals, in an otherwise interaction-free space. Implicit in this concept is an inverse proportionality between the IMFP and the atomic density.

On the basis of this concept of matter, a new IMFP- predictive equation has been formulated (henceforth designated as the G1 equation) which incorporates this inverse proportionality between IMFP and atomic density, but which also allows recognition to be given to the known predominance of inelastic interaction with

electrons in outer orbitals, as well as the possible exis- tence of distinct groups of IMFPs according to whether the attenuating medium is an element, an inorganic compound or an organic compound (as proposed by Seah and Dench3).

To this end the optical IMFPs of TPP for elements are systematized on the basis of the Periodic Table of elements. The underlying assumption is that the predominance of inelastic interaction with electrons in outer orbitals leads to a dependence of the IMFP on the position of the corresponding element in the Period- ic Table.

The limitation of the TPP equation to elements and stoichiometric compounds is overcome in the G1 equation through the absence of macroscopic physical input quantities that are difficult to come by. The local atomic composition, the atomic density and the electron energy suffice as physical input quantities. Two numerical parameters are required for fitting the G1 equation to optical or other experimental IMFPs. One of these parameters suffices for period classification of elements or chemical classification of compounds.

In the following, the orbital model is summarized which underlies the G1 equation. Then the equation is given with the parameter values for elements according to their electron configuration, as well as for inorganic and organic compounds. The G1 IMFPs are compared to the optical IMFPs of TPP as well as to other experimental IMFPs. The main points of difference between the G1 IMFPs and the T P P IMFPs are discussed.

The G1 equation was first presented in 1993" and was first documented in Ref. 2. The reliability of the G1 IMFPs depends on the reliability of the optical IMFPs of TPP. Recent calculations of IMFPs from proton stopping data suggest that the optical IMFPs of TPP may be twice as large as the proton data-based I M F P s . ~ ~ ~ ~ This discrepancy is unexplained at present.

~~

THE ORBITAL INTERACTION MODEL FOR ENERGY LOSS OF ANALYTICAL ELECTRONS

The orbital configuration model of bound electrons in atoms and between atoms is assumed to be well known and i s used as a departure point for the proposed orbital interaction model of energy loss of analytical electrons.

According to the orbital configuration model, bound electrons occupy regions of space known as orbitals. The orbitals are classed as s, p, d and f types. The occupancy of orbitals can be one, two (Pauli exclusion principle) or nil (empty orbital). The orbitals of an atom are filled according to the well-known rules of Hund.

At this stage it is helpful to recall the essential points of Bethe's findings" about the stopping power of an atom for high-energy electrons: The stopping power of an atom is the sum of those of the individual electron shells. The stopping power of an individual shell is proportional to the sum of the oscillator strengths of all transitions that the electrons in the shell are capable of. This causes the stopping power to increase with the number of electrons in a shell as well as overall. Dif- ferences in transition probabilities between inner and outer shells cause (electrons in) an outer shell to have a

40 W. H. GRIES

several times larger stopping power than (electrons in) an inner shell. In consequence, the overall stopping power increases with the atomic number Z, but slower than Z. A general expression for the dependence of the stopping power on Z cannot be provided (on the basis of the Bethe equation).

To assist the reader to follow more easily the arguments developed below, the Periodic Table is shown in Fig. 1. Periods are numbered Arabic 1-6 and groups Roman I-XVIII. Members of the same group feature the same type of outer orbital as well as the same occupancy. In Fig. 1, groups of elements with incompletely filled (outer) orbitals are denoted by the symbol for this type of orbital. Since orbitals of the same type are of identical or similar shapes, the outer orbitals (filled or partly filled) within the same group of the periodic table are of similar shape (and of the same occupancy).

Another property of importance is the spatial extent (volume) of the orbital. An approximate measure of the spatial extent of an outer orbital is the radius of (what is usually termed) the main maximum of the outer orbital. A plot of these dataI3 is shown in Fig. 2. As examples, the radii of outer orbitals of a group (group XIV, C to Pb) are marked, as well as the radii of outer orbitals of a subset of different atoms within a period (period 5, Zr to Ag). It is noted that the outer orbitals within a group are of about the same spatial extent if they fall within periods 3-6. The outer orbitals within a given period are seen to shrink with increasing atomic number.

The orbital interaction model is based on the widely accepted notion that for an inelastic interaction of a passing signal electron with a bound electron to take place, the associated wave packet must overlap the rele-

vant orbital to a sufficient extent and for a sufficient time interval for a (quantized) energy transfer to occur.

The orbital interaction model stipulates that the probability of an inelastic electron interaction depends on the shape (s, p, d or f), spatial extent and occupancy (1 or 2) of the orbital, and on the allowed (higher energy) states of electron@) in this orbital. In other words, orbitals of the same shape, spatial extent and occupancy offer the same probability for an inelastic interaction (equal probability rule).

Because inelastic interaction of analytical electrons is predominantly with electrons in the outer orbitals, the probability of inelastic interaction is strongly influenced by the shape, spatial extent and occupancy of outer orbitals. And since outer orbitals within a group of the Periodic Table are of similar shape, of about the same spatial extent and of identical occupancy, the probability of an inelastic interaction of an analytical electron is expected to be roughly the same for different atoms within the group (at least if they fall within periods 3-6). If the analytical electrons are of sufficient energy to permit inelastic interaction also with electrons of inner orbitals, the overall probability of an inelastic interaction is bound to increase with the atomic number within a given group. This increase may be significant, since the increase in atomic number within a group occurs in large steps.

For different atoms within a given period the overall probability of an inelastic interaction is difficult to predict, because the spatial extent and occupancy of the outer orbitals change from one atom to the next (i.e. from one element to the next). The change is gradual, though, as is the increase in atomic number. An increase

IS - 1 H

1 IIS

2

X x I x n d d d

' I 4 1

37 Rb 38 Sr 39 Y 43 Tc 44 Ru 45 Rh 5

6 II

XYQIP

61 TI 82 Pb 83 Bi 84 85 86

- Figure 1. The Periodic Table of elements, where periods are numbered 1-6 and groups I-XVIII. Groups of elements with incompletely or just filled outer orbitals are denoted by the symbol for this type of orbital (s, p and d). Elements for which 'optical' IMFPs have been published by TPP and which are used in this paper are flagged with filled triangles. Hatched triangles denote other elements for which experiment-derived IMFPs exist and which are used in this paper.


d

0 d

of atomic number leads to an increased occupancy of the outer orbitals, which (in turn) is accompanied by a shrinkage of these orbitals (as shown in Fig. 2). The overall effect may well be a slow change of overall probability of inelastic interaction. Evidence in support of this speculation is provided below.

The probability of an interaction is commonly expressed as a cross-section. For conversion of the cross-section for inelastic interaction to the IMFP, the atomic density has to be brought in. In the orbital interaction model, matter is considered to consist of a non- uniform distribution of interaction-prone regions (orbitals) in an otherwise interaction-free space. The interaction-prone regions cluster around atomic nuclei, leaving relatively wide interaction-free gaps between these clusters.

The analogy to the interaction of neutrons with atomic nuclei is apparent. There, as in other cases of exponential attenuation of particle flux, the cross- section a is defined by

1/A = na (1) where A is the IMFP and n is the spatial density of interaction centres. The analogy to the attenuation of analytical electrons is perfect. Any inelastic interaction removes an analytical electron from the total flux and each such interaction is associated with an atom. Hence, the above relationship between A, n and a holds and states that the IMFP of analytical electrons changes inversely with the atomic density. This inverse proportionality between the IMFP and the atomic density is an important implication of the orbital interaction model and is a major point of difference to the TPP equation.

A few relevant data are quoted from two of the TPP papers,8b*d which illustrate this point. In these papers, TPP have varied the input values of the material density into the TPP-2 equation. They then plotted the predicted IMFP (here denoted by ATPPJ as a function

s o r b l l a 1 s A D O r b i t a l s

.c

g P.1 Q b o r b i t a l s

5s L

a 4

200

c 0

E .L

X 0 z c 100 d x L 0

.e

.c

.c U 0 (L 0

R tomi c Numbe r

Figure 2. The radii of the main maxima of the outer orbitals as a function of the atomic number, according to tabulated values.’’

of energy, with the density (p) as parameter. The relationship between A T p p - 2 and p is conveniently expressed in the form A T P P - Z ( P ~ ) / A T P P - ~ ( P ~ ) = (pJpl)X, where the subscripts 1 and 2 denote different input values for the material density. At a given energy, x not only assumes non-unity values, but sometimes the values of x change with the density. For instance, at 2000 eV the following values pertain

for A1 ( p = 2.7 g ~ m - ~ ) :

x = 0.2 for &/pi) = (2.7/5)

x = 0.4 for (p2/pl) = (2.7/20)

for Au (p = 19.3 g ~ m - ~ ) :

x = 0.6 for (p2/pl) = (10/19.3)

x = 0.4 for (p2/pl) = (2/19.3)

for GaAs:

x w 0.3 for all p1 and p2

In other words, the TPP-predicted IMFPs are not even approximately proportional to the inverse of the atomic density.

The qualitative arguments advanced above as to the pattern of the overall probability of inelastic interaction can be tested on the basis of Eqn (1). For this purpose the optical IMFPs of TPP are used for A. First, however, the available data must be surveyed.

In Fig. 1, those elements are flagged (with filled triangles) for which optical IMFPs have been published by TPP. The majority (22) of these elements are transition elements (i.e. they have incomplete d-type outer orbitals, groups 111-XI), and only five are main group elements. Four of the latter have incomplete p-type outer orbitals (groups XIII-XV). All of them have complete s-type outer orbitals (refer to Fig. 2). Hence, the coverage of the Periodic Table is rather incomplete and is heavily biased towards transition elements. This implies that this author’s proposal that the probability of inelastic interaction is patterned on the Periodic Table of elements has to rely mainly on evidence from the transition elements. To this end, the elements have been subdivided into four sets: three sets of transition elements (periods 4, 5 and 6, without yttrium) and one set of elements from the main groups (made up of the remainder).

The optical IMFPs of 18 of the 22 transition elements have been chosen (i.e. those forming a complete matrix of groups us. periods) for calculation of atomic cross- sections (for inelastic interaction) by means of Eqn (1). The values obtained for a as well as for u/,/Z are shown in Table 1.

The values of a in Table 1 are seen to be significantly smaller (by one order of magnitude) than the geometric cross-section. The values of a (particularly in period 4) also indicate that the atomic cross-section of transition elements for inelastic interaction with 2 keV electrons tends to change rather little within a given period. Within a group, the values of a step up with 2 (with the exception of either or both Zr and Hf). These findings confirm the qualitative arguments developed above, concerning the overall probability of inelastic interaction. In particular, the Z dependence within a group

42 W. H. GRIES

Table 1. Atomic cross-sections for inelastic interaction of 2 keV electrons derived from ‘optical’ IMFPs of Tanuma, Powell and PennEb

Elements

(period)

Group (4) (5)

IV Ti Zr V V Nb VI Cr Mo Vl l l Fe Ru X Ni Pd XI Cu Ag

Cross-sections

u [lo-’’ cm?] a@ [ lo-” cm-”1

(6) (4) ( 5 ) ( 6 ) (4) (5 ) ( 6 )

Hf 4.4 6.5 6.0 0.93 1.03 0.71 Ta 3.8 4.7 7.3 0.80 0.73 0.85 W 4.2 5.6 7.1 0.86 0.86 0.82 0 s 4.3 5.2 6.5 0.83 0.79 0.75 Pt 4.3 4.8 6.9 0.82 0.70 0.78 Au 4.3 7.2 7.8 0.79 1.04 0.88

Period averages: 4.2 5.6 6.9 0.84 0.86 0.80

points towards a systematic dependence on the total number of bound electrons.

This dependence has been investigated for the transition elements for which optical IMFPs are available, and it was found that to a good (average) approximation 0 rises as JZ. If ,/Z is interpreted as the effective number of bound electrons per atom available for inelastic interaction, the quantity a/JZ assumes the meaning of an (average) cross-section per interaction- prone electron. Values of this quantity are also shown in Table 1, as calculated from Eqn (1) and the optical IMFPs. The standard deviations from the period averages are 6%, 17% and 8% for periods 4, 5 and 6, respectively.

The orbital interaction model does not allow quantitative predictions to be made as to the energy dependence of the IMFP. Qualitatively, it can be argued that a higher energy of a passing analytical electron is associated with a shorter duration of overlapping of the wave packet with a given orbital. In consequence, the probability for small energy transfer is reduced in accordance with the uncertainty principle. Large energy transfers are not affected. Overall, the probability for inelastic interaction (for a given orbital) is reduced, causing the IMFPs to increase with energy. This is known (both from optical IMFPs and from experimental EALs) to be correct for energies in excess of - 50 eV, but not below.

In summary, the orbital interaction model predicts a dependence of the IMFP on the atomic density, on certain properties of the orbitals and on the total number of bound electrons (i.e. on 2). The dependence, on the atomic density is one of inverse proportionality. The dependence on orbital properties concerns the shape, spatial extent and occupancy of the outer orbitals. The information contained in the optical IMFPs of TPP with regard to interaction cross-sections does not contradict the orbital interaction model. The former can be used, therefore, to quantify the dependence of the IMFP on 2 and on the energy of the analytical electrons. The dependence of the IMFP on Z is of the form 1/,/Z. The dependence on the energy E of the analytical electron was found to be of the general form proposed earlier, e.g. by Penn:14 E/(logE - k), where k is a constant.

Of cardinal importance for the general applicability of the IMFP-predictive equation to be formulated is that the orbital interaction model must apply also to compounds. In other words, the dependence of IMFP

on the atomic density, on certain properties of the orbitals, on the total number of bound electrons and on the energy of the analytical electron found for elements must apply also to compounds.

If the well-known linear combination of atomic orbitals (LCAO) model’5 is used to describe the bonding between atoms by formation of molecular orbitals, then the above orbital interaction model should indeed be directly applicable also to compounds. The LCAO model predicts the total number of orbitals to remain constant, although the occupancy and shape of the valence orbitals is changed. Obviously, there must be a prevalence of double occupancy in molecular orbitals, while other orbitals are empty. Also, molecular orbitals span the interatomic space (contrary to atomic orbitals). There is no change, however, in the (average) number of electrons in the outer orbitals, and the binding energies of these electrons remain practically the same. Hence, the (average) effective number of interaction-prone electrons remains invariant and the (average) atomic cross-section for inelastic interaction should not be affected to any significant degree. Also, Eqn (1) remains valid and the inverse proportionality between the IMFP and the atomic density is retained. This argumentation in favour of the general applicability of the orbital interaction model is supported by the finding (reported and discussed below) that the equation developed for elements with the assistance of the optical IMFPs for elements does indeed apply to the optical IMFPs for compounds as well as to experimentally derived IMFPs from sources other than TPP.

THE IMFP-PREDICTIVE EQUATION G1 AND THE OPTICAL IMFP DATA OF TPP

The G1 equation was first presented at a conference in 1993,” but has not been published until recently.2 It expresses the IMFP (denoted by h G 1 ) as

AGI = kl(E/z*)E/(lg - k 2 ) (2) where V , is the atomic volume in cm3 mol-’, E is the energy of the analytical electron in eV, lg E stands for log,,E and k, and k, are numerical fitting parameters (the former magnitude-adaptive, the latter energy- adaptive) derived from the optical IMFPs of TPP. The value of AG1 is in units of nm if k , is of magnitude 0.00xy (as given below); Z* is a real number which can

A UNIVERSAL IMFP-PREDICTIVE EQUATION FOR XPS AND AES

1 . 4 I I

.c

d 2 1 .0

a

T i

C r Fe N i V cu

0 200 500 1000 I500 2000 E l e c t r o n e n e r g y / eV

Figure 3. The ratios of AQl/Aopl for transition elements of the 4th period over the energy range 100-2000 eV. Here Aopt designates the optical IMFPs of Tanuma, Powell and Penn,8b and A,, is calculated from Eqn (2) with (k,, k 2 ) = (0.0020, l .30).

be regarded as the nominal ‘effective’ number of interaction-prone electrons per atom. [The energy quotient E/(lg E - k2) is simply a convenient mathematical construct for fitting the energy dependence of the G1 equation to that of the optical IMFP. The expression can be replaced, for instance, by a polynomial (with more coefficients) or some other approximating function. In order to emphasize the fact that no physical meaning is attached to the quotient, the logarithmic term has been chosen to the base 10 rather than to the base ‘e’. Despite this, it is interesting to note that the energy quotient is equivalent to that of the Bethe equation if the latter is written in the form:” E/ln(yE).]

The atomic volume V, is used to express the proposed inverse proportionality between the IMFP and the atomic density. In analogy, the quotient V,/Z* is inversely proportional to the average density of interaction-prone electrons in matter space.

As mentioned above, the substitution of JZ for Z* has been found to provide a good general fit to the optical IMFP of TPP for elements. The identity has

I4 ki = 0.0019 1 k Z = 1.35 t I \

I- I L

0.6 i

43

;?

MO Rh

R u

Nb Pd

0 200 500 I000 I500 2000 Electron e n e r g y / eV

Figure 4. The ratios of A,,/A,,, for transition elements of the 5th period over the energy range 100-2000 eV. Here A,,, designates the optical IMFPs of Tanuma, Powell and Penn,8b and A,, is calculated from Eqn (2) with (k,, k,) - (0.001 9, l .35).

been derived from a trial-and-error procedure in which both the Z dependence (in the form Zx) and the energy dependence of the optical IMFP of the transition elements were varied until the standard deviation of the optical IMFP with respect to the universal equation was found to pass through a minimum. This minimum was found to lie in the vicinity of x = 0.5. The substitution of JZ for Z* has proved to be a good approximation also for the other elements as well as for use with compounds (as is shown below). The author is unable at present to explain the fortuitously simple relationship between Z* and Z.

For compounds, Z* pertains to an (imaginary) average atom and is (in analogy to an element atom) given by

Z* = (pJz, + qJzB + . . . + r Jzc)/(p + q + * * . + r )

(3) for a compound of stoichiometry A,B, . . . C , and corresponding atomic numbers Z,, ZB, . . . , Z , .

Table 2. Nomenclature of organic compounds according to the number code of Fig. 7, and (reduced) stoicbiometries and densities (g cmd3) used for calculation of variables V, and Z* in the G1 equation

Code no Compound Stoichiometry Density

1 Polyacetylene CH -0.93 2 Diphenyl-hexatriene C,H8 0.99 3 Polyimide (Kapton) C22Hl 1.36 4 Poly(2-vinyl pyridine) C,H,N 1.01 5 Adenine CHN 1.35

7 Polystyrene CH 1.05 8 Guanine C6H6N60 1.58

10 Polyethylene CH2 0.93

6 DNA C3,H46N,6P4024 1’35

9 /3-Carotene C6H7 0.99

11 Poly(butene, 1 -sulphone) C4H,S02 1.39 12 PM MA C6H802 1.19 13 2611-paraffin C13H27 0.99

44 W. H. GRIES

kl = 0.0019

kz = 1.45

1 . 2 Re

0 .C

-.' 2 1.0

0.a

t i

RU l a U

P t 0s H f I r

0.6 I I I I I 0 200 so0 I000 I so0 2000

Electron energy / eV Figure 5. The ratios of AOl/Aopt for transition elements of the 6th period over the energy range 100-2000 eV. Here bOpt designates the optical IMFPs of Tanuma, Powell and Penn," and A,, is calculated from Eqn (2) with ( k l , k,) - (0.001 9, l .45).

The atomic volume for the same compound, of density p (g cm-3) and mass numbers MA, M , , . . . , M c is obtained from

V, = O M , + 4 M , + . . * + rM,)/p(p + 4 + . . + r ) (4) The numerical fitting parameters k , and k , were

determined by fitting Eqn (2) to the optical IMFPs of TPP* in the energy range 200-2000 eV. Within this energy range the optical IMFPs were found to be classi- fiable into six material categories according to a common energy dependence, i.e. the optical IMFPs within each of the six categories gave rise to the same

1 . 4 I I

I I ki = 0.0014 I

I k p = 1.10

0.6 I I I I I 0 200 so0 1000 IS00 2000

Electron energy / eV Figure 6. The ratios of AOl/AOpt for main group elements and Y over the energy range 100-2000 eV. Here A,,, designates the optical IMFPs of Tanuma, Powell and Penn,8b and A,, is calculated from Eqn (2) with (k,, k2) = (0.0014, 1 .lo). Labels C1.5 and C1.8 refer to carbon with densities of 1.5 and 1.8 g ~ m - ~ , respectively.

t I 0.6 I I 1 I I I

0 200 500 1000 1500 2000 Electron energy / eV

Figure 7. The ratios of AGl/AOpt for organic compounds over the energy range 100-2000 eV. Here A,,, designates the optical IMFPs of Tanuma, Powell and Perm,'' and A,, is calculated from Eqn (2) with ( k , , k,) - (0.001 8, 1 .OO). The compounds are numbered in accordance with the code used in Table 2. Labels C1.5 and C1.8 refer to elemental carbon with densities of 1.5 and 1.8 g ~ m - ~ , respectively.

value of the energy-adaptive parameter k , . The magnitude-adaptive parameter k , was found to vary from one element to the next and from one compound to the next. Rather than giving the value of k, for every element and compound separately, it was decided to use an average value of k , for each category.

In summary, for electron energies between 200 and 2000 eV, the following fitting parameters (kl, k,) lead to a best fitting of the G1 equation to the optical IMFPs of TPP in the material categories as shown, where the

1 . 4 , I

kl = 0.0019 kz = 1.30

lnfls InSb

/

tl 0.6 L I I I I

0 200 so0 I000 I so0 2000 Electron energy / eV

Figure 8. The ratios of AGl/Aopt for inorganic compounds over the energy range 100-2000 eV. Here Aopt designates the optical IMFPs of Tanurna, Powell and Perm,'' and AGi is calculated from Eqn (2) with (k , ,k , ) - (0.0019. 1.30).

A UNIVERSAL IMFP-PREDICTIVE EQUATION FOR XPS AND AES 45

Table 3. Optical IMFPs for water by Ashley16 vs. IMFPs from the G1 equation with fitting parameters (kl , k,) for inorganic compounds (0.0019, 1.30); E is the electron energy

E (keV) 0.2 0.3 0 4 0.5 0.6 0 8 1 .o 1.5 2 0

A,, (nm) 1.47 1.79 2.13 2.46 2.78 3.41 4.03 5.49 6.90 Aor (nm) 1.42 1.81 2.18 2.53 2.88 3.54 4.17 5.66 7.08

chosen magnitude of k, gives rise to units of nm for AG1

(0.0020, 1.30) for transition elements of

the 4th period (here, Ti to Cu)


the 5th period (here, Zr to Ag)


the 6th period (here, Hf to Au)

(0.0014, 1.10) for main group elements

(0.0018, 1.00) for organic compounds

(0.0019, 1.30) for inorganic compounds

(here, without the outliers InP and KC1)

Values of the ratio AGl/Aopt vs. energy E (where Aopt denotes the optical IMFP of TPP) are shown in Figs 3-8 for each of the six material categories.

Attention is drawn to the following: (1) The 5th period transition element Y (group 3) is

shown with the main group elements (Fig. 6) instead of with the other 5th-period transition elements (Fig.

(2) The main group element C is shown both with the main group elements (Fig. 6) and with the organic compounds (Fig. 7). In each figure it is shown twice,

4).

1.5

1.0

0 .5

0

because of uncertainty over the real identityldensity of the material for which the optical IMFPs were calculated by TPP. C1.5 pertains to a density of 1.5 g cm-3 (typical of glassy carbon), while C1.8 pertains to a density of 1.8 g cm-3 (typical of graphite).

(3) The organic compounds in Fig. 7 are identified by numbers. The coding is shown in Table 2.

(4) The compounds InP and KCl are not shown (in Fig. 8); they are outliers and fall into the ratio interval 1.4-1.7.

The implications of Figs 3-8 are discussed later.

THE GI EQUATION AND IMFP DATA FROM SOURCES OTHER THAN TPP

The validity of the G1 equation and the values of fitting parameters k, and k , has so far been established only for a limited set of elements and compounds, and for these only within the energy range 200-2000 eV. The general applicability is derived from successful testing against IMFP data from sources other than TPP and (if available) for IMFP data for elements and compounds with orbital configurations not covered by TPP. Also the apparent limitation to energies > 200 eV requires to be investigated.

All these aspects were investigated by comparison of the predicted G1 IMFP against the following three sets of IMFP data:

0 25 50 75 100

Electron energy 1 eV

Figure 9. The IMFP data for Be for energies <70 eV scaled from a graph in Ref. 18 (derived from intensity ratios of bulk and surface signals). The broken line is the locus of the IMFPs predicted by the GI equation, with fitting parametersk, andk, as shown (i.e. for main group elements).

W. H. GRIES

Q _____. Q '. _ _ _ _ _ _ _ _ - - - - - - - L i

46

1 . 5

1.0

0.5

0 ~

1.0 E

E 1

0

0 .5

cs Cm

0 0 50 100

E l e c t r o n e n e r g y / eV Figure 10.. The IMFP data for the alkalis Li to Cs for energies < lo0 eV scaled from graphs in Refs 18 and 1 9 (derived from intensity ratios of bulk and surface signals). The broken line is the locus of the IMFPs predicted by the G1 equation, with fitting parameter k , = 1.10 throughout and fitting parameter k , as follows: 0.00065 for Li, 0.00115 for Na, 0.0013 for K, and 0.00095 for both Rb and Cs.

(1) Optical IMFP data for water for the above energy range, by Ashley;16

(2) IMFP data for Be for energies <70 eV, derived from intensity ratios of bulk and surface signal^;'^

(3) IMFP data for the alkalis (Li to Cs) for energies < 100 eV, derived (as for Be) from intensity ratios of bulk and surface

The outcome for wzter is shown in Table 3, where denotes the optical IMFPs of Ashley and AG1 the IMFPs calculated from the G1 equation by use of the fitting parameters for inorganic compounds. Over the energy range 200-2000 eV the two sets of IMFP values were found to be within +4% of one another (deviating below 200 eV). The significance of this test lies in the fact that: the fitting parameters k, and k , were deduced for inorganic compounds of much higher Z* value than that of water; these compounds did not include hydrides; and hydrogen is the dominant element in water and the Z* value of water is, therefore, close to the lower extreme value of unity (in fact, Z* = 1.6). The close agreement between AGl and AAsh supports the

claim of the general applicability of Eqn (2) and of the values chosen for k , and kz at least for optical IMFPs in the energy range 200-2000 eV.

The result for Be is shown in Fig. 9. The experimental points have been scaled from Fig. 8 in Ref. 18. The broken line represents the IMFPs predicted by the G1 equation, with fitting parameters as shown (i.e. for main group elements). Although the G1 curve does not Jretrace the dip shown by the experimental points at the minimum, the fit is acceptable in general. In particular, the sharp upward turn on the low-energy side of the minimum is realistically predicted. This result indicates that the G1 equation does not only apply for electron energies beyond 200 eV, but also below. Also, Be is close to the extreme lower end of the Periodic Table. It has only four electrons in two s-type orbitals. Still, the identity Z* = ,/Z applies as it does for elements with many orbitals and with outer orbitals of a different type.

The fitting of the G1 equation to the experimental IMFPs of the alkalis is not as straightforward as to those of Be. The magnitude-adaptive fitting parameter k , for the alkalis is generally lower than for other main group elements, and an energy correction had to be introduced for the energy positions of the IMFP minima of the G1 equation to coincide with those of the experimental data. The latter adaptation was made by replacing the energy E in Eqn (2) by a corrected energy value E , = E + 10. Best fittings with k , adaptations only are shown in Fig. 10. The value of k , for Cs has been assumed to be the same as that for Rb. Adapta- tions of k , alone are seen to suffice for energies beyond 50 eV for Rb, K and Na (and probably for Cs), and for energies beyond 30 eV for Li.

DISCUSSION

The G1 equation was formulated on the basis of the orbital interaction model for elements, with fitting parameters for the optical IMFP of TPP for three sets of transition elements and for energies of 200-2000 eV. The equation has been shown to apply also for other elements as well as for organic and inorganic compounds for which optical IMFPs of TPP are available. The equation has been shown to apply also for water for which optical IMFPs are available from Ashley. The equation has also been shown to apply at energies below 200 eV, in general to 50 eV and sometimes lower, namely for elements (Be and the alkalis) for which experimental IMFPs other than optical IMFPs are available in this energy range.

The simplicity of the G1 equation derives from the proposed proportionality of the IMFP to the quotient V,/Z*. Although the proportionality of the IMFP to the atomic volume V, is a logical consequence of the orbital interaction model as formulated above, it is desirable to experimentally verify the proportionality of the IMFP to either quantity V, or l/Z*. For verification of the proportionality to V , one would have to measure the IMFP in density modifications of a given material (i.e. for fixed Z*) . Density differences of the solid state are seldom in excess of the uncertainty margin on the measured IMFP. [A rare exception is carbon, which occurs in the modifications diamond ( p = 3.5 g cmP3), graphite

41 A UNIVERSAL IMFP-PREDICTIVE EQUATION FOR XPS A N D AES

(p = 1.7-1.85 g ~ m - ~ ) and glassy carbon ( p = 1.4-1.55 g cm-3). For the last two, Lesiak et have derived IMFPs from measurements of the elastic peak of electron spectroscopy. Taking averages of their results (which show a considerable spread) one obtains IMFP ratios of glassy carbon to graphite of 1.3-1.4 at four energy positions from 500 to 2000 eV. The density values measured by Lesiak et al. are given as 1.8 g cmP3 for graphite and 1.5 g cm-3 for glassy carbon. Hence, the IMFP ratio should be 1.2 according to the G1 formula. The agreement is insufficient to serve as experimental proof for the proportionality of IMFP and atomic volume.] The author is not aware of IMFP measurements on a material in both the solid and the gaseous state. There is, however, stopping power data available for protons in various materials in both states. These data can be used in the present argument. In Ref. 2 it has been shown how the stopping power for protons is related to that for electrons and how the latter in turn can be related to the IMFP. [The relationship between the stopping powers for electrons and protons is given by:, s, = s, - 90.4 Z / E , with E = Ep/1836, where s, is the electron stopping power at electron energy E in eV and s, is the proton stopping power at proton energy E, in eV. Hence, the stopping power for, say, a 1 keV electron is derived from that of a 1.836 MeV proton. The value of the constant factor depends on the choice of units for s, and s, (here eV lo-,' atoms ern-,); Z is the atomic number. The stopping power for electrons can be related to the IMFP by:, A = E / s , , where E is the mean energy loss per collision.]

The stopping power for protons, if measured in electron-volts per matter thickness (e.g. eV atoms cm- 2), has been reported to be affected relatively little by a change of the physical state of most materials at the energies of interest here.22 For instance, at a proton energy of 2 MeV the stopping power of the gas phase is larger than that of the solid phase by generally not more than 5% for organic materials and water; no significant differences are reported between the two phases for argon and for chlorine and bromine. In accordance with the equations in Ref. 2 and above, and the fact that at this energy the difference term in the stopping-power equation is negligible and the proton is fully stripped, one can associate these findings for protons also with electrons of 1 keV energy. Therefore, it can be concluded that the corresponding IMFP, if measured in matter thickness, should be affected to the same extent, as long as the mean energy loss per collision ( E ) stays constant. On the latter provision, this means that, to a good approximation, the IMFP in units of length is directly proportional to the atomic volume (and, hence, inversely proportional to the atomic density).

For an independent verification of the proportionality of the IMFP to the quantity l/Z*, one has to compare IMFP data of materials of largely differing Z* and closely similar atomic volume. Such combinations are, for instance, Y and Bi in Fig. 6, and V and 0 s in Figs 3 and 5, respectively. The data for these combinations confirm the proposed proportionality to a good approximation. More significance is attached, though, to the fact that, on adoption of the proportionality of IMFP and the atomic volume, the identity Z* = ,/Z derived for the transition elements turns out to be of validity throughout the Periodic Table and in

the general form of Eqn (3) also for a wide range of compounds.

Factor Z* varies from a high of 9.1 for Bi to a low of 2 for the element Be and is 1.5 for the hydrogen-rich compounds polyethylene and 26n-paraffin. Over the same range of elements and compounds the atomic volume [as derived from Eqn (4)] varies from a high of 68 em3 mol-' for Cs to a low of 4.9 cm3 mol-' for the element Be and is 4.6 cm3 a-mol-' for the compound 26n-parafin (where a-mol stands for an Avogadro number of atoms, not molecules). Hence, overall the G1 equation accommodates a factor 6 difference in Z* and more than an order of magnitude difference in atomic volume. The fact that the fitting parameter k, has a modifying effect on the proportionality of the IMFP to the quotient VJZ* should be noted, but it is seen to be of secondary importance. In fact, for the elements in Fig. 6, inclusive of Be (Fig. 9), for which identical fitting parameters k, and k, have been used, we have a factor 6 difference in Z* and a factor 4.5 difference in atomic volume. These differences are far in excess of the experimental uncertainty in the optical IMFP.

One should note that the identity Z* = JZ is in line with the qualitative prediction of Bethelo for high- energy electrons, according to which the stopping power should show a slower increase than Z (cf. above under 'the orbital interaction model').

The general applicability of the G1 equation depends to a large degree on the predictability of the two fitting parameters k, and k,. These parameters will be discussed now on the basis of the data in Figs 3-10. There- after, the G1 equation is compared to the IMFP-predictive equation of TPP, with regard to the observed deviations of the predicted IMFP from the experimental values.

The fact that few values of the energy-adaptive fitting parameter k, suffice to cover all elements and compounds is very helpful. Only the magnitude-adaptive fitting parameter k, varies from one material to the next. But this variation is neither extreme nor entirely unsystematic (as pointed out below).

The values of the fitting parameters k, and k, used in Figs 3-8 were obtained by first adjusting k, until the ratios AGl/Aopl appeared constant on visual inspection. Then k , was adjusted for the average of all ratios to be as close to unity as possible over the widest possible energy range (within the interval 200-2000 eV).

While the gradient of the ratio AGl/Aoql us. energy is not overly sensitive to k , , the sensitivity is sufficient to justify subdividing the transition elements into the three categories shown in Figs 3-5. In these figures the majority behaviour is seen to be that the values of AGl/AOpl show a downward trend at low energies. There are a few exceptions, however, namely for the elements Ti and V in Fig. 3, Zr in Fig. 4 and Hf in Fig. 5. Interestingly, Ti, Zr and Hf belong to the same group (group IV) and, hence, have the same configuration of outer electrons and orbitals.

If Figs 3-5 are inspected for a possible correlation between the ratio AGl/Aop,. (i.e. the actual value of k,) and the electron configuration (resp. the position of the element in the Periodic Table), one finds such a correlation to be evident in Fig. 3. There, the ratios hGl/Aopl are arranged top to bottom in order of their atomic numbers, with the exception of V. This type of corre-

48 W. H. GRIES

lation is not evident in Figs 4 and 5. In Fig. 6, this type of correlation is found again for the elements Mg, Al, Si, Y and Bi.

In Fig. 6, values of AGl/Aopt are shown for elements belonging to main groups of the Periodic Table. It should be noted that values of k , and k, have been determined using only four of the elements in Fig. 6: the three 3rd-period elements Mg, A1 and Si and the 6th- period element Bi. One notices that the k, value is not optimal for Bi. The k, value for C is optimal. The AGl/Aopt ratio for C is acceptable for a density of 1.5 but not 1.8 g ~ m - ~ . The data on Be in Fig. 10 suggest that Be can be expected to fit in Fig. 6.

Yttrium is included in Fig. 6 because it is seen to belong there, rather than with the transition elements of Fig. 3. It is expected that, in analogy to Y, other transition elements from group I11 (Sc, La) will be found to belong with the main group elements.

The data in Fig. 10 should be seen in conjunction with those in Fig. 6, because the alkalis are main group elements and the k, value for main group elements has been used in Fig. 10. The k, values of the alkalis are seen to be consistently below the average for the other main group elements (in Figs 6 and 9). The average k, values of the alkalis is 0.001 (i.e. 30% below that for the other main group elements). The shorter IMFPs for alkalis may be due to the extreme spatial extent of their valence orbitals.

The AG1/Aopt ratios of the organic compounds (for which optical IMFP were calculated by TPP) are shown in Fig. 7 (where the compounds are identified by numbers, decoded in Table 2). The energy dependence of the compounds is seen to be very similar throughout. Again, the values of AG1/Aopt show a downward trend at low energies, without exception. From the stoichiometry (Table 2) emerges the interesting fact that the compounds are ordered according to the ratio of hydrogen to carbon: H/C w 1 for seven of the first eight compounds (exception no. 3) and H/C w 2 for the last four. For no. 3 (polyimide), H/C = 0.45. It is concluded that the organic compounds can be further divided into (at least) two sub-sets with different average values of k,

(k,, k,) = (0.00165, 1.00) for H/C x 1

(kl, k,) = (0.0022, 1.00) for H/C w 2

Another point of interest is that the element carbon fits in not only with the main group elements but also with the organic compounds (as shown). The energy dependence is not quite as optimal as in Fig. 6.

Two remarks can now be made on the significance of the data in Fig. 7. The first remark concerns the rather excellent predictability of the optical IMFP of TPP by the G1 equation. This is taken to be proof of the general validity of the G1 equation and the Z* identity [Eqn (3)], using the argument that (as in the case of water) there is a strong to dominating participation of hydrogen in organic compounds and that the Z* value is close to the lower extreme value of unity (1.5 for H/C w 2 and 1.7 for H/C w 1).

The second remark concerns the fact that the predicted IMFP (as well as the optical IMFP) is an average value for the organic molecule as a whole. In the case of a large molecule, the analytical electron is unlikely to see more than a small fraction thereof before experi-

encing an inelastic interaction. Therefore, the IMFP is determined by the atomic composition in the vicinity of the source atom of the analytical electron rather than by that of the molecule as a whole. In other words, the IMFP changes with local composition across the molecule. This change is expected to be particularly significant for Langmuir-Blodgett films. Once the problem is identified, the local IMFP can be predicted by the G1 equation if Z* is calculated [from Eqn (3)] for the local composition.

The AGl/Aopt ratios of the inorganic compounds for which optical IMFPs were calculated by TPP are shown in Fig. 8, except for two compounds (InP and KCl, not shown) which are obvious outliers. The energy-adaptive fitting parameter k, is seen not to be optimal in all cases. Again, the values of AGl/Aopt show a downward trend at low energies, with a few exceptions. The author has been unable to detect a correlation between the, ratio AGl/Aopt and the electron configuration. Also, the fact that the (k,, k,) values of the inorganic compounds are about the same as those of the 4th- and 5th-period transition elements must remain unexplained at present. The spread of the AG1/Aopt ratios is larger than for the organic compounds. Even if the larger spread were due to the extremely different character of the bonding that these compounds represent, one would have to conclude that the character of the bonding is of minor significance (as has been suggested above).

The fact that for many elements and compounds the values of AGl/Aopt show a downward trend below x 200 eV can be interpreted as a limitation of the G1 equation to the energy range 200-2000 eV. Alternatively, it may be that the optical IMFPs become unreliable below 200 eV. The non-optical IMFPs for Be and the alkalis point to the latter cause. More data are required to resolve this issue.

The discussion is now turned to the observed deviations of the predicted IMFPs from the experiment- derived values. These deviations are brought about by the use of an average value for the magnitude-adaptive fitting parameter k, instead of individual values. Full agreement with experiment-derived IMFPs can be obtained, of course, by allocating a characteristic value of k , (for appropriate k,) to every element and compound. This would be possible for many elements, but only for a very limited number of compounds. There- fore, the uncertainty associated with the use of average values of k , will have to be accepted for all other materials.

The use of an average value of k, smoothes out the effect of changing electron and orbital configuration which (according to the orbital interaction model) is present in the optical IMFP. In addition, systematic and random errors of measurement or of evaluation of the optical IMFPs contribute to the discrepancy. If the random error is of minor significance, the discrepancy cannot be expected to be sampled from a Gaussian distribution and the standard deviation is not an appropriate measure of the discrepancy. Nonetheless, the standard deviation is used here for convenience and to allow comparison with the deviations reported by TPP for their predictive equations (TPP report root-mean- square deviations, which are believed to equal the standard deviation). The standard deviations are reported


here only for the energy value 2000 eV. For the G1- predicted IMFP ratio AGl/Aopt they are (with respect to the respective average values of k , )

6% for transition elements of the

4th period (here, Ti to Cu)


5th period (here, Zr to Ag)


6th period (here, Hf to Au)

10% for the main group elements

(here, only for Mg, Al, Si, Bi)

10% for the organic compounds

12% for the inorganic compounds (without the

outliers InP and KCl; 20% with the outliers included)

The root-mean-square deviations quoted by TPP8e for their IMFP ratios ATPP-ZM/Aopt are

10% for the elements

9% for the organic compounds

19% for the inorganic compounds

(InP and KCl included)

As a point of interest, the standard deviation within each sub-set of the organic compounds (i.e. for H/C x 1 and for H/C x 2) is practically the same as that for the organic compounds combined.

The deviations are seen to be very similar for the G1 set and the TPP set of predicted IMFPs in each of the three (TPP) material categories : elements, organic compounds and inorganic compounds. This author con- siders this similarity to indicate that the observed deviations are due primarily to variations of electron configuration and to a lesser extent to errors of measurement. It is also concluded that the more elaborate IMFP-predictive equation of TPP offers no advantage over the (more widely applicable) G1 equation.

The discussion is concluded with a note on the final- ity of the fitting parameters k , and k , . In particular, individual and average values of k , are likely to change as more (and better) experiment-derived IMFP values become available. In Ref. 2, this author has shown how IMFP can be derived from measured stopping powers of sub-MeV to MeV protons. First results indicate2 that the proton-derived IMFPs are only about half as large as the optical IMFPs of TPP (at comparable energy dependence). Therefore, the optical IMFPs are not nec- essarily close to the true IMFPs.

a1 to the atomic density. The GI equation is the mathematical expression of the orbital interaction model, which is based on a concept of matter in which the latter consists of clusters of interaction-prone regions, identified with the orbitals of an atom, in an otherwise interaction-free space.

Values of IMFPs for 50-2000 eV electrons in 27 elements, derived by Tanuma, Powell and Penn from published optical data, were used by this author to derive the energy dependence of the G1 equation and to obtain best values for two fitting parameters k , and k , .

The general form of the G1 equation is as given by Eqn (2). Factor Z* is regarded as the nominal effective number of interaction-prone electrons per atom; Z* equals ,/Z for elements and is as given by Eqn (3) for compounds and other clusters of atoms.

Six categories of materials have been specified for which five different energy dependences apply (as expressed by the fitting parameter k, ) : four categories of elements (as specified above) and one category each of organic compounds and inorganic compounds. Average values of the fitting parameter k , have been specified for each of these material categories. There is evidence that the organic compounds can be subdivided into two more categories (with the same energy dependence), according to whether the atom ratio H/C is w 1 or w2. There is also evidence that the main group elements may have to be subdivided into alkali elements and non-alkali elements.

The orbital interaction model implies that if the G1 equation applies to regular phases of elements and compounds (as it has been shown to do), it must also apply for any other arrangement and assembly of atoms. The local atomic composition, the atomic density, the electron energy and some knowledge of the chemical state of the atoms suffice for the IMFPs to be calculated by the G1 equation.

This is a major point of difference to the IMFP- predictive equation of TPP (for which the free-electron plasmon energy and the bandgap energy of the material are required to be known). Another fundamental difference between the G1 equation and the TPP equation is that the latter does not lead to an inverse proportionality between IMFP and atomic density. Another aspect of importance is that, on average, the TPP-equation- predicted IMFPs are not closer to the optical-data- derived IMFPs than are the G1-equation-predicted IMFPs. In summary, owing to the simplicity of the orbital interaction model, the G1 equation is much more widely applicable than the TPP equation, at equal reliability.

According to optical-data-derived IMFPs, the applicability of the G1 equation would seem to be restricted to electron energies 2200 eV. Non-optical IMFPs for Be and the alkalis point to a general applicability also between 50 and 200 eV.

SUMMARY AND CONCLUSIONS

A new IMFP-predictive equation (the G1 equation) has been formulated which essentially states that the IMFP of an electron traversing matter is inversely proportion-

Acknowledgements

Cedric Powell of NIST is thanked for keeping the author supplied with the latest tables of optical IMFPs of Tanuma, Powell and Penn. Susi Krell has kindly generated the figures.

so W. H. GRIES

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6. A. Jablonski, Surf. Sci. 188, 164 (1987). 7. P. J. Cumpson, Surf. Interface Anal. 20, 727 (1 993). 8. S. Tanurna, C. J. Powell and D. R. Penn, (a) Surf. Interface

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