Relative Cross Sections for L- and M-Shell Ionization by Electron Impact

Relative Cross Sections for L- and M-Shell Ionization by Electron Impact

Xavier Llovet1;�, Claude Merlet2, Jos�e M. FernaÂndez-Varea3, and Francesc Salvat3

1 Services CientõÂ®co-T�ecnics, Universitat de Barcelona. LluõÂs Sol�e i SabarõÂs, 1 ± 3, 08028 Barcelona, Spain2 ISTEEM, FU 160, CNRS, Universit�e de Montpellier II, Sciences et Techniques du Languedoc. Pl. E. Bataillon, 34095 Montpellier

Cedex 5, France3 Facultat de FõÂsica (ECM), Universitat de Barcelona. Societat Catalana de FõÂsica. Diagonal 647, 08028 Barcelona, Spain

Abstract. Results from measurements and calcula-

tions of relative L- and M-shell ionization cross sec-

tions by electron impact are presented. Measurements

were performed for elements Te, Au and Bi on an

electron microprobe with specimens consisting of

extremely thin ®lms of the studied element deposited

on thin, self-supporting, carbon layers. The relative

variation of the ionization cross section was obtained

by counting the number of characteristic X-rays from

the considered element and shell, for varying incident

electron energies, from the ionization energy up to

40 keV. Measured data were corrected to account for

the energy-dependent spread of the electron beam

within the active ®lm and for the ionization due to

the electrons backscattered from the carbon layer,

using Monte Carlo simulation. Cross sections were

evaluated in the Born approximation using an optical-

data model with numerically evaluated dipole photo-

electric cross sections. Calculated ionization cross

section were converted to vacancy production cross

sections, which can be directly compared with our

experimental data.

Key words: L- and M-shell ionization; electron impact; X-rayemission; cross sections.

Reliable cross sections for inner-shell ionization by

impact of keV electrons are required for quantitative

Auger electron spectroscopy (AES), electron probe

microanalysis (EPMA) and electron energy loss

spectroscopy (EELS). In spite of this need, simple

systematic methods to calculate ionization cross

sections from ®rst principles remain to be found.

The usual practice consists in using semi-empirical

formulae, which have limited ranges of validity and

normally lead to signi®cantly different results. In the

energy range considered here, 1 ± 50 keV, experimen-

tal measurements have been mostly carried out for K-

shell ionization; cross section data for L- and M-shells

are, unfortunately, very scarce. As far as we know,

absolute L-shell ionization cross sections have been

reported only for the elements P, S and Cl [1], Ar [2 ±

5], Kr [6], Xe [6, 7] and Au [8 ± 11] (see also refs. [12,

13]). Comprehensive reviews of experimental mea-

surements (as well as theoretical calculations and

semi-empirical formulae) of inner-shell ionization

cross sections have been published by Powell [14,

15, 16]. Most of the reported cross sections were

based on the measurement of characteristic X-rays

which are emitted in the de-exitation of ionized atoms

in the gas phase. Because of the lack of knowledge of

the corresponding ¯uorescence yields and Coster-

Kronig decay yields, experimental results are affected

by large uncertainties. Measurements using thick solid

samples have also been performed, but the results are

obscured by multiple scattering within the target. It is

thus dif®cult to establish the reliability of semi-

empirical formulae and theoretical calculations by

comparison with the available experimental data. This

dif®culty is found not only regarding the absolute

value of the cross section, but also for its energy

dependence.

The determination of the (relative) energy depen-

dence of the ionization cross section is of interest for

Mikrochim. Acta 132, 163±171 (2000)

� To whom correspondence should be addressed

practical applications in EPMA and AES and also

from a theoretical point of view, especially near the

ionization threshold. The asymptotic limit of the

Bethe formula, derived in the ®rst Born approximation

(see e.g. [15]), gives the correct behaviour of the cross

section far from the ionization threshold. However,

near the threshold, this approximation does not hold.

It is worth pointing out that this is in some cases the

energy range of interest in EPMA and AES.

Unlike measurements intended to provide absolute

values of the cross sections, the determination of their

energy dependence does not require knowledge of

some experimental and atomic parameters that are

usually poorly known (e.g. detector ef®ciency and

solid angle of detection, ¯uorescence and Coster-

Kronig yields, line fractions, etc.). Therefore, for the

relative measurements, it is possible to use the poorly-

characterized crystal spectrometers, which have sub-

stantially better energy resolutions than solid-state

detectors. For this reason, the determination of the

energy dependence of the cross section can be

performed with relatively high accuracy, using a

conventional electron microprobe with appropriate

targets. Relative measurements of inner-shell ioniza-

tion cross sections have been previously reported for

some elements, e.g. the L3-shell of Zr [17], the L-

shells of W [18], the M5-shell [19] and L3-shell of Au

[20], the L3-shell of Cu and M5-shell of Ag and W

[21] and for the L3-shell of Ti and Cu and the M5-shell

of Au [22]. Nevertheless, important discrepancies are

still found when comparing the energy dependence of

the ionization cross section of the same element and

shell measured by different authors.

We have recently shown [23] that measurements of

the energy-dependence of K-shell ionization cross

sections of Ni and Cu were in good agreement with

the results of an optical-data model proposed by

Mayol and Salvat [24]. This agreement motivated the

extension of the study to other bound shells. In the

present work, we describe new measurements of the

relative variation of the L3- and M5-shell ionization

cross sections for elements Te, Au and Bi. The

experiments were performed on an electron microp-

robe using a crystal spectrometer, with specimens

consisting of an extremely thin layer of the studied

element deposited on a thin, self-supporting, backing

carbon layer. In this way, multiple scattering effects

inside the active layer are minimised. The dependence

of the ionization cross section on the electron energy

was obtained from the counting rate of characteristic

X-rays emitted from the specimen. We have also

calculated ionization cross sections using an optical-

data model similar to that proposed by Mayol and

Salvat [24], suitably extended to describe L- and M-

shell ionization. Experimental measurements have

been compared with results of our new calculations as

well as with results of simple analytical formulae of

common use in EPMA and AES.

Measurement of Relative Ionization Cross Sections

Preliminary Considerations

For the sake of simplicity, we shall describe the

experimental procedure for L-shells. The extension to

M-shells is straightforward. To determine the energy

dependence of L3-subshell ionization cross sections,

we have measured the intensity of the corresponding

L� X-rays (M5-L3 transitions) emitted from the

specimen bombarded with electrons of energy E from

the ionization energy up to 40 keV.

For an homogeneous ®lm of thickness t and a

normally incident electron beam the relationship

between the L3-shell ionization cross section and the

X-ray intensity of the L� X-rays, NL� , is given by (see

e.g. [10])

�L��E� �ÿM5ÿL3

ÿtotalÿL3

!L3��L3�E� � f23�L2

�E�

� � f13 � f12 f23��L1�E��

� NL��E��N tNe

�

4�"

�; �1�

where �L� is the cross section for emission of L� X-

rays, N is the atomic target density (atoms per unit

volume), Ne is the number of incident electrons, �is the solid angle subtended by the X-ray spectro-

meter, " is the spectrometer ef®ciency, !L3is the

¯uorescence yield, fij are Coster-Kronig transition

probabilities between subshells Li and Lj and ÿM5ÿL3

and ÿtotalÿL3are the X-ray emission rates for the M5 ±

L3 transition and for the totality of transitions from

higher shells to the L3 shell, respectively. Unlike

vacancies in K-shells, vacancies in L-shells can be

produced not only by direct ionization, but also by

Coster-Kronig transitions between subshells. There-

fore, it is dif®cult to evaluate L3-shell ionization cross

sections by measuring counting rates NL� , without any

assumption regarding the cross sections of the other

sub-shells. For this reason, we will limit our study to

164 X. Llovet et al.

determining the cross section for vacancy production,

�vL3

,

�vL3�E� � �L3

�E� � f23�L2�E�

� � f13 � f12 f23��L1�E� � C NL��E�;

�2�

where C is an energy-independent scaling factor.

Therefore, for a given kinetic energy E, �vL3

is

proportional to the measured X-ray intensity NL�

and the energy-dependence of the vacancy production

cross section is thus simply given by NL��E�.

Experimental Procedure

In eq. (1) it has been assumed that electrons penetrate the ®lmfollowing a straight trajectory without losing energy. In order tomake sure that this assumption holds, we have selected elementswith large ionization energies for the shells of interest and we haveproduced very thin ®lms of these materials. Owing to their smallthicknesses, a backing ®lm was required to hold the active ®lms.Carbon was chosen as the backing material, due to its low atomicnumber (small backscattering effect) and large tensile strength.The studied samples of Te, Au and Bi were produced as follows.Firstly, 30-nm-thick (nominal) carbon ®lms were thermallyevaporated on layers of mica, stripped from them by immersionin distilled water and mounted on electron-microscope grids. Then,Te, Au and Bi ®lms were evaporated on these self-supportingcarbon ®lms and, in the same runs, on thick polished pure targetsof similar atomic number which were used for thicknessdetermination as follows.

Film thicknesses were determined by electron microprobeanalysis. We measured X-ray intensities from the Au/Pt, Au/W,Bi/Pt, Bi/W, and Te/Sb targets and the intensities of the samepeaks from pure Au, Bi and Te standards. The resulting intensityratios were analyzed with the help of the X-FILM software [25],which computes the thickness and concentration of a multilayertarget by fitting Merlet's analytical X-ray emission model [25] tothe experimental data (see Fig. 1). Owing to the fact that films andsubstrates had similar atomic numbers, the accuracy of thisthickness determination is expected to be better than � 5%. Indeed,thickness measurements of the same films deposited on differentsubstrates were found to agree within � 4%. The determinedaverage thicknesses were 6.7, 3.9, and 7.5 nm for the Te, Au and Bifilms, respectively.

X-ray measurements were carried out using a wavelength-dispersive spectrometer with beam energies varying from theionization threshold up to 40 keV. Electron currents were selectedto meet a certain compromise between X-ray counting rate andfilm damage, with typical values of about 100 nA. The diameter ofthe incident beam was set to � 20mm to minimize film damage,which occurs preferentially at low incident energies. Measure-ments were performed on the wavelength channel corresponding tothe maximum of the characteristic peak, using PET, TAP and LiFcrystals. The bremsstrahlung background was subtracted by linearinterpolation of the spectral background on both sides of the peak.Counting times were typically 120 s for each peak and 60 s for thebackground. In order to minimize sample contamination duringmeasurements, an anticontamination device was used. For eachbeam energy and each sample, at least 10 measurements wereperformed at different positions on the sample. Therefore, thestandard deviation for the set of measurements not only accounts

for statistical uncertainties, but also for errors arising from possibleinhomogeneities in the active film thickness. Figure 2 shows anexample of M X-ray spectrum obtained with 20 keV electronsimpinging on a Au sample (active thin film plus backing). Relativeuncertainties due to counting statistics, sample non-uniformity andbackground subtraction are estimated to range from � 1 to 5%.

Data Correction

Although specimens are very thin, incident electrons

undergo a certain scattering and do not penetrate

straight through the ®lm. Hence, the mean track

length of transmitted electrons is somewhat larger

than the actual ®lm thickness. Unfortunately, this

Fig. 1. Variation of the k-ratio of X-ray intensities for Au M� andPt M� lines emitted from a 3.9-nm-thick ®lm of Au deposited onPt, as a function of the electron incident energy. Intensity ratios arerelative to pure Au and Pt. Symbols are experimental data. Linescorrespond to the X-ray emission model of Merlet [25], ®tted tothe corresponding thickness (see text)

Fig. 2. Typical X-ray spectrum near the M lines, obtained by bom-barding a thin target of Au with 20 keV electrons and using acrystal spectrometer

Relative Cross Sections for L- and M-Shell Ionization by Electron Impact 165

effect depends on the incident electron energy and

consequently it affects the shape of the cross section.

The mean track length of electrons that are trans-

mitted with energy larger than the considered

ionization threshold, was estimated with the aid of

Monte Carlo simulation. Our simulation code com-

bines elastic scattering cross sections computed from

partial wave analysis [26] with inelastic collisions

computed from a simple model of the generalised

oscillator strength density [27]. In the simulations, the

electron was assumed to be absorbed in the medium

(i.e. the simulation is stopped) when its energy

becomes less than the ionization energy of the

considered shell. Fig. 3a displays the relative excess

track length of electrons transmitted through the Au

target, with absorption energies equal to the corre-

sponding L3- and M5-shell ionization energies,

respectively. In the former case, the relative excess

path length ranges from 1% at high electron energies

up to � 3% at low incident electron energies. In the

latter case, differences between the thickness and the

mean track length of transmitted electrons vary from

1 ± 2% at high electron energies up to � 7 ± 12%, near

the ionization threshold, where the lateral beam

spread is larger. To account for this effect, the ®lm

thickness t in Eq. (1) was increased in an amount

equal to the energy-dependent excess track length of

transmitted electrons le(E )/t, estimated as described

here.

On the other hand, electrons lose energy due to

inelastic collisions within the ®lm and, as a con-

sequence, their `̀ effective'' energy is somewhat

smaller than the energy E of the incident beam. To

estimate the effective energy, we calculated the

average energy loss �E of electrons transmitted

through the ®lm, as a function of the incident energy,

using the same Monte Carlo code. Figure 3b displays

the average energy loss (in % percent) for the Au

target, with absorption energies set equal to the L3-

and M5-shell ionization energies. It is seen that �E

varies from � 0.1% to 4% of the incident energy. In

order to account for the energy loss of electrons in the

®lm, the effective electron energy for each measure-

ment was set equal to E ± �E /2.

Finally, to evaluate the increase of ionization due to

electrons backscattered from the carbon backing

layer, we performed simulations of electron transport

in the complete specimen (active ®lm � carbon

backing) combining Monte Carlo simulation of the

electron tracks with total ionization cross sections

calculated from the optical-data model described

below. From these simulations, the ratio of the X-

ray intensity of the active ®lm � carbon backing to

the intensity of the active ®lm was evaluated. This

ratio, to be referred to as Ib/I, was found to depart

from unity in � 1 ± 3% and was used as a correction to

the measured X-ray intensities.

In summary, the cross section for vacancy produc-

tion was determined from the measured X-ray

intensity by using the simple relation

�vL3

�E ÿ�E

2

�� C NL��E�

I�E�Ib�E�

1

le�E�=t: �3�

It is worth stressing that the most accurate method for

checking the reliability of a given cross section is to

Fig. 3. Simulated relative excess path length (a) and relativeenergy loss (b) of electrons transmitted through unsupported Au®lms, as a function of the incident electron energy. Thecorresponding absorption energies are indicated by the arrows.Simulated results are represented by symbols, joined by straightsegments for visual aid


use it in a Monte Carlo code and simulate the

emission of X-rays from the real sample (active ®lm

� backing). However, since the total corrections

amounted up to a maximum of 12% in the worst case,

we have prefered to report corrected experimental

data, which can be directly compared with theoretical

calculations. In order to verify the consistency of our

approach we have simulated the emission of X-rays

from the real sample by using a cross section derived

from a ®t of the experimentally determined cross

section. The agreement between the energy depen-

dence of the simulated X-ray intensity with measured

(raw) experimental intensities was found to be

satisfactory (see Fig. 4).

Optical-Data Model Calculations

In this section, we brie¯y describe the extension of

Mayol and Salvat's optical-data model to the ioniza-

tion of L- and M-shells. A more detailed description of

the complete calculation procedure will be published

elsewhere.

Consider the ionization of a given atomic inner

shell (with Zi electrons and binding energy Ei) by an

incident electron with kinetic energy E and velocity v.

Neglecting for the moment relativistic and exchange

effects, the ®rst Born approximation yields the

following expression for the differential cross section

[28]

d2�

dQ dW� 2�e4

mv2

1

WQ

df �Q;W�dW

; �4�

where W is the energy loss of the incident electron,

and the recoil energy Q� q2/2m is customarily used

instead of the momentum transfer q; m and e are

the electron mass and charge, respectively. Notice

that d2�/(dQdW) is proportional to the generalized

oscillator strength (GOS) df (Q,W)/dW, also known as

the `̀ Bethe surface''. Within the Born approximation,

the GOS characterizes the response of the target to the

perturbation caused by the projectile and can be

computed numerically from the initial and ®nal

atomic wave functions. The total ionization cross

section of the considered shell is then obtained by

integrating over the kinematically allowed region in

the (Q,W ) plane,

�i�E� ��Wmax

Ei

dW

�Q�

Qÿ

d2�

dQ dWdQ; �5�

where Q� � ��Ep � ��

E ÿWp �2 and Wmax is the

maximum allowed energy loss in the ionizing

collision; Wmax�E if the incident electron is regarded

as distinguishable from the target electrons (see

below).

As a useful alternative to these lengthy calculations,

simpli®ed optical-data models may be formulated

which take advantage of the following asymptotic

behaviour of the Bethe surface [28]. In the optical

limit, Q� 0, the GOS is proportional to the dipole

photoelectric cross section,

df �Q � 0;W�dW

� mc

2�2e2�h�ph�W�: �6�

On the other hand, for Q�Ei, the Zi electrons in the

shell behave as if they were free and at rest. Then

df �Q;W�dW

� Zi��W ÿ Q�; �7�

and the GOS vanishes except at the `̀ Bethe ridge''

W�Q.

Fig. 4. Experimental X-ray intensities, raw and corrected, of the BiM� line, as functions of incident electron energy. The continuousline (actually, discrete data joined by straight segments) is theresult of a Monte Carlo simulation in which the corrected crosssection (open circles) was used to reconstruct the measured rawdata (dots)


In the optical-data model of Mayol and Salvat [24],

the GOS is expressed as

df �Q;W�dW

��1

0

df �Q � 0;W 0�dW 0

F��W 0; Q;W�dW 0;�8�

i.e. the Bethe surface is generated by extending the

photoelectric cross section to Q 6� 0 with the disper-

sion relation

F��W 0; Q;W� � ��W ÿW 0��W 0 ÿ Q�� W ÿ Q��QÿW 0�: �9�

The total cross section can then be written as

�i�E� ��1

0

df �Q � 0;W 0�dW 0

�1�W 0�dW 0; �10�

where

�1�W 0� ��Wmax

Ei

dW

�Q�

QÿdQ

2�e4

mv2

1

WQF��W 0; Q;W�: �11�

The simple, but physically plausible, extension

algorithm F� yields an analytical expression for

�1(W 0) and, hence, �i is reduced to a single numerical

quadrature.

Another advantage of the model is the possibility of

including exchange and relativistic effects in a

consistent manner. Exchange effects are accounted

for by means of the Ochkur correction, and the

maximum energy loss of the incident electron is set to

Wmax�(E�Ei)/2. Notice that, in the original version

of the model [24], the Ochkur correction was only

applied to the close collisions (those modeled by the

second term in the right-hand-side of eq. (9)). On the

other hand, relativistic effects can be included using

v��c in eqs. (4) and (11), and replacing the

Rutherford energy-loss cross section, which corre-

sponds to the close collision term in eq. (11), by the

Mùller cross section.

Our optical-data model also incorporates an

empirical Coulomb correction that accounts for the

fact that the kinetic energy of the projectile increases

when it penetrates the potential well of the target atom

(which is disregarded in the plane-wave Born

approximation). In the case of close collisions, we

assume that this energy gain by the incident electron

is equal to the average potential energy of the atomic

electrons in the shell, as obtained from a Hartree-

Fock-Slater calculation. The original optical-data

model overestimated this low-energy correction,

partially compensating for the neglect of exchange

effects in the distant collisions.

To further facilitate the calculations in the case of

K-shell ionization, Mayol and Salvat [24] used a

hydrogenic model that provides a simple and

suf®ciently accurate analytical expression for �ph.

However, hydrogenic photoelelectric cross sections

are not adequate for L- and M-shells of elements with

intermediate or high atomic numbers. To extend the

calculations to these shells, in the present work we

have computed �ph within the single active electron

approximation, disregarding the relaxation of the

passive electrons [29]. The atomic potential was

obtained by means of the self-consistent Hartree ±

Fock ± Slater method. Use has been made of the

RADIAL subroutine package [30] to solve numerically

the Schr�odinger equation for the bound and free

orbitals of the active electron moving in the atomic

potential. Finally, the dipole photoelectric cross

sections are evaluated in both the length and velocity

forms (see e.g. [31]), yielding �ph(E) values with

relative differences that are less than 10ÿ9, which

provides an indication of the robustness of the

numerical procedures employed.

Results and Discussion

It is of interest to compare our experimental results

with the predictions of classical models or semi-

empirical formulae. We consider the Gryzinski

formula [32]

�i�E�E2i � �e4Zig�Ui�; �12�

where

g�Ui� � 1

Ui

�Ui ÿ 1

Ui � 1

�3=2�1� 2

3

�1ÿ 1

2Ui

�ln�2:7� �Ui ÿ 1�1=2

�; �13�

and the Worthington-Tomlin [33] formula

�i�E�E2i � �e4Zi0:35

ln�Ui�Ui

ln

�4Ui

1:65� 2:35 exp�1ÿ Ui��: �14�

Here Ui�E /Ei is the over-voltage.


Calculated ionization cross sections have been

converted to vacancy production cross sections by

using Coster-Kronig decay yields taken from the

compilation of Krause [34] and from the recent

measurements of Werner and Jitschin [35]. Unfortu-

nately, the conversion to vacancy production cross

section was not possible for the M-shell data because

of the lack of information on the Coster-Kronig decay

rates for those shells.

In Fig. 5 we compare our relative measurements of

L 3-shell vacancy production cross sections of Au and

Bi with the predictions of the optical-data model and

the two semi-empirical formulae. The calculated

curves have been rescaled to their respective maxima.

It is seen that, for the L3-shell of these two elements,

the vacancy production cross section that results from

the Worthington-Tomlin formula is seen to agree with

the experiments. This fact has already been reported

by Murata and Sugiyama [20].

Figure 6 displays our relative measurements of M5-

shell vacancy production cross sections of Au and Bi

and of L3-shell for Te, and the results of the optical-

data model as well as those from the two analytical

formulae. We see that the energy dependence of

measured cross sections agrees with the predictions of

the optical-data model (Born approximation) for

E� 5Ei. However, when E approaches the ionization

threshold, mesured data are seen to depart increas-

ingly from the results of the optical-data model, as

expected from the physical assumptions underlying

the Born approximation. This tendency is consistent

with the one found for the L3-shell of Au and Bi.

Figure 7 shows our cross sections for the various

elements and shells, normalized to their correspond-

ing maxima, as functions of the overvoltage U. The

continuous curve in this ®gure represents the function

�=�max�U� ��

a1 � a2

U

�1

Ua3ln�U�; �15�

where a1� 2.56, a2� 0.14 and a3� 0.97 are para-

meters, which have been determined from a numerical

®t to the totality of measured data. It is seen that,

within experimental uncertainties, this function

describes reasonably well all the measured elements

and shells. On the other hand, the spread of the

experimental data about this curve, indicates the

accuracy of our measurements.

In conclusion, we have performed accurate relative

measurements of vacancy production cross sections

Fig. 5. Energy dependence of the measured and calculated vacancyproduction cross sections for the L3 shells of Au (a) and Bi (b),normalized to the maximum height. The continuous curves areresults from the optical-data model proposed in this work; dashedcurves, Gryzinski [32] formula; long dashed curves, Worthington-Tomlin [33] formula. Circles are the results from the presentmeasurements


and included, in a consistent manner, corrections for

the effects of the ®nite thickness of the specimen and

the backing material. Our analysis cautions against the

risks of uncritically using analytical formulae beyond

the `̀ regions'' where they are known to apply.

Regarding simple theoretical approaches, it seems

possible to improve the accuracy of the Born

approximation near the ionization threshold by means

of semi-empirical Coulomb and exchange corrections.

Work along this line is in progress.

Fig. 6. Energy dependence of the measured and calculated vacancyproduction cross section for the M5-shell of the elements Au (a)and Bi (b) and the L3-shell of Te (c), normalized at 40 keV. Otherdetails are the same as in Fig. 5


Acknowledgement. We are thankful to E. Benedito for his assistancewith the Monte Carlo simulations. Financial support from thePICASSO programme (`̀ Acciones Integradas entre Espa~na y Fran-cia''), project HF1997-0033, is gratefully acknowledged. This workhas been partially supported by Comisi�on para la Investigaci�onCientõÂ®ca y T�ecnica (Spain), project no. PB95-0271-C02-01.

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Fig. 7. Scaled cross section as a function of the overvoltage.Symbols represent measured elements and shells (indicated in theFig.). The continuos line is the function given by eq. (12), with theparameter values indicated in the text


Relative Cross Sections for L- and M-Shell Ionization by Electron Impact

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