Copyright(c)JCPDS-International Centre for Diffraction Data 2001,Advances in X-ray Analysis,Vol.44
COMPARISON OF THREE UNIVERSAL CURVES FOR THE ESCAPE
PROBABILITY
OF X-RAY EXCITED ELECTRONS - I. THEORY
Horst Ebel’, Robert Svagera’, Maria F. Ebel’ and Wolfgang S.M. Werner2
Technische Universittit Wien, Wiedner HauptstraJe 8-l 0, A 1040 Wien, Austria
ABSTRACT
An essential quantity in quantitative surface analysis by x-ray induced electron emission is the
sampling depth or escape probability of electrons. In his investigations of a universal curve for
escape probabilities in total electron yield XAS Schroeder’ describes the escape probability P(x)
of electrons from a depth x in units of the Bethe range RB by
The expression holds for an electron emission into a 2x-geometry, electron energies from 2.96 to
7.40 keV and elements from K to Zn. Ebel et al2 describe the escape probability by
P(x) = A, . exp[-2]-A2
.exp[-:]
whith amplitudes Al, AZ and mean free paths 21 and & obtained from least squares fits to the
results of Monte Carlo calculations2’3’4’5’6. The solid angle of electron acceptance 0 is also
recognized in this concept. This approach covers electron energies from 0.5keV to 30keV and
elements from Z=l 1 to Z=79. For practical applications we propose the following algorithm for
escape probabilities
P(x) = A1 . exp[-2.
p)-
A, .exp[-k.p]
The dimension of I/p/z is cm2/g and the meaning is identical with the usual mass attenuation
coefficient ,L@ in x-ray analysis. The least squares fits of log(l/p/l)
and log(l/@z) versus
log(Eki,) of the electron energy do not depend on A, 2 and L?
PHOTOELECTRIC
ABSORPTION
In the course of photoelectric absorption of monoenergetic x-rays, electrons of kinetic energy
Ekin=h I+E~ are released from the atoms. The probability w describes the first step of the
rearrangement of the atoms towards the original electronic configuration by emission of
characteristic x-radiation. Further electrons can be released from the atoms with probability l-u
by an emission of one or even more Auger electrons. Auger electrons are characterized by
defined kinetic energies. Thus, photoelectric absorption of monochromatic x-rays is responsible
for the emission of a series of monoenergetic electrons from atoms. Besides photoelectric
absorption, Compton scattering causes an additional contribution to electron emission. This kind
of interaction of x-rays with matter is neglected due to the comparably small probability in the
photon energy range from 0.5 keV up to approximately 30 keV.
I Institutfir
’ Institutfiir
Angewandte und Technische Physik
Allgemeine Physik
380
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hv
PE
&in
= hv - EbK
w
1 vacancy
1 “t$
QJK
1 vacancy
in K
in L
2 vacancies
in L
Fig.1
Photoelectric absorption in
the K-shell, emission of
K-photoelectron, followed
by emission of either
characteristic K radiations
or KLL Auger electrons,
followed by emission of
characteristic L radiations
LMM
Auger
and/or
electrons.
Tuning the x-ray photon energy causes a systematic variation of the energy of photoelectrons,
whereas the energy of Auger electrons remains unchanged.
L WV)
Fig.2
Dependence of kinetic
energy of photo- and
Augerelectrons of copper
on the photon energy of
incident x-radiation
TRANSPORT
OF ELECTRONS
IN MATTER
The following considerations are dedicated to solid matter. An exponential law describes the
depth distribution of the number of monoenergetic electrons released by photoelectric absorption.
Depending on the composition of the specimen, the kinetic energy of the electrons, the initial
direction of the electrons and the distance between the point of origin and the surface only few
electrons reach the surface without any kind of scattering in the specimen. Much more electrons
leave the surface after numerous scattering events and consequently, with reduced kinetic energy,
and most of the electrons do not reach the surface.
380
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C
Fig.3
A. Escape of electron
without scattering event
B. Escape of electron
after numerous scattering
events
C. No escape of the
electron.
In our experiments we use nondispersive electron detection. The production of secondary
electrons is not within the frame of our theoretical concept and therefore, the detection of
electrons with kinetic energies of less than 50 eV is suppressed in our experiments by a dc-bias
between specimen and detector. Consequently, photoelectrons with energies of less than 50 eV
are also excluded from detection. In order to describe measured electron signals from solids
irradiated with monochromatic x-radiation, the transport of electrons with given start energy has
to be quantified. Monte Carlo calculation offers an excellent tool for the solution of this problem.
Our mode12’4’5’6closely follows the one described in detail by Shimizu and Ding Ze-jun3. Since
most of the detected electrons are detected after multiple scattering we assume an isotropic
angular distribution of directions of the trajectories from the point where the electron starts.
Escape energies of at least 25, 50 and 100 eV were chosen. Our electron detector is a channeltron
and allows us to detect only electrons within a given solid angle. Thus, the solid angle of electron
detection L? asks for an additional variable in the Monte Carlo algorithm. Typical results of the
Monte Carlo calculations can be seen from Figs.5 to 11.
Fig.4
Description of the solid
angle of electron detection
L2 by angles LX~and a~.
‘MC’
l
‘fit’ ‘global’
- - -
200
depth
250
(nm)
Fig.5
of
Escape probability
electrons versus depth for
Cu, 5 keV, bias -50 V,
acceptance angle 0” to 20”
(data points give MC, full
response gives 1.s.f. of
data points and broken
the global
curve is
description of the escape
probability).
381
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0.4
‘MC’
a
‘fit’
‘global’
150
200
depth
250
(nm)
300
350
400
l
-
- - -
Fig.6
As Fig.5 for an acceptance
angle 0” to 40”.
450
Fig.7
As Fig.5 for an acceptance
angle of 0” to 90”.
iril~
0
50
100
150
200
depth
250
(nm)
300
350
400
450
Fig.8
As Fig.5 with kinetic
electron energy 0.5 keV.
depth
(nm)
Fig.9
As Fig.5 with kinetic
electron energy 30 keV.
0
1000
2000
3000
4000
depth (nm)
5000
6000
7000
4
6000
382
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0.14
a
ST
‘MC’
0.12 .,
(‘1
l
‘fit’ ‘global’
- - .
I
Fig.10
As Fig.5 for Na instead of
cu.
depth
(nm)
‘MC’ .
‘fir ‘global’
- - -
Fig.11
As Fig.5 for Au instead of
cu.
-0
50
150
100
depth
200
250
(nm)
We describe the escape probability by
P(x)= A1 .exp[---&.pr)-dz
.exp[---&a-)
with amplitudes Al and AZ, and cross sections l/&
and l/p/22 in cm’/g. The values of Al, AZ,
l/,&l and l/p/22 have been evaluated by least squares fits (1.s.f.) of P(X) to the results of Monte
Carlo calculation for elements Li, Be, Na, Al, Si, K, V, Cr, Fe, Co, Ni, Cu, Se, Nb, MO, Rh, Pd,
Ag, Te, Ta, W, OS, Ir, Pt, Au, kinetic electron energies 0.15, 0.20, 0.25, 0.30, 0.40, 0.50, 0.60,
0.70,0.80, 0.90, 1.00, 1.20, 1.40, 1.60, 1.80,2.00, 3,4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
18, 19,20,21,22,23,
24,25,26, 27,28, 29, 30 keV, bias -25, -50, -100 V and acceptance angles
0” to lo”, 20”, 30”, 40”, 50”, 60”, 70”, 80” and 90”. The total number of combinations is
25.44.3.9 = 29700. A systematic investigation of the results led to the following conclusions:
1
The bias dependence ofAl , AZ, l/p/z1 and l/p/zz is only significant at Ekin < 0.5 keV.
Amplitudes Al and A2 depend on 0 and on Ekin.
2
Amplitudes
A1 and AZ do not depend on 2.
3
l/p/z1 and l/p/22 depend primarily on Ekin.
4
Cross sections l/p/z1 and l/p/22 do not depend on 2.
5
A first restriction for a global fit is the exclusion of the low 2 elements Li and Be.
6
A second restriction is the exclusion Of Ekin < 0.5 keV.
7
A, = exp C C,,j . azi-’ + log(E,,).
C C,,j . a~‘-’
j=1+4
j=1+4
5
1
383
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A, = exp c
384
3856
C,,i . azi-’ + log(E,, ). c C,, j *
j=1+4
j=1+4
q=O", a2 is given in degrees and can be chosen from 10” to 90”
$
= exp C Cl,j ’ log(E,, )‘-’ 7
1
+
j=1+4
aJ-1
C3.i
C4.i
C5.i
C6.i
j=1+4
-J-3
-l.l18E-01
-l.O86E-01
-2.240E-03
8.096E-05
-3.116E-03
1.275E-04
aJ-2
-1.214E+OO
-1.203E+OO
1.529E-01
-6.483E-03
1.858E-01
-9.934E-03
1.284E+O 1
1.367E+Ol
-3.417E+OO
1.724E-01
-4.050E+OO
2.438E-01
C1.i
C2.i
= exp C C2,j ’ l”g(E& >j-’
2
Table 1 Coefficients Cij for the global description of Al, AZ, l//Al
aJ-4
8.395E-03
7.830E-03
l.O74E-05
-2.863E-07
1.547E-05
-3.908E-07
and
lipA2
The global fits of A1 and A2 and of l/p/z1 and l/p/22 enable us to describe escape probabilities of
electrons by a universal curve. The quality of the global fits can be seen from the comparison of
the escape probabilities from Monte Carlo calculation (dots) with the broken curves (global) in
Figs. 5 to 11.
COMPARISON
OF UNIVERSAL
CURVES
Schroeder’ describes the escape probability P(X) of electrons from a depth x for an electron
emission into a 2x-geometry, electron energies from 2.96 to 7.40 keV and elements from K to Zn
in units of the Bethe range Rg.
P(x)=O.,,(,-,,)exp;-2.7.:)
We describe P(x) for electron energies from 0.5 keV to 30 keV, elements from Z=l 1 to Z=79 and
variable solid angle of electron acceptance by the much more general approach
P(x)= A1 .exp[-&.p]-A
.exp[-2-p)
whith amplitudes A and cross-sections I/p/ (expressed in cm2/g) obtained from least squares fits
to the results of Monte Carlo calculations2.3’4’5’6.Since the meaning of I/p/ is identical with the
usual mass attenuation coefficient ,A@ in x-ray analysis we propose our approach for an
application in the field of quantitative analysis by x-ray excited electron emission
REFERENCES
[l]
[2]
[3]
[4]
Schroeder, S.L.M., Solid State Comm. 1996,98,405-409
Ebel, H., R. Svagera, W.S.M. Werner, M.F. Ebel, Adv.X-Ray Anal. 1999,41 , 367-378
Shimizu, R., Ding Ze-jun, Rep.Prog.Phys. 1992,55,487-531
Ebel, H., R. Svagera, M.F. Ebel, N. Zagler, W.S.M. Werner, H. St&i, M. Griischl,
Proc. ECASIA 95 (edited by H.J.Mathieu, B.Reihl and D.Briggs) J.Wiley &Sons 1996
TD4,99 l-998
[5] Werner, W.S.M., Phys.Rev., 1997, B55, 14925-14930
[6] Wagner, H.W., W.S.M.Wemer, X-Ray Spectrom. 1998,27, 373-380
6