Spherical geometric model for absorption electrons

Vacuum 86 (2012) 1855e1859
Contents lists available at SciVerse ScienceDirect
Vacuum
journal homepage: www.elsevier.com/locate/vacuum
Rapid communication
Spherical geometric model for absorption electrons/positrons phenomenon:
Application to the mean penetration depth calculation
A. Bentabet*
Bordj-Bou-Arreridj University Center, Institute of Sciences and Technology, 34000, Algeria
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 11 April 2012
Received in revised form
13 April 2012
Accepted 14 April 2012
I present a spherical geometric model for the slowing down and transport of electrons and positrons in
solids. This latter is mainly based on the Continuous Slowing Down Approximation (CSDA) scheme. In
advantage, I have developed an analytical expression of the mean penetration depth at normal and
oblique incidence by combination between my model and the Vicanek and Urbassek theory. For this, I
have used the Relativistic Partial Wave Expansion Method (RPWEM) and the optical dielectric model to
calculate the elastic cross-sections and the ranges respectively. Good agreement was found with the
experimental and theoretical data.
Ó 2012 Elsevier Ltd. All rights reserved.
Keywords:
Continuous slowing down approximation
Mean penetration depth
Monte Carlo simulation
Electron
Normal incidence
Oblique incidence
Positron
Vicanek and Urbassek theory
1. Introduction
The electron and positron material interaction has a great
importance in so many domains of analytical techniques of the
material such as electron probe microanalysis, electronenergy-lossspectroscopy, Auger electron spectroscopy, positron annihilation
spectroscopy, etc. It is important to know that Monte Carlo method
(MC) has become a powerful tool for doing such a study [1,2].
Frequently the Continuous Slowing Down Approximation (CSDA)
[3,4] is applied. In this approximation, it is assumed that an electron
loses energy continuously along the length of its trajectory. In
Ref. [4], the validity of the CSDA for describing electron trajectories
in MC simulations at energies relevant to auger electron spectroscopy (AES) was analyzed and the performance of Monte Carlo
simulations was evaluated calculating the backscattering factors for
AES [4]. The mean penetration depth (Z1/2) is one of the most
important functions of the absorption transport phenomena.
Analytical model of light ion backscattering coefﬁcients from solid
targets have been made by Vicanek and Urbassek [5]. I have already
studied Z1/2 in my previous works either in order to demonstrate
the validity of the proposed differential cross-section
* Fax: þ213 35782145.
E-mail address: [email protected].
0042-207X/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved.
doi:10.1016/j.vacuum.2012.04.023
approximation [6e9] or to demonstrate the reﬂection coefﬁcient
theory implications [10]. All these works are based on a stochastic
model (Monte Carlo method). However, in the present work, I
developed a new mathematical expression (deterministic model)
for Z1/2. In advantage, in the best of my knowledge, relatively no
analytical model exit for electron or positron mean penetration
depth in solid targets. In this work, I have proposed a simple
geometric spherical model of absorbed particles based on CSDA
scheme. Then, I have developed an analytical expression of the
mean penetration depth (Z1/2) reﬂecting one of the absorption
phenomena, as application of my model.
2. The model
Before describing my model, it is worth noting that when we use
Monte Carlo method (MC) based in individual scattering event
scheme, each absorbed particle has a speciﬁc penetration length in
the interior of the target. So, the statistical mean value of all
obtained penetration lengths is the range. However, when we use
MC based in CSDA scheme, all absorbed particles have the same
penetration length with a length equal to the range of penetration.
On the basis of the above considerations, I proposed a spherical
model of absorbed particles impinging in the solid target which is
essentially based on CSDA scheme.
1856
A. Bentabet / Vacuum 86 (2012) 1855e1859
Consequentially, I can write Z1/2 as follow:
Z1=2 ¼
1000
900
700
600
500
400
i
300
200
Zi
0
-800
-600
-400
-200
0
200
400
600
800
xi ¼ x ¼
Fig. 1. The distribution of the trajectory endpoints, projected onto the yez plane, for
2 keV electrons impinging on A1. The arrow denotes the entrance position [11].
So, in the present study, solid targets are in the form of thick
thin ﬁlm (or semi-inﬁnite solid) attacked by slow electrons or
positrons. In other word, I suppose that the target is taken as being
a parallelepiped with inﬁnite length and width and thickness (half
entire space). The coordinates of the particles are located using
a (OXYZ) reference so that axis (OX) and (OY) belong to the exterior
surface outside of the target (entry surface) and (OZ) is directed
toward the interior of the solid. The angles q and 4 represent the
polar and the azimuthal angles which can be varied as follow:
0 q p and 0 4 2p. When I associate for each incidental
particle the index “i”, the location of the absorbed particle will be
deﬁned as follow: qi, 4i and Zi (or ri) where Zi is the normal distance
between the absorbed particle “i” and the exterior surface and ri is
the projected range’s absorbed particle “i” (in other word Zi ¼ ri
cos qi).
As example, Fig. 1 represents the absorbed electrons in Al target
done by [11] (I have added only the axis and the location coordinates particle).
By deﬁnition, the mean penetration depth (Z1/2) is the average
distances which separate the absorbed particles and the entry
surface, and it is given by [8]:
Z1=2 ¼ hZiA ¼
NA
1 X
ðZ Þ
NA i ¼ 1 i A
(1)
where the index A indicate “absorbed phenomenon”.
Elsewhere, I can write Zi as follow:
Zi ¼ ri cos qi ¼ R$cos ai
Z0
dE
SðEÞ
NA
N0
(5)
Let’s put nowPðai Þ as the probability that the polar angle a takes
the value ai. Therefore, when ai is greater than p/2, the particle will
be considered as backscattered (not absorbed). In other term:
Pðai Þ ¼
Pai s0 for 0 ai p=2
0 for ai p=2
(6)
So, hcos aiA can be written, in probabilistic form, as follow:
hcos aiA ¼ x
NA
X
cos ai Pðai Þ
(7)
i
Or in continuous form as:
hcos aiA ¼ x
Zp=2
0
cos a
dPðaÞ
da
da
(8)
Among the simpler scheme used in Monte Carlo code is that
based in Continuous Slowing Down Approximation (CSDA). For
this, the distribution of the scattering angle is that of elastic collision. Therefore, the scattering angle qs after each elastic collision is
calculated assuming that the probability Pel(qs) of elastic scattering
within an angular range from 0 to qs is a random number uniformly
distributed in the range [0, 1] given by [11]:
Pel ðqs Þ ¼
1
sel
Zqs
2psinðqÞsðqÞdq
(9)
0
Consequently, the density of Pel ðqs Þ is given by:
(2)
where ai is the polar angle verifying the equation (2) and R is the
range given by:
R ¼
(4)
where hcos aiA is the mean value of cos a of absorbed particles.
Equation (4) showed that I have simpliﬁed the distribution of
the trajectory endpoints (the absorbed particles) by another system
where all endpoints (virtual absorbed particles) are distributed on
the surface of half sphere with a radius equal exactly the range (see
Fig. 2).
Elsewhere, all incident particles have the same probability to be
absorbed by the target (they have the same incidence energy with
the same incidence direction). Let’s put xi as the probability that the
particle “i” will be absorbed by the target, N0 as the total number of
incident particles and NA as the total number of absorbed particle.
Consequently, xi can be written as follow:
800
100
NA
NA
1 X
1 X
ðri cos qi ÞA ¼
ðRcos ai ÞA ¼ Rhcos aiA
NA i¼1
NA i¼1
(3)
E0
where E0 and dE/ds are the primary energy and the energy lost per
unit length of primary particle respectively.
Del ðqs Þ ¼
dPel
1
¼
2psinðqs Þsðqs Þ
sel
dqd
(10)
where sel and sðqs Þ are the total and the differential elastic crosssections respectively.
In equation (10), qs is the scattering angle which can be varied
from 0 to p (for each scattered particle either absorbed or backscattered). In other word, in the interior of the target, the scattering
angle depends to the incident direction just before collision. When
all initial directions are considered as random uniform events, I
conclude that 50% of scattering event will be done during the
return to the surface of the target.
A. Bentabet / Vacuum 86 (2012) 1855e1859
1857
Fig. 2. Spherical geometric model.
In fact, according to the CSDA scheme, I suppose (as hypothesis)
that the distribution probability of the polar angles qi (or the corresponded angle ai) can be described by the same expression that of
the elastic scattering angle qs by taking in consideration the passage
from the scattering angle type (varied from 0 to p) to the polar
angle type (varied from 0 to p/2 for absorbed particles).
In short, I conclude that the mean value of a functionf ðaÞ linked
to a transport quantity, in the interior of the target, is given by:
hf ðaÞi ¼
Zp=2
0
f ðaÞ
dPðaÞ
1 2p
da ¼
da
2 sel
Zp
f ðqs Þsin qs sðqs Þdqs
Or, it is clear that:
2p
sel
sel
sTr
The factor ½ of equation (11) reﬂects the scattering symmetry
angle vis-à-vis the normal of the target, comparatively to that of the
polar angle.
f ð2qÞsin 2qsð2qÞdq ¼
0
1 2p
2 sel
Zp
f ðqÞsin qsðqÞdq
(12)
0
By
(11)
0
Zp=2
taking
in
consideration
the
equation
(11),
Z p
¼ 2p
sin qs sðqs Þdqs and the transport cross-section
Z0 p
¼ 2p
ð1 cos qs Þsin qs sðqs Þdqs I conclude that:
0
hcos aiA ¼
x
2
s
1 Tr
(13)
sel
In other part, the absorption probability and the backscattering
coefﬁcient are related by the following equation:
Table 1
Mean penetration depth (A ) as a function of incident electron/positron energy for semi-inﬁnite Al, Cu, Ag and Au by using equation (19).
E0 (keV)
1
2
3
4
5
6
7
8
9
10
Al
Cu
Ag
Au
Electron
Positron
Electron
Positron
Electron
Positron
Electron
Positron
121
365
694
1095
1566
2108
2718
3397
4140
4946
123
337
630
1013
1490
2058
2707
3423
4195
5008
47
136
258
408
585
785
1008
1253
1520
1808
54
144
266
415
587
780
993
1227
1483
1763
39
113
211
329
467
626
803
998
1210
1436
46
126
228
351
496
661
847
1050
1268
1499
31
80
145
222
311
410
520
639
769
907
39
102
180
272
377
495
624
763
910
1065
1858
A. Bentabet / Vacuum 86 (2012) 1855e1859
Table 2
Mean penetration depth (A ) as a function of incident electron/positron energy and angle of incidence q0 for semi-inﬁnite Al, Cu, Ag and Au by using equation (19).
q0 (Deg)
E0 (keV)
1
Al
20
40
60
80
20
40
60
80
20
40
60
80
5
10
Cu
Ag
Positron
Electron
Positron
Electron
Positron
Electron
Positron
118
107
79
27
1533
1405
1057
375
4849
4467
3406
1220
122
116
97
38
1485
1436
1280
810
4925
4595
3618
1327
45
38
26
9
562
481
323
111
1736
1487
998
345
53
49
39
14
572
517
377
132
1713
1529
1091
379
38
32
22
7
447
379
252
87
1372
1157
765
265
45
42
34
13
484
438
320
112
1454
1289
907
315
30
25
17
5
294
244
159
55
858
706
459
160
38
36
29
10
366
325
230
79
1026
889
605
209
xþh ¼ 1
(14)
Therefore, the mean penetration depth can be written as:
Z1=2 ¼
s
R
ð1 hÞ 1 Tr
sel
2
(15)
Elsewhere Vicanek and Urbassek [5] showed that h has been
calculated analytically at normal angles of incidence is expressed
by:
h¼
1 þ a1
1
n2
1
1
1
1
þ a2 þ a3 3 þ a4 2
n
1=2
(16)
n
n2
pﬃﬃﬃﬃ
pﬃﬃﬃﬃ
with:
a1 ¼pﬃﬃﬃ
6= p; a2 ¼ 27=p; a3 ¼ 27= pðð4=pÞ 1Þ and
2
a4 ¼ ðð3=2Þ 2Þ :
In expression (14), n is the mean number of wide angle collisions
deﬁned as,
n ¼ NRstr
(17)
where N is the number of atoms per unit of volume in the solid
target given by:
N ¼
Au
Electron
NAv
Ar
(18)
where A, NAv, and r are the atomic mass, the Avogadro constant and
the mass density respectively.
Finally, when I substitute the expression (16) in (15), Z1=2 can be
written analytically as follow:
)
(
s
R
1
1
1
1 1=2
Z1=2 ¼
1 Tr
1 1þa1 1 þa2 þa3 3 þa4 2
n
sel
2
n
n2
n2
(19)
Consequently, my spherical geometric model permits the
analytical calculation of the mean penetration depth at normal
incidence through the knowledge of R, sel and sTr .
Important remark: I have taken into account only the elastic
effect in the determination of the scattering angle because several
works showed that the use of both algorithms (one simulating individual electron scattering events and the other implementing CSDA)
were in satisfactory agreement for primary energies exceeding 1 keV
[2]. At lower energies there were deviations up to 10% occurred due to
numerical approximations. Since the CSDA considerably simpliﬁes the
simulation algorithm and decreases the computation time required to
obtain a particular precision [2].
The validity of my model will be closely connected to the
experimental and theoretical database. Indeed, the next section
will be shown aiming this object. In fact, I have calculated Z1/2 of
both electron and positron impinging in Al as light element, both
Cu and Ag as intermediate elements and Au as heavy element, in
Electron:
PW1
Ref [15]
Positron:
PW2
Ref [16]
6
5
4
Ref [9]
Exp1
Ref [16]
Exp2
3
2
Aluminium
1
0
0
2
4
6
8
Mean penetration depth (100 A°)
Mean penetration depth (1000A°)
7
10
Electron:
PW1
Ref.[14]
Ref.[9]
Exp
Positron:
PW2
Ref.[16]
Ref.[16]
Ref.[14]
8
6
4
Gold
2
10
Energy (keV)
0
2
4
6
8
10
Energy (keV)
Fig. 3. Mean penetration depth of electron and positron as a function of the incident
energy for semi-inﬁnite Al. PW1 and PW2: present work by using equation (19). Exp1
and Exp2 (experimental points): estimated from the experimental data reported in Ref.
[14].
Fig. 4. Mean penetration depth of electron and positron as a function of the incident
energy for semi-inﬁnite Au. PW1 and PW2: present work by using equation (19). Exp
(experimental points): estimated from the experimental data reported in Ref. [14].
A. Bentabet / Vacuum 86 (2012) 1855e1859
the energy range up to 10 keV where the range (R) was performed
by integrating the inverse of the best ﬁt of the stopping power
numerical results given by Ashley [12] (see equation (3)). The
integration was performed from the primary energy E0 to 100 eV
instead of 0 eV. The residual range was neglected because very low
energy electrons/positrons generally are not included in the
experimental evaluation of backscattering coefﬁcient.
Also, sel and sTr are calculated by using ELSEPA code developed
by Salvat et al. [13] where I tacked only the exchange and correlation potential, in the electron case, in consideration.
Tables 1and 2 represent the mean penetration depths in function of primary energy obtained via equation (17) for a normal
incidence. I preserve the same remarks and comments mentioned
in the previous works [7,8] for the physical aspects of these results.
Figs. 3 and 4 show my analytical mean penetration depths Z1/2
as a function of electron/positron incident energy over the range
1e10 keV at normal incidence for Al and Au, respectively. For
comparison, theoretical data [9,14e16] and values estimated
from the experimental data reported in Ref. [14] are also presented. My calculated values are in reasonable agreement with
the available experimental ones. The slight discrepancy between
theory and experiment is believed to be due to the experimental
results that have been estimated from the thin ﬁlm transmission
data [7,8].
3. Summary
In summary, the transport of absorbed particles normally
incident, impinging in solid targets, is modeled within a spherical geometric model where the radius length of the sphere is
1859
proposed equal to the range. On the basis of my model and the
Vicanek and Ubassek theory, I have developed a mathematical
expression for the mean penetration depth depending of the
range (R), the total elastic cross-section ðsel Þ and the transport
cross-section ðsTr Þ characterizing the interaction between the
particle projectile and the target. My calculated values are in
reasonable agreement with the available experimental and
theoretical data.
It is worth noting that in the present work the attention was only
restricted the mean penetration depth. The possibility of extending the
present study to other transport quantities such as the stopping proﬁle
and the transmission probability using the same models is in progress
and will be reported in due course, which may throw more light on the
validity of the models used.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
Bentabet A, Fenineche N. Appl Phys A 2009;97:425.
Bentabet A. Nucl Instrum Methods Phys Res B 2011;269:774.
Jablonski A, Powell CJ, Tanuma S. Surf Interface Anal 2005;37:861.
Zommer L, Jablonski A, Gergely G, Gurban S. Vacuum 2008;82:201.
Vicanek M, Urbassek HM. Phys Rev B 1991;44:7234.
Bouarissa N, Deghfel B, Bentabet A. Eur Phys J Appl Phys 2002;19:89.
Bentabet A, Bouarissa N. Appl Phys A 2007;88:353.
Bentabet A, Fenineche N, Loucif K. Appl Surf Sci 2009;255:7580.
Bentabet A, Chaoui Z, Aydin A, Azbouche A. Vacuum 2010;85:156.
Bentabet A. Mod Phys Lett B 2012;26:1150022.
Valkealahati S, Nieminen RM. Appl Phys A 1983;32:95.
Ashley JC. J Electron Spectrosc Relat Phenom 1990;50:323.
Salvat F, Jablonski A, Powell CJ. Comput Phys Commun 2005;165:157.
Valkealahti S, Nieminen RM. Appl Phys A 1984;35:51.
Aydin A. Appl Radiat Isot 2009;67:281.
Ghosh VJ, Aers GC. Phys Rev B 1995;51:45.

Download Report