Appl. Phys. A 35, 51-59 (1984)
Applied
,,,,,,,
Physics A Surfaces
"n"
9 Springer-Verlag 1984
Monte Carlo Calculations of keV Electron
and Positron Slowing Down in Solids. II
S. Valkealahti and R. M. Nieminen
Department of Physics, University of Jyv/iskyl/i, SF-40100 Jyv/iskyl/i 10, Finland
Received 17 January 1984/Accepted 17 March 1984
Abstract. An improved Monte-Carlo simulation technique has been used to investigate
positron and electron slowing down in solid matter. Elastic scattering is based on exact
cross sections of effective crystalline potentials and inelastic processes are described by
Gryzinski's semiempirical expression for each core and valence electron excitation.
Calculations with normal and oblique angles of incidence have been made for positrons and
electrons impinging on semi-infinite aluminium, copper, tungsten, and gold. Interesting
differences have been found between positron and electron penetration and backscattering
features.
PACS: 61.80, 78.70, 29.70
In a recent publication [-1] we have presented results of
Monte Carlo (MC) simulations of electron and
positron implantation in solid targets in the keV
energy range. The results are useful for a variety of
purposes, e.g. slow positron beam studies [2], x-ray
and Auger microprobes, electron microscopy, etc. The
implantation properties obtained in [1] are equal for
positrons and electrons. This is based on the
assumption that in the relevant energy range both the
elastic and inelastic cross sections are very similar for
the two particles. While this is a plausible first
approximation, it is of interest to explore the electronpositron differences in the slowing down process.
Furthermore, the calculations reported in [1] used the
screened Rutherford cross section for elastic scattering.
It is known that this cross section is not accurate for
scattering off atoms at low energies, and it is therefore
desirable to use more realistic cross sections based on
crystalline atom potentials.
In this paper we report results of extended and
improved MC simulations of slowing down and
backscattering for 1-10keV positrons and electrons
incident on solid targets. The differences with and
extensions from [1] are as follows: (i)the elastic
scattering cross sections are calculated exactly using
single-particle crystalline atom potentials for both
electrons and positrons, (ii) the small but
nonvanishing differences between the inelastic cross
sections for electrons and positrons are accounted for,
and (iii) the angle of beam incidence on the surface is
varied.
The simulation sequences are used to extract the
statistical distributions of overall and projected
implantation
ranges,
slowing
down
times,
backscattering energy and angular distributions and
total backscattering yields, for incident energies
between 1 and 10keV. Special attention is paid to
mapping out the electron-positron differences.
Comparison with experiment, where possible, shows
generally good agreement. In particular, the
backscattering yields obtained are much closer to
experimental estimations than those reported in [-1].
Analytic fits are presented for the stopping profiles and
their Laplace transforms. These should prove useful in
analyzing, e.g., re-emission data of implanted,
thermalized particles.
1. Calculation Procedure
The main ideas of the MC simulation have been
outlined in [1], and more practical details have been
presented in, e.g., [3]. Improving on [1], the elastic
scattering processes are here calculated using accurate
atomic cross sections and the inelastic processes are
52
S. Valkealahti a n d R. M. Nieminen
calculated separately for each energy level using the
Gryzinski's excitation function [4]. The effect of lattice
structure is approximately included in the elastic
collisions via modification of atomic cross sections, but
channelling processes are still excluded.
1.l. Elastic Scattering
The differential scattering cross section for a central
potential can be calculated from the partial wave sum
as [5]
da
I
dr2 - k 2 l=o
k [(2l+1) eia' sin31Pl(cos0)] 2,
CD
(1)
0.0
5
0.2
Ener-'g~
where k is the wave number, Pg is the Legendre
function, and 6~is the phase shift of the/th partial wave.
The total elastic scattering cross section is obtained by
integrating over angles as
&
O-el= j' ~ dO.
(2)
The Legendre functions can be effectively generated
using their recurrence relation [6]. The phase shifts are
obtained by numerically solving the radial
Schr6dinger equation
[
d2
2/~
-~r~r~+~V(r)-~
I(1+1)
r2
]
k 2 u,,k=0
(3)
and matching the solution to the asymptotic form
sin(kr- 17c/2+ (~l)"
V(r) in (3) is the effective single-atom scattering
potential in condensed matter. We evaluate it as
follows. Firstly, free atom potentials and electron
densities are calculated selfconsistently by the densityfunctional method [7, 8]. The exchange and
correlation between the electrons are included by using
a
local-density-dependent exchange-correlation
potential [8]. Secondly, an approximation for the
crystalline charge density and Coulomb potential is
built up by superimposing atomic charge densities in a
lattice. The spherical average of the charge density and
Coulomb potential around an atom are then
evaluated. The exchange-correlation potential for
electrons is added in the local density approximation.
For positrons, only the spherically averaged Coulomb
potential (with the reversed sign) is included. The
scattering potentials are normalized to zero at the
Wigner-Seitz radius. Equation (3) is then solved using
the effective central crystalline potential for each
partial wave by numerical integration. Inserting the
calculated phase shifts in (1, 2) the differential and the
total elastic scattering cross section can be obtained.
The da(E,O)/dO-data, calculated separately for
positrons and electrons, is evaluated for values of
10
[ keY )
Fig. 1. Total elastic scattering cross section for copper crystalatom. - ..... positrons;
electrons;
screened
Rutherford
energy and angle from 20eV to 10keV and from 0 to
180 degrees, respectively. In the simulations, the
differential cross section for an arbitrary value of
incident particle energy is calculated by interpolation
in the calculated data set. The scattering angle 0 is
determined by selecting a uniform random number
0<RI_-< 1 and finding the value of 0 from the cross
section data which satisfies
Rl=2rcid~
,
o ~ sin 0,dO/O'el
(4)
The total elastic scattering cross section of positrons
and electrons in metallic Cu as a function of energy is
shown in Fig. 1. Electrons generally have a larger
elastic scattering cross section than positrons. The
difference between electrons and positrons decreases as
a function of energy and increases as a function of
atomic number. Larger elastic scattering cross sections
for electrons result from two different phenomena.
Firstly, electrons encounter an attractive potential,
which makes large angle scattering more probable,
whereas positrons sense a repulsive potential.
Secondly, the exchange-correlation potential for
electrons makes the atomic potential larger in
amplitude in the outer region, which increases the
small angle scattering. At low energies, the electron
cross section may show rapidly varying (resonant)
behavior and may actually be smaller than the
positron cross section (Figs. 1-3). For comparison, the
screened Rutherford cross section (used in El]) is also
shown in Fig. 1. As can be seen, the Rutherford formula
overestimates the elastic scattering for small energies,
more so for heavier elements. Note, however, that the
overestimation mainly occurs at small angles and thus
the effect on slowing down is not as dramatic. The use
Monte Carlo Calculations ofkeV Electron and Positron Slowing Down in Solids. II
e§ beam
e--beam
53
e§ beam
e-- beam
da
~
sr -1)
~, do" (/],2sr -1)
d
_
_
q
~
~/ (~2sr-I)
,1' dO (/~2sr -1)
-aft-
Fig. 2. Polar plot of the differential elastic scattering cross section
of 100 eV positrons and electrons off a Cu crystal-atom
Fig. 3. Polar plot ofthe differentialelasticscatteringcross section
of 1keV positrons and electronsoff a Cu crystal-aton:t
of the screened Rutherford differential scattering cross
section has also been discussed in [9, 10].
Another important difference between electrons and
positrons appears in the differential cross sections.
Electrons have a greater probability to scatter into
large angles than positrons, especially at small energies
(Fig. 2). Electrons generally also have a larger
probability to scatter into small angles (Fig. 3). This is,
again, fairly unimportant for slowing down, because
elastic forward scattering does not affect the
penetration properties. At low energies large-angle
scattering causes the differences between electron and
positron backscattering yields while at larger energies
the more frequent small angle scatterings lead to
smaller differences between the overall electron and
positron backscattering probabilities.
we have used the approximation
1.2. Inelastic Scatterin9
All contributing core and valence electron excitations
have been described in terms of Gryzinski's [4]
excitation function. For a target electron shell denoted
by i,
dai(AE)
roe4 EB~(
E
"]a/z
d( E) - ( d E ) 3 E
9 ~ffff~B(1-En]E)+~ln 2 . 7 + \
EB, ]
,(5)
where AE, EB,, and E are the energy loss, the mean
electron binding energy, and the primary projectile
energy, respectively. Equation (5) is basically valid only
for ionization processes of bound electrons. Therefore
dai(AE)
dai(AE) A~=E,,
d(AE) A~<=e,,~- d(AE)
(6)
for processes where the energy loss is smaller than the
binding energy of outermost shell electrons. The total
inelastic scattering cross section for shell i is
O'inel'i=
ET~da,(AE')
0
d(AE') d(AU),
(7)
where the maximum energy loss Emax is for positrons
the particle energy E before collision and for electrons
(E+E~)/2. The lower cutoff for electrons appears
because the most energetic particle is followed after the
collision of the indistinguishable particles [11]. This
effect increases the total path length of electrons by up
to 10% and the penetration depth by less than 5%.
Another phase-space effect occurs in inelastic
scattering off conduction electron gas, where the Pauli
principle restricts the final electron states to be above
the Fermi surface. This also weakens the inelastic
scattering of electrons as compared to positrons. In the
present case, the total inelastic cross section is obtained
by summing over shells i,
O'inel= ~ O'inel,i 9
i=1
(8)
At each inelastic scattering event, the energy loss is
calculated by selecting a uniform random number R2,
and then finding a value of AE which satisfies
~ax dai(AE,)
Re= JA~ d(~E~ d(AE')/ai'e1'~'
(9)
54
S. Valkealahti and R. M. Nieminen
Table 1. Basic data for the calculations. The electron configuration is the one included in
inelastic processes; for the lattice potential calculation the full atomic potential is used
Material
A1
Cu
W
Au
Atomic
no.
13
29
74
79
Density
Lattice
type
Electron
configuration
[g/cm3]
Lattice
constant
[A]
2.70
8.96
19.3
19.3
4.05
3.61
3.16
4.08
~c
fcc
bcc
fcc
2s22p63sZ3p63dl~ 1
3sZ3p6...5p65d46s 2
3sZ3p6...5p65dl~
ls22s22p63sZ3pl
where the energy loss AE is between 0 < A E < Emax. The
scattering angle co after an inelastic collision is
obtained from the binary collision model [-4] from
(1 O)
sin co = (A E/E) 1/2 .
1.3. The Simulation Procedure
The mean free path of a penetrating particle is
,~-
A
N AOa'
(11)
where A, NA, 0, and a are the atomic mass, the
Avogadro constant, the mass density, and the collision
cross section, respectively. The inverse of the total
mean free path 2r is a sum over the inverses of the
different mean flee paths,
1/2T= 1//~el-I- ~
i=1
1/2i,
(12)
where 2el is the elastic mean free path and 2~ is the
inelastic mean free path of a specific core (valence)
excitation, where i runs over the contributing electron
states. The distance travelled between collisions is
obtained from
S= -2rlnR3,
2. Results
We present the most interesting results of our
calculations for aluminum, copper, tungsten, and gold.
Comparison with experimental results has been done
wherever accurate enough experimental values are
available.
We concentrate here in the features of penetrating
positrons and in the differences between positrons and
electrons. Attention has also been paid to the effects of
changing the angle of incidence.
(13)
where R3 is a uniform random number. Another
random number R4 is used to determine whether an
individual scattering event is elastic or one of the
inelastic processes. The event is of type i
(i = 1.... , m + 1) provided:
O< L - ,
f~
= 1/2r < R4 <
= 1/2r <
- 1,
Rutherford cross section drastically overestimates the
forward elastic scattering cross section for small
particle energies.
The termination energy of 20eV is used [1]. The
slowing down time is of the order of 10-14 s. The
computing time for one "experiment" is typically
around one hour of CPU time in a UNIVAC 1000/60
computer, depending naturally on the incident particle
energy and the material in question. No appreciable
difference in computing time is observed between
positrons and electrons.
The input parameters [-12-14] for the calculations are
collected in Table 1.
(14)
where f o = 0 , fl=l/~-el, f 2 = l / 2 e ~ + l / 2 1 , ..., fm+l = 1
for the contributing scattering processes.
2000 particle histories are used in each of the normal
incidence calculations and 1000 particle histories for
other incidence angles. One particle history typically
contains a few hundred scattering events, which is
much less than in the calculations using the Rutherford
cross section. The reason for this is that the screened
2.1. Normal Incidence
Positron and electron penetration is simulated in semiinfinite aluminum, copper, tungsten, and gold for
normal incidence at energies between 1-10 keV.
An example of the distribution of the trajectory
endpoints is shown in Fig.4. The first difference
between positrons and electrons is in the penetration
depth (Figs. 5 and 6), with positrons going deeper than
electrons. In heavy elements positrons penetrate up to
30% deeper, whereas the difference in light elements is
only a few percent. This result agrees with earlier
theoretical and experimental results for high energies
[11, 15] and with qualitative expectations. The main
reason is in the different elastic scattering cross sections
of positrons and electrons (Figs. 2 and 3). Electrons
have a distinctly larger probability to backscatter from
Monte Carlo Calculations of keV Electron and Positron Slowing Down in Solids. II
55
CO
,,.,<
AI, e §
C9
CD
O
E = 5 keV, r
~
Au
--- 200-=<
CO
.J
X
xl--
++
+++
o<(
Do
+
++
+ + +
-r+
o
+
0._
Ld
I
N
Z
9
~< lO0+
+
+
.~
+
-2
-~
T
rY
FuJ
z
[lJ
.~+
2
X--axug
(
<-
'lOOO
~
Z
<
]
Fig. 4. The distribution of the trajectory endpoints, projected
onto y - z plane, for 5 keV positrons at normal incidence on
aluminum. The arrow denotes the entrance position
I
I
I
l
I
I
--L
I
I
(,,o)
,.d~ ; / '
][
'
0
I
0
5
ENERGY
r
T
'
I
'0
I0
(keY)
Fig. 6. Mean penetration depth vs. initial beam energy for semiinfinite gold zxMCe+; oMCe-; x [ 1 8 ] e - , thin film data;
+ [19]e , thin film data
i
AL
c~E 200-
--<.
<z
21_
o4:
o
O
o
212
l.Ck
Ld
NO
I
~:
~'~z~
~
Cu, e§
t\
o
CD
Z
O
100-
g
02
I-Ld
Z
Ld
[1_
2
Z
<CD
Ld
2
O-
0
r-
'~
'
'
l
*
5
ENERGY
'
I
I
l
.0
10
(keY)
D
PENETRATION
DEPTH
( t000
A ]
Fig. 5. Mean penetration depth vs. initial beam energy for semiinfinite aluminum, zxMCe+; o M C e - ; x [16]e +, thin film
data; + [17] e-, thin film data
Fig. 7. Stopping profiles for 3 and 5 keV positrons in semi-intinite
copper. The full curves denote fits of (16) to the Monte Carlo
simulation data and the dotted lines the thin film data for 3.1 and
5.0 keV positrons from [16]
atoms, which arises from the attractive force between
electron and atom, and from the lack of exchange
effects between positron and core electrons. On the
other hand, the indistinguishability of electrons
increases the electron penetration depth by up to ~ 5%
and thus decreases the difference between electrons
and positrons.
The mean penetration depth in a semi-infinite target
can be described rather well by the formula ~= c~E",
where E is the incident energy and c~ and n are
parameters, which are for electrons weakly dependent
on target material. For electrons rough estimates are
c(_~4 gg/cm z and n ~_ 1.5, respectively. For positrons,
the dependence on target material is somewhat
stronger (Figs. 5 and 6).
Calculated mean penetration depths fit rather well
with the experimental results. The mean positron
penetration depths of Mills and Wilson [16] for A1 and
Cu, estimated from thin film data, fit well with the MC
values. Experimental mean penetration depths for
electrons have been estimated from the film
transmission data [17-19] but are unfortunately quite
inaccurate.
The simulated positron stopping profiles are
compared with the experimentally estimated stopping
profiles [16] for Cu in Fig. 7. The maxima of the
56
S. Valkealahti and R. M. Nieminen
I
I
I
L
I
I
i
I
,
I
J
~
,
I
,
r
~
,
I
2>-
AI
m
n
0,5•
CO
<
CO
o
Au
b--
X
rn
<
rn
X x
++
0,2-
~-
~0
•
orY
x
+
"--0....8++
9
X •
O--
9
Z
.------'-- 0
~
~
X
0.4-
0./0-I-
•
EL
Z
/ o ~ O
x
/
O/
0.3-
Ld
Ld
I-b<
U
t.q
../
I A - <
zx /
U 0.1-
Zxl
j A
j
U
<
CO
U
<
CO
0
I
o
I
I
I
I
I
I
5
ENERGY (keY)
I
i
]
IO
0.2-
0.1-
0
'
~
'
o
~
l
I
,
5
ENERGY (keY)
,
'
I
IO
Fig. 8. Backscattering probability as a function of incident
energy from semi-infinite aluminum, zxMCe+; oMCe-;
x [20]e-; + [17]e-
Fig. 9. Backscattering probability as a function of incident
energy from semi-infinite gold. zxMCe+; 9 MCe-; x [2lie-
simulated distributions are higher than those of the
experimental estimates and the MC values near the
surface are lower, respectively. These differences are
due to the ignoring in the experimental analysis of the
increase of the backscattering probability as a function
of film thickness, which is important especially for thin
films. If we take the backscattering effect into account,
the experimental curves would be much lower near the
surface and thereby the maximum values would
increase due to normalization. A conclusion can be
drawn that the agreement is quite satisfactory.
Difficulties in estimating positron and electron
implantation profiles from thin film transmission
measurements have been discussed in more detail in
For light elements the backscattering probability of
electrons decreases as a function of energy whereas for
heavy elements it increases. The reason is that light
elements have a relatively larger scattering probability to large angles (see the lobes in Figs. 2 and 3).
These decrease in magnitude with increasing energy.
For heavy elements the probability distribution of
scattering to different angles remains very similar at
these energies (the backscattering lobes are negligible).
At energies above 10 keV the relative backscattering
probability begins to decrease also for heavy
elements.
Comparisons with experimental electron backscattering coefficients [17, 20, 21] have been made
and fair agreement is found. The difference between the
experimental and the MC coefficients is probably due
to the energetic secondary electrons. Their
contribution to the experimentally observed total
backscattering can be of the order of 10% [22]. For
positrons no experimental data is so far available.
[i].
A most interesting difference between positrons and
electrons is in the backscattering probability.
Positrons have a much smaller probability to scatter
back from semi-infinite targets than electrons (Figs. 8
and 9). Electrons at 1 keV have more than twice as
large a probability to scatter back as positrons. The
difference is largest for small energies and decreases as
a function of energy. The reason lies again in the elastic
scattering cross section of crystal atoms for positrons
and electrons. Electrons have a relatively larger
probability to undergo large-angle scattering events,
whereas positrons almost always scatter by less than
30 ~ (Fig. 2). Positrons also have a smaller total elastic
scattering cross section than electrons (Fig. 1), which of
course decreases the overall positron backscattering
probability.
2.2. Oblique Incidence
Effects of changing the angle of incidence are studied
for A1 and W at incident energies of 1 and 5 keV.
Positrons behave similarly to electrons as a function of
incidence angle. The effect of changing the angle is a
little larger for positrons than for electrons, for
example, in the penetration depth and in the
backscattering probability (Figs. 11 and 13). Generally
the electron-positron differences decrease with
Monte Carlo Calculations of keV Elec~:ron and Positron Slowing Down in Solids. II
00
i
c~
i
(.13
33
i
AI, e*
E=5 keV, r
57
L
_
_
A(
~o
J
J
=
~
(D
C)
0
13_
+
r~
.g
X
O
+
+
++
_/
+++
_~+
+
N
~
0
0
C)
d
X--OXLS
( "1OOO
Fig. 10. The distribution of the trajectory endpoints, projected
onto the y - z plane, for 5 keV positrons at 80 ~ angle of incidence
on aluminum. The arrow denotes the entrance position
AI
E=5 keY
O
IN
A~'~-~___
F~
O~
O~
O
~A~A
o
~o
~o
so
ANGLE
qb ( o )
Fig. 11. Mean penetration depth as a function of incidence angle
for 5 keV positrons and electrons impinging on semi-infinite A1.
Ae+~ Oe-
i
i
i
AI, e §
~<
o
;o
AngLe
/I,)
GD
C)
O
C O.5
90
( ~
Fig. 13. Dependence of the backscattering probability on angle of
incidence for aluminum at 5 keV. A e +, O e . The lines are fits of
(15) to the MC points
increasing angle. The distribution of the trajectory
endpoints for 5 keV positrons at the angle of incidence
of 80 ~ impinging on aluminum is shown in Fig. 10.
Between 0 ~ (normal incidence) and 30 ~ the change of
incidence angle does not have much effect on the
penetration properties (Fig. i1). Only with larger
angles do the results start to differ from those of normal
incidence. The decrease of the mean penetration depth
is about 25% for electrons and 35% for positrons when
the angle of incidence is changed from 0 ~ to 80 ~ This
decrease of the penetration depth naturally moves the
implantation profiles nearer to the entrance surface
(Fig. 12). The difference in penetration depth between
0 ~ and 80 ~ angle of incidence decreases with energy.
The most distinct effect of changing the angle of
incidence is on backscattering probability (Fig. 13).
Backscattering is almost constant between 0 ~ and 30 ~
but increases rapidly at larger angles. Only about half
of the particles absorbed in the material on normal
incidence are absorbed when the angle of incidence is
80 ~.
The backscattering probability B(q~) for a positron or
electron beam with the angle of incidence q~ (with
respect to the surface normal) can be presented as [23]
.J
'S
B(r = b
J
&
0
_+J
o~ 0 . 0 - - -
o
4
Penetrotion
~
depth
4
( '1ooo A)
Fig. 12. Fits of(16) to the stopping profiles (normalized to unit
area) of 5 keV positrons on aluminum for 0 ~ 40 ~ and 80 ~
incidence angles
,
(15)
where B• is the backscattering probability at normal
incidence and b is an experimentally determined
constant. Fits of (15) to the MC data give a value of
0.79 _+0.01 for b. The value of b is very slightly (=< 0.01)
larger for positrons, and it also increases by 0.02 for
both particles when the incident energy is changed
from 1 to 5 keV. Similar behaviour of backscattering
as a function entrance angle has been found for
58
S. Valkealahti and R. M. Nieminen
10-100 keV electrons [23, 24]. Kalef-Ezra et al. [23]
obtained for b the value of 0.89 at the 10-100 keV
region. In our calculations the b value increases as a
function of energy and by extrapolation agrees with
[23]. The reason for the increase of b with energy is
that at high energies (above 10 keV) forward scattering becomes relatively more dominating.
The backscattering energy distribution of electrons
and positrons also changes with the angle of incidence.
At normal incidence the maximum is in the region of
intermediate energies (0.4Eo < E <0.7Eo, where Eo is
the initial energy) and moves to higher values with
increasing angle of incidence. This is due to the increase
of the backscattering probability for the first few elastic
collisions. The mean backscattering energy also
increases slightly as a function of incident energy E 0
and target atomic number Z. The reason is that the
elastic collision probability increases with respect to
the inelastic one as a function of energy and atomic
number.
The mean value of the backscattering energy is about
10% higher for electrons than for positrons. The
reason is again that electrons have a larger probability
to scatter into large angles than positrons. Thus the
backscattered electrons have more probably suffered
a large angle scattering event, whereas positrons have
encountered several small angle scatterings and have
lost more energy during the longer traversed path in
the material.
2.3. Parametrization of the Implantation Profile
In [1] we discussed the parametrization of the range
distributions and presented an approximation for
calculating the re-emission probability of positrons.
The formula used for the stopping profile is [25, 26]
mzm- 1
P(z) = - -
zorn
exp [ - (Z/Zo)"] 9
(16)
where m and Zo are parameters. The shape parameter m
was found to be nearly constant (~1.9), slightly
decreasing as a function of atomic number. The
penetration parameter zo is a function of incident
energy as z0 = ~E", and is proportional to the mean
penetration depth i (Zo = 1.1M).
These results are confirmed by the present calculations
at normal incidence. The value of m for positrons is
little larger (by ~0.1) than that of electrons. (The
statistical uncertainty of m is typically less than 0.1.)
Thus the conclusions and estimates made in [1]
remain qualitatively valid. Especially useful is the
formula used for the overall re-emission probability of
positrons from the entrance surface [27]
f=v
(1
(26)1/2
xI,
(17)
where
I = O(_dzp(z)exp
- ~=~)
zj.
(18)
0
Above, D is the diffusion constant, 2 the depletion
(annihilation) rate in the medium, v the rate of emission
from the surface, and P(z) the implantation profile
(normalized to unit area).
Using (16), one finds for m = 1
t= - 1+
1
(19)
SZo
and for m = 2
I = 1 - ~ - SZo e (sz~
erfc(szo/2).
(20)
In (19) and (20), s = ()~/D) 1/2.
Changing the angle of incidence affects the shape of the
stopping profiles (Fig. 12). This is also seen as a
decrease of m as a function of incidence angle.
Positrons have larger values of m than electrons at
small angles, but the m value decreases much more
rapidly for positrons than for electrons with increasing
angle. The decrease of m as a function of incidence
angle decreases with increasing energy both for
electrons and positrons. For example, when the angle
of incidence on A1 changes from 0 ~ to 80 ~ m decreases
from 2.0 to 1.4 and from 1.9 to 1.6 for 1 keV positrons
and electrons, respectively. The decrease for 5 keV
positrons and electrons is from 2.0 tO 1.5 and from 1.9
to 1.7, respectively.
The positron re-emission parameter I does not change
much below the incidence angle of 30 ~. At larger angles
the decrease of the mean penetration depth as a
function of angle increases I, if P(z) is normalized to
unit area. If, however, the increase of backscattering as
a function of angle is taken into account, the reemission probability will decrease as a function of
angle.
3. D i s c u s s i o n
Electrons typically have more than one lobe in the
differential elastic scattering cross section (Fig. 2). This
feature is, of course, strongest at low energies. The
number of these lobes and their position depends on
the material and the energy of the incident electrons.
They arise from the attractive potential of an atom for
electrons, whereas the repulsive potential for positrons
only causes the forward peak.
While the scattering lobes have no drastic effects on the
stopping properties, some of their consequences
should be seen, e.g., in the backscattering energy or
angular distributions with good statistics and
Monte Carlo Calculations of keV Electron and Positron Slowing Down in Solids. II
accuracy. On the contrary, in low energy electron
diffraction (LEED), the effect of the peaks and zeros is
most important, and complicates the multiple
scattering calculation of the L E E D intensity curves.
On the other hand, in L E P D the analysis is much
simpler due to the much smoother angular variation of
the cross sections.
At energies around 1 keV the absorption probability of
positrons in solid targets is around 85-90%, which is
much larger than was expected on the basis of the
k n o w n electron absorption probabilities, The small
backscattering probability of positrons at keV energies
makes the use of low energy monoenergetic positron
beams more effective. This is beneficial for the use of
keV positrons as a probe to give information of the
surface or near-surface region.
Acknowledgements. This research has been in part supported by
the Academy of Finland. Useful discussions with Dr.
H. E. Hansen are gratefully acknowledged,
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