Average depths of electron penetration. II. Angular dependence and use to evaluate secondary-electron yield by photons
ABSTRACT In our previous paper [V. Lazurik, V. Moskvin and T. Tabata, IEEE
Trans. Nucl. Sci. 45, pp. 626-31 (1998)] the average depth of electron
penetration, Rav, has been introduced as the average of the
maximum depths on the trajectories of electrons passing through a
target. In the present work the dependence of Rav on the
angle of incidence of an electron beam has been studied. A
semi-empirical equation is derived to calculate Rav as a
function of angle of incidence. We extend the study of Rav
from using it to characterize the average behavior of electron beams in
a target to describing the generation of secondary electrons by photon
beams. It is shown that Rav can be used in a wide variety of
applications in which the characteristic size of the spatial region of
electron production is important
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ABSTRACT: A mixed algorithm for Monte Carlo simulation of relativistic electron and positron transport in matter is described. Cross sections for the different interaction mechanisms are approximated by expressions that permit the generation of random tracks by using purely analytical methods. Hard elastic collisions, with scattering angle greater than a preselected cutoff value, and hard inelastic collisions and radiative events, with energy loss larger than given cutoff values, are simulated in detail. Soft interactions, with scattering angle or energy loss less than the corresponding cutoffs, are simulated by means of multiple scattering approaches. This algorithm handles lateral displacements correctly and completely avoids difficulties related with interface crossing. The simulation is shown to be stable under variations of the adopted cutoffs; these can be made quite large, thus speeding up the simulation considerably, without altering the results. The reliability of the algorithm is demonstrated through a comparison of simulation results with experimental data. Good agreement is found for electrons and positrons with kinetic energies down to a few keV.Nuclear Instruments and Methods in Physics Research Section B Beam Interactions with Materials and Atoms 01/1995; · 1.19 Impact Factor
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ABSTRACT: The average depth of electron penetration is introduced as the physical quantity useful in electron beam irradiation. It is defined as the average of the maximum depths on the trajectories of electrons passing through finite, semi-infinite or infinite medium. The relation between the transmission coefficient as a function of slab thickness and the distribution of the maximum depths is analyzed, and a semiempirical equation to calculate the average depth of electron penetration is given for 0.1- to 50-MeV electrons incident on materials of atomic numbers from 4 to 92. It is shown that the quantity introduced is usable as the characteristic depths of energy and charge depositions in a target, and can possibly be generalized to the case of heterogeneous targets.IEEE Transactions on Nuclear Science 06/1998; 45(3):626 - 631. · 1.22 Impact Factor
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ABSTRACT: An empirical relation for the transmission coefficient of electrons impinging on aluminum absorbers is given as a function of absorber thickness x, incident energy T0, and angle of incidence θ. It has been formulated by incorporating the dependence upon θ in the empirical equation for the case of normal incidence reported previously by the present authors. Numerical constants in the relation have been determined through least-squares fit to the data for T0 = 0.5−10 MeV generated by the Monte Carlo code of Berger and Seltzer. The rms absolute deviation, evaluated over θ = 0°−75°, of the relation from the Monte Carlo data is about 0.03 in the entire energy region considered.Nuclear Instruments and Methods 08/1976; 136(3):533-536.
IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 46, NO. 4, AUGUST 1999
Average Depths of Electron Penetration (11): Angular Dependence and Use to Evaluate
Secondary-Electron Yield by Photons
Valentin Lazurik I, Vadim Moskvin ') Yuri Rogov I, Tatsuo Tabata
' Radiation Physics Laboratory, Kharkov State University, P.O.Box 60, Kharkov, 3 10052, Ukraine
* RIAST, Osaka Prefecture University, 1-2 Gakuen-cho, Sakai, Osaka 599-8570, Japan
In our previous paper [V. Lazurik, V. Moskvin and T.
Tabata, IEEE Trans. Nucl. Sei. 45, pp. 626-631 (1998)] the
average depth of electron penetration, R,,
as the average of the maximum depths on the trajectories of
electrons passing through a target. In the present work the
dependence of R, on the angle of incidence of an electron
beam has been studied. A semi-empirical equation is derived
to calculate R, as a function of angle of incidence. We extend
the study of Rav &om using it to characterize the average
behavior of electron beams in a target to describing the
generation of secondary electrons by photon beams. It is
shown that R, can be used in a wide variety of applications in
which the characteristic size of the spatial region of electron
production is important.
has been introduced
Wide spread usage of electron and photon beams in
application purposes motivates a necessity of developing quick
algorithms to calculate characteristics of the effect of beam
impact on a target. Quick algorithms can be developed by the
method of semiempirical modeling. One of the first problems
in such modeling is to find basic trends in the relation between
the characteristics of the transport of ionizing radiation and its
Considering the average behavior of an electron beam in a
target, the concept of the average depth of electron
penetration, R,, has been introduced in OUT previous paper [l].
This parameter is defined as the average of the maximum
depths on trajectories of electrons passing through a target, and
was proposed to be used as a characteristic depth of exposure
by an electron beam. It was shown that R, is determined by
integrating the transmission coefficient of electrons as a
function of slab thickness. A semiempirical equation to
calculate R, has been derived for the electron beam normally
incident on a semi-infinite target. It was found that the value of
R, for a semi-infinite target was almost the same as that of an
infinite target under normal incidence.
For application purposes, it is important to account for the
angular spread of an electron beam incident on a target. In the
present work, therefore, we study the dependence of R, on the
angle of incidence of electrons. The result obtained is applied
to the modeling of the generation of secondary electrons by
photon beams. Thus we extend the study of Rsv fiom using it to
characterize the average behavior of an electron beam in a
target to describing the characteristic size of the region in a
target for secondary electron generation by photon beams.
a. ANGULAR DEPENDENCE
A. Average Depth o f Electron Penetration R,, and
Projected Range R,.
Let us consider the quantities determined by representative
points of electron trajectories in a semi-infinite medium. The
maximum depth of electron penetration in a semi-infinite
medium is the largest depth an electron reaches during its
motion in the medium. In general, this is not the track-end
point of an electron trajectory. This is illustrated in Figure 1,
where x, is the maximum depth of electron penetration and
x,"d is the depth of track-end for a given trajectory.
I Depth x
Figure 1: Electron trajectories in a target.
For a set of electron trajectories, the distribution D,(x) of
the maximum depths of electron penetration can be defined.
Analogously to D&), the distribution Den&) is defined as the
depth-distribution of track-end points of electrons. This has
also been called range distribution, range-straggling
distribution or projected-range distribution by previous
For the general case of electrons of energy E incident at
angle 8 on a surface of a semi-infinite medium of atomic
number 2, the average depth of electron penetration,
R,(B,E,Z), is defined by
Rav(Q,E,Z) = IxDm(x,Q,E,Z)dx,
where D,,,(x, B,E,Z) represents the distribution of the maximum
depths of electron penetration in semi-infinite medium for the
angle of incidence B.
Analogously, the projected range of electrons, R,,(B,E,Z), is
0018-9499/99$10.00 0 1999 IEEE
where Dend(B, E, Z) represents the distribution of the track-end
points of electrons in semi-infinite medium for the angle of
For considering an infinite medium, let us suppose that a
source of electrons is placed at the point x=O. We can define
R,*(B, E, Z) in an infinite,medium according to Equation (1)
by replacing D,(x) by Dm (x) in the infinite medium; but ,the
lower limit of integration is kept the same, because Dm (x)
does not extend to the left half of the medium. On the other
hand, track-end points can lie in the leR half of the infinite
medium, so that the definition of Rpr(B, E, z) should be
mofified ffom Equation (2) to the following form:
where Den;(x) is the range-straggling distribution (the
distribution of the track-end points) in an infinite medium.
The transmission coefficient ~ ( x )
in an "experiment" on the passage of electrons through a slab
of a given thickness x. An electron is registered as the
transmitted particle if its maximum depth x, of penetration
satisfies the condition x, 2 x, and it is not registered as the
transmitted particle if x,,, < x (see Figure 1). Therefore, the
transmission coefficient q(x,eE,Z) of electrons of energy E
incident at angle B on a slab target of thickness x consisting of
the material with atomic number 2, is defmed as
V(X,e,E,z) = IDm(X,B,E,Z)A.
It was shown in our previous work [l] that the use of
Equations (1) and (4) allows to write the average depth of
electron penetration, Rnv, as the integral of the transmission
coefficient of electrons as a function of slab thickness:
of electrons be measured
Rav (8, E, Z) = I ~ ( x ,
4 E, z)& .
For the infinite medium, the "experiment" is modified. The
source of electrcys is placed on the surface of semi-infinite
medium x=O. R, is determined by integrating the transmission
coefficient as a function of the thickness of slabs that constitute
the surface regions of different thicknesses of this medium,
analogously to Equation (5).
B. Monte Carlo Calculation o f RAV,
The average depth of electron penetration, R,, in a medium
is calculated in Monte Carlo simulation by
where N is the number of simulated electron trajectories, x , ,
is the maximum depth on a given trajectory reached by an
electron in a semi-infinite medium, or in the right-half of an
The projected range of electrons, Rpr, is calculated in a
Angle o f Incidence 8, deg
30 60 90
Angle o f Incidence 8, deg
Angle of Incidence 8, deg
Figure 2: Dependence of the average depth of electron penetration,
R,, on the angle of incidence, 0, of an electron beam: solid circles,
Monte Carlo results for the semi-infinite medium (SIM); solid
triangles, Monte Carlo results for the infinite medium ( I M ) ; the solid
line, the semiempirical equation. Monte Carlo data on the projected
range of electrons, R,,(B), are also shown: open circles, the semi-
infinite medium SIM, open triangles, the infinite medium IM.
of R,, and Rpr are given in scaling units of CSDA electron range Ro
(data are taken from ).
where N, is the number of simulated electron trajectories
whose track-ends are located within a considered medium (for
a case of an infinite medium Na=N), and xend is the depth of the
track-end of an electron, i.e., the point at which the electron is
assumed to be stopped in the calculation.
To compute Re,,( 8) and Rpr( 8) for various initial angles B of
an electron beam for a semi-infinite and an infinite medium, a
Monte Carlo code developed on the basis of PENELOPE ’ 
package was used. The incident-electron energies considered
have been from 0.1 to 50 MeV, and the atomic numbers of
medium materials have been from 4 to 92. The number of
electron trajectories calculated for each angle have been
2x104. An angle of 89.9 deg has been chosen for the limiting
case of grazing angles.
Figure 2 shows the comparisons of average depths of
electron penetration, R,, and projected ranges, Rpr, in a semi-
infinite and an infinite medium. The values of R , and R,, are
given in units of the continuous slowing-down approximation
(CSDA) range Ro (data taken from ). Comparisons of
average depths of electron penetration in a semi-infinite and an
infinite medium have shown that their values are almost the
same except at the largest values of 8. The values of R,, for the
above two configurations of the medium are closer for low and
intermediate atomic numbers. It should be noted that Rpr takes
on considerably different values for the two configurations at
The dada presented in Figure 2 illustrate the important role
of electron scattering in forming the difference between R,
and Rpr for both the medium types considered. It is seen that
the difference between R, and Rpr increases with increasing
extent of scattering of electrons in the medium, i.e., with
increasing atomic number of material and decreasing initial
C. A Generalized Semiempirical Equation to
From the expression (5) for R,(O,E,Z)
semiempirical equation for the transmission coefficient
r,(x,t?E,Z) (an extended version for arbitrary Z of the one
published for Z=13 ), a generalized semiempirical equation
for R,(B,E,Z) is derived in the form
where so(E,Z) is the model parameter in the transmission
coefficient, R,(E,Z) is the extrapolated range of electrons
normally incident on a semi-infinite medium. Equations for
and R,,(E,Z) are given in [I]; formulas for parameters
a@) and a@), which explain the dependence of transmission
coefficient on 0, are available in  and given in Appendix of
D. Accuracy o f the Semiempirical Equation.
A semiempirical equation for RaV( 9 has been compared
with the data computed by the Monte Carlo technique. A
measure of goodness of fit of the semiempirical equation can
be given by the relative root-mean-square deviation Arms
where n is the number of data points in a single set of R, for a
given incident energy and material, RMc,, is the Monte Carlo
value of Ray for the ith data point, and Req,, is the value given by
The statistical uncertainty of the Monte Carlo values of
R,(9 is about 1.5% except at grazing angles. At the limiting
angle of 89.9 deg, the uncertainty takes on a rather large value
because of poor statistics, which in turn is caused by the large
extent of backscattering loss of incident electrons, Also, it is
known that model uncertainties can arise in calculations while
using Monte Carlo technique to simulate electron transport
near the target-vacuum interface [SI. Therefore, the values for
89.9 deg are presented in Figure 2 only for qualitative
comparison and illustration of general trends in R,(9 and
Rpr(9 for semi-infinite madium. The data at this angle were
excluded in evaluating Arms.
Figure 3: Relative root-mean-square deviation, in %, of Eq. (8) from
the Monte Carlo data.
Arms is illustrated in Figure 3 for the region considered of
the initial energy Eo and the atomic number Z of material. It is
seen from this figure that for materials with atomic number
from 6 to 79 Equation (8) well fits to the Monte Carlo data in
the energy region from 0.1 to 20 MeV. Figures 2 and 3 show
that the deviation of the equation is less than 5% in the
’ Package PENELOPE is available from NEA Data Bank, see
intermeddle region of Eo and 2, and increases to 10% in
intermediate region of 2 at 0.1 MeV energy.
The deviation of the equation is rather large for EO above
20 MeV and for 2 greater than 79 or less than 6. To minimize
the deviation of Equation (8) fiom the Monte Carlo data,
iniprovements of expressions for some of the parameters are
The moderate accuracy of the generalized semi-empirical
equation indicates that it is applicable to semiempirical
modeling of processes in irradiated materials, in particular in
electron-beam application to electronic device technology.
A. R,, and R,, in Describing the Behavior of an
In the previous paper [I] we have shown the useful
application of average depth of electron penetration, R,,
the projected range of electrons, Rp,, for describing the
characteristic depth of exposure of a semi-infinite medium to
an electron beam under normal incidence. It has been found
that R, well describes the region in a medium where 80% of
the total absorbed energy is deposited. The region in which
50% of total deposited charge collected is well described by
both Ra, and Rpn because of numerical closeness of these
quantities for normal incidence. In addition, R,,, according to
its definition, is appropriate to represent the depth of 50%
attenuation of the number of incident electrons.
In the following we consider the use of R,, and R, at
various angles of incidence, for describing the average
behavior of an electron beam in a medium, analogously to
what has been done in [I].
Figure 4: The use of R,, and Rpr as characteristic depths of energy
deposition DE(x) and charge deposition Dc(x) by an electron beam in
Figure 2 shows that the closeness of the values of R,, and
R,,, observed for normal incidence of electrons on a semi-
infinite target [I], breaks down when the angle of incidence B
increases and that they are considerably different at large
values of B from the Monte Carlo data obtained by the use of
the PENELOPE package. Figure 4 shows typical relations
between R,,(9 and Rp,(9 and profiles of charge and energy
An analysis has shown that Rav(9 corresponds to the depth
within which from 60 to 80% of the total absorbed energy is
deposited; this depth is situated in the region of rapid change
in the depth profile of integral energy deposition. This allows
us to use RaV( 9 to describe an approximate depth within which
a great fraction of the total energy is deposited in a medium.
On the other hand, Rpr( 8) well describes an approximate depth
in a target within which 50% of the total charge is deposited. It
is to be noted that R,(Q
describes the depth of 50%
attenuation of the number of incident electrons in a medium.
Thus the properties of R , ( 9 and Rpr( 9 make these quantities
useful for the analysis of processes in irradiated materials.
B. Yield of Secondary Electrons by Photon Beams
Let us consider the application of R,(9 to calculate the
forward yield v+ of secondary electrons fiom the target of
equilibrium thickness' exposed to photon beams (SE fonvard
yield), i.e., the number of electrons emitted ffom the target in
the fonvard direction per incident photon (see Fig. 5).
For simplicity, we consider the energy region of photons
where their interactions with materials essentially consist of
Compton scattering and photoelectric effect. Let dmJdcosB be
the cross-section for Compton scattering; and (dmddcos8), be
the cross-section for the photoelectric effect for the ith shell of
an atom. Let us assume that the density of the secondary
electrons produced is uniform along the depth of the target;
this is reasonable for the equilibrium thickness because it is
considerably less than the photon range.
Figure 5: Schematic diagram to explain the SE forward yield VI.
To derive an equation for v', we use the fact mentioned
before, i.e., that the values of R,,(9 in an infinite medium are
almost equal to those of R,(9 in a semi-infinite medium
except at grazing angles, and assume that R,(9 in a semi-
infinite medium is equal to that in an infinite medium. Under
this assumption, the probability for the secondary electrons of
energy E produced at depth x with the initial direction of
motion given by angle B to escape from the target of atomic
' The equilibrium thickness is the thickness equal to the range of
and charge the most energetic secondary electron produced by photons in a target
deposition within the depths of Rav( 9 and Rpr( 9 for various material considered.
number Z is described by the transmission coefficient by the average depth of electron penetration, Ray( 9, for which
q(@E,Z). Integration over the depth of the sources of the semiempirical equation (8) we have derived is useful.
secondary electrons (the points of photon interactions with Further the application given in this section indicates the
atoms in the target) gives a characteristic depth for the escape possibility that R, can be used in a wide variety of problems in
of secondary electrons. According to Equation (5), this depth which the characteristic size of the spatial region of electron
is equal to average depth of electron penetration, R,(L?E,Z).
Thus we can write the equation for vf as
production is relevant.
V. L., V. M. and Y. R. are gratell to STCU for personnel
support (grant N 1 15).
Rav(B,E, -E,,Z) dcosB
[l] V. Lamik, V. Moskvin and T. Tabata, "Average depths of
electron penetration: use as characteristic depths of
exposwe," IEEfi Trans. Nucl. scj., vol. 45, pp. 626-63 1,
 J. Baro, J. Sempau, J. M. Femandez-Varea, and F. Salvat
"PENELOPE: An algorithm for Monte Carlo simulation
of the penetration and energy loss of electrons and
positrons in matter," Nucl. Instr. and Meth. B, vol. 100,
DD. 31-46. 1995.
 ICRU, Stopping powers for electrons and positrons, ICRU
Report 37, Bethesda, MD, 1984.
 T. Tabata and R. Ito, "An empirical relation for the
transmission coefficient of electrons under oblique
incidence," Nucl. Instr. and Meth., vol. 136, pp. 533-536,
Vhre E(cosB) is the energy Of the
in the direction of 6, E, is the photon energy, and E, is the
binding energy of the ith shell.
Table 1: The secondary electron forward yield v',in units of lo3
electrons/photon, from targets consisting of various materials
exposed to photon beams with the energy E,. Values calculated by
semiempirical technique are compared with experimental data .
3 f 0.3
2.3 f 0.2
2.1 f 0.2
3.6 f 0.3
 A. F. Bielayew, D. W. 0. Rogers and A. E. Nahum, "The
Monte Carlo simulation of ion chamber response to 6oCo
- resolution of anomalies associated with interfaces."
Phys. Med. Biol., vol. 30, No.5, pp. 419-427, 1985.
 A. F. Merman, V. A. Botvin, M. Ya. Grudskii, V. V.
Smirnov, Ya. Chernov, M .V. Shilenkova, "Secondary
electron radiation f r o m different targets exposed by
photons with energy 0.1 to 3 MeV," Phys. Stat. Sol. B,
vol. 110, pp. 285-297, 1982.
Table 2: The secondary electron forward yield vt, in units of 10'
electrons/photon, from targets consisting of various materials
exposed to photon beams with the energy Ey. Values calculated by
semiempirical technique are compared with QUICKE 3  data.
(Z=13) (Z=29) (Z=50) (Z=79)
I QUICKE3 I 2.38 1 1.96 I 2.22 I 3.87
1000 I Equation I 5.16 I 4.00 1 3.67 I 4.11
I QUICKE3 I 5.130 I 4.01 1 3.61 [ 4.38
The values of v' have been computed for the targets of Z
from 6 to 79 irradiated by photon beams of energies from 20 to
10 MeV. The semiempirical equation (8) for A,(@E,Z) in a
semi-infinite target is used in Equation (12). Results obtained
are compared With experimental data  in Table 1 and With
the results of the QUICKE 3 code  in Table 2.
It is concluded that the characteristic depth of secondary
electron production by photons from a target can be described
 T. A. Dellin, R. E. Huddleston, C. J. MacCallum, "Second
generation analytical photo-compton current methods,"
IEEE Trans. Nucl. Sci., vol. NS-22, No. 6, pp. 2549-2555,
The expressions for az(,!i') and a3Q
developed in  are
u3(E) = 0.5237 / ro
where r,,=~,-,/rnc*, rnc2 is the rest energy of the electron. These
expressions were determined for the target of atomic number Z
equal to 13. In the present work, it has been assumed that they
are also applicable to targets with different values of 2, i.e.,
that the dependence of a 2 and u3 on Z is not significant.