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A study of primary electron charge deposition near the surface of a target by Monte Carlo simulation
B ELSEVIER
Nuclear Instruments and Methods in Physics Research B 134 (1998) 1-12
A study of primary electron charge deposition near the surface of a target by Monte Carlo simulation Valentin Lazurik, Vadirn M o s k v i n * Radiation Physics Laboratory, Kharkov State University, P.O. Box 60, 310052 Kharkov, Ukraine Received 16 July 1996; received in revised form 15 September 1997
Abstract
The Method of Trajectory Rotation [V. Lazurik, V. Moskvin, Nucl. Instr. and Meth. B 108 (1996) 276] is modified to compute the charge deposition density and the yield of an electron spectrum at given depths of targets irradiated by electron beams. Primary electron charge deposition near the boundaries of finite targets is studied. It is found that the depth-profiles of the primary electron charge deposition have a drastic nonlinear decrease near the target vacuum interface. The theoretical description of the charge deposition near the inhomogeneity in a target is discussed. The election of model parameters, i.e., the cutoff energy and the depth bin width, for accurate computer simulation with the use of the conventional Monte Carlo techniques is considered. © 1998 Published by Elsevier Science B.V.
PACS: 78.70.-g; 34.50.Bw; 87.53.Fs; 81.40.Wx Keywords: Electron transport; Computer simulation; Monte Carlo techniques; Primary electrons; Charge deposition; Stochastic wandering; Target-vacuum interface; Boundary effect; Electron spectrum
1. Introduction
Data on quantities describing the action of fast electrons on materials are necessary for the analysis of the electrophysical processes in objects irradiated by electron beams. The energy and charge deposition density are the fundamental quantities used in such analysis. At present, the computer simulation of electron transport with the use of Monte Carlo techniques
*Corresponding author.
is the basic method of obtaining these data. The general purpose Monte Carlo codes, such as ITS [1], G E A N T [2] and EGS [3], are widely used in practice. What all these codes have in common is that the transport of electrons is simulated for a region of their energies from an initial energy/:~ to a given cutoff energy gmin. It is assumed that electrons deposit their charge at the end of their tracks. Tracing the trajectories of electrons until the energy Emin is connected with the limits of validity of the physical models used in computation. Thus the cutoff energy is a model parameter that defines an accuracy of simulation. Notice that
0168-583X/98/$19.00 © 1998 Published by Elsevier Science B.V. All rights reserved. P I I S O 1 6 8 - 58 3 X ( 9 7 ) O 0 5 1 0 - 7
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the methods of the election of the cutoff energy for calculating the energy deposition distributions are known (see [4]). When using the conventional Monte Carlo techniques, the charge deposition density at a given depth of a target cannot be computed as such. The target is divided into a set of depth bins. The charge deposition density at a given depth z is evaluated on the number of electrons stopped in a depth bin with a given width Az. As a consequence, the result of computation is the average density of the charge deposited in this depth bin. The depth-profile of the charge deposition (also called the charge deposition distribution) is fitted using the data on the average density of the deposited charge computed for a set of depth bins. The depth bin width Az is a model parameter, whose value should be set before computation. Thus, the accuracy of data obtained by the Monte Carlo simulation depends on the values of model parameters, i.e., the energy cutoff and the depth bin width. The general method of the election of these parameters for the charge deposition computation has not been developed. At present, a systematic set of the data on depth-profiles of charge deposition in semi-infinite targets has been computed recently (see [5,6]). The depth-profiles of the primary electrons charge deposition can also be calculated using the data on projected range straggling of fast electrons presented in [7-9]. Besides, data on the primary electron charge deposition in slabs are available (see [10]). The data presented in the cited papers have been computed using the conventional Monte Carlo techniques. The model parameters, both the energy cutoff and the depth bin width, were elected for these calculations in the same way as which commonly used for calculations of the energy deposition. The validity of this election is not analyzed thoroughly in these papers. From the results described in [5,6] it follows that the total charge deposition has peculiarities in the region near the target-vacuum interface, i.e., near the boundary of a target. The total charge deposition density varies sharply near the target-vacuum interface and, moreover, the sign of the total charge deposited in this region can be changed. It is agreed that these peculiarities
are associated essentially with escaping knock-on electrons (secondary electrons) in a vacuum. However, the results presented in [10] suggest that the density of the primary electron charge deposition is subject to strong variation near the boundary of a target. The widths of depth bins used in computations presented in [5-10] are comparable to a characteristic size of the region where the total charge deposition changes drastically. Therefore, the available data are inadequate to determine uniquely the role of primary electrons in forming the profiles of the total charge deposition near the boundaries of finite targets. When using the data on the average density of the charge deposited in depth bins, further theoretical assumptions are necessary for fitting the depth-profiles of the primary electrons charge deposition. The use of simple approaches for fitting the profiles near the boundaries of semi-infinite targets cannot lead to advance. Moreover, the contradiction between the results presented in [7,8] was caused by such simple approaches used in the analysis of the results of computations. It seems likely that the improper handling of algorithms in studying charge deposition, along with the energy deposition calculations, can lead to anomalous results known as "interface artifacts" (see [11]). The aim of this paper is the study of the primary electron charge deposition near the targetvacuum interface and the analysis of the election of model parameters, i.e., the cutoff energy and the depth bin width, to compute data on charge deposition with the use of the conventional Monte Carlo techniques.
2. Method of calculations
Let us assume that a slab target consisting of a uniform material is placed in a vacuum. An electron beam with energy E0 impinges normally on the target surface (called here the left boundary) at a point (0, 0, 0). The z-axis is the normal to the surface. To calculate the charge deposition density at given depths of a target, we modified the Method of Trajectory Rotation. This modification is discussed below.
V. Lazurik, E Moskvin I Nucl. Instr. and Meth. in Phys. Res. B 134 (1998) 1-12
3
Let us begin by considering some features of the Method of Trajectory Rotation presented in [10]. Using the Monte Carlo procedure, an electron trajectory (here called the base trajectory) Cj is traced in an infinite medium. The trajectory is simulated from the initial energy E0 to a given cutoff energy Emin. The point of the trajectory where electron has a given energy E~ is taken as a rotation point. The direction of the electron motion in this point is taken as a rotation axis. The part of the base trajectory Cj, where electron has the energy E < Er is selected. A continuous set of trajectories is created by rotating this part of the base trajectory around the rotation axis (see Fig. 1). Assume that a layer with the thickness Az is placed at the depth z in a target. This layer is called a depth bin. The contribution of the base trajectory Cj to the average density of the charge deposited in the depth bin is given by an equation
the energy Emi n intersects the plane z in rotating, i.e., the end of the trajectory CJ can be placed in a given depth bin. It equals 0 in other case. ~J" is the region indicator. It equals 1 if all points of the trajectory C~j defined by rotating the base trajectory CJ belong to the target and it equals 0 in other case. m is the index to account for the dual intersection of the plane z by the base trajectory in rotating. Let us modify the method of trajectory rotation to calculate the charge deposition density at a given depth z in a target. To do this, we suppose that the depth bin width Az approaches zero. Going to a limiting case, the ratio (A~o/kz) converts to the derivative (d(o/dz). The derivative (dq~/dz) is derived from a formula
(1)
where Zr is the z-coordinate of the rotation point (i.e., the trajectory point with the energy Er), /¢ the radius-vector from the rotation point to the end of a traject~y, h'~ the rotational axis defined by the direction ~¢'~r~--- (~r, fir, ~2r) of electron motion in the rotation point, and (/~, ff~) the dot product of R and ff~o. Using the (d~o/dz) in an analytical form, Eq. (1) can be recast for the modified Method of Trajectory Rotation. The contribution of the base trajectory CJ to the density of electrons with energy Emi, at a depth z is given by
1 ~ zMp W~(z)=~
(.Z~j,.. ' .,=,
where ~J is the rotation indicator. It equals 1 if the point of the base trajectory where the electron has
Target surface
Target region
z = zr + ~r(R, ~ )
-~ ¢ ( R 2 -- (*~, nq~)2)(1 -- 7r) 2 COS q),
W~(z)=
1
1
~:3--,2 .
2re v/S~ _ $22(z) ~m=,
(2)
(3)
where
}//J\\ .
Eo ".,/ ._
\
S, = ¢ ( R : - (R, ff~,)2)(1 - yr)2,
(4)
S2(z ) : Z -- (Zr q- yr(/~, h'q~))
(5)
and the rotation indicator is given by
M_-)cJ
C
j e={10 Z
RegistmUonplane Fig. 1. The method of calculation.
if(S2-S2(z))>~O'
(6)
in other case. The charge deposition density in a given depth z is calculated by averaging the contributions over the N base trajectories.
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V. Lazurik, V. Moskvin / Nucl. Instr. and Meth. in Phys. Res. B 134 (1998) 1-12 l
N
D(z) = N Z
W~(z).
(7)
j-1
Based on the modified Method of Trajectory Rotation we can develop its modification to calculate the yield of a spectrum S(E; z) of primary electrons in a given depth z in a target. The contribution of the base trajectory Cj to the spectrum S(E; z) is defined by the derivative (do/dE), The derivative (dq~/dE) can be written in the form (dq~)
(-~)
I°/"l
(8)
where 7~ is the direction cosine of the electron motion related to z-axis. 7m is calculated for a trajectory C~,Jcreated by rotating the base trajectory Ci to bring the point in which an electron has the energy E with the registration plane z. (dE/dl)e is the stopping power of electrons with the energy E. Using Eqs. (3) and (8), an equation to calculate the contribution of the base trajectory CJ to the yield of an electron spectrum S(E; z) is obtained in a form
equals) 1. Therefore, the divergence is integrated and it can be removed by conventional procedures. The method of calculating the charge deposition density (7) depends only on a set of quantities, which describes the motion of an electron in a target. Consequently, the method can be applied to computation with the use of any Monte Carlo scheme for tracing the trajectories of electrons. When using the method to calculate the spectrum of electrons Eq. (10), electron trajectories should be traced taking into account the stopping power of electrons. Therefore, the presented method can be applied to calculations when the schemes of the condensed or the catastrophic collisions are used in Monte Carlo simulation (see [4,1 2]). The data presented in this paper have been computed using the catastrophic collision scheme. The physical model of an electron trajectory has been described in [10]. When using the Method of Trajectory Rotation, the rotation point is defined in the same way as in [10]. Namely, a point of a trajectory in which the electron has the energy Er = 2-5/32E0 has been taken as the rotation point.
wg(E;z) 3. Results and discussion
2
= 1 1 1 ~J(E)Z ]Tml~J'm(E)' (9) 2~ v/S~ S2(z) (~)E m=, where ~](E) is the rotation indicator. It equals 1 if the point of the base trajectory where the electron has the energy E intersects the plane z in rotating. It equals 0 in other case. ~J'm(E) is the region indicator. It equals 1 if all points of the part of the trajectory CmJ from E0 to E belong to the target. It equals 0 in other case. S~ and ~(z) are calculated for the trajectory point with the energy E using Eqs. (4) and (5). The value of S(E; z) is calculated by averaging the contributions over the N base trajectories 1 N
S(E;z) = ~ - - ~ W~(E;z).
(10)
j--I
Notice that, the contributions W~(z) and W~(E;z) tend to infinite as S2(z) approaches Sl. This is evident from Eqs. (3) and (9). However, a total contribution of the base trajectory CJ defined by the integral of Eq. (3) (or Eq. (9)) is less than (or
3.1. The election of the model parameters Let us define the problem of the model parameter election. This is the determination of the parameters, which should be set before Monte Carlo computation, in such a way that the data obtained in computation describe the studied physical quantity with a model uncertainty no greater than a given 6. We consider below the election of the model parameters, i.e., a cutoff energy Emir, and a depth bin width Az, for calculations of depth-profiles of charge deposition. When using the conventional Monte Carlo techniques, the model parameters Emin and Az can be elected with the use of following scheme. • A characteristic depth Rch(E0) of the variation of the charge deposition profile in a target is evaluated. • The depth bin width &z is elected. To yield the detail data on a profile, the value of Az should be a small value so that
V. Lazurik, V. Moskvin / Nucl. Instr. and Meth. in Phys. Res. B 134 (1998) 1-12
zXz <
(11)
The value of the Az is calculated taking into account the relative variation 6~ of a depth-profile in the limits of a depth bin. This relative variation can be defined by an equation Az 61 = Rob(E0) "
(12)
• The cutoff energy Emin is determined. The residual range R(Emin) of electrons is considered for this purpose. The cutoff energy can be elected using the condition R(Emin)<
(13)
It is seen from the presented algorithm that the characteristic depth Rch(E0) of the variation of a depth-profile of charge deposition in a target is a key parameter. The data on the Rch(E0) being given, the described scheme provides the Monte Carlo computation with a model uncertainty of no more than 6. fi = fil + 62.
(14)
When using the conventional Monte Carlo techniques a great computer time is required to calculate the data which are necessary to analyze the election of the model parameters. The point is that the analysis is based on a lot of data obtained with the variation of two parameters, i.e., Az and Emin. Notice that these data should be computed with a great statistical accuracy which is necessary to determine their dependence on the model parameters. When using the Method of Trajectory Rotation with the modification described in this paper, the model parameter Ernin is required only for calculations. This feature of the method allows to considerably reduce computer time. A set of data on profiles of the primary electron charge deposition in semi-infinite and infinite tar-
5
gets have been computed for various cutoff energies. Calculations were made for Emin of 0.75, 0.5, 0.2, 0.1, 0.05, and 0.02 MeV for electrons with the initial energies more than 1 Meg. 10 000 electron histories have been used in calculations. The statistical uncertainties of the obtained data are less than 6% (regarding the efficiency of the Method of Trajectory Rotation see [10]). Profiles computed for a set of the Emin are considered as a sequence of approximation profiles D ( z , Emin) of the charge deposition profile D(z) for a given atomic number of the target material and a given initial energy of electrons. As an example, the results of computation for carbon target irradiated by 2 MeV electrons are presented in Fig. 2(a). An analysis of the results of calculation shows that a set of the approximation profiles D(z, Emin) converges to D(z) for all depths z with the reduction of the cutoff energy Emin. However this convergence has peculiarities which are discussed below. Let us assume that the D(z, 0.02) differs little from a limiting profile D(z). This assumption can be used to evaluate the uncertainties in fitting the D(z) by the approximation profile D(z, Emin). It is found that the model uncertainty for a given Emin depends only slightly on a depth z in an infinite target or in deeper regions of a finite target (see Fig. 2(a)). The values of Rch(Eo) for these cases should be evaluated as the half-width of profile on its half-height. The Rch(Eo) can be calculated from the data on the charge deposition depth-profiles in semi-infinite targets. A systematic lot of data [5] or [6] can be used for this purpose. Considering a region near the surface of a finite target, it is found that the model uncertainty depends dramatically on the distance to a target boundary. Therefore, the values of the model parameters Emin and Az which are elected to compute the D(z) with a given accuracy defined by 6 have a strong dependence on z. Let us define a characteristic energy ECin(z) as a cutoff energy Emin for which the relative variation of a profile D(z, Emin) from D(z) equal to a at a given depth z. An analysis of the results shows that the D(z~Emin) approaches the D(z) with Emi~ approaches zero. Consequently, the uncertainty of the computed value of the charge deposition den-
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V. Lazurik, I4 Moskvin / Nucl. Instr. and Meth. in Phys. Res. B 134 (1998) 1-12
3
a)
E
o G)
C, E0= 2 MeV//~F..= = 0.75MeV surface
.c_ E UJ
0.5
2
0.2
E3 ¢o
1
8. l
¢D
E~
(I J~.
0
-0.2
(3
0
0.2
0.4
0.6
0.8
1
Scaled Depth z/R 0
3.2. Charge deposition near the surface oJa target
'7,
03 -~. uJ
a
0.2
o ¢/)
oQ. Q o
0.1
~_~ 2,.v t
e-
(3
0 -0.01
0
.
0.01 0,02 0.03 0.04 Scaled Depth z/R 0
that the ECin approaches zero at the surface of a target. Though, the Emi n c a n be made as small as one likes, the model uncertainty 6 of data computed for a region near the target surface with the use of a given depth bin width Az can be significant. It may considerably exceed any given statistical uncertainty which is set for computation with the use of the conventional Monte Carlo techniques. Thus the distance from the inhomogeneity in a target should be taken into account to accurate election of the model parameters Emin and Az for computation with the use of the conventional Monte Carlo techniques.
0.05
Fig. 2. (a, b) The depth-profiles D(2";Emin) of the primary electron charge deposition calculated for various cutoff energies Emm- Solid lines are the results obtained for semi-infinite targets. Dashed lines are the results obtained for infinite targets. Histogram is the data taken from [5].
sity at a given depth of a target is no greater than 6 when Emin < ECin (z). Fig. 2(b) shows that the profile D(z, Emi,) decreases drastically near the surface of a target with the cutoff energy Emin decrease. The value of the D(0, Emin) is small in comparison with the charge deposition density D~(0) in an infinite target. Thus, an analysis of the computed data shows that the value of the E~in(z) decreases drastically with the distance to a surface decrease. Hence it follows
Plane-paralM electron beams with energies from 0.1 to 10 MeV have been assumed to impinge normally on targets consisting of materials with the atomic numbers from 6 to 92. Using the modified Method of Trajectory Rotation described above, the depth-profiles of primary electron charge deposition near the surfaces of finite targets (target boundaries), semi-infinite targets and infinite targets have been computed. 5000 electron trajectories were used in calculations. The cutoff energy of electron trajectories Emin of 20 keV has been elected based on the analysis presented in the previous section. The statistical uncertainties of the obtained data are no more than 8%. Certain charge deposition profiles near the boundary of semi-infinite targets are shown in Figs. 3 and 4. A comparison of the profiles in semi-infinite targets (shown by solid lines in figures) and in the infinite targets (dashed lines) shows that the primary electron charge deposition declines rapidly with the decrease of the distance from the boundary of a semi-infinite target. It is seen from figures that the profiles computed for semi-infinite targets agree well with the data (histogram in figures) presented in [5] (or [6]) when the distance from the target boundary is greater than 0.05R0 (R0 is the CSDA range of electrons, see [131). Let us consider the small distances from a boundary, which are comparable to a depth bin width. The average values of charge deposition calculated from our data agree with the data pre-
V. Lazurik, V. Moskvin / Nucl. Instr. and Meth. in Phys. Res. B 134 (1998) 1-12
7
0.1 0.075
0.75
0.05 "T ¢:D o~ 0.025
0.5 0.25
E
E O (D
O (D ,,~ N
0
N
a
=
0.5 0
O ¢/) O Q. ~
0.5 0.25
(D
(D
o 15
¢f~
1 0.75
1
O
a
0
GI
0
(D
1.3
e0.75
lo
0.5 0.25
0 -0.02
0 -0.02 0
0.02
0.04
0.06
Scaled Depth z/R 0
0
0.02 0.04 0.06 0.08
0.1
Scaled Depth z/R o
Fig. 3. The depth-profiles of the primary electron charge deposition D(z) near the boundary of A1 target calculated for various initial energies E0 of electrons. Solid lines are the results obtained for semi-infinite targets. Dashed lines are the results obtained for infinite targets. Histograms are taken from [5].
Fig. 4. The depth-profiles of the primary electron charge deposition D(z) near the boundaries of targets consisting of Cu, Ag and Au irradiated by electrons with the initial energy 2 MeV. Solid lines are the results obtained for semi-infinite targets. Dashed lines are the results obtained for infinite targets. Histograms are taken from [5].
sented by histograms. However, the differences between the profile computed with the use of the modified Method of Trajectory Rotation and its approximation by the histogram are considerable for this region of a target. Therefore, the calculation of a profile using the data on the average density of the charge deposited in the depth bins requires the complementary information on the profile behaviour near the surface of a finite target.
A linear approximation of the data on the distribution of electron range straggling, i.e., the primary electron charge deposition, has been used to describe the profile near the boundary of a semi-infinite target in [7,8]. Fig. 5 shows the profiles of charge deposition calculated with the use of the results presented in [7] (curve 1) and [8] (curve 2). The profile computed by the Method of Trajectory Rotation is given for comparison (curve 3). As it can be seen from Fig. 5 (see also Figs. 3 and 4)
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ld Lazurik, V. Moskvin / Nucl. Instr. and Meth. in Phys. Res. B 134 (1998) 1-12
EI.r~TRON T R A N S P O R T •
5
®
Surface
1
........
4
*i
3
2
,-'..
)
AI
g
a
D
0
D z)
e-
0
CHAROE DEPOSITION D ( z )
in infmim medium
0
'
0
I
0.02
'
I
0.04
'
I
'
0.06
I
0.08
Scaled Depth z/R o
'\
D(z) - in
Fig. 5. The approximated and computed profiles of the primary electron charge deposition near the boundary of a semi-infinite target. Curve 1 and Curve 2 are the profile of charge deposition calculated with the use of the linear approximation of data computed by conventional Monte Carlo techniques. These results are presented in Refs. [7,8] respectively. Curve 3 is the profile computed by the Method of Trajectory Rotation.
the use of the linear approximation cannot lead to decrease the uncertainty in the profile description near the target boundary. What this means is that the dependence of the charge deposition density on a distance from a target surface is a considerable nonlinear function near the surface of a target. The drastic nonlinear decrease of the charge deposition density observed near the surface of a target is called the boundary effect in charge deposition. Apparently, the neglect of the boundary effect leads to contradiction between the results presented in [7,8]. Let us consider the following model to account for the charge boundary effect. Let z be a registration plane placed at a depth z in a target; Zb is a boundary of a target (see Fig. 6). The charge deposition density at a depth z in an infinite target and in a semi-infinite target are denoted by D~ (z) and D(z), respectively.
/
\ k \
sklb I
z
Itqzi,msion l ~ n © -_.
II
z -I
Z
Iz~,
Fig. 6. Theoretical model of the boundary effect in charge deposition.
During the motion of electrons from their initial energy E0 to the energy Emin, the scattering processes are growing in their importance for forming their trajectories. At some phase of the motion, electrons pass into a stage of stochastic wandering in the neighborhoods of their stopping points. It can be stated that such an energy ECin exists for which the charge deposition density has a weak dependence on Emin when Emin < ECin• Based on this statement, the quasi-equilibrium density of low-energy electrons can be supposed for a small region of an infinite target. Therefore, the D~(z) is defined by the transport of electrons before their stochastic wandering, i.e., by a macroscopic transport. Let us consider a semi-infinite target. A part of the electrons executing the stochastic wandering escapes from a target when the registration plane z is located near the target boundary Zb. Denote the probability of electron escaping from the target
V. Lazurik, V. Moskvin / Nucl. Instr. and Meth. in Phys. Res. B 134 (1998) 1-12
The quasi-equilibrium of low-energy electrons near the boundary of a semi-infinite target is disturbed. Consequently, a charge density D(z) in a semi-infinite target is less than the value of the D~(z) in an infinite target. Using the P(z, zb), the ratio of D(z) and D~(z) can be expressed by an equation: b y P ( z , Zb).
D(z) = D~(z)(1 - P(Z, Zb)).
(15)
Based on the results of analysis of the data on the profiles D~(z) and D(z) we assume that the probability P(Z, Zb) depends only on the distance from the boundary of a target, i.e.,
P(Z, Zb) = P(Iz - Zbl).
(16)
Fig. 7 shows the P(lz - Zb]) calculated using the data on the charge deposition in the semi-infinite and finite targets consisting of Au for electrons with the energy of 2 MeV and consisting of AI for electrons with the energies of 0.2, 2 and 5 MeV. It is seen from figure that P(lz - Zb]) increases drastically when the registration plane z approaches the boundary Zb. P(0) approaches one in the limiting case, i.e., when Emin approaches zero. Therefore, the charge deposition density in a surface of a target approaches zero.
.....
0.75
N a.
Eo= 2 MeV
\?q
o.s
\,~".. ~ AI" \ ~
.1o
e n
0.25
Au
A region near the target surface where the boundary of the target effects dramatically on the charge deposition distribution can be defined as a region of the boundary effect. It characterizes the region in a target where the depth-profile of charge deposition has a strong decrease. Using the P ( ] z - Zb[), a characteristic size Lb of this region can be evaluated. The value of the Lb can be defined as the distance from the surface of a target where P ( ] z - Zb]) approaches 0.5. An analysis of the computed data shows that the Lb approximately equals 0.01-0.05 of the CSDA range of electrons. When using the conventional Monte Carlo techniques for calculations of the charge deposition near the target surfaces, the charge boundary effect should be taken into account. The parameter Lb can be used as a characteristic depth Rch(Eo) of the variation of the charge deposition profile near the surface of a target (see Section 3.1). Notice that the conventional Monte Carlo calculations with the use of the any finite cutoff energy Emin and the depth bin width Az yield the finite average value of the density of the charge deposited in a first depth bin. The consideration of this value as a value of the primary electron charge deposition density in a surface of a target leads to an artifact, i.e., to a finite value of the primary electron charge deposition in a surface of a target. That contradicts the disturbance of the quasi-equilibrium of low-energy electrons near the boundary of a finite target shown below. Let us consider the charge deposition in a finite target, i.e., in a slab. We assume, as well as in discussion above, that the P(lz -Zb]) depends only on the distance from a surface of a target. This leads to the conclusion that the boundary effect near the right boundary of a slab should be observed. Denote that z* is a registration plane near the right boundary z~,. Notice that Iz* - z;I = fz - zbl,
0 0
I
I
I
I
0.02
0.04
0.06
0.08
Scaled Depth
0.1
9
(17)
so that
Iz-zd/R o P(lz* - z~,]) = P(Jz - Zb]).
Fig. 7. The probability of intersections of the interface in a target by wandering electrons as a function of the distance from their stopping points to the interface.
(18)
From here, it can be written on the basis of Eq. (15):
10
D(z*) _ D(z) D~(z*) D~(z)
I~ Lazurik, V. Moskvin / Nucl. Instr. and Meth. in Phys. Res. B 134 (1998) 1 12
1.5
(19)
Fig. 8 shows the profiles of charge deposition near the right boundary of a slab with the thickness z~ calculated using Eq. (19) (solid lines 1). For comparison, the depth-profiles of charge deposition near the right boundary of slabs (dashdot lines) and profiles in an infinite targets (dashed lines) computed by the Method of Trajectory Rotation are plotted in the figure. The curve 2 (solid lines 2) in Fig. 8 was calculated by normalizing the theoretical curve 1 to the computed value of the charge deposited in the region near the right boundary. It will be noticed that the scaled curves 2 agree well with the data calculated by the Monte Carlo technique. Whence it follows that the model outlined above can be used in analytical estimations of the depth-profiles near the right boundary of slabs with the various thicknesses. In this case, a semi-empirical scaled factor as a function of the target thickness should be used in Eq. (16). As follows from the theoretical description of the charge boundary effect, a deficit of low-energy electrons exists in the region near the surface of a target. Therefore, the influence of a boundary of a finite target should be expected in forming the low-energy part of the electron spectrum near the surface of the target. To check this, the electron spectra at various depths in a target have been computed with the use of the Method of Trajectory Rotation. Fig. 9 shows the results of computation for a target consisting of A1. It is seen from Fig. 9 that the yield of the low-energy part of the electron spectrum at a given depth z near the target surface (solid lines) is significantly less than the yield in a finite target (dashed lines). The differences between these spectra increase when the distance from the surface of a target decreases. The variations in an electron spectrum near the surface of a target should be taken into account in an analysis of radiation effects caused by electron irradiation. 4. Conclusions
The results of this paper show that the distribution of the primary electron charge deposition is
1
¢E~
0.5
1.8 N n v
N
121 e.g 0.5 ~ O Q. ID
0
Z~=0.6 R0
...............................................................................................
a
m
e-
O 0.25
0
0
~
0.02
Scaled Depth Iz*-
r
0.04
l/R0
Fig. 8. The primary electron charge deposition density D(z) near the right boundary of AI slab with thickness z~, as a function of the distance from the target-vacuum interface Iz* -z~,]. Solid lines 1 are the results of calculations with the use of the Eq. (19). Solid lines 2 are the scaled results of model calculations. D o t - d a s h e d lines are the data for the right boundaries of the slabs and dashed lines are for an infinite target calculated with the use of the Monte Carlo simulation.
very sensitive to an inhomogeneity in a target. The primary electron charge deposition density near the inhomogeneity is defined by stochastic wandering of electrons in a region near this inhomogeneity. The quasi-equilibrium of low-energy electrons near the inhomogeneity can be disturbed. Therefore, the density of the charge deposited near the target~acuum interface has a dramatic nonlin-
I~ Lazurik, V. Moskvin / Nucl. Instr. and Meth. in Phys. Res. B 134 (1998) 1-12
11
time to c o m p u t e the quantities describing the charge d e p o s i t i o n n e a r the i n h o m o g e n e i t i e s (near the interfaces). T h e j o i n t use o f the semi-empirical m e t h o d s for d e s c r i p t i o n o f the b o u n d a r y effect a n d the results o f M o n t e C a r l o c o m p u t a t i o n can be m o r e efficient for o b t a i n i n g d a t a on charge d e p o s i tion in i n h o m o g e n e o u s objects i r r a d i a t e d b y electrons.
0.1
=[ N
LU v if) E 2
0.01
Acknowledgements
4) Q.
O.
MeV
/
0.001
,
z=O
,
, ,,,,,i
0.01
, 0.1
Energy
,
, ,,,,, 1
E, M e V
Fig. 9. The electron spectra at various depths z (z = 0.2, 0.05,0.01 and 0 of the R0) in AI target irradiated by 2 MeV electrons. Solid lines are the results of computation for a semi-infinite target. Dashed lines are the results obtained for an infinite target.
W e w o u l d like to express o u r sincere t h a n k s to Prof. T a t s u o T a b a t a for his interest in o u r w o r k a n d useful discussion. A l s o we are grateful to Prof. P. A n d r e o a n d Prof. D . W . O . R o g e r s for reprints o f their p a p e r s which have helped us in f o r m i n g some general conclusions o f this p a p e r a n d to Dr. V. Y u c h t for help in English edition o f the manuscript. W e t h a n k the U k r a i n i a n Science a n d T e c h n o l o gy Center for s u p p o r t o f the p a r t o f this p a p e r (grant N 115).
References
e a r d e p e n d e n c e on the d i s t a n c e f r o m the surface o f a target ( b o u n d a r y effect in charge deposition). N o tice t h a t this leads to a drastic n o n l i n e a r decrease o f the profile describing the p r o j e c t e d r a n g e straggling n e a r the b o u n d a r y o f a semi-infinite target. W h e n using the c o n v e n t i o n a l M o n t e C a r l o techniques, the b o u n d a r y effect s h o u l d be t a k e n into a c c o u n t in the election o f the m o d e l p a r a m e ters which s h o u l d be set before M o n t e C a r l o c o m p u t a t i o n . This is necessary for an a c c u r a t e c a l c u l a t i o n o f the d e p t h - p r o f i l e o f charge d e p o s i tion n e a r an i n h o m o g e n e i t y in a target. T h e b o u n d a r y effect s h o u l d also be t a k e n into a c c o u n t in an analysis o f the results o b t a i n e d in c o m p u t a t i o n . N e g l e c t o f the m u l t i p l e intersections o f the interface in a target caused b y stochastic w a n d e r i n g o f electrons can lead to i n a c c u r a t e conclusions b a s e d on the artifacts o b t a i n e d in c o m p u t a t i o n . It will be n o t i c e d t h a t the use o f the c o n v e n t i o n al M o n t e C a r l o techniques requires c o n s i d e r a b l e
[1] J.A. Halbleib, R.P. Kensek, T.A. Mehlhorm, G. Valdez, S.M. Seltzer, M.J. Berger, ITS Version 3.0, The integrated TIGER series of coupled electron/photon Monte-Carlo transport codes. Report SAND 91-1634, Sandia Nat. Labs. (1992)/Reprinted, 1994, p.132. [2] R. Brun, F. Bruyant, M. Maire, A.L. McPherson, GEANT-3, Data handling division, Report DD/EE/84-1, CERN, 1987. [3] W.R. Nelson, H. Hirayama, D.W.O. Rogers, The EGS4 code system, Stanford Linear Accelerator Center, Report SLAC-265, Stanford, CA, 1985. [4] D.W.O. Rogers, A.F. Bielajew, Monte Carlo Techniques of Electron and Photon Transport for Radiation Dosimetry, in: K.R. Kase, B.E. Bjrngard, F.H. Attix (Eds.), The Dosimetry of Ionizing Radiation, Academic Press, New York. 1990, p. 427. [5] P. Andreo, R. Ito, T. Tabata, Tables of Charge- and Energy-Deposition Distributions in Elemental Materials Irradiated by Plane-Parallel Electron Beams with Energies between 0.1- and 100-MeV, Res. Inst. Adv. Sci. Tech. Univ. Osaka Pref. Technical Report RIAST-UOP-TP 1, 1992, p. 142. [6] T. Tabata, P. Andreo, R. Ito, Nucl. Instr. and Meth. B 94 (1994) 103.
12
V. Lazurik, ld Moskvin / Nucl. Instr. and Meth. in Phys. Res. B 134 (1998) 1 12
[7] A. Cengiz, A. Ozmutlu, Nucl. Instr. and Meth. B 84 (1994) 310. [8] P. Deb, U. Nundy, J. Phys. D 21 (1988) 763. [9] T. Tabata, P. Andreo, K. Shinoda, R. Ito, Nucl. Instr. and Meth. B 108 (1996) 11. [10] V. Lazurik, V. Moskvin, Nucl. Instr. and Meth. B 108 (1996) 276.
[11] A.F. Bielajew, D.W.O. Rogers, A.E. Nahum, Phys. Med. Biol. 30 (5) (1985) 419. [12] M.J. Berger, Methods in Computational Phys. 1 (1963) 135. [13] International Commission on Radiation Units and Measurements, Stopping Powers for Electrons and Positrons, ICRU Report 37, 1984.
Nuclear Instruments and Methods in Physics Research B 134 (1998) 1-12
A study of primary electron charge deposition near the surface of a target by Monte Carlo simulation Valentin Lazurik, Vadirn M o s k v i n * Radiation Physics Laboratory, Kharkov State University, P.O. Box 60, 310052 Kharkov, Ukraine Received 16 July 1996; received in revised form 15 September 1997
Abstract
The Method of Trajectory Rotation [V. Lazurik, V. Moskvin, Nucl. Instr. and Meth. B 108 (1996) 276] is modified to compute the charge deposition density and the yield of an electron spectrum at given depths of targets irradiated by electron beams. Primary electron charge deposition near the boundaries of finite targets is studied. It is found that the depth-profiles of the primary electron charge deposition have a drastic nonlinear decrease near the target vacuum interface. The theoretical description of the charge deposition near the inhomogeneity in a target is discussed. The election of model parameters, i.e., the cutoff energy and the depth bin width, for accurate computer simulation with the use of the conventional Monte Carlo techniques is considered. © 1998 Published by Elsevier Science B.V.
PACS: 78.70.-g; 34.50.Bw; 87.53.Fs; 81.40.Wx Keywords: Electron transport; Computer simulation; Monte Carlo techniques; Primary electrons; Charge deposition; Stochastic wandering; Target-vacuum interface; Boundary effect; Electron spectrum
1. Introduction
Data on quantities describing the action of fast electrons on materials are necessary for the analysis of the electrophysical processes in objects irradiated by electron beams. The energy and charge deposition density are the fundamental quantities used in such analysis. At present, the computer simulation of electron transport with the use of Monte Carlo techniques
*Corresponding author.
is the basic method of obtaining these data. The general purpose Monte Carlo codes, such as ITS [1], G E A N T [2] and EGS [3], are widely used in practice. What all these codes have in common is that the transport of electrons is simulated for a region of their energies from an initial energy/:~ to a given cutoff energy gmin. It is assumed that electrons deposit their charge at the end of their tracks. Tracing the trajectories of electrons until the energy Emin is connected with the limits of validity of the physical models used in computation. Thus the cutoff energy is a model parameter that defines an accuracy of simulation. Notice that
0168-583X/98/$19.00 © 1998 Published by Elsevier Science B.V. All rights reserved. P I I S O 1 6 8 - 58 3 X ( 9 7 ) O 0 5 1 0 - 7
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K Lazurik, V. Moskvin / Nucl. Instr. and Meth. in Phys. Res. B 134 (1998) 1 12
the methods of the election of the cutoff energy for calculating the energy deposition distributions are known (see [4]). When using the conventional Monte Carlo techniques, the charge deposition density at a given depth of a target cannot be computed as such. The target is divided into a set of depth bins. The charge deposition density at a given depth z is evaluated on the number of electrons stopped in a depth bin with a given width Az. As a consequence, the result of computation is the average density of the charge deposited in this depth bin. The depth-profile of the charge deposition (also called the charge deposition distribution) is fitted using the data on the average density of the deposited charge computed for a set of depth bins. The depth bin width Az is a model parameter, whose value should be set before computation. Thus, the accuracy of data obtained by the Monte Carlo simulation depends on the values of model parameters, i.e., the energy cutoff and the depth bin width. The general method of the election of these parameters for the charge deposition computation has not been developed. At present, a systematic set of the data on depth-profiles of charge deposition in semi-infinite targets has been computed recently (see [5,6]). The depth-profiles of the primary electrons charge deposition can also be calculated using the data on projected range straggling of fast electrons presented in [7-9]. Besides, data on the primary electron charge deposition in slabs are available (see [10]). The data presented in the cited papers have been computed using the conventional Monte Carlo techniques. The model parameters, both the energy cutoff and the depth bin width, were elected for these calculations in the same way as which commonly used for calculations of the energy deposition. The validity of this election is not analyzed thoroughly in these papers. From the results described in [5,6] it follows that the total charge deposition has peculiarities in the region near the target-vacuum interface, i.e., near the boundary of a target. The total charge deposition density varies sharply near the target-vacuum interface and, moreover, the sign of the total charge deposited in this region can be changed. It is agreed that these peculiarities
are associated essentially with escaping knock-on electrons (secondary electrons) in a vacuum. However, the results presented in [10] suggest that the density of the primary electron charge deposition is subject to strong variation near the boundary of a target. The widths of depth bins used in computations presented in [5-10] are comparable to a characteristic size of the region where the total charge deposition changes drastically. Therefore, the available data are inadequate to determine uniquely the role of primary electrons in forming the profiles of the total charge deposition near the boundaries of finite targets. When using the data on the average density of the charge deposited in depth bins, further theoretical assumptions are necessary for fitting the depth-profiles of the primary electrons charge deposition. The use of simple approaches for fitting the profiles near the boundaries of semi-infinite targets cannot lead to advance. Moreover, the contradiction between the results presented in [7,8] was caused by such simple approaches used in the analysis of the results of computations. It seems likely that the improper handling of algorithms in studying charge deposition, along with the energy deposition calculations, can lead to anomalous results known as "interface artifacts" (see [11]). The aim of this paper is the study of the primary electron charge deposition near the targetvacuum interface and the analysis of the election of model parameters, i.e., the cutoff energy and the depth bin width, to compute data on charge deposition with the use of the conventional Monte Carlo techniques.
2. Method of calculations
Let us assume that a slab target consisting of a uniform material is placed in a vacuum. An electron beam with energy E0 impinges normally on the target surface (called here the left boundary) at a point (0, 0, 0). The z-axis is the normal to the surface. To calculate the charge deposition density at given depths of a target, we modified the Method of Trajectory Rotation. This modification is discussed below.
V. Lazurik, E Moskvin I Nucl. Instr. and Meth. in Phys. Res. B 134 (1998) 1-12
3
Let us begin by considering some features of the Method of Trajectory Rotation presented in [10]. Using the Monte Carlo procedure, an electron trajectory (here called the base trajectory) Cj is traced in an infinite medium. The trajectory is simulated from the initial energy E0 to a given cutoff energy Emin. The point of the trajectory where electron has a given energy E~ is taken as a rotation point. The direction of the electron motion in this point is taken as a rotation axis. The part of the base trajectory Cj, where electron has the energy E < Er is selected. A continuous set of trajectories is created by rotating this part of the base trajectory around the rotation axis (see Fig. 1). Assume that a layer with the thickness Az is placed at the depth z in a target. This layer is called a depth bin. The contribution of the base trajectory Cj to the average density of the charge deposited in the depth bin is given by an equation
the energy Emi n intersects the plane z in rotating, i.e., the end of the trajectory CJ can be placed in a given depth bin. It equals 0 in other case. ~J" is the region indicator. It equals 1 if all points of the trajectory C~j defined by rotating the base trajectory CJ belong to the target and it equals 0 in other case. m is the index to account for the dual intersection of the plane z by the base trajectory in rotating. Let us modify the method of trajectory rotation to calculate the charge deposition density at a given depth z in a target. To do this, we suppose that the depth bin width Az approaches zero. Going to a limiting case, the ratio (A~o/kz) converts to the derivative (d(o/dz). The derivative (dq~/dz) is derived from a formula
(1)
where Zr is the z-coordinate of the rotation point (i.e., the trajectory point with the energy Er), /¢ the radius-vector from the rotation point to the end of a traject~y, h'~ the rotational axis defined by the direction ~¢'~r~--- (~r, fir, ~2r) of electron motion in the rotation point, and (/~, ff~) the dot product of R and ff~o. Using the (d~o/dz) in an analytical form, Eq. (1) can be recast for the modified Method of Trajectory Rotation. The contribution of the base trajectory CJ to the density of electrons with energy Emi, at a depth z is given by
1 ~ zMp W~(z)=~
(.Z~j,.. ' .,=,
where ~J is the rotation indicator. It equals 1 if the point of the base trajectory where the electron has
Target surface
Target region
z = zr + ~r(R, ~ )
-~ ¢ ( R 2 -- (*~, nq~)2)(1 -- 7r) 2 COS q),
W~(z)=
1
1
~:3--,2 .
2re v/S~ _ $22(z) ~m=,
(2)
(3)
where
}//J\\ .
Eo ".,/ ._
\
S, = ¢ ( R : - (R, ff~,)2)(1 - yr)2,
(4)
S2(z ) : Z -- (Zr q- yr(/~, h'q~))
(5)
and the rotation indicator is given by
M_-)cJ
C
j e={10 Z
RegistmUonplane Fig. 1. The method of calculation.
if(S2-S2(z))>~O'
(6)
in other case. The charge deposition density in a given depth z is calculated by averaging the contributions over the N base trajectories.
4
V. Lazurik, V. Moskvin / Nucl. Instr. and Meth. in Phys. Res. B 134 (1998) 1-12 l
N
D(z) = N Z
W~(z).
(7)
j-1
Based on the modified Method of Trajectory Rotation we can develop its modification to calculate the yield of a spectrum S(E; z) of primary electrons in a given depth z in a target. The contribution of the base trajectory Cj to the spectrum S(E; z) is defined by the derivative (do/dE), The derivative (dq~/dE) can be written in the form (dq~)
(-~)
I°/"l
(8)
where 7~ is the direction cosine of the electron motion related to z-axis. 7m is calculated for a trajectory C~,Jcreated by rotating the base trajectory Ci to bring the point in which an electron has the energy E with the registration plane z. (dE/dl)e is the stopping power of electrons with the energy E. Using Eqs. (3) and (8), an equation to calculate the contribution of the base trajectory CJ to the yield of an electron spectrum S(E; z) is obtained in a form
equals) 1. Therefore, the divergence is integrated and it can be removed by conventional procedures. The method of calculating the charge deposition density (7) depends only on a set of quantities, which describes the motion of an electron in a target. Consequently, the method can be applied to computation with the use of any Monte Carlo scheme for tracing the trajectories of electrons. When using the method to calculate the spectrum of electrons Eq. (10), electron trajectories should be traced taking into account the stopping power of electrons. Therefore, the presented method can be applied to calculations when the schemes of the condensed or the catastrophic collisions are used in Monte Carlo simulation (see [4,1 2]). The data presented in this paper have been computed using the catastrophic collision scheme. The physical model of an electron trajectory has been described in [10]. When using the Method of Trajectory Rotation, the rotation point is defined in the same way as in [10]. Namely, a point of a trajectory in which the electron has the energy Er = 2-5/32E0 has been taken as the rotation point.
wg(E;z) 3. Results and discussion
2
= 1 1 1 ~J(E)Z ]Tml~J'm(E)' (9) 2~ v/S~ S2(z) (~)E m=, where ~](E) is the rotation indicator. It equals 1 if the point of the base trajectory where the electron has the energy E intersects the plane z in rotating. It equals 0 in other case. ~J'm(E) is the region indicator. It equals 1 if all points of the part of the trajectory CmJ from E0 to E belong to the target. It equals 0 in other case. S~ and ~(z) are calculated for the trajectory point with the energy E using Eqs. (4) and (5). The value of S(E; z) is calculated by averaging the contributions over the N base trajectories 1 N
S(E;z) = ~ - - ~ W~(E;z).
(10)
j--I
Notice that, the contributions W~(z) and W~(E;z) tend to infinite as S2(z) approaches Sl. This is evident from Eqs. (3) and (9). However, a total contribution of the base trajectory CJ defined by the integral of Eq. (3) (or Eq. (9)) is less than (or
3.1. The election of the model parameters Let us define the problem of the model parameter election. This is the determination of the parameters, which should be set before Monte Carlo computation, in such a way that the data obtained in computation describe the studied physical quantity with a model uncertainty no greater than a given 6. We consider below the election of the model parameters, i.e., a cutoff energy Emir, and a depth bin width Az, for calculations of depth-profiles of charge deposition. When using the conventional Monte Carlo techniques, the model parameters Emin and Az can be elected with the use of following scheme. • A characteristic depth Rch(E0) of the variation of the charge deposition profile in a target is evaluated. • The depth bin width &z is elected. To yield the detail data on a profile, the value of Az should be a small value so that
V. Lazurik, V. Moskvin / Nucl. Instr. and Meth. in Phys. Res. B 134 (1998) 1-12
zXz <
(11)
The value of the Az is calculated taking into account the relative variation 6~ of a depth-profile in the limits of a depth bin. This relative variation can be defined by an equation Az 61 = Rob(E0) "
(12)
• The cutoff energy Emin is determined. The residual range R(Emin) of electrons is considered for this purpose. The cutoff energy can be elected using the condition R(Emin)<
(13)
It is seen from the presented algorithm that the characteristic depth Rch(E0) of the variation of a depth-profile of charge deposition in a target is a key parameter. The data on the Rch(E0) being given, the described scheme provides the Monte Carlo computation with a model uncertainty of no more than 6. fi = fil + 62.
(14)
When using the conventional Monte Carlo techniques a great computer time is required to calculate the data which are necessary to analyze the election of the model parameters. The point is that the analysis is based on a lot of data obtained with the variation of two parameters, i.e., Az and Emin. Notice that these data should be computed with a great statistical accuracy which is necessary to determine their dependence on the model parameters. When using the Method of Trajectory Rotation with the modification described in this paper, the model parameter Ernin is required only for calculations. This feature of the method allows to considerably reduce computer time. A set of data on profiles of the primary electron charge deposition in semi-infinite and infinite tar-
5
gets have been computed for various cutoff energies. Calculations were made for Emin of 0.75, 0.5, 0.2, 0.1, 0.05, and 0.02 MeV for electrons with the initial energies more than 1 Meg. 10 000 electron histories have been used in calculations. The statistical uncertainties of the obtained data are less than 6% (regarding the efficiency of the Method of Trajectory Rotation see [10]). Profiles computed for a set of the Emin are considered as a sequence of approximation profiles D ( z , Emin) of the charge deposition profile D(z) for a given atomic number of the target material and a given initial energy of electrons. As an example, the results of computation for carbon target irradiated by 2 MeV electrons are presented in Fig. 2(a). An analysis of the results of calculation shows that a set of the approximation profiles D(z, Emin) converges to D(z) for all depths z with the reduction of the cutoff energy Emin. However this convergence has peculiarities which are discussed below. Let us assume that the D(z, 0.02) differs little from a limiting profile D(z). This assumption can be used to evaluate the uncertainties in fitting the D(z) by the approximation profile D(z, Emin). It is found that the model uncertainty for a given Emin depends only slightly on a depth z in an infinite target or in deeper regions of a finite target (see Fig. 2(a)). The values of Rch(Eo) for these cases should be evaluated as the half-width of profile on its half-height. The Rch(Eo) can be calculated from the data on the charge deposition depth-profiles in semi-infinite targets. A systematic lot of data [5] or [6] can be used for this purpose. Considering a region near the surface of a finite target, it is found that the model uncertainty depends dramatically on the distance to a target boundary. Therefore, the values of the model parameters Emin and Az which are elected to compute the D(z) with a given accuracy defined by 6 have a strong dependence on z. Let us define a characteristic energy ECin(z) as a cutoff energy Emin for which the relative variation of a profile D(z, Emin) from D(z) equal to a at a given depth z. An analysis of the results shows that the D(z~Emin) approaches the D(z) with Emi~ approaches zero. Consequently, the uncertainty of the computed value of the charge deposition den-
6
V. Lazurik, I4 Moskvin / Nucl. Instr. and Meth. in Phys. Res. B 134 (1998) 1-12
3
a)
E
o G)
C, E0= 2 MeV//~F..= = 0.75MeV surface
.c_ E UJ
0.5
2
0.2
E3 ¢o
1
8. l
¢D
E~
(I J~.
0
-0.2
(3
0
0.2
0.4
0.6
0.8
1
Scaled Depth z/R 0
3.2. Charge deposition near the surface oJa target
'7,
03 -~. uJ
a
0.2
o ¢/)
oQ. Q o
0.1
~_~ 2,.v t
e-
(3
0 -0.01
0
.
0.01 0,02 0.03 0.04 Scaled Depth z/R 0
that the ECin approaches zero at the surface of a target. Though, the Emi n c a n be made as small as one likes, the model uncertainty 6 of data computed for a region near the target surface with the use of a given depth bin width Az can be significant. It may considerably exceed any given statistical uncertainty which is set for computation with the use of the conventional Monte Carlo techniques. Thus the distance from the inhomogeneity in a target should be taken into account to accurate election of the model parameters Emin and Az for computation with the use of the conventional Monte Carlo techniques.
0.05
Fig. 2. (a, b) The depth-profiles D(2";Emin) of the primary electron charge deposition calculated for various cutoff energies Emm- Solid lines are the results obtained for semi-infinite targets. Dashed lines are the results obtained for infinite targets. Histogram is the data taken from [5].
sity at a given depth of a target is no greater than 6 when Emin < ECin (z). Fig. 2(b) shows that the profile D(z, Emi,) decreases drastically near the surface of a target with the cutoff energy Emin decrease. The value of the D(0, Emin) is small in comparison with the charge deposition density D~(0) in an infinite target. Thus, an analysis of the computed data shows that the value of the E~in(z) decreases drastically with the distance to a surface decrease. Hence it follows
Plane-paralM electron beams with energies from 0.1 to 10 MeV have been assumed to impinge normally on targets consisting of materials with the atomic numbers from 6 to 92. Using the modified Method of Trajectory Rotation described above, the depth-profiles of primary electron charge deposition near the surfaces of finite targets (target boundaries), semi-infinite targets and infinite targets have been computed. 5000 electron trajectories were used in calculations. The cutoff energy of electron trajectories Emin of 20 keV has been elected based on the analysis presented in the previous section. The statistical uncertainties of the obtained data are no more than 8%. Certain charge deposition profiles near the boundary of semi-infinite targets are shown in Figs. 3 and 4. A comparison of the profiles in semi-infinite targets (shown by solid lines in figures) and in the infinite targets (dashed lines) shows that the primary electron charge deposition declines rapidly with the decrease of the distance from the boundary of a semi-infinite target. It is seen from figures that the profiles computed for semi-infinite targets agree well with the data (histogram in figures) presented in [5] (or [6]) when the distance from the target boundary is greater than 0.05R0 (R0 is the CSDA range of electrons, see [131). Let us consider the small distances from a boundary, which are comparable to a depth bin width. The average values of charge deposition calculated from our data agree with the data pre-
V. Lazurik, V. Moskvin / Nucl. Instr. and Meth. in Phys. Res. B 134 (1998) 1-12
7
0.1 0.075
0.75
0.05 "T ¢:D o~ 0.025
0.5 0.25
E
E O (D
O (D ,,~ N
0
N
a
=
0.5 0
O ¢/) O Q. ~
0.5 0.25
(D
(D
o 15
¢f~
1 0.75
1
O
a
0
GI
0
(D
1.3
e0.75
lo
0.5 0.25
0 -0.02
0 -0.02 0
0.02
0.04
0.06
Scaled Depth z/R 0
0
0.02 0.04 0.06 0.08
0.1
Scaled Depth z/R o
Fig. 3. The depth-profiles of the primary electron charge deposition D(z) near the boundary of A1 target calculated for various initial energies E0 of electrons. Solid lines are the results obtained for semi-infinite targets. Dashed lines are the results obtained for infinite targets. Histograms are taken from [5].
Fig. 4. The depth-profiles of the primary electron charge deposition D(z) near the boundaries of targets consisting of Cu, Ag and Au irradiated by electrons with the initial energy 2 MeV. Solid lines are the results obtained for semi-infinite targets. Dashed lines are the results obtained for infinite targets. Histograms are taken from [5].
sented by histograms. However, the differences between the profile computed with the use of the modified Method of Trajectory Rotation and its approximation by the histogram are considerable for this region of a target. Therefore, the calculation of a profile using the data on the average density of the charge deposited in the depth bins requires the complementary information on the profile behaviour near the surface of a finite target.
A linear approximation of the data on the distribution of electron range straggling, i.e., the primary electron charge deposition, has been used to describe the profile near the boundary of a semi-infinite target in [7,8]. Fig. 5 shows the profiles of charge deposition calculated with the use of the results presented in [7] (curve 1) and [8] (curve 2). The profile computed by the Method of Trajectory Rotation is given for comparison (curve 3). As it can be seen from Fig. 5 (see also Figs. 3 and 4)
8
ld Lazurik, V. Moskvin / Nucl. Instr. and Meth. in Phys. Res. B 134 (1998) 1-12
EI.r~TRON T R A N S P O R T •
5
®
Surface
1
........
4
*i
3
2
,-'..
)
AI
g
a
D
0
D z)
e-
0
CHAROE DEPOSITION D ( z )
in infmim medium
0
'
0
I
0.02
'
I
0.04
'
I
'
0.06
I
0.08
Scaled Depth z/R o
'\
D(z) - in
Fig. 5. The approximated and computed profiles of the primary electron charge deposition near the boundary of a semi-infinite target. Curve 1 and Curve 2 are the profile of charge deposition calculated with the use of the linear approximation of data computed by conventional Monte Carlo techniques. These results are presented in Refs. [7,8] respectively. Curve 3 is the profile computed by the Method of Trajectory Rotation.
the use of the linear approximation cannot lead to decrease the uncertainty in the profile description near the target boundary. What this means is that the dependence of the charge deposition density on a distance from a target surface is a considerable nonlinear function near the surface of a target. The drastic nonlinear decrease of the charge deposition density observed near the surface of a target is called the boundary effect in charge deposition. Apparently, the neglect of the boundary effect leads to contradiction between the results presented in [7,8]. Let us consider the following model to account for the charge boundary effect. Let z be a registration plane placed at a depth z in a target; Zb is a boundary of a target (see Fig. 6). The charge deposition density at a depth z in an infinite target and in a semi-infinite target are denoted by D~ (z) and D(z), respectively.
/
\ k \
sklb I
z
Itqzi,msion l ~ n © -_.
II
z -I
Z
Iz~,
Fig. 6. Theoretical model of the boundary effect in charge deposition.
During the motion of electrons from their initial energy E0 to the energy Emin, the scattering processes are growing in their importance for forming their trajectories. At some phase of the motion, electrons pass into a stage of stochastic wandering in the neighborhoods of their stopping points. It can be stated that such an energy ECin exists for which the charge deposition density has a weak dependence on Emin when Emin < ECin• Based on this statement, the quasi-equilibrium density of low-energy electrons can be supposed for a small region of an infinite target. Therefore, the D~(z) is defined by the transport of electrons before their stochastic wandering, i.e., by a macroscopic transport. Let us consider a semi-infinite target. A part of the electrons executing the stochastic wandering escapes from a target when the registration plane z is located near the target boundary Zb. Denote the probability of electron escaping from the target
V. Lazurik, V. Moskvin / Nucl. Instr. and Meth. in Phys. Res. B 134 (1998) 1-12
The quasi-equilibrium of low-energy electrons near the boundary of a semi-infinite target is disturbed. Consequently, a charge density D(z) in a semi-infinite target is less than the value of the D~(z) in an infinite target. Using the P(z, zb), the ratio of D(z) and D~(z) can be expressed by an equation: b y P ( z , Zb).
D(z) = D~(z)(1 - P(Z, Zb)).
(15)
Based on the results of analysis of the data on the profiles D~(z) and D(z) we assume that the probability P(Z, Zb) depends only on the distance from the boundary of a target, i.e.,
P(Z, Zb) = P(Iz - Zbl).
(16)
Fig. 7 shows the P(lz - Zb]) calculated using the data on the charge deposition in the semi-infinite and finite targets consisting of Au for electrons with the energy of 2 MeV and consisting of AI for electrons with the energies of 0.2, 2 and 5 MeV. It is seen from figure that P(lz - Zb]) increases drastically when the registration plane z approaches the boundary Zb. P(0) approaches one in the limiting case, i.e., when Emin approaches zero. Therefore, the charge deposition density in a surface of a target approaches zero.
.....
0.75
N a.
Eo= 2 MeV
\?q
o.s
\,~".. ~ AI" \ ~
.1o
e n
0.25
Au
A region near the target surface where the boundary of the target effects dramatically on the charge deposition distribution can be defined as a region of the boundary effect. It characterizes the region in a target where the depth-profile of charge deposition has a strong decrease. Using the P ( ] z - Zb[), a characteristic size Lb of this region can be evaluated. The value of the Lb can be defined as the distance from the surface of a target where P ( ] z - Zb]) approaches 0.5. An analysis of the computed data shows that the Lb approximately equals 0.01-0.05 of the CSDA range of electrons. When using the conventional Monte Carlo techniques for calculations of the charge deposition near the target surfaces, the charge boundary effect should be taken into account. The parameter Lb can be used as a characteristic depth Rch(Eo) of the variation of the charge deposition profile near the surface of a target (see Section 3.1). Notice that the conventional Monte Carlo calculations with the use of the any finite cutoff energy Emin and the depth bin width Az yield the finite average value of the density of the charge deposited in a first depth bin. The consideration of this value as a value of the primary electron charge deposition density in a surface of a target leads to an artifact, i.e., to a finite value of the primary electron charge deposition in a surface of a target. That contradicts the disturbance of the quasi-equilibrium of low-energy electrons near the boundary of a finite target shown below. Let us consider the charge deposition in a finite target, i.e., in a slab. We assume, as well as in discussion above, that the P(lz -Zb]) depends only on the distance from a surface of a target. This leads to the conclusion that the boundary effect near the right boundary of a slab should be observed. Denote that z* is a registration plane near the right boundary z~,. Notice that Iz* - z;I = fz - zbl,
0 0
I
I
I
I
0.02
0.04
0.06
0.08
Scaled Depth
0.1
9
(17)
so that
Iz-zd/R o P(lz* - z~,]) = P(Jz - Zb]).
Fig. 7. The probability of intersections of the interface in a target by wandering electrons as a function of the distance from their stopping points to the interface.
(18)
From here, it can be written on the basis of Eq. (15):
10
D(z*) _ D(z) D~(z*) D~(z)
I~ Lazurik, V. Moskvin / Nucl. Instr. and Meth. in Phys. Res. B 134 (1998) 1 12
1.5
(19)
Fig. 8 shows the profiles of charge deposition near the right boundary of a slab with the thickness z~ calculated using Eq. (19) (solid lines 1). For comparison, the depth-profiles of charge deposition near the right boundary of slabs (dashdot lines) and profiles in an infinite targets (dashed lines) computed by the Method of Trajectory Rotation are plotted in the figure. The curve 2 (solid lines 2) in Fig. 8 was calculated by normalizing the theoretical curve 1 to the computed value of the charge deposited in the region near the right boundary. It will be noticed that the scaled curves 2 agree well with the data calculated by the Monte Carlo technique. Whence it follows that the model outlined above can be used in analytical estimations of the depth-profiles near the right boundary of slabs with the various thicknesses. In this case, a semi-empirical scaled factor as a function of the target thickness should be used in Eq. (16). As follows from the theoretical description of the charge boundary effect, a deficit of low-energy electrons exists in the region near the surface of a target. Therefore, the influence of a boundary of a finite target should be expected in forming the low-energy part of the electron spectrum near the surface of the target. To check this, the electron spectra at various depths in a target have been computed with the use of the Method of Trajectory Rotation. Fig. 9 shows the results of computation for a target consisting of A1. It is seen from Fig. 9 that the yield of the low-energy part of the electron spectrum at a given depth z near the target surface (solid lines) is significantly less than the yield in a finite target (dashed lines). The differences between these spectra increase when the distance from the surface of a target decreases. The variations in an electron spectrum near the surface of a target should be taken into account in an analysis of radiation effects caused by electron irradiation. 4. Conclusions
The results of this paper show that the distribution of the primary electron charge deposition is
1
¢E~
0.5
1.8 N n v
N
121 e.g 0.5 ~ O Q. ID
0
Z~=0.6 R0
...............................................................................................
a
m
e-
O 0.25
0
0
~
0.02
Scaled Depth Iz*-
r
0.04
l/R0
Fig. 8. The primary electron charge deposition density D(z) near the right boundary of AI slab with thickness z~, as a function of the distance from the target-vacuum interface Iz* -z~,]. Solid lines 1 are the results of calculations with the use of the Eq. (19). Solid lines 2 are the scaled results of model calculations. D o t - d a s h e d lines are the data for the right boundaries of the slabs and dashed lines are for an infinite target calculated with the use of the Monte Carlo simulation.
very sensitive to an inhomogeneity in a target. The primary electron charge deposition density near the inhomogeneity is defined by stochastic wandering of electrons in a region near this inhomogeneity. The quasi-equilibrium of low-energy electrons near the inhomogeneity can be disturbed. Therefore, the density of the charge deposited near the target~acuum interface has a dramatic nonlin-
I~ Lazurik, V. Moskvin / Nucl. Instr. and Meth. in Phys. Res. B 134 (1998) 1-12
11
time to c o m p u t e the quantities describing the charge d e p o s i t i o n n e a r the i n h o m o g e n e i t i e s (near the interfaces). T h e j o i n t use o f the semi-empirical m e t h o d s for d e s c r i p t i o n o f the b o u n d a r y effect a n d the results o f M o n t e C a r l o c o m p u t a t i o n can be m o r e efficient for o b t a i n i n g d a t a on charge d e p o s i tion in i n h o m o g e n e o u s objects i r r a d i a t e d b y electrons.
0.1
=[ N
LU v if) E 2
0.01
Acknowledgements
4) Q.
O.
MeV
/
0.001
,
z=O
,
, ,,,,,i
0.01
, 0.1
Energy
,
, ,,,,, 1
E, M e V
Fig. 9. The electron spectra at various depths z (z = 0.2, 0.05,0.01 and 0 of the R0) in AI target irradiated by 2 MeV electrons. Solid lines are the results of computation for a semi-infinite target. Dashed lines are the results obtained for an infinite target.
W e w o u l d like to express o u r sincere t h a n k s to Prof. T a t s u o T a b a t a for his interest in o u r w o r k a n d useful discussion. A l s o we are grateful to Prof. P. A n d r e o a n d Prof. D . W . O . R o g e r s for reprints o f their p a p e r s which have helped us in f o r m i n g some general conclusions o f this p a p e r a n d to Dr. V. Y u c h t for help in English edition o f the manuscript. W e t h a n k the U k r a i n i a n Science a n d T e c h n o l o gy Center for s u p p o r t o f the p a r t o f this p a p e r (grant N 115).
References
e a r d e p e n d e n c e on the d i s t a n c e f r o m the surface o f a target ( b o u n d a r y effect in charge deposition). N o tice t h a t this leads to a drastic n o n l i n e a r decrease o f the profile describing the p r o j e c t e d r a n g e straggling n e a r the b o u n d a r y o f a semi-infinite target. W h e n using the c o n v e n t i o n a l M o n t e C a r l o techniques, the b o u n d a r y effect s h o u l d be t a k e n into a c c o u n t in the election o f the m o d e l p a r a m e ters which s h o u l d be set before M o n t e C a r l o c o m p u t a t i o n . This is necessary for an a c c u r a t e c a l c u l a t i o n o f the d e p t h - p r o f i l e o f charge d e p o s i tion n e a r an i n h o m o g e n e i t y in a target. T h e b o u n d a r y effect s h o u l d also be t a k e n into a c c o u n t in an analysis o f the results o b t a i n e d in c o m p u t a t i o n . N e g l e c t o f the m u l t i p l e intersections o f the interface in a target caused b y stochastic w a n d e r i n g o f electrons can lead to i n a c c u r a t e conclusions b a s e d on the artifacts o b t a i n e d in c o m p u t a t i o n . It will be n o t i c e d t h a t the use o f the c o n v e n t i o n al M o n t e C a r l o techniques requires c o n s i d e r a b l e
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