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Current Applied Physics 14 (2014) 132e138

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Current Applied Physics

journal homepage: www.elsevier .com/locate/cap

Computational method for calculating geometric factors ofinstruments detecting charged particles in the 5e500 keV energyrange with deflecting electric field

S. Park, J.H. Jeon, Y. Kim, J. Woo, J. Seon*

School of Space Research, Kyung Hee University, 1, Seocheon-dong, Giheung-gu, Yong-In, Gyeong-Gi 446-701, Republic of Korea

a r t i c l e i n f o

Article history:Received 20 April 2013Received in revised form8 October 2013Accepted 10 October 2013Available online 30 October 2013

Keywords:Geometric factorInstrument responseCharged particle detectorGeant4

* Corresponding author. Tel.: þ82 31 201 3282; faxE-mail address: jhseon@khu.ac.kr (J. Seon).

1567-1739/$ e see front matter � 2013 Elsevier B.V.http://dx.doi.org/10.1016/j.cap.2013.10.013

a b s t r a c t

A computational method for calculating the geometric factors of an instrument detecting charged par-ticles in the energy range of about 5e500 keV is presented. The method takes into account the presenceof electric or magnetic fields that are intentionally generated to clearly separate electrons from positiveions. The method first solves the distribution of electric or magnetic fields near the detectors, and thencalculates the trajectory of scattered and unscattered charged particles under the influence of thecalculated fields. The propagation of charged particles through the fields, their interaction with the in-struments, and energy deposition into the detectors are calculated with Geant4, whereas the electric ormagnetic fields are solved with SIMION. To geometrically model the shielding distribution of the in-strument, a novel method is introduced for interfacing with the sophisticated mechanical designsavailable from computer-aided design tools. A description of this computational method is provided,along with the results for a representative example. The calculation applied to the example clearlydemonstrates the necessity of proper accounting of interaction mechanisms such as scattering or sec-ondary emission. This procedure will demonstrate a precise method for calculating the geometric factorthat allows estimation of the fluxes of incident charged particles.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

In general, the counting rate of any detector that responds to theincident fluxes of charged particles critically depends on theeffective dimensions of the detector facing a certain solid angle inspace, as well as the efficiency of the detectors as a function ofparticle energy. The general expression by Sullivan [1] gives thefollowing relation for the counting rate from the detector:

C ¼ 1T

Zt0þT

t0

ZS

ZU

ZE

Xa

εaðE;s;u; tÞ

� JaðE;s;u; tÞdtðds$brÞdudE; (1)

where C ¼ counting rate (counts/s), T¼ duration of the observationstarting from time t ¼ t0, εa ¼ detection efficiency for the ath par-ticle species, Ja ¼ differential particle flux of the ath particle species[cm�2 s�1 sr�1 E�1], ds ¼ element of surface area of the detector

: þ82 31 204 2445.

All rights reserved.

[cm2], u ¼ element of solid angle [sr], br ¼ unit vector in the di-rection u, and E ¼ energy of the particles [keV], respectively. Theintegration corresponds to all relevant domains of S, U, and E,which respectively represent the surface area of the detectors, solidangles, and particle energies.

In the case of ideal detector responses, for which εa remainsconstant over the domain of u, a and t, i.e., εa ¼ εa(E), and time-independent isotropic incoming particle fluxes Ja ¼ J0(E), theexpression for the detector counting rates reduces to the followingsimpler form:

C ¼ZE

εðEÞ

264 Z

S

ds$br$ ZU

du

375J0ðEÞdE: (2)

The bracketed quantity in Equation (2) often depends only onthe geometry of the detector relative to the guiding structure of theinstruments, such as a collimator, aperture, or baffle, and isconventionally defined as the geometric factor. For circular orrectangular apertures, the geometric factor has been previouslycalculated with the assumption that particles pass through theinstruments on a straight trajectory [1e3]. To properly interpret thecollected measurements, it is important to account for the

Fig. 1. Procedure for geometric field factor calculation. The CAD instrument design isconverted to an appropriate format to solve electromagnetic fields near the instru-ment. The non-uniform electric field obtained from the solver is used for trajectorycalculation in Geant4. The CAD file is also converted for Monte Carlo calculation ofgeometric factors in Geant4.

S. Park et al. / Current Applied Physics 14 (2014) 132e138 133

scattering of electrons off the structures of the instruments or thedetectors. Laboratory experiments on monoenergetic electron ir-radiations and the subsequent detection of electrons by semi-conductors [4e7] strongly indicate that backscattering of electronsfrom the collimators and detectors must be considered for properdata analysis. If significant scatterings indeed occur, this rendersinvalid the geometrical factor previously obtained assumingstraight particle trajectories. The scattering also induces un-certainties in the energy measurements because only a fraction ofincident electron energies may be deposited within the detectors,yielding a long low-energy wing in the distribution of themeasured electron energies.

Modern in situ observations of charged particles in space havefound that the space environment in the vicinity of the Earth isfilled with charged particles, often trapped by Earth’s magneticfields [8]. Later observations further showed that these regions ofaccessible space are dynamically filled with charged particles thatare diverse in energy [9], species [10] and origin [11]. Numerousobservations have been made, but the clear determination of en-ergies from electrons and positive ions still remains a challengingtask, even in recent spaceborne experiments. For example,contamination of electron measurements by protons [12,13] andcontamination of protonmeasurements by electrons [14] have bothbeen reported in the energy range of the present study. Contami-nation from electrons can be partially reduced by applying appro-priate magnetic fields [15] or electric fields during the passage ofthe particles from the aperture to the detector. On the other hand,contamination from protons is often mitigated by applying a thickdead layer over the active volume of detectors, a strategy madepossible by the greater stopping powers of protons and heavy ionsrelative to electrons (See, for example [16]).

Therefore, any numerical method to calculate the geometricfactor of modern instruments should include 1) direct acceptanceof mechanical drawings of the instruments for precise modelingand processing of actual instrument designs, 2) consideration of theeffect of particle scatterings off the considered instrument, 3)consideration of any applied electromagnetic fields used to segre-gate positive ions and electrons, 4) calculation of particle trajectoryacross the applied fields, and lastly 5) calculation of energy depo-sition within the detectors. There have been several investigationsthat utilized various numerical methods to calculate the geomet-rical factors of instruments operating in energy ranges near that ofthe present study [17e22], but to our understanding, none of theseinvestigations has satisfied all the aforementioned requirementssimultaneously. The purpose of this paper is to describe a numericalmethod that simultaneously satisfies such requirements. The nu-merical method is given in Section 2; Section 3 applies the methodto a representative instrument design. Section 4 presents ourconclusions.

2. Numerical method

In the present work, Geant4 (Version 9.3) [23] with PENELOPEphysics model was used to calculate the geometric factor. Thephysics model includes Compton scattering, photoelectric effectand bremsstrahlung processes in the energy range of the presentstudy. Geant4 provides two ways to model instrumental data foruse in simulations: one is to manually input the instrument ge-ometry within the Geant4 simulation code, and the other is to inputa file containing the instrument’s geometrical information in theGeometrical Description Markup Language (GDML) format, aformat based on Extensible Markup Language (XML) [24]. It isdesirable to find the method to adapt the latter because the in-strument design is often carried out with a computer-aided design(CAD) tool in parallel with this simulation. In this study, we adapt

the method for interfacing the CAD with Geant4 that has beensuggested by Kim et al. [25] (Fig. 1).

As a model instrument to further explain the present method,an instrument design of a conventional solid-state detector with anapplied electric field in front of the detector is employed [26]. Themodel instrument includes an electric field because this feature isexpected to be desirable in future instruments. Traditionally, apermanent magnet has been placed in front of the detector toseparate electrons from the measured ions (See, for example [21],and references therein), but this may electromagnetically disturbneighboring devices; such a concern is especially important if theinstrument is to be deployed on a small satellite. To reduce theamount of incident and scattered light striking the detector, theaperture of the instrument is assumed to have a collimator with aset of blackened optical baffles. The shape of the entrance apertureis a rectangle with a size 39.52 mm � 3.10 mm, corresponding to afield-of-view of 65.9 � 18.2� relative to the detector plane posi-tioned at the other side of the collimator. Interior to the collimatoris a parallel plate electric field that serves as a deflector, bending thetrajectories of electrons and positive ions in opposite directions. Aset of knife-edge plates is inserted along the edges of the collimatorto suppress secondary electrons generated at the wall surfaces.There are thirty knife-edge plates stacked along the direction of thecollimator. The size of each plate is 41.66 mm � 3.18 mm andthickness 56 mm. Each plate is separated by 0.61 mm along thecollimator direction, whereas the distance between the tips ofpaired plates is 3.10 mm. The potential difference between thepaired plates generates the deflecting electric fields. Near the end ofthe deflector lies a solid-state detector that measures the energydeposited by the entering particles. The present calculation as-sumes a solid-state detector with four 3 mm � 3 mm pixels in alinear array [27]. The spacing between the four detectors is 0.10mmeach. The positive ions and electrons will be separated by theirrelative displacement in the direction of the applied electric field,depositing their energies on different pixels as illustrated in Fig. 2.

We have used SIMION (Version 8.1) [28] to solve the electricfields near the instruments. The electric field is calculated bysolving the Laplace equation with appropriate boundary condi-tions. The CAD files can be directly imported for precise and

Electrons

Ions

Cell 1

Cell 2

Cell 3

Cell 4

Fig. 2. Schematic of the instrument shown in cross section. A sample instrument fordemonstrating the present numerical method is shown. The instrument employselectric fields to separate positive ions and electrons [26]. A four-pixel detector [27] isplaced near the exit of the deflector to measure displacements of the particles alongthe direction of the electric fields. The trajectories of protons and electrons are not toscale and are exaggerated for the purpose of illustration.

S. Park et al. / Current Applied Physics 14 (2014) 132e138134

consistent modeling of the mechanical structure of the instrument.In our example, electrostatic deflector plates as designed with CADare saved as a Standard Template Library (STL)-format file and thenimported into SIMION. As an example to demonstrate the currentnumerical method, it can be assumed that positive and negativevoltages of 2000 V are imposed on the upper and lower deflectorplate, respectively. The spacing between the tips of each pair ofpositive and negative deflector blades is 3 mm. In addition, the restof the structures are assumed to be 2-mm thick aluminum con-ductors and are given a potential of 0 V. The grid size of thecalculated electric field is 0.1 mm in each dimension. With theseboundary conditions, SIMION solves the Laplace equations (Fig. 3).The iteration tolerance of the numerical scheme is set to 0.005 V. Itis found that the maximum magnitude of the electric field is about

Fig. 3. Distribution of non-uniform electric field. Non-uniform electric field in the vicinity ovector representation electric field is for the case in which þ2000 V and �2000 V are appexpected features are seen: uniform electric fields within the plates and non-uniform field

1300 V/mm near the center of deflector blades, whereas a non-uniform electric field of about 500 V/mm is generated at bothedges of the deflector plates. The maximum electric field isconsistent with intuitive expectations for a potential difference of4000 V applied across the tip distance of 3 mm.

The electric field generated by the deflector is in general non-uniform. However, Geant4 does not directly accommodate non-uniform electric fields by default. Therefore, we have developed aclass to directly accommodate non-uniform electric fields inGeant4. To validate this newly generated class for non-uniformelectric fields, the different trajectories of electrons at energiesE ¼ 50, 105, 200 and 400 keV were compared (Fig. 4), using cal-culations made with non-uniform fields obtained from the Laplacesolver. In Fig. 4, the initial position of electrons is at the left-mostborder, halfway between the plates. The initial velocity of elec-trons is given along the horizontal direction of the figure withoutany vertical components. As the electrons propagate, they willinteract with the electric fields near the deflecting plates and cangenerate vertical components that will change trajectories from theinitial straight lines. Geant4 calculations are performed with thenon-uniform electric field class in this study and are comparedwiththe SIMION calculations using the same electric fields. The analyticsolutions are just parabolic trajectories, obtained using the as-sumptions that the electric field is uniform along the vertical di-rection of Fig. 4 and that the field is well-confined within the spacebetween the deflecting plates. Horizontal distance refers to thedistance traveled by the electrons parallel to their initial direction.Vertical displacement corresponds to distance traveled perpen-dicular to the initial direction which provides separation of parti-cles of different charges. Electrons begin to escape the deflectingplates without any collisions when their energies are greater than105 keV. The Geant4 and SIMION calculations agree closely (Fig. 4),suggesting that the newly applied class for non-uniform electricfields in Geant4 is sufficiently accurate. These results are non-negligibly different from the results of analytic solutions, indi-cating that this consideration of non-uniform electric fields isnecessary to accurately determine geometric factors.

It should be noted that in this study, the electric fields fromSIMION are used as an input to the Geant4 simulation, but anyelectric fields obtained from other methods will work equally wellif the electric fields are given in a format compatible with this fieldclass.

f two plates, calculated by solving the Laplace equation with the SIMION program. Thislied on either plate, and the distance between the tip of each plate pair is 3 mm. Thes near the edges of the plates.

Fig. 5. Particle injection from hypothetical surface to simulate isotropic flux. The ge-ometry and angles are illustrated to simulate isotropic particle injections from a hy-pothetical surface. In the present study, 106 particles are ejected for each energy step.The polar angles are limited to p/4 because outside this region penetration of particlesthrough the structure is highly unlikely given the considered energy range of 5e500 keV.

Fig. 4. Comparison of calculated trajectories based on different electric fields. Electrontrajectories are compared for energies E ¼ 50, 105, 200, 400 and 500 keV. The initialposition of electrons is at the left-most border, halfway between the plates. The initialvelocity of electrons is given along the horizontal direction. Horizontal distance refersto the traveling distance of the electrons along the initial direction of travel; verticaldisplacement corresponds to the perpendicular distance away from the initial velocitythat gives rise to the separation of particles for different charges.

S. Park et al. / Current Applied Physics 14 (2014) 132e138 135

3. Geometric factor

The definition of geometric factorwith the bracketed quantity inEquation (2) is only dependent on the geometry of instruments andis valid strictly for particles and photons following the straighttrajectories. However, the definition of the term has been oftenextended to include all the terms in Equation (2) to designate theinstrument response that is a complicated function of energy anddirection [29e36]. For this extended meaning, the term “geometricfactor” is not strictly adequate andmay cause confusions in terms ofthe extent to which different terms determining the instrumentresponse are covered from the instrument structure to the detectorelectronics. In this paper, the meaning of the geometric factor isused in its extended form and indeed includes the efficiency of theinstrument in Equation (2) as a function of direction and energy ofentering particles. The efficiency in the definition of the geometricfactor in this paper only includes those related up to the structure ofthe instrument head, but does not include those related to de-tectors, the front-end electronics, and digital electronics.

With this extended definition, the geometric factor of an in-strument relates the isotropic fluxes of incident particles with de-tector counting rates. In order to simulate isotropic particle fluxestoward the instrument, we assume a hypothetical sphere with itscenter situated at the detector of the instrument. For our study, thehypothetical sphere has a radius off 7.5 cm with its center at aposition which coincides with the center of the detector in theinstrument. The isotropic flux of charged particle fluxes is gener-ated by shooting particles at random position on the surface towardrandom directions. Note that our interest is in particles in the 5e500 keV range. The ranges of protons and electrons at 500 keV inaluminum are about 5.6 � 10�3 and 8.3 � 10�1 mm, respectively.The mechanical structure of the instrument is usually at least 1 mmthick. Therefore, it is highly unlikely that 5e500 keV electrons andpositive ions will penetrate it. Accordingly, we have limited therandom shooting of the particles on the hypothetical sphere to aspherical sector in the angular range of 0 � q � p/4 and 0 � 4 � 2p,where q is the polar angle measured from the center of the in-strument’s field of view (FOV) and f is the azimuthal angle around

the center (see Fig. 5). This spherical sector includes that subtendedby the greatest angles corresponding to the opening of the apertureviewed from the detector and should be sufficient to includeentrance of the electrons arising from scattering processes at theinstrument’s baffle or collimator. Therefore, any particles withinitial angular positions outside this sector will intersect thestructure of the instrument and be stopped; these particles do notcontribute to the detector counts and thus do not affect the geo-metric factor. We also limit the particle shootings to those thattravel radially inward because outward-moving particles obviouslywill not contribute to the detector counts.

Let us suppose that there are N particles intersecting the hy-pothetical surface as the result of an isotropic flux j. This intersec-tion of particles from the ambient distributions at the position ofthe instrument is equivalent to shooting N particles in this simu-lation. The isotropic ejection of particles from the surface is simu-lated by generating random numbers that are uniformwith respectto d(cos q0), where q0 is the polar angle measured from the localnormal vector of the surface [1e3]. Among the N particles, someparticles enter the aperture of the instrument and travel to thedetectors, with or without scattering. Let us say that the number ofthese particles finally hitting the detector is n. We will assume thatthe particle injections from the hypothetical surface need to occuronly over a sector with solid angle of magnitude DU. For our case,DU ¼ ð2�

ffiffiffi2

pÞp because we consider only 0 � q � p/4 and

0 � 4 � 2p. It is a straightforward task to show that the geometricfactor G of the instrument and isotropic particle flux j incident onthe hypothetical sphere are related to N and n according to thefollowing equations [1e3,21]:

N ¼ j$4p2r2$�DU4p

�; (3)

therefore,

G ¼ nj¼ n

N$4p2r2$

�DU4p

�: (4)

It should be noted that only the particles moving inward of thesphere are included to contribute to the final calculation. The

S. Park et al. / Current Applied Physics 14 (2014) 132e138136

standard deviation of the geometric factor determined from thePoisson distribution associated with the success of counting is:

sGw4p2r2$�DU4p

�$

ffiffiffin

pN

: (5)

Note that this relation is consistent with those derived by Yandoet al. [21] in the case in which the particle shooting takes placesover the entire surface of the hypothetical sphere, for which DU/4p ¼ 1. The fraction of the solid angles, DU/4p, appears inexpression (4) because any particles in our energy range of interest,5e500 keV, cannot directly penetrate the instrument structure toarrive at the detector. Such penetration is possible only forconsiderably higher energies. If such higher-energy penetrationsindeed take place, this limitation on particle shootings should beremoved; the overall numerical procedure would remain almostidentical otherwise. However, in the present study, this limit savesa considerable amount of computation time.

Fig. 6. Geometric factors as calculated with Geant4 and SIMION. (a) Geometric factoras a function of energy, calculated with Geant4. (b) SIMION results, suggesting that asignificant fraction of the electrons undergo scattering or emission of secondaryelectrons. The horizontal dashed line in black represents the geometric factor corre-sponding to particles following the straight trajectories. Significant deviation of thegeometric factor from the black dashed line is found for lower energies (E �w100 keV)indicating the significance of the deflecting electric field. For higher energies, theproton geometric factor is close to the dashed line, whereas the electron geometricfactor is significantly deviated due to the scatterings or secondary emissions.

4. Results

Geometric factors for electrons and protons at energies 50, 100,200, 300, 400 and 500 keV are calculated according to the proce-dure given in Sections II and III. In the numerical simulations, thereare 106 particles isotropically ejected from the hypothetical surface.Fig. 6 summarizes the results of the calculation. The statisticaluncertainties associated with counting statistics are shown as errorbars for each calculation. For energies below approximately100 keV, the geometric factor is an increasing function of particleenergy because the applied electric fields bend the trajectory andblock the entrance of particles. The applied fields are effective forboth electrons and protons. The results also clearly verify that theelectric fields bend the positive ions and electrons in opposite di-rections, giving rise to more counts in cells 1 and 2 for electrons and3 and 4 for ions, respectively (See Fig. 2 for definitions of the fielddirection and cell numbers.).

The horizontally dashed line in Fig. 6, near 0.008 cm2 sr, cor-responds to the analytical expression of geometrical factor for theparticle detector assuming straight particle trajectories [1e3]. Forexample, Thomas and Willies [3] obtained the following approxi-mate expression for the geometrical factor:

Gx16X1Y1X2Y2

Z2; (6)

where X1,Y1,X2,Y2 are the dimensions associated with the collimatorand detector, respectively, and Z is the distance between the colli-mator and detector (For more complete expression, see equation(13) of Thomas and Willis [3].). Because this analytic expressionincorporates the assumption of straight particle trajectories, onemay expect that it is more accurate for higher-energy particles,which are less affected by the fields; conversely, the fields have agreater effect on the trajectories of lower-energy particles. Thistrend is consistent with the results presented in Fig. 6.

Geometric factors calculated by SIMION, which does not takeinto account the interaction of the particles with the instrument,are similar for electrons and protons (Fig. 6b). The geometric factorscalculated by Geant4 and SIMIONmatch well for protons, but differconsiderably for electrons. It has already been demonstrated thatthe particle trajectories calculated with Geant4 and SIMION arenearly identical (recall Fig. 4). Therefore, any significant differencein detector counts between the two methods should be attributedto the physical interactions, such as scatterings or secondaryemissions, that are not modeled by SIMION. The differences seen inFig. 6 suggest that these effects are important for electrons. The

scattering of electrons off the instrument and Bremsstrahlung ra-diation within its matter [4e7] are well-known mechanisms thatexplain the observed difference.

To study the effects of scattering and secondary electrons, wecalculated the kinetic energy distributions of incoming electronsjust prior to their intersection with the detectors, i.e., before theelectrons begin to transfer their kinetic energy through physicalinteractions. It should be noted that this kinetic energy is not thesame as the energy deposited in the detector as measured in thelaboratory [4e7]. In the energy range of the present study, electronscan backscatter or create Bremsstrahlung within the detectors.Therefore, the energy deposited within the detector can be sub-stantially less than the electron’s total kinetic energies. Laboratoryexperiments confirm that the amount of energy transfer is acomplicated function of energy and angle of incidence [4e7]. InFig. 7 we show the kinetic energy distributions of electrons forinitial energies at 50, 200, 300 and 400 keV. The initial energies aredefined on the injection hypothetical surface. As the electrons

Fig. 7. Histogram of kinetic energy distributions of electrons. Calculated kinetic energydistributions of incoming electrons just prior to arrival at the detectors. The kineticenergies of the incoming electrons are binned at 10 keV intervals. Note that the energygain or loss due to the potential difference of the deflecting electric field is at most4 keV because only �2000 V are applied to the knife-edge blades. Therefore, thoseelectrons found at lower energy bins than that corresponding to the initial electronenergy have lost kinetic energies through the scatterings or secondary emissions.

S. Park et al. / Current Applied Physics 14 (2014) 132e138 137

move toward the detector, two different types of particle trajec-tories are expected, one with and the other without scattering offthe instrument. In Fig. 7 the kinetic energies are binned at intervalsof 10 keV. Electrons with kinetic energies smaller than 5 keV whenthey enter the detectors are not included in the histogram becausesuch electrons are not easily measurable with laboratory detectorelectronics. If there is no scattering, the kinetic energy of electronsat their arrival at the detector will be the sum of their initial energyplus that corresponding to their acceleration along the electricfields. On the other hand, the energy gain or loss due to the po-tential difference of the deflecting electric field is at most 4 keVbecause only �2000 V are applied to the knife-edge blades.Therefore, those electrons found at lower energy bins than thatcorresponding to the initial electron energy have lost kinetic en-ergies through the scatterings or secondary emissions. Detailedcalculation with Geant4 by tracing the presence of scatterings orsecondary emissions along the electron trajectories indeed sup-ports the finding. In Fig. 7, the fraction of electrons that maintaintheir initial energy is also shown. In these calculations, electronsare less likely to maintain their initial kinetic energy when theirinitial energy is higher. For example, 88.4% of 50-keV electronsmaintain their initial energy, while only 58.4% of 200-keV electrons

and 34.6% of 400-keV electrons do. This suggests that at least 65.4%of 400-keV electrons are scattered before they arrive at the de-tectors. This interpretation is consistent with the previous results inwhich the geometrical factors computed by Geant4 and SIMIONdiffered increasingly as the electron energy becomes greater (recallFig. 6). It also shows that the scattered or secondary electronscontribute significantly to the calculation of the electron geometricfactors, clearly demonstrating the importance of proper accountingfor electron scattering and secondary emission.

5. Conclusion

A computational method for calculating the geometric factors ofcharged particle detectors has been presented. The method directlyimports instruments’mechanical design files into Geant4, ensuringprecision and maintaining consistent instrument design andmodeling among different software packages. Applied electricfields intentionally generated to separate positive ions from elec-trons were considered for detailed investigation; this anticipatedthe future use of electric fields in this application, presentingseveral advantages over the conventional use of magnetic fields,such as easier control of the fields and presumably less interferencewith neighboring electronics. The results from Geant4 Monte Carlosimulations were compared with the results from the SIMION ray-tracing method. A class in Geant4 that allows non-uniform electricfields to be included in the calculation of particle trajectory wasdeveloped, and the use of this class was validated for selectedtrajectories by its close agreement with SIMION results.

The isotropic flux of particles was simulated as the ejection ofparticles at random directions on a hypothetical surface. The rela-tion between the isotropic flux and the number of particles ejectedfrom the surface was calculated. The geometrical factor, which isthe ratio between the particle flux and the detector counts, wasthen calculated in a straightforward manner described in a previ-ous section of this paper. For protons, the geometrical factor was anincreasing function of energy up to about 100 keV, but it does notexceed the analytical expectation that has been previously estab-lished with an assumption of straight particle trajectory. For elec-trons, however, the geometric factors were considerably different.The geometric factor continuously increased with incident electronenergy. We calculated the kinetic energies of electrons just prior totheir arrival at the detectors to clearly show that an appreciableamount of electrons had already lost kinetic energy through scat-tering or the emission of secondary electrons, and that theseelectrons contributed significantly to the calculated geometricfactor for electrons.

Acknowledgment

This research was supported partially by the National Space Lab(No. 2009-0091569) and BK21þ program through the NationalResearch Foundation (NRF) funded by the Ministry of Education ofKorea.

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