Computational method for calculating geometric factors of instruments detecting charged particles in the 5–500 keV energy range with deflecting electric field

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    Accepted 10 October 2013Available online 30 October 2013


    of electric or magnetic elds that are intentionally generated to clearly separate electrons from positive

    y detec

    JaE;s;u; tdtds$brdudE; (1)where C counting rate (counts/s), T duration of the observationstarting from time t t0, a detection efciency for the ath par-ticle species, Ja differential particle ux of the ath particle species[cm2 s1 sr1 E1], ds element of surface area of the detector

    which a remains a(E), and time-s Ja J0(E), thees to the following

    E S U


    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 bafe, and isconventionally dened 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

    * Corresponding author. Tel.: 82 31 201 3282; fax: 82 31 204 2445.

    Contents lists availab

    Current Appl


    Current Applied Physics 14 (2014) 132e138E-mail address: (J. Seon).effective dimensions of the detector facing a certain solid angle inspace, as well as the efciency 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 1T






    aE;s;u; t

    angles, and particle energies.In the case of ideal detector responses, for

    constant over the domain of u, a and t, i.e., aindependent isotropic incoming particle uxeexpression for the detector counting rates reducsimpler form:

    C Z


    264 Z ds$br$ Z du

    375J0EdE:incident uxes of charged particles critically depends on the which respectively represent the surface area of the detectors, solidGeometric factorInstrument responseCharged particle detectorGeant4

    1. Introduction

    In general, the counting rate of an1567-1739/$ e see front matter 2013 Elsevier B.V. the trajectory of scattered and unscattered charged particles under the inuence of thecalculated elds. The propagation of charged particles through the elds, their interaction with the in-struments, and energy deposition into the detectors are calculated with Geant4, whereas the electric ormagnetic elds 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 uxes of incident charged particles.

    2013 Elsevier B.V. All rights reserved.

    tor that responds to the

    [cm2], u element of solid angle [sr], br unit vector in the di-rection u, and E energy of the particles [keV], respectively. Theintegration corresponds to all relevant domains of S, U, and E,Received in revised form8 October 2013 ions. The method rst solves the distribution of electric or magnetic elds near the detectors, and thenReceived 20 April 2013A 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 presenceComputational method for calculating ginstruments detecting charged particlesrange with deecting electric eld

    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

    a r t i c l e i n f o

    Article history:

    a b s t r a c t

    journal homepage: wwAll rights reserved.metric factors ofthe 5e500 keV energy

    eong-Gi 446-701, Republic of Korea

    le at ScienceDirect

    ied Physics

    lsevier .com/locate/cap

  • 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 eld in front of the detector is employed [26]. Themodel instrument includes an electric eld 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 bafes. The shape of the entrance apertureis a rectangle with a size 39.52 mm 3.10 mm, corresponding to aeld-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 eld that serves as a deector, 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 and

    lied Physics 14 (2014) 132e138 133scattering 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 signicant 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 islled with charged particles, often trapped by Earths magneticelds [8]. Later observations further showed that these regions ofaccessible space are dynamically lled 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 elds [15] or electric elds 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 elds used to segre-gate positive ions and electrons, 4) calculation of particle trajectoryacross the applied elds, 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 satised all the aforementioned requirementssimultaneously. The purpose of this paper is to describe a numericalmethod that simultaneously satises 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 le containing the instruments geometrical information in theGeometrical Description Markup Language (GDML) format, aformat based on Extensible Markup Language (XML) [24]. It isdesirable to nd the method to adapt the latter because the in-strument design is often carried out with a computer-aided design

    S. Park et al. / Current App(CAD) tool in parallel with this simulation. In this study, we adaptthickness 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 deecting electric elds. Near the end ofthe deector 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 eld,depositing their energies on different pixels as illustrated in Fig. 2.

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

    Fig. 1. Procedure for geometric eld factor calculation. The CAD instrument design isconverted to an appropriate format to solve electromagnetic elds near the instru-ment. The non-uniform electric eld obtained from the solver is used for trajectory

    calculation in Geant4. The CAD le is also converted for Monte Carlo calculation ofgeometric factors in Geant4.

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

    The electric eld generated by the deector is in general non-uniform. However, Geant4 does not directly accommodate non-uniform electric elds by default. Therefore, we have developed aclass to directly accommodate non-uniform electric elds inGeant4. To validate this newly generated class for non-uniformelectric elds, the different trajectories of electrons at energiesE 50, 105, 200 and 400 keV were compared (Fig. 4), using cal-culations made with non-uniform elds 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 gure withoutany vertical components. As the electrons propagate, they willinteract with the electric elds near the deecting plates and cangenerate vertical components that will change trajectories from theinitial straight lines. Geant4 calculations are performed with thenon-uniform electric eld class in this study and are comparedwith



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    S. Park et al. / Current Applied Physics 14 (2014) 132e138134consistent modeling of the mechanical structure of the instrument.In our example, electrostatic deector plates as designed with CADare saved as a Standard Template Library (STL)-format le 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 deectorplate, respectively. The spacing between the tips of each pair of

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

    Fig. 3. Distribution of non-uniform electric eld. Non-uniform electric eld in the vicinity ovector representation electric eld is for the case in which 2000 V and 2000 V are appexpected features are seen: uniform electric elds within the plates and non-uniform eldthe SIMION calculations using the same electric elds. The analyticsolutions are just parabolic trajectories, obtained using the as-sumptions that the electric eld is uniform along the vertical di-rection of Fig. 4 and that the eld is well-conned within the spacebetween the deecting 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 deectingplates 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 electricelds in Geant4 is sufciently accurate. These results are non-negligibly different from the results of analytic solutions, indi-cating that this consideration of non-uniform electric elds is...


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