Designing a toroidal top-hat energy analyzer for low ... · energy analyzer for space particle measurement. In this re-port, a top-hat-type energy analyzer is taken and its design

An Introduction to Space Instrumentation,Edited by K. Oyama and C. Z. Cheng, 181–192.

Designing a toroidal top-hat energy analyzer forlow-energy electron measurement

Y. Kazama

Plasma and Space Science Center, National Cheng Kung University, Tainan, Taiwan

This report is a brief introduction of designing an electrostatic energy analyzer by taking a toroidal top-hatenergy analyzer. First, a top-hat analyzer is parameterized with four parameters: the deflection angle, the shellelectrode offset, the upper collimator height and the lower collimator thickness. Then three steps of parametersurveys are made by particle trajectory tracing simulations. In the surveys, the requirements of g-factor, azimuth-angle resolution and field of view are taken into accout. After the surveys, we confirm that the final designmeets the requirements. In addition, UV photon suppression performance is also evaluated by photon tracintgsimulations. The expected maximum photon count rate 1.59 count/sec is acceptable, compared to the MCP’sbackground count rates. The analyzer design investigated in this report is to be taken over by a low-energyelectron instrument (LEP-e) on the radiation-belt observation satellite ERG.Key words: Space instrumentation, plasma instrument basics, low-energy electron instrument, top-hat energyanalyzer.

1. IntroductionThe purpose of this report is to give an introduction for

those who are not familiar with designing an electrostaticenergy analyzer for space particle measurement. In this re-port, a top-hat-type energy analyzer is taken and its design isinvestigated by parameter surveys. After determining the fi-nal design, performances of particle measurement and pho-ton suppression are estimated. The top-hat analyzer dis-cussed here will be a prototype for the low-energy electroninstrument (LEP-e) onboard the radiation-belt observationmission ERG (Shiokawa et al., 2006). Detailed descrip-tions on a wide spectrum of space particle instrumentationare found in Pfaff et al. (1998a, b) and Wuest et al. (2007).

2. Basics of Electrostatic Analyzers2.1 Energy-voltage relation

To learn basics of electrostatic energy analyzers, we startwith an ideal analyzer of spherical type. Assuming two con-centric spherical electrodes and a particle moving exactlyalong the center of the electrode gap, we can express thekinetic energy of the charged particle K0 as:

K0

q= Vo R2

o − Vi R2i

R2o − R2

i

, (1)

where Vo, Vi, Ro and Ri are the potentials and the radii of theouter and inner electrodes, respectively. Note that K0 is thekinetic energy of the particle at infinity where no potentialexists.

First of all, it is obvious that an electrostatic energy an-alyzer cannot measure masses of charged particles becausethere is no mass dependence in the equation. To measuremasses, another method is necessary, such as magnetic-field

Copyright c© TERRAPUB, 2013.

deflection, time variation of electric fields, time-of-flightmeasurement, and so on.

Second, K0 and q only appear in the form of K0/q. Thismeans that we cannot measure a particle energy itself; AHe+ ion with a kinetic energy of 10 keV behaves exactlysame as He++ of 20 keV, and one cannot discriminate themby an electrostatic analyzer alone. For this reason, someliterature expresses energy in keV/q.

An analyzer is usually used by the voltage setting Vo =−Vi, or either Vo or Vi = 0. In this case, the energy ofcharged particles passing the analyzer simply becomes pro-portional to the single voltage. This proportionality indi-cates trajectory invariance; The trajectory of a 5-keV par-ticle with 1-kV voltage setting is identical to the one of a10-keV particle with 2-kV voltage setting.

Finally, the energy-voltage relation does not change byscaling Ro and Ri by a factor a (Ro, Ri → a Ro, a Ri) withkeeping the voltage. This means that particle trajectoriesscale as the geometry scales. Therefore, analyzer’s prop-erties related to particle energy and trajectory are constantover the scaling. It is, however, obvious that other factors,for example, timing of particles and a size of an aperture arescaled and changed.2.2 Geometric factor

Here we describe a sensitivity of an analyzer, that is,how many particles pass through the analyzer per unittime. A number of passing particles C during a time in-terval �t is proportional to a particle differential flux J asC = GE J ε �t , where ε contains other factors such as a de-tection efficiency of the detector device. The coefficient GE

[cm2 sr keV] is called “energy geometric factor” and rep-resents the sensitivity of the analyzer. Roughly speaking,GE can be expressed as ∼ S � �K , where S is the effec-tive aperture area, � is the effective solid angle of the fieldof view, and �K is the energy passband of the analyzer.Geometric factor is discussed in Appendix A. A detailed

181

182 Y. KAZAMA: DESIGNING TOROIDAL TOP-HAT ANALYZER

Fig. 1. Schematic illustration of the top and cross-sectional views ofa toroidal top-hat analyzer. Typical particle trajectories are drawn toindicate azimuth-angle focusing and wide particle acceptance.

description is found in Wurz et al. (2007).In the case of electrostatic analyzers, �K is pro-

portional to a tuned energy K0 because of the pro-portionality of electrostatic analyzer. Therefore, anenergy-independent value, “geometric factor” (hereafter “g-factor”) G [cm2 sr keV/keV] = GE/K0 is often used for suchsystems.

3. Top-Hat-Type AnalyzerA top-hat-type electrostatic analyzer was first investi-

gated by Carlson et al. (1983). The analyzer consists of twoconcentric hemispherical electrodes, and the top part of theouter electrode is cut out as a particle entrance. Analyzersof this type are sometimes referred to as “spherical top-hatanalyzer”, and have been employed for space plasma mea-surements (e.g. Paschmann et al., 1985; Lin et al., 1995;Reme et al., 1997).

Based on spherical top-hat analyzer, Young et al. (1988)proposed “toroidal top-hat analyzer”, in which two toroidalshells are used instead of two concentric spherical shells,as schematically illustrated in Fig. 1. The two parallelplates on the top form a collimator to define a field ofview of the analyzer. A charged particle enters the analyzerfrom the aperture into the collimator, and is deflected by anelectric field into the gap of the shell electrodes. Particleswith energies matched to the shell voltage can pass throughthe shells, which results in energy selection of the chargedparticles. As seen in the figure, initial velocity directions ofparticles can be distinguished by exit particle positions.

As pointed out by Young et al. (1988), a toroidal top-hat analyzer is characterized by long focal length in the az-

imuth direction and large sensitivity, relative to a sphericaltop-hat analyzer. In the case of spherical top-hat analyzers,azimuth-direction focusing occur inside the shell electrodesdue to its short focal length, and particles have started de-focusing at the analyzer exit. Relatively large sensitivity oftoroidal top-hat analyzer is an advantage for space instru-ments; We can shrink the analyzer to decrease the mass ofthe instrument while keeping the sensitivity requirements.

Figure 2 is a three-dimensional cut-away view of thetoroidal top-hat-type electrostatic energy analyzer to be dis-cussed in this report. We take the coordinate systems (x , y,z) and two velocity angles of elevation (EL) and azimuth(AZ ) as shown in the figure. The analyzer structure is ro-tationally symmetric with respect to the z axis. We assumethat the analyzer is mounted on the spacecraft such that thesymmetric axis is perpendicular to the spacecraft spin axis.Note that the definitions of elevation/azimuth angles are dif-ferent from those in Young et al. (1988).

The analyzer has two parallel plates on the top, whichconstitute a collimator to define the field of view of theanalyzer. A positive voltage is applied to the inner electrodeof toroidal shape (shown in red in the figure) for electronmeasurement. Only electrons with energies matched to theapplied voltage can pass through the shells (as shown bythe red line in the figure), and the others are lost by hittinganalyzer walls. An energy spectrum of electrons is taken bysweeping the voltage.

The energy-selected electrons are finally detected bymulti-channel plates (MCPs) shown in blue. An MCP isa thin lead glass plate with capillaries with a diameter from∼1 to ∼ tens µm. A high voltage is applied perpendicu-lar to the plate and the electric field produces cascading ofsecondary electrons initially triggered by a particle input.This multiplication finally creates an electron charge cloud,which is detectable as a current pulse by a front-end circuit.Usually two or three MCPs are stacked in an MCP assem-bly (green in the figure) to gain sufficient charges (typically∼107 electrons by three-stacked MCPs.) See Wiza (1979)for the details of MCP.

Because MCP detection process depends on secondaryelectron release, MCP is sensitive to any particle whichproduces secondary electrons, such as charged particles,UV photons and radiations. For this reason, in the caseof space instruments, we have to pay attention to effectsdue to radiations and UV photons. UV photon effects arediscussed later in this report.

Since a top-hat analyzer has a 360-deg field of view inthe azimuth direction, electrons simultaneously come in tothe analyzer in all the azimuth directions. In principle,a parallel electron flux at some azimuth angle focuses onone point after 90-deg deflection due to analyzer’s focus-ing property. As a result, resolving azimuth directions ofelectrons is made by position sensing on the MCP. There-fore, azimuth-angle resolution is determined by analyzer’sfocusing performance and MCP’s positioning resolution.

The field of view in the elevation direction is, on theother hand, as narrow as several degrees, as seen in Fig. 2.Accordingly, the elevation-angle resolution of this analyzeris identical to the field of view. Hence three-dimensionalmeasurement is made by sweeping this narrow field of view

Y. KAZAMA: DESIGNING TOROIDAL TOP-HAT ANALYZER 183

+Z

+X

-Y

AZEL

VV

MCPinner electrode

MCP assemmbly

uppper collimator

lower collimator e-

Fig. 2. Cut-away view of the toroidal top-hat analyzer to be discussed in this report. An electron trajectory is shown in red. Elevation angle EL andazimuth angle AZ are defined as indicated.

Z [m

m]

R*sign(X) [mm]

Cross Section

-20

-10

0

10

20

30

40

-40 -30 -20 -10 0 10 20 30 40

Fig. 3. Parameters (a, b, c, d) to parameterize a top-hat-type electrostatic analyzer. The modeling conditions are also shown in the figure.

over 180 degrees around the spacecraft spin axis (here weassume the spin axis perpendicular to the z axis.)

4. Instrument RequirementsThe purpose of our analyzer is to observe suprathermal-

to-hot electrons populated in the Earth’s inner magneto-sphere. To fulfill the observation, instrument performanceis supposed as follows:

1) energy range from ∼10 eV to >∼15 keV,2) energy resolution of <15%,3) elevation-angle resolution of <∼4 deg,

4) azimuth-angle resolution of ∼22.5 deg (16 channelsover 360 deg),

5) g-factor of (5±1)×10−4 cm2 sr keV/keV per 22.5-degazimuth angle.

In addition to the electron measurement requirementsstated above, the instrument has practical requirements:

1) to limit the field of view to the elevation angle of> +3 deg,

2) to be radiation hard for instrument operation in theradiation belts,

3) to suppress photon count rates down to


Table 1. Summary of the parameters c, K0 and zf for given a and b.

b [mm] K0 [keV] a [deg]

70 71.25 72.5 75 77 80

0.0 5.60 c [mm] — — — 2.0 1.8 1.5

zf [mm] — — — −3.0 2.1 7.7

2.5 5.10 c [mm] 3.9 — 3.8 3.9 4.8

zf [mm] −10.2 — −6.5 −2.2 8.7

5.0 4.80 c [mm] — 4.8 4.9 5.0 6.5

zf [mm] — −11.3 −9.4 −5.9 3.1

7.5 4.30 c [mm] 6.9 7.4 8.5 —

zf [mm] −12.0 −8.9 −5.8 —

Table 2. G-factor G and focusing performance W80% as a function of theupper collimator height c.

c G W80%

mm cm2 sr keV/keV deg

3.5 5.35 × 10−4 6.8

3.7 6.35 × 10−4 6.6

3.9 7.29 × 10−4 6.8

4.1 8.24 × 10−4 7.2

4.3 8.88 × 10−4 7.4

Table 3. G-factor G, focusing performance W80% and elevation-anglelimit of the field of view ELmax as a function of the lower collimatorthickness d .

d G W80% ELmax

mm cm2 sr keV/keV deg deg

1.0 6.4 × 10−4 7.2 +4.0

1.4 6.6 × 10−4 5.9 +3.5

1.6 5.9 × 10−4 5.4 +3.4

1.8 5.2 × 10−4 4.6 +3.1

2.0 4.5 × 10−4 4.2 +2.9

<∼10 counts/sec per channel.

About the first point, we must avoid artificial electronssuch as photoelectrons from the spacecraft body. Thenwe limit the field of view in the elevation direction below+3 deg. In this condition, the analyzer does not see thespacecraft body within the radius of ∼1 m if the apertureheight is 50 mm from the spacecraft surface.

Radiation hardness is crucial for low-energy plasma ob-servation in the Earth’s radiation belts. It is a challenge forthe present analyzer to minimize radiation effects withoutcoincidence/anti-coincidence techniques. This topic is be-yond the scope of this report and will be discussed else-where.

The last point, photon suppression, is requested for elec-tron signals not to be masked by photon signals. Becausetypical count rates of electrons populated in the inner mag-netosphere can be several thousands count/sec per channel,the target photon count rate of <10 count/sec keeps a goodsignal-to-noise ratio. We estimate photon count rates andthe results are discussed later.

5. Designing of a Top-Hat AnalyzerTo design an analyzer, we take two steps of numerical

analysis: (1) to find the optimum design for electron mea-

surement and (2) to evaluate photon count rates. For thenumerical analysis, we use a cylindrical-coordinates poten-tial solver and a particle tracer programs on a personal com-puter. A potential field is obtained by solving the Poissonequation by the successive over-relaxation (SOR) method.Tracing of a particle is made using the traditional 4th-orderRunge-Kutta method with adaptive time step control. Thetime step control and the SOR method are described in Ap-pendix A.5.1 Parameterization of the design

A toroidal top-hat analyzer design is parameterized withfour parameters a, b, c, d, as shown in Fig. 3. Here a isthe deflection angle of the energy analyzer, b is the offsetof the shell center from the symmetric axis. c is the heightof the upper collimator measured from the topmost edge ofthe outer shell, and d is the thickness of the lower collimatorplate.

First we set the gap between two shells to 3 mm. Thenthe diameter of the exit aperture, which corresponds to thesize of the MCP placed on the bottom, is determined to be70 mm. This is because we suppose to use 77-mmφ MCPsprovided by Hamamatsu Photonics. The voltage of theinner electrode is fixed at 1 kV in the following parametersurvey.5.2 Deflection angle and offset of the shells

In this section, we describe the first step of the parametersurvey, in which the parameters a, b and c are surveyed inthe viewpoint of azimuth focusing depth and of g-factor. Asdiscussed previously, a parallel beam of electrons focusesafter exiting the shells in theory. To have a better azimuth-angle resolution, the focusing position should be as close tothe MCP input surface at z ∼ −13.5 mm as possible. Atthe same time, a higher g-factor is also essential not only tomeet the measurement requirement but also to keep roomfor further design improvements.

The first survey is made to find the optimum parameterset (a, b, c) under d = 0. For given a and b, we find centerenergy K0, with which an electron being back-traced fromthe bottom (z = −17 mm) goes along the center of theshell gap. Then we find c which lets that electron go out ofthe analyzer in the horizontal direction along the collimatorplates. A trajectory example is shown in Fig. 3 as the redline.

Table 1 gives the survey’s result, in which c, K0 and zf aretabulated for given a and b. Here zf is a z-position wherea parallel beam focuses after exiting the shells. a is takenfrom 70 deg to 80 deg, and b from 0 mm to 7.5 mm. In


Fig. 4. Cross-sectional view of the final analyzer design based on a = 72.5 deg, b = 5 mm, c = 4.1 mm, and d = 2.0 mm.

the table, blanks mean that no survey is made, and dashesindicate that no electrons can go out of the analyzer parallelto the collimator plates.

In the table, center energy K0 only depends on b. Thisis because K0 is determined only by the curvature radius ofthe shells defined by b. In terms of higher-energy electronmeasurement, higher K0 is preferable. According to theresult, K0 decreases as b increases: 5.60 keV at b = 0 mm,down to 4.30 keV at b = 7.5 mm. This can be explained asfollows; For larger b, the curvature radius becomes smaller,since the exit aperture diameter of 70 mm is fixed. If thecurvature radius is small, the center energy must be smallto go along the shells.

The result shows that large b has large c. This is becausea large b has a small K0 as mentioned above, and an electrondoes not need a large downward electric field by which theelectron enters the shell gap. Looking at a certain b, one cansee that c increases as a increases. This can be explained bythe inlet angle of the shells; Since the inlet angle in a larger-a model is closer to the horizontal direction, less electricfield is necessary for a horizontally-moving electron to gointo the shell gap. The only exception is at b = 0 mm,where c decreases from 2.0 mm (a = 75 deg) to 1.5 mm(a = 80 deg). This is due to no sufficient space for anelectric field to deflect an electron near the z axis in thecase of a = 80 deg.

About focus height zf, smaller a results in a lower zf, be-cause focusing of a parallel beam occur at 90-deg deflectionin principle, and the beam virtually needs more deflectionangle to focus in the smaller-a cases.

It is expected that large c has large g-factor due to its largeaperture, because large-b designs can have space where anelectric field deflects electrons toward the shell gap. There-

fore, larger b is better in the point of view of large g-factor.In summary, it is preferable to take (1) smaller b

for higher energy measurement, (2) smaller a for betterazimuth-angle resolution (focusing closer to the MCP), and(3) larger b for larger g-factor. Here we decide to take theparameters a = 72.5 deg and b = 5 mm as the best com-promise. Based on this geometry, we check the parametersc and d in the following sections.5.3 Upper collimator position

Based on the model with a = 72.5 deg and b = 5 mm,the next parameter to survey is for the upper collimatorheight c. The parameter c controls both g-factor and az-imuth focusing. Therefore, the goal of defining c is to finda good balance between g-factor and focusing.

To evaluate g-factor, here we assume more realistic an-alyzer geometry: (1) The MCP input surface is located atz = −13.5 mm and +200 V is applied, (2) a mesh is placedabove the MCP and −12 V is applied, (3) a slit at the shellexit is added to suppress scattering particles, and (4) thelower collimator thickness d is set to 1 mm. The positionof the MCP input surface is determined due to the size ofthe MCP assembly. In fact, we need space for an assemblysubstrate, insulators, mesh support, etc. The input surfacevoltage of +200 V accelerates electrons above 200 eV, atwhich an MCP has the maximum detection efficiency forelectrons.

The mesh works as a repeller for eV-range electrons; Themesh’s negative voltage prevents stray electrons existing inthe energy analyzer from reaching the MCP input surface.Simultaneously, the mesh repels secondary electrons escap-ing from the MCP back to the input surface to enhance MCPdetection efficiency. It is noted that this mesh voltage lim-its the minimum energy of electron measurement, 12 eV in


Fig. 5. Energy-elevation response for the analyzer voltage of 1 kV.

this case.To eliminate scattering electrons, a slit is added at the

exit of the shells. Electrons easily scatter on the electrodesurfaces and may reach the MCP after multiple scattering.The width of the slit is 2 mm, 1 mm narrower than the gapof the shells.

Table 2 gives g-factor G and focusing performance W80%

for each c. Here W80% is a width of azimuth angle whichcovers 80% of the total azimuth-angle response for a paral-lel (AZ = 0) beam input. Naturally smaller W80% stands forbetter focusing performance. G-factor is hereafter definedfor one MCP channel (22.5-deg wide in azimuth) unlessotherwise noted. It is seen that the g-factor G decreases as cdecreases because the aperture size becomes smaller, as ex-pected. However, W80% has a peak at c = 3.7 mm. This canbe explained as follows: At c = 4.9 mm, zf is −9.4 mm ac-cording to the previous result in Table 1. This indicates thatelectrons focus above the MCP at z = −13.5 mm. When cbecomes smaller, the focusing point moves downward andapproaches the MCP. This is probably due to the shift ofelectron’s deflection center by changing the velocity direc-tion at the entrance of the shells. Accordingly, focusing ismade at the MCP position when c = 3.7 mm.

Although the best focusing is seen at c = 3.7 mm, herewe take c = 4.1 mm, considering the g-factor which is more

sensitive to c than focusing performance. This high g-factoris necessary for further improvements which will diminishg-factor.5.4 Lower collimator plate thickness

Finally, the lower collimator thickness d is to be defined.The purpose of defining d is to limit the elevation-anglefield of view not to see a spacecraft body. We calculatedg-factor G, focusing performance W80% and maximum ele-vation angle ELmax by changing d from 1 mm to 2 mm. Theresult is tabulated in Table 3. Here ELmax is defined as theelevation angle which includes 99.9% of the elevation-angleresponse.

As d increases, the g-factor G decreases obviously be-cause of limiting the field of view. On the other hand, W80%

improves as d increases. This improvement comes fromelectrons entering the analyzer from the side fringes of theaperture, that is, electrons with large |yinitial| values (remem-ber that the analyzer’s symmetric axis is parallel to z and thebeam comes along x .) These electrons focus on z-positionshigher than the MCP surface and degrade the focusing per-formance due to their defocusing at the MCP. Increasing dmakes a thicker “neck” of the analyzer, which blocks theseelectrons more. This results in the better focusing seen atlarger d.

To satisfy the requirement that the elevation field of view


Fig. 6. Focusing of an electron input with a fixed azimuth direction. Theangle TH ′

1 represents final positions of electrons on the MCP.

should be limited to < +3 deg, we take d of 2.0 mm, atwhich ELmax = +2.9 deg. The design with d = 2.0 mmhas the g-factor of 4.5 × 10−4 cm2 sr keV/keV, which satis-fies the required g-factor of (5 ± 1) × 10−4 cm2 sr keV/keVper 22.5 deg.5.5 Detailed performance estimation

In the previous section, all the parameters of the analyzerhave been defined as a = 72.5 deg, b = 5 mm, c =4.1 mm and d = 2.0 mm. Based on this parameter set,the final design of the analyzer is determined, as illustratedin Fig. 4. For the final design, two further modificationsare made in terms of UV photon suppression: (1) extensionof the collimator radius and (2) fine serration photon trap.First, the collimator radius is extended to 45 mm (originally42 mm) to block the UV photon paths directly toward theouter shell surface from the aperture. This point will bediscussed later. Second, fine serration structures are madeon the collimator plates to effectively absorb UV photons.A sawtooth structure traps photons in its deep “valley” ofthe structure. The serrations are made on (1) the innersurface of the upper collimator plate (R > 5.0 mm), and(2) the inner surface of the lower collimator plate (R >

27.5 mm). A serration is assumed to be 0.5 mm in depth,0.5 mm in pitch, and the cut angle is 45 deg in the presentcase. This final analyzer design is verified by simulationsand the performance parameters evaluated are summarizedin Table 4. The energy-elevation response and the focusingperformance are described below.

Figure 5 shows energy-elevation response obtained bythe simulation. It is seen that energy and elevation angleare not independent. For example, higher-energy electronsenter the aperture downward to draw larger radii of trajec-tories inside the analyzer, and vice versa. By integrating theenergy-elevation response, the g-factor is evaluated to be4.2×10−4 cm2 sr keV/keV, which meets the requirement of∼ (5 ± 1) × 10−4. It is noted that the g-factor is slightlysmaller than that in Table 3. This is probably due to theextension of the collimator radius which limits more elec-

Table 4. Summary of analyzer performance parameters.

Parameter Value

G-factor 4.2 × 10−4 cm2 sr keV/keV

Analyzer constant 4.97 keV/kV

Energy resolution 8.0 % (FWHM)

Field of view

azimuth 360 deg

elevation 2.91 deg (FWHM)

Angular resolution

azimuth 22.3 deg (FWHM)

elevation 2.91 deg (FWHM)

trons. The mean energy 〈k〉 is 4.97 keV for 1 kV of theinner shell voltage. Since a mean energy is proportional toan analyzer voltage in the case of electrostatic energy ana-lyzer, we can define an analyzer constant (ratio of a meanenergy per analyzer voltage), 4.97 keV/kV in this analyzercase. Assuming a safety field strength of 1 kV/mm, we canapply 3 kV to the analyzer electrode, since the shell gap is3 mm. Hence the maximum energy in electron measure-ment is about 15 keV. This maximum energy limit can berelaxed if no discharge is confirmed at >3 kV in laboratorytests. The energy passband is calculated to be 0.40 keVFWHM at 5 keV, and the relative energy resolution �k/〈k〉becomes 8.0%, better than the requirement. The elevation-angle field of view is 2.91 deg FWHM. Note that this fieldof view is identical to the elevation-angle resolution of theanalyzer. One can see that the elevation response goes tozero at +3 deg, as indicated by ELmax = +2.9 deg.

Figure 6 shows a distribution of final electron positionsto verify the focusing performance of this analyzer design.The final positions on the MCP are expressed as a functionof angle TH ′

1 = tan−1(yfinal/xfinal). The distribution is ob-tained by integrating over energy, area and elevation anglewith keeping AZ=0. The figure indicates that the electronsfocuses well on the MCP. The shaded area shows the regionwhich includes 80% of the total response, and the width ofthe 80% region W80% is 4.0 deg in the present case. This ismuch smaller than 22.5 deg of the width of a channel, andthus does not affect the azimuth-angle resolution. In fact theevaluated azimuth-angle resolution is 22.3 deg, very closeto 22.5 deg. Note that the width of 4.0 deg here is 0.2-degdifferent from 4.2 deg in Table 3. This small disagreementprobably comes from an error in W80% calculation and/orfrom the modifications made for the final design.

6. Estimation of UV Count RatesSince MCPs are sensitive to UV photons as well as

charged particles, it is important to estimate effects due toUV photons. In this section, UV photon count rates are es-timated by photon tracing simulations. Here we focus onUV photons with wavelengths shorter than approximately120 nm, because a photon detection efficiency of a bareMCP decreases as a wavelength goes above ∼120 nm (Mar-tin and Bowyer, 1982).

The Sun is obviously the main source of photons in Earthorbits. According to the reference solar UV spectrum byHeroux and Hinteregger (1978), the Lyman-alpha (Ly-α)


Fig. 7. Evaluated total photon count rates over all the MCP channels as a function of initial EL of photons.

line at 121.6 nm occupies ∼80% of the solar photon fluxesbelow 125 nm. Hence we take the Ly-α flux as the solarUV flux for an approximation. As a typical Ly-α intensityat 1 AU, a flux of 3×1011 /sec cm2 is often adopted for aero-nomical purposes (e.g. Vidal-Madjar, 1975). Lyman-alpha,or Ly-α (121.6 nm). As a typical Ly-α intensity at 1 AU, aflux of 3 × 1011 /cm2 sec is often adopted for aeronomicalpurposes (e.g. Vidal-Madjar, 1975). In near-Earth orbits,the second UV source is Earth’s geocorona (exospheric hy-drogen atoms), by which solar photons are scattered backto space. The UV fluxes coming from the geocorona werereported to be 22–35 kR in the daytime (Chakrabarti et al.,1983) and to be 1.7–3.6 kR in the nighttime (Chakrabartiet al., 1984). Here R is a photon flux unit “Rayleigh” and1 R is equivalent to 106/4π photons/sec cm2 sr. The geo-corona’s photon flux becomes ∼ 5 × 109 photons/sec cm2

if one assumes 10 kR as geocorona’s photon flux and thegeocorona’s solid angle as 2π . This value is approximatelytwo orders of magnitude smaller than the solar Ly-α flux.Therefore, in this estimation, we take the Ly-α solar UVphotons as an photon input flux.

Because the solar UV photon flux is extremely large com-pared to magnetospheric plasma fluxes, it is essential toprevent photons from reaching the MCP for correct plasmameasurement. In terms of this importance, photon suppres-sion should be taken into account at a designing stage ofanalyzer development. On the assumption of designing ananalyzer, a light-tight analyzer housing is necessary for pho-tons not to enter the inside of the analyzer. Then we can as-sume that all the photons come into an analyzer through itsaperture. Suppression of photons entering through the aper-ture is made by an electrostatic energy analyzer itself, inwhich charged particles are electrically deflected but pho-tons go straight and hit structure surfaces. Some photonsare then absorbed and the others reflect with a reflectioncoefficient. To suppress as many photons as possible, wemust increase a number of reflections of photons and im-

prove photon absorption on surface. In the present case, wedesigned the analyzer such that photons have at least two re-flections before reaching the MCP. Furthermore, in additionto fine serration structures, inner surfaces of the collimatorand the inner/outer electrodes are to be blackened by coppersulfide process.

Using photon tracing simulations, we estimate how manyphoton counts still remains to be detected. To save compu-tation time, the simulation uses a “weighted” photon. Eachphoton has a reflection count and it is incremented on eachreflection. If the photon reaches the MCP or the countreaches its limit, then that photon is removed and next trac-ing is started. Using this method, a UV photon count ratecan be estimated as:

Cph

�t=

(1

Ntotal

∑i

r nref,i

)jph S εph, (2)

where Ntotal is a total number of photons, and nref,i is areflection count for each photon. Diffuse (randomly di-rected) reflection is assumed with the reflection coefficientr of 2 × 10−2 for copper-sulfide surface according to Zur-buchen et al. (1995). The solar Ly-α flux jph is 3×1011 pho-tons/sec cm2, the MCP’s efficiency for photons εph is 1%,and the aperture area S is 2.79 cm2. Note that the fine serra-tion structures are fully reproduced in the simulations. SeeFig. 4 for the structure and the positions of the serrations.

The result of the photon simulation is given in Fig. 7 asa function of initial EL of photons. The count rates shownhere are integrated over all MCP channels. The calculationswere made at every 1 deg of EL from −10 deg to +10 deg.The initial AZ of photons is assumed to be zero. Note thatthe apparent diameter of the Sun is in fact ∼0.5 deg at 1 AU.However it does not change the results because the initialangular deviations are vanished by diffuse reflection.

According to the results, when the analyzer looks downat the Sun at EL = +2 deg, the analyzer receives thelargest UV count rate of 25.4 count/sec, which corresponds


Fig. 8. Trajectories of twice-reflected photons with EL = +2 deg.

Fig. 9. Trajectories of photons reflecting three times with EL = −2 deg.


to 1.59 count/sec per channel under the assumption thatthe final photon positions are equally distributed over chan-nels. At EL = +2 deg, a rejection ratio of photons (a ra-tio of a number of photons reaching the MCP to a num-ber of photons entering the aperture) is 3.0 × 10−9. Inspace electron measurement, electron count rates reach anorder of 104 count/sec per channel. Assuming that a typ-ical MCP dark count rate (a count rate with no input) is∼1 count/sec cm2, we expect the dark count rate to be∼0.4 count/sec per channel. Because the UV count rateis close to the MCP dark count level, we conclude that theestimated performance of UV photon suppression is suffi-cient.

It is worth to know how this rejection performance isachieved. Figure 8 illustrates trajectories of photons atEL = +2 deg. Note that the horizontal axis is a radialdistance in the X -Y plane with a sign of X component todraw three-dimensional trajectories. The result indicatesthat no photons can reach the MCP with one reflection, andthus trajectories of twice-reflected photons are shown in thefigure. Photons are reflected on the center part of the uppercollimator, then move down to be reflected on the outerelectrode, and finally reach the MCP. Accordingly, surfacestructure and process of the upper collimator plate and theouter electrode are important to reduce photon suppression.

Figure 9 gives trajectories of photons with EL = −2 degwhere the second highest peak is seen. No twice-reflectedphotons exist in this case, and the trajectories shown inthe figure are those of photons reflecting three times beforereaching the MCP. All of the first reflections happen at thetopmost edge of the outer shell; This is because we extendedthe collimator radius to block such paths directly towardthe outer shell electrode. The reflected photons are thenreflected twice on the outer shell and reach the MCP. Asa result of eliminating twice-reflected photon paths, thiscount rate is not as high as the one at EL = +2 deg.

In conclusion, the present analyzer design has potentialto suppress solar UV photons down to a few count/sec perchannel. This performance is achieved in combination with(1) copper sulfide surface processing, (2) fine serration pho-ton trap, and (3) analyzer design to block as many photonsas possible. It should be however emphasized that final con-firmation must be made by laboratory UV tests using a testmodel. The simulation is much simplified, and some of as-sumptions such as a reflection coefficient and an MCP de-tection efficiency are unpredictable.

7. SummaryIn this report, basics of designing an electrostatic energy

analyzer were described through computer simulations ofa toroidal top-hat energy analyzer. We modeled a top-hatanalyzer with four parameters, and parameter survey wasmade to find the optimum design. First, the deflection angleand the shell offset were determined to be 72.5 deg and5 mm, respectively, in the point of view of the locationof electron focusing. Second, we determined the uppercollimator height of 4.1 mm for focusing performance, andthen selected the lower collimator thickness of 2.0 mm tolimit the elevation-angle field of view. The g-factor of theanalyzer was also carefully considered during the parameter

survey.Finally, performances of electron measurement and UV

photon suppression were estimated by particle tracing sim-ulations. The measurement energy range and energy reso-lution, the g-factor, the field of view, and the angular res-olutions were evaluated and were confirmed to satisfy therequirements. According to the photon tracing simulations,the maximum photon count rate expected is 1.59 count/secper channel when the Sun is at EL = +2 deg. This countrate is roughly comparable to an MCP’s dark count rate, andis acceptable.

Appendix A.

. Geometric FactorsA particle count C in one energy channel during one

sampling time is expressed as:

C = −∫

T (K , �, x) (J(K , �, t) · dS) d� dK dt (A.1)

where K is an energy, J is a vector of the differential fluxof particles, x is a position on the aperture. Here, T isthe function that expresses detection of particles with acondition of (K , �, x).

T (K , �, x) ={

1 if detected0 otherwise.

(A.2)

Figure A.1 illustrates how this integration is made.Assuming that the flux is constant over (1) time of sam-

pling, (2) field of view of the analyzer and (3) energy withinthe energy bin, the count is then simplified to:

C = −J�t∫

T (K , �, x)(j · n)dS d� dK , (A.3)

where dS = ndS and J = J j.Here the integral in the expression

G E (k) ≡ −∫

T (K , �, x)(j · n) dS d� dK (A.4)

is called “energy geometric factor” in cm2 sr keV, which isdefined only by analyzer’s geometry.

Then, the count rate C/�t becomes

C

�t= J GE. (A.5)

Note that GE changes if a tuned energy is changed. In thecase of an electrostatic energy analyzer, G E is proportionalto an energy. Therefore it can be written as:

GE = G 〈k〉, (A.6)

where G [cm2 sr keV/keV] is “geometric factor” (g-factor),and a mean energy 〈k〉 is defined by

〈k〉 =∫

kT (K , �, x)(j · n) dSd�dK∫T (K , �, x)(j · n) dSd�dK

. (A.7)

Taking elevation angle α and azimuth angle β of a ve-locity vector as defined in Fig. 2, we can denote dS =


Fig. A.1. Schematic illustration for g-factor integration. A g-factor iscalculated by integrating a transmission function T over energy K ,aperture S and field of view �.

Rd�dz, d� = cos αdαdβ, (j · n) = − cos α cos(β − �).Here R is the radius of the aperture in the x-y plane (con-stant, 45 mm in the present case), and � is an angle of anaperture position defined by tan � = y/x .

Thus an energy geometric factor can be

GE

= R∫

T (K , α, β, �, z) cos2 α cos(β − �) dK dαdβd�dz.

(A.8)

By using the Monte-Carlo integration method, the energygeometric factor can be obtained by summing up trajectoryparameters as

GE ∼(

V

N

)R

N∑i

Ti cos2 αi cos(βi − �i ), (A.9)

where V is a total integration volume in the phase space of(K , α, β, �, z), and N is a number of points taken in thephase space. After calculating the average energy 〈k〉, wecan obtain the g-factor G = GE/〈k〉.

. Numerical Potential CalculationThis section explains how to calculate potential fields

numerically in the cylindrical coordinates. Here we assumesame grid spacing in the r and z directions.

By taking a difference of the Poisson equation inthe cylindrical coordinates ∂2φ/∂r2 + (1/r)(∂φ/∂r) +∂2φ/∂z2 = 0, we obtain the difference equation

φi, j =

1

4

(φi−1, j + φi, j−1 + φi+1, j + φi, j+1

)+ 1

8i

(φi+1, j + φi−1, j

)i �= 0

1

4

(φ1, j + φ0, j−1 + φ1, j + φ0, j+1

)i = 0

(A.10)where i and j are indexes of grids in the r and z axes,respectively. Note that φ0, j is located exactly on the z axis.To suppose the +r and ±z boundaries as free boundary

where ∂φ = 0, one takes a beyond-boundary potential valuesame as on-boundary one.

To solve the difference equation, we employ the succes-sive over-relaxation (SOR) method, in which a potential φi, j

is renewed in series with neighboring potentials in doubly-nested loops over i and j . A new potential φ

(n+1)i, j is given

as

φ(n+1)i, j = 1

4

(φ

(n+1)

i−1, j + φ(n+1)

i, j−1 + φ(n)

i+1, j + φ(n)

i, j+1

)+ 1

8i

(φ

(n)

i+1, j + φ(n+1)

i−1, j

). (A.11)

Here suffix (n) represents a calculation count, and (n + 1)

and (n) mean a new value and a current value, respectively.It should be emphasized that a new value is calculated withboth new and current neighboring values. As a result, everycalculation step is made in-place and no temporary memoryis required.

A new potential value at (i, j) is finally given as:

φ(n+1)i, j = ω

(φ

(n+1)i, j − φ

(n)i, j

)+ φ

(n)i, j , (A.12)

where ω is a constant to accelerate converging and 1.95 wastaken in the present case. This calculation is repeated untilthe condition is satisfied:

max

{ |φ(n+1)i, j − φ

(n)i, j |

|φ(n+1)i, j | + φtol

}< εacc. (A.13)

In the equation, εacc is an accuracy of calculation (for exam-ple, 10−4) and φtol is a tolerance voltage which correspondsto the maximum absolute potential not to affect the result(for example, 0.1 V).

. Time Step Control in Tracing TrajectoriesWe use the traditional 4th-order Runge-Kutta method

with adaptive time step control to numerically solve theequation of motion. A time step is determined in every step-ping by target accuracy εacc and maximum step size Smax.The basic idea of controlling a time step in this section isbased on “Numerical Recipes in C” by Press et al. (1993).

Here x0, x1, v0, v1 and �t denote initial/final positionsand velocities and a time step, respectively. The maximumdifference between one stepping and two half-steppings rel-ative to a step size is written as:

δ = max

∣∣∣x(2· 12 )

1 − x(1)

1

∣∣∣|x0| +

∣∣∣∣dxdt

∣∣∣∣ �t,

∣∣∣v(2· 12 )

1 − v(1)

1

∣∣∣|v0| +

∣∣∣∣dvdt

∣∣∣∣ �t

, (A.14)

where x(2· 1

2 )

1 , v(2· 1

2 )

1 , x(1)

1 , v(1)

1 are final positions and veloci-ties by two half-steppings and one stepping, respectively.

If δ ≤ εacc and |x1 − x0| < Smax, the next time step �t ′

can be increased as:

�t ′ = min

{(δ

εacc

)1/5

�t, 4�t

}, (A.15)

where the expansion is limited by a factor of 4 for stability.


Otherwise, recalculation is made with the decreased timestep until the time step increase has occurred. The newdecreased time step is calculated as:

�t ′ = min

{(δ

εacc

)1/4

�t, 1.051/4�t, 0.95Smax

|x1 − x0|�t

}.

(A.16)Here 1.05 is a minimum shrinking factor, and 0.95 is amargin for safety.

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Designing a toroidal top-hat energy analyzer for low ... · energy analyzer for space particle measurement. In this re-port, a top-hat-type energy analyzer is taken and its design