Download Designing a toroidal top-hat energy analyzer for low-energy electron measurement Y. Kazama

An Introduction to Space Instrumentation,
Edited by K. Oyama and C. Z. Cheng, 181–192.
Designing a toroidal top-hat energy analyzer for
low-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-hat
energy analyzer. First, a top-hat analyzer is parameterized with four parameters: the deflection angle, the shell
electrode offset, the upper collimator height and the lower collimator thickness. Then three steps of parameter
surveys are made by particle trajectory tracing simulations. In the surveys, the requirements of g-factor, azimuthangle resolution and field of view are taken into accout. After the surveys, we confirm that the final design
meets the requirements. In addition, UV photon suppression performance is also evaluated by photon tracintg
simulations. The expected maximum photon count rate 1.59 count/sec is acceptable, compared to the MCP’s
background count rates. The analyzer design investigated in this report is to be taken over by a low-energy
electron instrument (LEP-e) on the radiation-belt observation satellite ERG.
Key words: Space instrumentation, plasma instrument basics, low-energy electron instrument, top-hat energy
analyzer.
1.
Introduction
The purpose of this report is to give an introduction for
those who are not familiar with designing an electrostatic
energy analyzer for space particle measurement. In this report, a top-hat-type energy analyzer is taken and its design is
investigated by parameter surveys. After determining the final design, performances of particle measurement and photon suppression are estimated. The top-hat analyzer discussed here will be a prototype for the low-energy electron
instrument (LEP-e) onboard the radiation-belt observation
mission ERG (Shiokawa et al., 2006). Detailed descriptions on a wide spectrum of space particle instrumentation
are found in Pfaff et al. (1998a, b) and Wüest et al. (2007).
2.
Basics of Electrostatic Analyzers
2.1 Energy-voltage relation
To learn basics of electrostatic energy analyzers, we start
with an ideal analyzer of spherical type. Assuming two concentric spherical electrodes and a particle moving exactly
along the center of the electrode gap, we can express the
kinetic energy of the charged particle K 0 as:
Vo Ro2 − Vi Ri2
K0
,
=
q
Ro2 − Ri2
(1)
where Vo , Vi , Ro and Ri are the potentials and the radii of the
outer and inner electrodes, respectively. Note that K 0 is the
kinetic energy of the particle at infinity where no potential
exists.
First of all, it is obvious that an electrostatic energy analyzer cannot measure masses of charged particles because
there is no mass dependence in the equation. To measure
masses, another method is necessary, such as magnetic-field
c TERRAPUB, 2013.
Copyright deflection, time variation of electric fields, time-of-flight
measurement, and so on.
Second, K 0 and q only appear in the form of K 0 /q. This
means that we cannot measure a particle energy itself; A
He+ ion with a kinetic energy of 10 keV behaves exactly
same as He++ of 20 keV, and one cannot discriminate them
by an electrostatic analyzer alone. For this reason, some
literature 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 of
charged particles passing the analyzer simply becomes proportional to the single voltage. This proportionality indicates trajectory invariance; The trajectory of a 5-keV particle with 1-kV voltage setting is identical to the one of a
10-keV particle with 2-kV voltage setting.
Finally, the energy-voltage relation does not change by
scaling Ro and Ri by a factor a (Ro , Ri → a Ro , a Ri ) with
keeping the voltage. This means that particle trajectories
scale as the geometry scales. Therefore, analyzer’s properties related to particle energy and trajectory are constant
over the scaling. It is, however, obvious that other factors,
for example, timing of particles and a size of an aperture are
scaled and changed.
2.2 Geometric factor
Here we describe a sensitivity of an analyzer, that is,
how many particles pass through the analyzer per unit
time. A number of passing particles C during a time interval t is proportional to a particle differential flux J as
C = G E J t, where contains other factors such as a detection efficiency of the detector device. The coefficient G E
[cm2 sr keV] is called “energy geometric factor” and represents the sensitivity of the analyzer. Roughly speaking,
G E can be expressed as ∼ S K , where S is the effective aperture area, is the effective solid angle of the field
of view, and K is the energy passband of the analyzer.
Geometric factor is discussed in Appendix A. A detailed
181
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Y. KAZAMA: DESIGNING TOROIDAL TOP-HAT ANALYZER
Fig. 1. Schematic illustration of the top and cross-sectional views of
a toroidal top-hat analyzer. Typical particle trajectories are drawn to
indicate azimuth-angle focusing and wide particle acceptance.
description is found in Wurz et al. (2007).
In the case of electrostatic analyzers, K is proportional to a tuned energy K 0 because of the proportionality of electrostatic analyzer.
Therefore, an
energy-independent value, “geometric factor” (hereafter “gfactor”) G [cm2 sr keV/keV] = G E /K 0 is often used for such
systems.
3.
Top-Hat-Type Analyzer
A top-hat-type electrostatic analyzer was first investigated by Carlson et al. (1983). The analyzer consists of two
concentric hemispherical electrodes, and the top part of the
outer electrode is cut out as a particle entrance. Analyzers
of this type are sometimes referred to as “spherical top-hat
analyzer”, and have been employed for space plasma measurements (e.g. Paschmann et al., 1985; Lin et al., 1995;
Rème et al., 1997).
Based on spherical top-hat analyzer, Young et al. (1988)
proposed “toroidal top-hat analyzer”, in which two toroidal
shells are used instead of two concentric spherical shells,
as schematically illustrated in Fig. 1. The two parallel
plates on the top form a collimator to define a field of
view of the analyzer. A charged particle enters the analyzer
from the aperture into the collimator, and is deflected by an
electric field into the gap of the shell electrodes. Particles
with energies matched to the shell voltage can pass through
the shells, which results in energy selection of the charged
particles. As seen in the figure, initial velocity directions of
particles can be distinguished by exit particle positions.
As pointed out by Young et al. (1988), a toroidal tophat analyzer is characterized by long focal length in the az-
imuth direction and large sensitivity, relative to a spherical
top-hat analyzer. In the case of spherical top-hat analyzers,
azimuth-direction focusing occur inside the shell electrodes
due to its short focal length, and particles have started defocusing at the analyzer exit. Relatively large sensitivity of
toroidal top-hat analyzer is an advantage for space instruments; We can shrink the analyzer to decrease the mass of
the instrument while keeping the sensitivity requirements.
Figure 2 is a three-dimensional cut-away view of the
toroidal top-hat-type electrostatic energy analyzer to be discussed 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 rotationally symmetric with respect to the z axis. We assume
that the analyzer is mounted on the spacecraft such that the
symmetric axis is perpendicular to the spacecraft spin axis.
Note that the definitions of elevation/azimuth angles are different from those in Young et al. (1988).
The analyzer has two parallel plates on the top, which
constitute a collimator to define the field of view of the
analyzer. A positive voltage is applied to the inner electrode
of toroidal shape (shown in red in the figure) for electron
measurement. Only electrons with energies matched to the
applied voltage can pass through the shells (as shown by
the red line in the figure), and the others are lost by hitting
analyzer walls. An energy spectrum of electrons is taken by
sweeping the voltage.
The energy-selected electrons are finally detected by
multi-channel plates (MCPs) shown in blue. An MCP is
a thin lead glass plate with capillaries with a diameter from
∼1 to ∼ tens µm. A high voltage is applied perpendicular to the plate and the electric field produces cascading of
secondary 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 assembly (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 secondary
electron release, MCP is sensitive to any particle which
produces secondary electrons, such as charged particles,
UV photons and radiations. For this reason, in the case
of space instruments, we have to pay attention to effects
due to radiations and UV photons. UV photon effects are
discussed later in this report.
Since a top-hat analyzer has a 360-deg field of view in
the azimuth direction, electrons simultaneously come in to
the analyzer in all the azimuth directions. In principle,
a parallel electron flux at some azimuth angle focuses on
one point after 90-deg deflection due to analyzer’s focusing property. As a result, resolving azimuth directions of
electrons is made by position sensing on the MCP. Therefore, azimuth-angle resolution is determined by analyzer’s
focusing performance and MCP’s positioning resolution.
The field of view in the elevation direction is, on the
other hand, as narrow as several degrees, as seen in Fig. 2.
Accordingly, the elevation-angle resolution of this analyzer
is identical to the field of view. Hence three-dimensional
measurement is made by sweeping this narrow field of view
Y. KAZAMA: DESIGNING TOROIDAL TOP-HAT ANALYZER
183
upp
per collimator
e-
lower collimator
MCP assem
mbly
+Z
V
EL
inner electrode
MCP
AZ
+X
-Y
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 and
azimuth angle AZ are defined as indicated.
Cross Section
40
30
Z [mm]
20
10
0
-10
-20
-40
-30
-20
-10
0
10
R*sign(X) [mm]
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 we
assume the spin axis perpendicular to the z axis.)
4.
Instrument Requirements
4) azimuth-angle resolution of ∼22.5 deg (16 channels
over 360 deg),
5) g-factor of (5±1)×10−4 cm2 sr keV/keV per 22.5-deg
azimuth angle.
In addition to the electron measurement requirements
The purpose of our analyzer is to observe suprathermalstated
above, the instrument has practical requirements:
to-hot electrons populated in the Earth’s inner magnetosphere. To fulfill the observation, instrument performance
1) to limit the field of view to the elevation angle of
is supposed as follows:
> +3 deg,
2)
to
be radiation hard for instrument operation in the
1) energy range from ∼10 eV to >∼15 keV,
radiation
belts,
2) energy resolution of <15%,
3)
to
suppress
photon count rates down to
3) elevation-angle resolution of <∼4 deg,
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Y. KAZAMA: DESIGNING TOROIDAL TOP-HAT ANALYZER
b [mm]
K 0 [keV]
0.0
5.60
2.5
5.10
5.0
4.80
7.5
4.30
Table 1. Summary of the parameters c, K 0 and z f for given a and b.
a [deg]
70
71.25
72.5
75
c
[mm]
—
—
—
2.0
zf
[mm]
—
—
—
−3.0
c
[mm]
3.9
—
3.8
3.9
zf
[mm]
−10.2
—
−6.5
−2.2
c
[mm]
—
4.8
4.9
5.0
zf
[mm]
—
−11.3
−9.4
−5.9
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 the
upper collimator height c.
c
mm
3.5
3.7
3.9
4.1
4.3
G
cm2 sr keV/keV
5.35 × 10−4
6.35 × 10−4
7.29 × 10−4
8.24 × 10−4
8.88 × 10−4
W80%
deg
6.8
6.6
6.8
7.2
7.4
Table 3. G-factor G, focusing performance W80% and elevation-angle
limit of the field of view ELmax as a function of the lower collimator
thickness d.
d
mm
1.0
1.4
1.6
1.8
2.0
G
cm2 sr keV/keV
6.4 × 10−4
6.6 × 10−4
5.9 × 10−4
5.2 × 10−4
4.5 × 10−4
W80%
deg
7.2
5.9
5.4
4.6
4.2
ELmax
deg
+4.0
+3.5
+3.4
+3.1
+2.9
<∼10 counts/sec per channel.
About the first point, we must avoid artificial electrons
such as photoelectrons from the spacecraft body. Then
we limit the field of view in the elevation direction below
+3 deg. In this condition, the analyzer does not see the
spacecraft body within the radius of ∼1 m if the aperture
height is 50 mm from the spacecraft surface.
Radiation hardness is crucial for low-energy plasma observation in the Earth’s radiation belts. It is a challenge for
the present analyzer to minimize radiation effects without
coincidence/anti-coincidence techniques. This topic is beyond the scope of this report and will be discussed elsewhere.
The last point, photon suppression, is requested for electron signals not to be masked by photon signals. Because
typical count rates of electrons populated in the inner magnetosphere can be several thousands count/sec per channel,
the target photon count rate of <10 count/sec keeps a good
signal-to-noise ratio. We estimate photon count rates and
the results are discussed later.
5.
Designing of a Top-Hat Analyzer
To design an analyzer, we take two steps of numerical
analysis: (1) to find the optimum design for electron mea-
77
1.8
2.1
80
1.5
7.7
4.8
8.7
6.5
3.1
—
—
surement and (2) to evaluate photon count rates. For the
numerical analysis, we use a cylindrical-coordinates potential solver and a particle tracer programs on a personal computer. A potential field is obtained by solving the Poisson
equation by the successive over-relaxation (SOR) method.
Tracing of a particle is made using the traditional 4th-order
Runge-Kutta method with adaptive time step control. The
time step control and the SOR method are described in Appendix A.
5.1 Parameterization of the design
A toroidal top-hat analyzer design is parameterized with
four parameters a, b, c, d, as shown in Fig. 3. Here a is
the deflection angle of the energy analyzer, b is the offset
of the shell center from the symmetric axis. c is the height
of the upper collimator measured from the topmost edge of
the outer shell, and d is the thickness of the lower collimator
plate.
First we set the gap between two shells to 3 mm. Then
the diameter of the exit aperture, which corresponds to the
size of the MCP placed on the bottom, is determined to be
70 mm. This is because we suppose to use 77-mmφ MCPs
provided by Hamamatsu Photonics. The voltage of the
inner electrode is fixed at 1 kV in the following parameter
survey.
5.2 Deflection angle and offset of the shells
In this section, we describe the first step of the parameter
survey, in which the parameters a, b and c are surveyed in
the viewpoint of azimuth focusing depth and of g-factor. As
discussed previously, a parallel beam of electrons focuses
after exiting the shells in theory. To have a better azimuthangle resolution, the focusing position should be as close to
the MCP input surface at z ∼ −13.5 mm as possible. At
the same time, a higher g-factor is also essential not only to
meet the measurement requirement but also to keep room
for further design improvements.
The first survey is made to find the optimum parameter
set (a, b, c) under d = 0. For given a and b, we find center
energy K 0 , with which an electron being back-traced from
the bottom (z = −17 mm) goes along the center of the
shell gap. Then we find c which lets that electron go out of
the analyzer in the horizontal direction along the collimator
plates. A trajectory example is shown in Fig. 3 as the red
line.
Table 1 gives the survey’s result, in which c, K 0 and z f are
tabulated for given a and b. Here z f is a z-position where
a parallel beam focuses after exiting the shells. a is taken
from 70 deg to 80 deg, and b from 0 mm to 7.5 mm. In
Y. KAZAMA: DESIGNING TOROIDAL TOP-HAT ANALYZER
185
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 dashes
indicate that no electrons can go out of the analyzer parallel
to the collimator plates.
In the table, center energy K 0 only depends on b. This
is because K 0 is determined only by the curvature radius of
the shells defined by b. In terms of higher-energy electron
measurement, higher K 0 is preferable. According to the
result, K 0 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 as
follows; For larger b, the curvature radius becomes smaller,
since the exit aperture diameter of 70 mm is fixed. If the
curvature radius is small, the center energy must be small
to go along the shells.
The result shows that large b has large c. This is because
a large b has a small K 0 as mentioned above, and an electron
does not need a large downward electric field by which the
electron enters the shell gap. Looking at a certain b, one can
see that c increases as a increases. This can be explained by
the inlet angle of the shells; Since the inlet angle in a largera model is closer to the horizontal direction, less electric
field is necessary for a horizontally-moving electron to go
into 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 an
electric field to deflect an electron near the z axis in the
case of a = 80 deg.
About focus height z f , smaller a results in a lower z f , because focusing of a parallel beam occur at 90-deg deflection
in principle, and the beam virtually needs more deflection
angle to focus in the smaller-a cases.
It is expected that large c has large g-factor due to its large
aperture, because large-b designs can have space where an
electric 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 better
azimuth-angle resolution (focusing closer to the MCP), and
(3) larger b for larger g-factor. Here we decide to take the
parameters a = 72.5 deg and b = 5 mm as the best compromise. Based on this geometry, we check the parameters
c 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 collimator
height c. The parameter c controls both g-factor and azimuth focusing. Therefore, the goal of defining c is to find
a good balance between g-factor and focusing.
To evaluate g-factor, here we assume more realistic analyzer geometry: (1) The MCP input surface is located at
z = −13.5 mm and +200 V is applied, (2) a mesh is placed
above the MCP and −12 V is applied, (3) a slit at the shell
exit is added to suppress scattering particles, and (4) the
lower collimator thickness d is set to 1 mm. The position
of the MCP input surface is determined due to the size of
the MCP assembly. In fact, we need space for an assembly
substrate, insulators, mesh support, etc. The input surface
voltage of +200 V accelerates electrons above 200 eV, at
which an MCP has the maximum detection efficiency for
electrons.
The mesh works as a repeller for eV-range electrons; The
mesh’s negative voltage prevents stray electrons existing in
the energy analyzer from reaching the MCP input surface.
Simultaneously, the mesh repels secondary electrons escaping from the MCP back to the input surface to enhance MCP
detection efficiency. It is noted that this mesh voltage limits the minimum energy of electron measurement, 12 eV in
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Y. KAZAMA: DESIGNING TOROIDAL TOP-HAT ANALYZER
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 electrode
surfaces and may reach the MCP after multiple scattering.
The width of the slit is 2 mm, 1 mm narrower than the gap
of the shells.
Table 2 gives g-factor G and focusing performance W80%
for each c. Here W80% is a width of azimuth angle which
covers 80% of the total azimuth-angle response for a parallel (AZ = 0) beam input. Naturally smaller W80% stands for
better focusing performance. G-factor is hereafter defined
for one MCP channel (22.5-deg wide in azimuth) unless
otherwise noted. It is seen that the g-factor G decreases as c
decreases because the aperture size becomes smaller, as expected. However, W80% has a peak at c = 3.7 mm. This can
be explained as follows: At c = 4.9 mm, z f is −9.4 mm according to the previous result in Table 1. This indicates that
electrons focus above the MCP at z = −13.5 mm. When c
becomes smaller, the focusing point moves downward and
approaches the MCP. This is probably due to the shift of
electron’s deflection center by changing the velocity direction at the entrance of the shells. Accordingly, focusing is
made at the MCP position when c = 3.7 mm.
Although the best focusing is seen at c = 3.7 mm, here
we take c = 4.1 mm, considering the g-factor which is more
sensitive to c than focusing performance. This high g-factor
is necessary for further improvements which will diminish
g-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-angle
field of view not to see a spacecraft body. We calculated
g-factor G, focusing performance W80% and maximum elevation angle ELmax by changing d from 1 mm to 2 mm. The
result is tabulated in Table 3. Here ELmax is defined as the
elevation angle which includes 99.9% of the elevation-angle
response.
As d increases, the g-factor G decreases obviously because of limiting the field of view. On the other hand, W80%
improves as d increases. This improvement comes from
electrons entering the analyzer from the side fringes of the
aperture, that is, electrons with large |yinitial | values (remember that the analyzer’s symmetric axis is parallel to z and the
beam comes along x.) These electrons focus on z-positions
higher than the MCP surface and degrade the focusing performance due to their defocusing at the MCP. Increasing d
makes a thicker “neck” of the analyzer, which blocks these
electrons more. This results in the better focusing seen at
larger d.
To satisfy the requirement that the elevation field of view
Y. KAZAMA: DESIGNING TOROIDAL TOP-HAT ANALYZER
187
Table 4. Summary of analyzer performance parameters.
Parameter
G-factor
Analyzer constant
Energy resolution
Field of view
azimuth
elevation
Angular resolution
azimuth
elevation
Fig. 6. Focusing of an electron input with a fixed azimuth direction. The
angle TH 1 represents final positions of electrons on the MCP.
should be limited to < +3 deg, we take d of 2.0 mm, at
which ELmax = +2.9 deg. The design with d = 2.0 mm
has the g-factor of 4.5 × 10−4 cm2 sr keV/keV, which satisfies the required g-factor of (5 ± 1) × 10−4 cm2 sr keV/keV
per 22.5 deg.
5.5 Detailed performance estimation
In the previous section, all the parameters of the analyzer
have 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 illustrated
in Fig. 4. For the final design, two further modifications
are made in terms of UV photon suppression: (1) extension
of the collimator radius and (2) fine serration photon trap.
First, the collimator radius is extended to 45 mm (originally
42 mm) to block the UV photon paths directly toward the
outer shell surface from the aperture. This point will be
discussed later. Second, fine serration structures are made
on the collimator plates to effectively absorb UV photons.
A sawtooth structure traps photons in its deep “valley” of
the structure. The serrations are made on (1) the inner
surface 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 present
case. This final analyzer design is verified by simulations
and the performance parameters evaluated are summarized
in Table 4. The energy-elevation response and the focusing
performance are described below.
Figure 5 shows energy-elevation response obtained by
the simulation. It is seen that energy and elevation angle
are not independent. For example, higher-energy electrons
enter the aperture downward to draw larger radii of trajectories inside the analyzer, and vice versa. By integrating the
energy-elevation response, the g-factor is evaluated to be
4.2 × 10−4 cm2 sr keV/keV, which meets the requirement of
∼ (5 ± 1) × 10−4 . It is noted that the g-factor is slightly
smaller than that in Table 3. This is probably due to the
extension of the collimator radius which limits more elec-
Value
4.2 × 10−4
4.97
8.0
cm2 sr keV/keV
keV/kV
% (FWHM)
360
2.91
deg
deg (FWHM)
22.3
2.91
deg (FWHM)
deg (FWHM)
trons. The mean energy k is 4.97 keV for 1 kV of the
inner shell voltage. Since a mean energy is proportional to
an analyzer voltage in the case of electrostatic energy analyzer, we can define an analyzer constant (ratio of a mean
energy per analyzer voltage), 4.97 keV/kV in this analyzer
case. Assuming a safety field strength of 1 kV/mm, we can
apply 3 kV to the analyzer electrode, since the shell gap is
3 mm. Hence the maximum energy in electron measurement is about 15 keV. This maximum energy limit can be
relaxed if no discharge is confirmed at >3 kV in laboratory
tests. The energy passband is calculated to be 0.40 keV
FWHM at 5 keV, and the relative energy resolution k/k
becomes 8.0%, better than the requirement. The elevationangle field of view is 2.91 deg FWHM. Note that this field
of view is identical to the elevation-angle resolution of the
analyzer. One can see that the elevation response goes to
zero at +3 deg, as indicated by ELmax = +2.9 deg.
Figure 6 shows a distribution of final electron positions
to verify the focusing performance of this analyzer design.
The final positions on the MCP are expressed as a function
of angle TH 1 = tan−1 (yfinal /xfinal ). The distribution is obtained by integrating over energy, area and elevation angle
with keeping AZ =0. The figure indicates that the electrons
focuses well on the MCP. The shaded area shows the region
which includes 80% of the total response, and the width of
the 80% region W80% is 4.0 deg in the present case. This is
much smaller than 22.5 deg of the width of a channel, and
thus does not affect the azimuth-angle resolution. In fact the
evaluated azimuth-angle resolution is 22.3 deg, very close
to 22.5 deg. Note that the width of 4.0 deg here is 0.2-deg
different from 4.2 deg in Table 3. This small disagreement
probably comes from an error in W80% calculation and/or
from the modifications made for the final design.
6.
Estimation of UV Count Rates
Since MCPs are sensitive to UV photons as well as
charged particles, it is important to estimate effects due to
UV photons. In this section, UV photon count rates are estimated by photon tracing simulations. Here we focus on
UV photons with wavelengths shorter than approximately
120 nm, because a photon detection efficiency of a bare
MCP decreases as a wavelength goes above ∼120 nm (Martin and Bowyer, 1982).
The Sun is obviously the main source of photons in Earth
orbits. According to the reference solar UV spectrum by
Heroux and Hinteregger (1978), the Lyman-alpha (Ly-α)
188
Y. KAZAMA: DESIGNING TOROIDAL TOP-HAT ANALYZER
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 fluxes
below 125 nm. Hence we take the Ly-α flux as the solar
UV flux for an approximation. As a typical Ly-α intensity
at 1 AU, a flux of 3×1011 /sec cm2 is often adopted for aeronomical purposes (e.g. Vidal-Madjar, 1975). Lyman-alpha,
or Ly-α (121.6 nm). As a typical Ly-α intensity at 1 AU, a
flux of 3 × 1011 /cm2 sec is often adopted for aeronomical
purposes (e.g. Vidal-Madjar, 1975). In near-Earth orbits,
the second UV source is Earth’s geocorona (exospheric hydrogen atoms), by which solar photons are scattered back
to space. The UV fluxes coming from the geocorona were
reported to be 22–35 kR in the daytime (Chakrabarti et al.,
1983) and to be 1.7–3.6 kR in the nighttime (Chakrabarti
et al., 1984). Here R is a photon flux unit “Rayleigh” and
1 R is equivalent to 106 /4π photons/sec cm2 sr. The geocorona’s photon flux becomes ∼ 5 × 109 photons/sec cm2
if one assumes 10 kR as geocorona’s photon flux and the
geocorona’s solid angle as 2π . This value is approximately
two orders of magnitude smaller than the solar Ly-α flux.
Therefore, in this estimation, we take the Ly-α solar UV
photons as an photon input flux.
Because the solar UV photon flux is extremely large compared to magnetospheric plasma fluxes, it is essential to
prevent photons from reaching the MCP for correct plasma
measurement. In terms of this importance, photon suppression should be taken into account at a designing stage of
analyzer development. On the assumption of designing an
analyzer, a light-tight analyzer housing is necessary for photons not to enter the inside of the analyzer. Then we can assume that all the photons come into an analyzer through its
aperture. Suppression of photons entering through the aperture is made by an electrostatic energy analyzer itself, in
which charged particles are electrically deflected but photons go straight and hit structure surfaces. Some photons
are then absorbed and the others reflect with a reflection
coefficient. To suppress as many photons as possible, we
must increase a number of reflections of photons and im-
prove photon absorption on surface. In the present case, we
designed the analyzer such that photons have at least two reflections before reaching the MCP. Furthermore, in addition
to fine serration structures, inner surfaces of the collimator
and the inner/outer electrodes are to be blackened by copper
sulfide process.
Using photon tracing simulations, we estimate how many
photon counts still remains to be detected. To save computation time, the simulation uses a “weighted” photon. Each
photon has a reflection count and it is incremented on each
reflection. If the photon reaches the MCP or the count
reaches its limit, then that photon is removed and next tracing is started. Using this method, a UV photon count rate
can be estimated as:
Cph
1 n ref,i
=
jph S ph ,
r
(2)
t
Ntotal i
where Ntotal is a total number of photons, and n ref,i is a
reflection count for each photon. Diffuse (randomly directed) reflection is assumed with the reflection coefficient
r of 2 × 10−2 for copper-sulfide surface according to Zurbuchen et al. (1995). The solar Ly-α flux jph is 3×1011 photons/sec cm2 , the MCP’s efficiency for photons ph is 1%,
and the aperture area S is 2.79 cm2 . Note that the fine serration structures are fully reproduced in the simulations. See
Fig. 4 for the structure and the positions of the serrations.
The result of the photon simulation is given in Fig. 7 as
a function of initial EL of photons. The count rates shown
here are integrated over all MCP channels. The calculations
were made at every 1 deg of EL from −10 deg to +10 deg.
The initial AZ of photons is assumed to be zero. Note that
the apparent diameter of the Sun is in fact ∼0.5 deg at 1 AU.
However it does not change the results because the initial
angular deviations are vanished by diffuse reflection.
According to the results, when the analyzer looks down
at the Sun at EL = +2 deg, the analyzer receives the
largest UV count rate of 25.4 count/sec, which corresponds
Y. KAZAMA: DESIGNING TOROIDAL TOP-HAT ANALYZER
Fig. 8. Trajectories of twice-reflected photons with EL = +2 deg.
Fig. 9. Trajectories of photons reflecting three times with EL = −2 deg.
189
190
Y. KAZAMA: DESIGNING TOROIDAL TOP-HAT ANALYZER
to 1.59 count/sec per channel under the assumption that
the final photon positions are equally distributed over channels. At EL = +2 deg, a rejection ratio of photons (a ratio of a number of photons reaching the MCP to a number of photons entering the aperture) is 3.0 × 10−9 . In
space electron measurement, electron count rates reach an
order of 104 count/sec per channel. Assuming that a typical 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 rate
is close to the MCP dark count level, we conclude that the
estimated performance of UV photon suppression is sufficient.
It is worth to know how this rejection performance is
achieved. Figure 8 illustrates trajectories of photons at
EL = +2 deg. Note that the horizontal axis is a radial
distance in the X -Y plane with a sign of X component to
draw three-dimensional trajectories. The result indicates
that no photons can reach the MCP with one reflection, and
thus trajectories of twice-reflected photons are shown in the
figure. Photons are reflected on the center part of the upper
collimator, then move down to be reflected on the outer
electrode, and finally reach the MCP. Accordingly, surface
structure and process of the upper collimator plate and the
outer electrode are important to reduce photon suppression.
Figure 9 gives trajectories of photons with EL = −2 deg
where the second highest peak is seen. No twice-reflected
photons exist in this case, and the trajectories shown in
the figure are those of photons reflecting three times before
reaching the MCP. All of the first reflections happen at the
topmost edge of the outer shell; This is because we extended
the collimator radius to block such paths directly toward
the outer shell electrode. The reflected photons are then
reflected twice on the outer shell and reach the MCP. As
a result of eliminating twice-reflected photon paths, this
count rate is not as high as the one at EL = +2 deg.
In conclusion, the present analyzer design has potential
to suppress solar UV photons down to a few count/sec per
channel. This performance is achieved in combination with
(1) copper sulfide surface processing, (2) fine serration photon trap, and (3) analyzer design to block as many photons
as possible. It should be however emphasized that final confirmation must be made by laboratory UV tests using a test
model. The simulation is much simplified, and some of assumptions such as a reflection coefficient and an MCP detection efficiency are unpredictable.
survey.
Finally, performances of electron measurement and UV
photon suppression were estimated by particle tracing simulations. The measurement energy range and energy resolution, the g-factor, the field of view, and the angular resolutions were evaluated and were confirmed to satisfy the
requirements. According to the photon tracing simulations,
the maximum photon count rate expected is 1.59 count/sec
per channel when the Sun is at EL = +2 deg. This count
rate is roughly comparable to an MCP’s dark count rate, and
is acceptable.
7.
Note that G E changes if a tuned energy is changed. In the
case of an electrostatic energy analyzer, G E is proportional
to an energy. Therefore it can be written as:
Summary
In this report, basics of designing an electrostatic energy
analyzer were described through computer simulations of
a toroidal top-hat energy analyzer. We modeled a top-hat
analyzer with four parameters, and parameter survey was
made to find the optimum design. First, the deflection angle
and the shell offset were determined to be 72.5 deg and
5 mm, respectively, in the point of view of the location
of electron focusing. Second, we determined the upper
collimator height of 4.1 mm for focusing performance, and
then selected the lower collimator thickness of 2.0 mm to
limit the elevation-angle field of view. The g-factor of the
analyzer was also carefully considered during the parameter
Appendix A.
. Geometric Factors
A 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 flux
of particles, x is a position on the aperture. Here, T is
the function that expresses detection of particles with a
condition of (K , , x).
1 if detected
T (K , , x) =
(A.2)
0 otherwise.
Figure A.1 illustrates how this integration is made.
Assuming that the flux is constant over (1) time of sampling, (2) field of view of the analyzer and (3) energy within
the 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 is
defined only by analyzer’s geometry.
Then, the count rate C/t becomes
C
= J G E.
t
G E = G k,
(A.5)
(A.6)
where G [cm2 sr keV/keV] is “geometric factor” (g-factor),
and a mean energy k is defined by
kT (K , , x)(j · n) dSddK
k = .
(A.7)
T (K , , x)(j · n) dSddK
Taking elevation angle α and azimuth angle β of a velocity vector as defined in Fig. 2, we can denote dS =
Y. KAZAMA: DESIGNING TOROIDAL TOP-HAT ANALYZER
191
where ∂φ = 0, one takes a beyond-boundary potential value
same as on-boundary one.
To solve the difference equation, we employ the successive over-relaxation (SOR) method, in which a potential φi, j
is renewed in series with neighboring potentials in doublynested loops over i and j. A new potential φ̃i,(n+1)
is given
j
as
1 (n+1)
(n+1)
(n)
(n)
φ̃i,(n+1)
=
+
φ
+
φ
+
φ
φ
j
i, j−1
i+1, j
i, j+1
4 i−1, j
1 (n)
(n+1)
+
(A.11)
φi+1, j + φi−1,
j .
8i
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 with
Fig. A.1. Schematic illustration for g-factor integration. A g-factor is both new and current neighboring values. As a result, every
calculated by integrating a transmission function T over energy K , calculation step is made in-place and no temporary memory
aperture S and field of view .
is required.
A new potential value at (i, j) is finally given as:
Rddz, d = cos αdαdβ, (j · n) = − cos α cos(β − ).
φi,(n+1)
= ω φ̃i,(n+1)
− φi,(n)j + φi,(n)j ,
(A.12)
j
j
Here R is the radius of the aperture in the x-y plane (constant, 45 mm in the present case), and is an angle of an
where ω is a constant to accelerate converging and 1.95 was
aperture position defined by tan = y/x.
taken in the present case. This calculation is repeated until
Thus an energy geometric factor can be
the condition is satisfied:
(n+1)
GE
|φi, j − φi,(n)j |
max
< acc .
(A.13)
= R T (K , α, β, , z) cos2 α cos(β − ) dK dαdβddz.
|φi,(n+1)
| + φtol
j
(A.8) In the equation, acc is an accuracy of calculation (for exam-
ple, 10−4 ) and φtol is a tolerance voltage which corresponds
By using the Monte-Carlo integration method, the energy to the maximum absolute potential not to affect the result
geometric factor can be obtained by summing up trajectory (for example, 0.1 V).
parameters as
V
N
N
. Time Step Control in Tracing Trajectories
We use the traditional 4th-order Runge-Kutta method
with adaptive time step control to numerically solve the
equation of motion. A time step is determined in every stepwhere V is a total integration volume in the phase space of ping by target accuracy acc and maximum step size Smax .
(K , α, β, , z), and N is a number of points taken in the The basic idea of controlling a time step in this section is
phase space. After calculating the average energy k, we based on “Numerical Recipes in C” by Press et al. (1993).
can obtain the g-factor G = G E /k.
Here x0 , x1 , v0 , v1 and t denote initial/final positions
and velocities and a time step, respectively. The maximum
. Numerical Potential Calculation
difference between one stepping and two half-steppings relThis section explains how to calculate potential fields ative to a step size is written as:
numerically in the cylindrical coordinates. Here we assume


1

(2· 12 )
(2· 2 )
same grid spacing in the r and z directions.
(1) (1) 

 x1 − x1 
v1 − v1 
By taking a difference of the Poisson equation in
δ
=
max
,
, (A.14)
dx 

the cylindrical coordinates ∂ 2 φ/∂r 2 + (1/r )(∂φ/∂r ) +

t |v0 | + dv t 


|x
|
+
0
dt dt ∂ 2 φ/∂z 2 = 0, we obtain the difference equation
 (2· 1 )
(2· 1 )
1
(1)

where x1 2 , v1 2 , x(1)

φi−1, j + φi, j−1 + φi+1, j + φi, j+1
1 , v1 are final positions and veloci

4

ties by two half-steppings and one stepping, respectively.
1 φi, j =
+
i = 0
φi+1, j + φi−1, j
If δ ≤ acc and |x1 − x0 | < Smax , the next time step t 
8i


can be increased as:

 1 φ1, j + φ0, j−1 + φ1, j + φ0, j+1 i =0
4
δ 1/5
(A.10)
t = min
t, 4t ,
(A.15)
where i and j are indexes of grids in the r and z axes,
acc
respectively. Note that φ0, j is located exactly on the z axis.
To suppose the +r and ±z boundaries as free boundary where the expansion is limited by a factor of 4 for stability.
GE ∼
R
i
Ti cos αi cos(βi − i ),
2
(A.9)
192
Y. KAZAMA: DESIGNING TOROIDAL TOP-HAT ANALYZER
Otherwise, recalculation is made with the decreased time Press, W. H., B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling
(1993), Numerical Recipes in C (Japanese edition), Cambridge Universtep until the time step increase has occurred. The new
sity Press, Cambridge, U.K.
decreased time step is calculated as:
Rème, H., J. M. Bosqued, J. A. Sauvaud, A. Cros, J. Dandouras,
C. Aoustin, J. Bouyssou, Th. Camus, J. Cuvilo, C. Martz, J. L. Mèdale,
1/4
H. Perrier, D. Romefort, J. Rouzaud, C. D’Uston, E. Möbius,
δ
S
max
t = min
t, 1.051/4 t, 0.95
t .
K. Crocker, M. Granoff, L. M. Kistler, M. Popecki, D. Hovestadt,
acc
|x1 − x0 |
B. Klecker, G. Paschmann, M. Scholer, C. W. Carlson, D. W. Curtis,
R. P. Lin, J. P. Mcfadden, V. Formisano, E. Amata, M. B. Bavassano(A.16)
Cattaneo, P. Baldetti, G. Belluci, R. Bruno, G. Chionchio, A. Di Lellis,
Here 1.05 is a minimum shrinking factor, and 0.95 is a
E. G. Shelley, A. G. Ghielmetti, W. Lennartsson, A. Korth, H. Rosenmargin for safety.
bauer, R. Lundin, S. Olsen, G. K. Parks, M. Mccarthy, and H. Balsiger
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Y. Kazama (e-mail: [email protected])