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WDS'08 Proceedings of Contributed Papers, Part III, 44–49, 2008.
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Modelling of Trochoidal Electron Monochromator:
Influence of the Deceleration Potential on Electron
Beam
J. Matúška and Š. Matejčı́k
Department of Experimental Physics, Comenius University Bratislava, Mlynská dolina F2,
84248 Bratislava, Slovakia
Abstract. Trochoidal electron monochromator (TEM) which uses crossed homogeneous electric and magnetic fields has been investigated in a numerical simulation
using the SIMION 8.0 package. The electron trajectory calculations have been
carried out for large number of initial electrons with finite energy spread and
angular distribution function. The electron energy distribution function (EEDF) of
the TEM has been studied as a function of the monochromator parameters.
Introduction
The beams of mono-energetics electrons are required in many experimental studies concerning electron interactions with the molecules in the gas phase or on surfaces. Electron beams with
well defined and narrow distribution functions (high energy resolution of the electron beams)
can be formed with the help of electron monochromators.
There exist several types of the electron monochromators which operation is based on
different physical principles. The most popular types are the electrostatic monochromators
(in cylindrical or spherical geometry) [Roy and Burrow, 1975; Jost, 1979] and the trochoidal
electron monochromator (TEM) [Stamatovic and Schulz, 1970]. At low electron energy, the
effect of electrostatic repulsion of the electrons may play a significant role in the electrostatic
monochromators. This problem can be avoided if axial magnetic field is applied, however, the
presence of the magnetic fields is harmful for the operation of the electrostatic monochromators.
On the other hand, the trochoidal electron monochromator is based on the dispersive properties
of the perpendicular homogeneous electric and magnetic fields. The magnetic field is parallel to
the axis of the monochromator. For this reason the TEM is popular in applications where low
energy electron beams are required [Johnston and Burrow., 1983; Cloutier and Sanche, 1989;
Zubek, 1994; Allan, 1982; Illenberger, 1992; Matejcik et al., 1997].
The principle of operation of the TEM has been investigated earlier [Roy, 1972; Romanyuk
and Shpenik, 1994; Williams and O’Neill, 1995]. These analytical studies were performed only
for the simplified case neglecting transversal velocities of the electrons and fringing field. Grill
et al. [2001] carried out three dimensional study of the TEM, however, only in very limited range
of electron energies and angles. In present paper, we discus the basic attributes of the TEM
operation on the basis of numerical simulation of the TEM. Using the SIMION 8.0 3D charge
particle optics program, detailed three dimensional numerical calculations of the electrostatic
fields in the TEM have been performed. The electric fields were calculated for the real geometry
of the electrodes. The electric field thus includes the fringing fields at apertures of the electrodes.
The trajectories have been calculated for a large number of electrons in different initial state
(kinetic energy and direction) simulating the thermionic emission of the electrons from the
hot filament. The geometry of the simulated TEM was identical with the TEM used in our
laboratory [Matejcik et al., 2003].
There is no systematic study of the influence of the deceleration voltage in the crossed field
region on the TEM’s parameter, however this is commonly used technique to improve the full
width on half maximum (FWHM). The aim of this study was to determine the dependence of
the FWHM, electron current and the shape of the electron energy distribution function (EEDF)
on the deceleration voltage inside the selector of TEM to improve functionality of the TEM.
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MATÚŠKA AND MATEJČÍK: MODELLING OF TROCHOIDAL ELECTRON MONOCHROMATOR
Physical principles of electron separation
The dispersive element of the TEM is shown in Figure 1. The separation of the electrons
~ and magnetic B
~
occurs in the space with crossed perpendicular homogeneous electrostatic E
fields. The electric field produced by a pair of parallel plain electrodes (length L in the axial
direction) has only the z component. The magnetic field formed by the Helmholtz coils has
only the x component and is axially aligned with the original direction of the electrons. The
velocity v~0 of the electrons entering the dispersive element is parallel to the magnetic field i.e.,
to has x axis. The trajectory of the electrons in such fields can be obtained as a solution of the
differential equation:
d2~r
~ − e~v × B
~
= −eE
(1)
dt2
There exist analytical solution for this system of equations [Stamatovic and Schulz, 1970;
Grill et al., 2001]. In the y direction exists a constant drift velocity v~D , which depends only on
the magnitude of the electric and magnetic fields and is velocity independent:
me
~ ×B
~
E
(2)
B2
The drift velocity can be used to separate electrons according to the magnitude of the x
component of the initial velocity. The fast electrons spend relatively short time in the dispersive
element and thus at the end of the element only small deflection in y direction is observed. On
other hand the slow electrons spend more time in the dispersive element and thus their deflection
in y direction is larger. The apertures at the beginning and at the end of the crossed field space
have different y coordinates. Thus only electrons with proper initial velocity and with narrow
energy spread may leave the dispersive element. More details on the analytical theory of the
TEM are given in papers [Stamatovic and Schulz, 1970; Grill et al., 2001].
The influence of the TEM on EEDF is schematically illustrated in Figure 2. There are
a initial EEDF (exponential curve) and a final EEDF (Gaussian curve) after passing of the
dispersive element. The base principle of the TEM is that TEM is cutting out a part of the
initial electrons to obtain the narrow energy distribution. In ideal case, the final EEDF is a
Gaussian distribution. In fact, the final EEDF in TEM has a shape similar to Figure 3. Here
we can define the full width and the FWHM. The ratio of the full width to the FWHM is equal
v~D =
Figure 1. The dispersive element of the TEM monochromator.
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MATÚŠKA AND MATEJČÍK: MODELLING OF TROCHOIDAL ELECTRON MONOCHROMATOR
Figure 2. The schematic change of the EEDF after passing monochormator. The initial EEDF
is a exponencial curve and the final EEDF is a Gaussian curve. The final EEDF is created from
the initial EEDF by cutting off the part of the electrons.
Figure 3. The final EEDF obtained by calculation. Two important parameters are showed.
The FWHM anf the full width of the EEDF.
2 for a well designed instruments [Stamatovic and Schulz, 1970], but the value up to 3 is also
acceptable for practical use.
This analytical model illustrates only basic attributes of TEM monochromator and can
not describe all properties like the influence of the inhomogeneities in the electric fields, the
trajectories of the electrons with different initial angular orientation and the initial energy
distribution function of the electrons. In order to get better understanding of the TEM operation
we have decided to perform numerical trajectory simulation.
Numerical simulation
The numerical simulation of the TEM has been performed using the charge particle optics
program SIMION 3D 8.0. This software is a powerful instrument for numerical calculations of
the electrostatic fields for a system of electrodes and is able to calculate the ion and electron
trajectories in the system of electrostatic and magnetic fields.
The EEDF of the electron beam formed in the TEM is determined (i) by the initial distribution function of electrons emitted from the filament and (ii) by the dispersive properties of
the electron monochromator.
The electron source of the real TEM is directly heated filament. (The electron source of
the real TEM is a hairpin filament directly heated by passing electric current.) The electrons
are emitted from the tip of the filament by thermionic emission. In present simulation we have
approximated the filament by a point and we assume that the initial distribution functions of the
electrons is determined by the thermionic emission. The initial EEDF has thus an exponential
character.
Ek
(3)
f (Ek ) ≈ exp(− )
kT
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MATÚŠKA AND MATEJČÍK: MODELLING OF TROCHOIDAL ELECTRON MONOCHROMATOR
Figure 4. The picture of the simulated trochoidal electron monochromator.
where f (Ek ) is the relative abundance of the electrons with kinetic energy Ek , T is the temperature of metal. The angle was changed from 90 down to -90 degrees in both directions with
step 8 degrees. We have assumed that temperature of the hairpin filament is 2000 K.
The simulated TEM has the geometry and the dimensions of the TEM built at Department
of Experimental Physics, Comenius University [Stano et al., 2003]. The detailed design including
reaction chamber is shown in Figure 4. The electrodes made from stainless steel are 1.2 mm
thick and the distances between them are 1.0 mm. The apertures of the electrodes in the model
are cylindrical in contrast to the funnel shape of the apertures in the real monochromator.
The funnel shape apertures are used in order to decrease the charging of the apertures due to
adsorption of the molecules on them. In present model we neglect this effect. The diameter of
the apertures is 1.0 mm. The dispersive element of TEM is 20 mm long. The entrance aperture
of the dispersive element is 2 mm off axis. The exit aperture is exactly in the centre of the
electrode (Figure 1).
The electric and magnetic field in SIMION is modeled as boundary value problem solution
of the Laplace equation. The method used to solve is finite difference technique called overrelaxation. Simulation has been carried out on a grid (10 points per mm) and each point
represents the value of the electric and magnetic field in the area around it. The the value
of the electric and magnetic field is used to calculate the acceleration of the particle on their
position. Then the trajectory of the particle is obtained using the fourth order Runge-Kutta
integration algorithm of the motion equation.
In the simulations we have not considered the mutual repulsion (Coulomb repulsion) of the
electrons. We neglect this effect due to the fact that we have been using magnetic field in axial
direction which prevents the repulsion.
Results and discussion
In present work we have performed trajectory calculations of the electrons trough the TEM.
The number of the electrons which reach reaction chamber and their energy have been recorded
and evaluated. A histogram obtained on this way correspond to the EEDF of the TEM in
reaction chamber.
On the base of previous consideration we have decided to investigate the influence of the
deceleration potential on the exit electrode of the TEM dispersive element on final EEDF.
We used tree parameters to describe the final EEDF. In Figure 5 we see the depencence of the
calculated FWHM, electron current and the shape (characterised by ratio of the full width to the
FWHM) on deceleration voltage. This dependences was calculated for four diferent dimensions
of the entrance apertures. The primary atribute of the TEM is the FWHM. The dependence
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MATÚŠKA AND MATEJČÍK: MODELLING OF TROCHOIDAL ELECTRON MONOCHROMATOR
of the FWHM on deceleration voltage is showed in Figure 5a. There are minima for for each
diameter of the entrance aperture at the value -0.10 V. The minimum is more eminent for the
smallest entrance aperture. When the deceleration voltage -0.10 V is applied, the FWHM is
improved by approximately 20%. Only in the case of the smallest entrance aperture we obtained
more eminent improvement. The only disadvantage of this procedure is very significant decrease
of the transmitted electron current (Figure 5b). According the Figure 5a, FWHM can achieve
lower values like value for the deceleration voltage -0.10 V, but it is not so clear. The Figure 5c
showed that the ratio of the full width to the FWHM obtain values over 3 or under 2 and the
transmission electron current very low. These values mean, that the shape of the final EEDF is
irregular. Thus, for practical use the maximum deceleration voltage is -0.10 V. We obtain the
value of the FWHM
Figure 5. The dependence of the properties of the EEDF on the voltage of the exit electrode in
TEM. The value of magnetic induction is 50 gauss and the electric field is 1.67 V/m. Acceleration
potential is 1 V
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MATÚŠKA AND MATEJČÍK: MODELLING OF TROCHOIDAL ELECTRON MONOCHROMATOR
Conclusion
Using the numerical simulation we have calculated the EEDF of the electrons in the electron
beam formed in the TEM. Our results showed that the optimal deceleration potential across
the dispersive element in TEM on Comenius university is -0.10 V.
Acknowledgments. This work was supported by Science and Technology Assistance Agency
under contract No. APVT-20-007504.
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