The column of electron electrostatic accelerator is one of the critical components in electrostatic accelerator. The geometrical design of such accelerator must be as such that in the case of applying voltage to its electrodes, not only should its equipotential surfaces and its gradient accelerate the beam particles up to desired energy, but also it should focus the beam and hinder broadening of energy distribution of accelerated particles. The immersed electrodes in the field are, geometrically, perpendicular to optical axis around the medial plane. Numerous models that can be used in the distribution of axial potential, have been presented and linear model, analytical model, double-column electrode model and polynomial electrode model are among them. In this paper, series expansions based on Bessel functions is used to obtain the axial potential distribution of immersed accelerator electrodes in double-electrode field and it is then compared to the mentioned models by solving the final equation via the least square method. Finally, by using CST Studio software and the information we obtained from the axial potential, the column of electron accelerator with its energy distribution and its optimal electron output beam radius is designed and simulated.
Application of accelerator is common in many fields such as research, education, industry, medicine [
In addition to identifying electrostatic lenses with their electrodes, they can also be verified by their axial potential distribution; this is due to the fact that majority of the focusing elements which have axial symmetry in ion optics or electron optics can be denoted by a scalar potential function [
d 2 σ d z 2 + 3 16 [ U ′ ( z ) U ( z ) ] 2 σ = 0 (1)
which U ( z ) and U ′ ( z ) are axial potential and field, respectively. σ denotes the path of particles [
In order to calculate the axial potential of a lens immersed in field, we consider two flat planes which each of them have holes in them with the diameter of D and they are separated from each other with distance A. The applied voltages to these planes (electrodes) are V 1 and V 2 which have been calculated relative to potential of electrons with zero kinetic energy. In the upcoming equations the thickness of planes have been neglected (if the thickness is smaller than A and D, the dependence of lens parameters to thickness will be negligible). It is also required to consider three hypothetical cylinders with diameter D ′ which cylinders in regions I and III have potential V 1 and V 2 whereas the middle cylinder possess the linear potential running from V 1 to V 2 in its length.
It might be argued that having the middle cylinder in a uniform potential V m (which, for instance, is equal to V 1 or V 2 or their average) is more in accordance with real lenses; however this is not the case for two reasons: firstly, if D ′ ≲ 2 A , then such middle cylinder will alter the axial potential distribution between the holes and it would be necessary to calculate the lens parameters for different values of D ′ and V m ; secondly, if D ′ ≳ 2 A , the lens parameters would partly be analogous to the case in which the potential of middle cylinder is changing linearly, but the calculations will get longer and the accuracy will deplete.
Despite the fact that solution procedure of Laplace equation is a familiar simplification method, more precise and reliable results can be obtained faster with high degree of accuracy by modification of its expansion. It seems that the method used in these calculations will serve more precisely and strongly for wide range of problems with cylindrical symmetry. The series expansion for the three regions in
V I ( r , z ) = V 1 + ∑ n A n exp ( k n z ) J 0 ( k n r ) (2)
#Math_28# (3)
V III ( r , z ) = V 2 + ∑ n C n exp ( − k n z ) J 0 ( k n r ) (4)
Which the equations above are written in cylindrical coordinate and z is measured form central point of the lens. J 0 ( x ) is the zero order Bessel function. Each term of these expansions is a solution of Laplace equation with no boundary conditions. The signs which are associated to exponential variables are chosen in a manner that the boundary conditions in z = ± ∞ are automatically
satisfied. Boundary condition in r = 1 2 D ′ would be fulfilled if the values of
K n become limited which then we obtain:
J 0 ( 1 2 k n D ′ ) = 0 (5)
Now the only task that remains is to meet the boundary conditions in
z = ± 1 2 A , in addition we will have to make sure that the potential stays conti-
nuous all over the two holes.
From orthogonally condition for Bessel functions and from the condition:
V I ( r , − 1 2 z ) = V II ( r , − 1 2 z ) for 0 ≤ r ≤ 1 2 D ′ (6)
the following will result for all values of n:
A n exp ( − 1 2 k n A ) = B n exp ( 1 2 k n A ) + B ′ n exp ( − 1 2 k n A ) (7)
This equation will also guarantee that the radial component of the field remain continuous. The boundary condition for the left-hand-side electrode for
1 2 D ≤ r 2 ≤ 1 2 D ′ will obviously be [
∑ n { B n exp ( 1 2 k n A ) + B ′ n exp ( − 1 2 k n A ) } J 0 ( k n r 2 ) = 0 (8)
And for 0 ≤ r 1 ≤ 1 2 D ′ we will have [
∑ n B n k n exp ( 1 2 k n A ) J 0 ( k n r 1 ) = V 2 − V 1 2 A (9)
The same is also applied to second hole. Due to the symmetry around central point, equations B ′ n = − B n and C n = − A n will be true. Although Equations (8) and (9) are precisely satisfied when infinite number of terms is considered, we can still hope that finite number N of terms, the difference between the right and left side of these equations will rationally approach zero. Thus, N number of R will be selected within 0 to D ′ and N obtained equation of (8) and (9) will be used to calculate N coefficient of B n . This process will not converge by increasing N but instead, will lead to. For r values amongst the chosen ones, Equations (8) and (9) will poorly meet.
To overcome this problem many useful methods can be used but the least square method proves to be the most effective one and thus was used. In our problem, the least square method will contain forming the quantity.
S = ∑ r 2 [ ∑ n = 1 N B n { exp ( 1 2 k n A ) − exp ( − 1 2 k n A ) } J 0 ( k n r 2 ) ] 2 + ∑ r 1 { ∑ n = 1 N B n k n exp ( 1 2 k n A ) J 0 ( k n r 1 ) − V 2 − V 1 2 A } 2 (10)
And then obtaining N equation from ∂ S ∂ B n = 0 condition for 1 ≤ n ≤ N :
∑ r 2 [ ∑ j = 1 N 2 B j { 2 sinh ( 1 2 k j A ) } J 0 ( k j r 2 ) ] { { 2 sinh ( 1 2 k i A ) } J 0 ( k i r 2 ) } + ∑ r 1 2 { ∑ j = 1 N B j k j exp ( 1 2 k j A ) J 0 ( k j r 1 ) − V 2 − V 1 2 A } k i exp ( 1 2 k i A ) J 0 ( k i r 1 ) = 0 (11)
Next for calculating B n coefficients the equation can be solved. By writing a Fortran code, which can be generalized to more electrodes, and selecting V 1 = 100 V , V 2 = 60 V , d = 3 cm and N = 20 unknown coefficients will be attained numerically via the above equations and by substituting them in Equations (1) and (3), the potential will be specified for double-electrode electrostatic lens which is shown in
It should be noted that the values of r 1 ( 0 , 1 2 D ) and r 2 = ( 1 2 D , 1 2 D ′ ) must
be greater than N (about 3N in practice). This will result in converging solutions for Equations (8) and (9) and axial potential V ( 0 , z ) which in this case will converge by increasing N.
In potential distribution of double-electrode systems, there exists a bending point Z m which its axial component of field ( | U ′ | max ) reaches to its maximum pure value. This point can be located in either the geometrical center or any other point and this is the reason why those two symmetric lenses can be defined [
precisely coincident with the average value of a and b, some models have been proposed to specify the asymmetric potential distribution around Z m . Different models have been presented for axial potential distribution of this system which linear model, analytical model, double-cylinder lens model, and polynomial lens model are some to mention.
Axial potential distribution for immersed lens in field has been displayed in
In order to simulate the column of accelerator, the outer radius of electrodes has taken the value of 15 centimeters; therefore we consider the distance between electrodes to be 5 centimeters so as to have the optimal state for column of the accelerator. The other restrictive factor in choosing the distance between electrodes is formation of arc and electric refraction. The distance between the electrodes can be determined by Equation (12) [
V = C X α (12)
where V is the potential difference of the two consecutive electrodes, α is a constant value between 0.1 and 1.1 and C is also a constant value which depends on the intensity of the electric field, distance between electrodes and their material. Majority of systems assumes α to be between 0.5 and 0.7. Considering the applied potential to the planes and by obeying the 5 cm-rule between the electrodes, one can avoid electric discharge. Insulators made of borosilicate glass
or electrical ceramics (alumina-based) can be put in between the electrodes of column of accelerator to avoid occurrence of electric discharges. Moreover, they can generate the required vacuum (about 10 - 5 torr in lab) for the column of accelerator.
Simon and Superfish computer codes [
Since we want the beam current to be in the order of a few microamperes in
this accelerator column, Perveance beam K = ( I V 3 2 ) is negligible. Therefore,
we can disregard the spatial charge effects in designing electrodes. In
i.e. the first, third, fourth, …, last electrodes will have potentials of −300 kV, −275 kV, −250 kV, …, 0 kV relative to earth, respectively.
The second electrode which acts as the focusing lens of the column has variable voltage of +10 kV relative to the first one. Titanium has the lowest field emission, micro-discharge, arc-driven surface abrasion, best stable voltage, most work function and electric gradient in low distances. Due to these reasons a titanium-based material can be determined in designing electrodes. Nevertheless, the machining and burnishing process of titanium is more costly and burdensome in comparison with other metals. Thus, aluminum is suitable option in designing. The overall number of 14 electrodes will be implemented. For
Trajectory of particles is shown in
As it can be seen from
In the second geometry for optimizing of application of accelerator column, the electrodes can be considerated ridged that is shown in
The path of the particles for the second geometry in
In case of needing an electron beam with radius of lower than 6 mm, an electric quadrupole lens can be located at the exiting direction of electron beam to control the output beam radius. In this software, complete vacuum has been regarded for beam trajectory and since this is not achievable in reality, it would be better to set the total voltage of accelerator column to more than −300 kV.
For designing and manufacturing the accelerator tube should have comprehensive
information of the electrostatic lenses for electrons, protons and ions under the influence of an electrical potential gradient that these lenses create, will be accelerated and reaches to final desired speed. As well as focusing the beam of charged particles is also the responsibility of these lenses. In the case of controlling
low-energy beam, two electrodes electrostatic lenses are used.
Using an axial potential distribution which can be characterized by an electrostatic lens, can predict how charged particle beams move. Numerical calculations show that the best situation for lens that makes output electron beam energy maximum and focal is when electrodes diameter are the same and the rate of distance between electrodes to diameter is 0.1 to 0.3. First geometry that is designed reaches to 8 mm of output beam diameter and 270 keV of energy but when we use ridged electrodes, diameter of output beam will decrease to 6 mm and energy will increase to 300 keV. Of course it is predictable by comparing equipotential surfaces in
Karanian, S.I., Mogaddam, R.R., Seifnouri, E. and Amiri, M. (2017) Conceptual Design of Beam Tube of 300 KeV Electron Electrostatic Accelerator. Journal of Applied Mathematics and Physics, 5, 1763-1775. https://doi.org/10.4236/jamp.2017.59149