This work reports a numerical method for single charged particle trajectories computation in 2D electrostatic

and magnetostatic stationary fields, in other words, fields that do not change in time. This is approached by

the finite element method domain discretisation, and numerical computation of particle trajectory, calculated

by the two step centred in time method, which calculates the particle position on the next step using a dummy

step in order to increase the accuracy for the same step size. Given particle's coordinates, the finite element

that contains that particle is found based on Lohner's algorithm. The examples used to test the method are a

electric deflector for the electric case and cyclotron for the magnetic case. Both are very important devices to

science and technology, being used in a variety of domestic and industrial appliances and in several scientific

and technologic researches. Other particle optics devices can benefit of the method proposed in this paper, as

beam bending devices and spectrometers, among others. This method can be easily extended for particle

trajectories computation in 3D domains, can be extended also for dynamic fields and for the relativistic case,

which is ideal for the typical speed involved when working with particles near the atomic level

Carvalho, Mairo Cunha de

Jospin, Reinaldo Jacques

Carpe dIEN

2009 International Nuclear Atlantic Conference – INAC 2009
Rio de Janeiro, Rj, Brazil, September 27 to October 2, 2009
ASSOCIAÇÃO BRASILEIRA DE ENERGIA NUCLEAR – ABEN
ISBN: 978-85-99141-03-8
PARTICLE TRAJECTORY TRACING FOR ELECTROSTATIC AND 
MAGNETOSTATIC FIELDS.
Mairo Cunha de Carvalho¹ and Reinaldo J. Jospin¹
¹Instituto de Engenharia Nuclear (IEN / CNEN - RJ)
Rua Hélio de Almeida n. 75 – Cid. Universitária – Ilha do Fundão – Cx.P: 68.550
21945-970 Rio de Janeiro, RJ
mairocc@gmail.com
rj.jospin@ien.gov.br 
ABSTRACT
This work reports a numerical method for single charged particle trajectories computation in 2D electrostatic 
and magnetostatic stationary fields, in other words, fields that do not change in time. This is approached by 
the finite element method domain discretisation, and numerical computation of particle trajectory, calculated 
by the two step centred in time method, which calculates the particle position on the next step using a dummy 
step in order to increase the accuracy for the same step size. Given particle's coordinates, the finite element 
that contains that particle is found based on Lohner's algorithm. The examples used to test the method are a 
electric deflector for the electric case and cyclotron for the magnetic case. Both are very important devices to 
science and technology, being used in a variety of domestic and industrial appliances and in several scientific 
and technologic researches. Other particle optics devices can benefit of the method proposed in this paper, as 
beam bending devices and spectrometers,  among others.  This method can be easily extended for particle 
trajectories computation in 3D domains, can be extended also for dynamic fields and for the relativistic case, 
which is ideal for the typical speed involved when working with particles near the atomic level.
1. INTRODUCTION
This work presents the two step centred time method to calculate the trajectory described 
by charged particles under the influence of an electric or magnetic field. These fields are 
calculated using the finite element method [2] with first order triangular elements that leads 
to the discrete  formulation  of the corresponding Maxwell  differential  equation [3].  The 
field  values  in  the  2D  cartesian  referential  systems  are  obtained  using  the  Lohner’s 
algorithm [1]. The method is a numeric one that was chosen inspired on the Yasmara De 
Poli work [4].
The  method  presented  here  can  calculate  particle  trajectories,  and  other  variables,  like 
momentum and speed with a good precision. To ascertain this paper, it will be compared 
with the already tested analytical method [5].
In section 2, the method will be presented with it's algorithm and equations for each case 
studied.  The  results  obtained  in  the  numerical  examples  will  be  compared  with  those 
obtained with the analytical method [5]. In section 3, these results will be commented and 
some future work will be enhanced.
2. ALGORITHM AND RESULTS
In this chapter the two step centred in time method [6] is explained and the numerical results 
are presented in some test cases.
2.1. Algorithm
The two steps centred in time method follows a simple principle, very similar to the Euler 
method [6]. Knowing the starting point and the equation of the trajectory tangent in any 
point, the trajectory can be approached by a succession of straight lines, each one leading to 
a new point, where another tangent value will be used etc.
The starting point is the starting position of the particle and the equation for the tangent is 
an equation for the speed of the particle in any point. The position is given and the speed is 
achieved using the given force field value and the particle properties.
For  the  calculations  to  become  faster  and  to  require  a  lesser  computational  effort,  the 
momentum equation is used instead of speed.
With the starting position and the starting momentum, a dummy step will be calculated at a 
step of t/2, where a new momentum will be assigned and used to calculate the next position. 
This new position will be plotted in the graph, while the half step will be ignored, being 
used just to increase the precision of the method.
2.2. Equations for the Method
The  formulation  of  this  method  is  similar  for  both  electric  and  magnetic  cases.  The 
essential  differences  like  the  speed  dependence  and  the  tendency  of  not  changing  the 
absolute momentum value that occurs in the magnetic case is not present in the electric 
case.
2.2.1. Electric case
For the electric case, each direction's movement is independent to the others, the following 
equations are used for each one of them in each step. Dummy steps are used but only the 
real steps are kept.
The classical value of force is given below [7]:
                                                                   F=q E                                                              (1)
where q the charge value and E the electric field.
From (1), the momentum is calculated as:
                                                               p= p0q E t                                                         (2)
where p is the momentum and t is the time step length for the presented method.
One defines a step sufficiently small so that the momentum can be considered constant, so, 
the position with respect to the momentum is:
                                                                  r=r0
p
m
t                                                          (3)
for a particle mass m.
2.2.2. Magnetic case
In the magnetic case, the force field will affect the direction of the particle trajectory. That 
means, each component of the momentum will interact with the force applied to the other 
particles [7]:
                                                               F=q v×B                                                             (4)
for a particle with speed v under influence of a magnetic field B.
From the force, the momentum can be deduced [4]
                                               px=p0
q
c [ p0ym B z− p0zm B y]t                                            (5)
                                               py= p0
q
c [ p0zm Bx− p0xm B z]t                                            (6)
                                               pz= p0
q
c [ p0ym Bx− p0xm B y]t                                            (7)
Where   is the Lorentz transformation factor.
The z-axis momentum equation is used in this formulation to show how easy the method 
can be converted to a 3D environment. In the 2D case, the same equation can be used, since 
the z-momentum component is considered null.
Dependence between different components may be noticed, as it was said before. Again, 
the  momentum  can  be  considered  constant  for  small  steps,  and  the  position  for  any 
component will be:
                                                                 r=r0
p
m
t                                                           (8)
2.3. Results
In the following, we present the results obtained with the two steps centred in time method, 
for the electric and magnetic cases and the comparison between them and results obtained 
through the analytical method.
2.3.1. Electric case
An  example  of  electrostatic  deflector  was  used  in  order  to  test  the  method  in  the 
electrostatic  case.  Electrostatic  deflectors  are  charged  structures  inside  which  an 
electrostatic field will interact with the charged particles that pass through there, increasing 
the momentum component orthogonal to the walls of the deflector.
The studied case is a simplest  rectangular  two-dimensional  deflector.  The difference of 
potential in the walls of this deflector is 5000V, the lower and upper walls are 0.02m distant 
from each other. The particle within is an electron and the positively charged wall is the 
lower one.
Fig. 1 was plotted from the analytical method while Fig. 2. and Fig. 3. were plotted from 
the two steps centred in time method.
Figure 1. Trajectory of a charged particle in a 
deflector plotted from the analytical method
Figure 2. Trajectory of a charged particle in a 
deflector plotted form the two steps centred in 
time method with a small step size
Comparing Fig. 1 and Fig. 2 a significant difference in the particle deflection can be seen. 
A sensitive part of this difference is due to the numeric instability that is bigger when the 
step  size  increases.  In  this  example  the  effects  of  the  instability  do  not  cause  major 
problems, this is because the trajectory is short enough so the deviation is not so intense.
To solve this problem the step size must be decreased. In this case, the precision of the 
method will  increase.  There  is  an ideal  step size so that  the precision is  good and the 
numerical instability is not significant. This size tends to be as the same order so that to 
reach the required precision. Fig. 3 shows the trajectory plotted for such step size.
Figure 3. Trajectory of a charged particle in a 
deflector plotted form the two steps centred in 
time method with a proper step size
Comparing  Fig.  1  and  Fig.  3  the  difference  perceived  in  Fig.  2.  is  largely  vanished, 
suggesting the two methods exhibit a good agreement. In order to check the accordance, 
follows some numerical results.
Table 1. Numerical results to the deflector case
Method Deflection
Analytical 4.90271. 10−3
Time Centred Two Steps 4.68434.10−3
Table 1 compares the results of each method. In the simulation, a 478 elements and 280 
nodes mesh was used. The difference in the results is small, so both methods have a good 
agreement.
2.3.2. Magnetic case
To test  the two steps centred in  time method for this  case a cyclotron  was used as an 
example. A cyclotron is a structure that causes charged particles to describe a trajectory that 
resembles a circumference under the influence of a magnetic field, making possible to keep 
the  particle  accelerating  for  a  long  time.  For  it  is  constrained  in  a  limited  area  in  a 
predictable trajectory.  Being so, the compelling potential can be applied only where the 
particle is, saving space and increasing the efficiency of the compelling force.
Fig. 4 was plotted from the analytical method while Fig. 5 was plotted from the two steps 
centred in time method.
Figure 4. Trajectory of a charged particle in a 
cyclotron plotted from the analytical method. 
Figure 5. Trajectory of a charged particle in a 
cyclotron plotted from the two steps centred in 
time method with the proper step size.
Comparing Fig. 4 and Fig. 5 a good agreement can be noticed. The curve is approximately 
the same.
3. CONCLUSIONS
The two steps centred in time method presents a good approach to particle trajectories like 
the analytical method. The good precision of the results allows this numerical method to be 
applied so that the analytical method was used with approximately the same results.
As shown in section 2.2., the method is easily converted to a 3D environment as a possible 
and interesting application for future work. Another possible application is to convert both 
cases to the relativistic particles case.
ACKNOWLEDGEMENTS
We acknowledge CNPQ for the financial support provided, Domingos Eugênio de Sá Nery 
for the help in this paper and Altivo Monteiro and Rafael Araujo for the support at the 
development of this work.
REFERENCES
1.J.  Peraire,  M.  Vahdati,  K.  Morgan,  O.  C.  Zienkiewicz,  “Adaptative  remeshing  for 
compressible flow computation”, J.Comput. Phys., Vol. 72, pp. 449-466 (1987).
2.G. Dhatt, G. Touzot, Une Présentation de la Méthode des Éléments Finis, Maloine S. A. 
Éditeur, Paris, France (1981).
3.S.  J.  Salon,  Finite  Element  Analysis  of  Electrical  Machines,  Kluwer  Academic 
Publishers, Boston, MA (1995). 
4.Y.  C.  Polli;  Cálculos  de  Trajetórias  de Elétrons  em Estruturas  Magnéticas,  Tese  de 
Mestrado, Universidade de São Paulo, 1999.
5.C.  A.  F.  Jorge,  R.  J.  Jospin,  J.  A.  D.  Furlanetto,  “Particle  Trajectory  Tracing  and 
Computational  Electromagnetics  for Accelerator  and Related  Technologies”,  Journal  of  
Microwaves  and  Optoelectronics,  vol.  6,  number  1,  pp.  278-294, 
http://www.sel.eesc.usp.br/jmo/issues/vol_6/v6_n1/v6_n1_paper_pdf/momag2006/V6n1a2
3.pdf (2006).
6.W. H. Press, B. P. Flannery, S. A. Teulousky, W. T. Vetterling, Numerical Recipes: The 
Art of Scientific Computing, Cambridge University Press, United States of America (1987).
7.J.  P.  A.  Bastos,  Eletromagnetismo  para  Engenharia:  Estática  e  Quase-Estática, 
Florianópolis: Editora da UFSC, 2004.


Particle trajectory tracing for electrostatic and magnetostatic fields

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Particle trajectory tracing for electrostatic and magnetostatic fields

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