Asymptotic and Numerical Analysis of Charged Particle
Beams
M. Asadzadeh
Department of Mathematics
Chalmers University of Technology
SE-412 96 Götenborg, Sweden
Abstract. We study a model for a single particle moving through a homogeneous background medium. Starting with a
transport process for the electron particles we derive (i) the Fokker-Planck equation using asymptotic expansions, (assuming
small mean free path), then (ii) the Fermi equation by yet another asymptotic expansion assuming (in addition to small mean
free path) small angular deviations. Finally (iii) we discuss some numerical solutions of a simplified 2-dimensional Fermi
equation.
INTRODUCTION
To study the beam particles, (e. g. electrons), we start from a simplified balance equation: the transport equation which
describes the probability of a particle with a certain kinetic energy being at a certain position at a certain time and
moving with a certain velocity (phase-space-time coordinate). Electrons in the beam will encounter a high number
of collisions with the background medium, we refer to this process as scattering. The times between collisions are
independent of each others, we thus have to look at a mean value which will be depending on, e.g., the pre-collision
energy. As the particles travel with high velocity a collision will only result in a small change of kinetic energy
and direction. The small directional variation is referred as forward-peakedness of the beam. Between collisions we
assume that the particles move with a constant velocity on a straight
path. Furthermore
we assume isotropic media,
more precisely the scattering depends on the angular cosine
ω
ω
of
the
incoming
ω
and outgoing ω directions of an
electron only, and not! on the individual directions ω and ω . An electron is said to be absorbed when it has lost all its
kinetic energy.
Now we consider a slab of tickness L in a Cartesian coordinate system Oxyz and assume that a monodirectional
pencil
beam enters
this slab at the origin O. An electron entering the2slab with kinetic energy E will have the velocity
v E ω , where
v
E is the speed of electron and ω µ η ξ S is its direction. E and v E are related through
E 12 m v E 2 , where m is the mass of an electron.
We introduce the mean free time τ E to be the mean time
between collisions, the mean free path is thus τ E v E . The fact that lots of collisions occur is expressed in the mean
free path being very small in comparison to the tickness of the slab L.
As the collisions occur at random times and result in random energy and
direction changes the phase space
coordinates of the particle, i.e. its position x x y z 3 , its direction ω E S2 , and its kinetic energy E 0
at time t are random variables.
probability density will be denoted by f x ω E t . A particle entering a collision
Their
in position x at time t with ω E -coordinates will have a probability of
1 P ω ω E E 2π
for leaving the collisio site, with ω E -coordinates. Here the isotropic material enters our consideration as the
probability is not depending on ω and ω individually but on their scalar product ω ω . The
forward-peakedness
of the beam can thus be expressed as the probability density being close to zero unless ω ω is close to one, in other
words, the change of direction is assumed to be very small, but not! negligible.
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
THE CHARGED PARTICLE MODELS
To derive a transport equation we need to state a balance between the streaming
absorption , viz
, the total scattering
, and the
More specifically, suppressing t and x dependence, we get the equation for the probability density function f , as:
f :
ft ω E v ω ∇x f ω E Q0 f
f ω E
:
τ E
f f
(1)
where ∇x is the gradient operator in x and Q0 f is the total amount of particles encountering a collision at position x
and having an, after collision, energy E and direction of velocity ω . In other words, in order to get the gain term, (here
corresponding to total
scattering), we consider
the electrons of all possible entry energy levels and angles leaving the
collision site x x y z with the velocity v E ω :
Q0 f
thus we have
ft ω E 1
2π
∞
ω
0
P ω ω E E
S2
f ω E
τ E
v ω ∇x f ω E ∞
1
2π
ω
0
f ω E
τ E
d ω dE
P ω ω E E
S2
(2)
f ω E
τ E
d ω dE
(3)
Since the speed of the electrons is very high the steady state occurs very fast and hence the time dependence may be
neglected.
The particle beam problems are usually formulated for the current function:
Ψ x ω E v E f x ω E Then with the scattering cross section σt E defined by
L σt E 1
v E τ E
(4)
(5)
the corresponding equation for the current reads as follows: find the current Ψ satisfying the balance equation,
ω ∇x Ψ ω E L σt E Ψ ω E ∞
1
2π
0
ω
P ω ω E E L σt E Ψ ω E d ω dE
S2
(6)
To be concise we consider a simplified monoenergetic beam where we assume negligible energy losses. The equation
simplifies correspondingly to:
ω ∇x Ψ ω L σt Ψ ω 1
2π
ω
S2
P ω ω L σt Ψ ω d ω
Now introducing the differential cross section
σs ω ω
1 P ω ω σt 2π
we obtain the final transport equation and its boundary conditions viz,
ω ∇x Ψ x ω L σt Ψ x ω L
ω
S2
σs ω ω Ψ x ω δ µ 1
Ψ 0 y z ω δ y δ z 2π
Ψ L y z ω 0 1
x
µ
µ 0
0
x y z 1
3
0
x
L
(7)
(8)
(9)
The first two factors in the first boundary condition describe a beam entering at the origin of the Oxyz-coordinate
system and the third factor describes the fact that the velocity of the entering electrons is in x direction only. The
second boundary condition prescribes that no back scattering is allowed, i.e., even close to the exiting surface of the
slab, the change of direction of the electron’s velocity is not big enough to make electrons move backwards.
The Fokker-Planck and Fermi pencil beam equations
To obtain pencil beam equations we perform asymptotic expansions of the transport equation with respect to the
smallness parameters: ε L1σ , the mean free path, and ∆ Lσtr . We can interpret ∆, physically, as the angular
t
deviation caused by a collision. Observe that as the number of collisions are very high, (σt is very large), ε is going to
be very small. In our first asymptotic expansion we let ε
0, and neglect the ε 2 terms. In this way we derive the
Fokker-Planck equation. Here we introduce a stretched version of σ s :
p t
2πε ∆2 σs 1
σs 0
satisfying
1
Now since
µ0
ε∆ t 2
2
ε ∆2
p t dt
0
σ s µ0 2
ε ∆2
0
σs 0
1 µ0
p
2πε ∆2
ε ∆2
t p t dt
1
(10)
1, it is natural to expand σs using the Legendre polynomial basis in the interval
∑ 2n4π 1 σsn pn
∞
σ s µ0 µ0 1 1 , so
(11)
(12)
n 0
with the Legendre coefficients σsn obtained through the following relation:
1
1
∑ 2n4π 1 σsn
∞
σ s µ0 p k µ0 d µ 0
n 0
1 1
1
p n µ0 p k µ0 d µ 0
Hence, using (13), we have
σ s µ0 ∞
∑
2n
2
n 0
1
1
σs t pn t dt pn µ0 σs0 ∞ 2n 1
ε ∆2 n∑0 4π
1
1
1
σs
2π k
pn t p
(13)
1
t
dt pn µ0 ε ∆2
Inserting σs µ in the transport equation (7) we get the scaled transport equation
ω ∇x Ψ x ω 1 Ψ x ω ε
1
ε 2 ∆2
∞
ω
2n 1
∑ 4π
n 0
S2
1
1
pn t p
1 t
ε∆ dt pn µ0 2
Ψ x ω dω
Because of the factor 1 ε we have a large absorption term in the left hand side of the equation. This corresponds to
a large number of out-scattering particles (the corresponding in-scattering part is hidden in the integrand on the right
hand side). The first step in our asymptotic derivation is to get rid of these terms from both sides of the equation. To
this end we Taylor expand the Legendre polynomial on the integrand, about t 1, to obtain
1
ε ∆2
1
1
pn
1 t
t p
ε∆ 2
dt
1
p 1
t 1
n
1 pn
1 t
1 p
ε∆ ε ∆ p 1
p 1 ε ∆ p 1 ε ∆ 1 pn 1 2
ε ∆2
p u du
0
2
n
2
ε ∆2
2
n
up u du
0
ε∆
2 4
n
dt
ε ∆2
2
2n
n
ε 2 ∆4 1 2
ε 2 ∆4 where we have used (11). Inserting in the transport equation and also using the fact that we can expand the solution in
the Legendre polynomials as
Ψ x ω ∑ 2n4π 1
∞
n 0
ω
S2
we get the relationship
ω ∇x Ψ x ω 1 Ψ x ω ε
1 Ψ x ω ε
∆2 ∞ nn
2 n∑0
pn ω ω Ψ x ω d ω
1
2n 1
4π
ω
(16)
S2
pn ω ω Ψ x ω d ω
(17)
In this way the absorption term on the left hand side of the scaled transport equation will be cancelled by the first term
after the Taylor expansion on the right hand side, so that the large in- and out-scattering terms will disappear.
Finally, due to the fact that Ψ satisfies the partial differential equation of surface harmonics given by
∂ 1
∂µ
µ 2
∂
∂µ
∂2
∂θ2
1
1
µ2
nn
we get using (16) the Fokker-Planck equation
ω ∇x Ψ x ω Lσtr ∂ 1
2
∂µ
with the boundary conditions
Ψ 0 y z µ η ξ Ψ L y z µ η ξ ∂
µ ∂µ
2
1 Ψ x ω µ
2
δ µ 1
δ y δ z
2π
0
1 µ 0
0
(18)
∂2
Ψ x ω 2
∂θ
1
1
µ
0
(19)
1
(20)
(21)
Observe that (in addition to the fact the large in- and out-scattering terms are cancelled) the right hand side of the
Fokker-Planck equation is now the Laplacian operator on the unit sphere rather than an integral operator (which was
in the transport equation that we start with). See also Pomraning [5] for further details.
Now we invoke the forward peakedness and further simplify the Fokker-Planck equation in order to obtain the Fermi
equation. To do so we seek a good approximation for the Fokker-Planck equation for small angular deviations i.e., in
the vicinity of µ η ξ 1 0 0 . To this approach we project the Fokker-Planck operator from acting on the right
half of the unit sphere into the tangent plane at the point 1 0 0 to the unit sphere. See Larsen [4] for further details.
In this way we obtain a new asymptotic expansion due to small angular deviations and the final equation is the Fermi
equation:
∂Ψ
∂x
∂Ψ
η
∂y
∂Ψ
ξ
∂z
Lσtr
2
∂2
∂ η2
∂2
Ψ x ω ∂ξ2
η ξ
2
(22)
Thus the corresponding operator for the Fermi equation is the Laplacian on the tangent plane described above and the
boundary conditions are now reformulated for µ 1, and reduced to an initial condition viz,
Ψ 0 y z η ξ δ y δ y δ η δ ξ with
η ξ
2
(23)
A two dimensional model
In this section we derive a two dimensional model problem for the Fokker-Planck and Fermi pencil beam equations.
This model is for the numerical investigations below, where the full three dimensional problem is both costly and very
much complex. Therefore
we consider the two dimensional transport equation, (corresponding to our three dimensional
problem above), with x y denoting position variables and ω the directional variable, viz:
ω ∇ψ
ψ 0 y ω S1
1 δ y δ 1
2π
ω1 ω1
σs ω ω ψ x y ω
ψ x y ω d ω
0
ψ L y ω 0
ω1
0
where x 0 L y and ω ω1 ω2 : cos θ sin θ S1 , with S1 being the unit circle. Further σs and ψ are the
2D version of the same quantities defined previously and σt S1 σs ω d ω
FP
In the 2D case we denote the Fokker-Planck solution
π πby ψ . Now introducing the forward current function
j x y θ cos θ ψ FP and the scaling z : tan θ θ 2 2 , we obtain the Fokker-Planck equation for j:
jx
z jy
σtr 1
z2 ∂
∂z
1
z2 ∂
1
∂z
z2 j (24)
where we have used the following obvious relation
∂ 2Φ
∂θ2
Since cos2 θ
j x y tan 1 z 1 z2
1
1 z2
we have that ψ FP
1
1
z2 ∂
∂z
1
z2 ∂Φ
∂z
z2 j. We now define the scaled forward current function J as J x y z . With these substitutions we can rewrite the right-hand side of the Fokker-Planck equation (24) as
1
∂
σ
2 tr ∂ z
1
z2 ∂ 1
∂z
z2 3 2
J
The corresponding right hand side for the Fermi equation is: 12 σtr ∂∂z2 J. Thus we have the canonical form of the pencil
beam equations viz,
∂J
∂J
1
z
ε AJ ε
σ (25)
∂x
∂y
2 tr
with the operator A defined by A
∂
∂z
a z
∂
∂z
b z
2
, where a z 1 z2 and b z Planck equation, whereas the corresponding Fermi equation has simply A dimensional Fermi pencil beam equation is reduced to: J 0 y z δ y δ z on [1] and [2] and the references therein.
∂2
∂ z2
1
z2 3 2 ,
for the Fokker-
. The boundary condition for the two
. Results of this part are based
y z NUMERICAL INVESTIGATIONS
We study some fully discrete schemes for the numerical solution of a pencil beam model in two space dimensions. We
will sketch the standard Galerkin (SG) and semi-streamline diffusion (SSD) finite element methods and formulate some
fully discrete schemes. For the corresponding detailed convergence analysis we refer to [3]. In our model problem we
consider the Fermi equation (the corresponding Fokker-Planck equation is treated analogously), in a slab of thickness
L, x Ix : 0 L , with a symmetric cross section I : Iy Iz : y0 y0 z0 z0 , for y0 z0 . Thus the
physical domain Ω : Ix I is now three dimensional (Figure1). The corresponding Fermi equation is modelling
the penetration (in x-direction) of narrowly focused pencil beam particles incident at the transversal boundary of the
considered isotropic slab, and entering into the slab at the point x y z 0 0 0 .
In this setting our model problem for numerical investigations is thus formulated as follows: given the incident
source intensity f at x 0, find the current J defined on the domain Ω satisfying the Fermi pencil beam equation,
Jx zJy ε Jzz Jz x y z0 0 J 0 x f x J x x 0 in Ω Ix I for x y Ix Iy for x I on Γ x x Γ : ∂ Ω ñ β̃
β̃
(26)
0
recall that z tan θ x : y z 2ε σtr x y . Further β̃ 1 z 0 and ñ : ñ x x is the outward unit normal to
Γ at x x Γ.
This problem corresponds to a convection-diffusion problem which can be interpreted as a time-dependent (with
x corresponding to the time variable) degenerate (convection in y, diffusion in z) forward-backward (z takes both
positive and negative values), convection dominating (ε small) problem. For the convection dominated problems,
having hyperbolic nature, the SG method would have convergence of order O h k , (versus O hk 1 for elliptic and
parabolic problems), provided that the exact solution is in the Sobolev space H k 1 .
To speed up the convergence of the SG for hyperbolic type problems it is necessary to include artificial viscosity
terms, e.g. by adding some amount of diffusion in the equation. Here, using a SSD method, through a modified form
of the test functions we automatically add a proper amount of viscosity resulting in smoothing effects on the equation.
Note that SSD method is performed only on the x variable whereas the usual streamline-diffusion (SD) finite
element method includes also the discretization on x variable in the same variational formulation as that of x . In
our approach, however, the penetration variable x is interpreted as a time variable and treated by the usual time
discretization techniques such as: discontinuous Galerkin (DG), backward Euler (BE) and Crank-Nicolson (CN)
methods. To get a fully discrete problem we combine SG or SSD schemes for I with a time discretization method for
the penetration interval Ix , see [3] for the details.
z
u z= 0
u(0,y,z)=f(y,z)
z0
−
u(x,y,z)=0
0 y0
n=(0,−1,0)
n=(0,1,0)
u(x,y,z)=0
y
0
x
u z= 0
FIGURE 1. 2D-Model
In this part we discretize x the rectangular domain I Iy
y z using a finite element approximation based on quasi-uniform triangulation of
Iz with a mesh size h. To this approach we define the inflow (outflow) boundary
The Standard Galerkin Method
Γβ :
x Γ : ∂ I : n x and introduce a discrete, finite dimensional, function space Vh β
such that, v Hβ1 I H r I ,
inf
χ Vh β
v
χ
j
v Hβ1 I
Chα
j
v
H 1 I
α
β
0
Hβ1 I
0
z 0
and
(28)
1
α r
An example of such Vh β is the set of sufficiently smooth piecewise polynomials P x boundary conditions.
Now the objective is to find Jh Vh β , such that
Jh x χ zJh y χ ε Jh z χz 0 χ
Vh β (27)
with
: v 0 on Γβ j 0 1
β
Jh 0 x
(29)
of degree
fh x r, satisfying the
where fh is a finite element approximation of f and the mesh size h is related to ε according to: h 2
(30)
ε
h.
A Smoothing Petrov Galerkin Method
Below we introduce a SSD approach which includes a diffusion generating test function in the y direction over the
usual SG procedure. Its smoothing effects are due to symmetry in diffusivity in y and z, and its convergence rates,
as can be seen in numerical tests in Tables 1 and 2, are at least as good as those of the SG method. Note that the
degenerate nature of our original problem also contributes to the anisotropic character in the diffusion term. Using the
SSD method we obtain a non-degenerate type convection dominated, convection-diffusion equation with somewhat
improved regularity in the y direction.
We let vβ β ∇ v with ∇ being the gradient operator on x and β z 0 . Then the SSD test functions having
the form v δ vβ automatically add the extra diffusion term, δ vβ vβ , to the variational formulation which, combined
with v ε vzz ε vz vz term, gives a non-degenerate equation with a diffusion term of order O ε , if δ ε .
Multiplying the differential equation by v δ vβ , integrating over I , and using the boundary conditions yields,
Jx v δ Jx vβ Jβ v δ Jβ vβ ε Jz vz δ ε Jz vβ z 0
(31)
The discrete version is now obtained by replacing J by a suitable Jh , having the desired approximation properties.
Numerical Implementation
In this part we test the convergence rates of both SG and SSD through implementing a numerical test example.
Our implementations are performed over four different initial data: modified Dirac, hyperbolic, Maxwellian, and
cone functions, approximating our data, the δ -function. The procedure is split into two steps. First we discretize
the
two dimensional domain I Iy Iz by means of continuous piecewise linear Galerkin approximation: cG 1 , and
establish a mesh there in order to obtain a semidiscrete solution. Subsequently, we apply one of the time discretization
methods (BE, CN or DG), to step advance in x direction. The cG 1 basis functions have the form, φ i a1 y a2 z a3 .
In some special cases, e.g., for ε not depending on x , the closed form exact solution of our problem is given by,
J x y z 2 3
e
πε x2
4 3 y x 2 3 y 2 z z2 ε x This allows us to draw some limited comparisons in term of the actual error, e h : J
in the tables below and measured in a weighted L2 norm defined by
ϕ
L̃2
1 3 i
τi ∑ ϕ ζ k 3∑
τ
k 1
i
2
1 2
(32)
Jh , for some initial data indicated
where τi are the triangles in the mesh and ζki denote the midpoints of the edges of τi . Tables 1 and 2 below show the
convergence rates of both SG and SSD methods, associated with the time discretization methods BE, CN and DG,
implemented for the reference domain 0 1 1 1 2 and extrapolated in the mesh parameter h 0 025. The step
size k for the discretizations in x variable is taken in the range h k 0 0005, and we have chosen δ ε 0 005.
As a result of our numerical tests we can see the convergence of each scheme as the step size is reduced, as well as a
somewhat pronounced improvement in using SSD over SG.
In Figure 2, the dose intensity (the amount of deposited energy per unit volume, per unit time) radiating an elliptic
target is shown at the collision site x 0 3 and with a σtr 0 1. The figure shows, e.g. that our asymptotic expansions
have produced a reliable pencil beam model.
CONCLUSION
Our studies in this paper split into two parts. The first part is devoted to asymptotic expansions of a linear transport
procedure leading (assuming small mean free path) to the Fokker-Planck equation, as an intermediate step, and then
to the Fermi pencil beam equation by further assuming small angular deviation of the particle (electron) in each
collision. Our considered Fermi equation is a degenerate type, convection dominated, convection-diffusion equation
with a mixed inflow boundary condition, which is widely used in radiation cancer therapy. The second part of
our study concerns numerical approaches to solve pencil beam problems.
In this part we discuss combined space
(standard Galerkin and semi-streamline diffusion) discretizations in y z and time (backward-Euler, Crank-Nicolson
and discontinuous Galerkin) discretizations in, penetration direction, x for solving the Fermi equation. We have derived
some limited convergence estimates for different initial data approximating the Dirac δ function.
TABLE 1.
Standard Galerkin discretization in x
Dirac
Time discretization for x
Backward-Euler
Crank-Nicolson
Discontinuous Galerkin
TABLE 2.
e2 h
Hyperbolic
e4h e2h
Maxwellian
e4h e2h
13.63-1.806
13.73-1.814
13.40-2.065
.064-.013
.065-.014
.064-.012
.123-.042
.122-.041
.117-.043
e4h
Semi-Streamline Diffusion in x
Dirac
Time discretization for x
Backward-Euler
Crank-Nicolson
Discontinuous Galerkin
FIGURE 2.
y z
.115-.047
.115-.047
.110-.051
y z
e2 h
Hyperbolic
e4h e2h
Maxwellian
e4h e2h
13.33-1.801
13.44-1.806
13.28-2.068
.063-.014
.063-.015
.063-.014
.118-.041
.117-.040
.117-.042
e4h
Cone
e2 h
e4h
Cone
e2 h
e4h
.110-.045
.110-.045
.110-.049
The solution of the 2-d Fermi equation for fixed x and σtr
REFERENCES
1.
2.
3.
4.
5.
M. Asadzadeh On Convergence of FEM for the Fokker-Planck Equation, Proceedings of 20th RGD, Beijing, ed by C. Shen,
Peking Univerrsity Press, (1996), 309–314.
M. Asadzadeh Characteristic Methods for Fokker-Planck and Fermi Pencil Beam Equations, Proceedings of 21st RGD,
Marseille, ed by R. Brun et al, CÉPADUÈS ÉDITIONS, 2 (1998), 205–212.
M. Asadzadeh and A. Sopasakis On fully dicrete schemes for the Fermi pencil beam equation, to appear in Comp. Methods
Appl. Mech. Eng.
E. W. Larsen, The amplitude and radius of a radiation beam, Proceedings of the International Conference on Latest
Developments and Fundamental Advances in Radiative Transfer (Los Angeles, CA, 1996). Transport Theory Statist. Phys. 26
(1997), no. 4-5, 533–554.
G. C. Pomraning, The Fokker-Planck operator as an asymptotic limit, Math. Models, Methods Appl. Sci. 2 (1992), no. 1,
21–36.