Energy-dependent analytical solutions for the …Analytical solutions were compared to numerical results using the Finite Element Method, showing reasonable agreement. Tobias Geba¨ck

BackgroundEnergy distribution correction

Conclusions

Energy-dependent analytical solutions for the charged particletransport equation

Tobias Geback and Mohammad Asadzadeh

Mathematical Sciences, Chalmers University of Technology, GoteborgDepartment of Oncology-Pathology, Karolinska Institute, Stockholm

Sweden

ICTT-22, Portland, OregonSeptember 15, 2011

Tobias Geback and Mohammad Asadzadeh Energy-dependent analytical solutions for charged particle transport


Conclusions

Overview

Background Derivation of analytical results

Ignoring the straggling term Including the straggling term

Comparison with numerical results

Conclusions



Conclusions

The Boltzmann equation under CSDA

Under the continuous slowing down assumption (CSDA), the stationaryBoltzmann equation for the fluence f of charged particles with energy E

traveling in direction Ω ∈ S2, given an incident particle beam into thehalf-space z ≥ 0, is

8>>>>><>>>>>:

Ω · ∇f (x,E ,Ω) + σa(E)f (x,E ,Ω) − ∂(S(E)f )

∂E− 1

2

∂2(ω(E)f )

∂E 2=

Z

S2

σs(E ,Ω · Ω′)`f (x,E ,Ω′) − f (x,E ,Ω)

´dΩ′

f |z=0 = f0(x , y ,Ω)G(E)

(1)

Here, σa(E) is the absorption cross-section, while σs(E ,Ω ·Ω′) is the scatteringcross-section. S(E) is the stopping power, and ω(E) is the energy-lossstraggling coefficient.



Conclusions

The Fermi-Eyges equation (1)

Enrico Fermi suggested the equation below for narrow beams in the late 30’s(Rossi and Greisen, 1941). Eyges (1948) extended it to T = T (z) and derivedan analytical solution when f0(x , y , θx , θy ) = δ(x)δ(y)δ(θx)δ(θy).

∂ψ

∂z+ θx

∂ψ

∂x+ θy

∂ψ

∂y=

T (z)

2

„∂2ψ

∂θ2x

+∂2ψ

∂θ2y

«(2)

It can be viewed as the projection of the Fokker-Planck equation onto thetangent plane at Ω = [0, 0, 1] (Borgers and Larsen, 1996).



Conclusions

The Fermi-Eyges equation (2)

The analytical solution for an incident Gaussian beam G1(x , y , θx , θy ) is(Brahme, 1975):

hFE(x , y , z , θx , θy )

= C(z) exp“−a(z)(x2 + y

2) − 2b(z)(xθx + yθy ) − c(z)(θ2x + θ

2y )”,

(3)

where a(z), b(z), c(z) and C(z) may be computed explicitly from the firstthree moments of T (z) (formulas omitted here).



Conclusions

Derivation of energy dependence

Using the Fermi-Eyges approximation for the scattering term, and the CSDAassumption for energy dependence, we get the equation

8>>>>><>>>>>:

∂f

∂z+ θx

∂f

∂x+ θy

∂f

∂y+ σa(E)f − ∂

∂E(S(E)f ) − 1

2

∂2

∂E 2(ω(E)f )

= T (z)

„∂2f

∂θ2x

+∂2f

∂θ2y

«,

f |z=0 = G1(x , y , θx , θy )G2(E)

(4)

where

T (z) =

Z 1

−1

σs(E(z), µ) · µ2dµ (5)

and E(z) is the average energy at depth z (which is not yet determined).Equation (4) is separable and we will now use this fact to derive analyticalsolutions.



Conclusions

No straggling (1)In the case when the straggling term is ignored (ω(E) ≡ 0), the ansatz

f (r, θx , θy ,E) = g(E)h(r, θx , θy ). (6)

yields the two equations

∂

∂E(S(E)g(E)) − σa(E)g(E) = λg(E) (7)

∂h

∂z+ θx

∂h

∂x+ θy

∂h

∂y− T (z)

„∂2h

∂θ2x

+∂2h

∂θ2y

«= λh(r, θx , θy ), (8)

for some constant λ. These can now be solved separately.Equation (7) is solved using an integrating factor to yield

g(E) = g(E0)S(E0)

S(E)exp

„λ (R(E) − R(E0)) −

Z E0

E

σa(E′)

S(E ′)dE

′

«, (9)

where R(E) =R E

0dE ′

S(E ′)is the CSDA range.

Equation (8) is the Fermi-Eyges equation with a right-hand side and has thesolution

h(r, θx , θy ) = eλz

hFE(r, θx , θy ), (10)

where hFE is the Fermi-Eyges solution.



Conclusions

No straggling (2)

Combining the two solutions, and introducing the relationship

z(E) = R(E0) − R(E), (11)

or equivalently, E = E(z), to satisfy the boundary condition (4) withG2(E) = δ(E − E0) and get a single energy at each depth, the dependence onλ disappears and

f (r, θx , θy) =S(E0)

S(E(z))exp

−Z E0

E (z)

σa(E′)

S(E ′)dE

′

!hFE(r, θx , θy ). (12)

is a solution to (4) with ω(E) ≡ 0 for mono-energetic particle beams.NB: This solution was suggested by (Kempe and Brahme, 2010) withoutrigorous derivation.



Conclusions

With energy-loss straggling

When we include the straggling term (ω(E) > 0), we make a more generalansatz:

f (r, θx , θy ,E) = hFE(r, θx , θy ) · Z (z ,E) 6= 0. (13)

With this ansatz, equation (4) yields

ΥFE [hFE] · Z + hFE

“∂Z

∂z− ∂(S(E)Z )

∂E− 1

2

∂2(ω(E)Z )

∂E 2+ σa(E)Z

”= 0, (14)

where ΥFE [hFE] stands for the Fermi-Eyges equation and is identically zero forthe solution hFE.Therefore, we get the following equation for Z (z ,E)

8><>:

∂Z∂z

− ∂(SZ)∂E

− 12

∂2(ωZ)

∂E2 + σa(E)Z = 0, (z ,E) ∈ [0,∞) × [0,∞),

Z (0,E) = G2(E), E ≥ 0,

Z (z , 0) = 0, z ≥ 0

(15)



Conclusions

Analytical result with NESAIn order to get an analytical solution, we wish to apply the Fourier transform inE , and therefore wish to extend the problem to E ∈ R. In order to retain theboundary condition Z (z , 0) = 0, we look for solutions Z that are odd in E , sothat Z (z ,−E) = −Z (z ,E), z ≥ 0.Furthermore, we invoke the narrow energy spectrum approximation (NESA), i.e.

F“w(E) · Z (E)

”≈ w(E(z)) · bZ (z , ξ). (16)

Solving the resulting equation in the Fourier domain yields the result

Z (z ,E) =C0p

2πΩ(z)exp (−Σa(z))

“exp

“− 1

2

(E − E(z))2

Ω(z)

”

− exp“− 1

2

(E + E(z))2

Ω(z)

””,

(17)

where

Σa(z) =

Z z

0

σa(E(z ′)) dz′

, E(z) = E0 −Z z

0

S(E(z ′)) dz′

,

Ω(z) = Ω0 +

Z z

0

ω(E(z ′)) dz′

. (18)



Conclusions

Analytical result with NESA (2)

Note that (17) satisfies the boundary condition in (4) with

G2(E) =C0√2πΩ0

„exp

„−1

2

(E − E0)2

Ω0

«− exp

„−1

2

(E + E0)2

Ω0

««, (19)

that is, a Gaussian energy distribution for E > 0 if Ω0 is small enough so thatthe mirrored part has negligible influence for E > 0.



Conclusions

FEM computation

In order to investigate the accuracy of the NESA we have performed numericalcomputations for equation (15) using the finite element method.We treated z as a time variable with a backward Euler scheme, and withpiece-wise linear (CG1) elements in E . A 800 × 800 non-uniform grid was used.Cross-sections for electrons in water were taken from (Luo and Brahme, 1992).Parameters used:

(1) E0 = 50 MeV, Ω0 = 4.4 MeV2, C0 =

√2πΩ0,

(2) E0 = 20 MeV, Ω0 = 0.7 MeV2, C0 =

√2πΩ0,



Conclusions

Comparison - results (1)

z (cm)

E (

MeV

)

FEM

0 2 4 6 8 10 12 14 160

10

20

30

40

50

60

70

0.1

0.2

0.3

0.4

0.5

0.6

z (cm)

E (

MeV

)

Explicit (NESA)

0 5 10 150

10

20

30

40

50

60

70

0.1

0.2

0.3

0.4

0.5

0.6

Figure: Level curves for the numerical solution to equation (15) (left), and theanalytical solution (17) under the narrow energy spectrum approximation (NESA)(right). The incident beam has mean energy E0 = 50 MeV. The cross-sections forelectrons in water were used. The dashed lines are the curves E = E (z), and the redlines are the averages energies for the respective solutions.



Conclusions

Comparison - results (2)

z (cm)

E (

MeV

)

FEM

0 1 2 3 4 5 60

5

10

15

20

25

0.1

0.2

0.3

0.4

0.5

0.6

z (cm)

E (

MeV

)

Explicit (NESA)

0 1 2 3 4 5 60

5

10

15

20

25

0.1

0.2

0.3

0.4

0.5

0.6

Figure: Level curves for the numerical solution to equation (15) (left), and theanalytical solution (17) under the narrow energy spectrum approximation (NESA)(right). The incident beam has mean energy E0 = 20 MeV. The cross-sections forelectrons in water were used. The dashed lines are the curves E = E (z), and the redlines are the averages energies for the respective solutions.



Conclusions

Summary/conclusion

We have derived energy-dependent analytical solutions for the Fermi-Eygesequation under a continuous slowing down assumption.

In the straggling case, the NESA was used to obtain an analytical solution.

Analytical solutions were compared to numerical results using the FiniteElement Method, showing reasonable agreement.



Conclusions

Acknowledgments

Anders Brahme, Karolinska Institute

Luo Zhengming for inspiration



Conclusions

Thank you!



Conclusions

References

(Brahme, 1975) A. Brahme, Investigations of the Application of

Microtron Accelerator for Radiation Therapy, Ph. D. Thesis Institute ofRadiation Physics (Stockholm University), Stockholm, 1975.

(Borgers and Larsen, 1996) C. Borgers and E. W. Larsen, Asymptotic

derivation of the Fermi pencil beam equation, Nucl. Sci. Eng. 123, No. 3,343-358 (1996).

(Eyges, 1948) L. Eyges, Multiple Scattering With Energy Loss, Phys. Rev.74, 1534 (1948).

(Kempe, 2010) J. Kempe and A Brahme, Solution of the Boltzmann

equation for primary light ions and the transport of their fragments, Phys.Rev. Special Topics - accelerators and beams, 13, No. 10, (2010)

(Luo and Brahme, 1992) Z. Luo and A. Brahme, High-energy electron

transport, Phys. Rev. B 46, No. 24, (1992), 15 739-15 752.

(Rossi and Greisen, 1941) B. Rossi and K. Greisen, Reviews of Modern

Physics, 13, chapter Cosmic-ray theory, pages 241–315, (1941)


Documents

Energy-dependent analytical solutions for the …Analytical solutions were compared to numerical results using the Finite Element Method, showing reasonable agreement. Tobias Geba¨ck