0

0

2

0

1

,expexpd2

2

k

k

ztikzDkt kk

(25)

The boundary parameter k0 is given by k0 = 1/z0. The

evaluation of the integral (18) provides Equation (26):

2

12

,exp4erferf

F

ztU tzDtss

(26)

The parameters of this equation are given by:

0

10

20

42

12

12

F

FF

FF

Ut zDt

s

zizDt Dt

s

zizDt Dt

(26a)

With the help of Equation (26) we consider the solu-

tion:

1

22

00 0

22

10 0

,,,,exp

cos 4

sin4F

Exyzt ztt

AxyDt

A

xy Dt

(27)

The sine and cosine function appears by forming linear

combinations of solutions of Equation (21), since a ac-

cording to Equation (23) is an imaginary parameter and

the theorem for complex exponential functions can be

applied. It has to be mentioned that the cosine as well as

the sine are solutions, and both may form linear combi-

nations according to Equation (26). We should account

for that the function

in Equation (18) has not to be

restricted to the simple sine and cosine, but we can also

use the general solution manifold according to Equations

(26) and (27). Before we shall study some properties of

Equation (27), a comparison with the Schrödinger equa-

tion is indicated.

Before we shall study some properties of Equation

(27), a comparison with the Schrödinger equation is in-

dicated:

22it m

(28)

This equation assumes the character of an irreversible

transport equation, if the substitution ti

is carried

out. By that, the diffusion constant is given in terms of

the Planck’s constant: 2.

F

Dm

However, the solution (27) is not the only possible one,

and we are able to obtain a spectrum of solutions and

their linear combinations. The complete solution spec-

trum is given by the two different types.

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Powers of even order:

222 222

22222 22

0

20

cos 4sin4

21,1,2,3

nmm

nmnFnm nFn

m

n

A

Dxy tBDxy t

nn n

(29)

Powers of odd order:

21 2221 22

212 12121212121

0

21 0

cos 4sin4

210,0,1, 2,3

nmm

nmnFn mnFn

m

n

A

Dxy tBDxy t

nnn

(30)

Please note that superpositions of different order and

related eigen-frequencies are also possible solutions.

Thus we can perform a linear combination of all solu-

tions, e.g. a fast oscillating solution with a slow oscillat-

ing solution can be combined to form beat oscillations.

At first, we look at the connection between diffusion and

the quantum mechanical Schrödinger equation with ex-

ternal magnetic fields. The following aspects should be

emphasized: The resonance conditions for ω0 are in both

cases identical, this appears to be rather noteworthy. In a

formal sense, we have only to substitute the real time t by

an imaginary time it

, and the reversibility of the

Schrödinger equation goes lost. This behavior is also

known from the path integral formulation according to

Feynman et al. [6], which represents a further possible

way to solve the complicated task of scatter and the role

of magnetic fields by perturbation theory.

With respect to the eigen-frequency 0

and its de-

pendence on the related parameters e, m and B we are

able to make the following statements:

The z-part of the solution has also the character of an

oscillator due to the complex argument yielding nodes

(see e.g. the book of Abramowitz and Stegun 1970).

Only for sufficient large time t a homogeneous

charge distribution will be reached. The x-y-part does not

allow broadening by diffusion. The behavior is compara-

ble to that of a magnetic lense. Let us now consider an

example of a magnetic bifurcation. Assume an oscillating

propagation in the x-y plane with the highest frequency

0

given by the magnetic field strength B0. Thus a sud-

den change of the magnetic field strength from +B0 to

00 0

BBB

leads to a magnetic bifurcation, and, in

particular, the antisymmetric sine functions change the

sign, when the argument becomes negative. Such an ef-

fect may be induced by an inhomogeneous magnetic

field yielding changes of the field strength (amount and

orientation). The symmetry is spontaneously broken. The

same fact may also happen under a lot of similar external

influences: The change of the homogeneity of the mag-

netic field yields a change of the diffusion constant DF; a

change of the energy distribution E may require the for-

mation of complete different patterns and oscillation fre-

quencies, etc.

A principal result of the Bethe-Heitler theory is that

the energy loss due to creation of BS is proportional to

the actual electron energy. The differential equation for

the radiation loss reads (in one dimension):

1

dd

brert bre

EzXE

(31)

A theory of the creation of “BS” can be formulated via

propagator method [6]. The above mentioned phenome-

nological description summarizes all these parameters

resulting from the quantum theoretical treatment by the

radiation length Xrl according to Equation (31).

By iteration of Equation (31) we obtain a second order

differential equation, and the extension to 3D can readily

carried out, i.e. the Laplace operator Δ appears. This ex-

tension has the advantage that the resulting equation can

be added to further phenomenological equations con-

taining the Laplace operator:

1

brert bre

EXE

(32)

A further advantage results from the previous Figure 3:

If the amount of Tungsten sublayers is high, and, by that,

the distance between them is small (e.g. 1 mm in the

cone target), it is possible to solve equation (32) under

continuum conditions. The total Tungsten mass can be

divided by the cone volume to obtain the medium density

ρt. Step-by-step calculations (we do not report them here)

showed that for 1 mm distances between the plates and

identical overall mass a continuum approximation can be

justified. In a phenomenological theory, we can summa-

rize the complete problem by including both, energy loss

by radiation loss (Bethe-Heitler theory) and energy dis-

sipation (Fermi-Eyges theory):

22 2222

2

F

Fbrecolrlbrecolcol

Et DEemcEeDmcEEEXEXE

AA (33)

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In Equation (33) the parameter Xrl is referred to as the

radiation length, which is proportional to Z2, NA and AN,

whereas Xcol refers to the energy absorption by the cone

wall (collimator), which is proportional to Z, NA and AN.

The nuclear charge is denoted by Z, the nuclear mass

number by AN, and NA is the Avogadro number. The in-

fluence of the magnetic field can be accounted by the

following solution expansion:

2

22 2

0

2

21 2121

00

,cos4

cos 4

Larmor frequency

N

nnF n

n

nnFn

n

ExyAD r

BDr

neBmcne BB

(34)

B' refers to as a correction of B0 by ΔB0, since the

magnetic induction must not be constant in the volume

under consideration. In principle, we have to account for

N → ∞, which is impossible in numerical calculations.

There are two possible procedures, which we have

worked out:

Solution of the scatter problem by a proper magnetic

field acting between the subtargets and determination of

the corresponding phase space for Monte-Carlo calcula-

tions (GEANT4) with respect to collision interaction (Be-

the-Bloch) and BS (Bethe-Heitler theory, see e.g. [6]).

Complete solution of the above differential equation

containing all 3 components using the tools given by

deconvolution and inclusion of magnetic fields. In such a

situation we have to put: Ebre = Ecol = E.

The mathematical problem of scatter removal by de-

convolution operators has been presented [7-9]; the ap-

plication with inclusion of magnetic fields for scatter re-

moval has been analyzed previously [2].

2.3.3. M onte-Carlo C al c ulations w i th GEANT4

GEANT4 [3] represents an open system of a Monte-

Carlo code. Significant features with regard to our prob-

lem are creation of BS, multiple scatter according to Mo-

lière, heat production (Bethe-Bloch equation), energy

straggling (Gaussian-Landau-Vavilov), Compton scatter

of γ-radiation, and the actual energy/momentum of the

electron after interactions leading to energy loss and

change of the momentum. More sophisticated applica-

tions with regard to the focusing of a multi-layered Tung-

sten target and back scatter of the cone walls (Tungsten,

Tantalum, Lead) under boundary conditions require the

explicit use of the differential cross-section formula q(θ)

with the form-factor function F(θ). A further feature is

the implementation of the magnetic field B (i.e. vector

potential A) to account for the Lorentz force along the

track of the electrons according to Equation (3). In order

to obtain a reliable statistical foundation, each Monte-

Carlo run has been performed with 500×106 histories.

Figure 7 presents the back scatter properties (wall re-

flectance) of 9 MeV electrons at a high Z wall (W, Ta,

Pb); the corresponding properties of 6, 18 or 20 MeV are

rather similar.

Since the Figures 7 and 8 have methodical character,

we should like to show them already in this section. In

particular, Figure 7 has a fundamental meaning in this

study, namely angle-dependence of the reflectance (back

scatter) of fast electrons at wall consisting of high Z ma-

terial (Tungsten, Tantalum, and Lead). Although Pb

shows the high Z value, the density is much smaller than

that of W or Ta, and therefore according to Figure 7 we

prefer Tungsten as the wall material for focusing. In par-

ticular, Figure 7 represents the essential properties used

in Figure 3.

Figure 8 shows the scatter behavior of fast electrons in

air. In contrast to γ-radiation the scatter of electrons in air

is not negligible. The initial condition in all 3 figures is

an infinitesimally thin pencil ray of electrons. A conse-

quence of these figures is that the multi-target has to be

located in a vacuum in order to keep the lateral scatter

Figure 7. Backscatter (reflection) of fast electrons in depen-

dence of the impinging angle θ.

Figure 8. Comparison of air scatter: 6 MeV and 20 MeV

electrons.

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of electrons as small as possible.

We should also point out that cross-section formulas

used in the theoretical part of the foregoing publication

[2] would lead to a wrong behavior of reflection of elec-

trons (e.g. for angles smaller than 20˚), if the form factor

function F(θ) would have been omitted, since q(θ) would

then be highly diverging. On the other side, it is our par-

ticular interest to exploit small angle back scatter at the

Tungsten wall. At this place it should also be mentioned

that a smaller focusing effect in the multilayer cone is

obtained by the Compton scatter of the γ-radiation, if the

γ-quanta are scattered inside the cone. However, the fo-

cusing of fast electrons is much more significant. Since

the focusing via wall scatter works best with Tungsten,

we do, in general, not present calculations with other

material such as Tantalum or Lead. The only exception

with a Ta/Pb combination of the cone wall is restricted to

one case in order to verify the preference of a Tungsten

wall. In all figures of the section results we have adjusted

the impinging electron beam to real conditions: The ra-

dial distribution at target surface is assumed to be a

Gaussian with σ = 1 mm:

22

0exp 2Ir Ir

(35)

2.3.4. Remarks to the Measuremen t Configuration

The measurements have been carried out at a Varian

Clinac, which has been subjected to demounting at the

hospitals ‘Rhön-Klinikum’ in Frankfurt/München-Pasing,

Germany. A free place in the carousel served as the

source for positioning and testing the multi-target. The

preparation of the Tungsten plates and wall with/without

surrounding magnet has been handled in the machine

shop Feuchter (Backnang, Germany) equipped with high

technology facilities necessary for preparing of the mea-

surements. The expenses have been paid by the author

without any further support.

3. Results

The succeeding Figure 9 serves as a reference standard

for all other figures; this figure has been taken from the

previous study [2] and serves as a comparison standard.

The BS production according to Figure 1 (blue curve,

standard target) is scored along the plane immediately

below the Tungsten target. The height at the central axis

(x = y = 0) is normalized to “1”, and the whole behavior

of the intensity distribution shows all disadvantages of

the conventional target, since it decreases slowly, and

even at a radius of 7 cm a noteworthy intensity has been

scored. Thus the domain with r > 1 cm results from mul-

tiple electron scatter in the target with no benefit for any

application and requires a lot of shielding material. The

behavior in the domain r < 1 cm gives raise to study a

multitarget cone with a radius of 1 cm at the end of the

Figure 9. Comparison between standard target (Figure 1)

and multi-layer target, electron energy E = 6 MeV.

cone. The cone consists of 20 layers (distance 5 mm per

layer), total depth: 10 cm, the thickness of the wall

amounts to 0.02 mm Tantalum (inside) and 10 mm Lead

(outside) in contrast to all other cases, where 2 mm Tung-

sten have been used.

It should be pointed out that the application of E = 18

MeV electron energy instead of E = 6 MeV leads to

rather similar properties as shown in Figure 9. Therefore

we do not report them. With regard to all forthcoming

Figures we use standard conditions of the cone wall,

which consists of 2 mm Tungsten (with and without ex-

ternal magnetic field). It should be noted that in all re-

sults we had to assume air between the plates, the cones

were not positioned in vacuum.

Now we want to turn the interest to the three cases ac-

cording to Section 2. We should add that for comparison

we have also considered the case, where the Tungsten

wall has been replaced by Lucite. Figure 10 presents the

difference in the energy spectrum between one single

target (standard case) and the multi-target (case 1 with

1.2 T according to Figure 11). The shift to a higher en-

ergy spectrum in the non-standard case is obvious. Fig-

ures 11-13 show calculated results immediately below

the exit of the photon beam at the cone end, where meas-

urements were impossible.

Figures 11-13 clearly show the role of wall reflec-

tance of scattered electrons, if we consider Tungsten in-

stead of Lucite, i.e. the focusing effect of Tungsten (high

Z material) is significantly improved.

Figures 14-16 present the situation at a distance of 90

cm from the end of the cone (calculations and measure-

ments).

The normalization of the fluence has been taken such

that the maximum case according to Figure 14 is “1”,

which is also valid with regard to the following Figures

17-19 related to measurement data.

The rather small opening and the focusation by the

magnetic fields can obviously compensate the (small)

asymmetry of the incoming electron beam, which ap-

pears to be a consequence of the measurement condi-

tions. With regard to the larger field sizes the compa-

rison between measurement and calculation appears to

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156

Figure 10. Relative energy fluence spectrum of the BS of 6

MeV electrons: The standard target refers to the condition

presented in [7], i.e. below the flattening filter. In contrast

to this condition the multi-target spectrum is scored at the

end of the cone. The flattening filter is superfluous.

0

10

20

30

40

50

60

70

‐1.2 ‐0.8 ‐0.400.4 0.8 1.2

Fluence

Radial distancerincm

case1:Lucitewall

case1:Tun g s ten

wall

case1:Tun g s ten

wall+0.6T

case1:Tun g s ten

wall+1.2T

Figure 11. Fluence distribution at the end plate of the cone.

The cone diameter at this position amounts to 0.5 cm.

0

5

10

15

20

25

‐2‐1.5 ‐1‐0.50 0.5 1 1.5 2

Fluence

Radialdistancerincm

case2:Lucitewall

case2:Tungs te nwal l

case2:Tungs te nwal l +

0.6T

Figure 12. The diameter of the cone at the end plate now

amounts to 0.7 cm, the case with the magnetic field strength

1.2 T has been omitted, since it is not necessary for in-

creased field sizes.

0

1

2

3

4

5

6

‐2‐1.5 ‐1‐0.500.511.52

Fluence

Radialdistancerincm

case3:Lucitewall

case3:Tung st en wal l

Figure 13. The diameter of the cone at the end plate now

amounts to 1.3 cm (magnetic fields have been omitted).

0

0.25

0.5

0.75

1

‐40 4

Re lati ve fluence

Radial distancerincm

Lucite Tung s ten0TTungs te n 0.6TTu n g s ten 1.2T

Figure 14. Case 1 at z = 90 cm (diameter: 6 cm).

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

‐9‐6‐30 3 6 9

Re lat i vefluence

Radialdistancerincm

Tun gs te n 0.6T

Lucite

Tun gs te n

Figure 15. Case 2 at z = 90 cm (diameter: 12 cm).

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

‐20 ‐15 ‐10 ‐5 05 101520

Re lativefluence

Radialdistancerincm

Lucite

Tung s ten 0T

Figure 16. Case 3 at z = 90 cm (diameter: 30 cm).

Figure 17. Measurement data for the case 1 with 6 cm di-

ameter.

be more important (Figures 18 and 19).

In particular, the last case shows best the asymmetry in

measurements. However, in spit of this fact the agree-

ment between theory noteworthy.

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157

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

‐10 ‐50 510

RelativeFluence

Radial distanceincm

Measurement

Measurement

Calculation0T

Calculation0.6T

Figure 18. Calculations and measurements for the case with

a field size diameter of 12 cm.

Figure 19. Calculations and measurements for the case with

a field size diameter of 30 cm.

4. Discussion and Conclusion

It could be shown that in conventional linear accelerators

used in medicine a multi-target consisting of a Tungsten

wall (thickness of the wall at least 2 mm) and 31 very

thin plates (thickness of a plate: ca. 0.01 mm) is superior

to the standard accelerator. The BS beam (inclusive di-

vergence) can be formed according to the desired proper-

ties. The energy spectrum is significantly increasing even

in the absence of a focusing magnetic field and is even

better than a conventional beam, which has passed a flat-

tening filter. Thus the omission of such a filter provides a

further yield of the factor 3 - 4. The optional amplifica-

tion of the focusing effect by suitable external magnetic

fields (with regard to the required properties, see e.g.

Figure 6) can be taken into account, in particular, if the

outcoming γ-beam should be very efficient by restricting

rather small fields. These properties are important for

scanning methods, stereotaxy, IMRT or tomography. It is

possible to reach some essential progress in the domain

of linear accelerators in radiotherapy, since the modern

irradiation techniques such as IMRT, stereotaxy, etc. do

not require large field sizes, e.g. a 40 × 40 cm2 at a dis-

tance of 100 cm from the focus. This progress can be

achieved by exploiting small angle reflectance of fast

electrons at a Tungsten wall. The wall has to map the

desired divergent properties of the beam. A further aspect

of this study is that we are able to save heavy high

Z-material for the shielding of the accelerator head. The

attached appendix deals with stopping power and heat

production of high energy electrons. By that, we have

been able to estimate the heat production in each thin

plate, which turned out to be lower 20˚C per 600 MUs.

Thus the systems even works without further cooling of

plates, if the rate of MUs will be increased to 1000 or more.

REFERENCES

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[4] L. Eyges, “Energy Loss and Scatter of Neutrons and

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[6] R. P. Feynman and A. R. Hibbs, “Quantum Mechanics

and Path Integrals,” Mac Graw Hill, New York, 1965.

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position/Convolution Algorithm and Its Foundation on

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Applications to Proton/Photon Dosimetry and Image

Processing,” Inverse Problems, Vol. 26, No. 8, 2010, Ar-

ticle ID: 085002.

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[9] W. Ulmer, “Deconvolution of a Linear Combination of

Gaussian Kernels by Liouville-Neumann Series Applied

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[11] M. J. Berger, J. S. Coursey and M. A. Zucker, “ESTAR,

PSTAR and ASTAR: Computer Programs for Calculating

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Stopping-Power and Range Tables for Electrons, Protons

and

-Particles (Version 1.2.2),” National Institute of

Standards and Technology, Gaithersburg, 2000.

[12] I. Kawrakow and D. O. Rogers, “The EGSnrc Code Sys-

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Appendix: Collision Interaction of Electrons

with Matter

The purpose of this appendix is to provide tools for the

determination of the heat production of electrons in

Tungsten and to circumvent Bethe-Bloch equation at

analytical calculations. In a previous publication [10] we

have applied the generalized Bragg-Kleeman rule to

proton dosimetry. Therapeutic electrons always satisfy

E0 >> mc2 = 0.511 MeV. An optimum adaptation of rela-

tion (A1) to RCSDA of electrons [11]) shows (Figure A1)

that, for E0 ≤ mc2, p(E0) ≥ 1; for energies above mc2, p(E0)

< 1. For electrons, the factor A(water) = 0.238552 cm/

MeVp is also rather different to that for protons. The pa-

rameters for the calculation of p(E0) with Formula (1) are

given in Table A1.

Please note that the abscissa of Figure A1 refers to cm,

whereas the normalized ordinate is either stopping power

or dose absorption. According to [10], the Bethe-Bloch

equation describing the collision interaction of charged

particles can be summarized for electrons by the equa-

tion:

22

00

22

001 10

2

220

water 2

2exp2

exp 2

p

CSDA

l

RA EEmc

ppcE mc pqE mc

pqEmc

(A1)

A graphical representation of p(E) as a function of the

energy E is given in Figure A2. This equation is only

valid for water. Since the factor A according to equation

(A1) is proportional to

www

A

Z

, we are able to mo-

dify it by the substitution:

Figure A1. Stopping-power function of 20 MeV electrons

according to Formula (A3) and determination of the stop-

ping-power, obtained in the CSDA framework (red solid).

Measurement data have been obtained for a standard Var-

ian Clinac (“Golden Beam Data”).

Table A1. The table values of the dimensionless parameters

of Formula (A1).

p0 p1 p2 cl q1 q2

0.655 0.6344 0.2616 0.0023494 3.060 0.311

Figure A2. Function p(E0) determined by Formula (A1) and

ESTAR [11].

mediumwater mww wmm

A

AAZAZ

(A1a)

The meaning of the parameters (water: reference val-

ues) of the substitution (A1a) Aw = 18, Zw = 10, ρw = 1

g/cm3, and for other media we have to substitute the cor-

responding parameters Am, Zm, ρm (e.g. Tungsten: Zm = 74,

ρm = 19.25 g/cm3, Am = 183.84).

The inversion of Formula (A1) provides the stopping

power S in dependence of the residual energy:

112

111 2

0

1p

pp

SuRuRumc A

(A2)

We denote IB(z) the decreasing contribution of the im-

pinging BS produced in the double scatter layer of the

accelerator. The depth dose curve of an electron beam is

then given by the formula:

0

01

exp

,, d

B

B

Iz Iz

DzSuKssu zu Iz

(A3)

The stopping-power formulas (A1-A3) have to be used

for therapeutic electrons; in this case, the length contrac-

tion is not a negligible effect. It is also noteworthy that,

for p(E0) ≤ 1, the singularity of E(s) at s = RCSDA is re-

moved. We have used Formula (A3) for the depth-dose

calculation of 20-MeV electrons and subjected it to con-

volutions. The measured and calculated (including the

BS effects) curves are shown in Figure A1; the kernel,

used in the convolution, and related parameters are dis-

played in Table A1. The stopping-power formulas (A1-

A3) have to be used for therapeutic electrons; in this case,

the length contraction is not a negligible effect. It is also

noteworthy that, for p(E0) ≤ 1, the singularity of E(s) at s

= RCSDA is removed. We have used Formula (A3) for the

depth-dose calculation of 20-MeV electrons and subject-

ed it to convolutions. The measured and calculated (in-

cluding the BS effects) curves are shown in Figure A1;

the kernel, used in the convolution, and related parame-

ters are displayed in Table A1.

The CSDA stopping-power is shown in Figure A1.

With regard to accounting for BS, we have only consid-

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ered the contribution resulting basically from the dou-

ble-scatter foil. This contribution is determined by the

software EGSnrc, see [12]. The low-energy BS (produc-

ed by the electrons in water), its multiple scatter and ab-

sorption can be easily explained by the energy-range/

straggling (this is basically relativistic and a single Gaus-

sian is not sufficient). In order to include lateral scatter of

the electron beam, we have to add a further scatter kernel

in Equation (A3), which may based on the principles

developed in Section 2.3.2. By using the methods work-

ed out in a previous publication with regard to photon

scatter [7], an efficient and fast superposition/convol-

ution calculation model can readily be developed.