f3 fs14 fc0 sc0 ls0 ws0">by evaluation of the above integral (23):
 

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
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
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.
W. ULMER
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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
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
W. ULMER
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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
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
21.5 10.50 0.5 1 1.5 2
Fluence
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
21.5 10.500.511.52
Fluence
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
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
9630 3 6 9
Re lat i vefluence
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
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.
W. ULMER
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0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
10 50 510
RelativeFluence
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.
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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-
W. ULMER
Open Access IJMPCERO
160
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.