International Journal of Medical Physics, Clinical Engineering and Radiation Oncology, 2013, 2, 147-160
Published Online November 2013 (
Open Access IJMPCERO
Creation of High Energy/Intensity Bremsstrahlung by a
Multi-Target and Focusing of the Scattered Electrons by
Small-Angle Backscatter at a Cone Wall and a Magnetic
Field—Enhancement of the Outcome of Linear
Accelerators in Radiotherapy
W. Ulmer1,2
1Department of Radiotherapy, Klinikum München-Pasing, Göttingen, Germany
2MPI of Biophysical Chemistry, Göttingen, Germany
Received September 21, 2013; revised October 20, 2013; accepted November 2, 2013
Copyright © 2013 W. Ulmer. This is an open access article distributed under the Creative Commons Attribution License, which per-
mits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The yield of bremsstrahlung (BS) from collisions of fast electrons (energy at least 6 MeV) with a Tungsten target can be
significantly improved by exploitation of Tungsten wall scatter in a multi-layered target. A simplified version of a pre-
viously developed principle is also able to focus on small angle scattered electrons by a Tungsten wall. It is necessary
that the thickness of each Tungsten layer does not exceed 0.04 mm—a thickness of 0.03 mm is suitable for accelerators
in medical physics. Further focusing of electrons results from suitable magnetic fields with field strength between 0.5
Tesla and 1.2 Tesla (if the cone with multi-layered targets is rather narrow). Linear accelerators in radiation therapy
only need to be focused by wall scatter without further magnetic fields (a standard case: 31 plates with 0.03 mm thick-
ness and 1 mm distance between the plates). We considered three cases with importance in medical physics: A very
small cone with an additional magnetic field for focusing (the field diameter at 90 cm depth: 6 cm), a medium cone with
an optional magnetic field (field diameter at 90 cm depth: 13 cm) and a broad cone without a magnetic field (the field
diameter at 90 cm depth: 30 cm). All these cases can be positioned in a carousel. Measurements have been performed in
the existing carousel positioned in the plane of the flattening filter and scatter foils for electrons.
Keywords: Multitarget; Bremsstrahlung; Wall Scatter; Focusing by Magnetic Field
1. Introduction
A principal problem in the creation of BS of linear ac-
celerators used in radiotherapy is the lack of efficiency,
since only a rather small part of the created BS is avail-
able for applications. Even for a 40 × 40 cm2 field (dis-
tance of 100 cm from the focus), the BS yield is small and
most of it goes lost at the primary collimator and jaws.
We have usually to deal in radiotherapeutic applications
with much smaller field sizes than 40 × 40 cm2. Thus the
effectivity in IMRT and stereotaxy are much smaller. A
further source of BS loss is the flattening filter, which is
used to homogenize transverse profiles. To circumvent
some of the above disadvantages, a linear accelerator
using multiple Beryllium targets has been suggested [1],
with electron energies of the order of 80 - 100 MeV to
yield a spectrum compared to a conventional machine
with 4 - 6 MeV and a single Tungsten target. However,
this concept has the disadvantage that electrons deceler-
ated down to ca. 45 - 50 MeV have to be removed by a
magnetic field, since they would produce low BS ener-
gies in further Beryllium targets.
In this communication, we present experimental re-
sults of modified configurations of a multitarget system
considered previously [2] consisting of very thin Tung-
sten layers, with a thickness << 0.1 mm to create BS in a
much more efficient way by the ordinarily used electron
energies between 6 and 20 MeV. This way of BS crea-
tion exploits two physical effects, which can be used to
focus on scattered electrons, namely the wall scatter of
high Z materials (Tungsten), a further option, and a suit-
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able external magnetic field.
2. Material and Methods
2.1. Schematic Representation of BS by a Linear
The following Figure 1 shows the essential component
modules of a linear accelerator. It starts with the imping-
ing electron current on the BS target (Tungsten). The
foregoing modules of the beamline such as klystron,
modulator, acceleration tube, deflection of the electron
current by a bending magnet are not of importance here.
In the target two competition processes occur, namely BS
creation and multiple scatter of electrons by simultaneous
production of heat.
This multiple scatter and heat production must be re-
garded as the reason, that the BS creation does not show
any preference direction. The beam line according to
Figure 1 indicates that only that part of the BS can be
used, which can pass through the opening of the primary
collimator. The flattening filter immediately below the
vacuum window affects the shape of the beam, which is
further controlled by the jaws to obtain the desired field
size. However, the flattening filter significantly attenu-
ates the intensity of the photon beam, and due to the in-
evitable Compton scattering, it also acts as a second
source, which affects the shape of the profiles of larger
field sizes (e.g. the penumbra).
As a resume, we can conclude that the present linear
accelerators do not provide a high efficiency, in particu-
lar with regard to the novel irradiation techniques such as
Stereotaxy, Rapid Arc, and IMRT, where very small field
sizes are required and most of the produced BS goes lost
by shielding of the accelerator head. Therefore the ques-
tion arises, in which way we can significantly improve
the yield of BS and reduce the required shielding mate-
rial in the accelerator head.
Figure 1. Schematic representation of a linear accelerator.
2.2. Qualitative Considerations with Regard to
Focusing of Electron Scatter
Figures 2 and 3 indicate that by restriction to a single
Tungsten target there is no mean to prevent the scatter of
electrons within the target material.
The thickness of the target of standard linear accelera-
tors amounts to ca. 1 mm Tungsten and immediately be-
low 1 mm Copper in order to increase the removal of the
produced heat from the target. The BS spectrum created
in Copper is significantly lower than that of Tungsten; it
is most widely absorbed in the flattening filter. Since the
multiple scatter and heat production are responsible for
the low efficiency and for a lot of necessary shielding of
the accelerator head, we consider at first an alternative
way to exploit BS by a multi-target. This consists of a
configuration of very thin Tungsten layers. There thick-
ness should not exceed 0.04 mm. We assumed previously
a thickness of 0.01 mm for each plate we need 100 plates
to reach an overall thickness of 1 mm with an effective
Target (Tungsten, Copper)
Bremsstrahlung (BS)
Electron current:
Gaussian profile
Figure 2. Multiple scatter and creation of BS in the target.
Figure 3. Small angle backscatter of electrons at a Tungsten
wall (reflection) induces focusing of electrons. The backsca-
tter is amplified by a cone configuration of the Tungsten
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depth of 10 cm [2]. This configuration requires a com-
pletely new construction of the accelerator head. The
present study is based on measurement data is based on
measurement data with only a slight modification of the
head, since the new target could have been tested by
placing in a free position of the carousel. However, with
regard to clinical accelerators it is possible to work with
a smaller number of plates. Figure 3 qualitatively shows
that a tilted wall makes the reflection angle small and
therefore the backscatter can be increased. The left-hand
side of this figure presents the consequence of this effect,
namely a cone configuration of the wall, which embeds
31 Tungsten plates. The heat production in each plate is
negligible and no additional removal of heat is needed.
Thus we can verify that the configuration below makes
the primary collimator itself to a cone target consisting of
ca. 100 plates and the created BS shows the preference
direction of a cone. This configuration implies two ad-
vantages: The total depth of cone is assumed to amount
to 100 mm, and the 100 plates with 1 mm distance be-
tween the plates can be regarded as a continuum, i.e., a
Tungsten density ρt of the multi-target part of the cone
can be assumed. Therefore theoretical calculations can be
simplified much. Otherwise, we have to perform com-
plicated numerical step-by-step calculations (this is only
possible with regard to Monte-Carlo calculations). The
second advantage refers to the direction of the created BS.
If the electron energy is much higher than 0.511 MeV
(rest energy of the electrons), then the direction of the
created BS approximately agrees with the direction of the
impinging electrons. Thus BS with large angles cannot
be completely avoided, but significantly reduced to a mi-
nimum contribution. In a second order, there exists also a
focusing effect of BS at the Tungsten wall due to the
small-angle part of the Compton effect.
According to Figure 4 we can exploit and, by that,
amplify the focusing influence obtained by wall scatter,
namely by an additional external magnetic field, which
must have the property that a permanent gradient of the
field component Br perpendicular to the propagation axis
(z-axis) is present. Thus the complete configuration re-
presents a similarity to features of an electron micro scope.
Tungsten plate s
Inner boundary of the
cone: Tungsten wall
Figure 4. Schematic representation of a multi-layered target
with an additional magnetic field for focusing.
The influences of the additional magnetic field are the
reduction of the reflection angle at the Tungsten wall and
the corresponding decrease of the impinging angle of the
inner electrons at each Tungsten plate.
The additional magnetic field makes sense, if the cone
is rather narrow, i.e. the opening angle is small and a
very narrow radiation beam with extremely high intensity
is required. However, the application of an additional
magnetic field for focusing is optional, since many con-
ceptual designs of medical accelerators can already be
improved by a configuration without external magnetic
field (Figure 3). It should be pointed out that in both
Figures 3 and 4 we have used a qualitative presentation
based on forestalled results of succeeding sections. The
new conceptions of designing linear accelerators can be
justified by the following synopsis:
For electrons with energy >> mc2 (0.511 MeV) the di-
rection of the created γ-quanta agrees with the actual di-
rection of motion of electrons. This implies for electrons,
which suffered large-angle scattering, that the created BS
has unfortunately not the preference direction of the in-
cident beam. There is additionally a non-negligible amount
of energy loss of electrons and heat production as a con-
sequence of multiple electron scatter.
A multi-layer target is suitable to reduce multiple scat-
ter significantly by exploiting focusing effects by the
cone wall (high Z material, e.g. Tungsten) and optionally
by a proper magnetic field. Thus we have to split the
conventional Tungsten target (thickness: usually 1 mm)
in, at least, 10 sub-targets whose thickness is of the order
0.1 mm and the distance between the layers should then
amount to ca. 1 cm. However, a thickness of 0.012 mm
(these layers can be purchased) and a distance of ca. 1 - 2
mm between each appears to be much more convenient.
The geometrical configurations of the three cases un-
der consideration are:
Case 1 (magnetic field necessary)
Diameter of the Tungsten plate at entrance: 3 mm, at
the end of the cone with z = 3 cm: 5 mm; diameter of the
circular field size at z = 90 cm: ca. 6 cm.
Case 2 (magnetic field optionally possible)
Diameter of the Tungsten plate at entrance: 3 mm, at
the end of the cone with z = 3 cm: 7 mm; diameter of the
circular field size at z = 90 cm: ca. 12 cm.
Case 3 (magnetic field not required)
Diameter of the Tungsten plate at entrance: 3 mm, at
the end of the cone with z = 3 cm: 13 mm; diameter of
the circular field size at z = 90 cm: ca. 30 cm.
2.3. Theoretical Calculations and Monte-Carlo
Calculations with GEANT4
Monte-Carlo calculations including new boundary condi-
tions (wall scatter by some proper materials, influence by
magnetic fields) have been carried out by specific modi-
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fications of the Monte-Carlo code GEANT4 [3], which
have been described in detail in the previous study [2].
Therefore we intend to give here only a brief report on
the underlying physical principles in order to provide a
qualitative understanding.
2.3.1. Some Physical Toolkits
General properties and requirements
With respect to both calculation procedures (GEANT4
and analytical calculations) we have to use the relativistic
energy-momentum relation:
224 2
Wmc cp (1)
The relativistic energy E of a particle (without rest en-
ergy) is given by:
1Emc mc
The relativistic energy-momentum relation in the pres-
ence of a magnetic field (vector potential A) reads:
224 2
Wmcec cpA (3)
Equation (3) represents a quantum-mechanical equa-
tion, if the transition
is carried out.
The geometrical configuration is shown in Figure 5.
On the left-hand side: Small circle of the cone entrance
(radius r0) for impinging electron beam. Right-hand side:
Larger circle at the cone exit (radius rf). The inner circle
symbolizes the area of the entrance. The angle θ shows
the opening angle of the cone, which is usually small in
those cases, where a strong magnetic field is required.
This will be verified in the result section.
The components of the magnetic induction B with
div·B = 0 are given by:
xy z
zx y
The field strength has to satisfy, at least, the following
 
22 22
00 0
Bzr BrzLBBrr
rr zxy
 
 
For the reason of symmetry the following condition
Figure 5. Schematic representation of the geometrical pro-
perties of the multilayer target.
has to be valid:
BB (6)
In the above equation Br(z) means the radial compo-
nent as a function of z, r0 is the field radius at the en-
trance of the beam (z = 0), and rf the related radius at the
end (z = L). The possible devolution of some cases of
interest is shown in Figure 6. However, the properties of
this figure do not represent a rigid scheme, since it
mainly depends on the design of the target, whether other
lengths L of the cone (central axis) are required. We want
also point out that the magnetic field does not disappear
at z = L. This length has only to agree with the focusing
part of the magnetic field, whereas a small area with a
defocusing part cannot be avoided. It may serve to re-
move those electrons with sufficiently low energy, where
the production of BS is no longer desired.
2.3.2. Fermi-Eyges Theory, Multiple Scatter Theory
of Molière and Inclusion of Magnetic Fields
Fick’s law of diffusion plays a key role in a lot of physi-
cal/chemical/physiological processes; it is also used for
the description of scatter and absorption of electrons,
protons or neutrons in a medium such as Tungsten and is
referred to as Fermi-Eyges age equation [4]. A more ac-
curate theory of scatter has been given by Molière [5].
with respect to Fick’s law D is the diffusion coefficient, ρ
the particle (electron) density (concentration) and Δ the
Laplace operator. In the case of the Fermi-Eyges theory,
the particle density ρ has to be replaced by an energy-
distribution E, which reads:
Since both equations formally agree, we now intro-
duce the amplitude U in order to be independent of the
actual meaning. The same fact is also valid with regard to
the constant factor D, which may either be identified
with a diffusion constant D or with a parameter DF in
Fermi-Eyges theory. Equations (7) and (8) follow from
the property that a gradient of the density is connected
with a current:
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3 cm
Ferromagnet + internal Solenoid
cooling via air flow
Ferromagnet + internal Solenoid
(cooling via air flow)
Figure 6. Increase of the radial component Br along the
surface of the cone. M1 = case 1: B0 = 0.4 Tesla, Bf = 1.2
Tesla; case 2: B0 = 0.2 Tesla, Bf = 0.6 Tesla. (a) Schematic
representation of the considered configuration; (b) Section
of the magnet with rotational symmetry.
jDU  (9)
In addition, a balance equation has also to hold:
div jvU
The term v represents the scalar product of the 3D
differential operator with a 3D velocity v. If the par-
ticles have the charge q (for electrons: q = e) and a mag-
netic field, described by the vector potential A, is present,
then an additional momentum/velocity due to the mag-
netic field (Lorentz force) has to be accounted for:
divA A
 
The velocity v results from the division of the mo-
mentum p by the particle mass M (= electron mass m)
and c is the velocity of light. Now Equations (9) and (10)
jDAU Udivj
 
 (12)
 
The last Equation (12a) can also be written in the form:
div 0
 
If we replace U by a probability distribution P, then
Equation (13) turns out to be the Kolmogorov forward
equation, and we regard v as an effect of a magnetic field.
However, Equations (12) and (13) are not yet fully gauge
invariant, since we have only modified the diffusion cur-
rent j by the magnetic interaction. The magnetic effect is
not yet included in the balance equation. In order to com-
plete this condition.
We have to write:
mcD mc
 
 
 
 
22 22
eA Dmc U
 
Besides gauge invariance a further reason for obtain-
ing Equation (14) is the transition to the Schrödinger
equation with magnetic field, since the simple Fick’s law
of diffusion can be transformed to a Schrödinger equa-
tion without external fields. Equation (14) provides a cor-
respondence in the presence of an external magnetic field.
It should be noted that the Kolmogorov forward equation
is contained as in the special case of Equation (26) by
setting A2 = 0. According to the Fermi-Eyges/diffusion
theory we can always put:
2or 2
 (15)
The parameter
can replace the arbitrary time vari-
able t, if scatter/diffusion only occurs in a short time in-
terval. We consider the case, where the z-component B =
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Bz = B0 of a constant magnetic field is responsible for the
motion of electrons and pass to the general case thereaf-
ter. Since the magnetic field strength B0 is given by
rotBAxA, we can choose A as follows:
2, 2
xy yx
zy x
 
By that, Equation (14) becomes (U = E):
22222 2
EtD EeBmcDxyE
eB cmyxxyEE
 
 (17)
Since in z-direction motion without magnetic interac-
tion is allowed, the following separation of the above
equation is possible:
, ,expexpexp.
 (18)
This provides the following equation:
DxyD t
 
The following parameters are given by:
eB mc
 (20a)
It has to be pointed out that the Larmor frequency ω0
according to Equation (20) is identical with those ob-
tained by a Schrödinger or Dirac equation with an exter-
nal magnetic field. The basis solution (generating func-
tion) from which we can construct all other solutions
(see e.g. comparison with Schrödinger equation) is gi-
ven by:
,,exp2 exp
yta xyt
 
 (22)
  (22a)
The parameters a and λ' represent complex values;
however, we can form a linear combination to get the
real solution: Before we shall consider this property, we
note that with respect to the z-coordinate a set of solu-
tions of the diffusion equation are permitted [2]. They
result from the Fourier transform of the specified z-de-
pendent function:
  
01 2
ztA kikzk D tk
Akaak akak
 
Every power of k of the expansion of A(k) represents a
solution, if the integral is carried out. We therefore de-
nn F
ztPztzDt (24)
The polynomials Pn(z, t) have been previously defined
by evaluation of the above integral (23):
 
ztikzDkt kk
 
The boundary parameter k0 is given by k0 = 1/z0. The
evaluation of the integral (18) provides Equation (26):
ztU tzDtss 
The parameters of this equation are given by:
Ut zDt
zizDt Dt
zizDt Dt
 
With the help of Equation (26) we consider the solu-
00 0
10 0
cos 4
Exyzt ztt
xy Dt
 
 
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-
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.
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
cos 4sin4
nmnFnm nFn
Dxy tBDxy t
nn n
 
Powers of odd order:
 
21 2221 22
212 12121212121
21 0
cos 4sin4
210,0,1, 2,3
nmnFn mnFn
Dxy tBDxy t
  
 
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
given by the magnetic field strength B0. Thus a sud-
den change of the magnetic field strength from +B0 to
00 0
 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):
brert bre
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:
brert bre
  (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
  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:
22 2
21 2121
cos 4
Larmor frequency
nnF n
ExyAD r
neBmcne BB
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
Open Access IJMPCERO
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:
0exp 2Ir Ir
 
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-
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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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.
1.2 0.8 0.400.4 0.8 1.2
Radial distancerincm
case1:Tun g s ten
case1:Tun g s ten
case1:Tun g s ten
Figure 11. Fluence distribution at the end plate of the cone.
The cone diameter at this position amounts to 0.5 cm.
21.5 10.50 0.5 1 1.5 2
case2:Tungs te nwal l
case2:Tungs te nwal l +
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.
21.5 10.500.511.52
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).
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).
9630 3 6 9
Re lat i vefluence
Tun gs te n 0.6T
Tun gs te n
Figure 15. Case 2 at z = 90 cm (diameter: 12 cm).
20 15 10 5 05 101520
Re lativefluence
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-
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.
Open Access IJMPCERO
10 50 510
Radial distanceincm
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.
[1] R. Svensson and A. Brahme, “Effective Source Size,
Yield and Beam Profile from Multi-Layered Bremss-
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41, No. 8, 1996, pp. 1353-1379.
[2] W. Ulmer, “On the Creation of High Energy Bremss-
trahlung and Intensity by a Multi-Target and Repeated
Focusing of the Scattered Electrons by Small-Angle Back-
scatter at the Wall of a Cone and Magnetic Fields—A
Possible Way to Improve Linear Accelerators in Radio-
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tion Physics and Chemistry, Vol. 81, No. 4, 2012, pp.
[3] “GEANT4 Documents,” 2005.
[4] L. Eyges, “Energy Loss and Scatter of Neutrons and
Charged Particles,” Physical Review, Vol. 74, 1948, pp.
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ing through Matter,” Zeitschrift für Naturforschung, Vol.
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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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[8] W. Ulmer, “Inverse Problem of Linear Combinations of
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Processing,” Inverse Problems, Vol. 26, No. 8, 2010, Ar-
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[9] W. Ulmer, “Deconvolution of a Linear Combination of
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[10] W. Ulmer and E. Matsinos, “Theoretical Methods for the
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[11] M. J. Berger, J. S. Coursey and M. A. Zucker, “ESTAR,
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Open Access IJMPCERO
Stopping-Power and Range Tables for Electrons, Protons
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Open Access IJMPCERO
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-
001 10
water 2
exp 2
ppcE mc pqE mc
 
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
, 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
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:
 
111 2
SuRuRumc A
 
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:
 
,, d
Iz Iz
DzSuKssu zu Iz
 
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-
Open Access IJMPCERO
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.