ISSUES WITH THE HIGH ENERGY RADIOGRAPHY
SIMULATIONS
FeyziInane
Iowa State University Center for Nondestructive Evaluation Ames, IA 50010, USA
ABSTRACT. Although most of the industrial radiography relies on x-ray tubes, in some cases,
gamma rays or linear accelerators form the interrogating radiation beam. Due to high energy
levels of photons in such cases, new physics start to accompany radiation absorption, scattering
events. Most of these new physics are interactions of charged particles produced by high-energy
photons. Charged particles with sufficient energy can produce photons. High-energy
radiography simulations need to account for both photon and charged particle transport for
proper computation of photon fluxes. This work will outline various physical mechanisms that
need to be taken into consideration and challenges that come with designing deterministic
algorithms that can handle the mathematics related to these new physical mechanisms.
INTRODUCTION
Although industrial radiography and medical diagnostic radiography employ x-ray
tubes as the interrogating beam sources, there are other cases where the radiation source is
not a x-ray tube but either a radioisotope emitting high-energy photons or a linear
accelerator tube. In such cases, the photon beam will be composed of either multiple
high-energy photon lines or a spectrum that can go from a few keV to many MeV levels. In
the x-ray tube based implementations, the photon energy levels dictate that only photon
probable interaction mechanisms are photoelectric absorption, coherent scattering and
incoherent scattering. Since the primary photons are already low energy photons, any
secondary photon resulting from such interactions will not be instrumental in the outcome
of the radiography because of their very low energy levels and relatively small quantities.
With the increase in the primary photon energy levels, another photon interaction
mechanism, pair production, becomes possible and photoelectric absorption loses its
dominance over the other interaction mechanisms. One byproduct of this trend is
increasing amount of energy transferred to charged particles through scattering, pair
production and photoelectric absorption interactions. Since amount of energy transferred
to the charged particles in terms of kinetic energy is instrumental in production of
secondary photons, secondary photons start to increase both in quantity and importance.
The major difference between the primary photons and secondary photons is that
secondary photons are less energetic compared to the primary photons. Since the photon
detector usually are more sensitive to lower energy photons than the higher energy ones,
estimation of secondary photon fluxes incident upon the detectors becomes an important
issue in the high-energy radiography. In this article, we will provide some insight into the
issues that are important in developing a deterministic algorithm that can be used in
simulating high-energy radiography procedures.
CP657, Review of Quantitative Nondestructive Evaluation Vol. 22, ed. by D. O. Thompson and D. E. Chimenti
© 2003 American Institute of Physics 0-7354-0117-9/03/$20.00
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LITERATURE REVIEW
Early Monte Carlo charged particle transport computations started 1960's [1].
Monte Carlo computations were used for photon transport even before that date [2]. After
those initial attempts, Monte Carlo based transport calculations became widely available
with the advent of computers. Today, there are many general purpose Monte Carlo codes
that can be used for various forms of transport calculations. Some of the well-known
general-purpose Monte Carlo codes are MCNP [3,4] and EGS4 [5]. These codes or their
derivatives are routinely used in various transport calculations and dosimetry. In contrast
with the Monte Carlo methods, research for development of deterministic transport tools
for charged particle transport started much later and gained speed in the recent years. In
1967, Zerby et. al. provided an extensive review of the electron transport theory and some
experimental and computational results [6]. Most of the deterministic methods use discrete
ordinates method. The first one has been reported by Bartine et.al. in early 1970's [7,8]
and they implemented an extended transport correction for handling forward peaking
scattering cross section. This has been followed up by a series of articles reporting
implementation of Fokker-Planck operators [9-15]. Following these initial reports, there
have been many reports studying various aspects of the Boltzmann-Fokker-Planck
equations used in the charged-particle transport. Some of the later ones have been
implemented to solve the pencil beam or collimated beam problems that are typical to
medical radiotherapy implementations [16-18].
MOTIVATION
Detection efficiency is quite important in radiography for improving spatial
resolution. Regardless of the detector type, detection efficiency is a strong function of the
photon energy. Since higher energy levels result in lower interaction rates, detectors will
detect the incoming photons with lower efficiency levels. As a result, photon detection
efficiency is much better for lower energy levels because of higher interaction coefficients.
Very energetic photons transfer significant amounts of energy to charged particles emitted
as a result of photon object interactions. These charged particles, in turn, generate
secondary photons. These photons are likely to have energy levels low compared to the
primary photons that caused emission of charged particles. Introduction of these
secondary photons into the overall photon spectrum is likely to cause distortions in the
lower portions of the overall photon spectrum. A typical example of such a case is
provided by Takeuchi and et. al. [19]. One of the problems they worked on is made up of
a 10 cm. thick lead plate with 6.2 MeV photons incident on one side. They provide the
photon flux spectrum on the other side of the lead plate with and without bremsstrahlung
photons. According to their finding, two cases provide similar results above 3 MeV.
Around 1 MeV, case with bremsstrahlung photons provides a spectrum that is about 60%
larger than the case without bremsstrahlung. Those numbers are about 40, 30, 15 and 10 %
for 1.5, 2, 2.5 and 3 MeV respectively. In another problem, they study the photon flux
inside the lead shield. In this case they analyze the flux at 4 and 10 mean free path (mfp)
distances from the boundary with a 8 MeV photon source. The differences at 4 mfp are
500, 350, 220, 100 and 50 % at 1, 1.5, 2, 2.5 and 3 MeV respectively. There is not much
difference above 5 MeV. For 10 mfp case, these numbers are about 160, 100, 50, 25 and
10 % for the same energy levels. As seen from these numbers, introduction of charged
particle transport is likely to introduce significant amounts of secondary photons at energy
levels where the detectors are more sensitive. For lower atomic number materials, levels
of the secondary photon fluxes are likely to be much lower than the ones given for lead.
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GOVERNING EQUATIONS AND ISSUES
One of the major differences between x-ray tube and high energy radiography
mathematical formulation is that high-energy radiography requires a set of simultaneous
transport equations representing photon transport and charged particle transport rather than
only photon transport. If we use the integral transport equation notation, we can write
these equations for a high-energy photon source as given below.
R"
-l^(r-R"Q,E)clR"
RT
0
Ik(r,E,G)=\qk(r-KQ,E,Q)e
KT
-JZ* (r-R"Q,E)dR"
dR+Ik(rr,E,O)e °
k = 1,2,3
(1)
where Ik(r,E,Q) is photon, electron and positron flux respectively for k=l,2,3. Since the
interrogating radiation in radiography is usually an external photon source, the second term
in the right hand side of equation (1) will be nonzero only for k=l. The source term in
equation (1) is given in equation (2).
(2)
The first term in the RHS of equation (2) is Boltzmann operator, the second is continues
angular deflection term and the third is continues slowing down operator. The fourth term
is the source term induced by the other types of particles. The second and third terms are
usually known as Fokker-Planck operators used for cases where the scattering is strongly
forward peaked and energy loss per interaction is very small. Photon transport equation in
equation (1) does not have Fokker-Planck operators.
The fourth term in the RHS of
equation set (1) provides the coupling among three equations. Photons contribute to
electron flux through photoelectric absorption, incoherent scattering and pair production.
In the photoelectric absorption, photoelectron carries out the photon energy minus binding
energy. In the incoherent scattering, recoiled electron receives the energy that scattered
photon looses. In the pair production interaction, electron and positron receive their shares
of energy that is left over after the pair production energy is taken off from the incident
photon. This is the only mechanism that causes a positron flux. Electrons contribute to
photon flux through bremsstrahlung and fluorescence radiation following an ionization
event. Positrons contribute to photon flux though bremsstrahlung, fluorescence and
annihilation processes. All these mechanisms form the source term in the RHS of equation
(2). While photons and charged particles interact with the matter and contribute the fluxes
of other types, some of the interaction mechanisms that were not listed here contribute to
the flux of its own particle or photon type.
Photon Transport Issues
The major issue with the high-energy photon transport is severity of the anisotropy
seen in the incoherent scattering. With the level of anisotropy shown in figure 1, the
sampling of the scattering kernel becomes quite challenging. The conventional Legendre
polynomials approximation used in most of the deterministic methods face convergence
difficulties for scattering kernels that are highly anisotropic for such cases.
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Scattering Angle
FIGURE 1. Behavior of incoherent scattering kernel with incident photon energy levels.
Issues With Charged Particles
One of the issues that makes charged particle transport different than the photon
transport is amount of energy lost through charged particle interactions. While photons
can slow down to photoelectric absorption dominant energy levels through a few scattering
events with relatively large energy losses, charged particles slow down MeV levels to a few
keV energy levels only after thousands of interactions. Since energy groups can not be
chosen that small with the current computational power levels, conventional multigroup
approach is insufficient to represent this type of energy loss. The continuous slowing
down operator in equation (1) is meant to handle this energy loss process. Through that
operator, charged particle is assumed to loose energy continuously. Continuous slowing
down term requires scattering kernels to be decomposed into two components where one
of the components represent the cases where the energy loss is relatively large enough to
cause ionization and the other component represents the cases where the energy loss is
small. The slowing down operator is constructed on the basis of the component where the
energy losses are assumed to be small. The other component of the scattering kernel is
kept with the Boltzmann integral operator. The stopping power term, Sk(E), in equation
(2) is given below [20].
£,
S(E) = fa
(3)
where ass(E—>E',jio) represents scattering interactions with small energy losses. The
decomposition of charged particle scattering kernels into two components is not a unique
process and it is an issue that needs to be dealt carefully. Momentum transfer, a^E), in
continues angular deflection term in equation (2) is given by equation (4) [20].
cKE) =
(4)
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Computational Issues
The most important issue in the computational aspect of the photon-charged
particle transport is how to solve three integral equations simultaneously. Solving these in
a simultaneous manner with the current computational resources poses a major difficulty.
Luckily, it is possible to be able separate these three equations from each other and
implement an iterative algorithm that would provide the answer sought in the
computations. Usually, the nature of the problem dictates the character of the iterative
algorithm. In this specific case, an interrogating photon beam induces charged particle
fluxes. Those charged particle fluxes, in return, end up contributing to the photon flux.
This secondary photon flux contributed by the charged particle interactions is quite small
compared to the primary photon flux. Therefore, any charged-particle flux resulting from
the interactions of the secondary flux will be negligibly small. Therefore, all we need to
compute is the primary photon flux, primary charged particle fluxes and then secondary
photon flux. Therefore, the iteration will start with solving the photon transport equation
by using the external source. Once the photon fluxes are computed, electron and positron
source distributions through the problem domain will be calculated. Electron and positron
charged particle transport equations will be solved by using these source distributions.
Once the charged particle fluxes are known, they will be used for calculating
bremsstrahlung, annihilation and fluorescence photon source distributions. Secondary
photon flux will be done based on these secondary photon source distributions. The
resulting photon fluxes will be obtained by summing the primary and secondary photon
fluxes.
Another issue with the photon problem is sampling of the incoherent scattering
kernel. As it is seen in figure 1, mean scattering angle at high energy levels is very small.
To avoid any potential sampling problems, we plan to use a dynamic approach to the
approximation of scattering integral by quadratures. The scattering angle will be divided
into intervals and a different quadrature will be used in each interval. Partitioning of the
scattering angle domain will vary with the energy levels. Since a direct integration
approach will be used for solving the integral transport equations, this approach should
prove to be efficient enough to sample scattering kernel appropriately.
Another issue that needs is attention is presence of slowing down terms in the RHS
of the charged particle transport equations. Photons do not gain energy through scattering.
Therefore, the usual approach in the photon transport is to start with the highest energy
group equation and then work down to the lowest energy group equation. Because of the
slowing down terms in the charged particle transport, this approach is not feasible. In the
multigroup approximation, the slowing down term and the flux is to be expressed in terms
of the slowing down and flux terms in the higher and lower energy groups. This generates
a strong coupling between adjacent group equations. Therefore, instead of a top down
approach, either a simultaneous solution technique or an iterative technique should be
adapted for solving the energy group equations.
Another issue that is not very visible with the photon transport is the convergence
of the scattering source iterations. The scattering source iteration is a very popular
approach in deterministic transport based algorithms. This approach is very efficient in
solving the photon transport equation. Because high-energy photons can escape the object
boundaries or slow down to photoelectric absorption dominant energy levels very rapidly
through large energy loss incoherent scattering, scattering source iteration converges very
rapidly. In contrast with the photon transport, energy loss in the charged particle transport
is usually very small. In addition, charged particles do not travel far and escape from the
boundaries is minimal. Therefore, the convergence of scattering source surfaces as a major
issue in the computations. This issue has been studied for discrete ordinates methods
exhaustively and various algorithms developed for enhancing the convergence rate of
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scattering source iterations are reviewed in an article [21]. Although that article addresses
the issue for discrete ordinates method, facing similar difficulties with integral transport
based methods should not be surprising.
Data Issues
Gathering and arranging data for high-energy radiography computations is an
important issue. Although data for all types of interactions are available in the literature,
they need to be organized into proper forms for utilization in the computations. In the
photon transport, the most crucial data are scattering and pair production cross section
data. Since coherent scattering and binding effects are negligible for such high energy
levels, Klein-Nishina formula can provide incoherent scattering cross sections. Pair
production and photoelectric absorption cross sections are readily available in the literature
[22].
Most of the charged-particle interaction cross sections are described in a report by
Lorence et al. [23]. We hope to make use of that document in generation of the cross
section to be used in the computations.
CONCLUSIONS
As seen from the discussions in the previous sections, high-energy radiography is
radically different than the x-ray tube based radiography. Therefore, it is quite difficult to
adapt the x-ray tube based radiography simulation codes to high-energy radiography cases
with small modifications. One reason for such a radical shift is that number of physical
interactions in the high-energy regimes is much larger than the physical interactions seen
in the lower energy levels. Therefore, simulation codes should be able to accommodate the
physics introduced at higher energy levels. In addition, charged particles play a role in the
higher energy levels. Although bremsstrahlung generation starts to dominate chargedparticle physics at higher levels, it may still have enough contribution to the overall photon
flux to distort the flux in the lower end of the spectrum where the detectors are more
sensitive to the incoming radiation. One other thing that needs to be kept in mind that
there are three transport equation in high energy radiography simulations in contrast with
the one transport equation in the x-ray tube radiography cases. This increases
computational efforts significantly. Therefore, quick and easy solutions should not be
expected for high-energy radiography simulations. Such simulations can be performed
only with considerable computational resources that may include multiple processor
platforms. Such computational resources may not be available to typical users who are
planning to simulate high-energy radiography scenarios.
ACKNOWLEDGMENTS
This manuscript has been authored by Iowa State University of Science and
Technology under Contract No. W-7405-ENG-82.
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