AN EVALUATION OF BREMSSTRAHLUNG CROSS-SECTIONS FOR keV TO GeV
ELECTRONS
By
ANNE-SOPHIE T. LECLERE
A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
UNIVERSITY OF FLORIDA
2001
Copyright 2001
by
Anne-Sophie T. Leclere
ACKNOWLEDGMENTS
The author would like to thank the members of her supervisory committee:
Professor Samim Anghaie, Dr. Wesley Bolch, and Dr. Jatinder Palta. A special thank you
is extended to Professor S. Anghaie for his guidance and his enthusiasm for this project.
The author would also like to thank the staff of the Innovative Nuclear Space
Power and Propulsion Institute where this work took place: Dr. Travis Knight and Lynne
Schreiber for their support, and Dr. Gary Chen for his assistance with numerical
integration and Stanford Graphics.
TABLE OF CONTENTS
page
A C K N O W L E D G M E N T S ............................................................................................. iii
L IS T O F T A B L E S .................................................................................................... v i
LIST O F FIG U RE S ........................................................................................ .......vii
CHAPTERS
1. IN T R O D U C T IO N ....................................................................................... ........ 1
2. BREMSSTRAHLUNG CROSS-SECTIONS ............. ......................................4
Introduction ................. ......... ....... ........ ... ....................... 4
Born Approximation and Coulomb Field Approximation..................... ...................6
Coulomb Corrections .................................... ........................... ........... 13
H igh-frequency Lim it Corrections ....................................................................... 13
Screening C corrections ...................................... ......................... .............. 14
Calculations using Sommerfeld-Maue Wave Functions ..........................................14
Partial-wave Formulation and Numerical Calculations .............................................15
3. UTILIZATION OF BREMSSTRAHLUNG CROSS-SECTION........................... 19
Singly Differential Cross-Section: Seltzer and Berger Data Set................................. 19
Electron Energies Below 2 MeV.................. ............. ....................20
Electron Energies Above 50 M eV ................................................ ..............20
Interm ediate Electron Energies ............................................ .......................... 21
Results and Comparison with Analytical Formulas ..............................................22
A n gu lar D istrib u tion ..................................... ... ................ ...... .. ............... . 2 3
Bremsstrahlung Cross-Section Production in Monte Carlo Computational Codes ......28
E G S 4 ...................................... .................................................. . 2 9
E G SN R C .................. ........... ........... ... ....... ............. .............. . 3 1
M C N P ............................................................................... 3 1
4. M EASUREM ENT TECHNIQUES...................................... ............................ 33
M ethods and D ifficulties.................................................... .............................. 33
Transm mission Spectrom etry.......................................................... ............. 34
Photoactivation M ethod ........... .................. ........................... .............. 34
D irect M easurem ents ............ ...... .................... ................... ...... ........ 35
Compton (Incoherent Scattering) Spectrometry.................................... ................35
Scintillator and Germ anium D etectors................................... ....................... 36
Direct Measurements ........................ ................... .................37
Incoherent scattering spectrom etry..................................................... ...................45
5. CONCLUSIONS AND RECOMMENDATION FOR FUTURE WORK ................... 50
L IST O F R E FE R E N C E S ............................. .................. ........................................... 53
BIO GRAPH ICAL SK ETCH .................................................. ............................. 56
v
LIST OF TABLES
Table Page
1. Sym bols and constants............................................... 5
2. Comparison of the scaled differential electron-nucleus bremsstrahlung cross-sections,
p02 d7
k u-(mb), interpolated by Berger and Seltzer (BS) with the results of
Z2 dk
partial-wave numerical calculations by Tseng and Pratt (TP) for Al and W at
5 and 10 M eV incident electron energies. .................................. .................22
LIST OF FIGURES
Figure Page
1. Schematic diagram of electron bremsstrahlung in an atomic field. ................................6
2. Comparison of the Schiff angular distribution (formula 2BS) with the Sauter angular
distribution (formula 2BN)(dashed lines) for W a) at a photon energy of 10%
of the incident electron energy for electron incident energy of 0.511 MeV,
5.11 MeV, 10.22 MeV, and 25.55 MeV; b) at an electron energy of 0.511
MeV for photon energy of 10%, 50%, 75%, 90% of the electron incident
energy .......................................................................... 9
3. Comparison of the Sauter formula with a non-relativistic approximation and an
extreme-relativistic approximation. ......................................... .............12
4. Comparison of the bremsstrahlung energy spectrum calculated using partial-wave
method (points) with the Born approximation (continuous lines) for 1 and 2
M eV electrons. ................ .................. ........ ....... ....... .......... 17
5. Comparison of differential electron-nucleus bremsstrahlung cross-sections interpolated
by Berger and Seltzer (dashed lines) with formula 3BS with y oc (semi-
dashed lines) and formula 3BNb (continuous line) in Koch and Motz paper for
Al and W at 5 and 10 MeV incident electron energies. ........................... 24
6. Comparison of differential electron-nucleus bremsstrahlung cross-sections interpolated
by Berger and Seltzer with those from the Beithe-Heitler theory (formula 3BS
with Thomas-Fermi model screening) for Au at 5 and 10 MeV incident
electron energies. ................... ................. ................. .............. 25
7. Angular dependence of the thick target bremsstrahlung intensity integrated over photon
energy for beryllium and gold. Experimental results published by Buechner et
al. ...............................................................................2 6
8. Singly differential cross-section integrated over photon energy ....................................27
9. Schematic diagram of Buechner et al. experiment .................. ............................ 38
10. Integrated x-ray intensity as a function of the atomic number of the target for different
electron energies as presented by Buechner et al .....................................39
11. Corrected measured bremsstrahlung spectra along the beam axis (from 0 to 0.20) for
10 and 25 MeV electrons incident on a) Al targets and b) Pb targets. The
target thicknesses were nominally 110% of the CSDA electron range...............42
12. Bremsstrahlung spectra generated at angles of 00, 1, 20, 40, 100, 300, 600, and 900 by
15 MeV electrons incident on a) a 11.67g/cm2 thick, 6.72 g/cm2 radius Be
target, b) a 9.74g/cm2 thick, 9.81 g/cm2 radius Al target, and c) a 9.13g/cm2
thick, 17.95 g/cm2 radius Pb target. The solid lines are measured
bremsstrahlung yield, the dashed lines are calculations done using EGS4, and
the dotted lines are the spectral shapes determined from the Schiff spectra
with target attenuation, normalized to the measured values of the integrated
bremsstrahlung yield at the corresponding angles. .........................................43
13. Schematic geometry of the bremsstrahlung source used by Stritt et al. .......................44
14. Experimental setup used by Stritt et al. (1) linear accelerator; (2) ending magnets; (3)
bremsstrahlung source; (4) lead collimator; (5) 12cm thick tungsten
collimator; (6) additional lead shielding; (7) 170 cm3 Ge detector .................45
15. Measured spectrum at 7.400+0.020 after background subtraction by Stritt et al.............45
16. Schematic view of the measurement setup by Anghaie and coworkers.......................47
17. Measured 900 scattered spectrum of an orthovoltage machine by Anghaie et al.
(private communication)..................... ...... .............................. 47
Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science
AN EVALUATION OF BREMSSTRAHLUNG CROSS-SECTIONS
FOR keV TO GeV ELECTRONS
By
Anne-Sophie T. Leclere
August 2001
Chairman: Professor Samim Anghaie
Major Department: Nuclear and Radiological Engineering
The purpose of this study is to evaluate and analyze the experimental data and
theoretical basis of bremsstrahlung process and cross-sections for x-ray production. A
detailed discussion and a comparative evaluation of theoretical models are presented. The
majority of theoretical models for bremsstrahlung are based on low order approximations
that are valid for certain ranges of electron energies or x-ray emission energy and angle.
A primary focus of this study is the x-ray energy range of interest to radiation therapy,
which is generated by 1 to 30 MeV electrons impinging on thick high Z targets. Most
theoretical treatments using a variety of approximation methods have been limited to very
low (<100's keV) or very high (>100's MeV) x-rays. In intermediate energy range, the
first Born approximation has been predominantly used for development of energy and
angular dependent bremsstrahlung cross sections. However no reliable experimental data
or rigorous theory is available for high-energy bremsstrahlung differential cross-sections.
At the present time, the first Born approximation and the method of partial waves provide
the only basis for high-energy bremsstrahlung modeling. The first Born approximation
with correction for screening provides the best available estimate of bremsstrahlung
differential cross-section in the energy range of 1 to 30 MeV, while the method of partial
waves truncated at low values of angular momentum quantum number is only valid for
low-energy bremsstrahlung x-rays. The study also considered the case of thick target,
where multiple collisions of electrons could occur.
Bremsstrahlung process is the primary source of x-ray production in electron-
photon transport codes. A discussion of the electron transport modeling including both
ionizing and radiative interactions is presented. In particular, stochastic modeling of
electron transport that includes random sampling of bremsstrahlung x-rays is discussed.
Current Monte Carlo sampling approaches used in all major codes use crude
approximation and treat bremsstrahlung cross-section linearly separable in terms of
energy and emission angle. The bremsstrahlung sampling theory in three different Monte
Carlo transport codes, MCNP, EGS4, and EGSnrc is discussed.
A discussion of previous experimental approaches to measure bremsstrahlung
cross-sections in intermediate energy range is presented. Results of a review of different
measurement techniques including the advantage and disadvantage of different
techniques are presented. A comparative analysis of the direct measurement method
versus the incoherent and coherent scattering spectroscopy methods using high energy
resolution HPGe system and low resolution NaI(Tl) scintillation system is provided. Key
technical issues and a few recommendations for performing experiments leading to
validation of bremsstrahlung theory are discussed.
CHAPTER 1
INTRODUCTION
Bremsstrahlung radiation is of basic interest in many fields, and accurate cross-
section data over a wide range of energy and for all materials is required. The
fundamental characteristic of an x-ray machine is the spectral distribution of the photon
beam originating in the target. In diagnostic radiology, knowledge of the x-ray spectrum
is important to determine image quality, as well as patient dose. In radiation therapy,
accurate data is required for accelerator design and treatment planning (i.e. patient dose
distribution), and the spectrum-averaged quantities required by quality assurance
protocols are calculated from measured photon spectra. Unfortunately direct spectrum
measurement of radiation from a medical linear accelerator is hardly possible because of
high intensity and energy range (1 MeV to 50 MeV).
While the primary focus of this study is the x-ray energy range of interest to
radiation therapy, which is generated by 1 to 30 MeV electrons impinging on thick high Z
targets, the theory of bremsstrahlung production will be presented for electrons energies
from keVs to GeVs. The production of bremsstrahlung photons results from two different
types of interaction: incident electron with the nucleus, and incident electron with atomic
electrons. The contribution of electron-electron bremsstrahlung is small compare to the
one of electron-nucleus bremsstrahlung. Therefore only electron-nucleus bremsstrahlung
will be discussed in this thesis.
The elementary process of bremsstrahlung is described by the triply and doubly
differential cross-sections (i.e. differential in photon energy and photon and electron
emission angle and differential in photon energy and photon angle). The photon spectrum
is described by the differential in photon energy, while the angular distribution is
described by the doubly differential cross-section. The differential in photon energy
describes the photon spectrum but the angular distribution. In principle, the interaction of
an electron with the atomic field of a nucleus can be described exactly using exact
solutions of the Dirac wave equation. However, it is not possible to solve the Dirac
equation in closed form for a free electron in a Coulomb field. Therefore approximate
wave functions or numerical methods have been used.
Quantification of the bremsstrahlung process started in the mid 1900's. Majority
of theoretical models for bremsstrahlung are based on low order approximations that are
valid for certain ranges of electron energies or x-ray emission energy and angle. In 1959,
Koch and Motz1 published a comprehensive review of bremsstrahlung cross-section
formula and related data, which has become the reference in bremsstrahlung cross-
sections. This paper includes the first Born approximation theory, the extreme-relativistic
theories, Coulomb corrections, high-frequency limit corrections, and screening
corrections. Later work published in 1985 by Seltzer and Berger2'3 was to provide a
practical review of the Koch and Motz paper by combining the different available
theories in order to derive the bremsstrahlung cross-section, differential in emitted photon
energy, for electrons in the range 1 keV to 10 GeV incident on neutral atoms with atomic
numbers Z=1 to 100. For low energies, Seltzer and Berger relied on the method of
partial-waves to obtain the cross-section for bremsstrahlung in the field of the screened
nucleus for energies below 2 MeV. For energies above 50 MeV Seltzer and Berger made
use of the Born approximation with unscreened nucleus, modified to take into account the
screening of the nuclear charge by the atomic electrons and the Coulomb correction. In
the intermediate energy range of 2 to 50 MeV, Seltzer and Berger obtained cross-section
through an interpolation procedure. Also work on the method of partial-waves has been
and is undergoing mainly by the Pratt group.4-7 In 1985 Pratt and Feng8 published an
excellent and complete review of the phenomena for incident electron with kinetic
energies in the 100 eV to 10 MeV (and especially for 1 keV to 1MeV). Recently Quarles9
summarized the current experimental work on the doubly differential cross-section for
electron energies less than 1 MeV.
For incident electron energies utilized in the medical field (1 MeV to 30 MeV) no
reliable work, theoretical or experimental, has been realized in a satisfactory manner.
Very few measurements have been conducted for comparison and benchmark of the
current theories, and almost no studies of angular distribution of bremsstrahlung radiation
have been conducted. While several methods of measurement have been tried, none
resulted in accurate data. Therefore a purpose of this study will be to discuss the different
techniques of measurement and to provide some recommendations for future work.
Bremsstrahlung process is the primary source of x-ray production in electron-
photon transport codes. A discussion of the electron transport modeling including both
ionizing and radiative interactions will be presented. In particular, stochastic modeling of
electron transport that includes random sampling of bremsstrahlung x-rays will be
discussed, along with the bremsstrahlung sampling theory in three different Monte Carlo
transport codes, MCNP, EGS4, and EGSnrc.
CHAPTER 2
BREMSSTRAHLUNG CROSS-SECTIONS
Introduction
In 1959, Koch and Motz1 published a comprehensive review of bremsstrahlung
cross-section formula and related data, which has become the reference in bremsstrahlung
cross-sections. This paper includes the following analytical theories: (a) the Born
approximation theory of Bethe and Heitler, Sauter, Racah, Schiff, Sommerfeld, and
Hough;10-18 (b) the extreme-relativistic theory of Davies, Bethe, Maximon and Olsen19-23
(DBMO) with Coulomb correction and screening; (c) Coulomb corrections for
nonrelativistic, intermediate, and extreme-relativistic energies; high-frequency limit
corrections; and screening corrections. The DBMO formulas have also been reviewed by
Tsai24 with emphasis on the form-factor screening corrections.
In 1985, Seltzer and Berger2'3 provided a data set for the bremsstrahlung cross-
section, differential in emitted photon energy, for electrons in the range 1 keV to 10 GeV
incident on neutral atoms with atomic numbers Z=1 to 100. This data set was a
combination of several theories. For low energies, Seltzer and Berger referred to the
theory of Tseng and Pratt4'5 who used numerical partial-wave calculations to obtain the
cross-section for bremsstrahlung in the field of the screened nucleus for energies below 2
MeV. For energies above 50 MeV Seltzer and Berger made use of the Bethe-Heitler
formula for the case of unscreened nucleus, obtained in the Born approximation,
modified to take into account the screening of the nuclear charge by the atomic electrons
and the Coulomb correction. In the intermediate energy range of 2 to 50 MeV, Seltzer
and Berger obtained cross-section through an interpolation procedure.
The symbols and constants used throughout this paper are shown in Table 1, and
Fig. 1 is a diagram of the electron-nucleus bremsstrahlung process. All energies and
moment are expressed in electron rest mass units (moc2 units and moc units
respectively). Most of the formulas presented in this chapter are from the review of Koch
and Motz1 and will be referred throughout this thesis by the name given in this review.
TABLE 1. Symbols and constants
Symbols
Eo,E Initial and final energy of the electron
po,p Initial and final momentum of the electron
To,T Initial and final kinetic energy of the electron
k, k Energy and momentum of the emitted photon
60, 0 Angles of po and p with respect to k (polar angles)
( Angle between the planes (po,k) and (p, k) (azimuthal angle)
d'k Element of solid angle, sin0od0od), in the direction of k
d~p Element of solid angle, sinOdOd), in the direction of p
q Momentum transferred to the nucleus
Po, P
a
q = Po -p-k
q2 = p + p2 + k2 2pokcos0 + 2pkcosO 2pop(cosO cos0 + sin 0 sin 0 coso)
Ratio of the initial and final electron velocity to the velocity of light (v/c)
ro
2x137
E, p, T \ k
Atom, Z
e- Eo,p o,T
FIG. 1. Schematic diagram of electron bremsstrahlung in an atomic field.
Born Approximation and Coulomb Field Approximation
The first approximation is the Born approximation considering free-particle wave
27, Z
functions perturbed to the first order in Z. It requires that the term - is small
137 P
compared to one. Then, the Born approximation becomes less reliable as the atomic
number of the target increases, the initial electron energy decreases, and the photon
energy approaches the high-frequency limit. Bethe and Heitler have calculated the
corresponding triply differential cross-section (formula 1BS1):
d Z r [lF(qZ)]2 1 p 1 sin20 (4E q2)
ro [1- F(q, Z) 4E q
dkdQkdQp 137 2)cL k p, q4 E(E- pcosO)
+ sin Li0 4E q) 2pposn0sisinOcos(4EE- q )
(E p, cos0)2 (E- pcosO)(E p, cos0,)
2k2(p2 sin2 0 + p2 sin2 08 2pp, sin 0 sin 80 coso)
(E-pcosOXEO-pocos0o)
Screening of the atomic nucleus by atomic electrons, which reduces the bremsstrahlung
cross-section, is taken into account by the factor [1- F(q, Z)]2, where F(q, Z) is the
atomic form factor.
For non-relativistic case with no screening this formula simplifies into the
following (formula 1BN1):
d'( Z2 (r4 1 p 1 .2 2 2 200
---- =- Q j P lp sin o + p sin B, 2pp, sin 0 sin 0cos44
dkdQkdQp 137 jc k p q
where q2 = p2 + p2 2pop(cosO cosO0 + sin 0 sin 00 cos0).
Two general formulas (and associated approximations) for the doubly differential
cross-section have been derived from the Bethe-Heitler expression of differential in
photon energy and in photon and electron emission angles. They are direct integration
over the electron emission angle with different assumptions as the integration of the
screening contribution is difficult to achieve.
Schiff15 approximated nuclear screening by an exponential function. He also
considered only the extreme relativistic and small angles case, and there is an
approximation in the integration over electron angle, the result is not accurate for
Z1/3
00 < -- Schiff formula (2BS1) is the following:
111Eo
d2o 4Z2r y 16y2E (E + E)2 E E2 4y2E
dkdy 137 k (y2 +y1)4EO ( )2 y2+ 2 (y2+1 )4(E
1 k Z
where y = E0o, M -y K 2E + Z-11 -2.
M(y) 2EOE 1 +1
Both ends of the photon energy range are outside of Schiff s approximation, since at the
high-energy end the scattered electron must have low energy, and at the low-energy end
the electron emission angle must be small. Nevertheless this formula has been used
extensively in the medical physics field as shown in chapter II.
Sauter12 has derived the second formula (2BN1) for an unscreened potential:
d2 Z2 r 1 p 8sin 20(2E +1) 2(5E2 + 2EE, + 3) 2(p2 -k2 4E
dkdak 87r137 k po pA pAA Q Ao po A
+ L 4Esin 3k pE) 4E(E + E2) 2 2(7E 3EEo + E2) 2k(E2 + EE, 1)
244Eosin'j3 2(E 2EE E2)j2(E EE0
PPop 0 pA 0 Ao poA0
pA, pQ Ao A0, A, 0
,EEo -1+ppo, E oO, E+P o =lQ+P
where L = In EE A, = E,- p cosO e = In eQ = In
L EE -1 ppo E p Q p
Q2 = p2 +2 2p0kcosO .
The two analytical angular distributions, the Schiff formula (with screening,
2BS1) and the Sauter formula (without screening, 2BN1), are compared in Fig. 2. One can
notice that the two formulas are closer at higher energy. However, they differ near the
high-frequency limit: with formula 2BN the cross-section tends to zero in the forward
direction, while it reaches a maximum with the Schiff formula. At higher electron energy
one need to be closer the maximum photon energy to see this trend.
Two approximations of the Sauter formula exist for extreme-relativistic energies.
They are valid only in limited cases not very useful to characterize the outgoing photon
over the entire spectrum. Sommerfeld18 obtained the following formula at small angles
(2BNal):
d2o Z2r2 1 E f16(OEE2, E (E,+E)2E0 2lnEE i(E2 +E )E, 40E4
dkdaQk 137 k E,, 1 EE) E( +E k E(L (++ 0 )4
1.E+05
1.E+04
a) k = 0.1TO,
1.E+03
1.E+02 0.51e
--.-. ... .. = 0.511.. .
1.E+01 -
dkda
-dd (b/steriieV)1 .E+00
1.E-01 T= 5.11iMe
1.E-02 . . . .
1.E-03 ~= 10.221~ .5e -- ---...
1.E-04 To= 25.55MI-eV ..--
1.E-05
0 20 40 60 80 100 120 140 160 180
0 (degrees)
1.E+03
1.E+02 b) o 0.511 eV
1.E+01 .. k = 0.1To
S(b/sterMeV) ...
dkda ,".
1.E+00 o '* i
k = 0.5TO
1.E-01
k 0.95 To
1.E-02
0 20 40 60 80 100 120 140 160 180
0 (degrees)
FIG. 2. Comparison of the Schiff angular distribution (formula 2BS) with the Sauter
angular distribution (formula 2BN)(dashed lines) for W a) at a photon energy of 10% of
the incident electron energy for electron incident energy of 0.511 MeV, 5.11 MeV, 10.22
MeV, and 25.55 MeV; b) at an electron energy of 0.511 MeV for photon energy of 10%,
50%, 75%, 90% of the electron incident energy.
The same result is obtainable from the Schiff formula by setting Z = 0 in M(y). It
implies the formula is valid for small values of Z. At large angles Hough14 obtained
formula 2BNb1:
d o Z2r2 E 1 (E +EE 2E0E 5E + 2E
---- ---- -7 -t ---- ^ r i ---- sm ) In -- ---
dkdak 47137 E k(1-cos00)L EEo ) k E,
2E2- 2E2 In k [
Ic 0 E EE EQ +EE(k-E0cos0)]
+______ (l-cos ,)-
EE EQ2(- cos ,)
Ek( -cosOO)[Q2 + E(E + k)]lnQ+E
EQ' Q J-E
where Q2 = E2 + 2kE (1- cos00).
The photon spectrum is characterized by the cross-section differential in photon
energy. It can be obtained by integrating the triply differential cross-section over the
emission angles of the photon and electron. As for the doubly differential cross-section,
the difficulty is to integrate the screening factor. Bethe and Heitler obtained a formula
(3BS1) for the extreme-relativistic case with screening:
do 4Z22 1 1 E 2 E 02)_ W- 1
d 14 1 + E201 In Z(Y) In Z1
dk 137 k} E) L4 3 ] 3 E 4 3 J
100k
where y = 0E Different expressions of 4i and 42 exist; for arbitrary screening, they
EOEZ '
are given by 0 (y)= 4J(q )2(1- F(q))2 dq+ 4+ InZ
Sq 3
(y)=4 q-6 5qln +35 q-453(1 -Fq)) 3 + InZ
k yZ1/3
where 8 =- and F(q) is the atomic form factor.
2EoE 200
Bethe and Heitler also derived a non-screened formulation (3BN1), integration of
137 EoE
the Sauter formula (2BN1). It is valid for >> --E
Z1/3 k
da Z22 p {4 E p2+p eoE EEo o
dk 137 k po 3 2 P A p3 PoP
LEoE k2EE2 +pp2) k ((EoE+p (EoE +p2 2kEoE
SPP3 0 3 2 2
t3pop POP -pop pg P o I P p2 ) \
where L = 21n EE -1+ pp, = In ln, g = In E +p
L E- p LEo- P
k I _E-p _E -Po_
From this formula two approximations can be derived. The non-relativistic
S da Z2 02 16 1 1 po P+P
approximation (3BNa ) is given by d Z2r 16 11 ln o -+ and the extreme-
dk 137 3 k \Po -
.. da 4Z2 r02 1 E E2 2E I f2EoE) 1]
relativistic one (3BNb ) is given by 1d 4Zr 1 + - In 2E .
dk 137 k E, 3 Eo k 2_
This last expression results directly from formula 3BS when y -> o. Fig. 3 shows
formulas 3BN and 3BNa for an electron energy of 0.0511 MeV, and formulas 3BN and
3BNb for an electron energy of 5.11 MeV. Both approximations are very good in their
respective domain of application, but near the electron rest mass energy. The
discrepancies near the high-frequency limit are computation artifacts.
Corrections may be applied to the previous formulas: Coulomb corrections, high-
frequency limit corrections, and screening corrections. In each case, the correction is
12
restricted to a particular energy region, and is intended to apply only to a particular
differential form of cross-section.
1.E+06
1.E+05
d (b/Me V)
dk
1.E+04
1.E+03
1.E+02
1.E+03
1.E+02
a (b/MeV)
dk
1.E+01
1.E+00
1.E-01
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
k (unitless)
--3BN
- - extreme-relativistic approximation
To = 5.11MeV
0 1 2 3 4 5
k (unitless)
6 7 8 9 10
FIG. 3. Comparison of the Sauter formula with a non-relativistic approximation and an
extreme-relativistic approximation.
--3BN
- - - non-relativistic approximation
T, = 51.keV
Coulomb Corrections
For non-relativistic energies, Elwert17'25 has estimated a multiplicative Coulomb
correction factor for the formula 3BNa1. It was derived from a comparison between the
non-relativistic Born approximation and the non-relativistic Sommerfeld calculations,
and is then restricted to non-relativistic electron energies. The Elwert factor can be
P 1- exp(- 21caZ P,)
written as f 1 exp( 2ra ) This factor is valid only if
S1- exp(- 2raZ /P)
(Z/137)(P 1 / ) << 1, thus not near the high-frequency limit. Nevertheless, the Elwert
factor has been found to be applicable even for relativistic incident electron energies and
for (Z/137)(f/3 /o1) not small.2'8
At extreme-relativistic energies, the singly differential cross-section can be
corrected by the addition of the DBMO1'26 Coulomb correction
M = (Z) 1- 2E+ E with f(Z)= a 2Z n(n2 a2Z
3 EQ, E0 n=l
This additive term is independent of the type of screening approximation that is used.
High-frequency Limit Corrections
The previous formulas are derived with certain approximations, which do not
permit an evaluation of the cross-section at the high-frequency limit. The Born
approximation (expansion in powers of Z/137P and Z/1373) cross-section becomes
zero at this limit. However it has been shown that this cross-section has a finite value at
this limit. Fano et al.27 calculated a finite value using the Sauter approximation
(expansion in powers of Z/137P, and Z/137). Note that the Elwert factor has often been
used to correct the Born approximation near the high-frequency limit.
Screening Corrections
Screening effects are more important at extreme-relativistic and non-relativistic
electron energies. Koch and Motz1 showed that in the Born approximation, unscreened
differential cross-section formulas may be corrected for screening effects by including
the multiplicative factor [1- F(q, Z)]2. F(q, Z) can be evaluated by two models: the
Hartree self-consistent field model and the Thomas-Fermi model. The Thomas-Fermi
model has been the most extensively applied, though the Hartree self-consistent filed
model is more accurate.1 Bethe calculations of the cross-section formulas were performed
using the Thomas-Fermi model, while Schiff calculations were based on the complete
screening condition (y = 0) and an approximate exponential screened atomic potential.
Calculations using Sommerfeld-Maue Wave Functions
Results have been obtained for high-energy incident electrons by Bethe,
Maximon, Olsen and others19-23 using approximate Coulomb (Sommerfeld-Maue) wave
functions and including screening corrections. Those formulas are valid for all Z, but
only above 50 MeV. They are found to be the Beithe-Heitler formula multiplied by
another factor, which is only important for the smallest values of the nuclear recoil
momentum q.28 However screening reduces the effect of small values of q. So in the case
of complete screening, the Born approximation turns out to be a better approximation
than expected for high Z materials and high energies. Expressions derived by Bethe,
Maximon, Olsen and Davies (for the differential in photon energy) have been detailed by
Koch and Motz.1
Elwert and Haug25 also utilized Sommerfeld-Maue wave functions to obtain
formulas valid in the point Coulomb potential with no screening for aZ <<1 at all
energies, neglecting higher order in aZ, or for all aZ at sufficiently high incident and
outgoing energy, using the fact that the terms with higher order in aZ are negligible for
energies large compared with the electron rest mass energy, and for small angles, as
shown by Bethe and Maximon.19 Corrections for screening effects still need to be
included. While showing some improvement over the Born approximation results for
intermediate and high Z, discrepancies between results and experiments are important for
large Z at energies near the electron rest mass energy.
Partial-Wave Formulation and Numerical Calculations
As noticed by Nakel,29 for the relativistic energy domain, exact solutions of the
Dirac equation including the Coulomb field have only been found in series form as a
summation over quantum number of the angular momentum. Tseng and Pratt4 have
developed a numerical method based on a relativistic multiple and partial-wave
expansion, which is successful for calculation of the doubly differential cross-section,
average over the initial electron spin and both directions of the polarization of the photon.
However this method is not easily applicable for calculation of the triply differential
cross-section, and Tseng and Pratt obtained results only for an energy range from 5 keV
to 1 MeV. Results are obtained in the form of radial integrals; the wave functions and
integrals are evaluated numerically and the radial matrix elements summed over angular
momentum states. Convergence in the numerical evaluation is delicate and convergence
of the sums is slow at high energy. Tseng and Pratt found the angular distribution to have
the same shape as the Born approximation without screening, where the differential
cross-section decreases to a minimum in the forward direction. The deviation of the
partial-wave results from the Born approximation was believed to be due to the number
of partial waves used. An accuracy of 2% would require twice as many partial waves as
Tseng and Pratt used in their work. Therefore, for energies around the electron rest mass
energy, Tseng and Pratt found similar results to those of the Born approximation, and
failed to proved a real improvement.
Tseng and Pratt4 calculated the electron-nucleus bremsstrahlung cross-section
differential in energy for selected Z, To, and k/To values, and for k=0 they used a low-
energy theorem which connects the low-frequency limit of the spectrum with the electron
elastic-scattering cross-section. Pratt et al. interpolate those data to provide complete
coverage for 2 < Z < 92, for 0 < k/To < 1, and for lkeV < To < 2MeV Fig. 4 shows
those results for tungsten at 100 keV, 500 keV, 1 MeV, and 2 MeV along with the Born
approximation (Beithe-Heitler formula 3BN1).
Recent calculations of the triply differential cross-section by Schaffer, Tong, and
Pratt6'7 rely on the same method but with a different summation technique. The results for
high-Z cases have been compared with Bethe-Heitler and Elwert-Haug predictions.
While the Beithe-Heitler results give the same qualitative features of the angular
distribution and spectrum but quantitative agreement (cross-sections may differ by more
than 100%), Elwert-Haug results are closer than expected to the numerical computations
of Schaffer, Tong, and Pratt. The discrepancy of the Beithe-Heitler theory with the
27< Z
partial-wave formulation is understandable since the - << 1 criteria for the validity
137 P
of the Beithe-Heitler result are not satisfied. However, experimental data do not support
either Elwert and Haug theory or Schaffer, Tong, and Pratt theory.
10000
1000
d(b/Me V)
dk 100oo
10 -
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
k(MeV)
FIG. 4. Comparison of the bremsstrahlung energy spectrum calculated using partial-wave
method5 (points) with the Born approximation (continuous lines) for 1 and 2 MeV
electrons.
In summary the different theories can be classified according to the energy
domain where they best apply. The theory of Bethe, Maximon, Olsen and others using
approximate Coulomb (Sommerfeld-Maue) wave functions is designed to be valid at
extreme-relativistic energies (To >> 50 MeV). It is considered a very good approximation
of the problem as it is validated by experiment. At non-relativistic energies, the Born
approximation gives good results when corrected for screening. The intermediate energy
domain from 1 keV to 50 MeV is more complicated to characterize. Extensive work by
Pratt and co-workers has been done for energies from 1 keV to 2 MeV using numerical
partial-wave methods. They also focused on the contribution of polarization
bremsstrahlung to the total bremsstrahlung spectrum. However, very few theoretical
18
studies have been conducted for the energy range 2 MeV to 50 MeV, which are in the
range of energy useful in medicine. Instead Seltzer and Berger2 used interpolation
scheme to create data for this energy domain.
CHAPTER 3
UTILIZATION OF BREMSSTRAHLUNG CROSS-SECTION
As seen in Chapter I, none of the analytical theories to date are adequate to
describe accurately the bremsstrahlung cross-section over a wide range of energies. It is
necessary to combine the results of several theories to obtain accurate cross-sections. The
singly differential cross-section is used in computational codes to determine the energy of
the outgoing photon after interaction. The angular distribution necessary to determine the
emission angle is always handled separately. Due to the complexity of the analytical
formulation of the bremsstrahlung cross-section, one prefers the use of tabulations
obtained from the combination of analytical and numerical theories. Seltzer and Berger2
have prepared a comprehensive set of bremsstrahlung cross-sections, which is the basis
for bremsstrahlung photon production in ETRAN codes and the Integrated Tiger Series
and EGS systems. This chapter will discuss this data set and its use in different
computational codes. This study will present the EGS4 and MCNP codes and compare
the methods used for bremsstrahlung photon production.
Singly Differential Cross-Section: Seltzer and Berger Data Set
The analytical theories used by Seltzer and Berger2'3 include the following: (a) the
Born-approximation theory of Bethe and Heitler and others with a form-factor screening
correction; (b) the Elwert Coulomb correction factor derived from Sommerfeld's non-
relativistic theory for an unscreened Coulomb potential; (c) the high-energy theory of
Davies, Bethe, Maximon and Olsen (DBMO), which includes both screening and
Coulomb corrections but fails near the high-frequency limit; (d) theories of the high-
frequency limit of the bremsstrahlung spectrum available for very high electron energies.
Electron Energies Below 2 MeV
Seltzer and Berger choose the results of Pratt, Tseng and co-workers,4'5 which
were obtained in the field of the screened nucleus by numerical partial-wave calculations.
Those results are the most recent and reliable data set available. As seen earlier, the
number of partial waves that must be included is large and increases rapidly with electron
energy, so that comprehensive results are available only for electron energies up to 2
MeV. Pratt et al.5 interpolated those data to provide complete coverage for 2 < Z < 92,
for 0 <; ki/T < 1, and for lkeV *
*
extrapolation to extend their results to the cases Z = 1 and Z = 93 to 100. Pratt et al.
estimated an overall uncertainty of 10% for their tabulated cross-section values. This
uncertainty is due to the model assumptions in their calculations, the choice of atomic
potentials used, the errors inherent in the partial-wave calculations, and the fitting
procedures used to generate the tables from a limited amount of directly calculated
results.
Electron Energies Above 50 MeV
The singly differential cross-section was evaluated using the Bethe-Heitler, Born
approximation formula (3BN1) for an unscreened nucleus with no energy approximation.
This formula is modified to take into account the screening of the nuclear charge by the
atomic electrons, and a Coulomb correction. As shown by Olsen et al.21-23 these two
corrections are nearly independent and additive at high energies. The cross-section can
then be expressed by the following formula: da = 4rZ2 ( c reend+ + o)
dk 137 Born
The screening correction was obtained as the difference between the cross-section
with and without screening. The Bethe, Born approximation high-energy formula for a
screened nucleus (3BSb1) was the basis for this calculation. The results with screening
have been evaluated with Hartree-Fock atomic form factors from Hubbell et al.30'31
The Coulomb correction combines several theories: the Coulomb correction from
DBMO theory away from the high-frequency limit, the Elwert factor in the region near
the high-frequency limit, and specialized theories at the high-frequency limit. Seltzer and
Berger came up with the following formula:
3 {fE,~F~.( Pl2"r ]{iunscreened + 0)(T.5DBMO
6Cou = fEexp 1- 02-1 J "Born +(oT coul
where fE is the Elwert factor, and DBO is the DBMO Coulomb correction. The function
o(T) allows to switch to the DBMO Coulomb correction as the Elwert term goes to zero
i.e. to turn off the DBMO Coulomb correction near the tip where it is no longer
applicable.
Intermediate Electron Energies
The gap region between 2 MeV and 50 MeV was carried out by fitting least-
squares cubic splines to the cross-sections at lower and higher energies. Seltzer and
Berger compared their results with those from the partial-wave numerical calculations of
Tseng and Pratt for 5 and 10 MeV electrons in Al and U. They found good agreement
between the two results as shown in Table 2.
TABLE 2. Comparison of the scaled differential electron-nucleus bremsstrahlung cross-
sections, k d- (mb), interpolated by
z2 dk
partial-wave numerical calculations by
MeV incident electron energies.
z k/To
13 BS
TP
92 BS
TP
13 BS
TP
92 BS
TP
13 BS
TP
92 BS
TP
10 13 BS
TP
92 BS
TP
0
12.64
12.6
11.63
11.62
13.17
13.21
11.44
11.45
13.57
13.74
10.9
11.14
13.7
13.81
10.56
10.81
Seltzer and Berger3 (BS) with the results of
Tseng and Pratt32 (TP) for Al and W at 5 and 10
0.2
7.528
7.54
8.287
8.3
8.184
8.24
8.092
8.1
9.353
9.53
8.012
8.21
10.05
10.24
8.07
8.39
0.4
4.934
4.93
6.17
6.16
5.489
5.49
5.92
5.92
6.632
6.8
5.979
6.13
7.654
7.75
6.239
6.59
0.6
3.215
3.22
4.742
4.74
3.678
3.67
4.457
4.45
4.79
4.85
4.605
4.62
5.808
5.91
5.013
5.21
0.8
1.907
1.91
3.674
3.67
2.214
2.21
3.341
3.34
3.183
3.16
3.493
3.42
4.18
4.2
3.948
3.87
0.95
1.001
1
2.948
2.95
1.083
1.08
2.434
2.43
1.511
1.47
2.297
2.33
2.084
2.04
2.539
2.53
Results and Comnarison with Analytical Formulas
Fig. 5 shows the Seltzer and Berger results compared with the Beithe-Heitler
theory (formula 3BS1 with y -> oo) and the Born approximation without screening
(formula 3BN1) for Al and W at 5 and 10 MeV incident electron energies, i.e. in the
To
(MeV)
1
2
5
domain where data have been interpolated. Both ends of the spectrum are singular point:
at the low end, the singly differential cross-section approaches infinity, but it is not
considered a problem since -k- is a finite value, and as discussed previously the
Z2 dk
high-frequency region is not described well by the formula 3BN. With exception to the
high end of the spectrum, the Beithe-Heitler formula without screening (3BS with
y -> o or 3BNb) gives the same results as formula 3BN and is much simpler to use. As
expected Seltzer and Berger results are qualitatively consistent with those formula, but
with a difference of up to 100%.
Seltzer and Berger compared also the interpolated bremsstrahlung cross-sections
with the results from the Beithe-Heitler theory with screening (formula 3BS with the
Thomas-Fermi model for the calculation of the form factor).33 The comparison indicated
differences on the order of 10% (Fig. 6).
Angular Distribution
While the bremsstrahlung spectrum, depending on the three variables Z, To, and k,
is relatively simple to discuss, the bremsstrahlung angular distributions and
dkdQak
dkdk e, which also depend on the photon scattering angle and the electron scattering
dkdafdaf,
angle for the triply differential cross-section, are much more complicated to describe and
to use. Moreover, this becomes more complicated for the case of a thick target, which is
the situation encountered in most experiments. The angular distribution will be presented
here through the thick target situation.
do
-(b/MeV)
dk
1.0E+05
1.0E+04
1.0E+03
1.0E+02
1.0E+01
1.OE+00 - - -- -
1.OE-01 .
1.0E-02
1.0E-03
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
k (MeV)
1.0E+05
10 MeV incident electron
1.0E+04
1.0E+03
1.0E+02 T ungsten
(b/MeV) .OE+01
dk Aluminum
I .OE+00
1.OE-01
1.0E-02
1.0E-03
1.0E-04
0 1 2 3 4 5 6 7 8 9 10
k (MeV)
FIG. 5. Comparison of differential electron-nucleus bremsstrahlung cross-sections
interpolated by Seltzer and Berger3 (dashed lines) with formula 3BS with y -> (semi-
dashed lines) and formula 3BNb (continuous line) in Koch and Motz paper1 for Al and W
at 5 and 10 MeV incident electron energies.
9- \
\ Z = 79
S \\ --- BETHE-HEITLER THEORY
-- PRESENT WORK
0 7-
N
c T=10 MeV
cU
U)
0 \3
33-
1 2 3 4 6 6 7 8 9 10
energies.3
At non-relativistic and intermediate energies, no analytical or empirical formulas
have been derived for estimating the bremsstrahlung angular distribution for thick target,
and only a few experiments are available. Koch and Motz1 noticed that the radiation
intensity is important at large angles and is about the same order of magnitude at both
zero and ninety degrees. However, the absorption of the low energy bremsstrahlung
photons in the target is large, thus the angular distribution is greatly dependent on the
target geometry in specific experimental situations. Some experiments have been
designed that correct for the geometry and absorption. Buechner et al.34 have conducted
one of them. Their goal was to quantify angular dispersion and calculate the efficiency of
x-ray production. The results of their experiments are cited in the Koch and Motz paper
L \
1 2 3 4 5 6 7 8 9 10
BREMSSTRAHLUNG PHOTON ENERGY, MeV
FIG. 6. Comparison of differential electron-nucleus bremsstrahlung cross-sections
interpolated by Seltzer and Berger with those from the Beithe-Heitler theory (formula
3BS with Thomas-Fermi model screening) for Au at 5 and 10 MeV incident electron
energies.33
At non-relativistic and intermediate energies, no analytical or empirical formulas
have been derived for estimating the bremsstrahlung angular distribution for thick target,
and only a few experiments are available. Koch and Motz1 noticed that the radiation
intensity is important at large angles and is about the same order of magnitude at both
zero and ninety degrees. However, the absorption of the low energy bremsstrahlung
photons in the target is large, thus the angular distribution is greatly dependent on the
target geometry in specific experimental situations. Some experiments have been
designed that correct for the geometry and absorption. Buechner et al.34 have conducted
one of them. Their goal was to quantify angular dispersion and calculate the efficiency of
x-ray production. The results of their experiments are cited in the Koch and Motz paper
and are shown in Fig. 7 for aluminum and tungsten. A discussion of this experiment and a
computational analysis can be found in a project report prepared by Hower.37
10 20 eo 20
S00 -SO 0
S3al.36
impossible in the forward direction, but rather at 20
-J
45 45
S2. M v .35 Mew
2.00 Mew
..75 Mev 1.75 Mv
W I.0Me 1.50 Mv
,,0 Mev 60
5 200 1.25 Nov
z
0 BERYLLIUM Z 4 GOLD ZzT9
t-
FIG. 7. Angular dependence of the thick target bremsstrahlung intensity integrated over
photon energy for beryllium and gold. Experimental results published by Buechner et
al.36
At high energies, Koch and Motz1 considered some simplifying approximations.
First the thin-target spectrum (formula 3BS1) is assumed to represent the spectrum shape
for any angle. Then the intrinsic angular spread (formula 2BS1) is neglected at large
angles where a >> Eo1 but not at small angles where a < Eo1. Representation the
angular distribution 2BS can be found in Fig. 2 for several incident electron energies and
photon energies. For a given electron energy, the angular distribution is more forward at
the low end of the energy spectrum, while near the high-frequency limit, the cross-section
reaches a minimum, i.e. a complete transfer of energy from the electron to the photon is
impossible in the forward direction, but rather at 200.
Another interesting way to analyze the angular distribution is to consider the
singly differential cross-section integrated over the photon energy. Because of the
complication due to integration of the screening factor, this evaluation considers only the
integration of the Sauter formula 2BN without screening. While it is possible to perform
analytic integration of this formula over photon emission angle, the integration over the
photon energy must be perform numerically. Because the doubly differential cross-
section is infinite at the low end of the spectrum, one must use a cutoff energy for the
numerical integration. This translates into a flat line at 00 when the singly differential
cross-section over photon angle reaches a maximum, as shown in Fig. 8. As expected, the
emission of photon, disregards to their energy, is essentially forward, and is highly
peaked at higher electron energy.
10000
1000
100 / \ \
/ // \ N
daB (b/ster) 10-
dQ /
0.1 tau=To/511kev
tau=0.1
tau=1.0
0.01 tau=10
tau=20
tau=50
0.001
-200 -150 -100 -50 0 50 100 150 200
0(degrees)
FIG. 8. Singly differential cross-section integrated over photon energy.
The energy spectrum and angular distribution discussed previously have been
used in computational codes to determine the energy and emission angle of the
bremsstrahlung photons. Energies and angles can be picked following the Monte Carlo
method. Most of the transport codes utilize this method, but each one of them has its own
specificity. Two codes are mainly used in the field of medicine, EGS4 and MCNP. The
production of bremsstrahlung photon by theses transport codes will be presented here.
Bremsstrahlung Cross-Section Production in Monte Carlo Computational Codes
Before discussing the production of bremsstrahlung photon in Monte Carlo codes,
it is necessary to understand how electron transport is treated. This has been the subject
of numerous publications; among them is "Monte Carlo Transport of Electrons and
Photons"33 by T.M. Jenkins, W. R. Nelson, and A. Rindi, to which one can refer for
further detail on this subject. The many Coulomb interactions that occur during the
passage of a high-energy electron through matter are too numerous to simulate directly.
Most of the Monte Carlo transport codes are based on the approach that Berger and
Seltzer developed in the ETRAN code. The electron trajectories are divided into many
segments in each of which enough collisions occur to use multiple scattering theories. For
each segment, the net angular deflection and the net energy loss are sampled from
relevant multiple scattering distributions.33 This condensed history or path-segment
model is adaptable to Monte Carlo methodology. The ETRAN code has been enhanced
over the years and has become the basis for the Integrated TIGER Series, a system of
general purpose and application-oriented electron/photon codes. The electron physics in
MCNP and EGS4 are essentially that of the Integrated TIGER Series. The radiative
energy loss for the electron is determined by sampling the production of bremsstrahlung
photons using a dataset of bremsstrahlung cross-section, differential in photon energy.
The database has changed over the years. The current database includes stopping powers
and ranges from ICRU Report 37, and bremsstrahlung cross-sections given in Seltzer and
Berger.2'3 For each sub-step, bremsstrahlung production is sampled from a Poisson
distribution, and the photon energy is subtracted from the energy of the electron. If the
sampled photon energy is greater than a cutoff value, the photon history is traced. The
starting position for the photon is chosen at random along the sub-step. The intrinsic
photon emission angle, relative to the primary electron, is sampled from an angular
distribution derived from a combination of Beithe-Heitler cross-sections, differential in
photon energy and angle. The direction of the primary electron is taken to be that at the
beginning or end of the sub-step, depending on which end is closer to the chosen
production point. Angular deflections of the primary electron associated with the
emission of the more probable low-energy photons are assumed to be included in the
elastic-scattering distribution. Angular deflections due to the emission of high-energy
photons are ignored on the assumptions that such events are relatively unlikely, and the
effect is small compared to that of elastic scattering.
The sampling of the energy and angle of the bremsstrahlung photons will be now
discuss for two different Monte Carlo electron-photon transport codes: EGS4 and MCNP,
and for the last update of EGS4, EGSNRC.
EGS4
The energy sampling in the EGS4 system is based on formula 3BS with arbitrary
screening.26 To avoid confusion, it should be noted that the authors of the code, Nelson,
Hirayama and Rogers, used a slightly different notation than Koch and Motz,1 in their
manual the variable y is denoted 6 and y = 6 The functions 0, (y) and 2 (y) have
136
been evaluated using the Thomas-Fermi form factors. An empirical multiplicative
correction factor A' modifies formula 3BS. For E0 > 50MeV, A'= 1 since the Coulomb
corrected formula is used. For E0 < 50MeV, values interpolated in Z from the curves of
Koch and Motz are preferred. Also at high energy, a Coulomb correction term that was
derived by Davies, Bethe, and Maximon is added. Some additional corrections discussed
previously have been neglected. First, the differential cross-section goes to zero at the
maximum photon energy, whereas the real value is non-zero at the high frequency limit.
Second, the Elwert factor has been ignored at low energy (below 2 MeV). This
approximation is justified by the fact that radiative yield increases rapidly by electron
energy. The energy loss is then negligible for a low-energy electron compared to that of a
high-energy electron.
The angular sampling of the bremsstrahlung photons is extremely simple: all
newly created photons are set at a fixed angle 0 = 1/Eo with respect to the initiating
electron direction. This angle represents an estimate of the expected average scattering
angle. The justification for such an approximation was that at high energy the distribution
is strongly peaked in the forward direction, and at low energy multiple scattering of the
electron will overwhelm any photon distribution. It is recognized that this assumption
would break down for thin target. However, Bielajew, Mohan, and Chui34 note that this
assumption breaks down even for thick targets at about ten MeV for narrow beams. They
proposed the use of formula 2BS and a related sampling technique that still need to be
validated by experimental data.
EGSNRC
This new version of EGS4 released in 2001 by the National Research Council of
Canada (NRCC) has a complete new electron transport algorithm that will not be
discussed here.35 The physics of bremsstrahlung has also been modified. Two choices are
possible for energy sampling: the modeling used in EGS4 and based on the first born
approximation, and the modeling based on the Seltzer and Berger database. The angular
sampling was also modified. It is no longer a fixed angle, the angular selection scheme
based on formula 2BS in Koch and Motz1 proposed by Bielajew, Mohan, and Chui34 was
adopted with slight modification.
MCNP
MCNP code features two possible evaluations for sampling of bremsstrahlung
photons, ell and el3. In the ell evaluation, the oldest one, MCNP relies on the Seltzer and
Berger database that have been used in ETRAN. Tables include bremsstrahlung
production probabilities, photon energy distributions, and photon angular distributions as
discussed previously. In the el3 evaluation, the production cross-sections are from the
evaluation by Seltzer and Berger. No major changes have been made to the tabular
angular distributions, which are calculated in the ell evaluation. The procedure for
sampling the bremsstrahlung photons at each sub step is the same as the one mentioned
earlier. However the ell evaluation allows either none or one bremsstrahlung in a sub
step, while the el3evaluation allows the sampling of none, one or more photons along the
32
step according to a Poisson distribution. There is an alternative to the tabular data for the
angular distribution of bremsstrahlung photons: the material-independent probability
distribution p(pu)dy
1- 32 dc where p = cos This sampling method is of
2(1 3)2
interest only in the context of detectors and is not recommended otherwise.
CHAPTER 4
MEASUREMENT TECHNIQUES
Methods and Difficulties
Numerous experiments have been conducted to acquire the bremsstrahlung
spectrum, the angular distribution, and even the electron polarization. Koch and Motz1
describe the Buechner experiment36 on thick target, Quarles9 presents a variety of
experiments with thin solid target and gas target, and electron-photon coincidence
experiments have been detailed by Nakel.29 The experiments conducted since the earliest
studies on bremsstrahlung theory were realized with electron energies less than 1MeV.
However very few experiments have been made at the energy range that is utilized in
medicine. Measurements of the energy spectrum and angular distribution of high-energy
x-rays is delicate due to the energy level and the intensity of the beam. First, the
efficiency of the current detectors breaks down as the x-ray energy increases. At energies
above several megavolts, the absolute efficiency is close to zero. Secondly, spectrometers
can analyze up to a certain count rate of 105 to 106 photons per second for current
semiconductor spectrometers; this limit is much lower than the high intensity of medical
x-rays beams required in radiography or therapy. Several methods have been developed
to retrieve spectral data from x-ray machines. The main ones, which will be briefly
outlined below, are the transmission method, the photoactivation method, the incoherent
scattering (or Compton) spectrometry, and the direct measurements of primary photons.
Two other methods exist, but are not appropriate for this discussion due to their threshold
energy: pair production (1.02 MeV) and photodisintegration of deuterium (2.22 MeV).
Transmission Spectrometry
Because transmission of photon beams through an absorber depends on the
energy spectrum, transmission spectrometry consists of the reconstruction of the original
energy spectra from a transmission dose curve using an iterative least-square technique
and Laplace transform pairs. Several reconstruction models have been developed for
different energy regions.38-41 The limitations of this method include the measurement
accuracy of the transmission curve and the uncertainty inherent to curve fitting.
Therefore, the transmission method is an effective technique for specifying the beam
quality, but only approximates the true spectrum.
Photoactivation Method
Photons of sufficient energy can excite a nucleus and produce radioactive
isotopes. Those photonuclear reactions have energy-dependent cross-sections, and when
the same bremsstrahlung beam irradiates a series of foils, the resulting activities depend
strongly on the photon spectrum. The measurements are corrected for detector efficiency;
number of y-rays per nuclear decay; and the effects of finite irradiation time, waiting time
between activation and measurements, and finite counting time. An orthonormal
expansion technique is employed for the unfolding of the x-ray spectrum from the
measured activity data. The spectral reconstruction is not possible for the low-energy end
of the spectrum due to the high threshold energies. Also, the contributions of competing
reactions are difficult to assess and thus to eliminate. Prior knowledge of the spectral
shape is also required in order to obtain a stable solution to the unfolding problem. The
accuracy of this method is limited by the accuracy of the photoactivation cross-section
data of the nuclides irradiated, and the energy resolution (at best 1 MeV) depends
strongly on the choice of foils.
Direct Measurements
Despite the difficulties mentioned previously, direct measurements have been
tried with scintillation detectors and germanium detectors.4244 The photon beam from the
target is collimated to limit the count rate and improve the angular resolution. One of the
main problems encountered when measuring high intensity beam produced by a linear
accelerator is the saturation of the detector. Therefore one needs to have special
conditions to be able to perform those measurements. The bremsstrahlung spectrum is
related to the absorbed energy spectrum through the detector response and the detector
efficiency. Reconstruction of the bremsstrahlung spectrum can be realized by numerous
methods that vary in processing speed, degree of sophistication, and accuracy. HPGe
detectors have the advantage of having a better resolution and less secondary effects than
NaI(Tl) scintillators. Experiments with both types of detectors will be presented with
more details in this chapter.
Compton (Incoherent Scattering) Spectrometry
In this method, a thin-foil target is put into the beam to scatter the bremsstrahlung
photons and a spectrometer is used to measure the energy of the scattered beam at given
angles. The original spectrum is deduced from the energy and scattering angle. The
scattering process reduces the intensity of the beam incident on the detector by several
orders of magnitudes. Moreover, a more suitable detecting energy range is achieved
through incoherent scattering that reduces the energy of the scattered photons. The
angular distribution of photons can be measured by scanning the small scatter in the
radiation field. This method, though, suffers from a reduction of resolution due the
decrease in probability of incoherent scattering as the photon energy increases. This
method has been tried in the past to deduce energy spectra45-47 and will be presented in
more detailed below.
Scintillator and Germanium Detectors
Photons incident onto the detector material interact with the atomic electrons of
the material by photoelectric effect, Compton scattering, or by producing electron-
positron pairs. The created charged particles deposit most of their energy in the material.
In any normal detector, the measured spectrum consists of a photopeak, a Compton edge,
a Compton continuum, and the continuum between photopeak and Compton edge, which
is due to multiple scattering. Energy might be lost if secondary electrons, bremsstrahlung
photons, or characteristic x-rays escape. The observed spectrum can be more complicated
with the presence of back-scattered peak, annihilation peak, or characteristic x-ray peak
due to materials surrounding the detector. The response function of a gamma-ray detector
will depend on the size, shape, and composition of the detector, and also on the geometric
details of the irradiation. If the radiation beam incident on the detector is not
monoenergetic, the response function will be much more complicated. In gamma-ray
spectroscopy, it is preferred to use a large volume detector as the response function is
simpler and higher gamma rays can be detected with greater efficiency.
In the case of scintillator, the absorbed energy is converted into low energy
photons. Then a photomultiplier tube is used to collect those photons. Under ideal
operating conditions the resulting electron pulse height at the anode is linearly related to
the energy absorbed by the scintillator. In the case of semi-conductor detectors, electron-
hole pairs are created along the path of the charged particles. Their motion in an applied
electric field generates the basic electrical signal from the detector. NaI(Tl) scintillators
have been used extensively, but their main drawback is the poor energy resolution. The
recent high-purity germanium detector (HPGe) has the advantages of excellent energy
resolution and high detecting efficiency.
Some examples of direct measurements and incoherent scattering spectrometry
will now be detailed has they characterize experiments that have been undergoing since
the 1950s on bremsstrahlung spectrum and angular distribution created by electron
energies from 2 MeV to 30 MeV.
Direct Measurements
One of the first direct measurements was realized by Buechner et al.36 in 1948. It
was an investigation into the thick-target production of bremsstrahlung radiation using a
variety of target materials and electron energies ranging between 1.25 MeV and 2.35
MeV. The goal of their experiment was to quantify angular dispersion and calculate the
efficiency of x-ray production. The experiment (Fig. 9) was designed to eliminate the
effect of target geometry or differ path lengths within the target. The electron source
beam was generated from an electrostatic accelerator. The electrons were impinged on a
target located at the center of a vacuum chamber. This target was mounted on a
motorized arm that allowed tilting the target at specified angles. This was done to keep a
constant target thickness with respect to the detector at each angle of interest. The
intensity of the bremsstrahlung photons was measured by an ionization chamber of
cylindrical shape and was shielded with lead, except in the direction of the target. Fig. 7
shows the angular dependence of the thick target bremsstrahlung intensity integrated over
photon energy for beryllium and gold. In order to compare with the theory, Buechner et
al. integrated the measured intensity over the total solid angle surrounding the target for a
given voltage with varying materials. They found the total radiation flux to be linear with
the atomic number (Fig. 10). The accuracy of this experiment is not mentioned. Also
Buechner et al. applied unquantified self-absorption corrections. Nevertheless those
experimental results are still published in modem radiation texts to depict the energy
dependent angular distribution of bremsstrahlung photons.
From accelerating tube
Focused electron beam
Target
Ionization chamber
FIG. 9. Schematic diagram of Buechner et al. experiment.
s' W H
FIG. 10. Integrated x-ray intensity as a function of the atomic number of the target for
different electron energies as presented by Buechner et al.36
Recent experiments in 1990 and 1991 have been conducted by Faddegon et
al.42'43'48 They measured the forward-directed bremsstrahlung of 10 to 30 MeV electrons
incident on thick targets of Al and Pb, and the angular distribution of bremsstrahlung
from 15 MeV electrons incident on thick targets of Be, Al, and Pb. They used a Nal
scintillation detector, which required low currents that were several orders of magnitude
lower than produced by linear accelerators in radiotherapy. They used a research linear
accelerator operated at 240 Hz with a nominal pulse width of 2[ts. The spectrometry
system employed was limited to mean photon-detection rates of 0.5 photons per beam
pulse. Then for an electron beam of 30 MeV, current was restricted below 25 fA, or 700
electrons per beam pulse. Higher currents were tolerated at lower incident electron
energies or for off-axis measurements due to lower bremsstrahlung cross-sections.
Forward-directed measurements were made for electron energies of 10, 15, 20, 25, and
30 MeV. The Al and Pb target had nominal thicknesses of 110% of the CSDA range. A
transmission-beam current was used for monitoring the beam current. A lead shield with
collimators of various diameters was placed before the Nal detector. Detailed studies of
the pile-up, detector response, background, attenuation of the beam in post-target material
(Al and air), and collimator effect have been realized, and the measured bremsstrahlung
spectra (Fig. 11) have been corrected for these effects. Bremsstrahlung yield integrated
over energy has also been calculated. Faddegon et al. considered almost every aspects of
the measurement, and their study is valuable for that. However, they found the energy
resolution of the detector to be, from 0.2 to 30 MeV, F = 0.077E 60 where E and F are in
MeV. This gives a FWHM of 77 keV at 1MeV and 0.3 MeV at 10 MeV that is rather
large compared to other detectors.
Angular distribution measurements for 15 MeV electron beam have been made
with the arrangement shown in Fig. 13 and with Be, Al and Pb targets. The accelerator
was operated at currents ranging from 0. IpA for measurements along the beam axis to
0.2 nA for background measurements at 900. The same technique of measurement than
that of the forward measurements has been used. The Nal detector was modified to
improve the longitudinal uniformity of the detector sensitivity, and a new photomultiplier
tube was installed. The detector was found to be linear from 0.15 to 15 MeV and the
FWHM of the broadening function were found to be F = 0.117E063, where E and F are
in MeV. The bremsstrahlung yield was measured at 00, 1, 20, 40, 100, 300, 600, and 900
relative to the beam axis. The accuracy of the angle measurement was +0.070 at 00, and
+0.20 for off-axis measurements. Corrections were applied for pulse pile-up, drift and
noise in the electronics, background, attenuation of the bremsstrahlung in the air and in
the wall of the target chamber, the detector efficiency, and the collimator effect. Electron
contamination was found to be negligible (<1%), and photon scatter was also taken into
account. Shown in Fig. 12 are the measured spectra, calculations done using EGS4, and
spectral shapes determined from the Schiff spectra. The bremsstrahlung cross-section
formula of Schiff was used to calculate the thin-target spectral distributions, which were
then corrected for attenuation by the total thickness of the targets. Faddegon et al.
assumed that the major contribution to the bremsstrahlung yield is from electrons near the
surface of the target. For the smaller angles (0 to 100) are similar in shape to the
measured spectra. At large angles, 0 >> moc2 /E the Schiff formula for the thin-target
spectra is not valid. Moreover, the bremsstrahlung spectra at large angles are largely
composed of bremsstrahlung from multiple-scattered electrons and from photons
scattered in the target. Thus the simple attenuation of the thin-target spectrum is not a
correct approach. Faddegon et al. recognized that improvement in the analytical theories
of thick-target bremsstrahlung spectrum is still to come.
42
11 10
a) Al targets
1 r--
1 10
Energy (MeV)
FIG. 11. Corrected measured bremsstrahlung spectra along the beam axis (from 0 to 0.2)
for 10 and 25 MeV electrons incident on a) Al targets and b) Pb targets. The target
thicknesses were nominally 110% of the CSDA electron range.48
1 10
Energy (11eV)
FIG. 11. Corrected measured bremsstrahlung spectra along the beam axis (from 0 to 0.20)
for 10 and 25 MeV electrons incident on a) Al targets and b) Pb targets. The target
thicknesses were nominally 110% of the CSDA electron range.48
43
--- a) Be
1010
1o
10
Energy (MeV)
o so
t io
1 10 1 tO
Energy (MeV) Energy (MeV)
FIG. 12. Bremsstrahlung spectra generated at angles of 00, P1, 20, 40, 100, 300, 600, and
900 by 15 MeV electrons incident on a) a 11.67g/cm2 thick, 6.72 g/cm2 radius Be target,
b) a 9.74g/cm2 thick, 9.81 g/cm2 radius Al target, and c) a 9.13g/cm2 thick, 17.95 g/cm2
radius Pb target. The solid lines are measured bremsstrahlung yield, the dashed lines are
calculations done using EGS4, and the dotted lines are the spectral shapes determined
from the Schiff spectra with target attenuation, normalized to the measured values of the
integrated bremsstrahlung yield at the corresponding angles.48
Another example of direct measurement but with a germanium detector has been
realized in 1996 by Stritt, Bertschy, Jolie, and Mondelaers.44 Ten MeV electrons were
incident on a 2 mm-thick tantalum target surrounded by graphite. Figs. 13 and 14
represent the experimental set-up used and a schematic diagram of the bremsstrahlung
source. The linear accelerator operated at 4 kHz and delivered a 10 MeV electron beam
with intensity in the order of few [tA. The photon detection was carried out with a 170
cm3 Ge detector placed 11 m from the target. The spectrum was measured at three
different angles ac (+7.400, 00, -1.900) relative to the direction of the beam. The
acquisition time was three hours. Fig. 15 shows the measured photon spectrum at 7.400
angle after background subtraction. The authors noticed several difficulties. The first one
was an uncertainty (+1.00) in the angle of incidence of the electron beam. The alignment
of the electron beam was realized with an ionization chamber, by changing the incidence
angle of the electron beam to maximize the intensity of the photon flux at the center of
the photon beam. The second difficulty was to create a perfect shielding for the detector
against the background. This shielding was required considering the low current
necessary to avoid dead-time problems in the Ge detector. The third difficulty was the
detector effects introduced by the continuous spectrum. The measured spectra was been
corrected for detector response, attenuation, pile-up, but was rather normalized to the
total number of photons per spectrum and compared with calculations done using EGS4
and taking into account the detector response.
Scale in cm
___0 8.25110.
I n cD S :Vacuum
', ~ ~ ,-rI, 2 : Graphite
.. .. : Tantalum
05 3.45 : Inox
0.1k : Water
3.45 2.51I O : Air
1.4.*H,,
.3 .3
1.5
FIG. 13. Schematic geometry of the bremsstrahlung source used by Stritt et al.44
0 1 2 3m
FIG. 14. Experimental setup used by Stritt et al.44 (1) linear accelerator; (2) ending
magnets; (3) bremsstrahlung source; (4) lead collimator; (5) 12cm thick tungsten
collimator; (6) additional lead shielding; (7) 170 cm3 Ge detector.
S6000
S5000
4000
o 3000
2000
Ca
S1000
0
0 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Photon Energy in MeV
FIG. 15. Measured spectrum at 7.400+0.020 after background subtraction by Stritt et al.44
Incoherent scattering spectrometry
This technique has been studied in 1976 by Levy and coworkers45'46 using a
NaI(Tl) scintillation detector system, and more recently by Anghaie and coworkers (49
and private communication, 1993) using a HPGe detector. In this last study, the validity
and the different aspects of the method have been assessed for a cobalt machine, and
recommendations have been made for further utilization in the measurement of linear
accelerator photon beams.
As for direct measurements, one needs to pay attention to detector response,
collimator effects, and choice of shielding material. A problem specific to this method is
the choice of the scattering angle. The incoherent scattering process shrinks the original
spectrum to a narrower range. The reduction in energy depends on the scattering angle;
the larger is the angle, the lower is the final energy. Thus a smaller angle is preferred to
minimize this effect. However, a lower energy range is more suitable because suitable
gamma sources for detector efficiency calibration are not available for higher energies,
and the detection efficiency of germanium spectrometers decreases rather rapidly for
photons of energy above a few MeVs. Therefore a compromise must be reached. The
choice of scattering material is also important. Li showed that low Z materials are
preferable since Compton scattering is predominant at high energies and large angles for
those materials. Fig. 16 represents a schematic view of the experiment realized by
Anghaie and coworkers. One of their first measurements was made on an orthovoltage
machine with a scattering angle of 900 and is shown in Fig. 17. The orthovoltage machine
gave a continuous spectrum with a maximum energy of 250 keV. The measured spectrum
is composed of the spectrum coherently scattered and, superimposed, the spectrum
incoherently scattered with a maximum energy of 169 keV (250 kV incoherently
scattered at 900).
bremsstrahlung source
Lead
shield
Scatterer
Collimator
FIG. 16. Schematic view of the measurement setup by Anghaie and coworkers.49
Misr pIus
Markr: 174 = 124.151 kIu I
D isplay
MCBi
MCBt# 1 Seg# 1
Full1/E
Ut : Log
Hz :512
ROI :Mark
Presets
Rl Tm
Lu Tm
ROI Cnt
ROI Int
Time
SR Tm 618
Lu Tm
Dead Tm 18.5x
84-11-93
82:02:57
nni+ t: 1A4i33 ECA ORTEC
r ,l,'-- .l~ Fl I I II I I I IIII
FIG. 17. Measured 900 scattered spectrum of an orthovoltage machine by Anghaie et al.
(private communication, 1993).
From the measured scattered spectrum from cobalt-60 machine, Li49 noticed a
broadening of the scattered peaks. This broadening is due mainly to the finite size of the
photon source, and the size of the scatterer, which makes the scattering angle not strictly
900. Another but less important contribution for this broadening effect is the finite
aperture of the collimator. Another broadening effect is also introduced while
reconstructing the original spectrum. This broadening effect inherent to the
reconstruction process is a consequence of the conservation of energy and momentum in
the collision. At a given scattering angle, the higher the incident photon energy, the larger
the broadening effect.
Several attempts to measure the photon spectrum of a medical linear accelerator
have been made. Two major problems arose. The first one is the production of neutrons
by high-energy x-rays. For most relevant materials, the neutron production threshold
occurs at 8-10 MeV. Neutrons that originate in the primary collimator, target, and
flattering filter contaminate the photon beam. Neutrons will activate Ge, Cu and Al in or
around the HPGe detector when it is used to measure the photon spectrum from the linear
accelerator, thus neutron shielding is necessary in high energy cases. The second problem
is related to the intensity of the beam and the detection system. For a linear accelerator
operating at 6 MeV at dose rate of 400 centigrays per minute, the average beam current is
around 300 pulses per second, and the pulse duration is 3.5 hts. The pulse shaping of the
amplifier is several microseconds, so the detection system can pick only one or two
photons per pulse. Therefore, it would take many hours to obtain a spectrum of
acceptable statistics, which would cause overheating of the linear accelerator.
Nevertheless, the incoherent scattering method has been validated for linear accelerator
by means of Monte Carlo simulation. The primary photon spectrum outside the treatment
head was simulated using MCNP code. Then the scattered flux spectrum was predicted
using MCNP, and was used to reconstruct the primary flux spectrum. Good agreement
was found between this spectrum and the MCNP predicted primary spectrum.
Nevertheless, secondary effects have to be taken into account in real experiments:
neutron production and shielding complications can be minimized; detector response can
be obtained by Monte Carlo technique; line broadening effect can be minimize by using
smaller collimator size, but at the expense of detecting efficiency and resolution. Also, at
high energy and with a large scattering angle, the reconstructed spectrum is very sensitive
to resolution changes in the scattered spectrum.
CHAPTER 5
CONCLUSIONS AND RECOMMENDATION FOR FUTURE WORK
Actual theories on electron-nucleus bremsstrahlung have been reported in detail.
Those theories are the first Born approximation, calculations using Sommerfeld-Maue
wave functions and the method of partial-waves. Those theories can be classified
according to the energy domain where they best apply. At non-relativistic energies, the
first Born approximation and the method of partial-waves provide the best results. The
Sommerfeld-Maue wave functions theory applies at extreme-relativistic energies.
Calculations with the partial-waves method were performed for energies from 1 keV to 2
MeV, but computation becomes difficult and time consuming at higher energies. Also, at
energies of few MeVs, those calculations provide similar results to the Born
approximation, and no proof of real improvement has been produced yet. Very few
theoretical studies have been conducted for the energy domain 2 MeV to 50 MeV, and
interpolation techniques between the non-relativistic and extreme-relativistic energy
domains have been preferred. Considering the comparison with the partial-waves method,
it is believed that the oldest and simplest theory, the first Born approximation, could
describe accurately the bremsstrahlung process in this energy domain. Some corrections
are necessary since the first Born approximation fails to describe correctly the high end of
the bremsstrahlung spectrum: the cross-section differential in photon energy falls to zero
at the high-frequency limit, while it is known to be a finite value. The Elwert factor,
derived from Sommerfeld-Maue wave functions theory, is often used to correct the Born
approximation cross-section at the high-frequency limit of the bremsstrahlung spectrum.
The validation of the theory by the experiment in the energy domain 1 MeV to 30
MeV encounters several difficulties. The first one is the complications due to the thick
target effect. A target is considered "thin" when the thickness is less than one mean-free-
path of the incident electrons. But, in order to avoid electron contamination, the target
needs to be thicker than one mean-free-path. And the bremsstrahlung spectrum becomes
more complicated with the multiple scattering of electrons and the attenuation of the low-
energy photons. These effects need to be corrected in any measurement, especially when
measuring the angular distribution of the bremsstrahlung photons. A way to correct for
the thick-target geometry in angular distribution measurements would be the possibility
of tilting the target as in the Buechner experiment. Also measurement of the photon beam
produced by a medical linear accelerator is not suitable for the validation of any theory
since multiple scattering from the head, and electron and neutron contamination
complicates the spectrum.
The second difficulty is that at these energies, most detectors have poor energy
resolution or are ineffective. The achievement of high-purity germanium detectors with
higher energy resolution and the use of incoherent scattering spectrometry (ISS) method
might overcome this problem. The advantages and limitations of the ISS method have
been discussed with the review of other methods. The ISS method, by using a small
target scattering the primary beam to a collimated energy sensitive detector, shifts the
energy range of the photon spectrum to a detectible range for high energy resolution
detectors. However, the energy resolution of the ISS method is reduced by the
broadening effect of the scattering process; but this effect can be predicted and corrected.
The third difficulty is due to the detection system. One needs to minimize pile-up
and to collect sufficient photons for good accuracy. Medical linear accelerators produced
electron beam pulse with short duration. The charge collection time of HPGe detectors is
such that only one photon per beam pulse can be detected, which leads to several hours of
measurements. Because medical linacs are calibrated and cannot support the overheating
caused by an increased time of operation, the application of the ISS method with HPGe
detector is limited. NaI(Tl) scintillation detectors have been used in the past: since they
have a better efficiency than HPGe detectors, the time of measurement is reduced.
However energy resolution should be preferred. Then measurement using incoherent
scattering method with HPGe detectors would be preferable. Also Betatron or linear
accelerators with adaptable intensity and pulse duration are needed.
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BIOGRAPHICAL SKETCH
Anne-Sophie Leclere was born on September 13, 1976, in Troyes, France. In
spring 1997, after two years of undergraduate program, she received her Diplome
Universitaire de Technologie in Physics and Techniques of Measurements in Paris,
France. She entered the Ecole Nationale Superieure de Physique de Grenoble, a French
engineering school, in September 1997. In July 2000, she received her engineering
diploma in physics and nuclear engineering. She came to the University of Florida during
summer 1999. Since then, she has been pursuing the degree of Master of Science in
nuclear and radiological engineering while working as a Graduate Teaching and Research
Assistant.
*
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