ing Calculations of Compton Scattering for Molecules, Doppler Broadening Calculations of Compton Scattering for Molecules,
Plastics, Tissues, and Few Biological Materials in the X-Ray Region:
An Analysis in Terms of Compton Broadening and Geometrical
Energy Broadening
D. V. Rao,
a R. Cesareo, and A. Brunetti
Istituto di Matematica e Fisica, Universita di Sassari, Via Vienna 2, I-07100 Sassari, Italy
G. E. Gigante
Dipartimento di Fisica, Universita di Roma La Sapienza, Ple A. Moro 2, 00185 Roma, Italy
T. Akatsuka
Faculty of Engineering, Yamagata University, Yonezawa, Yamagata 992-8510, Japan
T. Takeda and Y. Itai
Institute of Clinical Medicine, University of Tsukuba, Tsukuba, Ibaraki 305-8575, Japan
Received 23 April 2002; revised manuscript received 16 December 2002; accepted 28 July 2003; published online 19 July 2004
Relativistic and nonrelativistic Compton prole cross sections for H, C, N, O, P, and
Ca and for a few important biological materials such as water, polyethylene, lucite,
polystyrene, nylon, polycarbonate, bakelite, fat, bone and calcium hydroxyapatite are
estimated for a number of K
x-ray energies and for 59.54 keV Am-241
photons.
Energy broadening and geometrical broadening ( G) is estimated by assuming
min
and
max
are symmetrically situated around
90°. FWHM of J( P
Z
) and FWHM of Comp-
ton energy broadening are evaluated at various incident photon energies. These values are
estimated around the centroid of the Compton prole with an energy interval of 0.1 and
1.0 keV for 59.54 keV photons. Total Compton, individual shell, and Compton energy
absorption scattering cross sections are evaluated in the energy region from 0.005 to 0.5
MeV. It is an attempt to know the effect of Doppler broadening for single atoms, many of
which constitute the biological materials.
© 2004 American Institute of Physics.
DOI: 10.1063/1.1614814
Key words: biological materials; Compton broadening; geometrical energy broadening, Compton energy
absorption cross sections; Compton scatter tomography and imaging; Doppler broadening, relativistic and
nonrelativistic Compton prole cross sections; total Compton and individual shell cross sections.
Contents
1.
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
628
2.
Basic Theoretical Methods. . . . . . . . . . . . . . . . . . . .
629
3.
Theory: Compton Scattering from Bound
Electrons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
629
4.
Geometrical Broadening. . . . . . . . . . . . . . . . . . . . . .
632
5.
Results and Discussion. . . . . . . . . . . . . . . . . . . . . . .
632
6.
Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
638
7.
Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . .
638
8.
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
638
List of Tables
1.
X-ray source and energy. . . . . . . . . . . . . . . . . . . . .
629
2.
Name of the material, density and chemical
formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
632
3.
Double differential scattering cross sections
m
2
sr
1
keV
1
nonrelativistic for single
atoms in the atomic region 1
Z
20; energy
interval: 0.1 keV; scattering angle ( )
90°. . . . .
6
39
4.
Double differential scattering cross sections
m
2
sr
1
keV
1
relativistic for single atoms
in the atomic region 1
Z
20; energy interval:
0.1 keV; scattering angle ( )
90°. . . . . . . . . . . . .
6433
5.
Double differential scattering cross sections
m
2
sr
1
keV
1
relativistic for few biological
materials; scattering angle ( )
90°. . . . . . . . . . . .
647
6.
Geometrical energy broadening ( G);
min
and
a
Electronic mail: dvrao@ssmain.uniss.it
© 2004 American Institute of Physics.
0047-2689
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2004
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J. Phys. Chem. Ref. Data, Vol. 33, No. 3, 2004
627 max
are symmetrically situated around
90°. . .
655
7.
FWHM of J( P
Z
) and FWHM of Compton
broadening CB for biological materials in
the energy region from 17.44 to 68.18 keV;
scattering angle ( )
90°. . . . . . . . . . . . . . . . . . . .
694
8.
Gaussian t of FWHM of J( P
Z
) and FWHM of
Compton broadening CB for biological
materials at 59.54 keV for few biological
materials; scattering angle ( )
90°. . . . . . . . . . . .
709
9.
Total and individual shell Compton scattering
cross sections for single atoms in the energy
region from 0.005 to 0.5 MeV using Compton
prole data of Biggs et al.
10
and comparison
with Strom and Israel values
21
and XCOM
1987 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711
List of Figures
1.
a Variation of geometrical energy broadening
G(
1
) (keV) in the energy region from 1 to
500 keV around the centroid of the Compton
prole with an interval of 1°. b Contribution of
geometrical energy broadening in the energy
region from 1 to 50 keV around the centroid of
the Compton prole with an interval of 1°.
Same as in b but for the region up to c 100
keV; d 150 keV; e 200 keV; f 250 keV;
g 300 keV; h 350 keV; i 400 keV; j 450
keV; and k 500 keV. . . . . . . . . . . . . . . . . . . . . . .
633
2.
a Compton broadening for primary photon
energy keV/sr in the angular region 1°
180° from 1 to 50 keV. Same as in a but for
the region up to b 100 keV; c 150 keV; d 200
keV; e 250 keV; f 300 keV; g 350 keV;
h 400 keV; i 450 keV; and j 500 keV. . . . . . .
635
3.
Comparison of FWHM of J( P
Z
) and FWHM of
Compton broadening in the energy region 17.44
68.18 keV for a water; polyethylene; c
lucite; d polystyrene; e nylon; f
polycarbonate; g fat adipose tissue ; and h
calcium hydroxyapatite. . . . . . . . . . . . . . . . . . . . . .
636
4.
Gaussian t of the FWHM of J( P
Z
) and
FWHM of Compton broadening for few materials
in the energy region 17.44 68.18 keV for a
water; b lucite; c fat adipose tissue ; and d
bone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
637
5.
Comparison of Compton-energy absorption cross
sections for single atoms with theoretical
estimates in the energy region from 0.005 to 0.5
MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
637
6.
Comparison of Compton-energy absorption cross
section ratios for single atoms with theoretical
estimates in the energy region from 0.005 to 0.5
MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
638
1. Introduction
Incoherent Compton scattering is an important mode of
interaction with atoms for photons in the energy region
2
MeV. Although the KleinNishina theory
1
gives a satisfac-
tory explanation of this process over most of the energy re-
gion where it is predominant, the incoherent scattering cross-
section is found to deviate from its predicted value for low
photon energies. This can be understood in terms of the in-
creasing inuence of the bound electrons of the scattering
medium. An attempt to quantify these deviations has been
made under the impulse approximation with the introduction
of the so-called incoherent scattering function S(X,Z) as a
multiplicative factor to the free electron differential cross
section. Impulse approximation refers to large transfers of
energy and momentum. However, if the electrons are in mo-
tion, as we know to be the case, there is a Doppler effect
related to the projected velocities of the electrons. This
causes the modied Compton line to spread out and be-
come a bond. The electron momentum distribution can be
determined from the spreading out of the Compton line.
The motion of the atomic electrons around the atomic
nucleus gives rise to a Doppler broadening of the apparent
energy of the incident photon, resulting in a corresponding
broadening of the Compton modied line for a given de-
ection angle of the outgoing scattered photon. The shape of
this broadened line is called the Compton prole. The
shape of the prole is sensitive to the valencies of the elec-
trons in the atoms and can also give some information about
molecular structure.
2
Although it is used to determine the
electron momentum distributions in condensed matter
physics,
3
little attention is focused for its use in medical ap-
plications.
In dosimetry calculations, use is frequently made of the
Compton energy absorption cross section per electron (
en
),
which expresses the probability of transfer of energy from a
photon to an electron by the Compton process. It is equal to
the total Compton scattering cross section per electron (
TC
)
times the fraction f of photon energy, which is converted to
kinetic energy of the recoil electrons in a single collision,
averaged over all directions of electron recoil (
en
TC
f ). Since the range of the recoil electron is small, the
Compton energy absorption cross section per electron is a
measure of the total energy communicated locally to the ab-
sorbing medium by the Compton process. It is interesting to
estimate these cross sections by means of double differential
scattering cross sections based on impulse approximations.
One particular area of interest is in Monte Carlo simulation
of photon transport in applications of medical physics and
industrial radiography.
4,5
The Compton scattered photon energy is broadened by the
precollision motion of the electron. Due to the Doppler
broadening electron binding effect, a part of the broadened
photon spectrum is suppressed and the image reconstruction
will be tedious with Compton scattering spectral data. The
electrons moving in the target cause an energy broadening of
the scattered photons. At energies below 100 keV, FWHM
628
628
RAO
ET AL.
J. Phys. Chem. Ref. Data, Vol. 33, No. 3, 2004 of the broadening over most of the
is greater than the
FWHM res