Download Chapter 5 Effective atomic numbers and electron

Chapter 5
Effective atomic numbers
and electron densities for some
fatty acids
Effective atomic numbers and electron densities for some fatty acids...
147
In section 5.1 of this chapter the materials and motivation for the present
study are given. In sections 5.2 and 5.3, the analysis and discussion of results on
photon interaction parameters for some fatty acids are presented.
5.1. Materials and motivation
Biological membranes are organized assemblies of lipids and proteins with
small amounts of carbohydrates. The lipids are heterogeneous organic compounds
that are important constituents of the animal and plant cells, and are the most
important energy storage units in animal cells. The fatty acids are the major
components in lipids. The fatty acids are carboxylic acids with long-chain
hydrocarbon side groups. They are rarely free in nature, but occur in esterified form
as the main components in various lipids [1]. The fatty acids play an important role
in the life and death of cardiac cells because they are essential fuels for the
mechanical and electrical activities of the heart. The human body can produce all
the fatty acids except two: linoleic acid and linolenic acid. Since the human body
cannot synthesize, they are called essential fatty acids (EFAs). EFAs are essential
for all mammals and must be obtained through diet. All other fatty acids can be
derived from EFAs. The human body needs EFAs to manufacture and repair cell
membranes, enabling the cells to obtain optimum nutrition and expel harmful waste
products. A primary function of EFAs is the production of prostaglandins, which
regulate body functions such as heart beat rate, blood pressure, blood clotting,
fertility, conception, and play a role in immune function by regulating
inflammation and encouraging the body to fight against infection. The EFAs
deficiency is linked with serious health conditions such as heart attack, cancer
Effective atomic numbers and electron densities for some fatty acids...
148
(colon, breast and prostate cancer), insulin resistance, asthma, lupus, schizophrenia,
depression, postpartum depression, accelerated aging, stroke, obesity, diabetes,
arthritis, attention deficit/hyperactivity disorder (ADHD) etc [2].
The chemical formulae of the fatty acids used in the present work, are listed in
table 5.1.
TABLE 5.1: Chemical formulae (or repeat units) of fatty acids (#1–7) studied in
the present work. S.N is the sample number. <Z> is the mean atomic number
calculated from the chemical formula of the fatty acids.
S.N Fatty acids
<Z>
1. Tridecyclic acid (C13H26O2)
2.93
2. Ceroplastic acid (C35H70O2)
2.77
3. Lacceroic acid (C32H64O2)
2.77
4. Margaric acid (C17H34O2)
2.86
5. Pentadecanoic acid (C15H30O2)
2.89
6. Propanoic acid (C3H6O2)
3.63
7. Heptanoic acid (C7H14O2)
3.13
Since radioactive sources are used in biological studies, radiation sterilization
and industry [3], a thorough knowledge of the interaction of photons with
biologically important substances such as fatty acids is desirable. The photons in
the keV range are important in radiation biology as well as in medical diagnostics
and therapy. Photons in the MeV range are vital for radiography and medical
imaging (CT scans), and photons in the GeV range are of interest in astrophysics
and cosmology.
Effective atomic numbers and electron densities for some fatty acids...
149
Several investigators have made extensive studies on effective atomic numbers
for photon interaction, Zeff (or ZPI, eff), in human organs/tissues and other biological
materials [417]. The corresponding studies on effective atomic numbers for
photon energy absorption, ZPEA, eff, appears to be limited [1719]. In literature, the
available experimental data on Zeff for fatty acids [2022] are restricted to discrete
energies between 81 keV and 1332 keV. So far, to the best knowledge of the
authors, no theoretical or experimental study has been done for these materials at
lower and higher energies. The studies on ZPEA,
eff
for any of the biological
molecules mentioned are completely missing. The Zeff for total and partial gamma
ray interactions are equally important, however, no further information is available
on Zeff of these materials for the partial interaction processes, viz. photoelectric
absorption, coherent and incoherent scattering and pair and triplet production. This
prompted us to undertake a rigorous and exhaustive calculation of the Zeff and Ne, eff
for total and partial photon interactions over an extended energy range 1 keV–100
GeV, and ZPEA, eff in the energy range 1 keV to 20 MeV.
5.2. Analysis and discussion of results on effective atomic numbers
and electron densities for total and partial photon interaction
processes of some fatty acids and kerma
In this section, the results on effective atomic numbers, Zeff, and electron
densities, Ne,
eff,
are presented for some fatty acids (table 5.1). Calculations have
been carried out in the extended energy region 1 keV100 GeV for total and partial
photon interaction processes by using a computer program, WinXCom [23, 24],
based on a modern and accurate database of photon interaction cross-sections, and
Effective atomic numbers and electron densities for some fatty acids...
150
a comprehensive and systematic set of formulas derived from first principles. The
variations of Zeff and Ne,
eff
with energy are shown graphically for all photon
interactions. One more parameter, called, kerma relative to air is calculated. The
relevance of the single values of Zeff and Ne provided by the program XMuDat [25]
is also discussed. Wherever possible, the calculations are compared with
experimental results.
In the present work, mass attenuation coefficients and photon interaction crosssections were generated for elements and the biological molecules, in the energy
range from 1 keV to 100 GeV using the WinXCom program. This program uses the
same underlying cross-section database as that of the well-known tabulation of
Hubbell and Seltzer [26]. The effective atomic numbers were calculated for total
and partial photon interaction using the relation (2.14), and the mean atomic
number, <Z>, using equation (2.12) [Chapter 2, Section 2.1.1]. Using these Zeff
values, the effective electron density, Ne, eff, of the fatty acids were calculated from
the relation (2.25), and the average electron density, <Ne>, using equation (2.24)
[Chapter 2, Section 2.1.2]. Single values of Zeff and Ne are also obtained using the
XMuDat program [25] and kerma of a biological molecule relative to air is
obtained using equation (2.34).
5.2.1. The effective atomic number for total and partial photon
interaction processes
The results on Zeff are shown graphically in figures 5.15.12 for total and partial
photon interaction processes. From these figures it can be observed that the
variation in Zeff for total and partial gamma ray interaction depends on the spread in
Effective atomic numbers and electron densities for some fatty acids...
151
the atomic numbers of which the biological molecule is composed off. The Zeff
value of a biological molecule varies within a range with lowest and highest atomic
numbers of its constituent elements as limits. The present results clearly confirm
the comment by Hine [27] that, the Zeff of a multielement material cannot be
represented by a single number throughout an extended energy range. The variation
of Zeff with energy for total and individual photon interactions are discussed in the
next paragraphs.
5.2.1a. Total photon interaction (with coherent scattering)
The energy dependence of Zeff for total photon interaction is shown in
figure 5.1. All fatty acids have almost the same behaviour, since they consist of
hydrogen, carbon and oxygen in about the same proportions. This is also evident
from the fact that the mean atomic number, <Z>, is about 2.8 for samples #1-5, 3.6
for sample #6 and 3.1 for sample #7 (table 5.1). Figure 5.1 mirrors the relative
importance of the partial photon interaction processes, viz. photoelectric
absorption, Rayleigh scattering, Compton scattering and pair production. In figure
5.1, one can clearly distinguish three energy regions, in each of which Zeff is almost
constant. The three energy regions are approximately E < 0.01 MeV, 0.05 < E < 5
MeV and E > 200 MeV. Between these regions, there are transition regions with a
steep variation of Zeff. The effective atomic number is largest at low energies where
photoelectric absorption dominates, and less at high energies where scattering and
pair production dominates. The lowest values of Zeff occur at intermediate energies
where Compton scattering is the main photon interaction process. This is in
Effective atomic numbers and electron densities for some fatty acids...
152
conformity with Sastry and Jnanananda [28], who have reported that Zeff of
composite material for photoelectric interaction is greater than other processes.
In the low-energy region E < 0.01 MeV, photoelectric absorption is the main
photon interaction process. For a given material, the maximum value of Zeff is
found in this low-energy range, since the Z 45 dependence of the photoelectric
absorption cross-section gives a heavy weight to the element with the highest
atomic number in the material. The Zeff decreases rapidly with increase of energy to
its lowest value typical for Compton scattering in the transition region between
0.01 MeV and 0.05 MeV.
At intermediate energies, between about 0.05 MeV and 5 MeV, incoherent or
Compton scattering is the main interaction process, and again, Zeff is almost
constant. For a given material, the minimum value of Zeff is found in this
intermediate energy range, since the Compton scattering cross-section of a given
element is proportional to Z. This minimum value of Zeff, in the intermediate energy
range, is very close to the mean atomic number of the material, <Z>
(table 5.1). In the transition region from 5 MeV to 200 MeV, Zeff increases with
increase in energy as pair production gradually becomes dominant.
Above 200 MeV, Zeff assumes an almost constant value determined by pair
production. The value of Zeff is smaller than that obtained for photoelectric
absorption. This is due to the fact that the pair production cross-section is
proportional to Z2, giving less weight to the higher-Z elements than the
photoelectric absorption cross-section.
Effective atomic numbers and electron densities for some fatty acids...
153
From the above discussion it can be concluded that, in the low and high energy
regions, Zeff is a weighted mean, where the element with the highest atomic number
has the greatest weight. Therefore, the weighted mean is larger than the simple
mean at intermediate energies.
8.0
Total photon interaction (Coherent)
1
2
3
4
5
6
7
7.5
7.0
6.5
6.0
ZPI, eff
5.5
5.0
4.5
4.0
3.5
3.0
2.5
2.0
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
Energy in MeV
FIG. 5.1: Energy dependence of effective atomic number, Zeff, of fatty acids
(samples # 1–7) for total photon interaction (with coherent). Sample numbers are as
for table 5.1.
The present theoretical results are in line with the result of Lingam et al. [29]
and Parthasararadhi [30] in their covered energy regions. Parthasararadhi [30] has
reported the decrease in total Zeff with increase in energy from 100 to 662 keV.
Effective atomic numbers and electron densities for some fatty acids...
154
5.2.1b. Photoelectric absorption
Figure 5.2 shows the energy dependence of the calculated effective atomic
number for the partial interaction process of photoelectric absorption. Figure 5.2 is
evidence to the fact that, the constant value of Zeff for this partial process is equal to
the maximum value of Zeff for total photon interaction (figure 5.1). For example, the
maximum value of Zeff for total photon interaction in margaric acid is 6.41 (figure
5.1) and this value is close to 6.53 as read from figure 5.2, similarly for other
biological molecules. It is seen in figure 5.2 that, the Zeff for photoelectric
absorption increases slightly with increase in energy up to 800 keV and it remains
constant thereafter.
Similar results were also obtained by Perumallu et al. [31] in multielement
materials of biological importance, who reported that Zeff for photoelectric
absorption increases with an increase in energy from 30 to 150 keV.
5.2.1c. Compton scattering or incoherent scattering
Figure 5.3 shows the energy dependence of Zeff for incoherent scattering. It can
be seen that, apart from a steep increase at the lowest energies up to 300 keV, the
Zeff is constant at high energies i.e. independent of energy for all fatty acids. As
mentioned in section 5.2.1a, the Compton scattering cross-section is proportional to
the atomic number, and in this special case, Zeff equals the mean atomic number of
the material. Consider for example margaric acid. The constant value of Zeff based
on Compton scattering is 2.87 (figure 5.3). The same value (= 2.87) is also found
for Zeff based on the total photon interaction (figure 5.1), at say 1 MeV, where
Effective atomic numbers and electron densities for some fatty acids...
155
Compton scattering is the main interaction process. Moreover, the value 2.86 is in
perfect agreement with the value of the mean atomic number, <Z>, calculated from
the chemical formula of margaric acid (table 5.1).
7.4
7.3
Photoelectric
1
2
3
4
5
6
7
7.2
7.1
7.0
6.9
ZPI, eff
6.8
6.7
6.6
6.5
6.4
6.3
6.2
6.1
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
Energy in MeV
FIG. 5.2: Energy dependence of Zeff of fatty acids for photoelectric absorption.
Notations as for table 5.1
The present theoretical results are similar to the theoretical results of Bhandal
and Singh [32] who have reported similar types of variation of Zeff for Compton
scattering in some biological samples. In the present study, the Zeff for Compton
scattering in the biological molecules is independent of photon energy only above
300 keV but depends on photon energy below 300 keV.
Effective atomic numbers and electron densities for some fatty acids...
156
3.6
Incoherent
3.4
3.2
3.0
ZPI, eff
2.8
2.6
1
2
3
4
5
6
7
2.4
2.2
2.0
1.8
1.6
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
Energy in MeV
FIG. 5.3: Energy dependence of Zeff of fatty acids for incoherent scattering.
Notations as for table 5.1
5.2.1d. Rayleigh scattering or coherent scattering
Figure 5.4 shows that variation of Zeff with energy for coherent scattering
have similar energy dependence as that of Zeff for incoherent scattering. From
figure it is clear that, Zeff increases with increasing energy up to 300 keV for all
fatty acids. From here onwards Zeff remains constant with increase in energy i.e.
independent of energy. The present theoretical results are similar to the
experimental findings of Parthasaradhi [30] who has reported the constancy of Zeff
for coherent scattering in the energy range 100662 keV for some alloys.
Effective atomic numbers and electron densities for some fatty acids...
157
6.8
Coherent
6.6
6.4
6.2
ZPI, eff
6.0
5.8
1
2
3
4
5
6
7
5.6
5.4
5.2
5.0
4.8
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
Energy in MeV
FIG. 5.4: Energy dependence of Zeff of fatty acids for coherent scattering.
Notations as for table 5.1
It should be mentioned, however, that coherent scattering never plays any major
role when calculating Zeff for total interaction, since the cross-section for coherent
scattering is appreciable only at low energies where photoelectric absorption is by
far the most important interaction process.
5.2.1e. Pair production in nuclear and electron field
The variation of Zeff for pair production in nuclear field with photon energy is
shown in figure 5.5. It can be seen that Zeff slightly decreases with increasing
photon energy from 1.25 to 200 MeV and then it is almost independent of energy.
It is due to the fact that pair production cross-section in nuclear field is Z 2
dependent. It should be noted that, the threshold energy for pair production
Effective atomic numbers and electron densities for some fatty acids...
158
is 1.022 MeV, but the calculations Zeff for this interaction have been done from
1.25 MeV. It is also observed that, the Zeff for pair production is in between Zeff for
photoelectric absorption and Zeff for Compton scattering. Figure 5.6 show the
energy dependence of Zeff for pair production in electron field, also called triplet
production. From figure it is clear that, Zeff is independent of photon energy from
3–30 MeV. From 30 MeV, Zeff decreases slightly with increase of photon energy up
to 30 GeV and thereafter it is independent of energy. It should be noted that, the
threshold energy for triplet production is 2.044 MeV, but the calculations of Zeff for
this interaction have been done from 3 MeV.
Pair production (nuclear)
6.0
1
2
3
4
5
6
7
5.8
ZPI, eff
5.6
5.4
5.2
5.0
4.8
4.6
10
0
10
1
10
2
10
3
10
4
10
5
Energy in MeV
FIG. 5.5: Energy dependence of Zeff of fatty acids for pair production in the nuclear
field. Notations as for table 5.1
Effective atomic numbers and electron densities for some fatty acids...
159
Pair production (electric)
3.6
3.4
1
2
3
4
5
6
7
ZPI, eff
3.2
3.0
2.8
2.6
2.4
10
1
10
2
10
3
10
4
10
5
Energy in MeV
FIG. 5.6: Energy dependence of Zeff of fatty acids for pair production in the field of
electron. Notations as for table 5.1
5.2.2. Effective electron density for total and partial interaction processes
Figures 5.7–5.12 show the energy dependence of the effective electron density,
Ne, eff, for total and partial interaction processes. The behaviour is very similar to
that of Zeff since the two parameters are related through equation (2.25)
[Chapter 2, Section 2.1.2] and can be explained in a similar manner. As that for Zeff,
the maximum value of Ne,
eff
occurs in the low-energy range determined by
photoelectric absorption and the minimum value occurs at intermediate energies,
where Compton scattering is the main photon interaction process.
Table 5.3 show the mean electron density, <Ne>, is approximately 3×1023 e/g
for all fatty acids, since <Z>/<A> is about ½. This is in conformity with the
statement made in the section 2.1.2.
Electron Density(Ne,eff)
Effective atomic numbers and electron densities for some fatty acids...
8.0x10
23
7.5x10
23
7.0x10
23
6.5x10
23
6.0x10
23
5.5x10
23
5.0x10
23
4.5x10
23
4.0x10
23
3.5x10
23
3.0x10
23
2.5x10
23
160
Total photon interaction (Coherent)
10
-3
10
-2
10
-1
10
0
10
1
10
2
1
2
3
4
5
6
7
10
3
10
4
10
5
Energy in MeV
Electron Density(Ne,eff)
FIG. 5.7: Energy dependence of effective electron density, Ne, eff, of fatty acids for
total photon interaction (with coherent). Notations as for table 5.1
7.8x10
23
7.6x10
23
7.4x10
23
7.2x10
23
7.0x10
23
6.8x10
23
6.6x10
23
6.4x10
23
Photoelectric
1
2
3
4
5
6
7
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
Energy in MeV
FIG. 5.8: Energy dependence of Ne, eff of fatty acids for photoelectric absorption.
Notations as for table 5.1
Effective atomic numbers and electron densities for some fatty acids...
4.0x10
23
3.8x10
23
3.6x10
23
3.4x10
23
3.2x10
23
3.0x10
23
2.8x10
23
2.6x10
23
2.4x10
23
2.2x10
23
161
Electron Density(Ne,eff)
Incoherent
1
2
3
4
5
6
7
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
Energy in MeV
FIG. 5.9: Energy dependence of Ne,
Notations as for table 5.1
7.6x10
23
7.4x10
23
7.2x10
23
7.0x10
23
6.8x10
23
6.6x10
23
6.4x10
23
6.2x10
23
6.0x10
23
5.8x10
23
eff
of fatty acids for incoherent scattering.
Electron Density(Ne,eff)
Coherent
1
2
3
4
5
6
7
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
Energy in MeV
FIG. 5.10: Energy dependence of Ne,
Notations as for table 5.1
eff
of fatty acids for coherent scattering.
Effective atomic numbers and electron densities for some fatty acids...
6.2x10
162
23
Electron Density (Ne,eff)
Pair production (nuclear)
6.0x10
23
5.8x10
23
5.6x10
23
5.4x10
23
5.2x10
23
1
2
3
4
5
6
7
10
0
10
1
10
2
10
3
10
4
10
5
Energy in MeV
FIG. 5.11: Energy dependence of Ne, eff of fatty acids for pair production in the
field of nucleus. Notations as for table 5.1
3.5x10
23
3.4x10
23
3.4x10
23
3.3x10
23
3.3x10
23
3.2x10
23
3.2x10
23
3.1x10
23
3.1x10
23
3.0x10
23
3.0x10
23
Electron Density(Ne,eff)
Pair production (electric)
1
2
3
4
5
6
7
10
1
10
2
10
3
10
4
10
5
Energy in MeV
FIG. 5.12: Energy dependence of Ne, eff of fatty acids for pair production in the
field of electron. Notations as for table 5.1
Effective atomic numbers and electron densities for some fatty acids...
163
5.2.3. Comparison of calculated Zeff and Ne, eff values for total photon
interaction with the values from XMuDat program
For a given material, the XMuDat program [25] calculates a single-valued
effective atomic number, Zeff,
XMuDat,
and a single-valued electron density,
Ne, XMuDat. The Zeff, XMuDat and Ne, XMuDat values predicted by XMuDat program are
energy-independent. In tables 5.2 and 5.3 these values are compared, for fatty acids
with (Zeff)max, (Zeff)min, (Ne,
eff)max
and (Ne,
eff)min,
i.e. the maximum and minimum
values of the effective atomic number and the electron density obtained in the
present calculations using WinXCom in the energy range from 1 keV to 100 GeV.
Tables also contain the average value, median and standard deviation data for Zeff
and Ne, eff in the energy region of 1 keV to 100 GeV. The percentage differences
(P.D) between (Zeff)max and Zeff, XMuDat is also provided.
XMuDat actually calculates the effective atomic number according to equation
(2.30). However, it is obvious from the data in table 5.2 and 5.3 that, Zeff, XMuDat and
Ne, XMuDat are not related by equation (2.25), since for each material Zeff, XMuDat is
close to (Zeff)max whereas Ne,
XMuDat
is in perfect agreement with (Ne,
eff)min.
As
discussed in section 5.2.1, (Zeff)max occurs in the low-energy region, where
photoelectric absorption is the main interaction process, and (Ne,
eff)min
occurs at
intermediate energies, where Compton scattering is dominant. It follows that
XMuDat calculates Zeff, XMuDat by assuming that photoelectric absorption is the main
interaction process. In contrast, Ne,
XMuDat
has been calculated assuming that
Compton scattering is dominant, i.e. XMuDat calculates the average electron
density, Ne, XMuDat = <Ne>, cf. equation (2.31). Thus, users of XMuDat should be
Effective atomic numbers and electron densities for some fatty acids...
164
aware that the calculations of the single values of Zeff, XMuDat and Ne, XMuDat are based
on two different assumptions.
5.2.4. The effect of chemical bonding
The present theoretical calculations are based upon atomic interaction crosssections. In the present approximation, Zeff and Ne,
eff
are independent of any
chemical effects. A much larger and higher-dimensional database would be
required to accommodate the molecular and other matrix environments of the target
atom [40]. Very careful experiments would be required to study any possible effect
of chemical bonding on the photon interaction cross-sections and the effective
atomic numbers of the biological molecules. Such studies could, however,
stimulate further theoretical developments, in particular close to absorption edges.
TABLE 5.2: Statistics of Zeff data (dimensionless) for fatty acids listed in table 5.1
for the energy range 1 keV to 100 GeV and Zeff, XMuDat values. Numbers are as for
table 5.1
(Zeff)Max
Zeff, XMuDat
(Zeff)Min
Average
value
Median
value
1.
6.51
6.03
2.93
4.09
4.26
2.
6.22
5.74
2.77
3.90
3.98
3.
6.23
5.76
2.78
3.88
3.99
4.
6.41
5.93
2.87
4.01
4.17
5.
6.46
5.97
2.89
4.05
4.21
6.
7.22
6.79
3.64
4.97
5.19
7.
6.79
6.31
3.13
4.36
4.56
S.N
Effective atomic numbers and electron densities for some fatty acids...
165
TABLE 5.3: Statistics of Ne, eff data (in units of 1023 electrons/g) for fatty acids
listed in table 5.1 for the energy range 1 keV to 100 GeV and Ne, XMuDat values.
Numbers are as for table 5.1. <Ne> is the mean electron density calculated from the
chemical formula.
S.N <Ne>
(Ne, eff)Max
Ne, XMuDat
(Ne, eff)Min
Average
value
Median
value
1.
3.37
7.50
3.37
3.37
4.72
4.09
2.
3.41
7.66
3.41
3.41
4.80
4.90
3.
3.41
7.65
3.41
3.41
4.76
4.90
4.
3.39
7.57
3.38
3.39
4.07
4.17
5.
3.38
7.54
3.38
3.38
4.73
4.92
6.
3.25
6.45
3.25
3.25
4.44
4.64
7.
3.33
7.23
3.33
3.33
4.64
4.85
5.2.5. Kerma relative to air
The energy dependence of kerma relative to air is shown in figure 5.13 for fatty
acids. The energy dependence of Kerma indicates the relative importance of
photoelectric absorption, Compton scattering and pair production. For the energy
range 3 keV- 40 keV, photoelectric absorption is the main interaction process. Then
there is a sharp rise between 40 keV and 200 keV. Kerma is constant from 200 keV
to 3 MeV and then it slowly decreases with increasing energy. The data reported on
kerma in the present work can be utilized for the interpretation of absorbed dose in
the radiation therapy and medical dosimetry.
Effective atomic numbers and electron densities for some fatty acids...
166
1.2
1.1
1.0
Kerma (Ka)
0.9
1
2
3
4
5
6
7
0.8
0.7
0.6
0.5
0.4
10
-3
10
-2
10
-1
10
0
10
1
Energy in MeV
FIG. 5.13: Energy dependence of kerma relative to air. The notation is
as for table 5.1.
5.2.6. Conclusions
1) The effective atomic number, Zeff, and the corresponding effective electron
density, Ne, eff, of some fatty acids have been calculated in the extended energy
region from 1 keV to 100 GeV. The calculations are performed using
WinXCom program [23, 24], based on a modern database of atomic photon
interaction cross-sections [26] and a comprehensive and consistent set of
formulas derived from first principles [Chapter 2, Section 2.1.1]. These
formulas, which can be of great practical utility, are valid for all types of
materials and for all photon energies greater than 1 keV.
Effective atomic numbers and electron densities for some fatty acids...
167
2) One can distinguish three energy regions, in each of which Zeff and Ne, eff are
nearly constant. The three energy regions are approximately E < 0.01 MeV,
0.05 MeV < E < 5 MeV and E > 200 MeV. The main photon interaction
processes in these regions are photoelectric absorption, incoherent (Compton)
scattering and pair production, respectively. Between these energy regions,
there are transition regions with a rapid variation of Zeff and Ne, eff.
3) The maximum values of Zeff and Ne,
eff
are found in the low-energy range,
where photoelectric absorption is the main interaction process.
4) The minimum values of Zeff and Ne,
eff
are found at intermediate energies,
typically 0.05 MeV < E < 5 MeV, where Compton scattering is dominant. In
this case, Zeff is equal to the mean atomic number of the biological molecule
calculated from its chemical formula.
5) The single values of the effective atomic number and the electron density
provided by the XMuDat program are found to be unrelated. XMuDat
calculates the effective atomic number assuming that photoelectric absorption
is the main interaction process. The electron density, on the other hand, has
been calculated assuming that Compton scattering is dominant.
6) The present calculations of Zeff, Ne, eff and kerma have thrown new light on the
underlying radiation physics and will hopefully be useful in medical and
biological applications e.g. for the interpretation of absorbed dose.
Effective atomic numbers and electron densities for some fatty acids...
168
5.3. Analysis and discussion of results on the energy dependence of
the effective atomic numbers for photon energy-absorption and
for photon interaction in some fatty acids
In this section, the results on effective atomic numbers for photon energyabsorption, ZPEA, eff, and for photon interaction, ZPI, eff, are presented for some fatty
acids (table 5.1). Calculations have been carried out in the photon-energy region
from 1 keV to 20 MeV, since photons of energy 51500 keV have found immense
applications in radiation biology especially during diagnostics and therapy [33].
The procedure of calculating ZPEA, eff and ZPI, eff and the ratio ZR, eff is described in
Section 2.1.1 of Chapter 2. The ZPEA, eff values are compared with calculated ZPI, eff
data. The ZPEA,
eff
and ZPI,
eff
are changing with energy and composition of the
biological molecule. The energy dependence of ZPEA,
and ZPI,
eff
eff
is shown
graphically as well as in tabular form and has similar variation. Significant
differences of 2–38% occur between ZPI,
eff
and ZPEA,
eff
in the energy region
3–100 keV. The reasons for these differences and for using ZPEA,
ZPI,
eff
eff
rather than
in calculations of the absorbed dose in radiation therapy and in medical
radiation dosimetry are also discussed.
5.3.1. The effective atomic numbers for photon energy-absorption and
for photon interaction
The energy dependence of the effective atomic numbers, ZPEA, eff, ZPI, eff and the
ratio ZR, eff are shown in figure 5.14 (sample #1-6), and is almost similar for all the
biological molecules studied. The ZPEA,
eff,
ZPI,
eff
and ZR,
eff
values of biological
molecules are given in table 5.4. The energy ranges discussed for the mass
attenuation and mass energy-absorption coefficients are seen also for the effective
Effective atomic numbers and electron densities for some fatty acids...
169
atomic numbers. This is natural, since each elementary attenuation process depends
on the atomic number, Z, in its own way. Thus, for a compound, the effective
atomic number will have different values in different energy regions, depending on
the dominating attenuation process. The effective atomic number is approximately
constant in the intermediate energy region, where Compton scattering is the
dominant attenuation process, because of the linear Z-dependence of incoherent
scattering.
Inspection of table 5.4 shows that the values of ZPEA,
eff
and ZPI,
eff
generally
agree very well at energies below about 3 keV and above about 100 keV. The
percentage difference (P.D) at these energies is mostly less than 1%, which is
insignificant. In the photon energy range 3–100 keV, however, the differences are
considerable and as large as 38%.
The ZPEA, eff, ZPI, eff and the ratio ZR, eff steadily increase up to 810 keV and then
they steadily decrease up to 100200 keV after which they almost remain constant
up to 25 MeV, for all the biological molecules studied. From 5 MeV, the values
increase with increase in energy up to 20 MeV. Significant differences exist
between ZPEA, eff and ZPI, eff in the energy region of 5100 keV for all selected fatty
acids.
Effective atomic numbers and electron densities for some fatty acids...
170
0.9
7.0
Effective atomic number
0.8
5.5
0.7
5.0
4.5
0.6
4.0
0.5
3.5
ZPI,eff
ZPEA,eff
ZR
6.0
Effective atomic number
ZPI,eff
ZPEA,eff
ZR
6.0
Effective atomic number ratio
6.5
Ceroplastic
6.5
0.9
5.5
0.7
5.0
0.6
4.5
4.0
0.5
3.5
0.4
3.0
3.0
0.8
Effective atomic number ratio
Tridecyclic acid
0.4
2.5
2.5
0.3
-3
10
-2
10
-1
10
0
10
1
10
-3
10
-2
Energy in MeV
0.9
6.5
10
0
10
1
0.8
0.7
5.0
4.5
0.6
4.0
0.5
3.5
3.0
0.4
Effective atomic number
5.5
Margaric acid
ZPI,eff
ZPEA,eff
ZR
6.0
Effective atomic number ratio
ZPI,eff
ZPEA,eff
ZR
6.0
Effective atomic number
-1
Energy in MeV
Lacceroic acid
6.5
10
5.5
5.0
0.9
0.8
0.7
4.5
0.6
4.0
0.5
3.5
Effective atomic number ratio
10
3.0
0.4
2.5
2.5
0.3
10
-3
10
-2
10
-1
10
0
10
10
1
-3
10
-2
-1
10
0
10
1
7.5
Pentadecanoic acid
Propanoic acid
0.9
6.0
0.8
5.5
0.7
5.0
4.5
0.6
4.0
0.5
3.5
ZPI,eff
ZPEA,eff
ZR
7.0
Effective atomic number
ZPI,eff
ZPEA,eff
ZR
Effective atomic number ratio
6.5
6.5
1.0
0.9
6.0
0.8
5.5
0.7
5.0
4.5
0.6
Effective atomic number ratio
7.0
Effective atomic number
10
Energy in MeV
Energy in MeV
4.0
3.0
0.5
0.4
3.5
2.5
10
-3
10
-2
10
-1
Energy in MeV
10
0
10
1
10
-3
10
-2
10
-1
10
0
10
1
Energy in MeV
FIG. 5.14: The energy dependence of the effective atomic numbers for photon
energy absorption, ZPEA, eff, for photon interaction, ZPI, eff, and the effective atomic
number ratio, ZR, eff, of some fatty acids.
Effective atomic numbers and electron densities for some fatty acids...
171
TABLE 5.4: Effective atomic numbers for photon energy absorption, ZPEA, eff, for
photon interaction, ZPI, eff, percentage difference, P.D, and effective atomic number
ratio, ZR, eff, of some fatty acids(samples #1-6).
Tridecyclic acid
Energy
(MeV)
ZPEA
ZPI
0.001
0.002
0.003
0.004
0.005
0.006
0.008
0.010
0.015
0.020
0.030
0.040
0.050
0.060
0.080
0.100
0.150
0.200
0.300
0.400
0.500
0.600
0.800
1.000
1.250
1.500
2.000
3.000
4.000
5.000
6.000
8.000
10.000
15.000
20.000
6.47
6.51
6.53
6.54
6.55
6.56
6.56
6.56
6.52
6.37
5.64
4.63
3.88
3.46
3.12
3.01
2.95
2.93
2.93
2.93
2.93
2.93
2.92
2.92
2.92
2.92
2.93
2.95
2.98
3.02
3.05
3.13
3.19
3.35
3.46
6.47
6.51
6.51
6.50
6.47
6.42
6.24
5.97
5.09
4.32
3.50
3.21
3.09
3.03
2.98
2.96
2.94
2.93
2.93
2.93
2.93
2.93
2.93
2.93
2.93
2.93
2.93
2.96
2.98
3.01
3.04
3.10
3.17
3.31
3.44
P.D
0.00
0.08
0.28
0.66
1.27
2.19
4.99
9.06
21.84
32.20
37.81
30.71
20.47
12.45
4.47
1.73
0.18
0.05
0.08
0.08
0.09
0.08
0.09
0.10
0.12
0.16
0.20
0.14
0.01
0.20
0.36
0.67
0.87
1.01
0.83
Ceroplastic acid
ZR
ZPEA
ZPI
0.89
0.90
0.90
0.87
0.87
0.87
0.87
0.87
0.86
0.83
0.74
0.60
0.51
0.46
0.42
0.41
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.41
0.41
0.41
0.42
0.43
0.44
0.46
0.47
6.20
6.22
6.23
6.24
6.24
6.25
6.25
6.24
6.18
6.01
5.22
4.23
3.56
3.20
2.92
2.83
2.78
2.77
2.77
2.77
2.77
2.77
2.77
2.77
2.76
2.76
2.76
2.77
2.79
2.82
2.85
2.88
2.95
3.01
3.15
6.20
6.22
6.21
6.19
6.15
6.09
5.89
5.59
4.69
3.96
3.25
3.00
2.90
2.85
2.81
2.79
2.78
2.77
2.77
2.77
2.77
2.77
2.77
2.77
2.77
2.77
2.77
2.77
2.79
2.82
2.84
2.87
2.93
2.99
3.12
P.D
0.00
0.00
0.32
0.80
1.44
2.56
5.76
10.42
24.11
34.11
37.74
29.08
18.54
10.94
3.77
1.41
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.36
0.36
0.36
0.00
0.00
0.00
0.35
0.35
0.68
0.66
0.95
Lacceroic acid
ZR
ZPEA
ZPI
0.85
0.86
0.86
0.83
0.83
0.83
0.83
0.83
0.82
0.79
0.68
0.55
0.47
0.42
0.39
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.39
0.39
0.40
0.41
0.41
0.43
0.45
6.21
6.24
6.25
6.26
6.26
6.27
6.27
6.26
6.21
6.03
5.25
4.25
3.58
3.22
2.93
2.84
2.79
2.78
2.78
2.78
2.78
2.78
2.78
2.77
2.77
2.77
2.78
2.80
2.83
2.86
2.89
2.96
3.02
3.16
3.27
6.20
6.22
6.21
6.19
6.15
6.09
5.89
5.59
4.69
3.96
3.25
3.00
2.90
2.85
2.81
2.79
2.78
2.77
2.77
2.77
2.77
2.77
2.77
2.77
2.77
2.77
2.77
2.79
2.82
2.84
2.87
2.93
2.99
3.12
3.23
P.D
0.29
0.40
0.65
1.09
1.81
2.90
6.14
10.72
24.36
34.35
38.12
29.44
18.92
11.32
4.15
1.83
0.50
0.38
0.22
0.25
0.26
0.28
0.30
0.17
0.24
0.12
0.22
0.26
0.39
0.59
0.68
0.94
1.12
1.33
1.12
ZR
0.86
0.86
0.86
0.83
0.83
0.83
0.83
0.83
0.82
0.79
0.68
0.55
0.47
0.42
0.39
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.39
0.39
0.40
0.41
0.42
0.43
0.45
Effective atomic numbers and electron densities for some fatty acids...
172
TABLE 5.4: (Continued.)
Margaric acid
Energy
(MeV)
ZPEA
ZPI
0.001
0.002
0.003
0.004
0.005
0.006
0.008
0.010
0.015
0.020
0.030
0.040
0.050
0.060
0.080
0.100
0.150
0.200
0.300
0.400
0.500
0.600
0.800
1.000
1.250
1.500
2.000
3.000
4.000
5.000
6.000
8.000
10.000
15.000
20.000
6.38
6.42
6.43
6.44
6.45
6.46
6.46
6.46
6.41
6.25
5.50
4.49
3.77
3.37
3.05
2.95
2.89
2.88
2.87
2.87
2.87
2.87
2.87
2.86
2.87
2.86
2.87
2.89
2.92
2.96
2.99
3.06
3.13
3.27
3.39
6.38
6.41
6.41
6.40
6.36
6.31
6.12
5.84
4.96
4.19
3.41
3.13
3.02
2.96
2.92
2.90
2.88
2.88
2.87
2.87
2.87
2.87
2.87
2.87
2.87
2.87
2.88
2.90
2.92
2.95
2.98
3.04
3.10
3.24
3.36
P.D
0.00
0.08
0.30
0.70
1.34
2.34
5.29
9.53
22.67
32.92
37.93
30.20
19.85
11.98
4.27
1.70
0.20
0.05
0.11
0.09
0.08
0.06
0.03
0.17
0.09
0.21
0.13
0.07
0.05
0.26
0.35
0.61
0.79
1.00
0.78
Pentodecanoic acid
ZR
ZPEA
ZPI
0.88
0.89
0.89
0.86
0.86
0.86
0.86
0.86
0.85
0.82
0.72
0.59
0.50
0.45
0.41
0.40
0.39
0.39
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.41
0.42
0.43
0.45
0.46
6.42
6.46
6.48
6.49
6.50
6.50
6.51
6.51
6.46
6.30
5.56
4.55
3.82
3.41
3.08
2.98
2.91
2.90
2.89
2.89
2.89
2.89
2.89
2.89
2.89
2.89
2.90
2.92
2.95
2.98
3.02
3.09
3.15
3.31
3.42
6.42
6.45
6.46
6.44
6.41
6.36
6.17
5.90
5.02
4.25
3.45
3.17
3.05
2.99
2.94
2.92
2.91
2.90
2.90
2.90
2.90
2.89
2.89
2.89
2.89
2.90
2.90
2.92
2.95
2.98
3.01
3.07
3.13
3.27
3.39
P.D
0.00
0.08
0.30
0.68
1.31
2.28
5.17
9.32
22.31
32.61
37.90
30.42
20.13
12.22
4.38
1.75
0.22
0.05
0.11
0.09
0.08
0.06
0.03
0.17
0.09
0.21
0.13
0.07
0.05
0.26
0.36
0.61
0.79
1.01
0.78
Propanoic acid
ZR
ZPEA
ZPI
0.88
0.89
0.89
0.86
0.86
0.86
0.86
0.86
0.85
0.82
0.73
0.59
0.50
0.45
0.41
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.41
0.41
0.42
0.43
0.43
0.45
0.47
7.15
7.20
7.23
7.24
7.25
7.26
7.27
7.27
7.25
7.16
6.68
5.85
5.04
4.48
3.96
3.79
3.67
3.65
3.64
3.64
3.64
3.64
3.64
3.63
3.63
3.63
3.64
3.67
3.71
3.76
3.80
3.89
3.98
4.17
4.31
7.15
7.20
7.22
7.21
7.20
7.17
7.06
6.89
6.23
5.49
4.51
4.08
3.89
3.80
3.72
3.69
3.66
3.65
3.64
3.64
3.64
3.64
3.64
3.64
3.64
3.64
3.65
3.67
3.71
3.75
3.79
3.87
3.95
4.13
4.28
P.D
0.01
0.04
0.16
0.37
0.70
1.24
2.83
5.23
14.10
23.34
32.56
30.18
22.70
15.26
6.16
2.64
0.41
0.07
0.10
0.12
0.10
0.08
0.05
0.18
0.11
0.22
0.18
0.07
0.03
0.30
0.38
0.62
0.81
1.06
0.80
ZR
0.99
0.99
1.00
0.96
0.96
0.96
0.96
0.96
0.96
0.94
0.87
0.76
0.66
0.59
0.53
0.51
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.51
0.51
0.52
0.54
0.55
0.57
0.59
Effective atomic numbers and electron densities for some fatty acids...
173
A maximum difference of 37.81%, 37.74%, 38.12%, 37.93%, 37.90%, 32.56%
and 35.72% is observed at 30 keV for Tridecyclic, ceroplastic, Lacceroic, Margaric,
Pentodecanoic, Propanoic and Heptonoic respectively. There is a shift in the energy
position at which maximum values of ZPEA, eff and ZPI, eff occur, this can be seen in
figure 5.14 and table 5.4.
The discussion can be exemplified by the case of Margaric acid (figure 5.14).
At low energies, where photoelectric absorption is dominating, the effective atomic
number is about 6.41 and weakly increasing with increasing energy. Above 100
keV, the value of Zeff has dropped to a constant value of about 2.87 typical for
Compton scattering (table 5.1 and 5.2). Finally, Zeff again increases with increasing
energy in > 5 MeV range, where the absorption is mainly due to pair production.
The ZPI, eff should be used in diagnostics, where one is generally looking at the
attenuation of the X-ray beam by imaging techniques; including X-ray tomography
(CAT scans). On the other hand, ZPEA, eff should be used in radiation dosimetry and
radiotherapy by means of X-ray beams, where the energy deposition of photons is
important. Radiotherapy, however, is usually performed with photons in the MeV
energy range, where the difference between ZPEA, eff and ZPI, eff is insignificant as
discussed, especially for low- and medium-Z materials e.g. human organs/tissues.
Thus, it follows from the present work that in radiotherapy one can very well use
ZPI, eff instead of ZPEA, eff. This has the advantage that ZPI, eff can be easily measured
experimentally for different tissues. The use of ZPEA, eff is important, however, when
dealing with the absorbed dose due to photons in the energy range 3–100 keV.
Effective atomic numbers and electron densities for some fatty acids...
174
The large percentage differences in the 3–100 keV range have a simple physical
explanation. A major part of the Compton scattered radiation escapes the absorbing
medium. Thus, while contributing significantly to the attenuation of the incident
beam, Compton scattering contributes only little to the energy-absorption.
Therefore, the transition from photoelectric absorption to Compton scattering as the
dominating absorption process is shifted towards higher energies for the mass
energy-absorption coefficient as compared with the mass attenuation coefficient.
These differences are mirrored in the effective atomic number as seen in figure
5.14. Upon increasing energy, the value of ZPI, eff begins to drop from the high value
of about 6.41, typical for photoelectric absorption. The value of ZPEA, eff on the other
hand remains at about 4.96 up to about 15 keV before dropping to the lower value
of about 2.95 typical for Compton scattering. This energy behavior of ZPEA, eff and
ZPI, eff explains the large percentage differences observed in the energy range 3–100
keV.
5.3.2. Conclusions
1) The effective atomic numbers for photon energy-absorption, ZPEA,
eff,
for
photon interaction, ZPI, eff, and their ratio, ZR, eff have been calculated based on
the use of tabulated data for the mass energy-absorption and mass attenuation
coefficient. The method has been applied to some selected fatty acids (table
5.1) for the photon-energy range from 1 keV to 20 MeV.
2) The energy dependence of ZPEA,
eff
and ZPI,
eff
reflects the dominating
absorption processes, which are changing from photoelectric absorption in the
Effective atomic numbers and electron densities for some fatty acids...
175
low-energy range (< 10 keV), to Compton scattering in the medium energy
range, and pair production in the high energy (> 5 MeV) range.
3) Significant differences up to 38% between ZPEA,
eff
and ZPI,
eff
occur in the
3–100 keV range. The reason for these differences is that the transition from
photoelectric absorption to Compton scattering as the dominating absorption
process is shifted to higher energy for the mass energy-absorption coefficient
as compared with the mass attenuation coefficient.
4) It has been shown that the difference between ZPEA, eff and ZPI, eff for biological
molecules is insignificant (less than 1%) at photon energies below about 3 keV
and above about 100 keV. Therefore, ZPI, eff can be used instead of ZPEA, eff in
radiotherapy, where photons in the MeV range are used. The use of ZPEA, eff is
important, however, when dealing with the absorbed dose due to photons in
the 3–100 keV energy range.
5) The data on ZPEA, eff and ZPI, eff reported here for some biological molecules,
should be useful in radiation therapy and in medical radiation dosimetry.
Effective atomic numbers and electron densities for some fatty acids...
176
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