chapter 5 effective atomic numbers and electron densities for...

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


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.


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 [417]. The corresponding studies on effective atomic numbers for

photon energy absorption, ZPEA, eff, appears to be limited [1719]. In literature, the

available experimental data on Zeff for fatty acids [2022] 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 keV100 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


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 cross-

sections 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 5.15.12 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


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


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 4 5Z 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.


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.

10-3

10-2

10-1

100

101

102

103

104

105

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5

8.0

Total photon interaction (Coherent)

ZP

I, e

ff

Energy in MeV

1

2

3

4

5

6

7

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.


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


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).

10-3

10-2

10-1

100

101

102

103

104

105

6.1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

7.0

7.1

7.2

7.3

7.4

Photoelectric

ZP

I, e

ff

Energy in MeV

1

2

3

4

5

6

7

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.


156

10-3

10-2

10-1

100

101

102

103

104

105

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

Incoherent

ZP

I, e

ff

Energy in MeV

1

2

3

4

5

6

7

FIG. 5.3: Energy dependence of Zeff of fatty acids for incoherent scattering.


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 100662 keV for some alloys.


157

10-3

10-2

10-1

100

101

102

103

104

105

4.8

5.0

5.2

5.4

5.6

5.8

6.0

6.2

6.4

6.6

6.8

Coherent

ZP

I, e

ff

Energy in MeV

1

2

3

4

5

6

7

FIG. 5.4: Energy dependence of Zeff of fatty acids for coherent scattering.


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 2Z

dependent. It should be noted that, the threshold energy for pair production


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.

100

101

102

103

104

105

4.6

4.8

5.0

5.2

5.4

5.6

5.8

6.0 Pair production (nuclear)

ZP

I, e

ff

Energy in MeV

1

2

3

4

5

6

7

FIG. 5.5: Energy dependence of Zeff of fatty acids for pair production in the nuclear

field. Notations as for table 5.1


159

101

102

103

104

105

2.4

2.6

2.8

3.0

3.2

3.4

3.6Pair production (electric)

ZP

I, e

ff

Energy in MeV

1

2

3

4

5

6

7

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.


160

10-3

10-2

10-1

100

101

102

103

104

105

2.5x1023

3.0x1023

3.5x1023

4.0x1023

4.5x1023

5.0x1023

5.5x1023

6.0x1023

6.5x1023

7.0x1023

7.5x1023

8.0x1023

Total photon interaction (Coherent)

Ele

ctro

n D

ensi

ty(N

e,ef

f)

Energy in MeV

1

2

3

4

5

6

7

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

10-3

10-2

10-1

100

101

102

103

104

105

6.4x1023

6.6x1023

6.8x1023

7.0x1023

7.2x1023

7.4x1023

7.6x1023

7.8x1023

Photoelectric

Ele

ctro

n D

ensi

ty(N

e,ef

f)

Energy in MeV

1

2

3

4

5

6

7

FIG. 5.8: Energy dependence of Ne, eff of fatty acids for photoelectric absorption.



161

10-3

10-2

10-1

100

101

102

103

104

105

5.8x1023

6.0x1023

6.2x1023

6.4x1023

6.6x1023

6.8x1023

7.0x1023

7.2x1023

7.4x1023

7.6x1023

Coherent

Ele

ctro

n D

ensi

ty(N

e,ef

f)

Energy in MeV

1

2

3

4

5

6

7

10-3

10-2

10-1

100

101

102

103

104

105

2.2x1023

2.4x1023

2.6x1023

2.8x1023

3.0x1023

3.2x1023

3.4x1023

3.6x1023

3.8x1023

4.0x1023

IncoherentE

lect

ron

Den

sity

(Ne,

eff)

Energy in MeV

1

2

3

4

5

6

7

FIG. 5.9: Energy dependence of Ne, eff of fatty acids for incoherent scattering.


FIG. 5.10: Energy dependence of Ne, eff of fatty acids for coherent scattering.



162

101

102

103

104

105

3.0x1023

3.0x1023

3.1x1023

3.1x1023

3.2x1023

3.2x1023

3.3x1023

3.3x1023

3.4x1023

3.4x1023

3.5x1023

Pair production (electric)

Ele

ctro

n D

ensi

ty(N

e,ef

f)

Energy in MeV

1

2

3

4

5

6

7

100

101

102

103

104

105

5.2x1023

5.4x1023

5.6x1023

5.8x1023

6.0x1023

6.2x1023

Pair production (nuclear)E

lect

ron

Den

sity

(N

e,ef

f)

Energy in MeV

1

2

3

4

5

6

7

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

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


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


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 cross-

sections. 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

S.N (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


165

TABLE 5.3: Statistics of Ne, eff data (in units of 1023

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.


166

10-3

10-2

10-1

100

101

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

Kerm

a (

Ka)

Energy in MeV

1

2

3

4

5

6

7

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.

FIG. 5.13: Energy dependence of kerma relative to air. The notation is

as for table 5.1.


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.


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 energy-

absorption, 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 51500 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, eff and ZPI, 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, eff rather than

ZPI, eff 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


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 810 keV and then

they steadily decrease up to 100200 keV after which they almost remain constant

up to 25 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 5100 keV for all selected fatty

acids.


170

10-3

10-2

10-1

100

101

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

0.4

0.5

0.6

0.7

0.8

0.9

Eff

ecti

ve

ato

mic

nu

mb

er r

ati

o

ZR

Pentadecanoic acid

Eff

ecti

ve

ato

mic

nu

mb

er

Energy in MeV

ZPI,eff

ZPEA,eff

10-3

10-2

10-1

100

101

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5

0.5

0.6

0.7

0.8

0.9

1.0

Eff

ecti

ve

ato

mic

nu

mb

er r

ati

o

ZR

Propanoic acid

Eff

ecti

ve

ato

mic

nu

mb

er

Energy in MeV

ZPI,eff

ZPEA,eff

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.

10-3

10-2

10-1

100

101

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

0.4

0.5

0.6

0.7

0.8

0.9

Eff

ecti

ve

ato

mic

nu

mb

er r

ati

o

ZR

Tridecyclic acid

Eff

ecti

ve

ato

mic

nu

mb

er

Energy in MeV

ZPI,eff

ZPEA,eff

10-3

10-2

10-1

100

101

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

Eff

ecti

ve

ato

mic

nu

mb

er r

ati

o

0.3

0.4

0.5

0.6

0.7

0.8

0.9

ZR

Lacceroic acid

Eff

ecti

ve

ato

mic

nu

mb

er

Energy in MeV

ZPI,eff

ZPEA,eff

10-3

10-2

10-1

100

101

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

Eff

ecti

ve

ato

mic

nu

mb

er r

ati

o

0.3

0.4

0.5

0.6

0.7

0.8

0.9

ZR

Ceroplastic

Eff

ecti

ve

ato

mic

nu

mb

er

Energy in MeV

ZPI,eff

ZPEA,eff

10-3

10-2

10-1

100

101

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

0.4

0.5

0.6

0.7

0.8

0.9

Eff

ecti

ve

ato

mic

nu

mb

er r

ati

o

ZR

Margaric acidE

ffec

tiv

e a

tom

ic n

um

ber

Energy in MeV

ZPI,eff

ZPEA,eff


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).

Energy

(MeV)

Tridecyclic acid Ceroplastic acid Lacceroic acid

ZPEA ZPI P.D ZR ZPEA ZPI P.D ZR ZPEA ZPI P.D ZR

0.001 6.47 6.47 0.00 0.89 6.20 6.20 0.00 0.85 6.21 6.20 0.29 0.86

0.002 6.51 6.51 0.08 0.90 6.22 6.22 0.00 0.86 6.24 6.22 0.40 0.86

0.003 6.53 6.51 0.28 0.90 6.23 6.21 0.32 0.86 6.25 6.21 0.65 0.86

0.004 6.54 6.50 0.66 0.87 6.24 6.19 0.80 0.83 6.26 6.19 1.09 0.83

0.005 6.55 6.47 1.27 0.87 6.24 6.15 1.44 0.83 6.26 6.15 1.81 0.83

0.006 6.56 6.42 2.19 0.87 6.25 6.09 2.56 0.83 6.27 6.09 2.90 0.83

0.008 6.56 6.24 4.99 0.87 6.25 5.89 5.76 0.83 6.27 5.89 6.14 0.83

0.010 6.56 5.97 9.06 0.87 6.24 5.59 10.42 0.83 6.26 5.59 10.72 0.83

0.015 6.52 5.09 21.84 0.86 6.18 4.69 24.11 0.82 6.21 4.69 24.36 0.82

0.020 6.37 4.32 32.20 0.83 6.01 3.96 34.11 0.79 6.03 3.96 34.35 0.79

0.030 5.64 3.50 37.81 0.74 5.22 3.25 37.74 0.68 5.25 3.25 38.12 0.68

0.040 4.63 3.21 30.71 0.60 4.23 3.00 29.08 0.55 4.25 3.00 29.44 0.55

0.050 3.88 3.09 20.47 0.51 3.56 2.90 18.54 0.47 3.58 2.90 18.92 0.47

0.060 3.46 3.03 12.45 0.46 3.20 2.85 10.94 0.42 3.22 2.85 11.32 0.42

0.080 3.12 2.98 4.47 0.42 2.92 2.81 3.77 0.39 2.93 2.81 4.15 0.39

0.100 3.01 2.96 1.73 0.41 2.83 2.79 1.41 0.38 2.84 2.79 1.83 0.38

0.150 2.95 2.94 0.18 0.40 2.78 2.78 0.00 0.38 2.79 2.78 0.50 0.38

0.200 2.93 2.93 0.05 0.40 2.77 2.77 0.00 0.38 2.78 2.77 0.38 0.38

0.300 2.93 2.93 0.08 0.40 2.77 2.77 0.00 0.38 2.78 2.77 0.22 0.38

0.400 2.93 2.93 0.08 0.40 2.77 2.77 0.00 0.38 2.78 2.77 0.25 0.38

0.500 2.93 2.93 0.09 0.40 2.77 2.77 0.00 0.38 2.78 2.77 0.26 0.38

0.600 2.93 2.93 0.08 0.40 2.77 2.77 0.00 0.38 2.78 2.77 0.28 0.38

0.800 2.92 2.93 0.09 0.40 2.77 2.77 0.00 0.38 2.78 2.77 0.30 0.38

1.000 2.92 2.93 0.10 0.40 2.77 2.77 0.00 0.38 2.77 2.77 0.17 0.38

1.250 2.92 2.93 0.12 0.40 2.76 2.77 0.36 0.38 2.77 2.77 0.24 0.38

1.500 2.92 2.93 0.16 0.40 2.76 2.77 0.36 0.38 2.77 2.77 0.12 0.38

2.000 2.93 2.93 0.20 0.40 2.76 2.77 0.36 0.38 2.78 2.77 0.22 0.38

3.000 2.95 2.96 0.14 0.41 2.77 2.77 0.00 0.38 2.80 2.79 0.26 0.38

4.000 2.98 2.98 0.01 0.41 2.79 2.79 0.00 0.39 2.83 2.82 0.39 0.39

5.000 3.02 3.01 0.20 0.41 2.82 2.82 0.00 0.39 2.86 2.84 0.59 0.39

6.000 3.05 3.04 0.36 0.42 2.85 2.84 0.35 0.40 2.89 2.87 0.68 0.40

8.000 3.13 3.10 0.67 0.43 2.88 2.87 0.35 0.41 2.96 2.93 0.94 0.41

10.000 3.19 3.17 0.87 0.44 2.95 2.93 0.68 0.41 3.02 2.99 1.12 0.42

15.000 3.35 3.31 1.01 0.46 3.01 2.99 0.66 0.43 3.16 3.12 1.33 0.43

20.000 3.46 3.44 0.83 0.47 3.15 3.12 0.95 0.45 3.27 3.23 1.12 0.45


172

TABLE 5.4: (Continued.)

Energy

(MeV)

Margaric acid Pentodecanoic acid Propanoic acid

ZPEA ZPI P.D ZR ZPEA ZPI P.D ZR ZPEA ZPI P.D ZR

0.001 6.38 6.38 0.00 0.88 6.42 6.42 0.00 0.88 7.15 7.15 0.01 0.99

0.002 6.42 6.41 0.08 0.89 6.46 6.45 0.08 0.89 7.20 7.20 0.04 0.99

0.003 6.43 6.41 0.30 0.89 6.48 6.46 0.30 0.89 7.23 7.22 0.16 1.00

0.004 6.44 6.40 0.70 0.86 6.49 6.44 0.68 0.86 7.24 7.21 0.37 0.96

0.005 6.45 6.36 1.34 0.86 6.50 6.41 1.31 0.86 7.25 7.20 0.70 0.96

0.006 6.46 6.31 2.34 0.86 6.50 6.36 2.28 0.86 7.26 7.17 1.24 0.96

0.008 6.46 6.12 5.29 0.86 6.51 6.17 5.17 0.86 7.27 7.06 2.83 0.96

0.010 6.46 5.84 9.53 0.86 6.51 5.90 9.32 0.86 7.27 6.89 5.23 0.96

0.015 6.41 4.96 22.67 0.85 6.46 5.02 22.31 0.85 7.25 6.23 14.10 0.96

0.020 6.25 4.19 32.92 0.82 6.30 4.25 32.61 0.82 7.16 5.49 23.34 0.94

0.030 5.50 3.41 37.93 0.72 5.56 3.45 37.90 0.73 6.68 4.51 32.56 0.87

0.040 4.49 3.13 30.20 0.59 4.55 3.17 30.42 0.59 5.85 4.08 30.18 0.76

0.050 3.77 3.02 19.85 0.50 3.82 3.05 20.13 0.50 5.04 3.89 22.70 0.66

0.060 3.37 2.96 11.98 0.45 3.41 2.99 12.22 0.45 4.48 3.80 15.26 0.59

0.080 3.05 2.92 4.27 0.41 3.08 2.94 4.38 0.41 3.96 3.72 6.16 0.53

0.100 2.95 2.90 1.70 0.40 2.98 2.92 1.75 0.40 3.79 3.69 2.64 0.51

0.150 2.89 2.88 0.20 0.39 2.91 2.91 0.22 0.40 3.67 3.66 0.41 0.50

0.200 2.88 2.88 0.05 0.39 2.90 2.90 0.05 0.40 3.65 3.65 0.07 0.50

0.300 2.87 2.87 0.11 0.40 2.89 2.90 0.11 0.40 3.64 3.64 0.10 0.50

0.400 2.87 2.87 0.09 0.40 2.89 2.90 0.09 0.40 3.64 3.64 0.12 0.50

0.500 2.87 2.87 0.08 0.40 2.89 2.90 0.08 0.40 3.64 3.64 0.10 0.50

0.600 2.87 2.87 0.06 0.40 2.89 2.89 0.06 0.40 3.64 3.64 0.08 0.50

0.800 2.87 2.87 0.03 0.40 2.89 2.89 0.03 0.40 3.64 3.64 0.05 0.50

1.000 2.86 2.87 0.17 0.40 2.89 2.89 0.17 0.40 3.63 3.64 0.18 0.50

1.250 2.87 2.87 0.09 0.40 2.89 2.89 0.09 0.40 3.63 3.64 0.11 0.50

1.500 2.86 2.87 0.21 0.40 2.89 2.90 0.21 0.40 3.63 3.64 0.22 0.50

2.000 2.87 2.88 0.13 0.40 2.90 2.90 0.13 0.40 3.64 3.65 0.18 0.50

3.000 2.89 2.90 0.07 0.40 2.92 2.92 0.07 0.40 3.67 3.67 0.07 0.50

4.000 2.92 2.92 0.05 0.40 2.95 2.95 0.05 0.41 3.71 3.71 0.03 0.51

5.000 2.96 2.95 0.26 0.40 2.98 2.98 0.26 0.41 3.76 3.75 0.30 0.51

6.000 2.99 2.98 0.35 0.41 3.02 3.01 0.36 0.42 3.80 3.79 0.38 0.52

8.000 3.06 3.04 0.61 0.42 3.09 3.07 0.61 0.43 3.89 3.87 0.62 0.54

10.000 3.13 3.10 0.79 0.43 3.15 3.13 0.79 0.43 3.98 3.95 0.81 0.55

15.000 3.27 3.24 1.00 0.45 3.31 3.27 1.01 0.45 4.17 4.13 1.06 0.57

20.000 3.39 3.36 0.78 0.46 3.42 3.39 0.78 0.47 4.31 4.28 0.80 0.59


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.


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


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.


176

References

[1] D. Voet, J. G. Voet, Biochemistry, 2nd

edition, John Wiley and Sons, London, 1995.

[2] Website: http://www.umm.edu/altmed/ConsSupplements/Omega3FattyAcidscs.html and

http://goodfats.pamrotella.com/

[3] D. F. Jackson, D. J. Hawkes, Phys. Rep., 70 (1981) 169233.

[4] F. W. Spiers, Brit. J. Radiol., 19 (1946) 5263.

[5] J. Weber, D. J. V. D. Berge, Brit. J. Radiol., 42 (1969) 378383.

[6] C. A. Jayachandran, Phys. Med. Biol., 16 (1971) 617623.

[7] D. R. White, Phys. Med. Biol., 22 (1977) 219228.

[8] B. V. T. Rao, M. L. N. Raju, B. M. Rao, K. L. Narasimham, K. Parthasaradhi, Med. Phys., 12

(1985) 745748.

[9] A. Perumallu, A. S. N. Rao, G. K. Rao, Physica C, 132 (1985) 388394.

[10] N. C. Yang, P. K. Leichner, W. G. Hawkins, Med. Phys., 14 (1987) 759766.

[11] K. Parthasaradhi, B. M. Rao, S. G. Prasad, Med. Phys., 16 (1989) 653654.

[12] H. Özyol, Radiat. Phys. Chem., 44 (1994) 573577.

[13] S. G. Prasad, K. Parthasaradhi, W. D. Bloomer, Med. Phys., 24 (1997) 883885.

[14] T. K. Kumar, K. V. Reddy, Radiat. Phys. Chem., 50 (1997) 545553.

[15] C. R. Murthy, A. S. N. Rao, G. K. Rao, Appl. Radiat. Isot., 51 (1999) 335339.

[16] N. Koç, H. Özyol, Radiat. Phys. Chem., 59 (2000) 339345.

[17] Shivaramu, Med. Dosi., 27 (2002) 1–9.

[18] Shivaramu, V. Ramprasath, Nucl. Instrum. Methods B, 168 (2000) 294304.

[19] Shivaramu, R. Vijayakumar, L. Rajasekaran, N. Ramamurthy, Radiat. Phys. Chem., 62 (2001)

371377.

[20] V. Manjunathaguru, T. K. Umesh, J. Phys. B: At. Mol. Opt. Phys., 39 (2006) 3969–3981.

[21] G. S. Bhandal, I. Ahmed, K. Singh, Appl. Radiat. Isot., 43 (1992) 11851188.

[22] G. K. Sandhu, K. Singh, B. S. Lark, L. Gerward, Radiat. Phys. Chem., 65 (2002) 211215.

[23] L. Gerward, N. Guilbert, K. B. Jensen, H. Levring, Radiat. Phys. Chem., 60 (2001) 23–24.

[24] L. Gerward, N. Guilbert, K. B. Jensen, H. Levring, Radiat. Phys. Chem., 71 (2004) 653–654.

[25] R. Nowotny, XMuDat: Photon attenuation data on PC, International Atomic Energy Agency,

Vienna, 1998.

[26] J. H. Hubbell, S. M. Seltzer, Tables of X-ray mass attenuation coefficients and mass energy-

absorption coefficients 1 keV-20 MeV for elements Z = 1 to 92 and 48 additional substances

of dosimetric interest, NISTIR 5632, 1995.

[27] G. J. Hine, Phys. Rev., 85 (1952) 725.

[28] K. S. R. Sastry, S. Jnanananda, J. Sci. Ind. Res. B, 17 (1958) 389–343.

[29] S. C. Lingam, K. S. Basu, D. V. K. Reddy, Ind. J. Phys. A, 58 (1984) 285287.


177

[30] K. Parthasararadhi, Ind. J. Pure Appl. Phys., 6 (1968) 609613.

[31] A. Perumallu, A. S. N. Rao, G. K. Rao, Physica C, 132 (1985) 388394.

[32] G. S. Bhandal, K. Singh, Appl. Radiat. Isot., 44 (1993) 505510.

[33] J. H. Hubbell, Phys. Med. Biol., 44 (1999) R1R22.

chapter 5 effective atomic numbers and electron densities for...

Documents