Shani G. Radiation Dosimetry: Instrumentation and Methods

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130 Radiation Dosimetry: Instrumentation and Methods

The radiation exposure from an x-ray tube can be

measured by an ion chamber as

(3.81)

(3.82)

where

X is exposure measured in C/kg, M is meter reading

in C, N

is the calibration factor (R/C) of an ion chamber

for standard beam energy from a national or accredited

dosimetry calibration laboratory (ADCL), K is a compos-

ite correction factor, k

is the energy correction factor, k

is the correction for the air density due to the temperature

and pressure, k

pol

is correction for the polarity, and P

ion

the ion recombination correction factor. An ADCL pro-

vides values of N

for each user’s beam quality condition

for an ion chamber and electrometer combination. How-

ever, the dose rate might be a variable and the ADCL may

not be able to match the user’s exposure condition. Even

though for routine use, a single factor might be useful,

scientiﬁcally, separation of each component of the cali-

bration factor individually as shown in Equation (3.82) is

desired. In such situations, an ADCL would prefer to

provide individual factors for each beam condition. Ide-

ally, ionization chambers should be independent of an

electrometer such that chamber factors can be provided

by an ADCL very similar to the factors for chambers used

in radiation therapy applications. If a chamber is cali-

brated by an ADCL with a polarity that is used by a

user,

pol

is 1.00 and further correction is not needed.

However, if calibration and measurement polarities are

different, k

pol

is needed for accurate exposure measure-

ment.

For an x-ray machine, the exposure is dependent upon

the applied voltage, V; beam current, mA (milliampere);

time, s (seconds); and distance, d. A rule of thumb expres-

sion for exposure can be given as: [45]

(3.83)

where k and n are constants for a machine. The typical

values of k and n are 1.5 and 2.5, respectively, in the

kilovoltage range. Hence, the polarity and ion recombina-

tion effects in general may depend on these parameters.

For the continuous radiation beam, the ion-recombi-

nation correction P

ion

can be calculated by the half-voltage

method as suggested by Attix. If Q

and Q

are the charges

collected at the chamber potential of V and V/2, respec-

tively, P

ion

is calculated as

(3.84)

The polarity effect depends strongly on the medium

between the plates of chamber and is the highest for air

( 30%) and the lowest ( 0.3%) for dielectric medium.

If the charges collected at negative and positive potentials

are denoted by Q



and Q



, respectively, then k

pol

is deﬁned

as the absolute value of the ratio.

(3.85)

The actual measured charge Q for the measurement of

exposure is the absolute value of mean of the two charges:

(3.86)

For the measurements, if only a single polarity is used,

the error in

Q will be

(3.87)

(3.88)

The P

ion

and k

pol

were measured with various exposure

factors kVp, mA, s, and d for all chambers in the diag-

nostic energy range by changing one parameter at a time

while keeping the others constant.

Figure 3.45 shows the variation of the P

ion

vs. x-ray

tube voltage in mammography range. Data are shown for

various ion chambers at a focus-to-chamber distance

(FCD) of 25 cm and 50 cm in Figure 3.45a and b, respec-

tively. The value of P

ion

is relatively independent of the

tube voltage for small-volume (150 cm

) ion chambers

within 1%; however, for the large-volume ion chambers,

the magnitude of P

ion

is much higher and rises steadily

with the tube potential. In general, P

ion

is greater for the

higher exposures. The slope of curve for the 180-cm

chamber was anomalously higher. Independent repeated

measurements with this chamber showed the same results

within 1.5%.

The effect of beam current for a given exposure (ﬁxed

kVp and mAs) is shown in Figures 3.46a and b for the

ion

and k

pol

, respectively, for the 125-kVp and 0.1-s expo-

sure typically used in diagnostic radiology. The value of

ion

rises linearly with beam current for most chambers,

but the effect is more pronounced for the large-volume

ion chambers. For small-volume chambers, the P

ion

is con-

stant within 1.0%. It suggests that P

ion

is signiﬁcant only

at high-exposure rate measurements with large chambers.

The polarity effect, on the other hand, is constant over a

wide range of beam current. The magnitude of the k

pol

Exposure

X() MN

K

pol

ion



R()exp

mAs

--------------------



ion

---

------







1



pol





-------









-----------------------



%error

pol

1

pol

1

------------------





100 for Q





%error

1 k

pol



1 k

pol



------------------





100 for Q





Ch-03.fm(part 1) Page 130 Friday, November 10, 2000 11:58 AM

Ionization Chamber Dosimetry 131

nearly constant within 4% with respect to the beam

current for all chambers, except for the 600-cm

ion cham-

ber, which has the value of 1.16. This indicates that k

pol

is independent of the exposure conditions and depends

only on the chamber characteristics.

ion

is higher for higher kilovoltage and rises with expo-

sure time for a 600-cm

chamber. In contrast, the 180-cm

chamber shows the reverse effect. The polarity effect has

a similar trend, as noted for the P

ion

, except that the slope

is higher for higher exposure time.

For low exposure and scatter radiation measurements

such as those used in diagnostic radiology, the large-

volume ion chambers are used for a better signal; however,

when large-volume chambers are used, the ion recombi-

nation and polarity effect become signiﬁcant and must be

counted for accurate exposure measurements. In general,

FIGURE 3.45 Ion recombination correction, P

ion

, vs. tube voltage (kVp) for different ion chamber in mammography range at a focal

to chamber distance (FCD) of 25 cm (a) and 50 cm (b). (From Reference [45]. With permission.)

Ch-03.fm(part 1) Page 131 Friday, November 10, 2000 11:58 AM

132 Radiation Dosimetry: Instrumentation and Methods

the k

pol

depends on the area and thickness of the electrodes,

window thickness, the potential across the electrodes, and

the depth of chamber in phantom. It is shown that k

pol

signiﬁcant for air exposure measurements with large vol-

ume chambers.

Volume recombination parameter in ionization cham-

bers was investigated by Boutillon. [46] The parameter

governing the volume recombination in ionization

chambers was measured under conditions which allow

strict application of the basic theory. The method consists

of measuring the ratio of ionization currents and

obtained at two given voltages V

and V

as a function of

. The value of m

is derived from a linear extrapolation

to zero current. Several pairs of voltages (V

, V

) were

used. The value of m

obtained in that work is 3.97 

s with a relative uncertainty of 1.7%.

FIGURE 3.46 Effect of beam current on P

ion

(a) and k

pol

(b) for a 125-kVp beam for various ion chambers and a ﬁxed exposure

time of 0.1 s. (From Reference [45]. With permission.)

1

Ch-03.fm(part 1) Page 132 Friday, November 10, 2000 11:58 AM

Ionization Chamber Dosimetry 133

The dependence of m

on atmospheric conditions has also

been investigated.

The various processes by which ions recombine in an

ionization chamber have been reviewed by Boag. [42]

Near saturation, the ratio between the saturation current

and the ionization current I

, measured at voltage V, can

be expressed, neglecting terms of higher order, by the

basic equation

(3.89)

with

where A is a constant depending on the chamber type,



is the recombination coefﬁcient under continuous irradi-

ation, e is the electron charge, k



and k



are the mobilities

of the positive and negative ions, and g is a factor depend-

ing on the chamber geometry. The ﬁrst variable term on

the right-hand side of Equation (3.89) describes the initial

recombination and diffusion. The second term describes

the volume recombination.

The effect of the free electrons on the recombination

can be neglected for the low air-kerma rates.

The geometric factor g is given by Boag [47] for a

parallel-plate free-air chamber as

(3.90)

where L is the plate separation and D is the length of the

collecting plate in the direction of the photon beam. This

relation is based on the assumption of a sharp Gaussian

distribution of the ionization around the beam axis.

Two parallel-plate free-air chambers were used by

Boutillon for the determination of m

, one of them in a

low-energy x-ray beam and the other in a medium-energy

x-ray beam.

Figure 3.47 shows the data for chamber B. The exper-

imental values for the ratio I

I

V/n

have been corrected for

space charge according to Boag [42] and for variations in

the atmospheric conditions. The slope of each curve,

divided by (n

 l)g gives an estimate of m

. The cor-

rection applied for space charge amounts to about 0.01%

for an ion-recombination loss of 1%.

Ion recombination corrections for the NACP parallel-

plate chamber in a pulsed electron beam was studied by

Burns and McEwen. [48] The NACP electron chamber is

one of three parallel-plate chambers recommended for use

in the UK. Measurements with this chamber type have

indicated a problem in determining the recombination cor-

rection. This is due to a variation of the ionization current

I with polarizing voltage V, which deviates from the

accepted Boag theory. It is shown that there is a chamber-

dependent threshold voltage below which the NACP

chamber follows the Boag theory. Above this voltage the

chamber should be used with caution, although it is still

possible to correct for the dependence of the chamber

response on the dose per pulse. The existence of such

deviations from theory demonstrates the usefulness of the

1/I against 1/V plot and the limitations of the Boag two-

voltage analysis.

Since the component of initial recombination k

init

independent of the dose per pulse, k

ion

will tend to k

init

the dose per pulse, and hence u, tends to zero. Thus, the

practical measurement is better represented by

(3.91)

FIGURE 3.47 Ratio of the ionization currents I

and I

V/n

, measured in chamber B with voltages V and V/n, respectively, as a function

, for several values of n (100–250 kV). (From Reference [46]. With permission.)

I

1 AVm

gV

()I





ek







2



D

ion

init

---



Ch-03.fm(part 1) Page 133 Friday, November 10, 2000 11:58 AM

134 Radiation Dosimetry: Instrumentation and Methods

where the value of k

ion

) at the normal user polarizing

voltage V

(3.92)

It follows that if

ion

) is measured using the 1/I

against 1/V method at each of a series of doses per pulse

air,p

, then a plot of k

ion

) against D

air,p

should be approx-

imately a straight line with intercept k

ion

) and gradient

gen

) deﬁned by

(3.93)

The parameter C

gen

) describes the effect of general re-

combination on a chamber response and is referred to here

as the coefﬁcient of general recombination.

A typical plot is shown in Figure 3.48 for chamber

N3009 at a moderate D

air,p

of 0.43 mGy. The y-axis is

normalized by the best estimate of the saturation current

, as determined by a ﬁt using the full Boag equation. The

solid line is a simple linear ﬁt to the data. The value of

the intercept given by the linear ﬁt is within 0.01% of the

best estimate of I

using the Boag ﬁt, thereby justifying

the linear approximation at this dose per pulse.

For all four Scanditronix chambers, the value of

ion

(100 V) evaluated as above was measured as a function

air,p

. According to Equation (3.92), this dependence

should be approximately linear, as was found to be the

case for all chambers (including the Calcam chambers).

Figure 3.49 shows the results for chamber N3701. The

solid line shows a linear ﬁt. The intercept of the linear ﬁt

is taken to be the value of the correction for initial recom-

bination k

init

(100 V). The gradient is the coefﬁcient of

general recombination C

gen

(100 V). As one would expect,

the value of k

ion

(100 V) is dependent only on the dose per

pulse and not the electron energy.

It was found that for all the chambers tested (including

the Scanditronix chambers), the close agreement with the

Boag theory does not extend to voltages signiﬁcantly

beyond 100 V.

A measurement of C

gen

) allows one to extract a

value for the effective plate separation d

eff

for each cham-

ber consistent with Equation (3.93). (This is not necessar-

ily the actual plate separation

d.) For a polarizing voltage

of 100 V and assuming the values



equal to 3.02 

V m C

1

(ICRU 1982), (W

air

e) equal to 33.97 JC

1

and



air

equal to 1.205 kg m

3

(at standard temperature

and pressure), Equation (3.93) gives

(3.94)

where

gen

(100 V) is in Gy

1

and d

eff

is given in m.

ion

()k

init

()





air

e()

-----------------------------

air p,



gen

()





air

e()

------------------------------



eff

air

e()C

gen

()



air

---------------------------------------------------

4.32 10

4

gen

100V()

FIGURE 3.48 The inverse of the chamber current I as a function of the inverse of the polarizing voltage V for chamber N3009 at

a dose per pulse of 0.43 mGy to air. The chamber current is normalized by the saturation current

as determined by a full Boag

analysis. The solid line is a linear ﬁt to the data. (From Reference [48]. With permission.)

Ch-03.fm(part 1) Page 134 Friday, November 10, 2000 11:58 AM

Ionization Chamber Dosimetry 135

Ion-recombination correction factor k

sat

for spherical

ion chambers irradiated by continuous photon beams

was investigated by Piermattei et al. [49] When large-

volume ionization chambers are used, the ion-recombina-

tion correction factor k

sat

has to be determined. Three

spherical ion chambers with volumes ranging from 30 to

have been irradiated by photons of a

192

Ir source

to determine the k

sat

factors. The k

sat

values for large-

volume ionization chambers obtained by considering the

general ion recombination as predominant (Almond’s

approach) are in disagreement with the results obtained

using methods that consider both initial and general ion-

recombination contributions (Niatel’s approach). Such

disagreement can reach 0.7% when high currents are mea-

sured for a high-activity source calibration in terms of

reference air-kerma rate. A ‘‘two-voltage” method, inde-

pendent of the voltage ratio given by a dosimetry system,

is proposed for practical dosimetry of continuous x- and

gamma-radiation beams. In the case where the Almond

approach is utilized, the voltage ratio V

should be less

than 2 instead of Almond’s limit of V

 5.

For continuous radiation, such as x- and gamma-ray

beams of high dose rate, generally used in external radio-

therapy treatments, the general recombination dominates

the total recombination losses. In this case, the trend of

the reciprocal of the observed ionization current, i, against

the voltage,

V, can be described by

(3.95)

with b a constant that summarizes the recombination

coefﬁcient, the electron charge, and ion mobilities. The

ion-recombination correction factor k

sat

, used in several

dosimetric protocols, is the reciprocal of the collection

efﬁciency:

(3.96)

Almond [50] and Weinhous and Meli [51], using Equa-

tions (3.95) and (3.96), obtained for

sat

the expression

(3.97)

with i

and i

the currents obtained using voltages V

(the

normal operating bias voltage) and V

 V

), respec-

tively. Equation (3.97), known as a ‘‘two-voltage” technique,

can be written replacing the currents with the collected

charge Q

and Q

A new ‘‘two-voltage” method to determine the

p(i

sat

,V) factor was proposed. If V

and V

, with V



n  1, are the only two voltages given by the dosimetry

FIGURE 3.49 The total recombination correction k

as a function of the absorbed dose to air per pulse for chamber N3701 at a

polarizing voltage of 100 V. The correction at each dose per pulse was evaluated using a Boag ﬁt over the polarizing voltage range

40–100 V. The solid line is a linear ﬁt to the data. (From Reference [48]. With permission.)

1i 1i

sat

bV



sat

1 fi

sat

i

i

sat



sat

V

()

1[] V

V

()

i

()[]

Ch-03.fm(part 1) Page 135 Friday, November 10, 2000 11:58 AM

136

Radiation Dosimetry: Instrumentation and Methods

system and

and

are the relative ionization currents,

then [49]

(3.98)

(3.99)

and the difference between Equations (3.98) and (3.99)

can be approximated for by

(3.100)

If the experimental data (



)

as a function of

can be ﬁttted by a straight line, then the ordinate value

for



0, and the slope value

of the line are given by

Then

sat

(

sat

) can be evaluated as

(3.101)

which, near the saturation condition , can be

rewritten as

(3.102)

The

sat

values for the ion chambers examined,

obtained using the three methods, are reported in Figure

3.50 as a function of the

values and the air-kerma rates.

The arrows indicate the maximum air-kerma rate values

related to minimum source-chamber distances used in the

calibration of

192

Ir (740 GBq) when multiple measure-

ments (at different source-chamber distances) are made.

In particular, the

sat

average values, obtained using Equa-

tion (3.102) with varying

values, are reported with the

maximum variations shown by bars; these data are in good

agreement with the results obtained by applying the Niatel

approach. [49]

The collection efﬁciency of a 5.7-cm-diameter spher-

ical ionization chamber was measured by Biggs and

Nogueira. [52] Due to its large volume, the collection

efﬁciency is low when placed in the primary beam at

isocenter and, hence, the correction factor is large. By

measuring the primary ﬂuence with the machine operating

at a low dose rate and at a distance where the collection

efﬁciency is high, a measure of the dose with this chamber

at isocenter can be obtained by extrapolating back to the

isocenter using the inverse square law.

Although Boag’s theory was derived for parallel-plate

geometry, he extended the calculation to cylindrical and

spherical chambers by providing suitable correction factors.

The collection efﬁciency of this chamber was mea-

sured using three approaches. The ﬁrst approach was to

use the conventional two-voltage technique based on

Boag’s theory. In this case measurements were made at

two voltages (



being the standard operating

voltage) and the collection efﬁciency was calculated from

the formalism outlined below. The second approach was

to measure the collection efﬁciency using the voltage ex-

trapolation technique. Here, the voltage across the cham-

ber was varied between 50 V and 1000 V and the inverse

of the collected charge, normalized to unity at the standard

collection voltage, was plotted against the inverse of the

voltage. The collection efﬁciency was then derived by

extrapolating this plot to 1/

Geleijns et al. [53] showed that for large chambers,

up to 600 cm

, recombination effects due to transit-time

effects can be quite signiﬁcant.

The chamber used by Biggs and Nogueira was a Shonka

(Model A5, Exradin, Lisle, IL 60532) 100-cm

spherical

ionization chamber with a 5.7-cm inner diameter. The

leakage current for this chamber is less than 1 fA. A cubic,

PMMA build-up cap with a minimum thickness of 1 cm

was made for this chamber. This shape was chosen to

provide build-up for the chamber in the primary beam and

to allow for easy incorporation into a lead-shielding

arrangement with adequate mechanical protection during

the scattered radiation measurements. The build-up was

found to be adequate for the 4-MV primary beam, but

additional build-up was necessary for the 10-MV beam.

Measurements were also taken with a 0.6-cm

Farmer-

type chamber at 100% and 50% bias at the same distances

as the Shonka chamber. Since the efﬁciency of a Farmer

chamber when calibrating at isocenter is generally



98%,

this chamber acts as a useful reference.

Geleijns et al. gave the formula for the transit time for

ions in an ionization chamber as

(3.103)

where

is the ion mobility,

is the collecting potential,

and

is the effective electrode separation for a spherical

chamber given by

(3.104)

Here,

and

are the radii of the air volume and central

electrode, respectively, and

sph

is a constant, given by

(3.105)

sat

,()aV

sat

V



sat

, V

n()naV

sat

V



sat



()/i

n 1()a/V

1()bi



An1()a/V



1()b/V



sat

,()1 pi

sat

() 1 aV

sat

V



sat



sat

,()1 A n 1()Bi

/ n

1()

→



()

-------------



sph

ab()

sph

---











Ch-03.fm(part 2) Page 136 Friday, November 10, 2000 11:59 AM

Ionization Chamber Dosimetry

137

FIGURE 3.50

A comparison of the

sat

values obtained for three spherical ion chambers. Exradin model A4 (with



380 V), Exradin

model A6 (with



380 V), and OFZS model LS10 (with



1000 V) as a function of the ionization current,

. Results obtained

using Almond (1981) (O) (with

3.5) and Equation (3.102) (



) (the bars show the maximum variation obtained changing the

ratio) are compared with the results obtained by applying the Niatel (1967) method (



). The arrows show the air-kerma rate

values related to minimum source-chamber distances used in the

192

Ir (740 GBq) calibration. (From Reference [49]. With permission.)



Ch-03.fm(part 2) Page 137 Friday, November 10, 2000 11:59 AM

138

Radiation Dosimetry: Instrumentation and Methods

For this particular chamber,



2.84 cm and



0.3 cm, giving a value of

sph



1.88 and



4.78 cm.

Using the value for

of 1.58





given by

Geleijns et al., one obtains a value for



of approximately

0.05 s.

-V

OLTAGE

ECHNI

QUE

The ion-collection efﬁciency was calculated using the

two-voltage technique based on the Boag formalism. One

ﬁrst takes the ratio,

(3.106)

where

100%

corresponds to the charge collected at 100%

bias and

50%

corresponds to the charge collected at 50%

bias. One then calculates the value of u that satisﬁes the

equation

(3.107)

and the ion-collection efﬁciency f is then given by

(3.108)

GELEIJNS’ METHOD:

In this method, which is based on the Boag theory, a plot

is made of the inverse of the charge measured by the

Shonka chamber vs. the inverse of the charge measured

by the Farmer chamber. The various points correspond to

measurements made at different distances from the source;

the line is a least-squares ﬁt to the data with an

value

of 0.9999. This linearity derives from an approximation

to the Boag formula. The chamber collection efﬁciency

at any distance, is then given by

(3.109)

where S is the slope of the plot, Q is the charge measured

by the Shonka chamber, and Q

mon

is the charge measured

by the Farmer chamber.

Figure 3.51 shows the charge measured by the Farmer

chamber at 4 MV multiplied by the square of the distance

from the source plotted against the source distance.

The general collection efﬁciency in pulsed radiation

was studied by Johansson et al. [54] for isooctane (C

)

and tetramethylsilane (Si(CH

)

). These two liquids were

used as sensitive media in a parallel-plate liquid ionization

chamber with a 1-mm sensitive layer. Measurements were

carried out using 20-MV photon radiation from a linear

100%

50%

-------------



2ln 1 u()

ln 1 2u()

---------------------------



---

ln 1 u()

mom

-----------



FIGURE 3.51 Variation of the charge measured by the Farmer chamber multiplied by the square of the distance from the target for

10-MV x-rays. This function, which is constant to within

1.3% over the distance from 1 to 4.5 m, veriﬁes the applicability of the

inverse-square law for this beam over that distance. (

) 100 MU/min; () 200 MU/min; () 400 MU/min; () 600 MU/min. (From

Reference [52]. With permission.)

Ch-03.fm(part 2) Page 138 Friday, November 10, 2000 11:59 AM

Ionization Chamber Dosimetry 139

accelerator with a pulse repetition frequency of 30 pulses/s

and a pulse length of 3.5



s. The aim of the work was to

examine whether the theoretical equation for general col-

lection efﬁciency for gases in pulsed radiation, derived by

Boag (1969), can be used to calculate the general collec-

tion efﬁciency for isooctane and tetramethylsilane.

The body of the parallel-plate liquid ionization cham-

ber used for the experiments was made of Rexolite

, a

styrene copolymer with density 1.05 g cm

1

, and the elec-

trodes were made of pure graphite. The chamber had

cylindrical symmetry without any guard.

The theory neglects the space-charge screening effect

and assumes that recombination loss during the radiation

pulse is negligible; i.e., the pulse length is short. Further-

more, it assumes that the densities of the ions of different

sign are equal; i.e., the negative charge is carried by neg-

ative ions, and that diffusion loss can be neglected.

For two ions of opposite sign at a separation distance

R and approaching each other with a velocity v, the recom-

bination rate constant



in a nonpolar molecular liquid is

given by [55]

(3.110)

where r

is the Onsager escape distance [56], the separa-

tion distance at which the Coulombic potential energy is

equal to the diffusion kinetic energy of the ions; D is the

sum of the diffusion coefﬁcients of the ions;



is the per-

mittivity of free space; and



is the permittivity of the

dielectric liquid in the chamber (



 1.94 for isooctane and



 1.84 for tetramethylsilane). For dielectric media with

a low permittivity, the ratio r

/R is large and the recom-

bination rate constant can be approximated by the equa-

tion derived by Debye:

(3.111)

Using the Debye equation for



in the Boag equation

gives

(3.112)



is thus a characteristic for the general recombination

rate in the sensitive media in the ionization chamber, deter-

mined by the permittivity of the dielectric liquid.

The relation between the ionization current in a dielec-

tric liquid and the electric ﬁeld strength has been shown

by several authors, experimentally and by computer sim-

ulations, to follow a linear equation in a limited interval

of the electric ﬁeld strength, where the general recombi-

nation can be ignored. The linear relation is given by

(3.113)

where

i is the ionization current, c

is a constant charac-

teristic of the dielectric liquid and determined by the prob-

ability for initial recombination escape due to diffusion,

is a constant characteristic of the dielectric liquid and

determined by the probability for initial recombination

escape due to the external electric ﬁeld strength, E is the

external electric ﬁeld strength, and is the dose rate.

Figure 3.52 shows schematically the relation between the



()



-------------------------

---------

----







exp

----







exp

1





()



-------------------------







--------



E()D



FIGURE 3.52 The characteristic relation of the ionization current from a dielectric liquid and the electric ﬁeld strength in a liquid

ionization chamber. The ionization current is the sum of the two components , where

is the fraction of ions that

escape initial recombination due to diffusion,

is the fraction of ions that escape initial recombination due to the external electric

ﬁeld strength,

E is the electric ﬁeld strength, and is the dose rate. (From Reference [54]. With permission.)



Ch-03.fm(part 2) Page 139 Friday, November 10, 2000 11:59 AM