Shani G. Radiation Dosimetry: Instrumentation and Methods

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120 Radiation Dosimetry: Instrumentation and Methods

If the loss due to these phenomena is small, the collection

efﬁciency, f( i/i

), can be expressed as

where

, f

, and f

are collection efﬁciencies corresponding

to initial recombination, back-diffusion, and volume

recombination, respectively.

The two main ways in which ions recombine in an air-

ﬁlled ionization chamber are by ‘‘initial recombination”

and ‘‘general recombination,” the total recombination

being the sum of these two effects. Initial recombination

takes place when positive and negative ions formed in the

same collision event meet and recombine along their ini-

tial track. This is a dose rate independent effect, unless

the ion density becomes so great that the ﬁeld strength

responsible for ion collection is reduced. The initial

recombination is small with radiation encountered in clin-

ical use.

As irradiation continues, the initial trade structure in

the chamber cavity is lost by drift or diffusion of ions and

the recombination effect evolves into what is called gen-

eral recombination. In this process ions produced by dif-

ferent ionizing particles along distinct tracks encounter

each other and recombine as they move toward the two

electrodes. This effect depends on ion density and, thus,

on the dose per pulse and is dominant in pulsed radiation.

Measurements concerning the collection efﬁciency

were made by Takata and Matiullah [36] for a parallel-

plate ionization chamber which has a variable space

FIGURE 3.35 Typical 6-MV x-ray buildup curves in polystyrene for (a) 5  5 cm

60°, (b) 20  20 cm

60°, and (c) 5  5 cm

75° angled beams. The data is taken at 100-cm SSD with no blocking trays in the beam. %DD on the y axis is normalized to the

dose at the reference depth (

 1.6 cm) for a normally incident beam. Depth along CA represents the slant depth in polystyrene

from the phantom surface to the point of measurement of the plane-parallel chamber. Relative response vs. the extrapolation chamber

is shown on the right axis. (From Reference [35]. With permission.)



Ch-03.fm(part 1) Page 120 Friday, November 10, 2000 11:58 AM

Ionization Chamber Dosimetry 121

between the polarizing electrode and collector. To do so,

the chamber was exposed to



-rays at several different

exposure rates and m  (



eK





)

1/2

, where



is the

recombination coefﬁcient, e is the charge per ion, and K



and K



are the mobilities of positive and negative ions,

respectively. It was deduced from the collection efﬁcien-

cies, which were determined at various applied voltages.

It was found that m depends upon the lifetime of ions in

the chamber (i.e., it increases from 17.1 to 18.5 MVA

1/2

, with a decrease in the ratio of the voltage to the

square of the chamber space from 600 to 20 kVm

2

Figure 3.37 shows the collection efﬁciencies of the

chamber for a ﬁxed chamber space of 15 mm at different

exposure rates as a function of

U/d

1/2

(U is the applied

voltage, d is the chamber space, and q is the ion charge).

They are normalized to each saturation current and only

the values in the range of

U/d

1/2

less than 1000 MVA

1/2

are plotted. Theoretical values, calculated with

various equations for



 3.99,



 0.163, and m 

17.5 MVA

1/2

are also shown in the ﬁgure.

Values for m were calculated from experimental data

on f( i/i

) using Greening’s equation both with and with-

out taking the initial recombination and diffusion losses

into consideration. Figure 3.38 shows the results obtained

without taking these losses into consideration (i.e., assum-

ing f

 f

 1).

The procedure recommended by radiation dosimetry

protocols for determining the collection efﬁciency f of an

ionization chamber assumes the predominance of general

recombination and ignores other charge-loss mechanisms

such as initial recombination and ionic diffusion. [37] For

continuous radiation beams, general recombination theory

predicts that f can be determined from a linear relationship

between 1Q and 1V

in the near saturation region (f 

0.7), where Q is the measured charge and V is the applied

chamber potential. Measurements with Farmer-type cylin-

drical ionization chambers exposed to

Co gamma-rays

reveal that the assumed linear relationship between 1/Q

and 1/V

breaks down in the extreme near-saturation region

( f  0.99) where Q increases with V at a rate exceeding

the predictions of general recombination theory. [37]

Boag and Wilson [38] developed the following rela-

tionship for the charge-collection efﬁciency f

(V) for gen-

eral recombination as a function of applied potential V in

FIGURE 3.36 The angular response of the various chambers relative to the response of the extrapolation chamber with 0-mm plate

separation. Measurements were done for (a) 6-MV and (b) 24-MV x-rays in a 3

 5-cm

ﬁeld. (From Reference [35]. With permission.)

Ch-03.fm(part 1) Page 121 Friday, November 10, 2000 11:58 AM

122 Radiation Dosimetry: Instrumentation and Methods

an ionization chamber containing an electronegative gas

and irradiated by a continuous radiation ﬁeld with a con-

stant dose rate:

(3.62)

where



contains chamber and air parameters and is

proportional to the dose rate of the continuous beam. The

relationship was found to be valid in the near-saturation

region (

 0.7) and may also be written in the following

form:

(3.63)

where



and



are related through



 

Q

sat

, making



independent of the dose rate.

A simpliﬁed version of this approach is the so-called

two-voltage technique, in which the collected charge is

measured at only two voltages, V

and V

, assuming that

Equation (3.63) is valid in the region spanned by the two

FIGURE 3.37 Diagram showing the collection efﬁciency, f, measured for a chamber space of 15 mm at exposure rates of , 3.6;

, 31; and O, 310



Ckg

1

2

. Saturation curves, f

, calculated with various equations for



 3.99,



 0.163, and m  17.5

MVA

1/2

are also represented. (From Reference [36]. With permission.)

FIGURE 3.38 Values of m obtained without taking into account the initial recombination and diffusion losses. Chamber spaces: , 2;

, 5; , 15; and •, 30 mm. Exposure rates: ..., 3.6; ---, 31; and —, 310 (145 for •)



Ckg

1

. (From Reference [36]. With permission.)

V()

QV()

sat

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

1 



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



----

sat

---------



------



Ch-03.fm(part 1) Page 122 Friday, November 10, 2000 11:58 AM

Ionization Chamber Dosimetry 123

voltage points. If Q

and Q

are the charges measured at

and V

, respectively, then f

) can be written as: [37]

(3.64)

In the AAPM-TG21 protocol which deals with the

calibration of high-energy photon and electron beams,

Equation (3.64) is further simpliﬁed by using

 2V

(typically, V

 300 V and V

 150 V), so that the

collection efﬁciency for general recombination in contin-

uous radiation beams f

) may be found from

(3.65)

Jaffe’s theory of initial recombination reduces to a

problem of simple Brownian motion under the inﬂuence

of two forces: the Coulomb attraction between oppositely

charged ions produced in the same charged-particle ion

track and the applied electric ﬁeld in an ionization cham-

ber. Jaffe found that the collection efﬁciency for initial

recombination f

in an ion chamber with a collecting ﬁeld

normal to the ion track of E (V/cm) could be expressed

as follows:

(3.66)

where g is a constant and

(3.67)

with  . For large x, Equation (3.67) has

the asymptotic approximation , which is

accurate enough to permit, for large polarizing potentials,

Equation (3.66) to be written in the following form:

(3.68)

where



is a parameter incorporating various chamber

and gas parameters and is independent of the dose rate.

Ritz and Attix [39] showed that for initial recombination,

one should ﬁnd that

(3.69)

Collection efﬁciencies for ion loss due to initial

recombination and back-diffusion were measured by

Takata [40] for several humidities using a parallel-plate

cavity ionization chamber irradiated with



-rays. It

was shown, from measurements in a range of inverse

electric ﬁeld strengths from 0.05 to 14 mm V

1

, that initial

recombination took place both in clusters and columns

of ions produced along the path of the secondary elec-

trons ejected by the



-rays. The ion loss due to recom-

bination in clusters was found to increase with humidity,

but that in columns did not. Effects of ion-clustering

reactions on recombination may be reduced after longer

periods of ion drift, when recombination in columns

takes place.

Figure 3.39 shows the collection efﬁciencies mea-

sured for relative humidities of 1.5  0.3, 30.7  0.5, and

74.2  0.7%, respectively. These are indicated as func-

tions of l/E with the applied voltage U as a parameter.

Solid lines in the ﬁgures are obtained by the least-squares

method for each result at different applied voltages. They

were all measured at temperatures in a range from 21.6°C

to 21.8°C. The ﬂuctuation of the temperature during each

measurement for one saturation curve was less than

0.04°C. The pressure in the chamber during measurements

was in the range of 991–1018 hPa, and the average was

1006 hPa. Values of the ordinate at the intercepts corre-

spond to the ion loss due to back-diffusion. Arrows at the

ordinate show theoretical values for each applied voltage

T  21.7°C.

Figure 3.40 shows the results of calculations for initial

recombination in clusters (solid lines) and for columns

(broken lines). The values of the parameter h (ion clusters

merging parameter) for solid lines

A, B, C, and D were

, 5.7, 2.85, and 5.7, respectively. The value of v

(initial

number of ion pairs in a cluster) was 1.5 for line D. The

value of h was 5.7 and was 16.7 and 33.4 mm

1

for

broken lines E and F, respectively.

Ion-recombination corrections for plane-parallel and

thimble chambers in electron and photon radiation was

discussed by Havercroft and Klevenhagen. [41] The aim

of the work was to investigate the collection efﬁciency of

several popular ionization chambers used in high-energy

electron and photon dosimetry. The recombination effect

was evaluated in plane-parallel-type ion chambers

(Markus, NACP, Calcam, and Vinten-631) and in a thim-

ble-type chamber (NE2571). The chambers’ response was

studied in a pulsed accelerator beam for both photons and

electrons, and in continuous radiation from a

machine. Two main conclusions from the work may be

drawn here. First for the Farmer NE2571 0.6-cm

chamber

used here, the correction factors required are on the order

of 0.1% for

Co radiation, 0.5% for 5- and 8-MV x-ray

and 10- and 12-MeV electrons, and 0.9% for 15-MV

x-ray and 15- and 18-MeV electrons, all at typical clinical

dose rates of 200–400 cGy min

1

(0.017 cGy pulse

1

Second, the recombination effect for the plane-parallel

chambers was found to be smaller than for the thimble

()

sat

--------

Q

V

()



1 V

V

()



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



()

sat

--------

---

----------



sat

--------

1 gh x()

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



hx() e



------





1()

ix()

 V d()

hx()



/2x→

sat

--------

1 

/V

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



----

sat

--------



----



Ch-03.fm(part 1) Page 123 Friday, November 10, 2000 11:58 AM

124 Radiation Dosimetry: Instrumentation and Methods

chamber, with values of  0.1% in continuous radiation

and 0.2–0.4% in pulsed radiation.

It is convenient to express the collection efﬁciency

(for pulsed radiation), f, in terms of q, the charge collected

per pulse, i.e., that which is being measured by the ion-

ization chamber. The expression for collection efﬁciency

is as follows: [42]

(3.70)

FIGURE 3.39 Collection efﬁciencies measured at humidities of (a) 1.5%, (b) 31%, and (c) 74%. Solid lines were obtained by the

least-squares method for data measured at the applied voltage indicated in the ﬁgure. Data from left to right, for each applied voltage,

correspond to chamber distance 1, 3, and 7 mm, respectively. (From Reference [40]. With permission.)

fv/(exp v() 1 )

Ch-03.fm(part 1) Page 124 Friday, November 10, 2000 11:58 AM

Ionization Chamber Dosimetry 125

where

and

Where



is the ionic recombination coefﬁcient, e is the

electronic charge, k

and k

are the mobilities of the posi-

tive and negative ions, respectively, q is the initial charge

density of the positive and negative ions collected by the ion

chamber during irradiation, d is the effective electrode

spacing (for plane-parallel chambers, this is the actual elec-

trode spacing, but for cylindrical or spherical chambers the

equivalent spacing must be calculated), and V is the col-

lecting voltage.

Burns and Rosser (1990) [40a] employed the expan-

sion of exp(

v) to simplify Equation (3.70), neglecting all

terms after v

and assuming that the dose per pulse does

not exceed 0.2 cGy with linear accelerators. Thus, the

formula for collection efﬁciency, in a form convenient for

numerical evaluation, becomes

(3.71)

The correction factor to compensate for lost (recom-

bined) ions, for general recombination in pulsed radiation,

is the reciprocal of the collection efﬁciency

(3.72)

The equivalent electrode separation for the farmer-

type chamber was determined using Boag’s expressions.

Due to the chamber shape, it was considered to consist of

two parts, a cylindrical main volume and a hemispherical

section at the tip (see Figure 3.41). The equivalent plate

separations for the cylindrical and hemispherical chamber

sections, with weighting calculated as a percentage of the

total volume, were found for this chamber to be

cyl



2.984 mm, weighting 86% and d

hem

 5.168 mm, weight-

ing 14%. Substituting these values, along with the polar-

izing potential V  244 V and



 3.02  10

1

V (ICRU 1982), into Equation (3.70), the following values

for v were obtained:

where the units of

q are Cm

3

By applying the given weightings, one obtains

The charge, q, can be expressed in terms of the

dose per pulse, D

, by utilizing the calculated chamber

volume of 6.95  10

7

and calibration factor of

FIGURE 3.40 Calculated collection efﬁciencies for initial recombination in clusters (solid lines) and in columns (broken lines). The

value of the parameter

h is 5.7 in general but  and 2.85 for lines A and C, respectively. The value of m was set equal to 3 except

for line

D, where it was 1.5. The value of was 16.7 mm

1

and 33.4 mm

1

for broken lines B and F, respectively. (From Reference

[40]. With permission.)



q/V



/e() k

()[]

f 1



---

v







ion gen()

1 f 1

---

v

cyl

1102qv

hem

3306q

v 1411q

Ch-03.fm(part 1) Page 125 Friday, November 10, 2000 11:58 AM

126 Radiation Dosimetry: Instrumentation and Methods

4.48  10

cGy C

1

where

is the dose (cGy) to water per pulse, corrected

for temperature and pressure, as measured by the dosim-

eter. Combining these equations, the recombination cor-

rection factor’s dependence on pulse density used in a

pulsed beam is found: [41]

(3.73)

for an NE2570/1 (Farmer-type) dosimeter with a standard

polarizing potential of approximately 250 V, where D

is in centigrays.

For continuous radiation, the ﬁeld-ionization cham-

bers used in electron dosimetry are calibrated against a

reference chamber, usually in a local

Co beam and

hence in low-dose-rate continuous radiation in which the

initial recombination is dominant. Boag’s theory applies

also to continuous radiation and provides an equation for

collection efﬁciency, f, which is as follows:

(3.74)

where

d is the electrode separation (effective for a thimble

chamber), V is the polarizing potential, m is a constant

depending on the nature of the gas (2.01  10

V m

1/2

1/2

1/2

for air), and is the rate of charge collected per

unit volume of the gas (C m

3

1

Employing the same derivation method as in the case

of pulsed radiation, a correction for general recombination

in continuous radiation was obtained for a Farmer chamber:

(3.75)

Therefore

This is a very small correction and difﬁcult to verify

experimentally. [41]

The correction factor for ion recombination for a

chamber irradiated in

Co radiation was obtained using

the two-voltage technique in conjunction with the analy-

tical expression of Almond [43]

(3.77)

where Q

is the charge collected at the normal operating

voltage V

and Q

is the charge collected at the lower bias

voltage V

Table 3.13 shows correction factors measured exper-

imentally for plane-parallel chambers. With factors

between 1.002 and 1.004, it is clear that the experimentally

determined correction is larger than the theoretically

FIGURE 3.41 Calculation of equivalent plate separation for NE2571 chamber using an equivalent hemisphere. The chamber is

considered in two sections: a cylinder and a hemisphere, equivalent in volume to the conical end. (From Reference [41]. With

permission.)

v 0.453D



ion gen()

1 0.23D



f 1







6()d

q˙V



q˙

ion gen()

1/ f 1/ 1





()1





for



 1

ion gen()

1 1.0 10

6

 1.00015

for D

150 cGymin

1



ion tot()

()1[]/ V

()Q

()

(3.76)

Ch-03.fm(part 1) Page 126 Friday, November 10, 2000 11:58 AM

Ionization Chamber Dosimetry 127

derived value. The latter was for general recombination

only; however, measured values of initial recombination

were too small to give recombination of experiment and

theory.

The other major point to note is a slight energy depen-

dence of the recombination correction factor. By looking

at the saturation curve for each chamber individually, it

can be seen that the 18-MeV curves consistently fall off

faster than the 10-MeV and then the 4-MeV curves, as the

polarizing voltage decreases. An example is shown in

Figure 3.42. At present, no explanation of this phenome-

non has been suggested. [41]

The effect of recombination in plane-parallel cham-

bers was found to be smaller than in thimble chambers,

particularly for pulsed radiation.

General ion recombination has been studied by

Geleijns et al. [44] under irradiation conditions relevant for

diagnostic radiology and with four different ionization

chambers. Recombination was estimated by measuring the

collected charge at various collecting potentials of the ion-

ization chamber. An alternative approach is to vary the

exposure or exposure rate over a wide range at a constant

collecting potential.

For an ionization chamber exposed to continuous radi-

ation, a linear relation between 1/f and 1/V

was derived

by Boag under the condition that f  0.7:

(3.78)

where



is the recombination coefﬁcient, q is the charge

liberated in the ionization chamber and escaping initial

recombination per unit volume and per unit time, and e is

the elementary charge. The mean numerical value of the

coefﬁcient



/6ek

in air is estimated by Boag as (6.73 

0.80)  10

s m

1

TABLE 3.13

Experimentally Determined Recombination Correction Factors for Plane-Parallel

Chambers in Electron Beams (using electrometer settings of 244 V and 61 V)

Chamber 4 MeV 6 MeV 8 MeV 10 MeV 12 MeV 15 MeV 18 MeV

Markus 1.0025 — — 1.0029 — — 1.0043

NACP 1.0036 1.0031 1.0033 1.0031 1.0031 1.0045 1.0045

Calcam 1.0031 — — 1.0024 — — 1.0043

Vinten-631 1.0017 — — 1.0024 — — 1.0025

Source: From Reference [41]. With permission.

FIGURE 3.42 Saturation curves for Markus chamber in electron beams, i.e., variation of electrometer reading with polarizing voltage.

(From Reference [41]. With permission.)

---



6ek

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





------







Ch-03.fm(part 1) Page 127 Friday, November 10, 2000 11:58 AM

128 Radiation Dosimetry: Instrumentation and Methods

For an ionization chamber exposed to pulsed radia-

tion, the collection efﬁciency f for pulsed radiation has

been derived by Boag as

(3.79)

where

u is



/V, r is the charge liberated and escaping

initial recombination per unit volume and per pulse, and



[







e(k

+ k

)] is a constant that depends only on

properties of the ions in the ionization chamber. In this

study, a value for



of 3.19  10

mVC

1

was used. At

relatively high collecting potentials and short pulses, the

inverse of the collection efﬁciency (1/

f ) can be approxi-

mated by a linear equation:

(3.80)

Thus, under these conditions, a linear relation between 1/f

and 1/V is derived.

Examples of general recombination measured with the

Keithley 600-cm

ionization chamber at two collecting

potentials are given in Figures 3.43a and b. These ﬁgures

illustrate recombination due to continuous radiation and

f 1/u() ln 1 u()

---

 1



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







FIGURE 3.43 The collection efﬁciency of the 600-cm

ionization chamber. (a) Measurements at a constant exposure rate of about

3.0

 10

3

mC kg

1

(0.011 R s

1

); and (b) measurements at a constant exposure (pulse) of about 0.74  10

3

mC kg

1

(2.9 

3

R). Collecting potential 200 V:  and 300 V: . (From Reference [44]. With permission.)

Ch-03.fm(part 1) Page 128 Friday, November 10, 2000 11:58 AM

Ionization Chamber Dosimetry 129

due to pulsed radiation. Collecting potentials of 200 and

–300 V were used. Under these conditions, ion transit

times for this ionization chamber are about 100–200 ms.

In Figure 3.44, measurement results are summarized

for two ionization chambers. The NE 35-cm

ionization

chamber is exposed to several exposure rates between

0.27 mC kg

1

(1.0 Rs

-1

) and 2.8 mCkg

1

(11 Rs

1

)

at exposure times that correspond with the continuous

radiation model (Figure 3.44a). The Keithley 600-cm

ionization chamber is exposed to pulses of 0.50 

4

mC kg

1

(0.19  10

3

R) to 4.1  10

4

mC kg

1

(1.6  10

3

R) at exposure times, corresponding with the

pulsed radiation model (Figure 3.44b). During the mea-

surements the collecting potential varied from 200 to

400 V. Linear extrapolation of charge values measured

at different collecting potentials yielded the initial

charge from which, subsequently, the collection efﬁ-

ciency was calculated.

Ion recombination and polarity effect of ionization

chambers in kilovoltage x-ray exposure measurements

was investigated by Das and Akber. [45] Exposure mea-

surements with ionization chambers are dependent on the

correction factors related to the beam energy (k

), temper-

ature and pressure (k

), ionization recombination (P

ion

and polarity (k

pol

) effects. Six different chambers com-

monly used in diagnostic radiology were investigated for

the P

ion

, and k

pol

at various exposure rates by changing the

tube voltage, beam current, exposure time, and distance.

FIGURE 3.44 Recombination measurements at different collecting potentials and different exposure rates or exposures. (a) NE

35-cm

chamber and continuous radiation model. : 0.27 mC kg

1

(1.0 R s

1

); •: 0.56 mC kg

1

(2.2 R s

1

); : 1.2 mC kg

1

(4.6 R s

1

); : 2.4 mC kg

1

(9.4 R s

1

); and : 2.8 mC kg

1

(11 R s

1

). (b) Keithley 600-cm

chamber and pulsed

radiation model.

: 0.050  10

3

mC kg

1

(0.19  10

3

R), •: 0.095  10

3

mC kg

1

(0.37  10

3

R): :0.19  10

3

mC kg

1

(0.75  10

3

R); and : 0.41  10

3

mC kg

1

(1.6  10

3

R). (From Reference [44]. With permission.)

Ch-03.fm(part 1) Page 129 Friday, November 10, 2000 11:58 AM