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Is the response of the Fricke dosimeter constant for high energy electrons and photons?
Radiat. Phys. Chem. Vol. 47, No. 4, pp. 637~47, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0969-806X(95)00062-3 0969-806X/96 $15.00 + 0.00
Pergamon
IS THE RESPONSE OF THE FRICKE DOSIMETER CONSTANT FOR HIGH ENERGY ELECTRONS A N D PHOTONS? M. A. HILL and F. A. SMITH Department of Physics, Queen Mary and Westfield College, University of London, Mile End Road, London El 4NS, U.K. (Received September 1994; accepted Janua O, 1995) Al~traet--An attempt has been made to theoretically model the chemical response of the Fricke dosimeter using track structure concepts and to indicate areas in which the modelling can be improved. The Monte Carlo technique has been used to investigate the differences in the fundamental chemical response of the ferrous sulphate Fricke dosimeter to photon and electron irradiation. Calculations of the fractional energy loss in the track entities (spurs, blobs and short tracks) were made, together with more quantitative descriptions of these clusters which were used to estimate the final ferric yield. Photons and electrons of various energies were considered for an infinite medium and for a water equivalent ampoule in a water scattering medium, using four secondary electron energy distributions. Emphasis was placed on the variation in yields rather than in the absolute values. The results show that the relative differencesin the Fricke G-value, not only between electrons and photons, but also with primary energy are critically dependent on the variation in the mean energy of the secondary electron followingionisation with primary energy. This dependence is due to an expected strong variation in Fricke yield for low energy electrons (< 100 eV).
I. INTRODUCTION The motivation for this work was the investigation of possible fundamental differences in the response of the Fricke dosimeter between photon and electron irradiation, since in dosimetry practice it is tacitly assumed that any dependence of the Fricke G-value is of no consequence and that a primary standard calibrated with high energy photons for example can be used for routine work with high energy electrons. The investigation has been carried out using Monte Carlo codes written at QMW to calculate the variation in radical and ferric yields and the variation in track entities with energy for photons and electrons. The aim was to try to understand the link between the initial physical interaction of the photons and electrons, the subsequent distributions of radicals produced and the final ferric yield. The present knowledge of the physical and chemical events at very early times following irradiation is insufficient to enable the computation of accurate absolute yields. However, Monte Carlo calculations of this type are still useful for following the relative variations in yields between electrons and photons which deposit energy via electrons. The ferrous sulphate Fricke dosimeter has been recommended by both the AAPM (1983) and ICRU (1984) as an alternative method for determination and calibration of absorbed dose in water. It is used
by the National Research Council, Canada (NRC) and the Physikalische Teschnische Bundesantalt of Germany (PTB) as part of a primary standard and by the National Physical Laboratory, U.K. (NPL) and the National Institute for Standards and Technology, U.S.A. (NIST) as part of a calibration service for clinical accelerators. Exposure to radiation results in the oxidation of the ferrous ions to ferric ions. This is directly proportional to the absorbed dose if certain conditions are met (ICRU, 1969) and can be measured spectrophotometrically, Of = AA /(plEmG )
(1)
where D r is the dose to the Fricke solution, AA is the increase in absorbance due to irradiation, p the density of the Fricke solution, l the length of the light path in the photometer cell, E,, the molar linear absorption coefficient and G the radiation chemical yield of ferric ions. A constant value of EmG, equal to 352 x 10-6m2kg -l Gy -l (G = 1.60/~ molJ -1 assuming em=219.5m2mol-~), is recommended for both photons and electrons with mean elecron energies from 1 MeV up to at least 30 MeV (ICRU, 1984). The mean absorbed dose in water can then be obtained by multiplying with correction factors which account for a non-water equivalent detector and wail. These have recently been investigated by Ma and Nahum (1993a, b) for high energy photons and electrons and help explain discrepancies between experimental results.
637
638
M.A. Hill and F. A. Smith
The purpose of this paper is to quantify the likely errors incurred in Fricke Dosimetry due to the assumption of a constant G-value, when all aspects of radiation track chemistry are taken into account. Although there has been considerable support for a Fricke G-value which is constant to within 1% or better for energies about 1 MeV (ICRU, 1984), extensive re-evaluation of recent experimental results do not rule out an energy dependent G(Fe 3+) and may explain the discrepancies between experimental data and Monte Carlo calculations of certain correction factors (Ma and Nahum, 1993a, b; Ma et al., 1993). However these discrepancies could also be explained by systematic errors in the ionisation chamber dosimetry. A theoretical analysis of the ferrous sulphate Gvalue was developed by Burch (1959). Using experimental values of G(Fe 3+) for 21°po~- and 6°Co y-radiations and the Jaff6 recombination formulae, a relationship between G(Fe 3÷) and LET100 was used to define "local" energy deposition, with all energy transfers greater than 100 eV being treated as &-rays with their own rate of energy loss. This relationship was used with the local energy dissipation spectra to compute G(Fe 3+) using different values of LET100 and weighting them over the spectrum of local energy deposition. A similar approach was used by Nahum et al. (1981) using Monte Carlo simulations of photon and electron transport to derive the local energy deposition spectra which was used to empirically derive, by "trial and error", a G(Fe 3÷) vs LET100 relation. Because the approach used was empirical, the computed G(Fe 3÷) yields for high energy photons and electrons h a d to agree well with the experimental values to which they had been fitted: X- and v-rays for low LET and ct particles for high LET. The computed variations in these yields with energy was a 0.8% increase with electron energies from 5 to 30 MeV and a 1% decrease with mean energy of primary photons from 0.66 to 14.5 MeV. However, the above approach does not take into account the complex mechanisms leading to changes in chemical yields in the Fricke dosimeter, especially when considering low energy ( < 5 keV), high LET secondary electrons, through which a large fraction of energy is deposited. Calculations by Pimblott et al. (1990) found that for a single collision of a 1 MeV electron in liquid water 50% of all energy loss is due to events of 43 eV or less, 10% of the total energy loss is due to events of greater than 600eV and <0.2% to events greater than 5 keV. Due to increased scatter at low energies and decrease mean free paths, an increasingly spherical distribution of interactions is obtained, which are close enough together to allow reactions to take place between chemical species produced at these different sites and which are critically dependent on their spatial distributions. For these low energy electrons the chemical response is dependent on the whole track leading to a significant variation with energy. In this regard the determi-
nation of the chemical response will require a cut-off energy much greater than 100 eV. Low energy electrons cannot be expected to show the same chemical response as ct particle of an equivalent LET due to the differences in the shape of the track structure. Therefore the approaches of Burch (1959) and Nahum et aL (1981) will be inaccurate in modelling the high LET electron component. 2. THE PETS CODE
The QMW Monte Carlo computer codes (Hill and Smith, 1994) were written in order to model the complete evolution of events, from the initial interactions to the "initial" chemical yields formed at ~ 10 -12 s, before diffusion reactions take place and finally to the primary chemical yields defined as those observable in steady state experiments at 10-6s. These codes were used to determine the sensitivity of the yield and time dependence of chemical yields to different assumptions and approximations. For the present investigation the codes were extended to incorporate photon interactions together with the possibility of bremsstrahlung production at the higher energies, details of which can be found in the Appendix. The PETS (Photon and Electron Track Structure) code follows the spatial and temporal co-ordinates of the primary and all secondary photons and electrons, interaction by interaction, as they lose energy. For photons these interaction include: the photoelectric effect, Compton scatter and pair production, and for electrons: elastic scatter, excitations, ionisations and bremsstrahlung production. Multiple scattering routines have also been included to transport the electrons through a water phantom to a volume of interest, i.e. the ampoule, where the electrons are followed interaction by interaction. 2.1. Electron interactions
The distance travelled between interactions is determined by the exponential random number with an average equal to the mean free path given by the reciprocal of the total cross section. The relative magnitudes of the partial cross sections determine the interaction type at the new position. Cross sections for elastic interactions were taken from Zaider et al. (1983) and are based on water vapour data. Above 0.2 keV the angular scatter was described using the Rutherford formula with a modified screening parameter (Grosswendt and Waibel, 1978). Below this value the analytical expression of Porter and Jump (1978) with fitting parameters from Brenner and Zaider (1983) was used. The inelastic cross sections were obtained from an analysis of the optical absorption data of liquid water and include the possibility of collective excitations (plasmons). The first ionisation potential for liquid water was taken to be 8.8 eV. For energies above 1 keV cross sections were calculated using a straight line extrapolation on a
639
Is the response of the Fricke dosimeter constant?
4BiG~
Fano plot corrected for relativistic effects (Schutten
et al., 1966). 2.1. I. Secondary electron energy distributions. Comparisons were made between the four energy distribution of secondary electrons emitted following ionisation events due to Jain and Khare (1976), Green and Sawada (1972), Vriens (1966, 1969) and Gryzinski (1965). When the incident electron energy is high and the energy transfer small, the Born-Bethe approximation is valid with the cross section given by
4rca2oR2 1 Of(W, O)
SsB(T, E) = - -
T
W
- ~W
In CT
4r~a~R2 [1 =
T
S
1
1
E2 E(T-E------~)q (~r - E )
]
where S is the number of electrons available for ionisation. The leading term gives the Rutherford cross section and the next two representexchangeand interference contributions. The single relation proposed by Jain and Khare (1976) to describe the energy spectra of secondary electrons at all energies of the incident and ejected electrons is given by
SjK( T, E) = f~(T, E)SBB(T, E) =f2(T, E)SM(T, E) (4)
E )I+C(T-I) TZI InCT E3 f2(T, E) = E3 + E03(1 - 1/T) 1/~(1
(5)
1+
(6)
where E0(70 eV) is a parameter. Experimental values of Of/OW were taken from Tan et al. (1978) and extrapolated to higher values of W using the relation A e x p ( - B W ) to obtain fitted values of A and B. The semi-empirical formula of Green and Sawada (1972) is described by
I"
1
S~s(T, E) = A (T) (E -- To)2 + r ~
(7)
where the parameters A, To and F are all dependent on the primary energy T and x = 1. The binary encounter model of collisions was developed by Vriens (1966, 1969) to take into account interference between direct and exchange terms.
Sv(T, E)
N, ¢7o
x
(8)
q~,= cos{[R/(T + Bi)]I/2In(T/Bi)}, ao = 4na2oR2, N~ the number, Bi the binding energy and G~ where
the ratio of the orbital kinetic energy to the binding energy, of electrons in the ith subshell of the target. Gryzinksi (1965) also used the binary encounter approximation in addition to some other approximations to give
SG(T, E)
y.,
NiPi
o-o~ (E + Bh3(I +p~)3/2 x~'E + B~F1
(3)
f,(T,E)
]
(2)
where W is the energy loss suffered by the incident electrons in ionizing collisions i.e. W = E + I, a0 is the Bohr radius, R is the Rydberg energy, Of/OW the optical oscillator strength and C (0.075 e V - ' ) a collision parameter. For hard collisions the cross section is given by Mott's formula.
SM(T, E)
(Pi
q 3(T ~ - E) 4 (E + B,)(T - Ej
@ r + Bi(l + ai)
1 4B~ G i 1 ( E + B~)2 + 3(E " - - - -+~ 3B~) -t ( T - E) 2
xle+FT-E-B,]'/2] L bTkT j j
x[l
E-I'-Bi]G'/(a'+I+E/ai)],~ j j,
(9)
where p~ = ( r / a , Oi) ':2 2.1.2. Multiple scattering. In order to model the passage of radiation through a scattering medium to a volume of interest, such as an ampoule, in an economic fashion, only those interactions leading to electrons or photons with energy larger than 5 keV (chosen to be consistent with cut-off used inside the ampoule) were considered. The paths of the electrons between each interaction were divided into smaller steps, each with negligible energy loss, in which the electrons were assumed to follow a straight line. The energy loss of the electrons was determined from the restricted collision stopping power (ICRU, 1984) with an energy cutoff of 5 keV, and multiple scattering was taken into account by changing the direction of the electron at the end of each step using an algorithm presented in Kuhn and Dodge (1992). This algorithm is an approximation to the analytic description of multiple scattering by M oli6re (1948) and provides a proper treatment of large angles as well as a fast generation of random scatter angles. The step sizes were kept small enough to avoid the introduction of significant errors and were determined either by the distance to the next 6-ray (> 5 keV), where the energy and angular descriptions of primary and secondary electrons used sampling methods which were based on those given by Messel and Crawford (1970) for M~ller scattering, or by the distance to the next bremsstrahlung interaction ( > 5 keV). Inside the volume of interest, the electrons were followed interaction by interaction and not by the use of multiple scatter transport. 2.2. Photon interactions The distance travelled by each photon before interaction was determined by the total mass attenuation coefficient for water taken from Hubbell (1977), while the type of interaction taking place was determined by the relative magnitudes of partial cross sections
640
M.A. Hill and F. A. Smith
(Hubbell, 1977) for water calculated from the values for atomic hydrogen and oxygen using the independent atom model.
Direct calculations of the ferric yields G(Fe3:+ ) for 1 MeV short track sections calculated by PETS (Hill and Smith, 1994) were found to have a wide range of values depending on the assumptions made.
3. FRICKE D O S I M E T E R
• 15.7-17.2 for vapour compared to liquid cross sections. • 14.5-17.2 using spatial distributions of hydrated electrons and radicals. • 16.2-20.5 using different vibrational cross sections. • 14.9-17.2 when the ionisation potential is reduced by ~ 4 eV for liquid water compared to the use of water vapour potentials. • 13.2-18.5 for a decrease of pH from 7 to 0.
The ampoule was modelled simply as a volume of water with a radius, ra of 0.6 cm, positioned at a depth, da, of 5 cm for photons and I cm for electrons in a 25 cm thick water phantom. The electron or photon beam was assumed to have infinite width and to travel parallel to the x-axis with the ampoule, positioned parallel to the z-axis and centred on the x-axis at a depth, da. Electrons at energies greater than 5 keV (Mozumder and Magee, 1966) produce isolated clusters of chemical species at inelastic interaction sites of the primary electron. Reactive species produced in one cluster are therefore very unlikely to react with species from another. Since the csda range of a 5keV electron is < 0 . 8 # m in water, 5keV provides a suitable cut-off energy below which electrons outside the ampoule can be ignored, while clusters produced inside the ampoule by electrons with energy < 5 keV are assumed to remain within the ampoule. The broad beam data were obtained from single tracks by using a geometric weighting factor which was applied to individual events at depths corresponding to that of the ampoule, i.e. from d~ - ra to da + r~ to give the relative probability that the event will be inside the ampoule. The weighting factor wg= [ r 2 - ( x - d ~ ) ~ ] 1/2 is related to the probability that an event produced by a track randomly positioned along the y-axis will actually be within the ampoule. The quality of the beam can be expected to change slightly with depth, however with a water equivalent ampoule charge particle equilibrium is expected perpendicular to the beam. Radiation lost from one side will be compensated by incoming radiation on the other, therefore the infinite width assumption is justified. 4. C A L C U L A T I O N O F F R I C K E Y I E L D S F R O M PRIMARY YIELDS
5. I S O L A T E D C L U S T E R S A N D FERRIC G - V A L U E S
Due to the complexity of, and sensitivity to, the assumptions involved in the above calculation of primary yields, a more analytical approach was used here. This allowed a much more detailed analysis of the reasons behind the above variations because of the use of better statistics. The method uses the theoretical ferric energy dependence of Magee and Chatterjee (1978) who used the prescribed diffusion model and attempted to include the variation in the initial spatial distribution of chemical species with energy by separating the tracks into events with energy < 100 eV, 100-1600 eV, 1600-5000 eV and >5000eV (Fig. 1). Also included in Fig. 1 are calculations by Yamaguchi (1987, 1989) for electrons and photons, who also used the prescribed diffusion model and track structure concepts along with exper2 °5
....... 1
...... 1
...... 1 .....
t
,~, 2 . 0 co
~ -~
Ferric ion yields in the presence and absence of oxygen can be calculated from the primary yields ( ~ 10 -7 s) following irradiation, using the relations (Magee and Chatterjee, 1978):
. . . . . . -i
1.5
ol _~ ~ 1.0
G(Fe3+)N2 = 2GH2o2+ GOH + (GH + G~_) (10) G(Fe3+)o2 = 2GH:o~ + GoH + 3(GH + Ge-)
(11)
Figure 1 shows the variation in the Fricke yield with energy previously calculated (Hill and Smith, 1994) by following the passage of electrons to sub-excitational energies and then following the subsequent chemistry of the radicals produced up to 10 -7 s. The calculations shown are for complete and short track sections where the energy loss of the initial electron is minimal.
0.5 101
10 2
10 3
10 4
10 5
0s
Energy (eV) Fig. 1. Energy dependence of G(Fe3+) yields for complete tracks: Magee and Chatterjee (1978) (--), Yamaguchi electrons (1987) (---), photons.(1989) (. . . . -), Hill and Smith (1994) ( . . . . ), (O complete tracks, • short track sections); and experimental A, [] ICRU (1970), ~7 Freyer (1989) and <> Hoshi et al. (1992)
641
Is the response of the Fricke dosimeter constant? imental values for X-rays (ICRU, 1970; Feyer et al, 1989; Hoshi et al., 1992). Mozumder and Magee (1966) originally compared different energy depositions, AE, in terms of spurs ( < 100 eV), blobs (100 eV ~< AE < 500 eV) and short tracks (500 eV ~
Calculations have been carried out to investigate possible differences in the chemical response with energy of the Fricke dosimeter between photon and electron beam irradiation. A description of clusters in terms of the excitation type and the secondary elecRPC 47:4--1
tron energies, has been used both in the calculation of the final ferric yield and also in the determination of the importance of different cluster types. The following situations were considered: • Complete tracks and an ampoule in a water scattering medium. • The dependence on initial photon and electron energies, especially for 6°Co 5'-rays, was determined for the two cases above. (Energy was assumed to be a 6-function containing none of the scattered radiation found in practice.) • The importance of the energy distribution of the secondary electrons from ionisation events. • Ferric energy dependence of Magee and Chatterjee (1978) compared to that of Yamaguchi (1987). 7. R E S U L T S
7.1. Complete tracks Calculations were carried out for photon and electron beams with incident energies ranging from 5 keV up to l0 MeV, assuming complete absorption of the beam. Using the assumptions made in Section 5, the associated variations in the calculated ferric yields with incident energy for photons and electrons are shown in Fig. 2, along with the relative contribution of each of the cluster types to the total ferric yield and the fractional energy loss to track entities (spurs, blobs and short tracks). The calculated ferric yields for incident electrons increases with incident energy, the energy dependence being more marked at the lower energies. At high and low energies the variation in the calculated yield for incident photons is very similar to that of electrons. However, the absolute values for photons are consistently lower than those for incident electrons, although the difference becomes less significant at higher energies. Between 20 keV and 2 MeV the calculated ferric yields for photons are significantly lower than those for electrons, reaching a maxmum at ~ 30 keV followed by a minimum at ~ 80 keV. This feature results from changes in the mean initial electron energy brought about by the importance of Compton scatter in this energy region. Below 20 keV, photoelectric absorption dominates, producing electrons with energies nearly equal to the photon energy. At higher energies, Compton scatter becomes important and the average initial electron energy produced by photon interactions decreases due to the lower energy of the Compton electrons compared with that of the photo-electrons. The average electron energy increases again at higher photon energies due to the increase in the mean Compton electron energy with a greater fraction of the photon energy being transferred to the electron. Changes in the secondary electron distributions lead to similar variations in the calculated ferric yields, but vary in their absolute values. The increase in the ferric G-value with incident energy, calculated
642
M.A. Hill and F. A. Smith 2.0 1.8
"'"1
. . . . . . . .
I
. . . . . . . .
I
(o)
1.0
. . . . . . . .
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....
!
. . . . . . . .
I
. . . . . . . .
!
. . . . . . . .
I (b)
0.8 o
-~o 1.4
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spurs
0.7
~"
1.2
0.6
=
1.0
0.5
0.8
0.4
0
I 0
-~ 0 I=
"
0.6
o
4
0.3
t,
0.4
0.2
0.2
0.1
0.0
0.0
10 4
10 5 Incident
10 s
10 7
energy (eV)
10 4
10 5
10 e
10 7
Incident energy (eV)
Fig. 2. Calculations of complete electron (--) and photon ( ' . . ) tracks (6°Co= A) using Jain-Khare (1976) secondary electron energy distribution (a) G(Fe3÷) yields [contributions from (1) e-E < 8.8 eV; (2) e-8.8eV< E < 5keV; (3) isolated excitations excluding plasmons; (4) plasmon excitations; and (5) "isolated" ionisations], (b) fractional energy loss to spurs blobs, and short tracks.
using the Green and Sawda (1972) distribution, is smaller than that obtained using the Jain and Khare (1976) distribution, and this also results in a smaller difference between photons and electrons. As the initial electron energy increases, more of the energy is deposited by lower energy secondary electrons and isolated excitations and this is reflected in an increase in spurs and blobs at the expense of short tracks. The contribution of clusters produced by electrons in the energy range 8.8 eV-5 keV (line 2) falls with increasing incident energy but this is more than compensated by the increase in the contribution by isolated interactions (lines 3 and 5), the plasmon in particular (line 4), and clusters produced by low energy electrons (line 1). 7.2. Ampoule in water scattering medium
The ferric yields calculated for photons and electrons of various incident energies (1.0-25.1 MeV) for an ampoule in a water scattering medium using the four different secondary electron energy distributions are compared in Fig. 3(a). The variations in the fractional energy loss to each track entity, along with the cluster contributions to the calculated ferric yield using the Jain-Khare (1976) distribution are shown graphically in Fig. 4. The differences in the calculated total ferric yields between 10 MeV electrons, 10 MeV photons and 6°Co ?-rays are given in Table 1 along with results obtained using the Yamaguchi (1987) Fricke yield vs energy relationship. Yields calculated using the Jain-Khare (1976) and the Gryzinski (1965) distributions (Fig. 3) rise with increasing incident energy, with photon yields being
lower than those of electrons. Yields calculated using the Green-Sawada (1972) and Vriens (1966, 1969) distributions are relatively independent of incident energy and the values obtained for photons and electrons are approximately equal. It should be noted that the distributions of Vriens and Gryzinski, which are both based on the binary encounter model, lead to very different results. It is clear from Fig. 3 however that the secondary electron energy distribution is important not only in determining the absolute yields with incident energy ( ~ 15%) but also the relative variation in these yields with incident energy as well as the relative difference in yields between photon and electron irradiation. The four energy distributions for a 1 MeV primary electron are compared in Fig. 5(a). The variation with primary energy, T, of the mean secondary electron energy, E2, and the mean energy of cluster-forming secondary electrons (E < 5 keV), Ecj, resulting from a single ionisation event is shown for the four energy distributions in Fig. 5(b). (Below ~ 10 keV the two curves for a particular distribution are identical since the maximum energy of the secondary electron is defined as ½(T - I) where I is the ionisation potential.) Distributions which lead to a reduction in the average energy Eel with increasing primary energy (e.g. Jain-Khare, 1976; Gryzinski, 1965) show increases in the calculated ferric yield with incident energy, as well as differences in the yields calculated for photons and electrons. The yields calculated for the Green-Sawada (1972) and Vriens (1966, 1969) distributions are relatively constant and so are the values of Ec~ for primary energies above 1 MeV.
643
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25
I
10 e
10 7 incident energy
I
I
10 e
10 7
(eV)
Incident energy
(eV)
Fig. 3. Calculation following electron (--) and photon (.. • ) tracks (6°Co = &) of varying incident energies through a scattering medium into an ampoule, using the secondary electron distributions of O Jain-Khare (1976), [] Green-Swada (1962), O Vriens (1966, 1969) and V Gryzinski (1965); (a) G(Fe 3÷) yields, (b) mean energy of cluster forming electrons (~< 5 keV) inside ampoule.
The variation with incident radiation energy of the calculated mean energy of cluster forming electrons ( < 5 keV) which enter, or are produced directly inside, the ampoule is shown in Fig. 3(b). The number of Fe 3+ ions produced by a particular cluster can be obtained using the Magee 0.75
I
.
.
.
.
.
.
.
.
and Chatterjee (1978) distribution (Fig. 1). The change in the average secondary electron energy is clearly important at these low energies since the ferric yield of clusters produced by electrons increases significantly for smaller and smaller energies (Fig. 1). 0.80
I
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Fig. 4. Calculation following electron (--) and photon (. •. ) tracks (6°Co = &) of varying incident energies through a scattering medium into an ampoule, using the Jain and Khare (1976) energy distribution. (a) Fractional energy loss to track entities. (b) Contributions of clustes to calculated G(Fe 3+) yield [(I) e - E < 8.8 eV; (2) e-8.8 eV < E < 5 keV; (3) isolated excitations excluding plasmons; (4) plasmon excitations; and (5) "isolated" ionisations].
644
M. A. Hill and F. A. Smith
Table 1. Comparison o f ferric yields between photon and electron beams calculated in an ampoule for the four secondary electron energy distributions used with the ferric energy relationships o f Magee and Chatterjee (1978) and Yamaguchi (1987). Also shown are the percentage differences between the photon yields and those for I0 MeV electrons
10 MeV electrons (/~ mol J ')
Distribution Magee and Chatterjee (1978) Jain-Khare Green-Sawada Vriens Gryzinski Yamaguchi (1987) Jain-Khare Green-Sawada Vriens Gryzinski
10 MeV photons (# mol J i)
1.7183 1.5480 1.6465 1.8782
+ 0.0007 -t- 0.0005 + 0.0006 _+ 0.0007
1.704 1.548 1.647 1.869
-~: 0.002 ± 0.002 + 0.002 ± 0.002
- 0.8% 0.0% 0.0%° -0.5%
1.667 1.545 1.647 1.841
+ 0.003 ± 0.002 +_ 0.002 ± 0.003
- 3.0% -0.2% 0.0% -2.0%
1.5937 1.5059 1.6036 1.7781
± ± ± ±
1.584 1.506 1.604 1.771
_+ 0.005 ± 0.004 _+ 0.004 + 0.005
-0.6% 0.0% 0.0% -0.4%
1.561 1.504 1.603 1.750
+ 0.005 _+ 0.004 ± 0.004 ± 0.006
-2.1%o -0.1% 0.0% - 1.6%
0.0005 0.0003 0.0005 0.0005
What is important is the energy spectrum of electrons ( > 5 keV) entering, or produced in, the ampoule, which we refer to as encounter electrons. The encounter electron spectra for an incident electron beam will still be relatively high in energy and fairly monoenergetic, even after taking into account the collision losses and the energy straggling during its passage to the ampoule. By comparison, the encounter electron energies for a photon beam will cover a range from zero up to an energy just below the incident photon energy, with a mean which is less than that for electrons of the same primary energy. This is mainly a result of the dominance of Compton scatter events which themselves produce a broad spectrum of electrons in addition to the fact that these electrons may be created anywhere inside or outside the ampoule. For decreasing incident energies ( < 4 MeV) the average energy of the electrons entering the ampoule i 0 -2
. . . . . . . .
i
. . . . . . . .
6°Co ),-rays (# mol J ')
i
. . . . . . . .
beings to fall significantly due to the loss of energy in the water phantom and the finite size of the ampoule, For example, 2 MeV incident electrons with a csda (continuous slowing down approximation) range of 0.98 cm will enter the ampoule with energies ranging from zero to ~ 1.3 MeV and are likely to be completely stopped within the ampoule. From Fig. 1 this would lead to a corresponding reduction in the Fricke yield with decreasing incident energy. For photons in this energy region this is not the case however, due to the relatively large distances travelled between photon interactions which result in a spectrum of electrons entering the ampoule from zero to just below that of the incident photon. Since, at the higher incident energies (>4MeV), the average energy of the encounter electrons for photons is less than that for electrons, the energy variation of the ferric yield for photons should also reflect the ferric yield response for electrons at lower 120
,
(a)
........ i
........ !
....... 1
. . . . . . . -.
.......
- - -||.d
........ = ....... J
....... J
.......
(b) 100 I) C
,-
80
U e,,
"~
o
'~
0,.
10 - 3
60
0
\
i.
,°
/ ,
g ~
20
> 0
10-1
.... |
10o
........
i
101
....
~1..i
102
eecondary electron energy (eV)
103
O
103
10 ~
10 5
10 8
10 7
primary electron energy (4V)
Fig. 5. Comparison of the secondary electron energy distributions for single ionisation events of Jain-Khare (1976) (--), Green-Sawada (1972) (---), Vriens (1966, 1969) (...) and Gryzinski (1965) (-. - . -): (a) probability distributions for a 1 MeV primary electron; (b) variation in mean secondary electron energy with primary energy, the lower of the two lines for each distribution indicates the mean energy of cluster forming electrons (<5 keV).
10 8
Is the response of the Fricke dosimeter constant? energies. Therefore if the ferric yield for an electron beam increases with incident energy before becoming energy independent, it would be expected that a photon beam would behave in a similar manner, but with yields which are lower than those for an electron beam of equivalent incident energy which would approach the energy independent value for electrons at a higher energy. This reasoning is borne out by ferric response graphs shown in Fig. 3(a). 8. DISCUSSION
Calculations of the energy dependence in total ferric yield following complete absorption of all primary and secondary radiation in a Fricke solution are different for electrons and photons. Electrons show an increase with incident energy, becoming less energy dependent at higher energy [Fig. 2(a)]. Yields for photons are consistently lower than those for electrons, the variations being similar to that for electrons but showing a minimum at ~ 80 keV before rising at higher energies to give values approaching those for electrons. Absolute values of the yields and their energy variation depend on the secondary electron energy distributions which are used to model the ionisation events. The greater the variation in the ferric yield with energy for electron beams, the greater the differences between photon and electron beams. The main contribution to the yield at high incident energies comes from the clusters produced by low energy secondary electrons ( < 100 eV) and isolated plasmon excitations induced by high energy electrons ( > 5 keV). Ferric yields calculated for an ampoule positioned in a phantom are slightly higher than those obtained for complete tracks of the same energy. Using secondary electron distributions of Jain-Khare (1976), Green-Sawada (1972), Vriens (1966,1969) and Gryzinski (1965), the differences between calculated yields in an ampoule for 10 MeV incident energy photon and electron beams are 0.8, 0, 0 and 0.5% respectively; and between 6°Co 7-rays and 10 MeV electrons, 3.0, 0.2, 0 and 2.0%, the electron yield being higher in all cases. The variations of the ferric yield with incident energy is critically dependent on the way the secondary electron distribution varies with primary electron energy. Distributions which produce a reduction in the average secondary energy of cluster-forming electrons ( < 5 keV) with increasing primary energy (e.g. Jain-Khare and Gryzinski) cause an increase in the calculated ferric yield with incident energy and produce differences in the yields for photons and electrons. This sensitivity is probably the result of a large negative energy dependence of the ferric yield at low energies (Fig. 1) and the prominence of low energy electrons ( < 100 eV). The energy response of the ferric yield for a photon beam will be similar to that for electrons at lower
645
energies since photon beams produce electrons over a wide range of energies with an average which is lower than the incident photon energy, due to the importance of Compton interactions. The photon results show no evidence of a maximum at energies in the region of 0.66 MeV associated with the mCs 7rays as predicted by Nahum et al. (1981). The finite size of the ampoule is an important factor in the energy dependence of the ferric yield for electrons entering the ampoule with energies of the order of 4 MeV and below. One area of uncertainty in our calculated ferric yields is the reliabillity of the energy dependence of the theoretical ferric G-value calculated by Magee and Chatterjee (1978), especially at the lower energies (<100eV). This uncertainty is illustrated by the difference in the Magee and Chatterjee curve to that calculated by Yamaguchi (1987) (Fig. 1). Although the absolute yields obtained are different, a similar variation with energy and similar differences between photon and electrons are observed (Table 1). Simplifying assumptions made in the present calculations have also ignored the possibility of recombination of dissociation events due to the "cage effect" where the surrounding molecules help prevent separation. All secondary electrons with energy below the ionisation potential (~<8.8 eV) are asssumed to produce 3 Fe 3~ ions, and in reality, a fraction of these will recombine with their parent ions and lead to a reduction in the calculated G-value. Successful modelling of the chemical response following irradiation of high energy electrons and photons requires an accurate description of the chemical development of the clusters of low energy electrons from < 5 - 1 0 k e V down to sub-excitational energies to determine the energy variation of the ferric yield at these lower energies. This can be achieved either by using an analytic or a Monte Carlo track structure approach and should ideally be compared to experimental results at these low energies. This approach could be further used in conjunction with "condensed-history" Monte Carlo codes to transport electrons into a walled chamber in which the chemistry is followed. 9. CONCLUSIONS
This work has attempted to investigate the link between the actual track structure, as opposed to the mean LET, of an electron and the chemical (radical) yields in determining the response of the Fricke dosimeter. Four secondary electron distributions have been compared and shown to alter the differences in the Fricke response, not only between electrons and photons but also for different primary energies. The relative variation in the Fricke G-value is critically dependent on the variation on the mean energy of the secondary electron following ionisation with primary energy and, more specifically, on the corresponding variation in the mean energy of cluster
646
M . A . Hill and F. A. Smith
forming electrons (i.e. energy < 5 keV) in the ampoule. This dependence is due to a n expected strong variation in Fricke yield for low energy electrons ( < 100 eV). A question still remains o n the size of the effect, b u t experimental results to date do not rule o u t variations o f up to 1%. The possibility o f the varia t i o n in G ( F e 3+) with b e a m quality must be b o r n e in m i n d w h e n using or analyzing the results o f Fricke dosimetry. F u r t h e r experimental investigtions are therefore needed to try a n d resolve the secondary electron spectrum variations. Acknowledgements--The authors are grateful to the National Physical Laboratory for their financial support under EMRA contract NPL 82/A/0523. REFERENCES
AAPM (1983) A protocol for the determination of absorbed dose from high-energy photon and electron beams. Med. Phys. 10, 741. Bethe H. and Heitler W. (1934) On the stopping of fast particles and on the creation of positive electrons. Proc. R. Soc. A 146, 83. Brenner D. J. and Zaider M. (1983) A computationally convenient parameterization of experimental angular distibutions of low energy elastically scattered off water vapour. Phys. Med. Biol. 29, 443. Burch P. R. J. (1959) A theoretical interpretation of the effects of radiation quality on yields in the ferrous and cerric sulphate dosimeters. Radiat. Res. 11, 481. Davisson C. M. and Evans R. S. (1952) Gamma-ray absorption coefficients. Rev. Mod. Phys. 24, 79. Freyer J. P., Schillaci M. E. and Raju M. R. (1989) Measurement of the G-value for a 1.5 keV X-rays. Int. J. Radiat. Biol. 56, 885. Green A. E. S. and Sawada T. (1972) Ionisation cross sections and secondary electron distributions. J. Atmos. Terr. Phys. 34, 1719. Grosswendt B. and Waibel E. (1978) Transport of low energy electrons in nitrogen and air. Nucl. Instrum. Methods 155, 145. Gryzinski M. (1965) Two-particle collisions. II. Coulomb collisions in the laboratory system of coordinates. Phys. Rev. A 138, 322. Hill M. A. and Smith F. A. (1994) Calculation of initial and primary yields in the radiolysis of water. Radiat. Phys. Chem. 41, 265. Hoshi M., Uehara S., Yamamoto O., Sawada S., Asao T., Kobayashi K., Maezawa H., Furusawa Y., Hieda K. and Yamada T. (1992) Iron (II) sulphate (Fricke solution) oxiation yields for 8.9 and 13.6 keV X-rays from synchrotron radiation. Int. J. Radiat. Biol. 61, 21. Hubbell J. H. (1977) Photon mass attenuation and energyabsorption coefficients for H, C, N, O, Ar, and seven mixtures from 0.1 keV to 20 MeV. Radiat. Res. 70, 58. Hubbell J. H., Veigele Win. J., Briggs E. A., Brown R. T., Cromer D. T. and Howerton R. J. (1975) Atomic form factors, incoherent scattering functions, and photon scattering cross sections. J. Phys. Chem. Ref. Data 4, 471. ICRU (1969) Radiation Dosimetry: X-rays and Gamma Rays with Maximum Photon Energies Between 0.6and 50 MeV. ICRU Report 14, International Commission on Radiation Units and Measurements, Bethesda, Md, U.S.A. ICRU (1970) Radiation Dosimetry: X-rays Generated at a Potential o f 5-150kV. ICRU Report 17, International Commission on Radiation Units and Measurements, Bethesda, Md, U.S.A. ICRU (1984) Radiation Dosimetry: Electron Beams with Energies Between I and 50 MeV. ICRU Report 35, Inter-
national Commission on Radiation Units and Measurements, Bethesda, Md, U.S.A. Jain D. K. and Khare S. P. (1976) Ionizing collisions of electrons with CO 2, CO, H20, CH 4 and NH 3. J. Phys. B: Atom. Molec. Phys. 9, 1429. Kuhn S. E. and Dodge G. E. (1992) A fast algorithm for Monte Carlo simulations of multiple Coulomb scattering. Nucl. lnstrum. Methods Phys. Res. A322, 88. Ma C.-M. and Nahum A. E. (1993a) Dose conversion and wall correction factors for Fricke dosimetry in highenergy photon beams: analytical model and Monte Carlo calculations. Phys. Med. Biol. 38, 93. Ma C.-M. and Nahum A. E. (1993b) Correction factors for Fricke dosimetry in high-energy electron beams. Phys. Med. Biol. 38, 423. Ma C.-M., Rogers D. W. O., Shortt K. R., Ross C. K., Nahum A. E. and Bielajew A. F. (1993) Wall-correction and absorbed-dose conversion factors for Fricke dosimetry: Monte Carlo calculations and measurements. Med. Phys. 20, 283. Magee J. T. and Chatterjee A. (1978) Theory of the chemical effects of high-energy electrons. J. Phys. Chem. 82, 2219. Messel H. and Crawford D. F. (970) Electron-Photon Shower Distribution Function. Pergamon Press, Oxford. Moli6re G. Z. (1948) Theorie der streuung schneller geladener teilchen II. Mehrfach-und vielfachstreuung. Z. Naturforsch 3a, 78. Mozumder A. and Magee J. L. (1966) Model of tracks of ionizing radiation for radical reaction mechanisms. Radiat. Res. 28, 203. Nahum A. E., Svensson H. and Brahme A. (1981) The ferrous sulphate G-value for electron and photon beams: A semi-empirical analysis and its experimental support. Seventh Symp. on Microdosimetry (Edited by Booz J., Ebert H. G. and Hartfiel H. D.) p. 841. Harwood Academic. Pimblott S. M., LaVerne J. A., Mozumder A. and Green N. J. B. (1990) Structure of electron tracks in water. 1. Distribution of energy deposition events. J. Phys. Chem. 94, 488. Porter H. S. and Jump F. W. (1978) Computer Science Corp. Report, CSC/TM-78/017. Seltzer S. M. and Berger M. J. (1986) Bremsstrahlung energy spectra from electrons with kinetic energy 1 keV-10 GeV incident on screened nuclei and orbital electrons of neutral atoms with Z = 1-100. Atomic Data Nuclear Data Tables 35, 345. Schutten J., de Heer F. J., Moustafa H. R., Boerbom A. J. H. and Kistemaker J. (1966) Gross and partial cross sections for electrons on water vapour in the energy range 0.1-20 keV. J. Chem. Phys. 44, 3924. Tan K. H., Brion C. E., Van der Leeuw Ph. and Van der Wiel M. J. (1978) Absolute oscillator strengths (10~0 eV) or the photoabsorption, photoionisation and fragmentation of H20. Chem. Phys. 29, 299. Vriens L. (1966) Electron exchange in binary encounter collision theory. Proc. Phys. Soc. (London)89, 13. Vriens L. (1969) Case Studies in Atomic Physics VoL L p. 337. North Holland, Amsterdam. Yamaguchi H. (1987) A prescribed diffusion model of a many-radical system considering electron track structure in water. Radiat. Phys. Chem. 30, 279. Yamaguchi H. (1989) A spur diffusion model applied to estimate yields of species in water irradiated by mono-energetic photons of 50 eV=2 MeV. Radiat. Phys. Chem. 34, 801. Zaider M., Brenner D. J. and Wilson W. E. (1983) The applications of track calculations to radiobiology I. Monte Carlo simulation of proton tracks. Radiat. Res. 95, 231. Zerby C. D. (1963) A Monte Carlo calculation of the response of gamma ray scintillation counters. Meth. Comp. Phys. 1, 89.
Is the response of the Fricke dosimeter constant?
is ignored since the fluorescence yield for oxygen is less than 1%.
APPENDIX Brief details of the extension to the Monte Carlo code (Hill and Smith, 1994) to incorporate photon interactions are given below. Bremsstrahlung High energy electrons passing close to an atomic nucleus may be decelerated with the production of bremsstrahlung photons having a continuous energy, k, ranging from zero up to the incident electron energy E. A low energy cutoff, k e = 8.8 eV, has been used to represent the lowest ionisation threshold in liquid water, since only photons with greater energy will lead to ionisation via the photoelectric effect. The dependence on incident electron energy of the differential bremsstrahlung cross sections was obtained for atomic hydrogen and oxygen from tables given in Seltzer and Berger (1986) and the relevant cross sections and distributions for water were calculated by assuming the independent atom model (i.e. all2o = 2Oa + a0). These data were used to give the appropriate photon energy distributions from k~ to E from which the total cross section was obtained by integration over the photon energy range. trb~m = f f c ~ da dk
(AI)
where k is the photon energy. The angular distribution was assumed to be forward peaked with a mean polar angle given by 0brem =
moC2/(mo)C2 + T
647
(A2)
Photoelectric effect The photoelectric effect involves the total absorption of a photon and the emission of an electron with an energy described by the Einstein equation, E = hv - I B, where hv is the incident photon energy and I Bthe binding energy of the molecular electron involved in the interaction. Photons with energy greater than the binding energy of the oxygen K-shell electrons, I K = 539.7eV, were assumed to interact solely with these electrons. Otherwise a binding energy of 1L = 19.6eV averaged over the remaining electrons was used. Values of 1K and 1L were taken from the appropriate ionisation thresholds for water given in Zaider et al. (1983). The angular distribution of low energy photon-electrons (B < 0.9) were obtained using non-relativistic distributions due to Fischer, and for higher energies the relativistic distributions of Sauter (Davisson and Evans, 1952). Following the emission of a K-shell electron, de-excitation of the highly excited water ion was assumed to take place via the isotropic production of an Auger electron with energy of 520.1 eW (1K - 21L). Characteristic X-ray emission
Cornpton scatter The absorption of the initial photon leads to the production of an electron and a second photon with reduced energy and a changed direction. The simulation of Compton scatter events was based on the Klein-Nishina formula multiplied by the incoherent scattering function to account for electron binding. dtr~omp _ da~ N S(x) dO dO
(A3)
The incoherent scattering function S(x), where x = sin(0/2)/2(/~,), was obtained from Hubbell et al. (1975) for atomic hydrogen and oxygen and calculated for water using the independent atom model. It was also assumed in the program that S(x ), which for water varies between 0 and 10, indicates the number of electrons available for interaction. The probability of a photon interacting with an oxygen K-shell electron was assumed to be given by [S(x) - 8]/10, while the energy of the Compton electron was reduced by the appropriate binding energy, either 539 eV or 19,6eV. Following the emission of a K-shell electron, de-excitation was again assumed to take place via the isotropic production of an Auger electron with energy 520.1 eV. Pair production Pair production becomes possible for photons with energy greater than 2moc: and results in the disappearance of the photon and the creation of an electron-positron pair in the fields of a nucleus. The distribution of the remaining energy E;.-2mo c2 between the electron and positron was determined using the method given by Zerby (1963) which gives a very good approximation to the Bethe-Heitler expression for pair production below 10MeV. The polar scattering angle 0+ of the positron and electron with respect to the direction the photon was taken from the simplistic relation (Bethe and Heitler, 1934) 0+_ = moc2/(rno c2 + T+ )
(A4)
The azimuthal angle was assumed to be uniformly distributed and differed by n for the electron and the positron. The positron were followed in an identical manner to the electrons and annihilation assumed to take place only after the positron has come to rest. Two 0.511 MeV photons are assumed to be emitted, isotropically distributed and opposite to one another. Triplet production occurring in the held of an atomic electron was treated in an identical manner to that of pair production.
Pergamon
IS THE RESPONSE OF THE FRICKE DOSIMETER CONSTANT FOR HIGH ENERGY ELECTRONS A N D PHOTONS? M. A. HILL and F. A. SMITH Department of Physics, Queen Mary and Westfield College, University of London, Mile End Road, London El 4NS, U.K. (Received September 1994; accepted Janua O, 1995) Al~traet--An attempt has been made to theoretically model the chemical response of the Fricke dosimeter using track structure concepts and to indicate areas in which the modelling can be improved. The Monte Carlo technique has been used to investigate the differences in the fundamental chemical response of the ferrous sulphate Fricke dosimeter to photon and electron irradiation. Calculations of the fractional energy loss in the track entities (spurs, blobs and short tracks) were made, together with more quantitative descriptions of these clusters which were used to estimate the final ferric yield. Photons and electrons of various energies were considered for an infinite medium and for a water equivalent ampoule in a water scattering medium, using four secondary electron energy distributions. Emphasis was placed on the variation in yields rather than in the absolute values. The results show that the relative differencesin the Fricke G-value, not only between electrons and photons, but also with primary energy are critically dependent on the variation in the mean energy of the secondary electron followingionisation with primary energy. This dependence is due to an expected strong variation in Fricke yield for low energy electrons (< 100 eV).
I. INTRODUCTION The motivation for this work was the investigation of possible fundamental differences in the response of the Fricke dosimeter between photon and electron irradiation, since in dosimetry practice it is tacitly assumed that any dependence of the Fricke G-value is of no consequence and that a primary standard calibrated with high energy photons for example can be used for routine work with high energy electrons. The investigation has been carried out using Monte Carlo codes written at QMW to calculate the variation in radical and ferric yields and the variation in track entities with energy for photons and electrons. The aim was to try to understand the link between the initial physical interaction of the photons and electrons, the subsequent distributions of radicals produced and the final ferric yield. The present knowledge of the physical and chemical events at very early times following irradiation is insufficient to enable the computation of accurate absolute yields. However, Monte Carlo calculations of this type are still useful for following the relative variations in yields between electrons and photons which deposit energy via electrons. The ferrous sulphate Fricke dosimeter has been recommended by both the AAPM (1983) and ICRU (1984) as an alternative method for determination and calibration of absorbed dose in water. It is used
by the National Research Council, Canada (NRC) and the Physikalische Teschnische Bundesantalt of Germany (PTB) as part of a primary standard and by the National Physical Laboratory, U.K. (NPL) and the National Institute for Standards and Technology, U.S.A. (NIST) as part of a calibration service for clinical accelerators. Exposure to radiation results in the oxidation of the ferrous ions to ferric ions. This is directly proportional to the absorbed dose if certain conditions are met (ICRU, 1969) and can be measured spectrophotometrically, Of = AA /(plEmG )
(1)
where D r is the dose to the Fricke solution, AA is the increase in absorbance due to irradiation, p the density of the Fricke solution, l the length of the light path in the photometer cell, E,, the molar linear absorption coefficient and G the radiation chemical yield of ferric ions. A constant value of EmG, equal to 352 x 10-6m2kg -l Gy -l (G = 1.60/~ molJ -1 assuming em=219.5m2mol-~), is recommended for both photons and electrons with mean elecron energies from 1 MeV up to at least 30 MeV (ICRU, 1984). The mean absorbed dose in water can then be obtained by multiplying with correction factors which account for a non-water equivalent detector and wail. These have recently been investigated by Ma and Nahum (1993a, b) for high energy photons and electrons and help explain discrepancies between experimental results.
637
638
M.A. Hill and F. A. Smith
The purpose of this paper is to quantify the likely errors incurred in Fricke Dosimetry due to the assumption of a constant G-value, when all aspects of radiation track chemistry are taken into account. Although there has been considerable support for a Fricke G-value which is constant to within 1% or better for energies about 1 MeV (ICRU, 1984), extensive re-evaluation of recent experimental results do not rule out an energy dependent G(Fe 3+) and may explain the discrepancies between experimental data and Monte Carlo calculations of certain correction factors (Ma and Nahum, 1993a, b; Ma et al., 1993). However these discrepancies could also be explained by systematic errors in the ionisation chamber dosimetry. A theoretical analysis of the ferrous sulphate Gvalue was developed by Burch (1959). Using experimental values of G(Fe 3+) for 21°po~- and 6°Co y-radiations and the Jaff6 recombination formulae, a relationship between G(Fe 3÷) and LET100 was used to define "local" energy deposition, with all energy transfers greater than 100 eV being treated as &-rays with their own rate of energy loss. This relationship was used with the local energy dissipation spectra to compute G(Fe 3+) using different values of LET100 and weighting them over the spectrum of local energy deposition. A similar approach was used by Nahum et al. (1981) using Monte Carlo simulations of photon and electron transport to derive the local energy deposition spectra which was used to empirically derive, by "trial and error", a G(Fe 3÷) vs LET100 relation. Because the approach used was empirical, the computed G(Fe 3÷) yields for high energy photons and electrons h a d to agree well with the experimental values to which they had been fitted: X- and v-rays for low LET and ct particles for high LET. The computed variations in these yields with energy was a 0.8% increase with electron energies from 5 to 30 MeV and a 1% decrease with mean energy of primary photons from 0.66 to 14.5 MeV. However, the above approach does not take into account the complex mechanisms leading to changes in chemical yields in the Fricke dosimeter, especially when considering low energy ( < 5 keV), high LET secondary electrons, through which a large fraction of energy is deposited. Calculations by Pimblott et al. (1990) found that for a single collision of a 1 MeV electron in liquid water 50% of all energy loss is due to events of 43 eV or less, 10% of the total energy loss is due to events of greater than 600eV and <0.2% to events greater than 5 keV. Due to increased scatter at low energies and decrease mean free paths, an increasingly spherical distribution of interactions is obtained, which are close enough together to allow reactions to take place between chemical species produced at these different sites and which are critically dependent on their spatial distributions. For these low energy electrons the chemical response is dependent on the whole track leading to a significant variation with energy. In this regard the determi-
nation of the chemical response will require a cut-off energy much greater than 100 eV. Low energy electrons cannot be expected to show the same chemical response as ct particle of an equivalent LET due to the differences in the shape of the track structure. Therefore the approaches of Burch (1959) and Nahum et aL (1981) will be inaccurate in modelling the high LET electron component. 2. THE PETS CODE
The QMW Monte Carlo computer codes (Hill and Smith, 1994) were written in order to model the complete evolution of events, from the initial interactions to the "initial" chemical yields formed at ~ 10 -12 s, before diffusion reactions take place and finally to the primary chemical yields defined as those observable in steady state experiments at 10-6s. These codes were used to determine the sensitivity of the yield and time dependence of chemical yields to different assumptions and approximations. For the present investigation the codes were extended to incorporate photon interactions together with the possibility of bremsstrahlung production at the higher energies, details of which can be found in the Appendix. The PETS (Photon and Electron Track Structure) code follows the spatial and temporal co-ordinates of the primary and all secondary photons and electrons, interaction by interaction, as they lose energy. For photons these interaction include: the photoelectric effect, Compton scatter and pair production, and for electrons: elastic scatter, excitations, ionisations and bremsstrahlung production. Multiple scattering routines have also been included to transport the electrons through a water phantom to a volume of interest, i.e. the ampoule, where the electrons are followed interaction by interaction. 2.1. Electron interactions
The distance travelled between interactions is determined by the exponential random number with an average equal to the mean free path given by the reciprocal of the total cross section. The relative magnitudes of the partial cross sections determine the interaction type at the new position. Cross sections for elastic interactions were taken from Zaider et al. (1983) and are based on water vapour data. Above 0.2 keV the angular scatter was described using the Rutherford formula with a modified screening parameter (Grosswendt and Waibel, 1978). Below this value the analytical expression of Porter and Jump (1978) with fitting parameters from Brenner and Zaider (1983) was used. The inelastic cross sections were obtained from an analysis of the optical absorption data of liquid water and include the possibility of collective excitations (plasmons). The first ionisation potential for liquid water was taken to be 8.8 eV. For energies above 1 keV cross sections were calculated using a straight line extrapolation on a
639
Is the response of the Fricke dosimeter constant?
4BiG~
Fano plot corrected for relativistic effects (Schutten
et al., 1966). 2.1. I. Secondary electron energy distributions. Comparisons were made between the four energy distribution of secondary electrons emitted following ionisation events due to Jain and Khare (1976), Green and Sawada (1972), Vriens (1966, 1969) and Gryzinski (1965). When the incident electron energy is high and the energy transfer small, the Born-Bethe approximation is valid with the cross section given by
4rca2oR2 1 Of(W, O)
SsB(T, E) = - -
T
W
- ~W
In CT
4r~a~R2 [1 =
T
S
1
1
E2 E(T-E------~)q (~r - E )
]
where S is the number of electrons available for ionisation. The leading term gives the Rutherford cross section and the next two representexchangeand interference contributions. The single relation proposed by Jain and Khare (1976) to describe the energy spectra of secondary electrons at all energies of the incident and ejected electrons is given by
SjK( T, E) = f~(T, E)SBB(T, E) =f2(T, E)SM(T, E) (4)
E )I+C(T-I) TZI InCT E3 f2(T, E) = E3 + E03(1 - 1/T) 1/~(1
(5)
1+
(6)
where E0(70 eV) is a parameter. Experimental values of Of/OW were taken from Tan et al. (1978) and extrapolated to higher values of W using the relation A e x p ( - B W ) to obtain fitted values of A and B. The semi-empirical formula of Green and Sawada (1972) is described by
I"
1
S~s(T, E) = A (T) (E -- To)2 + r ~
(7)
where the parameters A, To and F are all dependent on the primary energy T and x = 1. The binary encounter model of collisions was developed by Vriens (1966, 1969) to take into account interference between direct and exchange terms.
Sv(T, E)
N, ¢7o
x
(8)
q~,= cos{[R/(T + Bi)]I/2In(T/Bi)}, ao = 4na2oR2, N~ the number, Bi the binding energy and G~ where
the ratio of the orbital kinetic energy to the binding energy, of electrons in the ith subshell of the target. Gryzinksi (1965) also used the binary encounter approximation in addition to some other approximations to give
SG(T, E)
y.,
NiPi
o-o~ (E + Bh3(I +p~)3/2 x~'E + B~F1
(3)
f,(T,E)
]
(2)
where W is the energy loss suffered by the incident electrons in ionizing collisions i.e. W = E + I, a0 is the Bohr radius, R is the Rydberg energy, Of/OW the optical oscillator strength and C (0.075 e V - ' ) a collision parameter. For hard collisions the cross section is given by Mott's formula.
SM(T, E)
(Pi
q 3(T ~ - E) 4 (E + B,)(T - Ej
@ r + Bi(l + ai)
1 4B~ G i 1 ( E + B~)2 + 3(E " - - - -+~ 3B~) -t ( T - E) 2
xle+FT-E-B,]'/2] L bTkT j j
x[l
E-I'-Bi]G'/(a'+I+E/ai)],~ j j,
(9)
where p~ = ( r / a , Oi) ':2 2.1.2. Multiple scattering. In order to model the passage of radiation through a scattering medium to a volume of interest, such as an ampoule, in an economic fashion, only those interactions leading to electrons or photons with energy larger than 5 keV (chosen to be consistent with cut-off used inside the ampoule) were considered. The paths of the electrons between each interaction were divided into smaller steps, each with negligible energy loss, in which the electrons were assumed to follow a straight line. The energy loss of the electrons was determined from the restricted collision stopping power (ICRU, 1984) with an energy cutoff of 5 keV, and multiple scattering was taken into account by changing the direction of the electron at the end of each step using an algorithm presented in Kuhn and Dodge (1992). This algorithm is an approximation to the analytic description of multiple scattering by M oli6re (1948) and provides a proper treatment of large angles as well as a fast generation of random scatter angles. The step sizes were kept small enough to avoid the introduction of significant errors and were determined either by the distance to the next 6-ray (> 5 keV), where the energy and angular descriptions of primary and secondary electrons used sampling methods which were based on those given by Messel and Crawford (1970) for M~ller scattering, or by the distance to the next bremsstrahlung interaction ( > 5 keV). Inside the volume of interest, the electrons were followed interaction by interaction and not by the use of multiple scatter transport. 2.2. Photon interactions The distance travelled by each photon before interaction was determined by the total mass attenuation coefficient for water taken from Hubbell (1977), while the type of interaction taking place was determined by the relative magnitudes of partial cross sections
640
M.A. Hill and F. A. Smith
(Hubbell, 1977) for water calculated from the values for atomic hydrogen and oxygen using the independent atom model.
Direct calculations of the ferric yields G(Fe3:+ ) for 1 MeV short track sections calculated by PETS (Hill and Smith, 1994) were found to have a wide range of values depending on the assumptions made.
3. FRICKE D O S I M E T E R
• 15.7-17.2 for vapour compared to liquid cross sections. • 14.5-17.2 using spatial distributions of hydrated electrons and radicals. • 16.2-20.5 using different vibrational cross sections. • 14.9-17.2 when the ionisation potential is reduced by ~ 4 eV for liquid water compared to the use of water vapour potentials. • 13.2-18.5 for a decrease of pH from 7 to 0.
The ampoule was modelled simply as a volume of water with a radius, ra of 0.6 cm, positioned at a depth, da, of 5 cm for photons and I cm for electrons in a 25 cm thick water phantom. The electron or photon beam was assumed to have infinite width and to travel parallel to the x-axis with the ampoule, positioned parallel to the z-axis and centred on the x-axis at a depth, da. Electrons at energies greater than 5 keV (Mozumder and Magee, 1966) produce isolated clusters of chemical species at inelastic interaction sites of the primary electron. Reactive species produced in one cluster are therefore very unlikely to react with species from another. Since the csda range of a 5keV electron is < 0 . 8 # m in water, 5keV provides a suitable cut-off energy below which electrons outside the ampoule can be ignored, while clusters produced inside the ampoule by electrons with energy < 5 keV are assumed to remain within the ampoule. The broad beam data were obtained from single tracks by using a geometric weighting factor which was applied to individual events at depths corresponding to that of the ampoule, i.e. from d~ - ra to da + r~ to give the relative probability that the event will be inside the ampoule. The weighting factor wg= [ r 2 - ( x - d ~ ) ~ ] 1/2 is related to the probability that an event produced by a track randomly positioned along the y-axis will actually be within the ampoule. The quality of the beam can be expected to change slightly with depth, however with a water equivalent ampoule charge particle equilibrium is expected perpendicular to the beam. Radiation lost from one side will be compensated by incoming radiation on the other, therefore the infinite width assumption is justified. 4. C A L C U L A T I O N O F F R I C K E Y I E L D S F R O M PRIMARY YIELDS
5. I S O L A T E D C L U S T E R S A N D FERRIC G - V A L U E S
Due to the complexity of, and sensitivity to, the assumptions involved in the above calculation of primary yields, a more analytical approach was used here. This allowed a much more detailed analysis of the reasons behind the above variations because of the use of better statistics. The method uses the theoretical ferric energy dependence of Magee and Chatterjee (1978) who used the prescribed diffusion model and attempted to include the variation in the initial spatial distribution of chemical species with energy by separating the tracks into events with energy < 100 eV, 100-1600 eV, 1600-5000 eV and >5000eV (Fig. 1). Also included in Fig. 1 are calculations by Yamaguchi (1987, 1989) for electrons and photons, who also used the prescribed diffusion model and track structure concepts along with exper2 °5
....... 1
...... 1
...... 1 .....
t
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~ -~
Ferric ion yields in the presence and absence of oxygen can be calculated from the primary yields ( ~ 10 -7 s) following irradiation, using the relations (Magee and Chatterjee, 1978):
. . . . . . -i
1.5
ol _~ ~ 1.0
G(Fe3+)N2 = 2GH2o2+ GOH + (GH + G~_) (10) G(Fe3+)o2 = 2GH:o~ + GoH + 3(GH + Ge-)
(11)
Figure 1 shows the variation in the Fricke yield with energy previously calculated (Hill and Smith, 1994) by following the passage of electrons to sub-excitational energies and then following the subsequent chemistry of the radicals produced up to 10 -7 s. The calculations shown are for complete and short track sections where the energy loss of the initial electron is minimal.
0.5 101
10 2
10 3
10 4
10 5
0s
Energy (eV) Fig. 1. Energy dependence of G(Fe3+) yields for complete tracks: Magee and Chatterjee (1978) (--), Yamaguchi electrons (1987) (---), photons.(1989) (. . . . -), Hill and Smith (1994) ( . . . . ), (O complete tracks, • short track sections); and experimental A, [] ICRU (1970), ~7 Freyer (1989) and <> Hoshi et al. (1992)
641
Is the response of the Fricke dosimeter constant? imental values for X-rays (ICRU, 1970; Feyer et al, 1989; Hoshi et al., 1992). Mozumder and Magee (1966) originally compared different energy depositions, AE, in terms of spurs ( < 100 eV), blobs (100 eV ~< AE < 500 eV) and short tracks (500 eV ~
Calculations have been carried out to investigate possible differences in the chemical response with energy of the Fricke dosimeter between photon and electron beam irradiation. A description of clusters in terms of the excitation type and the secondary elecRPC 47:4--1
tron energies, has been used both in the calculation of the final ferric yield and also in the determination of the importance of different cluster types. The following situations were considered: • Complete tracks and an ampoule in a water scattering medium. • The dependence on initial photon and electron energies, especially for 6°Co 5'-rays, was determined for the two cases above. (Energy was assumed to be a 6-function containing none of the scattered radiation found in practice.) • The importance of the energy distribution of the secondary electrons from ionisation events. • Ferric energy dependence of Magee and Chatterjee (1978) compared to that of Yamaguchi (1987). 7. R E S U L T S
7.1. Complete tracks Calculations were carried out for photon and electron beams with incident energies ranging from 5 keV up to l0 MeV, assuming complete absorption of the beam. Using the assumptions made in Section 5, the associated variations in the calculated ferric yields with incident energy for photons and electrons are shown in Fig. 2, along with the relative contribution of each of the cluster types to the total ferric yield and the fractional energy loss to track entities (spurs, blobs and short tracks). The calculated ferric yields for incident electrons increases with incident energy, the energy dependence being more marked at the lower energies. At high and low energies the variation in the calculated yield for incident photons is very similar to that of electrons. However, the absolute values for photons are consistently lower than those for incident electrons, although the difference becomes less significant at higher energies. Between 20 keV and 2 MeV the calculated ferric yields for photons are significantly lower than those for electrons, reaching a maxmum at ~ 30 keV followed by a minimum at ~ 80 keV. This feature results from changes in the mean initial electron energy brought about by the importance of Compton scatter in this energy region. Below 20 keV, photoelectric absorption dominates, producing electrons with energies nearly equal to the photon energy. At higher energies, Compton scatter becomes important and the average initial electron energy produced by photon interactions decreases due to the lower energy of the Compton electrons compared with that of the photo-electrons. The average electron energy increases again at higher photon energies due to the increase in the mean Compton electron energy with a greater fraction of the photon energy being transferred to the electron. Changes in the secondary electron distributions lead to similar variations in the calculated ferric yields, but vary in their absolute values. The increase in the ferric G-value with incident energy, calculated
642
M.A. Hill and F. A. Smith 2.0 1.8
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Fig. 2. Calculations of complete electron (--) and photon ( ' . . ) tracks (6°Co= A) using Jain-Khare (1976) secondary electron energy distribution (a) G(Fe3÷) yields [contributions from (1) e-E < 8.8 eV; (2) e-8.8eV< E < 5keV; (3) isolated excitations excluding plasmons; (4) plasmon excitations; and (5) "isolated" ionisations], (b) fractional energy loss to spurs blobs, and short tracks.
using the Green and Sawda (1972) distribution, is smaller than that obtained using the Jain and Khare (1976) distribution, and this also results in a smaller difference between photons and electrons. As the initial electron energy increases, more of the energy is deposited by lower energy secondary electrons and isolated excitations and this is reflected in an increase in spurs and blobs at the expense of short tracks. The contribution of clusters produced by electrons in the energy range 8.8 eV-5 keV (line 2) falls with increasing incident energy but this is more than compensated by the increase in the contribution by isolated interactions (lines 3 and 5), the plasmon in particular (line 4), and clusters produced by low energy electrons (line 1). 7.2. Ampoule in water scattering medium
The ferric yields calculated for photons and electrons of various incident energies (1.0-25.1 MeV) for an ampoule in a water scattering medium using the four different secondary electron energy distributions are compared in Fig. 3(a). The variations in the fractional energy loss to each track entity, along with the cluster contributions to the calculated ferric yield using the Jain-Khare (1976) distribution are shown graphically in Fig. 4. The differences in the calculated total ferric yields between 10 MeV electrons, 10 MeV photons and 6°Co ?-rays are given in Table 1 along with results obtained using the Yamaguchi (1987) Fricke yield vs energy relationship. Yields calculated using the Jain-Khare (1976) and the Gryzinski (1965) distributions (Fig. 3) rise with increasing incident energy, with photon yields being
lower than those of electrons. Yields calculated using the Green-Sawada (1972) and Vriens (1966, 1969) distributions are relatively independent of incident energy and the values obtained for photons and electrons are approximately equal. It should be noted that the distributions of Vriens and Gryzinski, which are both based on the binary encounter model, lead to very different results. It is clear from Fig. 3 however that the secondary electron energy distribution is important not only in determining the absolute yields with incident energy ( ~ 15%) but also the relative variation in these yields with incident energy as well as the relative difference in yields between photon and electron irradiation. The four energy distributions for a 1 MeV primary electron are compared in Fig. 5(a). The variation with primary energy, T, of the mean secondary electron energy, E2, and the mean energy of cluster-forming secondary electrons (E < 5 keV), Ecj, resulting from a single ionisation event is shown for the four energy distributions in Fig. 5(b). (Below ~ 10 keV the two curves for a particular distribution are identical since the maximum energy of the secondary electron is defined as ½(T - I) where I is the ionisation potential.) Distributions which lead to a reduction in the average energy Eel with increasing primary energy (e.g. Jain-Khare, 1976; Gryzinski, 1965) show increases in the calculated ferric yield with incident energy, as well as differences in the yields calculated for photons and electrons. The yields calculated for the Green-Sawada (1972) and Vriens (1966, 1969) distributions are relatively constant and so are the values of Ec~ for primary energies above 1 MeV.
643
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The variation with incident radiation energy of the calculated mean energy of cluster forming electrons ( < 5 keV) which enter, or are produced directly inside, the ampoule is shown in Fig. 3(b). The number of Fe 3+ ions produced by a particular cluster can be obtained using the Magee 0.75
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and Chatterjee (1978) distribution (Fig. 1). The change in the average secondary electron energy is clearly important at these low energies since the ferric yield of clusters produced by electrons increases significantly for smaller and smaller energies (Fig. 1). 0.80
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Fig. 4. Calculation following electron (--) and photon (. •. ) tracks (6°Co = &) of varying incident energies through a scattering medium into an ampoule, using the Jain and Khare (1976) energy distribution. (a) Fractional energy loss to track entities. (b) Contributions of clustes to calculated G(Fe 3+) yield [(I) e - E < 8.8 eV; (2) e-8.8 eV < E < 5 keV; (3) isolated excitations excluding plasmons; (4) plasmon excitations; and (5) "isolated" ionisations].
644
M. A. Hill and F. A. Smith
Table 1. Comparison o f ferric yields between photon and electron beams calculated in an ampoule for the four secondary electron energy distributions used with the ferric energy relationships o f Magee and Chatterjee (1978) and Yamaguchi (1987). Also shown are the percentage differences between the photon yields and those for I0 MeV electrons
10 MeV electrons (/~ mol J ')
Distribution Magee and Chatterjee (1978) Jain-Khare Green-Sawada Vriens Gryzinski Yamaguchi (1987) Jain-Khare Green-Sawada Vriens Gryzinski
10 MeV photons (# mol J i)
1.7183 1.5480 1.6465 1.8782
+ 0.0007 -t- 0.0005 + 0.0006 _+ 0.0007
1.704 1.548 1.647 1.869
-~: 0.002 ± 0.002 + 0.002 ± 0.002
- 0.8% 0.0% 0.0%° -0.5%
1.667 1.545 1.647 1.841
+ 0.003 ± 0.002 +_ 0.002 ± 0.003
- 3.0% -0.2% 0.0% -2.0%
1.5937 1.5059 1.6036 1.7781
± ± ± ±
1.584 1.506 1.604 1.771
_+ 0.005 ± 0.004 _+ 0.004 + 0.005
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+ 0.005 _+ 0.004 ± 0.004 ± 0.006
-2.1%o -0.1% 0.0% - 1.6%
0.0005 0.0003 0.0005 0.0005
What is important is the energy spectrum of electrons ( > 5 keV) entering, or produced in, the ampoule, which we refer to as encounter electrons. The encounter electron spectra for an incident electron beam will still be relatively high in energy and fairly monoenergetic, even after taking into account the collision losses and the energy straggling during its passage to the ampoule. By comparison, the encounter electron energies for a photon beam will cover a range from zero up to an energy just below the incident photon energy, with a mean which is less than that for electrons of the same primary energy. This is mainly a result of the dominance of Compton scatter events which themselves produce a broad spectrum of electrons in addition to the fact that these electrons may be created anywhere inside or outside the ampoule. For decreasing incident energies ( < 4 MeV) the average energy of the electrons entering the ampoule i 0 -2
. . . . . . . .
i
. . . . . . . .
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i
. . . . . . . .
beings to fall significantly due to the loss of energy in the water phantom and the finite size of the ampoule, For example, 2 MeV incident electrons with a csda (continuous slowing down approximation) range of 0.98 cm will enter the ampoule with energies ranging from zero to ~ 1.3 MeV and are likely to be completely stopped within the ampoule. From Fig. 1 this would lead to a corresponding reduction in the Fricke yield with decreasing incident energy. For photons in this energy region this is not the case however, due to the relatively large distances travelled between photon interactions which result in a spectrum of electrons entering the ampoule from zero to just below that of the incident photon. Since, at the higher incident energies (>4MeV), the average energy of the encounter electrons for photons is less than that for electrons, the energy variation of the ferric yield for photons should also reflect the ferric yield response for electrons at lower 120
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103
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10 8
Is the response of the Fricke dosimeter constant? energies. Therefore if the ferric yield for an electron beam increases with incident energy before becoming energy independent, it would be expected that a photon beam would behave in a similar manner, but with yields which are lower than those for an electron beam of equivalent incident energy which would approach the energy independent value for electrons at a higher energy. This reasoning is borne out by ferric response graphs shown in Fig. 3(a). 8. DISCUSSION
Calculations of the energy dependence in total ferric yield following complete absorption of all primary and secondary radiation in a Fricke solution are different for electrons and photons. Electrons show an increase with incident energy, becoming less energy dependent at higher energy [Fig. 2(a)]. Yields for photons are consistently lower than those for electrons, the variations being similar to that for electrons but showing a minimum at ~ 80 keV before rising at higher energies to give values approaching those for electrons. Absolute values of the yields and their energy variation depend on the secondary electron energy distributions which are used to model the ionisation events. The greater the variation in the ferric yield with energy for electron beams, the greater the differences between photon and electron beams. The main contribution to the yield at high incident energies comes from the clusters produced by low energy secondary electrons ( < 100 eV) and isolated plasmon excitations induced by high energy electrons ( > 5 keV). Ferric yields calculated for an ampoule positioned in a phantom are slightly higher than those obtained for complete tracks of the same energy. Using secondary electron distributions of Jain-Khare (1976), Green-Sawada (1972), Vriens (1966,1969) and Gryzinski (1965), the differences between calculated yields in an ampoule for 10 MeV incident energy photon and electron beams are 0.8, 0, 0 and 0.5% respectively; and between 6°Co 7-rays and 10 MeV electrons, 3.0, 0.2, 0 and 2.0%, the electron yield being higher in all cases. The variations of the ferric yield with incident energy is critically dependent on the way the secondary electron distribution varies with primary electron energy. Distributions which produce a reduction in the average secondary energy of cluster-forming electrons ( < 5 keV) with increasing primary energy (e.g. Jain-Khare and Gryzinski) cause an increase in the calculated ferric yield with incident energy and produce differences in the yields for photons and electrons. This sensitivity is probably the result of a large negative energy dependence of the ferric yield at low energies (Fig. 1) and the prominence of low energy electrons ( < 100 eV). The energy response of the ferric yield for a photon beam will be similar to that for electrons at lower
645
energies since photon beams produce electrons over a wide range of energies with an average which is lower than the incident photon energy, due to the importance of Compton interactions. The photon results show no evidence of a maximum at energies in the region of 0.66 MeV associated with the mCs 7rays as predicted by Nahum et al. (1981). The finite size of the ampoule is an important factor in the energy dependence of the ferric yield for electrons entering the ampoule with energies of the order of 4 MeV and below. One area of uncertainty in our calculated ferric yields is the reliabillity of the energy dependence of the theoretical ferric G-value calculated by Magee and Chatterjee (1978), especially at the lower energies (<100eV). This uncertainty is illustrated by the difference in the Magee and Chatterjee curve to that calculated by Yamaguchi (1987) (Fig. 1). Although the absolute yields obtained are different, a similar variation with energy and similar differences between photon and electrons are observed (Table 1). Simplifying assumptions made in the present calculations have also ignored the possibility of recombination of dissociation events due to the "cage effect" where the surrounding molecules help prevent separation. All secondary electrons with energy below the ionisation potential (~<8.8 eV) are asssumed to produce 3 Fe 3~ ions, and in reality, a fraction of these will recombine with their parent ions and lead to a reduction in the calculated G-value. Successful modelling of the chemical response following irradiation of high energy electrons and photons requires an accurate description of the chemical development of the clusters of low energy electrons from < 5 - 1 0 k e V down to sub-excitational energies to determine the energy variation of the ferric yield at these lower energies. This can be achieved either by using an analytic or a Monte Carlo track structure approach and should ideally be compared to experimental results at these low energies. This approach could be further used in conjunction with "condensed-history" Monte Carlo codes to transport electrons into a walled chamber in which the chemistry is followed. 9. CONCLUSIONS
This work has attempted to investigate the link between the actual track structure, as opposed to the mean LET, of an electron and the chemical (radical) yields in determining the response of the Fricke dosimeter. Four secondary electron distributions have been compared and shown to alter the differences in the Fricke response, not only between electrons and photons but also for different primary energies. The relative variation in the Fricke G-value is critically dependent on the variation on the mean energy of the secondary electron following ionisation with primary energy and, more specifically, on the corresponding variation in the mean energy of cluster
646
M . A . Hill and F. A. Smith
forming electrons (i.e. energy < 5 keV) in the ampoule. This dependence is due to a n expected strong variation in Fricke yield for low energy electrons ( < 100 eV). A question still remains o n the size of the effect, b u t experimental results to date do not rule o u t variations o f up to 1%. The possibility o f the varia t i o n in G ( F e 3+) with b e a m quality must be b o r n e in m i n d w h e n using or analyzing the results o f Fricke dosimetry. F u r t h e r experimental investigtions are therefore needed to try a n d resolve the secondary electron spectrum variations. Acknowledgements--The authors are grateful to the National Physical Laboratory for their financial support under EMRA contract NPL 82/A/0523. REFERENCES
AAPM (1983) A protocol for the determination of absorbed dose from high-energy photon and electron beams. Med. Phys. 10, 741. Bethe H. and Heitler W. (1934) On the stopping of fast particles and on the creation of positive electrons. Proc. R. Soc. A 146, 83. Brenner D. J. and Zaider M. (1983) A computationally convenient parameterization of experimental angular distibutions of low energy elastically scattered off water vapour. Phys. Med. Biol. 29, 443. Burch P. R. J. (1959) A theoretical interpretation of the effects of radiation quality on yields in the ferrous and cerric sulphate dosimeters. Radiat. Res. 11, 481. Davisson C. M. and Evans R. S. (1952) Gamma-ray absorption coefficients. Rev. Mod. Phys. 24, 79. Freyer J. P., Schillaci M. E. and Raju M. R. (1989) Measurement of the G-value for a 1.5 keV X-rays. Int. J. Radiat. Biol. 56, 885. Green A. E. S. and Sawada T. (1972) Ionisation cross sections and secondary electron distributions. J. Atmos. Terr. Phys. 34, 1719. Grosswendt B. and Waibel E. (1978) Transport of low energy electrons in nitrogen and air. Nucl. Instrum. Methods 155, 145. Gryzinski M. (1965) Two-particle collisions. II. Coulomb collisions in the laboratory system of coordinates. Phys. Rev. A 138, 322. Hill M. A. and Smith F. A. (1994) Calculation of initial and primary yields in the radiolysis of water. Radiat. Phys. Chem. 41, 265. Hoshi M., Uehara S., Yamamoto O., Sawada S., Asao T., Kobayashi K., Maezawa H., Furusawa Y., Hieda K. and Yamada T. (1992) Iron (II) sulphate (Fricke solution) oxiation yields for 8.9 and 13.6 keV X-rays from synchrotron radiation. Int. J. Radiat. Biol. 61, 21. Hubbell J. H. (1977) Photon mass attenuation and energyabsorption coefficients for H, C, N, O, Ar, and seven mixtures from 0.1 keV to 20 MeV. Radiat. Res. 70, 58. Hubbell J. H., Veigele Win. J., Briggs E. A., Brown R. T., Cromer D. T. and Howerton R. J. (1975) Atomic form factors, incoherent scattering functions, and photon scattering cross sections. J. Phys. Chem. Ref. Data 4, 471. ICRU (1969) Radiation Dosimetry: X-rays and Gamma Rays with Maximum Photon Energies Between 0.6and 50 MeV. ICRU Report 14, International Commission on Radiation Units and Measurements, Bethesda, Md, U.S.A. ICRU (1970) Radiation Dosimetry: X-rays Generated at a Potential o f 5-150kV. ICRU Report 17, International Commission on Radiation Units and Measurements, Bethesda, Md, U.S.A. ICRU (1984) Radiation Dosimetry: Electron Beams with Energies Between I and 50 MeV. ICRU Report 35, Inter-
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Is the response of the Fricke dosimeter constant?
is ignored since the fluorescence yield for oxygen is less than 1%.
APPENDIX Brief details of the extension to the Monte Carlo code (Hill and Smith, 1994) to incorporate photon interactions are given below. Bremsstrahlung High energy electrons passing close to an atomic nucleus may be decelerated with the production of bremsstrahlung photons having a continuous energy, k, ranging from zero up to the incident electron energy E. A low energy cutoff, k e = 8.8 eV, has been used to represent the lowest ionisation threshold in liquid water, since only photons with greater energy will lead to ionisation via the photoelectric effect. The dependence on incident electron energy of the differential bremsstrahlung cross sections was obtained for atomic hydrogen and oxygen from tables given in Seltzer and Berger (1986) and the relevant cross sections and distributions for water were calculated by assuming the independent atom model (i.e. all2o = 2Oa + a0). These data were used to give the appropriate photon energy distributions from k~ to E from which the total cross section was obtained by integration over the photon energy range. trb~m = f f c ~ da dk
(AI)
where k is the photon energy. The angular distribution was assumed to be forward peaked with a mean polar angle given by 0brem =
moC2/(mo)C2 + T
647
(A2)
Photoelectric effect The photoelectric effect involves the total absorption of a photon and the emission of an electron with an energy described by the Einstein equation, E = hv - I B, where hv is the incident photon energy and I Bthe binding energy of the molecular electron involved in the interaction. Photons with energy greater than the binding energy of the oxygen K-shell electrons, I K = 539.7eV, were assumed to interact solely with these electrons. Otherwise a binding energy of 1L = 19.6eV averaged over the remaining electrons was used. Values of 1K and 1L were taken from the appropriate ionisation thresholds for water given in Zaider et al. (1983). The angular distribution of low energy photon-electrons (B < 0.9) were obtained using non-relativistic distributions due to Fischer, and for higher energies the relativistic distributions of Sauter (Davisson and Evans, 1952). Following the emission of a K-shell electron, de-excitation of the highly excited water ion was assumed to take place via the isotropic production of an Auger electron with energy of 520.1 eW (1K - 21L). Characteristic X-ray emission
Cornpton scatter The absorption of the initial photon leads to the production of an electron and a second photon with reduced energy and a changed direction. The simulation of Compton scatter events was based on the Klein-Nishina formula multiplied by the incoherent scattering function to account for electron binding. dtr~omp _ da~ N S(x) dO dO
(A3)
The incoherent scattering function S(x), where x = sin(0/2)/2(/~,), was obtained from Hubbell et al. (1975) for atomic hydrogen and oxygen and calculated for water using the independent atom model. It was also assumed in the program that S(x ), which for water varies between 0 and 10, indicates the number of electrons available for interaction. The probability of a photon interacting with an oxygen K-shell electron was assumed to be given by [S(x) - 8]/10, while the energy of the Compton electron was reduced by the appropriate binding energy, either 539 eV or 19,6eV. Following the emission of a K-shell electron, de-excitation was again assumed to take place via the isotropic production of an Auger electron with energy 520.1 eV. Pair production Pair production becomes possible for photons with energy greater than 2moc: and results in the disappearance of the photon and the creation of an electron-positron pair in the fields of a nucleus. The distribution of the remaining energy E;.-2mo c2 between the electron and positron was determined using the method given by Zerby (1963) which gives a very good approximation to the Bethe-Heitler expression for pair production below 10MeV. The polar scattering angle 0+ of the positron and electron with respect to the direction the photon was taken from the simplistic relation (Bethe and Heitler, 1934) 0+_ = moc2/(rno c2 + T+ )
(A4)
The azimuthal angle was assumed to be uniformly distributed and differed by n for the electron and the positron. The positron were followed in an identical manner to the electrons and annihilation assumed to take place only after the positron has come to rest. Two 0.511 MeV photons are assumed to be emitted, isotropically distributed and opposite to one another. Triplet production occurring in the held of an atomic electron was treated in an identical manner to that of pair production.