Calculation of the energy dependence of dosimeter response to ionizing photons

Int. J. Appl. Radiat. Isot. Vol. 33, pp. 1299 to 1310, 1982 Printed in Great Britain

0020-708X/82]111299-12503.00/0 Pergamon Press Ltd

Calculation of the Energy Dependence of Dosimeter Response to Ionizing Photons A R N E M I L L E R and W. L. M c L A U G H L I N * Accelerator Department, Rise National Laboratory, DK 4000 Roskilde, Denmark

(Received I March 1981) Using a program in BASIC applied to a desk-top calculator, simplified calculations provide approximate energy dependence correction factors of dosimeter readings of absorbed dose according to BraggGray cavity theories. Burlin's general cavity theory is applied in the present calculations, and certain limitations of the theory are considered. Examples of the use of the program are given for 6°Co }.-ray irradiation of a LiF dosimeter held in aluminum and for evaluation of the influence of changes in broad ~'-ray spectra on the response of several dosimeters. The BASIC program and typical data plots as given here are available for certain dosimeter probe materials and combinations used with intermediate energy photon spectra 10.01. 100 MeV).

Introduction THE ABSORBED DOSE, DM, in a medium irradiated by

(b) The cavity size is small compared with the range of the most energetic secondary electrons:

energetic photons, as from a bremsstrahlung or a }.-ray source, can be determined by analysis of a dosimeter placed in the medium. The medium and the dosimeter generally consist of different materials with different radiation absorption properties, and the energy absorbed in the dosimeter may therefore not be the f being equal to the ratio of the electron mass collisame as that absorbed by the medium, had the dosi- sion stopping powers for cavity C and medium M. ~t~ in each case, the photon absorption coefficients or meter not been there. The relation between the two doses may be electron stopping power values must be evaluated over the energy spectrum of photons or secondary expressed as: electrons at the position of the dosimeters. I D M = ~.D c (I) General cavity theories have been proposed in order to treat the situation where the size of the cavity where DM is the dose in the medium and Dc is the is comparable to the range of secondary electrons, i.e. dose in the dosimeter, which is referred to here as the the absorbed dose in the dosimeter is due both to "cavity". electrons generated in the medium outside the dosiCavity theories have been developed in order to meter, and to those generated within the dosimeter find expressions for the correction factor f These itself. Burlin's general cavity theory ~2~ may be written theories are well established in two extreme situations aS:(3) depending on the size of the cavity relative to the f--d.S c+(I-d)'pc (4) range of the secondary electrons: (a) The cavity size is lar.qe compared with the range of the most energetic secondary electrons, in this case the correction factor is:

f being equal to the ratio of the photon mass energy absorption coefficients for cavity C and medium M. *Visiting scientist from the Center for Radiation Research, National Bureau of Standards, Washington, DC 20234, U.S.A. , T See theorem by Tomkeiff for spheres,t'*~ applicable to any convex volume.°)

where d is a weighting factor calculated as

I -e -k d = - - ;

@

(5)

p being the effective mass attenuation coefficient for electrons and g the average pathlength through the dosimeter, often determined as four times the dosimeter volume divided by its surface area.t Burlin's theory has been modified by several authors, ~5-7~ mainly because of the way in which/~ is calculated, but also on a more conceptual basis,tm Burlin first suggested that ~ be calculated as: 16 ,8 = (E,,,a" _ 0.036)t. 4

(6)

but later BURLIN and CHAN 19) in calculating//for the

1299

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Arne Miller and W. L. McLaughlin

ferrous sulfate (Fricke) dosimeter used the relation e -p'a = 0.01,

(7)

where R is the extrapolated range of the secondary electrons. Different methods of calculation of R have been used ~1°~ without significantly changing the final correction factor. JANSSENS et al.ts~ found improved agreement between experiment and theory when they used: e -p'R = 0 . 0 4

and PALIWALand ALMOND~It} used 14 -~ (Emx)l.O9.

(8)

TABLE I. Expressions for the correction factor f = Dc/D M depending on the size of the cavity (tc) and the thickness of the wall (t,) compared with the range of the secondary electrons. R. The contribution from the wall is not readily determined in the case where tw ~ R

tc<< R tc ~ R tc ~, R

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d" SM c + (t - ,~" ~,% ~w. [d" Sc + (I - d). ~Cw'l i~c

pc

Whereas detailed discussions of energy and particle fluence, radiation equilibrium, and cavity theories may be found in the Aim Carlsson and Carlsson article in this issue, pp. 953-965, ~zs~ our calculations SIMONS and EMMONSct'~ compared various ways of make certain assumptions contrary to strict adhercalculating p for the dose interpretations with LiF dosimeters encased in various materials. They found ence to modified cavity theory rules/I~ The present the best agreement between theory and experiment calculations work best for the Compton effect spectral using equation (7). BF.g~LSSO~7~ suggested that two regions of intermediate-energy photon spectra (e.g. experimentally-determined weighting factors be used 6°Co, 13~Cs), and it should be emphasized that there instead of d and (1 - d) in Burlin's theory. BRAI-IME(6} are certain sources of error that are ignored. Theresuggested restricted energy-absorption coefficients be fore the present simplified calculations are only apused, thereby eliminating the weighting factors. proximations suited mainly for applications in indusSeveral authors ~t3-ts~ have used Burlin's theory with- trial applications and radiation effects studies not out finding disparities between theory and experi- requiring high accuracy. Since this work is concerned mainly with condensed-state "cavities", it is assumed ment.* In many situations the dosimeter may be surrounded that we can ignore the Spencer-Attix refinements of by a material different from both the material of the cavity theory, which account for energy loss from medium and that of the dosimeter (cavity). The thick- gaseous cavities by high-energy secondary electrons, ness of the wall may be very thin compared with the and which make use of restricted stopping power range of the secondary electrons, and its influence on ratios. Instead we use the classical Bragg-Gray forthe energy absorbed by the dosimeter is then negli- mulation, which involves use of unrestricted stopping gible; alternatively, the wall thickness may be suffi- power ratios. In addition, perturbations associated ciently large that the influence of the wall material with the replacement of phantom material by the cannot be neglected. If the wall thickness is much material of the probe are not treated in detail here. smaller or much larger than the range of the secondary The errors due to changes in fluence and the mass electrons,f can be expressed in rather simple terms as density of condensed dosimeter probe materials are in shown in Table 1, where Burlin's cavity theory is the most cases relatively small. A notable exception basis (or other suitable cavity theories might also be would be the case of large disparities in fluenc¢ and applied). When the dimensions are large, attenuation radiation equilibria across boundaries of probe of the primary photon beam must also be considered materials differing greatly from their surroundings. We have made a BASIC program for a desk-top in determining the dose. calculator (HP 9830 A) allowing the calculation of Whatever the size of the cavity or wall, stopping stopping power and absorption coefficient ratios power ratios or absorption coefficient ratios must be known as a function of energy, for proper interpreta- weighted over given photon and electron spectra, to be used in a general cavity theory.t The energy range tion of the dose measurement. Usually these data are extends from 10 keV to 100 MeV. We chose Burlin's available in the form of tabulations, or as analytical general cavity as one of the more widely-used theories expressions requiring sophisticated computato calculate the final correction factor, but the protion/ t9-24~ Simple expressions often yield less accugram may easily be adapted to other forms of the rate results but may be sufficient for many purposes. cavity theory. * It should be noted that this subject is of interest to many different workers in this field, and some published Ratio of Stopping Powers papers have not been included. ~"The detailed procedures used in the program are pubFor the calculation of the ratio of electron stopping lished as Rise-M-2345 and may be obtained by writing to the authorsYit) powers,~ the tabulated data by BEROEg and Newer improved data are given by SVJ.TZEa and SELTZER(21) or by PAGES et al/TM were fitted to a BEROERin this volume/2°j but were not available in time for our computations. In the intermediate energy region of ninth-order polynomial. The values of mass collision interest, inaccuracies in the older stopping-power values stopping powers calculated from this polynomial are not very large. deviate less than +0.3% from the tabulated values. (9)

Energ7 dependence of doshneter response to ionizinO photons

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stopping power for water tabulated by Berger and Seltzer is 20~ lower at 100MeV than SHoO= k~" SH + k2"So when Sn and So are taken for ~ases as was done here.* The difference diminishes towards lower energies so that at I MeV it is less than i~, as shown in Fig. 2. For calculations of electron stopping powers due to S~,,mp = ~ k f S, (I0) primary photon irradiations with 6°Co ),-rays, whose where ki is the weight fraction of the ith component secondary electrons are generally below 1 MeV, the and Sj is its mass collision stopping power. The addi- additivity rule may therefore be used; also the error is tivity rule does not strictly apply, especially at the still fairly small at 10 MeV. If higher accuracies are higher energies, (26) because the important density required, however, the tabulated values for comeffect correction must be evaluated for that actual pounds should be used. The tabulated data for water compound. (2m Water is typical in this respect. The have been entered as a special "element" to be used in these calculations, and other relevant compounds * The density effect and other departures from additivity might be added. are discussed in Ref. 20; in making subsequent calculations The ratio of stopping powers that should be used in with our program, it is recommended that values of Sco,p the general cavity theory (e.g. equation 4) must be be based as much as possible on values of St for condensed elemental forms, especially when higher radiation energies weighted with the spectrum of the secondary elecare involved. trons. For calculating with this program, the spec-

The energy dependence of mass collision stopping powers is shown in Fig. I for several elements, as computed and plotted by the calculator. When determining stopping powers for composite materials and compounds, the additivity r u l e (26) is applied:

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F]o. 3. Electron energy dependence of ratios of stopping powers ~ for a polyethylene dosimeter in various materials: !. water: 2. carbon; 3. aluminum: 4. iron; 5. lead. plotted as a function of energy. Figure 4 shows that for several dosimeter materials in water, in most cases, the ratio of stopping powers vs energy is a slowly varying function.

trum must be entered as a histogram with a maximum of 29 intervals and with an arbitrary height of each level representing the energy flux of this energy interval. The weighted ratio of stopping power is then calculated as: ~A = YAj(S¢)j

Ratio o f M a s s Energy-absorption

(1 I)

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For calculating photon mass energy-absorption coefficients, the data tabulated by STOgM and ISRAEL(zT) the factors in parentheses are the average stopping were used. The accuracy of these data is estimated to powers for this interval. The histogram must be be of the order of 10%. The data by HUBBELL('3) are chosen so that it fits reasonably well with the esti- more accurate, but fewer elements were tabulated.* It mated spectrum, but the spectrum should be more was difficult to fit these data to a simple expression, as carefully delineated at energies where the ratio of was done for the electron stopping-power values. stopping power changes drastically. It may therefore Instead, we loaded the tabular data sets into the calbe helpful to plot stopping powers as a function of culator and interpolated for the values desired. For composite or compound materials, the addienergy. Figure 3 provides an example where the ratio of stopping powers for several elements and a typical tivity rule(24) was again applied. Figure 5 shows mass polymeric dosimeter material (polyethylene) have been energy-absorption coefficients as a function of photon energy for several elements, as they have been calculated and plotted by the calculator. * A newer work by HUBe~LL('9) includes more elements, The ),-ray spectrum may be entered, as was the elecbut was not available at the time of our present computations. These new data will be used in the future to im- tron spectrum for the stopping-power calculation, and the same considerations are applied. The weighted prove the accuracy of energy-dependence data. where Aj is area o f the j t h interval of the energy spect r u m relative to the total area of the spectrum, and

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Emrgy MeV F]O. 4. Ratios of stopping powers ~ for various dosimeter materials in water as a function of electron energy: I. polyethylene; 2. polyvinyl chloride; 3. Perspex; 4. LiF; 5. glass.(23)

Energy dependence of dosimeter response to ionizing photons

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FIG. 5. Mass energy absorption coefficients as a function of photon energy for several elements and for water as calculated by interpolation from tabulated data of STORM and ISR^EL.~27~

ratio of mass energy-absorption coefficients is calculated in the same manner as was the weighted ratio of the stopping powers.

p^ - ZAi(PM)J.

(12)

Figure 6 shows the ratio of mass energy-absorption coefficients as a function of photon energy for polyethylene in various media, and Fig. 7 shows the ratio of absorption coefficients as a function of photon energy for several dosimeter materials in water, t23"2s) It is clear that large variations in the ratio of absorption coefficients may occur, particularly when heavier elements are involved. ''

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As a first example, the correction factor f is calculated for a LiF dosimeter in A! irradiated with 6°Co ,/-rays. Cavity (LiF):

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weight ~oLi: 26.8~ weight ~ F : 73.2~o Medium (Al): weight ~oAi: 100.0% y-Ray spectrum: we have used a typical spectrum ~2m with a large fraction of primary photons and only relatively small fraction of scattered secondary radiation. Secondary-electron spectrum: we have assumed a spectrum with E,~x equal to the maximum energy of the photons, with a most probable energy somewhat

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Energy MeV FIG. 6. Ratios of mass energy absorption coel~cients pc as a function of photon energy for a polyethylene dosimeter held in the same elements as in Fig. 3: I. water: 2. carbon; 3. aluminum: 4. iron: 5. lead. A.R.I. 33/I I v

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lower (~0.5 MeV), and with a lowest energy at 0.01 MeV} 3°~

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are plotted in Fig. 8b along with the energy spectrum of secondary electrons in arbitrary units. The weighted ratio of average values of collision stopping powers is calculated to be "~A = i.039. Assuming the density of LiF to be 2.675 g/cm 3 and the size of the dosimeter to be (5 x 5 x 2)ram s, and applying equations (6) and (5): d = 0.163, and from equation (4): f = __De= 0.97.

DM

This means that the dose in the aluminum is equal to the dose as measured by the LiF dosimeter divided by 0.97. In a second example, this program was used to determine to what extent ?-ray irradiations, with different spectral distributions, would affect the response of the dosimeters considered as candidates for use in a planned International Atomic Energy Agency Dose Assurance Service for high doses. {s'; These considerations were made because in all the preliminary IAEA high-dose intcrcomparison studies, an apparent small systematic deviation was found between two groups of dosimeters.

In dosimetry intercomparison studies using large industrial O°Co irradiators, if the energy spectrum of the calibration irradiation is different from that of the test irradiation, there may be systematic differences in the dosimeter response. The magnitude of the resulting correction factors would depend (assuming no energy dependence of intrinsic dosimeter response) on cavity-theory corrections and would be appreciable only if components of such correction factor expressions are energy dependent. The dosimeters previously considered in the IAEA high-dose intercomparison study °l~ were: Solid

Liquid

!. Nylon-base radiochromic dye films 2. Alanine (amino acid) in paraffin 3. Ceric--cerous aqueous HzSO4 solution 4. Ethanoi--chlorobenzene solution.

A fifth dosimeter was included in these considerations, namely: 5. glutamine (solid-phase amino acid), since it is a promising candidate system in future IAEA intercomparison studies for food irradiation. In calculating the cavity-theory correction factor, it is assumed in all cases that the dosimeters are surrounded by a layer of polymethyl methacrylate ("Perspex") thick enough to provide approximate secondary electron equilibrium, i.e. the dosimeters are considered "cavities" giving dose readings in Perspcx, so that for the correction factor, f = Dc/Da, Da is the dose in the Perspex, and Dc is the dose in the cavity (i.e. the dosimeter probe). 1. Radiochromic dye films. °z~ The film dosimeters placed between Perspex are very thin (50/~m), and the only cavity-theory correction would therefore be in terms of stopping power ratios, ~33~ as indicated in Table 1. The dosimeter consists of 6% hexa (hydroxyethyl) pararosaniline in nylon, which has the following

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1306

Arne Miller and W. L. McLauohlin

elemental composition: dye dosimeter: 9.6% hydrogen 63.8% carbon 12.3% nitrogen 14.3% oxygen. The surrounding material is as follows: Perspex medium: 8.0% hydrogen 60.0°/0 carbon 32.0°/0 oxygen. The stopping powers and the stopping power ratio are plotted in Fig. 9 by the calculator, showing that the ratio has virtually no energy dependence over the energy region of interest. 2. Alanine. °'tj The dosimeter consists of 80% alanine in 20% paraffin. It is made as pellets 4 mm in diameter and 7.5 mm long, surrounded by polyethylene. The wall thickness is not specified, but assuming I mm thickness, these sizes indicate that both ratios of electron mass collision stopping power and photon mass energy absorption coefficient for medium, wall, and dosimeter have to be taken into account (see Table I).

In Figs 10 and 11 mass energy absorption coefficients and ratios for alanine to polyethylene and for polyethylene to Perspex, respectively are plotted. These data show that alanine and Perspex are virtually identical, with respect to radiation interactions over the indicated photon energy range. A small perturbation might arise from the wall material, polyethylene, but only with respect to the absorption coefficient ratio, as the stopping power ratios are not appreciably energy dependent. Since spectral influences from the two components are inverse, it is apparent that they tend to balance each other (see curves numbered 3 in Figs 10 and 11). For the alanine dosimeter there should also be no appreciable energy dependence in the photon energy region from 0.01 to 100 MeV. 3. Ceric-cerous solution. °s) The dosimeter consists of an aqueous solution of: 15 mM Ce(SO4)z 15 mM Cez(SO4h 0.8 N H2SO4 it is irradiated in 2-ml glass ampoules of ,,, 10 mm i.d. held in Perspex for electron equilibrium, and as such can be considered a relatively large dosimeter probe. Referring to Table t and assuming that the effect of thin glass walls can be neglected, ~36) only the ratio of photon energy absorption coefficients pc, needs to be considered (see equation 2).

Elemental Composition

alanine dosimeter: 7.5% hydrogen 51.2% carbon 12.6% nitrogen 28.7% oxygen polyethylene wall: 14.4% hydrogen 85.6% carbon

Elemental Composition

ceric--cerous dosimeter: 0.1% hydrogen 3.0% oxygen 1.5% sulfur 0.6% cerium

Perspcx medium: 8.0% hydrogen 60.0°/0 carbon 32.0% oxygen l O

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FIG. IO. Photon mass energy absorption coefficientsfor (I) the dosimeter material alanine; (2) polyethylene; and (3l their ratio, as a function of energy.

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spectra have been approximated by 13-interval histograms with interval widths of 0.1 MeV (see Fig. 13).

8.0% hydrogen 60.0% carbon

32.0% oxygen. Spectrum at

Figure 12 shows that the absorption coefficient ratio increases considerably towards lower energies. The ratio, f, of the absorption coefficients of ceri¢--cerous solution and Perspex has been calculated for y-ray spectra degraded at various depths of water, and originating from an infinite 6°Co plaque source. °7~ The

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FIG. 13. The }'-ray photon spectrum from a plane parallel 6°Co source of infinite lateral size, at 10-cm depth in water. The continuous curve is from Ref. (37), while the histogram is the approximated spectrum used in these calculations. Similar approximations have been made for 2, 5, and 20 cm depths. The relative fractions of the primary and degraded parts of the spectrum are:(2s) Primary Degraded At 2 cm depth 0.83 0.17 At 5 cm depth 0.73 0.27 At 10cm depth 0.53 0.47 At 20 cm depth 0.28 0.72

Thus, if a considerable portion of the irradiation is performed with a degraded spectrum, the response of the dosimeter relative to the low atomic number surrounding medium increases appreciably. The f values above have the same trend as those predicted in calculations by McLAUGHLIN(3s) for an aqueous acid solution of 15 mM eerie ammonium sulfate irradiated with typical degraded 6°Co y-ray spectra. 4. Ethanol-chlorobenzene. (39) The dosimeter solution consists of 100 ml 40 mi 0.4 ml 859.6 ml

chlorobenzene water acetone ethanol.

it is irradiated in 2-ml glass ampoules similar to the ceric-cerous dosimeter, and thus is a large detector probe. Again assuming the wall effect of the thin glass is negligible, only the ratio of photon energy absorption coefficients between ethanol-chlorobenzene solution and Perspex needs be considered (see equation 2). Elemental Composition ethanoi--chlorobenzene 11.7% hydrogen dosimeter: 51.2~ carbon 29.9% oxygen 3.2% chlorine 4.0% water

Perspex medium: 8.0% hydrogen 60.0% carbon 32.0% oxygen. In Fig. 14 (curve 3) it is seen that this ratio, # c increases slightly towards the lower energies. When a calculation of the correction factor f is carried out as for the ceric-cerous dosimeter, the following values are obtained for different degraded 6°Co y-ray spectra:°~) Spectrum at

f

2 cm depth 5 cm depth 10 cm depth 20 cm depth

1.038 1.038 1.038 1.044

Such degraded photon spectra in water absorbers apparently have only a slight energy dependence effect on the response of the ethanol-chlorobenzene dosimeter. 5. Glutamine.('°) The dosimeter is typically irradiated in a capsule of polyethylene ~ 10 mm in diameter and 30 mm long. That size can be treated as a large probe, and thus only the photon energy absorption coefficient ratio between glutamine and Perspex needs be considered. Figure 15 shows that this ratio varies so little with energy that no correction is needed.

Energy dependence of dosimeter response to ionizing photons

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Elemental Composition

glutamine dosimeter: 6.9% hydrogen 41.1% carbon 19.2% nitrogen 32.8% oxygen Perspex medium: 8.0% hydrogen 60.0% carbon 32.0°/,, oxygen

Summary The accurate calculation of weighted stopping power and weighted absorption coefficient ratios are 10

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tedious without the availability of large computers and sophisticated computer codes. This paper shows that a desk-top calculator may be employed for an approximate calculation, with errors being introduced mainly in the determination of stopping power ratios at the higher energies (> 10MeV).(41) For the calculation, the secondary electron and photon energy spectra must be known approximately. An extension of the simple program is planned, for calculating the spectra at defined depths in given homogeneous media, knowing for example the primary photon spectrum and the geometry of radiation incidence. I

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1310

Arne Miller and W. L. McLaughlin

Acknowledgement--The assistance of ERLING BUGGE CHRISTENSEN,who skillfully programmed the HP 9830 A calculator, is gratefully acknowledged.

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