Correction factors for two new reference beta radiation fields

Several correction factors for two new reference beta-particle radiation fields complementing ISO 6980 have been measured and calculated for both primary beta dosimetry and the operational quantities in radiation protection. The following correction factors for primary beta dosimetry for the determination of the absorbed dose to tissue, Dt, have been determined by calculations: kba for backscatter from the collecting electrode, kpe for perturbation by the chamber’s side walls, kih for inhomogeneity inside the collecting volume, kSta for the stopping power ratio at different phantom depths, and kSA for the use of the Spencer-Attix theory, while the following have been measured: kbr for the effect of bremsstrahlung and kabs for variations in the attenuation and scattering of beta particles between the source and the collecting volume due to variations from reference conditions and for differences of the entrance window to a tissue-equivalent thickness of 0.07 mm. Furthermore, calculations were undertaken to determine the following correction factors to assess the operational quantities Hp(0.07), Hp(3), H′(0.07), and H′(3): krod(0.07) for the rod instead of the slab phantom, kcyl(3) for the cylinder instead of the slab phantom, as well as k′(0.07) and k′(3) for the ICRU sphere instead of the slab phantom, while the correction factor for oblique radiation incidence has been measured. The newly determined correction factors were determined in the same way as those for several well-established beta-particle radiation fields described in two earlier publications by the author. Furthermore, they are ready to be implemented in an updated version of the ISO 6980 series and in the software of the Beta Secondary Standard BSS 2.


Introduction
Beta dosimetry in radiation protection is based on the ISO series ISO 6980 [1][2][3]. In ISO 6980-1 [1], the methods of production of the reference beta-particle radiation fields are described, in ISO 6980-2 [2], the primary dosimetry for the absorbed dose to tissue, D t , in a slab phantom is outlined, while in ISO 6980-3 [3], the determination of the operational quantities, H, is described. The radiation fields described in Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
ISO 6980 show a gap with respect to mean energy between radiation from 85 Kr sources (0.24 MeV mean energy) and 90 Sr/ 90 Y sources (0.8 MeV mean energy). Therefore, two new energy-reduced radiation fields from 90 Sr/ 90 Y sources were suggested resulting in a fluence weighted mean energy of about 0.6 MeV [4][5][6]. They are defined at a distance of 20 cm from a 90 Sr/ 90 Y source, i.e. a reference distance of y 0 = 20 cm, behind an absorber made of polymethyl methacrylate (PMMA) of 3 mm or 4 mm thickness positioned at 4 cm distance from the source.
To determine D t and H, several correction factors are required. For the radiation qualities already contained in ISO 6980, these correction factors are given in part 2 and 3 of that standard for D t and H, respectively. Therefore, for the two new radiation fields all these correction factors were determined within this work in the same way as described in two earlier publications by the author [7,8].
Section 2 gives a short overview of the basic relations in beta dosimetry, section 3 describes the determination of the correction factors for primary dosimetry, while section 4 lays down the methods to obtain the correction factors for the operational quantities.

Absorbed dose to tissue
The primary quantity in beta dosimetry is the reference betaparticle absorbed dose rate to tissue,Ḋ R,β [2], i.e. the absorbed dose rate to tissue in a tissue equivalent slab phantom at 0.07 mm depth due to beta-particles. It is determined by measurement with extrapolation chambers using the following equation according to ISO 6980-2 [2]: with (W 0 /e), the quotient of the mean energy required to produce an ion pair in air under reference conditions and the elementary charge e, with a recommended value of (33.88 ± 0.05) J/C, ϱ a,0 , the density of air at the reference conditions of temperature, pressure and relative humidity, a, the effective area of the collecting electrode, s t,a , the ratio of the mean mass-electronic stopping powers in tissue to air, k ′ , the product of the correction factors which are independent of the chamber depth, k, the product of the correction factors which vary with the chamber depth and [ d dl {k · k ′ · I (l)} ] l=0 , the limiting value of the slope of the corrected current versus chamber depth l.
The factors varying with the chamber depth are: k abs correction factor for variations in the attenuation of beta particles between the source and the collecting volume due to variations from reference conditions, k ac correction for the attenuation of beta particles in the collecting volume, k ad correction factor for the variations of the air density in the collecting volume from reference conditions, k de correction factor for the radioactive decay of the betaparticle source, k di correction factor for the axial non-uniformity of the beta-particle field, k pe correction factor for the perturbation of the beta-particle flux density by the side walls of the extrapolation chamber, and k sat correction factor for ionisation losses due to ionic recombination.
The factors independent of the chamber depth are: k ba correction factor for the difference in backscatter between tissue and the material of the collecting electrode, k br correction factor for the effect of bremsstrahlung from the beta-particle source, k el correction factor for the electrostatic attraction of the entrance window due to the collecting voltage, k hu correction factor for the effect of humidity of the air in the collecting volume on the average energy required to produce an ion pair, k in correction factor for interface effects between the air in the collecting volume and the adjacent entrance window and collecting electrode, and k ra correction for the radial non-uniformity of the beam, i.e. perpendicular to the beam axis.
These correction factors and a few more will be treated in section 3 of this work.
For simplification, in the following equations the dose, D, instead of the dose rate,Ḋ, will be used.
From the reference beta-particle absorbed dose, D R,β , the reference absorbed dose (due to beta-particles and photons), D R , results according to As for betas and photons the quality factor is unity, Q = 1 Sv Gy −1 , and as D R is valid for a reference depth of 0.07 mm in a tissue equivalent slab phantom, the operational quantity personal dose equivalent at 0.07 mm depth in a slab phantom, H p (0.07) slab , results to For different source nuclides and geometries, source, as well as for normal and oblique radiation incidence, H p (0.07; source; α) slab results from the conversion coefficient h p,D (0.07; source; α) slab to H p (0.07; source; α) slab = D R · h p,D (0.07; source; α) slab (4) Ring dosemeters are calibrated and irradiated on a rod phantom. The correction factor k rod (0.07; source; α) leads to The personal dose equivalent at 3 mm depth in a slab phantom, H p (3; source; α) slab , results from the conversion coefficient h p,D (3; source; α) slab to Eye lens dosemeters are calibrated and irradiated on a cylinder phantom. The correction factor k cyl (3; source; α) leads to Area dosemeters are calibrated and irradiated free in air. The corresponding quantity directional dose equivalent at 0.07 mm depth, H ′ (0.07), is defined in the ICRU sphere [9]. Thus, the correction factor k ′ (0.07; source; α) leads to The area quantity to estimate the eye lens dose is H ′ (3). The correction factor k ′ (3; source; α) leads to The conversion coefficients and correction factors described in this subsection will be treated in section 4.

General remarks
The correction factors described in this section were determined for several other reference beta-particle radiation fields within an earlier work [8] based on their field characteristics supplied earlier [10]. Therefore, the corresponding descriptions are only rephrased briefly in this work. Regarding details as well as information on the radiation transport simulations performed, the reader is referred to my previous work [8]. For comparison, the figures shown in this work also contain the results obtained for the well-established sources [8] together with the results for the two new radiation fields. The tables in appendix A of this work only contain the results for the two new source types. The Beta Secondary Standard BSS 2 [11,12] which is commercially available [13], can be used to realize all these radiation fields. As in my previous work, all correction factors were determined for two primary extrapolation chambers used at PTB: the Böhm chamber [14] (Beta Primary Standard, BPS1) and a new extrapolation chamber (BPS2) [8]. Both BPS chambers were constructed at PTB and regarding their details such as dimension and material, the reader is referred to the given references [8,14].
The use of the correction factors treated in this section is explained in section 2.1 of this work.

Correction factors adopted from earlier work
The following corrections factors can be taken from ISO 6980-2:2004 [2]: correction for the electrostatic attraction of the entrance window, k el correction for ionization losses due to recombination, k sat correction for the effect of humidity of the air in the collecting volume on the average energy required to produce an ion pair, k hu correction for interface effects between the air in the collecting volume and the adjacent entrance window and collecting electrode, k in correction for radioactive decay of the beta-particle source, k de The following corrections factors can directly be taken from the previous work mentioned above [8]: correction for the air density in the collecting volume, k ad correction for the source to chamber distance at different phantom depth, k ph Values for all these correction factors are compiled in the previous work [8].

Correction for backscatter from the collecting electrode, k ba
To correct the backscatter of the chamber's material to that of ICRU tissue (as the dose to ICRU tissue is to be determined) the following ratio is used: with D tissue , the calculated dose in the active volume with a tissue back in the chamber and D chamber , the calculated dose in the active volume with the real chamber material.
The values of k ba are independent of the chamber depth l (250 µm up to 2500 µm). Therefore, figure 1 shows the calculated values (mean for ten chamber depths in steps of 250 µm) of k ba depending on the source, i.e. on the radiation's field mean energy, and the angle of incidence, α, for both BPS1 and BPS2; the corresponding data are listed in the tables A1 and A2 in appendix A. The values for the two new radiation fields fit perfectly into the former ones.

Correction for perturbation by the chamber's side wall, kpe
To correct the perturbation of the chamber's side wall to that of air (as the collecting volume is made of air) the following ratio is used: with D wallless chamber , the calculated dose in the active volume with the chamber wall replaced by air and D chamber , the calculated dose in the active volume with the real chamber material.
The resulting correction factor depends on the chamber depth l. The data points can be fitted by a 2nd order polynomial regression resulting for l = 0 µm in k pe = 1.0, i.e. through the point (0/1): with f 7 and f 8 , parameters obtained from a 2nd order polynomial regression through (0/1) and l, the chamber depth. The same equation and parameter names are used in ISO 6980-2 [2]. The parameters f 7 and f 8 obtained are listed for the two new radiation fields in tables A3 and A4 for BPS1 and BPS2, respectively, in appendix A. To get an overview of all sources, figure 2 shows k pe for the largest chamber depth, i.e. for l = 2500 µm, depending on the radiation's field mean energy and the angle of incidence α. For both BPS1 and BPS2, the behaviour of k pe is similar to larger effects for BPS1 which is plausible as the chamber diameters are 6 cm and 8 cm, respectively, for the BPS1 and BPS2.

Correction for inhomogeneity inside the collecting volume, k ih
To correct the inhomogeneous dose within the finite active chamber volume to the dose in a small point, the following ratio is used: with D part of act. vol. , the dose in the centre front part of the active volume of 0.5 cm diameter and 100 µm depth, and D total act. vol. , the dose in the total active volume of 3 cm diameter and the whole chamber depth l. D part of act. vol. . was used as a representative for the dose in a single centre point at the front of the chamber.
The resulting correction factor linearly depends on the chamber depth l. Therefore, a linear regression was used to fit the values: with b and m, parameters obtained from a linear regression and l, the chamber depth.
The parameters b and m obtained from the regression are listed for all sources in tables A5 and A6 for BPS1 and BPS2, respectively, in appendix A. To get an overview of all sources, figure 3 shows the mean of k ih for all chamber depths together with its largest value (which it always takes for the largest l = 2500 µm) and its smallest value (which it always takes for the smallest l = 250 µm) depending on the radiation's field mean energy and the angle of incidence α. For both BPS1 and BPS2, the behaviour of k ih is the same with larger values for BPS1 as already seen earlier [8]. Calculated correction factor k ih for BPS1 (top) and BPS2 (bottom) depending on the source, i.e. on the radiation's field mean energy and the angle of incidence, α, see legend. The data points represent the mean for the ten chamber depths from l = 250 µm … 2500 µm, while the vertical bars represent the maximum and minimum value for the ten chamber depths.

Correction factor for the effect of bremsstrahlung from the beta-particle source, k br
The contribution of photons to the total dose was measured by placing an absorber made of PMMA in front of the extrapolation chamber, sufficiently thick to stop all beta particles and by comparing the corresponding ionization current with the one at 0.07 mm tissue depth: with I (0.07), the ionization current interpolated to 0.07 mm tissue depth and I phot , the ionization current behind a thick PMMA absorber. I (0.07) is considered to represent the total dose at reference conditions, D R , while I phot is interpreted as contribution due to photons, D br = (D R − D R,β ). In an earlier work their ratio, τ br = D br /D R = I phot /I (0.07), was given for several other radiation fields [12]. Table A7 in appendix A lists values of k br for the two new radiation fields.

Correction factor for variations in the attenuation and scattering of beta particles between the source and the collecting volume due to variations from reference ambient conditions and for differences of the entrance window to a tissue-equivalent thickness of 0.07 mm, k abs
The correction factor k abs accounts for variations in the attenuation and scattering of beta particles between the source and the collecting volume due to variations from reference ambient conditions and for the difference of the entrance window to a tissue-equivalent thickness of 0.07 mm [2]. It is given by with d 0 the reference thickness of the entrance window of the extrapolation chamber, 0.07 mm or 3 mm ICRU tissue equivalent thickness, ρ a0 = 1.1974 • 10 −3 g cm −3 the reference air density, ρ a the air density during the measurement, η m,t •ρ m •d m /ρ t the tissue-equivalent thickness of a window of medium m, thickness d m and density ρ m and η a,t •[ρ a −ρ a0 ]•y 0 /ρ t the tissue-equivalent difference from the reference air path y 0 with y 0 = 20 cm for the two new radiation fields.
The measured depth dose curves due to beta radiation, i.e. the transmission functions T(d), are adequately represented by functions of the form [12,15,16] with X (d) = 2 · log 10 (d+δ)−log 10 (d min +δ) log 10 (dmax+δ)−log 10 (d min +δ) − 1 a variable transformation from d to X (d) ∈ [−1; 1] and T i for i = 0…8 nine parameters, and τ br the bremsstrahlung contribution. See table A7 in appendix A and section 3.6 for k br and further details.
Values for the parameters T i , i = 0…8, as well as for d min , d max and δ for the two new radiation fields are shown in table A8 in appendix A They were obtained as fits of measurements of transmission through polyethylene terephthalate (PET) foils and PMMA absorbers [15,16]. The values for α = 0 • are to be applied to extrapolation curve measurements at all angles of incidence, i.e. at α = 0 • and at α̸ = 0•.

Correction factor for the stopping power ratio at different phantom depths, k Sta
The stopping power ratio, s t,a , accounts for the fact that the measurement volume is filled with air, not tissue. The value of s t,a is equal to the ratio of the energy transfer of electrons in tissue and air. It is not a constant as it depends on the energy E of the electrons which decreases continuously as they pass deeper into matter due to the energy loss of the electrons along their path. To obtain the stopping power ratios at different depths in a tissue equivalent slab phantom, d t , s t,a (d t ) BG , the spectral fluences at depth d t were folded with the stopping power ratios of tissue and air for monoenergetic electrons. With that, the correction factor can be formulated [15] k Sta = s t,a (0) BG + a · (d t /µm) b s t,a (0) BG (18) with s t,a (0) BG the stopping power ratio due to Bragg-Gray at the phantom's front; their values are given in table A9 in appendix A The spectral electron fluences were taken from literature [6]. The parameters a and b were determined with the method described in literature [15] and are given in table A10 in appendix A

Use of the Spencer-Attix theory via correction factor, k SA
The Spencer-Attix cavity theory is considered to be more accurate than the Bragg-Gray theory as it accounts for the variation in the response measured as a function of cavity dimension l, whereas the Bragg-Gray theory does not [17]. From the two new radiation fields, the spectral information is freely available [6]. From these data, corresponding Spencer-Attix stopping power ratios, s t,a (l) SA , depending on the chamber depth, l, were calculated using the method described above [17], the results will be published in [18]. Their ratio to the Bragg-Gray stopping power ratio, s t,a (0) BG , see table A9 in appendix A, can be interpreted as correction factor k SA to apply the Spencer-Attix theory: This ratio can be fitted to a 2nd order polynomial function depending on the chamber depth l of the form with c 0 , c 1 and c 2 , parameters obtained from a 2nd order polynomial regression and l, the chamber depth. The parameters c 0 , c 1 and c 2 obtained from the regression are listed for the two new radiation fields in table A11 in appendix A for the primary extrapolation chamber of PTB, i.e. a Böhm chamber (BPS1) [14].

General remarks
The conversion coefficients and corrections factors described in this section were determined for several other reference beta-particle radiation fields (sources) earlier [7,12]. Therefore, the corresponding descriptions are only rephrased briefly in this work. Regarding details as well as information on the measurements and on the radiation transport simulations performed, the reader is referred to the previous publications. For comparison, the figures shown in this section also contain the results obtained earlier for the well-established sources together with the results for the two new radiation fields. The tables in appendix B of this work contain the results for the two new radiation fields. The use of the conversion coefficients and

Conversion coefficient from D R to Hp(0.07) for angle α and source, h p,D (0.07;source,α) slab
The conversion coefficient from D R to H p (0.07; source; α) slab was measured using the primary extrapolation chamber of PTB, i.e. a Böhm chamber (BPS1) [14], for several angles of incidence α. The conversion coefficient is given by the ratio of the ionization current at α ̸ = 0 • and at α = 0 • , both interpolated to 0.07 mm tissue equivalent depth: The interpolation method of the measured depth dose curves to 0.07 mm is outlined in section 3.7. The results from the equation (21) are shown in figure 4 and listed in table B1 in appendix B. Figure 4 shows that the behaviour of the two new radiation fields fits perfectly into the well-established radiation fields, namely: the smaller the mean energy the less prominent the dose build-up effect and the steeper the drop of h p,D (0.07; source; α) slab with increasing angle of incidence.
The use of h p,D (0.07; source; α) slab to determine H p (0.07; source; α) slab is outlined in equation (4) in section 2.2.

Conversion coefficient from D R to Hp(3) for angle α and source, h p,D (3;source,α) slab
The conversion coefficient from D R to H p (3; source; α) slab was determined in the same way as described in the previous  figure 5 and listed in table B2 in appendix B. They also fit perfectly to those of the well-established radiation fields, namely: the smaller the mean energy the smaller the values of h p,D (3; source; α) slab and the steeper their decrease with increasing angle of incidence due to the stronger scattering and absorption of electrons in the 3 mm tissue material. Furthermore, no dose build-up effect is present as this is completed at depths smaller than 3 mm.
The use of h p,D (3; source; α) slab to determine H p (3; source; α) slab is outlined in equation (6) in section 2.2.

Correction factors to account for the phantom shape and depth, k rod (0.07), k cyl (3), k ′ (0.07), and k ′ (3)
The four correction factors were determined by simulations calculating the dose at 0.07 mm and 3 mm depth in a slab phantom made of ICRU tissue and comparison with the dose in 0.07 mm of an ICRU tissue rod phantom for k rod (0.07), by comparison with the dose in 3 mm of an ICRU tissue cylinder phantom for k cyl (3), and by comparison with the dose in 0.07 mm and 3 mm of an ICRU tissue sphere for k ′ (0.07) and k ′ (3), respectively [7]:   For a detailed discussion of their dependence on the energy and angle of incidence, the reader is referred to the publication mentioned above [7]. The values for the two new radiation fields are given in appendix B in tables B3-B6, for k rod (0.07), k cyl (3), k ′ (0.07), and k ′ (3), respectively. It shall be noted that several values are, within their standard uncertainties, compatible with 1.0. In those cases, in the corresponding tables a value of unity together with its standard uncertainty is given. This is especially the case for k ′ (0.07), see right part of figure 6.

Conclusions
For two new reference beta-particle radiation fields both a complete set of correction factors for primary beta dosimetry as well as conversion coefficients and correction factors for the determination of the operational quantities were determined, for both normal and oblique radiation incidence. With the data determined in this work it is possible to implement the two new radiation fields in the current revision of the ISO 6980 series [1][2][3] and especially the tables B1, B2 C1-C4. in ISO 6980-3 [3].

Acknowledgments
Special thanks go to Gert Lindner (PTB Berlin) for his enormous help in using the high-performance computer cluster (HPC) of PTB on which the many simulations were carried out. Further thanks go to Phil Brüggemann (PTB) for the many Table A1. Calculated values of the backscatter factor k ba for BPS1 (PMMA back electrode): mean and standard deviation of the values of the ten chamber depths (from l = 250 µm up to 2500 µm in steps of 250 µm), see section 3.3  Table A2. Calculated values of the backscatter factor k ba for BPS2 (PEEK back electrode): mean and standard deviation of the values of the ten chamber depths (from l = 250 µm up to 2500 µm in steps of 250 µm), see section 3.3  Table A3. Values of the fit parameters f 7 and f 8 for the calculation of the perturbation factor, kpe, for the BPS1, see equation (12) in section 3.4 The parameters were determined via 2nd order polynomial regression through (0/1) of the calculated perturbation factor, kpe.   Table A4. Values of the fit parameters f 7 and f 8 for the calculation of the perturbation factor, kpe, for the BPS2, see equation (12) in section 3.4. The parameters were determined via 2nd order polynomial regression through (0/1) of the calculated perturbation factor, kpe.   Tables with the calculated and  measured correction factors   Tables A1 and A2 list the   Values of the fit parameters m and b for the calculation of the inhomogeneity factor, k ih , for the BPS1, see equation (14) in section 3.5. The parameters were determined via linear regression of the calculated inhomogeneity factor, k ih . Values of the fit parameters m and b for the calculation of the inhomogeneity factor, k ih , for the BPS2, see equation (14) in section 3.5. The parameters were determined via linear regression of the calculated inhomogeneity factor, k ih .  (16), to calculate, k abs , see equation (17) in section 3.7     90 Sr + 90 Y with 3 mm PMMA; y 0 = 20 cm 1.000 0.50% 1.000 0.50% 1.000 0.50% 1.000 0.50% 1.000 0.50% 1.000 0.50% 90 Sr + 90 Y with 4 mm PMMA; y 0 = 20 cm 1.000 0.50% 1.000 0.50% 1.000 0.50% 1.000 0.50% 1.000 0.50% 1.000 0.50% Table B6. Calculated values of the correction factor k ′ (3), see equation (26) in section 4.4 for its determination and equation (9)