Collection efficiencies of ionization chambers in pulsed radiation beams: an exact solution of an ion recombination model including free electron effects

Objective. Boag et al (1996) formulated a key model of collection efficiency for ionization chambers in pulsed radiation beams, in which some free electrons form negatively charged ions with a density that initially varies exponentially across the chamber. This non-uniform density complicates ion recombination calculations, in comparison with Boag’s 1950 work in which a collection efficiency formula, f, was straightforwardly obtained assuming a uniform negative ion cloud. Boag et al (1996) therefore derived collection efficiency formulae f′, f″ and f′″ based on three approximate descriptions of the exponentially-varying negative ion cloud, each uniform within a region. Collection efficiencies calculated by Boag et al (1996) using these formulae differed by a maximum of 5.1% relative (at 144 mGy dose-per-pulse with 212 V applied over a 1 mm electrode separation) and all three formulae are often used together. Here an exact solution of the exponentially-varying model is obtained. Approach. The exact solution was derived from a differential equation relating the number of negative ions collected from within some distance of the anode to numbers of ions initially located within that region. Using the resulting formula, f exp, collection efficiencies were calculated for a range of ionization chamber properties and doses-per-pulse, and compared with f, f′, f″ and f″′ values and results from an ion transport code. Main results. f exp values agreed to 5 decimal places with ion transport code results. The maximum relative difference between f exp and f″′, which was often closest to f exp, was 0.78% for the chamber properties and doses-per-pulse studied by Boag et al (1996), rising to 6.1% at 1 Gy dose-per-pulse and 2 mm electrode separation. Significance. Use of f exp should reduce ambiguities in collection efficiencies calculated using the approximate formulae, although like them f exp does not account for electric field distortion, which becomes substantial at doses-per-pulse ≥100 mGy.


Introduction
Ion recombination effects in radiotherapy dosimetry are currently receiving much attention, stemming from increased doses-per-pulse of around 0.8 mGy in photon beams generated by flattening filter-free linear accelerators (linacs), 10-100 mGy in electron beams used intra-operatively, and 10-5000 mGy in FLASH radiotherapy treatments, in comparison to around 0.3 mGy in conventional photon beams generated by linacs with flattening filters ( Recombination can be classified as initial, between charge carriers generated within a single ion track (Jaffé 1913(Jaffé , 1929, and volume, between carriers from different ion tracks (Thomson and Thomson 1928). In photon and electron beams initial recombination (sometimes called columnar recombination) is usually ignored, with volume (or general) recombination considered to be the dominant process (Christensen et al 2016). In these beams a pulse of radiation initially liberates free electrons and positive ions, and within the sensitive volume of an ionization chamber the free electrons drift rapidly towards the anode. In fact, compared to the positive ions the electrons move sufficiently fast that they are often all considered to have either reached the anode or formed negative ions by becoming attached to oxygen molecules, before the positive ions have moved from the locations in which they were created. After this early phase the positive and negative ions drift past each other more slowly, reaching the electrodes or recombining. To minimize recombination, distances d between electrodes are limited to millimetres over which potential differences V of hundreds of volts are applied to generate high electric fields. Boag et al (1996) formulated a landmark model of ion recombination for parallel plate ionization chambers in pulsed radiation beams (Kranzer et al 2021). In an earlier publication Boag (1950) had assumed that free electrons liberated in the chamber volume travel negligible distances before forming a uniform cloud of negative ions that drifts slowly through the uniform cloud of positive ions moving in the opposite direction. But in the 1996 study Boag et al instead took the free electrons to have a probability exp(-as) of rapidly drifting a distance s or more through air before becoming attached to oxygen. Correspondingly, a fraction = --( ( ) ) ( ) ( ) p a d a d 1 exp , 1 / of the free electrons were expected to be collected at the anode and the rest to form negative ions with a number density In these equations n 0 is the uniform number density of positive ions (or electrons) originally liberated by the radiation pulse; x is the distance from the cathode; t = 0 denotes an early time before the ions have moved significantly; and a depends on the electrical field strength and properties of air filling the detector sensitive volume (Thomson and Thomson 1928). Throughout the rest of this work number (not charge) density is succinctly referred to as 'density'. The total number of recombinations that occur when uniform clouds of positive and negative ions drift past each other can be calculated straightforwardly enough, because the resulting recombination rates are also uniform throughout the overlapping region (Boag et al 1996). However, when the negatively charged cloud is initially non-uniform, recombination rates vary throughout the overlap, causing the positive ion cloud also to become non-uniform and making the number of recombinations more difficult to determine. As a result, Boag et al (1996) obtained three approximate formulae f′, f″ and f″′ for ionization chamber collection efficiency, defined as the ratio of the charge collected by a chamber to the charge liberated in its sensitive volume. The formulae are based on three uniform approximations to the non-uniform initial negative ion density of equation (2): n -′, a uniform density -( ) p n 1 0 extending from one electrode to the other; n -′′, a uniform density n 0 extending from = x pdto d; and n -′′′, a uniform density l -( )n 1 0 extending from ld to d where l = -p 1 1 (figure 1). The resulting approximate collection efficiency formulae differ from each other and from the earlier formula f obtained by Boag (1950) which ignored free electron effects. Boag et al (1996) expected that f″′ would be more accurate than f′′ and f′′ than f′, because approximations n -′, n -′′ and n -′′′ visually appear to provide incrementally improving descriptions of the exponential variation of ion density with x (figure 1). However, collection efficiency calculations continue to be made using all three approximate formulae (Laitano et al 2006, Christensen et al 2016, Gotz et al 2017, McManus et al 2020 and f′ has also been used alone due to its algebraic properties (Di Martino et al 2005).
In fact, it is possible to obtain an analytical solution for recombination given the initial exponential distribution of negative ions described in equation (2), as set out in the appendix. This leads to an exact collection efficiency formula f exp , which can be used to determine the relative accuracies of the approximate formulae of Boag et al (1996) and which potentially removes the need to make three different approximate calculations of collection efficiency. It should be noted, though, that the model formulated by Boag et al does not account for distortion of the applied electrical field by charge imbalance. Consequently, neither the exact f exp formula nor any of the approximate formulae account for such distortions. These become substantial at very high doses-perpulse, one hundred mGy or higher (Boag et al 1996), and are greater at chamber voltages 150 V due to increasing numbers of free electrons reaching the anode (Gotz et al 2017, Petersson et al 2017. In the main body of this work the collection efficiency formula f exp is set out alongside the approximate formulae of Boag et al (1996) and the original 1950 Boag formula. Collection efficiency values calculated using all the formulae are compared in tables and graphs. Details are also provided of a one-dimensional numerical scheme for calculating ion transport, recombination and collection efficiency, which was used to check the correctness of f exp whose derivation is lengthy.

Nomenclature
Quantities appearing outside the derivation of f exp presented in section 2.2 and the appendix are denoted by the following symbolsf, f′, f″, f″′, f exp , ionization chamber collection efficiency formulae p, fraction of free electrons collected at the anode d, electrode spacing a, drift coefficient for free electrons γ, rate-constant for attachment of free electrons to oxygen n 0 , initial uniform density of positive ions or free electrons  n , density distributions of positive and negative ions n -′, n -′′, n -′′′, approximations ton at t = 0 used to derive f′, f″ and f″′ V, potential difference across the chamber k 1 , k 2 , k e , mobilities of positive and negative ions and electrons v + , v − , v e , drift velocities of positive and negative ions and electrons α, rate-constant for recombination between positive and negative ions = dimensionless combination of parameters Notice that the composite parameter u varies with n 0 and thus with dose-per-pulse. For ease of reference the symbols used to represent quantities that appear in Boag et al (1996) are nearly identical to those in the 1996 publication. Quantities used only in the derivation of f exp are described as they are introduced.

Derivation of f exp
The derivation is set out in the appendix and has two key steps. First, the number of negative ions initially located in a thin slab lying a distance x from the cathode that eventually recombine with positive ions is related to + N , x the total number of positive ions through which the slab of negative ions passes on its way to the anode (equation (A6)). This step achieves little on its own since + N x is not known ab initio. However, at the second step it is recognized that + N x is equal to = + ( ) N x t , 0 ,the known total number of positive ions initially lying between the slab and the anode immediately after the radiation pulse, minusR , x the number of these positive ions that have recombined with negative ions before the slab passes through them (equation (A10)). By definition,R x is also equal to the integral of all the negative ions lying in thin slabs initially located > x from the cathode that have recombined with positive ions en route to the anode.
These considerations allow a differential equation (A15) to be constructed that links the total number of negative ions eventually reaching the anode to quantities related to the known initial density distributions of positive and negative ions. This equation is solved to obtain a formula for the total number of negative ions Figure 1. A plot of the initial, t = 0, densityn of negative ions versus distance from the cathode in the sensitive volume of an ionization chamber, according to the model of Boag et al (1996) allowing for free electron effects. Density has been normalized to the initial uniform density of positive ions, n , 0 and is plotted for a chamber with a 1 mm distance between electrodes in which 40% of free electrons are collected and 60% form the negative ions. Also plotted normalized to n 0 are the approximate density distributions ¢ -n ,  -n and ¢ -n used to derive the collection efficiency formulae f′, f′′ and f′′′ of Boag et al (1996). collected, expressed in terms of an integral of a function ( ) g x related to the initial ion density distributions (equation (A22)). For a uniform initial positive ion distribution and the exponential negative ion distribution of equation (2), f exp can then be obtained in closed form from equation (A22).
In fact, the same process can be followed up to equation (A22) for any positive and negative ion distributions, with different distributions leading to different ( ) g x functions. Analytical collection efficiency formulae can then be obtained for those distributions that yield ( ) g x functions with closed form integrals, while for other distributions collection efficiencies can be calculated by numerical integration of equation (A22), a faster method than numerical simulation of the whole ion transport, recombination and collection process.

Collection efficiency calculations
Three sets of calculations have been performed. First, using the f exp formula collection efficiencies were calculated directly for generic ionization chambers with the same set of u and p values for which Boag et al (1996) compiled f, f′, f″ and f′″ values in their table 1. It is easy to evaluate f exp using a pocket calculator, given values of the exponential integral function which can be obtained from tables or online resources (e.g. Casio's Keisan site). However, exponential integral values can also be obtained using functions available in languages such as MATLAB (Mathworks, Natick, MA) and R (R Foundation, Vienna, Austria). And since the collection efficiency formulae were evalulated many times in this work, code was written in R version 4.0.2 to repeatedly and rapidly compute them.
Using all five formulae a second set of collection efficiencies was calculated for chambers with the specific properties listed in table 1. The tabulated parameter values describe properties of air and the widths of the gaps between the parallel plate electrodes in the dosimeters being studied, namely 1 mm representing an Advanced Markus chamber (PTW 34045, Freiburg, Germany) and 2 mm representing the classic Markus chamber (PTW 23343) and also the Roos (PTW 34001), NACP Plane Parallel (IBA, Louvain-la-Neuve, Belgium) and Exradin A10 (Standard Imaging, Madison WI) chambers. Calculations were carried out for doses-per-pulse of 1, 10, 100 and 1000 mGy and chamber voltages across the range 100-1000 V.
The air-related parameter values in table 1 were taken from Gotz et al (2017), except for the n 0 -per-Gy factor which was obtained from the (W/e) value of 33.97 eV required to generate each electron/ion pair in dry air (Boutillon and Perroche-Roux 1987) and the 1.204 kg m −3 density of dry air at 20°C temperature and 101.325 kPa pressure. The tabulated k e value was estimated from figure 2(a) of Gotz et al (2017), while the γ function describing the electron attachment rate-constant and its dependence on electric field-strength was obtained by fitting data plotted in figure 2(b) of the same publication. The drift coefficient for free electrons, a, was calculated from γ using / is the electron drift velocity. The relationship between a and g follows because these coefficients are the constants of proportionality linking free electron density to its rates of loss with distance travelled and time respectively (Boag et al 1996, Gotz et al 2017. Next, p was calculated from the values of a and d via equation (1). Then u values were calculated from α, d, k 1 and k 2 plus the chamber voltage and initial ion density, n 0 , which was obtained from the dose-per-pulse and the n 0 -per-Gy factor in table 1. Finally, ionization chamber collection efficiencies were calculated for these u, p and (ad) values.
In a third set of calculations the transport of positive and negative ions towards the cathode and anode was computed using a one-dimensional numerical scheme, starting from a uniform positive charge cloud and the distribution of negative ions described by equation (2). The positive and negative ion drift velocities v + and vwere determined from the ion mobilities and electric field-strength (V/d), and transport calculations were carried out in a rest-frame in which the negative ions were static and the positive ions moved past them with a Table 1. Ionization chamber parameter values used in collection efficiency calculations that are not based directly on u and p.

Parameter
Value a a All values derived from Gotz et al (2013) except for n 0 -per-Gy which follows directly from (W/e) and the density of dry air at 20°C and 101.325 kPa. b E is electric field-strength measured in V m −1 , equal to the applied potential difference divided by the electrode separation. velocity / This is the reverse of the rest-frame used in the derivation of the f exp formula, in which the positive ions were stationary (see appendix).
To travel right through the negative ion cloud, positive ions need to move a maximum of 1 mm in the chosen rest-frame. This movement was computed in 5 × 10 4 steps, equivalent to a distance-per-step of 20 nm, and ion number densities were discretized in corresponding 20 nm bins. At each step the positive ion density distribution was shifted 20 nm through the negative ion distribution, and recombination during the step was accounted for by depleting the densities of positive and negative ions within each bin by a D + -n n t where Δt is the time-interval between steps, equal to´+ - or 1.68 × 10 −10 s. Finally, the collection efficiency was determined by summing the numbers of positive ions in bins emerging from the cathode side of the negative ion cloud and dividing by the initial total number of positive ions.
This scheme provides a check of the f exp formula, and was used to independently compute collection efficiencies for the generic chamber with u = 0.5 and p = 0.1, 0.3 and 0.6. These values were chosen because collection efficiencies calculated by Boag et al (1996) for u = 0.5 differed substantially from one and varied notably with p. So far as possible the parameter values of table 1 were used in the numerical calculations. However, for p to equal 0.1, 0.3 and 0.6, the product (ad) had to be set to 10.0000, 3.1971 and 1.1263 respectively rather than being computed from γ, k e , V and d. To achieve the u value of 0.5 at a chamber voltage of 300 V, n 0 was set to 9.138 46 × 10 16 m −3 equivalent to a dose-per-pulse of 41.3 mGy, rather than being calculated for a pre-specified dose-per-pulse.

The f exp formula
The collection efficiency formula derived analytically in the appendix is and E 1 is the exponential integral function (Gradshteyn and Ryzhik 2007). Alternatively, f exp can be written as 1 exp exp . 5 1 1 Figure 2. The exponential integral function E 1 (ζ) graphed over the range ζ = 10 −3 to 1.5. E 1 tends to infinity as ζ approaches zero.
For comparison, the formulae obtained by Boag (1950) and Boag et al (1996) where l = -p 1 1 . Whereas the collection efficiency formula f ignores free electron effects and is determined by u alone, f′, f″ and f″′ approximately account for these effects and depend on both u and p. At first sight f exp appears to depend on u and (ad) instead. However, p and (ad) are determined uniquely by each other (equation (1)) and therefore f exp can equivalently be viewed as being determined by u and p.
The exponential integral function E 1 in equation (3) is defined as and has the widely tabulated form shown in figure 2. A difference between two E 1 values also appears in a collection efficiency formula obtained by Jaffé for ion tracks that run parallel to the electric field in an ionization chamber, based on his theory of initial recombination (Jaffé 1913, Jaffé 1929, Kanne and Bearden 1936, Christensen et al 2016. Table 2 shows collection efficiencies calculated by Boag et al (1996) using the f, f′, f″ and f′″ formulae for generic ionization chambers having a range of (u, p) values, together with corresponding collection efficiencies calculated using f exp . When determining f exp , values of (ad) corresponding to the values of p were first obtained by numerically inverting equation (1) and were then used alongside u in equation (3). Across all the (u, p) combinations values of f, f′, f″ and f″′ have the consistent order f″ > f″′ > f′ > f. The maximum relative difference between f′, f″ and f′″ values is 5.1%, seen at u = 0.5 with p = 0.5, equivalent to 144 mGy per pulse with 212 V applied over an electrode gap of 1 mm. Boag et al (1996) expected f″′ to be the most accurate of the approximate formula, and f″′ values do indeed lie closest to the exact f exp values over the range p = 0.3-0.6. However, for p values of 0.1-0.2, f″ lies closer to f exp . The greatest difference between f″′ and f exp values is seen at u = 0.5 with p = 0.2, where f″′ = 0.8596 and f exp = 0.8663. The correction factors corresponding to these collection efficiencies are 1.163 and 1.154 which differ by 0.78% relative.

Variation of f exp with u and p for generic chambers
To make the tabulated values of p more tangible, the electrode separation d to which each p value corresponds has been calculated by dividing the equivalent (ad) value by a, which in turn was calculated from g, k e and the electric field strength (V/d). These d values are shown in table 2 for field strengths of 100 and 300 V mm −1 . Similarly, doses-per-pulse corresponding to the tabulated levels of u have been calculated using the k 1 , k 2 , a and n 0 -per-Gy values listed in table 1, assuming applied potential differences of 100 and 300 V and electrode separations of 1 and 2 mm. The dose-per-pulse values are listed in table 3.
Values of the h(u, ad) (or equivalently h(u, p)) term of equation (4) are shown in table 4 for the same (u, p) combinations for which collection efficiencies are listed in table 2. The degree to which h exceeds one is a measure of the effect of free electrons on collection efficiency, since when h equals one the f exp formula reduces to f. Across all the (u, p) combinations h varies between 1.0011 and 1.1987, exceeding 1.10 for u = 0.3 and p 0.5, and for u = 0.5 and p 0.3. Figure 3 shows Jaffé plots describing the variation of (1/collection efficiency) with (1/V ) for a parallel plate ionization chamber with the air-related properties listed in table 1 and the 1 mm electrode separation of the Advanced Markus chamber. The figure comprises four graphs showing results for doses-per-pulse of 1, 10, 100 and 1000 mGy. Each graph includes lines representing collection efficiencies calculated across a potential difference range of 100-1000 V according to the f exp , f, f′, f″ and f″′ formulae. Collection efficiency values have the consistent order f″ > f″′ > f″′ > f, the same pattern seen in table 2. To assist further comparisons with the results shown in table 2 it is useful to note that for the 1 mm electrode gap, chamber voltages of 100, 200 and 300 V correspond to p values of 0.163, 0.515 and 0.738, and to u values of 7.275, 3.638 and 2.425 when the doseper-pulse is 1 Gy, u scaling linearly with dose-per-pulse.

Variation of f exp with voltage and dose-per-pulse for Advanced and classic Markus chambers
Plots of (1/f ) are roughly linear in figure 3, whereas the (1/f exp ), (1/f′), (1/f″), and (1/f″′) plots initially curve upwards before also rising approximately linearly. The f″′ and f″ curves lie closer to f exp than do f or f′. Generally f″′ is closer to f exp than is f″, but at 100 V (corresponding to p = 0.1633) f″ is closer to f exp . The (1/f exp ) and (1/f″′) Table 2. Ionization chamber collection efficiencies calculated for sensitive volumes with various u and p values and corresponding (ad) and d values. The collection efficiencies have been computed according to Boag's original formula f which ignores the collection of free electrons, the formulae f′, f″ and f″′ obtained for uniform approximations to the exponential distribution of negative ions described by equation (2), and f exp obtained exactly for the exponential distribution.  curves cross in each graph, (1/f″′) lying below (1/f exp ) at higher voltages and above it at lower ones. At doses-perpulse of 1, 10, 100 and 1000 mGy maximum relative differences between (1/f″′) and (1/f exp ) are 0.014%, 0.14%, 1.1% and 4.9%. Figure 4 shows corresponding Jaffé plots for detectors with an electrode separation of 2 mm, for example the classic Markus and NACP Plane Parallel chambers. Collection efficiencies are substantially lower (higher reciprocals) for this separation, because the ions have further to travel to reach the electrodes and they drift more slowly since the same applied voltage produces a weaker electric field. When comparing these collection efficiencies with those of table 2, chamber voltages of 100, 200 and 300 V correspond to p values of 0.027, 0.082 and 0.179 for the 2 mm electrode gap, and to u values of 29.100, 14.550 and 9.700 when the dose-per-pulse is 1 Gy.

Computational check of the f exp formula
The numerical scheme for calculating ion transport and recombination described in the Materials and methods was used to check three of the f exp values calculated for the generic ionization chamber. The scheme was run with parameters mostly set to the values in table 1 but with free electron transport adjusted to achieve p values of 0.1, 0.3 and 0.6, and with the dose-per-pulse selected as 41.2 mGy to achieve a u value of 0.5. The computed collection efficiencies were 0.841 41, 0.887 18 and 0.939 17 for the 0.1, 0.3 and 0.6 values of p respectively, in agreement to 5 decimal places with collection efficiencies calculated from the f exp formula and shown to 4 decimal places in table 2.

Discussion
A collection efficiency formula, f exp , has been obtained by analytically solving the model of Boag et al (1996) which describes ion transport and recombination in parallel plate ionization chambers and allows for free electron effects. Because the derivation of f exp is quite lengthy the formula was independently checked by running an in-house ion transport and recombination code, producing collection efficiency values in very close agreement with those obtained from f exp .
Across the (u = 0.01-0.50, p = 0-0.60) range of chamber properties studied by Boag et al (1996), the maximum relative difference between collection efficiency values calculated using three approximate formulae obtained in the 1996 publication is 5.1%. Values of the approximate formula f″′ lie closest to those of f exp over much of the (u, p) range studied, although for p 0.2 values of f″ are closer to f exp . The greatest difference between f″′ and f exp is 0.78% relative. Given access to tabulated exponential integral values f exp can easily be evaluated using a hand calculator, and since f exp represents an exact solution of the model of Boag et al its use should reduce some of the ambiguity introduced into collection efficiency estimation by the multiple approximate formulae.
Greater differences between collection efficiencies calculated using the various formulae can be seen in figures 3 and 4, because the graphs in the figures effectively extend to higher u values than those of table 2. In the graphs the largest difference between correction factors based on f″′ and f exp is some 6.1% relative. For chamber potential differences above 200 V, f″′ values are closest to the exact f exp values. However, at 100 V with an electrode separation of 1 mm, f″ is closest to f exp and this is also the case at 100-200 V with a separation of 2 mm. These voltages and electrode separations correspond to p values of 0.026-0.163.
The model of Boag et al (1996) ignores electric field distortions resulting from imbalances between the spatial distributions of positive and negative charges, an effect that becomes substantial at doses-per-fraction 100 mGy. Consequently, while f exp represents an exact solution of the model, like the approximate collection efficiency formulae it does not account for field distortion effects at very high doses-per-pulse.
The approach used to derive f exp is more intricate than the method used by Boag (1950) and Boag et al (1996) to obtain the f, f′, f″ and f″′ formulae, but accounts exactly for evolving ion cloud non-uniformity during transport. It would be useful to further extend this analytical approach to include spatial variations in the electric field-strength and drift velocities of free electrons and ions. This is challenging but would potentially allow collection efficiency formulae to be obtained for cylindrical ionization chambers in which electric fields are intrinsically non-uniform.
If the effects of electric field non-uniformities could be described using analytical methods, it might also be possible to include in the analysis changes in the field due to evolving imbalances in charge distributions. This would perhaps allow a closed-form collection efficiency formula to be obtained that accounts for free electron effects, electric field non-uniformity and field distortion at high doses-per-pulse. Such a formula would provide further insights into how the various transport and recombination factors collectively influence detector performance when electric field distortion is substantial. It could also be used to calculate collection efficiency values for this circumstance faster and more conveniently than by running computational transport codes (Gotz et al 2017, Kranzer et al 2021.

Conclusions
A collection efficiency formula f exp has been derived, based on an exact solution of the model of Boag et al (1996) which describes recombination in parallel plate ionization chambers and includes free electron effects but excludes electric field distortion. Calculated values of f exp are in excellent agreement with collection efficiencies obtained computationally using an ion transport and recombination code.
Of the three approximate collection efficiency formulae developed by Boag et al (1996) values of f″′ most often lie closest to the exact f exp values, although when only 10%-20% of free electrons reach the anode, f″ rather than f″′ values lie closest to f exp . Over the (u = 0.01-0.50, p = 0-0.6) range studied by Boag et al (1996), the maximum relative difference between f″′ and f exp is 0.78%. For larger u values corresponding to doses per pulse of 100 and 1000 mGy and an electrode separation of 2 mm, maximum relative differences between correction factors based on f″′ and f exp reach 1.5% and 6.1% respectively.
Use of the exact f exp formula should reduce ambiguities in collection efficiency estimates currently obtained from three different approximate formulae. It must be remembered, though, that like the approximate formulae, f exp does not account for electric field distortion which becomes substantial at doses-per-pulse 100 mGy.

Appendix
Consider an air-filled plane-parallel ionization chamber with the geometry shown in In equation (A1) n 0 is the uniform density of positive ions initially generated by the radiation pulse andx is distance to the right of the cathode in the rest-frame of the chamber. Figure A1. A schematic illustration of the cavity of a plane parallel ionization chamber showing in gold the cathode and anode separated by a distance d, and in green a slab of negative charge of thickness dx. In the rest-frame of the detector the cathode and anode havex-coordinates 0 and d and the green slab is initially located atx. In the rest-frame of the positive ions the electrodes have x coordinates that at time t = 0 are also 0 and d, and the position x of the slab in this rest-frame also equalsx at this time. In real detectors the cavity thickness d is less than the diameter of the electrodes.
In the absence of a net flow of ions into or out of a small region, the densities of both positively and negatively charged ions in the region decrease at a rate a + -n n where a is the recombination rate-constant. From this it is possible to calculate charge recombination for positive and negative ions as they move past each other, travelling to the cathode and anode respectively with velocities + v andv . For ease of analysis, in the rest of the appendix the rest-frame is shifted to one in which the positive ions have zero velocity and the negative ions move past them to the right with speed = + + - Locations in the shifted frame are denoted by x rather thanx, and the originsx = 0 and x = 0 are coincident at t = 0. Now focus on a thin slab of negative ions initially located at x → d + x xas it moves to the right ( figure A1). Denote as -( ) n t x the density of negative ions remaining in the slab at time t, when the slab is at + x v t.
tot Throughout the appendix  ( ) n x t , is used to describe the density of positive or negative ions at position x at time t, whereas  ( ) n t x describes density at time t in a thin slab that was located at x when t = 0. Consequently  ( ) n x t , and  ( ) n t x are different quantities but equivalent at t = 0. The rate of change of -( ) n t x tot / at which point the slab has passed right through the cloud of positive ions initially to its right in the chamber, and changing variable to = + ¢ y is the total number of positive ions per area passed through by the slab initially located at x. Thus, the number of recombination events that occur per unit area as the thin slab of negative charge passes completely through the cloud of positive ions to its right is Defining R x as the total number of recombinations per area after all the slabs initially located between 0 and x pass completely through the positive ions to their right, then Now, denote as + ( ) N x, 0 the total number of positive ions per area that are initially positioned to the right of x at time t = 0 whereR x is the total number of positive charges per area initially lying to the right of x that have already recombined with negative ions before the slab originally located at x travels through them. These recombination events are due to the negative ions located to the right of the slab having moved through the positive charge cloud ahead of the slab, and consequentlyR x is given by From equations (A7), (A8), (A10) and (A11) in which -( ) N x, 0 is the total number of negative ions per unit area located to the right of x at time zero, andC x is the number of these negative ions per area that pass through the positive ions and are collected at the anode. Differentiating equation (A13) with respect to x and comparing with (A12) Equation (A15) can be written as whereC d describes the number of negative ions per area collected at the anode that were initially located to the right of d, and is zero since this range of locations lies outside the region between the electrodes.C 0 is the total number of negative ions per area that eventually arrive at the anode regardless of their original x position in the chamber, and together with the known number of free electrons per area collected at the anode it defines the total negative charge collected per unit cross-sectional area. To evaluateC , 0 all that remains is to solve the integral


The collection efficiency, f, of the ionization chamber is the sum of the total numbers of free electrons and negative ions collected per unit cross-sectional area, divided by n d, 0 the total number of electrons (or positive ions) initially generated per area. Therefore Finally, by extracting the factor -( ) up exp from inside the logarithm, a constant term -p is added outside it cancelling the +p term, and the equation becomes