The ion-recombination correction factor is used to correct the response of an ionization chamber for the lack of complete charge collection, which is due to the recombination of ions exhibiting opposite charges during transit to each electrode. For positive ions, the ion-collection efficiency f is given by the ratio of the collected charge M to the total charge produced, M₀: f=M/M₀. Then, the ion-recombination correction factor is defined as k_s=1/f [6,10].

Because interactions between the ions themselves—or between ions and neutral molecules—in an electric field are complicated, determining f or k_s exactly is difficult, so k_s is currently obtained from empirical formulas [10]. In the IAEA code of practice (the TRS-398 protocol) for pulsed radiation beams, the ion-recombination correction factor k_s is given by [11]

where a₀, a₁, and a₂ are constants that depend on the ratio V₁/V₂ of the two polarizing voltages. For V₁/V₂=2, the values of the constants are a₀=2.337, a₁=–3.636, and a₂=2.229. The quantity M₁/M₂ is the ratio of the charges measured at the two polarizing voltages V₁ and V₂. Similarly, in the AAPM TG-51 protocol, the ion-recombination correction factor P_ion (=k_s) is given by [12]

Both of these methods are based on Boag’s early model (1950), which assumes a linear dependence of 1/M on 1/V [11]. According to that model, the correction factor is determined by extrapolating the 1/M vs. 1/V plot to determine the value 1/M₀ obtained when 1/V=0, where M₀ is the saturated charge for which f=1. By estimating M₀, one can thus obtain the ion-correction efficiency f=M₁/M₀ or the ion-recombination correction factor k_s=M₀/M₁ [13].

The quantity f declines from 1 to 0 with increasing DPP, and for clinical electron beams (DPP<1 mGy), the recombination correction factor can be determined using the current methods that apply the dosimetry protocols of the IAEA and AAPM. However, because free electrons that do not contribute to the production of negative ions increase with increasing DPP, eventually the value of f does not continue to decrease significantly. Thus, the current methods are not appropriate for cases with high DPP, especially DPP>20 mGy [6,7].

Therefore, in order to incorporate free electrons into the determination of the ion-collection efficiency, Boag proposed three improved models, where the third model (denoted by f''') was introduced as a practical correction method for high-DPP electron beams [6]. The ion collection efficiency (f’’’) in Boag’s third model is given by [6]

where $\lambda =1\sqrt{1-p}$, p is the free-electron fraction of the total electrons produced by a beam pulse, the dimensionless parameter u=µrd²/V, where µ depends on the ionic recombination coefficient and the ion mobility, r is the charge density of positive ions initially generated by the beam pulse, d is the electrode spacing, and V is the polarizing voltage [6,7].

Equation (3) cannot be solved directly due to the difficulty in determining the parameter u. However, if f₁ and f₂ are the ion-collection efficiencies for the charges M₁ and M₂ measured at the two polarizing voltages V₁ and V₂, respectively, then the ratio f₁/f₂=M₁/M₂ can be expressed a function of u₁ by using u₁/u₂=V₂/V₁:

Then, one can determine u₁ iteratively using a computer program and the measured value of M₁/M₂. The correction factor k_s=1/f1 can be determined by substituting the resulting value of u₁ into the expression for f₁. This is the method proposed by Boag et al. (1996) [6] and implemented by Laitano et al. (2006) [7]. However, although this method can be used to determine ks, applying in practice is inconvenient.

Conversely, because the single variable u₁ determines both M₁/M₂ and f₁ simultaneously, the correction factor k_s=1/f₁ can be determined as a function of M₁/M₂ without iteration by utilizing pre-calculated values of M₁/M₂ and f₁ as functions of u₁. Thus, the quantity f₁ can be obtained as a function of M₁/M₂ for practical use by fitting. Our study used this new method with the new equations based on the Boag model, to measure high-DPP electron beams produced by an electron LINAC.

Laitano et al. [7] investigated Boag’s three improved models and published the ion-recombination factors determined for six types of commercial plane-parallel ionization chambers for various polarization voltages. For comparison with our method, we selected 22 of the data points obtained by Laitano et al. [7] for comparison. The selected data cover the range 0.1–70 mGy/pulse and include values of M₁/M₂ up to about 2, as measured at 10 voltage ratios with an Exradin A12 ionization chamber. We solved equation (4) for the given M₁/M₂ values and two polarization voltages, and we fitted the k_s values as a function of M₁/M₂ with second-order polynomial functions. We employed the same physical parameters used by Laitano et al. [7] in our calculations of equation (4).

To determine the reference dose in water at source-to-surface distance (SSD)=100 cm, we performed dosimetry for electron beams from a 6 MeV electron LINAC using an Advanced Markus^® ionization chamber, as shown in Fig 1. The electron LINAC used in this study is the prototype developed by the Dongnam Institute of Radiological and Medical Science in collaboration with the Pohang Accelerator Laboratory [14]. Because the electron energy depends on the heating current of the electron gun, this experiment was performed at about 6 MeV by adjusting the heating current [15].

An electron-irradiation device was used for the electron-beam irradiations, as shown in Fig. 1. It consists of specially designed scattering foils and collimators to generate an optimal electron beam for FLASH preclinical studies. We do not discuss its detailed geometry in the present work because the irradiation device is still under study.

The ionization chamber was calibrated in terms of water equivalents using a Co-60 reference beam. We applied the TRS-398 protocol to determine the dose in water [11]. The reference point of the ionization chamber was positioned at z_ref=0.6R₅₀−0.1 cm, where R₅₀ is the half-value depth in water [11]. In this work, we determined R₅₀ by measuring the percentage depth dose (PDD) curve using a radiochromic film. The ionization measurements were performed at polarization voltages of 400 V and 200 V for 100 electron-beam pulses at the repetition rate of 50 Hz using pulse-mode control of the pulse-modulator system [16]. We applied the values of k_s calculated with the new method to correct the measured charge, and we also applied environmental and polarity corrections according to the TRS-398 protocol [11].

Fig. 2 shows f₁/f₂ and 1/f₁ calculated as a function of u₁ for V₁/V₂=400/200 using equations (3) and (4) for the Advanced Markus^® chamber, a plane-parallel ionization chamber with an electrode spacing of 1 mm. The value of k_s can be determined as a function of M₁/M₂, because k_s=1/f₁ and M₁/M₂=f₁/f₂.

In this manner, we calculated k_s for the 22 selected data points, and the calculated results are shown in Fig. 3 together with the results of Laitano et al. [7]. Here, Fig. 3a and Fig. 3b, respectively, are for cases with small and with large slopes of k_s as a function of M₁/M₂ [7]. Comparing the calculated and published values of ks shows good agreement, i.e., to within 0.1%.

The calculated values of k_s for the Advanced Markus chamber we used to measure the dosimetry of high-DPP electron beams is shown in Fig. 4. This figure shows the two k_s curves at two voltage ratios obtained with the new method (based on Boag’s model) and the two corresponding k_s curves calculated using current methods (those of the IAEA and the AAPM). The k_s curves obtained using the new method can be fitted with the following function for practical use:

Here, b₀=1.3890, b₁=−0.9705, and b₂=0.5819 for polarization voltage ratio V₁/V₂=400/200 and b₀=1.0413, b₁=−0.1549, and b₂=0.1157 for V₁/V₂=300/100. In Fig. 4, the values of k_s obtained with the currently used IAEA TRS-398 and AAPM TG-51 protocols are similar, but they are higher than those obtained with Boag’s model. Compared to the k_s value for M₁/M₂=1.10 at V₁/V₂=400/200, the results from the IAEA TRS-398 and AAPM TG-51 protocols differed by about 8.9% and 8.2%, respectively, although no significant difference was found in the range M₁/M₂<1.01.

The PDD curve measured with radiochromic film in water to determine the beam-quality index R₅₀ (=2.4 g/cm²) is shown in Fig. 5. The mean electron energy at the phantom surface, $\overline{E}$=2.33 R₅₀=5.6 MeV, and the most probable energy, E_p=0.22+1.98 R_p+0.0025 R_p² ≈ 6.4 MeV, can be determined from the PDD plot in Fig. 5, where R_p (=3.1 g/cm²) is defined as the practical range of the electron beam in water [17]. These parameters are similar to those of a clinical electron beam with a nominal energy of 6 MeV [17].

We took into account the 0.2-cm-thick window made of polymethyl methacrylate, with a density of 1.19 g/cm³, in determining the measurement depth. The reference depth, z_ref=1.38 g/cm², and the beam-quality factor as functions of R₅₀ were taken from the tabulated value for the Markus chamber in the TRS-398 protocol. The determined dose in water at the reference depth is listed in Table 1. Because the measured dose D (z_ref) in the table is for 100 electron-beam pulses at a 50 Hz repetition rate, the average dose rate is 5.75 Gy/s, and DPP (z_ref)=115 mGy/pulse. The combined standard uncertainty in this measurement is estimated to be 2.4% (coverage factor k=1), which includes contributions from the measured value (0.1%), the ion-recombination correction factor (2.0%), a polarity correction factor (0.4%), an air-density correction factor (0.2%), an ionization-chamber calibration factor (0.6%), a quality factor (1.0%), and others (0.5%). Here, we assumed the uncertainty in the ion-recombination correction factor to be the value given by Boag [6].