Calculation of the effects of electric fields on ionization by electrons emitted in a parallel plate ionization chamber irradiated with 60Co γ-rays

Radiation Physics and Chemistry 170 (2020) 108623

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Calculation of the effects of electric fields on ionization by electrons emitted in a parallel plate ionization chamber irradiated with 60Co γ-rays

T

Nobuhisa Takata Department of Radiological Sciences, IPU, Ami, Ibaraki, 300-0394, Japan

A R T I C LE I N FO

A B S T R A C T

Keywords: Ionization chamber Applied voltage Electric field effect Polarity effect Signal current 60 Co γ-rays

The effects of applied voltage on a parallel plate ionization chamber irradiated with 60Co γ-rays were investigated by performing calculations using EGS5 code, which incorporates the electromagnetic field. It was confirmed that the signal current from the chamber increases or decreases linearly beyond the recombination region, when positive or negative voltage is applied to the polarizing electrode, respectively, which is in agreement with the existing measurements. It was also confirmed that the voltage dependence is due to the stopping power of air, which increases or decreases as the secondary electrons are decelerated or accelerated in the electric field, respectively, because the electrons are emitted forward more than backward. The calculated rate of increase in the signal current was smaller than the experimental ones, and the calculated rate of decrease was larger than the experimental ones. Numerical calculations were performed to investigate the effects of the electric field on the number of ion pairs produced by electrons with energy less than 1 keV, which are emitted in the ionization chamber; this effect is not accounted for by the EGS5 code. It was revealed that the number of ion pairs increases with an increase in the electric field strength. The increase cannot be attributed to the Townsend primary ionization mechanism and takes place even at very low electric field strength. This theoretical correction improves the agreement of the calculation with experiment for both the positive and negative applied voltages. The contribution of the charge of secondary electrons which exit and alive at electrodes or the ionization volume on the signal current from the ionization chamber was also discussed.

1. Introduction It is known that signal currents from a parallel plate ionization chamber irradiated with 60Co γ-rays from the front side (polarizing electrode) linearly increases and decreases with the positive and negative applied voltages, respectively, beyond the recombination region. The increase or decrease of the current is reversed when the chamber is irradiated from the rear side (collector). The rate of the increase is larger for a smaller separation between the two electrodes. On the contrary, the dependence of the rate of decrease on the separation is insignificant in the range 2–30 mm. The increase or decrease was attributed to the change in the stopping power of air due to deceleration or acceleration of secondary electrons in the applied electric field (Takata and Matsumoto, 1991). This is because secondary electrons passing between the electrodes are expected to be emitted forward more than backward. However, it is unclear why the signal current dependence on the electrode separation is different depending on the polarity of the applied voltage. In the present work, the number of ion pairs produced inside an ionization chamber filled with air is calculated for various separations

between the electrodes and at various applied voltages to understand the reason for the increase or decrease of the signal current and the dependence on the separation. The EGS5 code (Hirayama et al., 2005) which incorporates the electromagnetic field (Torii and Sugita, 2007), is used for performing the calculation. In addition to the number of ion pairs, the numbers of secondary electrons that exit and arrive at the electrodes or the ionization volume are obtained. This is needed because a signal current could be affected by secondary electrons if the numbers of the secondary electrons vary with the applied voltage. In the calculations performed using the EGS5 code, the cutoff energy is set to 1 keV; therefore, electrons whose energy become less than 1 keV are assumed to stop and deposit the residual energy to the substance where they stop. The effects of electric fields on the ionization and diffusion of electrons with energy less than 1 keV are not well understood. In the present study, the effects of the electric fields, the changes in ionization and diffusion to the cathode or anode, are calculated from the data concerning the stopping power, range and W value of air for these low energy electrons. The effects of the electric field on the signal current from an ionization chamber due to low energy electrons are estimated

E-mail address: [email protected]. https://doi.org/10.1016/j.radphyschem.2019.108623 Received 11 September 2019; Received in revised form 17 November 2019; Accepted 26 November 2019 Available online 27 November 2019 0969-806X/ © 2019 Elsevier Ltd. All rights reserved.

Radiation Physics and Chemistry 170 (2020) 108623

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from an assumption about the energy distribution of the electrons emitted in the ionization volume. 2. Secondary electrons with energy larger than 1 keV 2.1. Calculation for the secondary electrons An ionization chamber with parallel flat electrodes was used for measuring the effects of the applied voltage on signal currents (Takata and Matsumoto, 1991). It has a collector (diameter: 10 cm) surrounded by a guard ring electrodes (diameter: 19 cm). The polarizing electrode is also 19 cm in diameter. The collector and the guard ring electrode are 8 mm thick, and the polarizing electrode is 3 mm thick. They are made of graphite. The separation between the polarizing electrode and the guard ring electrode (and thus the collector) was changed to 1.99 ± 0.02, 5.07 ± 0.02, 14.54 ± 0.05 and 30.66 ± 0.04 mm by replacing spacers of various thicknesses. To calculate the effect of the applied voltage on the signal current from an ionization chamber, the EGS5 code (Hirayama et al., 2005) which incorporates the electromagnetic field (Torii and Sugita, 2007) was used. To simplify the calculation, only two electrodes (polarizing electrode and the collector) were included. The diameters of these electrodes were fixed at 2 m, and their thicknesses were set to 3 mm. The density of the graphite material of the electrodes was assumed to be 1.85 g cm−3. The pressure and the temperature of the air between the electrodes were assumed to be 1 atm and 22 °C. These values are similar to the condition during the measurement of the effects of the applied voltage. The relative humidity was about 35%. In calculation, however, the air is assumed to consist of N, O and Ar with a mass ratio of 0.75575, 0.23143 and 0.01282, respectively. The outside of the chamber was assumed to have vacuum conditions. The cutoff energy was set to 1 keV both for photons and electrons. The γ-rays were perpendicularly irradiated at the center of the electrodes to their surfaces. The scattered γ-rays and the secondary electrons that escaped into the vacuum from the ionization chamber were neglected. Calculation was performed for 2 × 109 60Co γ-rays for each separation between the electrodes and the electric field. It was repeated four times using different seed states for random number generator. Mean values of the calculation results were always used. The number of ion pairs produced in the air between the electrodes was obtained from the energy deposited to the air divided by the W value of the air for electrons, i.e. 33.97 eV. In the calculation, the regions in which an electron is ejected and stopped are registered for all electrons. Electrons that are ejected and stopped in the polarizing electrode or in the collector do not contribute to the signal current. Electrons that are ejected from the polarizing electrode and have stopped in the collector contribute as a negative current. Furthermore, electrons that are ejected from the collector and have stopped in the polarizing electrode contribute as a positive current. The contribution of electrons that are ejected from the polarizing electrode or the collector and have stopped in the air depends on the polarity of the applied voltage because after stopping, they migrate to the electrode from which they are ejected or to the opposite side electrode depending on the polarity. The contribution of the secondary electrons that are ejected in the air also depends on the polarity of the applied voltage. This is because the electrons that stop in the air and the positive ions from which the electrons are ejected migrate to the polarizing electrode or the collector depending on the polarity of the applied voltage. The number of electrons that contribute as a positive current or as a negative current was counted, and the magnitude of the signal due to the charge of the secondary electrons was obtained from the difference between the numbers of electrons. The difference, i.e., the net number of secondary electrons for the current at the zero applied voltage, was obtained assuming that a considerably small positive or negative voltage is applied. The value of the current is necessary for normalizing currents at various applied voltages by the value.

Fig. 1. Variation in the normalized currents with the applied voltage for the listed electrode separations of the parallel plate ionization chamber. Here, the chamber is exposed to 60Co γ-rays from the polarizing electrode side. The empty symbols connected by the dotted lines show experimental results (Takata and Matsumoto, 1991) and filled symbols connected by solid lines show results obtained using EGS5.

2.2. Results of calculation for the secondary electrons In Fig. 1, the experimental results (Takata and Matsumoto, 1991) are compared with the present calculation results. The results showing a linear increase of the normalized current with the applied voltage are for positive applied voltages, whereas those showing a linear decrease of the normalized current with the applied voltage are for negative applied voltages. The numbers associated with the lines in the figure refer to the nominal values of electrode separation (in mm). The experimental values are normalized to the signal currents obtained by extrapolating each linear portion of data to zero applied voltage. The values obtained by calculation using EGS5 show the number of ion pairs produced in the region of air plus the net number of secondary electrons, which contribute to the signal current. The values are normalized by each value obtained for zero applied voltage. The symbols for the normalized values at the zero applied voltage and solid lines connecting the symbols are not shown in the figure for clarity. Table 1 summarizes the experimental and calculation results for the rates of increase at positive applied voltages (+) and the rates of decrease at negative applied voltages (−) in the signal current for various separations between the electrodes. Standard deviations of calculation results are added. Although the values for 2 mm and 5 mm separations are large, the calculation results show that the rate of increase of the signal current for positive applied voltage is smaller than the rate of decrease of the signal current for negative applied voltage. It can be also deduced from the results that both the rates of increase and decrease become larger for smaller separations between the electrodes. It may be because more secondary electrons pass through the separation between the electrodes without stopping or changing their direction and are fully accelerated or decelerated by the applied voltage in smaller separations. The experimental results show that the rate of increase of the signal current for positive applied voltage is larger than the rate of decrease of the signal current for negative applied voltage, especially for small separations. Further, the dependence of the rate of decrease on the separation for negative applied voltage is not clear. Table 2 lists the number of electrons that exit and arrive at the electrodes or the air between the electrodes. It summarizes the results for 2 mm separations (values above the broken line) and 30 mm separations (values below the line). Further, the table lists the net number of electrons that contribute to the increase in positive or negative signal currents and de/W, which is the number of ion pairs produced in the air between the electrodes. It is derived from the energy deposited to the 2

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Table 1 Rates of increases and decreases in signal currents from a parallel plate ionization chamber irradiated with are applied. They are shown in the unit of 10−6 V−1.

60

Co γ-rays when positive (+) or negative (−) voltages

Separation (mm)

2

5

15

30

Experiment (+) (−) Calculation (+) (−)

8.19 −1.72 2.42 ± 0.53 −4.00 ± 0.46

4.72 −2.38 2.49 ± 0.44 −3.20 ± 0.40

2.63 −2.11 2.17 ± 0.06 −2.53 ± 0.06

1.63 −1.57 1.84 ± 0.02 −2.14 ± 0.03

signal current in all cases except for positive applied voltages to electrodes with 2 mm separation. It was found that the change in the net number of electrons due to the applied voltage increases the rate of change in the signal current in all cases for positive and negative applied voltages except for negative applied voltage at 2 mm separation. In all cases, the number of electrons for a→a, which is dominant, increased with the increase in the positive applied voltage, and decreased in the negative applied voltage as shown in Table 2. In all cases, the contribution of the change in the net number to the signal current was less than 8 × 10−5 and the rate of increase or decrease in the signal current was less than 4 × 10−8 V−1. Consequently, the contribution of the change in the net number is too small to be attributed for the differences between the normalized signal currents obtained by measurements and calculations or for the differences between the rates of increases and decreases in the signal current, which are shown in Fig. 1 and Table 1, respectively. It was confirmed from the EGS5 calculation that the signal current from a parallel plate ionization chamber irradiated with 60Co γ-rays increases or decreases due to deceleration or acceleration of secondary electrons depending on the polarity of the applied voltage. This is due to the fact that the secondary electrons directed forward are more than those directed backward. From Fig. 1, it is notable that all normalized signal currents obtained by experiments are greater than the corresponding currents obtained by calculations except for positive applied voltage at 30 mm separation. This is also shown in Table 1 that calculated rate of increase in the signal current is smaller than the experimental ones except for 30 mm separation, and the calculated rate of decrease is larger than the experimental ones. It is expected that if there is a phenomenon that increases the signal current with the applied voltage or with the electric field strength for both polarities, the agreement between the results of the experiments and calculations becomes better. In the following section, the effects of the electric fields on ionization by electrons with energy less than 1 keV are discussed because ionization by these low energy electrons is not taken into consideration in the calculation described above.

air, de, divided by the W value. The letters p, a, and c indicate the polarizing electrode, air in the ionization volume, collector electrode, respectively. The symbol p→a indicates the number of electrons that are emitted in the polarizing electrode and have stopped in the air, whereas the symbol a→a indicates the number of electrons that are emitted and have stopped in the air. In both cases of electrode separations of 2 and 30 mm, p→c, a→a and c→p are greater than p→a, a→p, a→c and c→a. The standard deviation of the net number was less than 0.004 × 106 and 0.007 × 106 for 2 mm and 30 mm separation, respectively. The standard deviation of de/W was less than 0.4 × 106 and 3 × 106 for 2 mm and 30 mm separation, respectively. It is clear from Table 2 that the number of electrons emitted in the forward direction (p→a, p→c and a→c) is larger than the number of electrons emitted in the backward direction (a→p, c→p and c→a). This shows the cause of the electric field effect: an increase or decrease in the signal current due to the deceleration or acceleration of the secondary electrons depending on the polarity of the applied voltage. The net number of electrons for positive applied voltages at 2 mm separation is negative because the contribution of electrons for p→c is large and the positive signal current is decreased. The value of the net electron number for the negative applied voltage minus the net electron number for the positive applied voltage is equal to 2(p→c − c→p) + (p→a − a→p) + (a→c − c→a). This value corresponds to the difference between the absolute values of the negative and positive signal currents. The difference is a result of the Compton current: the signal for the negative applied voltage is increased, whereas that for the positive applied voltage is decreased by the current. The ratio of this value to the number of ion pairs produced by the secondary electrons at zero applied voltage was calculated for each separation. The ratio was 0.55, 0.22, 0.07 and 0.04% for the separation between the electrodes of 2, 5, 15 and 30 mm, respectively. These values are about 30% of respective values obtained in the experiments (Takata and Matsumoto, 1991). The reason for the discrepancy is not clear. It should be also noted that the contribution of the net number of electrons to the signal current is about −0.03 and 0.52%, respectively for positive and negative applied voltages to electrodes with 2 mm separation and, 0.13 and 0.35, 0.21 and 0.28, 0.22 and 0.26% for 5, 15, 30 mm separation, respectively. This means that the charge of the net number of electrons increases the

Table 2 Number of electrons that exit and arrive at the electrodes or the air between the electrodes with 2 mm (above the broken line) and 30 mm (bellow the broken line) separations. The first column shows the applied voltage. The numbers are shown in the unit of 106. (V)

p→a

p→c

a→p

a→a

a→c

c→p

c→a

Net number de/W

0 1000 2000 −0 −1000 −2000

0.064 0.065 0.067 0.064 0.062 0.061

4.535 4.529 4.522 4.535 4.540 4.548

0.048 0.051 0.055 0.048 0.046 0.043

1.024 1.028 1.030 1.024 1.020 1.016

0.065 0.062 0.060 0.065 0.068 0.071

3.260 3.265 3.266 3.260 3.257 3.256

0.047 0.047 0.045 0.047 0.048 0.049

−0.155 −0.139 −0.125 2.428 2.434 2.440

470.0 471.2 472.3 470.0 467.9 466.4

0 7050 14100 −0 −7050 −14100

0.266 0.277 0.287 0.266 0.254 0.241

4.134 4.108 4.076 4.134 4.167 4.195

0.142 0.157 0.173 0.142 0.129 0.117

16.694 16.928 17.146 16.694 16.453 16.176

0.265 0.252 0.240 0.265 0.279 0.294

2.979 2.987 2.995 2.979 2.970 2.965

0.139 0.136 0.132 0.139 0.141 0.143

15.820 16.100 16.369 18.381 18.183 17.941

7069.2 7161.3 7252.1 7069.2 6965.4 6854.0

3

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3. Electrons with energy less than 1 keV

where xΔ’ and yΔ’ are displacements of the electron position in the directions, respectively, parallel or perpendicular to the electric field while the electron passes through the step length in the electric field. The symbol q is the electron charge and F is the electric field strength. In the calculation program, E2’ and θ2’ are used as E1 and θ1, respectively, for the next step of the electron. The low energy electrons emitted in the ionization volume can be separated into two groups. One is the electrons that are emitted in a space distant more than their range from both the cathode and the anode. They lose their energy due to stopping power, and stop in the air. The other is the electrons that are emitted in the spaces near the cathode or the anode, and some of them strike on the surface of the electrodes. In the present work, the effects of the electric field on the electrons emitted in a space far from the electrodes are discussed first. In the calculation program of the effects of the electric field, it is assumed that electrons are emitted homogeneously and isotorpically in the air of the ionization volume with energy from 1 to 1000 eV at a 1 eV interval. The emission angle is set from 0° to 180° from the direction of the electric field at a 0.5° interval. In the calculations, the data concerning the stopping power of air and the range of electrons are required. The solid line in Fig. 2 indicates the theoretical values of the stopping power obtained by Khare and Kumar (1977) for nitrogen gas. The solid line in Fig. 3 indicates the range of electrons obtained via definite integration of the inverse of the stopping power shown in Fig. 2. The broken lines in Figs. 2 and 3 represent the theoretical values reported by Sugiyama (1974). The lines show the values calculated from equations obtained by fitting quartic, quintic and quartic equations to three groups of tabulated data in the range of energy from 20 to 80 eV, from 60 to 400 eV and from 300 to 1000 eV, respectively. The lines are connected at points where corresponding two equations become closest each other. The experimental values of the stopping power and range are also shown in Figs. 2 and 3 for reference. The dotted line and the chain line in the figures show the values calculated from equations obtained by Cole (1969) and Iskef et al. (1983), respectively. The equations for stopping power have been derived by differentiation of the equations for the range obtained by fitting to experimental results of the range for air or for various gases and solids, respectively. The range measured is the projected range, and therefore, the values obtained for stopping power show the energy loss of electrons along the projected track and not along the electron trajectory. In the present work these values are not used for the calculation of the effects of the electric fields on

3.1. Calculation for electrons with energy less than 1 keV It is considered that a large number of electrons with energy less than 1000 eV are produced in the air of the ionization volume. In the calculation performed with EGS5 used in the preceding section, these low energy electrons are assumed to stop and deposit their energy at that point. It is reasonable to expect that these low energy electrons are emitted homogeneously and directed isotropically in the air. The range of these electrons is much smaller than the electrode separation. When the low energy electrons are emitted in an electric field, they must gain energy from the electric field in total because their moving directions are easily changed from deceleration to acceleration in the electric field. Moreover, electrons that are accelerated gain energy through their extended paths. On the contrary, electrons that are decelerated are affected only through their shortened paths. In the derivation of equations for the motion of electrons in air under the effects of electric fields, it is assumed that electrons lose energy due to the stopping power of air and gain or lose energy from the electric field. The moving directions of electrons are changed only by the electric field and not by collision with gas molecules, i.e., electron flight path in the zero electric field is straight. If the electron energy before and after passing a small straight step length, h, is denoted by E1 and E2, respectively, they are expressed by the following equation: E1 – E2 = S h

(1)

where S is the stopping power of air for the electron. The energy obtained from or lost in the electric field is calculated from the change in the electron velocity parallel to the electric field direction. The change is calculated from the acceleration due to the electric field and the flight time for passing through the step length. The flight time is obtained from the average of the electron velocities that correspond to E1 and E2. The relativistic effect is neglected because the effect is less than 0.2% for 1000 eV electrons. Different equations were obtained depending on the order of the two components of the calculations concerning the electron velocity: (a) the change in the velocity due to the energy loss caused by the stopping power and (b) the change in the direction and velocity due to the electric field. The following are the equations obtained, where E1, E2’ are electron energy and θ1, θ2’ are the moving directions of electrons from the electric field before and after passing through the step length in the electric field, respectively. (a) The effect of the stopping power is concerned first: E2’ = ((E1 – S h)1/2 cosθ1 + h q F / (E11/2 + (E1 – S h)1/2))2 + (E1−S h) (sinθ1)2 (2) θ2’ = arccos(((E1 – S h)1/2 cosθ1 + h q F / (E11/2 + (E1 − h)1/2)) / E2’1/ 2 ) (3) xΔ’ = h cosθ1 + h2 q F / (E11/2 + (E1 – S h)1/2)2

(4)

yΔ’ = h sin θ1

(5)

(b) The effect of the electric field is concerned first: E2’ = (E11/2 cosθ1 + h q F / (E11/2 + (E1 – S h)1/2))2 + E1 (sinθ1)2 – S h (6) θ2’ = arccos((E11/2 cosθ1 + h q F / (E11/2 + (E1 –S h)1/2)) / ((E11/2 cosθ1 + h q F / (E11/2 + (E1 – S h)1/2))2 + E1(sinθ1)2)1/2) (7) xΔ’ = h cosθ2’ + h2 q F / (E11/2 + (E1 − S h)1/2)2

(8)

yΔ’ = h sinθ2’

(9)

Fig. 2. Stopping power for electrons. Solid line indicates the theoretical value reported by Khare and Kumar (1977) for nitrogen gas. Broken line indicates the theoretical value reported by Sugiyama (1974) for air. The dotted and chain lines by Cole (1969) and Iskef et al. (1983), respectively, show values of equations derived from experimental results for range. Filled circles indicate the experimental results reported by Waibel and Grosswendt (1981). 4

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Fig. 4. Rate of increase from the value at the zero electric field in the energy deposited by electrons that are isotropically emitted in various electric fields. The field strength is 1000, 800, 600, 400 and 200 kV/m for the lines in the order from top to bottom.

Fig. 3. Range of electrons. Solid line represents the values obtained from the theoretical values of stopping power reported by Khare and Kumar (1977) for nitrogen gas, and broken line represents the theoretical values reported by Sugiyama (1974) for air. The dotted and chain lines represent experimental results reported by Cole (1969) and by Iskef et al. (1983), respectively. The filled circles represent the experimental values reported by Waibel and Grosswendt (1981).

electrons that are emitted in air far from the electrodes are calculated for electric fields from 0 to 106 V/m with an interval of 2 × 105 V/m. In the calculation program, the energy deposited to the air is obtained as the sum of E1 – E2 for all steps from electron emission until the electron energy becomes smaller than the cutoff energy. When the electron energy becomes smaller than the cutoff energy, the residual energy is added to the deposited energy. The energy obtained from or lost in the electric field is not included in E2. The value E1 used in the calculation for the next step is equal to the value E2’, that is obtained from equation (2) or from equation (6). Fig. 4 shows the rate of increase from the value at zero electric field in the energy deposited to the air by electrons that are isotropically emitted in various electric fields. The rate becomes larger for lower energy electrons because the stopping power becomes smaller for lower energy electrons and the energy obtained from the electric field becomes relatively large. The results shown in Fig. 4 are obtained from the data reported by Khare and Kumar (1977). If the W values for electrons with energies E1 and E2 are denoted by W1 and W2, respectively, the number of ion pairs produced by the electrons while they pass the small step, h, is obtained using the following equation:

ionization because electron energy loss and gain used in the present calculation are those along their flight paths. The values of stopping power and range for nitrogen gas measured by Waibel and Grosswendt (1981) are also indicated in the figures by filled circles for reference. Although the results reported by Khare and Kumar (1977) are for nitrogen gas, no correction was made in the present work except adjusting the values for the density of air at 1 atm and 22 °C. The data reported by Sugiyama (1974) are summarized in a table for electron energy of 20 eV and above, the quartic equation obtained by fitting to the data was used also for lower energy in the present study. In the calculation program, first, the range is obtained for each emitted electron. The energy loss of the electron due to stopping power of air and the energy gained or lost due to the electric field are calculated for each moving step, h, using an interval of 1/400 of the range. Calculation is repeated until the electron energy becomes less than the half of the original energy, and the range is obtained for the residual energy of the electron. Subsequently, the calculation continues for each step with an interval of 1/400 of the new range. The calculation is repeated until the electron energy becomes smaller than the cutoff energy. When the electron energy becomes smaller than the cutoff energy, it is assumed that the electron stops moving and deposits the residual energy at the point, regardless of the data shown in the Figs. 2 and 3. The cutoff energy was fixed to 12.2 eV, which is the ionization potential of O2 molecules (Takata and Begum, 2008). It should be noted that the energy obtained from the largest electric field used in the present work, 106 V/m, exceeds the energy lost due to the stopping power for electrons with an energy smaller than 8.6 or 10.3 eV when the data reported by Khare and Kumar (1977) or Sugiyama (1974) for stopping power, respectively, were used. The electron energy, moving direction and the position, i.e., the distance along the electric field and that perpendicular to the field from the zero point where the electron is emitted are recorded for all steps of electrons emitted with various amount of energy and in various directions. In the present study, the distance perpendicular to the electric field was used only to check the electron flight path.

E1 E − 2 W1 W2

(10)

The sum of the number of ion pairs is obtained for all steps of each electron from the emission until the residual energy becomes smaller than the cutoff energy. The W value for electrons is calculated using the following equation (Takata and Begum, 2008), which was obtained by fitting to experimental data on a log–log scale for electron energy (ex) and W value (ey) in eV:

y = y0 + A1 e−(x − x 0)/ t1 + A2 e−(x − x 0)/ t2 + A3 e−(x − x 0)/ t3

(11)

where y0 = 3.51339, x0 = 2.50144, A1 = 2.12296, t1 = 0.45155, A2 = 11.34283, t2 = 0.103, A3 = 0.88307 and t3 = 1.48666. Fig. 5 shows the rate of increase in the number of ion pairs produced by electrons that are isotropically emitted in various electric fields. It shows the rate of increase from the value at zero electric field. The electron emitted is counted as one and added to the number of ion pairs because it contributes to the signal current of the ionization chamber. In the calculation, the data reported by Khare and Kumar (1977) were used. The rate of increase becomes a maximum near 35 eV and shows minimum values near 420 eV. This is because lower energy electrons obtain relatively more energy from the electric field, as shown in Fig. 4. Fig. 4 also shows shallow minima near 550 eV. This is also due to the fact that electrons lose less energy per flight length because the

3.2. Deposited energy and number of ion pairs produced by low energy electrons The deposited energy and the number of ion pairs produced by 5

Radiation Physics and Chemistry 170 (2020) 108623

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Fig. 6. Rate of increase from the value at the zero electric field in the energy deposited by electrons emitted with energy less than 1000 eV in a space far from the electrodes. It is assumed that the electron energy distribution follows equation (12). The filled circles are obtained from the data reported by Khare and Kumar (1977) for stopping power, and the empty squares are obtained from the data reported by Sugiyama (1974).

Fig. 5. Rate of increase from the value at the zero electric field in the number of ion pairs produced by electrons that are isotropically emitted in various electric fields. The field strength is 1000, 800, 600, 400 and 200 kV/m for the lines in the order from top to bottom.

stopping power decreases with an increase in the amount of electron energy in the range larger than about 150 eV and obtain relatively more energy from the electric field. It is evident from Figs. 4 and 5 that the rate of increase in the energy deposited by electrons with energy less than 200 eV is much larger than the rate of increase in the number of ion pairs. Results similar to those shown in Figs. 4 and 5 were obtained from the data reported by Sugiyama (1974), although the values in the range below 200 eV were about twice of those shown in Figs. 4 and 5. The effects of the electric field on the deposited energy and the number of ion pairs due to electrons that are emitted in a space far from the electrodes are obtained by the summation of the values shown in Figs. 4 and 5, weighted with deposit energy and number of ion pairs produced by each electron plus 1, respectively. Moreover, the values are weighted with an energy distribution of the electrons. The following equation proposed by Opal et al. (1971) is used for the calculation of the energy distribution:

C

σ(Es) = 1+

Es 2.1 14.1

( )

Fig. 7. Rate of increase from the value at the zero electric field in the number of ion pairs produced by electrons emitted with energy less than 1000 eV in a space far from the electrodes. It is assumed that the electron energy distribution follows equation (12). The filled circles are obtained from the data reported by Khare and Kumar (1977) for stopping power, and the empty squares are obtained from data reported by Sugiyama (1974).

(12)

where σ(Es) is the cross section for emitting an electron with energy Es and C is the normalization constant that depends on the primary electron energy. Although the equation is presented for nitrogen molecules, for a primary electron energy of 2000 eV, and Es in the range less than 200 eV, it is assumed in the present work that the equation can be used for the energy distribution of electrons emitted in the air in the ionization chamber irradiated with 60Co γ-rays and for energy Es up to 1000 eV. In the calculation, the value C was fixed so that the total number becomes unity for electrons emitted with energy in the range 1–1000 eV with an interval of 1 eV. Figs. 6 and 7 show, respectively, the results of calculation for the rate of increases in the deposited energy and that in the number of ion pairs produced by electrons emitted with energy less than 1000 eV in a space far from the electrodes. They are shown as the rate of increases from each value at the zero electric field. The filled circles and empty squares show the results obtained from the data reported by Khare and Kumar (1977) and from the data reported by Sugiyama (1974), respectively. The following equations were obtained by fitting a cubic equation to the respective results shown in Fig. 7:

where R is the rate of increase in the number of ion pairs produced by electrons emitted with energy less than 1000 eV and F is the electric field strength (V/m). An inspection of Figs. 6 and 7 also reveals that (a) the values of the deposited energy and the number of ion pairs obtained from the data reported by Sugiyama (1974) are larger than those obtained from the data reported by Khare and Kumar (1977) and (b) the rate of increase in the deposited energy is much larger than that in the number of ion pairs produced. 3.3. Electrons that strike the surface of the cathode or anode Some electrons that are emitted near the cathode or anode strike the surface of the electrodes. This section considers the effects of the electric field on the deposited energy and the number of ion pairs produced by the emitted electrons. In the calculation program, it is assumed that the spaces near the cathode or the anode have the same thickness as the electron range in the zero electric field. The simulated electrons are uniformly and isotropically emitted from the points that are aligned perpendicular to the electrode at equal intervals, 1/200 of

R = 1.334 × 10−22 × F 3 + 1.937 × 10−15 × F 2 + 1.515 × 10−11 × F (13)

R = 7.459 × 10−22 × F 3 + 3.440 × 10−15 × F 2 + 9.339 × 10−11 × F (14) 6

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in Fig. 5. Calculations based on the data reported by Khare and Kumar (1977) and equation (12) for an energy distribution up to 2 keV yielded a value 2.21 × 10−3 for the rate of increase in the number of ion pairs in the electric field of 1000 kV/m in comparison with the value of 2.09 × 10−3 for an energy distribution up to 1000 eV, as shown in Fig. 7. The difference is not so large because the distribution of the high energy electrons obtained from equation (12) decreases rapidly. To estimate the effect of the differences in an energy distribution up to 1 keV, the change of the signal current from the ionization chamber was calculated for both equation (12) multiplied by 1 + 9 × Es/1000 and that divided by 1 + 9 × Es/1000. Values of 2.26 × 10−3 and 1.82 × 10−3 were obtained, respectively, when the data by Khare and Kumar (1977) were used. The signal current does not change so much from the value 2.09 × 10−3 which is the value for the energy distribution given by equation (12). This is also because the distribution of larger energy electrons obtained from equation (12) multiplied by 1 + 9 × Es/1000 is still small. The values obtained for the deposited energy and the number of ion pairs were slightly different depending on which equations (2)–(4) or (6)–(8) are used. The difference became smaller for calculations with a smaller step size and a smaller interval of the emission angle. The differences between the values obtained in the present program were less than 0.9% of each value; average values were always used in this study. Both the value of the stopping power and the value of the range reported by Khare and Kumar (1977) were larger than those reported by Sugiyama (1974) for electrons with energy lower than 180 eV. This is probably because the value of the range obtained from the value of stopping power reported by Khare and Kumar (1977) is the result of integration of the inverse of the stopping power from the energy where the value of the stopping power becomes zero. On the contrary, the range reported by Sugiyama (1974) was obtained by ignoring the values for energy lower than 10 eV. The range given by the equation obtained by fitting to his data, however, extends to the region lower than 10 eV, as shown in Fig. 3. In the present work, values of the range were used to find appropriate step sizes for the calculation of the effects of the electric field on electrons with energy less than 1 keV. On the contrary, the range was also used in the calculation to estimate the fraction of ion pairs produced by electrons emitted near the electrodes. The present research showed that the electron migration length parallel to the electric field does not exceed the range of the electrons, even in the largest electric field, i.e., 106 V/m. It is conceivably because the cutoff energy was fixed to 12.2 eV. This means that electrons emitted at a distance farther than the range from an electrode do not strike on the surface of the electrode even if accelerated in the electric field. The growth of photoelectric currents from cathodes in air has been measured by several authors in studies focusing on electrical breakdown. The growth is in relatively high electric fields and is caused by electrons that have not enough energy for ionization gain sufficient energy for ionization from the electric fields. It is known that the growth is proportional to exp(αd), where α is the Townsend primary ionization coefficient which represents the number of ion pairs produced on average by each drifting electron per unit length along the electric field and d is the drift length of the electron. Fig. 9 shows the reduced coefficient α/n, where n is the number density of particles of air, as a function of F/n, where F is the electric field strength. The values shown in the figure are all apparent primary ionization coefficients and are not corrected for the effects of attachment and detachment of electrons. It is expected from the figure that α/n exponentially changes with F/n in the range smaller than 10 × 10−20 Vm2. To confirm whether the differences between the normalized signal currents obtained by experiments and by EGS5, as shown in Fig. 1, are due to Townsend primary ionization, the reduced ionization coefficient was obtained from the differences. In the calculation, the value of d was set equal to half that of the electrode separation because low energy electrons are expected to be uniformly distributed in the separation and

Fig. 8. Ratio of the increase from the value at the zero electric field in the number of ion pairs produced by electrons emitted in a space near the cathode (empty triangles) and that near the anode (empty squares) to the total number of ion pairs produced in an ionization chamber with a 2 mm electrode separation. Filled circles show the sum of the ratios.

the range, from the surface. The deposited energy and the number of ion pairs produced are summed up until either the electron energy becomes smaller than the cutoff energy or the electron collides with the surface of the cathode or the anode. When the electron energy becomes smaller than the cutoff energy, the residual energy is added to the deposited energy. When the electron stops in the air or strikes the surface of the anode, the number of the ion pairs is increased by unity. If the electron strikes the surface of the cathode, the number of ion pairs is not increased because the electron charge does not contribute to the signal current. Using equation (12) for electron energy distribution, and weighting with deposit energy and number of ion pairs produced by each electron plus 1, respectively, the effects of the electric field on the deposited energy and the signal current from an ionization chamber due to electron emission near the electrodes were calculated. It was found that both the deposited energy and the number of ion pairs due to electron emission near the cathode increase with the electric field and those due to emission near the anode decrease with the electric field. Fig. 8 shows the ratio of the increase from the value at the zero electric field in the number of ion pairs produced by electrons emission near the cathode and the anode to the total number of ion pairs produced in the zero electric field in an ionization chamber with a 2 mm electrode separation. It also shows the sum of the ratios, i.e. net contribution by electrons. emitted near electrodes. They are obtained from the data reported by Khare and Kumar (1977) for stopping power and equation (12) for electron energy distribution. A similar result but with a 1.9 times larger net contribution was obtained from the data reported by Sugiyama (1974). It is evident from Fig. 8 that the effect of the electric field due to electron emission near the electrodes is small for the signal current from the ionization chamber. Besides, the effect is approximately proportional to the inverse of the separation between the electrodes. Consequently, it can be concluded that the effect of the electric field on the signal current from an ionization chamber is mainly due to electrons emitted in a space distant more than their range from the electrodes.

4. Discussion The calculations revealed that ionization by electrons with energy less than 1 keV is increased by the electric field. The rate of increase has a maximal value at about 35 eV and a minimal value at about 420 eV (see Fig. 5). The rate of increase becomes larger for electrons with higher energy. The rate of increase for electrons with 2 keV is approximately twice of the value for electrons with 1 keV, which is shown 7

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due to Townsend primary ionization. 5. Comparison between the experimental and calculation results Fig. 10 shows the experimental and calculation results for the normalized signal currents from the ionization chamber irradiated with 60 Co γ-rays as functions of the electric field. As the effects of the electric fields on the number of ion pairs produced by electrons emitted in the spaces near the electrodes are considerably small, the calculation results are not corrected for the effects. The corrections made herein show an improved agreement of the values obtained using EGS5 with those obtained via experiments, although the amount of correction is not sufficient. The discrepancy between the experimental and calculation results is larger for smaller separation between the electrodes at positive applied voltage. On the contrary, it is larger for larger separation at negative applied voltage. The change in the signal current due to the effects of the electric field obtained using EGS5 linearly increases with the applied voltage, i.e., with the electric field. On the contrary, the change in the signal current due to the effects of electric field on ionization by electrons with energy less than 1 keV increases more rapidly than the linear relation with the electric field.

Fig. 9. Townsend primary ionization coefficient divided by the number density of particles of air (α/n) as a function of the electric field divided by the number density F/n. The symbols in the low-electric-field region show values obtained based on an assumption that the differences between the normalized signal currents obtained by experiment and EGS5 are due to Townsend primary ionization. The numbers by the symbols show electrode separation; the symbols + for solid line and − for dotted line show the positive and negative applied voltages, respectively. Symbols in high-electric-field region show the values obtained by experiments in dry air: △ by Masch (1932), □ and solid line by Sanders (1932), ● by Sanders (1933), ◆ by Prasad (1959), ✕ and ✱ (for humid air) by Dutton et al. (1963).

6. Conclusion The calculations performed using EGS5 which incorporated the electromagnetic field, confirmed that the increases and decreases in the signal current from the parallel plate ionization chamber with the applied voltage were due to deceleration and acceleration of the secondary electrons, respectively, that are emitted more in the forward direction. The calculation results showed improved agreement with the experimental results when corrected for the effects of the electric fields on the number of ion pairs produced by electrons emitted with energy less than 1 keV. However, small differences still remain between the calculation and experimental results. It was also discussed about the contribution of the charge of secondary electrons which exit and alive at electrodes or the ionization volume on the signal current from the ionization chamber. This study revealed that the signal current from the ionization chamber increased with the electric field because the electrons emitted uniformly and isotropically in the ionization volume with energy less than 1 keV gain energy from the electric field and produced more ion pairs than those produce in the zero electric field. This phenomenon is different from Townsend primary ionization. The increase in the number of ion pairs occurs even for electrons with energy as low as several tens of electron volts and even in considerably low electric fields. This affects signals from every type of ionization chambers. Moreover, this study showed that the rate of increase in the deposited energy due to electric fields was much larger than that in the number of ion pairs produced by electrons emitted with energy less than 1 keV.

Fig. 10. Variation in the current with the electric field for the listed electrode separations of the parallel plate ionization chamber exposed to 60Co γ-rays. The solid lines show linear parts of the experimental results shown in Fig. 1. Filled circles show the results obtained using EGS5, which are also shown in Fig. 1. Empty triangles and squares show the results corrected for the rate of increase in the number of ion pairs produced by electrons with energy less than 1000 eV. The values of the rate were calculated using equations (13) and (14) which have been obtained from the data reported by Khare and Kumar (1977) and by Sugiyama (1974), respectively.

Declaration of competing interest There is no conflict of interest.

the average drift length of the electrons becomes half the value of electrode separation when the number of electrons produced by ionization is small. The calculated values are shown in the low-electric-field region of Fig. 9. The values obtained from the differences at positive applied voltages for 30 mm separation are not shown in the figure as they are negative. It is evident from Fig. 9 that the values obtained in the present work are quite different from those obtained via extrapolation higher-electric-field region. These values exhibit a dependence on the drift length. Consequently, it can be concluded that the differences between the normalized signal currents obtained by experiment and EGS5 are not

References Cole, A., 1969. Absorption of 20-eV to 50,000-eV electron beams in air and plastic. Radiat. Res. 38, 7–33. Dutton, J., Harris, F.M., Llewellyn, J.F., 1963. The determination of attachment and ionization coefficients in air. Proc. Phys. Soc. 81, 52–64. Hirayama, H., Namito, Y., Bielajew, A.F., Wilderman, S.J., Nelson, W.R., 2005. The EGS5 Code System. SLAC-R-730. and KEK Report 2005-8 (2005). Iskef, H., Cunningham, J.W., Watt, D.E., 1983. Projected ranges and effective stopping powers of electrons with energy between 20 eV and 10 keV. Phys. Med. Biol. 28, 535–545. Khare, S.P., Kumar Jr., A., 1977. Mean energy expended per ion pair by electrons in molecular nitrogen. J. Phys. B At. Mol. Phys. 10, 2239–2251.

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positrons. Bull. Electrotech. Lab. (Jpn.) 38, 351–362. Takata, N., Begum, A., 2008. Corrections to air kerma and exposure measured with free air ionisation chambers for charge of photoelectrons, Compton electrons and Auger electrons. Radiat. Prot. Dosim. 130, 410–418. Takata, N., Matsumoto, T., 1991. The effect of the applied voltage on ionization in a parallel plate ionization chamber. Nucl. Instrum. Methods Phys. Res. A302, 327–330. Torii, T., Sugita, T., 2007. Incorporation the electromagnetic field in the EGS5 code. 14th EGS Users' Meeting in Japan. KEK Proc 2007 5, 43–49. Waibel, E., Grosswendt, B., 1981. Stopping power and energy range relation for low energy electrons in nitrogen and methane. In: Proc. 7th Symp. on Microdosimetry EUR. 7147. CEC, Luxembourg, pp. 267–276.

Masch, K., 1932. Über Elektronenionisierung von Stickstoff, Sauerstoff und Luft bei geringen und hohen Drucken. Arch. Elektrotechnik 26, 587–596. Opal, C.B., Peterson, W.K., Beaty, E.C., 1971. Measurements of secondary-electron spectra produced by electron impact ionization of a number of simple gases. J. Chem. Phys. 55, 4100–4106. Prasad, A.N., 1959. Measurement of ionization and attachment coefficients in dry air in uniform fields and the mechanism of breakdown. Proc. Phys. Soc. 74, 33–41. Sanders, F.H., 1932. The value of the Townsend coefficient for ionization by collision at large plate distances and near atmospheric pressure. Phys. Rev. 41, 667–677. Sanders, F.H., 1933. Measurement of the Townsend coefficients for ionization by collision. Phys. Rev. 44, 1020–1024. Sugiyama, H., 1974. Tables of energy losses and ranges of low energy electrons and

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