The air kerma is the basic physical quantity for measuring the X-ray dose for radiation protection purposes in both medical and the industrial fields. The air kerma standard of the X-ray beams is realized by the free air chamber (FAC), known as the “primary standard device” if it is used to establish the air kerma standard in the national metrology institute (NMI). Most NMIs have two FACs: one for measuring the low-energy X-ray and the other for measuring the medium-energy X-ray. According to the NMI of Korea, we used two FACs to realize the primary standards of air kermas for low- and medium-energy X-ray beams, including the mammography X-ray beams specified in the International Organization for Standardization (ISO) [1] and International Electrotechnical Commission (IEC) reference documents [2]. We manufactured the FACs with our own designs, mainly because they were not commercially available. The structural information of the FAC is critical in realizing the unit of air kerma, and is obtained with sufficient accuracy by measuring it during the manufacturing and assembly stages of the FAC.

The air kerma from the X-ray beams in the reference plane is measured by the FAC using the following equation:

where $\dot{K}$ is the kerma rate, (W_air/e) is the mean energy required to create an ion pair in dry air, I is the ionization current measured by the FAC, m_air is the air mass in the collection volume of the FAC, (1-g) is the mean energy loss in air due to the bremsstrahlung of the secondary electrons generated by the incident X-ray beams, and k_i is various correction factors introduced to correct for the limitations of the air kerma measurement using the FAC in view of the definition of air kerma. The air mass m_air in the collection volume of FAC is given by the following:

where ρ_air is the dry air density, V₀ is the measurement volume of air kerma, r₀ is the aperture diameter of the diaphragm, and l is the length of the FAC collector electrode.

Given that the air mass is usually determined by multiplying the dry air density by the measurement volume, as given in Eq. (2), the measurement volume of the FAC is an essential physical factor in conducting the absolute measurement of the air kerma using the FAC. As the air kerma is defined as the mean sum of the initial kinetic energies of all charged particles liberated in air by the incident photons divided by the mass of air [3], the volume of the FAC should be carefully evaluated when wishing to develop the primary standard air kerma ensuring traceability to the International System of Units.

We have recently found that the existing mechanically determined volume of the Korea Research Institute of Standards and Science (KRISS) FAC L1 for the low-energy X-ray measurement is different from the value deduced from the capacitance measurement. In this study, considering the result obtained from the finite element method (FEM), the change in the collector length due to the electric field distortion was evaluated from the capacitance measurement and, as a result, the effective volume of the KRISS FAC L1 was determined.

The cross-sectional view of the KRISS FAC L1 for measuring the low-energy X-ray is presented in Fig. 1. The base plate, in which the collecting electrode is located, was in the same plane as the collecting electrode, and within 5 μm. The distance between the guard strips was maintained using four sapphire disks with a nominal thickness of 2 mm. The difference in thickness in a layer composed of four sapphires was <1 μm. After piling all the guard strips over the base plate, we checked the assembled heights with a 3-dimensional coordinate measuring machine (3-d CMM) on the upper high-voltage (HV) plate. The maximum difference in height on the upper surface of the HV plate measured from the base plate was <10 μm. The front wall was comprised of a nominal 10-mm-thick stainless steel, the face of which was in contact with the rear plane of the diaphragm and used for the reference plane of the measurement. The flatness of the reference plane was within 1 μm. The subtended angle of the front face from the base plate was 90.0057±0.0007 degrees, with a 68% confidence level (k=1).

Figure 1. Cross-sectional view of the KRISS FAC L1 for low-energy X-ray measurement. The front wall was composed of 10-mm-thick stainless steel. The diaphragm was supported by a tungsten holder. Front view (a) and side view (b). Note that the units are given in mm and the figures are not to scale. KRISS FAC, Korea Research Institute of Standards and Science free air chamber.

The mechanical dimensions were measured using the 3-d CMM (Model UMM 500, Carl Zeiss, Oberkochen, Germany) with the typical measurement uncertainty at a 68% confidence level (k=1) given by the following:

where l is the measured length (mm). Using the 3-d CMM, the measurement uncertainty of the mechanical dimensions of the KRISS FAC L1 would be within 0.5 μm given that the maximum length of the FAC was <200 mm. The aperture diameter of the diaphragm used for defining the beam size of the incident low-energy X-ray was measured by the home-made ring gauge calibrator (Model KRISS-DM1, KRISS, Daejeon, Korea) [4], the typical measurement uncertainty of which was given as a 68% confidence level (k=1) by the following:

where D is the measured diameter of the aperture (μm) [4]. The nominal aperture diameter of the diaphragm we used was 10 mm, for which the uncertainty level of the diameter measurement was approximately 0.1 μm. The aperture diameter was measured in accordance with the KRISS calibration procedure [5]. Before measurement we waited for >6 hours to reach thermal equilibrium. The ring gauge calibrator was installed in the thermal shield case to reduce the temperature variation inside.

The high voltage was loaded onto the HV plate in the FAC (see Fig. 1) during the ionizing current measurement. We loaded −2000 V to the HV plate of the KRISS FAC L1. If the electric field is distorted, it influences the measurement volume of the FAC. To evaluate how the electric field is distorted, we analyzed the electric field formed in our FAC using the two-dimensional FEM using the code QuickField v6.4. Based on the measured dimensions by the 3-d CMM, the FAC was modeled realistically in the QuickField and the electric field line was drawn along the height direction starting from the air gap between the collector and base palate, with the execution script named electric_field_lines.hta, which was included in the QuickField v6.4.

The capacitance of our FAC was measured with the charge mode of the Keithley 6517 electrometer (Keithley, Cleveland, OH, USA). Before the capacitance measurement, we calibrated the charge mode using a standard air capacitor and the constant voltage source according to the KRISS calibration procedure of the electrometer [6]. The uncertainty of the capacitance measurement was within 0.02% (k=1). In a parallel plate capacitor, the capacitance is proportional to the area:

where C is the capacitance of the parallel plate capacitor, ε₀ is the permittivity of the vacuum, ε_air is the dielectric constant of air [7], A is the area of the capacitor, and d is the height between the parallel plates. In this way, the effective area of the extrapolation chamber was determined [8]. However, we used a capacitance bridge to measure the capacitance of the extrapolation chamber. If the area of the capacitor A is increased to, A_eff it is expected that the capacitance will also be increased.

If the area of the plate (in a rectangular shape) with the length l and width w is increased from A to A_eff then the amount of elongation in the collector length of the FAC, ∆l, can be assessed by the following equation:

where ∆w is the elongated width of the collector electrode.

The collector dimensions of the KRISS FAC L1 were measured using the 3-d CMM. The collector length, including the air gap between the collector and base plate, was 15.8003±0.0014 mm at a 68% confidence level (k=1), with a collector length of 15.5879±0.0013 mm and an air gap of 0.2124±0.0005 mm. The values were obtained from >10 points uniformly distributed along the width of the collector electrode. The aperture diameter of the diaphragm was measured at 10.0021±0.0002 mm (k=1), which was the average over the measured values in two orthogonal directions. During the measurement of the aperture diameter of the diaphragm, the temperature was varied from 20.47°C to 20.51°C. At five depths at a diaphragm thickness of 5 mm and in six directions, the values of the aperture diameter were checked and it was confirmed that all measured values agreed with the value above the maximum difference of <0.3 μm. The resulting mechanical volume of the KRISS FAC L1 was then 1.2415±0.0006 cm³ at a 68% confidence level (k=1), which has been used so far.

The electric field lines are shown in Fig. 2. The electric field line in Fig. 2a is drawn starting from the center of the air gap in the collector width, while that in Fig. 2b is drawn from the air gap along the collector length.

Figure 2. Field line was drawn from the center of the air gap in the width of the collector electrode (a) and the field line drawn from the air gap in the length of the collector electrode (b). The X-ray beam axis is perpendicular to the figure plane (a), whereas the beam is incident from the left and exits right (b).

We found that the distortion of the electric field line was mainly originated from the thickness of the guard strip. The distortion becomes larger as the distance from the center is relatively close to the guard strips. The field line starting from the air gap shown in Fig. 2 looks straight but it bends toward the guard strips. The field lines in Fig. 2 are enlarged in Fig. 3. The elongation peaked on the beam axis, the value of which was 0.08225 mm, with an average value of 0.0433 mm in the length of the collector. In terms of the width, the peak value was 0.16950 mm, with an average of 0.0892 mm. The total length increase in the collector electrode evaluated by the FEM was obtained by adding the peak elongations in both sides of the collector, i.e., 0.1645 mm. If the elongation is constant along the height, the average elongated length of the collector electrode is 0.0865 mm. The elongated length evaluated from the FEM analysis showed a maximum variation of approximately 0.01 mm with the number of nodes we selected.

Figure 3. Electric field line drawn along the height at the center of the air gap. The surface plane of the collector electrode was 0 mm-high, and the HV plate was 70 mm-high. The thin line indicates the air gap in the width of the collector, and the thick line indicates the air gap in the length of the collector. The ratio (∆w⁄∆l) in Eq. (6) was 2.06.

The collector area determined by the capacitance measurement was 0.83% larger than the area determined by the 3-d CMM measurement and the ∆l was 0.0895 mm from Eq. (6) of (∆w/∆l) 2.06=0.0892/0.0433, as shown in Fig. 3. If we assume that the shape of elongation is the same as that in the FEM analysis, the ∆l of 0.0895 mm would correspond to 0.170 mm, with a peak elongation length of 0.085 mm at both sides of the collector; this was 3 μm larger than the result of the current 2-dimensional FEM analysis obtained from QuickField v6.4. During the evaluation, the height d between the collector electrode and the HV plate was measured using 3-d CMM, and was found to be 69.285±0.005 mm (k=1). The effective collector length was then obtained by adding two times the peak elongation length to the mechanically determined length, which was 15.970±0.021 mm at a 68% confidence level (k=1).

Because the elongated volume was smaller than the collection volume in the cuboid shape, it was necessary to evaluate and correct the electron energy lost outside the elongated region. For evaluating the energy deposition in the elongated region by the Monte–Carlo method, we modeled 14 air slabs, each of 5-mm thickness. Their lengths and widths were the same as those of the collector electrode. The air slabs were piled and placed around the middle of the air box with the same dimension as the KRISS FAC. In the simulation model, a low-energy X-ray beam 10-mm in diameter was incident on the front face of the air box. The beam axis was at the center of the air box, which was 35 mm-high from the bottom of the air box. The centers of the slabs were 77-mm away from the front face of the air box. The air slabs were paired in plane symmetry with the beam axis plane parallel to the air slabs as the symmetry plane. The energy deposition portion of the paired air slabs was calculated using the code EGSnrc [9] and the statistical uncertainty of the calculation was <0.03% (k=1). The simulation geometry is shown in Fig. 4.

Figure 4. Simulation geometry for calculating the energy deposition in the air slabs. An X-ray beam with a diameter of 10 mm was incident on the center of the front face of the air box. Fourteen air slabs with a thickness of 5-mm were piled from the bottom to the top of the air box and the energy deposited portion in each slab was calculated.

The calculated result is given in Table 1 and 2. As expected, the portions of energy deposition in the slabs placed outer from the beam center were minor. The portion of energy deposition was >90% in the 1st paired slabs adjacent to the beam center for all beam codes in the investigation, with the exception of the beam codes Bureau International des Poids et Mesures (BIPM) 50b and BIPM50a.

Table 1 . Energy deposition portions in the paired air slabs for the low-energy X-ray beams of BIPM quality.

Location of the paired air slabs			Energy deposition portion (%)
Upper slab (mm)	Lower slab (mm)		Beam code
			BIPM10	BIPM30	BIPM25	BIPM50b	BIPM50a
35–40	30–35		98.87	93.81	93.06	79.56	65.88
40–45	25–30		1.07	6.01	6.85	17.01	23.91
45–50	20–25		0.02	0.14	0.07	2.72	7.50
50–55	15–20		0.01	0.01	0.01	0.55	2.06
55–60	10–15		0.01	0.01	0.01	0.12	0.51
60–65	5–10		0.01	0.01	0.00	0.03	0.11
65–70	0–5		0.00	0.00	0.00	0.01	0.03

BIPM, Bureau International des Poids et Mesures..

The air slabs were piled symmetrically with the beam axis plane placed 35-mm-high from the collector electrode as the symmetry plane. The low-energy X-ray beams of BIPM quality were incident on the front face of the air box (see Fig. 4) and the beam size was 10-mm in diameter. The lengths and widths of the slabs were the same as those of the collector electrode, 15.8 mm and 70.0 mm, respectively..

Table 2 . Energy deposition portions in the paired air slabs for the mammography X-ray beams.

Location of the paired air slabs			Energy deposition portion (%)
Upper slab (mm)	Lower slab (mm)		Beam code
			M20	M23	M25	M28	M30	M35	M40
35–40	30–35		94.39	93.34	92.78	92.11	91.70	90.93	90.35
40–45	25–30		5.56	6.61	7.16	7.82	8.20	8.89	9.34
45–50	20–25		0.02	0.03	0.03	0.05	0.07	0.15	0.25
50–55	15–20		0.01	0.01	0.01	0.01	0.01	0.02	0.04
55–60	10–15		0.01	0.01	0.01	0.01	0.01	0.01	0.01
60–65	5–10		0.00	0.00	0.01	0.00	0.00	0.00	0.00
65–70	0–5		0.00	0.00	0.00	0.00	0.00	0.00	0.00

The air slabs were piled symmetrically with the beam axis plane placed 35-mm-high from the collector electrode as a symmetry plane. The low-energy mammography X-ray beams were incident on the front face of the air box (see Fig. 4). The number appearing in the beam code indicated the high voltage loaded on the X-ray tube with a molybdenum anode. The added filtration was 0.03 mm molybdenum, and the inherent filtration was 1 mm beryllium..

Note that the shape of the elongated volume defined by the electric field distortion is in the plane symmetry with the beam axis plane as the symmetry plane. Due to the plane symmetry and the charged particle equilibrium in the axial direction of the incident X-ray beam in the elongated region, the energy deposition portion in the thinly sliced elongated region is proportional to the cross-sectional area of the thin slice. We cut the elongated volume into 14 sections, each of which had a thickness of 5 mm. They were also paired symmetrically with the beam axis plane as the symmetry plane, as depicted in Fig. 5. In the figure, the sum of the cross-sectional areas of the pared slices, where A_i is given relative to that of the cross-sectional areas of the paired slabs, and A₀ (=0.85 mm²) corresponds to the peak elongation of 0.085 mm.

Figure 5. Ratio of Aⁱ, the sum of the cross-sectional areas of the paired 5-mm-thick slices in the elongated volume to A₀ (=0.85 mm²) that of the cross-sectional areas of the paired 5-mm-thick slabs with the same length equal to the peak elongation of 0.085 mm. KRISS FAC, Korea Research Institute of Standards and Science free air chamber.

To correct the electron energy loss conveyed outside the elongated region, we redefined the electron loss correction of the KRISS FAC L1 as follows:

where k_e,new is the newly determined energy loss correction, ε_tr is the energy transferred by the incident X-ray beam to air in the measurement volume of the FAC, ε_l is the energy deposited in the extended volume outside the collection volume of the FAC by the secondary electrons and their progeny produced in the interaction with the incident primary photon, ε_el is the energy deposited in the collection volume of the FAC outside the elongated volume by the secondary electrons and their progeny above, and ε_d is the energy deposited in the collection volume by the secondary electrons and their progeny above. We used the relationship ε_d=ε_tr−ε_l and assumed that the collection volume of the FAC was bounded by the planes of the collector length and width inside the FAC. In practice, ε_tr can be calculated by adding to ε_d the energy deposited in the extended volume, which is the volume outside the collection volume bounded by the planes in both sides of the collector length and in which the guard strips, the base and HV plates bounded by the planes of the collector length are included. The factor (ε_tr/ε_d) is the same as the electron loss correction k_e in other studies [10-12]. For better understanding of the elongated volume, the collection volume, and the effective collection volume, we portrayed the cross-sectional view of these volumes in Fig. 6. Note that the relationship between the measurement volume and the effective measurement volume of the FAC is also explained at the end of this section.

Figure 6. Cross-sectional view of the collection volume, the elongated volume, and the effective collection volume.

From Fig. 4–6, we have the following relationship:

where l_p is the peak-elongated length; l_e is the effective collector length given by l_e=2l_p+l_m, with the collector length l_m measured by 3-d CMM; ε_i is the energy deposition in the i-th pair of regions with the cross-sectional sub-area of; and A_i and f_i are the energy deposition fractions in the i-th pair of the air slabs given in Table 1 and 2. Using eqs. (7) and (8), we can rewrite:

with k_e,el=1/(1−ε_el/ε_d). In Table 3 and 4, we present the values of k_e,el. As can be seen in the tables, the values of k_e,el slowly varied in the range from 1.0002 to 1.0007 for the beam qualities investigated. The relative standard uncertainty of k_e,el was negligible given that the factor (2l_p/l_e) in Eq. (8) was about a percentage.

Table 3 . Energy deposition fraction in the elongated region and the correction factor ke,el of the low-energy X-ray beams of BIPM quality.

Quantity	Beam code
	BIPM10	BIPM30	BIPM25	BIPM50b	BIPM50a
∑ i=1 Ai A0fi  in Eq. (8)	0.9852	0.9810	0.9806	0.9630	0.9363
ke,el	1.0002	1.0002	1.0002	1.0004	1.0007

BIPM, Bureau International des Poids et Mesures..

Table 4 . Energy deposition fraction in the elongated region and the correction factor ke,el of the low-energy X-ray beams of the molybdenum anode mammography quality.

Quantity	Beam code
	M20	M23	M25	M28	M30	M35	M40
∑ i=1 Ai A0fi  in Eq. (8)	0.9817	0.9809	0.9804	0.9798	0.9795	0.9787	0.9780
ke,el	1.0002	1.0002	1.0002	1.0002	1.0002	1.0002	1.0002

The effective volume measurement of our FAC will be obtained by replacing l with l_e when deriving the measurement volume V₀ in Eq. (2). We have used l_m for the measurement volume so far. Considering the mechanically determined volume separately from the effective measurement volume of FAC, the effective measurement volume can be treated as the correction factor for correcting the effect of the electric field distortion on the mechanically determined volume of FAC given by the following:

where k_d is the correction factor and V_e is the effective volume. The k_d was 0.9894, with a relative standard uncertainty of 0.15% with a 68% confidence level (k=1).

The FEM analysis on the electric field of KRISS FAC L1 demonstrated that there was elongation in the measurement volume of the FAC due to electric field distortion caused by the thickness of the guard strips surrounding the collector electrode. This electric field distortion cannot be avoided as long as guard strips are used, and they are necessary in the current design of the parallel plate FAC. The amount of elongation was evaluated from the measured capacitance of the FAC. The energy loss correction for the elongated volume was also estimated.

Most NMIs have their own parallel plate-type FACs as the primary standard devices, and, without exception, there are guard strips inside; thus, the effect of the electric field distortion on the measurement volume of the FACs should be evaluated. We expect that the proposed methodology will be useful in the evaluation of the elongated volume.

We evaluated the effective volume of the FAC based on the precision measurement for the collector length and the aperture diameter of the diaphragm of the KRISS FAC L1. During the evaluation, the elongated length of the collector electrode was determined from the capacitance measurement of the FAC considering the FEM analysis result. The effective volume of the KRISS FAC L1 was 1.2548±0.0019 cm² with a 68% confidence level (k=1) with an elongated length of 0.170±0.021 mm. When the effective length was treated as the correction factor k_d, the value was 0.9894 with a relative standard uncertainty of 0.15% at a 68% confidence level (k=1).

In this study, we evaluated an essential physical parameter, the effective volume of the KRISS FAC L1, for the low-energy X-ray air kerma measurement. The electron loss corrections due to the elongated volume were evaluated thoroughly, with the values presented in Table 3 and 4. The effective volume will replace the existing mechanically determined volume and will be used to establish and maintain the primary standard of the low-energy X-ray air kerma of KRISS. Therefore, our findings contribute to the successful implementation of the air kerma metrology.

This work was supported by the Korea Research Institute of Standards and Science (KRISS) under the project “Development of Measurement Standards for Medical Radiation (2022).” The authors express gratitude to Dr. D.T. Burns in the Bureau International des Poids et Mesures (BIPM) for suggesting the introduction of (∆w/∆l) Eq. (6), which gives a 0.033-mm-reduction in the elongated length (originally, it was evaluated to be 0.203 mm, with an assumption of ∆l=∆w). We note that this study was motivated from by the discussion between the author (Yi) with Dr. D.T. Burns and Dr. C Kessler during his stay in the BIPM dosimetry laboratory for the BIPM-CCRI(I) Key Comparison measurement of the low-energy X-ray air kermas between KRISS and BIPM in 2017.

The authors have nothing to disclose.

The data that support the findings of this study are available on request from the corresponding author.

Conceptualization: Chul-Young Yi. Data curation: Chul-Young Yi, Yun Ho Kim, and Don Yeong Jeong. Formal analysis: Chul-Young Yi. Funding acquisition: none. Investigation: Chul-Young Yi and Yun Ho Kim. Methodology: Chul-Young Yi. Project administration: Chul-Young Yi. Resources: Chul-Young Yi and Don Yeong Jeong. Software: Chul-Young Yi and Yun Ho Kim. Supervision: Chul-Young Yi. Validation: Chul-Young Yi, Yun Ho Kim, and Don Yeong Jeong. Writing–original draft: Chul-Young Yi. Writing–review and editing: Chul-Young Yi, Yun Ho Kim, and Don Yeong Jeong.