Study of equivalent efficiency calibration method for radioactive gas source measured on HPGe detectors

ABSTRACT

In view of the complexity of current detection efficiency calibration of radioactive gas sources, a method using solid planar sources to be equivalent to gas sources was studied. For the 50 mL gas source box, an optimal equivalent scheme was selected by Monte Carlo Simulations. Then, the full-energy-peak efficiency curve of gas sources at the measurement position of 25 cm, with source-to-detector distance of 25 cm, was fitted by measuring solid planar sources with known activity. To verify the accuracy of the efficiency curve, ⁴¹Ar, ¹³³Xe and ⁸⁷Kr gases were produced and determined by length-compensated method. Then, their full-energy-peak efficiencies at 25 cm position away from the detector were directly calibrated. The percentage efficiency deviations between interpolation from the efficiency curve and direct calibration are all less than 2.5%, which proves the accuracy of the equivalent method. This calibration method is a general one and can be also used for some other radioactive sample measurements, such as non-destructive analysis of gaseous fission product samples with a suitable source-to-detector distance.

KEYWORDS:

• Efficiency calibration
• planar source
• gas source
• equivalent method
• efficiency curve

1. Introduction

Activity determinations of radioactive gases (⁸⁵Kr, ¹³³Xe, ¹³⁵Xe, ^131mXe, ^133mXe, et al.) play important roles in nuclear reactor safety monitoring and the Comprehensive Nuclear-Test-Ban Treaty (CTBT). To do this, these gases should be collected and prepared for accurate measurement, usually with a high-resolution gamma-ray spectroscopy spectrometry. Accurate calibrations of gamma-ray detection efficiency for gas sources measured on the high purity germanium (HPGe) detectors are the key but not easy work. The most reliable approach is directly direct calibrated with radioactive standard gases. The standard activity can be determined by internal gas counters [1,2] or β-γ coincidence techniques [3–5]. But this method is generally limited to partial nuclides with beta decay mode, suitable half-lives and can be easily produced by (n,γ) or (n,2n) reactions, such as ⁴¹Ar,¹³³Xe,¹³⁵Xe, et al. Another general method is by interpolating from the efficiency curve calibrated by solid or liquid volume sources [6–8]. It is appropriate for most calibration of gas sources. However, the self-absorption correction cannot be neglected and the calibration source preparation procedures are often complex [9,10]. Efficiency can be also obtained from Monte Carlo simulations, which is perfect to calculate the relative the relative efficiency values of sources with different geometric shapes [11,12–14]. But it is difficult to get exact values for the challenge of precise construction for simulation models [15]. Therefore, this method is mostly used in the measurement of environmental samples with low accuracy requirements.

Another widely employed method is efficiency transfer method [12,16–20]. Its basic principle is to obtain the detection efficiency of target sources with different geometries by using the experimental efficiency value of point sources and efficiency transfer factors. The transfer factors between the efficiency of reference point sources and target sources are usually obtained by Monte Carlo simulation or semi-empirical calculation. Saegusa [21] and co-workers detailed described the method for determination of efficiency curves of volume samples by the combination of Monte Carlo simulations and a single-point calibration using a standard point source. It is a general method for most volume source efficiency calibrations, but the efficiency-curves group around the detectors which must be initially obtained is rather a lot of work.

To overcome these drawbacks in this article, a method using solid planar sources to be equivalent to gas sources has been developed. In this article, we firstly describe the principle of the equivalence and selected the optimal equivalent calibration scheme for our measurement system by Monte Carlo Simulations. Then, the processes of efficiency calibration and verification are also described in detail. This method is a simple, general and accurate method and suitable for the efficiency calibration of most regular-shaped gas sources. Strictly speaking, this method also belongs to the efficiency transfer method. However, the equivalent process of this method is simpler, the efficiency transfer factor value is about 1.0 and the uncertainty is small. At the same time, the source box material’s absorption correction for gamma rays is also negligible because we can use the same source box material and thickness. This method is suitable for the efficiency calibration of most regular-shaped gaseous sources.

2. Principle and simulations

2.1. Principle of equivalent efficiency calibration method

The mainly ideal of the equivalent calibrating method is that for every cylindrical volume gaseous source, we can find a planar source with the same cross-section to be equivalent to it. As shown in Figure 1, a homogeneous cylindrical gas source named S with height of h was placed at the position of H (H≫h) apart from the upper face of the detector. Let S be divided into equal n pieces (n≫1), named S_k (k = 1, 2 … n) respectively. Then, the distance between the bottom of each piece and the upper surface of the detector can be written as:

(1) $H^{\prime }=H+x=H+\left(k-1\right)\cdot h/n\left(k=1,2,\dots \dots ,n\right)$(1)

Figure 1. Schematic diagram of the equivalent method.

The detection efficiency at energy of E_γ of S and S_k are regard as ε_S and ε_k. We can easily know ε₁≥ ε₂≥ ... ≥ε_n−1≥ ε_n and ε₁≥ ε_S≥ε_n for the inverse relation between the detection efficiency value and source-to-detector distance. When n tends towards infinite, every piece above can be deemed to a planar source and there must be one satisfied formula: ε_k≈ε_S. So, the key for the equivalence algorithm is to find appropriate measuring position (H’).

2.2. Simulations

Geant4 simulation toolkit, a software that can simulate the physical process of particle transport in matter [15,22], was used for our simulations. Figure 2 shows the section diagrams of two types of source boxes we used in our simulation models and efficiency calibrations. They have the same diagram diameter of 76 mm and the same bottom and sidewall thicknesses of 1 mm. For the gaseous source box, the air space height inside is 11 mm. Hollow steel pipes were filled in the steel piping between the valve and the box body to reduce the proportion of dead volume to less than 0.1% compared with the volume of the whole box. Then, the dead volume had been neglected in our simulation. The tube between the valve and the main body of the source box is filled with a hollow steel column such that the ratio of the tube’s volume to the volume of the source box is less than 0.1%. Therefore, during the simulation, we ignored this small volume of gas. The carrier of nuclides in the planar source box shown in Figure 2(b) was made of filter paper with less than 0.1 mm thickness. So the self-absorption of it can be neglected.

Figure 2. Section diagrams of two types of source boxes.

A coaxial HPGe detector model was established and shown in Figure 3. The detailed parameters mainly provide by manufacturer’s instructions are listed in Table 1. Precise parameters are non-essential for we are only interested in the relative efficiency ratios between solid and gas sources at different energy.

Table 1. Mainly parameters of the HPGe detector model.

Parameters	Size (cm)
Aluminum shell thickness	0.1
Air gap thickness	0.4
Ge dead layer thickness	0.1
Ge crystal length	7.11
Ge crystal diameter	6.90
Cooling rod hole length	6.29
Cooling rod hole diameter	0.91

Figure 3. Schematic diagram of the coaxial HPGe detector.

The full-energy efficiencies of the gaseous volume source at position of 25 cm and the solid planar sources at multiple positions H′ = 25 cm+x (x= 3, 4, …, 8 mm) were simulated. The γ-ray full-energy peaks were chosen from 60 keV~2.8 MeV, 20 points in total. The relative standard uncertainties of all simulated values are about 1% when ignored the inaccuracy of geometric models. Figure 4 shows the percentage deviations of the full-energy peak efficiency between the volume source and planar sources with multiple values of x. Obviously, when measuring at a high position apart from the detector, the slight change of measuring positions would bring insignificant variations of efficiency. When x = 5 mm, the deviations for all energy peaks are less than 0.6%, which are within the limits of simulation uncertainties. It indicates that solid planar sources measured at position of 25.5 cm can be used to calibrate the full-energy peak efficiency curve of the gaseous volume source measured at position of 25 cm equivalently.

Figure 4. Percentage efficiency deviations between gaseous volume source and planar sources with multiple values of x.

3. Experimental

3.1. Planar sources preparation and measurement

Radioactive solutions of ¹⁵²Eu, ⁷⁵Se, ¹³³Ba, ⁶⁰Co, ¹³⁴Cs, ¹²⁴Sb and ²⁴Na were prepared and their specific activity was determined by the standardized measurement system, respectively. The relative standard uncertainties of the specific activity were about 1.0% (k = 1), except ⁷⁵Se (1.5%, k = 1). They were dropped on the filter papers of the planar source boxes shown in Figure 2(b) quantitatively and homogeneously. After drying, the filter paper was impacted by sealing plug made of Polymethyl methacrylate (PMMA). Silica gel was used to seal the tiny gap between the plug and the box bodies. For every nuclide, a planar source was prepared.

The planar sources were measured at the position of 25.5 cm distance from the top of a coaxial type HPGe detector with a relative efficiency of 70% and FWHM of 1.9 keV at 1332keV of ⁶⁰Co. A digital multichannel analyzer (ORTEC DSPECPLUS) and a computer were used to record the γ-ray spectra. The ORTEC software GAMMAVISION was used to extract the necessary data from the spectra. The count rate of the ith ith full-energy peak E_i can be written:

(2) $n_i=\frac{N_{Si}}{T_{\mathrm{L}}}\frac{\lambda T_{\mathrm{R}}}{1-\mathrm{e}\mathrm{x}\mathrm{p}(-\lambda T_{\mathrm{R}})}$(2)

where N_Si is the peak net counts at energy E_i, T_L and T_R are live time and real time of the count, λλ is the decay constant for the measured nuclide.

The coincidence summing effects are negligible when measured at a high position (such as 25.5 cm). So the full-energy peak efficiency at energy E_i can be computed by,

(3) ${\epsilon }_i=\frac{n_i}{P_i\cdot M\cdot a\cdot F}$(3)

where P_i is the emission probability of gamma-ray at energy of E_i, a (Bq/g) is the specific activity of the radioactive solution of the measured nuclide, M (g) is the mass of injected solution when planar source prepared, F is the self-absorption factor, for the thin filter paper, it equals 1.0.

The relative standard uncertainty of ε_i is:

(4) $u^2({\epsilon }_i)=u^2(n_i)+u^2(a)+u^2(M)+u^2(P_i)+u^2(F_i)$(4)

where u(n_i) is the statistical uncertainty of n_i for multiple measurements, u(M) is about 0.5%, u(P_i) is given by the evaluations nuclear database ENDF/B-VII.1 [16] [23]. u (a) is relative standard uncertainty of a: for ⁷⁵Se, it is 1.5%; and for others, it is 1.0%; u(F_i)≈0.

The main γ-ray peak energy, emission probabilities, measured efficiencies and their relative standard uncertainties of these nuclides are listed in Table 2.

Table 2. Mainly γ-ray peak energy, emission probabilities, measured efficiencies, and their relative standard uncertainties.

Nuclides	Ei (kev)	Pi (%)	εi	u(εi) (%)	Nuclides	Ei (kev)	Pi (%)	εi	u(εi) (%)
^133Ba	81.0	34.06(26)	1.21E-03	1.3	^152Eu	121.8	28.67(15)	2.12E-03	1.2
	276.4	7.16(2)	1.85E-03	1.2		244.7	7.61(4)	1.97E-03	1.2
	302.9	18.33(6)	1.75E-03	1.2		344.3	26.56(51)	1.62E-03	2.2
	356.0	62.05(19)	1.57E-03	1.2		411.1	2.24(2)	1.44E-03	1.6
	383.8	8.94(3)	1.48E-03	1.2		444.0	3.16(3)	1.35E-03	1.5
^134Cs	563.2	8.34(1)	1.17E-03	1.5		778.9	12.96(14)	9.31E-04	1.6
	569.3	15.37(2)	1.16E-03	1.3		867.4	4.26(3)	8.61E-04	1.3
	604.7	97.62(11)	1.08E-03	1.1		964.1	14.65(7)	8.07E-04	1.2
	795.9	85.46(6)	9.03E-04	1.1		1112.1	13.69(7)	7.43E-04	1.3
	802.0	8.69(2)	8.98E-04	1.2		1408.0	21.07(10)	6.22E-04	1.2
^60Co	1173.3	99.85(3)	7.12E-04	1.1	^75Se	96.7	3.35(3)	1.73E-03	1.8
	1332.5	99.98(0)	6.52E-04	1.1		121.1	16.86(33)	2.14E-03	2.5
^24Na	1368.6	99.99(0)	6.45E-04	1.1		136.0	57.7(70)	2.19E-03	2.0
	2754.0	99.86(1)	3.69E-04	1.1		198.6	1.46(4)	2.11E-03	3.3
^124Sb	602.7	97.79(31)	1.09E-03	1.2		264.7	58.75(35)	1.86E-03	1.7
	645.8	7.42(3)	1.04E-03	1.2		279.5	24.89(13)	1.80E-03	1.7
	722.8	10.76(5)	9.64E-04	1.2		303.9	1.31(1)	1.70E-03	1.7
	1691.0	47.57(18)	5.40E-04	1.2		400.7	11.39(9)	1.42E-03	1.8
	2090.9	5.49(3)	4.56E-04	1.3		–	–	–	–

In this paper, all uncertainties are given with a coverage factor k= 1.

3.2. Efficiency curve fitting

The full-energy efficiency function was derived as fifth-order polynomials in double logarithmic scale [24]:

(5) $\mathrm{L}\mathrm{n}({\epsilon }_i)=\sum _{k=0}^5{B}_k{\left[\mathrm{L}\mathrm{n}(E_i)\right]}^k$(5)

where B_k(k = 0,1,2,3,4,5) is the polynomial coefficient; the unit of E_i is MeV, i = 1,2,3, … 37 (shown in Table 2). The values of ε_i and E_i (i = 1, 2, 3 … 37) are listed in Table 2, but here the unit of E_i is MeV.

The weighted least square method was used to fit Equation (5)(5) $\mathrm{L}\mathrm{n}({\epsilon }_i)=\sum _{k=0}^5{B}_k{\left[\mathrm{L}\mathrm{n}(E_i)\right]}^k$(5) . The independent variable matrix X can be written as,

(6) $X=\left(\begin{array}{cccccc}1& \mathrm{L}\mathrm{n}{E}_1& {(\mathrm{L}\mathrm{n}{E}_1)}^2& {(\mathrm{L}\mathrm{n}{E}_1)}^3& {(\mathrm{L}\mathrm{n}{E}_1)}^4& {(\mathrm{L}\mathrm{n}{E}_1)}^5\\ 1& \mathrm{L}\mathrm{n}{E}_2& {(\mathrm{L}\mathrm{n}{E}_2)}^2& {(\mathrm{L}\mathrm{n}{E}_2)}^3& {(Ln{E}_2)}^4& {(\mathrm{L}\mathrm{n}{E}_2)}^5\\ \dots & \dots & \dots & \dots & \dots & \dots \\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ 1& \mathrm{L}\mathrm{n}{E}_{36}& {(\mathrm{L}\mathrm{n}{E}_{36})}^2& {(\mathrm{L}\mathrm{n}{E}_{36})}^3& {(Ln{E}_{36})}^4& {(\mathrm{L}\mathrm{n}{E}_{36})}^5\\ 1& \mathrm{L}\mathrm{n}{E}_{37}& {(\mathrm{L}\mathrm{n}{E}_{37})}^2& {(\mathrm{L}\mathrm{n}{E}_{37})}^3& {(\mathrm{L}\mathrm{n}{E}_{37})}^4& {(\mathrm{L}\mathrm{n}{E}_{37})}^5\end{array}\right)$(6)

And the Y matrix is:

(7) $Y={\left[\begin{array}{ccccc}\mathrm{L}\mathrm{n}({\epsilon }_1)& \mathrm{L}\mathrm{n}({\epsilon }_1)& \cdots & \mathrm{L}\mathrm{n}({\epsilon }_{36})& \mathrm{L}\mathrm{n}({\epsilon }_{37})\end{array}\right]}^{\mathrm{T}}$(7)

Use the reciprocal value of the square of relative standard uncertainties listed in Table 2 to make up the weight matrix W:

(8) $W=\left(\begin{array}{cccccc}\frac{1}{u^2({\epsilon }_1)}& \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}\\ \hspace{0.17em}& \frac{1}{u^2({\epsilon }_2)}& \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}\\ \hspace{0.17em}& \hspace{0.17em}& \ddots & \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}\\ \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& \ddots & \hspace{0.17em}& \hspace{0.17em}\\ \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& \frac{1}{u^2({\epsilon }_{36})}& \hspace{0.17em}\\ \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& \hspace{0.17em}& \frac{1}{u^2({\epsilon }_{37})}\end{array}\right)$(8)

Then, the polynomial coefficient matrix and its covariance matrix are given by the following equations:

(9) $B={\left[\begin{array}{cccccc}{B}_0& B_1& B_2& B_3& B_4& B_5\end{array}\right]}^{\mathrm{T}}={\left(X^{\mathrm{T}}WX\right)}^{-1}{X}^{\mathrm{T}}WY$(9)

(10) $V_B={\left(X^{\mathrm{T}}WX\right)}^{-1}$(10)

The fitting efficiency curve was and parameters are shown in Figure 5. Where R² is the goodness of fit.

Figure 5. Efficiency curve fitted by weighted least square method.

For arbitrary energy Ex, its full-energy peak efficiency ε (Ex) and covariance matrix of efficiency$V_{\epsilon (E_x)}$are obtained by:

(11) $\left\{\begin{array}{c}{X}_x=\left[1\mathrm{L}\mathrm{n}{E}_x{(\mathrm{L}\mathrm{n}{E}_x)}^2{(\mathrm{L}\mathrm{n}{E}_x)}^3{(\mathrm{L}\mathrm{n}{E}_x)}^4{(\mathrm{L}\mathrm{n}{E}_x)}^5\right]\\ \epsilon (E_x)=exp(X_{x}B)\\ V_{\epsilon (E_x)}=X_x{V}_B{X_x}^{\mathrm{T}}\cdot {\epsilon }^2(E_x)\end{array}\right.$(11)

3.3. Verification

To verify the accuracy of the efficiency curve above, the full-energy-peak efficiencies of ¹³³Xe, ⁸⁷Kr and ⁴¹Ar were also directly calibrated with radioactive gases and then compared with the values interpolated by Equation (11)(11) $\left\{\begin{array}{c}{X}_x=\left[1\mathrm{L}\mathrm{n}{E}_x{(\mathrm{L}\mathrm{n}{E}_x)}^2{(\mathrm{L}\mathrm{n}{E}_x)}^3{(\mathrm{L}\mathrm{n}{E}_x)}^4{(\mathrm{L}\mathrm{n}{E}_x)}^5\right]\\ \epsilon (E_x)=exp(X_{x}B)\\ V_{\epsilon (E_x)}=X_x{V}_B{X_x}^{\mathrm{T}}\cdot {\epsilon }^2(E_x)\end{array}\right.$(11) Figure 5. The major characteristic γ-ray energy peaks of these three radioactive gases are 81 keV, 402 keV, and 1294 keV, respectively.

3.3.1. Gas source preparation

Three types of polyethylene targets containing high purity target gas were prepared. Each contains about 1 mL of ¹³²Xe, ⁸⁶Kr or ⁴⁰Ar in volume in standard condition. Then, they were irradiated in the pneumatic rabbit tube of Xi’an Pulsed Reactor with thermal neutron flux about 10¹³ n/(cm²·s) for several minutes, to produce enough ¹³³Xe, ⁸⁷Kr and ⁴¹Ar, respectively. After irradiation and cooling, the targets were used to prepare gas sources. Figure 6 shows the schematic diagram of the gas source preparation platform. Where Y₁~ Y₇ are valves.

Figure 6. Schematic diagram of the gas source preparation platform.

where Y₁~ Y₁₁ are valves; V₁,V₂, and V₃ are three sections of pipeline sealed by corresponding valves; GS1 is the 50 ml gas source box for γ measurement; PC1, PC2, and PC3 are proportional counters with different volumes and used for specific activity measurement. The carrier gases were natural xeon gas for ¹³³Xe, natural krypton gas for ⁸⁷Kr and P10 (90% Ar +10% CH₄) gas for ⁴¹Ar.

The relative activity ratio (R_V) between GS1 and PC3 can be derived from the volumes of pipelines, three counters, and GS1.

Take the preparation of ⁸⁷Kr sources as an example to illustrate the preparation process. Put the ⁸⁶Kr target into target breaker. Vacuumize the whole platform (except the carrier gas bottle) and close all the valves. Break the target and open Y₂, Y₄ and Y₁₁ so that ⁸⁷Kr can be carried to the buffer tank by the carrier gas (P10 gas: 90% ⁴⁰Ar+10% CH4). Close Y₂ and Y₄, vacuumize V₃. Then, open Y₁, Y₄, Y₅, and Y₆ after waiting for 3 min. Close Y₁ and Y₆. By now, the 50 ml gas source (GS1) has been prepared.

Open Y₈ and let the gas in V₁ diffuse into PC3, close Y₈ after 3 min. vacuumize V₁, V₂, V₃ and target breaker. Then, close Y₁₀ and guide the carrier gas into V₁ and adjust the pressure to 400 kPa. Open Y₈ and fill PC3 with 400 kPa carrier gas. Close Y₈ and Y₆, vacuumize V₁ for 10 min. Close Y₁₀, open Y₇, Y₈ and Y₉ for 5 min. Close Y₇, Y₈ and Y₉ and now three sources for proportional counter measurement have been prepared.

Preparations of ¹³³Xe and ⁴¹Ar sources are on the whole the same. The relative activity ratio (R_V) between GS1 and PC3 can be derived from the volumes of pipelines, three counters, and GS1:

(12) $R_{\mathrm{V}}=\frac{V(\mathrm{G}\mathrm{S}1)}{V_1}\cdot \frac{V_1+V(\mathrm{P}\mathrm{C}3)}{V(\mathrm{P}\mathrm{C}3)}\cdot \frac{V(\mathrm{P}\mathrm{C}1)+V(\mathrm{P}\mathrm{C}2)+V(\mathrm{P}\mathrm{C}3)+V_1}{V(\mathrm{P}\mathrm{C}3)}$(12)

where V(GS1) is the volume of GS1; V(PC1), V(PC2) and V(PC3) are the volumes of three counters, respectively.

3.3.2. Direct efficiency calibration of ¹³³Xe, ⁸⁷Kr, and ⁴¹Ar

Let’s continue with take the full-energy-peak efficiency calibration of ⁸⁷Kr (402 keV) as an example. The proportional counters were sent to metering station to determine the specific activity of ⁸⁷Kr by length-compensated method. The determination principle, internal gas proportion counters system, and data analysis processes are described in reference [6] in detail. GS1 was measured at 25 cm position distance from the top of the detector mentioned before. The efficiency at energy of 402 keV can be calculated by:

(13) ${\epsilon }_{\mathrm{d}}=\frac{n}{P_{\gamma }\cdot a\cdot V(\mathrm{P}\mathrm{C}3)\cdot R_{\mathrm{V}}\cdot F}$(13)

where n is the full-energy-peak net counting rate at reference time, P_γ is the emission probability of 402 keV, a is the specific activity of ⁸⁷Kr at reference time, F is the self-absorption factor, F≈ 1.0 when the source pressure is not very high.

The direct calibration results of these three peaks are shown in Table 3.

Table 3. Directly calibrated efficiency (ε_d) of 81 keV, 402 keV, and 1294 keV.

Nuclides	γ-ray Energy (keV)	Pγ (%)	Counting rate (cps)	Activity (Bq)	εd
^133Xe	81.0	38.27(70)^[a]	2.52	5.34E+03	1.24E-03
^87Kr	402.7	49.5(9)^[b]	64.1	8.74E+04	1.47E-03
^41Ar	1293.6	99.16(1)^[a]	216	3.22E+05	6.75E-04

^[a]Ref [23] ^[b] Ref [6]

The major contributions to the relative standard uncertainties of directly calibrated efficiencies are listed in Table 4.

Table 4. Major contributions to the uncertainties of directly calibrated efficiencies.

Parameters	Relative uncertainty (%)
	^133Xe	^87Kr	^41Ar
n	0.5	0.5	0.5
Pγ	1.8	1.9	0
RV	1.0	1.0	1.0
a	1.5	2.0	1.5
V(PC3)	1.0	1.0	1.0
Total	2.8	3.0	2.2

3.3.3. Comparison of the efficiencies calibrated by equivalent and direct method

Table 5 shows the efficiencies and their uncertainties of 81 keV, 402 keV, and 1294 keV interpolated by the equivalent efficiency curve. The percentage efficiency deviations from the values calibrated by direct method (ε_d) are also shown. The results of two methods are consistent in the uncertainty range, indicating the accuracy of the equivalent calibration method.

Table 5. Efficiency calibrated by equivalent method and comparison with direct calibration results.

		Equivalent calibration
			Relative uncertainty (%)^[a]			Direct calibration
Nuclide	Energy (keV)	Efficiency	Curve-fitting	Equivalence	Total	Efficiency	Uncertainty (%)	Relative error (%)
^133Xe	81.0	1.24E-03	1.3	1.0	1.6	1.24E-03	2.8	0.1
^87Kr	402.7	1.44E-03	0.4	1.0	1.1	1.47E-03	3.0	-2.1
^41Ar	1293.6	6.62E-04	0.4	1.0	1.1	6.75E-04	2.2	-2.0
[a] The uncertainties of half-lives and γ-ray peak energy are negligible.
		Equivalent calibration
			Relative uncertainty (%)^[a]			Direct calibration
Nuclide	Energy (keV)	Efficiency	Curve-fitting	Equivalence^[b]	Total	Efficiency	Uncertainty (%)	Relaitve error E/D-1 (%)
^133Xe	81.0	1.24E-03	1.3	1.0	1.6	1.24E-03	2.8	0.1
^87Kr	402.7	1.44E-03	0.4	1.0	1.1	1.47E-03	3.0	-2.1
^41Ar	1293.6	6.62E-04	0.4	1.0	1.1	6.75E-04	2.2	-2.0

[a] The uncertainties of half-lives and γ-ray peak energy are negligible.

[b] The relative deviation of simulation results of surface source and gaseous volume source are mainly considered.

4. Conclusion and discussion

The exact detection efficiency calibration is a foundational but not easy work for gaseous sample measurements. In this paper, a new efficiency calibration method for gas sources was investigated and verified. With GEANT4 simulation toolkit, we choose the best measurement position of the planar source to be equivalent to the gas source measurement. Then, the equivalent efficiency curve was fitted by the measurement of several planar sources. The full-energy peak efficiencies of ¹³³Xe (81 keV), ⁸⁷Kr (402 keV) and ⁴¹Ar (1294 keV) were also directly calibrated with their activities determined by length-compensated method. The efficiencies calibrated by the two methods are consistent well in the uncertainty range, indicating the accuracy of the equivalent calibration method.

Most of the nuclides we used in preparation of the planar sources have much longer half-lives than that of the radioactive gases. Thus, they can be easily prepared and preserved. This equivalent efficiency calibration ideal is a general one and can be also used for some other radioactive sample measurements. For example, when do non-destructive analyses of the gaseous fission products sealed in small quartz tubes and measured with a bit higher source-to-detector distance compared with the length of the tubes [25], we can use the equivalent efficiency curve calibrated by solid point-like point sources. It should be pointed out that the self-absorption factors should be considered if necessary.

In addition to the detection efficiency calibration of radioactive gas sources, another purpose of establishing this method is to calibrate the detection efficiency of fission products in fission ionization chambers for our future fission yield measurement projects. Our ionization chamber is similar in shape to that in reference [26] (see its Figure 1), but is completely sealed in structure, allowing non-destructive measurements of both gaseous and solid fission products after irradiation.

Disclosure statement

No potential conflict of interest was reported by the authors.