Study of Gas Amplification Impact on Plateau Curve Characteristics in Boron-Coated Proportional Counters

Abstract

The characteristics of the plateau curve, specifically its length and slope, in boron-coated proportional counters are key performance indicators that impact the detector’s overall performance. Currently, the lack of research on the plateau curve of boron-coated proportional counters is holding back progress in engineering design and scientific research. This study harnesses the Diethorn formula, a calculation method for gas gain, to explore the relationship between gas amplification and the plateau curve in a boron-coated proportional counter. Based on the analysis of experimental data, this study proposes methods for improving the characteristics of the plateau curve in a proportional counter, including modifications to the counter structure and gas pressure, along with an evaluation method for assessing the effectiveness of these improvements. First, the causes of the plateau curve in a boron-coated proportional counter are analyzed through a physical process perspective, identifying factors that influence gas amplification and subsequently affect the plateau curve. Building upon this foundation, the Diethorn formula is utilized to explore the effects of structural parameters and gas pressure on gas multiplication. Finally, experimental validation is conducted, resulting in the proposal of three methods for improving the characteristics of the plateau curve and an evaluation method for assessing the effectiveness of improvements.

1. Introduction

The boron-coated proportional counter, with its high anti-interference capability and strong adaptability, is widely utilized in various fields such as scientific research, radiation measurement, nuclear energy, and environmental monitoring. In reactor nuclear instrumentation systems today, boron-coated proportional counters play a crucial role, particularly in monitoring the neutron flux rate within the source range. These counters are vital equipment for ensuring the safe operation of the reactor. In nuclear instrumentation systems, the boron-coated counting tube must adhere to certain standards, one of which specifies its plateau curve characteristics. The characteristic of its plateau curve represents a crucial performance indicator that determines the applicable working scenarios for the detector.

So far, there have been many studies on the gas amplification and platform curve characteristics of proportional counters. Since the 1940s, several calculation formulas for gas amplification [1,2,3,4] have been proposed. Among these, the Diethorn formula [1] stands out as the most straightforward and has therefore been widely adopted.

The Diethorn formula, since its introduction, has been at the forefront of characterizing various types of gas detectors, including Geiger–Müller counters [5,6] and proportional counters [7,8,9,10]. Researchers have applied this formula to study the characteristic parameters of a diverse range of gas mixtures, including ethanol–argon, methane–argon, krypton–isopentane, helium–isobutane, and argon–isobutane. The extensive validation across these diverse gas mixtures has firmly established the applicability and versatility of the Diethorn formula in studying gas amplification in these detectors.

From the late 20th century, the Diethorn formula has also been employed in the characteristic studies of detectors used in large-scale experimental setups. M. Aleksa et al. [11] conducted a detailed discussion on the performance of Ar–CO₂ gas mixtures as drift gases in large detectors. They measured the gas gain of various Ar–CO₂ mixtures at a pressure of 3 bar and fitted the data using the Diethorn formula, thereby obtaining Diethorn parameters for different gas mixtures. E. Dané et al. [12] conducted a thorough investigation into the gain characteristics of multi-wire proportional chambers (MWPCs) used in the LHCb muon detector system. Through precise current measurements, they studied the gain performance of MWPCs under different operating voltages and gas densities, comparing the results with predictions made using the Diethorn formula. D. Pinci et al. [13] explored the quantitative relationship between the gas gain of MWPCs and gas pressure, as well as temperature, through theoretical analysis and experimental verification. They also proposed a feedback mechanism to stabilize the gain. Giovanni Passaleva et al. [14] proposed a gain monitoring method for MWPCs based on the average time response. This method, which is efficient, online, and highly accurate, has been applied to monitor the aging effects of LHCb MWPCs.

In the study of the influence of gas pressure, gas temperature, and gas density on detector performance, the Diethorn formula has played a significant role. Yu.I. Davydov [15] conducted research on the gas gain of gases at extremely low pressures below 100 Torr and discovered that gas amplification intensifies as the anode diameter increases. This phenomenon is associated with the form of the first Townsend coefficient under high reduced electric field strengths. H. Loyola1 et al. [16] investigated the gas amplification factor in a proportional counter filled with high-purity ethylene within a pressure range of 10–40 Torr. Xu-ai Z. et al. [17] studied the gas gain in tubes under varying pressures and compared their findings with the Diethorn formula. They also explored the relationship between the output pulse signal and the distance from the charged particle track to the anode wire, attributing this relationship to different aggregation levels of primary ionized electrons during their transit. Yuanyuan M. et al. [18] investigated the gas gain of helium-based gas mixtures under 5.9 keV X-rays from ⁵⁵Fe, further exploring how gas gain varies with operating voltage, gas temperature, gas pressure, and gas proportions. Lei H. et al. [19] conducted a comparative study on the performance of three types of flow-gas proportional counters: ZJ-LD-240, RTM860, and a self-developed detector by the Taiyuan Institute of Radiation Medicine. They compared the physical parameters, plateau curves, and efficiencies of these counters. Guoyun C. et al. [20] delved into the mechanism of gas amplification in cylindrical proportional counters. By applying the Diethorn formula, they found that utilizing fine anode wires, small cathode tube diameters, and low gas pressures can reduce the minimum operating voltage of proportional counters while also optimizing their amplification performance. Juncheng W. et al. [21] conducted experimental research on how gases affect the performance of boron-lined proportional counters, and obtained plateau curves for boron-coated proportional counters under different gas compositions and pressures.

In some studies, higher-precision instruments have been employed to measure gas gain. P. Gianotti et al. [22] investigated the absolute gas gain measurement of a straw tube detector filled with an Ar–CO₂ (90-10) mixture, along with variations in gas gain with operating voltage, gas parameters, and anode radius. By adjusting the source intensity and using a high-precision ammeter, Özkan Şahin et al. [23] studied the variation in Penning transfer rates and photon feedback parameters with concentration and pressure in Ar–CO₂ mixtures, covering a CO₂ concentration range of 1–50% and a gas gain range from 1 to 5 × 10⁵. T.Z. Kowalski [24] measured the gas gain factors of methane-based and propane-based tissue-equivalent gas mixtures, determining the highest stable gas gain, the second ionization Townsend coefficient, and the size of electron avalanches for these mixtures. Ö. Şahin et al. [25] conducted high-precision gas gain measurements to investigate the Penning energy transfer rates and secondary processes in Ne–CO₂ mixtures, aiming to obtain accurate gain data and optimize the performance of gas detectors. Mohamed Fares et al. [26] developed a boron-coated proportional counter and conducted experimental studies on its performance. They obtained results such as the detector’s plateau curve, neutron energy deposition spectrum, and gas amplification factor. At higher voltages, the logarithmic value of the gas amplification factor exhibited an approximately linear relationship with the operating voltage.

Based on the current research, the understanding of gas amplification in proportional counters is relatively mature. The applicability of the Diethorn formula has been validated and widely used in studying the performance of detectors. Measurements of characteristic parameters for some common gases have been completed. There are well-established conclusions regarding the relationships between gas amplification and factors such as gas temperature, gas pressure, gas composition, and operating voltage. However, in studies related to plateau curves, while some preliminary factors influencing plateau curve characteristics have been analyzed, the underlying principles remain unclear. There is a lack of in-depth analysis combining gas amplification in proportional counters with plateau curves. Furthermore, due to the absence of research on the impact of gas amplification in boron-coated proportional counters on plateau curves, it is challenging to effectively support both the forward engineering design and scientific research efforts of detectors.

Starting from the Diethorn formula, this study analyzes the underlying causes and influencing factors of plateau curves in proportional counters. By examining variations in gas amplification across different structural parameters and gas pressures, it provides a comprehensive investigation of the phenomenon. Through the analysis of experimental data, three methods are proposed to improve the characteristics of plateau curves, along with a computational approach to evaluate the effectiveness of these improvements. These contributions significantly enhance the design and optimization of boron-coated proportional counters, thereby possessing substantial engineering value and scientific significance for the development of detectors.

2. Analysis of Factors Influencing the Proportional Counters Plateau Curve

2.1. Diethorn Formula

Assuming that the anode radius and cathode radius of the proportional counter tube are a and b, respectively, the anode potential and cathode potential are V_a and V_b, respectively, and the applied operating voltage is V₀ = V_a − V_b, the electric field strength at the distance r from the anode can be expressed as:

$$\begin{array}{c}E\left(r\right)={\displaystyle \frac{V_0}{r ln(b/a)}}\end{array}$$

(1)

The reduced field strength ε(r) is the ratio of the electric field strength E(r) to the gas pressure p, measured in V·m⁻¹·Pa⁻¹.

The Diethorn formula can be expressed as follows:

$$\begin{array}{c}lnM={\displaystyle \frac{a p \epsilon \left(a\right) ln2}{\mathsf{\Delta }V}}·ln{\displaystyle \frac{\epsilon \left(a\right)}{\mathrm{K}}}\\ ={\displaystyle \frac{V_0 ln2}{ln(b/a) \mathsf{\Delta }V}}·ln{\displaystyle \frac{V_0}{\mathrm{K} \mathrm{p} \mathrm{a} \mathrm{l}\mathrm{n}(b/a)}}\end{array}$$

(2)

In the formula, M is the gas amplification factor; ΔV is the energy obtained by a single electron from the electric field between two consecutive collisions, eV; and K is the minimum reduced field strength at which multiplication occurs, V·m⁻¹·Pa⁻¹.

For given gases, ΔV and K are constants. The ΔV and K of the operating gas in common proportional counters are shown in

Table 1. Diethorn parameters of common gases in PC [20].

Gas Mixtures	K (V·m−1·Pa−1)	ΔV (eV)
90% Ar + 10% CH4 (P10)	48.4	23.6
95% Ar + 5% CH4 (P5)	45.0	21.8
90% Xe + 10% CH4	36.2	33.9
95% Ar + 5% CH4	36.6	31.4
96% He + 4% C4H10(isomer)	14.8	27.6
100% CH4	69.0	36.5
100% C3H8	100.0	29.5
75% Ar + 15% Xe + 10% CO2	51.0	20.2
64.6% Ar + 24.7% Xe + 10.7% CO2	60.0	18.3
90% Ar + 10% CH4 (P10)	48.4	23.6

.

According to Equation (2), in order for gas multiplication to occur, M needs to be greater than 1. Therefore, we can deduce:

$$\begin{array}{c}{V}_{min}>K p a ln(b/a)\end{array}$$

(3)

In the formula, V_min is the minimum operating voltage at which amplification occurs, V.

2.2. Formation and Influencing Factors of Plateau Curve

The variation in the counting rate and pulse amplitude of the proportional counter with voltage is shown in Figure 1, revealing the cause of the plateau curve.

In Figure 1, as the operating voltage of the proportional counters increases, the recombination of electron ion pairs is suppressed, and a saturated working region II appears in curve a, where the pulse amplitude remains basically unchanged with the operating voltage. When the operating voltage is further increased, gas amplification occurs, and the pulse amplitude gradually increases. By selecting a reasonable threshold, the counting rate gradually increases with the increase in operating voltage, and a plateau region will appear in curve b. As the operating voltage continues to increase, the gas amplification factor becomes too large. At this time, the counting rate rapidly increases, and the detector enters a region of limited proportionality where the output pulse amplitude is not proportional to the ionization of charged particles.

The voltage value V₁ is defined with its corresponding counting rate denoted as n₁, and the voltage value V₂ is defined with its corresponding counting rate denoted as n₂. Therefore, the plateau length V_L is V₂ − V₁, and the formula for calculating the plateau slope P is:

$$\begin{array}{c}P={\displaystyle \frac{100\times 2(n_2-n_1)}{(n_2+n_1)(V_2-V_1)}}\times 100\%\end{array}$$

(4)

When the detector’s threshold is selected, the particles are detected and the amplification factor are determined, subsequently fixing the energy spectrum. The proportion of signals exceeding this threshold remains constant, resulting in a constant counting rate. As the amplification factor varies, so does the energy spectrum and, consequently, the counting rate. Specifically, a higher amplification factor leads to a higher counting rate. According to Equation (2), an increase in operating voltage corresponds to a greater amplification factor, which leads to a higher counting rate. The plateau curve illustrates how the detector’s counting rate varies with the operating voltage. If the amplification factor increases slightly with the increase in operating voltage, indicating that the gas amplification factor remains similar across different operating voltages, the particles’ energy spectrum, the proportion of threshold signals, and the counting rate will all be similar. At this point, the plateau curve, plotted using the counting rate and working voltage, will appear nearly horizontal, signifying a gradual increase in the counting rate with an increase in the operating voltage. It is usually hoped that the operating voltage of the detector will not be too high, with a longer plateau length and a smaller plateau slope. This corresponds to a slower variation in gas amplification factor with operating voltage. Therefore, according to Equation (2), the main factors affecting gas amplification and thus the plateau curve characteristics are:

• Dimensions of detector anode and cathode;
• Composition and pressure of gas.

3. Research on Influencing Factors

3.1. Dimensions of Anode and Cathode

3.1.1. V_min

Firstly, the dimensions of the cathode and anode will jointly determine the minimum operating voltage V_min. For Equation (3), calculate the partial differential derivatives of V_min with respect to b and a, respectively:

$$\begin{array}{c}{\displaystyle \frac{\partial V_{min}}{\partial b}}=K p {\displaystyle \frac{a}{b}}\end{array}$$

(5)

$$\begin{array}{c}{\displaystyle \frac{\partial V_{min}}{\partial a}}=K p (ln{\displaystyle \frac{b}{a}}-1)\end{array}$$

(6)

Taking the example of filling with (30 kPa) P10 gas to calculate the minimum operating voltage for several common combinations of cathode radius b and anode radius a, the results are shown in

Table 2. Minimum operating voltage at different parameters.

	b	6.35 mm	10 mm	12.7 mm	13.5 mm	15 mm	17.5 mm	21.5 mm	25.4 mm	35 mm
a
6 μm		60.7	64.6	66.7	67.2	68.2	69.5	71.3	72.8	75.5
8 μm		77.6	82.8	85.6	86.3	87.5	89.3	91.7	93.7	97.4
10 μm		93.7	100.3	103.8	104.7	106.2	108.4	111.4	113.8	118.5
12.5 μm		113.1	121.3	125.7	126.8	128.7	131.5	135.2	138.2	144.1
15 μm		131.7	141.6	146.8	148.2	150.5	153.8	158.3	161.9	168.9
20 μm		167.3	180.5	187.4	189.2	192.2	196.7	202.7	207.5	216.9
25 μm		201.0	217.5	226.2	228.4	232.2	237.8	245.3	251.3	263.0
30 μm		233.3	253.0	263.5	266.1	270.7	277.4	286.4	293.7	307.6
50 μm		351.7	384.7	402.0	406.4	414.1	425.3	440.2	452.3	475.6

.

The data unit in Table 2 is V. According to Table 2 combined with Equations (5) and (6), it can be seen that when the gas pressure is constant, the minimum operating voltage V_min gradually increases with the increase in cathode radius b, and the degree of change increases with the increase in anode radius a. As the anode radius a increases, the minimum operating voltage V_min gradually increases, and the degree of change increases with the increase in cathode radius b.

A smaller operating voltage is beneficial for the miniaturization of secondary instruments, and a lower minimum operating voltage V_min is a prerequisite for achieving a smaller operating voltage. According to Table 2, V_min is all below 500 V. Choosing a combination of small cathode radius and anode radius parameters is beneficial for achieving the minimum operating voltage V_min.

3.1.2. Gas Amplification

Taking the example of filling with (30 kPa) P10 gas to calculate the variation law of lnM with an operating voltage (V₀) under several common combinations of cathode radius and anode radius by Equation (2), Figure 2 and Figure 3 are compiled.

As shown in Figure 2, when the anode radius is given, as the cathode radius decreases, the change in M with V₀ becomes more intense, and the degree of change slows down as the anode radius increases.

As shown in Figure 3, when the cathode radius is given, as the anode radius decreases, the change in M with V₀ becomes more intense, and the degree of change slows down as the cathode radius increases.

Based on the discussion in Section 2.2, it can be concluded that within a specific range, the slower the gas amplification factor changes with the operating voltage, the longer the plateau length, and the smaller the plateau slope. The experimental data in [19,24,25] provide support for the conclusions in this section. Taking data from [19] as an example, three types of proportional counters were measured in the experiment. The anode radius of the self-developed detector is the same as that of the ZJ-LD-240 detector, and the cathode radius of the self-developed detector is 1.7 times that of the ZJ-LD-240 detector, with a plateau length extension of 20 V. The cathode radius of the self-developed detector is 0.9 times that of the RTM-860 detector, and the anode radius of the self-developed detector is 0.5 times that of the RTM-860 detector, resulting in a reduction of 30 V in plateau length.

In summary, in order to achieve better plateau curve characteristics of the detector, a larger cathode radius and a larger anode radius should be selected.

3.2. Composition and Pressure of Gas

3.2.1. V_min

For Equation (3), calculate the partial differential derivatives of V_min with respect to p:

$$\begin{array}{c}{\displaystyle \frac{\partial V_{min}}{\partial p}}=K a ln{\displaystyle \frac{b}{a}}\end{array}$$

(7)

Taking b = 25.4 mm and a = 12.5 μm as examples of cathode radius and anode radius, the minimum operating voltage at different gas pressures can be calculated when the filling gas is P10 gas. The results are shown in

Table 3. Minimum operating voltage at different gas pressures.

p (kPa)	15	20	25	30	35	40	45	50
Vmin (V)	69.1	92.2	115.2	138.2	161.3	184.3	207.4	230.4

.

According to Table 3 combined with Equation (7), the minimum operating voltage V_min is directly proportional to the gas pressure p, and the degree of change is related to the gas composition, anode radius, and the selection of cathode radius. Choosing the appropriate gas composition and smaller gas pressure is beneficial for achieving the minimum working voltage V_min.

3.2.2. Gas Amplification

Taking b = 25.4 mm and a = 12.5 μm as examples of cathode radius and anode radius, the variation in M with V₀ at different gas pressures can be calculated using Equation (2) when the filling gas is P10 gas. Figure 4 is obtained.

As shown in Figure 4, with the decrease in gas pressure, the change in M with V₀ becomes more intense. Combining with the discussion in Section 2.2, it can be concluded that in order to make the gas amplification factor change more slowly with the operating voltage within a specific range, a larger gas pressure should be selected. From Equation (2), it can be seen that selecting a gas composition with a larger K and ΔV can also slow down the change in gas amplification factor. The experimental data in [5,6,7,8,9,10,11,17,18,21,22,23,24,25] provide support for the conclusions in this section. Taking data from [21] as an example, the research conclusion on the gas composition of 85% Ar + 15% CO₂ in the experiment is that when the gas pressure is very low, there is no obvious plateau curve in the detector. When the pressure is 30 kPa, a more obvious plateau curve appears. As the gas pressure increases, the plateau curve of the detector becomes longer, and the plateau slope becomes smaller. By changing the proportion of CO₂, it was found that as the proportion increased, the plateau curve gradually appeared, the plateau length gradually increased, and the plateau slope decreased.

In summary, in order to achieve better plateau curve characteristics of the detector, an appropriate gas composition and pressure should be selected.

4. Improvement Methods and an Evaluation Method for Plateau Curve Characteristics

4.1. Improvement Methods for Plateau Curve Characteristics

As discussed in Section 3.1.1 and Section 3.2.1, the minimum operating voltage V_min can be reduced by decreasing the cathode radius, decreasing the anode radius, and decreasing the gas pressure. As discussed in Section 3.1.2 and Section 3.2.2, it is possible to improve the plateau curve characteristics by increasing the cathode radius, anode radius, and gas pressure to slow down the gas amplification factor within a specific range. Therefore, in order to achieve good plateau curve characteristics, there is no choice but to increase the operating voltage. This study focuses on improving the plateau curve characteristics while ensuring that the operating voltage is not too high.

The cathode radius of a detector directly determines its size and significantly impacts its sensitivity, and it is generally related to the detector’s usage scenario and measurement requirements. Therefore, improving the characteristics of the plateau curve mainly starts with the anode radius and gas pressure.

This study proposes three improvement methods:

• Increasing the anode radius;
• Increasing the gas pressure;
• Increasing the anode radius while increasing the gas pressure.

Experimental verification on the boron-coated proportional counter is conducted, and counter parameters are shown in

Table 4. Boron-coated proportional counter parameters.

Part	Physical Parameters	Value or Description
cathode	radius (including thickness)	30 mm
	length	1000 mm
	thickness	1 mm
	material and density	TA2 4.51 g/cm3
	potential	grounding
anode	radius	12.5 μm or 20 μm
	length	1000 mm
	material	gold-plated tungsten wire
	potential	positive high voltage
boron	material	98% 10B + 2% 11B
	mass thickness	0.6 mg/cm2
	composition	85% Ar + 15% CO2
gas	pressure	25 kPa, 30 kPa, 35 kPa
		40 kPa, 45 kPa, 50 kPa

.

The experimental setup is a neutron sensitivity calibration device for the secondary dose station, as shown in Figure 5. In the experiment, the detector height is aligned with the active section of the line neutron source, and the calibration value of the thermal neutron flux at the detector position is 550 n·cm⁻¹·s⁻¹. Connect the detector to the secondary instrument according to Figure 6, where the amplification factor of the preamplifier is ×5 and the amplification factor of the linear amplifier is ×800.

At different parameters, when the detector is within its operating voltage range and the threshold of the calibrator is measured to be 300 mV, the detector basically does not output the counting rate with the neutron source off. Therefore, the threshold of 300 mV is sufficient to eliminate background signals. Therefore, the experiment uses 500 mV as the threshold, firstly to ensure that the increase in operating voltage caused by changing parameters is less affected by dark current, and secondly to make the effect of changing parameters on the plateau curve more intuitive and convenient for research. The experiment changes the operating voltage at intervals of 20 V, and counts the total count for 60 s at each voltage point to calculate the counting rate. The plateau curve obtained from the experiment is shown in Figure 7.

Due to the high threshold, the counting rate in Figure 7 closely approximates a straight line as the operating voltage varies. As depicted in Figure 7, as the anode radius and gas pressure increase, the minimum operating voltage gradually rises. This trend aligns with the analysis presented in Equation (3).

When the anode radius remains constant, the counting rate gradually slows down with the variation in operating voltage as the gas pressure increases. When the gas pressure remains constant, the counting rate gradually slows down with the variation in operating voltage as the anode radius increases. This is consistent with the expectations discussed in Section 3.1 and Section 3.2, indicating that as the anode radius and gas pressure increase, the gas amplification factor decreases with the variation in operating voltage.

Using a slope of 40% as the standard, the length of the slope that meets the slope requirement can be calculated according to Equation (4). According to Figure 7, when the gas pressure is less than 40 kPa, the data with the second highest counting rate are basically representing the inflection point when the detector enters the region of limited proportionality. In order to ensure that the calculated plateau does not enter the region of limited proportionality, the two data with the highest counting rate are discarded. The calculation results are shown in

Table 5. Plateau curve characteristics at different parameters.

Gas Pressure	Plateau Length (Anode Radius 12.5 μm)	Plateau Slope (Anode Radius 12.5 μm)	Plateau Length (Anode Radius 20 μm)	Plateau Slope (Anode Radius 20 μm)
25 kPa	1080 V~1100 V:20 V	53.76% (Unmatched)	1200 V~1240 V:40 V	39.70%
30 kPa	1200 V~1220 V:20 V	40.18% (Unmatched)	1200 V~1340 V:140 V	39.64%
35 kPa	1240 V~1320 V:80 V	40.00%	1260 V~1460 V:200 V	38.54%
40 kPa	1300 V~1420 V:120 V	38.47%	1320 V~1560 V:240 V	39.79%
45 kPa	1340 V~1500 V:160 V	38.95%	1380 V~1640 V:260 V	39.30%
50 kPa	1380 V~1580 V:200 V	38.03%	1420 V~1720 V:300 V	39.42%

.

According to Table 5, among the three methods for improving plateau characteristics, changing the gas pressure has the worst improvement effect on plateau characteristics, but the operational difficulty is the lowest. Changing the anode radius has a good effect on improving the plateau characteristics, but it is difficult to operate. Changing the gas pressure and anode radius at the same time has the best improvement effect on the plateau characteristics, but the operation difficulty is the highest. In practical applications, it is necessary to select appropriate methods to improve the characteristics of the detector according to different usage scenarios.

4.2. Evaluation of Improvement Effect on Plateau Curve Characteristics

When the gas pressure is high enough (for this experiment, the pressure should be greater than 30 kPa), the amplitude spectrum of the signal is basically consistent under repeated experiments. Assuming that only the anode radius of the detector is changed, and the influence of the electric field on the particle migration rate is ignored, the same counting rate level corresponds to the same gas amplification factor. Therefore, Equation (2) can be used to predict and evaluate the improvement effect.

Select data with a counting rate greater than 300 s⁻¹ (excluding the two data with the highest counting rate) for line fitting, ensuring that the original total count remains above 18,000, which implies that the fluctuation in the data is less than 1%. The formula is shown in

Table 6. Linear fitting at different parameters.

Gas Pressure	Fitting Line Formula (Anode Radius 12.5 μm)	R-Squared (Anode Radius 12.5 μm)	Fitting Line Formula (Anode Radius 20 μm)	R-Squared (Anode Radius 20 μm)
25 kPa	y = 87.553x − 82,105	0.9985	y = 77.864x − 79,190	0.997
30 kPa	y = 73.781x − 73,494	0.9985	y = 67.452x − 71,065	0.994
35 kPa	y = 64.759x − 68,321	0.9987	y = 57.762x − 64,634	0.9954
40 kPa	y = 58.255x − 64,589	0.9996	y = 53.654x − 64,052	0.9979
45 kPa	y = 53.758x − 62,491	0.9992	y = 51.107x − 64,197	0.9989
50 kPa	y = 51.608x − 62,473	0.9995	y = 48.663x − 63,896	0.9996

.

In Table 6, y is the counting rate, s⁻¹, and x is the operating voltage, V. By fitting the linear formula, the operating voltage of the detector at different counting rate levels can be calculated as the true value for further analysis. R² value in Table 6 is extremely close to 1, suggesting a certain degree of reliability in data processing.

If gas amplification factors under different parameters with the same counting rates are equal, then Equation (2) can be used to obtain:

$$\begin{array}{c}{\displaystyle \frac{V_1 ln2}{ln(b/a_1) \mathsf{\Delta }V}}·ln{\displaystyle \frac{V_1}{\mathrm{K}\mathrm{p}{a}_1\mathrm{l}\mathrm{n}(b/a_1)}}={\displaystyle \frac{V_2 ln2}{ln(b/a_2) \mathsf{\Delta }V}}·ln{\displaystyle \frac{V_2}{\mathrm{K}\mathrm{p}{ a}_2\mathrm{l}\mathrm{n}(b/a_2)}}\end{array}$$

(8)

In the formula, V₁ is the operating voltage that reaches a specific counting rate under parameter a₁, and V₂ is the operating voltage that reaches the same counting rate under parameter a₂.

Equation (8) can be simplified as:

$$\begin{array}{c}{\displaystyle \frac{V_1 ln2}{ln(b/a_1)}}·ln{\displaystyle \frac{V_1}{\mathrm{K}\mathrm{p}{ a}_1\mathrm{l}\mathrm{n}(b/a_1)}}={\displaystyle \frac{V_2 ln2}{ln(b/a_2)}}·ln{\displaystyle \frac{V_2}{\mathrm{K}\mathrm{p}{a}_2\mathrm{l}\mathrm{n}(b/a_2)}}\end{array}$$

(9)

Therefore, only the parameter K in the Diethorn formula is needed for prediction. The Diethorn parameters for Ar–CO₂ mixtures given in [11] are shown in

Table 7. Diethorn parameters of gas mixtures in PC [11].

Gas Mixtures	K (V·m−1·Pa−1)	ΔV (eV)
80% Ar + 20% CO2	22	49
90% Ar + 10% CO2	20	43
92% Ar + 8% CO2	19	42
93% Ar + 7% CO2	24	34
96% Ar + 4% CO2	23	34

.

The data of the K values in Table 7 are provided by the author [11] with an error of 10%. The average of the K values under 80% Ar + 20% CO₂ and 90% Ar + 10% CO₂ in Table 7 (K = 21 V·m⁻¹·Pa⁻¹) is taken for prediction.

Taking the counting rate levels of 6000 s⁻¹ and 16,000 s⁻¹ as examples for explanation: under different gas pressures, with an anode radius of 12.5 μm, the operating voltage is substituted into the left side of Equation (9) to provide the predicted operating voltage at each pressure with an anode radius of 20 μm, as shown in

Table 8. Plateau curve characteristics prediction at different gas pressures with anode radius 20 μm at 6000 s⁻¹ counting rate.

Gas Pressure (kPa)	Measured Value (V)	Predicted Value (V)	Absolute Error Between Predicted and Measured Values (V)
30	1142.5 V	1152.3 V	9.8 V
35	1222.8 V	1205.8 V	17.0 V
40	1305.6 V	1304.2 V	1.4 V
45	1373.5 V	1374.9 V	1.4 V
50	1436.3 V	1435.6 V	0.7 V

and

Table 9. Plateau curve characteristics prediction at different gas pressures with anode radius 20 μm at 16,000 s⁻¹ counting rate.

Gas Pressure (kPa)	Measured Value (V)	Predicted Value (V)	Absolute Error Between Predicted and Measured Values (V)
30	1290.8 V	1251.4 V	39.4 V
35	1396.0 V	1350.0 V	46.0 V
40	1492.0 V	1481.6 V	10.4 V
45	1569.2 V	1567.3 V	1.9 V
50	1641.8 V	1636.3 V	5.5 V

.

According to Table 8 and Table 9, it can be seen that changing the anode radius under the same pressure has good accuracy in evaluating the improvement effect of plateau curve characteristics, which is of good reference value.

5. Conclusions

This study establishes the relationship between the plateau curve and gas amplification through the discussion of the plateau curve of proportional counters, and analyzes the influence of structural parameters and gas pressure on gas amplification by combining Diethorn’s formula. We propose three methods to improve the plateau curve characteristics: (1) increasing the anode radius; (2) increasing the gas pressure; (3) increasing the anode radius while increasing the gas pressure. We conducted experimental verification and compared the three methods. The simplest operation is to change the gas pressure, but the improvement effect is the worst. Changing the anode radius is a complex operation, but the improvement effect is more pronounced. Changing the anode radius and gas pressure at the same time is the most complex operation and has the best improvement effect. A calculation method for evaluating the improvement effect has been proposed, which has good reference value when the gas pressure is the same. This study provides a reference and guidance for the design and improvement of boron-coated proportional counters.