Dead time calculation formula derivation

The principal contributor to the dead time of the ³He neutron detector is the detector itself. In the process of detecting neutrons, because neutrons are not charged and cannot directly cause the “ionization” or “excitation” of the material, the neutron signal pulse is a neutron reaction with ³He to generate protons, emit tritium nuclei in the opposite direction, and cause the ³He gas to ionize to form a large number of ion pairs (electrons and positive ions). A ³He tube with a high voltage of the anode filament and ³He tube shell form an electric field. Under the action of the electric field, the electrons and positive ions drift to the anode filament, to the wall of the cathode tube, and, ultimately, through a series of anode filament reactions to produce an effective neutron signal. Therefore, the pulsed signal from a neutron must undergo a series of processes before it can be recorded. Figure 1 illustrates the reason for dead time generation: if the ³He detector system comes with another neutron pulse signal during signal processing, it will produce pulse superposition. Therefore, the pulse signal cannot be processed in time and misses counting, which is the main reason for non-expanded dead time generation. Scholars have proven that most of the contribution of the dead time of the ³He detector comes from non-expanded dead time. Related scholars have proven that most of the dead time of the ³He detector comes from the nonexpanded dead time. Therefore, the model used in this experiment refers to the nonexpanded dead time. The ³He detector can be synchronized with the pulse signal of the pulsed neutron generator through the digital multichannel collector and record the time distribution curve of the number of neutrons after the neutron pulse is discharged, which ultimately forms the required neutron time spectrum. The counts at each moment, corresponding to time t₀, are converted to count rates before processing the neutron time spectrum. The conversion equation is represented by Eq. (1). $n_i=\frac{N_i}{t\times f\times t_0}i=\mathrm{1,2,3}\cdots$ (1) where ni is the count rate normalized to the i channel, Ni is the count corresponding to the i channel in the neutron time spectrum, t is the acquisition time of the detector, f is the frequency of the pulsed neutron source, and t₀ is the time corresponding to the i channel. After processing Eq. (1), we obtain a revised neutron time spectrum.

Fig. 1Full-screen View

Dead time generation process

In the neutron field of a pulsed source, the average number of true neutrons incident on the ³He detector per unit time is m, the number of neutrons recorded per unit time is n, and the dead time of the detector is τ. The dead time of the ³He detector is denoted by nτ, and the total number of counts lost owing to dead time is mnτ, which is the number of counts lost. Subsequently, τ can be obtained from Eq. (2) and is written as: $\tau =\frac{m-n}{mn}.$ (2) As shown in Fig. 2, two neutron time spectra detected by the ³He ratio counting tube were selected, and two windows of the same size were set up with window widths defined as t_ww, which can be set to 2 μs, 8 μs, 16 μs, 24 μs, or 32 μs. The initial time of window 1 (W1) should start when the count rate is high and unrelated to the source neutrons. The time W1 from the peak of the neutron time spectrum is the delay time T0. The initial time of window 2 (W2) should be far away from W1, and the interval between the two time windows is denoted by t_wd. The ratio of the count rates of the two time spectra within the window is denoted by f_d, used to measure the height of the time window, and subsequently discussed in detail. The position of W1 is at moment t1 of the neutron time spectrum. Then, t1 is equal to t2, and the corresponding neutron count rates of the high- and low-flux neutron time spectra are n_t1 and n_t2, respectively. Moreover, $f_{\text{d}1}=n_{t1}/n_{t2}$, and the corresponding real count rates are m_t1 and m_t2. The position of W2 is at the t3 moment of the neutron time spectrum. Then, t3 is equal to t4, corresponding to the high- and low-flux neutron time spectra of the neutron count rates of n_t3 and n_t4, respectively. Then, $f_{\text{d2}}=n_{t3}/n_{t4}$, and the corresponding true count rate is m_t3,m_t4.

Fig. 2Full-screen View

(Color online) Parameters related to setting the time window on the neutron time spectrum

Evidently, in a neutron time spectrum devoid of dead-time effects, the ratio of the neutron count rates for any two captured windows should be equal. The ratio of the true count rate can be expressed as: $\frac{m_{t1}}{m_{t2}}=\frac{m_{t3}}{m_{t4}}.$ (3) Combining Eqs. (3) and (2), the τ term can be written as: $\tau =\frac{n_{t1}{n}_{t2}-n_{t3}{n}_{t4}}{n_{t1}{n}_{t4}\left(n_{t2}+n_{t3}\right)-n_{t3}{n}_{t2}\left(n_{t1}+n_{t4}\right)}.$ (4) The only unknown parameter is the true count rate m. The detector dead time can be determined by equating the two sets of count rates from the experimental tests in Eqs. (4).

Dead time correction

Finally, the formula for the dead time τ is introduced into the equation for the relationship between the true count n and count m detected by the ³He detector as well as the dead time τ described in Eq. (2) and combined with Eq. (1). The correction formula for the true count mi can be obtained as follows: $\begin{array}{l}{m}_i=\frac{n_i}{1-n_i\tau }=\\ \frac{n_i}{1-n_i\frac{n_{t1}{n}_{t2}-n_{t3}{n}_{t4}}{n_{t1}{n}_{t4}\left(n_{t2}+n_{t3}\right)-n_{t3}{n}_{t2}\left(n_{t1}+n_{t4}\right)}}i=\mathrm{1,2,3}\cdots \end{array}$ (5) where ni corresponds to the count corresponding to each tract site in the neutron time spectrum and mi is the corrected true count for that tract site.

Experimental parameters

The experiment uses a variety of specifications of small, sealed neutron tubes that were independently developed and produced. Moreover, this experiment adopts the Penning Ion Source Technology, which vacuum packages the ion source, acceleration system, target, etc., and accelerates the target after the ionization of deuterium and tritium mixed gases. The deuterium and tritium nuclear reaction occurs on the target membrane to generate neutrons. Hence, the neutron flux generated can be controlled by controlling the size of the target's high voltage. The neutron yield of the D-T neutron tube is up to 1×10⁹ n/s, which meets the requirements of this experiment. Moreover, the working life under the maximum yield is ≥300 h, which can withstand a 175 ℃ high-temperature environment. The neutron generator has two modes of operation: pulse and direct current(DC). This experiment used a pulse mode with a pulse period of 2000 μs and pulse width of 200 μs that can stably emit 14.1 MeV fast neutrons in the 4π direction. The neutron detector used in this experiment was a Global Nucleonics 15He3-304-38FS-type ³He positive-ratio counting tube. The ³He tube and neutron generator, together with the polyethylene barrel, were set into a 163 cm × 203 cm × 166 cm polyethylene shielding body at the center of the hole with a depth of 120 cm and radius of 6 cm. The shield can quickly slow the emitted fast neutrons. Its geometric position is shown in Fig. 3a. This experiment uses a digital multi-channel acquisition board (DCMA) for the acquisition of the neutron time spectrum [41, 42]. The neutron generator pulse synchronization signal is connected to the multi-channel board synchronization signal, and the detection time is synchronized with the pulse neutron emission time of the neutron generator so that it can collected neutrons emitted within 200 μs within 2000 μs of the neutron attenuation. The multi-channel acquisition board sends the data to the upper computer software via RS485, the upper computer software completes the acquisition and preservation of the neutron time spectrum, and Fig. 3b [43] shows the neutron time spectrum acquisition process. In this study, Monte Carlo N-Particle (MCNP) software was used to simulate the neutron transport and interaction process inside a polyethylene shield, which is a powerful Monte Carlo simulation tool that is widely used in the fields of nuclear physics, medical radiation therapy, and radiation protection. This method is used to simulate the neutron time spectrum of the experimental environment in an ideal situation for comparison with experimental and corrected results [44]. Thus, the neutron time spectrum was used to compare and verify the experimental and corrected results. Finally, a neutron time spectrum correction experiment of the PFNUL instruments was conducted in a standard model well in a uranium mine.

Fig. 3Full-screen View

(Color online) (a)Physical diagram of the experimental apparatus. (b)Schematic of the experimental setup

Experiment procedure

The experimental measurements involved neutron time spectrum measurements under five different neutron fluxes by controlling a D-T neutron generator. The parameters for each experimental group were as follows. After setting the target high pressure for the first experimental group, neutron time spectrum data collection was initiated when the neutron generator's target high pressure was stabilized. A data collection duration of 300 s was employed, and three sets of neutron time spectrum data were collected at the same target high pressure. The average of the three sets was used as the effective test result for that particular group. Following the completion of each experiment, the target high pressure was adjusted for the next experimental group, and the process was repeated accordingly. Table 1 lists the parameter configurations of each experimental group.

Figure 3a shows the experimental setup. The MCNP method was used to establish the Monte Carlo model. The model illustrated in Fig. 3b consists mainly of a D-T neutron source, polyethylene shielding box, sleeve, and ³He detector. The D-T neutron source was approximated as a point source emitting 14.1 MeV fast neutrons in the 4π direction with a pulse period of 2000 μs and a pulse width of 200 μs, aligned with the parameters of the actual experimental setup. The simulation utilized element cross-section data from the NDF/B-VIII.0 database, and ultimately, theoretically simulated data for the experimental thermal neutron time spectrum were obtained.

By employing a ³He detector, neutron counts were recorded at varying target voltages. The discrepancies in the neutron decay coefficients within the neutron time spectrum under different target voltages were ascribed to count losses stemming from the dead time. The detector dead time was estimated using Eq. (4).

Effect of window settings on dead time

4.1.2

W1 initiation time

Considering that the pulse width of the neutron emitter is 200 μs, the simulation results demonstrated that it takes approximately 150 μs for all the source neutrons to be converted into thermal neutrons. Therefore, the initiation time of W1 is set to approximately 400 μs to avoid the influence of the source neutrons. The initiation time of W1 is shown in Fig. 5, which shows that after 400 μs, the dead time calculation results of the combinations tend to stabilize. Hence, it is recommended to set the initiation time of W1 to 400 μs.

Fig. 5Full-screen View

Changes in dead time with an initial time shift of W1

4.1.4

Dead time calculations

Based on Eq. (4), the dead time was calculated for different t_ww by varying f_d as mentioned above. Figure 6a presents the results. Different t_ww values have little effect on the dead time at the same height. Moreover, #4 + #5 and other groups of calculation results differ because the two high voltages are too close, and the dead time calculation method utilizes the difference between the effect of the dead time on the time spectrum and the time spectrum without the effect of dead time. The time spectrum at a high voltage of 70 kV is relatively more affected by dead time, which causes minimal differences in the two neutron time spectra, ultimately resulting in inaccurate findings. Therefore, according to the size of the f_d parameter of each group, the selection of the height must be constrained. That is, the f_d should be greater than 2 as far as possible.

Fig. 6Full-screen View

(Color online) (a) Dead time comparison for different window widths at four window heights.(b) Changes in dead time with gradually increasing spacing between two time windows for different values of f_d

The effect of two-window spacing t_wd on dead-time calculations

Here, t_wd denotes the time interval between W1 and W2. The initiation channel of W1 was set to 400 μs to mitigate the effect of statistical fluctuations in the data when the count rate was extremely low. The channel of W2 was incremented from 400 μs to 1000 μs in steps of 10 μs, with the t_ww fixed at 8 μs. Figure 6b elucidates the corresponding variation in dead time for two window intervals ranging from 0 μs to 600 μs. These results are consistent with those previously mentioned. Moreover, the distance between two windows should not be extremely small. Notably, the dead time tended to stabilize when the interval between the two windows exceeded 200 μs. Table 5 lists the average values of the dead time in the stable interval. The determined dead time of the ³He detector system aligns closely with the ³He tube dead time calculated by Hashimoto using the variance-mean method (Feynman-α).

Correction of neutron time spectrum and MCNP validation

Under identical experimental conditions, the neutron decay process is expressed as follows: $N=N_0{e}^{-\frac{t}{\mu }},$ (6) where N₀ is the neutron density after a delay time of T₀ after a single neutron emission and the delay time of T₀ eliminates the influence of the aperture effect such that the decay of neutrons is only affected by a single stratum. Additionally, t is the beginning time of the T₀ moment, and μ is the neutron decay constant in microseconds.

Then, m can be written as: $\{\begin{array}{c}{m}_{t1}=N_1{e}^{-\frac{t1}{\mu 1}}\\ m_{t2}=N_2{e}^{-\frac{t2}{\mu 2}}\\ m_{t3}=N_1{e}^{-\frac{t3}{\mu 1}}\\ m_{t4}=N_2{e}^{-\frac{t4}{\mu 2}}\end{array}$ (7) It follows from Eq. (3) that $\frac{m_{t1}}{m_{t2}}=\frac{m_{t3}}{m_{t4}}\stackrel{\text{equivalence}}{\Rightarrow }\frac{m_{t1}}{m_{t3}}=\frac{m_{t2}}{m_{t4}}.$ (8) Combining Eqs. (7) and (8), the equation can be written as: $e^{-\frac{t1-t3}{\mu 1}}=e^{-\frac{t2-t4}{\mu 2}}.$ (9) Here, t1-t3=t2-t4. Thus, μ1=μ2, and the attenuation coefficients of our two measurements for different numbers of neutrons should be equal.

Based on the relationships expressed in Eq. (5) between the actual counts, observed counts, and dead time, a detailed correction of the observed counts was performed using the previously calculated dead-time values. Figures 7(a–d) illustrate the correction effects. Notably, the dead time correction result for Height IV exhibits a negative outlier, according to the dead time formula and related theories, Hence, the dead time results do not match the theoretical results [45]. This theoretical derivation implies that the neutron attenuation coefficients with varying fluxes should be uniform [46]. Accordingly, a negative exponential fit was employed, and the fitting equations are shown in Eq. (10) on the linear decay curves in linear coordinates. This fit was conducted both before and after dead-time correction, considering different heights. The fitting range was chosen as 400 μs–2000 μs to eliminate the influences of source neutrons [47].

Fig. 7Full-screen View

(Color online) (a) Height 1 dead time correction results. (b) Height 2 dead time correction results. (c) Height 3 dead time correction results.(d) Height 4 dead time correction results

Figures 8(a-d) present the fitting results, where Fig. 8(a) illustrates the results before correction and Figs. 8(b–d) illustrate the fitting outcomes after dead-time correction for heights I-III. Moreover: $y=N_0{e}^{-\frac{t}{\mu }}+y_0.$ (10) The addition of y₀ to Eq. (10) represents the background noise.

Fig. 8Full-screen View

(Color online) (a) Fitting results before correction. (b) Fitting results after the dead time correction for Height 1. (c) Fitting results after the dead time correction for Height 2. (d) Fitting results after the dead time correction for Height 3

The attenuation coefficient of the time spectrum fit for the MCNP simulation is 237.5±0.8, and the precision of the correction effect is quantified by computing the relative deviation (RD) of the attenuation coefficients of the neutron time spectrum before and after correction from the MCNP simulation, as indicated in Eq. (11) [48, 49]: $RD=\frac{\left|X_{\text{av}}-X_{\text{mc}}\right|}{X_{\text{mc}}}\times 100\%$ (11) where X_av is the value of μ before and after dead-time correction and X_mc is the value of μ_mc for MCNP simulation.

In the data presented in Table 6, it is evident that the decay coefficients of each curve closely align after correction, aligning with the conclusion that the deduced decay coefficients are consistent. Comparing the pre-correction and post-correction attenuation coefficients with the MCNP simulation results, it is apparent that height I, height II, and height III exhibit favorable correction effects. The RD is 12.4% before correction, reduced to approximately 5% after correction.

Application in PFNUL

PFNUL instrument is used for pulsed neutron uranium quantitative logging, using pulsed neutrons to trigger the fission reaction of uranium (²³⁵U) element in the formation [50-52], detecting and extracting the transient neutron time spectrum generated by the fission of uranium, and then realizing the rapid detection of uranium in drilling holes and uranium quantification. Especially for the method technology of uranium fission information based on the ratio of Epithermal neutrons to Thermal neutrons with a double neutron time spectrum, the accuracy of the detection of thermal neutrons will have a direct impact on the accuracy of uranium quantification [53-56]. Finally, the corrected method and the dead time results were made in the uranium fission transient neutron logging instrument, as shown in Fig. 9, for the collection of two different sources under high target pressure in the same uranium model well with a neutron generator under controlled target pressure. The neutron time spectrum under neutron flux are original spectrum 1 and original spectrum 2, respectively, and it can be seen that the count rate of the original neutron time spectrum 1 and the original neutron time spectrum 2 before correction is low due to the dead time problem, for a period of time at the end of the pulse, since the neutron count rate is high at this time, again the effect of dead time is the largest, so a significant correction effect of dead time can be observed for this period of time, and as the neutron count rate decreases, the effect of dead time also decreases, so the correction effect of dead time decreases with the decrease of the neutron count rate. According to Eq. (10), the attenuation coefficient μ* is fitted to the neutron time spectrum before correction and the neutron time spectrum after correction, and the fitting results are shown in Table 7, which shows that the attenuation coefficients μ* of the two neutron time spectrums have a big difference due to the dead time before correction, and the attenuation coefficients of the two curves are close to the same after correction. According to Eq. (11), the two-time spectrum's μ* before and after correction were computed and compared. The results indicated a significant reduction in RD from 29% to 0.49%, which conforms to the theory of neutron attenuation, and it proves that this correction method has a better effect on the correction of the neutron time spectrum of the PFNUL instrument. It proves that this correction method has a good effect on the neutron time spectrum acquisition correction of the PFNUL instrument, which can greatly improve the quantitative measurement accuracy of the uranium logging site.

Fig. 9Full-screen View

(Color online) Experimental time-spectrum correction results for PFNUL in model wells