# A New Temperature Correction Method for NaI(Tl) Detectors Based on Pulse Deconvolution

^{*}

## Abstract

**:**

^{137}Cs 662 keV peak was less than 3 keV, and the corresponding resolution at 662 keV of the sum spectra ranged from 6.91% to 10.60% with the trapezoidal width set from 1000 ns to 100 ns. The DTSAC method corrects the temperature effect via pulse processing, and needs no reference peak, reference spectrum or additional circuits. The method solves the problem of correction of pulse shape and pulse amplitude at the same time, and can be used even at a high counting rate.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Method Description

#### 2.1.1. Pulse Model and Deconvolution

_{i}and λ

_{j}are the decay constants of the exponential functions, and φ

_{i}and ϕ

_{j}are the proportions of each exponential components (calculated by area), which satisfy

_{1}(t), is a Gaussian function. The other two, f

_{2}(t) and f

_{3}(t), are the sum of several exponential functions. f

_{1}(t), f

_{2}(t) and f

_{3}(t) are all normalized functions of area instead of amplitude. The commonly used uni-exponential model, bi-exponential model, Gauss-exponent convolution model and others can be regarded as special versions of this model. Due to the influence of temperature, even for the same energy deposition, the parameters A, σ, τ

_{i}, λ

_{j},φ

_{i}and ϕ

_{j}are all slowly varying functions of temperature, T. Model parameters at specific temperature can be obtained by fitting the average pulse. The relationship between model parameters and temperature can be established via function fitting or interpolation.

_{s}is the sampling period. As developed in in our previous work [16], the original pulse, f(t), can be deconvolved into a narrow Gaussian pulse, A∙f

_{1}(t), with an invariant area, and its z-domain expression is

_{k}is the polynomial coefficient determined only by model parameters τ

_{i}, λ

_{j}, φ

_{i}and ϕ

_{j}. When M = 2, N = 1, δ is very close to 0, and a simplified model such as (7) can be obtained. It is the convolution between an exponent function and the sum of two exponent functions.

_{0}is the rising time constant, τ

_{1}and τ

_{2}represent the time constant of the fast component and the slow component, respectively, and φ is the proportion of the fast component.

#### 2.1.2. Trapezoidal Shaping and Amplitude Correction

_{a}and n

_{b}are trapezoidal parameters, and t

_{Top}= (n

_{b}− n

_{a})T

_{s}and t

_{Width}= (n

_{b}+ n

_{a})T

_{s}are the time width of the flat-top and bottom, respectively. In order to avoid a ballistic deficit caused by σ(T), it is required that t

_{Top}> 6σ

_{max}, which can be easily met since σ is generally about 10 ns. Trapezoidal parameters can be selected according to the pulse count rate to achieve the balance between pulse throughput and energy resolution.

_{T}, of the trapezoidal pulse will be equal to the original pulse area, A. Considering the inaccuracy of the pulse model, the model error correction factor, κ, is introduced to make κH

_{T}= A, and κ is a function of temperature and trapezoidal parameters, denoted as κ(T,n

_{a},n

_{b}). The more accurate the model is, the closer to 1 the value of κ would be. The area, A, of the original pulse for γ-rays with specific energy is mainly determined by the light yield of the scintillator and the PMT gain, both of which are affected by temperature [12]. The area of the original pulse at a temperature, T, is denoted as A(T), and the one at reference temperature, T

_{0}, is denoted as A

_{0}. Equation (12) can be obtained while introducing the area correction factor ε(T) = A

_{0}/A(T).

_{a},n

_{b}) = ε(T)κ(T,n

_{a},n

_{b}) is the amplitude correction factor. The corrected trapezoidal height is equal to the area of the original pulse at the reference temperature, which is independent of temperature, so as to achieve spectrum stabilization. The correction result calculated by ε has some deviation due to the model error, while the one calculated by ε′ is more accurate. However, ε is easy to use, since it is independent of trapezoidal parameters, while ε′ should be calculated according to trapezoidal parameters.

#### 2.2. Pulse Data Acquisition and Preprocessing

^{137}Cs point source was located near to its front end. The experiments were carried out in a KOWINTEST KW-TH-100X-PC thermostatic chamber (Guangdong Kewen Test Equipment Co., Dongwan, Guangdong, China; temperature control range: −60~150 °C; temperature control accuracy: ±0.5 °C). The pulses were collected at several temperature points from −20 °C to 50 °C with a step of 10 °C. The temperature changing rate was controlled within 5 °C/h in order to protect the NaI(Tl) crystal. Each temperature point was kept constant for 8 h to ensure thermal balance, and the pulses were collected in the last 30 min using a CAEN N6730 digitizer [18]. The pulse data recording length was 8 μs and 1 μs before triggering was reserved as its baseline. The baseline of each pulse was estimated and then subtracted from the recorded pulse. After discarding the data with pulses truncated by the digitizer because of the excessive amplitude and the data with multiple pulses within the acquisition time, 200,000 single pulses were obtained at each temperature point. Pulses from the 662 keV photoelectric peak were picked out according to the pulse area.

## 3. Results

#### 3.1. Pulse Model Parameter Fit Result

_{0}, the pulse model function f(t) should be modified to f(t − t

_{0}), where t

_{0}is also treated as a fit parameter. In order to make the area of the deconvolution pulse equal to that of the original pulse, the pulse area parameter, A, was set as the value of average pulse area, which means that A does not need to be determined by fitting. The fit results of the first two models were not sensitive to initial values due to the few fit parameters, so the Levenberg–Marquardt algorithm with roughly selected initial values was used for fitting. The fit results of the last two models were sensitive to the initial values due to there being too many fit parameters. An iterative process of alternating optimization was used, and the fit results of the first two models were adopted as the initial values. The R

^{2}values fitted by each model are shown in Figure 3. It can be seen that the exponent-double exponent sum convolution model is the best, with its R

^{2}values ranging from 0.9980 to 0.9994. The fit parameters are shown in Table 1.

#### 3.2. Temperature Correction Result

_{Width}.

^{2}values are all above 0.9993.

_{Width}= 100~1800 ns and a flat-top ratio of 0~0.9 were used for trapezoidal shaping. The energy resolutions of the 662 keV peak are shown in Figure 8.

_{Width}. However, even if t

_{Width}is reduced to 100 ns when ε′ is used for amplitude correction, the energy resolution worsens only to 10.6% (t

_{Top}= 10 ns). Thus, this temperature correction method can work well for a high count rate.

## 4. Discussion

^{2}values of the exponent-double exponent sum convolution models fitted to the average pulse at different temperatures were both greater than 0.9980, showing the best performance among the compared models. However, there was a certain deviation between the model and the real response. The high R

^{2}values were partly due to the smoothness of the average pulses, and the fitting residual mainly reflected the model error. In fact, it can be seen in Figure 5 that there are certain differences between ε and ε′ for different trapezoidal parameters, which is caused by the model error. The actual scintillator luminescence process is very complicated [19,20,21], and is difficult to describe with mathematical functions. The pulse models used in practice are models that are simplified for practical purposes. In principle, a more complex model can be used to improve the accuracy of fitting, but it will increase the difficulty of fitting and subsequent deconvolution. From the practical point of view, the complexity and accuracy of the model should be considered comprehensively. In this paper, the exponent-double exponent sum convolution model was selected because of its moderate complexity. Although there was some deviation between the pulse model and the actual response, the consistency of pulse shapes after deconvolution and trapezoidal shaping at different temperatures shown in Figure 4 reflects the effectiveness of this method in pulse shape correction. Compared with previous temperature correction methods [5,6,7,8,9,10,11,12], this method can be used at a high count rate due to the correction of pulse shape and adjustable trapezoid width.

_{Width}= 1000 ns and t

_{Top}= 500 ns had a resolution of 6.91%@662 keV(ε′), which verifies the spectral stabilization effectiveness of the proposed method. It should be noted that this result was obtained under ideal conditions: firstly, the pulse model construction and spectral drift evaluation were all aimed at the 662 keV gamma rays; secondly, constant temperature conditions in the experiment ensured the consistency of the temperature in the whole NaI(Tl) crystal and PMT; thirdly, the temperature values used for calculating the model parameters were the same as those used for pulse processing, which prevented additional errors introduced by the model parameter estimation using temperature.

_{Width}, and flat-top ratio, which is because trapezoid parameters affect the SNR of the trapezoid pulse. Figure 8 can provide a basic reference for selecting trapezoidal parameters at a specific count rate. The energy resolution deteriorates with the decrease in t

_{Width}, which can be attributed to the fact that the fewer data points are used to calculate the pulse amplitude information with the narrower trapezoidal width. Therefore, a relatively wide t

_{Width}value should be adopted at a low count rate, and a smaller t

_{Width}value should be adopted with the increase in the count rate in order to reduce the pile-up effect of the trapezoidal pulse. When the t

_{Width}is small, the energy resolution becomes better with the decrease in the flat-top ratio. Therefore, it is advisable to adopt a smaller flat-top ratio at a high count rate. When using ε′ for amplitude correction, even if the t

_{Width}is reduced to 100 ns, an energy resolution of 10.60% can still be obtained by adjusting the flat-top ratio, which preliminarily verifies the effectiveness of this method at a high count rate. It should be noted that, although the pulse pile-up effect can be reduced by using a smaller t

_{Width}at a high count rate, the statistical fluctuations inherent in the pulse signals cannot be eliminated, which is reflected in the random fluctuation in the baseline on the right side next to the trapezoidal pulse in Figure 4c,d, and it will reduce the SNR of the trapezoid pulse. Therefore, the actual energy resolution at a high count rate will be worse than that shown in Figure 8.

^{137}Cs 662 keV was verified. The consistency of the Compton edge in the energy spectra at different temperatures in Figure 6d can provide partial verification for the validity of the spectrum stabilization at other energy points. Due to the nonlinear energy response of the NaI(Tl) crystal [20,21], the hypothesis that the pulse shape is independent of energy cannot be perfectly confirmed in practice, so the effectiveness of this method with other energy points needs to be further verified.

## 5. Conclusions

_{Width}and t

_{Top}, and the amplitude H

_{T}of the trapezoidal pulse can be corrected to be equal to the pulse area, A

_{0}, at the reference temperature (20 °C). Compared to the existing spectrum stabilization methods based on adjusting the PMT’s high voltage or amplifier gain, the proposed method process scintillation pulses rather than the energy spectrum. This method can be used for field applications, including situations in which there is a high counting rate. Through adjusting the trapezoidal parameters, an acceptable energy resolution can be obtained. The results show that the proposed method can effectively correct the temperature response of NaI(Tl) detectors, and the position of the 662 keV peak changes by less than 3 keV with a temperature ranging from −20 °C to 50 °C. The trapezoidal parameters can affect the final energy resolution, but even when the trapezoidal pulse width is reduced to 100 ns, an energy resolution of 10.60% can still be ensured.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Gilmore, G. Scintillation Spectrometry. In Practical Gamma-Ray Spectroscopy, 2nd ed.; John Wiley & Sons Nuclear: Warrington, UK, 2008; p. 207. [Google Scholar]
- Schweitzer, J.S.; Ziehl, W. Temperature dependence of NaI(Tl) decay constant. IEEE Trans. Nucl. Sci.
**1983**, 30, 380–382. [Google Scholar] [CrossRef] - Moszyński, M.; Nassalski, A.; Syntfeld-Każuch, A.; Szczęśniak, T.; Czarnacki, W.; Wolski, D.; Stein, J. Temperature dependences of LaBr
_{3}(Ce), LaCl_{3}(Ce) and NaI(Tl) scintillators. Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrometers Detect. Assoc. Equip.**2006**, 568, 739–751. [Google Scholar] [CrossRef] - Knoll, G.F. Inorganic Scintilltors. In Radiation Detection and Measurement, 4th ed.; John Wiley & Sons: Hoboken, NJ, USA, 2010; p. 241. [Google Scholar]
- Dudley, R.A.; Scarpatetti, R. Ṡtabilization of a gamma scintillator spectrometer against zero and gain drifts. Nucl. Instrum. Methods
**1963**, 25, 297–313. [Google Scholar] [CrossRef] - Patwardhan, P.K. Spectrum stabilization with variable reactance gain control. Nucl. Instrum. Methods
**1964**, 31, 169–172. [Google Scholar] [CrossRef] - Stromswold, D.C.; Meisner, J.E. Gamma-ray spectrum stabilization in a borehole probe using a light emitting diode. IEEE Trans. Nucl. Sci.
**1979**, 26, 395–397. [Google Scholar] [CrossRef] - Tsankov, L.T.; Mitev, M.G. A simple method for stabilization of arbitrary spectra. In Proceedings of the Sixteen International Scientific and Applied Science Conference ELECTRONICS ET, Sozopol, Bulgaria, 19–21 September 2007; pp. 3–8. [Google Scholar]
- Chen, Y.; Li, J.; Zhang, Y.; Xiao, W. Gamma spectrum stabilization method based on nonlinear least squares optimization. Appl. Radiat. Isot.
**2021**, 169, 109515. [Google Scholar] [CrossRef] [PubMed] - Pommé, S.; Sibbens, G. Concept for an off-line gain stabilisation method. Appl. Radiat. Isot.
**2004**, 60, 151–154. [Google Scholar] [CrossRef] [PubMed] - Casanovas, R.; Morant, J.J.; Salvadó, M. Temperature peak-shift correction methods for NaI(Tl) and LaBr
_{3}(Ce) gamma-ray spectrum stabilisation. Radiat. Meas.**2012**, 47, 588–595. [Google Scholar] [CrossRef] - Pausch, G.; Stein, J.; Kreuels, A.; Lueck, F.; Teofilov, N. Multifunctional application of pulse width analysis in a LED-stabilized digital NaI(Tl) gamma spectrometer. In Proceedings of the IEEE Nuclear Science Symposium Conference Record, Fajardo, PR, USA, 23–29 October 2005. [Google Scholar]
- Stein, J.; Neuer, M.J.; Herbach, C.M.; Pausch, G.; Ruhnau, K. Radiation detector signal processing using sampling kernels without bandlimiting constraints. In Proceedings of the 2007 IEEE Nuclear Science Symposium Conference Record, Honolulu, HI, USA, 26 October–3 November 2007. [Google Scholar]
- Xiao, W.; Farsoni, A.T.; Yang, H.; Hamby, D.H. A new pulse model for NaI(Tl) detection systems. Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrometers Detect. Assoc. Equip.
**2014**, 763, 170–173. [Google Scholar] [CrossRef] - Radeka, V. Trapezoidal filtering of signals from large germanium detectors at high rates. Nucl. Instrum. Methods
**1972**, 99, 525–539. [Google Scholar] [CrossRef] - Xiao, W.; Farsoni, A.T.; Yang, H.; Hamby, D.M. Model-based pulse deconvolution method for NaI(Tl) detectors. Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrometers Detect. Assoc. Equip.
**2015**, 769, 5–8. [Google Scholar] [CrossRef] - Photomultiplier CR173 with an End Window. Available online: http://www.bhphoton.com/upload/1/content/1662094262625.pdf (accessed on 9 May 2023). (In Chinese).
- N6730/N6725 8-Channel 14-bit 500/250 MS/s Waveform Digitizer. Available online: https://www.caen.it/?downloadfile=7499 (accessed on 9 May 2023).
- Alexandrov, B.S.; Ianakiev, K.D.; Littlewood, P.B. Branching transport model of NaI (Tl) alkali-halide scintillator. Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrometers Detect. Assoc. Equip.
**2008**, 586, 432–438. [Google Scholar] [CrossRef] - Payne, S.A.; Moses, W.W.; Sheets, S.; Ahle, L.; Cherepy, N.J.; Choong, W.S. Nonproportionality of scintillator detectors: Theory and experiment. II. IEEE Trans. Nucl. Sci.
**2011**, 58, 3392–3402. [Google Scholar] [CrossRef] - Khodyuk, I.V.; Dorenbos, P. Trends and patterns of scintillator nonproportionality. IEEE Trans. Nucl. Sci.
**2012**, 59, 3320–3331. [Google Scholar] [CrossRef]

**Figure 4.**Actual pulses from 662 keV at different temperatures and their processing results. (

**a**) Original pulse; (

**b**) deconvolution pulse; (

**c**) trapezoidal pulse (t

_{Width}= 200 ns, t

_{Top}= 100 ns); (

**d**) trapezoidal pulse (t

_{Width}= 1000 ns, t

_{Top}= 500 ns).

**Figure 5.**Amplitude correction factors at different temperatures. Trapezoid parameter 1: t

_{Width}= 500 ns, t

_{Top}= 100 ns; trapezoid parameter 2: t

_{Width}= 500 ns, t

_{Top}= 250 ns; trapezoid parameter 3: t

_{Width}= 1000 ns, t

_{Top}= 200 ns; trapezoid parameter 4: t

_{Width}= 1000 ns, t

_{Top}= 500 ns.

**Figure 6.**Spectra obtained with different methods at different temperatures. (

**a**) Gated integration at 1000 ns; (

**b**) gated integration at 7000 ns; (

**c**) the proposed method (t

_{Width}= 1000 ns, t

_{Top}= 500 ns, ε); (

**d**) the proposed method (t

_{Width}= 1000 ns, t

_{Top}= 500 ns, ε′).

**Figure 7.**Sum spectra obtained via different processing methods. (

**a**) Gated integration at 1000 ns; (

**b**) gated integration at 7000 ns; (

**c**) the proposed method (t

_{Width}= 1000 ns, t

_{Top}= 500 ns, ε); (

**d**) the proposed method (t

_{Width}= 1000 ns, t

_{Top}= 500 ns, ε′).

**Figure 8.**The 662 keV resolution of sum spectra with different correction factors and trapezoidal parameters. (

**a**) Amplitude corrected by ε; (

**b**) amplitude corrected by ε′.

Temperature/°C | A | τ_{0}/ns | τ_{1}/ns | τ_{2}/ns | φ | t_{0}/ns |
---|---|---|---|---|---|---|

−20 | 3.382 × 10^{5} | 9.784 | 160.8 | 842.9 | 0.2682 | 933.3 |

−10 | 3.302 × 10^{5} | 9.343 | 218.8 | 603.0 | 0.3202 | 933.3 |

0 | 3.194 × 10^{5} | 11.67 | 271.2 | 404.9 | 0.3778 | 937.4 |

10 | 3.059 × 10^{5} | 18.05 | 281.1 | 281.3 | 0.8297 | 930.6 |

20 | 2.935 × 10^{5} | 23.05 | 236.2 | 236.4 | 0.8120 | 929.4 |

30 | 2.785 × 10^{5} | 25.34 | 199.7 | 200.0 | 0.9993 | 929.3 |

40 | 2.625 × 10^{5} | 25.24 | 174.0 | 200.0 | 1.000 | 929.4 |

50 | 2.453 × 10^{5} | 23.93 | 155.9 | 200.0 | 1.000 | 929.7 |

a_{0} | a_{1} | a_{2} | R^{2} | ||
---|---|---|---|---|---|

ε | −1.537 × 10^{−5} | −4.398 × 10^{−3} | 1.095 | 0.9993 | |

ε′ | t_{Width} = 500 ns, t_{Top} = 100 ns | −2.928 × 10^{−5} | −3.335 × 10^{−3} | 1.068 | 0.9997 |

t_{Width} = 500 ns, t_{Top} = 250 ns | −4.033 × 10^{−5} | −2.900 × 10^{−3} | 1.078 | 0.9993 | |

t_{Width} = 1000 ns, t_{Top} = 200 ns | −3.214 × 10^{−5} | −3.345 × 10^{−3} | 1.077 | 0.9998 | |

t_{Width} = 1000 ns, t_{Top} = 500 ns | −2.861 × 10^{−5} | −3.563 × 10^{−3} | 1.078 | 0.9998 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Xie, J.; Yang, L.; Li, J.; Qi, S.; Chen, W.; Xu, H.; Xiao, W.
A New Temperature Correction Method for NaI(Tl) Detectors Based on Pulse Deconvolution. *Sensors* **2023**, *23*, 5083.
https://doi.org/10.3390/s23115083

**AMA Style**

Xie J, Yang L, Li J, Qi S, Chen W, Xu H, Xiao W.
A New Temperature Correction Method for NaI(Tl) Detectors Based on Pulse Deconvolution. *Sensors*. 2023; 23(11):5083.
https://doi.org/10.3390/s23115083

**Chicago/Turabian Style**

Xie, Jianming, Liu Yang, Jinglun Li, Sheng Qi, Wenzhuo Chen, Hang Xu, and Wuyun Xiao.
2023. "A New Temperature Correction Method for NaI(Tl) Detectors Based on Pulse Deconvolution" *Sensors* 23, no. 11: 5083.
https://doi.org/10.3390/s23115083