Energy Calibration and Inverse-Square Law of Radiation in Gamma-Ray Measurement Using a Scintillation Detector for Laboratory Experiments

Over the past two years, third-year undergraduate students from Toho University’s Department of Physics conducted laboratory experiments on gamma-ray measurement using a scintillation detector. Approximately, 30 experimental data points were collected and subsequently analyzed. Te analysis focused on the energy calibration method and the inverse-square law of radiation. Results revealed that employing quadratic or cubic function fts for energy calibration yielded more than twice the accuracy compared to the conventional linear function ft. Regarding the deviation from the inverse-square law of radiation, a correction method utilizing a correction parameter was compared with a power function ft method. Te discussion encompassed the correction parameter and the exponent of the power function.


Introduction
Gamma-ray measurement holds signifcance in nuclear physics as it pertains to fundamental physics [1][2][3] and radiation measurement [4][5][6][7][8][9].Te Fukushima Daiichi Nuclear Power Plant accident, triggered by the Great East Japan Earthquake of March 11, 2011, led to the release of radioactive materials.Consequently, a substantial area, centered on Fukushima Prefecture, remains contaminated.Tis incident has heightened the interest in the radiation's fundamental knowledge and measurement technology [10][11][12][13].Measurement of the relationship between gammaray intensity and distance is essential for radiation protection, while energy spectrum measurement is vital for radiation object identifcation [14].
Te scintillation detector presents several advantageous characteristics, including afordability, stability, and userfriendliness [15].It serves as a fundamental instrument for gamma-ray measurement and fnds applications in laboratory experiments across numerous universities [16][17][18][19][20][21][22].Trough these experiments, students not only acquire gamma-ray measurement techniques but also gain insights into the interaction between gamma-ray and matter.Toho University's Department of Physics has conducted laboratory experiments on gamma-ray measurement using the scintillation detector for several years, specifcally targeting third-year undergraduates [23].Since these experiments are conducted annually, it is suitable for the radioactive source [24,25] to possess a half-life of several years or more.Accordingly, Toho University employs three radioactive sources: 137 Cs, 60 Co, and 22 Na.Within the laboratory experiment, measurements of gamma-ray spectrum and intensity with distance were conducted.Spectral measurements [26] involved obtaining energies of positron annihilation peaks from 22 Na through energy calibration using a linear function ft.Te photopeaks at 662 keV of 137 Cs, 1173 and 1333 keV of 60 Co, and 1275 keV of 22 Na were utilized [7,8,27].Te efciency of the NaI (Tl) detector has been extensively studied for various radioactive sources by the Alexandria University research group [28][29][30][31][32][33][34][35][36][37].However, the experimental positron annihilation peak energy of 22 Na consistently exhibited a deviation of more than 10% from the theoretical value.In the intensity measurement with distance, the deviation from the inverse-square law of radiation intensity [38][39][40] was analyzed by introducing a correction parameter.However, the utilization of this correction parameter complicated students' understanding of the inversesquare law of radiation.Furthermore, neither the comprehension of the correction parameter itself nor the recommended value for it is currently known.
Tis study aims to collect approximately 30 experimental data points obtained by students during laboratory experiments conducted in the third-year of Toho University's Department of Physics.Te collected data will be analyzed to assess the deviation in conventional energy calibration and the deviation from the inverse-square law of radiation.Improved fts for energy calibration and recommended parameters for the inverse-square law of radiation are examined and discussed.

Experimental Setup and Methods
Te experiments utilized three radioactive point sources: 137 Cs, 60 Co, and 22 Na. Figure 1(a) illustrates a typical sealed radioactive point source of 60 Co, including its container.Te scintillation detector used (OKEN SP-20) is shown in Figure 1(b), which consists of a 2-inch NaI (Tl) scintillator, a photomultiplier tube, and a preamplifer.Figure 1(c) depicts the arrangement during measurements, with the source and scintillation detector positioned on a wooden straight rail within a box, ensuring they remain in line and facilitating accurate distance measurements.
Te complete experimental setup is presented in Figure 1(d).Te photomultiplier tube received a voltage of 500-700 V from a high voltage supply (OKEN 714-1C).Te output from the scintillation detector was amplifed using a linear amplifer (OKEN 704-4B).Te radioactivity intensity, or gamma-ray count, was measured with a scaler (OKEN 711-6), while the gamma-ray spectrum was measured using a multichannel analyzer (MCA) (HOSHIN ELECTRONICS HE1442).
Figure 2 displays the decay schemes of the three sources [7,8,27]. 137Cs has a half-life of 30.0 years and emits a 662 keV gamma-ray through β − decay. 60Co has a half-life of 5.27 years and emits two gamma rays of 1173 and 1333 keV through β − decay. 22Na has a half-life of 2.60 years and emits a 1275 keV gamma-ray through β + decay.Te characteristics and long half-lives of the three sources of 137 Cs, 60 Co, and 22 Na render them highly suitable for laboratory experiments.Te initial intensities used for these sources in this experiment were 3.7 × 10 5 Bq.

Results and Discussion
3.1.Energy Calibration. Figure 3 presents typical energy spectra of 137 Cs, 60 Co, and 22 Na, as measured by a student.In the 137 Cs spectrum, the full width at half maximum of the 662 keV photopeak is approximately 80 keV.Tis resolution sufces for laboratory experiments, as the double peaks of 60 Co are clearly distinguished.Te backscatter peak was frequently observed in the 137 Cs spectrum but rarely in the 60 Co and 22 Na spectra.
During laboratory experiments, the Gaussian distribution photopeaks at 662 keV of 137 Cs, 1173 and 1333 keV of 60 Co, and 1275 keV of 22 Na were employed for energy calibration of the MCA channels.A least-squares ft with a linear function was used to establish the relationship between channel and energy by measuring the central channel numbers of the aforementioned four photopeaks.Figure 4 displays a typical linear function ft, revealing a substantial intercept.By using the ftted linear function, the experimental energy value E e ex of the positron annihilation peak from 22 Na was determined.However, the obtained experimental value E e ex consistently deviated by more than 10% from the theoretical value E e th = 0.511 MeV.Considering that the MCA ofset is typically adjusted to pass through the origin, this study examined the deviation of the experimental value E e ex by ftting the data using a linear function, a quadratic function, and a cubic function, while considering the origin.Figure 4 illustrates the typical lines resulting from least-squares fts to the experimental data.Te conventional linear function ft without considering the origin (L1) is compared with the linear function ft considering the origin (L10), the quadratic function ft considering the origin (L20), and the cubic function ft considering the origin (L30).
Te energy calibration line was evaluated using the gammaray photopeak from 22 Na positron annihilation, and the energy value E e ex was obtained from the central channel number of the peak using the calibration line.By comparing the experimental value E e ex with the theoretical value E e th , a relative deviation S was calculated using the following formula: All three functions, along with the conventional method shown in Figure 4, were ftted to the data collected by thirdyear students over the past two years, and the relative deviation S was determined.Figure 5 presents the distribution of S for the collected data.Te range of S is observed to be 9%-13% for L1 and L10, and 2%-6% for L20 and L30.Te average values of S are as follows: S � 10.5 (2)% for L1, S � 11.6 (2)% for L10, S � 4.7 (2)% for L20, and S � 3.6 (3)% for L30.It is noteworthy that the quadratic function (L20) and cubic function (L30) fts yielded more than a twofold improvement in the relative deviation.

Inverse-Square Law of Radiation.
In the experiment, counts were recorded by varying the distance R from the detector to the radioactive source, ranging from 2 to 48 cm.By subtracting the background counts and considering only counts from the radioactive source, the count rate K was determined.Figure 6 presents typical data showing the count rate K as a function of distance R, as measured by the students.Te experimental data were ftted with an inverse-square function (equation ( 2)) that includes a constant term K 0 : 2 Journal of Engineering Two separate fttings were performed: one using data from 2 to 48 cm and another using data from 12 to 48 cm.Te ftted lines are plotted in Figure 6.It is evident from Figure 6 that the data from 2 to 48 cm deviate signifcantly from the inversesquare function.However, the data from 12 to 48 cm exhibit a closer alignment with the inverse-square function, although some deviation remains.Similar trends are observed for 137 Cs and 60 Co sources.Tis suggests that gamma-ray intensity does not strictly follow the inverse-square law within a distance of 10 cm, and particularly at short distances of a few cm, it deviates considerably from the inverse-square law.
Dead time of the detector may be one of the reasons for this deviation from the inverse-square law in Figure 6.Infuence of dead time will become strong with loss of count for strong count rate. 137Cs and 60 Co have a large diference of count rate: count rate of 137 Cs is about one order of magnitude larger than that of 60 Co.Terefore, the infuence of dead time, i.e., possibly induced deviation from the inverse-square law, should be strong for 137 Cs.Since the dead time in our detector has an order of magnitude of μs, the infuence on count, i.e., loss of count, can be neglected for present lower count rate (∼10 3 cps) as shown in Figure 6.For 60 Co, the coincidence summing efect due to the cascade gamma-rays of 1173 and 1333 keV may be another reason for the deviation from the inverse-square law.Te coincidence summing efect is dependent on the detector type and geometry and becomes strong at close distance between the source and detector.However, since the total counts from the source, rather than the photopeak, were measured in this experiment using a scaler infuence from the coincidence summing efect should be greatly reduced.
To check the infuence of the dead time and coincidence summing efect on deviation from the inverse-square law, the relative deviation K D of the experimental count rate (K exp ) from the ftted line of equation ( 2 Results of K D calculated from Figure 6 for experimental data of K from 2 to 48 cm are shown in Figure 7.It can be seen from Figure 7 that K D of 137 Cs agrees well with that of 60 Co with a same trend.Tis shows that infuence of the dead time and coincidence summing efect on the deviation from the inverse-square law is negligibly small.Terefore, detailed corrections of the dead time and coincidence summing efect were not performed in this experiment.Tese possible infuences, very weak as shown in Figure 7, can be included in the correction method using a parameter shown as follows. Te main reason for the deviation from the inversesquare law is considered to be from the measured distance.Te measured distance R from the center of the source to the front surface of the detector difers from the actual distance (R + a) from the emitted point of gamma rays to the measured point.Here, a mainly accounts for this diference between the measured and actual distances.To correct the deviation from the inverse-square law in laboratory experiments, a correction parameter a was introduced, and the data were ftted with a modifed inverse-square function (equation ( 4)) that includes a constant term K 1 : Figure 8 illustrates a typical example of ftting the modifed inverse-square function to the data from 2 to 48 cm, demonstrating suitable agreement between the ftted function and the experimental data for both 137 Cs and 60 Co sources.Te recommended values for the correction parameter a were obtained by collecting data taken by students over the past two years, from 2 to 48 cm. Figure 9 presents the distribution of the correction parameter a.As observed, the values of a for 137 Cs and 60 Co sources are distributed in the range of 3-5 cm.Te average values are determined as follows: a � 3.91(7) cm for 137 Cs, a � 3.81(9) cm for 60 Co.

(5)
Te obtained values of a for 137 Cs and 60 Co sources agree within experimental uncertainties, indicating that the efect of the diference in sources on the parameter a is negligibly small.Additionally, the obtained a values of approximately 4 cm are reasonable for the 2-inch NaI (Tl) scintillator used in the experiment.
In this study, a direct ftting approach was applied using a power function (equation ( 5)) without incorporating the correction parameter.
Here, b is the exponent of the power function and K 2 is a constant.Figure 10 displays typical results of the leastsquares fts using the power function for the 137 Cs source.
It is notable that the obtained values of b for 137 Cs and 60 Co agree within experimental uncertainties.Tis suggests that the exponent b, similar to the correction parameter a, is independent of the radioactive source.

Conclusions
In this study, the focus was on investigating energy calibration and the inverse-square law in gamma-ray measurements using data collected by students in laboratory experiments.
Energy calibration was examined by comparing diferent ftting functions for the gamma-ray photopeak of 22 Na positron annihilation.Te conventional linear function ft was compared with quadratic and cubic function fts.It was observed that both the quadratic and cubic function fts signifcantly improved the relative deviation between the experimental and theoretical values of the 22 Na positron annihilation peak compared with the linear function ft.However, there was no a substantial diference between the quadratic and cubic function fts.Regarding the inversesquare law of radiation, a large deviation was observed in the experimental data, with the deviation increasing as the distance decreased.To correct this deviation, a correction parameter was introduced.Te correction parameter allowed for a better ft of the experimental data to the inverse-square law.Te distribution of the correction parameter was obtained from the student data, and the recommended value was provided based on the mean value.Furthermore, it was found that the experimental data could be well-ftted by directly applying a power function without using the correction parameter.Te power function ftting provided an appropriate representation of the data without the need for additional corrections.Te distribution of the power function exponent was obtained from the student data, and the recommended value was provided based on the mean value.Overall, this study demonstrated the importance of energy calibration and the challenges associated with the inverse-square law in gamma-ray measurements.Te use of alternative ftting functions and the introduction of correction parameters or direct power function fts can signifcantly improve the accuracy and reliability of the measurements.

Figure 1 :
Figure 1: Sealed radioactive point source of 60 Co (a), the scintillation detector (b), the arrangement of the source and the scintillation detector (c), and the entire experimental setup (d).S denotes source, PMTdenotes photomultiplier tube, HV denotes high voltage, and MCA denotes multichannel analyzer.

Figure 3 :Figure 4 :Figure 6 :Figure 5 :Figure 7 :
Figure 3: Typical measured gamma-ray energy spectra of 137 Cs (a), 60 Co (b), and 22 Na (c).All sources were placed at a distance of 2 cm from the detector.(a) Measured for 10 min and (b) and (c) measured for 120 min.

Figure 10 (Figure 8 :Figure 9 :
Figure 8: Typical lines of the least-squares ft with the modifed inverse-square function of equation (4) to experimental data of K from 2 to 48 cm for 137 Cs (a) and 60 Co (b).Experimental statistical errors of K are within the symbols.

Figure 11 :Figure 10 :
Figure 11: Distribution of the exponent b for 137 Cs (a) and 60 Co (b).