Study on the correlation between radioactive counting time and measurement uncertainty of ceramic materials

ABSTRACT This paper sketched the basic principles of ceramic materials radioactivity measurement and mathematical calculation formula of testing time. When the sample measurement conditions and the way of data analysis are constant, using sodium iodide gamma spectrometer for getting the radioactive spectra data of 5 groups of samples. In condition of the background count time tb or sample measurement time ts increasing gradually, study the radioactivity specific activity measurement error range of all samples. It is derived when the background count rate of nb and sample counting rate ns is constant, the measurement error ν2 and the sum of reciprocal of the background count time tb and reciprocal of the sample measuring time ts for approximately linear positive correlation. The measurement error of 226Ra, 232Th, 40K radioactive specific activity of different ceramic samples presents the corresponding fluctuations as their initial energy spectrum count rate differences. This study for coordination of the contradiction between ceramic materials radioactive measurement time and the uncertainty has certainly theoretical and practical value.


Introduction
In the process of complex geochemical evolution, uranium element usually exists in the form of isomorphism in the blank and glaze of calcite, fluorite and zircon [1]. The addition of 226 Ra, 232 Th, 40 K and other radionuclides has brought some potential radioactive pollution. Studies show that radiation causes harm to human body through two aspects, namely external irradiation and internal irradiation [2]. External exposure is the irradiation of radionuclides on the outer surface of the human body. Internal irradiation refers to the ionization of the basic molecular structure of human cells by natural radioactive nuclei entering the human body through diet or breathing, which damages the molecular structure and cells. In vivo radiation is mainly radon gas through breathing into the human body, the decay of short life radioactive nuclear decay, the release of α particles to the human body is the most damage, can make the respiratory system epithelial cells exposed to radiation. Long-term internal radiation may cause local tissue damage and even induce lung cancer and bronchial cancer. If the level of radiation exceeds a certain level, it can cause harm to people. Current studies believe that when indoor radon concentration is less than 200 Bq/m 3 (equivalent annual effective dose of 2.5 msV) and external radiation dose rate is less than 0.5 MGy/h (equivalent annual effective dose of 2.5 msV), it is acceptable for human body. When the effective dose exceeds this value, remedial action is required. The radiation of ceramic products is a kind of chronic irradiation, and its harm to human body is a random effect produced by long-term chronic irradiation with a small dose rate, that is, the probability of cancer and genetic effect is proportional to the dose of radiation. In addition, over the service life or artificial damage of the dust off the glaze fabric is inhaled or ingested by the human body will also cause internal irradiation.
Because the radioactivity of radioactive elements cannot be changed after high temperature, high pressure, electromagnetic or dissolution, combination and other physical and chemical processes, so the United States, Germany, Russia and other countries have a clear limit on the exemption value of natural radioactive substances in ceramic products [3]. China has also issued GB 6566-2010 "Limits of Radionuclides in Building Materials" [4], HJ/T 297-2006 "Technical Requirements for Environmental Marking Products sanitary Ceramics" [5], SN/T 1570.2-2005 "Inspection procedures for export building sanitary Ceramics Part 2: Sanitary ceramics [6] and other standards put forward specific requirements for the radioactivity level of building sanitary ceramics materials. In the standard, there are clear provisions on testing instruments, sample form and quantity, testing methods, etc., but there is no clear description of the testing time for different products (different radioactive levels). The intensity of radioactive rays is not constant, so it is necessary to determine the detection time according to experience and different product types. The radioactive activity of ceramic samples is generally low, so the measurement time is relatively long. However, if the data acquisition time is too long, the stability of the instrument and the economy of the experiment can not be guaranteed. If the measurement time is too short, the statistical error will increase and the accuracy of test results will be reduced [7].

Parameters for the measurement of radioactivity
In theory, radioactive activity concentration (C Ra , C Th , C K ) is usually used to evaluate the radioactive level of elements. This index is expressed by the quotient of the radioactive activity of a nuclide divided by the weight of the substance, and the unit is Bq/kg [8]. In the actual measurement process, I γ (external exposure index) and I Ra (internal exposure index) are often used to measure the radioactivity of a material. The relationship between radioactivity concentration and internal exposure index and external exposure index is: I Ra = C Ra /200; I γ = C Ra /370 + C Th /260 + C K /4200 [9]. 226 Ra, 232 Th and 40 K are the main γ radiators in ceramic materials, and the γ radiation dose of ceramic materials is proportional to their radioactive activity concentration.

The relationship between measurement error and measurement time
The measurement length of the sample is determined by the energy resolution of the gamma spectrometer, the background value, the type of the sample to be measured, the requirement of the measurement accuracy and the level of radioactive activity of the sample, etc. Assuming that n s and n b are the sample count (including background count) in the period of t s and background count in the period of t b respectively [10], then the net count rate n 0 of the sample to be tested is: In the formula, n s and n b are the sample count rate and background count rate respectively. According to the error transfer theory, the standard deviation σ n0 of the net count rate n 0 can be obtained from formula (1): The relative standard error (ν) of radioactive sample counting rate is: In the process of measurement, the counting rate error of radioactive samples is minimized and the collection time of radioactive data of samples is shortened to the maximum within the given measurement uncertainty range. When the total sample count rate and background count rate are known, according to the error requirements, it can be deduced from the above two equations that the total sample count rate (including background count rate), background count rate and measurement time have the following relationship [11]: From equations (2) -(4), the mathematical relationship between measurement error and sample measurement duration (t S ) can be deduced:

Detector lead shielded chamber and data analysis system
The instrument used in this test was Scinbi SPEC -3 sodium iodide γ spectrometer manufactured by Target, Germany. The NaI crystal size was φ 76 mm × 76 mm, the instrument resolution was 6.41% at 15°C, and the short -term stability was 0.07%. The background counting rate of the lead shielding chamber was 0.18 cps, as shown in Figure 1, where the number of channels in the multi-channel pulse amplitude analyzer was 1024 [12]. Data analysis was carried out by the trace-by-channel least squares fitting method.

Voluminal standard radioactive source
The specific activity of the volume standard source used in this experiment is shown in Table 1 [13].

The sample
The test sample used in the experiment was the standard sample, which was made of white glazed ceramic tile with a size of 300 mm × 300 mm and a particle size no larger than 0.16 mm. After constant treatment, the standard sample was put into a hard plastic sample box with a diameter of 75 mm × 70 mm and preserved in paraffin seal, so that uranium, radium and their short-lived daughters in the sample could reach radioactive decay equilibrium and be weighed (accurate to 0.1 g). The standard sample used in this experiment is the volume standard source, and only the volume is considered, not the quality. Therefore, the same sample box is used for the three standard samples.

Test conditions and measurement uncertainty
In order to prevent the scintillation probe crystal from experiencing sudden temperature changes, the room temperature is controlled at 15 ± 2°C. The voltage of the data acquisition system was set as 802 V and the amplification factor was set as 1.001 times. When the instrument count rate does not exceed 1000 cps, the spectral data of a 137 Cs point source was collected at room temperature for 10 min. By selecting the average value of two discontinuous measurements, the channel address corresponding to the γ ray omni-power peak of 661.64 keV was determined to be 296.57.

Theoretical discussion and experimental verifications of the correlation between the error of radioactivity measurement and the shortest measurement time
The national standard GB 6566-2010 does not specify the measurement time of radioactive samples. Due to the influence of measuring equipment, sample differences and other factors, different experimental conditions and measurement processes often correspond to different measurement errors. After verifying the measurement results of zirconium silicate samples from 4000 to 45,000 s, Hao Xiaoyong [14] concluded that the accuracy of measurement results tended to be stable when the measurement time was no less than 4000 s. Chang Wenfang [15] proposed that when the activity concentration of 226 Ra and 232 Th in the sample is greater than 37 Bq/kg, and the activity concentration of 40 K is greater than 200 Bq/kg, the measurement time of the energy spectrometer which calculates the activity concentration by the method of the least-squares fitting is more than 9000 s, it can ensure that the measurement error meets the capability verification requirements of the radioactive activity concentration. At present, theoretical research on the quantitative relationship between the acquisition time of radioactive data and the measurement uncertainty is relatively lagging behind in the industry. Only Qu Jinhui [16] deduced the mathematical formula between the shortest measurement time of samples and the measurement uncertainty based on the continuous counting characteristics of the measurement results of radioactive samples. The mathematical formula is T min = {n b × ν 2 × [(n s /n b ) 1/2 -1] 2 } -1 。It has been proved that when the collection time of radioactive spectra is extended to a certain stage, the full spectrum counting rate ns of background, body source and sample will tend to be stable. The variation trend of counting rate of the background, 226 Ra, 232 Th, 40 K and standard samples 1 # -5 # under different counting time conditions in this experiment is shown in Figure 2 - Figure 6. The n s were 0.18 cps, 12.30 cps, 7.13 cps, 1.10 cps and 3.21 cps, 4.44 cps, 5.18 cps, 6.74 cps, 10.45 cps, respectively. In addition, it can be seen from 2.2 and equations (2) -(5) that for lead chambers with good shielding effect, the background counting rate n b is relatively stable. Under the condition that the background counting time (t b ) is given, the theoretical values of the corresponding volume of standard source and the counting time ts of the sample and the measurement relative error ν can be calculated in turn. The specific values are shown in Table 2.
According to equation (5), it can be seen that under the condition that the measurement error ν, background counting rate n b and background counting time t b remain unchanged. The t s is approximately negatively correlated with n s 1/2 . When the values of n b , t b and n s are determined, the relationship between t s and ν 2 is approximately negative line. As shown in Table 2 Ra standard source increased from 1.10 cps to 12.30 cps, and the count time of body standard source and standard sample increased from 4449.72s to 14,879.52s (without considering the difference of radioactivity level). As the national standard GB 6566-2010 clearly stipulates the measurement uncertainty ν should not exceed 20%, in addition, verified ceramic sample radioactivity counting rate n s is usually between 0 cps -6 cps.
Therefore, when 1/ν 2 is not less than 25 and n s 1/2 is less than 3 cps, the effect of measurement error ν on the sample counting time t s is obviously much greater than n s . Due to the limitations of the experimental data in     Table 2, as well as the restriction of the conditions of radioactivity measurement and the characteristics of the data analysis of the comparison method, the correlation between the radioactivity count rate of the sample and the measurement error of specific activity cannot be concluded.  The voluminal standard source, samples and measurement conditions used in the experiment refer to sections 2.3-2.5 explanatory. The background count rate of lead shielding enclosures n b is 0.18 cps and the counting time t b is 1800 s.Regarded the t s theoretical calculation values as the actual count time of voluminal standard radioactive sources and samples and based on the spectral data of background and voluminal standard radioactive sources described in 3.2, then analyzed the 226 Ra, 232 Th, 40 K radioactivity specific activity of standard samples 1 #~5# . Calculating the error between radioactivity specific activity measurement value and calibration value of standard samples 1 #~5#. Radioactivity specific activity calibration values of each standard sample are in Table 3.

The influence of background counting time on sample measurement uncertainty and its causes
Sodium iodide γ spectrometer was used to measure the lead chamber background in sequence from 1800 s, 3600 s, 7200 s, 14,400 s, 21,600 s, 28,800 s, 36,000 s and 43,200 s in accordance with the measuring conditions specified in Section 2.3. At the same time, the measuring time of 226 Ra, 232 Th and 40 K voluminal standard radioactive source is set as 21,600 s, and the measuring time of radionuclide of standard sample 1 # -5 # with knew nominal activity concentration is set as 14,400 s. After the database was established with the background and body standard source spectra of different counting hours, the samples were analyzed for radioactivity. The measurement results and measurement error data are shown in Table 3.
As showed in Figure 2  K voluminal standard radioactive source and the 5 groups of samples are relatively stable. Therefore, with the extension of the background counting time for database analysis, the background radioactivity count increases from 324 to 15,552. In view of the characteristic of "automatic background deduction" of the least square fitting comparison method, the measured values of the specific activity of C Ra , C Th and C K gradually increase. Table 3 shows that the measured values of the specific activity of the standard samples 1 # to 5 # also increase with time. At the same time, according to equations (4) and (5), under a certain background counting rate nb, the measurement error ν is approximately negatively correlated with the square root of sample measurement time t s , background counting time t b and sample counting rate n s . Therefore, when the measuring time t s of standard sample 1 # -5 # is 14400s, with the increase of the radioactivity counting rate n s , the measurement error ν corresponding to the eight different background counting times in 1800s, 3600s, 7200s, 14400s, 21600s, 28800s, 36000s, 43200s shows a decreasing trend; In addition, the 226 Ra, 232 Th and 40 K energy spectrum counting rates of the standard sample 1 # to 5 # change in ascending order at each t b counting time node. Therefore, the experimental data fully verify the conclusion of theoretical analysis. In addition, it can also be deduced from equations (4) and (5) that when the sample measurement time t s and its counting rate n s are constant, the measurement error ν is approximately negatively correlated with the square root of the background counting time t b . The data of 97.5% of the 120 measurement results in Table 3 proves that: With the extension of t b , the measured values of 226 Ra, 232 Th and 40 K specific activity of standard sample 1 # -5 # gradually approached their calibration values, and the measurement error gradually decreased.

The influence of different sample data collection time on measurement uncertainty and its cause analysis
Sodium iodide γ spectrometer was used to collect radioactivity spectra of 43,200 s and 21,600 s with background and 226 Ra, 232 Th and 40 K body standard sources according to the measuring conditions specified in 2.3, and the database was built to analyze the sample measurement data.
Then, the specific activity of the standard sample 1 # -5 # was determined in the counting time conditions of 1800 s, 3600 s, 7200 s, 14,400 s, 21,600 s, 28,800 s, 36,000 s and 43,200 s successively. The result data and measurement error are shown in Table 4. The change in amplitude of the full spectrum count rate of the standard sample 1 # -5 # under different measurement time conditions is shown in Figure 7.
According to the data in Table 4, when the spectrum collection time of background and body standard sources are fixed, the extension of sample counting time does not directly lead to the rise of measured values of 226 Ra, 232 Th and 40 K specific activity, and the fluctuation of values within a certain range is inevitable. When the measurement time is continuously extended, the C Ra , C Th measurement error ν of the standard sample 1 # is "first decrease and then increase" trend change, while the measurement error of C K is "increase -decreaseincrease" change characteristics, in which the sample count of 14,400 s, 36,000 s and 28,800 s after the C Ra , C Th and C K measurement error respectively reach the minimum value. The measurement errors of C Ra , C Th and C K of standard sample 2 # and 3 # show the change mode of "decrement --increase --decrease", "first increase and then decrease" and "first decrease and then increase" successively. After counting 14,400 s, the measurement errors of C Ra are minimized, and the errors of C Th and C K are minimized after 28,800 s. The C Ra measurement errors of the standard samples 4 # and 5 # follow the "increase first and decrease later" mode, but the counting time corresponding to the minimum error values is 43,200 s and 1800 s respectively. The sample counting time corresponding to the minimum measurement error of C Th is 14,400 s for both of them, but the fluctuation of the error value shows two different characteristics: "increasedecrease -increase" and "decrease first and then increase". On the contrary, the measurement error of C K changes in the ways of "reduce -increase -decrease" and "increase first and then decrease", while the minimum value corresponding to the counting time 3600 s and 1800 s is relatively close. As showed in formula (5), when the background counting rate n b and its counting time t b are constant, the square of the measurement error ν is approximately negatively correlated with the product of the sample measuring time t s and its counting rate n s . Figure 7 shows the variation trend of energy spectrum counting rate n s of 226 Ra, 232 Th and 40 K of standard sample 1 # -5 # in different counting time range. It can be seen from the trend that the sample radioactivity count early, if the initial count rate and the count rate value in each period of time is almost the same, the measurement error ν will decrease with the extension of the sample count time t s . Therefore, the measured values of C Ra , C Th and C K of standard sample 2 # and 3 # and C Ra and C Th of standard sample 1 # showed a trend of "increase first and then decrease", while the measurement error was "decrease first and then increase". In general, although the measurement time ts corresponding to the minimum error of the specific activity of different samples or different nuclides in the same sample is different, it is not less than 14,400 s. From the above samples 226 Ra, 232 Th, 40 K energy spectrum count rate in different measurement time range changes relatively gently, the measurement error ν is mainly determined by the measurement time t s . In a certain sample counting period, ν and t s show the reverse to change. Because in the process of sample measurement, the gamma rays interact with lead between the various material of the indoor photoelectric effect, Compton scattering effect may occur and the phenomenon such as electron pair effect, lead to the end of the spectrum count the volatility, and the experiment of the least -square fitting software used to take "automatic deducting background data analysis made after deducting background count sample measurement error has been changing.
The "first decrease and then increase" of the measured results of the specific activity of the standard sample 4 # and 5 # and the "first increase and then decrease" of the measurement error are derived from the relatively high counting rate n s of the corresponding nuclide energy spectrum in the early counting period of the sample, which leads to the relatively large measured value of the specific activity. Therefore, the initial measurement error is mainly controlled by the sample count rate n s . But with the extension of the measurement time, n s will gradually tend to be consistent, at this time the measurement error ν will be dominated by t s and show an orderly decreasing trend.

Conclusion
The calculation formula of radioactivity counting time and measurement error of the ceramic sample can reveal the qualitative relationship between t s and ν. Radioactivity measurement results of ceramic materials are significantly affected by the background spectrum data of database. When the background count rate n b and sample measurement time t s , counting rate n s is certain, the measurement error ν 2 and background count time t b are approximately a linear negative correlation, so the reasonable determination of background spectrum collection time is very important to improve the measurement accuracy. When the background counting time (t b ) and the counting rate (n b ) are determined, the measurement error of specific activity of different ceramic materials varies with the energy spectrum counting rate (n s ) and the sample counting time (t s ), which depends on the difference of the energy spectrum counting rate (n s ) of each element in the initial period of sample counting (t s is less than 7200s). In this case, n s with poor stability will inevitably lead to a large measurement error of the specific activity of the sample, and the measurement error can be effectively reduced by extending the measurement time of the sample t s . Due to the difference of the composite energy spectrum counting rate n s and its corresponding radioactivity specific activity levels in different ceramic samples, and the difference of the radioactivity counting rate between different nuclides in the same sample, the measurement error of radioactivity counting rate and the measurement uncertainty of radioactivity specific activity are different.