Radioactive Source-Detector System: Design and Monte Carlo Opinion

In the current research, a computer simulation program was designed and written according to the Monte Carlo method to serve as a virtual practical system instead of a real one. The program has been statistically, geometrically and numerically tested for virtual radioactive source-detector setup. The simulation program is carried out for NaI(Tl) detector, and once for Gieger-Muller counter, for a range of energy up to 10 MeV. The Law of Large Numbers and the Central Limit Theorem were used to test the accuracy and precision of the program’s workflow and an indication of how the results are close to their averages and, statistically, how they tend to a normal distribution. Generally, results of a number of detector efficiency types showed a high agreement with published experimental and several global codes results within a percentage error of ~ 0.02-5% (i.e. the accuracy ~ 95-99.98%) and the significance level reflects the precise of the algorithm of simulation. The accurate and precise estimation of the current simulation gives it the desired reliability. The current simulation program also showed flexibility and effectiveness in designing any nuclear source-detector system and providing the relevant workers or experimenters with indicators that help in the optimal design of a system in terms of equipment and geometrical configuration with the least time. It may take a few seconds to a few minutes of execution time for a personal computer with normal specifications. Unlike laboratory experiments which may take from several minutes to several hours. In addition, it provides an ideal work environment that is completely free of radiation hazards. Also, the current simulation provides a deep understanding of the interactions that occur in a real physical practical system.


Introduction
Designing and testing a particular detection system usually brings some financial costs and time considerations 1 .Numerical Simulation can be an effective and economical alternative tool.Monte Published Online First: March, 2024 https://doi.org/10.21123/bsj.2024.2288P-ISSN: 2078-8665 -E-ISSN: 2411-7986 Baghdad Science Journal can reproduce, flexibly, any experimental circumstances, whatever complicated 4 .Another clear advantage of this approach is the short computation time.
As can be done by experimental 5,6 or theoretical 7,8 approaches, Monte Carlo technique has been used in gamma detection techniques which have a fundamental role in the field of gamma-rays spectroscopy 9,10 applied in nuclear physics, radiation measurement of environmental samples radiation dosimetry [11][12][13][14][15] , medical radiography 16 , neutron activation analysis 17 , well logging 18 , and study of cosmic rays 19 .
Monte Carlo simulation computer program, as a virtual setup, was designed and written to be used instead of a real experimental system.The achievements concerning time assumption and flexibility versus the complicated were verified.A number of statistical, geometrical and characteristic parameters concerning the detection system have been estimated.

Materials and Methods
The proposed configuration for the current simulation is shown in Fig. 1 including, a 'virtual', radioactive source and detector (for example type of NaI(Tl)), in the coordinates system.The source lies within the fixed dimensions system with Cartesian coordinate axes (x,y,z) 20 to determine the position of the source with respect to the axial z-axis.While the space outside of the source volume towards the detector lies within the reference dimensions system with spherical coordinate axes (r,θ,φ) 20 to follow the random geometric projections of emitted photons from the source on the planes of the detector and to determine the random probabilities of whether or not a photons they were to fall and be detected.The distance between the source and the detector can change.Based on the suggested configuration, a Fortran95 program for Monte Carlo simulation has been developed.It was designed and written to depict the interaction processes that occurred when the photon beam came into the detector, the results of which are that only the photons that are incident within a solid angle covered by the detector are to be impinging with the front face of the detector.These photons, later, are to be registered (counted) after satisfying a number of concern conditions as will be explained.These registered events allowed us to study "virtually" the characteristics of the experimental system concerning geometrical characteristics of a system.The present program is executed by Compaq Visual Fortran Professional edition 6.6.0,2000 compiler package under windows 10.
The attenuation in the material of the radioactive source and in the air between the source and detector are neglected.The particles are transferred using the eq. 1 21 : where:   ⃗⃗⃗⃗⃗ is initial position,  ⃗ is a new position with  ⃗⃗ direction and  is the distance that the particle travels before it intersects with a plane of a specific region.
In calculating particle transport in Monte Carlo as well as ray-tracing algorithms, a common problem is finding the distance a particle must travel in order to intersect a particular surface.Therefore the distribution function method can be used to sample the distributed photon path length.The probability function is given by Eq. 2 22 : where:   is the linear attenuation coefficient for a certain medium.
Briefly, the history of photons can be illustrated in the following algorithm: 1. Based on simple linear congruential generators (LCG) 21 , generate two random numbers Rn1 and Rn2 into the interval (0, 1).
2. In fixed coordinates, using Eq. 1 to determine the random gamma emitting position from a radioactive source, with rs Radius, (xs,ys,zs) 22 : where rnx is off-axial location of source.Then a certain counter called emitting photon is increasing.In the reference dimensions system, Fig. 1 If   >   , this means that the radius of the spherical projection of the photon is greater than the radius of the front face of the detector.Then the photon was rejected, a certain counter called non -incident is increasing, and come back to step 1, else a certain counter called incident photon is increasing, then continue.
8. Generate one random number  5 : 0 ≤  5 ≤ 1 9. Solve the probability function 22 , Eq. 2, to determine the free path-length of gamma photon within/not the active medium of detector gives: 10. From steps (5 and 9), estimate the photon interaction position location within the detector, that is, (   ,   ,   ), then 22 : 11. Calculate the spherical projection radius of a gamma-ray on the detector planes,   .Where: 12. With exception when   <   and   <    < (  + ℎ  ), the photon must be rejected and a certain counter called unregistered photons are increasing, then come back to step 1, otherwise a certain counter called registered photons are increasing, then continue.Compton scattering or pair production) depending on the probability of occurrence of that interaction.
14. Repeat the above steps of the algorithm by the number of emitted photons from the radioactive source.
15. Ordering the results in particular files.
The total efficiency of the detector was estimated by comparing the number of registered photons and those that were emitted from the source.While, the intrinsic efficiency was estimated by the ratio of registered photons number to those that hit the front face of the detector.The geometric efficiency was estimated by comparing the number of photons that hit the front face of the detector and those that were emitted from the source.
The values of mass attenuation coefficients for the active medium of detector were calculated using the XCOM program 23 .
For more Accurate validation, GM-counter system (type ABG, CAT: PA1885-020-030) as shown in Fig. 2 was used to validate, experimentally, some findings of the present simulation.

Results and Discussion
The random nature of radiation interaction with matter must be explained and interpreted according to probabilistic terms.Two of the main results in probability: are the Law of Large Numbers (LLN) 24 and the Central Limit Theorem (CLT) 24 .Both are related to sums of independent random varibles 6 that are included in the above algorithm.So, to complete the calculations with an accurate estimation of the value to be calculated, it must be re-implemented for a suitable large number of photon histories based on the LLN.The experimental curve in Fig. 4 is the result of statistical repetitions of series of one hundred 1 minute counts of a Cs-137 source made with G-M laboratory counter set up which is shown in Fig. 2. While , the simulated curves in Fig. 4 are the result of a series of one hundred 10, 10 2 and 10 6 history of photons for 662 keV that was executed to mimic the experimental setup that is shown in Fig. 2. The distribution curve of 10 history of photons is not normal, right another hump is formed.As for the curve of 100 history of photons is not precise enough to be normal.While the identical is clear between the experimental and simulated 10 6 history of photons curves and the variation between them is, axiomatically, due to random error introduced by radionuclide decay that is a randomly varying quantity.Therefore, repetition of more measurements, the measurements tend to be a normal distribution, as stated in the central limit theorem.Thats is, the values close to the expected value are more frequent than values that are far from them.
The average value for the results of series of 1 minute counts of a Cs-137 source made with G-M counter was 1940 C/1mint.with 40.4 of standard deviation and chi square (χ²) 24.43.It took about 1.5 hours.Whereas for a series of one hundred 10 6 history of photons for 662 keV, the average was 250196 count with 449.5 of standard devaition and chi square 79.942.It took about 5 minutes.From the table of χ² values, the experimental and simulated significance level was 0.706 and 0.77 respectively.This indicates that the results reflect feasible instrument operation and the precision of the performance of the measurement system and the algorithm of simulation.The accurate and precise estimation of the current simulation gives it the desired reliability.Varying types of efficiency, as numerical factors, were mimicked to validate the present simulation.
The outcomes, tabulated compared with published results 26 for For 2″×2″ of NaI(Tl) detector at particular distances 0.001, 5,10 and 15 cm for different energies 150-3000 keV as shown in Table 1.The comparison exhibits high agreement within a percentage error of ~ 0.02-5%.For further distances between the source and detector, the program has been implemented as it showed a clear match whether with the experimental or calculated results as shown in Table 2.  To demonstrate the effect of the source dimensions, the simulation was carried out for a radiant source with a disk shape.The results were compared with the published results of the international code Geant4 based GATE simulation program 29 , which clearly showed quite well congruence, see Fig. 9.
Another Validation was implementing the present simulation to scrutinize the detection efficiency versus different 2Rd/Hd ratio of NaI(Tl) detectors, so that the volume of detector remains fixed.The incident monoenergetic energies of γ-rays are 0.662, 1.331 and 4.438 MeV, experimentally, emitted from 137 Cs, 60 Co and 241 Am radioactive sources respectively 30 .As shown in Fig. 10, the result reveals that the detector efficiency dependent on the mentioned ratio of the NaI(Tl) detector and the incident energy of γ-photon.At a very low of 2Rd/Hd, the total detector efficiency increases rapidly and tends to be stable while the intrinsic of which is on the contrary.Anyway, the ratio from 1-2, (i.e. with an average 1.5, matching with ref. 30) is an interesting and meaningful point.It is valuable to design detectors for detecting γ-rays that have optimal dimensions.

Figure 1 .
Figure 1.The suggested design for the present source-detector systems simulation, the trajectory t represents a propagating "beam" of γ-ray that passes through point r in many of random directions.

Fig. 3 .
Exhibits a smaller number of trials for 10, 10 2 and 10 3 do not effectively estimate value conformity with a standard or observed value (one of values of Hoang 25 ).While, the result of 10 4 and 10 5 trials produce a values are close to observed value.Concern 10 6 trials, the resultant are the closest with the smallest percentage error rate and highest accuracy.

Figure 3 .
Figure 3.A clarification of the law of large numbers using a particular run of the simulation.These accuracy values refer to how Monte Carlo values get closer to the observed values up to 99.9%.However this test is insufficient to assess the performance of present simulation, because these values are based, in their calculation, on independent

Fig. 5
Fig.5depicts an additional representation of the total and partial attenuation coefficients for γ-rays, which were obtained through the current simulation of a 3"×3" NaI(Tl) detector at specific energies of a radioactive source.The attenuation coefficient, in probabilistic terms, describes how radiation interacts with matter.As a function of energy, Fig.5and Fig.6are identical.

Figure 5 .
Figure 5. Comparing the number of detected photons that interact by photoelectric, Compton scattering and pair production effects at particular energy.

Figure 6 .
Figure 6.Total and partial mass attenuation coefficients of NaI(Tl) detector material. photons

Fig. 7
Fig.7shows the dependence of the registered count by detector versus the distance between the radioactive source and detector face (Dsd) for varied volume values of NaI(Tl) detector.When Dsd increases, the registered count decreases, and after passing through minimum, then it increases again.Since mean free path length of gamma rays is the inverse of the total attenuation coefficient as shown in Fig.8.Therefore, as a function of geometry, Fig.7is upside down to Fig.8and vice versa.Similar results have been reported by Hoang 25 , Urkiye 26 , Ogundare27 and Jehouani28 .

Figure 7 .
Figure 7. Variation of the registered count rate of different size of NaI(Tl) detectors as a function of source to detector distance for 662 keV of gamma rays.

Figure 8 .
Figure 8. Mean free path length of various volume NaI(Tl) detectors as a function of Source-to-Detector distance at 662 keV of gamma rays.

Figure 9 .
Figure 9.Total efficiency as a function of γ-ray comparison for point and disc radioactive source (p.s. and d.s.respectively).

Figure 10 .
Figure 10.Intrinsic and total efficiency as a function of diameter to height ratio of 3 ″  3 ″ NaI(Tl) detector at different values of γ-ray energy.
Calculate the spherical projection radius of a gamma-ray on the front detector face   , where: