INTRODUCTION
In experimental nuclear physics research, a high-purity germanium (HPGe) detector is generally used to measure gamma-ray spectra [1–3]. The accuracy of physical measurement is directly affected by the detector’s efficiency. Accurate efficiency is important to gamma-ray spectral analysis, efficiency calibration, and radioactivity calculations [4–6]. Some progress has been made, including determination of the efficiency function and its parameters for point sources [7], analysis of the relationship between the efficiency and height of the body source by an HPGe spectrometer [8], study of the crystal size and detection efficiency of an HPGe detector [9], investigation of the effect of source self-absorption, high voltage, cladding materials, and detection distance on HPGe detector efficiency [10], and the establishment of an absolute full-energy peak efficiency [11, 12].
The gamma point source efficiency represents the relationship between the full-energy peak count and the position in space of a gamma point source [5]. The main factors affecting the gamma point source efficiency are the radial angle, detection distance, and gamma-ray energy. However, the efficiency functions of gamma point sources either include the parameters of the spatial position without the parameters of the gamma-ray energy [7, 8] or include the detection distance and gamma-ray energy without the radial angle [13]. It is difficult to determine the nonlinear relationship between impact factors by constructing a perfect model of the efficiency function. Therefore, it is worth finding a new method as an alternative for quickly evaluating the gamma point source efficiency with satisfactory accuracy.
A back-propagation (BP) neural network model has advantages in solving multi-parameter nonlinear problems; thus, it has been applied in many fields [14, 15]. Currently, a BP neural network algorithm is one of the most widely used network algorithms [16]. A BP neural network is a feed-forward neural network that is applied by a BP algorithm. A BP neural network has characteristics such as induction, fault tolerance, and nonlinear processing. It is also able to learn and store extremely large mapping relations of input and output modes without prior disclosure and description of the mathematical equations for the mapping relations [17]. Using a BP neural network to analyze the effect of multiple nonlinear parameters on the calibration of the gamma point source efficiency is work of great significance.
GAMMA POINT SOURCE EXPERIMENT
Efficiency Calculation
The full-energy peak efficiency is defined as the probability that a pulse residing in the full-energy peak of the spectrum is produced when a photon strikes the detector. When the full-energy peak area is obtained from the measured spectrum, the intrinsic efficiency of the gamma point source for an HPGe detector can be calculated. The full-energy peak efficiency of the gamma point source at an energy E can be determined by [18]
where ε(E) is the full-energy peak efficiency, N(E) is the full-energy peak count, A0 is the activity of the source at the time of standardization, λ is the decay constant, t is the time from standardization to measurement of the source, P(E) is the photon emission probability at energy E, and T is the measuring time.
By neglecting the influence of the time (t and T) and detection distance (d), the uncertainty in the experimental full-energy peak efficiency σε can be determined by the uncertainties in N(E), A0, λ, and P(E), and it can be calculated as
where σN, σA0, σλ, and σP are the uncertainties associated with the quantities N(E), A0, λ, and P(E), respectively, which are obtained from the gamma point source. The uncertainty in the count is always set to less than 0.1%; the uncertainty in the activity of the gamma point source is dominated by the initial activity [13] and is less than 1.0%. The uncertainty in the decay constant is dominated by the half-life of the radionuclide [19] and is less than 1.0%, and the uncertainty in the photon emission probability is always set to be less than 0.1%.
Inevitably, when a detecting platform has been constructed with a detector in a nuclear physics experiment, we need a calibration of the detector’s efficiency at each spatial position for diverse gamma-ray energies. For example, in the energy range 121.782–1408.011 keV and a gamma point source position of 10 cm in front of the detector cap, the calibration curve of the HPGe detector efficiency is determined and shown in Fig. 1.
Experimental Measurement
The gamma-ray spectrum is acquired from the spectrometer system, which consists of an HPGe detector and a multichannel analyzer (DSP-jr2.0, ORTEC B.V., USA). An electrical refrigeration P-type coaxial HPGe detector is used, and its structure is shown in Fig. 2. The diameter of the HPGe crystal is 7 cm, and its length is 8.26 cm. The diameter of the copper cold finger is 0.9 cm, and its length is 6.9 cm. The high voltage is 2600 V, the operating temperature is 100 K, and the range of measured gamma-ray energies is 4 keV to 10 MeV. Other parameters of the HPGe detector structure are shown in Table 1.
| Name | Material | Size (mm) |
|---|---|---|
| Dead layer in front of crystal | Li | < 0.015 |
| Crystal | Ge | Ø70×82.6 |
| Cold finger | Cu | Ø9×6 |
| Dead layer on inner side of crystal | B | 0.0003 |
| Dead layer on broad side of crystal | Li | 0.7 |
| Covering layer of crystal | Al | 1.5 |
| Cap layer of detector | C | 1.6 |
Gamma point sources include 137Cs and 60Co: (a) 137Cs (3.343 × 105 Bq), with a photon emission probability at 661.661 keV of 85%; and (b) 60Co (2.070 × 105 Bq), with a photon emission probability at 1173.238 keV of 99.87% and at 1332.513 keV of 99.982%. The small cylindrical samples of these two gamma sources have a radius of 0.3 cm and a height of 0.8 cm. Compared with the spatial scale of the source position, the size of the sources is negligible, so they can be regarded as point sources.
The position of the gamma point source is generally determined in the space of a rectangular coordinate system [20]. In this work, we first adopt the polar coordinate system to determine the position of the point source, as shown in Fig. 2.
When the detection distance is less than 15 cm, the measurement accuracy of the point source efficiency is strongly influenced by the large detector dead time because of the high photon count rate. Normally, the distance from the point source to the detector is less than 55 cm [9], so the point source is located at a distance d in the range 15–55 cm in front of the detector cap, at intervals of 5 cm. The radial angle θ ranges from 0 to 11π/24 at intervals of π/24. A spectrum can be obtained in 180 s for every measurement condition. The gamma point source efficiency at 661.661 keV for 137Cs and at 1173.238 and 1332.513 keV for 60Co is listed in Tables 2, 3 and 4, respectively.
| Angle (rad) | Distance (cm) | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | |
| 0 | 5.586×10−3 | 3.307×10−3 | 2.261×10−3 | 1.581×10−3 | 1.158×10−3 | 9.027×10−4 | 7.261×10−4 | 5.980×10−4 | 4.951×10−4 |
| π/24 | 5.497×10−3 | 3.269×10−3 | 2.209×10−3 | 1.571×10−3 | 1.159×10−3 | 8.983×10−4 | 7.320×10−4 | 5.998×10−4 | 4.935×10−4 |
| 2π/24 | 5.522×10−3 | 3.333×10−3 | 2.223×10−3 | 1.552×10−3 | 1.147×10−3 | 8.982×10−4 | 7.251×10−4 | 5.971×10−4 | 4.955×10−4 |
| 3π/24 | 5.498×10−3 | 3.280×10−3 | 2.209×10−3 | 1.526×10−3 | 1.130×10−3 | 8.934×10−4 | 7.201×10−4 | 5.916×10−4 | 4.912×10−4 |
| 4π/24 | 5.435×10−3 | 3.180×10−3 | 2.136×10−3 | 1.454×10−3 | 1.080×10−3 | 8.442×10−4 | 6.785×10−4 | 5.584×10−4 | 4.671×10−4 |
| 5π/24 | 5.075×10−3 | 2.900×10−3 | 1.936×10−3 | 1.294×10−3 | 9.710×10−4 | 7.484×10−4 | 5.983×10−4 | 4.881×10−4 | 4.024×10−4 |
| 6π/24 | 4.665×10−3 | 2.714×10−3 | 1.786×10−3 | 1.172×10−3 | 8.726×10−4 | 6.696×10−4 | 5.307×10−4 | 4.308×10−4 | 3.512×10−4 |
| 7π/24 | 4.387×10−3 | 2.405×10−3 | 1.489×10−3 | 9.943×10−4 | 7.278×10−4 | 5.540×10−4 | 4.335×10−4 | 3.460×10−4 | 2.835×10−4 |
| 8π/24 | 3.584×10−3 | 1.823×10−3 | 1.140×10−3 | 7.694×10−4 | 5.530×10−4 | 4.178×10−4 | 3.190×10−4 | 2.585×10−4 | 2.094×10−4 |
| 9π/24 | 2.697×10−3 | 1.353×10−3 | 7.986×10−4 | 5.408×10−4 | 3.793×10−4 | 2.822×10−4 | 2.154×10−4 | 1.724×10−4 | 1.520×10−4 |
| 10π/24 | 1.870×10−3 | 8.394×10−4 | 4.572×10−4 | 2.899×10−4 | 2.168×10−4 | 1.546×10−4 | 1.114×10−4 | 8.550×10−5 | 6.849×10−5 |
| 11π/24 | 5.887×10−4 | 2.704×10−4 | 1.253×10−4 | 7.501×10−5 | 4.403×10−5 | 2.975×10−5 | 2.317×10−5 | 2.000×10−5 | 1.375×10−5 |
| Angle (rad) | Distance (cm) | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | |
| 0 | 4.278×10−3 | 2.626×10−3 | 1.792×10−3 | 1.234×10−3 | 9.334×10−4 | 7.290×10−4 | 5.921×10−4 | 4.809×10−4 | 3.880×10−4 |
| π/24 | 4.281×10−3 | 2.680×10−3 | 1.763×10−3 | 1.221×10−3 | 9.202×10−4 | 7.189×10−4 | 5.830×10−4 | 4.789×10−4 | 3.966×10−4 |
| 2π/24 | 4.242×10−3 | 2.602×10−3 | 1.758×10−3 | 1.223×10−3 | 9.027×10−4 | 7.191×10−4 | 5.867×10−4 | 4.793×10−4 | 3.996×10−4 |
| 3π/24 | 4.261×10−3 | 2.582×10−3 | 1.755×10−3 | 1.214×10−3 | 9.045×10−4 | 7.098×10−4 | 5.708×10−4 | 4.709×10−4 | 3.896×10−4 |
| 4π/24 | 4.245×10−3 | 2.547×10−3 | 1.710×10−3 | 1.172×10−3 | 8.751×10−4 | 6.793×10−4 | 5.378×10−4 | 4.486×10−4 | 3.697×10−4 |
| 5π/24 | 4.139×10−3 | 2.409×10−3 | 1.618×10−3 | 1.081×10−3 | 8.163×10−4 | 6.264×10−4 | 4.963×10−4 | 4.077×10−4 | 3.347×10−4 |
| 6π/24 | 3.944×10−3 | 2.249×10−3 | 1.487×10−3 | 9.868×10−4 | 7.289×10−4 | 5.594×10−4 | 4.397×10−4 | 3.602×10−4 | 2.942×10−4 |
| 7π/24 | 3.639×10−3 | 2.021×10−3 | 1.253×10−3 | 8.537×10−4 | 6.170×10−4 | 4.773×10−4 | 3.715×10−4 | 2.992×10−4 | 2.460×10−4 |
| 8π/24 | 3.187×10−3 | 1.612×10−3 | 1.028×10−3 | 6.860×10−4 | 4.989×10−4 | 3.702×10−4 | 2.926×10−4 | 2.336×10−4 | 1.871×10−4 |
| 9π/24 | 2.633×10−3 | 1.307×10−3 | 8.148×10−4 | 5.390×10−4 | 3.826×10−4 | 2.793×10−4 | 2.121×10−4 | 1.685×10−4 | 1.349×10−4 |
| 10π/24 | 2.007×10−3 | 9.217×10−4 | 5.129×10−4 | 3.292×10−4 | 2.474×10−4 | 1.753×10−4 | 1.294×10−4 | 1.067×10−4 | 8.132×10−5 |
| 11π/24 | 6.991×10−4 | 3.867×10−4 | 2.422×10−4 | 1.582×10−4 | 1.052×10−4 | 7.354×10−5 | 5.924×10−5 | 5.007×10−5 | 3.771×10−5 |
| Angle (rad) | Distance (cm) | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | |
| 0 | 3.911×10−3 | 2.415×10−3 | 1.642×10−3 | 1.132×10−3 | 8.619×10−4 | 6.751×10−4 | 5.422×10−4 | 4.400×10−4 | 3.571×10−4 |
| π/24 | 3.916×10−3 | 2.473×10−3 | 1.627×10−3 | 1.136×10−3 | 8.571×10−4 | 6.658×10−4 | 5.351×10−4 | 4.436×10−4 | 3.644×10−4 |
| 2π/24 | 3.875×10−3 | 2.390×10−3 | 1.615×10−3 | 1.136×10−3 | 8.412×10−4 | 6.657×10−4 | 5.385×10−4 | 4.405×10−4 | 3.698×10−4 |
| 3π/24 | 3.910×10−3 | 2.380×10−3 | 1.614×10−3 | 1.125×10−3 | 8.371×10−4 | 6.535×10−4 | 5.251×10−4 | 4.350×10−4 | 3.619×10−4 |
| 4π/24 | 3.913×10−3 | 2.336×10−3 | 1.574×10−3 | 1.072×10−3 | 8.140×10−4 | 6.290×10−4 | 4.929×10−4 | 4.139×10−4 | 3.445×10−4 |
| 5π/24 | 3.805×10−3 | 2.226×10−3 | 1.494×10−3 | 1.007×10−3 | 7.517×10−4 | 5.752×10−4 | 4.591×10−4 | 3.734×10−4 | 3.122×10−4 |
| 6π/24 | 3.630×10−3 | 2.071×10−3 | 1.376×10−3 | 9.132×10−4 | 6.793×10−4 | 5.185×10−4 | 4.109×10−4 | 3.342×10−4 | 2.744×10−4 |
| 7π/24 | 3.378×10−3 | 1.886×10−3 | 1.168×10−3 | 7.941×10−4 | 5.817×10−4 | 4.472×10−4 | 3.486×10−4 | 2.792×10−4 | 2.253×10−4 |
| 8π/24 | 2.993×10−3 | 1.522×10−3 | 9.620×10−4 | 6.504×10−4 | 4.669×10−4 | 3.563×10−4 | 2.697×10−4 | 2.181×10−4 | 1.775×10−4 |
| 9π/24 | 2.495×10−3 | 1.244×10−3 | 7.684×10−4 | 5.170×10−4 | 3.687×10−4 | 2.681×10−4 | 2.025×10−4 | 1.586×10−4 | 1.325×10−4 |
| 10π/24 | 1.954×10−3 | 8.926×10−4 | 5.038×10−4 | 3.270×10−4 | 2.406×10−4 | 1.715×10−4 | 1.291×10−4 | 1.036×10−4 | 8.193×10−5 |
| 11π/24 | 7.423×10−4 | 4.068×10−4 | 2.569×10−4 | 1.691×10−4 | 1.124×10−4 | 8.011×10−5 | 6.288×10−5 | 5.317×10−5 | 4.182×10−5 |
BP NEURAL NETWORK PREDICTION
BP Neural Network Model
An artificial neural network (ANN) algorithm is established by referring to the structure and characteristics of the human brain, in which numerous simple processing units are interconnected. The BP neural network algorithm is one of the most widely used ANN algorithms and consists of the input layer, hidden layer, and output layer. In the input layer, the input values are conveyed to the network, and these neurons transmit the information to the next layer as a value. In the hidden layer, the experimental problem determines the number of layers and neurons present. The hidden layer is placed between the input and output layers. In the output layer, the output values of the network are generated [21]. A BP neural network includes two processes for signal forward-propagation and error BP. By analyzing the relative error between the sample value and the output value calculated by the BP neural network, the network weight coefficients can be corrected tautologically, and then the output layer can obtain the expected value.
In this work, a BP neural network is first applied to predict the gamma point source efficiency. We know that the gamma point source efficiency is related to the radial angle, detection distance, and gamma-ray energy by nonlinear relationships. A BP neural network can be used effectively to resolve those nonlinear problems, so a nonlinear prediction model is constructed with the BP neural network using MATLAB software (MathWorks, USA). The structure of this nonlinear prediction model is shown in Fig. 3.
Model Parameters
The node number of the input layer n is 3, and that of the output layer m is 1. The number of neurons in the hidden layer is determined by
where l is the number of neurons, and a is a constant in the range 1–10. According to practical training using the experimental data, the optimal number of neurons in the hidden layer was determined to be six. When a finite number of gamma point source efficiencies are experimentally measured and the nonlinear prediction model is constructed using MATLAB, the number of gamma point source efficiencies can be predicted.
The prediction model of the BP neural network is trained by sample data with different radial angles (0, π/24, 2π/24, 3π/24, 4π/24, 6π/24, 7π/24, 8π/24, 9π/24, 10π/24, and 11π/24), detection distances (15, 20, 25, 30, 40, 45, 50, and 55 cm), and gamma-ray energies (661.661, 1173.238, and 1332.513 keV) in Tables 2–4. The training goal of the BP neural network is 10−9, and Fig. 4 shows that the training error curve reached the goal of 10−9 after 1690 epochs.
RESULTS AND DISCUSSION
When the BP neural network model is trained, the gamma point source efficiency can be determined by inputting the radial angle, detection distance, and gamma-ray energy. The predicted results of the gamma point source efficiency for 661.661, 1173.238, and 1332.513 keV are shown in Fig. 5.
When the HPGe detector distance and the detection distance from the gamma point source to the detector cap are fixed, with increasing radial angle, the solid angle between the detector and gamma point source decreases as a cosine curve, so the gamma point source efficiency decreases slowly at small angles and decreases quickly at large angles. When the HPGe detector distance and radial angle are fixed, with increasing detection distance, the solid angle of the detector decreases quickly, and the gamma rays are attenuated in the air, so the gamma point source efficiency decreases quickly. The predicted efficiency changes less at angles of 0 to 4π/24 and decreases almost linearly at angles of 4π/24 to 11π/24 in Fig. 5. The predicted efficiencies decrease almost exponentially with increasing detection distance from 15 to 55 cm in Fig. 5. The predicted efficiency values are in good agreement with theoretical analysis.
The rest of the sample data can then be used to validate the accuracy of the trained BP neural network with different gamma-ray energies (661.661, 1173.238, and 1332.513 keV), a radial angle of 5π/24, and a detection distance of 35 cm. The error between the BP neural network prediction and experimental measurement of the gamma point source efficiency can be calculated as
where δ is the error, εBP is the efficiency predicted by the BP neural network, and εEX is the experimentally measured efficiency. The experimental measurements and BP neural network predictions are compared in Tables 5, 6 and 7.
| Number | Energy (keV) | Angle (rad) | Distance (cm) | Experiment | BP neural network | Error (%) |
|---|---|---|---|---|---|---|
| 1 | 661.661 | 5π/24 | 15 | 5.075×10−3 | 5.150×10−3 | 1.460 |
| 2 | 661.661 | 5π/24 | 20 | 2.900×10−3 | 2.964×10−3 | 2.190 |
| 3 | 661.661 | 5π/24 | 25 | 1.936×10−3 | 1.906×10−3 | −1.544 |
| 4 | 661.661 | 5π/24 | 30 | 1.294×10−3 | 1.286×10−3 | −0.627 |
| 5 | 661.661 | 5π/24 | 35 | 9.710×10−4 | 9.669×10−4 | −0.424 |
| 6 | 661.661 | 5π/24 | 40 | 7.484×10−4 | 7.599×10−4 | 1.547 |
| 7 | 661.661 | 5π/24 | 45 | 5.983×10−4 | 6.091×10−4 | 1.797 |
| 8 | 661.661 | 5π/24 | 50 | 4.881×10−4 | 4.786×10−4 | −1.946 |
| 9 | 661.661 | 5π/24 | 55 | 4.024×10−4 | 4.067×10−4 | 1.080 |
| 10 | 661.661 | 0 | 35 | 1.158×10−3 | 1.174×10−3 | 1.391 |
| 11 | 661.661 | π/24 | 35 | 1.159×10−3 | 1.162×10−3 | 0.247 |
| 12 | 661.661 | 2π/24 | 35 | 1.147×10−3 | 1.151×10−3 | 0.394 |
| 13 | 661.661 | 3π/24 | 35 | 1.130×10−3 | 1.123×10−3 | −0.566 |
| 14 | 661.661 | 4π/24 | 35 | 1.080×10−3 | 1.085×10−3 | 0.407 |
| 15 | 661.661 | 6π/24 | 35 | 8.726×10−4 | 8.760×10−4 | 0.389 |
| 16 | 661.661 | 7π/24 | 35 | 7.278×10−4 | 7.347×10−4 | 0.956 |
| 17 | 661.661 | 8π/24 | 35 | 5.530×10−4 | 5.521×10−4 | −0.164 |
| 18 | 661.661 | 9π/24 | 35 | 3.793×10−4 | 3.761×10−4 | −0.848 |
| 19 | 661.661 | 10π/24 | 35 | 2.168×10−4 | 2.158×10−4 | −0.442 |
| 20 | 661.661 | 11π/24 | 35 | 4.403×10−5 | 4.568×10−5 | 3.732 |
| Number | Energy (keV) | Angle (rad) | Distance (cm) | Experiment | BP neural network | Error (%) |
|---|---|---|---|---|---|---|
| 1 | 1173.238 | 5π/24 | 15 | 4.139×10−3 | 4.104×10−3 | −0.843 |
| 2 | 1173.238 | 5π/24 | 20 | 2.409×10−3 | 2.441×10−3 | 1.329 |
| 3 | 1173.238 | 5π/24 | 25 | 1.618×10−3 | 1.605×10−3 | −0.789 |
| 4 | 1173.238 | 5π/24 | 30 | 1.081×10−3 | 1.098×10−3 | 1.557 |
| 5 | 1173.238 | 5π/24 | 35 | 8.163×10−4 | 8.176×10−4 | 0.156 |
| 6 | 1173.238 | 5π/24 | 40 | 6.264×10−4 | 6.316×10−4 | 0.840 |
| 7 | 1173.238 | 5π/24 | 45 | 4.963×10−4 | 5.005×10−4 | 0.838 |
| 8 | 1173.238 | 5π/24 | 50 | 4.077×10−4 | 4.071×10−4 | −0.144 |
| 9 | 1173.238 | 5π/24 | 55 | 3.347×10−4 | 3.309×10−4 | −1.135 |
| 10 | 1173.238 | 0 | 35 | 9.334×10−4 | 9.236×10−4 | −1.041 |
| 11 | 1173.238 | π/24 | 35 | 9.202×10−4 | 9.204×10−4 | 0.014 |
| 12 | 1173.238 | 2π/24 | 35 | 9.027×10−4 | 9.155×10−4 | 1.413 |
| 13 | 1173.238 | 3π/24 | 35 | 9.045×10−4 | 9.156×10−4 | 1.226 |
| 14 | 1173.238 | 4π/24 | 35 | 8.751×10−4 | 8.695×10−4 | −0.641 |
| 15 | 1173.238 | 6π/24 | 35 | 7.289×10−4 | 7.260×10−4 | −0.389 |
| 16 | 1173.238 | 7π/24 | 35 | 6.170×10−4 | 6.230×10−4 | 0.988 |
| 17 | 1173.238 | 8π/24 | 35 | 4.989×10−4 | 4.956×10−4 | −0.659 |
| 18 | 1173.238 | 9π/24 | 35 | 3.826×10−4 | 3.815×10−4 | −0.297 |
| 19 | 1173.238 | 10π/24 | 35 | 2.474×10−4 | 2.511×10−4 | 1.482 |
| 20 | 1173.238 | 11π/24 | 35 | 1.052×10−4 | 1.057×10−4 | 0.481 |
| Number | Energy (keV) | Angle (rad) | Distance (cm) | Experiment | BP neural network | Error (%) |
|---|---|---|---|---|---|---|
| 1 | 1332.513 | 5π/24 | 15 | 3.805×10−3 | 3.861×10−3 | 1.465 |
| 2 | 1332.513 | 5π/24 | 20 | 2.226×10−3 | 2.254×10−3 | 1.285 |
| 3 | 1332.513 | 5π/24 | 25 | 1.494×10−3 | 1.515×10−3 | 1.360 |
| 4 | 1332.513 | 5π/24 | 30 | 1.007×10−3 | 1.002×10−3 | −0.552 |
| 5 | 1332.513 | 5π/24 | 35 | 7.517×10−4 | 7.449×10−4 | −0.909 |
| 6 | 1332.513 | 5π/24 | 40 | 5.752×10−4 | 5.781×10−4 | 0.506 |
| 7 | 1332.513 | 5π/24 | 45 | 4.591×10−4 | 4.668×10−4 | 1.658 |
| 8 | 1332.513 | 5π/24 | 50 | 3.734×10−4 | 3.740×10−4 | 0.155 |
| 9 | 1332.513 | 5π/24 | 55 | 3.122×10−4 | 3.130×10−4 | 0.285 |
| 10 | 1332.513 | 0 | 35 | 8.619×10−4 | 8.582×10−4 | −0.424 |
| 11 | 1332.513 | π/24 | 35 | 8.571×10−4 | 8.453×10−4 | −1.371 |
| 12 | 1332.513 | 2π/24 | 35 | 8.412×10−4 | 8.528×10−4 | 1.378 |
| 13 | 1332.513 | 3π/24 | 35 | 8.371×10−4 | 8.395×10−4 | 0.290 |
| 14 | 1332.513 | 4π/24 | 35 | 8.140×10−4 | 8.149×10−4 | 0.110 |
| 15 | 1332.513 | 6π/24 | 35 | 6.793×10−4 | 6.814×10−4 | 0.310 |
| 16 | 1332.513 | 7π/24 | 35 | 5.817×10−4 | 5.811×10−4 | −0.110 |
| 17 | 1332.513 | 8π/24 | 35 | 4.669×10−4 | 4.731×10−4 | 1.326 |
| 18 | 1332.513 | 9π/24 | 35 | 3.687×10−4 | 3.619×10−4 | −1.847 |
| 19 | 1332.513 | 10π/24 | 35 | 2.406×10−4 | 2.436×10−4 | 1.234 |
| 20 | 1332.513 | 11π/24 | 35 | 1.124×10−4 | 1.121×10−4 | −0.217 |
Tables 5, 6 and 7 compare the predicted and experimental results for the gamma point source efficiency. The predicted results show that the maximum error is 3.732% at 661.661 keV, 11π/24, and 35 cm, whereas that of the other results is less than 3%. Overall, the predicted efficiency of the BP neural network is in good agreement with the experimental data in Tables 5–7. This method, based on the BP neural network, is feasible for determining the gamma point source efficiency, and the predicted efficiency values are accurate and effective. Therefore, the gamma point source efficiency can be determined quickly and accurately using the BP neural network.
CONCLUSION
A method of determining the gamma point source efficiency using a BP neural network was proposed and validated. Compared with the conventional method, it is simple and convenient. Error analysis shows that the predicted results are in good agreement with the experimental data; namely, that the maximum error is 3.732% at 661.661 keV, 11π/24, and 35 cm, whereas that of the other results is less than 3%. This method was reliably and practicably applied to determine the gamma point source efficiency. This method also quickly determines the nonlinear relationships between the efficiency and the radial angle, detection distance, and gamma-ray energy. With increasing radial angle, the gamma point source efficiency decreases as a cosine curve. With increasing detection distance, the gamma point source efficiency decreases as an exponential curve. For gamma-rays with three energies (661.661, 1173.238, and 1332.513 keV), the gamma point source efficiency is inversely proportional to the gamma-ray energy. This method can further predict the gamma point source efficiency for arbitrary nuclides and their position in front of the HPGe detector cap, which is important not only for efficiency calibration experiments but also for the quantitative and qualitative analysis of radionuclides.
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