1 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 ef ficiency[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 dif ficult 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 signi ficance.

2 Gamma point source experiment

2.1 Ef ficiency calculation

The full-energy peak efficiency is de fined 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,A0is 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.

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 2,3 and 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 10-9after 1690 epochs.

where σN，σA0，σλ，and σPare 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%.

你之所以成为现在的你，正是之前所有的兴趣、热情、思想、行动的集合，而有质量的思想和行为必定是以浓厚的兴趣为基础的。

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.

2.2 Experimental measurement

The gamma-ray spectrum is acquired from the spectrometer system,which consists of an HPGe detector and a multi-channel 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–10 MeV.Other parameters of the HPGe detector structure are shown in Table 1.

这天午后，柳红在镇上遇到白玉儿——也就是白天明的妹妹，烂眼阿根的儿媳妇，她丈夫张翔也在城里打工。白玉儿把柳红拉到一边，不无神秘地悄悄问她：“你知不知道你丈夫他们干的好事？”柳红知道她说的是苏石被几个城里的小流氓打了的事。白玉儿就是这种女人，就喜欢看到别人的不好，别人难过她就开心。但白玉儿说才不是这么回事呢。前几天张翔打电话回家，她听到他在电话里和她公公说，苏石和她哥白天明是在洗脚屋里被抓的。

Fig.1 Ef ficiency of HPGe detector versus energy

Fig.2(Color online)Geometry of HPGe detector with a gamma point source

Table 1 Parameters of HPGe detector structure

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 include137Cs and60Co:(a)137Cs(3.343×105Bq),with a photon emission probability at 661.661 keV of 85%;and(b)60Co(2.070×105Bq),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.

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.

3 BP neural network prediction

3.1 BP neural network model

When the detection distance is less than 15 cm,the measurement accuracy of the point source efficiency is strongly in fluenced 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 ef ficiency at 661.661 keV for137Cs and at 1173.238 and 1332.513 keV for60Co is listed in Tables 2,3 and 4,respectively.

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.

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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.

Fig.3(Color online)Prediction model of BP neural network

Fig.4 Training error curve of BP neural network

3.2 Model parameters

When the BP neural network model is trained,the gamma point source efficiency can be determined by inputting the radial angle,detection distance,and gammaray energy.The predicted results of the gamma point source efficiency for 661.661,1173.238,and 1332.513 keV are shown in Fig.5.

90例骨质疏松性椎体压缩骨折患者中男29例、女61例,年龄54-89岁、平均(74.23±1.17)岁,发病至手术间隔时间1-3d、平均(1.69±0.21)d。利用随机数字表法将入选90例骨质疏松性椎体压缩骨折患者均分为研究组(n=45)、对照组(n=45),各组上述相关数据对比P>0.05(具有可比性)。

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Fig.5(Color online)Ef ficiency of BP neural network prediction for a137Cs(661.661 keV),b60Co(1173.238 keV),and c60Co(1332.513 keV)

By neglecting the in fluence of the time(t and T)and detection distance(d),the uncertainty in the experimental full-energy peak efficiency σεcan be determined by the uncertainties in N(E),A0,λ,and P(E),and it can be calculated as

4 Results and discussion

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

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 gammarays are attenuated in the air,so the gamma point source efficiency decreases quickly.The predicted efficiency changes less at angles of 0–4π/24 and decreases almost linearly at angles of 4π/24–11π/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 differentgamma-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

根据上文对于医药B2C平台顾客忠诚度影响因素的分析，本文提出如下提升医药B2C平台顾客忠诚度的策略，来帮助医药B2C平台更好地进行经营管理。

Table 5 Comparison between experimental measurement and BP neural network prediction(661.661 keV)

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

Table 6 Comparison between experimental measurement and BP neural network prediction(1173.238 keV)

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

where δ is the error,εBPis the efficiency predicted by the BP neural network,and εEXis the experimentally measured efficiency.The experimental measurements and BP neural network predictions are compared in Tables 5,6 and 7.

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%at661.661 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 5,6 and 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.

Table 7 Comparison between experimental measurement and BP neural network prediction(1332.513 keV)

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

5 Conclusion

A method of determining the gamma point source ef ficiency 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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