Monte Carlo simulation of the PGNAA device

Based on the theory of neutron-gamma distribution, a PGNAA measurement model was established using the Monte Carlo method. As shown in Fig. 2, the model barrel, made of polyethylene, has a cylindrical structure. The PGNAA device was located in the center of the model barrel and consisted of a deuterium tritium (D-T) neutron generator, a BGO detector, and a shielding layer. The dimensions of the neutron generator were 7 cm×70 cm (diameter × height). The BGO crystals had dimensions of 7.62 cm×7.62 cm (diameter × height). To improve the performance of the PGNAA device, parameters including the size of the neutron moderator, the thermal neutron absorption material, and the dimensions of the device must be optimized.

Fig. 2Full-screen View

(Color online) Schematic of the Monte Carlo simulation model

Neutrons entering a gamma detector cause activation noise [31]. To minimize the activation noise, a combination of heavy metals and the Li₂CO₃ scheme was employed for neutron shielding. Heavy metals were used to thermally decelerate the 14 MeV energy neutrons produced by the D-T neutron generator. Subsequently, thermal neutrons are absorbed by Li₂CO₃.

Tungsten is an ideal material for slowing neutrons via inelastic scattering. These reactions can rapidly reduce the neutron energy and slow fast neutrons to thermal neutrons [32, 33]. The fast neutron flux was statistically computed by varying the thickness of tungsten. The decrease in fast neutron flux with increasing tungsten thickness is shown in Fig. 3. When the tungsten thickness reached 7 cm, the shielding effect on fast neutrons reached 90%. Considering the moderating effect and model dimensions, the tungsten thickness was set to 7 cm.

Fig. 3Full-screen View

(Color online) Relationship between fast neutron flux and W thickness

Li₂CO₃, as a thermal neutron absorption material, was placed between the tungsten and the gamma detector to minimize neutron entry into the gamma detector [34]. The detector neutron fluxes with different Li₂CO₃ thicknesses were calculated using the F4 card, as shown in Fig. 4. As the Li₂CO₃ thickness increased, the neutron flux gradually decreased. At a thickness of 5 cm, the neutron flux was significantly reduced. The detector was effectively protected when its thickness was 5 cm. Therefore, an Li₂CO₃ thickness of 5 cm was used.

Fig. 4Full-screen View

(Color online) Relationship between neutron flux and Li₂CO₃ thickness

The radius and height of the model barrel were varied to determine the optimal structure. The characteristic peak region of copper in the gamma-ray spectrum ranged from 7.0 MeV to 8.5 MeV. The gamma ray count in this region obtained using the BGO detector verified this result. Based on the simulation, the count of the characteristic peak region of copper increased and then leveled off at approximately 70 cm, as shown in Fig. 5(a). The height of the setup was therefore set to 70 cm. Figure 5(b) shows the relationship between the count of the characteristic peak regions of copper and the radius of the model barrel. The count rapidly increased at first and then became saturated at approximately 50 cm. Therefore, the radius of the setup was set as 50 cm.

Fig. 5Full-screen View

(Color online) Relationship between the gamma-ray counts and device dimensions: (a) height; (b) radius

ViT model

Transformer models, which represent a neural network architecture based on self-attention mechanisms, have been widely applied to natural language processing (NLP) and other sequence modeling tasks [35]. The ViT model was found to surpass a convolutional neural network (CNN) model on extremely large-scale datasets by entirely replacing the convolutional structure with a transformer structure in classification tasks [36].

The ViT model structure is illustrated in Fig. 6. A one-dimensional gamma energy spectrum was used as the input dataset for the model. The entire spectrum was divided into 32 segments, where each segment corresponded to positional encoding from 0 to 31 using the tokenization method from NLP. Subsequently, the data were fed into the encoder module for calculation. The encoder consisted of six identical stacked transformer blocks. Each transformer block included a multihead attention mechanism sublayer and a multilayer perceptron sublayer. In the multi-head attention mechanism sub-layer, multiple self-attention mechanism modules were employed in parallel to enhance the attention diversity and improve the expressive capacity of the model. Before the data entered each sublayer, they were normalized to ensure that the mean and variance were relatively stable. This step can accelerate the model training process. When the computation within the sublayer was completed, the residual connections that retain the original input information were employed to directly merge the outputs and inputs of the sublayer. All transformer blocks in the ViT encoder were configured with the same output dimensions. The output of each transformer block was encoded and fed into the MLP classification head, which ultimately resulted in category prediction of the input spectrum.

Fig. 6Full-screen View

(Color online) Schematic of the ViT model

Hyperparameter tuning

To evaluate the classification performance of the ViT model, two additional classifiers (Long short-term memory (LSTM) and CNN) were built in Python based on the PyTorch library. All three models were trained and tested using the same input dataset. Because the selection of hyperparameters significantly effects the ViT model for grade identification, the hyperparameters must be optimized. A grid search is an effective hyperparameter optimization strategy that can exhaustively search for the optimized hyperparameters of a model. During the grid-search process, a multidimensional grid with successive candidate parameter values was generated, allowing the enumeration of combinations of tunable parameters. Subsequently, the performance metrics of the model were calculated for different parameter combinations. The parameter combination with the best performance metrics was selected as the final model parameter. Table 1 lists the hyperparameter optimization results for each model.

Model evaluation

To compare and determine the best-performing classification model, the input dataset was divided into training and test sets. After training with the training set, the models were evaluated using the test set to determine the number of true positives (TPs), false positives (FPs), true negatives (TNs), and false negatives (FNs). The following evaluation metrics were calculated to measure the performance of the classifier models [37]:

Accuracy refers to the proportion of samples correctly predicted by the model relative to the total number of samples. Accuracy represents the overall predictive accuracy of the classifier for all samples, which can be calculated using Eq. (1). $Accuracy=\frac{\text{TP}+\text{TN}}{\text{TP}+\text{FP}+\text{TN}+\text{FN}}$ (1) Precision is the proportion of all samples predicted to be positive by the classifier that are true positives. Recall is the proportion of all TP samples that the classifier correctly predicts as positive. Precision measures how accurately a model predicts a positive class, whereas recall assesses the coverage of the classifier and the risk of losing positive samples. These metrics can be calculated using Eqs. (2) and Eqs. (3), respectively. Typically, there is a trade-off between these metrics, where increasing precision often tends to decrease recall, and vice versa. Therefore, when evaluating the performance of a model, it is necessary to select an appropriate threshold or adjust the strategy of the classifier according to the actual requirements to achieve optimal results. $Recall=\frac{\text{TP}}{\text{TP}+\text{FN}}$ (2) $Precision=\frac{\text{TP}}{\text{TP}+\text{FP}}$ (3) The F₁ score combines the precision and recall of the positive class by calculating their harmonic mean of the precision and recall as the evaluation metric. The F₁(-) score was used to measure the classification performance of the model for the negative class; by calculating the F₁ score for negative samples. The F₁ and F₁(-) scores are calculated using Eqs. (4) and Eqs. (5), respectively, and these metrics provide a more comprehensive assessment of model performance in different classes. $F_1=2\times \frac{1}{\frac{1}{Precision}+\frac{1}{Recall}}=\frac{\text{2TP}}{\text{2TP}+\text{FP}+\text{FN}}$ (4) $F_1(-)=\frac{\text{2TN}}{\text{2TP}+\text{FP}+\text{FN}}$ (5) Macro averaging is an evaluation method used for multiclass problems that uses metrics such as precision, recall, F₁ score from each class to calculate their arithmetic mean as the final metric value. For example, the macro-average precision can be calculated using Eq. (6). The macro average provides the average performance of the model across all classes, which does not consider differences in the number of samples in each class, but treats each class as equally significant. $MacroPrecision=\frac{1}{n}{\displaystyle \sum _{i=1}^n}Precisio{n}_i$ (6)

Dataset

To construct the dataset required for machine learning, the gamma spectrum of copper ore was obtained using MCNP. The material card of the MCNP was set according to the actual copper mineral composition [38]. The primary elements in the Cu ore and their contents are listed in Table 2. In total, 4400 energy spectrum data points were obtained. Based on the copper content, the data were divided into five categories: gangue (0 to 0.2%), industrial-grade copper ore (0.2 to 0.5%), low-grade copper ore (0.5 to 1.5%), medium-grade copper ore (1.5 to 2%), and high-grade copper ore (2 to 3%).

Cu is associated with minerals such as pyrite (FeS₂), sphalerite (ZnS), galena (PbS), and cobaltite (CoAsS) [39]. Therefore, copper ores often contain associated minerals, such as Pb, Zn, Fe, and Co. Machine-learning models were used to identify the presence of associated minerals in the copper ore. Ores containing Pb, Zn, Fe, and Co below the cut-off grade were designated as gangue, whereas those above the cut-off grade were designated as minerals, based on the content of the associated elements in the ore. The cut-off grades for each associated element are listed in Table 3.

Gamma spectrum analysis of copper ore

The gamma ray spectra of copper ores with different grades obtained by Monte Carlo simulations are shown in Fig. 7. The energy range was 0–10.0 MeV. Characteristic peaks of major elements such as Ca (1.94 MeV), Si (3.54 MeV), and O (6.13 MeV) were observed. The characteristic peaks of Cu (7.63 MeV and 7.91 MeV) were inconspicuous, primarily because of the lower copper content in the ore and the low detection efficiency of BGO at high energies. From Fig. 7(b), the gamma-ray spectrum counts for different copper ore grades exhibited differences in the energy range of 7.0–8.5 MeV. This was mainly due to the contribution of the captured gamma rays produced by the reaction between the copper and neutrons.

Fig. 7Full-screen View

(Color online) Gamma energy spectrum of copper ores with different grades: (a) energy range of 0-10 MeV; (b) energy spectrum in the energy range of 6-9 MeV

Identification of copper ore grade

The proposed ViT model was used for copper ore grade identification based on the model parameters obtained through the grid search method. To evaluate the classification performance of the proposed ViT model, the LSTM and CNN models were chosen for comparison. The datasets were randomly divided into ten subsets. One subset was selected to validate the model, and the other nine subsets were selected to train the model. A five-repetition ten-fold cross-validation method [40] was used to ensure the robustness of the classification results. Table 4 presents the results of copper ore grade identification using the LSTM, CNN, and ViT models.

As shown in Table 4, the identification provided by the ViT model was better than those of the CNN and LSTM models. Among them, LSTM had the lowest accuracy, precision, recall, and F₁ score. Although LSTM can learn nonlinear models, the identification results were unsatisfactory. The ViT model exhibited the best evaluation metrics, with average accuracy, precision, recall, F₁ score, and F₁(-) score values of 0.9795, 0.9637, 0.9614, 0.9625, and 0.9942, respectively. The results of the CNN model were similar to ViT model, with average accuracy, precision, recall, F₁ score, and F₁(-) score values of 0.9659, 0.9396, 0.9450, 0.9422, and 0.9899, respectively. LSTM performed the worst of the three models, with accuracy, precision, recall, F₁ score, and F₁(-) score values of 0.9455, 0.9210, 0.9272, 0.9236, and 0.9854, respectively. This analysis indicates that the ViT model is the most effective of the three methods for copper ore grade identification.

Figure 8 shows the confusion matrix for the copper ore grade identification. The diagonal elements of the confusion matrix denote the proportions of accurately classified copper ore grades. Combined with Table 4, it can be inferred that the highest accuracy rate was achieved for gangue and industrial-grade copper ores. All three models had accuracies of greater than 96%. Such high accuracy is attributed to the significant mineral differences between gangue and industrial-grade copper ores as well as their typical ore structures. High-grade copper ores exhibited the highest rate of misidentification and were often misclassified as medium-grade ores. There are two reasons for this finding. The first is that the difference between high- and medium-grade ores is small and easily confused, and the second is that high-grade copper ores often contain Fe. Because the characteristic peaks of Fe (7.631 MeV and 7.645 MeV) are similar to those of copper (7.64 MeV and 7.91 MeV), Fe affects the identification of high-grade copper ores. In identifying medium- and high-grade copper ores, the ViT model outperformed the LSTM and CNN models with a prediction accuracy of more than 92%.

Fig. 8Full-screen View

(Color online) Confusion matrices for the copper ore grade identification of different models: (a) LSTM; (b) CNN; (c) ViT

Identification of associated minerals

In the identification of associated minerals, the positive class was associated with minerals and the negative class was associated with gangue. Minerals associated with copper mines include Pb, Zn, Fe, and Co. Table 5 shows the identification results of the LSTM, CNN, and proposed ViT models for the minerals associated with copper ore. As shown in Table 5, the identification of the ViT model of associated minerals was significantly better than those of the other two models, and the performance ranking was ViT > CNN > LSTM. ViT had the best evaluation indicators, with average accuracy, precision, recall, F₁ score, and F₁(-) score values of 0.9222, 0.9278, 0.8919, 0.9091, and 0.9308, respectively. This analysis shows that ViT is the most effective method for classifying and identifying associated minerals and gangue.

Figure 9 shows the confusion matrix of the associated mineral identification using the LSTM, CNN, and ViT models. This figure shows that Fe is most accurately identified among all elements. A possible reason for this is that the capture cross-section of Fe is large and the boundary grade is high. Co has the highest rate of misclassification. The LSTM, CNN, and ViT models often failed to identify the presence of Co in minerals. This is because the cutoff grade of Co was low (0.01%). Therefore, the low intensity of the characteristic gamma rays induced by the reaction between Co and neutrons makes it difficult to determine the presence of Co in the ore. The ViT model outperformed the other classification methods in identifying Co. In general, the ViT method achieved the best classification performance, particularly for Fe. The main reason for this is that, compared with other competing classifiers, the ViT model employs a self-attention mechanism to establish a global association of energy spectrum data, thus allowing the model to capture information from the entire gamma energy spectrum and improve the accuracy of identification.

Fig. 9Full-screen View

Confusion matrix plots for the associated mineral identification of three models: (a)–(d) LSTM; (e)–(h) CNN; (i)–(l) ViT