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The geometry of uranium components is one of the key characteristics and strictly confidential. The geometry identification of metal uranium components was studied using ^{252}Cf source-driven correlation measurement method. For the 3 uranium samples with the same mass and enrichment, there are subtle differences in neutron signals. Even worse, the correlation functions were disturbed by scatter neutrons and include “accidental” coincidence, which is not conductive to the geometry identification. In this paper, we proposed an identification method combining principal component analysis and least-square support vector machine (PCA-LSSVM). The results based on PCA-LSSVM showed that the training precision was 100% and the test precision was 95.83% of the identification model. The total precision of the identification model was 98.41%, which indicated that the identification model was an effective way to identify the geometry properties with the correlation functions.

The geometry of uranium components is one of the key characteristics and strictly confidential [

Correlation measurements used for in the field of nuclear material identification have been developed in recent decades [^{252}Cf sources-driven correlation measurement method is a popular one [

In this paper, the research work of the geometry identification to the metal uranium components has been conducted. The uranium components have the same mass, enrichment, and different geometry. Just because the uranium components only have geometric difference, the geometry identification is very difficult by the correlation functions simply. By analyzing the features and differences of the correlation functions, we proposed an identification method combining principal component analysis and least-square support vector machine (PCA-LSSVM) [

In ^{252}Cf sources-driven correlation measurement method, ^{252}Cf is employed as an interrogating neutron source.^{ 252}Cf source, uranium component, neutron detectors, and relevant data acquisition and display system are included in the measurement setup. Figure ^{252}Cf source, ^{235}U goes through fission readily within 10^{−14} seconds in average and emits gamma rays, neutrons, and photons [^{252}Cf source and the uranium component are chiefly in three possible processes: penetration, scattering, and induced fission. Then the neutrons and gamma rays are detected and recorded by the detector. This method takes advantage of innumerable neutron counts and makes inferences about the properties of uranium components observed by the correlation function between the ^{252}Cf source and detector.

Flowchart of the proposed identification system.

As shown in Figure ^{252}Cf source is placed in an ionization chamber called channel 1 (CH1). The chamber’s operation mode is typically pulse mode so that individual source fission event is observable. Neutrons from the ^{252}Cf source interacting with the uranium component are detected by a scintillator detector, which is placed straightly behind the uranium component and called channel 2 (CH2). The uranium component is placed between the ^{252}Cf source and the scintillator detector.

Thus, we can obtain signals

Figure ^{252}Cf source and a detector measured by ORNL [^{252}Cf spontaneous fission. (b) shows the directly transmitted gamma rays recorded by the detector. (c) means the scattered gamma rays and directly transmitted neutrons. (d) corresponds to the scattered neutrons, the neutrons, and gamma rays from induced fission that are detected lately (e). Some of these divisions are not distinct in time but overlap. For a long time, slow decay (f) will reduce to the background level (g).

Correlation function between ^{252}Cf source and a detector (

Principal component analysis (PCA) is a powerful multivariate statistical technique which usually applied to data treatment of high dimensionality. It is based on the reduction of the variable numbers for a lower value of the p-original ones, in order to represent the characteristics of this data set, however, without loss of information. Therefore, it can cover the combination among the variables as well as grouping of samples. Least squares-support vector machine (LS-SVM) is a method based on the Statistical Learning Theory (SLT) that employs a least-square linear system as a cost function, resembling a regularized network. PCA-LSSVM deals with combination of the feature extraction and classification. Geometry identification of the uranium components is separated into two phases:

Considering high radioactivity of high-enrichment uranium components, the simulation method is commonly used in the first place instead of making some experiments. In fact, the simulation research will help us further improve the performance of real-data experiments. Correlation measurements were carried out by application of MC simulation. Figure ^{252}Cf source and the detector is fixed for all measurements.

Structure of the simulation setup.

The particle transport code, Geant 4, was used for all simulations. Figure ^{252}Cf spontaneous fission, directly transmitted neutrons, the scattered neutrons, and the neutrons from induced fission all are detected by the detector, the peaks appeared in the curve.

Simulation results of the sphere uranium component.

For the neutrons detected by our detectors, we choose part of them whose energy covers 0~20 MeV as the objects to be processed. The average energy of ^{252}Cf source spontaneous fission neutron is 2.13 MeV. The average energy for the induced neutrons is about 2 MeV. Furthermore, neutron energy decreases when neutrons collide with uranium nuclei, which is beneficial to improving the probability of ^{235}U fission reaction. Thus, neutrons with energy less than 2 MeV account for the vast majority. Neutrons with energy ranging 2~20 MeV are very few. Neutrons with energy larger than 20 MeV are rare. That is why we choose 0~20 MeV.

Compared with Figures ^{252}Cf source and detector.

Three kinds of uranium components with the same mass (16.9736 kg) and same enrichment (93.15%) but different geometry are used. Table

Parameters of the 3 uranium components.

Geometry | Dimension | Volume | Surface area |
---|---|---|---|

Sphere | Radius: 6 cm | 9.05 × 10^{2} cm^{3} | 452.39 cm^{2} |

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Cylinder | Radius: 6 cm | 9.05 × 10^{2} cm^{3} | 527.79 cm^{2} |

| |||

Cuboid | Length: 10 cm | 9.05 × 10^{2} cm^{3} | 561.91 cm^{2} |

Figure

Simulation results of neutron correlation function.

To analyze the parameters of 3 uranium components, which have the same enrichment, mass and volume, so they have the same neutron net production. The 3 samples have different geometry, so they have different superficial area. Different superficial areas lead to different neutron leakages. The bigger superficial area has more possibility of stimulating the heavy nuclei and to induce easily more fission. Considering the above, the geometry identification of the 3 samples is very difficult, because there are subtle differences in neutron signals. Even worse, the correlation functions include “real” coincidence and “accidental” coincidence as shown in Figure

Simulation results of “real” and “accidental” coincidence.

In order to identify the geometry of uranium components efficiently, we proposed a method combined principal component analysis and least-square support vector machine (PCA-LSSVM) to build an identification model based on the correlation functions of the 3 different geometry properties. In the above research, the correlation functions were divided into different groups with an energy interval of 0.1 MeV. The sample number of each geometry property was 21. So 21 × 3 = 63 groups of correlation functions were obtained. The 63 groups of correlation functions were shown in Figure

63 groups of correlation functions of 3 different geometry properties.

In Figure

A geometry identification model based on PCA-LSSVM was constructed. Firstly, PCA was applied to extract the feature components from the 63 groups of correlation functions. The feature components were the linear combination of the correlation functions and they were able to characterize the original data well. According to the extraction through PCA, the redundant information was deleted and the dimension was reduced dramatically compared with the original data. The number of the feature components was related to the including information percentage. Normally, an information percentage of no less than 85% contained in feature components was acceptable. In the experiment, the information percentage was selected as 95% and 3 feature components were obtained. The coefficients of the first feature component were shown in Figure

Coefficients of the first feature component.

The coefficients of the first feature component were the weights in characterizing the geometry properties with the correlation functions. When building the identification model of the geometry properties, the feature components were applied to replace the correlation functions as the input of the identification model. The labels of the 3 geometry properties were sets 1, 2, and 3 as output of the model corresponding to sphere, cylinder, and cuboid, respectively. Then LSSVM was applied to train the identification model of the geometry properties. In the model training process, the correlation functions were divided into training set and test set with a proportion of 13 : 8. Radial basis function was selected as the kernel function. The regularization parameter

Identification model of the geometry properties trained by LSSVM.

The training precision was 100% and the test precision was 95.83% of the identification model. The total precision of the identification model was 98.41%, which indicated that the identification model was an effective way to identify the geometry properties with the correlation functions. However, the number of the sample data shown in this paper was not rich enough. In the future work, more data is expected to be achieved to test the reliability and precision of the identification model. Moreover, the identification model can be optimized for the possibility of identifying the geometry properties with different uranium masses and enrichments.

This paper describes a set of simulation experiments for the 3 uranium samples which have the the same mass and enrichment but different geometry. The experiments were performed using ^{252}Cf sources-driven correlation measurement method, based on the correlation functions. However, the correlation functions are disturbed by “accidental” coincidence chiefly from scatter neutrons and showed disagreement in different energy intervals. Therefore, we cannot realize the geometry identification by the correlation functions directly. We proposed further a method combined principal component analysis and least-square support vector machine (PCA-LSSVM). The results based on PCA-LSSVM showed that the training precision was 100% and the test precision was 95.83% of the identification model. The total precision of the identification model was 98.41%, which indicated that the identification model was an effective way to identify the geometry properties with the correlation functions. Obviously, this correlation function based identification method should be feasible not only for these three geometries but also for other geometries; it is one of our follow-up studies.

The authors declare that they have no conflicts of interest.

This work was partially supported by the National Natural Science Foundation of China (no. 11605017) and the Fundamental Research Funds for the Central Universities (no. 10611CDJXZ238826).

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