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Hall sensor angle error and relative position calibrations for cryogenic permanent magnet undulator of high energy photon source test facility (HEPS-TF)

  • Ling-Ling Gong
  • Wan Chen
  • Wen Kang
  • Shu-Chen Sun
  • Zhi-Qiang Li
  • Lei Zhang
  • Yu-Feng Yang
  • Hui-Hua Lu
  • Xiao-Yu Li
  • Shu-Tao Zhao
  • Xiang-Zhen Zhang
  • Ya-Jun Sun
Review

Abstract

Purpose

A new in-vacuum three-dimensional Hall probe magnetic measurement system is under fabrication for characterizing the magnetic performance of the Cryogenic Permanent Magnet Undulator (CPMU). In order to fit the small gap (5 mm) of magnetic structure and vacuum environment, a small three-dimensional Hall probe has been manufactured. The angular and positional misalignment errors of the Hall sensors play an important role in the measurement accuracy of the CPMU. In order to minimize the misalignment errors, a method of calibrating angle error and relative assembly displacements of a three-dimensional Hall probe is carried out.

Methods

The angle error of Hall sensors will be calibrated by a standard dipole magnet and a five-dimensional Hall bench. The standard dipole magnet will generate a single direction and uniform magnetic field. And the five-dimensional Hall bench is used to rotate the Hall probe which is put in the center of magnet. Based on the relationship between angle and magnetic field strength, the angle error of each Hall sensor will be obtained. The relative position between the sensitive areas of the Hall sensors will be calibrated by a two-dimensional magnetic field undulator section. Based on Maxwell’s equations, through the calculation of measurement magnetic field strength, the relative assembly displacements of the three Hall sensors can be derived.

Results

The details of the calibration methods and the data processing of angle error and relative assembly displacements of a three-dimensional Hall probe are presented. The three-dimensional magnetic fields of a cryogenic permanent magnet undulator can be received accurately by correcting these angle errors and position errors of Hall sensors.

Conclusions

This paper illustrates the relative position and angle calibration procedures and the data processing of a three-dimensional Hall probe. Now the design of a smaller Hall probe is in process. The calibration of the angle errors and position errors will be carried out after the fabrication of the standard dipole magnet.

Keywords

Magnetic measurement system Cryogenic permanent magnet undulator Three-dimensional Hall probe Calibration of Hall probe 

PACS

07.85.Qe 41.60.Ap 52.59.Px 

Introduction

Chinese High Energy Photon Source Test Facility (HEPS-TF) is a 6 GeV third-generation synchrotron radiation facility with ultralow emittance and extremely high brightness [1, 2]. A Cryogenic Permanent Magnet Undulator (CPMU) will be installed to deliver a high-performance X-ray [1, 3, 4]. The CPMU is a full-scale in-vacuum undulator with a period of 13.5 mm and a magnetic length of 2 m. By adopting a gap of 5 mm and cryogenic temperatures of 85 K, the target peak field achieved will be 1 T with the RMS of Phase errors less than \(3{^{\circ }}\) and the first field integrals less than \(100\mathrm{Gs}\cdot \mathrm{cm}\) [5, 6].

Since the RMS phase error and the field integrals of CPMU have a strong influence on the spectral flux and the closed orbit, the precise measurement of the magnetic quantities is essential to characterize magnetic errors [7]. A new in-vacuum Hall probe measurement system which is used to measure the CPMU magnetic field is under development [6].

As the gap is limited to 5 mm and the operating environment is vacuum condition, a new three-dimensional Hall probe, which is shown in Fig. 1, has been manufactured instead of the old one which is used in atmospheric environment [8]. Three AKM HG-116C [9] Hall sensors are welded on a ceramic printed circuit board. The dimensions of Hall sensor are 2.5 mm by 1.5 mm by 0.6 mm. In order to accurately measure the three orthogonal field components of the same point of the CPMU, the misalignment angle error and relative position of the Hall sensors sensitive areas should be calibrated and readjusted exactly [10, 11, 12]. The angle error of Hall sensors will be calibrated by a standard dipole magnet. The relative position between the sensitive areas of the Hall sensors will be calibrated by a two-dimensional magnetic fields undulator section.
Fig. 1

The three-dimensional Hall probe with ceramics printed circuit board. From top to bottom are HX, HY, and HZ hall sensors which are used to measure the horizontal, vertical, and longitudinal magnetic field, respectively. The PT100 is a platinum resistance thermometer sensor

Fig. 2

Schematic diagram of the angle error \({\theta }\) between Hall sensor and magnetic field

Hall sensors angle error calibrations

As the output voltage of a Hall sensor is proportional to the magnetic field strength in the vertical direction of this sensor, so once there is angle error \({\theta }\) between Hall sensor and magnetic field, shown as in Fig. 2, the measured magnetic field of Hall sensor is only the component of magnetic field strength:
$$\begin{aligned} { B}_\mathrm{m} = B\cdot \mathrm{cos}( \theta ) \end{aligned}$$
(1)
where \(B_\mathrm{m}\) is the measured magnetic field of Hall sensor, and B is the real magnetic field. The angle error \({\theta }\) will lead to a relative magnetic measurement error:
$$\begin{aligned} {\sigma }=\left| {\frac{B_\mathrm{m} -B}{B}} \right| =1-\mathrm{cos}\left( \theta \right) \end{aligned}$$
(2)
As the error of the peak field measurement is not more than \(2\times 10^{-4}\) T [6], the angle error \({\theta }\) should be less than \(1.15^{^{\circ }}\), which is difficult to realize for the manual welding Hall probe. Therefore, the angle error of Hall sensors must be calibrated.
Equation 1 illustrates the relationship between \(B_\mathrm{m}\) and B of a Hall sensor placed in a one-dimensional magnetic field space. However, in the measurement application, the output voltage of the Hall sensor is a collection of the three magnetic field components where there exist angle errors between Hall sensors and magnetic fields. The relationship between the measured magnetic fields and the real ones can be indicated as:
$$\begin{aligned} B_{\mathrm{mx}}= & {} a_{11} \cdot B_\mathrm{x} +a_{12} \cdot B_\mathrm{y} +a_{13} \cdot B_\mathrm{z} \end{aligned}$$
(3)
$$\begin{aligned} B_{\mathrm{my}}= & {} a_{21} \cdot B_\mathrm{x} +a_{22} \cdot B_\mathrm{y} +a_{23} \cdot B_\mathrm{z} \end{aligned}$$
(4)
$$\begin{aligned} B_{\mathrm{mz}}= & {} a_{31} \cdot B_\mathrm{x} +a_{32} \cdot B_\mathrm{y} +a_{33} \cdot B_\mathrm{z} \end{aligned}$$
(5)
where \(B_{\mathrm{mx}} \), \(B_\mathrm{my}\), and \(B_{\mathrm{mz}}\) are the measured magnetic field of HX, HY, and HZ Hall sensors, respectively. \(B_{\mathrm{x}}\), \(B_{\mathrm{y}}\) and \(B_{\mathrm{z}}\) correspond to the horizontal, vertical and longitudinal magnetic field of a space.
Table 1

The main parameters of standard dipole magnet which is used to implement the angular calibration of Hall probe

Parameter

Value

Gap

82 mm

Maximum field

1.3 T

Good field region from vertical and horizontal plane

\(\pm 15\,\hbox {mm}\)

Quality of good field

\(5\times 10^{-4}\)

Fig. 3

The schematic diagram of Hall sensors angle error calibrations

These angle errors \(a_{11} \) to \(a_{33} \) form a matrix T:
$$\begin{aligned} T=\left( {{\begin{array}{c@{\quad }c@{\quad }c} {a_{11} }&{}\quad {a_{12} }&{}\quad {a_{13} } \\ {a_{21} }&{}\quad {a_{22} }&{}\quad {a_{23} } \\ {a_{31} }&{}\quad {a_{32} }&{}\quad {a_{33} } \\ \end{array} }} \right) \end{aligned}$$
(6)
To obtain these angle errors, a standard dipole magnet is required. Table 1 shows the parameters of the standard dipole magnet which is under processing. The first step is to align and adjust the plane of the Hall probe perpendicular to the magnetic field direction of dipole magnet by a five-dimensional Hall bench [8]. Suppose that the Hall probe is placed in a space with only magnetic field of \(B_\mathrm{y} \) as shown in Fig. 3a. The relationship between the measured magnetic field and the real values is:
$$\begin{aligned} \left( {{\begin{array}{l} {B_{\mathrm{mx}} } \\ {B_\mathrm{my} } \\ {B_\mathrm{mz} } \\ \end{array} }} \right) =\left( {{\begin{array}{c@{\quad }c@{\quad }c} {a_{11} }&{}\quad {a_{12} }&{}\quad {a_{13} } \\ {a_{21} }&{}\quad {a_{22} }&{}\quad {a_{23} } \\ {a_{31} }&{}\quad {a_{32} }&{}\quad {a_{33} } \\ \end{array} }} \right) \cdot \left( {{\begin{array}{l} 0 \\ \mathrm{B} \\ 0 \\ \end{array} }} \right) \end{aligned}$$
(7)
where B is the magnetic field of dipole magnet. The angle errors, \(a_{12},a_{22}\) and \(a_{32}\) can be obtained by:
$$\begin{aligned} a_{12}= & {} \frac{B_{\mathrm{mx}} }{B}\end{aligned}$$
(8)
$$\begin{aligned} a_{22}= & {} \frac{B_{\mathrm{my}} }{B}\end{aligned}$$
(9)
$$\begin{aligned} a_{32}= & {} \frac{B_{\mathrm{mz}} }{B} \end{aligned}$$
(10)
Then, the Hall probe is rotated \(90^{\circ }\) around the \({z}^{\prime }\) axis of itself. Suppose the Hall probe is placed in a space with only magnetic field of \(\hbox {B}_{\mathrm{x}} \) as shown in Fig. 3b. The relationship of measurement magnetic field and real magnetic field is:
$$\begin{aligned} \left( {{\begin{array}{l} {B_\mathrm{mx} } \\ {B_\mathrm{my} } \\ {B_\mathrm{mz} } \\ \end{array} }} \right) =\left( {{\begin{array}{c@{\quad }c@{\quad }c} {a_{11} }&{}\quad {a_{12} }&{}\quad {a_{13} } \\ {a_{21} }&{}\quad {a_{22} }&{}\quad {a_{23} } \\ {a_{31} }&{}\quad {a_{32} }&{}\quad {a_{33} } \\ \end{array} }} \right) \cdot \left( {{\begin{array}{l} B \\ 0 \\ 0 \\ \end{array} }} \right) \end{aligned}$$
(11)
the angle errors, \(a_{11} , a_{21} \) and \(a_{31} \) can be obtained by:
$$\begin{aligned} a_{11}= & {} \frac{B_{\mathrm{mx}} }{B} \end{aligned}$$
(12)
$$\begin{aligned} a_{21}= & {} \frac{B_{\mathrm{my}} }{B} \end{aligned}$$
(13)
$$\begin{aligned} a_{31}= & {} \frac{B_{\mathrm{mz}} }{B} \end{aligned}$$
(14)
Finally, the Hall probe is rotated \(90^{\circ }\) around the \({y}^{\prime }\) axis of itself. Suppose the Hall probe is placed in a space with only magnetic field of \(B_{\mathrm{z}} \) as shown in Fig. 3c. The relationship of measurement magnetic field and real magnetic field is:
$$\begin{aligned} \left( {{\begin{array}{l} {B_{\mathrm{mx}} } \\ {B_\mathrm{my} } \\ {B_\mathrm{mz} } \\ \end{array} }} \right) =\left( {{\begin{array}{c@{\quad }c@{\quad }c} {a_{11} }&{}\quad {a_{12} }&{}\quad {a_{13} } \\ {a_{21} }&{}\quad {a_{22} }&{}\quad {a_{23} } \\ {a_{31} }&{}\quad {a_{32} }&{}\quad {a_{33} } \\ \end{array} }} \right) \cdot \left( {{\begin{array}{l} 0 \\ 0 \\ \mathrm{B} \\ \end{array} }} \right) \end{aligned}$$
(15)
the angle errors, \(a_{13} \), \(a_{23} \) and \(a_{33} \) can be obtained by:
$$\begin{aligned} a_{13}= & {} \frac{B_{\mathrm{mx}} }{B}\end{aligned}$$
(16)
$$\begin{aligned} a_{23}= & {} \frac{B_{\mathrm{my}} }{B}\end{aligned}$$
(17)
$$\begin{aligned} a_{33}= & {} \frac{B_{\mathrm{mz}} }{B} \end{aligned}$$
(18)
The matrix T is obtained through the above calibration method. After finishing the measurement of the CPMU, the real magnetic field of CPMU is calculated by the measured magnetic field with the formula:
$$\begin{aligned} \left( {{\begin{array}{c} {B_\mathrm{x}} \\ {B_\mathrm{y}} \\ {B_\mathrm{z}} \\ \end{array} }} \right) = \mathrm{T}^{-1}\cdot \left( {{\begin{array}{c} {B_\mathrm{mx} } \\ {B_\mathrm{my} } \\ {B_\mathrm{mz} } \\ \end{array} }} \right) \end{aligned}$$
(19)

Hall sensors relative position calibrations

In order to obtain the accuracy magnetic field of CPMU using the mathematical expression (19), the three Hall sensors with overlapped sensitive areas is preferred. However, in the actual situation, the three Hall sensors have to be placed apart due to the fabrication and welding. The three Hall sensors are arranged along the Z axis. There are spacing, \({\delta }_{\mathrm{{z}^{\prime }}(\mathrm{H}1-\mathrm{H}2)} \) (H1, H2 = HX, HY, HZ), among three Hall sensors. Owing to the manual welding error, there are small distances, \({\delta }_{\mathrm{{x}^{\prime }}(\mathrm{H}1-\mathrm{H}2)} \) and \({\delta }_{\mathrm{{y}^{\prime }}(\mathrm{H}1-\mathrm{H}2)} \), among three Hall sensors. Figure 4 shows the relative position between sensitive areas of the Hall sensors, which will be calibrated on a two-dimensional magnetic fields undulator section as shown in Fig. 5.
Fig. 4

The relative position between sensitive areas of Hall sensors. The coordinate system of Hall probe is (\({x}^{\prime },\,{y}^{\prime },\,{z}^{\prime })\). Use the HY Hall sensor as a reference

Fig. 5

View of the two-dimensional magnetic fields undulator section model

Fig. 6

The schematic diagram of position errors calibration. a The \(B_{{\mathrm{y}}}^{\prime }\) sensor is used to measure \({\hbox {B}}_{\mathrm{y}}\) field and the \({\hbox {B}}_{\mathrm{z}}^{\prime }\) sensor is used to measure \({\hbox {B}}_{\mathrm{z}}\) field. b The \({\hbox {B}}_{{\mathrm{x}^{\prime }}}\) sensor is used to measure \({\hbox {B}}_{{\mathrm{y}}}\) field and the \({\hbox {B}}_{{\mathrm{y}}^{\prime }}\) sensor is used to measure \(B_{\mathrm{z}}\) field. c The \({B}_{{\mathrm{x}}^{\prime }}\) sensor is used to measure \({B}_{{\mathrm{y}}}\) field and the \(B_{{{\mathrm{z}}^{\prime }}} \) sensor is used to measure \({B}_{{\mathrm{z}}}\) field

The magnetic field of an ideal two-dimensional magnetic fields undulator can be expressed as \(\textit{B} ({\mathrm{x}} ,{\mathrm{y}} ,{\mathrm{z}} ) = (0, {B}_{\mathrm{y}} ({\mathrm{y}} ,{\mathrm{z}} ),{ {B}}_{\mathrm{z}} (\mathrm{y} ,\mathrm{z}))\). The space magnetic field of the two-dimensional undulator should fulfill the Maxwell’s equations [7, 11, 13, 14]:
$$\begin{aligned}&\nabla \cdot \vec {B}(\vec {r}) = 0 \end{aligned}$$
(20)
$$\begin{aligned}&\nabla \times \vec {B}(\vec {r}) = 0 \end{aligned}$$
(21)
In the Descartes rectangular coordinate system, the magnetic field \((0, { B}_{\mathrm{y}}, { B}_{\mathrm{z}})\) of the two-dimensional magnetic fields undulator will meet the following formula:
$$\begin{aligned} f_1 \left( {y,z} \right)= & {} \frac{\partial B_\mathrm{y}}{\partial \mathrm{y}}+\frac{\partial B_\mathrm{z}}{\partial \mathrm{z}}=0 \end{aligned}$$
(22)
$$\begin{aligned} f_2 \left( {y,z} \right)= & {} \frac{\partial { B}_\mathrm{y} }{\partial \mathrm{z}}-\frac{\partial { B}_z }{\partial \mathrm{y}}=0 \end{aligned}$$
(23)
One defines the function:
$$\begin{aligned} f^{2}\left( {y,z} \right) =f_1^2 \left( {y,z} \right) +f_2^2 \left( {y,z} \right) =0 \end{aligned}$$
(24)
Assuming the misalignments between the three sensors can be ignored, the magnetic field measured in a rectangular region (S) as shown in Fig. 5 should meet the following equation:
$$\begin{aligned} {\uptau }=\sqrt{\frac{\mathop \int \nolimits _s f^{2}({y,z})\mathrm{d}{} { s}}{s}}=0 \end{aligned}$$
(25)
The Hall probe is placed primly in the undulator section as shown in Fig. 6a. The \({\tau }\) is obtained after mapping the magnetic field in the rectangular region (S). Due to the position errors \({\delta }_{{{\mathrm{y}}^{\prime }}({\mathrm{HZ}}-{\mathrm{HY}}) }\) and \({\delta }_{\mathrm{{z}^{\prime }}({\mathrm{HZ}}-\mathrm{HY})} \), the \({\tau }\) is unequal to zero. By moving the coordinate of \({ {B}}_\mathrm{mz}\), a set of solutions \( (\delta _1,\delta _2 )\) will meet the formulas (26) and (27) to minimize the function \({\tau }\).
$$\begin{aligned} f_1 \left( {y,z} \right)= & {} \frac{\partial B_\mathrm{my} }{\partial \mathrm{y}}+\frac{\partial B_\mathrm{mz} \left( {y-{\delta }_1 ,z-{\delta }_2 } \right) }{\partial \mathrm{z}} \end{aligned}$$
(26)
$$\begin{aligned} f_2 \left( {y,z} \right)= & {} \frac{\partial B_\mathrm{my} }{\partial \mathrm{z}}-\frac{\partial B_\mathrm{mz} \left( {y-{\delta }_1 ,z-{\delta }_2 } \right) }{\partial \mathrm{y}} \end{aligned}$$
(27)
The error \({\delta }_{{\mathrm{y}}^{\prime }({\mathrm{HZ}}-{\mathrm{HY}})} \) will obtained by calculating \(\delta _1\), and the error \({\delta }_{{\mathrm{z}}^{\prime }({\mathrm{HZ}}-{\mathrm{HY}})} \) will obtained by calculating \(\delta _2\). The data processing technique uses the MATLAB code. The detailed MATLAB computational flow diagram is shown in Fig. 7.
Fig. 7

The diagram of MATLAB computational flow. The parameter ‘step’ means magnetic field measurement step, parameter ‘a’ means maximum moving range of coordinate

Then, the Hall probe is rotated around the \({{z}^{\prime }}\) axis and \({x}^{\prime }\) axis of itself which is shown in Fig. 6b. By moving the coordinate of \(B_\mathrm{mx} \), one can also find a set of solutions \((\delta _1 ,\delta _2 )\) meet the formulas (28) and (29) to minimize the function \({\tau }\).
$$\begin{aligned} f_1 \left( {x,y} \right)= & {} \frac{\partial B_\mathrm{my} }{\partial \mathrm{y}}+\frac{\partial B_\mathrm{mx} \left( {x-{\delta }_1 ,y-{\delta }_2 } \right) }{\partial \mathrm{x}} \end{aligned}$$
(28)
$$\begin{aligned} f_2 \left( {x,y} \right)= & {} \frac{\partial B_\mathrm{my} }{\partial \mathrm{x}}-\frac{\partial B_\mathrm{mx} \left( {x-{\delta }_1 ,y-{\delta }_2 } \right) }{\partial \mathrm{y}} \end{aligned}$$
(29)
The error \({\delta }_{\mathrm{{x}^{\prime }}({\mathrm{HX}}-{\mathrm{HY}})} \) will be obtained by calculating \(\delta _1 \), and the error \({\delta }_{\mathrm{{y}^{\prime }}({\mathrm{HX}}-{\mathrm{HY}})}\) will be obtained by calculating \(\delta _2\).
Finally, rotate the Hall probe around the \({{ x}}^{\prime }\) axis of itself as shown in Fig. 6c. By moving the coordinate of \(B_\mathrm{mx} \) and the coordinate of \(B_\mathrm{mz}\), a set of solutions (\(\delta _1 ,\delta _2 \)) will meet the formulas (30) and (31) to minimize the function \({\tau }\).
$$\begin{aligned} f_1 \left( x,z \right)= & {} \frac{\partial B_\mathrm{mx} \left( x-{\delta }_{\mathrm{x}^{\prime }({\mathrm{HX}}-{ {\mathrm{HY}}})},z-{\delta }_1 \right) }{\partial \mathrm{x}} \nonumber \\&+ \frac{\partial {\textit{B}}_\mathrm{mz} \left( {x -{\delta }_{2},z - {\delta }_{{\text {z}^{\prime }}\left( {{\mathrm{HZ}}-{\mathrm{HY}}} \right) }}\right) }{\partial {\text {z}}} \end{aligned}$$
(30)
$$\begin{aligned} f_2 \left( x,z \right)= & {} \frac{\partial B_{mz} \left( {x-{\updelta }_2 ,z-{\delta }_{{\mathrm{z}}^{\prime }{(\mathrm{HZ}-\mathrm{HY})}}} \right) }{\partial \mathrm{x}} \nonumber \\&-\frac{\partial {\textit{B}}_{\mathrm{mx}} \left( {x - {\delta }_{{\text {x}^{\prime }}\left( {{\text {HX}} - {\text {HY}}} \right) } ,z - {\delta }_{1} } \right) }{\partial {\text {z}}} \end{aligned}$$
(31)
Since the errors \({\delta }_{{{{\mathrm{z}}^{\prime }}({\mathrm{HZ}}-{\mathrm{HY}})}}\) and \({\delta }_{{\mathrm{{x}^{\prime }}({ \mathrm{HX}}-{\mathrm{HY}})}}\) have already been acquired, the error \({\delta }_{{{{\mathrm{z}}^{\prime }}({\mathrm{HX}}-{\mathrm{HY}})}}\) will obtained by calculated \(\delta _1\), and the error \({\delta }_{{{\mathrm{x}}^{\prime }}({\mathrm{HZ}}-{\mathrm{HY}})}\) will obtained by calculated \(\delta _2 \).

The three-dimensional magnetic fields of CPMU can be received accurately by corrected these position errors of Hall sensors. After the field calibration of Hall sensors, the position errors calibration can be carried out immediately.

Conclusions

HEPS is developing a high-performance CPMU. In order to test its performance, a high precision vacuum hall measurement system is required. To adjust to the vacuum and the small gap conditions, we manufacture a new three-dimensional Hall probe. This paper illustrates the relative position and angle calibration procedures and the data processing of a three-dimensional Hall probe. Now the design of a smaller Hall probe is in process. The calibration of the angle errors and position errors will be carried out after the fabrication of the standard dipole magnet.

Notes

Acknowledgements

The authors would like to acknowledge Dr. Shu Guan, Dr. Wu Lei, Dr. Gu Kuixiang and Dr. Tang Zheng for great helpful discussion and suggestion.

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Copyright information

© Institute of High Energy Physics, Chinese Academy of Sciences; China Nuclear Electronics and Nuclear Detection Society and Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  • Ling-Ling Gong
    • 1
    • 2
  • Wan Chen
    • 1
  • Wen Kang
    • 3
    • 4
  • Shu-Chen Sun
    • 1
  • Zhi-Qiang Li
    • 1
  • Lei Zhang
    • 1
  • Yu-Feng Yang
    • 1
  • Hui-Hua Lu
    • 1
  • Xiao-Yu Li
    • 1
  • Shu-Tao Zhao
    • 1
  • Xiang-Zhen Zhang
    • 1
  • Ya-Jun Sun
    • 1
  1. 1.Laboratory of Particle Acceleration Physics and TechnologyInstitute of High Energy Physics Chinese Academy of SciencesBeijingChina
  2. 2.University of Chinese Academy of ScienceBeijingChina
  3. 3.China Spallation Neutron Source (CSNS), Institute of High Energy Physics (IHEP)Chinese Academy of Sciences (CAS)DongguanChina
  4. 4.Dongguan Institute of Neutron Science (DINS)DongguanChina

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