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Journals >High Power Laser Science and Engineering >Volume 7 >Issue 3 >Page 03000e40 > Article

High Power Laser Science and Engineering
Vol. 7, Issue 3, 03000e40 (2019)

A demonstration of extracting the strength and wavelength of the magnetic field generated by the Weibel instability from proton radiography

Bao Du¹, Hong-Bo Cai^1,2,3,†, Wen-Shuai Zhang¹, Shi-Yang Zou¹..., Jing Chen^1,2,3 and Shao-Ping Zhu^1,4,5,†|Show fewer author(s)

Author Affiliations

¹Institute of Applied Physics and Computational Mathematics, Beijing 100094, China

²HEDPS, Center for Applied Physics and Technology, Peking University, Beijing 100871, China

³IFSA Collaborative Innovation Center, Shanghai Jiao Tong University, Shanghai 200240, China

⁴STPPL, Research Center of Laser Fusion, China Academy of Engineering Physics, Mianyang 621900, China

⁵Graduate School, China Academy of Engineering Physics, Beijing 100088, China

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DOI: 10.1017/hpl.2019.30 Cite this Article Set citation alerts

Bao Du, Hong-Bo Cai, Wen-Shuai Zhang, Shi-Yang Zou, Jing Chen, Shao-Ping Zhu, "A demonstration of extracting the strength and wavelength of the magnetic field generated by the Weibel instability from proton radiography," High Power Laser Sci. Eng. 7, 03000e40 (2019) Copy Citation Text

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Abstract

The Weibel instability and the induced magnetic field are of great importance for both astrophysics and inertial confinement fusion. Because of the stochasticity of this magnetic field, its main wavelength and mean strength, which are key characteristics of the Weibel instability, are still unobtainable experimentally. In this paper, a theoretical model based on the autocorrelation tensor shows that in proton radiography of the Weibel-instability-induced magnetic field, the proton flux density on the detection plane can be related to the energy spectrum of the magnetic field. It allows us to extract the main wavelength and mean strength of the two-dimensionally isotropic and stochastic magnetic field directly from proton radiography for the first time. Numerical calculations are conducted to verify our theory and show good consistency between pre-set values and the results extracted from proton radiography.

Keywords

magnetic field plasma diagnostics proton radiography Weibel instability

\begin{array}{r} \boldsymbol u \approx u_{y} \boldsymbol e_{y} \approx \frac{q}{\unicode [S T I X] x 1 D 6 F E m_{p}} \int_{0}^{l_{z}} B_{x} d z \boldsymbol e_{y}, \end{array}

(1)

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\begin{array}{r} u_{y} = - \frac{u_{z}}{L_{D}} \int_{y_{0}}^{y} \unicode [S T I X] x 1 D 6 F F n / n_{0} d y, \end{array}

(2)

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\begin{array}{r} R_{i j} (\boldsymbol p, \boldsymbol p^{'}) = ⟨ \boldsymbol B_{i} (\boldsymbol p) \boldsymbol B_{j} (\boldsymbol p^{'}) ⟩, \end{array}

(3)

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\begin{array}{r} {\hat{R}}_{i j} (\boldsymbol k) = \frac{1}{(2 \unicode [S T I X] x 1 D 70 B)^{2}} \iint R_{i j} (\boldsymbol r) e^{- i \boldsymbol k \cdot \boldsymbol r} d^{2} \boldsymbol r, \end{array}

(4)

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\begin{array}{r} R_{i j} (\boldsymbol r) = \iint {\hat{R}}_{i j} (\boldsymbol k) e^{i \boldsymbol k \cdot \boldsymbol r} d^{2} \boldsymbol k . \end{array}

(5)

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\begin{array}{r} R (\boldsymbol r) = R_{i i} (\boldsymbol r) = R_{x x} (\boldsymbol r) + R_{z z} (\boldsymbol r) . \end{array}

(6)

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\begin{array}{r} R (\boldsymbol r = 0) = \int_{0}^{2 \unicode [S T I X] x 1 D 70 B} \int_{0}^{\infty} {\hat{R}}_{i i} (k) d k \cdot k d \unicode [S T I X] x 1 D 711, \end{array}

(7)

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\begin{array}{r} R (\boldsymbol r = 0) = \int_{0}^{\infty} 2 \unicode [S T I X] x 1 D 70 B k {\hat{R}}_{i i} (k) d k . \end{array}

(8)

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\begin{array}{r} E_{B} (k) = 2 \unicode [S T I X] x 1 D 70 B k \hat{R} (k), \end{array}

(9)

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\begin{array}{r} E_{B} (k_{x}) = 2 \unicode [S T I X] x 1 D 70 B k_{x} \hat{R} (k_{x}, k_{z} = 0) . \end{array}

(10)

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\begin{array}{r} M_{i j} (\boldsymbol p, \boldsymbol p^{'}) = ⟨ \boldsymbol u_{i} (\boldsymbol p) \boldsymbol u_{j} (\boldsymbol p^{'}) ⟩ . \end{array}

(11)

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\begin{array}{r} M_{y y} (r_{x}) = {(\frac{q}{\unicode [S T I X] x 1 D 6 F E m_{p}})}^{2} \int_{0}^{l_{z}} d z \int_{0}^{l_{z}} d z^{'} R_{x x} (\boldsymbol r) . \end{array}

(12)

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\begin{array}{r} M_{y y} (r_{x}) = l_{z} {(\frac{q}{\unicode [S T I X] x 1 D 6 F E m_{p}})}^{2} \int_{- z}^{l_{z} - z} d r_{z} R_{x x} (\boldsymbol r) . \end{array}

(13)

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\begin{array}{r} M_{y y} (r_{x}) = \frac{l_{z}}{2} {(\frac{q}{\unicode [S T I X] x 1 D 6 F E m_{p}})}^{2} \int_{- \infty}^{\infty} d r_{z} R (\boldsymbol r) . \end{array}

(14)

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\begin{array}{r} left : {\hat{M}}_{y y} (k_{x}) = \frac{1}{2 \unicode [S T I X] x 1 D 70 B} \int_{- \infty}^{\infty} M_{y y} (r_{x}) e^{- i k_{x} \cdot r_{x}} d r_{x} \end{array}

(15)

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\begin{array}{r} right : \unicode [S T I X] x 1 D 70 B l_{z} {(\frac{q}{\unicode [S T I X] x 1 D 6 F E m_{p}})}^{2} \hat{R} (k_{x}, k_{z} = 0), \end{array}

(16)

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\begin{array}{r} E_{u} (k_{x}) = \unicode [S T I X] x 1 D 70 B l_{z} {(\frac{q}{\unicode [S T I X] x 1 D 6 F E m_{p}})}^{2} \hat{R} (k_{x}, k_{z} = 0) . \end{array}

(17)

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\begin{array}{r} M_{y y} (r_{x}) = \int {\hat{M}}_{y y} (k_{x}) e^{i k_{x} r_{x}} d k_{x} . \end{array}

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\begin{array}{r} 2 \unicode [S T I X] x 1 D 70 B \int {\hat{u}}_{y}^{2} (k) d k_{x} = \int u_{y}^{2} (x) d x \approx ⟨ u_{y}^{2} (x) ⟩ l_{x}, \end{array}

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\begin{array}{r} \int {\hat{M}}_{y y} (k_{x}) d k_{x} = \frac{2 \unicode [S T I X] x 1 D 70 B}{l_{x}} \int {\hat{u}}_{y}^{2} (k) d k_{x} . \end{array}

(20)

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\begin{array}{r} E_{u} (k_{x}) = \frac{2 \unicode [S T I X] x 1 D 70 B}{l_{x}} {\hat{u}}_{y}^{2} (k_{x}) . \end{array}

(21)

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\begin{array}{r} E_{B} (k_{x}) = \frac{4 \unicode [S T I X] x 1 D 70 B}{l_{z} l_{x}} {(\frac{\unicode [S T I X] x 1 D 6 F E m_{p}}{q})}^{2} k_{x} {\hat{u}}_{y}^{2} (k_{x}), \end{array}

(22)

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\begin{array}{r} B_{rms}^{2} = \int_{0}^{\infty} d k E_{B} (k_{x}) . \end{array}

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\begin{array}{r} \unicode [S T I X] {x 1 D 706}_{| B |} = \unicode [S T I X] {x 1 D 706}_{{| B |}^{2}} . \end{array}

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\begin{array}{r} \iint B^{2} (x, z) d x d z = 4 \unicode [S T I X] {x 1 D 70 B}^{2} \int | \hat{B} (k) |^{2} d^{2} k, \end{array}

(25)

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Bao Du, Hong-Bo Cai, Wen-Shuai Zhang, Shi-Yang Zou, Jing Chen, Shao-Ping Zhu, "A demonstration of extracting the strength and wavelength of the magnetic field generated by the Weibel instability from proton radiography," High Power Laser Sci. Eng. 7, 03000e40 (2019)

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