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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 Cai1、2、3、†, Wen-Shuai Zhang1, Shi-Yang Zou1, Jing Chen1、2、3, and Shao-Ping Zhu1、4、5、†
    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[J]. High Power Laser Science and Engineering, 2019, 7(3): 03000e40 Copy Citation Text show less

    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

    $$\begin{eqnarray}\boldsymbol{u}\approx u_{y}\boldsymbol{e}_{y}\approx \frac{q}{\unicode[STIX]{x1D6FE}m_{p}}\int _{0}^{l_{z}}B_{x}\,\text{d}z\boldsymbol{e}_{y},\end{eqnarray}$$(1)

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    $$\begin{eqnarray}u_{y}=-\frac{u_{z}}{L_{D}}\int _{y_{0}}^{y}\unicode[STIX]{x1D6FF}n/n_{0}\,\text{d}y,\end{eqnarray}$$(2)

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    $$\begin{eqnarray}R_{ij}(\boldsymbol{p},\boldsymbol{p}^{\prime })=\left\langle \boldsymbol{B}_{i}(\boldsymbol{p})\boldsymbol{B}_{j}(\boldsymbol{p}^{\prime })\right\rangle ,\end{eqnarray}$$(3)

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    $$\begin{eqnarray}\hat{R}_{ij}(\boldsymbol{k})=\frac{1}{(2\unicode[STIX]{x1D70B})^{2}}\iint R_{ij}(\boldsymbol{r})e^{-i\boldsymbol{k}\cdot \boldsymbol{r}}\,\text{d}^{2}\boldsymbol{r},\end{eqnarray}$$(4)

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    $$\begin{eqnarray}R_{ij}(\boldsymbol{r})=\iint \hat{R}_{ij}(\boldsymbol{k})e^{i\boldsymbol{k}\cdot \boldsymbol{r}}\text{d}^{2}\boldsymbol{k}.\end{eqnarray}$$(5)

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    $$\begin{eqnarray}R(\boldsymbol{r})=R_{ii}(\boldsymbol{r})=R_{xx}(\boldsymbol{r})+R_{zz}(\boldsymbol{r}).\end{eqnarray}$$(6)

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    $$\begin{eqnarray}R(\boldsymbol{r}=0)=\int _{0}^{2\unicode[STIX]{x1D70B}}\int _{0}^{\infty }\hat{R}_{ii}(k)\,\text{d}k\cdot k\,\text{d}\unicode[STIX]{x1D711},\end{eqnarray}$$(7)

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    $$\begin{eqnarray}R(\boldsymbol{r}=0)=\int _{0}^{\infty }2\unicode[STIX]{x1D70B}k\hat{R}_{ii}(k)\,\text{d}k.\end{eqnarray}$$(8)

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    $$\begin{eqnarray}E_{B}(k)=2\unicode[STIX]{x1D70B}k\hat{R}(k),\end{eqnarray}$$(9)

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    $$\begin{eqnarray}E_{B}(k_{x})=2\unicode[STIX]{x1D70B}k_{x}\hat{R}(k_{x},k_{z}=0).\end{eqnarray}$$(10)

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    $$\begin{eqnarray}M_{ij}(\boldsymbol{p},\boldsymbol{p}^{\prime })=\left\langle \boldsymbol{u}_{i}(\boldsymbol{p})\boldsymbol{u}_{j}(\boldsymbol{p}^{\prime })\right\rangle .\end{eqnarray}$$(11)

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    $$\begin{eqnarray}M_{yy}(r_{x})=\left(\frac{q}{\unicode[STIX]{x1D6FE}m_{p}}\right)^{2}\int _{0}^{l_{z}}\,\text{d}z\int _{0}^{l_{z}}\,\text{d}z^{\prime }R_{xx}(\boldsymbol{r}).\end{eqnarray}$$(12)

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    $$\begin{eqnarray}M_{yy}(r_{x})=l_{z}\left(\frac{q}{\unicode[STIX]{x1D6FE}m_{p}}\right)^{2}\int _{-z}^{l_{z}-z}\,\text{d}r_{z}R_{xx}(\boldsymbol{r}).\end{eqnarray}$$(13)

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    $$\begin{eqnarray}M_{yy}(r_{x})=\frac{l_{z}}{2}\left(\frac{q}{\unicode[STIX]{x1D6FE}m_{p}}\right)^{2}\int _{-\infty }^{\infty }\,\text{d}r_{z}R(\boldsymbol{r}).\end{eqnarray}$$(14)

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    $$\begin{eqnarray}\text{left}:\hat{M}_{yy}(k_{x})=\frac{1}{2\unicode[STIX]{x1D70B}}\int _{-\infty }^{\infty }M_{yy}(r_{x})e^{-ik_{x}\cdot r_{x}}\,\text{d}r_{x}\end{eqnarray}$$(15)

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    $$\begin{eqnarray}\text{right}:\unicode[STIX]{x1D70B}l_{z}\left(\frac{q}{\unicode[STIX]{x1D6FE}m_{p}}\right)^{2}\hat{R}(k_{x},k_{z}=0),\end{eqnarray}$$(16)

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    $$\begin{eqnarray}E_{u}(k_{x})=\unicode[STIX]{x1D70B}l_{z}\left(\frac{q}{\unicode[STIX]{x1D6FE}m_{p}}\right)^{2}\hat{R}(k_{x},k_{z}=0).\end{eqnarray}$$(17)

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    $$\begin{eqnarray}M_{yy}(r_{x})=\int \hat{M}_{yy}(k_{x})e^{ik_{x}r_{x}}\,\text{d}k_{x}.\end{eqnarray}$$(18)

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    $$\begin{eqnarray}2\unicode[STIX]{x1D70B}\int \hat{u} _{y}^{2}(k)\,\text{d}k_{x}=\int u_{y}^{2}(x)\,\text{d}x\approx \langle u_{y}^{2}(x)\rangle l_{x},\end{eqnarray}$$(19)

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    $$\begin{eqnarray}\int \hat{M}_{yy}(k_{x})\,\text{d}k_{x}=\frac{2\unicode[STIX]{x1D70B}}{l_{x}}\int \hat{u} _{y}^{2}(k)\,\text{d}k_{x}.\end{eqnarray}$$(20)

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    $$\begin{eqnarray}E_{u}(k_{x})=\frac{2\unicode[STIX]{x1D70B}}{l_{x}}\hat{u} _{y}^{2}(k_{x}).\end{eqnarray}$$(21)

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    $$\begin{eqnarray}E_{B}(k_{x})=\frac{4\unicode[STIX]{x1D70B}}{l_{z}l_{x}}\left(\frac{\unicode[STIX]{x1D6FE}m_{p}}{q}\right)^{2}k_{x}\hat{u} _{y}^{2}(k_{x}),\end{eqnarray}$$(22)

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    $$\begin{eqnarray}B_{\text{rms}}^{2}=\int _{0}^{\infty }\,\text{d}kE_{B}(k_{x}).\end{eqnarray}$$(23)

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    $$\begin{eqnarray}\unicode[STIX]{x1D706}_{\left|B\right|}=\unicode[STIX]{x1D706}_{\left|B\right|^{2}}.\end{eqnarray}$$(24)

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    $$\begin{eqnarray}\iint B^{2}(x,z)\,\text{d}x\,\text{d}z=4\unicode[STIX]{x1D70B}^{2}\int |\hat{B}(k)|^{2}\,\text{d}^{2}k,\end{eqnarray}$$(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[J]. High Power Laser Science and Engineering, 2019, 7(3): 03000e40
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