Heat flow of laser-ablated gold plasma in inertial confinement fusion hohlraum
Authors: Zhang En-Hao , Cai Hong-Bo , Du Bao , Tian Jian-Min , Zhang Wen-Shuai , Kang Dong-Guo , Zhu Shao-Ping ,   Graduate School, China Academy of Engineering Physics, Beijing 100088, China,   Institute of Applied Physics and Computational Mathematics, Beijing 100094, China,   Center for Applied Physics and Technology, Peking University, Beijing 100871, China,   Science and Technology on Plasma Physics Laboratory, Laser Fusion Research Center, China Academy of Engineering Physics, Mianyang 621900, China
Published: December 2020

When evaluating the plasma parameters in inertial confinement fusion, the flux-limited local Spitzer-Härm (S-H) model in radiation hydrodynamics simulations may be invalid when electron temperature gradient is too large. In other publications, the electron distribution function (EDF) could be explained by comparing the energy equipartition rate <inline-formula><tex-math id="M7">\begin{document}$R_{\rm eq}=\dfrac{1}{2}m_{\rm e}v_{\rm te} ^2\nu_{\rm ee}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M7.png"/></alternatives></inline-formula> with the heating rate <inline-formula><tex-math id="M8">\begin{document}$R_{\rm heat}=\dfrac{1}{2}m_{\rm e}v_{\rm os} ^2\nu_{\rm ei}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M8.png"/></alternatives></inline-formula>. When the condition <inline-formula><tex-math id="M9">\begin{document}$R_{\rm heat}\sim R_{\rm eq}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M9.png"/></alternatives></inline-formula> is satisfied, the EDF deviates from Maxwell equilibrium distribution, and is well fitted to the super-Gaussian distribution <inline-formula><tex-math id="M10">\begin{document}$f({{ v}})=C_m{\rm e}^{-(v/v_m)^m}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M10.png"/></alternatives></inline-formula> with the index <i>m </i>(<inline-formula><tex-math id="M11">\begin{document}$2<m<5$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M11.png"/></alternatives></inline-formula>). The number of energetic electrons of the super-Gaussian distribution is less than that of the Maxwell distribution, which plays an important role in electron heat flux, especially for electrons of 3.7<inline-formula><tex-math id="M12">\begin{document}$v_{\rm te}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M12.png"/></alternatives></inline-formula>. So electron heat flux of the super-Gaussian distribution is smaller than that of the Maxwell distribution. In this paper, EDF and electron heat flux in laser-produced Au plasma are simulated by using 1D3V PIC code (Ascent). It is found that in the coronal region, the laser intensity is larger, and the electron temperature is lower than the high-density region. So <inline-formula><tex-math id="M13">\begin{document}$\alpha=Z(v_{\rm os}/v_{\rm te})^2>1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M13.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M14">\begin{document}$R_{\rm heat}>R_{\rm eq}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M14.png"/></alternatives></inline-formula>, the EDF is well fitted to super-Gaussian distribution, where the index <i>m</i> is evaluated to be 3.34. In this region, the large electron temperature gradient leads to a small temperature scale length (<inline-formula><tex-math id="M15">\begin{document}$L_{\rm e}=T_{\rm e}/(\partial T_{\rm e}/\partial x)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M15.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M15.png"/></alternatives></inline-formula>), but the low e-e and e-i collision frequencies lead to a large electron mean-free-path (<inline-formula><tex-math id="M16">\begin{document}$\lambda_{\rm e}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M16.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M16.png"/></alternatives></inline-formula>). So the Knudsen number <inline-formula><tex-math id="M17">\begin{document}$\lambda_{\rm e}/L_{\rm e}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M17.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M17.png"/></alternatives></inline-formula> is evaluated to be 0.011, which is much larger than the critical value <inline-formula><tex-math id="M18">\begin{document}$2\times10^{-3}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M18.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M18.png"/></alternatives></inline-formula> of the S-H model, flux-limited local S-H electron heat flux is invalid. As a result, the limited-flux S-H predicts too large an electron heat flux, which results in much higher electron temperature of radiation hydrodynamics simulation than that of SG experiments. This heat flux inhibition phenomenon in coronal region cannot be explained by the flux-limited local S-H model, and non-local electron heat flux should be considered. In the high density region, the laser intensity is weaker, and the electron temperature is higher, so <inline-formula><tex-math id="M19">\begin{document}$\alpha=Z(v_{\rm os}/v_{\rm te})^2<1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M19.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M19.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M20">\begin{document}$R_{\rm heat}<R_{\rm eq},$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M20.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M20.png"/></alternatives></inline-formula> but EDF is still well fitted to super-Gaussian distribution, where the index m is evaluated to be 2.93. In this region, <inline-formula><tex-math id="M21">\begin{document}$L_{\rm e}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M21.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M21.png"/></alternatives></inline-formula> is larger, <inline-formula><tex-math id="M22">\begin{document}$\lambda_{\rm e}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M22.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M22.png"/></alternatives></inline-formula> is smaller, so the Knudsen number is smaller, which is evaluated to be <inline-formula><tex-math id="M23">\begin{document}$7.58\times10^{-4}<2\times10^{-3}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M23.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M23.png"/></alternatives></inline-formula>. As a result, The flux-limited local S-H electron heat flux is valid. However, the electron heat flux depends on the flux limiting factor (<inline-formula><tex-math id="M24">\begin{document}$f_{\rm e}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M24.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20191423_M24.png"/></alternatives></inline-formula>) that varies with laser intensity and electron temperature.