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The plasma dispersion function $Z(s)$ is a fundamental complex special integral function widely used in the field of plasma physics. The simplest and most rapid, yet accurate, approach to calculating it is through rational or equivalent multi-pole expansions. In this work, we summarize the numerical coefficients that are practically useful to the community. Besides the Pade approximation to obtain coefficients, which are accurate for both small and large arguments, we also employ optimization methods to enhance the accuracy of the approximation for the intermediate range. The best coefficients provided here for calculating $Z(s)$ can deliver twelve significant decimal digits. This work serves as a foundational database for the community for further applications., Comment: 10 pages, 10 figures

In fusion plasmas, where electron temperatures $T_e$ range from keV to hundreds of keV, Bremsstrahlung radiation constitutes a significant energy loss mechanism. While various thermal average fitting formulas exist in the literature, their accuracy is limited, particularly for $T_e \leq 20$ keV with error $>10\%$. Additionally, non-relativistic fitting formulas become invalid for $T_e \geq 50$ keV. The accurate calculation of Bremsstrahlung radiation is important for fusion gain study, especially of advanced fuels fusion. In this work, we develop new but still simple fitting formulas that are valid for electron temperatures ranging from small ($\lesssim0.1$ keV) to extreme relativistic range, with errors of less than 1\% even for high atomic number ions (e.g., $Z=30$), which could be sufficient for fusion plasma applications. Both electron-ion (e-i) and electron-electron (e-e) Bremsstrahlung radiations are considered. Both polynomial fitting with truncated ($t=k_BT_e/m_ec^2\leq10$) and one-fit-all formulas are provided. Additionally, we offer fitting formulas for e-i and e-e specifically for electron energies, which could prove useful in non-Maxwellian Bremsstrahlung radiation studies., Comment: 15 pages, 8 figures

Ion cyclotron emission (ICE) holds great potential as a diagnostic tool for fast ions in fusion devices. The theory of magnetoacoustic cyclotron instability (MCI), as an emission mechanism for ICE, states that MCI is driven by a velocity distribution of fast ions that approximates a drift ring beam. The influence of key parameters on the linear MCI is systematically investigated using the linear kinetic dispersion relation solver BO (Xie H. 2019 Comput. Phys. Comm. 244 343). The computational spectra region considered extends up to 40 times the ion cyclotron frequency. By examining the influence of these key parameters on MCI, several novel results have been obtained. In the case of MCI excited by super-Alfv\'enic fast ions, the parallel velocity spread significantly affects the bandwidth of harmonics and the continuous spectrum, while the perpendicular velocity spread has a decisive effect on the MCI growth rate. As the velocity spread increases, the linear relationship between the MCI growth rate and the square root of the number density ratio transitions to a linear relationship between the MCI growth rate and the number density ratio. This finding provides a linear perspective explanation for the observed linear relation between fast ion number density and ICE intensity in JET. Furthermore, high harmonics are more sensitive to changes in propagation angle than low harmonics because a decrease in the propagation angle alters the dispersion relation of the fast Alfv\'en wave. In the case of MCI excited by sub-Alfv\'enic fast ions, a significant growth rate increase occurs at high harmonics due to the transition of sub-Alfv\'enic fast ions to super-Alfv\'enic fast ions., Comment: 14 pages, 21 figures

Non-Maxwellian distributions of particles are commonly observed in fusion studies, especially for magnetic confinement fusion plasmas. The particle distribution has a direct effect on fusion reactivity, which is the focus of this study. We investigate the effects of three types of non-Maxwellian distributions, namely drift-ring-beam, slowing-down, and kappa super-thermal distributions, on the fusion reactivities of D-T (Deuterium-Trillium) and p-B11 (proton-Boron) using a newly developed program, where the enhancement of fusion reactivity relative to the Maxwellian distribution is computed while keeping the total kinetic energy constant. The calculation results show that for the temperature ranges of interest to us, namely 5-50 keV for D-T and 100-500 keV for p-B11, these non-Maxwellian distributions can enhance the fusion reactivities. In the case of the drift-ring-beam distribution, the enhancement factors for both reactions are affected by the perpendicular ring beam velocity, leading to decreased enhancement in low temperature range and increased enhancement in high temperature range. However, this effect is favorable for p-B11 fusion reaction and unfavorable for D-T fusion reaction. In the slowing-down distribution, the birth speed plays a crucial role in both reactions, and increasing birth speed leads to a shift in the enhancement ranges towards lower temperatures, which is beneficial for both reactions. Finally, the kappa super-thermal distribution results in a relatively large enhancement in the low temperature range with a small high energy power-law index {\kappa}. Overall, this study provides insight into the effects of non-Maxwellian distributions on fusion reactivity and highlights potential opportunities for enhancing fusion efficiency., Comment: 12 pages, 18 figures

We have developed a new global eigenvalue code, Multiscale Analysis for plasma Stabilities (MAS), for studying plasma problems with wave toroidal mode number n and frequency omega in a broad range of interest in general tokamak geometry, based on a five-field Landau-fluid description of thermal plasmas. Beyond keeping the necessary plasma fluid response, we further retain the important kinetic effects including diamagnetic drift, ion finite Larmor radius, finite parallel electric field, ion and electron Landau resonances in a self-consistent and non-perturbative manner without sacrificing the attractive efficiency in computation. The physical capabilities of the code are evaluated and examined in the aspects of both theory and simulation. In theory, the comprehensive Landau-fluid model implemented in MAS can be reduced to the well-known ideal MHD model, electrostatic ion-fluid model, and drift-kinetic model in various limits, which clearly delineates the physics validity regime. In simulation, MAS has been well benchmarked with theory and other gyrokinetic and kinetic-MHD hybrid codes in a manner of adopting the unified physical and numerical framework, which covers the kinetic Alfven wave, ion sound wave, low-n kink, high-n ion temperature gradient mode and kinetic ballooning mode. Moreover, MAS is successfully applied to model the Alfven eigenmode (AE) activities in DIII-D discharge #159243, which faithfully captures the frequency sweeping of RSAE, the tunneling damping of TAE, as well as the polarization characteristics of KBAE and BAAE being consistent with former gyrokinetic theory and simulation. With respect to the key progress contributed to the community, MAS has the advantage of combining rich physics ingredients, realistic global geometry and high computation efficiency together for plasma stability analysis in linear regime., Comment: 40 pages, 21 figures

Calculating fusion reactivity involves a complex six-dimensional integral of the fusion cross section and ion velocity distributions of two reactants. We demonstrate a simple Monte Carlo approach that efficiently computes this integral for arbitrary ion velocity distributions with a time complexity of $O(N)$, where $N$ is the number of samples. This approach generates random numbers that satisfy the reactant velocity distributions. In cases where these numbers are not readily available, we propose using Gaussian random numbers with weighted factors. For cases where only a small number of $N$ samples are available, a $O(N^2)$ method can be used. We benchmarked this approach against analytical results for drift bi-Maxwellian distributions and provided examples of drift ring beam and slowing down distributions. Our results show that the error can be less than 1\% with $N\sim10^4$ samples for our standard approach., Comment: 8 pages, 5 figures

The calculation of fusion reactivity involves a complex six-dimensional integral that takes into account the fusion cross-section and velocity distributions of two reactants. However, a more simplified one-dimensional integral form can be useful in certain cases, such as for studying fusion yield or diagnosing ion energy spectra. This simpler form has been derived in a few special cases, such as for a combination of two Maxwellian distributions, a beam-Maxwellian combination, and a beam-target combination, and can greatly reduce computational costs. In this study, it is shown that the reactivity for two drift bi-Maxwellian reactants with different drift velocities, temperatures, and anisotropies can also be reduced to a one-dimensional form, unifying existing derivations into a single expression. This result is used to investigate the potential enhancement of fusion reactivity due to the combination of beam and temperature anisotropies. For relevant parameters in fusion energy, the enhancement factor can be larger than 20\%, which is particularly significant for proton-boron (p-B11) fusion, as this factor can have a significant impact on the Lawson fusion gain criteria., Comment: 12 pages, 10 figures, 1 supplementary material for detail derivation

Here we report the generation of MeV alpha-particles from H-11B fusion initiated by laser-accelerated boron ions. Boron ions with maximum energy of 6MeV and fluence of 10^9/MeV/sr@5MeV were generated from 60-nm-thick self-supporting boron nanofoils irradiated by 1J femtosecond pulses at an intensity of 10^19W/cm^2. By bombarding secondary hydrogenous targets with the boron ions, 3*10^5/sr alpha-particles from H-11B fusion were registered, which is consistent with the theoretical yield calculated from the measured boron energy spectra. Our results demonstrate an alternative way toward ultrashort MeV alpha-particle sources employing compact femtosecond lasers. The ion acceleration and product measurement scheme are referential for the studies on the ion stopping power and cross-section of the H-11B reaction in solid or plasma.

Most tokamak devices including ITER exploit the D-T reaction due to its high reactivity, but the wall loading caused by the associated 14MeV neutrons will limit the further development of fusion performance at high beta. To explore p-11B fusion cycle, a tokamak system code is extended to incorporate the relativistic bremsstrahlung since the temperature of electrons is approaching the rest energy of electron. By choosing an optimum p-11B mix and ion temperature, some representative sets of parameters of p-11B tokamak reactor, whose fusion gain exceeds 1, have been found under the thermal wall loading limit and beta limit when synchrotron radiation loss is neglected. However, the fusion gain greatly decreases when the effect of synchrotron radiation loss is considered. Helium ash also plays an important role in the fusion performance, and we have found that the helium confinement time must be below the energy confinement time to keep the helium concentration ratio in an acceptable range., Comment: 9 figures, 2 tables

A significant number of confined energetic electrons have been observed outside the Last Closed Flux Surface (LCFS) of the solenoid-free, ECRH sustained plasmas in the EXL-50 spherical torus. Several diagnostics have been applied, for the first time, to investigate the key characters of energetic electrons. Experiments reveal the existence of high-temperature low density electrons, which can carry relatively a large amount of the stored energy. The boundary between the thermal plasma and the energetic electron fluid appears to be clearly separated and the distance between the two boundaries can reach tens of centimeters (around the size of the minor radius of the thermal plasma). This implies that the Grad-Shafranov equilibrium is not suitable to describe the equilibrium of the EXL-50 plasma and a multi-fluid model is required. Particle dynamics simulations of full orbits show that energetic electrons can be well confined outside the LCFS. This is consistent with the experimental observations.

Although the accurate description of the wave propagation and absorption in plasmas requires complicated full wave solutions or kinetic simulations, the local dispersion analysis can still be helpful to capture the main physics of wave properties. Plasma wave accessibility informs that whether a wave can propagate to a region, which usually depends on the wave frequency, wave vector, and the local plasma density and magnetic field. In this tutorial paper, we describe the wave accessibility beyond usual textbooks and especially highlight the warm plasma effects. Useful numerical models and methods are provided, which can be used to obtain a quick view of the wave accessibility parameters space. The thermal effects are modeled by both multi-fluid model with isotropic pressure term and kinetic model with Maxwellian velocity distribution function. All cold plasma waves from high frequency electron cyclotron waves, intermediate frequency low hybrid waves to low frequency ion cyclotron waves, as well as kinetic ion and electron Bernstein waves, are presented. The questions that how many plasma wave modes exist and how to find the solutions are also discussed. It is interesting to find that the warm multi-fluid model, though incapable of reproducing the Bernstein modes, can provide a quick way to determine whether the thermal effects are important. To show the kinetic thermal effects, the ray tracing calculations of the mode conversion from cold plasma waves to kinetic waves are also provided, i.e., from the slow X electron cyclotron wave to electron Bernstein wave and from the ion cyclotron fast wave to ion Bernstein wave., Comment: 17 pages, 12 figures. This is a tutorial paper of plasma waves accessibility and would be submitted for formal publish in the near future. To help to improve it and make it more useful to the community, comments and suggestions are welcome. The numerical codes used in the manuscript would be open sourced in the future or be available on request

Differential flows among different ion species are often observed in the solar wind, and such ion differential flows can provide the free energy to drive Alfv\'en/ion-cyclotron and fast-magnetosonic/whistler instabilities. Previous works mainly focused on ion beam instability under the parameters representative of the solar wind nearby 1 au. In this paper we further study proton beam instability using the radial models of the magnetic field and plasma parameters in the inner heliosphere. We explore a comprehensive distribution of proton beam instability as functions of the heliocentric distance and the beam speed. We also perform a detailed analysis of the energy transfer between unstable waves and particles and quantify how much the free energy of the proton beam flows into unstable waves and other kinds of particle species (i.e., proton core, alpha particle, and electron). This work clarifies that both parallel and perpendicular electric fields are responsible for the excitation of oblique Alfv\'en/ion-cyclotron and oblique fast-magnetosonic/whistler instabilities. Moreover, this work proposes an effective growth length to estimate whether the instability is efficiently excited or not. It shows that oblique Alfv\'en/ion-cyclotron instability, oblique fast-magnetosonic/whistler instability, and oblique Alfv\'en/ion-beam instability can be efficiently driven by proton beams drifting at the speed $\sim$ 600-1300 km s$^{-1}$ in the solar atmosphere. In particular, oblique Alfv\'en/ion-cyclotron waves driven in the solar atmosphere can be significantly damped therein, leading to the solar corona heating. These results are helpful for understanding proton beam dynamics in the inner heliosphere and can be verified through in situ satellite measurements., Comment: 23 pages, 20 figures, 1 table. v2: corrected typos and minor errors; published in ApJ, see the link https://doi.org/10.3847/1538-4357/ac133c (there is a misunderstanding for the journal in v1)

A new tool (GSEQ-FRC) for solving two-dimensional (2D) equilibrium of field-reversed configuration (FRC) based on fixed boundary and free boundary conditions with external coils included is developed. Benefiting from the two-parameter modified rigid rotor (MRR) radial equilibrium model and the numerical approaches presented by [Ma et al, Nucl. Fusion, 61, 036046, 2021], GSEQ-FRC are used to study the equilibrium properties of FRC quantitatively and will be used for fast FRC equilibrium reconstruction. In GSEQ-FRC, the FRC equilibrium can be conveniently determined by two parameters, i.e., the ratio between thermal pressure and magnetic pressure at the seperatrix $\beta_s$, and the normalized scrape of layer (SOL) width $\delta_s$. Examples with fixed and free boundary conditions are given to demonstrate the capability of GSEQ-FRC in the equilibrium calculations. This new tool is used to quantitatively study the factors affecting the shape of the FRC separatrix, revealing how the FRC changes from racetrack-like to ellipse-like., Comment: 10 pages, 11 figures

A new equilibrium pressure profile extending the Rigid-Rotor (RR) model with a simple unified expression $P=P(\psi;\beta_{s},\alpha, \sigma)$ for both inside and outside the separatrix is proposed, in which the radial normalized field-reversed configuration (FRC) equilibrium profiles for pressure, magnetic field, and current can be determined by only two dimensionless parameters $\beta_s\equiv P_s/2\mu_0B_e^2$ and $\delta_s\equiv L_{ps}/R_s$, where $P_s$ is the thermal pressure at the separatrix, $B_e$ is the external magnetic field strength, $L_{ps}$ is the pressure profile scale length at the separatrix, and $R_s$ is the separatrix radius. This modified rigid rotor (MRR) model has sufficient flexibility to accommodate the narrow scrape of layer (SOL) width and hollow current density profiles, and can be used to fit experimental measurements satisfactorily. Detailed one-dimensional (1D) characteristics of the new MRR model are investigated analytically and numerically, and the results are also confirmed in two-dimensional (2D) numerical equilibrium solutions., Comment: 10 pages, 9 figures, submitted to Nucl. Fusion

Electron temperature anisotropies and electron beams are nonthermal features of the observed nonequilibrium electron velocity distributions in the solar wind. In collision-poor plasmas these nonequilibrium distributions are expected to be regulated by kinetic instabilities through wave-particle interactions. This study considers electron instabilities driven by the interplay of core electron temperature anisotropies and the electron beam, and firstly gives a comprehensive analysis of instabilities in arbitrary directions to the background magnetic field. It clarifies the dominant parameter regime (e.g., parallel core electron plasma beta $\beta_{\mathrm{ec\parallel}}$, core electron temperature anisotropy $A_{\mathrm{ec}}\equiv T_{\mathrm{ec\perp}}/T_{\mathrm{ec\parallel}}$, and electron beam velocity $V_{\mathrm{eb}}$) for each kind of electron instability (e.g., the electron beam-driven electron acoustic/magnetoacoustic instability, the electron beam-driven whistler instability, the electromagnetic electron cyclotron instability, the electron mirror instability, the electron firehose instability, and the ordinary-mode instability). It finds that the electron beam can destabilize electron acoustic/magnetoacoustic waves in the low-$\beta_{\mathrm{ec\parallel}}$ regime, and whistler waves in the medium- and large-$\beta_{\mathrm{ec\parallel}}$ regime. It also finds that a new oblique fast-magnetosonic/whistler instability is driven by the electron beam with $V_{\mathrm{eb}}\gtrsim7V_{\mathrm{A}}$ in a regime where $\beta_{\mathrm{ec\parallel}}\sim0.1-2$ and $A_{\mathrm{ec}}<1$. Moreover, this study presents electromagnetic responses of each kind of electron instability. These results provide a comprehensive overview for electron instability constraints on core electron temperature anisotropies and electron beams in the solar wind.

Microinstabilities and waves excited at moderate-Mach-number perpendicular shocks in the near-Sun solar wind are investigated by full particle-in-cell (PIC) simulations. By analyzing the dispersion relation of fluctuating field components directly issued from the shock simulation, we obtain key findings concerning wave excitations at the shock front: (1) at the leading edge of the foot, two types of electrostatic (ES) waves are observed. The relative drift of the reflected ions versus the electrons triggers an electron cyclotron drift instability (ECDI) which excites the first ES wave. Because the bulk velocity of gyro-reflected ions shifts to the direction of the shock front, the resulting ES wave propagates oblique to the shock normal. Immediately, a fraction of incident electrons are accelerated by this ES wave and a ring-like velocity distribution is generated. They can couple with the hot Maxwellian core and excite the second ES wave around the upper hybrid frequency. (2) from the middle of the foot all the way to the ramp, electrons can couple with both incident and reflected ions. ES waves excited by ECDI in different directions propagate across each other. Electromagnetic (EM) waves (X mode) emitted toward upstream are observed in both regions. They are probably induced by a small fraction of relativistic electrons. Results shed new insight on the mechanism for the occurrence of ES wave excitations and possible EM wave emissions at young CME-driven shocks in the near-Sun solar wind.

A unified numerically solvable framework for dispersion relations with arbitrary number of species drifting at arbitrary directions and with Krook collision is derived for linear uniform/homogenous kinetic plasma, which largely extended the standard one [say, T. Stix, {\em Waves in Plasmas}, AIP Press, 1992]. The purpose of this work is to provide a kinetic plasma dispersion relation tool not only the physical model but also the numerical approach be as general/powerful as possible. As a very general application example, we give the final dispersion relations which assume further the equilibrium distribution function be bi-Maxwellian and including parallel drift, two directions of perpendicular drift (i.e., drift across magnetic field), ring beam and loss-cone. Both electromagnetic and electrostatic versions are provided, with also the Darwin (a.k.a., magnetoinductive or magnetostatic) version. The species can be treated either magnetized or unmagnetized. Later, the equations are transformed to the matrix form be solvable by using the powerful matrix algorithm [H. S. Xie and Y. Xiao, Plasma Science and Technology, 18, 2, 97, 2016], which is the first approach can give all the important solutions of a linear kinetic plasma system without requiring initial guess for root finding and thus can be extremely useful to the community. To our knowledge, the present model would be the most comprehensive one in literature for the distribution function constructed bases on Maxwellian, which thus can be applied widely for study waves and instabilities in space, astrophysics, fusion and laser plasma. We limit the present work to non-relativistic case., Comment: 34 pages, 3 figures