Particle-in-cell (PIC) methods have a long history in the study of laser-plasma interactions. Early electromagnetic codes used the Yee staggered grid for field variables combined with a leapfrog EM-field update and the Boris algorithm for particle pushing. The general properties of such schemes are well documented. Modern PIC codes tend to add to these high-order shape functions for particles, Poisson preserving field updates, collisions, ionisation, a hybrid scheme for solid density and high-field QED effects. In addition to these physics packages, the increase in computing power now allows simulations with real mass ratios, full 3D dynamics and multi-speckle interaction. This paper presents a review of the core algorithms used in current laser-plasma specific PIC codes. Also reported are estimates of self-heating rates, convergence of collisional routines and test of ionisation models which are not readily available elsewhere. Having reviewed the status of PIC algorithms we present a summary of recent applications of such codes in laser-plasma physics, concentrating on SRS, short-pulse laser-solid interactions, fast-electron transport, and QED effects.

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Contemporary particle-in-cell approach to laser-plasma modelling

Many-particle charged-particle plasma simulations using spatial meshes for the electromagnetic field solutions, particle-in-cell (PIC) merged with Monte Carlo collision (MCC) calculations, are coming into wide use for application to partially ionized gases. The author emphasizes the development of PIC computer experiments since the 1950s starting with one-dimensional (1-D) charged-sheet models, the addition of the mesh, and fast direct Poisson equation solvers for 2-D and 3-D. Details are provided for adding the collisions between the charged particles and neutral atoms. The result is many-particle simulations with many of the features met in low-temperature collision plasmas; for example, with applications to plasma-assisted materials processing, but also related to warmer plasmas at the edges of magnetized fusion plasmas. >

Particle-in-Cell Charged Particle Simulations, plus Monte Carlo Collisions with Neutral Atoms, PIC-MCC

We review the data and models describing the production of the electrons, termed secondary electrons, that initiate the secondary and subsequent feedback avalanches required for the growth of current during breakdown and for the maintenance of low-current, cold-cathode discharges in argon First we correlate measurements of the production of secondary electrons at metallic cathodes, ie the yields of electrons induced by Ar+ ions, fast Ar atoms, metastable atoms and vuv photons The yields of electrons per ion, fast atom and photon vary greatly with particle energy and surface condition Then models of electron, ion, fast atom, excited atom and photon transport and kinetics are fitted to electrical-breakdown and low-current, discharge-maintenance data to determine the contributions of various cathode-directed species to the secondary electron production Our model explains measured breakdown and low-current discharge voltages for Ar over a very wide range of electric field to gas density ratios E/n, ie 15 Td to 100 kTd We review corrections for nonequilibrium electron motion near the cathode that apply to our local-field model of these discharges Analytic expressions for the cross sections and reaction coefficients used by this and related models are summarized

https://iopscience.iop.org/article/10.1088/0963-0252/8/3/201/pdf

Cold-cathode discharges and breakdown in argon: surface and gas phase production of secondary electrons

The high power impulse magnetron sputtering (HiPIMS) discharge is a recent addition to plasma based sputtering technology. In HiPIMS, high power is applied to the magnetron target in unipolar pulse ...

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High power impulse magnetron sputtering discharge

. The knowledge of the lightning-induced nitrogen oxides (LNOx) source is important for understanding and predicting the nitrogen oxides and ozone distributions in the troposphere and their trends, the oxidising capacity of the atmosphere, and the lifetime of trace gases destroyed by reactions with OH. This knowledge is further required for the assessment of other important NOx sources, in particular from aviation emissions, the stratosphere, and from surface sources, and for understanding the possible feedback between climate changes and lightning. This paper reviews more than 3 decades of research. The review includes laboratory studies as well as surface, airborne and satellite-based observations of lightning and of NOx and related species in the atmosphere. Relevant data available from measurements in regions with strong LNOx influence are identified, including recent observations at midlatitudes and over tropical continents where most lightning occurs. Various methods to model LNOx at cloud scales or globally are described. Previous estimates are re-evaluated using the global annual mean flash frequency of 44±5 s−1 reported from OTD satellite data. From the review, mainly of airborne measurements near thunderstorms and cloud-resolving models, we conclude that a "typical" thunderstorm flash produces 15 (2–40)×1025 NO molecules per flash, equivalent to 250 mol NOx or 3.5 kg of N mass per flash with uncertainty factor from 0.13 to 2.7. Mainly as a result of global model studies for various LNOx parameterisations tested with related observations, the best estimate of the annual global LNOx nitrogen mass source and its uncertainty range is (5±3) Tg a−1 in this study. In spite of a smaller global flash rate, the best estimate is essentially the same as in some earlier reviews, implying larger flash-specific NOx emissions. The paper estimates the LNOx accuracy required for various applications and lays out strategies for improving estimates in the future. An accuracy of about 1 Tg a−1 or 20%, as necessary in particular for understanding tropical tropospheric chemistry, is still a challenging goal.

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The global lightning-induced nitrogen oxides source

The stochastic law that prescribes the velocities of simulated molecules after a small time increment was derived from the Boltzmann equation. The scheme to determine the velocity of a molecule after a small time increment is divided into three steps. The first step gives the collision probability of the molecule without specifying its collision partner. The second step gives a conditional probability distribution. If the molecule is accepted in the first step as a colliding molecule, its collision partner is sampled from this probability distribution. The last step gives a probability density from which the direction of the relative velocity after collision is sampled, and hence the step gives the post-collision velocity of the molecule. It is shown that the use of the present simulation scheme gives an exact solution of the Boltzmann equation.

Direct Simulation Scheme Derived from the Boltzmann Equation. I. Monocomponent Gases

A succession of small-angle binary collisions can be grouped into a unique binary collision with a large scattering angle. The latter is called a cumulative collision. This makes it possible to treat the cumulative collision like a collision between neutral molecules. A significant feature of the cumulative collision is that the probability density function for a deflection angle depends on the time spent by a charged particle while engaged in the cumulative collision. Here a simple analytic expression for the function is proposed which is easy to use together with the Monte Carlo method. The validity of the present theory is ascertained by calculating various relaxation phenomena in plasmas. The theory is best suited to particle simulation of plasmas.

Theory of cumulative small-angle collisions in plasmas

Weighted Particles in Coulomb Collision Simulations Based on the Theory of a Cumulative Scattering Angle

A time-explicit formula that describes the time evolution of velocity distribution functions of gases and plasmas is derived from the Boltzmann equation. The formula can be used to construct collision simulation algorithms. Specialization of the formula to the case of the Coulomb interaction shows that the previous method [K. Nanbu, Phys. Rev. E 55, 4642 (1997)] for a Coulomb collision simulation is a solution method of the Landau-Fokker-Planck equation in the limit of a small time step. Also, a collision simulation algorithm for multicomponent plasmas is proposed based on the time-explicit formula derived.

Theory of collision algorithms for gases and plasmas based on the boltzmann equation and the landau-fokker-planck equation

The structures of the CF4 radio-frequency discharge between parallel electrodes are clarified by the use of the particle-in-cell/Monte Carlo method. The simulation is performed based on the most reliable collision data, i.e., detailed cross-section data for electron–CF4 collision, measured rate for positive–negative ion recombination, and the newly developed ion–CF4 collision model for endothermic reactions. Reactive collisions between positive ions (especially CF3+) and CF4 molecules are found to be important. The major loss process of negative ions is the recombination with positive ions. It is also found that the discharge sustaining mechanism is the ionization, not the electron detachment from negative ions.

Kenichi Nanbu

Papers

Direct Simulation Scheme Derived from the Boltzmann Equation. I. Monocomponent Gases

Theory of cumulative small-angle collisions in plasmas

Weighted Particles in Coulomb Collision Simulations Based on the Theory of a Cumulative Scattering Angle

Theory of collision algorithms for gases and plasmas based on the boltzmann equation and the landau-fokker-planck equation

Self-consistent particle simulation of radio-frequency CF4 discharge with implementation of all ion–neutral reactive collisions