## Plasma Physics

Plasma physics is the study of ionised gases. Plasma is the fourth state of matter, after solid, liquid and gaseous. If sufficient ennergy is applied to a neutral gas, the outer electrons of the atoms or molecules of the gas can be detached from the gas particle. This process is called ionisation. The negative electrons and the positive ions (the remains of a gas particle after one or more electrons have left) are now free to move around individually. If a sufficient number of gas particles has been ionised we talk of a plasma. Most of the baryonic matter in the universe is made from plasmas. The stars are giant balls of hot plasma that sustain nuclear fusion in their centres. The matter between the planets and the stars, and even between the galaxies is a (very) dilute plasma. But even on Earth plasmas are produced naturally and artificially. Lightning bolts, the arcs of arc welders, and the glow inside a modern energy saving light bulb are all examples of plasmas we encounter in our everyday lives.

The behaviour of plasmas is very different from that of a gas. Like a gas, a plasma will fill all the available space that is available to it. Unlike a gas, plasmas can conduct electric current because the is is made of charged particles and these charged particles can move around freely. Plasmas react to electromagnetic field. The charge carriers are accelerated by electric and magnetic fields. On the other hand any seperation of charges in a plasma will generate strong electric fields and the currents flowing in a plasma will generate magnetic fields. This interdependency of the plasma flow and the electromagnetic fields makes the physics of plasmas an extremely rich and interesting, albeit difficult topic.

There two main approaches to modelling plasmas. On large scales plasmas behave like fluids that generate electric currents and react to the influence of electromagnetic fields. At each point the plasma is uniquely described by the macroscopic quantities, such as the density, the flow velocity, and the pressure. Collisions within the plasma are assumed to always restore *local thermodynamic equilibrium*. The theory that describes plasmas at this level is called Magnetohydrodynamics (MHD). In it’s basic form it treats the plasma as an infinitely conducting fluid.

On spatial or time scales where collisions are not able to restore local thermodynamic equilibrium the MHD description breaks down. The kinetic description of plasmas allows to model plasmas in these regimes. Here the plasma is modelled by distribution functions for each type of particle. A distribution function provides the probability of finding a particle at a given position and with a given velocity. In general, the distribution function is a function of six parameters, the three coordinates in space and the three components of the velocity. For collisionless plasmas the Vlasov equation describes the evolution of the distribution function. If collisions are present a collision term is added to the equation and, depending on the type of collisions, the resulting equations is known as the Fokker-Planck equation of the Boltzmann equation.

## High Intensity Laser Plasma Interactions

When a high intensity pulse hits a plasma target many interesting things can happen. If the density of the target is low the laser pulse will enter the plasma and generate waves or accelerate particles in the plasma. If, however, the density exceeds the so called critical density of the plasma for the laser frequency the pulse can not enter the plasma and is partially reflected by the plasma. The critical density is determined by the relation of the laser frequency to the plasma frequency. The plasma frequency is the frequncy at which electrostatic disturbances in the plasma oscillate. If the plasma frequency is larger than the laser frequency the laser can not penetrate the plasma and is reflected.

### Fast Electron Transport

In the case of solid density targets, where the laser is reflected by the plasma a considerable amount of the energy of the laser pulse is deposited in the target. The energy is used to accelerate electrons to relativistic energies. These electrons are so fast that they can fly through the solid density target suffering almost no collisions with the background plasma. There are many applications where such high energy electrons are useful. One example is the use of these electrons to provide the heating of the fuel core in the Fast Ignition approach to Inertial Confinement Fusion. Another example is to use the electrons to locally heat the plasma target itself. In both cases, the control of the fast electron transport is crucial for the success of the experiment.

The fast electrons are accelerated at the front surface of the plasma target. Various experiments have found that the angular spread of these fast electrons dramatically increases with laser intensity, once the intensity exceeds some 10^{18} W cm^{-3}. This means that, by the time the electrons have travelled some distance they are spread over a large area and are not effective at heating a small region. Some control over the fast electrons is desirable.

It turns out that electrons can be controlled using magnetic fields. These fields don’t have to be imposed externally but can be generated by the fast electrons themselves. One promising approach is the use of resistivity gradients. At the interface between two materials of different resistivity the fast electron current can generate a magnetic field strong enough to completely reflect and trap most of the fast electrons in the high resistivity region. This can be used to design *magnetic mirror optics*. Using well known principles from ordinary mirror optics can be used to guide and focus the flow of fast electrons in the target.

## Computational Plasma Physics

Both the fluid equations and the kinetic equations are coupled to the electromagnetic field equations. In general these are Maxwell’s equations. This makes the modelling of plasmas extremely complicated. Only simple physical systems can be computed directly. For a large number of problems analytic techniques do not yield any results and computers have to be used to simulate the behaviour of plasmas. This is is the field of Computational Plasma Physics. Many different simulation approaches exist, each having their distinct advantages and disadvantages.

### Particle-In-Cell Method

The most widely used kinetic simulation method used in plasma physics is the Particle-In-Cell (PIC) method. Here the plasma is described by a large number of *simulation particles*. Each of these particles is determined by its position and momentum (or velocity). In addition the particles carry an electric charge and have a defined mass. Because computers are not large enough to follow every physical particle in the system, each simulation particle represents a large number of real particles. The number of real particles that the simulation particle represents is called the *weight* of the particle. Particles can have different *species* for example electons or ions. The particles move freely under the influence of the electromagnetic fields caused by all the other particles.

It would be too computaionally expensive to directly calculate the force that one particle sees as a result of all the other perticles in the system. For this reason the electromagnetic fields are evaluated on a numerical grid. The particle charges and currents are weighted onto the numerical grid. Then Maxwell’s equations are used to calculate updated values of the electric and magnetic fields. After this the fields are interpolated back to the particle positions and the forces are calculated for each particle. The particles are accelerated and moved according to the interpolated forces and the velocity of the particles.

The reason for this differnt treatment of particles and fields is not only for computational efficiency but also has a deeper physical justification. The kinetic plasma equations include averaged fields. These fields have been averaged over a large number of particles but over a scale length smaller than any other scale length of interest in the plasma. Thus the fields that enter for example Vlasov’s equation are not the microscopic interparticle fields but always field that are averaged over small volume elements. The Particle-In-Cell method models the averaging by introducing a smallest volume over which the electromagnetic fields can change.

### Vlasov Method

In contrast to the PIC method, the Vlasov method does not follow individual particle trajectories. Instead the particle *distribution function* stored directly and the kinetic equations are used to integrate the evolution of the distribution function. There are a number of ways of storing the distribution function. The two most widely used approaches are to either store the value of the distribution function on a grid in space and velocity or to approximate the distribution function by expanding it in spherical harmonics. The first of these approaches requires a high dimensional grid that spans all the axes of the position **x** and the velocity (momentum) **v** (**p**). If all the components are retained this results in a 6-dimensional grid which is extremely memory consuming and computaionally expensive. For this reason, Vlasov simulations on a phase space grid are usually carried out on lower dimansional systems in which some coordinates are integrated over.

The other approach is faster and requires less memory. However, because the approach stores deviations from an equilibrium distribution, it is best suited when these deviations are small and the system is near the local equilibrium.