On the inhomogeneous magnetised electron gas

In this thesis, we investigate an inhomogeneous gas of charged particles in the presence of a hard wall. From the point of view of physics one would like to study a `real' plasma, taking into account the inter-particle (Coulomb) interactions. Unfortunately, this would be a very difficult task. Since the interaction-less case could serve as a valuable reference system, and various aspects of it have not been studied before, a study of a gas of charged non-interacting particles in a magnetic field close to a hard wall seems appropriate.
As we see in this thesis, this study turns out to yield some surprising results that describe a richer structure than one would expect for such a rather simple model.

In chapters 2 and 3 we calculate charge and current density profiles for the completely degenerate magnetised free-electron gas near a wall. The expressions for these density profiles are asymptotic expansions in terms of the distance from the wall and give information on how the profiles decay towards their bulk values.

Chapter 2 uses a Green function approach. In chapter 3 we use a path integral technique. The final result, the aforementioned asymptotic expressions for charge and current density, are very similar, but nevertheless, both methods are of interest. The Green function approach makes it straightforward to calculate more terms in the
asymptotic expansions for the densities. It also yields an interesting integral relation for parabolic cylinder functions. The path integral approach from chapter 3 on the other hand, makes it possible to give an intuitive derivation of the multiple-reflection expansion. We see that the higher-order terms in this expansion
correspond to paths reflecting from the wall a number of times. Using the multiple-reflection expansion, we then derive the density profiles for Maxwell-Boltzmann statistics, and via an inverse Laplace transform technique, the profiles for the completely degenerate case.

The density profiles derived in chapters 2 and 3 have the form of a sum over Landau levels. For weak magnetic fields the number of Landau levels will become very large, and the region where the asymptotic expressions for the profiles are valid, shifts further and further away from the wall. In chapter 4, we investiaget this weak field
limit, by looking first at the much simpler case where the magnetic field is perpendicular to the wall. After a comparison of our results with numerical data we treat the original geometry, where the magnetic field is parallel to the wall, in a similar way.

Finally, in the last chapter, we derive correlation functions for the inhomogeneous electron gas. Before we calculate those correlation functions in the magnetised case, we derive the correlation functions for the field-free case in order to be able to compare the magnetised case with the field-free case. When the external magnetic field is present, the full two-point correlation function, i.e. the correlation function where the components of both coordinates are all different, is difficult to evaluate. However, the correlation function for points at the same distance from the wall, with the difference vector either parallel or perpendicular to the magnetic field, is much more manageable. We derive asymptotic expansions for these correlation functions both for small and large distances from the wall.