13.1 Introduction

For many aspects of the analysis symmetry and other properties of the constitutive coefficients have been used, without giving a thorough discussion of the background of these relations. It is the aim of this chapter to do so. To first order in the non-locality of the response the constitutive relations were introduced in the third chapter, cf. eqs. (3.42) - (3.45). In the ( k , ?) representation these constitutive equations become

As discussed in chapter 3 this form of the constitutive relations was chosen such, that it agrees with the various symmetry relations. In order to show that this is indeed the case, and to discuss the general formulation of such relations, it is necessary and convenient to write the constitutive relations in a more general form.

For this purpose it is most appropriate to define a 6-dimensional field, which is a combination of the electric and the magnetic fields:

Similarly a 6-dimensional polarization/magnetization density is introduced

It should be noted that the N-field is in general discontinuous at the z = 0 surface, while the polarization/magnetization field P also contains a singular contribution at this surface. The general form of the linear relation between these fields is given by

The response function, which now is a 6 6 tensor, is the sum of bulk and surface contributions:

In using this expression it should be noted that the ?-functions should be used in the following manner

where 0 is an infinitesimally small...