1.1 Aims and Outline 3
which are the focus of this book, represent a specific form of condensed mat-
ter with a pronounced structure due to a characteristic long-range order. At
the beginning of Chap.
2, the Hamiltonian of a solid composed of ions (nuclei
and closed electro n shells) and valence electrons will be introduced. These
two kin ds of constituents, with masses differing by or ders of magnitude, can
be treated separately as independent subsystems after applying the adiabatic
or Born–Op penheimer approximation. Before doing so, the linear response
theory is introduced as the basic concept to describe the material properties
of a solid. In Chap.
3, the dynamics of the ions, the heavy constituents of
a solid, will be described as theory of latti ce vibrations. This will first be
done in a classical approach using the model of massive spheres connected
by springs; but in a second step, we turn to the quantum-mechanical concept
of phonons as elementary excitations of the lattice. Acoustic phonons will be
discussed in the context of heat capacity, elastic properties, and sound prop-
agation; optical phonons will be related to optical prop erties of solids in the
far-infrared spectral range. Examples of phonon dispersion curves for quite
different solids will be presented to illustrate the influence o f structure and
chemical composition. The next chapters (Chaps.
4–7) are devoted to electrons
and their properties. The basic concept of the Fermi surface and the funda-
mentals of the many-particle theory, like Fock r epresentation, Hartree–Fock
approximation, dielectric screening, and electronic correlation, for free elec-
trons in the jellium model (Chap.
4) will be introduced. The influence of the
periodic lattice structure on the electron states will be treated in the single-
particle approximation justified by the density–functional theory (Chap.
5).
In this chapter, we also present methods for calculating the band structure,
which are important to understand the material specific aspects of energy
bands, and discuss the properties of two-dimensional electron systems. As a
particular outcome of electron–electron interaction, the Heisenberg Hamilto-
nian will be the starting point in Chap.
6 to discuss spin waves as excitations
out of a ground state with ferromagnetic or anti-ferromagnetic ordering. This
Hamiltonian will also be used to demonstrate the molecular field app roxima-
tion and the ferromagnetic phase transition. Finally, the theory of itinerant
electron magnetism will be presented in this chapter. Electron–electron inter-
action is the focus also in Chap.
7 which is devoted to correlated electrons. For
the treatment of some asp ects in this field, we take advantage of using Green
functions, which have to be introduced for this purpose. They will be used to
deal with the Hubbard model, leading to the Mott–Hubbard metal-insulator
transition. We discuss the phenomenological concept of Fermi liquids and its
modification for one-dimensional electron systems. Finally, heavy fermion s
and the fractional quantum Hall states, both dominated by correlation, are
also introduced. In Chap.
8, we go beyond the adiabatic approximation and
study the electron–phonon interaction as a prototype of coupling between
fermions and bosons. It is relevant in scattering pro cesses, which are essential
for the electric conductivity, for re laxation and lifetime effects of free carriers,
but can also mediate an attractive electron–electron interaction that gives rise