In the previous chapters we have been dealing mainly with effects involving a single electron, such as the solution of the Schrödinger equation for a hydrogen-like atom. However, the majority of interesting problems in spectroscopy deal with systems that contain many electrons. In this chapter we shall see how to construct many-body wave functions from single-particle wave functions, and how to build a many-body Hamiltonian in matrix form and apply this to the Coulomb interaction for many-electron atoms.

Many-body wave functions

A many-body wave function needs to satisfy several characteristics. First, we need to ensure that the particles in the wave function are identical. This is a clear difference from classical physics, where we can distinguish one object from another. Second, it is with many-body wave functions that the distinction between fermions and bosons comes to the forefront. The construction of many-body basisfunctions starts by choosing a basis of one-particle basisfunctions. Let us denote these by ϕk(r), where k is a generic quantum number describing the quantum states (for example, momentum and spin k → kσ with spin projection σ = ↑, ↓ or k → nlmσ for hydrogen-like atomic orbitals). The basis can consist of eigenfunctions of the one-particle problem. In this case, the one-particle interactions H1 (r1) are already solved and we are only dealing with the two-particle interactions H2(r1, r2). However, this is not essential and we often choose a basis that is convenient to work with. For example, for atoms and ions, convenient basisfunctions are the hydrogen-like atomic orbitals. In the presence of spin-orbit coupling, these states are not eigenfunctions. In this case, both one- and two-particle interactions couple different many-body basisfunctions. In principle, the basis of many-body wave functions has to be complete. In practice, this is impossible since it requires an infinite number of basisfunctions.

Although almost all quantum mechanics textbooks consider absorption and emission, the discussion is usually limited to hydrogen-like atoms. This gives a somewhat limited view of the process. This book deals with aborption of high-energy X-rays. Let us look at a particular example to demonstrate the concepts that we will be dealing with throughout the book. Figure 1.1 shows a calculation of the L-edge of a divalent cobalt ion in a solid. In X-ray terminology, L-edge stands for the excitation of an electron from a 2p orbital into a 3d one. In fact, the calculated spectrum resembles closely the experimental X-ray absorption on CoO. Simply by looking at this spectrum, a number of salient features are apparent that will hopefully become clearer throughout the book.

First, one can start by asking the very basic question: what is a spectrum? Apparently, it is the absorption intensity as a function of energy. This implies that energy is a good quantum number. Therefore, in an absorption process energy is conserved. We therefore have to understand why certain quantities are conserved. We shall see that this is inherently related to the symmetry properties of the system. It is important to understand these properties since they tell us what quantities are exchanged between the incoming photon field and the material. By studying the changes in the photon field (in this case, how many photons are absorbed by the material), we can learn something about the material. We shall see that energy is not the only quantity that can be exchanged between the photons and the atoms. Linear and angular momentum are other examples of quantities that can be conserved in a spectroscopy experiment.

We already identified the spectrum in Fig. 1.1 as arising from a 2p → 3d transition. This implies that the basic atomic structure is still valid. We shall therefore review some aspects of the hydrogen-like atom in Chapter 2. The absolute energy scale for this transition is of the order of hundreds of electronvolts.

First thoughts of writing a textbook on X-ray spectroscopy followed the publication of a review article on resonant inelastic X-ray scattering (Ament et al., 2011). Jeroen van den Brink and I set up an outline of what basis was necessary to read the review article. The idea was that it should be accessible to graduate students or scientists new to the field with little prior knowledge of spectroscopy with an emphasis on the theoretical background. Since it is impossible to interpret spectroscopy without a model of the system that is being studied, it is also necessary to consider a certain amount of atomic and condensed-matter physics. The focus of the book is predominantly on materials that are strongly correlated, i.e. the interactions between the electrons are usually larger than the bandwidths. The book does not aim for completeness in theoretical approaches, experimental overview, or bibliography. Rather it aims to give the reader a basis for further study and an overview of the necessary ingredients to interpret X-ray spectra.

The book is divided into the following chapters. The first chapter gives a brief overview of what is needed to interpret a spectrum. Since the absorption and scattering of X-rays is, to a first approximation, a local process, Chapters 2-4 lookat the local electronic structure starting from atomic physics and introduce solid-state effects by crystal fields. These sections rely heavily on group theory necessary to understand the complex interactions between electrons in the atomic orbitals and the X-ray photons that carry angular momentum through the polarization vectors. Obviously, we can only skim the surface of these topics which are by themselves the subjects of entire books. Chapter 5 then discusses many-body effects, focusing on the description of the Coulomb interactions in terms of linear and angular momentum. The latter in particular is responsible for many characteristic features in the X-ray absorption spectra known as multiplet structures. Chapter 6 describes the interaction between the photons and the electrons.

How to deal with the Coulomb interactions between particles is one of the more difficult questions in physics. Without their presence, N-particle problems can generally be turned into N one-particle problems. An exact treatment of the Coulomb interaction requires knowledge of the positions of the particles in space. This would not be a problem if the particles were not moving or the velocities of particles differed by orders of magnitudes. The last assumption is generally valid for the Coulomb interaction between electrons and the nuclei. Since in most systems the nuclei are more or less fixed (say, in a solid or a molecule), we can replace the effect of the nuclei on the electrons by an effective potential. For the interaction between the electrons, we can separate the electrons into two types. Core electrons are strongly bound to the nuclei and generally have a binding energy of tens to thousands of electronvolts. If the atomic shells of these electrons are full, we can include their effect in the effective potential of the nucleus. On the other hand, this approach often fails for the valence electrons, a term that we loosely use to describe electrons in states that have a relatively low binding energy, such as the highest-occupied and lowest-unoccupied molecular orbitals (HOMO and LUMO, respectively) in a molecule, the states close to the Fermi level in a metal, or the valence and conduction bands in a semiconductor. However, even for these electrons, treating their interaction in terms of an effective potential often works surprisingly well.

The most commonly-used approach is the local-density approximation used in density-functional theory. However, for many systems, this theory has serious deficiencies. In Chapter 5, we looked at Coulomb multiplets for an atom/ion, which are often clearly visible in X-ray spectroscopy. These effects cannot be described within an effective independent-particle framework. For solids, materials that are known to be insulating are often predicted to be metallic when the interactions between particles are described in terms of an effective potential.