Introduction

Centrifugal barrier effects have their origin in the balance between the repulsive term in the radial Schrödinger equation, which varies as 1/r2, and the attractive electrostatic potential experienced by an electron in a many-electron atom, whose variation with radius differs from atom to atom because of screening effects. In order to understand them properly, it is necessary to appreciate the different properties of short and of long range potential wells in quantum mechanics.

As the energy of the incident photon is increased above the ionisation threshold, centrifugal barrier effects often come to dominate the response of many-electron atoms, which totally alters the spectral distribution of oscillator strength in the continuum from what might be anticipated by comparison with H. This applies not only to free atoms, but also to the same atoms in molecules and in solids: many of the changes due to centrifugal effects occur within a small enough radius that they are able to survive changes in the environment of the atom.

Since the centrifugal term is present in the radial Schrödinger equation for all atoms, we must explain why centrifugal effects only dominate the inner valence spectra of fairly heavy atoms. Centrifugal barrier effects are present even in H. However, they act differently in transition elements or lanthanides.

One reason is straightforward: the ground state of H has ℓ = 0, and therefore only p states are accessible directly by a dipole transition from the ground state.

Introduction

There are many connections between the physics of free atoms and that of solids which have been noted, in passing, several times already in the present volume. One should add that many-body theory and, especially, the concept of excitations as quasiparticles in free atoms, owe much to the theory of excitations in solids [590].

The theme of the present chapter is rather more specific: the intention is to present a number of effects which are counterparts of those we have studied in previous chapters, but for atoms in the solid rather than in the gaseous phase. Also, the intention is to set the scene for the last chapter, in which atomic clusters will be used in an attempt to bridge the gap from the atom to the solid experimentally. A highly excited atom in a solid will be taken as an atom excited close to or above the Fermi energy (including, of course, core excitation). There are some solid state systems for which electrons with energies close to the Fermi level behave like those in atoms. X-ray absorption and electron energy loss spectroscopy involving core excitation to empty electronic states can then be described by initial and final states possessing L, S and J quantum numbers, and the allowed transitions follow strict dipole selection rules. Examples include the d → f transitions of Ba in high Tc superconductors, and many instances involving transition metals and lanthanides.

Line strengths of discrete transitions

In the present chapter, we consider line strengths of transitions between bound states, i.e. of lines whose only form of natural broadening is radiative, and which lie below the first ionisation potential. The simplest situation is encountered in the photoabsorption or photoexcitation of an atom, initially in its ground state | i ≥, in which case one transition is observed to each excited final state | f ≥. The price one pays for this simplicity is that all excited states cannot be reached in this way because of selection rules.

The distribution of intensities is an essential property of a spectrum. We may consider: (a) the distribution in energy over the whole spectrum; or (b) the distribution within an individual spectral line.

It was noted in the previous chapter that spectral lines of interacting channels can differ greatly in intensity within a narrow energy range. However, with some significant exceptions, if one can approach the series limit closely enough to be clear of perturbers, the intensities of successive members decrease monotonically with increasing principal quantum number n. This is described as the normal course of intensity for a Rydberg series.

In the presence of perturbations, the course of intensities becomes far less regular than that of transition energies, and it is generally more likely that intensities will exhibit fluctuations or departures from the expected behaviour.

It took a very long time to write this book, especially to bring it to a relatively consistent and complete form. The journey of the reader to these final pages was also not easy and straightforward. What are your feelings after getting through the jungle of more than 1300 many-storeyed formulas? Perhaps, twofold. At first – relief and satisfaction: it is all over now, I made it! But secondly – are all these formulas correct? The answer is not so simple. I tried to do my utmost to be able to answer ‘yes’: compared with the original papers, deduced some of them again, checked numerically, looked for special cases, symmetry properties, etc. But I cannot assert that absolutely all signs, phases, indices, etc. are correct. Therefore, if you intend to do some serious research starting with one or other formula from the book, it is worthwhile carrying out additional checks, making use of one of the above mentioned methods.

Not all aspects of the theory are dealt with in equal depth. Some are just mentioned, some even omitted. For example, the method of effective (equivalent) operators deserves mentioning. It allows one to take into account the main part of relativistic effects but at the same time to preserve the LS coupling used for classification of the energy spectra of the atoms or ions considered.

While studying the energy spectra or other spectral quantities of atoms and ions having complex electronic configurations, one ought to consider the expressions for the matrix elements of the operators both within each shell of equivalent electrons and between each pair of these shells. For example, in order to find the energy spectrum of the ground configuration 1s22s2 of the beryllium atom, we have to calculate the interaction energy in each shell ls2 and 2s2 as well as between them. The last case will be discussed in this chapter. If there are more shells, then according to the two-particle character of interelectronic interactions, we have to account for this interaction between all possible pairs of shells.

Let us notice that momenta of each shell may be coupled into total momenta by various coupling schemes. Therefore, here, as in the case of two non-equivalent electrons, coupling schemes (11.2)–(11.5) are possible, only instead of one-electronic momenta there will be the total momenta of separate shells. To indicate this we shall use the notation LS, LK, JK and JJ. Some peculiarities of their usage were discussed in Chapters 11 and 12 and will be additionally considered in Chapter 30. Therefore, here we shall restrict ourselves to the case of LS coupling for non-relativistic and JJ (or jj) coupling for relativistic wave functions. We shall not indicate explicitly the parity of the configuration, consisting of several shells, because it is simply equal to the sum of parities of all shells.

It has taken a very long time for this book to appear. For many years I had in my mind the idea of publishing it in English, but practical implementation became possible only recently, after drastic changes in the international political situation. The book was started in the framework (a realistic one) of the former USSR, and finished soon after its collapse, after my motherland, the Republic of Lithuania, regained its independence.

Academician of the Lithuanian Academy of Sciences, Adolfas Jucys, initiated the creation of a group of scientists devoted to the theory of complex atoms and their spectra. Later it was named the Vilnius (or Lithuanian) school of atomic physicists, often called by his name. However, for many years the results of these studies were published largely only in Russian and, therefore as a rule they were not known among Western colleagues, particularly those in English-speaking countries. A large number of the papers were published in Russian in the main Lithuanian physical journal Lietuvos fizikos rinkinys – Lithuanian Journal of Physics, translated into English by Allerton Press, Inc. (New York) as Soviet Physics – Collection (since 1993 – Lithuanian Physics Journal).

Recently the situation has become incomparably better. There is no problem publishing the main ideas and results in prestigious international journals in English. However, it would be very useful to collect, to analyse and to summarize the main internationally recognized results on the theory of many-electron atoms and their spectra in one monograph, written in English. This book is the result of the long process of practical realization of that idea.

In previous chapters we considered the wave functions and matrix elements of some operators without specifying their explicit expressions. Now it is time to discuss this question in more detail. Having in mind that our goal is to consider as generally as possible the methods of theoretical studies of many-electron systems, covering, at least in principle, any atom or ion of the Periodical Table, we have to be able to describe the main features of the structure of electronic shells of atoms. In this chapter we restrict ourselves to a shell of equivalent electrons in non-relativistic and relativistic cases.

A shell of equivalent electrons

The non-relativistic wave function (1.14) or its relativistic analogue (2.15), corresponds to a one-electron system. Having in mind the elements of the angular momentum theory and of irreducible tensors, described in Part 2, we are ready to start constructing the wave functions of many-electron configurations. Let us consider a shell of equivalent electrons. As we shall see later on, the pecularities of the spectra of atoms and ions are conditioned by the structure of their electronic shells, and by the relative role of existing intra-atomic interactions.

N electrons with the same values of quantum numbers nili (LS coupling) or niliji (jj coupling) are called equivalent. The corresponding configurations will be denoted as nlN (a shell) or nl jN (a subshell). A number of permitted states of a shell of equivalent electrons are restricted by the Pauli exclusion principle, which requires antisymmetry of the wave function with respect to permutation of the coordinates of the electrons.

The wave function for the particular case of two equivalent electrons may be constructed, using vectorial coupling of the angular momenta and antisymmetrization procedure.