Atomic Structure

We can now extend this quantum approach and consider the way in which electrons are confined in an atom. Such electrons move in a potential well created by the electric field due to the positively charged atomic nucleus. The problem is now three dimensional as compared to the simple one dimensional situations we have considered so far and the associated mathematical solutions of the Schrodinger equation (i.e. the wave functions) are significantly more complicated, but the same type of result emerges, namely that, for well-behaved wave functions, only certain energy levels for an electron are allowed. The wave functions are now characterized by three quantum numbers, reflecting the fact that we are dealing with three dimensions. Considering a single electron, the allowed energy levels are precisely those conjectured by Bohr, except that the quantum number n does not specify the angular momentum of the electron: it gives information about the shape, not least the ‘wiggliness’, of the wave function. The electron orbital angular momentum is, neverthess, specified and its value and direction are given by two other quantum numbers (denoted by I and m) whose values, for a given n, are limited. Also, as Bohr conjectured, it is measured in units of (h = h/2π). The situation for a single electron is simple and the energy levels and wave functions of the hydrogen atom, which has only one electron, can be calculated very straightforwardly. The allowed frequencies of the electromagnetic radiation (referred to as spectral lines in the hydrogen spectrum) when the electron jumps from one level to another can therefore be readily calculated by equating the energy difference between the two levels to hv, the energy of the emitted photon. Experimental tests showed that, although there was fairly good agreement, some of the lines appeared to consist of two closely separated lines. Similar anomalies appeared in the spectrum emitted when the emitting hydrogen atoms were placed in a magnetic field. To account for these discrepancies, it was proposed in 1925 by Goudsmit and Uhlenbeck that the electron itself had an intrinsic angular momentum and was always spinning like a top. The nature of the anomalies could be accounted for if the electron spin had the value h/2 and we speak of the electron as having spin i. It also followed that, since the electron carries an electric charge, its rotation would lead to an electric current and, in turn, to the electron behaving as a miniscule magnet. The splitting of the spectral lines could then be understood in terms of small changes in magnetic energy depending on which way the electron spin was pointing in relation to the magnetic field produced by the orbital motion of the electron. Similar explanations also accounted for the behaviour of atoms in external magnetic fields. For atoms containing two or more electrons the situation becomes much more complicated, because not only is there the attractive electric force between the nucleus and the electrons but also there is electrical repulsion between the electrons themselves. This is, however, a complicated matter of detail. Another much more fundamental issue is consideration of which energy level each electron in a multi electron atom occupies. The natural assumption is that in a stable atom every electron is in the same energy state the lowest one possible; in a higher state an electron would be expected to drop to the lowest state, emitting radiation in the process. This would lead one to expect that all chemical elements, i.e. the different possible atoms, would be very similar. However, it is found that atoms differing even by one electron can have dramatically different chemical and physical properties. The solution to this basic problem was put forward by Pauli, also in 1925, who hypothesized that all electons are absolutely identical (not like two billiard balls with their small imperfections) and that no two identical particles of spin ½ can occupy the same quantum state; that is each electron in an atom must occupy a state with a different set of quantum numbers. This is known as the Pauli exclusion principle. It is now known to be correct and, as we shall, has a profound effect on the structure of all states of matter. In an atomic energy level whose energy is specified by the quantum number n (see above) only a certain number of electrons can be accommodated depending on how many different values of the quantum numbers 1 and m are available to it and in which direction the electron spin is pointing (colloquially referred to as ‘up’ or ‘down’). For example, the lowest level can only accommodate two electrons; it is then full and we speak of a filled shell. Atoms such as this (helium in this simplest of cases) with only completely filled levels are very stable and hardly interact; they include what are known as the inert gases (helium, neon, argon etc). Moving to an atom having a nucleus with three units of positive charge and three electrons, two will be in the lowest energy level and the third is alone in the next level up. In some respects the resultant atom is rather like a hydrogen atom having a single electron outside a tight core consisting of the nucleus and the two inner electrons which together have unit net positive charge (three positive charges on the nucleus and two negative charges on the two electrons). This atom is lithium and other atoms with filled inner shells and one spare electron are sodium, potassium etc, all having similar properties. And so we could go on to atoms with higher and higher positive charge on the nucleus and having larger and larger numbers of electrons. These electrons fill higher and higher energy levels according to the limitation on their numbers required by the exclusion principle. In moving through these different atoms further similarities in atomic structure and associated properties are found. It is in this way that we understand the periodic table of atoms first introduced on an empirical basis by Mendeleev in 1869, which exhibits the recurrently similar properties of the different chemical elements.