The Quantum Universe_ Everything That Can Happen Does Happen - Brian Cox [71]
How so? Let’s begin by considering a chain of atoms (as ever modelled by a chain of potential wells), but now suppose that each atom has several electrons bound to it. This, of course, is the norm – only hydrogen has just the one electron bound to a single proton – and so we are moving from a discussion of a chain of hydrogen atoms to the more interesting case of a chain of heavier atoms. We should also remember that electrons can come in two types; spin up and spin down, and the Pauli principle informs us that we can drop no more than two electrons into each allowed energy level. It follows that for a chain of atoms each containing just one electron per atom (i.e. hydrogen) the n = 1 energy band is half-filled. This is illustrated in Figure 8.7, where we have sketched the energy levels for a chain of 5 atoms. This means that each band contains 5 distinct allowed energies. These 5 energy states can accommodate a maximum of 10 electrons, but we only have 5 to worry about so, in the lowest energy configuration, the chain of atoms contains the 5 electrons occupying the bottom half of the n = 1 energy band. If we had 100 atoms in the chain then the n = 1 band could contain 200 electrons, but for hydrogen, we only have 100 electrons to deal with and so once again the n = 1 band is half filled when the chain of atoms is in its lowest energy configuration. Figure 8.7 also shows what happens in the case that there are 2 electrons for every atom (helium) or 3 electrons per atom (lithium). In the case of helium, the lowest-energy configuration corresponds to a filled n = 1 band, whilst for lithium the n = 1 band is filled and the n = 2 band is half filled. It should be pretty clear that this pattern of filled or half-filled continues such that atoms with an even number of electrons always lead to filled bands whilst atoms with an odd number of electrons always lead to half-filled bands. Whether a band is full or not is, as we shall very soon discover, the reason why some materials are conductors whilst others are insulators.
Figure 8.7. The way electrons occupy the lowest available energy states in a chain of five atoms when each atom contains one, two or three electrons. The black dots denote the electrons.
Let’s now imagine connecting the ends of our atomic chain to the terminals of a battery. We know from experience that if the atoms form a metal then an electric current will flow. But what does that actually mean, and how does it emerge from our story so far? The precise action of the battery on the atoms within the wire is, fortunately, something we don’t really need to understand. All we need to know is that connecting up the battery provides a source of energy that is able to kick an electron a little, and that kick is always in the same direction. A good question to ask is exactly how a battery does that. To say ‘it is because it induces an electric field within the wire and electric fields push electrons’ is not entirely satisfying, but it will have to satisfy us as far as this book is concerned. Ultimately, we could appeal to the laws of quantum electrodynamics and try to work the whole thing out in terms of electrons interacting with photons. But we would add absolutely nothing to the current discussion by doing this, so in the interests of brevity, we won’t.
Imagine an electron sitting in one of those states of definite energy. We will start by assuming that the action of the battery can only provide very tiny kicks to the electron. If the electron is sat in a low energy state, with many other electrons above it on the energy ladder (we have Figure 8.7 in mind when using this language), it will be unable to receive the energy kick from the battery. It is blocked, because the