Two is a big number – revisited

By Marcus Wilson 08/05/2014

In a long-in-the-past blog entries I commented on how two is a large number and three might be considered a working approximation to infinity. This kind of mathematics suits Benjamin (22 months old now). He's beginning to grasp what counting is about, but there's some way to go. I'm not exactly sure what's going on in his head when he counts things, but the end result is pointing to objects in turn and saying 'two', 'two', 'two', with the occasional 'eight'. A favourite is to count the stairs in our house as he goes up them. There are fourteen (counted conventionally), but, counting Benjamin-style, there are usually two.

In some ways it seems that 'two' is simply a term used to mean 'more than one'.   That will get him so far in life; for example he has two hands, two arms, two legs, and two rabbits. And as I'm discussing with my second-year solid-state physics class, two is quite sufficient when one is counting electrons.

These are the negatively charged particles often associated with atoms. The physics of electrons is extremely important – it is responsible for electricity and electronics doing what they do. The similarity in the names is no co-incidence.  In a crude model, we think of negatively charged electrons in orbit around a positively charged nucleus, but the reality is rather more complicated and very much more bizarre. In the quantum world, electrons can be found in energy levels. Every system (e.g. an atom, or a molecule, or a crystal) has it's own set of energy levels. If we were to give the system a minimum amount of energy, electrons would fill up these energy levels, from the lowest one upwards. One might think of energy levels as rungs on a ladder. But here's the important bit – for electrons an energy level can only hold a maximum of two electrons. The 'two' comes from a property of electrons called 'quantum spin'. A level might carry zero electrons, it might carry one electron. or it might carry two, but it won't have any more. This leads to something called the Fermi-Dirac distribution, which is a rather essential concept for solid-state physics. It tells you that the more electrons you put into a system, the higher the 'rungs' that they must occupy. It will also tell you that the electrons buried on the low rungs aren't going to do anything useful – they can't move because there's no empty spot for them to move to. Only electrons near the top of the occupied section of the ladder (the Fermi energy) can do anything useful and contribute to electronic properties such as electrical and thermal conductivity of materials. 

Benjamin is then ideally placed for studying solid-state physics. All he needs to do is to count up to two, and he can do just that. In fact, as he goes up the stairs ('two', 'two', 'two') he is, perhaps, counting electrons in energy levels…