All particles in three-dimensional space are either bosons or fermions. What distinguishes one from the other is not a simple matter: essentially it’s whether the particle’s spin is integer (e.g., 0, 1, and so forth), or half integer (e.g., 1/2, 3/2, and so forth). These two classes of particles are distinguished behaviorally by different statistics, which refer to the way physicists count the number of possible states in collections of different particles.
A main new insight of quantum mechanics is that the same fundamental entities, whether the same elementary particles or the same atoms, are indistinguishable.
Let’s say we have two boxes and two identical entities, which are indistinguishable. The problem is to count the states of that system because the number of states determines the probabilities of the entity being located in a given state. In terms of quantum mechanics, there are three possibilities or states: both entities can go into one box, or both into the other box or one can go into each box.
A perspicacious child might ask whether each of the entities could also be put in the other box, so that the number of possible states would be four. Classically, yes, but quantum mechanically, no, because the entities are indistinguishable, and there is no way to make the differentiation that allows for the possibility of putting each in one or the other box.
In the case of atoms, the boxes are akin to the positions of the atoms in a crystal. The number of ways of putting the atoms at positions in the crystal can be “counted,” and gives rise to different “statistics” that are different for atoms that are bosons and for atoms that are fermions.
Fermions that are indistinguishable can’t be in the same state. Two electrons of the same spin can’t be put on top of each other (the Pauli exclusion principle). So in the case of two spin up electrons and two boxes, there is only one possible state — each box contains one spin up electron.
In relativistic quantum mechanics, there is a deep connection between the spin of a particle and its statistics (the spin statistics theorem).
If the spin is one-half integer, like the spin of the electron or the quark, then the particle is a fermion. If the spin is integer, such as zero or one or two, then the particle is a boson.
An atom consists of a nucleus and orbiting electrons. Since a nucleus, except for the simplest hydrogen atom, is made out of protons and neutrons, both of which have spin one half, putting them together results in either a nucleus (and an atom) of integer spin for an even number of nucleons, or a nucleus (and an atom) of half integer spin for an odd number of nucleons. Helium-4 with two protons and two neutrons has an even number of nuclei and is a boson. Helium-3 with two protons and one neutron has an odd number of nuclei so that atom is a fermion. Though both atoms contain two electrons, the number of nuclear constituents determines that one is a boson and that the other is a fermion.
Interestingly, the properties of a fermionic gas are very different from the properties of a bosonic gas. In the fermionic helium-3 gas, electrons tend to repel each other. Bosons, on the other hand, like to be in the same state, which helps to account for the phenomenon of Bose/Einstein condensation or superfluidity.