Nuclear forces
We conclude this chapter with a brief discussion of the only other known forces, which are called nuclear forces. These forces are within the nuclei of atoms, and although they are much discussed, no one has ever calculated the force between two nuclei, and indeed at present there is no known law for nuclear forces. These forces have a very tiny range which is just about the same as the size of the nucleus, perhaps $10^{-13}$ centimeter. With particles so small and at such a tiny distance, only the quantum-mechanical laws are valid, not the Newtonian laws. In nuclear analysis we no longer think in terms of forces, and in fact we can replace the force concept with a concept of the energy of interaction of two particles, a subject that will be discussed later. Any formula that can be written for nuclear forces is a rather crude approximation which omits many complications; one might be somewhat as follows: forces within a nucleus do not vary inversely as the square of the distance, but die off exponentially over a certain distance $r$, as expressed by $F=(1/r^2)\exp(-r/r_0)$, where the distance $r_0$ is of the order of $10^{-13}$ centimeter. In other words, the forces disappear as soon as the particles are any great distance apart, although they are very strong within the $10^{-13}$ centimeter range. So far as they are understood today, the laws of nuclear force are very complex; we do not understand them in any simple way, and the whole problem of analyzing the fundamental machinery behind nuclear forces is unsolved. Attempts at a solution have led to the discovery of numerous strange particles, the $\pi$-mesons, for example, but the origin of these forces remains obscure.