# Why is the “fine structure” correction called that way?

I'm working on the fine structure correction to the Hydrogen atom. I have more of a conceptal, maybe historical question, why is this correction called this way? and why is the fine structure constant called this way?

We use the term "fine structure" to refer to the three terms in the Hamiltonian that contribute to the splitting of energy levels (the relativistic kinetic energy correction, the spin-orbit coupling term, and the Darwin term). As you may end up showing, these perturbative corrections are of order $$(Z\alpha)^2$$ times the unperturbed energy, where $$\alpha$$ is the fine structure constant. Given that - for the case of the hydrogen atom, where $$Z=1$$ - this product is $$\sim5\times10^{-5}$$, we see that the correction is indeed small compared to the unperturbed energy, and thus fine. We therefore call the resulting splitting fine structure to distinguish it from the energy levels obtained without the perturbations (the gross strucuture), which are comparatively large.
This of course then leads to the natural name of the hyperfine structure, which arises because of interactions between the magnetic moments of the proton and electron. The energy splitting is of the same order as the fine structure perturbations by multiplied by the ratio of the particles' masses, $$(m_e/m_p)\sim5\times10^{-4}$$, making it over three orders of magnitude smaller than the fine structure contributions, and thus hyperfine.