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For relativistic atomic two-body systems such as the hydrogen atom,positronium, and muon-proton bound states, a two-body generalisation of thesingle-particle Sommerfeld fine-structure formula for the relativisticbound-state energies is found. The two-body Sommerfeld bound-state energyformula is obtained from a two-body wave equation which is physically correctto order $(Z\alpha)^4$. The two-body Sommerfeld formula makes two predictionsin order $(Z\alpha)^6$ for every bound state and every mass ratio. With $N$ theBohr quantum number: (a) The coefficient of the $(Z\alpha)^6/N^6$ energy termhas a specified value which depends only on the masses of the bound particles,not on angular quantum numbers; (b) The coefficient of the $(Z\alpha)^6/N^4$energy term is a specified multiple of the {\em square} of the coefficient ofthe $(Z\alpha)^4/N^3$ energy term. Both these predictions are verified inpositronium by previous calculations to order $(Z\alpha)^6$ which usedsecond-order perturbation theory. They are also correct in the Coulomb-Diraclimit. The effect of the two-body Sommerfeld formula on calculations ofmuon-proton bound-state energies is examined.

Sommerfeld Fine-Structure Formula for Two-Body Atoms