Relativistic Corrections to the Fine Structure of Helium

Relativistic corrections to the fine structure of the hydrogen atom have been calculated by many authors, but the experimental confirmation of the theory, or alternatively the determination of the numerical value of the fine-structure constant $\alpha$, is limited to an accuracy of $\frac{1}{{10}^5}$ because of the short lifetime of $2{}_{}{}^2{}_{}{}^{}P$ states of hydrogen. However, in view of the fact that $2{}_{}{}^3{}_{}{}^{}P$ states of helium have a longer lifetime, we expect that the fine-structure intervals of the $2{}_{}{}^3{}_{}{}^{}P$ states of helium can be measured to an accuracy of $\frac{1}{{10}^6}$ or better. With this expectation, we attempted to evaluate semirelativistically the theoretical formula for the helium fine structure to the order ${\alpha }^4$ Ry.