Opening Angles of Electron-Positron Pairs

The cross section for production of a high-energy electron-positron pair of opening angle $\theta$ and electron energy ${\epsilon }_1$, $\frac{d^2\sigma (\theta , {\epsilon }_1)}{d\theta d{\epsilon }_1}$ is calculated. Comparison with available experimental data shows good agreement with the present theory. The cross section $\frac{d^2\sigma (\theta , {\epsilon }_1)}{d\theta d{\epsilon }_1}$ is shown to be closely related to the cross section for an angle ${\theta }_1$ between the photon and the electron, $\frac{d^2\sigma ({\theta }_1, {\epsilon }_1)}{d{\theta }_{1}d{\epsilon }_1}$. At high photon energies the functional dependence of $\frac{d^2\sigma (\theta , {\epsilon }_1)}{d\theta d{\epsilon }_1}$ on the variable $w=\left(\frac{{\epsilon }_1{\epsilon }_2}{k}\right)\theta$ is very nearly the same as the functional dependence of $\frac{d^2\sigma ({\theta }_1, {\epsilon }_1)}{d\theta d{\epsilon }_1}$ on the variable $u={\epsilon }_1{\theta }_1$. The experimental method of estimating the energy of a photon creating a pair from the opening angle of the pair is discussed. Formulas for the most probable photon energy for a measured opening angle, including the effect of multiple scattering, are given.