Electron-pair-production cross section in the tip region of the positron spectrum

The radial integrals for electron-pair production in a point Coulomb potential have been expressed by Sud, Sharma, and Sud in terms of the matrix generalization of the $\Gamma$ function. Two new partial differential equations in photon energy satisfied by the matrix $\mathit{\Gamma }$ function are obtained. We have obtained, on integrating the partial differential equations, accurate radial integrals as a function of photon energy for the pair production by intermediate-energy photons. The cross section in the tip region of the spectrum are calculated for photons of energy 5.0 to 10.0 MeV for ${}_{}{}^{92}{}_{}{}^{}\mathrm{U}$. The new technique results in extensive saving in computer time as the basic radial integrals in terms of the hyper-geometric function $F_2$ are computed at one photon energy for each pair of partial waves. The results of our calculations are compared with plane-wave Born-approximation results and with the calculations of Dugne and of Deck, Moroi, and Alling.