The Interaction ofγ-Rays with Mesotrons

Differential cross sections for the production of a mesotron pair by a $\gamma$-ray and for the bremsstrahlung of a mesotron in the electromagnetic field of a nucleus have been calculated. Both the Proca wave field and the Kemmer matrix formulations of the theory for mesotrons of unit spin and unit magnetic moment were used. These differential cross sections have been integrated in the limit where the energies of the mesotrons and photon are large compared to the rest energy of the mesotron. For a pure Coulomb field, the integrated cross sections are $\frac{A\alpha Z^2{e}^4{E}^2}{{\left(\mu c^2\right)}^4}$ where for pair production $A=\frac{19}{72}$ and $E$ is the $\gamma$-ray energy, and for bremsstrahlung $A=\frac{11}{72} \mathrm{and} E$ is the initial mesotron energy. Since in these processes the important impacts are much closer than the nuclear radius, these cross sections do not describe correctly the electromagnetic effects in the neighborhood of an actual nucleus. The cross sections in a field of the form $\left(\frac{\mathrm{Ze}}{r}\right)(1-e^{\frac{-r}{d}})$ where $d$ is the nuclear radius and is taken to be $\frac{5\hslash Z^{\frac{1}{3}}}{6\mu c}$, are $\frac{B^{\prime }\alpha Z^{\frac{5}{3}}{e}^{4}E}{{\left(\mu c^2\right)}^3}$ where $E$ is defined as before and $B^{\prime }=\frac{\pi }{6} \mathrm{and} \frac{\pi }{18}$ for pair production and bremsstrahlung, respectively. Since considerations of the validity of the Born approximation method used show that the leading terms in the cross sections lose their validity before they become dominant, terms of lower order which give cross sections increasing only logarithmically with the energy have been calculated by the method of virtual quanta. Further, to estimate the minimum values of the cross sections beyond the range of validity, the frequency integral in the virtual quantum calculation was cut off at $\frac{k}{\mu c^2}=\frac{\hslash c}{e^2}$.