Energies ofπ-Mesonic X-Rays

The energies of the $2p\to 1s$ x-rays from the $\pi$-mesonic atoms Li through F have been measured to ½% or better. All the measured energies were lower than those calculated from the Klein-Gordon equation assuming a point-charge Coulomb field and corrected for finite size effects and vacuum polarization. (These are the only non-negligible electromagnetic corrections.) This indicates that the net nuclear interaction between a meson in the $1s$ orbit and a nucleus is repulsive. The energy shifts varied in a fluctuating manner from 3.4% for Li to 11.7% for O and 11.4% for F. If we assume that the shift arises from a scattering type potential, and further, that the effects of the individual nucleons are simply additive, then values for the $s$-wave scattering lengths $a_1$ and $a_3$ can be obtained by a least squares fit to the data. The values of $a_1$ and $a_3$ obtained in this manner are $a_1={0.14}_5{\lambda }_{\pi }$ and $a_3=-0.10\mathrm{}{\lambda }_{\pi }$. The $K_{\alpha }$ energies of the isotopes ${\mathrm{B}}^{10}$ and ${\mathrm{B}}^{11}$ were measured separately, thus giving a direct determination of $a_3$ under the previous assumptions. This value for $a_3$ is $(-0.12\mathrm{}\pm 0.02){\lambda }_{\pi }$.