Electric-dipole-moment enhancement factor for the thallium atom, and a new upper limit on the electric dipole moment of the electron

Some time ago, an accurate upper limit on a possible permanent electric dipole moment of the thallium atom in the $6 {}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}$ ground state was obtained by Gould. The result was $D_{\mathrm{Tl}}=\left[\right(1.3\mathrm{}\pm 2.4)\times {10}^{-21\mathrm{}} \mathrm{cm}]e$. In connection with this value, we have carried out a calculation of the electric dipole enhancement factor $R_{\mathrm{Tl}}$, which is defined as the ratio $\frac{D_{\mathrm{Tl}}}{D_e}$, where $D_e$ is the corresponding upper limit on a possible electric dipole moment of the (valence) electron. A value $R_{\mathrm{Tl}}=700\mathrm{}$ was obtained, which leads to an upper limit $D_e=\left[\right(1.9\mathrm{}\pm 3.4)\times {10}^{-24\mathrm{}} \mathrm{cm}]e$. This result is comparable with the value $D_e<(3\mathrm{}\times {10}^{-24\mathrm{}} \mathrm{cm})e$ previously obtained by Weisskopf et al. from measurements on the cesium atom, and with the result of Player and Sandars of $\left[\right(0.7\mathrm{}\pm 2.2)\times {10}^{-24\mathrm{}} \mathrm{cm}]e$ obtained from the search for an electric dipole moment in the ${}_{}{}^3{}_{}{}^{}P_2^{}{}_{}{}^{}$ metastable state of xenon. All three results set a stringent upper limit on the amount of a possible violation of $T$ and $P$ invariance in electromagnetic interactions.