Neutron-activated sources of $^{103}$Ru and $^{103}$Pd both share the isomeric first-excited state in $^{103}$Rh as a daughter product. From independent measurements of both decays, we have measured the $K$-shell and total internal conversion coefficients, $alpha_K$ and $alpha_T$, for the 39.8-keV $E$3 transition, which de-excites that state in $^{103}$Rh, to be 141.1(23) and 1428(13), respectively. When compared with Dirac-Fock calculations, our new results disagree with the version of the theory that ignores the $K$-shell atomic vacancy, which is consistent with our conclusion drawn from a series of measurements on high multipolarity transitions in nuclei with higher $Z$. Calculations that include the atomic vacancy indicate that the transition actually has a small $M$4 component with mixing ratio $delta$ = 0.023(5).
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