We experimentally investigate the $4f_{5/2}\text{--}4f_{7/2}$ fine-structure splitting energies of Ag-like ions by optical emission spectroscopy of ${\mathrm{Lu}}^{24+}$, ${\mathrm{Hf}}^{25+}$, and ${\mathrm{Ta}}^{26+}$ using a compact electron beam ion trap. The measurements are enabled by an electron-impact metal injection technique, while the transition energies are determined with high accuracy using wavelength calibration based on reference emission from the same spatial region. The measured transition energies in the range $Z=71–73$ provide valuable benchmarks for relativistic many-electron calculations involving a $4f$ electron and, together with previous measurements, establish a consistent $Z$-dependent trend. Comparison with theoretical calculations over a wide range of $Z$ reveals systematic differences among the predictions.

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• Atomic & molecular structure
• Atomic spectra
• Electron correlation calculations for atoms & ions
• Electronic structure of atoms & molecules
• Fine & hyperfine structure
• Ions
• Optical spectroscopy
• Spectroscopy