As noted first by Sunyaev & Churazov ( 1984 ) , the 3.071 mm hyperfine line from ^ { 57 } Fe ^ { +23 } might be observable in astrophysical plasmas .
We assess the atomic processes which might contribute to the excitation of this line .
The distorted wave approximation was used to compute the direct electron collision strength between the two hyperfine sublevels of the ground configuration ; it was found to be small .
Proton collisional excitation was calculated and found to be negligible .
We determine the rate of line excitation by electron collisional excitation of more highly excited levels , followed by radiative cascades .
The branching ratios for hyperfine sublevels for allowed radiative decays and electron collisional excitation or de-excitation are derived .
We show that the dominant line excitation process is electron collisional excitation of the 2p levels followed by radiative decay , as first suggested by Sunyaev & Churazov ( 1984 ) .
We calculate an effective collision strength for excitation of the hyperfine line , including all of these effects and correcting for resonances .
Because the hyperfine line is near the peak in the Cosmic Microwave Background Radiation spectrum , induced radiative processes are also very important .
The effect of background radiation on the level populations and line excitation is determined .

We determine the intensity of the hyperfine line from an isothermal , coronal plasma in collisional ionization equilibrium .
Because of the variation in the ionization fraction of Fe ^ { +23 } , the emissivity peaks at a temperature of about 1.8 \times 10 ^ { 7 } K. We have also derived the hyperfine line luminosity emitted by a coronal plasma cooling isobarically due to its own radiation .
Comparisons of the hyperfine line to other lines emitted by the same ion , Fe ^ { +23 } , are shown to be useful for deriving the isotopic fraction of ^ { 57 } Fe .
We calculate the ratios of the hyperfine line to the 2s—2p EUV lines at 192 Å and 255 Å , and the 2s–3p X-ray doublet at 10.6 Å .