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The linear polarization of the characteristic photon emission fromfew-electron ions is studied for its sensitivity with regard to the nuclearspin and magnetic moment of the ions. Special attention is paid, in particular,to the K$\alpha_1$ ($1s 2p_{3/2} ^{1,3}P_{1,2} \to 1s^2 ^1S_0$) decay ofselected helium-like ions following the radiative electron capture intoinitially hydrogen-like species. Based on the density matrix theory, a unifieddescription is developed that includes both, the many-electron and hyperfineinteractions as well as the multipole-mixing effects arising from the expansionof the radiation field. It is shown that the polarization of the K$\alpha_1$line can be significantly affected by the mutipole mixing between the leading$M2$ and hyperfine-induced $E1$ components of $1s2p ^3P_2, F_i \to 1s^2 ^1S_0,F_f$ transitions. This $E1$-$M2$ mixing strongly depends on the nuclearproperties of the considered isotopes and can be addressed experimentally atexisting heavy-ion storage rings.

Hyperfine-induced effects on the linear polarization of the K$\alpha_1$ emission from helium-like ions