Reply to “Comment on ‘Hyperfine-induced modifications to the angular distribution of the $K{\alpha }_1$ x-ray emission’ ”

In a recent work [Phys. Rev. A 89, 022513 (2014)], the $K{\alpha }_1$ ($1s2p_{3/2}{}^{1,3}P{}_{1,2}\to 1s^2{}^1{S}_0$) x-ray emission following the radiative electron capture into initially hydrogen-like ions has been explored for ions with nonzero nuclear spin $(I\ne 0)$. A rather strong influence upon the angular distribution of the (hyperfine- and fine-structure averaged) $K{\alpha }_1$ radiation was found, especially for isotopes with nuclear spin $I=1/2$, while this effect are less important for isotopes with nuclear spin $I>1/2$. Two comments were made by Inal and Benmouna about this work with regard to (i) the incoherent summation of the individual hyperfine components of the $1s2p_{3/2}{}^1{P}_1\to 1s^2{}^1{S}_0$ transition and (ii) the treatment of the hyperfine-induced $E1\text{−}M2$ multipole mixing in the $1s2p_{3/2}{}^3{P}_2\to 1s^2,{}^1{S}_0$ fine-structure component. While we agree with the first comment and here provide updated anisotropy parameters, the hyperfine-induced modification of the $K{\alpha }_1$ emission remains valid and may help in the future to determine the nuclear parameters of radioactive isotopes. We also explain that the hyperfine-induced $E1\text{−}M2$ mixing has already been fully taken into account in our previous work.

Figure 1

Effective anisotropy parameter ${\beta }_2^{\mathrm{eff}}$ of the $K{\alpha }_1$ emission following the REC of (finally) helium-like isotopic ions with kinetic energy 50 MeV/u. Results are shown for ${}_{50}{}^A\mathrm{Sn}^{48+}$ (black points), ${}_{54}{}^A\mathrm{Xe}^{52+}$ (red [gray] squares), and ${}_{81}{}^A\mathrm{Tl}^{79+}$ (blue [dark gray] triangles) isotopes as well as for their zero-spin (i.e., no hyperfine structure) counterparts (shaded area), respectively. Lines are drawn to just guide the eyes.