Hyperfine-induced modifications to the angular distribution of the $K{\alpha }_1$ x-ray emission

The angular distribution of 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 hydrogenlike ions with nonzero nuclear spin has been studied within the density matrix theory and the multiconfiguration Dirac-Fock method. Emphasis is placed especially upon the hyperfine interaction and how this interaction of the magnetic moment of the nucleus with those of the electrons affects the angular properties of the $K{\alpha }_1$ radiation. Calculations were performed for selected isotopes of heliumlike Sn${}^{48+}$, Xe${}^{52+}$, and Tl${}^{79+}$ ions. A quite sizable contribution of the hyperfine interaction upon the $K{\alpha }_1$ angular emission is found for isotopes with nuclear spin $I=1/2$, while its effect is suppressed for (most) isotopes with nuclear spin $I>1/2$. We therefore suggest that accurate measurements of the $K{\alpha }_1$ angular distribution at ion storage rings can be utilized as a tool for determining the nuclear parameters of rare stable and radioactive isotopes with $I\ge 1/2$.

Figure 1

The fine- and hyperfine-level scheme for the $1s^2$ ${}^1{S}_0$ ground level and the $1s2p_{3/2}$ ${}^{1,3}{P}_{1,2}$ excited levels of heliumlike ions with nuclear spin $I=1/2$.

Figure 2

Effective anisotropy parameter ${\beta }_2^{\mathrm{eff}}$ of the $K{\alpha }_1$ characteristic emission as functions of the magnetic dipole moment ${\mu }_I$ following the REC into the two excited $1s2p_{3/2}{}^{1,3}{P}_{1,2}$ levels of (finally) heliumlike projectiles with kinetic energy $T_p=50$ MeV/u. Results are shown for the ${}_{50}^A$Sn${}^{48+}(A=119,113,121$; black points), ${}_{54}^A$Xe${}^{52+}(A=129,127,125,123$; red squares), and ${}_{81}^A$Tl${}^{79+}(A=187,205,207$; blue triangles) isotopes as well as their zero-spin counterparts (shadowed area), respectively. Lines are drawn as a guide to the eyes.

Figure 3

Effective anisotropy parameter ${\beta }_2^{\mathrm{eff}}$ of the $K{\alpha }_1$ characteristic emission as functions of the projectile energy $T_p$ following the REC into the two excited $1s2p_{3/2}{}^{1,3}{P}_{1,2}$ levels of (finally) heliumlike ions. Results are shown for the ${}_{50}^{119}$Sn${}^{48+}$ ($I=1/2$, ${\mu }_I=-1.047{\mu }_N$; left panel) and ${}_{81}^{207}$Tl${}^{79+}$ ($I=1/2$, ${\mu }_I=+1.876{\mu }_N$; right panel) isotopes, respectively. Computations for the two spin-1/2 isotopes (black solid lines) are compared with those for zero-spin isotopes of the same elements (blue dashed lines).

Figure 4

Angular distribution of the $K{\alpha }_1$ characteristic emission following the REC into the excited $1s2p_{3/2}{}^{1,3}{P}_{1,2}$ levels of (finally) heliumlike ions. Results for the spin-1/2 isotopes from Fig. 3 (black solid lines) are compared with computations for zero-spin isotopes (blue dashed lines) of the same element. All calculations were performed within the projectile frame and for projectile ions with kinetic energy $T_p=10$ MeV/u.

Figure 5

The same as Fig. 4, but for the projectile energy $T_p=50$ MeV/u and for three different xenon isotopes with nuclear spin $I\ge 1/2$ and comparable magnetic dipole moment ${\mu }_I$:   ${}_{54}^{129}$Xe${}^{52+}$ ($I=1/2,{\mu }_I=-0.778{\mu }_N$; left panel), ${}_{54}^{121}$Xe${}^{52+}$ ($I=5/2,{\mu }_I=-0.701{\mu }_N$; middle panel), and ${}_{54}^{137}$Xe${}^{52+}$ ($I=7/2,{\mu }_I=-0.968{\mu }_N$; right panel).