Investigation of two-photon $2s\to 1s$ decay in one-electron and one-muon ions

We study the radiative decay of the $2s$ state of one-electron and one-muon ions, where the two-photon mechanism plays an important role. Due to the nuclear size corrections, the radiative decay of the $2s$ state in the electron and muon ions is qualitatively different. Based on the accurate relativistic calculation, we introduce a two-parameter approximation, which makes it possible to describe the two-photon angular-differential transition probability for the polarized emitted photons with high accuracy. The emission of photons with linear and circular polarizations is studied separately. We also investigate the transition probabilities for the polarized initial and final states. The investigation is performed for ions with atomic numbers $1\le Z\le 120$.

Figure 1

Feynman graphs describing a two-photon transition in a one-electron ion. The double solid lines denote an electron in the potential of the atomic nucleus (the Furry picture). The wavy lines with the arrows describe the emission of a photon with momentum $k^{\mu }=(\omega ,\mathit{k})$ and polarization $\lambda$.

Figure 2

Differential transition probabilities $\frac{d{W}^{(e,\mu )}}{d{\omega }_1}$ (in ${\mathrm{s}}^{-1}{\mathrm{keV}}^{-1}$) for electron and muon ions for $Z=92$ as a function of the energy sharing fraction $x$ [see Eq. (15)]. The differential transition probabilities are given on a logarithmic scale as ${log}_{10}f\left(x\right)$, where $f\left(x\right)=\frac{d{W}^{(e,\mu )}}{d{\omega }_1}/\left(1{\text{s}}^{-1}1{\text{keV}}^{-1}\right)$.

Figure 3

Differential transition probabilities $\left(\frac{d{W}^{\left(\mu \right)}}{d{\omega }_1}\right)$ (in ${\mathrm{s}}^{-1}{\mathrm{keV}}^{-1}$) for muon ions as a function of the energy sharing fraction $x$ [see Eq. (15)]. The differential transition probabilities are given on a logarithmic scale as ${log}_{10}f\left(x\right)$, where $f\left(x\right)=\frac{d{W}^{\left(\mu \right)}}{d{\omega }_1}/\left(1{\text{s}}^{-1}1{\text{keV}}^{-1}\right)$. The data are presented for $Z=1,50,92,118$.

Figure 4

Ratios between the one-photon and two-photon transition probabilities $(W_{1\mathrm{ph}}^{(e,\mu )}/W_{2\mathrm{ph}}^{(e,\mu )})$ for electron (black solid curve) and muon (red dashed line) ions as a function of the atomic number $Z$. The ratios are given on a logarithmic scale as ${log}_{10}f\left(Z\right)$, where $f\left(Z\right)=W_{1\mathrm{ph}}^{(e,\mu )}/W_{2\mathrm{ph}}^{(e,\mu )}$.

Figure 5

Normalized differential transition probabilities $[f\left(\theta \right)=\frac{d{W}^{\left(e\right)}}{W^{\left(e\right)}sin\theta d\theta }]$ for one-electron ions as a function of the angle between the momenta of the emitted photons ($\theta$). The data are presented for $Z=1,64,92,118$.

Figure 6

Normalized differential transition probabilities [$f\left(\theta \right)=\left(\frac{d{W}^{\left(e\right)}}{sin\theta d\theta d{\omega }_1}\right){\left(\frac{d{W}^{\left(e\right)}}{d{\omega }_1}\right)}^{-1}$] for one-electron ions as a function of the angle between the momenta of the emitted photons ($\theta$) for $x=\frac{1}{2}$. The data are presented for $Z=1,64,92,118$.

Figure 7

Normalized differential transition probabilities [$f\left(\theta \right)=\left(\frac{d{W}^{\left(e\right)\left({\lambda }_1{\lambda }_2\right)}}{sin\theta d\theta d{\omega }_1}\right){\left(\frac{d{W}^{\left(e\right)}}{d{\omega }_1}\right)}^{-1}$] as a function of the angle between the momenta of the emitted photons ($\theta$) for $x=\frac{1}{2}$. The results for the photon linear polarizations ${\mathit{\epsilon }}^{\left(0^{\circ }\right)}$ and ${\mathit{\epsilon }}^{\left({90}^{\circ }\right)}$ considered in Sec. 3c1 are presented. The blue dashed line gives the angular dependence of the differential transition probability for the photons with polarizations ${\mathit{\epsilon }}_1^{\left({90}^{\circ }\right)}$ and ${\mathit{\epsilon }}_2^{\left({90}^{\circ }\right)}$ [see Eq. (34)]. The red solid line gives the angular dependence of the differential transition probability for the photons with polarizations ${\mathit{\epsilon }}_1^{\left(0^{\circ }\right)}$ and ${\mathit{\epsilon }}_2^{\left(0^{\circ }\right)}$ [see Eq. (33)]. The data are presented for $Z=1,64,92,118$.

Figure 8

Normalized differential transition probabilities [$f\left(\theta \right)=\left(\frac{d{W}^{\left(e\right)\left({\lambda }_1{\lambda }_2\right)}}{sin\theta d\theta d{\omega }_1}\right){\left(\frac{d{W}^{\left(e\right)}}{d{\omega }_1}\right)}^{-1}$] as a function of the angle between the momenta of the emitted photons ($\theta$) for $x=\frac{1}{2}$. The results for the photon linear polarizations ${\mathit{\epsilon }}^{\left({45}^{\circ }\right)}$ and ${\mathit{\epsilon }}^{\left({135}^{\circ }\right)}$ and the circular polarization considered in Secs. 3c2 and 3c3 are presented, respectively. In the case of the photon linear polarizations ${\mathit{\epsilon }}^{\left({45}^{\circ }\right)}$ and ${\mathit{\epsilon }}^{\left({135}^{\circ }\right)}$, the blue dashed line represents the angular dependence of the differential transition probability for the photons with equal polarizations [see Eq. (38)] and the red solid line gives the angular dependence of the differential transition probability for the photons with different polarizations [see Eq. (39)]. In the case of the circular photon polarization, the blue dashed line represents the angular dependence of the differential transition probability for the photons with different polarizations [see Eq. (43)] and the red solid line gives the angular dependence of the differential transition probability for the photons with equal polarizations [see Eq. (42)]. The data are presented for $Z=1,64,92,118$.