Isotopic effects in scattering and kinetics of the atomic cascade of excited ${\mu }^{-}p$ and ${\mu }^{-}d$ atoms

The quantum-mechanical calculations of the differential and integrated cross sections of the elastic scattering, Stark transitions, and Coulomb deexcitation at collisions of excited ${\mu }^{-}p$ and ${\mu }^{-}d$ atoms with hydrogen isotope atoms in the ground state are performed. The scattering processes are treated in a unified manner in the framework of the close-coupling approach. The basis used includes both open and closed channels corresponding to all exotic-atom states with principal quantum numbers from $n=1$ up to $n_{\max }=20$. The energy shifts of $ns$ states due to electron vacuum polarization and finite nuclear size are taken into account. The kinetics of the atomic cascade of ${\mu }^{-}p$ and ${\mu }^{-}d$ atoms are studied in a wide range of relative target densities ($\phi ={10}^{-8}–1$) within the improved version of the extended cascade model, in which the results of the numerical quantum-mechanical calculations of the cross sections for quantum numbers and kinetic energies of muonic atoms that are of interest for the detailed cascade calculations, are used as input data. Initial $(n,l,E)$ distributions of muonic atoms at the instant of their formation and the target motion are taken into account explicitly in present cascade calculations. The comparison of the calculated cross sections, the kinetic-energy distributions of muonic atoms at the instant of their $np\to 1s$ radiative transitions, as well as the absolute and relative x-ray yields for both muonic hydrogen and muonic deuterium reveals the isotopic effects, which, in principal, may be observed experimentally. The present results are mainly in very good agreement with experimental data available in the literature.

Figure 1

The partial-wave elastic cross section for ${\left({\mu }^{-}p\right)}_{2s}+\text{H}$ collision vs the laboratory kinetic energy $E_{\mathrm{lab}}$ calculated for four variants of the basis corresponding to $n_{\max }=8$ and different values $l_{\max }$ of the internal orbital momentum.

Figure 2

The same is in Fig. 1 for the partial-wave cross section of the Coulomb deexcitation $2s\to 1s$.

Figure 3

The energy dependence of the partial-wave elastic cross section for ${\left({\mu }^{-}p\right)}_{3s}+\text{H}$ collision calculated with the basis corresponding to different values $n_{\max }$.

Figure 4

The same is in Fig. 3 for the partial-wave cross section of the Coulomb deexcitation $3s\to 2s$.

Figure 5

The energy dependence of the integrated cross sections for ${\left({\mu }^{-}p\right)}_{3s}+\text{H}$ collisions calculated with different basis sets: $n_{\max }=3$ (dashed lines) and $n_{\max }=20$ (solid lines).

Figure 6

The partial-wave elastic $4f\text{−}4f$ and Stark $4f\to 4l(l=0–2)$ cross sections for ${\left({\mu }^{-}p\right)}_{4f}+\text{H}$ collision vs the total angular momentum $J$ at the laboratory kinetic energy $E_{\mathrm{lab}}=1$ eV. The calculations were performed with the basis set including all the exotic-atom states with principal quantum numbers $n\le 20$ ($\mathrm{\Delta }l=|l_i-l_f|$) for two variants of interaction potentials: $t_{\max }=14$ (solid lines) and $t_{\max }=1$ (dashed lines).

Figure 7

The partial-wave CD cross sections for ${\left({\mu }^{-}p\right)}_{4l}+\text{H}$ collisions vs the total angular momentum $J$ calculated at the laboratory kinetic energy $E_{\mathrm{lab}}=1$ eV with the basis including all the exotic-atom states with $n\le 20$.

Figure 8

The $l$-averaged cross sections for ${\left({\mu }^{-}p\right)}_n+\text{H}\to {\left({\mu }^{-}p\right)}_{n^{\prime }}+\text{H}$ (solid lines) and ${\left({\mu }^{-}d\right)}_n+\text{D}\to {\left({\mu }^{-}d\right)}_{n^{\prime }}+\text{D}$ (dashed lines) collisions ($n=4$) vs the laboratory kinetic energy $E_{\mathrm{lab}}$.

Figure 9

The same as in Fig. 8 but for $n=5$.

Figure 10

The same as in Fig. 8 but for $n=8$.

Figure 11

The $l$-averaged Coulomb $3\to 2$ deexcitation rates calculated at the liquid hydrogen density ($\phi =1$) for muonic hydrogen (solid line) and deuterium (dashed line) atoms vs the laboratory kinetic energy $E_{\mathrm{lab}}$. The curves with open circles are from Ref. [26]. The horizontal line shows the rate of the radiative $3p\to 1s$ transition in both muonic hydrogen and deuterium (see the text).

Figure 12

The same as in Fig. 11 for Coulomb $4\to 3$ deexcitation rates vs the laboratory kinetic energy $E_{\mathrm{lab}}$. The curves with open circles are from Ref. [26]. The horizontal line shows the rate of the radiative $4p\to 1s$ transition in both muonic hydrogen and deuterium.

Figure 13

The ratio of the external Auger effect and CD rates, ${\lambda }^{\text{Auger}}/{\lambda }^{\text{CD}}$, vs laboratory kinetic energy calculated for ${\mu }^{-}p$ and ${\mu }^{-}d$ atoms in the $3s$ state (solid and dashed lines, respectively) and in the $3p$ state (open circles and asterisk, respectively).

Figure 14

The same as in Fig. 13 but for ${\mu }^{-}p$ and ${\mu }^{-}d$ atoms in the $5s$ state (solid and dashed lines, respectively) and in the $5p$ state.

Figure 15

The density dependence of absolute yields for $K\alpha ,K\beta ,K\gamma$, and $K(\ge \delta )$ lines, and $Y_{\mathrm{tot}}$ calculated for ${\mu }^{-}p$ (lines) and ${\mu }^{-}d$ (lines with triangles) atoms at a target temperature $T=30$ K.

Figure 16

The temperature effect in the density dependence of the absolute x-ray yields for $K\alpha$ and $K\beta$ lines in muonic deuterium calculated at target temperature $T=30$ K (solid lines) and $T=300$ K (dashed lines).

Figure 17

The density dependence of the relative $K\alpha$-line x-ray yields for ${\mu }^{-}p$ (solid line) and ${\mu }^{-}d$ (dashed line) atoms. The experimental data are from Refs. [58, 59, 60, 61].

Figure 18

The density dependence of relative x-ray yields of $K\beta ,K\gamma$, and $K\delta$ lines in muonic hydrogen (solid lines) and deuterium (dashed lines). The experimental data are from Refs. [58, 59, 60, 61].

Figure 19

The density dependence of the $Y\alpha /Y\beta$ ratio for muonic hydrogen (solid line) and deuterium (dashed line), as well as their differences (dashed-dotted line). The experimental data are from Refs. [58, 60, 61].

Figure 20

The kinetic-energy distribution of the ${\mu }^{-}p$ atom at the instant of the radiative transition $2p\to 1s$ calculated at various relative target densities $\phi$ (target temperature $T=300$ K). The kinetic energies, which muonic atoms acquire after CD transitions $n\to n^{\prime }$, are shown by arrows.

Figure 21

The same as in Fig. 20 but for the ${\mu }^{-}d$ atom at the instant of the radiative $2p\to 1s$ transition.

Figure 22

The kinetic-energy distributions of ${\mu }^{-}p$ (thick lines) and ${\mu }^{-}d$ (thin lines) atoms at the instant of their radiative $3p\to 1s$ transition calculated at a few values of the relative target density $\phi$ (target temperature $T=300$ K).

Figure 23

The density dependence of the mean kinetic energy of ${\mu }^{-}p$ (solid lines) and ${\mu }^{-}d$ (dashed lines) atoms at the instant of their radiative transitions $2p\to 1s$ and $3p\to 1s$.