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Register a new userLong-range interactions of hydrogen atoms in excited states. II. Hyperfine-resolved 2S-2S system
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The interaction of two excited hydrogen atoms in metastable states constitutes a theoretically interesting problem because of the quasi-degenerate 2P_{1/2} levels which are removed from the 2S states only by the Lamb shift. The total Hamiltonian of the system is composed of the van der Waals Hamiltonian, the Lamb shift and the hyperfine effects. The van der Waals shift becomes commensurate with the 2S-2P_{3/2} fine-structure splitting only for close approach (R < 100 a_0, where a_0 is the Bohr radius) and one may thus restrict the discussion to the levels with n=2 and J=1/2 to good approximation. Because each S or P state splits into an F=1 triplet and an F=0 hyperfine singlet (eight states for each atom), the Hamiltonian matrix {em a priori} is of dimension 64. A careful analysis of symmetries the problem allows one to reduce the dimensionality of the most involved irreducible submatrix to 12. We determine the Hamiltonian matrices and the leading-order van der Waals shifts for states which are degenerate under the action of the unperturbed Hamiltonian (Lamb shift plus hyperfine structure). The leading first- and second-order van der Waals shifts lead to interaction energies proportional to 1/R^3 and 1/R^6 and are evaluated within the hyperfine manifolds. When both atoms are metastable 2S states, we find an interaction energy of order E_h chi (a_0/R)^6, where E_h and L are the Hartree and Lamb shift energies, respectively, and chi = E_h/L ~ 6.22 times 10^6 is their ratio.
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The theory is developed for one and two atom interactions when the atom has a Rydberg electron attached to a hyperfine split core state. This situation is relevant for some of the rare earth and alkaline earth atoms that have been proposed for experiments on Rydberg-Rydberg interactions. For the rare earth atoms, the core electrons can have a very substantial total angular momentum, $J$, and a non-zero nuclear spin, $I$. In the alkaline earth atoms there is a single, $s$, core electron whose spin can couple to a non-zero nuclear spin for odd isotopes. The resulting hyperfine splitting of the core state can lead to substantial mixing between the Rydberg series attached to different thresholds. Compared to the unperturbed Rydberg series of the alkali atoms, the series perturbations and near degeneracies from the different parity states could lead to qualitatively different behavior for single atom Rydberg properties (polarizability, Zeeman mixing and splitting, etc) as well as Rydberg-Rydberg interactions ($C_5$ and $C_6$ matrices).
Ab initio study of the density-dependent population and lifetime of the long-lived $(mu p)_{2s}$ and the yield of $(mu p)_{1s}$ atoms with kinetic energy 0.9 keV have been performed for the first time. The direct Coulomb $2sto 1s$ deexcitation is proved to be the dominant quenching mechanism of the $2s$ state at kinetic energy below $2p$ threshold and explain the lifetime of the metastable $2s$ state and high-energy 0.9 keV component of $(mu p)_{1S}$ observed at low densities. The cross sections of the elastic, Stark and Coulomb deexcitation processes have been calculated in the close-coupling approach taking into account for the first time both the closed channels and the threshold effects due to vacuum polarization shifts of the $ns$ states. The cross sections are used as the input data in the detailed study of the atomic cascade kinetics. The theoretical predictions are compared with the known experimental data at low densities. The 40% yield of the 0.9 keV$(mu p)_{1s}$ atoms is predicted for liquid hydrogen density.
Metastable ${2S}$ muonic-hydrogen atoms undergo collisional ${2S}$-quenching, with rates which depend strongly on whether the $mu p$ kinetic energy is above or below the ${2S}to {2P}$ energy threshold. Above threshold, collisional ${2S} to {2P}$ excitation followed by fast radiative ${2P} to {1S}$ deexcitation is allowed. The corresponding short-lived $mu p ({2S})$ component was measured at 0.6 hPa $mathrm{H}_2$ room temperature gas pressure, with lifetime $tau_{2S}^mathrm{short} = 165 ^{+38}_{-29}$ ns (i.e., $lambda_{2S}^mathrm{quench} = 7.9 ^{+1.8}_{-1.6} times 10^{12} mathrm{s}^{-1}$ at liquid-hydrogen density) and population $epsilon_{2S}^mathrm{short} = 1.70^{+0.80}_{-0.56}$ % (per $mu p$ atom). In addition, a value of the $mu p$ cascade time, $T_mathrm{cas}^{mu p} = (37pm5)$ ns, was found.
The hyperfine induced $2s2p ^3P_0, ^3P_2 to 2s^2 ^1S_0$ E1 transition probabilities of Be-like ions were calculated using grasp2K based on multi-configuration Dirac-Fock method and HFST packages. It was found that the hyperfine quenching rates are strongly affected by the interference for low-Z Be-like ions, especially for $2s2p ^3P_0 to 2s^2 ^1S_0$ transition. In particular, the trends of interference effects with atomic number $Z$ in such two transitions are not monotone. The strongest interference effect occurs near Z=7 for $2s2p ^3P_0 to 2s^2 ^1S_0$ E1 transition, and near Z=9 for $2s2p ^3P_2 to 2s^2 ^1S_0$ E1 transition.
In calculating the energy corrections to the hydrogen levels we can identify two different types of modifications of the Coulomb potential $V_{C}$, with one of them being the standard quantum electrodynamics corrections, $delta V$, satisfying $left|delta Vright|llleft|V_{C}right|$ over the whole range of the radial variable $r$. The other possible addition to $V_{C}$ is a potential arising due to the finite size of the atomic nucleus and as a matter of fact, can be larger than $V_{C}$ in a very short range. We focus here on the latter and show that the electric potential of the proton displays some undesirable features. Among others, the energy content of the electric field associated with this potential is very close to the threshold of $e^+e^-$ pair production. We contrast this large electric field of the Maxwell theory with one emerging from the non-linear Euler-Heisenberg theory and show how in this theory the short range electric field becomes smaller and is well below the pair production threshold.
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