Nuclear-structure corrections to the energy spectra of ordinary and muonic deuterium

One-loop nuclear-structure-induced corrections of order (Zα)⁵ to the Lamb shift and to the hyperfine structure of deuterium are calculated. The contribution of deuteron-structure effects to the (ep)-(ed) and (μp)-(μd) isotopic shifts for the 1S–2S splitting is obtained with the aid of modern experimental data on the electromagnetic form factors for the deuteron. A comparison with the analogous contributions to the Lamb shift for ordinary and muonic hydrogen shows that the relative contribution of corrections associated with the nuclear structure increases as we go over from the hydrogen to the deuterium atom owing to the growth of the nuclear size.     

Theory of the lamb shift in muonic deuterium

We present new calculation of the Lamb shift (2P _1/2 ? 2S _1/2), fine splitting of the 2P state and hyperfine splitting of the 2S state in muonic deuterium (µd) using the quasipotential method in quantum electrodynamics. The vacuum polarization, nuclear structure, and recoil effects are calculated with the account of contributions of orders α ³, α ⁴, α ⁵, and α ⁶. The obtained numerical value of the Lamb shift 202.4139 meV can be considered as a reliable estimate for the comparison with forthcoming experimental data.     

Recoil corrections to muonic atom energy levels

The relativistic correction due to nuclear recoil is calculated for a distributed nuclear charge using the effective potential model of Grotch and Yennie. For the lowest states the results differ substantially from the well-known point nucleus values and disagree somewhat with the recent calculation of Fricke.     

Nuclear fusion in charge-asymmetric muonic molecules

Nuclear fusion reactions in hydrogen-lithium muonic molecules, (where h=p,d,t) are considered and fusion rates from rotational states J=0 of the molecules are presented. Results obtained depend on the isotopic composition of the molecules and range between and . The upper limit for fusion rates from rotational states J=0 of hydrogen-helium muonic molecules, and , equal , is also found. Received: 4 December 1997 / Revised: 30 April 1998 / Accepted: 7 May 1998     

Theory of the muonic hydrogen-muonic deuterium isotope shift

Corrections of the α³, α⁴, and α⁵ orders are calculated for the Lamb shift of the 1S and 2S energy levels of muonic hydrogen μp and muonic deuterium μd. The nuclear structure effects are taken into account in terms of the charge radii of the proton r _p and deuteron r _d for one-photon interaction, as well as in terms of the electromagnetic form factors of the proton and deuteron for the case of one-loop amplitudes. The μd-μp isotope shift for the 1S-2S splitting is found to be equal to 101003.3495 meV, which can be treated as a reliable estimate when conducting the corresponding experiment with an accuracy of 10^?6. The fine-structure intervals E(1S)-8E(2S) in muonic hydrogen and muonic deuteron are calculated.     

Radiative corrections for level widths in light muonic atoms

The corrections, which are associated with electron vacuum polarization, for the radiation level widths and line intensities in light muonic atoms are examined. The total level widths, with allowance for the finite size of the nuclei, relativistic effects, and recoil are found. Zh. éksp. Teor. Fiz. 112, 805–817 (September 1997)     

Nuclear auger effect in muonic atoms

In a muonic atom electromagnetic transitions proceed via emission of X-rays, electrons from the atomic cloud (Auger electrons), or neutrons from the nucleus (nuclear Auger effect). We calculate the neutron spectrum for muonic ²⁰⁷Pb and ²⁰⁹Bi within a microscopic theory of nuclear reactions. The compound nucleus mechanism is dominant. Most of the neutrons arise from the E2 transition 3d → 1s. Agreement between experiment and calculation is achieved to within a factor of 2.     

Screening corrections in cold deuterium fusion rates

We have recalculated the fusion rates of the free deuterium molecule, ion and mesomolecule taking into account the shielding of the Coulomb barrier by the electrons and muon. Electron screening increases the rates by several orders of magnitude. The fusion of hydrogen nuclei is known to be the source of energy of stars occuring in their high-temperature, high-pressure interiors. In terrestrial conditions, one attempts currently to tap this source of nuclear fusion energy by alternative techniques, i.e. high-temperature plasmas or muon-catalyzed fusion. Very recently cold fusion in solids has achieved worldwide attention, since experimental evidence for increased hydrogenic fusion rates in palladium and titanium has been reported[1,2]. These reports have motivated us to reconsider the fusion probability of free hydrogen molecules, ions and mesomolecules, following closely the studies of Refs. [3,4,5]. Our aim however is to improve the previous estimates of the respective fusion rates by taking account of the screening effects of the electrons and muons on the nuclear Coulomb barrier which will turn out to be quite important. As in Ref. [5] we will restrict our numerical results to deuterium only.     

Nuclear polarization shifts in light muonic atoms

A consistent nonrelativistic expression for the energy shift in muonic atoms due to second-order processes is derived under the assumption that the muon is weakly bound. The transverse contribution is shown to be finite only if the two-photon (“seagull”) amplitude is taken into account as required by gauge invariance. Numerical results are presented for muonic ¹²C using a recently developed model for the nuclear response function. The total transverse contributions to the energy shift are found to be small although dependent to some extent on the detailed high-momentum behaviour of the seagull term.     

Nuclear finite-size effects in light muonic atoms

The finite nuclear size corrections to the s-state energy levels of light muonic (or hydrogenic) atoms are calculated analytically through order (Zα)⁶. In addition to the usual expression of order (Zα)⁴, the (Zα)⁵, (Zα)⁶ and (Zα)⁶ log(Zα) contributions have been determined. These corrections have been separated into terms of nonrelativistic and relativistic orders. The results have been checked by solving perturbatively the exact eigenvalue equations of the Schroedinger and Dirac problems of a particle orbiting in the Coulomb field of a uniform charge distribution of fixed radius. Application is made to the case of the μ-⁴He atom. Finite-size contributions to the hydrogen Lamb shift and the relativistic recoil effect are briefly discussed.     

Radiative corrections to the light muonic atoms decay rate

Radiative corrections to the decay rate in low-Z hydrogen-like muonic atoms are considered. The correction is due to the Uehling potential and it has the relative order of α. The numerical results are reported for the 2p → 1s transition for atoms with nuclear charge up to Z = 10.     

Correlation between muonic levels and nuclear structure in muonic atoms

A method that deals with the nucleons and the muon unitedly is employed to investigate the muonic lead, with which the correlation between the muon and nucleus can be studied distinctly. A “kink” appears in the muonic isotope shift at a neutron magic number where the nuclear shell structure plays a key role. This behavior may have very important implications for the experimentally probing the shell structure of the nuclei far away from the β-stable line. We investigate the variations of the nuclear structure due to the interaction with the muon in the muonic atom and find that the nuclear structure remains basically unaltered. Therefore, the muon is a clean and reliable probe for studying the nuclear structure. In addition, a correction that the muon-induced slight change in the proton density distribution in turn shifts the muonic levels is investigated. This correction to muonic level is as important as the Lamb shift and high order vacuum polarization correction, but is larger than anomalous magnetic moment and electron shielding correction.     

Nuclear polarization in muonic atoms of deformed nuclei

The treatment of nuclear polarization correction in muonic atoms belonging to deformed nuclei is analyzed. The geometrical factors involved are expanded into a series of multipoles and the exact expansion coefficients are calculated. It is shown that, using reasonable assumptions about the nuclear spectrum, the nuclear polarization correction may be expressed as a shift of all hyperfine components plus a renormalization of the even multipole hyperfine interaction constants. All nuclear excited states contribute to the shift, but the ground-state rotational band gives an over-whelming contribution to the multipole moment renormalization. The effect of the ground-state band is analyzed in detail. The radial coefficients are calculated and an approximate formula, applicable over a broad range of atomic numbers and deformations, is obtained. By comparing our results with exact calculations we conclude that this part of the nuclear polarization correction may be calculated with accuracy better than 10 %.     

Finite-size corrections to the energy levels of light muonic atoms

An analytic expression is developed in perturbation theory for the nuclear finite-size correction to the s-state energy levels of light muonic atoms. Using first-, second- and third-order perturbation theory, the finite-size corrections of order (Zα)⁴, (Zα)⁵, and (Zα)⁶ are calculated analytically for an arbitrary charge distribution. Application is made to the case of the μ? ⁴He atom, where the error in our finite-size expression is shown to be less than 10ppm.     

A mechanism of unexpected temperature dependence of muonic catalysis in solid deuterium

It is shown that, at sufficiently low temperatures, the elastic scattering of dµ mesic atoms (as well as slow neutrons) in solid deuterium proceeds on the whole crystal lattice without energy loss, whereas inelastic scattering with excitation of phonons is weak. For this reason, the resonant formation of ddµ mesic molecules in solid deuterium occurs before the thermalization of dµ mesic atoms, which explains the observed temperature independence of the ddµ-molecule formation rate and muonic catalysis.     

Nuclear excitation and prompt fission in muonic238U

A study of muonic²³⁸U has been performed in a combined (μ ^?,γ f) and (μ ^?,γγ) coincidence experiment to investigate the role of non-radiative transitions and their fission probabilities. An augmentation of the outer fission barrier ofΔE _b =(0.6±0.1) MeV due to the presence of the muon is deduced. A significant contribution to the prompt fission yield not only results from the (2p→1s) and (3d→1s) non-radiative transitions, but also from other radiationless transitions. Specifically, the measured fission probabilities of the transitions (2p→1s), (3d→1s), and (3p→1s) are (1.5±0.4)%, (5.7±1.7)%, and (5.3±1.9)%, respectively.     

Nuclear signatures in high-order harmonic generation from laser-driven muonic atoms

High-order harmonic generation from muonic atoms exposed to intense laser fields is considered. Our particular interest lies in effects arising from the finite nuclear mass and size. We numerically perform a fully quantum mechanical treatment of the muon-nucleus dynamics by employing modified soft-core and hard-core potentials. It is shown that the position of the high-energy cutoff of the harmonic spectrum depends on the nuclear mass, while the height of the spectral plateau is sensitive to the nuclear radius. We also demonstrate that gamma-ray harmonics can be generated from muonic atoms in ultrastrong VUV fields, which have potential to induce photonuclear reactions.     

Nuclear monopole polarization in muonic 208Pb

Nuclear polarization effects due to the presence of a muon in an atomic orbit are described in a self-consistent way. Within this framework the monopole part of the polarization energy is calculated by solving Hartree-Fock equations for the combined muon-nucleus system. The nuclear part is described by using the density dependent forces of both Skyrme and Moszkowski. The muon is assumed for the present to be in the 1s and 2s state, respectively, and is treated relativistically. The monopole part of the energy gain turns out to be -0.943 ± 0.070 keV for the 1s-muon and -0.169 ± 0.070 keV for the 2s-muon. The rms-charge-radii are decreasing by 0.20% and 0.036%, respectively.