Energy shift in (dtμ)e due to the finite size of the muonic molecular ion (dtμ)+

Abstract

The energy shift due to the presence of the extended (dt~)l I pseudonucleus (in its first excited state with one unit of angular momentum) in the quasihydrogenlike system (dtl~)~lei, is estimated using perturbation theory up to second order with two choices of zeroth order electron wavefunctions. The energy shift is found to be 0.50 meV using pure Coulomb electron wavefunctions and 0.58 meV using electron wavefunctions calculated with a potential modified to take partial account of the finite size of (dtll)H. In both cases, the perturbation Hamiltonian is expanded in multipoles, retaining terms up to and including octupole terms.

Effect ofdtμquasinucleus structure on energy levels of the(dtμ)Xeeexotic molecule

“…Thus, under the above approximations, the radial wave function Φ nl (ρ) in the potential (19) coincides with the harmonic-oscillator wave function and the multipole matrix elements (16), (17), (18) are reduced to l-independent…”

“…It is worthwhile to compare this result with the first ever elaborate six-body calculation of the (dtµ)dee energy shifts in the first-order PT [8]. In this paper, the molecular structure, i. e., the dependence on l, was explicitly taken into account in contrast with previous calcula-tions [9,10,18] where the l-independent energy shift was obtained by scaling the result for the atom-like four-body system (dtµ)e. As pointed out in this paper, the monopole contribution calculated in Ref. [8] depends on the choice of the coordinate system that does not allow a comparison.…”

“…The reason for this dependence is an increasing in the dipole matrix element (22) with increasing n, which, in turn, leads to an increasing in the second-order PT contribution. The cancellation effect was widely discussed, e. g., in [9,10,18]; nevertheless, the dependence on the molecular quantum state was beyond the scope of those papers where only the atom-like system (dtµ)e was calculated.…”

Alternative approach to the derivation of an expression for the cross section for resonant formation ofdtμ in the muon-catalyzed fusion cycle

“…(75), which corresponds to resonant level shifts. However, these terms are at least partially included in the resonance energy values used by Faifman et al [8,34], for they are analogous to the energy correction terms calculated for the (dry)»e resonant complex by Harston, Shimamura, and Kamimura [37] using perturbation theory to second order [38].…”

Four-body variational calculation of a hydrogen-like atom involving an excited muonic molecule

“…While the dtµ has a compact structure, the three-body system has a finite charge distribution, and might be deformed under the presence of the electron. Harston et al adopted a perturbative approach [8,9] to calculate ∆E (FS) ; however, the first order perturbation energy ∆E (1) = 18.253 meV of the ∆E (FS) is found to be comparable in opposite sign to the second order perturbation energy, ∆E (2) = −17.752 meV, which implies a slow convergence of the perturbative expansion of ∆E (FS) = ∑ i ∆E (i) . Thus, a full variational calculation is desirable.…”