We investigate the effects of the tensor polarizability of a nucleus on thebound-state energy levels, and obtain a general formula for the contribution of the tensor polarizability to the energy levels in two-body bound systems. In particular, it is demonstrated that the tensor polarizability leads to mixing between states with different orbital angular momenta. The effect of tensor polarizability is evaluated for the hyperfine-structure components of P states and for the mixing of S and D states in muonic deuterium.
Nuclear effects are very important for muonic bound systems in view of the smaller Bohr radius as compared to electronic bound systems [1][2][3][4][5]. In an early investigation on the subject (Ref. [6]), it was shown that, for a deuteron nucleus, the displacement of the constituent proton (inside the deuteron) relative to the constituent neutron gives rise to an energy shift which, for S states and in leading order, is proportional to the probability density at the origin. For non-S states, because of the vanishing probability density at the origin, the effect enters in higher order. In particular, for the 2P state of muonic deuterium, the effect has been shown to be proportional to the product of the nuclear polarizability constant and the matrix element ⟨1/r 4 ⟩ of the bound muonic state (here, r is the muon-deuteron distance, see Eqs. (7) and (10) of Ref. [1]). The nucleus, in this case, acts as a "polarizable core" which interacts with the bound muon in much the same way as a charged atomic core would otherwise interact with a "Rydberg electron" (the latter point of view is illustrated in Chap. 6 of Ref. [7]).
In this work, we explore an effect which transcends the simple picture of an electrically polarizable deuteron and takes into account its spin structure, namely, the tensor polarizability. In general, nuclei with spin quantum numbers higher than spin-1/2 can exhibit a nonvanishing tensor polarizability. Our aim is to derive a general formula for the energy shift experienced by a bound muon (or electron) bound to a nucleus which has a nonvanishing tensor polarizability. One notes that, in contrast to the scalar polarizability, the diagonal energy shift of S states due to the nuclear tensor polarizability vanishes after angular integration. For non-S states, we find that the effect of nuclear polarizability in muonic deuterium is small, not yet discernible in experimental measurements, but that it gives rise to a very interesting alteration of the hyperfine mixing manifold, even leading (somewhat surprisingly) to mixed states involving more than one value of the orbital angular momentum. For example, the nuclear tensor polarizability leads to a mixing of S and D states.
We use natural units with ℏ = c = ϵ 0 = 1 throughout this paper, except where factors of ℏ, c, and ϵ 0 are shown explicitly for emphasis.
We work with a generalized Bohr radius
where Z is the nuclear charge number, α is the finestructure constant and m r is the reduced mass. With Friar and Payne [8], we assume that the polarizability tensor (α P ) ij of the nucleus can be expressed as
.
Here, the constants α E , σ N and τ N parameterize the electric (scalar), vector (ℓ = 1) and tensor (ℓ = 2) components of the nuclear polarizability, respectively. We denote Cartesian indices by superscripts and use the Einstein summation convention. For the (static) scalar and tensor polarizabilities of the deuteron, the calculated results of Friar and Payne [8] read as follows, α E = 0.6330(13) fm These results are expressed in terms of a polarization volume, i.e., in units of fermi cubed. For reference, we indicate the conversion of the polarization volume to the SI polarizability, which, for the scalar polarizability, reads as follows,
where ϵ 0 is the vacuum permittivity. The numerical values in Eq. ( 3) have to be interpreted in terms of volume polarizabilities.
A comprehensive derivation of the polarizability correction of an atomic core for non-S states of a bound system is presented in Sec. 6.6 of Ref. [7]. A generalization of Eq. (6.174) of Ref. [7] for a polarizable nucleus shows that, for non-S states,
α r 4 , (5) where the second-rank coordinate and nuclear spin tensors are
The coupling of angular momenta in one-muon ions proceeds by first coupling the orbital angular momentum ⃗ L and the spin ⃗ S of the muon to form the total angular momentum of the muon according to ⃗ J = ⃗ L + ⃗ S µ . One then couples ⃗ J to the nuclear spin ⃗ S N , according to ⃗ F = ⃗ J + ⃗ S N . This is appropriate because the spinorbit coupling of the bound muon is stronger than the spin-spin coupling of the muon and the nucleus.
In a one-muon ion, one parameterizes the bound states as follows (with hyperfine resolution)
where the hyperfine-resolved spin-angular function is m N , respectively. Via angular reduction formulas, which are investigated in the next section, one arrives at the following, general result for the matrix element of an operator proportional to f (r) X ij
(2) S ij (2) , taken with hyperfineresolved bound-state wave functions,
Here, G
LJ is an angular factor calculated in Eq. ( 22). We take into account the mixing of the fine-structure and hyperfine structure by allowing for different orbital angular momentum numbers L ′ , L and total muon angular momentum numbers J ′ , J in the bra and ket states. Finally, in the manifold of states with given F and M F , the matrix of the energy perturbations by the nuclear polarizability is
We
This content is AI-processed based on open access ArXiv data.