In their 1947 theory of the capture of negative mesotrons (muons) in matter, Fermi and Teller described how a stopping negative muon loses energy to the electrons of the medium, is captured into a high-lying atomic orbit, and cascades down to the orbit nearest the nucleus, and they proposed that in chemical compounds the capture probability of each atomic species is roughly proportional to its atomic number Z. The paper itself flagged four approximation areas for further work: possible trapping of the muon in stable circular orbits in very light elements (about Z < 6) in insulators; the gas-phase slowing-down model built on collisional neutralization cycles with an ionization-energy scaling I=Ki2I = K i^2; the perturbative, circular-orbit treatment of the inner-shell region; and chemical-bonding and molecular-structure deviations from the Z law. The trapping, gas-kinetics, and inner-shell components are now settled: cascade times are measured at the tens-of-nanoseconds scale in dilute gases and capture in condensed matter is prompt, no long-lived trapped muonic states exist (although the analogous circular-orbit metastability was later discovered for antiprotons and pions in helium), and gas-phase kinetics is computed from ab initio cross sections in the extended cascade model. The surviving open core is the chemical fidelity of the capture stage: per-atom capture probabilities in compounds, mixtures, and molecular gases deviate from the Z law at the tens-of-percent level, existing models are semi-empirical, and the initial quantum levels populated at capture were measured for even the simplest molecules only recently, with mechanisms so far established only qualitatively. The problem posed is a quantitative, predictive account of negative-muon atomic capture in matter — per-atom capture probabilities and initial (n,l)(n,l) populations as functions of the chemical and physical state of the medium — with the settled components entering as constraints.

Background

By 1947 the particle now called the muon was still named the mesotron, and the experiment of Conversi, Pancini and Piccioni (1947) had just shown that its interaction with nuclei is far too weak for it to be the meson predicted by Yukawa; Fermi, Teller and Weisskopf (1947) drew the consequence for its lifetime in matter. Fermi and Teller then supplied the atomic physics behind those observations: they followed a negative mesotron from an energy of about 2000 eV down to the lowest quantum orbit around a nucleus, treating its energy loss to the electrons of the medium classically. Once the mesotron's velocity drops below that of the valence electrons, the medium can be approximated as a degenerate electron gas — a quantum Fermi gas in which all electron states below the Fermi momentum are already occupied, so only electrons near the Fermi surface can absorb energy — and, inside atoms, by the Thomas–Fermi statistical model, an early self-consistent description of the atom as an inhomogeneous electron cloud around its nucleus. With these approximations Fermi and Teller concluded that slowing down, capture into a high-lying orbit, and the subsequent de-excitation cascade to the innermost orbit take far less than the about 2×10⁻⁶ s natural mesotron lifetime: picoseconds in condensed matter and at most about 10⁻⁷ s in normal air. Capture into an atom turns it into a mesic atom (today called a muonic atom) — a hydrogen-like system in which the muon, 207 times heavier than the electron, occupies orbits about 200 times smaller than electronic ones; the cascade emits muonic X-rays (photons of keV-to-MeV energy characteristic of the capturing element) and ends in nuclear muon capture or muon decay. In chemical compounds Fermi and Teller argued by 'crude estimates' that the capture probability of each atomic species is roughly proportional to its atomic number Z — the Fermi–Teller Z law. The paper explicitly flagged four areas where the approximations could fail: (i) in insulators of very light elements (roughly Z < 6, e.g. lithium and beryllium), where electronic excitations require at least the band-gap energy (the Brillouin gap — the minimum energy an electron of an insulating crystal can absorb, set by the gap between the filled and the next empty electronic band), the muon might settle into a stable circular orbit and lose energy anomalously slowly; the authors argued this is 'probably' harmless but did not prove it; (ii) in gases, the slowing-down model assumed that the ion carrying the muon is neutralized in collisions with molecules and that its excitation energy grows with the ionization degree as I=Ki2I = K i^2 (Eq. (42) of the paper), with molecular dissociation treated only qualitatively; (iii) the innermost (K-shell) region was handled with circular-orbit approximations and perturbation theory grafted onto the statistical atom model; (iv) the Z law itself neglected deviations due to chemical bonding and molecular structure. The question raised is the quantitative refinement of these predictions. The subsequent literature built the field of exotic (muonic, pionic, antiprotonic) atoms. Early experimental tests of the Z law, using the decay-electron time spectra of negative muons stopped in compounds (Sens et al. 1958; Astbury et al. 1960; Eckhause et al. 1962; Baijal et al. 1963), found large deviations: in BiF₃, UF₄ and a Cu–Al alloy the measured capture ratios were 1.58±0.15, 1.52±0.15 and 2.08±0.25 where the Z law predicts 3.07, 2.56 and 1.11. The theory of the slowing-down and capture stage was refined in the 1970s: capture rates were computed for H, He and Li as functions of energy and final quantum numbers nn and ll (Haff and Tombrello 1974); transport-theory treatments of solids showed capture typically at a few tens of eV with a nearly statistical angular-momentum distribution and capture ratios scaling as (Z1/Z2)7/6(Z_1/Z_2)^{7/6} in the statistical model, modified by bond ionicity and shell structure (Vogel et al. 1975); models in which the weakly bound electrons dominate capture (Schneuwly, Pokrovsky and Ponomarev 1978) and atomic-radius effects (Daniel 1979) followed. The muonic-atom cascade — the sequence of Auger transitions (de-excitation accompanied by ejection of an atomic electron), radiative transitions, and collisionally induced Stark transitions (l-mixing in the electric field of a collision partner) by which the captured muon reaches the inner shells — was developed into quantitative cascade models for elements with Z > 2 (reviewed by Hartmann 1990), and the level structure of muonic atoms is now given to high precision by a mature combination of atomic, nuclear and QED (quantum electrodynamics) theory (Borie and Rinker 1982). In gases, the extended cascade model computes the kinetics of muonic hydrogen and deuterium from ab initio cross sections with the initial (n,l,E)(n,l,E) distribution taken explicitly into account (Popov and Pomerantsev 2017), and non-thermalized ('epithermal') muonic atoms are a standard ingredient of muon-catalyzed fusion — the process in which a negative muon binds two hydrogen isotopes into a short-lived molecule that fuses (Adamczak and Faifman 2012). Fermi and Teller's caution about stable circular orbits proved physically well-founded in a neighboring sector: antiprotons stopped in liquid helium were discovered trapped in long-lived (microsecond) high-nn, high-ll circular states, the so-called atomcules of antiprotonic helium (Iwasaki et al. 1991; reviewed by Yamazaki et al. 2002), and long-lived pionic helium has been confirmed by laser spectroscopy (Hori et al. 2020); for muons themselves, whose decay lifetime is 2.2 μs, no such trapping occurs — the measured cascade time of muonic hydrogen in dilute gas is 37±5 ns (Ludhova et al. 2007) and capture in condensed matter is prompt on accessible timescales. The one component that remains genuinely open is the chemical fidelity of the capture stage. Modern muonic X-ray analysis — non-destructive bulk elemental analysis applied to archaeological artifacts, lithium batteries, meteorites and steel (Terada et al. 2014; Ninomiya et al. 2024) — still relies on empirically calibrated capture yields, and dedicated experiments at J-PARC (the Japan Proton Accelerator Research Complex muon facility) measure capture ratios and initial quantum levels molecule by molecule, reporting systematic molecular-structure effects that existing models reproduce only qualitatively (Yoshida et al. 2014, 2019; Ninomiya et al. 2018a, 2018b, 2020).

Problem Statement

For negative muons stopping in matter — elemental condensed phases (conductors and insulators, including very light elements), atomic and molecular gases over the relevant density range, and chemical compounds and mixtures — establish a quantitative, predictive description of the atomic-capture stage of the Fermi–Teller theory: (i) the probability that the muon is captured by each atomic species, as a function of the chemical and physical state of the medium (stoichiometry, bond ionicity, molecular structure, electronic band structure, phase and density), including the deviations from the roughly-Z-proportional capture law that the source derived only by 'crude estimates'; and (ii) the distribution of initial capture states — principal quantum number nn, angular momentum ll, and kinetic energy — together with the earliest steps of the cascade that convert this initial population into the observed muonic X-ray yields and final muon destinations. The now-settled results of the source question enter as constraints: capture times are short (no long-lived trapping of muons in stable circular orbits; measured cascade times of tens of nanoseconds in dilute gases; prompt capture in condensed matter), the post-capture gas-phase kinetics of muonic hydrogen and deuterium is described by the extended cascade model with ab initio cross sections, and the inner-shell level structure is given by modern muonic-atom theory. For a representative set spanning the classes above — at least one elemental conductor, one very light-element insulator, one atomic gas and one molecular gas over the relevant density range, and one chemical compound and one mixture, the six representatives being distinct media — establish this account in terms of the observable defined for each class: for the single-species media (the elemental conductor, the very light-element insulator and the atomic gas) the absolute atomic-capture rate and the initial (n,l,E)(n,l,E) distribution; and for the multi-species media (the molecular gas, the chemical compound and the mixture) the per-atom capture probabilities, equivalently the deviations from the roughly-Z-proportional law, together with the initial (n,l,E)(n,l,E) populations. The account is to be accurate for each observable at the level of the existing measurements — for the multi-species capture probabilities, closely enough to reproduce the tens-of-percent Z-law deviations that motivate the problem — and, where an observable has no direct measurement (such as the initial (n,l,E)(n,l,E) distribution of a single-species medium), at the level of the corresponding ab initio benchmark (for example the capture-energy and angular-momentum distributions). Within (ii), the earliest post-capture steps concern media not covered by a settled cascade, while for gas-phase muonic hydrogen and deuterium those steps follow from the extended cascade model.

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