Muonium reaction in MgO: A showcase for the final steps of ion implantation

Referee 4
We do not agree with the assessment of the paper of Referee 4. We think the model presented in the paper has a solid foundation by the theoretical and experimental data presented in the manuscript.

Referee 4
The authors must set a solid experimental grounds before suggesting a model. For that: 1. The authors must carry out electric field experiments at various magnetic fields. 2. The authors must measure magnetic field dependences of the muonium amplitude and the initial phase of the muonium precession.

Our answer
In the present case, electric field measurements are not necessary. Such measurements cannot distinguish between the different models because the behavior of the weakly bound electron is the same in both models. The same applies to magnetic field measurements. The justification for our interpretation is based on the broadened diamagnetic line and the frequency shift.

Referee 4
After that is done: 3. The suggested model should be set against different models of Mu formation, in particular, the model of delayed Mu formation via an electron capture by the muon.

Our answer
In the introduction, we describe the competing models and mention the relevant literature. We also mention the controversy about these models and cite the references about it. In the other sections of the paper, a comparison of the models is implicitly addressed when we talk about the broadened diamagnetic-like line and the frequency shift, which shows that the paramagnetic interaction is present from the beginning.
In our opinion, the model is sufficiently well-founded experimentally and theoretically to be presented to the scientific community for discussion.

Referee 3 (Prof. Jess Brewer)
We thank Prof. Jess Brewer for this fair, thoughtful, critical and inspiring review of our paper. We appreciate his comments and questions and answer them as follows

Referee 3
The paper begins with a list of assorted applications of energetic ions (especially protons) in which (the authors declare) the final stages of the stopping process are poorly understood, despite being essential to effective utilization. Since no description is given of what is unknown in the several cases, the reader is left to either read all the references or just take the authors’ word for it.

Our answer
We have rephrased this part as: "Although the stopping process is rather well understood for the high-energy region, very little is known about the final thermalization process [8–12]. The reason for this is that conventional experimental methods are not able to investigate this process at the atomic level. Muon spectroscopy offers the unique possibility that the final steps can be observed directly on the implanted particle and thus detailed information about these final steps can be obtained.”

Referee 3
Footnote 1: For some reason they refer to the 1947 Fermi and Teller paper concerning µ− capture.

Our answer
We wanted to express that stopping of muons was already on the agenda at the beginning of muon research. But this mentioning is not necessary here and we have skipped this reference.


Referee 3
Footnote 2: At the bottom of p. 11 they say, “The inelastic reaction is caused by the force exerted by the squeezed muonium on the surrounding atoms.” If the Mu0 is “squeezed” rather than “expanded”, then why is the hyperfine interaction weaker than in vacuum?

Our answer
We skipped the word “squeezed” here and the word “compressed” in a similar context in the conclusion section.

Referee 3
I have some problems with this picture.
First, the “doorway state” is proposed to last for “hundreds of nanoseconds” before losing its electron completely, during which time it has a slightly shifted precession frequency due to the loosely associated, thermally-polarized electron, after which it precesses at the free muon Larmor frequency ωD. If so, then a Fourier transform omitting the first few hundred ns should yield a sharp line at ωD. Was this done? If not, it should be. If so, and it yielded the same broadened and shifted line as the full time spectrum, then this picture cannot be correct.

Our answer
Analyzing the time spectra separately in the earlier and later times regime gives similar information as a two-component fit with a fast and a slow relaxation. We have tried such fits and obtained of course a fraction with slow relaxation. But the data were not good enough to allow a reliable extraction of the lifetime. We discuss that in section IV.2, third paragraph.

Referee 3
Second, the energy scales seem inconsistent. A fully-bound, compact Mu0 state is bound by some substantial fraction of a Rydberg unless its electron is already shared with the lattice. (Weakly bound Mu0 states are frequently observed in solids, but they always involve the lattice intimately.) Therefore the picture of a epithermal weakly-bound Mu0 state diffusing rapidly through the lattice is self-contradictory. Such diffusion is only possible for a compact, strongly-bound Mu0 state, which therefore arrives at the crucial inelastic collision with a significant fraction of a Rydberg of energy. This energy is supposedly released in said inelastic process, contributing to the local “thermal spike” that raises the effective temperature of the local lattice (which must dramatically reduce the thermal equilibrium polarization of the weakly-bound electron in the “doorway state”). How is it then possible that the resultant frequency shift is so strongly temperature dependent?

Our answer
The energetics discussion is based on the DFT calculations and the data shown in Fig. 6. The diffusing muonium is a compact atom similar to ground state muonium at the interstitial site. At the top of the barrier, the “forward directed” kinetic energy is only a few meV or less and is fully transferred to the oscillating atoms after the inelastic reaction. The inelastic reaction and the subsequent relaxation of the lattice lead then to the long-lived doorway state. The total energy transfer to heat is about 1.2 eV (difference of the barrier heights in Fig. 6).

Referee 3
It may be that my difficulty in following the logic of this paper is due to the somewhat artificial separation of experimental observations from the theory that explains their interpretation; this is probably structurally mandatory, but I get confused when flipping back and forth between assertions and their separate justifications.

Our answer
We had the same difficulty when writing the manuscript, but found that the current structure, showing very clearly the experimental data first and then building the model, is the one which gives the reader the most clear separation of experimental data and theoretical model.

Sorry for the wrong citation. We have now corrected it.