Re-Evaluation of the Nuclear Magnetic Octupole Moment of ²⁰⁹Bi

Round 1

Reviewer 1 Report

The authors calculated the electronic structure factors which enter
the magnetic octupole HFS for a number of states in neutral 209Bi.
This was done by modifying the HFS92 code of the GRASP package.
The obtained results were compared with the related experimental data
to derive the nuclear octupole moments. The obtained values are found
to be in good agreement with other estimations and differ significantly
from the prediction of the single particle nuclear model. The discrepancy
with the single particle model is not surprising since a similar situation
occurs for the nuclear magnetic moment.

The obtained results may be of interest to theorists working on
relativistic calculations of atomic systems. I recommend the paper
for publication in Atoms.

Author Response

Thank you very much for your comments on our article.

Reviewer 2 Report

The authors present a short but highly interesting and relevant contribution related to the extension of GRASP to enable calculations of the atomic matrix elements required to evaluate the magnetic octupole moment contribution to atomic HFS. The authors chose 209Bi as an example to apply their computational extension of GRASP. This is a particularly well-chosen isotope, as the magnetic dipole moment is long known to pose a challenge to nuclear theory; the magnetic octupole moment appears to pose a similar puzzle for the shell model. Correct treatment of the atomic structure is essential to such a discussion. 

The paper is well written, clear, and concise. The results are appropriately summarized in tables, and compared to the relevant literature. The authors furthermore make welcome recommendations for future experimental work, which this paper and their theoretical developments will be able to support. I therefore strongly recommend this article for publication and hope it can trigger further work in the nuclear physics (experimental and theoretical) and atomic physics community.

I have a few comments and remarks the authors may wish to consider.

1. The introduction starts rather abruptly with the single-particle shell-model estimates of Omega. The authors may wish to add provide a few sentences of context; e.g. that the magnetic dipole moment is nowadays routinely measured to study nuclear structure, but that the magnetic octupole moment is comparatively studied a lot less.

2. Page 1, line 14: in the single-particle approximation shown in the text, the <r^2> is the mean-squared radius of the orbit of the valence proton, not the total nuclear radius. In practice, however, this quantity to be determined from theory, and therefore often the nuclear radius is used as an approximate estimate instead.

3. Sometimes one finds in the literature the intuitive idea that for the magnetic dipole moment, the A constant is proportional to the magnetic field at the site of the nucleus. For Q, the B constant is proportional to the electric field gradient. Would there be a similar intuition for the magnetic octupole, based on the expressions in the paper? Would it be e.g. proportional to the third derivative of the magnetic field? This is mostly a curiosity on my part, but perhaps a statement in the paper would be of interest for the reader.

4. It is interesting that the electron correlations are not so important for C/Omega. In the references 7,16 it seems the results are more sensitive to correlations (since the Dirac Hartree Fock values are very different from the full calculations). Can this be understood in terms of the intuitive argument on page 3, around line 70?

5. The experimental paper referenced by the authors lists the experimental hyperfine spacings. Would it be possible for the authors to calculate the matrix elements required to calculate the second order hyperfine shifts using GRASP? In doing so, they would enable a future experimental group to also re-analyse of the old experimental data, to see if the second-order HFS was correctly treated in literature. This would make the publication even more useful in the future, in my opinion.

6. The final numbers of magnetic octupole moment should include an uncertainty as well, due to experiment when possible, and due to the new theoretical atomic structure values.

7. Perhaps more of a comment related to the conclusions on page 4, line 103 onwards. Indeed, the single particle shell model performs poorly. It is interesting to note that for the magnetic dipole moment, the same is true. It seems the magnetic properties of 209Bi cannot be understood using a pure single-particle model. 

Author Response

Please see the attachment.

Author Response File: Author Response.pdf