Quantifying uncertainties in crystal electric field Hamiltonian fits to neutron data

1- The paper describes the protocol through which inelastic neutron scattering data can be applied to characterize and understand crystalline electric field spectra and the associated Hamiltonian, along with appropriate uncertainties. As such it is a useful contribution.

1- The author largely discusses the cases of Yb3+ based magnetic materials. This is fine, as Yb3+ is a Kramer's ion, and thus its crystal field ground state doublet must have a magnetic moment. It is also an interesting case, as, as the author points out, there are only three transitions from the ground state possible, so error bars are relatively large.

However the author then presents his results generalized to two families of rare-earth based magnetic materials, pyrochlores and those in the NaErSe2 structure. In so far as a calculation is concerned, this is fine, but the author should make clear that:

i) Some of the rare-earth pyrochlores that his calculations are relevant to, the ones with light rare earths, do not exist as rare earth titanates. For example, there is no Ce2Ti2O7 or Pr2Ti2O7 or Nd2Ti2O7 as a cubic pyrochlore. The author should make this clear in the manuscript so as to avoid undue confusion. The same may be true for some of the NaErSe2 family members as well.

ii) Some of the rare-earth pyrochlores (those based on Ho3+, Tm3+, Pr3+, and Tb3+) are non-Kramer's ions, and are not required by symmetry to have at least doubly degenerate CEF - they could have ground states that are non-magnetic singlets. There is nothing wrong with including these in the table and paper, but again it would be important to state that these results are relevant to this particular calculation. For example, Tm2Ti2O7 is known to possess a non-magnetic, singlet CEF ground state; see M.P. Zinkin, M.J. Harris, Z. Tun, R.A. Cowley, and B.M. Wanklyn, J. Phys. Condens. Matt. 8, 193 (1996). The association of gzz=8.65 is only meaningful within the context of this specific calculation - it is not relevant to Tm2Ti2O7 itself.

I strongly suggest the author include language that makes it clear that these results may not describe the actual materials listed in these two tables, and, indeed that some of these materials do not actually exist as equilibrium crystal structures - and also briefly discuss the reasons for this. I don't believe this need lengthen the paper very much.

The acceptance criteria could be met, if my comments can be appropriately addressed.

The two points under Weaknesses should be addressed.

A method is presented that determines the uncertainty in the crystal field parameters obtained from inelastic neutron scattering measurements and the consequences for other derived properties. This is important, but rarely (or never) done.

Absence of equations.

Inelastic neutron scattering is possibly the best experimental probe to determine crystal field parameters and hence the ground state of rare-earth ions in various compounds, but it is rare to estimate the related uncertainties. A method to do this is presented and benchmarked using simulated neutron spectra. The results are very interesting, in that a small uncertainty in e.g. the determined wave function may give rise to a large uncertainty in derived physical parameters (such as the g-tensor), and vice versa, which underlines the importance of such estimates.

Although the equations used in the present work are known and documented in the literature, the paper would become more self-contained if the basic equations used are given, such as the Hamiltonian, the expression for the neutron scattering cross-section, and the expression for the calculation of the g-tensor. This could be done in the description of the method, in the beginning of Section 2, before the Yb2Ti2O7 pyrochlore example; the latter perhaps then requiring a minor revision.

In Table 2, which shows eigenvectors for different eigenvalues, it is not clear to me why two entries (lines) are given per energy (eigenvalue), as the two entries (in my understanding) are related by symmetry and the corresponding coefficients are the same and have the same uncertainties.

The same remark applies to Tables 3 and 4, where each compound has two entries, which are related by symmetry (except for the case of accidentally nearly degenerate singlets in Table 4).

The rational for the order of the entries in Tables 3 and 4 is unclear to me. Why not in order of the atomic number?

In Section 4 on line 9 is said that the "peak shapes are asymmetric". This is not an intrinsic feature of crystal-field levels, but is sometimes seen on instruments on undermoderated pulsed sources with asymmetric resolution functions. I'd suggest to replace "are" by "may be".

The importance of additional data or constraints is mentioned in the 3rd paragraph of Section 3.3. This is indeed often necessary and rather commonplace, and perhaps a few more citations in this context could guide the reader, e.g. Galera et al, J. Phys.: Condens. Matter 30 (2018) 285802 (which also outlines an alternative to the standard Monte-Carlo search), but any other relevant work would also do.

Typos:
Page 2, 9 lines from the bottom: replace "based off" by "based on" (or "based of")
Section 4, line 11: replace "bad" by "severe" or similar.
Ref. [3]: DOI missing.
Ref. [15]: Journal name (or Editors/Publisher) missing.
Ref. [16]: Page number (356) and DOI missing.

Add basic equations.

1. Authors considers an important property, the relation between the crystal electric field levels that can be measured by neutron scattering and effective magnetic state of the low-energy components of the J-multiplet for a number of topical rate earth systems.
2. The presented calculation results can be useful for other researchers.
3. The manuscript is clearly written and is easy to read and understand. Figures and tables are clear. The length is appropriate.

1. The manuscript lacks expressions for neutron scattering cross-section that were used to obtain fits shown in the figures. These need to be included.
2. Because of the above, it is not clear how many independent parameters does a measurement of at most 4 peaks represent. With peak widths being fixed by the properties of the instrument, there are at most 8 independent parameters contained in 4 peaks (positions and intensities). Hence, it is obvious that a model with 9 parameters, such as author considers for Yb2O3, is over-parameterized and parameters cannot be refined. Author finds this result upon performing the fitting and chi-squared search and states it as a result of the work, "Thus, a Yb3+ CEF model with nine independent parameters definitely needs more information than just neutron scattering peaks to constrain a CEF fit." This should be revised and put in the context of relation between the number of free parameters and number of independent measured quantities contained in the crystal field peaks.
3. More generally, the measured intensities of the 4 peaks might be related to their positions via some type of general expressions, such as the sum rules, and therefore not represent independent measurements. It is therefore not even clear whether the positions and intensities of the 4 peak do represent 6 independent measured quantities, which are needed to refine 6 crystal field parameters for pyrochlores and delafossites cases.

The manuscript can be published in this journal after it is revised to address its weaknesses. I repeat these below:
1. The manuscript lacks expressions for neutron scattering cross-section that were used to obtain fits shown in the figures. These need to be included.
2. Because of the above, it is not clear how many independent parameters does a measurement of at most 4 peaks represent. With peak widths being fixed by the properties of the instrument, there are at most 8 independent parameters contained in 4 peaks (positions and intensities). Hence, it is obvious that a model with 9 parameters, such as author considers for Yb2O3, is over-parameterized and parameters cannot be refined. Author finds this result upon performing the fitting and chi-squared search and states it as a result of the work, "Thus, a Yb3+ CEF model with nine independent parameters definitely needs more information than just neutron scattering peaks to constrain a CEF fit." This should be revised and put in the context of relation between the number of free parameters and the number of independent measured quantities contained in the crystal field peaks.
3. More generally, the measured intensities of the 4 peaks might be related to their positions via some type of general expressions, such as the sum rules, and therefore not represent independent measurements. It is therefore not even clear whether the positions and intensities of the 4 peak do represent 6 independent measured quantities, which are needed to refine 6 crystal field parameters for pyrochlores and delafossites cases.

1. The manuscript lacks expressions for neutron scattering cross-section that were used to obtain fits shown in the figures. These need to be included.

2. The conclusions should be revised and put in the context of the relation between the number of free parameters of the CEF model and the number of independent measured quantities contained in the crystal field peaks. It should be clearly explained whether the positions and intensities of the 4 peak do represent at least 6 independent measured quantities, which are needed to refine 6 crystal field parameters for pyrochlores and delafossites cases, and that they certainly do not contain 9 independently measured quantities needed for the Yb2O3 CEF model.

3. >The CEF parameter uncertainties are straightforward, defined by the range of parameter fit values. For derived quantities like the ground state eigenkets or the g tensor, we calculate these quantities for each solution and then take the range of calculated values to be the uncertainty bounds. In this way the uncertainties are propagated through the CEF calculations.

Explain why not use mean square deviations of the set of the obtained values for the uncertainty?