Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The ionization potential depression (IPD) in plasmas is crucial to plasma modelling. This paper gives a detailed study of Pauli blocking on IPD in degenerate plasmas for the case of multi-electron He-like atoms, using the established quantum statistical approach. This work is an extension of the related studies on single-electron atoms. Therefore, I recommend the publication of this work in “Particles”. The author may consider shorten the paper, eg, the discussion on single-electron cases could be more brief with some citations of previously published works. 

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This manuscript studied the ionization potential depression and ionization balance of the composition of dense plasmas at which the free electrons are degenerate. The effect of Pauli blocking is addressed implemented by a quantum statistical approach. The proposed theoretical formalism is applied to one- and two-electron ions with the Mott effect being considered. Ionization potential depression and ionization balance of dense plasmas is of vital importance for investigations of charge state distributions and plasma spectroscopy. This topic is important and timely for readers. However, a few issues should be clarified before considering its publication.

(1) The title of this manuscript is “Ionization potential depression in degenerate plasmas and Pauli blocking of multi-electron ions”, which investigates the ionization potential depression in dense plasmas with special attention to Pauli blocking effect. However, this topic is not mentioned in the abstract, which is given below.

“The composition of partially ionised plasmas is investigated for densities and temperatures at which the free electrons are degenerate. Based on a quantum statistical approach, the effect of Pauli blocking is addressed. Specifically, one- and two-electron ions are studied. New results regarding the degree of ionisation and the Mott effect are presented. Standard codes for plasma properties do not take Pauli blocking effect into account and are therefore unable to explain the experiments in the high-density regime, where the electrons are degenerate.”

From the above abstract, the author discussed the ionization balance (composition) of dense plasmas at which the free electrons are degenerate. Similarly, the Pauli blocking effect seems to be restricted in the problem of ionization balance, not the ionization potential depression. Indeed, there is a relation between the ionization potential depression and ionization balance, however, these two subjects are not the same completely. The author should clarify the inconsistency between the title and abstract.

 

(2) Some most updated references are not mentioned in the manuscript. As is well known, the subjects of physical properties and processes in dense plasmas made great advances during the last decade. Some theoretical formalisms also take into account Pauli blocking effect in various other methods in investigations of ionization potential depression and ionization balance. For the readers’ convenience, the most updated references which deal with these two subjects of ionization potential depression and ionization balance should be introduced in the text.

 

(3) On page 21, the author gave Table A2 which shows the average ionization degree and other physical quantities of dense plasmas as a function of density. Are these physical quantities independent on temperature? In table caption and in the main text, one cannot find information of temperature. If these physical quantities are independent on plasma temperature, the author should explain this issue. From the theoretical formulas, in principle, these physical quantities should be temperature dependent.

 

(4) On page 2, the author stated that ”Due to Pauli blocking, the degree of ionisation at high densities is greater than predicted by standard approaches such as OPAL.” From a recent paper published in Nature, the experimental results showed that at high plasma densities, the degree of ionisation predicted by standard approaches such as OPAL is usually lower than the experimental measured value. If the degree of ionisation at high densities predicted by this work is greater than predicted by standard approaches such as OPAL, then the degree of ionization predicted by this work is much higher than the experimental measured value reported in the Nature paper. How can the author explain this?

 

(5) In the theoretical section, some redundant information might be deleted. For example, Debye screening was introduced and discussed too much, while the important text on the screening potential and screening effect which the present work take is inadequate and unclear.

 

Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The author investigates partially ionised plasmas, focusing on the role of Pauli blocking. The manuscript presents new results, including the treatment of the Mott effect and the degree of ionisation. While I find the study useful in the context of low-density plasmas, several points should be addressed before the work can be considered for publication, particularly to strengthen its applicability and generality.

First, in the Introduction the author refers to Debye theory in its standard point-like classical formulation. In the context of plasmas containing nuclei up to carbon, a word of caution is warranted. The point-charge approximation is only strictly valid for low atomic numbers (of order (A \lesssim 10)). For heavier nuclei, a finite-size charge distribution (e.g. Gaussian or more realistic profiles) leads to a reduction of the binding energy and modifies screening properties. This effect has been studied in higher-density plasmas and is well established within virial expansions (see, e.g., Barba et al., MNRAS 528, 3498 (2024), https://doi.org/10.1093/mnras/staf040). The author should comment on the limitations of the point-like assumption in the present framework.

Second, Eq. (1) is explicitly non-relativistic. This raises the question of the regime of validity of the results when applied to sufficiently dense plasmas, where relativistic effects may become non-negligible. In particular, it would be important to clarify whether the treatment of Pauli blocking remains consistent within a non-relativistic approximation, and to delineate the parameter space in which this description is reliable.

Furthermore, the manuscript would benefit from a broader discussion of physical environments where the considered densities and mechanisms may play a role. One relevant example is the magnetosphere of neutron stars. In such environments, relativistic effects are unavoidable, even if magnetic field effects are neglected at first order. While Debye screening provides an adequate description at small momentum transfer, deviations are expected at larger momenta. In particular, the behaviour of charge polarisation coefficients as a function of the electron Fermi momentum, (y = k/(2k_{F,e})), leads to modified potentials beyond (y \gtrsim 1/4). The author is encouraged to discuss how such effects may impact the validity of the present approach.

Finally, the manuscript would benefit from careful proofreading. There are several typographical errors (e.g. around line 608), and the notation for equations should be made consistent throughout (e.g. “Eq. (x)” vs “eq. (x)”).

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have addressed my previous comments. I recommend its publication in Particles.

Reviewer 3 Report

Comments and Suggestions for Authors

The author has satisfactorily addressed most of the comments and concerns raised.