Original e⁻ Capture Cross Sections for Hot Stellar Interior Energies

Round 1

Reviewer 1 Report

 

The paper `Original e−–Capture Calculations for hot stellar interior energies' presents QRPA calculations of the electron capture cross sections for several astrophysically  important isotopes in relevant energy range. The paper is well written, provides required information of the calculations along with references for more details.  The results of the paper are new, astrophysically important and thus I can recommend it for publication in Particles.

However, I would like to suggest an optional modification:

I believe that if a few illustration, presenting the reaction rates in astrophysical conditions, obtained by ‘appropriate convolution techniques assuming that leptons under stellar interior conditions follow Maxwell-Boltzmann energy distribution’  as well as comparison of these rates with widely applied models (e.g. TALYS) will be extremely useful for readers.

Minor points:

1) The results of the paper can be also important for accreting neutron stars (see, e.g., Lau, R., Beard, M., Gupta, S. S., et al. 2018, ApJ, 859, 62)

2) Line 147 has a misprint `momwntum’

Author Response

Reply to  Reviewers

Dear Reviewer, 

first of all, the authors wish to thank you very much for reading our manuscript and sent us your comments and suggestions which gave us the good opportunity to improve our first version of the paper.

Kindly please, find below our detailed answers to your comments. Also, please, in the attached  new version of our manuscript see all changes made (indicated with red characters and text in red).

After these improvements, we do hope that now our manuscript meets the standards of publication of J. “Particles” and we expect to be accepted for publication in this Journal.

Thanks a lot once again.
Kind regards
On behalf of all authors
Haris Kosmas

 

 

Referee 1: Comments and Suggestions for Authors
The paper `Original e−–Capture Calculations for hot stellar interior energies' presents QRPA calculations of the electron capture cross sections for several astrophysically important isotopes in relevant energy range. The paper is well written, provides required information of the calculations along with references for more details. The results of the paper are new, astrophysically important and thus I can recommend it for publication in Particles. 

Authors’ reply: Yes exactly. Thank you very much.

However, I would like to suggest an optional modification:
I believe that if a few illustration, presenting the reaction rates in astrophysical conditions, obtained by ‘appropriate convolution techniques assuming that leptons under stellar interior conditions follow Maxwell-Boltzmann energy distribution’ as well as comparison of these rates with widely applied models (e.g. TALYS) will be extremely useful for readers.

Authors’ reply: We fully agree with the Referee. Such convoluted results are already submitted for publication in the same Journal as a separate paper.  Also, we note that two of the manuscript’s authors are aware (since long ago) about the advantages of the TALYS algorithm.

Minor points:
1) The results of the paper can be also important for accreting neutron stars (see, e.g., Lau, R., Beard, M., Gupta, S. S., et al. 2018, ApJ, 859, 62)

Authors’ reply: We wish to thank very much the Reviewer for introducing us this quite important and interesting research work. It constitutes really a challenging work for future extensions of the present calculations and for their comparison with those of the above reference.

2) Line 147 has a misprint `momwntum’
Authors’ reply: We are sorry for this misprint and thanks a lot for noticing it. 

Author Response File: Author Response.pdf

Reviewer 2 Report

The authors study electron capture on selected nuclei using the pn-QRPA formalism. The goal is to derive stellar electron capture rates, which, however, are not presented in this manuscript. This can be done by folding the cross sections presented here with a proper stellar electron distribution (which must be of Fermi-Dirac and not Boltzmann type as stated in the conclusion). The rates should be included in the manuscript and compared to the previous work of others. Then the paper might be recommendable for publication.

Much of the results presented here are of little importance. It is well-known that Gamow-Teller transitions dominate the  capture at low electron energies (equivalent to low stellar densities and early collapse where the present nuclei occur in the stellar composition). The electron energy chosen for Figures 1-3 is too high for the stellar conditions at which the chosen nuclei are abundant. With increasing electron energy, other multipolarities contribute increasingly to the cross sections (see for example Ref. 7). This finding is considered in modern stellar electron capture rate tabulations (Juodagalvis et al, 2010). The distinction into axial and polar vector contributions is not needed and can be omitted.

The comparison to experimental data should not be restricted to the energies of strong transitions, but also to the strengths.

For N=Z nuclei like 32S and 28Si, the cross sections are dominated by Fermi (0+) and Gamow-Teller (1+) transitions. For nuclei with neutron excess there are no Isobaric Analogue States in the electron capture direction. Where are the strong 0+ transitions in nuclei like 56Fe coming from?

Some statements should be reworded. Here are some examples: Intermediate-mass stars (~8-12 solar masses) might not end up as regular type II. The work by Bethe et al. and Fuller did not predict that electron capture is dominated by nuclei over free protons. Bethe's review clearly discusses the expected blocking of the GT strength in neutron-rich nuclei which was shown by modern many-body approaches to be inadequate.  The references used to describe electron capture in type Ia in fact discuss capture in type II and should be quoted there. An appropriate reference for Type Ia is Brachwitz et al. (2000).

The statement that the present model "offers a reliable construction of all the accessible final states of the daughter nuclei" is too optimistic (not to say wrong). The QRPA uses a model space with restricted correlations and hence cannot produce all states. More importantly it does not describe all nuclear states accessible at low energies. The consequence is that the fragmentation of the GT strength observed in experiment is not fully reproduced by the QRPA. As shown in Ref. 19 the shell model is a significantly better method to describe GT distributions in nuclei than the QRPA. Hence it is also the better method to derive stellar electron capture rates for nuclei which are abundant at relatively low stellar densities (or electron Fermi energies). The strategy to describe stellar electron capture is derived by Juodagalvis et al. (2010), and summarized in the recent review by Langanke, Martinez-Pinedo and Zegers (Reports on Progress in Physics 84 (2021)). In this review the authors might find also a few relevant additional references, for example the improved shell model studies by Otsuka, Suzuki et al. 

Reference 7, despite of being a very impressive review, did not discover the significance of electron capture in supernovae. This was the milestone paper by Bethe, Brown, Lattimer and Applegate from 1979.

 

 

Author Response

Reply to  Reviewers

Dear Reviewer, 

first of all, the authors wish to thank you very much for reading our manuscript and sent us your comments and suggestions which gave us the good opportunity to improve our first version of the paper.

Kindly please, find below our detailed answers to your comments. Also, please, in the attached  new version of our manuscript see all changes made (indicated with red characters and text in red).

After these improvements, we do hope that now our manuscript meets the standards of publication of J. “Particles” and we expect to be accepted for publication in this Journal.

Thanks a lot once again.
Kind regards
On behalf of all authors
Haris Kosmas

 

 

 

Referee 2:  Comments and Suggestions for Authors
The authors study electron capture on selected nuclei using the pn-QRPA formalism. The goal is to derive stellar electron capture rates, which, however, are not presented in this manuscript. 

Authors’ reply: Yes exactly. Thanks a lot.
This can be done by folding the cross sections presented here with a proper stellar electron distribution (which must be of Fermi-Dirac and not Boltzmann type as stated in the conclusion). The rates should be included in the manuscript and compared to the previous work of others. Then the paper might be recommendable for publication.

Authors’ reply: Such calculations have already been done. Due to the size of the present paper they appear in our next paper submitted few days ago to the J. “Particles”.
Much of the results presented here are of little importance. It is well-known that Gamow-Teller transitions dominate the capture at low electron energies (equivalent to low stellar densities and early collapse where the present nuclei occur in the stellar composition). The electron energy chosen for Figures 1-3 is too high for the stellar conditions at which the chosen nuclei are abundant.
With increasing electron energy, other multipolarities contribute increasingly to the cross sections (see for example Ref. 7). This finding is considered in modern stellar electron capture rate tabulations (Juodagalvis et al, 2010). 

The distinction into axial and polar vector contributions is not needed and can be omitted.

Authors’ reply: These suggestions of the Reviewer 2 are applicable if the goals of the calculations were only the folded cross sections related to stellar evolution and supernova physics. But the absolute (original) cross sections of e-capture (throughout the periodic table), presented in this manuscript are useful for studies of other laboratory nuclear processes like single and double charge exchange reactions. For this reason, in the new version we keep these results (otherwise the structure of the article spoils completely, but this is authors’ choice). Furthermore, they are useful for the sake of comparisons of the present pn-QRPA calculations with various future ones and for fixing existing uncertainties and model parameters. Finally, only few publications of this kind with QRPA results appear in this Journal. 
The comparison to experimental data should not be restricted to the energies of strong transitions, but also to the strengths. For N=Z nuclei like 32S and 28Si, the cross sections are dominated by Fermi (0+) and Gamow-Teller (1+) transitions. For nuclei with neutron excess there are no Isobaric Analogue States in the electron capture direction. Where are the strong 0+ transitions in nuclei like 56Fe coming from?

Authors’ reply:  See our comment at the end of Sect. 3.2

Some statements should be reworded. Here are some examples: Intermediate-mass stars (~8-12 solar masses) might not end up as regular type II. The work by Bethe et al. and Fuller did not predict that electron capture is dominated by nuclei over free protons. Bethe's review clearly discusses the expected blocking of the GT strength in neutron-rich nuclei which was shown by modern many-body approaches to be inadequate. The references used to describe electron capture in type Ia in fact discuss capture in type II and should be quoted there. An appropriate reference for Type Ia is Brachwitz et al. (2000).

Authors’ reply:  The reviewer is fully right, in the new version of our paper we have improved writing on these statements.
The statement that the present model "offers a reliable construction of all the accessible final states of the daughter nuclei" is too optimistic (not to say wrong). The QRPA uses a model space with restricted correlations and hence cannot produce all states. More importantly it does not describe all
nuclear states accessible at low energies. The consequence is that the fragmentation of the GT strength observed in experiment is not fully reproduced by the QRPA. As shown in Ref. 19 the shell 
model is a significantly better method to describe GT distributions in nuclei than the QRPA. Hence it is also the better method to derive stellar electron capture rates for nuclei which are abundant at relatively low stellar densities (or electron Fermi energies). The strategy to describe stellar electron capture is derived by Juodagalvis et al. (2010), and summarized in the recent review by Langanke, Martinez-Pinedo and Zegers (Reports on Progress in Physics 84 (2021)). In this review the authors might find also a few relevant additional references, for example the improved shell model studies by Otsuka, Suzuki et al. 

Authors’ reply: Generally speaking, in our opinion, the Reviewer is somehow overestimating the advantages of the Shell Model and the disadvantages of the quasi-particle RPA! It is well known that, in nuclear theory so far we don’t have available “a very fine instrument” for extracting exact predictions throughout the chart of nuclids and for any nuclear process. In this spirit, quasi-particle RPA and Shell Model show advantages and disadvantages that have been extensively discussed in the published literature (many nuclear theorists like two of the authors have also used several refinements of the shell model too). In some of the specific calculations mentioned by the Reviewer, for example, the momentum transfer is totally neglected, by putting q=0 (in some cases, afterwards, several types of corrections are inserted). In addition, the quenching effect of the axial vector coupling is also neglected. In our pn-QRPA method, the q-dependence is explicitly taken into account, by keeping the Bessel functions j_l(qr), while a quenched axial-vector coupling constant for the bound nucleon inside the studied nuclear  system is employed (see paper’s text for other comments).
After the above, since absolute (original) cross sections of e-capture are needed throughout the periodic table, not only to fold them and discuss stellar evolution topics but also others within and beyond the standard model, nuclear theorists should encourage publication of such calculations (unless they are surely incorrect). Since, they are coming out of other methods at least they are useful for the sake of comparisons. Closing, the authors of this manuscript have appreciated (through citing) many shell model calculations (see reference list)! 

Reference 7, despite of being a very impressive review, did not discover the significance of electron capture in supernovae. This was the milestone paper by Bethe, Brown, Lattimer and Applegate from 1979.

Authors’ reply:  We thank very much the Reviewer for noticing it. We have now corrected this mistake.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

The revised manuscript has taken most of my comments into account. Although the paper's impact would strongly benefit from the presentation of capture rates and their comparison with previous work, I recommend publication of the manuscript. Nevertheless the authors should take the following comments into account.

1) Fuller et al. have not really performed microscopic calculations. However, they used experimental data whenever available.

2) Refs. 32-34 should be moved to the sentence discussing both types of supernovae. The shift in neutron richness found in Type Ia is discussed in Refs. 11 and 12 (and in later work of Mori et al.)

3) The concerns voiced in Ref. 62 about the blocking at the N=50 shell gap was convincingly answered in Ref.  25 and recently by some authors of Ref. 62: S. Giraud et al, PRC 105, 055801 (2022)

4) The tables 1 and 2 are in the wrong order, i.e. the first table is numbered Table 2. Both tables have the same information so that one table can actually be omitted.

5) I am still puzzled about the large 0+ contributions in nuclei with neutron excess. The authors state that there calculations give higher 0+ values than in other weak processes. Why is this? What is the reason for the high 0+ contributions.

6) The authors mention the Talys code. Is this used here? Why is it mentioned?

7) The authors call their calculation realistic although no comparison to data is given, except for some energies. For some of the nuclei the experimental GT distribution is known.

8) The word 'Original' in the title is puzzling. What does it mean? I recommend to cancel it.

 

 

 

Author Response

Dear Reviewer 2,

at first, all authors wish to thank you very much for the very careful reading of our manuscript (in both the initial and the revised versions) and the very useful comments and suggestions you sent us. They, really, gave us the best opportunity to clarify our findings and opinions as well as to provide more explanations and details on the points raised, of course, for the benefit of the readers and the science.

Kindly please, see below our answers in each individual comment.

Thanking you very much again,

With our best regards

The authors

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Reply to the comments and suggestions of Reviewer 2

The revised manuscript has taken most of my comments into account. Although the paper's impact would strongly benefit from the presentation of capture rates and their comparison with previous work, I recommend publication of the manuscript.

Authors’ reply: Yes, thanks a lot.

Nevertheless the authors should take the following comments into account.

1) Fuller et al. have not really performed microscopic calculations. However, they used experimental data whenever available.

2) Refs. 32-34 should be moved to the sentence discussing both types of supernovae. The shift in neutron richness found in Type Ia is discussed in Refs. 11 and 12 (and in later work of Mori et al.)

3) The concerns voiced in Ref. 62 about the blocking at the N=50 shell gap was convincingly answered in Ref. 25 and recently by some authors of Ref. 62: S. Giraud et al, PRC 105, 055801 (2022)

Authors’ reply: It is true that the authors didn’t check very carefully the works cited in the manuscript. We appreciate very much the careful reading of our paper by Reviewer 2 and in this version we corrected the afore mentioned inconsistencies. On the other hand, authors were more careful in the terminology employed so, the comments/suggestions # 6), 7), and 8) below are mostly undertaken as an opportunity to give (for the benefit of the readers) further explanation/emphasis rather than as a suggestion to remove or replace present terms (see below).

4) The tables 1 and 2 are in the wrong order, i.e. the first table is numbered Table 2. Both tables have the same information so that one table can actually be omitted.

Authors’ reply: Yes, Tables 1 and 2 appear in reversed order [due to LATEX space restrictions and our “wrong commands”: Table 1 to appear at the “top” of the page and Table 2 to appear at the “bottom” of the page). Thanks a lot for noticing. Now, they appear OK. Since Table 2 offers the advantage to compare percentages of the various multipolarities into the total rate (among the studied isotopes) and discuss their dominance as A increases, we decided to keep it. On the other hand, its omission doesn’t save essential space.

5) I am still puzzled about the large 0+ contributions in nuclei with neutron excess. The authors state that there calculations give higher 0+ values than in other weak processes. Why is this? What is the reason for the high 0+ contributions.

Authors’ reply: In single charge-changing processes (in the range of mass number A where the mentioned isotopes belong) some other pn-QRPA methods “fail completely” to reproduce the 0+ experimental contribution [see, e.g. Table V in the recent bublication Phys.Rev. C 100(2019)014619]. The pn-QRPA employed in the latter work is based on a two-body nucleon-nucleon interaction derived from the Bonn-A while in our pn-QRPA we used a well improved version, the Bonn-CD potential (see point 7 below).

6) The authors mention the Talys code. Is this used here? Why is it mentioned?

Authors’ reply: TALYS code is not used here, but it is mentioned (suggested by the other Reviewer) due to its relevance with the present calculations. Among its specific features we mention that with TALYS package one may perform: (i) complete calculations of astrophysical reaction rates, (ii) computation of astrophysical reaction rates using Maxwellian averaging, (ii) calculations of partial and total cross sections, energy spectra, etc. (see e.g. DOI: 10.1103/PhysRevC.91.044605 and references therein). One of the manuscript’s authors (TSK) employed in the past this code (it is available in the group).

7) The authors call their calculation “realistic” although no comparison to data is given, except for some energies. For some of the nuclei the experimental GT distribution is known.

Authors’ reply: In the present calculations, the nucleon-nucleon interactions employed (the Bonn-CD potential) are build upon rich physics at the nucleon-nucleon level and are of high-precision (they are derived within the context of the meson exchange theory) and the reproduction of p-p, p-n scattering data. In the literature, they are referred to as “realistic” (not schematic, etc.) interactions (see e.g. PRC 103(2021) 044305 and references therein). Furthermore, not all nuclear methods use nucleon-nucleon interactions with these features and, thus, not all nuclear methods require the same tests. As mentioned in the manuscript, our pn-QRPA method has been checked, in addition, on the reproducibility of the experimental muon-capture rates.

8) The word 'Original' in the title is puzzling. What does it mean? I recommend to cancel it.

Authors’ reply: The term “original” in the title (inside the text the term “original” or, alternatively, the term “absolute” appears several times) has been used to distinguish the cross sections under laboratory conditions from the folded (convoluted) cross sections [see, e.g. PRC 86(2012)044618, and references of same authors therein] related to the stellar environment. We however, improved the title by replacing the word “Calculations” with “Cross Sections”.

Author Response File: Author Response.pdf