Effect of the Ions on the Electron Collision Operator through Electronic Trajectory Modification

Round 1

Reviewer 1 Report

The article considers the possible interaction between the electron and the ionic electric field during the collision of this electron with a neutral emitter.
The impact of such interaction on the Stark line width.
The manuscript introduces many analytical results of rather tedious calculations. I can not trace them in detail. It gives the impression of solving a complex task from integral calculus.
Physical references to the problem seem to have a secondary role.

My remarks:

1. In the quasi-static approximation the ionic microfield is  uniform in the volume of emitter. However, it is not uniform in Debye's sphere, so one can only speak about some an effective force on electron moving through the Debye sphere, in the middle of which the emitter rests.
In my opinion this effective force is not simply equal to eF.
Incidentally, the expected value of the ionic distribution of micropolises is about twice as high as F_{Holts}.

2. Under the equation (8) "acceleration eF / m ... not very different from V ^ 2 / rho".
Why does the strength of the ionic micro field F depend on the electronic speed V and the parameter rho?

The same expression in the numerator is unchanged.

3. In the formula (10) the symbol F has been changed to E. The same typo appears in the next formulas and text.

4. "de De broglie" --> de Broglie

5. lambda_{th}  or lambda_T  ?

6. Equations (31) and (32): Comparison with Griem's results. Formula (31) actually corresponds to the Griem's formula, but the key thing is the value of z_ {min} and is completely different than that of Griem. What's more, Griem introduces a additional formula describing the participation of strong electron-emitter collisions in Stark broadening, the so-called Lorentz term. Thus, the presented comparison refers to the result that Griem would receive if he accepted the choice  z_ {min}  the same as the authors of the reviewed work and if he neglected the participation of strong collisions. We have no direct comparison here with Griem's results.

7. Much better physical content would have this work if the authors calculated the electronic width of even one He line, eg 388.9 nm.
The necessary atomic data can be found in NIST databases or Griem's works. Then, comparing the widths obtained by the authors with Griem and experimental results would make much more sense.

Author Response

Dear Referee, we have improved the paper avoiding all duplications.

The new paper is attached "bennaa1.pdf".

Responses to Referee 1

1)      Around formula (2) we have give some explanation

2)      We have justify more clearly our formula (9)

3)      E  replaced by F

4)      Broglie

5)      Lambda(T) at all the text

6)      We have specified our work concerns weak collision contribution to collision operator

Author Response File: Author Response.pdf

Reviewer 2 Report

Dear authors,

Excuse me for the delayed answer.

The work presented in this paper is an important in the microfield theory and in describing of spectral line form. The Helium has an tremendous importance for both the experimental investigation and theoretical describing of the plasma. The only remark, that could have more influence on further work could be the form of the presented graphs. Although there is a better understanding of the underlying processes for the theoretical physicist when the line profile is presented in the form of the colisional operator frequency dependence it could be nice to have an accomplaining graph of adequate line profile since an feel of the processes of an experimentator could be of an importance.

Although the approach of purely impact approximation may not be always adequate and influences the overall spectral line form, and symetrically, the colisional operator strength, I was amazed with the fact that the plasma with the strongest Coulomb coupling described was closest to the Griem values, although the ratio of De Broglie wavelength to the Debye radius is more desirable in other cases.

Author Response

We have write responses just before the conclusion in the revision (see the pdf revision attached)

Author Response File: Author Response.pdf

Reviewer 3 Report

1) Abstract is too short. It should be a synopsis of the main achievements. English should be improved and sentences must be to the point.

-- names of chemical elements are not capitalized, so replace 'Helium' by 'helium'

-- 'local electric field' is redundant to 'electric microfield', I recommend using the latter only as it is common.

-- 'plasma' should be 'macro-neutral plasma' to distinguish it from local departures from the macro-neutrality within Debye radius.

-- 'used till nowadays by many authors' is too general and should be removed. There is plethora of plasma studies that treat electron trajectories as Kepler problem, thus not straight lines as authors suggest.

2) Introduction

line 1: replace 'between' by 'among'

line 2: 'others' should be 'other'

lines 2-4: this sentence is not clear. Try something like: 'The line shape analysis relies on measurable macroscopic plasma parameters such are inventory of chemical species and their temperatures and number densities.'

lines 5-6: 'kind' should be 'types'; 'between the' should be 'among'

line 7: replace ', is the ion' by ', is due to the ion'

line 8: replace 'to the' by 'for'; replace 'hydrogen-like' by 'hydrogen-like ions'

line 9: 'Helium' should be 'helium'

line 10: replace 'familiar' by 'often used'; replace 'in the plasma' with 'in the plasma emission spectroscopy'

line 12: replace 'the spectral' by 'the emitted spectral'

line 13: replace 'shapes.' by 'shapes of neutral emitters.'

line 14: ', is subjected' should be ', being subjected'

line 15: change 'the straight' to 'the assumed straight'

line 15: 'As last' should be 'The last'

line 17: change 'many experiences' to 'numerous experiments'

line 18: and throughout text, replace 'Helium' with 'helium'

line 20-21: I will reformulate sentence starting with 'The newest in this work ...' as others have used microfield distribution effects on electron collision operator, see for example "S. Djurović et al, Measurements of Hβ Stark central asymmetry and its analysis through standard theory and computer simulations, Phys. Rev. E 79, 046402"

line 22: replace 'second section.' by 'Section 2.'; replace 'give our' by 'detail our'

3) Spectral Line Formalism

line 1-2: please reformulate this first sentence, and avoid words like 'understand' and 'must'. I would write instead: 'The foundations of the spectral line shape theory can be found in works of Baranger, Griem and collaborators [1-9] as well as Kogan, Lisitsa, Sholin, and collaborators [XXX]', where [XXX] should be at least acknowledged for decades of russian work pre-dating or at least contemporary to western school of spectral line shape theories. After all, it is Russian school that successfully described the asymmetry of peaks in Hbeta line that puzzled plasma theorists for so long.

For examples of [XXX] see the following list:

Kogan V.I., Lisitsa V.S., Sholin G.V. (1987) Spectral-Line Broadening in a Plasma. In: Kadomtsev A.B.B. (eds) Reviews of Plasma Physics. Springer, Boston, MA.

S. P. Zagorodnikov, Yu. E. Smolkin, E. A. Striganova, and G. V. Sholin, Dokl. Akad. Nauk SSSR, 195, 861 (1970).

L. P. Kudrin, Statistical Physics of Plasma [in Russian] , Atomizdat, Moscow (1973) .

G. V. Sholin, A. V. Demura, and V. S. Lisitsa, Zh. Eksp. Teor. Fiz., 64, 2097 (1973).

N. Tran Minh, N. Feautrier, and H. van Regemorter, J. Quant. Spectrosc. Radiat. Transfer, 16, 849 (1976).

line 2: 'was developed' should be 'were developed'

line 3: 'the difference' should be 'discrepancies'; 'experimentals' should be 'experimental'

line 5: replace 'by' with 'due'

line 6: 'the ion electric' should be 'the static ion electric'

line 7: 'the spectral line' should be 'the spectral line profile'

equation 1: Please provide a reference to this equation. If alpha and beta are initial and final levels, what are alpha' and beta'? If these are intermediate levels then some sort of summation/averaging procedure over these intermediate states is required? So where is the sum over alpha' and beta'? If Einstein convection of repeat indices is used, then state it clearly so readers will not be confused. Also which units are used? Natural system where hbar=h/2Pi is 1? Scalar product d.F has units of energy, so I guess division by hbar is in order and then Phi(F) will be collision frequency operator.

line 8: 'F is the' should be 'F is the time-averaged intensity of'

line 14-15: 'the the'; 'weak enough' should be 'sufficiently small'

line 16: 'does not follow' should be 'not following'

equation 2: delete '.' from quadratic term over time, confuses reader for dot product of two vectors; please comment about initial conditions, especially time 'zero'. Is the time 'zero' defined when rho vector is on the surface of Debye sphere drawn around neutral emitter?

3) Electron Collision Operator

line 1: 'obtain' should be 'describe', replace 'say some words about' with 'address the effect of'

line 3: 'with' should be 'to'

line 10: 'Maxwell distribution' should be 'Maxwell probability density'

equation for E(u) should have its own number; here either state which system of units you are using or add (4Pi*eps0) to the denominator.

line 12: remove 'well known' as you already cited ref [2].

line 17: replace 'changes' with 'substitution'

line 19: replace 'do' with 'carry out'; 'on' with 'over'; 'the last formula' with 'equation (7)'

line 20: replace 'broadening' with 'broadening phi_alpha'

line 22: replace 'In order to' with 'To'; 'practicable' with 'tractable'

line 23: remove 'we use'

line 24: remove 'to the'

line 25: replace 'the shift' with 'the line shift'

line 27-28: rewrite this sentence. Not clear what 'the second with x1x2' means.

line 33: replace 'on' with 'over'

line 34: replace 'distribution,' with 'probability density in equation (4),'; 'to the' with 'for'

line 35: replace 'electrons broadening as' with 'broadening by electrons'

line 37: replace 'to mention here,' with 'mentioning'

line 38-39: replace 'not occur' with 'are negligible'

line 39: replace 'Else' with 'Otherwise,'; 'atom' with 'emitter'.

line 40: replace 'tarjet' with 'target'; 'not follows' with 'does not follow'

line 43: replace 'has' with 'does'; 'on the trajectory of the electron' with 'electron trajectory'

line 49: replace 'a little change of variable' with 'variable substitution'

line 51: remove 'well known as'

line 52: replace 'the formula (24) writes in more condensed expression as' with 'equation (23) becomes'

line 53: replace ' In the same way ....' with 'Similarly, we write the second contribution given by equation (19) as'

line 54: remove 'the following way'

line 56: replace 'contribution is' with 'contribution, see equation (20), is'

lines 59-60: remove 'Finally,'; replace 'exactly computed' with 'expressed in closed form'

line 60: replace 'by the help of the' with 'using'

line 61: replace 'some' with 'certain'

line 62: remove 'the' in front to 'figures'; 'We clearly see that' is very confusing as I do not see it clearly. Where is it clearly shown that "phi_3^alpha,alpha'"

depends on E? Is it equation (29)? If so rewrite this paragraph to exactly point to relevant equations as it is right now, I would look into figures 1-3 for this dependence; also all six figures should be grouped into a single figure with a), b), etc insets. Label on x-axis should be uniform, non-capitalized, and contain mathematical symbol "omega^tilda_alpha,alpha'" because in the present form it is not obvious to which quantity this axis relates to. For example, Figures 4-6 have 'frequency separation' as x- axis and Figures 1-3 just 'frequency' and what is their difference? 

Same for y-axis, is this label for quantity given in equation (17)? Or is it only for  "phi_3^alpha,alpha'" given by equation (29). Please redo all figures and make axis labels self-explanatory. This is hinted in paragraph preceding 'Conclusion' that Figures 1-3 represent a sum of three contributions, but as I already mention, this info needs to be integral part of figures, either in figure caption or directly as axes labels. And here in this paragraph it is even more confusing, as authors give the x-axis as "omega_alpha,alpha'" and not its "tilda" variant in plasma frequency units. Which one is it? If labels had these mathematical symbols, readers would easily relate.

Please rewrite the last paragraph before conclusion to better describe all six figures, as this is the place for your discussion of results. When you say here that your results are more realistic, you should elaborate on it. For example, in equation (32) you show Griem analog of your broadening operator. How do you know you are more realistic. Which terms in your results, when neglected, will reduce your results to Griems? This is the crucial part of your analysis and I think it merits the separate discussion before you jump to conclusions. Especially because at your highest electron densities in figures 5 and 6, for low collision energies your results become lower than Griem's prediction. So which terms in your result at high densities change the sign and overtake the central part of the line profile?

4) Conclusion

line 1: replace 'the' with 'spectral'

line 2: replace 'lightening' with 'relaxing'; remove 'when you have assumed,'

line 3: add 'such that' in front of 'the electron'

line 8: replace 'then a best' with 'more complete'; 'the coupling' with 'the quadratic coupling' 

line 9: replace 'do' with 'carry out'

5) Appendix

line 1: replace 'following' with 'equivalent'

Acknowledgements: replace 'Pr:' with 'Prof.'; 'some' with 'supporting'

Author Contributions: remove 'were' and replace 'equivalently' with 'equaly'

Conflict of Interest: remove 'to signal'

Author Response

We have respond to all questions and remarks. See the revision in pdf attached.

Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

Please have a native English speaker read through your article and point out to grammar inconsistencies.

Author Response

Dear Reviewer, I have correct all suggestions you comment.

These corrections are in the attached file, the revised paper.

Author Response File: Author Response.pdf