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Low-temperature scattering with the R-matrix method: from nuclear scattering to atom-atom scattering and beyond

by T. Rivlin, L. K. McKemmish, K. E. Spinlove, J. Tennyson

Submission summary

As Contributors: Jonathan Tennyson
Arxiv Link: https://arxiv.org/abs/1911.00027v1
Date submitted: 2019-11-04
Submitted by: Tennyson, Jonathan
Submitted to: SciPost Physics Proceedings
Proceedings issue: 24th European Few Body Conference (University of Surrey, U.K.)
Discipline: Physics
Subject area: Quantum Physics
Approaches: Theoretical, Computational

Abstract

The R-matrix method is a fully-quantum time-independent scattering method, used to simulate nuclear, electron-atom, electron-ion, electron-molecule scattering and more. Here, a novel R-matrix method, RmatReact, is presented as applied to ultracold (sub-Kelvin) atom-atom scattering. This is in response to experimental methods which in recent decades have facilitated the routine use of ultracold atoms and small molecules in experiments. This project lays the groundwork for future codes to eventually simulate polyatom-polyatom reactive collisions. Results are presented in this paper for numerical comparisons to other established methods, and for resonances in the elastic scattering of ultracold argon atoms.

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Reports on this Submission

Anonymous Report 1 on 2019-11-14 Invited Report

Report

The manuscript presents recent developments on R-matrix calculations, applied to ultracold atom-atom scattering. The authors describe a new R-matrix framework, referred to as the "RmatReact method".
I have no doubt that the physics of ultracold systems is fundamental, and that it opens new perspectives in modern physics. However, as nuclear physicist and frequent user of the R-matrix method, I am rather puzzled by the paper. I do not want to bother the authors with minor details, but this text could be a good opportunity to clarify some technical aspects.

My main concerns are as follows:
1) I am surprised that the R-matrix framework needs a specific adaptation for ultracold systems. To my knowledge, going to lower energies is simple since there are less and less nodes in the wave function. I am therefore wondering why a "novel method" is necessary (and, what is "novel", compared to previous works)?

2) As a general statement, the authors emphasize that "RmatReac" is a new development, with several authors, many related publications, and a strong financial support.
In fact, what seems to be done is to solve a one-dimensional Schrodinger equation for scattering states. Unless the potentials present (very) unusual specificities, this looks rather trivial. It may be a misunderstanding on my side, but I think that the difficulties should be better underlined.

3) I am very surprised by Table 2, where standard deviations are given. It might be that the apparently strong differences between both methods are due to slightly different resonance energies. If this is the case, I do not think that the delta_RMSD are significant. If not, one the methods (at least) is very poor.

4) The numerical testing (sect.4.1) involves 400 points in the 'Small' inner region and 1600 grid points in the 'Big' inner region. Again, these huge numbers of basis functions are difficult to understand. It is really necessary? or is it a sledgehammer to crack a nut?
Additionally, where are iterations (4000!) necessary??

Minor questions/comments:
A) In Eq.(9) it seems that the eigenvalues E^J_k are different from those of eq.(1), although the same notation is used.
B) In eq.(15), a reference would be welcome.
C) The parameterization of the background phase shift (19) is somewhat arbitrary. How would the resonance properties (E_res and Gamma_res in Fig.2) be affected if another parameterization is adopted (e.g. only A_0, or A_0,A_1,A_2)? Are all figures significant?

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Author Jonathan Tennyson on 2019-12-20
(in reply to Report 1 on 2019-11-14)
Category:
answer to question

My main concerns are as follows: 1) I am surprised that the R-matrix framework needs a specific adaptation for ultracold systems. To my knowledge, going to lower energies is simple since there are less and less nodes in the wave function. I am therefore wondering why a "novel method" is necessary (and, what is "novel", compared to previous works)?

The introduction has been expanded to explain that while R-matrix studies on electron collisions are well established, the demands of ultracold physics requires development of methods for treating (very slow) nuclear collisions. Essentially this is new territory for the R-matrix methods and requires both new algorithms and new codes.

2) As a general statement, the authors emphasize that "RmatReac" is a new development, with several authors, many related publications, and a strong financial support. In fact, what seems to be done is to solve a one-dimensional Schrodinger equation for scattering states. Unless the potentials present (very) unusual specificities, this looks rather trivial. It may be a misunderstanding on my side, but I think that the difficulties should be better underlined.

The diatomic problems considered her provide testbeds for the new method rather than new physics. Again we have put more emphasis on why we are doing this in the introduction.

3) I am very surprised by Table 2, where standard deviations are given. It might be that the apparently strong differences between both methods are due to slightly different resonance energies. If this is the case, I do not think that the delta_RMSD are significant. If not, one the methods (at least) is very poor.

These RMSD values were, in fact, erroneous, and have been replaced with the correct numbers. The corrected RMSD values are far lower, in line with the commentary in the section.

4) The numerical testing (sect.4.1) involves 400 points in the 'Small' inner region and 1600 grid points in the 'Big' inner region. Again, these huge numbers of basis functions are difficult to understand. It is really necessary? or is it a sledgehammer to crack a nut? Additionally, where are iterations (4000!) necessary??

A major difference between electron scattering and so-called heavy particle scattering is the density of states. Thus many states are need to represent the inner region: we are designing the method to treat problems with maybe a thousand bound states in the inner region. This obviously requires large basis sets. Similarly at ultralow energies propagation distances are huge so many propagation steps (iteratations) is standard. The introduction has been expanded to cover these points.

Minor questions/comments: A) In Eq.(9) it seems that the eigenvalues E^J_k are different from those of eq.(1), although the same notation is used.

Good point. The Eq.(1) is changed to E^J_n.

B) In eq.(15), a reference would be welcome.

A reference had been added.

C) The parameterization of the background phase shift (19) is somewhat arbitrary. How would the resonance properties (E_res and Gamma_res in Fig.2) be affected if another parameterization is adopted (e.g. only A_0, or A_0,A_1,A_2)? Are all figures significant?

The background parameterisation was re-done with only A_0 and with A_0,A_1,A_2, as suggested. The results for the position and width of the resonance were found to not vary as a result of this. Commentary to this effect has been included, also addressing the question of significance.

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