Simulation of NMR spectra at zero and ultralow field from A to Z – a tribute to Prof. Konstantin L’vovich Ivanov
 ^{1}Univ Lyon, CNRS, ENS Lyon, UCBL, Université de Lyon, CRMN UMR 5280, 69100 Villeurbanne, France
 ^{2}Laboratoire des biomolécules, LBM, Département de chimie, École normale supérieure, PSL University, Sorbonne Université, CNRS, 75005 Paris, France
 ^{1}Univ Lyon, CNRS, ENS Lyon, UCBL, Université de Lyon, CRMN UMR 5280, 69100 Villeurbanne, France
 ^{2}Laboratoire des biomolécules, LBM, Département de chimie, École normale supérieure, PSL University, Sorbonne Université, CNRS, 75005 Paris, France
Abstract. Simulating NMR experiments may appear mysterious and even daunting for those who are new to the field. Yet, broken down into pieces, the process may turn out to be easier than expected. Quite to the opposite, it is in fact a powerful and playful means to get insights into the spin dynamics of NMR experiments. In this Tutorial Paper, we show step by step how some NMR experiments can be simulated, assuming as little prior knowledge from the reader as possible. We focus on the case of NMR at zero and ultralow field, an emerging modality of NMR in which the spin dynamics is dominated by spinspin interactions rather than spinfield interactions, as is usually the case of conventional highfield NMR. We first show how to simulate spectra numerically. In a second step, we detail an approach to construct an eigenbasis for systems of spin½ nuclei at zerofield. We then use it to interpret the numerical simulations. In this attempt to make NMR simulation approachable, the authors wish to pay a tribute to Prof. Konstantin L’vovich Ivanov, a great scientist and pedagogue who passed away on March 5th 2021.
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Quentin Stern and Kirill Sheberstov
Status: final response (author comments only)

RC1: 'Comment on mr202218', Anonymous Referee #1, 07 Nov 2022
The discussion by Stern and Sheberstov introduces to readers some basic principles by which to simulate NMR spectra of simplistic molecules during free evolution of their nuclear magnetization in a zero or ultralow magnetic field (ZULF). Specifically, the compounds must contain two sets of magnetically equivalent spins1/2, each with a different gyromagnetic ratio, for example a carbon13 and hydrogen1.
Technically speaking, the work is correct, but readers should note that the content is not so original, and in my opinion a narrow viewpoint on the topic of nearzerofield NMR. Butler et al. wrote a key paper back in 2013 describing the theory of zerofield NMR in not only AX_{n} systems as presented in this work, but more complex spin systems A_{m}X_{n} and AX_{m}B_{n} as well. Many of the results were derived analytically without simulations, for example, using perturbation theory.
Simulations of zeroandultralowfield NMR spectra in AX_{n}, A_{m}X_{n} and AX_{m}B_{n} are also presented in detail in several PhD theses from the early 2010s: see for example Dr Thomas Theis (2012, UC Berkeley, https://escholarship.org/uc/item/01d528kh ), Dr John Blanchard (2014, UC Berkeley, https://escholarship.org/uc/item/2mp738zn ) and Dr Tobias Sjolander (2017, UC Berkeley, https://escholarship.org/uc/item/2kj4v04n ). Readers are encouraged to consult these original sources.
I encourage the authors to revise the paper by including more original content. For example, simulating the zerofield NMR spectra of compound that has not yet been studied experimentally, or magnetic fields bordering the zerofield condition where the perturbation theory starts to break down. Alternatively, by going beyond summarizing the main results of Butler2013 and reviewing other works where zerofield NMR spectra are calculated. I believe this would be very useful to readers interested in simulation, not only to do justice to the works listed above. Perhaps a quick way is to use a table: compound or spin system, simulation approach (e.g. exact, perturbation theory), software, literature reference .
Additional comments:
 Authors and Readers should be aware of review article on zeroandultralowfield NMR, plus applications, published by Jiang M, Peng X et al. in 2021. This deserves to be mentioned: https://doi.org/10.1016/j.fmre.2020.12.007
 Some of the equations can provide valuable physical insight into how a ZULF NMR experiment works, but it is not explained. Let us take Equation 57 as an example. The total magnetization operator M_{z} = g_{I} I_{z} + g_{S} S_{z} is applied to an eigenstate of the zero field, that is F, m_{F}> in the case of AX_{n}. The result is immediately quoted in terms of ClebschGordan coefficients. There is a slightly different way to write the result where the operator is given as Mz = g_{I} (I_{z} + S_{z}) + (g_{S}  g_{I}) S_{z}. Here the first term on the righthand side of the (=) sign leads to the eigenvalue m_{F} g_{I}, while the second term leads to a sum over other operators, and overall proportional to (g_{S}  g_{I}) times the CG coefficient. The authors may want to mention that this second term leads to ZULF signals where the amplitude of peaks scale with (g_{S}  g_{I}).

AC1: 'Reply on RC1', Quentin Stern, 14 Nov 2022
The referee is right in saying that the content of our paper is “not so original”. It is in fact not original at all. Our aim in writing this paper was not to present original theory but rather to present known theory in a pedagogical way. We chose to focus on the simplest systems to make the paper easiest to follow for the newcomer. Yet, to make the paper more appealing to the expert, we will add a section showing the transition from ZULF to highfield for a CH3 group.
We agree with the referee that highly valuable and pedagogical material can be found in PhD dissertations. One may therefore argue that papers that are only meant to be pedagogical are not useful. However, PhD dissertations are usually not as easily accessible. The starting Master student or PhD student who wants to discover the field will not come across a PhD dissertation easily by searching online. It’s usually by word of mouth that people are advised to read PhD dissertations. That’s why we believe that a paper presenting the basic concepts in detail is useful and can serve as an entry point to newcomers. That being said, we will add references to the PhD dissertations of T. Theis, J. Blanchard, and T. Slojander. Not only this will be useful for the reader but, as pointed out by the referee, it will be a due acknowledgment of the work done by these authors. We will also mention clearly that Butler et al’s 2013 paper presents analytical solutions to similar cases (and more).
We would like to point out to the referee that we did not present numerical simulations only to understand the appearance of ZULF spectra. The primary aim was to show how to numerically simulate spectra. Again, whether such a paper is in itself useful is up for debate.
We thank the referee for the additional comments. We will take them into account in the next version of the paper (once we have received feedbacks from other referees)

CC1: 'Reply on RC1', Tom Barbara, 15 Nov 2022
Perhaps just as important as originality, perhaps even more so, is clairity of exposition. It may be the case that those thesis publications are not as clear as they could be. And as far as originality goes, I am sure that everything anyone needed to know could be found in Paul Corio's very complete monograph "Structure of High Resolution NMR specta" Academic Press 1966.

CC2: 'Reply on CC1', Tom Barbara, 16 Nov 2022
Late last night I recalled another aspect of the NMR spectra simulation problem. This method is not covered in Paul Corio's book and though old, it does appear to supply an "original" perspective not usually encountered in the NMR field. This is the method of Primakoff and Holstein and the close cousin invented by Schwinger, where bosonic states are used to map angular momentum states. A nice, relatively recent effort at expostion is offered by J.A. Gaymfi and this can be found on the physics arxiv:
arXiv:1907.07122v1The author works out the use of these for zero field Hamiltonians. Beware that it is "very mathematical".

CC2: 'Reply on CC1', Tom Barbara, 16 Nov 2022

RC2: 'Reply on RC1', Anonymous Referee #1, 17 Nov 2022
In response to the Author and Community comments so far:
I think that this paper could eventually end up being a good source of reference for ZULF NMR as the authors intend, but for now it lacks in both clarity and detail.
I still find the objective unclear. The abstract promises “detail, including the tricks that are usually omitted from research papers and assuming as little prior knowledge from the reader as possible”. However, the paper is not written clearly, several details are omitted, neither appear any tricks, and there are several inaccuracies (see the end) that can confuse readers regardless of their level of familiarity with theory of NMR.
Here is what I would expect the paper to contain for those entering the ZULF NMR field, or those with little experience of simulations, in this order:
 An overview of common situations where ZULF NMR simulations may be useful, maybe because analytical expressions are not available in: (a) complex spin systems, (b) free evolution near the edge of the ZULF regime, (c) evolution under pulses, e.g. dc pulses, composite pulses, (d) evolution during field sweeps, level crossings, such as those used in ZULF parahydrogeninduced polarization protocols, (e) evolution during dynamical decoupling sequences or the ZULFTOCSY type of spin locking experiments done by the Ivanov group, (f) relaxation, (g) errors in pulse sequences or spin Hamiltonian parameters, (h) simulation of geometric effects such as sample shape, detector position, or inhomogeneous fields. The authors do not have to cover all of these in detail, but it would help to specify what they are, and give literature examples for each.
 An overview of the simulation “choices” that can be made by beginners: (a) eigenvector/eigenvalue analysis in either Hilbert of Liouville space, so that for time intervals where the Hamiltonian is constant you simply take an exponential of the eigenfrequency, rather than the whole matrix, (b) use of other symmetryadapted bases, e.g. Zeeman vs. total angular momentum, and treat smaller subspaces one by one, (c) bruteforce propagation of the Liouville vonNeumann or Schrodinger equation, say in the Zeeman basis. In the manuscript, it appears that a combination of (a) and (c) has been used, but it is not completely clear when and how.
 A few illustrative examples from the list (a)(g) above. Novelty is not required if these examples reproduce the results of previous work with appropriate citation. Such as “In work (author et al., 20xx) this was shown. We will now show the reader how this can be simulated by …” and then proceed to do so, step by step, without leaving any gaps or diverting the reader to sources elsewhere. I will concede that the examples given by the authors are acceptable for a tutorial paper, but they do lack the complete starttofinish detail that would allow a graduate student with a basic knowledge of quantum mechanics to sit down at a computer and work through them without getting stuck.
In my opinion, the authors are assuming a lot more of the reader than they realize, so their work can only be followed by someone who already has an expert level of experience in spin dynamics simulations for NMR, and who is probably capable of looking up the PhD theses already mentioned in the previous review comment.
Below are some specific or technical comments that I recommend the authors address:
 Line 21: “Use frequency dispersion” instead of “chemical shift dispersion”
 Line 24: The reference “(Thayer and Pines, 1987)” does not seem an appropriate citation with the 1.2 GHz NMR magnet?
 Line 29: “Mumetal”, rather than “μmetal”
 Line 31: “does no longer influence the outcome of the experiment” is highly ambiguous. Consider replacing with “where the Larmor period is much longer than the coherence time, so that the field can be completely neglected”.
 Line 47: “at ZULF, there is no such intuition as the vector model”. Is that really true? The AX spin system can often partitioned into an isolated 2LS where cyclically commuting operators can be represented as a vector model picture, see for example https://doi.org/10.1063/PT.3.1948 or https://doi.org/10.1021/acs.jpca.6b04017 supporting information.
 Line 56 and 77: “using a theoretical framework coming from atomic physics”. I don’t think this is ok. You apply the rules of addition of angular momentum. This is not specific to atomic physics – it is just as widely used in molecular (e.g. rotational) spectroscopy and NMR.
 Line 69: “we assume the reader is familiar with general concepts of NMR but not necessarily with simulation”. This is not a clear sentence.
 Line 82: Spinach already has a large set of examples for zerofield NMR, e.g., http://spindynamics.org/wiki/index.php?title=Zerofield.m and http://spindynamics.org/wiki/index.php?title=Zulf_abrupt.m (sudden field drop), and probably SpinDynamica does too. These could be mentioned more directly, even though the above links are not permanent ones.
 Line 269: “is sometimes referred to as the sandwich formula” needs a reference. The only time I have read about “sandwich formulae” in the context of NMR is the rotation sandwich formulae in Levitt’s book “Spin Dynamics: basics of nuclear magnetic resonance”. These are specific to cyclically commuting triads of operators, i.e. fictitious spinhalf representation of a twolevel subspace
 Line 272: the “propagation operator” is a propagator
 Starting line 341: For the sake of clarity, let us not call signal or time a vector. Perhaps “list” or “array” of time points or timeamplitude pairs
 Line 505 507: The phenomenon described is not nutation. Better to refer to the excitation curves as “Rabi curves”
 Style comment: please cite equations as “…, see Equation (x)” rather than “see (x)”.

AC1: 'Reply on RC1', Quentin Stern, 14 Nov 2022

CC3: 'Review of mr202218', Bernhard Bluemich, 08 Dec 2022
General comment:
I really enjoyed reading this paper. It reminded me of my time as a PhD student, when I tried to understand the density matrix formalism and program the transverse magnetization response to some odd excitation using assembler code. In those days long gone I found the then recent paper by P.D. Buckley, K.W. Jolley, D.N. Pinder “Application of density matrix theory to NMR lineshape calculations”, PNMRS 10 (1975) 126 most helpful as it gave handson examples which I could adapt to my own case of interest. Compared to that old paper, the manuscript at stake is written even more in a tutorial style, working out the details of the A_{3}X system as an example. In my view this is a valuable guide to beginning PhD students interested in ZULF NMR. Clearly this manuscript is not a review, nor does it cover all common cases encountered with ZULF NMR, but I find it to be quite useful as a starting point for one’s own simulations. I recommend it to be published following revision. In particular, the corrections and comments voiced in the discussion so far should be considered and implemented so long as the length of the manuscript can largely be maintained.
Detailed additional comments:
Line 20 ff: “Increasing magnetic field strength boosts the sensitivity thanks to higher Boltzmann nuclear polarization and higher Larmor frequency”. Although stated many times in the literature, this only applies to highresolution spectroscopy at constant linewidth in frequency units. It is the peak amplitude and not the peak integral in the spectrum that defines the sensitivity. (This raises the question of the “homogeneity” of the zerofield and its impact on spectral resolution and sensitivity at a given polarization.) Perhaps one can write “increasing the field strength while maintaining the linewidth …”.
Line 133: Figure 1B
Line 170: What defines the directions of the axes at zero field?
Lines 224 and 226: Are “density operator” and “density matrix” used synonymously? That is confusing to the beginner.
Equation (19) and throughout the entire manuscript: The format of constants, variables, functions is inconsistent. This poses extra barriers for a student struggling to understand the math. The formatting rules apply independently to the quantity under consideration and its superscripts and subscripts. For example, the subscript “eq” in (19) is not a variable and should not be written italic.
Lines 335: What is a “Fourier transform function”? The Fourier transformation is an operation, the result of which is a transform. Both, input and output of the Fourier transformation are functions. See also line 345.
Lines 399, 416 “… multiply the frequency domain signal ….” Should probably read “… multiply the abscissa of the frequencydomain signal …”.
Line 435, “Hermitian”: Charles Hermite was a French mathematician. Consequently, the attribute referring to his name is correctly spelled “Hermitean”. Admittedly, contrary to the older literature, one often finds “Hermitian” in the modern literature. You may want to pay tribute to the correct spelling of his name in the manuscript.
Caption to Fig. 5 and elsewhere in the text: “with a zerofilling of 65’536 points” should be replaced by “with zero filling to 65,536 points”, because you did not fill in 65,536 zeroes.
Overall the manuscript is well written, with just a few language issues , which I am sure, the Copernicus editors will pick up.

RC3: 'Comment on mr202218', Bernhard Bluemich, 19 Dec 2022
General comment:
I really enjoyed reading this paper. It reminded me of my time as a PhD student, when I tried to understand the density matrix formalism and program the transverse magnetization response to some odd excitation using assembler code. In those days long gone I found the then recent paper by P.D. Buckley, K.W. Jolley, D.N. Pinder “Application of density matrix theory to NMR lineshape calculations”, PNMRS 10 (1975) 126 most helpful as it gave handson examples which I could adapt to my own case of interest. Compared to that old paper, the manuscript at stake is written even more in a tutorial style, working out the details of the A_{3}X system as an example. In my view this is a valuable guide to beginning PhD students interested in ZULF NMR. Clearly this manuscript is not a review, nor does it cover all common cases encountered with ZULF NMR, but I find it to be quite useful as a starting point for one’s own simulations. I recommend it to be published following revision. In particular, the corrections and comments voiced in the discussion so far should be considered and implemented so long as the length of the manuscript can largely be maintained.
Detailed additional comments:
Line 20 ff: “Increasing magnetic field strength boosts the sensitivity thanks to higher Boltzmann nuclear polarization and higher Larmor frequency”. Although stated many times in the literature, this only applies to highresolution spectroscopy at constant linewidth in frequency units. It is the peak amplitude and not the peak integral in the spectrum that defines the sensitivity. (This raises the question of the “homogeneity” of the zerofield and its impact on spectral resolution and sensitivity at a given polarization.) Perhaps one can write “increasing the field strength while maintaining the linewidth …”.
Line 133: Figure 1B
Line 170: What defines the directions of the axes at zero field?
Lines 224 and 226: Are “density operator” and “density matrix” used synonymously? That is confusing to the beginner.
Equation (19) and throughout the entire manuscript: The format of constants, variables, functions is inconsistent. This poses extra barriers for a student struggling to understand the math. The formatting rules apply independently to the quantity under consideration and its superscripts and subscripts. For example, the subscript “eq” in (19) is not a variable and should not be written italic.
Lines 335: What is a “Fourier transform function”? The Fourier transformation is an operation, the result of which is a transform. Both, input and output of the Fourier transformation are functions. See also line 345.
Lines 399, 416 “… multiply the frequency domain signal ….” Should probably read “… multiply the abscissa of the frequencydomain signal …”.
Line 435, “Hermitian”: Charles Hermite was a French mathematician. Consequently, the attribute referring to his name is correctly spelled “Hermitean”. Admittedly, contrary to the older literature, one often finds “Hermitian” in the modern literature. You may want to pay tribute to the correct spelling of his name in the manuscript.
Caption to Fig. 5 and elsewhere in the text: “with a zerofilling of 65’536 points” should be replaced by “with zero filling to 65,536 points”, because you did not fill in 65,536 zeroes.
Overall the manuscript is well written, with just a few language issues , which I am sure, the Copernicus editors will pick up.

RC4: 'Comment on mr202218', Meghan Halse, 22 Dec 2022
This is a well written and clearly presented tutorial paper. I think that a pedagogical paper supported by a welldocumented and accessible simulation code to bridge the gap between a basic NMR understanding using the vector picture and advanced product operator simulation packages like SPINACH and Spin Dynamica will be of benefit to the NMR community. This paper goes a long way to providing this link; however, I think that focus on ZULF NMR simulations, particularly in the earlier sections of the paper, may be confusing for many nonexperts and may therefore limit the audience for this very nice contribution. The final section of the paper is quite theoretically challenging for the nonexpert and will likely only be of interest to those in the ZULF community. Overall I think that this paper is publishable with minor revisions; however, the addition of some clearer links to highfield NMR simulations (see specific suggestions below) along with a slightly modified version of the MATLAB code to allow the user to perform a more familiar HF NMR simulation to compare with the ZULF simulations would make this a much more broadly useful contribution.
Specific Comments:
 There are a few concepts that are mentioned by not defined clearly and so may confuse a less knowledgeable reader
 Hilbert space (p6 line 154) is mentioned but not explicitly defined
 Braket notion is used in eq. 17 and 18 without being explained/introduced
 On a related note it would be helpful to define the alpha and beta kets with the matrix notation when they are first introduced. I don’t think it is made quite clear the relationship between these states and the columns/rows of the matrices.
 P8 line 201 is the first explicit mention of eigenstates. I think it would be useful when introducing the Zeeman Hamiltonian (eq 5) to explicitly define the alpha and beta kets as the eigenstates and to show the relationship between these states and the matrix representation. This would also be an opportunity to introduced the timeindependent Schrodinger equation for 1 spin before the introduction of the Liouville vonNeuman equation for the density matrix (see point 3 below).
 Eq. 11 – the use of I_{x}, I_{y} and I_{z} to denote the sum of the anguluar momentum operators over a number of spins is confusing as this same notation is used in eqs. 3 and 13 to denote just the 2x2 angular momentum matrices. I suggest using a different letter such as L to denote a sum over multiple spins. I had a similar issue with the definitions in eqs 4851, where I was unsure of the definitions of various terms and found the use of bold and italics unclear/inconsistent.
 Eq. 23 for the equilibrium matrix includes implicitly within it a twospinorder term that emerges from the product. It is common in NMR textbooks, when analysing pulse sequences using product operators to express the equilibrium starting state as just the sum of the Iz operators for the various spins of interest and to omit any higher spin order terms. I think it would be useful to explicitly show the expansion and explain why, and under what circumstances, the single spin operators are a reasonable approximation of the equilibrium density matrix at thermal equilibrium.
 I think it would be useful prior to Eq. 26 to explicitly give the Liouvillevon Newumann equation to which it is a solution.
 On p14 line 350 T2 is defined as the “coherence time constant”, which is a true definition but will be unfamiliar to most readers who will better know this as the spinspin relaxation or transverse relaxation time constant.
 In section 2.7, the focus is put on the acquisition parameters for ZULF spectra without any discussion of how these relate to the acquisition parameters for standard HF spectra, where the spectrum is acquired in the rotating frame and Larmor frequencies are defined as chemical shift offsets relative to a reference frequency. As the ppm scale and HF NMR is the most natural reference point for most readers, I think it is important to describe the two regimes and the relationship between them. Indeed, that is what I was expecting in the “comparison with highfield NMR” section.
 Due to the focus on ZULF NMR the nonstandard case of static field pulses is introduced but the standard representation of ideal RF pulses in the high field regime using rotation operators is not described. The first mention of a rotation operator is on line 599 (p26). This is potentially confusing as the role of rotation operators in NMR simulations has not been described previously. I think it would be helpful in the pedagogical spirit of this paper to include a brief description of this in the theory section.
 In the introduction to ZULF NMR the authors choose to define ULF NMR as where there is a Zeeman contribution but this is not dominant. This definition excludes NMR in the tens of uT regime, notably Earth’s field NMR. By excluding EFNMR a key step in the development of ZULF NMR is omitted, from the initial EFNMR experiments by Packard and Varian in 1954 (Phys Rev), where prepolarisation and nonadiabatic field switching was first used to the first pulsed EFNMR work of Callaghan and LeGros (Americal Journal of Physics, 1982) and up to the work by Appelt et al (Nature Physics 2006), who was the first to introduce the idea of using a Halbach for prepolarisation at a few Tesla before detection in uT fields.
Technical corrections
I noted a couple equation references that appear to be incorrect:
 I think on p17 line 411 it should be eq 36 while line 412 should be eq 37.
 I think on p18 line 456 it should be eq 35.
 There are a few concepts that are mentioned by not defined clearly and so may confuse a less knowledgeable reader

RC5: 'Comment on mr202218', Anonymous Referee #4, 29 Dec 2022
The work from Q. Stern and K. Sheberstov is a tutorial paper addressed to PhD students approaching NMR experiments simulations at ultralow or zero field.
Despite the manuscript is well written and over all correct, I struggle to see a real pedagogical aim in this work. In my honest opinion, I doubt that first year PhD students will be able to open their MATLAB and start to run NMR simulation thanks to this manuscript. I must agree with Reviewer 1 about the fact that the authors assume a lot more of the readers’ preknowledge than what they think. Through the text, apart some NMR theory that can be found in any topic specific textbook (e.g. Spin Dynamics from Prof Levitt), I could not see any real trick that would “carry by hand” the newbie from analytical formulas to computation. Moreover, especially at the beginning, jumping straight to ZULF simulation might be confusing for the nonexperts. The paper could be useful for a broader audience if a “standard/highfield” NMR section was added.
The following is just a suggestion, but, why not starting with a very simple system (noninteracting spins ½ at thermal equilibrium with Zeeman interaction only, no propagation of the density matrix yet just CW style) and show stepbystep how to go from the Hamiltonian to the spectrum in MATLAB, reporting even chunks of code into the main text. Then, we can add the Jcoupling and see how the spectrum changes, propagate the density matrix etc.
As a first year PhD student I would love to find in the literature something like this!
Below some minor details:
Line 27: the second “interaction” is redundant
Line 41: add a reference after “coils” about OPM detection for ZULF
Line 42: I would remove “This simple idealization of”
Line 427: “positioned” instead of “position”

CC4: 'Reply on RC5', Tom Barbara, 29 Dec 2022
I have now read the reviews of this paper and I can understand the conundrum of publishing a tutorial effort in a journal that emphasizes the primary aim is on works of originality. I confronted that same issue with my submission on aspects of relaxation theory where I compared the old classic papers to the new modern efforts, and I felt oblidged to asked one of the editors if the contents of my effort fell with the proper aims of the journal. It is true that the basics are covered in many fine monographs, mentioned by myself and the other reviewers. It is a classic slippery slope when one needs to consider just how basic one must go. In the wake of changes within the publishing world, I see that JMR Open now solicites "tutorial papers" along the lines of the more or less now defunct "Concepts" tradition. Perhaps MR should follow suite.

CC5: 'Reply on CC4', Gottfried Otting, 31 Dec 2022
The executive editors of MR recently decided to broaden the scope of the journal by opening it to educational articles as well as research articles and reviews.
https://www.magneticresonanceampere.net/about/manuscript_types.html states:
"Educational articles provide informative and original insights into topics of current interest within the scope of the journal. Before preparing and submitting an educational article, please contact an editor covering the relevant subject area and an executive editor."
For maximal impact, it is important that the article permits PhD students in the field of magnetic resonance to follow the arguments step by step and reproduce the results. A file provided as Supporting Information can go far in this regard without diluting the main text.
 CC6: 'Reply on CC5', Tom Barbara, 02 Jan 2023

CC5: 'Reply on CC4', Gottfried Otting, 31 Dec 2022

CC4: 'Reply on RC5', Tom Barbara, 29 Dec 2022
Quentin Stern and Kirill Sheberstov
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Fourier transform and simulations of spectra Quentin Stern and Kirill Sheberstov https://doi.org/10.5281/zenodo.7271319
Quentin Stern and Kirill Sheberstov
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