Analytical treatment of proton double-quantum NMR intensity build-up: multi-spin couplings and the flip-flop term

Fatkullin, Nail; Brekotkin, Ivan; Saalwächter, Kay

doi:https://doi.org/10.5194/mr-2024-15

Preprints

https://doi.org/10.5194/mr-2024-15

Preprints

19 Sep 2024

| 19 Sep 2024

Status: a revised version of this preprint was accepted for the journal MR and is expected to appear here in due course.

Analytical treatment of proton double-quantum NMR intensity build-up: multi-spin couplings and the flip-flop term

Nail Fatkullin, Ivan Brekotkin, and Kay Saalwächter

Abstract. A modified Anderson-Weiss approximation for describing double quantum (DQ) NMR experiments in systems with many I = 1/2 spins is proposed, taking inter-spin flip-flop processes into special consideration. In this way, an analytical result is derived for multi-spin systems for the first time. It is shown that in the initial stages of DQ intensity build-up, the probability of flip-flop processes in DQ experiments is half as large as in analogous Hahn-echo or free-induction-decay experiments. Their influence on the experimentally observed DQ NMR signal becomes dominant at times t > (9/2)^1/2 T₂^eff≈ 2.12T₂^eff, where T₂^eff is the effective spin-spin relaxation time measured by the Hahn echo. Calculations and a comparison with spin-dynamics simulations of small spin systems up to 8 spins reveal a satisfactory agreement.

Received: 04 Sep 2024 – Discussion started: 19 Sep 2024

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Nail Fatkullin, Ivan Brekotkin, and Kay Saalwächter

Status: closed

RC1:
'Comment on mr-2024-15', Anonymous Referee #1, 23 Sep 2024

The manuscript focuses on the application the Anderson-Weiss approximation for an interpretation of double-quantum (DQ) NMR experiments in many-spin systems (s = ½). The authors proposed new theoretical methods for the analysis of inter-chain flip-flop processes in such experiments. I believe that the manuscript will be interesting and useful for both experimentalists and theoreticians.
I suppose that the manuscript should be completed with materials showing connection of the developed approach with the theory of average Hamiltonian and the Floquet theory. From my point of view the manuscript deserves to be published in Magnetic Resonance.

Citation: https://doi.org/10.5194/mr-2024-15-RC1
- AC3: 'Reply on RC1', Nail Fatkullin, 26 Oct 2024
  
  Response to anonymous Referee #1
  The manuscript focuses on the application the Anderson-Weiss approximation for an interpretation of double-quantum (DQ) NMR experiments in many-spin systems (s = ½). The authors proposed new theoretical methods for the analysis of inter-chain flip-flop processes in such experiments. I believe that the manuscript will be interesting and useful for both experimentalists and theoreticians.
  I suppose that the manuscript should be completed with materials showing connection of the developed approach with the theory of average Hamiltonian and the Floquet theory. From my point of view the manuscript deserves to be published in Magnetic Resonance.
  
  We thank Reviewer 1 for supporting our work.
  If a mathematical equation is solved exactly, without approximations, then the final answer does not depend on the method of solution. If we are talking about approximations, of course such universality is lost, generally speaking. Average Hamiltonian theory and the Floquet formalism are to best of our knowledge the most popular approaches to problems in solid state NMR. In our case, the Floquet formalism is not really suitable as it is almost exclusively applied to solve problems in MAS NMR, i.e. cases with periodic modulations. Average Hamiltonian theory is in turn in fact used by us to derive eq. (3). In revised manuscript we will add some relevant references:
  (Haeberlen, U., High Resolution NMR in Solids- Selective Averaging, Supplement 1, Adv. Magn. Reson., Academic Press, New York, 1976 and Ivanov K.L., Mote K.R., Ernst M., Equbal A., K. Madhu P.K.: Floquet theory in magnetic
  
  Citation: https://doi.org/10.5194/mr-2024-15-AC3
AC1: 'Comment on mr-2024-15', Nail Fatkullin, 09 Oct 2024

Response to anonymous Referee #1, 23 Sep 2024
We are grateful to Reviewer 1 for supporting our work.
    If a mathematical equation is solved exactly, without approximations, then the final answer does not depend on the method of solution. If we are talking about approximations, of course such universality is lost, generally speaking. The average Hamiltonian theory and Floguet formalism to best of our knowledge are usually using to problem of solid state NMR. In our case by mentioned methods can be derived the Hamiltonian (3), in revised manuscript will be add some corresponding references:
    (Haeberlen, U., High Resolution NMR in Solids- Selective Averaging,     Supplement 1, Adv. Magn. Reson., Academic Press, New York, 1976
and Ivanov K.L., Mote K.R., Ernst M., Equbal A., K. Madhu P.K.: Floquet theory in magnetic resonance: Formalism and applications, Prog. NM Spectr. 126–127, 17-58, https://doi.org/10.1016/j.pnmrs.2021.05.002, 2021).
     We are trying to develop the variant of DQ theory when one can not neglects by spatial displacements of investigated spins during experiment motion of lattice variables can be considered classically. In contrast to the theoretical approaches known to us, we make the transition to the interaction representation (Dirac) in two steps.
    At the first stage, the lattice Hamiltonian is included along with the Zeeman interaction in the zero Hamiltonian of the interaction, which asymptotically leads to the DQ -Hamiltonian (15). In the Hamiltonian (15) we extract the part, see relation (16), responsible for the flip-flop transitions commuting with the z-projection of the total spin of the system, which is essentially an observable quantity in experiment, see relation (19).   Then, which to the best of our knowledge is done for the first time, we make another transition to the interaction representation by taking the previously extracted flip-flop part as the new zero-point Hamiltonian. On this basis we obtain the exact relations (23) and (24), with respect to which further approximations can be made in the future. In this paper the simplest approximations are investigated.    A nontrivial result is obtained already in the zero approximation with respect to the Hamiltonian of flip-flop transitions for the spin 1/2 system: an exactly solvable multispin system (eq. (27-29)). Next, we study the first non-zero corrections associated with flip-flop transitions and their Anderson-Weiss generalizations.


Citation: https://doi.org/10.5194/mr-2024-15-AC1
RC2:
'Comment on mr-2024-15', Anonymous Referee #2, 14 Oct 2024

The manuscript describes an analytical treatment of DQ excitation in solids in large spin systems where dynamics and spin flip-flop processes lead to time modulations of the dipolar couplings. In an iterative process of interaction-frame transformaions, it is shown that the DQ build up can be calculated analytically while taking the action of the dynamics and spin flip-flop transitions into account. For this an additional interaction-frame transformation with the part of the DQ Hamiltonian that commutes with I^z (flip-flop part) is performed. This leaves just a yy part of the Hamiltonian that commutes with itself at all times and allows the calculation of an analytical phase factor that describes the DQ intensity. Results are compared to numerical simulations in relatively small spin systems (6 spins) which shows reasonable agreement with the analytical calculations. Can the authors at least comment whether they expect better agreement if state of the art spin system sizes of 10-12 spins (doable without sparse matrix implementations) would give better agreement?
The manuscript is clearly inside the scope of MR and presents interesting new results that are of interest to the MR readership. However, I believe that the presentation could be improved and that there are a larger number of typos and very colloquial expressions that make it difficult to follow the text.
The authors start from a secular high-field truncated dipolar Hamiltonian (Eq (1)) which has the usual form. But then the authors say: "The Hamiltonian (1) in this

55 paper plays the role of the original spin-lattice interaction Hamiltonian. In the lowest order of perturbation theory, it induces in a spin system 0-quantum transitions (dephasing processes or 0-quantum coherence) by terms proportional to .... 1-quantum transitions (transitions with a flip of one of a pair of interacting spins or 1-quantum coherence) and 2-quantum transitions (coordinated transitions of two spins or 2-quntum coherence) by terms proportional ...." (equation cut out of text). Why would this (static) ZQ Hamiltonian induce DQ or SQ transitions. This would be true for the untruncated dipolar Hamiltonian but not for the high-field truncated one used here, or am I missing something?
There are a lot of very colloquial and not very clear and precise statements that could be improved. Examples are:

Line 74: "In solids, the joint effect of mentioned factors, in the limit Δ → 0 , where Δ is time interval between the nearest RF pulses" which joint effects are meant here?

Line 78: "Note, that n = 0 corresponds to the first mentioned version of DQ

experiment no phase change and n = 1 to the second version 90° phase shift." The English is wrong (replace "no" with "without") and it could be stated clearly that you refer to A-0 and A_1 signals here.

Line 84: "At time moment DQ t =t the operator (3) changes time for the case, when n = 1." A Hamiltonian cannot change time. I can change the sign.

Line 95: "where the role of the zero Hamiltonian is sum of the lattice Hamiltonian and the Hamiltonian of Zeeman interaction of investigated spins with an external magnetic field." What is a zero Hamiltonian? It is defined later but I find this a very colloquial expression. Maybe the "dominant part of the Hamiltonian" would be a better term.
There are more such unclear statements and the manuscript would profit from a carefull reading and checking for consistency of notation (see also minor points) as well as the English language.
I think the manuscript would benefit from making section 3 really a discussion without pages and pages of equations that could be moved to an appendix. At the moment section 3.1-3.3 is more a theory section that is very hard to follow and gives little insight for the non specialist.

Minor points - many typos (?) or unclear expressions in equations:

line 39: two full stops

line 58: quantum instead of quntum

Eq. (4): This should probably be a \omega_{ij}(t) and not \omega_{ij}^{(n)}(t) which is only defined in (5)

Line 102: Give a reference for AW approximation.

line 164: Are I_z and I^z the same or different quantities?

line 164: What is H_0 here? the Zeeman Hamiltonian or the flip-flop part of the DQ Hamiltonian?

Eq. 14 and 15: Is U_0(t) on the first line of Eq. 14 the same as U_0^{(DQ,n)} on the first line of Eq. 15?

Eq. 15: Is H^{(0)}_{DQ} the same as H^{(n)}_{DQ} in Eq. 16

Line 205: There is no section 2b

Eq. 32: Is {\tilde{I}_i^y\tilde{I}_j^y}^{fl}_t the same as the (I_i^yI_j^y)^{fl}_t on line 205 and Eq. 31? If not then please define.
There are probably many more that I did not see.

Citation: https://doi.org/10.5194/mr-2024-15-RC2
- AC2: 'Reply on RC2', Nail Fatkullin, 26 Oct 2024
  
  Response to anonymous Referee #2
  
  We are grateful to the second reviewer for his careful review of the manuscript and comments, which will incorporate in the new version of our paper.
  
  The manuscript describes an analytical treatment of DQ excitation in solids in large spin systems where dynamics and spin flip-flop processes lead to time modulations of the dipolar couplings. In an iterative process of interaction-frame transformations, it is shown that the DQ build up can be calculated analytically while taking the action of the dynamics and spin flip-flop transitions into account. For this an additional interaction-frame transformation with the part of the DQ Hamiltonian that commutes with I^z (flip-flop part) is performed. This leaves just a yy part of the Hamiltonian that commutes with itself at all times and allows the calculation of an analytical phase factor that describes the DQ intensity. Results are compared to numerical simulations in relatively small spin systems (6 spins) which shows reasonable agreement with the analytical calculations. Can the authors at least comment whether they expect better agreement if state of the art spin system sizes of 10-12 spins (doable without sparse matrix implementations) would give better agreement?
  
  To discuss possible improvements in the agreement between simulations and predictions, one should consider that our (still limited) approximation is for now limited to times (see lines 470-480); for longer times, additional work (e.g. evaluating higher-order corrections) is required. Generally, of course, we expect better agreement between simulations and either better predictions or experimental results for more spins. Our (outdated) code is unfortunately limited to 9 spins at present. But the main limitation is in fact the consideration of a realistic spin system, where our alkyl cut-outs are already a bit of a stretch. With nowadays’ improved simulation codes such as SPINACH, handling tens of spins, we shall in the future focus on trajectories from MD simulations for the most realistic modelling of fast local conformational dynamics and in particular inter-molecular couplings. We will add corresponding remarks in the paper.
  
  The manuscript is clearly inside the scope of MR and presents interesting new results that are of interest to the MR readership. However, I believe that the presentation could be improved and that there are a larger number of typos and very colloquial expressions that make it difficult to follow the text.
  The authors start from a secular high-field truncated dipolar Hamiltonian (Eq (1)) which has the usual form. But then the authors say: "The Hamiltonian (1) in this paper plays the role of the original spin-lattice interaction Hamiltonian. In the lowest order of perturbation theory, it induces in a spin system 0-quantum transitions (dephasing processes or 0-quantum coherence) by terms proportional to .... 1-quantum transitions (transitions with a flip of one of a pair of interacting spins or 1-quantum coherence) and 2-quantum transitions (coordinated transitions of two spins or 2-quntum coherence) by terms proportional ...." (equation cut out of text). Why would this (static) ZQ Hamiltonian induce DQ or SQ transitions. This would be true for the untruncated dipolar Hamiltonian but not for the high-field truncated one used here, or am I missing something?
  
  The reviewer is right: the properties of the full Hamiltonian of magnetic dipole-dipole interactions, which was written in this part of the paper in the very first version, are attributed to its secular part. Corresponding changes will be made in the revised version of the paper:
  “ The Hamiltonian (1) in this paper plays the role of the original spin-lattice interaction Hamiltonian. In the lowest order of perturbation theory, it induces in a spin system only 0-quantum transitions (dephasing processes or 0-quantum coherence) through terms proportional to and flip-flop processes proportional to . We also note that the non-secular part of the Hamiltonian of magnetic dipole-dipole interactions inducing the processes of spin-lattice relaxation describes the processes of 1-quantum and 2-quantum coherence. However, at the times of interest they can be neglected.”
  
  There are a lot of very colloquial and not very clear and precise statements that could be improved. Examples are:
  
  Line 74: "In solids, the joint effect of mentioned factors, in the limit Δ → 0 , where Δ is time interval between the nearest RF pulses" which joint effects are meant here?
  
  This will be improved in the following way:
  “ In solids, the joint effect of the BP sequence and the Hamiltonian (1) , in the limit , where is time interval between the nearest RF pulses, the spin system’s time evolution can be described in terms of an effective DQ Hamiltonian having the following structure:
  , (3)
  where is the Heaviside step function, . The case corresponds to the first mentioned version of DQ experiment without phase change, i.e, to the signal , and to the second version with 90° phase shift, i.e. to the signal . ”
  
  Line 78: "Note, that n = 0 corresponds to the first mentioned version of DQ experiment no phase change and n = 1 to the second version 90° phase shift." The English is wrong (replace "no" with "without") and it could be stated clearly that you refer to A-0 and A_1 signals here.
  
  This will be improved.
  
  Line 84: "At time moment DQ t =t the operator (3) changes time for the case, when n = 1." A Hamiltonian cannot change time. I can change the sign.
  
  This is considered in the above passage referring to eq. (3).
  
  Line 95: "where the role of the zero Hamiltonian is sum of the lattice Hamiltonian and the Hamiltonian of Zeeman interaction of investigated spins with an external magnetic field." What is a zero Hamiltonian? It is defined later but I find this a very colloquial expression. Maybe the "dominant part of the Hamiltonian" would be a better term.
  
  To be improved:
  “…zeroth-order Hamiltonian, i.e. the zero-point Hamiltonian or the dominant part of the Hamiltonian (see also expr. (8)), is sum of the lattice Hamiltonian and the Zeeman Hamiltonian of the investigated spins with an external magnetic field...”
  
  There are more such unclear statements and the manuscript would profit from a careful reading and checking for consistency of notation (see also minor points) as well as the English language.
  
  The revised version will be carefully edited.
  
  I think the manuscript would benefit from making section 3 really a discussion without pages and pages of equations that could be moved to an appendix. At the moment section 3.1-3.3 is more a theory section that is very hard to follow and gives little insight for the non-specialist.
  
  Citation: https://doi.org/10.5194/mr-2024-15-AC2
- AC4: 'Reply on RC2', Nail Fatkullin, 26 Oct 2024
  
  The comment was uploaded in the form of a supplement: https://mr.copernicus.org/preprints/mr-2024-15/mr-2024-15-AC4-supplement.pdf
  
  Citation: https://doi.org/10.5194/mr-2024-15-AC4

Status: closed

RC1:
'Comment on mr-2024-15', Anonymous Referee #1, 23 Sep 2024

The manuscript focuses on the application the Anderson-Weiss approximation for an interpretation of double-quantum (DQ) NMR experiments in many-spin systems (s = ½). The authors proposed new theoretical methods for the analysis of inter-chain flip-flop processes in such experiments. I believe that the manuscript will be interesting and useful for both experimentalists and theoreticians.
I suppose that the manuscript should be completed with materials showing connection of the developed approach with the theory of average Hamiltonian and the Floquet theory. From my point of view the manuscript deserves to be published in Magnetic Resonance.

Citation: https://doi.org/10.5194/mr-2024-15-RC1
- AC3: 'Reply on RC1', Nail Fatkullin, 26 Oct 2024
  
  Response to anonymous Referee #1
  The manuscript focuses on the application the Anderson-Weiss approximation for an interpretation of double-quantum (DQ) NMR experiments in many-spin systems (s = ½). The authors proposed new theoretical methods for the analysis of inter-chain flip-flop processes in such experiments. I believe that the manuscript will be interesting and useful for both experimentalists and theoreticians.
  I suppose that the manuscript should be completed with materials showing connection of the developed approach with the theory of average Hamiltonian and the Floquet theory. From my point of view the manuscript deserves to be published in Magnetic Resonance.
  
  We thank Reviewer 1 for supporting our work.
  If a mathematical equation is solved exactly, without approximations, then the final answer does not depend on the method of solution. If we are talking about approximations, of course such universality is lost, generally speaking. Average Hamiltonian theory and the Floquet formalism are to best of our knowledge the most popular approaches to problems in solid state NMR. In our case, the Floquet formalism is not really suitable as it is almost exclusively applied to solve problems in MAS NMR, i.e. cases with periodic modulations. Average Hamiltonian theory is in turn in fact used by us to derive eq. (3). In revised manuscript we will add some relevant references:
  (Haeberlen, U., High Resolution NMR in Solids- Selective Averaging, Supplement 1, Adv. Magn. Reson., Academic Press, New York, 1976 and Ivanov K.L., Mote K.R., Ernst M., Equbal A., K. Madhu P.K.: Floquet theory in magnetic
  
  Citation: https://doi.org/10.5194/mr-2024-15-AC3
AC1: 'Comment on mr-2024-15', Nail Fatkullin, 09 Oct 2024

Response to anonymous Referee #1, 23 Sep 2024
We are grateful to Reviewer 1 for supporting our work.
    If a mathematical equation is solved exactly, without approximations, then the final answer does not depend on the method of solution. If we are talking about approximations, of course such universality is lost, generally speaking. The average Hamiltonian theory and Floguet formalism to best of our knowledge are usually using to problem of solid state NMR. In our case by mentioned methods can be derived the Hamiltonian (3), in revised manuscript will be add some corresponding references:
    (Haeberlen, U., High Resolution NMR in Solids- Selective Averaging,     Supplement 1, Adv. Magn. Reson., Academic Press, New York, 1976
and Ivanov K.L., Mote K.R., Ernst M., Equbal A., K. Madhu P.K.: Floquet theory in magnetic resonance: Formalism and applications, Prog. NM Spectr. 126–127, 17-58, https://doi.org/10.1016/j.pnmrs.2021.05.002, 2021).
     We are trying to develop the variant of DQ theory when one can not neglects by spatial displacements of investigated spins during experiment motion of lattice variables can be considered classically. In contrast to the theoretical approaches known to us, we make the transition to the interaction representation (Dirac) in two steps.
    At the first stage, the lattice Hamiltonian is included along with the Zeeman interaction in the zero Hamiltonian of the interaction, which asymptotically leads to the DQ -Hamiltonian (15). In the Hamiltonian (15) we extract the part, see relation (16), responsible for the flip-flop transitions commuting with the z-projection of the total spin of the system, which is essentially an observable quantity in experiment, see relation (19).   Then, which to the best of our knowledge is done for the first time, we make another transition to the interaction representation by taking the previously extracted flip-flop part as the new zero-point Hamiltonian. On this basis we obtain the exact relations (23) and (24), with respect to which further approximations can be made in the future. In this paper the simplest approximations are investigated.    A nontrivial result is obtained already in the zero approximation with respect to the Hamiltonian of flip-flop transitions for the spin 1/2 system: an exactly solvable multispin system (eq. (27-29)). Next, we study the first non-zero corrections associated with flip-flop transitions and their Anderson-Weiss generalizations.


Citation: https://doi.org/10.5194/mr-2024-15-AC1
RC2:
'Comment on mr-2024-15', Anonymous Referee #2, 14 Oct 2024

The manuscript describes an analytical treatment of DQ excitation in solids in large spin systems where dynamics and spin flip-flop processes lead to time modulations of the dipolar couplings. In an iterative process of interaction-frame transformaions, it is shown that the DQ build up can be calculated analytically while taking the action of the dynamics and spin flip-flop transitions into account. For this an additional interaction-frame transformation with the part of the DQ Hamiltonian that commutes with I^z (flip-flop part) is performed. This leaves just a yy part of the Hamiltonian that commutes with itself at all times and allows the calculation of an analytical phase factor that describes the DQ intensity. Results are compared to numerical simulations in relatively small spin systems (6 spins) which shows reasonable agreement with the analytical calculations. Can the authors at least comment whether they expect better agreement if state of the art spin system sizes of 10-12 spins (doable without sparse matrix implementations) would give better agreement?
The manuscript is clearly inside the scope of MR and presents interesting new results that are of interest to the MR readership. However, I believe that the presentation could be improved and that there are a larger number of typos and very colloquial expressions that make it difficult to follow the text.
The authors start from a secular high-field truncated dipolar Hamiltonian (Eq (1)) which has the usual form. But then the authors say: "The Hamiltonian (1) in this

55 paper plays the role of the original spin-lattice interaction Hamiltonian. In the lowest order of perturbation theory, it induces in a spin system 0-quantum transitions (dephasing processes or 0-quantum coherence) by terms proportional to .... 1-quantum transitions (transitions with a flip of one of a pair of interacting spins or 1-quantum coherence) and 2-quantum transitions (coordinated transitions of two spins or 2-quntum coherence) by terms proportional ...." (equation cut out of text). Why would this (static) ZQ Hamiltonian induce DQ or SQ transitions. This would be true for the untruncated dipolar Hamiltonian but not for the high-field truncated one used here, or am I missing something?
There are a lot of very colloquial and not very clear and precise statements that could be improved. Examples are:

Line 74: "In solids, the joint effect of mentioned factors, in the limit Δ → 0 , where Δ is time interval between the nearest RF pulses" which joint effects are meant here?

Line 78: "Note, that n = 0 corresponds to the first mentioned version of DQ

experiment no phase change and n = 1 to the second version 90° phase shift." The English is wrong (replace "no" with "without") and it could be stated clearly that you refer to A-0 and A_1 signals here.

Line 84: "At time moment DQ t =t the operator (3) changes time for the case, when n = 1." A Hamiltonian cannot change time. I can change the sign.

Line 95: "where the role of the zero Hamiltonian is sum of the lattice Hamiltonian and the Hamiltonian of Zeeman interaction of investigated spins with an external magnetic field." What is a zero Hamiltonian? It is defined later but I find this a very colloquial expression. Maybe the "dominant part of the Hamiltonian" would be a better term.
There are more such unclear statements and the manuscript would profit from a carefull reading and checking for consistency of notation (see also minor points) as well as the English language.
I think the manuscript would benefit from making section 3 really a discussion without pages and pages of equations that could be moved to an appendix. At the moment section 3.1-3.3 is more a theory section that is very hard to follow and gives little insight for the non specialist.

Minor points - many typos (?) or unclear expressions in equations:

line 39: two full stops

line 58: quantum instead of quntum

Eq. (4): This should probably be a \omega_{ij}(t) and not \omega_{ij}^{(n)}(t) which is only defined in (5)

Line 102: Give a reference for AW approximation.

line 164: Are I_z and I^z the same or different quantities?

line 164: What is H_0 here? the Zeeman Hamiltonian or the flip-flop part of the DQ Hamiltonian?

Eq. 14 and 15: Is U_0(t) on the first line of Eq. 14 the same as U_0^{(DQ,n)} on the first line of Eq. 15?

Eq. 15: Is H^{(0)}_{DQ} the same as H^{(n)}_{DQ} in Eq. 16

Line 205: There is no section 2b

Eq. 32: Is {\tilde{I}_i^y\tilde{I}_j^y}^{fl}_t the same as the (I_i^yI_j^y)^{fl}_t on line 205 and Eq. 31? If not then please define.
There are probably many more that I did not see.

Citation: https://doi.org/10.5194/mr-2024-15-RC2
- AC2: 'Reply on RC2', Nail Fatkullin, 26 Oct 2024
  
  Response to anonymous Referee #2
  
  We are grateful to the second reviewer for his careful review of the manuscript and comments, which will incorporate in the new version of our paper.
  
  The manuscript describes an analytical treatment of DQ excitation in solids in large spin systems where dynamics and spin flip-flop processes lead to time modulations of the dipolar couplings. In an iterative process of interaction-frame transformations, it is shown that the DQ build up can be calculated analytically while taking the action of the dynamics and spin flip-flop transitions into account. For this an additional interaction-frame transformation with the part of the DQ Hamiltonian that commutes with I^z (flip-flop part) is performed. This leaves just a yy part of the Hamiltonian that commutes with itself at all times and allows the calculation of an analytical phase factor that describes the DQ intensity. Results are compared to numerical simulations in relatively small spin systems (6 spins) which shows reasonable agreement with the analytical calculations. Can the authors at least comment whether they expect better agreement if state of the art spin system sizes of 10-12 spins (doable without sparse matrix implementations) would give better agreement?
  
  To discuss possible improvements in the agreement between simulations and predictions, one should consider that our (still limited) approximation is for now limited to times (see lines 470-480); for longer times, additional work (e.g. evaluating higher-order corrections) is required. Generally, of course, we expect better agreement between simulations and either better predictions or experimental results for more spins. Our (outdated) code is unfortunately limited to 9 spins at present. But the main limitation is in fact the consideration of a realistic spin system, where our alkyl cut-outs are already a bit of a stretch. With nowadays’ improved simulation codes such as SPINACH, handling tens of spins, we shall in the future focus on trajectories from MD simulations for the most realistic modelling of fast local conformational dynamics and in particular inter-molecular couplings. We will add corresponding remarks in the paper.
  
  The manuscript is clearly inside the scope of MR and presents interesting new results that are of interest to the MR readership. However, I believe that the presentation could be improved and that there are a larger number of typos and very colloquial expressions that make it difficult to follow the text.
  The authors start from a secular high-field truncated dipolar Hamiltonian (Eq (1)) which has the usual form. But then the authors say: "The Hamiltonian (1) in this paper plays the role of the original spin-lattice interaction Hamiltonian. In the lowest order of perturbation theory, it induces in a spin system 0-quantum transitions (dephasing processes or 0-quantum coherence) by terms proportional to .... 1-quantum transitions (transitions with a flip of one of a pair of interacting spins or 1-quantum coherence) and 2-quantum transitions (coordinated transitions of two spins or 2-quntum coherence) by terms proportional ...." (equation cut out of text). Why would this (static) ZQ Hamiltonian induce DQ or SQ transitions. This would be true for the untruncated dipolar Hamiltonian but not for the high-field truncated one used here, or am I missing something?
  
  The reviewer is right: the properties of the full Hamiltonian of magnetic dipole-dipole interactions, which was written in this part of the paper in the very first version, are attributed to its secular part. Corresponding changes will be made in the revised version of the paper:
  “ The Hamiltonian (1) in this paper plays the role of the original spin-lattice interaction Hamiltonian. In the lowest order of perturbation theory, it induces in a spin system only 0-quantum transitions (dephasing processes or 0-quantum coherence) through terms proportional to and flip-flop processes proportional to . We also note that the non-secular part of the Hamiltonian of magnetic dipole-dipole interactions inducing the processes of spin-lattice relaxation describes the processes of 1-quantum and 2-quantum coherence. However, at the times of interest they can be neglected.”
  
  There are a lot of very colloquial and not very clear and precise statements that could be improved. Examples are:
  
  Line 74: "In solids, the joint effect of mentioned factors, in the limit Δ → 0 , where Δ is time interval between the nearest RF pulses" which joint effects are meant here?
  
  This will be improved in the following way:
  “ In solids, the joint effect of the BP sequence and the Hamiltonian (1) , in the limit , where is time interval between the nearest RF pulses, the spin system’s time evolution can be described in terms of an effective DQ Hamiltonian having the following structure:
  , (3)
  where is the Heaviside step function, . The case corresponds to the first mentioned version of DQ experiment without phase change, i.e, to the signal , and to the second version with 90° phase shift, i.e. to the signal . ”
  
  Line 78: "Note, that n = 0 corresponds to the first mentioned version of DQ experiment no phase change and n = 1 to the second version 90° phase shift." The English is wrong (replace "no" with "without") and it could be stated clearly that you refer to A-0 and A_1 signals here.
  
  This will be improved.
  
  Line 84: "At time moment DQ t =t the operator (3) changes time for the case, when n = 1." A Hamiltonian cannot change time. I can change the sign.
  
  This is considered in the above passage referring to eq. (3).
  
  Line 95: "where the role of the zero Hamiltonian is sum of the lattice Hamiltonian and the Hamiltonian of Zeeman interaction of investigated spins with an external magnetic field." What is a zero Hamiltonian? It is defined later but I find this a very colloquial expression. Maybe the "dominant part of the Hamiltonian" would be a better term.
  
  To be improved:
  “…zeroth-order Hamiltonian, i.e. the zero-point Hamiltonian or the dominant part of the Hamiltonian (see also expr. (8)), is sum of the lattice Hamiltonian and the Zeeman Hamiltonian of the investigated spins with an external magnetic field...”
  
  There are more such unclear statements and the manuscript would profit from a careful reading and checking for consistency of notation (see also minor points) as well as the English language.
  
  The revised version will be carefully edited.
  
  I think the manuscript would benefit from making section 3 really a discussion without pages and pages of equations that could be moved to an appendix. At the moment section 3.1-3.3 is more a theory section that is very hard to follow and gives little insight for the non-specialist.
  
  Citation: https://doi.org/10.5194/mr-2024-15-AC2
- AC4: 'Reply on RC2', Nail Fatkullin, 26 Oct 2024
  
  The comment was uploaded in the form of a supplement: https://mr.copernicus.org/preprints/mr-2024-15/mr-2024-15-AC4-supplement.pdf
  
  Citation: https://doi.org/10.5194/mr-2024-15-AC4

Nail Fatkullin, Ivan Brekotkin, and Kay Saalwächter

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Short summary

We believe that, in addition to nontrivial theoretical interest, the proposed work offers experimenters a reliable time interval in which the experimentally measured signal allows a relatively simple interpretation uncomplicated by contributions from three-particle dynamical correlations of having spins nuclei in condensed matter.


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