Effects of radial radio-frequency field inhomogeneity on MAS solid-state NMR experiments

Aebischer, Kathrin; Tošner, Zdeněk; Ernst, Matthias

doi:https://doi.org/10.5194/mr-2-523-2021

Articles | Volume 2, issue 1

https://doi.org/10.5194/mr-2-523-2021

© Author(s) 2021. This work is distributed under
the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/mr-2-523-2021

© Author(s) 2021. This work is distributed under
the Creative Commons Attribution 4.0 License.

Articles | Volume 2, issue 1

Research article

|

01 Jul 2021

Research article |

| 01 Jul 2021

Effects of radial radio-frequency field inhomogeneity on MAS solid-state NMR experiments

Kathrin Aebischer, Zdeněk Tošner, and Matthias Ernst

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Final revised paper (published on 01 Jul 2021)
Supplement to the final revised paper
Preprint (discussion started on 07 May 2021)
Supplement to the preprint

Interactive discussion

Status: closed

RC1:
'Comment on mr-2021-43', Malcolm Levitt, 20 May 2021

This is a very fine paper describing highly detailed state-of-the-art theory and simulations of “radial radio-frequency inhomogeneity” effects in MAS NMR. The most unusual aspect of this type of inhomogeneity is that since the phase of the applied rf field varies in space, and a volume element of a rotating sample traverses the inhomogeneous field, the rf field experience by the nuclear spins acquires a periodically time-varying phase, which can interfere with the performance of NMR experiments. The authors analyze these effects closely for a variety of sample rotors and pulse sequences, and conclude that the effects are usually too small to have serious consequences.

In this conclusion they may well be right. However they may have missed at least one prior report in the literature. In 1988 (doi.org/10.1002/ijch.198800039) rotary resonance recoupling experiments were reported in which an unanticipated peak appeared in the centre of the expected dipolar doublet (see Fig.4a in that paper). The following discussion, around equation 27, postulates a spinning-induced periodic phase modulation of the radio frequency field, due to inhomogeneity in the field direction, of the same type analysed in the current paper (this was actually 3 years before the Goldman article cited by the authors as the seminal reference). Simulations including a highly simplified model of this phase modulation did reproduce the central peak (figure 6). It would be interesting to know whether the far more sophisticated calculations performed by the authors verify — or disprove — this finding.

Apart from this comment which might require some minor changes to the paper the only criticism I have for this excellent piece of work is that the use of radians as the phase unit makes some of the figures needlessly hard to interpret. I suggest they plot the phase in units of degrees for ease of interpretation — or if they really do not like degrees, plot phase divided by pi, or similar.

Citation: https://doi.org/10.5194/mr-2021-43-RC1
- AC1:
  'Reply to review report', Matthias Ernst, 24 May 2021
  
  Since we have no special preference for radians, we have changed the units for the phase to degrees in figures 1 and 2.
  We were not aware of the effect of time-modulated rf fields in R3 recoupling and would like to thank you for bringing this to our attention. This is quite interesting and we have incorporated it in the introduction. We have modified the relevant part on page 2 to:
  
  "In solid-state NMR under magic-angle spinning (MAS) conditions, the radial component of the rf field gets modulated by time (Levitt et al., 1988; Tekely and Goldman, 2001; Goldman and Tekely, 2001; Tošner et al., 2017) leading to further potential complications in the experiments. Such MAS-induced time-dependent radio-frequency fields could give rise to additional or modified resonance conditions or to other changes in the effective Hamiltonians generated by the pulse sequence. The importance of such time-dependent terms was first described in rotary-resonance recoupling (Levitt et al., 1988) where it leads to changes in the observed line shape. Besides the appearance of additional sidebands in cross-polarization experiments (Tekely and Goldman, 2001; Goldman and Tekely, 2001), nutation spectra (Elbayed et al., 2005) and reported phase distortions and loss of magnetization in MLEV16 sequences under MAS (Piotto et al., 2001), there have been very few studies of the effects of such modulations of the amplitude and phase of the rf field caused by MAS rotation."
  We have simulated the R3 experiment and see the same effect of an additional peak appearing in the center of the powder pattern identical what is shown in the original reference.
  
  Citation: https://doi.org/10.5194/mr-2021-43-AC1
  - RC2: 'Reply on AC1', Malcolm Levitt, 24 May 2021
    
    Thanks to the authors for considering this early work and I hope it is possible for them to add the R3 simulations to the revised manuscript.
    
    Citation: https://doi.org/10.5194/mr-2021-43-RC2
    
    AC2: 'Reply on RC2', Matthias Ernst, 24 May 2021
    
    We are still running some more simulations but we plan on including some of them into the SI of the final version of the paper.
    
    Citation: https://doi.org/10.5194/mr-2021-43-AC2
RC3:
'Comment on mr-2021-43', Anonymous Referee #2, 28 May 2021
Reviewer comment: Aebischer et al. 2021

This very good paper characterizes the effect of radial rf field inhomogeneity on MAS solid-state NMR experiments in great detail, both in simulations and experimentally. This type of inhomogeneity leads to rotor-periodic amplitude and phase modulations within the sample under magic angle spinning. The authors could demonstrate that this does not have a significant effect on a number of commonly used experiments. They present their findings in a clear and very detailed manner.

Nevertheless, I still would like to propose some improvements:

First, in the section on nutation experiments, the authors attribute the lower intensity of the sidebands caused by amplitude modulations (compared to the phase modulation sidebands) partially to the lower magnitude of the amplitude modulations themselves. However, the magnitudes of both modulations do not seem too different when looking at Fig. 2 and taking the different scales of amplitude and phase into account. The second explanation provided by the authors (the broadening of the sideband by axial rf inhomogeneity) seems more likely. This could be tested in a simulation that only includes the radial amplitude modulation and not the axial inhomogeneity profile, which would remove the broadening of the sideband.

Second, in the section on cross-polarization, the difference of the rf field inhomogeneity of the two rf fields across the sample is held responsible for the restriction of the active sample volume. But the rf fields on the two channels do not need to have different rf profiles, the mere existence of rf inhomogeneity is sufficient to make the matching condition in Eq. 37 impossible to attain within the whole sample at the same time.

Furthermore, I want to suggest several changes to figures:

Fig. 2 may benefit from two changes. The first would put the panels showing z = +4 mm and z = -4 mm on the same vertical scales (they are very similar already, but not the same). The second would add another set of panels showing the largest radius of each axial position on the same scale to illustrate the difference in magnitude along the rotor axis.

In Fig. 3, vertical scales would be especially useful on the insets showing the sidebands caused by amplitude modulation, since they are barely visible in the main graphs, making an estimation of the scale all but impossible.

The meaning of the arrows in Fig. 11 is described in section 4.2, but not in the figure caption, where I was first looking to find out what they meant.

In addition, there are a small number technical errors that caught my eye:

The paper by Tošner et al. from 2017 does not include any actual optimal control calculations, as suggested by the citation in line 43 (their tm-SPICE pulses were presented in 2018).

The insets in panels b) and d) of Fig. 3 look exactly the same despite the relevant parts of the main graphs looking different, has there been an accidental duplication?

In line 437, Fig. 9c/d is referenced, but the authors likely mean to reference Fig. 9e/f.

The first sentence of the caption to Fig. 9 makes it look like the simulations for all panels were done at 150 MHz, which, according to the text and a later sentence in the same caption, is not the case.

In the last sentence of the caption to Fig. 10, the line splitting is placed in panel d) instead of c), and it can be attributed.

“NRM” in line 583 is probably supposed to read “NMR”.
Citation: https://doi.org/10.5194/mr-2021-43-RC3
- AC3: 'Reply on RC3', Matthias Ernst, 01 Jun 2021
  
  We would like to thank the reviewer for the very careful reading of the manuscript and the useful suggestions to improve the presentation.
  Q: First, in the section on nutation experiments, the authors attribute the lower intensity of the sidebands caused by amplitude modulations (compared to the phase modulation sidebands) partially to the lower magnitude of the amplitude modulations themselves. However, the magnitudes of both modulations do not seem too different when looking at Fig. 2 and taking the different scales of amplitude and phase into account. The second explanation provided by the authors (the broadening of the sideband by axial rf inhomogeneity) seems more likely. This could be tested in a simulation that only includes the radial amplitude modulation and not the axial inhomogeneity profile, which would remove the broadening of the sideband.
  A: We have looked into this in a bit more detail and this is correct. We have deleted the sentence on line 277: "The reduced intensity of these amplitude-modulation sidebands can be explained by the lower magnitude of amplitude modulations in comparison to phase modulations (see Fig. 2)." The reason is really the distribution of the sidebands over a larger spectral range which we have also confirmed in simulations as suggested by the reviewer.
  Q: Second, in the section on cross-polarization, the difference of the rf field inhomogeneity of the two rf fields across the sample is held responsible for the restriction of the active sample volume. But the rf fields on the two channels do not need to have different rf profiles, the mere existence of rf inhomogeneity is sufficient to make the matching condition in Eq. 37 impossible to attain within the whole sample at the same time.
  A: We agree with the reviewer that the presence of rf-field inhomogeneity alone will lead to a mismatch in parts of the probe. We have rephrased the relevant text after EQ. (37) to: "Due to the rf-field inhomogeneity across the sample, this condition cannot be fulfilled simultaneously in the entire sample volume and only certain parts of the sample will participate in the polarization transfer thus decreasing the resulting signal intensity."
  Q: Fig. 2 may benefit from two changes. The first would put the panels showing z = +4 mm and z = -4 mm on the same vertical scales (they are very similar already, but not the same). The second would add another set of panels showing the largest radius of each axial position on the same scale to illustrate the difference in magnitude along the rotor axis.
  A: We have adjusted the scaling of the two panels in Fig.2 but we have not added a separate panel with the largest radius. The reason for this is that the figure would become too small to recognise any details. In addition, the shape of the rf inhomogeneity is not something that we investigated but only copied and used from previous work.
  Q: In Fig. 3, vertical scales would be especially useful on the insets showing the sidebands caused by amplitude modulation, since they are barely visible in the main graphs, making an estimation of the scale all but impossible.
  A: We agree that this is a good idea and have added the vertical axis to the insets.
  Q: The meaning of the arrows in Fig. 11 is described in section 4.2, but not in the figure caption, where I was first looking to find out what they meant.
  A: We have added a sentence to the figure caption that explains the meaning of the arrows: "The arrows indicate the positions of the carrier frequency."
  
  Q: The paper by Tošner et al. from 2017 does not include any actual optimal control calculations, as suggested by the citation in line 43 (their tm-SPICE pulses were presented in 2018).
  A: We have corrected the citation for the Tosner 2017 and 2018 papers.
  Q: The insets in panels b) and d) of Fig. 3 look exactly the same despite the relevant parts of the main graphs looking different, has there been an accidental duplication?
  A: There was indeed a mixup and one of the insets was duplicated and wrong. Thanks a lot for spotting this. It has been updated in the revised version of the manuscript.
  Q: In line 437, Fig. 9c/d is referenced, but the authors likely mean to reference Fig. 9e/f.
  A: corrected
  Q: The first sentence of the caption to Fig. 9 makes it look like the simulations for all panels were done at 150 MHz, which, according to the text and a later sentence in the same caption, is not the case.
  A: corrected
  Q: In the last sentence of the caption to Fig. 10, the line splitting is placed in panel d) instead of c), and it can be attributed.
  A: corrected
  Q: “NRM” in line 583 is probably supposed to read “NMR”.
  A: corrected
  
  Citation: https://doi.org/10.5194/mr-2021-43-AC3

Peer review completion

AR: Author's response | RR: Referee report | ED: Editor decision | EF: Editorial file upload

AR by Matthias Ernst on behalf of the Authors (07 Jun 2021) Author's response Author's tracked changes Manuscript

ED: Publish as is (14 Jun 2021) by Bernd Reif

AR by Matthias Ernst on behalf of the Authors (15 Jun 2021) Manuscript

Short summary

The radio-frequency (rf) field amplitude in solid-state NMR probes changes over the sample volume, i.e. different parts of the sample will experience different nutation frequencies. If the sample is rotated inside the coil as it is typical for magic angle spinning in solid-state NMR, such a position-dependent inhomogeneity leads to an additional time dependence of the rf field amplitude. We show that such time-dependent modulations do not play an important role in many experiments.