20 May 2021
20 May 2021
Residual Linewidth in MagicAngle Spinning SolidState NMR
 Physical Chemistry, ETH Zürich, VladimirPrelogWeg 2, 8093 Zürich, Switzerland
 Physical Chemistry, ETH Zürich, VladimirPrelogWeg 2, 8093 Zürich, Switzerland
Abstract. Magicangle spinning is routinely used to average anisotropic interactions in solidstate NMR. Due to the fact, that the Hamiltonian of a stronglycoupled spin system does not commute with itself at different time points during the rotation, secondorder and higherorder terms lead to a residual line broadening in the observed resonances. Additional truncation of the residual broadening due to isotropic chemicalshift differences can be observed. We analyze the residual line broadening in coupled proton spin systems based on theoretical calculations of effective Hamiltonians up to third order using Floquet theory and compare these results to numerically obtained effective Hamiltonians in small spin systems. We show that at spinning frequencies beyond 50 kHz, secondorder terms dominate the residual line width leading to a 1/ω_{r} dependence of the second moment which we use to characterize the line width. However, chemicalshift truncation leads to a partial ω_{r}^{2} dependence of the line width which looks as if thirdorder effective Hamiltonian terms are contributing significantly. We show that secondorder contributions not only broaden the line but also lead to a shift of the center of gravity of the line. Experimental data reveals such spinningfrequency dependent line shifts in proton spectra in model substances that can be explained by line shifts induced by the secondorder dipolar Hamiltonian.
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Matías Chávez et al.
Status: final response (author comments only)

RC1: 'Comment on mr202145', Anonymous Referee #1, 02 Jun 2021
This is a very well written paper about an interesting and key topic in solidstate NMR, namely understanding the spinningfrequency dependence of 1H solidstate NMR lineshapes under magicangle spinning (MAS). Specifically, exact numerical simulations are compared to predictions from second or thirdorder Floquet theory effective Hamiltonian expressions. In this way, subtle effects relating a change from a 1/spinning frequency to 1/spinning frequency^2 dependence as well as small shifts in the lineshape centre of gravity are explored, notably considering the usual case of 1H spins with different 1H chemical shifts. Experimental 1H MAS NMR spectra up to 160 kHz MAS are also presented for orthophosphoLserine showing the same shifts in the lineshape centre of gravity are well explained by the theoretical predictions.
Minor improvements
Fig. 4 caption, 3^{rd} line: deviations which become
Fig. 6 explain the straight lines in the caption
Fig. 8 avoid repeat of bue and red in c to avoid confusion with colours used in b: how about green and orange instead?
Can the SI use S1 etc labels for page numbers, sections, Tables and Figures.
SI Figure 2: line 4, space between set and between

AC1: 'Reply on RC1', Matthias Ernst, 02 Jun 2021
Thank you for the positive comments and the detailed reading of the manuscript.
Q: Fig. 4 caption, 3rd line: deviations which become
A: corrected
Q: Fig. 6 explain the straight lines in the caption
A: The lines are a guide to the eye to show the linear correlation. They go through the point (0,0) and the data point for the lowest MAS frequency. We have added the following sentences to Figure caption 6: "The straight lines go through (0,0) and the line width at the slowest spinning frequency. They are meant as a guide to the eye for the linear correlation."
Q: Fig. 8 avoid repeat of bue and red in c to avoid confusion with colours used in b: how about green and orange instead?
A: We have changed the coloring scheme to avoid confusion.
Q: Can the SI use S1 etc labels for page numbers, sections, Tables and Figures.
A: We have changed the labelling of the SI to include the "S" for figures, pages and tables.
Q: SI Figure 2: line 4, space between set and between
A: corrected as well as another error in this caption.

AC1: 'Reply on RC1', Matthias Ernst, 02 Jun 2021

RC2: 'Comment on mr202145', Anonymous Referee #2, 04 Jun 2021
The manuscript by Chavez et al. reports on secondorder Hamiltonian analysis of proton resonances as a function of the MAS frequency. The authors find that the residual proton line width follows a 1/omega_r^1 dependence beyond 50 kHz of MAS. Chemical shift truncation can induce a 1/omega_r^2 dependence. This is a very interesting paper that should be published as soon as the authors have addressed the following issues.In the manuscript, the authors do a great job in discussing the MAS dependent 1H line width. Unfortunately, the contribution to the signal that is hidden in the Pake like pattern in the base line is not quantified. Can the authors give an estimate how the intensity is changed with MAS frequency using secondorder Hamiltonian arguments ? This would be extremely interesting, since most solidstate NMR experiments are sensitivity and not resolutionlimited. The manuscript is very similar to a recent paper by Xue et al. (J. Phys. Chem. C 2018, 122, 16437). The authors should discuss theoretical versus computational approaches to yield an understanding of proton resonances in the solidstate. In the presented approach, the geometry is restricted to very few angles and distances. Is it possible to derive general laws if only a few spins are considered ?While reading the paper, one gets the impression that a proton line can be infinitely narrow if only the MAS frequency is high enough. The authors should add an additional term to their equations which summarizes the contributions to line width that are not affected by MAS. At which MAS frequency does the 1/omega_r dependence break down ?I am missing a paragraph in which the theoretical considerations are compared with experimental data, and correlate theory and experiment. The author did this for the MAS induced frequency shift shown in fig 8 which is very nice. However, I am missing a discussion on the line width. The data for phosphoserine exist (at least in the MAS range 70160 kHz), and it should be straight forward to read out the line width after a line shape fit.

AC2: 'Reply on RC2', Matthias Ernst, 07 Jun 2021
Response to Reviewer 2:
Q: ABMS, like protonproton dipolar interactions, yield a MAS dependent line width. How do the authors differentiate between the two effects. The same applies to the MAS induced frequency shift (see e.g. Alla and Lippmaa, Chem. Phys. Lett. 1982, 87, 3033; Samoson et al. Solid State Nucl. Magn. Res. 2001, 20, 130136). While reading through the paper, one gets the impression that dipoledipole interactions dominate the proton line width. The abstract / title should be modified into something like "Residual line width resulting from proton dipolar interactions in MagicAngle Spinning SolidState NMR“ in order to avoid confusions. To differentiate, some experimental data on the field dependence of the MAS dependent line width would be highly appreciated.
A: We agree that the title should include the fact that we exclusively discuss broadening from dipolar contributions. Therefore, we have changed the title to "Residual Dipolar Linewidth in MagicAngle Spinning Proton SolidState NMR". We have also added dipolar in the abstract in two locations. The influence of ABMS on line position and line width has been proven to be elusive in the literature. The way we understand this effect is that ABMS originates from the shape and finite dimensions of the crystallites present in the powder sample. MAS averages ABMS contributions from the isotropic susceptibility but not ABMS effects originating from the anisotropic part of the susceptibility. The reason for this is the fact that they can be described as the product of two secondrank tensors that generate rank024 components. Therefore we expect a line shift due to anisotropic ABMS contributions, rank2 that are averaged out and rank4 that are scaled but not fully averaged out (see Alla and Lippmaa, Chem. Phys. Lett. 1982, 87, 3033 section 3). However, the ABMS line width should scale with P_4(cos(\theta_m)) and should, in principle, be MAS independent as should be the shift. They are both B_0 field dependent as is experimentally shown in Samoson et al. Solid State Nucl. Magn. Res. 2001, 20, 130136. If our understanding of the ABMS shifts is wrong, we would appreciate more pointers where we misunderstood the literature.
We have not included experimental data since we are mostly interested in the theoretical underlying mechanism of the changes in scaling of the MAS line width. We reference several papers that discuss this experimental finding in the introduction (around line 40): "Experimental observations of the residual homogeneous line width as a function of spinning frequency show that it can often be approximated by a linear correlation with the inverse of the spinning frequency with some deviation that indicate a partial inverse quadratic dependence (Nishiyama, 2016; Sternberg et al., 2018; Penzel et al., 2019; Schledorn et al., 2020). This has been attributed to thirdorder contributions to the effective Hamiltonian or to chemicalshift effects (Sternberg et al., 2018; Moutzouri et al., 2020)." We hope that this is sufficient.Q: In the manuscript, the authors do a great job in discussing the MAS dependent 1H line width. Unfortunately, the contribution to the signal that is hidden in the Pake like pattern in the base line is not quantified. Can the authors give an estimate how the intensity is changed with MAS frequency using secondorder Hamiltonian arguments ? This would be extremely interesting, since most solidstate NMR experiments are sensitivity and not resolutionlimited. The manuscript is very similar to a recent paper by Xue et al. (J. Phys. Chem. C 2018, 122, 16437). The authors should discuss theoretical versus computational approaches to yield an understanding of proton resonances in the solidstate. In the presented approach, the geometry is restricted to very few angles and distances. Is it possible to derive general laws if only a few spins are considered ?
A: The line width is easy to assess via an expansion of the moments but of course there are many distributions that produce the same (second) moment and an easy guess of the line height is not possible. Of course in the limit of many spins, we might get a Gaussian line and then we could predict the line height. However, to what extend this is true, we do not know yet and is part of our current research efforts.
The appendix (and also references cited in the paper) give analytical expressions for the second and thirdorder Hamiltonian. Based on these expressions, one can calculate analytical solutions of the line width. However, the expressions become very complex functions of the distances and relative orientations of the couplings and it is not easy to get much insight from them beyond the form of the spin operators. This is the reason that we opted for a numerical implementation of the effective Hamiltonians. The strength of this method is that we can distinguish which order of the effective Hamiltonian contributes which is not accessible from a purely numerical simulation as in the work by Xue.Q: While reading the paper, one gets the impression that a proton line can be infinitely narrow if only the MAS frequency is high enough. The authors should add an additional term to their equations which summarizes the contributions to line width that are not affected by MAS. At which MAS frequency does the 1/omega_r dependence break down ?
A: The homonuclear dipolar part of the line width that is discussed in the paper has no spinningfrequency independent term. Of course, other homogeneous parts of the line width, e.g., relaxation terms or chemical exchange and inhomogeneous terms, e.g., sample inhomogeneity or ABMS shifts might broaden the line in addition to the dipolar line width as discussed in the introduction (around line 25). It depends on the relative magnitude of the various contributions where the 1/omega_r dependence breaks down and no general rule can be given. Especially, sample inhomogeneity and ABMS contributions can vary over a large range of values.
Q: I am missing a paragraph in which the theoretical considerations are compared with experimental data, and correlate theory and experiment. The author did this for the MAS induced frequency shift shown in fig 8 which is very nice. However, I am missing a discussion on the line width. The data for phosphoserine exist (at least in the MAS range 70160 kHz), and it should be straight forward to read out the line width after a line shape fit.
A: The isotropic line shift can be compared quite well because it looks like the dipolar contribution to the isotropic line shift is the dominating one. As discussed above, there are many contributions to the line broadening and we do not claim that we can predict the line width from one simple threespin simulation. In addition, the focus of the paper is on understanding the spinningfrequency dependence of the line broadening and not predicting the line broadening. Therefore, we do not want to make comparison to experimental data of the line width but give references to published data that illustrate the spinning frequency dependence.

AC2: 'Reply on RC2', Matthias Ernst, 07 Jun 2021
Matías Chávez et al.
Matías Chávez et al.
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