the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Origin of the residual line width under frequency-switched Lee–Goldburg decoupling in MAS solid-state NMR

### Johannes Hellwagner

### Liam Grunwald

### Manuel Ochsner

### Daniel Zindel

Homonuclear decoupling sequences in solid-state nuclear magnetic resonance (NMR) under
magic-angle spinning (MAS) show experimentally significantly larger residual
line width than expected from Floquet theory to second order. We present an
in-depth theoretical and experimental analysis of the origin of the residual
line width under decoupling
based on frequency-switched Lee–Goldburg (FSLG) sequences. We analyze the effect of experimental pulse-shape errors (e.g., pulse
transients and *B*_{1}-field inhomogeneities) and use a Floquet-theory-based
description of higher-order error terms that arise from the interference
between the MAS rotation and the pulse sequence. It is shown that the
magnitude of the third-order auto term of a single homo- or heteronuclear
coupled spin pair is important and leads to significant line broadening
under FSLG decoupling. Furthermore, we show the dependence of these
third-order error terms on the angle of the effective field with the
*B*_{0} field. An analysis of second-order cross terms is presented that
shows that the influence of three-spin terms is small since they are
averaged by the pulse sequence. The importance of the inhomogeneity of the radio-frequency (rf) field
is discussed and shown to be the main source of residual line broadening
while pulse transients do not seem to play an important role.
Experimentally, the influence of the combination of these error terms is
shown by using restricted samples and pulse-transient compensation. The
results show that all terms are additive but the major contribution to the
residual line width comes from the rf-field inhomogeneity for the standard
implementation of FSLG sequences, which is significant even for samples with
a restricted volume.

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Protons are present in most materials and are one of the important nuclei in
nuclear magnetic resonance (NMR) spectroscopy in the study of biological
systems and materials. Besides the advantage of high sensitivity, protons
allow insights into molecular packing in solids as direct observations of
hydrogen bonding and C–H–*π* as well as *π*–*π* interactions are
possible
(Barfield,
2002; Berglund and Vaughan, 1980; Parker et al., 2006). While proton
detection is routine in solution-state NMR, it is not yet used routinely in
solid-state NMR due to the large proton–proton dipolar couplings that are
only partially averaged out by magic-angle spinning (MAS). At slow to medium
MAS frequencies, significant residual line broadening is observed. The
technical advances in the field of MAS during the past decade have resulted
in very fast spinning frequencies (up to 150 kHz); however, the obtained
resolution is still not sufficient for all applications
(Agarwal
et al., 2014; Andreas et al., 2016; Penzel et al., 2019; Stöppler et
al., 2018). The residual line width in a fully protonated protein sample is
still 100–200 Hz, making de novo resonance assignment and structure determination
from proton-detected spectra challenging.

To further reduce the residual line width, especially at slow and medium-range MAS frequencies (lower than 60 kHz), homonuclear decoupling sequences can be employed. The first strategy to average out homonuclear proton–proton interactions using radio-frequency (rf) fields in static samples was suggested by Lee and Goldburg (Goldburg and Lee, 1963; Lee and Goldburg, 1965). Since then, many homonuclear decoupling sequences have been developed including solid-echo-based sequences (WAHUHA; Waugh et al., 1968, MREV; Mansfield and Grannell, 1971; Rhim et al., 1973), BR-24 (Burum and Rhim, 1979a, b), time-reversal sequences (Rhim et al., 1971), C- and R-symmetry-based sequences (Levitt, 2007; Madhu et al., 2001; Paul et al., 2010), tilted magic-echo-based sequences (Gan et al., 2011; Lu et al., 2012; Nishiyama et al., 2012), and computer-optimized sequences like DUMBO (Grimminck et al., 2011; Halse and Emsley, 2012; Sakellariou et al., 2000; Salager et al., 2009). Each of these sequences has their own advantages and disadvantages in terms of robustness for different MAS regimes and requirements for the radio-frequency field strength. Several examples of these types of sequences and their performance can be found in a recent review (Mote et al., 2016) and will not be discussed in detail.

The theoretical basis of the Lee–Goldburg sequence relies on the manipulation of the spin interactions by an off-resonance irradiation such that the quantization axis of the effective field is along the magic angle in a rotating frame (Lee and Goldburg, 1965). This leads to the removal of all second-rank interactions in spin space, i.e., the dipolar coupling, to second order in static samples. The Lee–Goldburg pulse sequence has undergone a lot of modification to be more robust and to compensate pulse errors generated by the spectrometer hardware. It was later also combined with magic-angle spinning where interference between averaging in real and in spin space becomes an issue (Mote et al., 2016; Vinogradov et al., 2004). Well-known alterations of the pulse sequence include the frequency-switched Lee–Goldburg (FSLG) (Bielecki et al., 1990; Levitt et al., 1993; Mehring and Waugh, 1972a) or the phase-modulated Lee–Goldburg (PMLG) sequences (Vinogradov et al., 1999, 2000, 2001, 2004). Various super cycles have been developed to compensate higher-order terms and pulse errors. The most commonly used super cycles or alterations include an inversion of the phase ramp (PMLG $\mathrm{x}\stackrel{\mathrm{\u203e}}{\mathrm{x}})$ (Leskes et al., 2007; Paul et al., 2009) or a relative phase shift between two PMLG cycles with an inversion of the second cycle (LG-4) (Halse et al., 2014; Halse and Emsley, 2013). These super cycles have the disadvantage of reducing the scaling factors of the chemical shifts, leading to reduced separation of the lines assuming similar decoupling efficiency. The advantage of such super-cycled sequences is the suppression of artefacts (quadrature images, axial peaks, and spurious signals) in the spectrum (Bosman et al., 2004).

The theoretical description of Lee–Goldburg sequences can be done within the framework of average Hamiltonian theory (AHT) (Ernst et al., 1990; Haeberlen, 1976) or Floquet theory (Leskes et al., 2009, 2010; Scholz et al., 2010; Shirley, 1965). Such a description has been used to predict the first-order resonance conditions between the MAS frequency and the modulation frequency of the pulse sequence as well as the magnitude of the non-vanishing second-order dipole–dipole cross terms. The second-order cross terms are one source of the residual line width that is still observed after homonuclear decoupling, and they need to be minimized in order to obtain narrow spectral lines (Vinogradov et al., 2004). A further factor for the performance degradation in FSLG sequences was believed to be experimental imperfections caused by either rf-field inhomogeneity or pulse transients (Barbara et al., 1991; Mehring and Waugh, 1972b), which, in certain cases, were also used for the improvement of the performance of the pulse sequence (Vega, 2004). It was shown that changing the phase transients by changing the tuning of the probe can be used to improve the spectral quality obtained by S2-DUMBO sequences (Brouwer and Horvath, 2015).

In this article, we investigate the influence of multiple parameters on the
residual line width in FSLG decoupled spectra. We characterize the magnitude
of the broadening by pulse transients for a standard FSLG sequence and
compare the results to an implementation using transient-compensated pulses
(Hellwagner
et al., 2018; Tabuchi et al., 2010; Takeda et al., 2009; Wittmann et al.,
2015, 2016). Furthermore, we analyze the FSLG sequence theoretically using
Floquet theory up to the third order. We are able to show that third-order terms
play an important role in strongly coupled systems like CH_{2} groups. We
present FSLG-based experiments using transient-compensated pulses on model
compounds where homonuclear dipolar second-order terms are purposefully
minimized to illustrate the importance of the third-order contribution.
Additionally, we investigate the broadening due to rf-field inhomogeneity by
comparing results from restricted samples and numerical simulations taking
the rf-field distribution over the sample into account. We show that the
rf-field inhomogeneity is the main source of line broadening in FSLG-based
experiments due to the dependence of the chemical-shift scaling on the
rf-field amplitude. The distribution of rf fields leads to a difference in
scaled isotropic chemical shifts depending on the characteristics of the
generated *B*_{1} field and causes the dominant contribution to the residual
broadening FSLG decoupled spectra.

The Lee–Goldburg scheme is based on averaging the second-rank spin tensor of a homonuclear dipolar Hamiltonian by rotating it around a field oriented along the magic angle. Such an averaging scheme is related to the removal of the spatial second-rank tensors of the Hamiltonian by MAS. The magic-angle irradiation is achieved by applying an off-resonance rf Hamiltonian of the general form

where the off-resonance term Δ*ω* is defined as *ω*_{0}−*ω*_{rf} with *ω*_{0} denoting the Larmor frequency of protons. The combination of the offset and
a constant rf irradiation along the *x* axis generates a quantization axis
which is tilted by the angle *θ* defined by

It can be shown that in first-order average Hamiltonian theory the
homonuclear dipole–dipole interactions vanish if *θ*
is adjusted to the magic angle. As a consequence, the chemical-shift
Hamiltonian is also scaled down by a factor ${d}_{\mathrm{0},\mathrm{0}}^{\left(\mathrm{1}\right)}\left(-\mathit{\theta}\right)=\mathrm{cos}\mathit{\theta}$
which takes a value of around 0.577 at $\mathit{\theta}=\mathrm{54.7}{}^{\circ}$. An equivalent description of off-resonance irradiation can be achieved by
a phase modulation of the radio-frequency irradiation of the form

Here, *ω*_{1} represents a constant rf amplitude and *φ*(*t*) a linear phase ramp.

The arguments based on AHT only hold true in a static regime but to get a full understanding of the sequence under MAS, the interference with the sample rotation has to be considered. The analysis of Hamiltonians with multiple time dependencies that are not commensurate is done best using Floquet theory (Leskes et al., 2010; Scholz et al., 2010).

The Floquet analysis is done in an interaction frame of the rf-field Hamiltonian where the interaction-frame transformation is defined by the propagator

with the interaction-frame Hamiltonian given by

Here, $\widehat{T}$ represents the Dyson time-ordering operator
(Dyson, 1949). The spherical spin-tensor operators of
rank *r* under a generalized interaction-frame transformation will transform
according to

where *ω*_{m} is the modulation frequency of the sequence and *ω*_{eff} is the effective nutation frequency of a basic pulse element. Here, the
scaling factors ${a}_{r,s}^{(k,l)}$ are the
Fourier coefficients of the interaction-frame trajectory.

These two frequencies together with the MAS frequency *ω*_{r}
constitute the three basic frequencies that characterize the time dependence
of the Hamiltonian. Note that for an ideal FSLG sequence the effective
nutation frequency will be zero. Hence, the interaction-frame Hamiltonian
can be written as a Fourier series.

A set of possible resonance conditions can be derived for any set of
integers
(*n*_{0}*k*_{0}*l*_{0})
that fulfill the equation

Since decoupling sequences are mostly applied outside of any resonance conditions and the residual line width is dominated by residual couplings, we will only focus on non-resonant terms where the effective Hamiltonian is given by

with

the second-order effective Hamiltonian defined by

and the third-order component given by

The summations in the third-order term have to be restricted to values of $\left(\mathit{\nu}{\mathit{\nu}}^{\prime}\mathit{\kappa}{\mathit{\kappa}}^{\prime}\mathit{\lambda}{\mathit{\lambda}}^{\prime}\right)$ that fulfill the inequalities $\mathit{\nu}{\mathit{\omega}}_{\mathrm{r}}+\mathit{\kappa}{\mathit{\omega}}_{\mathrm{m}}+\mathit{\lambda}{\mathit{\omega}}_{\mathrm{eff}}\ne \mathrm{0}$ and ${\mathit{\nu}}^{\prime}{\mathit{\omega}}_{\mathrm{r}}+{\mathit{\kappa}}^{\prime}{\mathit{\omega}}_{m}+{\mathit{\lambda}}^{\prime}{\mathit{\omega}}_{\mathrm{eff}}\ne \mathrm{0}$. Note that the equation for the third-order contribution differs from the original paper due to a sign mistake in the original work (Ernst et al., 2005). To the best of our knowledge, only one theoretical description of third-order terms under simultaneous rf irradiation and MAS (Tatton et al., 2012) has been published. These terms were shown to cause a shift of the resonance frequency. Evaluation of these expressions for homonuclear dipolar coupled Hamiltonians under FSLG irradiation provides insight into terms that are not averaged out and can contribute to the residual line width of the spectrum. A detailed analysis of the individual contributions and their behavior under the pulse sequence is presented in the following section.

The analytical calculation of the second-order cross terms for a homonuclear
coupled three-spin system with two non-vanishing dipolar couplings *δ*_{1,2} and *δ*_{1,3} yields lengthy expressions that depend on the
powder angles *α* and *β* as well as the relative orientation of
the two couplings Φ and the angle of the effective field *θ*
with respect to the external magnetic field which is set to the magic angle
for a standard FSLG sequence. All of these terms scale linearly with the
product of the two dipolar-coupling constants *δ*_{1,2} and *δ*_{1,3}.
In order to illustrate the symmetry of the remaining terms, the Hamiltonian
is projected on all possible three-spin tensor operators. The three-spin
operators are defined according to Garon et
al. (2015) where they were first derived. The projections are calculated for
powder angles $\mathit{\alpha}=\mathit{\beta}=\mathrm{45}{}^{\circ}$ and a relative dipole orientation
$\mathrm{\Phi}=\mathrm{45}{}^{\circ}$ since many of the terms have a local maximum at this set
of angles. The modulation frequency of the pulse sequence was set to be 10
times larger than the MAS frequency to avoid any possible resonance
conditions in the calculations. The ratio of the modulation and the MAS
frequency will be noted as $z={\mathit{\omega}}_{\mathrm{m}}/{\mathit{\omega}}_{\mathrm{r}}$.
The second-order cross terms between two dipolar couplings with one spin in
common lead to spin-tensor operators of a rank of zero to three.
Figure 1 shows the dependence of these second-order
three-spin cross terms as a function of the effective-field angle of the
FSLG irradiation. The dominant terms are the

and the

and

tensor operators where we follow the definition and notation
introduced in Garon et al. (2015). They all
arise from commutators of one homonuclear dipolar coupling in the
interaction frame of the effective field with another dipolar coupling where
one spin is in common. The dependence on the effective-field angle *θ*
can be expressed by a combination of Legendre polynomials of the zeroth, second,
and fourth orders, indicating that the origin of the terms is indeed second
order and a product of two spatial second-rank tensors. In order to verify
the analytical calculations, full numerical simulations (using the spin
simulation environment GAMMA; Smith et al., 1994)
using the same set of parameters were run to calculate the effective
Hamiltonian numerically over a full rotor period. The numerical effective
Hamiltonian was also projected onto the relevant spin-tensor operators, which
led to very similar but not identical values for the coefficients of the
tensor operators (compare Fig. 1a, b showing the same tensor components). The
differences are explained as follows. In Fig. 1a
and c, all non-zero terms from numerical simulations are shown but
analytical calculations only result in *T*_{0,0}, and
*T*_{2,0} terms
(Fig. 1b). The tensor components
*T*_{1,m} and
*T*_{3,m} only appear in the
numerical simulations (Fig. 1c) and originate
from higher-order contributions that are not considered in the analytical
calculations. The significant contributions (*T*_{l,0}
terms) to the second-order three-spin terms are minimized around the
orientation of the effective field along the magic angle while the remaining
terms (*T*_{l,m} terms with *m*≠0) are all very small. The minimum
appears to be very broad and, therefore, it is expected that the sequence is
fairly robust towards maladjustments in the rf-field amplitude or the phase
ramp which would result in a change of the effective-field angle.

In order to investigate other possible contributions to the residual
line width that are not averaged out by the combination of MAS and the
FSLG-based pulse sequence, third-order terms of a single dipolar coupling in
a two-spin system were analyzed (for numerical expression of the third-order
effective Hamiltonian, see the Supplement). These terms are
expected to scale with ${\mathit{\delta}}_{\mathrm{1},\mathrm{2}}^{\mathrm{3}}$ since they result from double-commutator terms due to their third-order
origin. Under MAS without simultaneous rf irradiation, such terms are
averaged out since a single dipolar coupling commutes with itself at all
times. This is no longer true under rf irradiation. Evaluating the double
commutators for all non-resonant terms and analyzing the resulting effective
Hamiltonian, only terms with the tensor symmetry
*T*_{2,m} remain. The analytical expressions for
these terms can be found in the Supplement. The magnitude and
dependence on the effective-field angle are shown in
Fig. 2. It can be concluded from these
calculations that the terms are not averaged out by a FSLG irradiation with
the angle of the effective field set to the magic angle but rather around
60^{∘} for the *T*_{2,0} term and around
40^{∘} for the ${T}_{\mathrm{2},\pm \mathrm{2}}$ term. The ${T}_{\mathrm{2},\pm \mathrm{1}}$
term does not show a local minimum around typical effective-field angles but
calculations of the propagation of the density operator under such a term
show that it does not result in an effective line broadening but rather in a
shift of the resonance frequency. This fact does not hold true for the
*T*_{2,0} and the ${T}_{\mathrm{2},\pm \mathrm{2}}$
terms, which ultimately contribute to the line width under FSLG. The magnitude
and angle dependence of these spin-tensor elements are shown in
Fig. 2b, and they were again calculated for the
angle values $\mathit{\alpha}=\mathit{\beta}=\mathrm{45}{}^{\circ}$, $\mathrm{\Phi}=\mathrm{45}{}^{\circ}$, and *z*=10. The dipolar coupling was set to 40 kHz, which is
representative for a CH_{2} group that remains one of the biggest
challenges in homonuclear decoupling. The effective-field strength was set
to 125 kHz, and it can be shown that the magnitude of the third-order terms
scales down quadratically with the effective field assuming the same ratio
$z={\mathit{\omega}}_{\mathrm{m}}/{\mathit{\omega}}_{\mathrm{r}}$. However, rf-field amplitudes higher than 100 kHz are often
experimentally not feasible for many practical applications.

It is obvious from Fig. 2 that the third-order
terms (double commutator terms of a single homonuclear dipolar coupling in
the interaction frame of the effective field) do not vanish under
FSLG-irradiation and they are significant in size given that the
${T}_{\mathrm{2},\mathrm{0}}=\frac{\mathrm{1}}{\sqrt{\mathrm{6}}}\left(\mathrm{3}{I}_{\mathrm{1}z}{I}_{\mathrm{2}z}-\left({\mathit{I}}_{\mathrm{1}}\cdot {\mathit{I}}_{\mathrm{2}}\right)\right)$ and the ${T}_{\mathrm{2},\pm \mathrm{2}}=\frac{\mathrm{1}}{\mathrm{2}}\left({I}_{\mathrm{1}}^{\pm}\cdot {I}_{\mathrm{2}}^{\pm}\right)$
contribute directly to the residual line broadening. We believe that these
third-order terms contribute significantly to the residual line width under
FSLG decoupling, especially for strongly coupled spins as encountered in
CH_{2} groups. As in the case of the second-order cross terms, there is a
good agreement between the analytical (Fig. 2b) and the numerical
calculations (Fig. 2a) of the third-order terms.

The influence of heteronuclear dipolar couplings on the residual line
broadening can be analyzed theoretically in the same way. Again, third-order
terms from a single heteronuclear dipolar coupling are obtained and do not
vanish if the effective-field angle is set to the magic angle. These terms
have the same spin-tensor components as the third-order homonuclear terms
but the spatial dependence differs. The dependence on the effective-field
angle is shown in the Supplement (Fig. S2). The second-order
heteronuclear–homonuclear dipolar cross terms show a similar behavior as the
homonuclear–homonuclear terms. Their functional form is shown in Fig. S3
of the Supplement. They are minimized for an effective-field orientation around the
magic angle. Further line broadening during the FSLG sequence can come from
experimental errors like sample inhomogeneities, pulse transients, and
*B*_{1}-field inhomogeneities.

In order to compensate pulse transients during the FSLG pulse sequence,
small modifications have to be made to the implementation of the sequence.
On the spectrometer, the sequence is typically implemented as a rectangular
pulse of constant amplitude but with discrete phase steps. The new
spectrometer hardware can generate shape files with a time resolution of
50 ns and an almost perfect phase ramp can be realized in order to generate
a constant offset irradiation. Nevertheless, due to the finite bandwidth of
the resonance circuit, a finite rise time of the pulse is observed as well
as phase transients at the start of the pulse and at the positions of the
180^{∘} phase jump. To compensate for these pulse transients, a
finite edge of the pulse has to be introduced. As a consequence of this
pulse edge, the flip angle of the shaped pulse is no longer 2*π* and the
amplitude has to be corrected. Using a linear phase ramp will lead to an
effective-field angle that is not constant throughout the sequence.
Therefore, the phase ramp has to be calculated explicitly by numerical
integration of the required offset of the irradiation that is needed in
combination with the time-dependent rf-field amplitude to generate a
constant effective-field direction and a 2*π* rotation about the
effective field.

The phase of the shaped pulse is defined as

with the offset frequency Δ*ν* defined by

The rf-field amplitude *ν*_{1} is defined by the shape and the length of the pulse. The implementation of the pulse sequence for shaped and rectangular pulses
is shown in Fig. 3. The shape of the phase ramp
can be explained by considering the functional form of the pulse edge which corresponds to a sine function. Therefore, the
phase ramp during the pulse edges must correspond to a cosine function and
the slope in the constant part is steeper compared to rectangular pulses to
compensate for reduced effective rotation during the finite edge.

In order to validate the theoretical consideration of the pulse sequence,
numerical simulations in a homonuclear three-spin system were performed
with both pulse implementations shown in Fig. 3.
To validate the contributions of second- and third-order terms discussed in
the theory section, simulations using only one (*δ*_{12}≠0) and two
non-zero dipolar couplings (${\mathit{\delta}}_{\mathrm{12}}\ne \mathrm{0},{\mathit{\delta}}_{\mathrm{13}}\ne \mathrm{0})$ in
a three-spin system were performed. The average effective field was set to
be 125 kHz corresponding to a Lee–Goldburg pulse length for a full 2*π*
nutation of 8 µs. The MAS frequency was set to be 6.25 kHz leading to a
ratio of *z*=10. Powder averaging was used according to the Zaremba–Conroy–Wolfsberg (ZCW) scheme with
300 crystal orientations (Cheng et al., 1973). A
relative orientation of $\mathrm{\Phi}=\mathrm{45}{}^{\circ}$ was used for the two dipole
tensors. The residual splitting as a function of the effective-field direction is
shown in Fig. 4 without taking chemical-shift
correction into account. Therefore, the splitting represents the residual
effective coupling in the Hamiltonian. It is obvious that the shaped-pulse
implementation and the rectangular pulses lead to very similar line widths
and dependences on the effective-field angle *θ*. Furthermore, these
results demonstrate the fact that the second-order three-spin terms are
averaged out fairly well around the magic angle but that there are still
significant contributions from third-order terms. These third-order terms
are a combination of the functional forms shown in
Fig. 2 for the
*T*_{2,0} and the ${T}_{\mathrm{2},\pm \mathrm{2}}$ tensor operators. Considering the chemical-shift scaling for residual line widths, the
third-order terms are minimized around an effective-field angle of around
60^{∘}. This effective-field angle dependence was also demonstrated
experimentally and the results are shown in Fig. 5.

Experiments were performed on various glycine derivatives designed to
illustrate the different contributions to the residual line width under FSLG
decoupling. In order to avoid unexpected effects due to windowed PMLG
detection during the decoupling period
(Vinogradov et al., 2002), the experiments were
implemented as two-dimensional experiments with the FSLG decoupling in the indirect
dimension followed by either a long CP for carbon detection or direct proton
detection. The CP time was chosen to be 3 ms to ensure transfer from all
protons in natural abundance glycine and to minimize the effects of
heteronuclear dipolar couplings in natural abundance samples. Since proton
spin diffusion is very efficient at the low MAS frequency that was used in
our measurements, we expect that magnetization from protons bound to
^{12}C is also observed. Additionally, simple FSLG sequences without a super
cycle were used to benefit from the maximum chemical-shift scaling.
Therefore, quadrature images and axial peaks were observed in the indirect
dimension, which were discarded for the analysis.
Figure 5 shows a comparison of the decoupling
efficiency using transient-compensated pulses and conventional rectangular
pulses. The dependence of the line width on the effective-field angle was
investigated in the range from 40 to 65^{∘}. The
experiments were performed on a uniformly labeled ^{13}C-^{15}N-glycine
at an external magnetic field of 14.1 T using an effective field of 125 kHz
and MAS speed of 14 kHz. The quantity that was used to judge the
decoupling efficiency was the separation of the two proton signals of the
H_{2} group. The separation parameter is defined by the ratio of the
intensity between the two lines and the intensity of the two lines
$\mathit{\epsilon}=\frac{\mathrm{2}{I}_{\mathrm{min},\mathrm{H}}}{\left({I}_{\mathrm{max},\mathrm{H}\mathrm{1}}+{I}_{\mathrm{max},\mathrm{H}\mathrm{2}}\right)}$
(see Supplement Fig. S6 for a graphical representation of the
parameters). A value of 0 corresponds to baseline separation of the two
lines whereas a value of 1 represents indistinguishable lines.
Figure 5b shows this splitting as a function of the
effective-field angle, and it can be seen that the transient-compensated FSLG
sequences perform slightly worse than the conventional rectangular pulses
but the differences are small. Furthermore, it is shown in the figure that
the optimum decoupling efficiency is not exactly at the magic angle but
shifted to slightly higher angles. This agrees with the theoretical
predictions that the third-order terms, which are believed to be significant
in a CH_{2} group, are minimized around higher values of ca. 60^{∘}. A further observation of these experiments is the behavior of the
chemical-shift scaling. The use of compensated pulses leads to
chemical-shift scaling factors that agree very well with the theoretical
prediction of cos *θ* whereas rectangular pulses lead to higher chemical-shift scaling
(Fig. 5c).

A further contribution to the residual line width is the heteronuclear
dipolar coupling, which can be avoided to a large degree by using natural
abundance samples. The influence of the heteronuclear coupling was
investigated by recording the spectra of natural abundance glycine using
compensated and rectangular pulses. The resulting spectra for an
effective-field angle of *θ*=*θ*_{m} and 60^{∘} are shown in Fig. 6a and b,
respectively. The improvement is significant compared to the labeled
compound since in all implementations the separation of the CH_{2} group
is almost at the baseline (compare to Fig. 5a).
Quantifying the line width (without chemical-shift scaling), an improvement
of ∼60 Hz is observed going from fully labeled to unlabeled
samples. This agrees well with the theoretical calculations of heteronuclear
third-order terms shown in Fig. S2 and previous results
(Tatton et al., 2012).

It can be argued from the spectra shown in Fig. 6
that the compensation leads to slightly narrower CH_{2} resonances.
However, this improvement is still within the range of experimental
uncertainties. Note that the spectra are shown without post-processing,
i.e., chemical-shift scaling and relative referencing. It is interesting to
observe that the whole spectrum shifts to lower parts per million values for the
compensated implementation (Figs. 5a and 6). Based on numerical
simulations, we believe this to be due to the better compensation of the
effective nutation over a full FSLG cycle and the additional removal of
fictitious fields (second-order one-spin terms) by applying transient
compensation (Ernst et al.,
2005; Hellwagner et al., 2017). The effect of changing the effective-field
angle from the magic angle to 60^{∘} is very small and is hard to
judge from the spectra.

Experimental quantification of the relative size of the second- and
third-order terms was implemented by designing and synthesizing glycine
derivatives that contain an isolated two-spin system as well as a multi-spin
system. A deuterated d_{8}-2-^{13}C-^{15}N-glycine ethyl ester (see
Fig. 7a) with a protonated CH_{2} group was synthesized to represent an
isolated ^{1}H-^{1}H spin system. In full analogy, a deuterated
d_{5}-2-^{13}C-^{15}N-glycine ethyl ester with protonated CH_{2}
and
${\mathrm{NH}}_{\mathrm{3}}^{+}$ groups (see Fig. 7a) was used as a multi-spin model
system. Hahn-echo sequences with FSLG-based decoupling during the echo time
were recorded and the ${T}_{\mathrm{2}}^{\prime}$ times were extracted. The oscillations in the
decay curves have been observed before and could, according to the
literature, be removed by a double-echo sequence
(Paruzzo et al., 2018).

It can be seen from Fig. 7 that the influence of the second-order terms is very small and only contributes about 5%–10% of the effective proton ${T}_{\mathrm{2}}^{\prime}$ times. The dominating terms are identified to be the third-order auto term since they make up most of the non-refocusable residual line width when comparing a two-spin to a multi-spin system. Furthermore, the quantification of the decoupling performance leads to the conclusion that the pulse-transient compensation does improve the decoupling efficiency by 20 %–30 % in terms of refocusable line width. Nevertheless, as shown before, this effect is hardly visible in the directly detected spectra, but pulse-transient compensation leads to higher predictability of the sequence.

The huge discrepancy in directly detected line width (∼400 Hz
for the N–H peak of natural abundance glycine; see Fig. 6) and the
refocusable terms (55–80 Hz; see Fig. 7b and c) needs to be explained by
other effects. In order to study the effect of the time-independent part of
the rf-field inhomogeneity under MAS, samples in 2.5 mm o.d. Bruker rotors
were packed by filling different parts of the rotor. We have not included
phase or amplitude modulations generated by MAS due to rf-field
inhomogeneity in our investigation
(Goldman
and Tekely, 2001; Levitt et al., 1988; Tekely and Goldman, 2001; Tosner et
al., 2018). Five rotors each of adamantane and natural abundance glycine
were packed using a full rotor, the upper third (*up*), the bottom third
(*low*), the middle third (*mid*), and a very small part in the middle of the rotor
(*center*). The remaining rotor volume was filled with Teflon spacers. The
distribution of the ^{1}H rf field over the active sample volume was
determined by measuring nutation curves using the adamantane sample with
direct detection of the proton signal. Subsequent Fourier transformation of
the nutation curves yields a distribution profile of the rf fields within
the probe. These profiles are shown in the Supplement (Fig. S7). The maxima of the rf profile are consistently at higher values than the
calibrated rf-field amplitudes of 100 kHz. The calibration was done on a
full rotor by determining the first zero crossing of a *π* pulse. These
shifted maxima are due to the very broad distribution of rf-field amplitudes
in the full rotor and the large drawn-out foot towards low-rf fields. The
profiles of the restricted samples show that this foot is mostly observed in
the outer thirds of the rotor, whereas the middle as well as the center part
are narrowed down around the maximum. The sum of the middle, upper, and lower
parts of the rotor compares very well to the profile of the full rotor.
However, the integral of the distribution is slightly higher by a factor of
1.1, which is due to either spacers that do not cover exactly a third of the
rotor or looser packing in the full sample.

These profiles have been used in further studies to investigate the influence of the rf-field distribution on the decoupling efficiency. Numerical simulations were performed using an eight-spin system with characteristic couplings and shifts similar to the ones found in glycine (spin system details can be found in the Supplement, Tables S1 and S2) under FSLG decoupling using a distribution of rf-field amplitudes as observed in the nutation experiments. These simulations are compared with FSLG experiments that were performed on the restricted natural abundance glycine samples. The comparison of the numerical simulations and experimental results are shown in Fig. 8.

The measured line widths for the simulations and experiments are listed in Table 1. The full width at half maximum (FWHM) obtained from the simulations compares well with the experiments. The relative intensities of the ${\mathrm{NH}}_{\mathrm{3}}^{+}$ peaks for different packing are also reproduced fairly well with small discrepancies for the center-packed and the middle-third rotor. One problem in the experimental spectra is the phase correction as well as the baseline correction. Due to the very broad and drawn-out rf profile, the simulated peak has a large foot to higher chemical shifts. Such an asymmetric peak can be corrected by a zeroth-order phase correction to obtain a more symmetric-looking line shape. This phase correction can lead to a distortion of the relative intensity as well as the extracted line width.

Further data that can be extracted from the simulations are the inherent line width that remains due to insufficient decoupling strength at various rf fields. This can be interpreted and compared with an experimentally determined ${T}_{\mathrm{2}}^{\prime}$. The inherent line width from the numerical simulations was obtained by correcting the respective rf-field values for the chemical-shift scaling. The superimposed line width was fitted and was found to be about 35 Hz for the ${\mathrm{NH}}_{\mathrm{3}}^{+}$ peak of glycine corresponding to ${T}_{\mathrm{2}}^{\prime}=\mathrm{10}$ ms. This compares well to observed values in the literature (Paruzzo et al., 2018).

## 5.1 Material and methods

Numerical simulations were performed using the GAMMA spin-simulation environment (Smith et al., 1994). Different crystallites were simulated with 300 ZCW orientations for powder averaging (Cheng et al., 1973). The single-crystal orientations are specified in the text. The MAS frequency was set to 6.25 kHz for all the simulations and analytical calculations with an effective-field strength of 125 kHz.

The analytical calculations were done in Mathematica by transforming the
dipolar Hamiltonian into a tilted-frame with the effective-field angle.
Then, the rf-field interaction-frame transformation of the Hamiltonian was
calculated by (2*π*)(-2*π*) rotation around the tilted axis. The
effective Hamiltonians were calculated according to Eqs. (11) and (12). The
decomposition of the Hamiltonian was done by projecting the Hamiltonian on
the two- or three-spin spherical tensor operators.

The glycine ethyl ester derivatives were synthesized starting from
2-^{13}C-^{15}N-glycine (purchased from Sigma Aldrich) with
d_{6}-ethanol (anhydr.) by dropwise addition of SOCl_{2} at 0 ^{∘}C. After 2 h under reflux conditions, the reaction mixture was rinsed with
toluene and subsequent removal of the solvent under vacuum led to
d_{8}-2-^{13}C-^{15}N-glycine ethyl ester as a highly crystalline white
powder. In order to synthesize the
d_{5}-2-^{13}C-^{15}N-glycine ethyl ester, the d_{8}-glycine ethyl ester
was treated with methanol in an ultrasonic bath to exchange the amine
protons. Removal of residual solvent under vacuum yielded a white
crystalline powder.

The experiments were all carried out on a 14.1 T magnet (600 MHz proton resonance frequency) on a Bruker Avance III HD spectrometer using a 2.5 mm triple-resonance Bruker probe. The probe was modified with a pickup coil to perform transient compensation and the trap was removed to operate the probe in double-resonance mode. All rotors were completely filled without sample restriction except where stated explicitly. Processing was done in Topspin (Bruker Biospin, Rheinstetten, Germany) and zeroth- and first-order phase corrections were applied manually after Fourier transformation. All spectra were recorded as two-dimensional spectra with protons in the indirect dimension to employ windowless decoupling. The transient compensation was performed as described in Wittmann et al. (2016). The quality of the transient compensation was monitored by measuring the phase and the amplitude of the radio-frequency field using the pickup coil. Transient-compensated pulses showed only minor deviations in phase and amplitude from the intended pulse shape.

In conclusion, we investigated the different contributions to the residual line broadening in FSLG decoupled proton spectra and tried to quantify their magnitude. The most important factor was found to be the rf-field inhomogeneity that contributes to about 75 % of the line width even if the sample is restricted in the center of the rotor. This is a result of a distribution of chemical-shift scaling factors due to different effective-field directions in different parts of the sample. The outer parts of the rotor do not contribute much to the observed spectrum and typically represent themselves as a foot in the peak due to the low-rf fields at the edges of the coil. A further confirmation that the rf-field inhomogeneity is the main source of the residual line width is the fact that the use of higher effective fields does not result in better signal resolution. Second- and third-order error terms scale down linearly or quadratically with the effective-field strength but this was not observed experimentally. The relative rf-field distribution in the coil is always the same independently of the magnitude of the rf field. Therefore, the chemical-shift scaling and the resulting spectra are expected to show little change with an increase in the rf-field amplitude. Since the rf-field inhomogeneity is the main contribution to the residual line width, improved performance is only expected if the probe design is improved so that the rf-field profile is more homogeneous over the whole sample.

We have also shown that pulse transients do not contribute significantly to
the residual line width, but generate a shift of the spectrum, which makes it
more difficult to interpret the results and achieve a reliable frequency
calibration. Removal of phase transients and adaption of the pulse sequence
led to more predictable results in terms of chemical-shift scaling and
smaller variations in the shift on the frequency axis. Furthermore, it was
shown theoretically and by numerical simulations that homonuclear
third-order terms contribute strongly to the residual homogeneous line width.
These terms cannot be removed by altering the sequence, e.g., changing the
angle of the effective field, as they do not exhibit the same spatial
behavior as second-order three-spin terms. Small improvements were found by
changing the effective-field angle to slightly higher values of around
60^{∘}, which can be understood theoretically, but the spectral
quality still remains too broad to be useful for many practical
applications.

Experimental data and simulations are available upon request from the corresponding authors.

The supplement related to this article is available online at: https://doi.org/10.5194/mr-1-13-2020-supplement.

JH, ME, and BM designed the research. MO and DZ synthesized the specifically labeled compounds. LG and JH carried out all measurements and simulations. JH and ME wrote the manuscript and BM edited the manuscript with input from all other authors.

The authors declare that they have no conflict of interest.

We would like to thank Federico M. Paruzzo, Perunthiruthy K. Madhu, and Kaustubh Mote for insightful discussions about theory and experimental implementation of homonuclear decoupling.

This research has been supported by the Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (grant nos. 200020_169879, 200020_159707, and 200020_188988), the Eidgenössische Technische Hochschule Zürich (grant no. ETH-10 15-1), and the European Research Council (grant no. FASTER (741863)).

This paper was edited by Kay Saalwächter and reviewed by Malcolm Levitt and one anonymous referee.

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