¹⁷O NMR relaxation measurements for investigation of molecular dynamics in static solids using sodium nitrate as a model compound

Abstract

¹⁷O NMR methods are emerging as a powerful tool for determination of structure and dynamics in materials and biological solids. We present experimental and theoretical frameworks for measurements of ¹⁷O NMR relaxation times in static solids focusing on the excitation of the central transition of the ¹⁷O spin 5/2 system. We employ ¹⁷O-enriched NaNO₃ as a model compound, in which the nitrate oxygen atoms undergo 3-fold jumps. Rotating frame $\left(T_{1\rho }\right)$, transverse $\left(T_2\right)$ and longitudinal $\left(T_1\right)$ relaxation times as well as line shapes were measured for the central transition in the 280 to 195 K temperature range at 14.1 and 18.8 T field strengths. We conduct experimental and theoretical comparison between different relaxation methods and demonstrate the advantage of combining data from multiple relaxation time and line shape measurements to obtain a more accurate determination of the dynamics as compared to either of the techniques alone. The computational framework for relaxation of spin 5/2 nuclei is developed using the numerical integration of the Liouville – von Neumann equation.

Graphical Abstract

Introduction

Advancement in method development of ¹⁷O NMR led to significant progress in structural and dynamics studies of materials and biomolecules[1,2,3,4,5,6,7,8]. ¹⁷O has a spin 5/2 quadrupolar nucleus with a wide range of quadrupolar coupling constant $\left(C_{\text{q}}\right)$ values possible, up to tens of MHz. Both first order and second order quadrupolar interaction govern its spin physics, but the central transition (CT), i.e. the transition between the $m_z=+1/2$ and $m_z=-1/2$ states, is not affected by the first order contribution. Thus, it yields relatively narrow line shapes and most works have focused on detecting this transition[2].

¹⁷O nuclei are sensitive dynamics probes and distortions in quadrupolar line shapes induced by motions can be used to quantify the motions[2,9,10,11]. In addition, relaxation measurements can provide useful complementary information on a variety of time scales. We are particularly interested in the relaxation in the solid phase due to potential applications of relaxation approaches to a variety of non-soluble compounds, including biomolecular solids. Recent advances in ¹⁷O labeling approaches will likely enable numerous future applications of ¹⁷O solid state NMR relaxation methods. Examples of future application to biomolecular solids can include dynamics studies of backbone and side-chain motions probed at the carbonyl oxygen nuclei in proteins, as well as the dynamics of hydration shells.

The goal of this work is to compare dynamics information from several ¹⁷O NMR relaxation times measurements, such as rotating frame ($T_{1\rho }$), transverse $\left(T_2\right)$ and longitudinal $\left(T_1\right)$ as well as line shapes analysis and to assess whether the combined approach including all measurements can increase the accuracy of the molecular dynamics information in solids.

Quadrupolar relaxation in the solid state is a challenging topic from both experimental and modeling perspective. For the spin 5/2 nuclei, the base set spans 35 coherences and the relaxation, in general, is inherently non-exponential [12,13]. As a result, the initial conditions can heavily influence the apparent relaxation times, as shown by Haase et al. for longitudinal $T_1$ relaxation times[14]. Coherent evolution of half-integer quadrupoles under spin-lock has been considered in detail by Wimperis and co-workers[15,16]. Additionally, the laboratories of Werbelow and Veeman have presented an extensive body of work describing the theory of the nutation experiment and relaxation of multiple quantum coherences for half-integer nuclei[17,18]. Approaches based on the numerical solutions of the Liouville – von Neumann equation can provide means of parametrizing experimentally observed dynamics[19]. Spin-locking behaviors of quadrupolar nuclei were previously utilized in the design of several pulse sequences geared toward structural studies[20,21,22].

The model compound used here, ¹⁷O-labeled NaNO₃, was recently studied in detail by line shape measurements by Hung and coworkers[9] as well as by Beerwerth et al.[23]. The main motions are the 3-fold jumps of the nitrate group. Two motional transitions were seen in this compound, including the one at which the rate of motions matches the second order quadrupolar interaction. The dynamics were analyzed using the full Liouvillian approach by Hung et al.[9]. In addition to the quadrupolar interaction, the chemical shift anisotropy (CSA) interaction has to be taken into account.

We first present theoretical consideration of the ¹⁷O laboratory and rotating frame relaxation and examples of the 2-site jump simulations (Figure 1A) in solids using the Liouvillian approach. These simulations are performed for an axially symmetric quadrupolar (EFG) tensor with an arbitrary value of the quadrupolar coupling constant of 10 MHz and the jump angle of 104.5°. A relevant study using a similar 2-site jump model was performed by Dai et al. using ¹⁷O-enriched NaNO₂[24]. The theoretical description is followed by the experimental section on the relaxation rates and line shape measurements in the powder-state of NaNO₃. The geometry of the EFG and CSA tensors in NaNO₃, the notation for the tensor components as well as their values as determined in reference [9], are illustrated in Figure 1B.

Figure 1.

A) A schematic representation of an axially symmetric quadrupolar tensor undergoing a 2-site orientational exchange process with the exchange constant $k_{\text{ex}}$. In the theory section we use the jump angle of 104.5°, $V_{zz}=C_q=10\text{MHz}$ and $\eta =\frac{V_{yy}-V_{xx}}{V_{zz}}=0$. B) Ball-and-stick representation of the ${{\text{NO}}_3}^{-}$ ion, in which the oxygen atoms undergo the 3-fold jumps with the rate constant $k_{\text{ex}}$. The quadrupolar and CSA tensor orientations in their corresponding principal axes systems are shown explicitly, such that their major axes are aligned and pointing along the N-O bond. $V_{zz}=C_q=12.5\text{MHz},\eta =0.8,\left[{\delta }_{11},{\delta }_{22},{\delta }_{33}\right]=\left[\mathrm{250,400,550}\right]\text{ppm}$.

To our knowledge, ¹⁷O $T_{1\rho }$ relaxation times measurements have not yet been implemented to studies of molecular dynamics in solids, and we present in detail experimental and computational approaches for this technique. The simulations protocol to generate ¹⁷O NMR laboratory and rotating frame relaxation rates for solids undergoing stochastic jumps is presented and is generalizable in principle to any type of molecular motions in the solid state. Experimental $T_{1\rho },T_1$, and $T_2$ times as well as the line shape data are obtained in 280 to 195 K temperature range, and the rate constants of 3-fold jumps are obtained for all series. We then obtain the Arrhenius activation energy from the combined data set.

Results and Discussion

I. Theoretical and computational description of the ¹⁷O NMR relaxation in powders

Hamiltonian description for spin 5/2 system

In the presence of a static magnetic field and a radiofrequency field applied in the transverse plane the Hamiltonian for a single ¹⁷O nuclear spin $(I=5/2)$ in a frame rotating with Larmor frequency can be written as

$$H=H_Q+H_{RF}+H_{CS},$$ (1)

where $H_{RF}={\omega }_{RF}{I}_x$ is the spin-locking interaction applied on-resonance and $H_{CS}={\omega }_0{\sigma }_{zz}{I}_z$ is the chemical shift interaction. The secular part of the quadrupolar interaction tensor $H_Q$ includes the first (Q1) and second order (Q2) terms in the average Hamiltonian perturbation theory

$$H_Q=H_{Q,1}+H_{Q,2},$$ (2)

with the following definitions, using the notation and the choice of tensor definitions given in[25].

$$H_{Q,1}=\frac{eQ}{I(2I-1)\hslash }\sqrt{\frac{3}{8}}\left[I_Z^2-I(I+1)/3\right]V_0$$ (3)

$$H_{Q,2}=\frac{1}{{\omega }_0}{\left(\frac{eQ}{I(2I-1)\hslash }\right)}^2{I}_z\left(\left[I_z^2-I(I+1)/2+1/8\right]V_1{V}_{-1}+\left[I_z^2-I(I+1)+1/2\right]V_2{V}_{-2}/4\right)$$ (4)

${\omega }_0$ is the Larmor frequency and $V_i$ are the spherical components of the EFG tensor. The EFG tensor is defined in its principal axis system (PAS) as

$${\left(V_2,V_1,V_0,V_{-1},V_{-2}\right)}^{PAS}=eq(1/2,0,\sqrt{3/2},\mathrm{0,1}/2)$$ (5)

and then is transformed to the laboratory frame by the Wigner matrices. The overall strength of the first order quadrupolar interaction is given by the quadrupolar constant $C_q=\frac{e^{2}qQ}{h}$ and, for a particular orientation of the EFG tensor in the laboratory frame, by

$${\omega }_q=\frac{3}{4}\frac{2\pi C_q}{I(2I-1)}\left[\frac{3{\text{cos}}^2\theta -1}{2}+\frac{\eta }{2}{\text{sin}}^2\theta \text{cos}2\phi \right],$$ (6)

where $\theta$ and $\phi$ are polar and azimuthal angles of the transformations respectively[13].

Simulations for large-angle 2-site jumps motions

The coherent spin evolution described by the Hamiltonian of Eq. (1) can be represented in the form of a system of linear differential equations acting on the density matrix $\rho$ of spin $I$. The matrix of such a system is the Liouville operator, $L$. The changing magnetic environment of a spin can be described by the exchange between different orientations of EFG and CSA tensors. They correspond to different orientations of a given molecule within the laboratory frame. Therefore, the spin state of the system can be described by a direct product of density matrices for individual spin states and vectors describing the exchanging molecular orientations. The exchange between different orientations can be represented by a Markovian exchange matrix. We thus arrive at the general form of the Liouville – von Neumann equation for the combined density matrix

$$\frac{d{\rho }_{i,m}}{dt}=i{L}_{ij}^{\left(m\right)}{\rho }_{j,m}+K_{mn}{\rho }_{i,n}$$ (7)

where the indices $i$ and $j$ refer to the spin degrees of freedom and $m$ and $n$ to the orientational (spatial) degrees of freedom. The Markov exchange process leads to relaxation. The details of the inclusion of the exchange matrix are given in the “Materials and Methods” section.

In general, the density matrix for spin $I=5/2$ can be represented in the basis of 35 basic coherences., listed in Eq. (S1). Laboratory frame relaxation for half-integer quadrupolar nuclei has been considered in detail in the past and shown to be non-exponential in general, and in particular for powder solids[12,13,14,27,28,29]. There are three populations involved in the longitudinal $R_1$ relaxation. They correspond to three possible absolute values of magnetic number $\left|m_z\right|$: the central transition $I_Z\left(CT\right)$, the first satellite transition $I_z\left(ST1\right)$, and the second satellite transition $I_z\left(ST2\right)$. The longitudinal relaxation processes cause interconversions between these three coherences, creating non-exponential magnetization recovery/build-up curves. For the transverse magnetization with no spin-locking field applied and neglecting the CSA interaction, there are again three coherences corresponding to $\left|m_z\right|\leftrightarrow \left|m_z\right|-1$ transitions $I_x\left(CT\right)$, $I_x\left(ST1\right)$, and $I_x\left(ST2\right)$ defined in Eq. (S4). The relaxation process is non-exponential as well, due to the interconversions between these coherences.

Though inherently non-exponential, the relaxation decay and build up curves can be phenomenologically treated as nearly exponential in many situations. Precise characterization of nonexponentiality of the relaxation curves is difficult, unless there is a significant contribution from the components which differ in their relaxation rates by a factor of 5 or more. Because the focus of this work is not to study non-exponentiality of the relaxation itself, but to advance the methods of extracting dynamic parameters from relaxation experiments, we fit both experimental and simulated relaxation build up and decay curves with a monoexponential function with a baseline, $M\left(t\right)=M_0{e}^{-t/T}+B$. The baseline effectively removes contributions of very slowly relaxing components, while $T$ gives a good approximation to the values of $T_1,T_2$, and $T_{1\rho }$ for the respective experiments. This approximation should not lead to a bias in the fitted exchange rates when relaxation delay times used in simulations and experiments are chosen to be the same.

Rotating frame relaxation

Relaxation of half-integer quadrupolar nuclei’s transverse magnetization in the presence of a spin-lock field is less discussed in the literature and we will focus on it first. We will then compare its key features to the $R_1$ and $R_2$ experiments. We will first consider the coherent evolution of the density operator in the absence of any exchange or any other sources of relaxation. As the initial condition we take $\rho \left(0\right)$ corresponding to the single quantum transverse magnetization and select the spin-locking RF field strength in the interval $\frac{1}{{\omega }_0}{\left(\frac{2\pi c_q}{I(2I-1)}\right)}^2\ll {\omega }_{RF}\ll \frac{2\pi c_q}{I(2I-1)}$, such that only the central transition is effectively locked for the majority of the orientations, aside from the case when the polar angle $\theta$ matches the magic angle at which the first order quadrupolar interaction vanishes. This is the special case of a more general RF field range considered for the coherent evolution under spin-lock by Wimperis et al.[16]. The quadrupolar interaction induces mixing of the initial $I_x\left(CT\right)$ coherence with other basis coherences, including those corresponding to the satellite transitions. The locking efficiency is orientation-dependent because of the variations in the strengths of the first and second order quadrupolar interactions.

Figure 2A shows the projection coefficient of single quantum coherence $I_x\left(CT\right)$ on the subspace of locked coherences versus the quadrupolar frequency ${\omega }_q$ for a selected set of parameters (an axially symmetric EFG tensor with $C_q=10\text{MHz},\eta =0,{\omega }_0/2\pi =54.2\text{MHz},{\omega }_{RF}/2\pi =35\text{kHz}$) in the absence of motions at $t=\infty$. We also show individual calculations when only the first order or the second order quadrupolar interactions are taken into account. Note that the second order interaction, which is proportional to ${\omega }_q^2/{\omega }_0$, reduces the locking efficiency, as noted by Wimperis and coworkers[16]. Figure 2B shows three other time points in the 10 to $80\mu \text{s}$ range to demonstrate the time-dependence of coherent oscillations due to the oscillatory component of $I_x\left(CT\right)$ arising as the result of interconversions between other elements of the basis (a total of 20 coherences listed in Eq. S2). Figure S1 shows full time-dependent curves for selected crystallite orientations.

Figure 2.

Simulated transverse magnetization $I_x\left(CT\right)$, corresponding to the central transition of ¹⁷O spin-5/2 system with an axially symmetric EFG tensor with $C_{\text{q}}=10\text{MHz}$ under the spin-lock field of 35 kHz and 9.4 T magnetic field, as a function of orientation-dependent frequency ${\omega }_q$ under static (non-spinning) conditions. A) The locked component in the absence of motions for both the first and second order quadrupolar interactions Q₁+Q₂ (solid line), or the first order quadrupolar interaction only (black dashed line), or the second order quadrupolar interaction only (blue dashed line). B) After coherent evolution under the full quadrupolar interaction, with the evolution times shown directly on the panel. The black line corresponds to the locked component. C) Including 2-site jumps of the EFG tensor principal axis, using $k_{ex}={10}^5{\text{s}}^{-1}$ and the jump angle of 104.5°, averaged over the azimuthal angle of the second site. Shown for evolution times corresponding to fast (0.082 ms) and slow (2.2 ms) relaxation times of the magnetization decay curve, and averaged over crystallite orientations. The black line corresponds to the locked component of the coherent evolution. D) Including the 2-site jumps as in C) for the $k_{ex}$ and evolution time values shown on the panel. The normalization is performed to the maximum value of $⟨I_x\left(CT\right)⟩$ for each individual curve. The evolution times are chosen to approximate $T_{1\rho }$ times, using the single-exponential fits of the powder-averaged magnetization decay curves. For panels B)-D) the values of $I_x\left(CT\right)$, excluding the locked coherent lines, are averaged over the ±20 kHz intervals around the values ${\omega }_q$ shown on the $x$-axes, to increase smoothness. Simulations are based on 17711 crystallite orientations.

In the absence of motions, the locking efficiency for the case of Figure 2A ranges from about 30% for small ${\omega }_q$ values and reaches up to 100% for large values of ${\omega }_q$. The low efficiency for small values of ${\omega }_q$ is governed by dephasing due to Q2 term, which is the dominant term for small values of ${\omega }_q$. On the other hand, for large values of ${\omega }_q$ the Q2 interaction is less efficient than Q1, which induces locking of non-$I_x$ coherences during the spin-lock period.

Figure 2C shows an analogous situation in the presence of large-angle 2-site jumps with equal populations, the jump angle of 104.5° and the rate constant of $1\cdot {10}^5{s}^{-1}$. In the presence of motions, the results are shown for the time points 0.082 and 2.2 ms, corresponding to the apparent bi-exponential fit of the simulated magnetization decay curve (averaged over the powder pattern and shown in Figure S2). The $x$-axis represents the value of ${\omega }_q$ for one of the two exchanging site orientations, while the values of time-evolved $I_x\left(CT\right)$ coherence shown on the vertical axis are averaged over all possible orientations of the second site. Crystallite orientations with values of ${\omega }_q$ within the ± 20 kHz range were averaged, in order to decrease scatter in the data. Aside from the overall loss of intensity due to relaxation, one can observe a clear broadening of the pattern (Figure 2C blue and red lines, shown for $k_{ex}=1\cdot {10}^5{s}^{-1}$) compared to the one in the absence of motion (black line), indicative of mixing of the crystallite orientations due to motions. To probe the effect of $k_{ex}$ magnitude on this pattern (Figure 2D), we select the snapshots of the projected $I_x\left(CT\right)$ approximately corresponding to the value of $T_{1\rho }$ taken from the fast component (i.e., the initial decay) of magnetization decay curves. The patterns are normalized to the maximum of each curve, which visually de-emphasizes the effect of the reduction in the projected $I_x\left(CT\right)$ intensity and displays the distortions in the overall patterns. In the $k_{ex}$ range of 10² to ${10}^6{s}^{-1}$ the patterns are similar, but in the range between 10⁷ to ${10}^{10}{s}^{-1}$ the broadening becomes even more pronounced, such as the distinctions between different crystallite orientations are further blurred. This indicates the motional regime in which the Q1 contribution is substantially averaged. One notable exception is the case when one of the exchanging orientations is parallel to the magnetic field (high ${\omega }_q$ limit), for which the distinction remains.

To further define the dependence of rotating frame relaxation rates $R_{1\rho }$ as a function of $k_{ex}$, we first examine $R_{1\rho }$ rates for two selected initial crystallite orientations, 0 and 90° with respect to the magnetic field (Figure 3A). The orientations of the exchanging sites, constrained by the 104.5° jump angle, are averaged in the $R_{1\rho }$ rate calculations. A single exponential fit is assumed for this analysis. The $R_{1\rho }$ versus $k_{ex}$ pattern has two maxima, the positions of which are determined roughly by the conditions $k_{ex}={\omega }_q$ and $k_{ex}={\omega }_0$. The dependence on ${\omega }_0$ is considered in more detail in Figure S3, in which we calculate $R_{1\rho }$ rates for three crystallite orientations for 9.4 T and 5·9.4 T static magnetic field strength. The field-dependence is most pronounced when ${\omega }_q<k_{ex}<{\omega }_0$ and is also evident for low values of $k_{ex}$. The extent of field dependence varies with the crystallite orientation.

Figure 3.

Simulated $R_{1\rho },R_1$ and $R_2$ NMR relaxation rates corresponding to the central transition of ¹⁷O spin-5/2 system with an axially symmetric EFG tensor with $C_{\text{q}}=10\text{MHz}$ under the spin-lock field of 35 kHz as a function of $k_{ex}$ using the model of the 2-site jumps with the 104.5° jump angle and under static conditions. A) $R_{1\rho }$ rates for crystallite orientations with the initial principal axis of the quadrupolar tensor aligned either at 0° (black line) or 90° (red line) relative to the static magnetic field of 9.4 T. The vertical solid lines show the values of $\frac{2\pi C_q}{I(2I-1)}$ (green line) or the Larmor frequency ${\omega }_0$ (blue line). B) Powder-averaged $R_{1\rho }$ rates at two values of the magnetic field strengths, 9.4 (solid line) and 5·9.4 (dashed line). The additional vertical dashed line stands for the Larmor frequency at 5·9.4 T. C) Powder-averaged $R_{1\rho }$ (black line), $R_1$ (blue line) and $R_2$ (magenta line) rates at 9.4 T magnetic field strength. The additional orange vertical line corresponds to $\frac{1}{{\omega }_0}{\left(\frac{2\pi C_q}{I(2I-1)}\right)}^2$. The qualitative definitions of the motional regimes are shown under the graph. All graphs are on the log-log scale. The single exponential approximation was used for all fits of the relaxation rates. 610 crystallite orientations were employed in simulations of the powder-averaged values. The initial conditions are: CT with saturated ST1 and ST2 for $R_{1\rho }$ and $R_2$; inversion of CT only for $R_1$ with ST1 and ST2 at equilibrium.

Similar results are obtained when averaging over the entire powder pattern is considered and are shown in Figure 3B for the case of $C_q=10\text{MHz}$. The case of a much smaller values of $C_{\text{q}}$ is considered in Figures S4 and S5.

We now define the motional regimes (see also Figure 3C, bottom panel), using the magnitudes of powder-averaged first and second order quadrupolar interactions as well the Larmor frequency as markers. Thus, $k_{ex}\ll \frac{1}{{\omega }_0}{\left(\frac{2\pi C_q}{I(2I-1)}\right)}^2$ defines the ultraslow regime, $k_{ex}~\frac{1}{{\omega }_0}{\left(\frac{2\pi C_q}{I(2I-1)}\right)}^2$ defines the slow regime, $\frac{1}{{\omega }_0}{\left(\frac{2\pi C_q}{I(2I-1)}\right)}^2\ll k_{ex}~\frac{2\pi C_q}{I(2I-1)}\ll {\omega }_0$ is the intermediate regime, $\frac{2\pi C_q}{I(2I-1)}\ll k_{ex}~{\omega }_0$ defines the fast regime, and $k_{ex}\gg {\omega }_0$ is the ultrafast regime.

As the theoretical and computational approaches for spin ½ nuclei undergoing $R_{1\rho }$ relaxation are well developed, it is instructive to examine to what extent quadrupolar ¹⁷O CT $R_{1\rho }$ relaxation can be treated within this formalism. In this approach, the role of ST coherences is not taken into account, as well as the role of Q1 interaction which does not affect the CT transition.

Miloushev and Palmer[30] derived an analytical expression for $R_{1\rho }$ relaxation for the spin ½ nuclei undergoing isotropic chemical shift modulation within the 2-site jumps model, using the Bloch–McConnell formalism for the case of equal populations of the two sites (Eq. 33 in reference [30]). To adapt this approach to quadrupolar relaxation, we note that the modulating interaction becomes the Q2 term instead of the isotropic chemical shift interaction (Eq. S6). SI2 and Figure S6 present the analytical treatment and its comparison with the computation according to the full Liouvillian approach, either in the basis of the fictious spin ½ nucleus or in the full basis of 35 coherences of the spin 5/2 nucleus and using the full quadrupolar Hamiltonian. We conclude that the analytical expression in general is not applicable outside of the ultrafast regime. The largest deviations are seen in the intermediate regime, which is dominated by the Q1 contribution in the full treatment. In the slow and ultraslow regimes, the differences are less significant but are complex and reflect the relative magnitudes of the quadrupolar coupling constant and the Larmor frequency.

Comparison of information provided by ¹⁷O CT $R_2,R_1$, and ${\mathit{R}}_{1\mathit{\rho }}$ relaxation rates.

In the fast and ultrafast exchange limits, all three relaxation measurements, $R_1,R_2$, and $R_{1\rho }$ provide essentially the same information (Figure 3C) due to the dominance of non-secular quadrupolar interaction terms. However, in the intermediate, slow, and ultraslow regimes the behavior of these rates can be different. The behavior of $R_1$ is the most straightforward and is always dominated by the non-secular terms of the quadrupolar interaction. For the $R_2$ and $R_{1\rho }$ rates in the intermediate and slow regimes contributions from both secular and non-secular terms come into play. For the $R_{1\rho }$ relaxation the Q1 contribution dominates in these regimes (green line in Figure 3C), which is manifested as the plateau in $R_{1\rho }$ versus $k_{ex}$ dependence. At higher $k_{ex}$ values the Q1 contribution is effectively averaged between the two orientations and has a negligible effect on the relaxation. For the $R_2$ rate, it is the Q2 contribution that becomes dominant in the slow regime, which manifests as the peak at $k_{ex}=\frac{1}{{\omega }_0}{\left(\frac{2\pi C_q}{I(2I-1)}\right)}^2$ (orange line in Figure 3C). For higher $k_{ex}$ values the Q2 contribution is averaged.

Because of the non-monotonous dependence of the $R_2$ rate on $k_{ex}$ in the region given by $\frac{1}{{\omega }_0}{\left(\frac{2\pi C_q}{I(2I-1)}\right)}^2<k_{ex}<\frac{2\pi C_q}{I(2I-1)}$, the $R_{1\rho }$ measurements can be more reliable in defining the value of $k_{ex}$. One can also probe the dependence of $R_{1\rho }$ rates on the spin-locking field strength, which is well known to provide additional detailed characterization of the exchange processes [31]. An example of such relaxation dispersion behavior is shown in Figure S7 for the 2-site large-angle jumps model.

It is interesting to note the absence of the field dependence in the $R_2$ rates in the ultraslow regime (Figure S8). In contrast, the $R_{1\rho }$ rates can retain field dependence, but the effect can be reduced for small $C_{\text{q}}$ values (Figure 3B, S4, and S5).

Figure S9 demonstrates the comparative behavior of $R_{1\rho }$ and $R_2$ rates for the 2-site jumps with either 10° or 104.5° jump angle. It shows that the comparison of the two measurements can distinguish between small and large angle jumps in all regimes other than ultrafast.

All relaxation rates are sensitive to the values of $C_q$, except for the $R_2$ rate in the ultraslow limit (Figure S10). However, the effect of different $C_{\text{q}}$ values on the relaxation rates cannot be disentangled from the effect of different $k_{ex}$ values in the case of the $R_1$ rates, as well as for the $R_{1\rho }$ and $R_2$ rates in the ultrafast and fast limits. In the intermediate and slow regimes, the temperature dependence of $k_{\text{ex}}$ rate can, in principle, allow for the separation of the effects of $C_{\text{q}}$ and $k_{ex}$. The sensitivities of the $R_2$ and $R_{1\rho }$ rates to $C_{\text{q}}$ values are expected to be similar but will depend on the details of the model. In the ultraslow limit the line shapes are likely to yield the most detailed information on the quadrupolar tensor parameters due to their direct dependence on ${\omega }_q$ [12].

Due to the non-exponential nature of the quadrupolar relaxation, the effect of the initial conditions should be considered, as noted in a number of prior works[12,14,29,32]. The apparent ¹⁷O CT longitudinal relaxation rates can vary by a factor of 12 for the 2-site jumps model (Figure S11). For the CT $R_1$ relaxation the cleanest measurement appears to be the inversion recovery with both ST are at equilibrium, provided that the inversion pulse does not invert ST1. In contrast, ¹⁷O CT $R_2$ rates do not depend on whether the initial conditions for ST is in the saturated or equilibrium sate. ¹⁷O CT $R_{1\rho }$ rates are largely independent of the initial conditions when the phase cycle of Vega[33] is employed, which cancels the equilibrium population states. The details are discussed in SI3.

In summary, for the purposes of mechanistic interpretation the addition of $T_{1\rho }$ experiment is expected to be the most valuable in the intermediate to slow motional regimes. In this regime the motions are likely to be too slow for a reliable determination of the rate constant from the line shape experiments. Further, in this regime the $T_2$ relaxation times could be difficult to interpret because of non-monotonous dependence on the exchange rate, or even difficult to obtain experimentally because of their small values. Fits of the rate constant from the $T_1$ relaxation times in isolation requires postulating the exact model when the motions are much slower than the Larmor frequency, as well as precise control of the initial conditions.

II. Experimental implementation of ¹⁷O NMR relaxation times measurements

In the weak field limit when ${\omega }_{RF}\ll \frac{2\pi C_q}{I(2I-1)}$, the central transition nutates with the frequency of $3{\omega }_{RF}$. [34] Thus, the maximum excitation is achieved with the 30° pulse[16,34], while the inversion or refocusing can be accomplished with the 60° pulse. If a pure excitation of the CT transition is desired, the excitation pulse has to be selective enough to exclude all crystallites of the first satellite transition. The pulse bandwidth can be estimated by the expression $2.8/\pi {\tau }_p$, where ${\tau }_{\text{p}}$ is the pulse length[12]. A saturated state of the ST1 can be achieved using selective off-resonance irradiation centered at the ST1. We used the sequence consisting of 10 pairs of Gaussian pulses with alternating positive and negative offsets referred to as the “RAPT block” below (Figure 4), with the details listed in Materials and Methods section.

Figure 4.

The pulse sequence for measurements of ¹⁷O NMR CT $T_{1\rho }$ relaxation times: after the inter-scan delay $d_1$, the (optional) RAPT suppression of satellite transitions consisting of loops of off-resonance Gaussian pulses[38] applied at the offsets of ±300 kHz is employed, followed by the 30° excitation pulse ($2\mu \text{s}$ at 41.7 kHz), and then by the spin-lock period of the variable delay $\text{SL}\left({\tau }_D\right)$, applied at the center of the spectrum (on-resonance) with the RF field amplitude of 35 kHz. The FID collection immediately proceeds the spin-lock period. The phase cycle of Vega is employed[33].

The detection scheme for the wide powder pattern of ¹⁷O nuclei with the relatively low value of the gyromagnetic ratio needs to consider minimization of the acoustical ringing effects. This can be achieved either by the traditional Hahn echo scheme[35], or the recently developed TRIP sequence[36], consisting of three phase-cycled 30° pulses. The former scheme decreases signal intensity due to $T_2$ losses.

The CT line shapes were collected using the Hahn echo pulse sequence (${\tau }_{\text{p}}-\Delta -2\cdot {\tau }_{\text{p}}-\Delta$) with the echo delay $\Delta =25\mu \text{s}$ and ${\tau }_{\text{p}}=2\mu \text{s}$ at 41.7 kHz RF field, yielding the excitation bandwidth of 450 kHz. The spectral window was 400 kHz at temperatures above 245 K and 800 kHz at 235 K and below. The partial excitation of the ST1 enters into the line shapes as a raised baseline. For consistency all line shapes presented in Figure 5 are those collected with the Hahn echo scheme. The details are described in Figure S12 and its legend.

Figure 5.

Normalized experimental (left) and simulated (right) ¹⁷O line shapes of NaNO₃ at 14.1 T under static conditions. The Hahn echo detection scheme with the $25\mu \text{s}$ echo delay was used. Simulations included explicit 3-fold jumps of the ${{\text{NO}}_3}^{-}$ group using the tensor parameters described in the text and demonstrated in Figure 1.

As elaborated in SI3 and Figure S11, the choice of initial conditions in general affects the resulting relaxation times. The RAPT block preceding excitation of the transverse coherence helps in establishing a purer CT excitation, with less dependence on the excitation bandwidth. For the longitudinal coherence, the inversion recovery sequence without pre-saturation of ST1 is expected to yield the most accurate relaxation times measurements, which are least dependent on the extent of inversion and pre-saturation of ST1. For motions slower than the Larmor frequency, it is also expected to yield the shortest $T_1$ values. We, thus, choose it as our main approach for the temperature-dependence collection of $T_1$ times. At selected temperature points we also conduct a version of the $T_1$ saturation recovery measurements for $T_1$ times measurements to probe the theoretical expectations regarding the dependence of the relaxation times on the initial conditions. Pulse imperfection and bandwidth consideration can also be included directly into the simulations routine when needed.

$T_2$ relaxation times were measured with the Hahn echo pulse sequence preceded by the RAPT block ($\text{RAPT}-{\tau }_{\text{p}}-{\tau }_D/2-2\cdot {\tau }_{\text{p}}-{\tau }_D/2$) with ${\tau }_{\text{p}}=2\mu \text{s}$ (corresponding to 30° pulse at 41.7 kHz RF field) and a variable relaxation delay ${\tau }_D$. We use the notation of ${\tau }_D$ for discrete relaxation delays in the experiments, while the continuous time variable has the notation of $t.T_1$ times across the entire temperature range were obtained using the inversion recovery scheme ($2\cdot {\tau }_{\text{p}}-{\tau }_D-\text{detect}$) with ${\tau }_{\text{p}}=2\mu \text{s}$ as above. The detection scheme was chosen as TRIP to avoid $T_2$-related intensity losses. The measurements were repeated at two low temperatures with the echo detection scheme to confirm the results.

Rotating frame relaxation must involve a spin-locking field of sufficient strength to lock the desired interaction. For the central transition of the ¹⁷O nuclei the most efficient locking is expected for the spin-locking field strength of about 30 kHz (Figure S13, shown for 14.1 T, $C_{\text{q}}=12.5\text{MHz}$ and $\eta =0.8$). The locking efficiency is diminished for smaller values of the spin-locking field strengths below about 15 kHz, for which the second order quadrupolar interaction provides an effective exchange with coherences other than transverse magnetization. The locking efficiency then becomes optimal in the 25 to 60 kHz range, and slowly diminishes for much higher spin-locking fields due to introduction of non-central transitions into the locked transverse coherence. We employed the spin-lock RF field strength of 35 kHz, applied “on-resonance”, approximately in the middle of the CT line at 283 ppm for all temperatures. The phase cycle of Vega was employed[33], implemented by alternating the phases of the excitation pulse and the receiver simultaneously by 180°. Another potential modification of the $T_{1\rho }$ experiment can include the heat compensation block to account for the differences in RF-induced heating as a function of the variable relaxation delay [37]. In the case of NaNO₃ it was not needed but can become important if the method is applied to hydrated and/or biological samples. The main pulse sequence employed for the collection of the temperature-dependence of $T_{1\rho }$ times is shown in Figure 4. It involves saturation of ST1 for the suppression of the residual ST1 transverse magnetization using the RAPT block, and ${\tau }_{\text{p}}=2\mu \text{s}$ excitation pulse at 41.7 kHz RF field. The omission of the RAPT block is expected to have an effect only in the intermediate to fast motional regimes (see SI3). The detection immediately precedes the spin-lock period except for the lowest temperature, in which significant spectral distortions necessitated the use of the echo detection scheme.

III. Relaxation times and line shapes in the 280 to 195 K temperature range.

The relaxation times and line shapes (Figures 5, 6 and Figure S14) were measured in the 280 to 195 K temperature range at 14.1 T. The limits of the range were defined by the limits of feasibility of the $T_{1\rho }$ magnitudes measurement. $T_{1\rho }$ times lie between $9\mu \text{s}$ and 5.5 ms in this temperature range. Above 280 K $T_{1\rho }$ relaxation times are expected to be too short to be measured reliably, and below 195 K they are expected to be too long and would require relaxation delay times beyond the RF limit of the probe, which was about 20–25 ms using the low-E static probe [39] for the spin-lock field of 35 kHz. The single-exponential approximation was adequate to fit all of magnetization decay/build up curves (Figure S14). The fit utilized the version with the baseline, $M\left(t\right)=M_0{e}^{-t/T}+B$. The presence of baseline is due to inherently non-exponential nature of relaxation in solid state., Contributions from the orientations for which the decay is very slow cannot be reliably included in the overall single exponential decay fit but could manifest themselves as a baseline. These contributions didn’t exceed 10%. The resulting relaxation times are shown in Figure 6.

Figure 6.

A) Semilog plots of the ¹⁷O NMR relaxation times $T_1$ (blue circles), $T_{1\rho }$ (black circles) at 35 kHz spin-locking field, $T_2$ (magenta triangles) versus $1000/T$ collected for NaNO₃ at 14.4 T using the single-exponential fits. The experimental schemes are the following: $T_1$- inversion recovery of the CT line with the $4\mu \text{s}$ inversion pulse at 41.7 kHz RF, $T_{1\rho }$- using of the pulse sequence of Figure 4, the spin-lock field of 35 kHz, presaturation of ST1, and the $2\mu \text{s}$ excitation pulse at 41.7kHz RF, $T_2$- Hahn echo with the presaturation of ST1 and using the $2\mu \text{s}$ excitation pulse and $4\mu \text{s}$ refocusing pulse at 41.7 kHz RF. B) Simulated $T_1,T_{1\rho },T_2$ relaxation times according to the 3-site jumps model of Figure 1B versus $k_{ex}$ and the matching experimental points.

We have also probed the dependence of $T_1$ and $T_{1\rho }$ times on the initial conditions (SI3). As expected based on the theoretical calculations (Figure S11), a significant effect was observed for the $T_1$ times, but no differences were seen for the $T_{1\rho }$ times.

IV. Comparison of the rate constants obtained from ¹⁷O relaxation and line shape measurements

We used the quadrupolar and CSA tensor parameters reported in Hung et al.[9]: $C_q=12.5\text{MHz},\eta =0.8$ for the EFG tensor and $\left[{\delta }_{11},{\delta }_{22},{\delta }_{33}\right]=\left[\mathrm{250\; 400\; 550}\right]\text{ppm}$ for the CSA tensor, with relative orientation [0, 90°, 90°] of the two PAS frames. The definition of the tensor components and their relative orientations are shown in Figure 1.

The line shapes are consistent with the previously determined tensor parameters and yield the values of $k_{ex}$ shown in Figure 7 and the temperature-dependent isotropic chemical shift shown in Figure S15. There is no perfect fit of all features between the experiment and the simulations, probably due to experimental imperfections such as excitation bandwidth consideration [23], etc. In general, the sensitivity of the line shapes to the value of $k_{ex}$ appears to be lower than the ones resulting from the relaxation rates, and we state the values corresponding to the best visual comparison without the estimate of error bars.

Figure 7.

Fitted values of $k_{ex}$ versus $1000/T$ from all of the ¹⁷O $T_1,T_2,T_{1\rho }$ relaxation and line shape measurements in NaNO₃ at 14.1 T.

To obtain the values of $k_{ex}$ from the experimental relaxation times, simulations of the relaxation times as a function of $k_{ex}$ were performed for the 3-site jumps model of Figure 1B. The simulation procedure is described in detail in the “Materials and Methods” section, and it is important to note that the simulated magnetization decay curves were generated using the values of relaxations delays ${\tau }_D$ identical to those in the experiment to avoid skewing the simulated data due to the inherent non-exponentiality of relaxation. The experimental relaxation times were then matched with the simulated relaxation times (Figure 6B).

This procedure allows for unambiguous determination of $k_{ex}$ values from the $T_1$ and $T_{1\rho }$ experiments. The temperature dependence of $T_1$ clearly shows that the motions are much slower than the Larmor frequency, while the temperature dependence of the $T_{1\rho }$ times points to the slow regime at temperatures below about 245 K, followed by the onset of the intermediate regime, i.e., when $k_{ex}$ becomes on the order of significant averaging of the Q1 contribution, $k_{ex}=\frac{2\pi C_q}{I(2I-1)}=7.9\cdot {10}^6{\text{s}}^{-1}$. This is manifested as the beginning of the plateau region at around 255 to 245 K temperature range. The non-monotonic temperature dependence of the $T_2$ times indicates that the motions span slow and intermediate regimes. The onset of the slow regime (i.e, the Q2 contribution averaging) for the $T_2$ experiment is defined by the approach to the condition $k_{ex}=\frac{1}{{\omega }_0}{\left(\frac{2\pi C_q}{I(2I-1)}\right)}^2=1.2\cdot {10}^5{\text{s}}^{-1}$. Mapping of experimental $T_2$ times to the simulations unsurprisingly shows non-monotonic dependence and emphasizes the fact that the motions span a major subset of the slow and intermediate regimes. Thus, the fits of $T_2$ times needed to be constrained by the condition that $k_{ex}$ decreases with temperature. For two temperatures (260 and 225K), this constraint was not enough to resolve the ambiguity in the $k_{ex}$ values, which had to be resolved on the basis of the values obtained from the $T_1$ and $T_{1\rho }$ fits.

The summary of fitted $k_{ex}$ values from all measurements is demonstrated in Figure 7. The errors in the resulting $k_{ex}$ values were determined by propagation of the experimental errors in the relaxation times. There is a good overall agreement between all fitted $k_{ex}$ values obtained from all measurements. The values of the Arrhenius activation energies $E_{\text{a}}$ obtained from each of the series separately ($T_1:E_{\text{a}}=44.4\pm 0.5\text{kJ}/\text{mol},T_{1\rho }:E_{\text{a}}=47\pm 2\text{kJ}/\text{mol},T_2:E_{\text{a}}=49\pm 3\text{kJ}/\text{mol}$, line shape: $E_{\text{a}}=41\pm 2\text{kJ}/\text{mol}$) are also in a reasonable agreement with each other as well with the value obtained from the line shape measurement by Hung at el., 44.5 ± 2 kJ/mol, when the same temperature range of 280 to 195 K is taken for their 14.1 T data set. The weighted average of $E_{\text{a}}$ obtained from the individual data set is 44.4 ± 0.5 kJ/mol.

An additional confirmation of the results of $k_{ex}$ values can be obtained by comparing their consistency with the expected results at a different value of the magnetic field. Experimental $T_1$ and $T_{1\rho }$ relaxation times obtained at 18.8 T in the 265 to 200 K range (Figure S16) are in good agreement with the prediction based on rate constants originating from the 14.1 T relaxation times.

The combined approach of using multiple ¹⁷O NMR relaxation times and line shapes allows for a better control of issues such as the dependence of the rates on the initial conditions, pulse bandwidth considerations, and different sensitivity to motional regimes with respect to the Q1 and Q2 terms. While the $T_1$ experiment has a single well-defined minimum centered on the Larmor frequency, it is most sensitive to the effect of the initial conditions across the entire dynamics range and can be thus dependent on pulse bandwidth considerations. The $T_2$ experiment on the other hand is least sensitive to the initial conditions but has two broad minima given by $\frac{1}{{\omega }_0}{\left(\frac{2\pi C_q}{I(2I-1)}\right)}^2$ and by the Larmor frequency, which may complicate an unambiguous determination of the rate constants if used in isolation. The $T_{1\rho }$ experiments appear to be a good compromise between the minimal dependence on the initial conditions and the absence of ambiguities in fitting the $k_{ex}$ values in the intermediate regime. The line shapes are very sensitive to the EFG and CSA parameters but can be less sensitive to the $k_{ex}$ values due to the distortions caused by experimental imperfections, as discussed above. A possible extension of the experimental approach can involve the use of Gaussian pulses for enhancing the selectivity of the excitation or inversion of the central line.

In the case of NaNO₃ the multiple measurements may appear somewhat redundant, however they demonstrate the application of the combined approach to more complex systems with multiple motional modes and unknown tensor parameters. In this case, several relaxation rates may distinguish the details of the multi-modal mechanisms on different time scales and provide a more precise determination of the corresponding rate constant and populations of conformational states.

Materials and Methods

I. Sample details

The NaNO₃ sample in the crystalline phase was generously provided by Gang Wu and prepared as described in reference[9]. Additionally, the degree of ¹⁷O isotopic incorporation was measured by Beewerth et al. using the same preparation protocol to be 4.5%[23], thus rendering intra-molecular ¹⁷O-¹⁷O coupling a low occurrence. Homonuclear dipolar couplings between ¹⁷O nuclei are in general small, on the order of 60 Hz for a pair of intramolecular oxygens in NaNO₃.

II. NMR measurements

The measurements were performed using the 14.1 T and 18.8 T spectrometers at Maglab, equipped with the static low E probe of 5 mm coil diameter[39]. The temperature was calibrated using lead nitrate. Cooling was achieved with liquid nitrogen. The number of scans varied between 8 and 32 in the relaxation times measurements and 32 to 128 in the line shape measurements. The inter-scan delay was 1–2 s. 4 to 8 dummy scans were employed in the relaxation measurements. The RAPT block employed a pair of Gaussian pulses of $10\mu \text{s}$ duration with ±300 kHz offset, spaced at $1\mu \text{s}$ intervals (Figure 4). This RAPT sequence[38] was optimized for the saturation of ST and enhancement of the CT in the presence of MAS but can also be employed in the case of static solids. The method is an extension of the earlier work by Haase et al.[32] on manipulation of population transfers in quadrupolar nuclei of static solids using frequency swept RF fields. Other experimental parameters and schemes are stated in the Results and Discussion section.

III. Computational approach

Modeling of a relaxation experiment involves setting up the initial coherences, simulating the evolution during mixing period, and generating the detected signal in the form of free induction decay. For the line shape simulation the mixing period step is omitted. The idealized initial conditions take into account the non-zero equilibrium values of the coherences unaffected by the preparatory pulses.

Eq. (7) includes relaxation only due to the fluctuations of the secular terms of Hamiltonians in Eqs. (3) and (4). The relaxation from non-secular terms were calculated in the Redfield theory limit[13] and included explicitly into Eq. (7) [40], modifying it to

$$\frac{d{\rho }_{i,m}}{dt}=i{L}_{ij}^{\left(m\right)}{\rho }_{j,m}+K_{mn}{\rho }_{i,n}-R_{ij}\left({\rho }_{j,m}-{\rho }_{j,m}^0\right)$$ (8)

where ${\rho }_{j,m}^0$ indicates the equilibrium coherence.

Evolution during the mixing period was thus simulated by solving the differential equations Eq. (8) with matrix exponentiation using the internal Matlab algorithm[41,42]. The full set of 35 spin 5/2 coherences was used to represent the density matrix for a single orientation, Eqs. (S1) and (S2), supplemented by an extra state to maintain the equilibrium magnetization. Because there are three possible orientations for the N–O bond in the ${{\text{NO}}_3}^{-}$ ion, the overall size of ${\rho }_{j,m}$ vector is 108 components. The three orientations of the N–O bond vector in a single crystallite have equal occupation numbers, which is reflected in the symmetric exchange matrix

$$K=k_{ex}\left(\begin{array}{ccc}-2& 1& 1\\ 1& -2& 1\\ 1& 1& -2\end{array}\right).$$

The orientation-dependent parameters of CSA and EFG tensors were converted through a sequence of transformations starting with the CSA PAS frame into the EFG PAS frame to the molecular frame (with $z$-axis directed perpendicular to the ${{\text{NO}}_3}^{-}$ plane and the N–O bond orientations at 0°, 120° and 240°) and, finally, into the laboratory frame for individual crystallites using the corresponding Wigner matrices. The set up of motional frames, as well as the steps of the transformation of the tensors between different coordinate systems were taken from the EXPRESS program written by Vold and Hoatson[43].

Relaxation arising from the stochastic fluctuations of the CSA tensor and the secular parts of the EFG tensors is accounted for by the explicit Liouvillian term through the exchange between different N–O bond orientations. The non-secular CSA terms were not included due to their negligible contributions. The non-secular terms arising from the quadrupolar interactions oscillate at the Larmor and twice Larmor frequencies. We calculate these terms following the approach of EXPRESS program[43]. Further details are provided in SI4.

For the simulations of the $T_2$ experiment, the evolution during the mixing period was split into two halves with the 180° refocusing pulse in between the two. Relaxation delay times used in the simulations for the fits of the experimental data were matched with those employed in the experiment.

Following the evolution period, the detection block can be implemented explicitly: for the inversion or saturation recovery $T_1$ experiment, $I_z\left(CT\right)$ coherence was transformed into the transverse plane, while for the $T_2$ and the $T_{1\rho }$ simulations no additional transformations were applied before the acquisition period. However, it is possible to include other detection schemes such as the Hahn echo. The inclusion of Hahn echo block did not change the outcomes of the simulated relaxation times or line shapes.

The acquisition block included a single component of the $I_{-}$ operator corresponding to the central transition, with the details of the approach specified in the SI4. This reduced the 36-component coherence vector for a single site in the 3-fold jumps model down to a single coherence. The size of the Liouvillian matrix is reduced accordingly. Nonetheless, the relaxation during the acquisition period can still be taken into account, including the Redfield terms in Eq. (9) in addition to the Liouvillian terms. This procedure is clarified and extended from the approach presented in reference[43].

The powder averaged spectra as well as magnetization decay and build up curves in the relaxation measurements require averaging over the crystallite orientations. The choice of the number of discreet orientations included in the averaging is dictated by the stability of the fitted relaxation times and jitter-free line shapes. We found the commonly used ZCW scheme[44,45] adequate based on our tests. For the relaxation times simulations, the 610-orientations grid was chosen. It yields the results differing from the smaller 377-orientations grid by no more than 1%. For the line shape simulations, the 17711-orientations grid was sufficient. The latter grid was also used to produce the orientation-dependent (rather than powder averaged) relaxation rates.

Simulations of relaxation took about 3 minutes for one magnetization decay curve with 11 relaxation delays using the 610-orientations crystallite averaging and the 3-site jump model on Intel Core i7 2.60 GHz CPU with an ordinary laptop, and about 5 minutes for the 17711-orientations crystallite averaging for the line shapes simulations. The calculation times can be reduced by an order of magnitude if parallel processing for different crystallites is used on a dedicated CPU of the same type.

Conclusion

This work explored the combined approach using multiple ¹⁷O CT NMR relaxation times ($T_{1\rho },T_1$, and $T_2$) as well as CT line shape analysis to enhance the tools for studies of molecular dynamics. Each method in isolation has its own limitations, broadly defined by the sensitivity to the initial conditions on one hand, the potential ambiguities and complex behavior around the slow and intermediate regimes on the other hand, the general sensitivity to the motional and EFG/CSA tensors’ parameters for the relevant temperature ranges, and experimental limitations on the magnitudes of the relaxation times that can be measured.

We performed modelling of the relaxation experiments for an arbitrary system undergoing large-angle 2-site jumps to explore theoretical limitations and establish the modeling routines. In general, the Redfield approach may not be valid across the entire range of the rate constants for the $R_2$ and $R_{1\rho }$ relaxation rates and, thus, the numerical solutions of the Liouville – von Neumann equation remains the most general tool. We develop the protocols for simulations of the relaxation rates and line shapes of half-integer quadrupolar nuclei with the explicit inclusion of the motional model into the routine. We also examined the applicability of using the fictious spin ½ system for the analysis of ¹⁷O CT $R_{1\rho }$ relaxation using either direct calculations in the reduced basis for the Liouville – von Neumann equation, or the analytical expressions adopted from reference[30]. The analytical expression in general is not applicable outside of the ultrafast regime.

The $T_1$ times have a single minimum with respect to dependence on the rate constant, however they are the most dependent on the initial conditions. The $T_2$ times are the least dependent on the initial conditions but have a rather complex dependence on the rate constant in the broad slow to intermediate motional regime, which can create ambiguities if these measurements are used in isolation. The novel $T_{1\rho }$ experiment is shown to have a minimal dependence on the initial conditions and does not lead to ambiguities in the rate constant in intermediate regime region compared with the $T_2$ measurements. However, it requires the employment of the spin-locking field strong enough to lock the CT line, which invokes probe RF limits considerations, especially for long $T_{1\rho }$ times. The RF power limitations can be partially overcome by using the off-resonance conditions. Relaxation dispersion can also be performed for the $T_{1\rho }$ experiment, which provides additional information on the motional processes when the rate constant is on the order of Q1 contribution. The line shape measurements can have a limited sensitivity to the rate constant outside of the intermediate regimes but are the simplest to perform and model. They provide direct information about the EFG and CSA tensor parameters but can be less sensitive to the motional parameters outside of the intermediate regime. Relaxation rates are, of course, also sensitive to the EFG tensor parameters, but this dependence is difficult to disentangle from the dependence on the rate constants and the details of the model. In general, to resolve these ambiguities multiple measurements of all rates at different temperatures that sample intermediate and slow regimes are required.

We used NaNO₃ as a model compound under static (non-spinning) conditions to probe experimental advantages and limitations of each method and to demonstrate the combined approach using ¹⁷O CT line shapes, $T_1,T_2$, and $T_{1\rho }$ relaxation times measurements. The measurements were performed in the 280 to 195 K temperature range and at two static magnetic fields, 14.1 and 18.8 T. The combined approach of using multiple ¹⁷O NMR relaxation times and line shapes allowed for a better control of issues such as the dependence of the rates on the initial conditions and pulse bandwidth considerations. It was interesting to observe the power of the $T_{1\rho }$ measurements in the intermediate to slow regime, in which the $T_2$ rate constants could not be fitted unambiguously. We relied on the line shapes measurements for defining the EFG and CSA tensor parameters. The numerical approach was implemented for the 3-site jumps model of the N–O bond. A reasonable agreement in the resulting values of rate constants was obtained from all the measurements, and we discussed the complementarity of all the measurements to raise the accuracy of the rate constants and the resulting value of the activation energy.

The combined approach is expected to be especially useful for complex systems with unknown motional models and tensor parameters, and the ¹⁷O $T_{1\rho }$ relaxation times measurements constitute an important addition to the suite of the relaxation experiments in this regard. While the approach is presented here for static powders, it is extendable to rotating solids, for which the evolution under magic-angle spinning should be carefully taken into account in simulations, and the potential for rotary resonances considered in the choice of experimental parameters for the $T_{1\rho }$ measurements[46,47]. Magic-angle spinning will also reduce the contributions to non-exponentiality related to the orientation-dependence of the relaxation rates.

• Theoretical and experimental approaches for ¹⁷O relaxation in solid state are given

• Advantages of using combination of line shapes and $T_{1\rho },T_1,T_2$ times are shown

• Measurements using NaNO₃ are performed in 280 to 195 K range at two fields

• Dynamics modeling using Liouvillian approach is discussed