Modulation of cross polarization in motionally averaged solids by Variable Angle Spinning NMR

Abstract

In systems where the dipolar couplings are partially averaged by molecular motion, cross-polarization is modulated by sample spinning. The cross-polariation efficiency in Variable Angle Spinning (VAS) and Switched Angle Spinning (SAS) experiments on mobile samples is therefore strongly dependent on the spinning angle. We describe simulations and experimental measurements of these effects over a range of spinning angles from 0° to 90°.

1. Introduction

Cross-polarization (CP), in which magnetization is transferred from abundant, high gyromagnetic-ratio nuclei to rare, low gyromagnetic-ratio nuclei via the dipolar couplings [1], is a nearly ubiquitious building block in solid-state NMR experiments. In the static case, CP occurs when the RF fields for two nuclei, I and S, e.g. ¹H and ¹³C, are matched in the rotating frame according to the Hartmann-Hahn condition. In practice, CP is most useful under magic angle spinning (MAS) [2], where it is well known that the cross-polarization intensity is modulated by the spinning frequency, giving rise to a sideband pattern where |ω_1S ± ω_1I| = n ωr, where ω_1S and ω_1I are the RF fields applied to the S and I spins, and ω_r is the spinning frequency. At the magic angle, the n = ±1 and n = ±2 conditions dominate the matching profile and the center band is suppressed [3, 4].

Spinning off the magic angle reintroduces anisotropic interactions, providing an alternative to pulse sequence-based recoupling schemes. Variable angle spinning (VAS) [5, 6] and switched angle spinning (SAS) experiments yield correlations between isotropic spectra and anisotropic spectral information such as chemical shift anisotropy [7, 8, 9] or heteronuclear dipolar couplings [10]. VAS requires that the entire pulse sequence be repeated at each angle, and in many SAS experiments the preparation period and mixing time take place off the magic angle, with only the signal collection done under MAS. Most previous SAS studies have focused on strongly coupled systems, i.e. rigid solids where the modulation is negligible, particularly for slow to moderate spinning rates. In this case, an acceptable signal may be obtained at any spinning angle without altering the CP match condition. One exception is the dynamic angle spinning experiment using CP between a spin 1/2 nucleus and the central transition of a quadrupole at 0° followed by detection at the magic angle [11], since in this case the time dependence of the first-order quadrupolar interaction interferes with the polarization transfer when the spinning axis deviates from zero degrees. For systems with attenuated dipolar couplings, the CP match condition likewise strongly depends on the spinning axis as demonstrated by Sardashti and Maciel for SAS between 0° and 90° [12].

This type of measurement has the potential to be useful at the interface between solid-state and solution NMR, where the dipolar couplings are partially averaged, particularly in the context of new methods for assignment of proteins in oriented samples where it is beneficial to obtain spectra at both 0° and 90° [13]. VAS and SAS methods have been applied to strongly ordered liquid crystals, in which the director alignment changes with the spinning axis, modulating the anisotropic interactions [14, 15, 16, 17]. Scaled dipolar couplings and CSAs can also be measured using stretched polymer gels to provide weak alignment; in this case the variable angle experiment does not require sample spinning [18]. VAS experiments on bicelles have been performed with direct detection of ²H [19], or ³¹P [20, 21], and SAS experiments used to simplify the spectra of organic molecules in a liquid crystal [22, 23] and peptides in bicelles [24] have used direct detection of ¹⁹F or ¹H, respectively, without polarization transfer in the indirect dimension. It has been demonstrated that adiabatic-passage CP is effective for strongly oriented membrane systems, even at the magic angle [25], but for systems that are particularly sensitive to sample heating or where T_1ρ is short, a short CP time optimized for the spinning angle is preferable. In order for VAS and SAS methods to be applied more broadly to samples in the intermediate regime between rigid solids and oriented liquids, it is necessary to consider the behavior of CP at different spinning angles. Many interesting systems fall into this category, including highly mobile solids, semisolids, or hydrogels. However, these samples are complex, often having several components and a strong dependence of their phase behavior on temperature. Therefore, characterization with a well-established model system is necessary. In this letter, we describe simulations and measurements of CP in adamantane over the full range of angles between 0° and 90° in a VAS/SAS probe built for this type of experiment.

2. Theory

CP is most conveniently considered in a doubly rotating frame with the z–axis along the direction of the spin lock field. In this frame the total spin Hamiltonian Ĥ_IS for two coupled spins I and S is given by:

$${\widehat{H}}_{IS}={\omega }_{1I}{\widehat{I}}_z+{\omega }_{1S}{\widehat{S}}_z+{\omega }_{IS}2{\widehat{I}}_x{\widehat{S}}_x$$ (1)

where ω_1I and ω_1S are the RF nutation frequencies and ω_1S the heteronuclear dipolar coupling between an abundant I spin and a rare S spin.

For a sample spinning at a frequency ω_r and at an angle β_RL with respect to the static magnetic field B₀, the heteronuclear dipolar coupling becomes time dependent and is given by:

$${\omega }_{IS}(t)=b_{IS}\sum _{n=-2}^2{D}_{0n}^2({\mathrm{\Omega }}_{PR})d_{n0}^2({\beta }_{RL})e^{\text{i}n{\omega }_{r}t}$$ (2)

where $b_{IS}=(\hslash {\mu }_0/4\pi ){\gamma }_I{\gamma }_S{r}_{IS}^{-3}$ is the dipolar constant, $D_{0n}^2(\mathrm{\Omega })$ is an element of the second-rank Wigner matrix, $d_{n0}^2(\beta )$ is an element of the second-rank reduced Wigner matrix, and Ω_PR is a set of Euler angles describing the relative orientation between the internuclear vector and the rotor axis [26].

The Hamiltonian in Equation 1 can be decomposed into two spin subspaces, a zero-quantum subspace and a double-quantum subspace. In the case where ω_1I + ω_1S ≫ ω_IS and ω_1I − ω_1S ≪ ω_IS, the zero-quantum transition dominates the cross polarization process. The Hamiltonian Ĥ_IS in the zero-quantum subspace is given by:

$${\widehat{H}}_{ZQ}=({\omega }_{1I}-{\omega }_{1S}){\widehat{I}}_z^{ZQ}+{\omega }_{IS}(t){\widehat{I}}_x^{ZQ}$$ (3)

where

$${\widehat{I}}_z^{ZQ}={\widehat{I}}_z-{\widehat{S}}_z$$ (4)

and

$${\widehat{I}}_x^{ZQ}=\frac{1}{2}({\widehat{I}}_{+}{\widehat{S}}_{-}+{\widehat{I}}_{-}{\widehat{S}}_{+})$$ (5)

Transforming the Ĥ_ZQ into a frame rotating at the frequency ω_Δ = ω_1I − ω_1S we obtain:

$$\begin{array}{l}{\widehat{H}}_{ZQ}^T=exp\left(i{\omega }_{\mathrm{\Delta }}t{\widehat{I}}_z^{ZQ}\right){\widehat{H}}_{ZQ}exp\left(-i{\omega }_{\mathrm{\Delta }}t{\widehat{I}}_z^{ZQ}\right)\\ ={\omega }_{IS}(t)\left[cos({\omega }_{\mathrm{\Delta }}t){\widehat{I}}_x^{ZQ}-sin({\omega }_{\mathrm{\Delta }}t){\widehat{I}}_y^{ZQ}\right]\end{array}$$

As a result, we expect to find a non-vanishing heteronuclear dipolar coupling if

$$exp(in{\omega }_{r}t)exp(\pm i{\omega }_{\mathrm{\Delta }}t)=1$$ (6)

leading to the usual condition ±ω_Δ + nω_r = 0.

Moreover, taking into account the angle dependence of the rotor a non-vanishing heteronuclear dipolar coupling will be found if $d_{n0}^2({\beta }_{RL})\ne 0$, with:

$$\begin{array}{l}{d}_{00}^2({\beta }_{RL})=\frac{1}{2}\left(3{cos}^2({\beta }_{RL})-1\right)\\ d_{\pm 10}^2({\beta }_{RL})=\mp \sqrt{\frac{3}{8}sin(2{\beta }_{RL})}\\ d_{\pm 20}^2({\beta }_{RL})=\pm \sqrt{\frac{3}{8}{sin}^2({\beta }_{RL})}\end{array}$$

Interactions among these give rise to the CP match conditions as well as numerous recoupling conditions. A thorough Floquet treatment of the Hamiltonians involved is available [27], but in this case the relevant interactions can be seen by plotting the elements of the reduced Wigner matrices (Figure 1). At the magic angle, β_RL = arctan $\sqrt{2}\simeq 54.74°$ and the reduced Wigner matrix elements vanish for n = 0 explaining why the Hartmann-Hahn matching occurs only at n =1,2. For the n = 0 condition, transfer can occur via either the J-coupling, or the second-order cross terms between a homonuclear and a heteronuclear coupling. The latter effect has recently been exploited by Lange et al to deliver low-power CP at fast MAS rates [28]. In the case of a mobile sample such as adamantane at the low to moderate spinning speeds of interest here, the J-coupling effect dominates the n = 0 condition. Looking at some particular angles, when β_RL = 0°, $d_{10}^2(0°)=0,d_{20}^2(0°)=0$, such that the Hartmann-Hahn matching is equivalent to the static case when the sample is spinning parallel to the magnetic field B₀. On the other hand, when β_RL = 90°, $d_{10}^2(90°)=0$ and thus the odd-numbered sidebands disappear from the Hartmann-Hahn match curve.

Figure 1.

The elements of the reduced Wigner matrices are plotted as a function of rotor angle, with 1 being the maximum theoretical transfer efficiency, observed at 0° (the static case). At the magic angle, the $d_{-10}^2$ and $d_{+10}^2$ terms dominate, although the contribution from the $d_{20}^2$ term is also significant ( $d_{-20}^2$ is not plotted here because the low-power CP conditions used in the experiment did not permit observation of polarization transfer by this mechanism in the measured data). At 90°, the $d_{-10}^2$ and $d_{+10}^2$ terms vanish, leaving only the $d_{00}^2$ and $d_{+20}^2$ terms. Although the maximum theoretical transfer is not achieved away from 0°, it can easily be seen that the intensity obtained from $d_{20}^2$ at 90° is comparable to that of the $d_{-10}^2$ and $d_{+10}^2$ terms at the magic angle.

3. Materials and methods

Adamantane was purchased from Sigma-Aldrich. One-dimensional ¹H-¹³C CP NMR spectra were collected on a 11.7 T wide-bore magnet equipped with a Chemagnetics console and a homebuilt 3.2 mm rotor double-resonance pneumatic SAS NMR probe [29]. All experiments were performed at room temperature using SPINAL decoupling [30]. The CP experiments were done over a range of angles from 0° to 90° which were set using a HGT-3030 Hall effect sensor (Lakeshore) [31]. Simulations were performed using the SIMPSON software [32]. The powder averages were calculated using 320 orientations, selected according to the REPULSION algorithm [33].

4. Results and discussion

Adamantane has a long history in solid-state NMR because its molecular symmetry and fast motions result in its having no chemical shift anisotropy and strongly attenuated heteronuclear and homonuclear dipolar couplings [1, 34, 35]. In this experiment, the RF field applied to the ¹³C nuclei is held constant at 28 kHz and the RF field applied to the protons is arrayed from 15-55 kHz. The ¹³C signal intensity is plotted as a function of the applied proton field in Figure 2. The off-magic angle arrays were adjusted to align with the MAS data due to minor differences in probe tuning at different angles. As expected, at the magic angle the n = +2 and n = ±1 sidebands give the strongest signals. For the low-power condition explored here, the n = −2 sideband is weak because the efficiency of CP declines at RF amplitudes below 20 kHz. At angles smaller than the magic angle, the center band is more pronounced, and begins to dominate at small spinning angles, resembling the static case at 0°. As described by Sardashti and Maciel [12], the situation at 90° is quite different - the n = ±1 conditions disappear and the match array is dominated by the n = 0 and n = ±2 bands. The CP maxima at the n = 0 and n = 2 conditions are significantly broader than the n = ±1 conditions at the magic angle, making this condition easier to locate in a mobile sample. As expected, a similar set of experiments performed on ¹³C alanine at ω_r = 5 kHz showed greatly reduced modulation of the signal due to the much stronger ¹H-¹H dipolar couplings. In principle, CP transfer is most efficient at 0°. However, good signal intensity is also obtained with CP at 90°, and this may be preferable for some experiments, particularly in the context of separated local field spectra where the spectral width in the indirect dimension is limited by the magnitude of the couplings.

Figure 2.

Experimental ¹H-¹³C cross-polarization arrays of adamantane at different spinning angles. The ¹³C RF field strength was held constant at 28 kHz and the ¹H nutation frequency was arrayed from 15-55 kHz. The rotor frequency was 5 kHz at all angles. The ¹H 90° pulse length was 5.5 μs, corresponding to a ¹H power of 45.45 kHz).

CP match curves for adamantane were also simulated at different spinning angles. The dipole-dipole couplings can be derived assuming that the internal couplings of each adamantane molecule are completely averaged by its fast isotropic rotation, leaving sixteen uncoupled protons in the center of each molecule. These centers form a body-centered tetragonal lattice such that each center has twelve nearest neighbor-centers each at a distance of 6.54 Å [36]. Using the Root-Sum-Square dipolar coupling approximation [37], we can derive the effective dipolar couplings: D_HH = −5950 Hz and D_CH = −1500 Hz. The simulated CP match curves resulting from these parameters are shown in Figure 3.

Figure 3.

Simulated ¹H-¹³C cross-polarization arrays for the CH peak of adamantane as a function of spinning angle, under the same conditions as the experiment. Overall, the results are in agreement with the experimental data, except for the the n=-2 match condition, which is not observed in the measured spectra.

In the fast spinning regime, the ZQ and DQ Hartmann-Hahn conditions are better separated, and either can be used with low-power irradiation to provide frequency-selective CP [38, 39]. Recently, low-power CP with ultra fast MAS has been a topic of intense interest. This is useful in rigid solids where low-power experiments are desired, such as protein nano- or microcrystals. However, we are interested in low-power CP under slow to moderate MAS, as these are the conditions most amenable to semisolid materials and highly mobile solids. In addition to liquid crystals and membranes, this could include surfactants at low water concentration [40] as well as proteins containing both mobile and rigid regions [41]. At the low spinning speeds used here, low-power CP results in inefficient transfer, as observed in the experimental data shown in Figure 2. At values of the RF field strength at or below about 20 kHz, the signal intensity is dramatically reduced or eliminated. The probe used in these experiments is a homebuilt double-resonance VAS/SAS probe in which the B₁ field delivered is independent of the angle [29]. Although this probe can deliver higher RF amplitudes on both channels, the combination of low field and slow spinning is used here with the future intention of investigating membrane systems, hydrogels, and semisolids. In this type of sample, fast spinning is not required for effective decoupling. In fact, in many cases it is detrimental because the ordered components can be centrifuged to the center of the rotor, leading to complex inhomogeneities. These samples are also lossy, such that high RF power or the long pulses required for adiabatic CP may cause significant sample heating.

In order to optimize CP-VAS experiments, it is necessary to consider the time-dependence of the CP buildup as well as the intensity of the signal obtained. In the case of a single proton I dipolar coupled to one ¹³C S, the theoretical time-dependence of the transferred S-spin polarization is given by [4]:

$$s({\tau }_{CP})\stackrel{\sim }{=}{sin}^2\left[\frac{{\omega }_{IS}\times {\tau }_{CP}}{2}\right]$$ (7)

Thus, for a single crystal orientation the CP buildup curve follows:

$$s({\tau }_{CP})={sin}^2\left[\frac{{\tau }_{CP}}{2}{b}_{IS}{d}_{n0}^2({\beta }_{PR})d_{0n}^2({\beta }_{RL})\right]$$ (8)

After powder averaging the buildup curve becomes

$$s({\tau }_{CP})=\frac{1}{2}{\int }_0^{\pi }sin{\left[\frac{{\tau }_{CP}}{2}{b}_{IS}{d}_{n0}^2({\beta }_{PR})d_{0n}^2({\beta }_{RL})\right]}^{2}sin({\beta }_{PR})d{\beta }_{PR}$$ (9)

Eq. 9 provides a theoretical equation describing the CP buildup curves as a function of the spinning angle β_RL and the contact time τ_CP It has been shown that exact solutions to Eq. 9 can be analytically derived [42, 43]. For example, in the case where n = 0, Eq. 9 becomes:

$$s_n=0({\tau }_{CP})=\frac{1}{2}-\frac{1}{2x_0}(Fc(x_0)cos{\phi }_0+Fs(x_0)sin{\phi }_0)$$ (10)

where

$$x_0=\sqrt{\frac{3{\tau }_{CP}{b}_{IS}{d}_{00}^2({\beta }_{RL})}{\pi }}$$ (11)

$${\phi }_0=\frac{{\tau }_{CP}}{2}\left(b_{IS}{d}_{00}^2({\beta }_{RL})\right)$$ (12)

and Fc(x), Fs(x) are the cosine and sine Fresnel integrals

$$Fc(x)={\int }_0^{x}cos\left(\frac{\pi y^2}{2}\right)dy$$ (13)

$$Fs(x)={\int }_0^{x}sin\left(\frac{\pi y^2}{2}\right)dy$$ (14)

Similarly, in the case where n = ±2, Eq. 9 becomes:

$$s_{n=\pm 2}({\tau }_{CP})=\frac{1}{2}-\frac{1}{2x_2}(Fc(x_2)cos{\phi }_2+Fs(x_2)sin{\phi }_2)$$ (15)

where

$$x_2={\left(\frac{3}{2}\right)}^{1/4}\sqrt{\frac{{\tau }_{CP}{b}_{IS}{d}_{20}^2({\beta }_{RL})}{\pi }}$$ (16)

$${\phi }_2=\frac{{\tau }_{CP}}{2}\sqrt{\frac{3}{2}}\left(b_{IS}{d}_{20}^2({\beta }_{RL})\right)$$ (17)

In the case where n = ±1, Eq. 9 cannot be solved analytically, but must be evaluated numerically.

Fig. 4 compares the theoretical curves obtained form Eq. 9, for n = 0, ±1, ±2, with a series of numerical simulations in which the spinning angle β_RL is increased form 40 to 66 degrees. The agreement is excellent in this idealized case.

Figure 4.

Dependence of the CP signal intensity on the contact time τ_CP at different spinning angles β_RL, for two coupled spins I and S a) Numerical simulations with a dipolar coupling constant b_IS = −1500 Hz (-3000 π radians/second) for the CP conditions ω_Δ = ±2ω_r and spinning angles β_RL = 40° (blue circles), β_RL = Magic Angle (green circles) and β_RL = 66° (yellow circles). b) same as a) but for the CP conditions ω_Δ = ±ω_r. c) same as a) but for the CP conditions ω_Δ = 0. The theoretical curves obtained from Eq. 9, with a dipolar coupling constant b_IS = −1500 Hz, are shown by a solid green curve when β_RL = Magic Angle, by a dashed blue curve when β_RL = 40° and by a dot-dashed yellow curve when β_RL = 66°.

For a real sample, the situation is more complicated due to the presence of additional spins and the effects of relaxation. Experimental CP buildup curves for the CH₂ resonance of adamantane for the n = 0, n = 1, and n = 2 conditions were collected at different angles (Figure 5). These experimental curves show fewer oscillations than the idealized case of a single C-H pair, as the simulations for this ideal model neglected relaxation effects and ¹H-¹H dipolar couplings. Furthermore, in the magic angle case, the simulations predict zero intensity for n = 0 at the magic angle, which is not the case due to J-couplings and second-order effects neglected in the simulation. Nevertheless, the simulations are useful for making qualitative comparisons. As observed by Wu and Zilm [4], at the magic angle the buildup of magnetization on ¹³C occurs fastest at the n = 1 sideband condition, slightly slower at n = 2 and significantly slower at n=0. In agreement with the simulation, on either side of the magic angle at 36° and 66° the magnetization is most intense and has the fastest buildup at the n = 1 condition. Comparing the behavior of the n = 0 and n = 2 conditions away from the magic angle shows faster CP buildup at the n = 0 condition for 36° and at the n = 2 condition for 67°. At 36°, the difference between n=2 and n=0 is reduced, which can be explained by the matrix elements plotted in Figure 1 $d_{00}^2$ and $d_{20}^2$, which both approach zero at this angle. At 90°, the fastest buildup is obtained at the n = 2 condition. Overall, the slope of the curve for $d_{20}^2$ is less pronounced, making the match at this condition easier to achieve experimentally for a mobile sample.

Figure 5.

Contact time arrays for relatively short CP times at different spinning angles. The peak intensity of the CH₂ carbon resonance of adamantane is plotted as a function of contact time for (a) n = 2 (b) n = 1 and (c) n = 0.

5. Conclusion

We have measured and simulated the cross-polarization intensity and buildup dynamics over a range of angles from 0° to 90°. For mobile samples, such as adamantane, the CP match condition has a significant dependence on the spinning axis. This is relevant not only to VAS experiments, where CP is performed at every angle, but also to SAS experiments on mobile samples. Our findings show that for off-magic angle conditions, it may be advantageous to use the n = 0 or n = ±2 match conditions, as they are often easier to achieve and result in more a rapid build-up of S-spin polarization. As oriented membrane systems and many other important samples that fall in the regime between rigid solids and isotropic liquids are quite lossy and are therefore prone to overheating when subjected to high RF power or pulses of extended length, minimizing CP contact time is essential. Characterization of CP performance over a broad range of angles is a necessary first step for the development of novel SAS and VAS experiments. The ability to obtain equally good CP signal off the magic angle with proper calibration opens the door to a number of new multidimensional SAS or VAS experiments, since every angle switch entails storing the magnetization along z and losing signal. The ability to perform at CP any angle eliminates the need to switch angles before performing CP and again before detection, allowing more freedom to design multidimensional experiments correlating two anisotropic dimensions or observing the evolution of anisotropic terms in a ¹³C dimension.

Highlights

• In mobile solids, dipolar couplings are modulated as a function of spinning angle.

• The modulation affects the intensity and rate of cross polarization (CP).

• These effects have been simulated and measured from 0 degrees to 90 degrees.

• Good CP intensity can sometimes be more easily obtained away from the magic angle.

• CP must be optimized at different angles to develop new SAS and VAS experiments.