Broadband Adiabatic Inversion Pulses for Cross-Polarization in Wideline Solid-State Nuclear Magnetic Resonance Spectroscopy

Abstract

Efficient acquisition of wideline solid-state NMR powder patterns is a continuing challenge. In particular, when the breadth of the powder pattern is much larger than the cross-polarization (CP) excitation bandwidth, transfer efficiencies suffer and experimental times are greatly increased. Presented herein is a CP pulse sequence with an excitation bandwidth that is up to ten times greater than that available from a conventional spin-locked CP pulse sequence. The pulse sequence, broadband adiabatic inversion CP (BRAIN-CP), makes use of the broad, uniformly large frequency profiles of inversion chirped pulses, to provide these same characteristics to the polarization transfer process. A detailed theoretical analysis is given, providing insight into the polarization transfer process involved in BRAIN-CP. Experiments on spin-1/2 nuclei including ¹¹⁹Sn, ¹⁹⁹Hg and ¹⁹⁵Pt nuclei are presented, and the large bandwidth improvements possible with BRAIN-CP are demonstrated. Furthermore, it is shown that BRAIN-CP can be combined with broadband frequency-swept versions of the Carr-Purcell-Meiboom-Gill experiment (for instance with WURST-CPMG, or WCPMG for brevity); the combined BRAIN-CP/WCPMG experiment then provides multiplicative signal enhancements of both CP and multiple-echo acquisition over a broad frequency region.

1. Introduction

Solid-state nuclear magnetic resonance (SSNMR) spectroscopy has become a widely used technique for the characterization of atomic-level structure and dynamics, largely because of the extreme sensitivity of the NMR interactions to the local environment of each nucleus.^1–2 In many such instances, the anisotropic character of the chemical shift and/or quadrupolar interactions can lead to NMR powder patterns possessing spectral breadths ranging from hundreds of kHz to several MHz. Unfortunately, experimental difficulties associated with acquiring such broad, ultra-wideline (UW), powder patterns can prevent the determination of the NMR parameters and the detailed chemical information they encode. Development of specialized techniques and hardware for the acquisition of UW NMR spectra to provide new materials information has accordingly been the subject of intensive research efforts.³

Because the dilution of spectral intensity over large frequency regions leads to a reduction in the signal-to-noise ratio (S/N), methods for increasing the signal strength are often required in UW NMR spectroscopy. One technique for enhancing the S/N relies on preservation of the observable transverse-plane NMR signal, by applying a series of refocusing pulses, following the method of Carr/Purcell⁴ and Meiboom/Gill⁵ (CPMG). The CPMG method has proven to be extremely useful for acquiring wideline spectra from both spin-1/2 and quadrupolar nuclei.^6–7 When the effective bandwidth of the pulses or of the probe electronics is insufficient, the powder pattern can be mapped out with a series of acquisitions at different transmitter frequencies.⁸ Broadband frequency-swept pulses are also capable of exciting spin echoes;^9–12 for instance, it has recently been shown that a CPMG echo train can be acquired using wideband, uniform rate, smooth truncation (WURST) pulses.¹³ The ensuing WURST-CPMG (WCPMG for brevity) pulse sequence enables extremely broad frequency ranges to be observed simultaneously, providing the S/N advantage of a CPMG-like echo train in experiments involving both spin-½ and quadrupolar (spin > ½) nuclei.^14–15

Cross-polarization (CP) is another S/N-enhancing route that is applied extensively in SSNMR spectroscopy.^16–18 S/N gains are afforded by the transfer of polarization from proximate higher-γ nuclei. A further advantage over direct excitation methods often results from the faster repetition rates associated with the generally much shorter longitudinal relaxation times (T₁’s) of the high-γ nuclei. Unfortunately, the bandwidth over which CP is effective is often determined by the strength of the spin-locking fields employed, and the resultant excitation covers only a small fraction of the total NMR powder pattern when large anisotropic NMR interactions (and low-γ target nuclei) are investigated. Therefore, once again, either stepwise acquisition or broadband direct excitation methods must be employed, leading to lengthy total experimental times. The recent availability of commercial ultrahigh-field NMR spectrometers and increased interest in NMR studies of paramagnetic materials has placed a further emphasis on this subject. For example, Peng et al. have used frequency-swept adiabatic half passages to improve the efficiency of ¹H to ¹³C CP in MAS experiments on samples that have paramagnetically broadened ¹H spectra.^19–21 In recent years, a significant amount of research effort has gone into modifying polarization transfers involving commonly studied nuclides (¹³C, ¹⁵N, ²⁹Si, ³¹P, etc.) under MAS.²² Many of these studies have focused on improving transfer efficiencies by altering the classical spin-locking scheme with ramped or adiabatic passage modulations of the RF profiles.^23–26 Studies have also contemplated the effects and use of large anisotropic offsets.²⁷ However, improvements to the limited excitation bandwidth of conventional CP for use in wideline NMR remains an open challenge. To address this challenge, we discuss herein a CP experiment in which an adiabatic full passage is used in place of the conventional monochromatic S-nucleus spin-lock field (a standard rectangular spin-lock pulse is still employed on the high-frequency channel given the relatively narrow I = ¹H bandwidths of the samples investigated, but in principle I-spin waveforms could also be utilized). Since the characterizing feature of the resulting experiment is a polarization transfer using a broadband adiabatic inversion pulse, we have designated the pulse sequence BRAIN-CP. The application of adiabatic inversion pulses in BRAIN-CP provides two crucial characteristics: (i) the excitation profile is broad and uniform over bandwidths solely determined by the transmitting electronics, and (ii) the S-nucleus polarization is produced parallel (or anti-parallel) to the external field, which allows the magnetization to remain coherent across the broad powder pattern. A theoretical investigation of this polarization transfer process using average Hamiltonian theory and numerical simulations is presented. We also demonstrate the efficiency and broad excitation bandwidth of BRAIN-CP using ¹H nuclei to polarize ¹¹⁹Sn, ¹⁹⁹Hg and ¹⁹⁵Pt nuclei in model samples. Furthermore, we demonstrate that BRAIN-CP can be used in conjunction with WCPMG signal acquisition, leading to a completely broadband experiment exhibiting multiplicative S/N gains from both its components.

2. Pulse Sequence Design

BRAIN-CP: Overall Considerations

The theory of adiabatic inversions on which this study is based is well known,^28–32 so only a brief summary is presented before discussing their use in cross polarization. An adiabatic inversion can be brought about using a wide variety of pulse shapes (or by sweeping the external magnetic field), but the mechanism in all cases is similar, so we focus the discussion on one method, the WURST pulse,^13,33 for simplicity. A WURST pulse involves a linear chirp of the effective transmitter offset frequency, generated via quadratic phase modulation of the radiofrequency (RF) pulse, and a slow rise/fall of the RF field amplitude at the beginning/end of the pulse (Fig. 1a). While the magnetic field generated by a monochromatic pulse would appear stationary in the rotating frame defined by the transmitter frequency, the field generated by a linearly chirped pulse only appears stationary after a second rotational transformation applied at the instantaneous effective offset frequency shown in Fig. 1a; conventional nomenclature for this shifting frame is the frequency-modulated (FM) frame.^29–30,32 In the FM frame, the z-component of the effective magnetic field, B_eff, varies with time because the instantaneous offset between the effective transmitter frequency and the resonance frequency of a given nucleus continuously changes (Fig. 1b). Typical B_eff trajectories for spins that have different resonance frequencies (e.g., due to an anisotropic interaction) are shown in Fig. 1c. Although the details of these trajectories are different, a slow enough sweep of the effective transmitter frequency guarantees that spins initially polarized along the z-axis will remain locked along the magnetic field as it moves toward the z<0 hemisphere; we note that only cases where the frequency sweep and amplitude modulation are such that the magnetization ends up aligned (or nearly aligned) along the -z-axis are discussed here.

Figure 1.

Description of a WURST pulse as a representative adiabatic pulse in terms of: (a) effective transmitter frequency sweep and amplitude dependence versus time, and (b), (c) the concomitant path taken by the effective magnetic field, B_eff, during a typical adiabatic pulse as viewed in the FM frame. The behavior of the magnitude of B_eff (in red) and polar angle θ (in blue) is shown for cases where anisotropy shifts the resonance frequency of the spin to a lower frequency than the transmitter (ω_off < 0), to a higher frequency (ω_off > 0), or where the spin is on resonance (ω_off = 0). B_eff is at a minimum when it crosses the transverse plane (i.e., B_eff = B₁ when θ= π/2), which occurs at the point where the effective transmitter frequency equals the resonance frequency of the spin.

While WURST pulses are typically employed for manipulating existing nuclear spin polarization, a coherence transfer application is explored herein in which the variable-amplitude, variable-angle B_eff generated by a frequency swept pulse is used as the spin-locking field in a modified CP experiment. The BRAIN-CP pulse sequence in Figure 2a shows one possibility for applying this strategy. The I = ¹H portion of the experiment is unchanged from the conventional CP experiment,^16–18 while the contact pulse on the low-γ S-spin channel is replaced with a chirped RF sweep applied to the anisotropically broadened target powder pattern. The underlying rationale is foreshadowed in Fig. 1c (and further demonstrated below): at one or more points during the frequency sweep, a Hartmann-Hahn matching condition is fulfilled for each isochromat of the powder pattern. Polarization can then be transferred to S nuclei regardless of their precise resonance offset, and magnetization will build up for every crystallite at some instant throughout the sweep. Once generated, the S-spin polarization adiabatically follows B_eff, resulting in bulk spin polarization aligned along –z for all sections of the powder pattern at the end of sweep. An S-channel π/2 pulse can be used to convert this polarization stored along –z to observable transverse magnetization–we refer to this final pulse as the conversion pulse. Notice that the S-spin polarization can also be produced along +z, either by reversing the chirped pulse sweep direction or by phase-shifting the ¹H excitation by π/2 vis-à-vis the ensuing spin-lock pulse. This behavior was exploited by implementing a two-step phase cycle in all experiments presented herein, to filter out any initial S-spin polarization arising from thermal processes (which would lie along +z).

Figure 2.

Schematic pulse sequences for (a) BRAIN-CP, a cross-polarization sequence based on a broadband adiabatic full passage; (b) BRAIN-CP/echo, a version incorporating a hard π-pulse echo, (c) BRAIN-CP/CPMG, a multiple-hard-pulse echo version; and (d) BRAIN-CP/WCPMG, a fully broadband version of (c) which also incorporates adiabatic frequency sweeps during the acquisition period. These schematic representations assume a WURST envelope and a linear chirp—yet ample flexibility arises in these functions. See text for further details.

A variety of conversion schemes can be used to observe the ±S_z spin polarization produced by BRAIN-CP. For example, a single echo or a train of echoes can be acquired by using the BRAIN-CP-based sequences shown in Figures 2b and 2c, respectively. The use of hard π/2 and π S-pulses, however, could deprive these sequences of their “broad-bandedness.” Alternatively, the ±S_z polarization produced by BRAIN-CP can be excited and observed using broadband methods; for instance, via the WCPMG pulse sequence in which the conversion and refocusing pulses are also frequency swept. An example of the resulting BRAIN-CP/WCPMG sequence is shown in Figure 2d, where different labels denote the swept pulses used for polarization transfer (A), for conversion (B) and for refocusing (C). The BRAIN-CP/WCPMG pulse sequence allows for the S/N advantages of both CP and CPMG, and provides broadband characteristics in both these portions of the experiment.

Theoretical Analysis: CP during an Adiabatic Pulse

We consider the effects of the pulse sequence shown in Figure 2a on an isolated I,S spin pair during the contact period. The Hamiltonian may be defined in the doubly rotating frame, with spin I as the polarization source and spin S the target, as:

$$H(t)={\Omega }_I{I}_Z+{\Omega }_S{S}_Z+2{\omega }_D{I}_Z{S}_Z+{\omega }_{1,I}{I}_X+{\omega }_{1,S}(t)[S_X\text{cos}(\Psi (t)+S_Y\text{sin}(\Psi (t)]$$ (1)

Here, Ω_I and Ω_S are the rotating-frame resonance offsets of the I and S spins, ω_D is the dipolar coupling constant of the I,S spin pair, and the RF amplitudes applied to each spin are given by ω_1,I and ω_1,S. The phase profile for the S channel RF is set according to the explicit form of a WURST pulse:

$$\Psi (t)=\frac{R{t}^2}{2}-\frac{\Delta \omega }{2}t+{\Psi }_0$$ (2)

which generates an instantaneous frequency shift of the RF pulse on that channel of

$${\omega }_{RF}(t)=\dot{\Psi }(t)=Rt-\frac{\Delta \omega }{2}$$ (3)

where the time, t, runs from 0 to τ_RF, Ψ₀ is the initial RF phase, and the instantaneous irradiation frequency sweeps over the range Δω, from ±½Δω to ∓½Δω, at the rate R = Δω/(τ_RF). Inclusion of a WURST-like amplitude profile could have an effect on the spin dynamics of S nuclei with offset frequencies that are close to the edges of the frequency sweep (i.e., when Ω_S is near ±Δω/2); however, we expect that most of the behavior is captured by treating ω_1,S as a constant.

After applying consecutive rotations to transform first to the FM frame²⁹ and then to a B_eff parallel to S_Z′ frame,^30,32 in which the effective magnetic field is along the z′-axis, the Hamiltonian can be expressed as

$$\begin{array}{l}\tilde{H}(t)={\Omega }_I^{\mathit{\text{eff}}}{I}_Z+{\Omega }_S^{\mathit{\text{eff}}}(t)S_{Z^{\prime }}\\ -2{\omega }_D[I_Z\text{cos}({\theta }_I)-I_X\text{sin}({\theta }_I)][S_{Z^{\prime }}\text{cos}({\theta }_S(t))-S_{X^{\prime }}\text{sin}({\theta }_S(t))]\end{array}$$ (4)

where the effective resonance frequencies of the I and S nuclei are

$$\begin{array}{l}{\Omega }_I^{\mathit{\text{eff}}}=\sqrt{{\Omega }_I^2+{\omega }_{1,I}^2}\\ {\Omega }_S^{\mathit{\text{eff}}}(t)=\sqrt{{({\Omega }_S-{\omega }_{RF}(t))}^2+{\omega }_{1,S}^2}\end{array}$$ (5)

Notice that after these transformations the effective fields influencing I and S lie along the z′-axes of their respective B_eff frame, which subtend angles of θ_I and θ_S(t) with respect to the z-axes of a standard rotating frame, as described by

$$\begin{array}{l}\text{cos}({\theta }_I)={\Omega }_I/{\Omega }_1^{\mathit{\text{eff}}}\\ \text{cos}[{\theta }_S(t)]=[{\Omega }_S-{\omega }_{RF}(t)]/{\Omega }_S^{\mathit{\text{eff}}}(t)\end{array}$$ (6)

When the frequency sweep rate R is small, as in an adiabatic passage, the function θ_S(t) varies slowly and further time dependencies can be disregarded. Making the substitution

$$\tilde{U}(t)=\text{exp}\{-i{\Omega }_I^{\mathit{\text{eff}}}t{I}_Z-i{\displaystyle {\int }_0^t{\Omega }_S^{\mathit{\text{eff}}}(t)dt{S}_Z\}\tilde{\tilde{U}}(t)}$$ (7)

one obtains a Hamiltonian

$$\begin{array}{l}\tilde{\tilde{H}}(t)=-2{\omega }_D[I_Z\text{cos}({\theta }_I)-I_X\text{sin}({\theta }_I)\text{cos}({\Omega }_I^{\mathit{\text{eff}}}t)-I_Y\text{sin}({\theta }_I)\text{sin}({\Omega }_I^{\mathit{\text{eff}}}t)]\times \\ \times [S_Z\text{cos}({\theta }_S(t))-S_X\text{sin}({\theta }_S(t))\text{cos}({\displaystyle {\int }_0^t{\Omega }_S^{\mathit{\text{eff}}}(t)dt})-S_Y\text{sin}({\theta }_S(t))\text{sin}({\displaystyle {\int }_0^t{\Omega }_S^{\mathit{\text{eff}}}(t)dt})]\end{array}$$ (8)

An approximation to the dynamics can be obtained by considering the spins evolution under the average of Eq. 8

$${\tilde{\tilde{H}}}_{AV}=\frac{1}{{\tau }_{RF}}{\displaystyle \underset{0}{\overset{{\tau }_{RF}}{\int }}\tilde{\tilde{H}}(t)dt}$$ (9)

During the bulk of the irradiation period, the modulation frequencies ${\Omega }_S^{\mathit{\text{eff}}}(t)$ and ${\Omega }_I^{\mathit{\text{eff}}}$ impose fast oscillations on $\tilde{\tilde{H}}$ and therefore make no contributions to the mean Hamiltonian. However, whenever ${\Omega }_S^{\mathit{\text{eff}}}(t)\cong {\Omega }_I^{\mathit{\text{eff}}},$ the integral of $\tilde{\tilde{H}}$ may accumulate. This condition is analogous to the Hartmann-Hahn match,¹⁶ except that in the swept case the time-dependent match will occur at instants when the instantaneous frequency offset of the S-channel chirp fulfills

$${\omega }_{RF}(t)={\Omega }_S\pm \sqrt{{\Omega }_I^2+{\omega }_{1,I}^2-{\omega }_{1,S}^2}$$ (10)

Assuming the abundant spins are on resonance, Eq. (10) describes a simple situation for the case where ω_1,I = ω_1,S: a polarization transfer occurs when the S-channel sweep is on-resonance with the S spin (i.e., when ω_RF = Ω_S). On the other hand, when ω_1,I > ω_1,S, polarization transfer occurs at two distinct positions in the sweep that lie on either side of the Ω_S resonance offset. It is also possible that a single match condition occurs when ω_1,I > ω_1,S, depending on the exact range of the RF sweep with respect to Ω_S. These locations of these “match” criteria are further explored below with the aid of numerical simulations.

In order for this kind of broadband CP effect to be useful, it is important that the “matching” condition be maintained long enough for significant polarization to be transferred. Furthermore, for the experiment to be efficiently optimized, it is important to determine how the CP dynamics depend on each experimental parameter. It can be shown (see Appendix) that if the effective RF fields on both channels are equal, cross polarization can occur at each Ω_S position during a time interval τ_CP, given by

$${\tau }_{CP}=\sqrt[3]{24\pi {\omega }_{1,S}/R^2}$$ (11)

It is more difficult to provide a simple estimate of the effective CP matching time when unequal RF fields are involved; still, in the ω_1,I >> ω_1,S case, this can be approximated as

$${\tau }_{CP}=2\sqrt[4]{\frac{{\pi }^2}{R^1}\left(1-\frac{{\omega }_{1,s}^2}{{\omega }_{1,I}^2}\right)}$$ (12)

Not surprisingly, a longer period of polarization transfer can be obtained by using either ${\omega }_{1,S}^{\text{max}}={\omega }_{1,I}$ irradiation fields or by sweeping through the S frequency offsets at a slower rate.

BRAIN-CP: Numerical Simulations

Numerical simulations of the behavior of an isolated I,S spin system during the contact portion of the BRAIN-CP experiment were also performed. Simulations are presented for an initial density operator ρ(0) ∝ I_X propagated numerically with the Hamiltonian in Eq. 1. All simulations are for a 10 ms chirped RF pulse swept at a constant rate from -75 to +75 kHz and a WURST-40 amplitude profile with a maximum RF power level ${\omega }_{1,S}^{\text{max}}/2\pi =30\text{kHz}.$ In each simulation, the expectation values 〈I_X〉 (t) and 〈S_Z〉 (t) were calculated in order to monitor the polarization lost by spin I and gained by spin S during the contact period.

The polarization transferred under a range of I-channel RF power levels and S-spin frequency offsets is shown in Fig. 3. In each simulation, the point of polarization transfer from I to S is clear from the drop in 〈I_X〉 (t) ; the ensuing progression of the generated S magnetization toward the –z axis is also apparent. In Fig. 3a, where ${\omega }_{1,S}^{\text{max}}/2\pi ={\omega }_{1,I}/2\pi =30\text{kHz},$ the simulations evidence a single polarization transfer that occurs at the point where the frequency of the RF sweep equals each S-spin frequency offset (i.e., when ω_RF(t) = Ω_S), in agreement with the prediction in Eq. 10. In Fig. 3b, where ${\omega }_{1,S}^{\text{max}}\ne {\omega }_{1,I},$ two separate polarization transfers are evident for the on-resonance Ω_S = 0 case. By contrast, the same set of simulations shows that when Ω_S = 60 kHz, a single transfer happens as only one of the two offset conditions of Eq. 10 (ω_RF(t)/2π = 60 kHz ±33.5 kHz) lies within the ±75 kHz range of the RF sweep. This behavior is further illustrated in Fig. 3c, where the larger difference in RF power levels is such that two polarization transfers are possible only when Ω_S is near the center of the sweep.

Figure 3.

Numerical simulations of polarization transfers for single I,S spin pairs during frequency-swept CP contact. Conditions for each simulation are: (a) ${\omega }_{1,I}/2\pi ={\omega }_{1,S}^{\text{max}}/2\pi =30\text{KHz},$ ω_D/2π = 0.5 kHz; (b) ω_1,I/2π = 45 kHz, ${\omega }_{1,S}^{\text{max}}/2\pi =30\text{KHz},$ ω_D/2π = 1.3 kHz; and (c) ω_1,I/2π = 60 kHz, ${\omega }_{1,S}^{\text{max}}/2\pi =30\text{KHz},$ ω_D/2π = 1.5 kHz. For the initial conditions associated with each panel, results are calculated for anisotropic offsets Ω_S/2π= 0 kHz (red), 20 kHz (blue), 40 kHz (green), and 60 kHz (purple), while Ω_I was set to 0 in all simulations. The transfer is monitored via calculation of the expectation values <I_X> and <S_Z>, which are shown, respectively, at the top (with initial conditions <I_X> = 1) and middle (initial conditions <S_Z> = 0) of the frame.

Effects of the I-S dipolar coupling strength on the polarization transfers were also investigated. The S-spin polarization calculated at the end of the transfer period is shown in Fig. 4a as a function of ω_D under each of the three RF power conditions described above. The behavior under the lowest power setting clearly demonstrates that the polarization transfer is quenched when the dipolar coupling is higher than a certain threshold value. Most likely, this results from the spin-lock only being maintained when the RF power is of sufficient magnitude to dominate all interactions. The polarization transfer curves for ω_1,I/2π = 45 and 60 kHz power levels display an interesting oscillation as a function of ω_D (Fig. 4a), representing efficient or poor polarization transfer at various points. The origin of this behavior is made clear by time-dependent simulations (Figs. 4b, 4c), expanding on the behavior at selected positions of the ω_D-curve in Fig. 4a: the second of the two distinct transfer conditions associated with the frequency sweep may result in further polarization transfer from spin I to spin S (Fig. 4b), or in a reverse transfer, in which some or all of the polarization is returned to spin I (Fig. 4c). Oscillatory shuttling of polarization back and forth between spins in an isolated spin pair is well known in conventional CP,³⁴ but does not typically affect solid state results as the oscillation is often damped out by proton spin diffusion or powder averaging in transfers over long time periods.^35–37 We therefore expect BRAIN-CP to produce polarization transfers in a net I-to-S direction under usual experimental conditions; however, these results also suggest that the effectiveness of the transfer may vary somewhat from crystallite-to-crystallite.

Figure 4.

Numerical simulations of the dipolar coupling dependence of polarization transfers for single I,S spin pairs during BRAIN-CP. (a) Polarization transferred to the S spin at the end of the contact pulse versus I-S dipolar coupling constant, for three different RF power levels: ω_1,I /2π = 30 kHz, ${\omega }_{1,S}^{\text{max}}/2\pi =30\text{KHz}$ (yellow); ω_1,I /2π = 45kHz, ${\omega }_{1,S}^{\text{max}}/2\pi =30\text{KHz}$ (green); and ω_1,I /2π = 60 kHz, ${\omega }_{1,S}^{\text{max}}/2\pi =30\text{KHz}$ (purple). Two positions on the ω_1,I /2π = 45 kHz curve displayed in panel (a) are expanded upon in (b) and (c), in which the time evolution of <I_X> (red) and <S_Z> (blue) is shown in panel (b) for the dipolar coupling that yields the most complete transfer (ω_D/2π = 1.4 kHz) and in panel (c) for the dipolar coupling that yields the least effective transfer (ω_D/2π = 3.6 kHz).

This last feature was investigated further with a series of numerical SPINEVOLUTION³⁸ calculations probing the ability of BRAIN-CP to provide a faithful representation of a broad powder pattern dominated by chemical shift anisotropy (CSA). An ABCX spin system comprising one S and three I = ¹H nuclei was deemed to serve as a useful model system including CSA effects (modeled on ¹¹⁹Sn at 9.4 T) as well as both hetero- and homonuclear dipolar coupling (protons are assumed to be on resonance). The results of these simulations (Fig. 5) demonstrate that an accurate powder pattern is obtained when ω_1,I(¹H)/ω_1,S ≅2. Under these conditions, the simulations indicate a polarization transfer equal to approximately half of the γ_I/γ_S theoretical maximum. Given that this particular set of simulations only accounts for one of a multitude of possible spin systems and associated NMR parameters, we do not place much weight on this enhancement value; still, such simulations highlight the usefulness and validity of the pulse sequences presented here.

Figure 5.

Simulations of ¹H-to-S BRAIN-CP polarization transfers for an ABCX spin system with one ¹¹⁹Sn and three ¹H nuclei, using SPINEVOLUTION.³⁸ For each simulation, ${\omega }_{1,S}^{\text{max}}/2\pi (S{=}^{119}\text{Sn})$ is 24.5 kHz while the value of ω_1,I(I = ¹H) is as listed in the figure. The model system has S-spin chemical shift tensor components: δ₁₁ = 50 kHz, δ₂₂ = -12 kHz and δ₃₃ = -38 kHz; three equivalent ${\omega }_D^{1,S}/2\pi {(}^1\text{H}{,}^{119}\text{Sn})$ values of 3.3 kHz; and ${\omega }_D^{I,I}/2\pi {(}^1\text{H}{,}^1\text{H})$ values of 3, 3.7, and 21 kHz. The inset shows the idealized S-spin powder pattern, which is most accurately reproduced with BRAIN-CP when ω₁(¹H)/2π ≈ 40 kHz.

3. Experimental

NMR spectra were acquired using a Varian Infinity Plus console and a 9.4 T Oxford magnet at resonance frequencies of 149.0 MHz for ¹¹⁹Sn, 83.5 MHz for ²⁰⁷Pb, and 71.4 MHz for ¹⁹⁹Hg. All ¹¹⁹Sn experiments were performed using a Varian/Chemagnetics 4 mm HX MAS probe with samples packed in zirconia rotors, while ²⁰⁷Pb and ¹⁹⁹Hg experiments employed a Varian/Chemagnetics 5 mm HX static probe and samples packed in glass tubes. Power calibrations at each frequency were performed using separate experiments with liquid solution standards. ¹¹⁹Sn experiments were referenced to liquid tetramethylstannane at 0 ppm, ¹⁹⁹Hg experiments to a saturated aqueous solution of Hg(ClO₄)₂ at -2253 ppm (with respect to Hg(CH₃)₂ at 0 ppm), and ¹⁹⁵Pt experiments to a 1 M aqueous solution of Na₂PtCl₆ at 0 ppm. All spectra were recorded using proton decoupling power levels of ω₁/2π(¹H) = 30 kHz to 50 kHz.

Simulations of polarization transfers were performed using either custom made routines written for Matlab or with SPINEVOLUTION,³⁸ as noted in the text. Lineshape simulations of the total powder patterns were generated analytically with WSOLIDS.³⁹

4. Results and Discussion

¹¹⁹Sn NMR Experiments

¹¹⁹Sn has a relatively high gyromagnetic ratio (e.g., ν_L = 149 MHz at 9.4 T), and often exhibits broad powder patterns arising from large CSAs,⁴⁰ making ¹¹⁹Sn an ideal starting point for testing the broadbanded-ness of our methods. Dibutyltin(IV) oxide, whose 110 kHz simulated static spectrum is shown in Fig. 6a,^40–41 was employed as test sample. Spectra acquired using conventional CP are displayed in Figs. 6b and 6c. The use of a high RF power for the match condition results in a successful enhancement of nearly the entire width of the powder pattern whereas a low-power match results in excitation of only a fraction of the total powder pattern neighboring the on-resonance position. Evidently, the low power condition is not optimal, but mimics samples spanning the ultra-wide bandwidths alluded to earlier. By contrast, the chirped BRAIN-CP approach successfully enhances the entire breadth of dibutyltin oxide’s ¹¹⁹Sn powder pattern without the need for high RF power levels (Fig. 6d). The contact-pulse sweep used in this example to provide the broadband CP transfer involved a WURST envelope and a carrier offset swept from just above to just below the spectral edges, covering a total range of 125 kHz. Interestingly, the same optimum RF power is found for the ¹¹⁹Sn contact pulse in both conventional CP and BRAIN-CP experiments when the amplitude of the ¹H contact pulse is fixed at ω₁/2π(¹H) = 20 kHz for each experiment. Furthermore, the sensitivities to mis-sets of the match power are found to be similar in both methods. There is a slight difference in the optimum contact time (5 ms for conventional CP vs. 10 ms for BRAIN-CP). We ascribe the slower polarization transfer observed with BRAIN-CP to the fact that the frequency sweep brings the RF pulse into contact with each isochromat of the powder pattern during only a portion of the total WURST pulse.

Figure 6.

¹¹⁹Sn static NMR spectra of dibutyltin(IV) oxide. (a) Simulated spectrum based on literature chemical shift tensor parameters δ_iso = -173 ppm, Ω = 737 ppm (110 kHz at 9.4 T), and κ = 0.12;⁴⁰–⁴¹ these values, and those in subsequent figures, are given using the Maryland convention.¹ (b),(c) Echo spectra recorded using conventional CP with a 5 ms contact time. Spectrum (b) was acquired with a moderately high RF power [ω_1,I/2π(¹H) = 70 kHz, ω_1,S/2π(¹¹⁹Sn) = 65 kHz], and (c) with a low RF power [ω_1,I/2π = 20 kHz, ω_1,S/2π = 25 kHz]. (d) BRAIN-CP spectra recorded using a 10 ms contact time with low RF powers identical to those employed for spectrum (c). (e) CP/CPMG spectrum recorded using conventional CP with the contact time and RF powers set to equal those applied for the acquisition of spectrum (c). (f),(g). Hard-pulse and swept-pulse CPMG echo variants of the BRAIN-CP experiment, with contact time and RF powers identical to those employed for the experiment displayed in (d). Conventional rectangular pulses used in the echo and CPMG experiments used ω₁/2π(¹¹⁹Sn) = 55 kHz, while the frequency-swept WURST-B (echo-train) pulses were swept over a range of 1 MHz with a power level of ω₁/2π(¹¹⁹Sn) = 33 kHz. All echo train experiments (CPMG and WCPMG) were recorded as fifty 300 µs echoes.

Given the success of BRAIN-CP in increasing both the bandwidth of the polarization transfer, the viability of combining this process with CPMG-style acquisitions was tested. A conventional CP/CPMG spectrum, acquired under low-power CP matching conditions, is shown in Fig. 6e. While the initial ¹¹⁹Sn polarization generated by CP is equivalent, the S/N boost provided by the multi-echo acquisition of the CPMG loop allowed the CP/CPMG spectrum to be acquired using 1/10^th the number of scans. The BRAIN-CP/CPMG spectrum shown in Fig. 6f, collected using identical pulse powers (and for the acquisition portions, equal timings) as the CP/CPMG spectrum, demonstrates the successful combination of broadband polarization transfer with CPMG acquisition. Still, even for this test spectrum, the observable bandwidth limited by the inversion profile of the rectangular π pulses used in the CPMG train, as evidenced by the loss of intensity in the outer portions of the CPMG-derived powder pattern vis-à-vis its static BRAIN-CP counterpart. These distortions in the BRAIN-CP/CPMG spectrum highlight the need for broadband refocusing pulses in the CPMG train, if one is to obtain a truly ultra-wideline spectrum of a more challenging sample. The spectrum shown in Fig. 6g demonstrates that such broadband polarization/refocusing experiments are feasible if BRAIN-CP is used in conjunction with the S/N enhancement of the (also broadband) WCPMG pulse sequence, using the pulse sequence shown in Fig. 1d. It is also interesting to note that the ensuing experiment appears to be quite robust, as doubling the contact-pulse sweep range to 250 kHz produces an indistinguishable BRAIN-CP/WCPMG spectrum (not shown), with approximately the same optimum ¹¹⁹Sn RF power, and a slightly longer optimum contact time of 15 ms.

¹⁹⁹Hg NMR Experiments

Acquisition of ¹⁹⁹Hg solid-state NMR spectra represents an even more difficult challenge for broadband CP experiments, as ¹⁹⁹Hg has a lower gyromagnetic ratio than ¹¹⁹Sn (e.g., ν_L = 71 MHz at 9.4 T for ¹⁹⁹Hg), and is often associated with CSA-dominated powder patterns that are thousands of ppm wide.⁴⁰ Mercury acetate has previously been used to demonstrate the benefits of combining conventional CP with the CPMG protocol,^42–43 and is therefore a fitting material with which to benchmark BRAIN-CP. The ¹⁹⁹Hg CP/CPMG spectrum of mercury acetate is displayed in Figure 7b together with the ~130 kHz broad simulated spectrum. Despite the application of maximum RF power levels for the equipment, the excitation profile of conventional CP covers only a portion of the total spectrum, in agreement with previous reports which found 6 to 9 CP/CPMG sub-spectra were required to obtain the entire powder pattern.^42–43 By contrast, the BRAIN-CP/WCPMG sequence of Fig. 1d yields the mercury acetate spectrum shown in Fig. 7c, where the broader excitation profile is evident. Similar to the ¹¹⁹Sn experiments, conventional CP and BRAIN-CP have nearly identical optimum match conditions, and also similar sensitivity to mis-setting of these power levels. Furthermore, the optimum contact time is again slightly longer for BRAIN-CP (25 ms), as compared to conventional CP (15 ms). We also note that a nearly identical spectrum (not shown) could be obtained using BRAIN-CP/WCPMG with a 500 kHz contact-pulse sweep width, a slightly longer contact time (30 ms) and a similar optimum RF power, suggesting that even extremely broad ¹⁹⁹Hg powder patterns should be easily observable by this route. Moreover, whereas a recent report has demonstrated that direct ¹⁹⁹Hg excitation using frequency-swept pulses also allows for the acquisition of undistorted mercury acetate spectra,¹⁵ the WCPMG experiment was found to be quite inefficient due to the lack of CP’s ~ γ_H/γ_Hg signal enhancement and the extremely slow ¹⁹⁹Hg longitudinal relaxation: 5xT₁(¹⁹⁹Hg) ≈ 25 minutes. By comparison, the benefits of using BRAIN-CP/WCPMG are reflected in the fact that the spectrum in Fig. 7c was acquired ca. 50 times faster than a similar quality spectrum could be collected with direct excitation using the WCPMG method.

Figure 7.

Simulated and experimental ¹⁹⁹Hg SSNMR spectra of mercury acetate. (a) Simulated spectrum based on the literature parameters δ_iso = -2496 ppm, Ω = 1850 ppm (131 kHz at 9.4 T), and κ = 0.87.⁴² (b) CP/CPMG spectrum acquired with RF power levels of ω_1,I/2π(¹H) = 40 kHz and ω_1,S/2π(¹⁹⁹Hg) = 27.5 kHz during the 15 ms contact time and ω₁/2π(¹⁹⁹Hg) = 60 kHz refocusing pulses (c) BRAIN-CP/WCPMG spectrum acquired with a 250 kHz sweep and RF powers of ω_1,I/2π(¹H) = 40 kHz and ${\omega }_{1,S}^{\text{max}}/2\pi$ (¹⁹⁹Hg) = 30 kHz during the 25 ms contact time; 50 µs pulses swept over a range of 1 MHz at ω₁/2π(¹⁹⁹Hg) = 32 kHz were used for the WURST-B (echo-train) pulses. The spectra in both (b) and (c) were recorded as thirty 330 µs echoes.

¹⁹⁵Pt NMR Experiments

As a final and particularly challenging test of the BRAIN-CP experiment, its use in ¹⁹⁵Pt solid-state NMR spectroscopy was investigated. It is well known that spans of ¹⁹⁵Pt chemical shift tensors are generally thousands of ppm in square planar environments,^40,44 generating spectra that are several hundred kHz broad on moderate and high-field NMR spectrometers. Pt[NH₃]₄Cl₂ exhibits these characteristics, and was used as a test sample. The ¹⁹⁵Pt shift parameters of this compound are: δ_iso = -2540(60) ppm, Ω = 7250(100) ppm, and κ = -0.96(1); further details of the NMR spectroscopy and its interpretation will be reported in a forthcoming publication. The ¹⁹⁵Pt NMR spectrum of powdered Pt[NH₃]₄Cl₂ acquired using CP/CPMG is shown in Fig. 8a, evidencing that only a small fraction of the approximately 600 kHz wide powder pattern is excited. Acquisition of the entire powder pattern using CP/CPMG at multiple transmitter positions was not attempted, but it is estimated that ca. 35 sub-spectra would be required for its faithful representation. Obviously, the narrow excitation bandwidth of conventional CP is a severe impediment to ¹⁹⁵Pt ultrawide NMR spectroscopy.

Figure 8.

Experimental and simulated ¹⁹⁵Pt NMR spectra of polycrystalline Pt[NH₃]₄Cl₂. (a) CP/CPMG spectrum acquired using RF power levels of ω_1,I/2π(¹H) = 47 kHz and ω_1,S/2π(¹⁹⁵Pt) = 27.5 kHz during the 5 ms contact time; ). Seventy-five 200 µs echoes were recorded using refocusing pulses with ω₁/2π(¹⁹⁵Pt) = 100 kHz. (b) Direct-excitation WCPMG spectrum acquired with 96 scans (128 scans were employed for all CP spectra to account for the 6 s recycle delay necessary for direct ¹⁹⁵Pt excitation differing from the 4 s recycle delay of CP experiments). Seventy-five 200 µs echoes were recorded using 50 µs WURST-B pulses swept over 2 MHz at ω₁/2π(¹⁹⁵Pt) = 52 kHz. (c)-(e) BRAIN-CP/WCPMG experiments acquired using the indicated contact-pulse sweep widths, 20ms contact times, and RF power levels of ω_1,I/2π(¹H) = 47 kHz and ${\omega }_{1,S}^{\text{max}}/2\pi {(}^{195}\text{Pt})$ = 27.5 or 28.5 kHz. Each spectrum was recorded using identical echo train timings and refocusing pulse parameters to those employed in (b). (f) The total spectrum of Pt[NH₃]₄Cl₂ from the co-addition of 5 sub-spectra acquired by stepping the transmitter in 150 kHz intervals and employing 500 kHz contact-pulse sweep widths. The simulated spectrum is based on best-fit parameters δ_iso = -2540(60) ppm, Ω = 7250(100) ppm (620 kHz at 9.4 T), and κ = -0.96(1) ppm.

The WCPMG spectrum of Pt[NH₃]4Cl₂ in Fig. 8b demonstrates the broadband capability of this frequency swept acquisition method,⁴⁴ but also highlights the significantly lower signal intensity afforded by direct excitation in comparison to the CP/CPMG spectrum. In this case, the reduction in signal is due partly to the longer recycle delay associated with direct excitation (6 s) versus CP experiments (4 s) and the slightly poorer refocusing efficiency of the WURST pulses, yet the most important factor is the larger ¹⁹⁵Pt spin polarization afforded by the CP experiment.

¹⁹⁵Pt BRAIN-CP/WCPMG spectra of Pt[NH₃]₄Cl₂, optimized using three different contact-pulse sweep widths, are shown in Figs. 8c-8e. Experiments with all three contact-pulse sweep widths are found to require nearly identical power levels and optimal match conditions, and these are furthermore similar to the optimum power level for the conventional CP experiment. It is immediately apparent from this data that the BRAIN-CP/WCPMG spectra provides a broader excitation while retaining the S/N advantages of CP. When a 1 MHz contact-pulse sweep width is applied, some signal intensity is excited over nearly the entire width of the spectrum, and in fact, the same excitation bandwidth as the extremely broadband WCPMG method is achieved. In this case the limiting factor is more likely the excitation and/or receiving bandwidths of the probe, rather then the performances of either the BRAIN-CP or WCPMG portions of the experiments. The same contact time was used for each experiment; however, it should be noted that no intensity was lost when a 10 ms contact time was used with the 250 kHz sweep, and memory limitations of the waveform generator prevented the investigation of contact times longer than 20 ms. Therefore, the behavior of the signal intensity as a function of contact time in these BRAIN-CP ¹⁹⁵Pt experiments is not inconsistent with that found for the other nuclei. Given the extremely large excitation bandwidth of the BRAIN-CP/WCPMG pulse sequence, it is possible to obtain even the 650 kHz broad ¹⁹⁵Pt spectrum of Pt[NH₃]₄Cl₂ using a small number of sub-spectra and a short total experimental time (e.g., Fig. 8f). It has previously been noted that the broadband refocusing of WCPMG makes it a preferred method over CP/CPMG for samples with characteristics similar to those of Pt[NH₃]₄Cl₂; the present results demonstrate that BRAIN-CP/WCPMG represents a further increase in efficiency. We also note that broadband direct ¹⁹⁵Pt excitation experiments are often precluded by sample characteristics that are more challenging than those of Pt[NH₃]₄Cl₂: the sample may be available in limited quantities, have a dilute Pt content (Pt[NH₃]₄Cl₂ is nearly 60% Pt by weight), or possess unfavorable ¹⁹⁵Pt relaxation properties (¹⁹⁵Pt T₁ values are often 10x times greater and ¹⁹⁵Pt T₂ values much shorter than those of Pt[NH₃]₄Cl₂).

5. Conclusions

The BRAIN-CP pulse sequence, which uses frequency-swept adiabatic inversion pulses to achieve broad CP excitation profiles, has been introduced. Experimental results demonstrate that ¹H nuclei can be used to polarize ¹¹⁹Sn, ¹⁹⁹Hg and ¹⁹⁵Pt nuclei with high efficiency over very broad frequency regions. In particular, the execution of CP along an FM-frame spin-locking field allows broad powder patterns to become uniformly polarized, leading to up to a tenfold reduction in the number of required scans as compared to conventional CP. Moreover, these experiments are found to require similar optimum RF power levels to conventional CP, and their spectral quality is relatively insensitive to changes in the contact-pulse sweep width or contact time. A detailed theoretical analysis provides insight into the behavior of the pulse sequence; as noted above, polarization transfer can occur when the S-channel offset creates an effective magnetic field equaling that experienced by the spin-locked ¹H nuclei. Furthermore, we show here that this transfer can be satisfied for most isochromats in broad powder pattern by sweeping the S-channel offset over a suitable range. This mechanism of polarization transfer leads to a large increase in bandwidth—not at the expense of power, but rather with the price of extended contact times. Experimental results demonstrate that the cost of this increase is well worth paying, making the ensuing BRAIN-CP sequence an attractive alternative for the polarization of CSA-broadened signals. The limits of the resulting bandwidths appear to be governed by the probehead’s electronics rather than by spin-physics or waveform generating constraints. It is likely that the use of flat-Q circuits⁴⁵ could help overcome this particular hurdle. It is also worth noting the potential relevance of this scheme towards the polarization enhancement of quadrupole-broadened central and satellite transitions. This aspect, together with extensions of BRAIN-CP analogues for the case of magic-angle spinning, is actively being investigated. Finally, it is important to note that it is the adiabaticity of the X-channel contact pulse that ensures a homogeneous magnetization transfer across the broad powder pattern. Therefore, the same basic mechanism for CP could be taken advantage of with a wide variety of alternate adiabatic inversion shapes beyond simple linear frequency chirps. The results obtained using BRAIN-CP and frequency-swept WCPMG demonstrate high efficiency and ease of use, and the method shows clear promise for providing spectroscopic access to many nuclei with anisotropically broadened powder patterns in a wide array of materials.

Supplementary Material

Appendix Figure A1: Graphs of the phase function Φ(t), as well as its cosine and integrated cosine functions, that govern the polarization transfer conditions, see Appendix for further discussion.

Appendix

We derive here an estimate of the time interval during which polarization transfer can occur using approximations to the equations for an isolated I,S spin pair described in the Theory Section of the main text. Polarization transfer is possible when the instantaneous RF frequency satisfies Eq. 10, and therefore occurs at time(s) t_CP of the sweep

$$t_{CP}={\tau }_{RF}\left(\frac{1}{2}+\frac{{\Omega }_S}{\Delta \omega }\pm \sqrt{{\Omega }_I^2+{\omega }_{1,I}^2-{\omega }_{1,S}^2}\right)$$ (A.1)

provided that 0 ≤ t_CP ≤ τ_RF. Polarization transfer should be possible during the interval around t_CP for which the Hamiltonian of Eq. 8 is relatively unchanging. If the rapidly oscillating term

$$\begin{array}{l}-2{\omega }_D\text{cos}\left({\theta }_I\right)[-S_x\text{sin}\left({\theta }_S\left(t\right)\right)\text{cos}({\displaystyle {\int }_0^t{\Omega }_S^{\mathit{\text{eff}}}\left(t\right)dt})-2{\omega }_D{S}_Y\text{sin}\left({\theta }_S(t)\right)\text{sin}({\displaystyle {\int }_0^t{\Omega }_S^{\mathit{\text{eff}}}\left(t\right)dt})]I_z-\\ -2{\omega }_D\text{sin}({\Omega }_I^{\mathit{\text{eff}}}t)\text{cos}({\displaystyle {\int }_0^t{\Omega }_S^{\mathit{\text{eff}}}\left(t\right)dt})I_Y{S}_X+\text{cos}({\Omega }_I^{\mathit{\text{eff}}}t)\text{sin}({\displaystyle {\int }_0^t{\Omega }_S^{\mathit{\text{eff}}}\left(t\right)dt})I_Y{S}_X\end{array}$$ (A.2)

of Eq. 8 is neglected, the Hamiltonian during the match intervals can be written as

$$\begin{array}{l}\tilde{\tilde{H}}(t)\cong -2{\omega }_D\text{cos}\left({\theta }_I\right)\text{cos}\left({\theta }_S(t)\right)I_Z{S}_Z-\\ -2{\omega }_D\text{sin}\left({\theta }_I\right)\text{sin}\left({\theta }_S(t)\right)\frac{1}{2}\text{cos}({\Omega }_I^{\mathit{\text{eff}}}t-{\displaystyle {\int }_0^t{\Omega }_S^{\mathit{\text{eff}}}\left(t\right)dt})(I_X{S}_X+I_Y{S}_Y)\end{array}$$ (A.3)

We note that the effective Hamiltonian in Eq. A.3 is similar to that for conventional CP. Given that θ(t) is a slowly varying function, the only oscillations are from the cosine term whose phase is given by

$$\Phi (t)={\Omega }_I^{\mathit{\text{eff}}}t-{\displaystyle {\int }_0^t{\Omega }_S^{\mathit{\text{eff}}}(t)dt}$$ (A.4)

The change in Φ from t = t_CP to t_CP + t' is

$$\Phi (t_{CP}+t^{\prime })-\Phi (t_{CP})={\omega }_{1,I}{t}^{\prime }-{\displaystyle {\int }_{T_{CP}}^{t_{CP}+t^{\prime }}{\Omega }_S^{\mathit{\text{eff}}}(t)dt}$$ (A.5)

where, for simplicity, we have taken Ω_I = Ω_S = 0. Expanding Eq. A.5 to third order as a Taylor series yields the phase change as

$$\Phi (t_{CP}+t^{\prime })-\Phi (t_{CP})=\pm \frac{1}{2}\sqrt{1-\frac{{\omega }_{1,S}^2}{{\omega }_{1,I}^2}}R{t^{\prime }}^2+\frac{1}{6}\frac{{\omega }_{1,S}^2}{{\omega }_{1,I}^2}{R}^2{t^{\prime }}^3$$ (A.6a)

where the ± is related to the two possible match conditions of Eq. 12. In cases where ω_1,I = ω_1,S, the quadratic term in t' vanishes and the condition of Eq. A.6a becomes simplified to

$$\Phi (t_{CP}+t^{\prime })-\Phi (t_{CP})=\frac{R^2{t^{\prime }}^3}{6{\omega }_{1,S}}$$ (A.6b)

We define phase changes in Eqs. A.6 of < |π/2| as an approximation of the period during which the effective Hamiltonian A.3 varies little, allowing the mean Hamiltonian to build up and CP transfer to occur.

According to this analysis, when ω_1,I = ω_1,S CP occurs during the interval τ_CP given by

$${\tau }_{CP}=\sqrt[3]{24\pi {\omega }_{1,S}/R^2}$$ (A.7)

To better illustrate the behavior, the phase function Eq. A.4 (and its Taylor series approximation) are shown in Figure A1 for a BRAIN-CP transfer with Δω/2π = 150 kHz, τ_RF = 10 ms and ω_1,I/2π = ω_1,S/2π = 20 kHz. The period during which Φ(t) is relatively stable is apparent in the first panel, as are the rapid oscillations in cos[Φ(t)] in the middle panel. It is only near the center portion of the sweep that the rapid oscillation of the cosine terms does not average to zero, and the integral of cos[Φ(t)], rightmost panel of Figure A1, only builds up during this time. Estimation of τ_CP directly from Figure A1 gives τ_CP ≅ 0.7 ms, in excellent agreement with the estimation of 1.0 ms derived from Eq. A.7. A simple estimate analogous to Eq. A.7 is not possible when ω_1,I ≠ ω_1,S, however, if ω_1,I >> ω_1,S, the cubic contribution can be neglected such that the CP interval is approximated by

$${\tau }_{CP}=2\sqrt[4]{\frac{{\pi }^2}{R^2}\left(1-\frac{{\omega }_{1,S}^2}{{\omega }_{1,I}^2}\right)}$$ (A.8)

For the case where ω_1,I/2π = 2*ω_1,S/2π = 40 kHz, Δω/2π = 150 kHz and τ_RF = 10 ms, one gets τ_CP ≅ 0.5 ms which agrees with simulations performed for this matching condition. This analysis clearly demonstrates that contact between the spins is not an instantaneous moment, but rather occurs over a significant interval. We also note that τ_CP may be somewhat larger in experimental systems as the analysis presented is for a simple 2-spin system, for example, a range of Ω_S values would broaden the condition.