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- Abstract
- 1. Introduction
- 2. Theory
- 3. Numerical Simulations
- 4. Results
- 5. Chemical shift anisotropy selective inversion during cross-polarization
- 6. Conclusions
- References

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J Magn Reson. Author manuscript; available in PMC 2010 October 1.

Published in final edited form as:

Published online 2009 July 9. doi: 10.1016/j.jmr.2009.07.003

PMCID: PMC2745485

NIHMSID: NIHMS131600

Department of Chemistry and Francis Bitter Magnet laboratory, Massachusetts Institute of Technology, Cambridge Massachusetts 02139

The publisher's final edited version of this article is available at J Magn Reson

Magic Angle Spinning (MAS) is used in solid-state NMR to remove the broadening effects of the chemical shift anisotropy (CSA). In this work we investigate a technique that can reintroduce the CSA in order to selectively invert transverse magnetization. The technique involves an amplitude sweep of the radio frequency field through a multiple of the spinning frequency. The selectivity of this inversion mechanism is determined by the size of the CSA. We develop a theoretical framework to describe this process and demonstrate the CSA selective inversion with numerical simulations and experimental data. We combine this approach with cross polarization (CP) for potential applications in multi-dimensional MAS NMR.

Almost all approaches to selective excitation are related to the pioneering work of Alexander [1] who devised a method of pulsed excitation of one resonance while leaving a chemically shifted neighbor unaffected (subjected to rotation by 2π radians). Although modern selective inversion methods use amplitude- and/or phase-modulated pulses, the selectivity is nevertheless based upon differences in the chemical shift. In this work we discuss a magic-angle spinning (MAS) NMR experiment in which the selection criterion is the anisotropy of the chemical shielding, rather than the chemical shift itself.

The chemical shift anisotropy (CSA) is rotationally averaged at MAS frequencies higher than the span of the CSA. However, the CSA interaction can be reintroduced with rotary resonance recoupling by matching the RF field strength to a multiple of the MAS frequency [2]. When the resonance condition is met with a constant value of *ω*_{1} = *nω _{r}* (

This paper is organized as follows. In Section 2 we outline the theoretical framework for CSA selective inversion. In Section 3 we present numerical simulations, which relate the efficiency of inversion to parameters such as the anisotropy, spinning frequency, and the duration of the RF amplitude ramp. In section 4 we present experimental data to demonstrate some of these effects. Finally, in section 5 we show that CSA selective inversion can be combined with ramped cross-polarization, especially at such high MAS frequencies that the cross-polarization matching condition can be close to one of the rotary resonance recoupling conditions [4, 5].

The pulse sequence starts with a period of cross-polarization (CP) followed by a period during which the ^{13}C RF amplitude is swept through a CSA recoupling condition as shown in Figure 1b. Proton decoupling is applied during the ^{13}C RF amplitude sweep, at a power level high enough to avoid depolarization effects during this period [6, 7].

(a) Depiction of an adiabatic sweep of the Hamiltonian from *I*_{x} to −*I*_{x}. The initial spin state *σ*(−*τ*^{−}) is projected from *I*_{x} onto the Hamiltonian as the RF ramp begins. The polarization is spin-locked along the Hamiltonian **...**

We shall describe the process of creating an adiabatic passage through rotary resonance via average Hamiltonian theory (AHT). For simplicity, we assume a single spin and consider only the RF irradiation and the CSA terms of the Hamiltonian. We restrict the RF amplitude variation, *ω*_{1}(*T*), to be slow relative to the MAS rotation so that the Hamiltonian in the rotating frame takes the form of Equation (1).

$$H(t,T)={\omega}_{1}(T){I}_{x}+\sum _{k=-2}^{2}{c}_{k}{e}^{-ik{\omega}_{r}t}{I}_{z}$$

(1)

The Fourier expansion parameters, *c _{k}*, of the modulated chemical shift depend upon the CSA parameters and the orientation of the CSA tensor in the rotor frame and are given by Equation (2).

$${c}_{k}=\sqrt{{\scriptstyle \frac{2}{3}}}{\omega}_{0}\sum _{{k}^{\prime}=-2}^{2}{\rho}_{2.{k}^{\prime}}{D}_{{k}^{\prime},k}^{2}(\alpha ,\beta ,\gamma ){D}_{k,0}^{2}(0,{\theta}_{m},0)$$

(2)

where *ω _{0}* is the Larmor frequency,

In a doubly rotating frame, defined by the additional transformation operator *R = e*^{−}^{in}* ^{ωrtIx}*, the Hamiltonian becomes:

$${H}_{R}(T,t)=\mathrm{\Delta}{\omega}_{1}(T){I}_{x}+\frac{1}{2}\sum _{k=-2}^{2}{c}_{k}{e}^{-ik{\omega}_{r}t}(({I}_{z}+i{I}_{y}){e}^{in{\omega}_{r}t}+({I}_{z}-i{I}_{y}){e}^{-in{\omega}_{r}t})$$

(3)

where Δ*ω*_{1}(*T*) = *ω*_{1}(*T*) − *nω _{r}*. The zeroeth order average for

$${\stackrel{\sim}{H}}_{0}(T)=\mathrm{\Delta}{\omega}_{1}(T){I}_{x}+({\scriptstyle \frac{{c}_{-n}-{c}_{n}}{2i}}){I}_{y}+({\scriptstyle \frac{{c}_{-n}+{c}_{n}}{2}}){I}_{z}$$

(4)

The expression [(*c*_{−}* _{n}* −

For n=1,

$${c}_{\mathit{eff}}={\omega}_{0}\delta sin{\theta}_{m}cos{\theta}_{m}\sqrt{({\scriptstyle \frac{\eta}{4}}{d}_{2,-1}^{2}(\beta )cos(2\alpha )+\sqrt{{\scriptstyle \frac{3}{2}}}{d}_{1,0}^{2}(\beta ))({\scriptstyle \frac{\eta}{2}}{d}_{2,1}^{2}(\beta )cos(2\alpha )+\sqrt{{\scriptstyle \frac{3}{2}}}{d}_{1,0}^{2}(\beta ))}$$

(5)

For n= 2,

$${c}_{\mathit{eff}}=\frac{{\omega}_{0}\delta {sin}^{2}{\theta}_{m}}{2}\sqrt{({\scriptstyle \frac{\eta}{4}}{d}_{2,-2}^{2}(\beta )cos(2\alpha )+\sqrt{{\scriptstyle \frac{3}{2}}}{d}_{2,0}^{2}(\beta ))({\scriptstyle \frac{\eta}{4}}{d}_{2,2}^{2}(\beta )cos(2\alpha )+\sqrt{{\scriptstyle \frac{3}{2}}}{d}_{2,0}^{2}(\beta ))}$$

(6)

To estimate the size of the *c _{eff}* for a given

In order to adiabatically sweep the Hamiltonian through the recoupling axis we must satisfy two conditions: first minimize the angle of the initial and final projections of the spin polarization onto the Hamiltonian and second ensure the angular sweep rate of the Hamiltonian is slow enough to be adiabatic. The first condition is satisfied when Δ*ω*_{1}(±*τ*) ≥ 5** c_{eff}**[9]. The second condition is determined by the angular sweep rate (

For the following discussion, we will detail the construction of an adiabatic ramp through the *n* = 2 rotary resonance condition that selectively inverts a spin with *c _{eff}* of 2 kHz while leaving one of 0.5 kHz unchanged. We choose the total RF ramp width to be

At the *n* = 1 condition, the CSA tensor, the homonuclear [11], and the heteronuclear dipolar coupling tensor are all recoupled whereas at the *n* = 2 condition only the CSA tensor and the heteronuclear dipolar-coupling tensor are reintroduced [12]. This means that the *n* = 1 recoupling condition becomes very complicated and is unlikely to be useful except in situations where homonuclear dipolar couplings are negligible. Therefore, we restrict the remainder of the discussion to the *n* = 2 condition.

At the *n* = 2 condition, CSA and heteronuclear dipolar couplings are simultaneously recoupled. If the dipolar interaction is much smaller than the CSA, it amounts to only a slight perturbation of the larger CSA tensor. Dipolar coupling constants for directly bonded ^{13}C-^{15}N pairs range from approximately 1 to 2 kHz. For ^{13}C-^{14}N pairs, the larger spin of ^{14}N, *I* = 1, together with the smaller gyromagnetic ratio yield dipolar effects roughly comparable in magnitude. These dipolar couplings are smaller than most ^{13}C CSAs at high field and, more significantly, are smaller than the differences between large and small ^{13}C CSAs (e.g., between carbonyl and aliphatic), and therefore do not significantly affect the site-specific selectivity of CSA inversion.

For this reason, ^{15}N decoupling was not utilized during the CSA recoupling interval in this work. However, experiments involving selective inversion of ^{13}C sites with small CSA would likely require both ^{15}N labeling, to avoid ^{14}N-^{13}C couplings, together with ^{15}N decoupling. In such an experiment, the RF levels on ^{1}H, ^{13}C, and ^{15}N must be chosen with care to avoid interfering recoupling conditions such as CP and TSAR [13].

In the experimental data presented in this article, ^{1}H decoupling was accomplished by 100 kHz CW irradiation to avoid cross polarization/depolarization effects. In the absence of ^{1}H RF irradiation, the *n* = 1 and *n* = 2 conditions would recouple the very strong ^{1}H-^{13}C dipolar interaction (~25 kHz) and inversion with CSA selectivity would be impossible.

Although the expressions derived in the previous section provide theoretical insight into the spin dynamics of CSA selective inversion, we believe it helpful to examine numerical simulations of the relationship between the CSA, the experimental parameters and the efficiency of inversion. The simulations were generated with SPINEVOLUTION 3.3.3 with powder averaging over 300 triplets of the (α, β, γ) Euler angles [14]. The simulations ignore the effects of relaxation, but include the effects of ±10% RF inhomogeneity. Since the major parameters that affect the efficiency of CSA selective inversion are the MAS frequency and the RF power level, both of which are conveniently described in frequency units, the anisotropy is also described in frequency units, using the symbol *ω*_{0}*δ*, making the simulations independent of the Larmor frequency.

Figure 2 simulates of the effect of changing the duration of an RF ramp through the *ω*_{1} = 2*ω _{r}* recoupling condition, by plotting the resulting transverse magnetization against

Numerical simulations of the effect of the duration of an RF amplitude ramp on the transverse magnetization plotted against the CSA in frequency units with a fixed asymmetry (η = 1) at the *ω*_{1} = 2*ω*_{r} recoupling condition. The RF **...**

The simulations in Figure 3 demonstrate the effect of varying the MAS frequency while using a constant ramp rate. Since the ramp extends from 1.5*ω _{r}* to 2.5

Figure 4 shows the experimental result of changing the magnitude of the RF amplitude ramp after CP on a sample of [U-^{13}C, ^{15}N] L-tryptophan spinning at 22 kHz in a static magnetic field of 9.4 Tesla. After CP, the RF field amplitude was ramped down from 56.5 kHz (at time zero) to a final value of 31.5 kHz over a period of 2 ms. the ramp was interrupted at fixed intervals and ^{13}C spectra acquired. Thus, in Figure 4, the time axis corresponds to magnitude of the ramp. In essence, it demonstrates the behavior of the ^{13}C magnetization during a CSA recoupling experiment. We can see that the ramp has little effect on the magnetization before the 2*ω _{r}* recoupling condition (roughly midway through the ramp) whereas after the midpoint of the ramp, signals from

Experimental results showing the peak heights of the carbon sites of [U-^{13}C, ^{15}N] L-tryptophan at various points during a CSA-selective inversion ramp. The aliphatic sites (shown in blue) remain positive while the carbonyl and aromatic sites (shown in **...**

Figure 5 demonstrates the effect of changing the ramp rate on a sample of [U-^{13}C, ^{15}N] N-*acetyl*-valyl-leucine (VL) plotted against the duration of an RF ramp of 25 kHz, centered at the *ω*_{1} = 2*ω _{r}* CSA recoupling condition. The spectra were recorded at

The pulse sequence (shown in Figure 1b) used to generate Figures 4 and and55 explicitly separates the initial cross-polarization period from the CSA recoupling period. However, it is possible to combine these two periods into a modification of the amplitude swept CP experiment [16, 17]. Amplitude swept or ramped CP can either be thought of as a method of reducing the instrumental imperfections such as RF inhomogeneity [16, 18] or as a method of improving the polarization transfer via Adiabatic Passage Hartmann-Hahn (APHH) CP [17]. The theoretical gain of the APHH CP compared to conventional CP in large spin systems (*I*_{n}*S*) has been investigated. It was shown that APHH CP should outperform non-ramped CP by a factor of close to two [19]. This gain is predicted to occur when the width of the ramp is approximately equal to or greater than the MAS frequency.

There are 4 separate choices for implementation of ramped CP. The RF ramp can be applied to either channel, and RF power during the ramp can either increase or decrease. These choices can produce different effects.

Figure 6 shows effect of changing the direction of an RF amplitude sweep through a CSA rotary resonance condition during ramped CP on ^{13}C spectra of [U-^{13}C,^{15}N] N-*formyl* MLF-OH spinning at 31 kHz in a static magnetic field of 17.6 Tesla. Notice that there was no change in the magnitude of the RF amplitude sweep in Figure 6, only the direction. In order to produce CSA selective inversion during ramped CP, significant magnetization must have been built up via CP before the RF amplitude reaches the CSA recoupling condition. That is, the sweep must pass through a CP matching sideband [20] before reaching the CSA recoupling condition, and therefore the inversion only occurs with an amplitude sweep in the correct direction.

(Top) Ramped CP spectrum of [U-^{13}C, ^{15}N] N-*f*-MLF-OH (MLF) spinning at 31 kHz in a static magnetic field strength of 17.6 Tesla with a ^{13}C RF amplitude sweep passing through the *ω*_{1} = 2*ω*_{r} CSA recoupling condition before passing through the **...**

This combination of CSA selective inversion and CP can be put to use. In Figure 7 we show a 2D Proton Driven Spin Diffusion (PDSD) [21] ^{13}C–^{13}C correlation spectrum of [U-^{13}C, ^{15}N] N-*formyl* MLF-OH under similar conditions as Figure 6, in which the carbonyl and aromatic region were inverted with respect to the aliphatic region, simply by adjusting the parameters of ^{13}C RF ramp during the initial cross-polarization. Cross-peaks arising from the inverted spins appear negative (green) while those arising from non-inverted spins appear positive (red). This form of selective inversion can be achieved without increasing the duration of the pulse sequence, and/or without the addition of additional pulse sequence elements such as frequency selective soft pulses. This technique could also be thought of as the basis of a method of measuring CSAs in 2D correlation spectra at high MAS frequencies.

We have examined the process of chemical shift anisotropy selective inversion in solids undergoing MAS, both in theory and in practice. We have shown that it is relatively easy to invert carbonyl and aromatic sites (both of which have high anisotropies) while merely attenuating the aliphatic sites (which have low anisotropies) using ramps with various durations between 2 and 5 ms at static magnetic fields between 9.4 and 17.6 Tesla. Since the adiabaticity of the CSA inversion depends on the size of the CSA, performing these experiments at higher magnetic fields should allow sites with smaller CSA to be inverted. Higher MAS frequencies should also be beneficial, as it permits a larger range of sweep sizes and rates. The combination of higher static magnetic fields and higher MAS frequencies should allow for greater control of selective inversion as a function of CSA.

CSA selective inversion can be combined with CP by sweeping the RF amplitude through a sideband of the CP matching profile before extending the sweep to match a multiple of the spinning frequency. In this situation, the results are dependent not only upon the magnitude of the RF amplitude sweep but also the direction. In conditions where the CP matching power is near a multiple of the spinning frequency, this combination of CP and CSA inversion may be encountered inadvertently while attempting to perform ramped CP.

Since protons generally have small CSA values it might seem preferable to apply the amplitude ramp to the proton channel during ramped CP experiments in order to avoid CSA recoupling. However, with the increasing use of proton Larmor frequencies of 900 MHz and above, a proton CSA of 10 PPM produces a *ω*_{0}*δ* of 9 kHz. The simulations shown in Figures 2 and and33 demonstrate that this is quite large enough to cause CSA inversion. While reversing the direction of RF amplitude sweep can often reduce these effects, the only safe way to prevent CSA recoupling is to avoid all recoupling conditions (except CP) on both channels.

The research was supported by the National Institutes of Biomedical Imaging and Bioengineering through grants EB-003151 and EB-002026. We would like to thank Dr. M. Veshtort, and Dr. G. De Paepe for simulating discussions. We would also like to give special thanks to an unknown referee who alerted us to the work cited in reference [3] which was previously unknown to us.

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