^{1}, Paul Hodgkinson

^{2}, Beat H. Meier

^{1}and Matthias Ernst

^{1,a)}

### Abstract

A theoretical description of the two-pulse phase-modulated (TPPM) decoupling sequence in magic-angle spinning NMR is presented using a triple-mode Floquet approach. The description is formulated in the radio-frequency interaction-frame representation and is valid over the entire range of possible parameters leading to the well-known results of continuous-wave (cw) decoupling and XiX decoupling in the limit of a phase change of 0° and 180°, respectively. The treatment results in analytical expressions for the heteronuclear residual coupling terms and the homonuclear spin-diffusion terms. It also allows the characterization of all resonance conditions that can contribute in a constructive or a destructive way to the residual linewidth. Some of the important resonance conditions are described for the first time since they are not accessible in previous treatments. The combination of the contributions from the residual couplings and the resonance conditions to the effective Hamiltonian, as obtained in a Floquet description, is shown to be required to describe the decoupling behavior over the full range of parameters. It is shown that for typical spin system and experimental parameters a linewidth of approximately can be obtained for TPPM decoupling in an organic solid or a protein. This is a major contribution to the experimentally observed linewidths of around and indicates that decoupling techniques are still one of the limiting factors in the achievable linewidths.

Financial support was provided by the Swiss National Science Foundation and the ETH Zürich through the TH system. We would like to thank Herbert Zimmermann from The MPI für Medizinische Forschung, Heidelberg for the preparation of the labeled glycine ethylester sample.

I. INTRODUCTION

II. EXPERIMENTAL DATA

III. THEORY

IV. RESULTS AND DISCUSSION

V. CONCLUSIONS

### Key Topics

- Linewidths
- 33.0
- Tensor methods
- 16.0
- Protons
- 9.0
- Sequence analysis
- 8.0
- Nuclear magnetic resonance
- 7.0

## Figures

Schematic representation of the TPPM sequence.

Schematic representation of the TPPM sequence.

Experimental peak height (peak height after Fourier transformation without an apodization function) of the group in -glycine ethylester under TPPM decoupling as a function of the pulse length and the phase angle at different experimental conditions: (a) , , (b) , , (c) , , and (d) , . The experiments were run on a Varian with a proton Larmor frequency of using a 1.8 or a double-resonance MAS probe. The phase resolution of the measurements was 0.25° and the time resolution was . The rf-field amplitudes were determined using a proton nutation experiment. The position of the highest intensity is marked by a white +. Numerical values for the parameters at the peak maxima can be found in Table I.

Experimental peak height (peak height after Fourier transformation without an apodization function) of the group in -glycine ethylester under TPPM decoupling as a function of the pulse length and the phase angle at different experimental conditions: (a) , , (b) , , (c) , , and (d) , . The experiments were run on a Varian with a proton Larmor frequency of using a 1.8 or a double-resonance MAS probe. The phase resolution of the measurements was 0.25° and the time resolution was . The rf-field amplitudes were determined using a proton nutation experiment. The position of the highest intensity is marked by a white +. Numerical values for the parameters at the peak maxima can be found in Table I.

Plot of the complex phase and the magnitude of the Fourier coefficients for , , and for a rf field amplitude of for two different pulse lengths ((a)–(c)) and ((d)–(f)) and three different phase angles ((a) and (d)), 20° ((b) and (e)), and 30° ((c) and (f)). The diameter of the circles indicates the magnitude and the color the sign and complex phase of the Fourier coefficients. Blue indicates that the Fourier coefficient is positive and real, red that it is negative and real, black that it is positive and imaginary, and green that it is negative and imaginary.

Plot of the complex phase and the magnitude of the Fourier coefficients for , , and for a rf field amplitude of for two different pulse lengths ((a)–(c)) and ((d)–(f)) and three different phase angles ((a) and (d)), 20° ((b) and (e)), and 30° ((c) and (f)). The diameter of the circles indicates the magnitude and the color the sign and complex phase of the Fourier coefficients. Blue indicates that the Fourier coefficient is positive and real, red that it is negative and real, black that it is positive and imaginary, and green that it is negative and imaginary.

(a) Experimental peak height (peak height after Fourier transformation without apodization) of the group in -glycine ethylester under TPPM decoupling as a function of the pulse length and the phase angle at and . The experiments were run on a Varian with a proton Larmor frequency of using a double-resonance MAS probe. The phase resolution of the measurements was 0.25° and the time resolution was . The rf-field amplitude was determined using a proton nutation experiment. Numerical simulations of the peak height as a function of the pulse length and the phase angle using (b) a five-spin system and (c) three-spin -type spin system. The simulations were carried out using the PNMRSIM simulation package. The spinning frequency in all simulations was set to and the rf amplitude was set to . The phase resolution of the simulation was 0.25° and the time resolution was . All dipolar couplings were included in the simulations as well as isotropic and anisotropic chemical shifts. The peak height was simulated for a powder average of 1000 crystallite orientations.

(a) Experimental peak height (peak height after Fourier transformation without apodization) of the group in -glycine ethylester under TPPM decoupling as a function of the pulse length and the phase angle at and . The experiments were run on a Varian with a proton Larmor frequency of using a double-resonance MAS probe. The phase resolution of the measurements was 0.25° and the time resolution was . The rf-field amplitude was determined using a proton nutation experiment. Numerical simulations of the peak height as a function of the pulse length and the phase angle using (b) a five-spin system and (c) three-spin -type spin system. The simulations were carried out using the PNMRSIM simulation package. The spinning frequency in all simulations was set to and the rf amplitude was set to . The phase resolution of the simulation was 0.25° and the time resolution was . All dipolar couplings were included in the simulations as well as isotropic and anisotropic chemical shifts. The peak height was simulated for a powder average of 1000 crystallite orientations.

Numerical simulations of the peak height as a function of the pulse length and the phase angle using a three-spin system as used in Fig. 4(b) but with an extended range of parameters. (a) Full simulation including all interactions. (b) Simulation without the proton chemical-shift tensor. (c) Simulation without the proton homonuclear dipolar coupling. Otherwise, in all simulations the same parameters were used as for the simulation shown in Fig. 4(b), except that the phase resolution of the simulation was 0.5° and the time resolution was .

Numerical simulations of the peak height as a function of the pulse length and the phase angle using a three-spin system as used in Fig. 4(b) but with an extended range of parameters. (a) Full simulation including all interactions. (b) Simulation without the proton chemical-shift tensor. (c) Simulation without the proton homonuclear dipolar coupling. Otherwise, in all simulations the same parameters were used as for the simulation shown in Fig. 4(b), except that the phase resolution of the simulation was 0.5° and the time resolution was .

Contour plots (blue and red lines) of the coefficients for (a) , ; (b) , ; (c) , ; and (d) , . These contour plots are superimposed on a grayscale density plot of the numerical simulations of a spin system shown in Fig. 5(a). In addition, the resonance conditions (straight lines) and (curved lines) are shown using green lines.

Contour plots (blue and red lines) of the coefficients for (a) , ; (b) , ; (c) , ; and (d) , . These contour plots are superimposed on a grayscale density plot of the numerical simulations of a spin system shown in Fig. 5(a). In addition, the resonance conditions (straight lines) and (curved lines) are shown using green lines.

Contour plot of the sum of the heteronuclear coefficients for (a) and (c) and the sum of the homonuclear coefficients for (b) and (d) . These plots are superimposed on a grayscale density plot of the numerical simulations of a spin system shown in Fig. 5(a). In addition, the resonance conditions and are shown using green lines.

Contour plot of the sum of the heteronuclear coefficients for (a) and (c) and the sum of the homonuclear coefficients for (b) and (d) . These plots are superimposed on a grayscale density plot of the numerical simulations of a spin system shown in Fig. 5(a). In addition, the resonance conditions and are shown using green lines.

Plots of the resonance conditions encountered in TPPM decoupling. The location of the resonance conditions are plotted as lines (blue or color coded: zeroth-order resonance conditions: green: second-order or third-order resonance conditions) superimposed on a grayscale density plot of the numerical simulations of a spin system shown in Fig. 5(a). (a) Straight lines correspond to the resonance conditions which recouple heteronuclear dipolar couplings; curved lines correspond to resonance conditions which also recouple heteronuclear dipolar coupling. The strength of the resonance condition is given by the magnitude of the and Fourier coefficient, respectively. The magnitude of some of the zeroth-order resonance conditions has been color coded on the lines. (b) resonance conditions which result in zeroth order ( and 2) in a purely homonuclear dipolar Hamiltonian. Again, the magnitude of the resonance terms has been color coded on the line. (c) resonance condition. (d) resonance condition.

Plots of the resonance conditions encountered in TPPM decoupling. The location of the resonance conditions are plotted as lines (blue or color coded: zeroth-order resonance conditions: green: second-order or third-order resonance conditions) superimposed on a grayscale density plot of the numerical simulations of a spin system shown in Fig. 5(a). (a) Straight lines correspond to the resonance conditions which recouple heteronuclear dipolar couplings; curved lines correspond to resonance conditions which also recouple heteronuclear dipolar coupling. The strength of the resonance condition is given by the magnitude of the and Fourier coefficient, respectively. The magnitude of some of the zeroth-order resonance conditions has been color coded on the lines. (b) resonance conditions which result in zeroth order ( and 2) in a purely homonuclear dipolar Hamiltonian. Again, the magnitude of the resonance terms has been color coded on the line. (c) resonance condition. (d) resonance condition.

Contour plots of the decoupling efficiency shown in Fig. 2 ((a) , , (b) , , (c) , , and (d) , ) with the location of the theoretical minimum of the second-order cross terms plotted as a white dotted line. The resonance conditions are shown as white lines for purely homonuclear resonance conditions and as black line for heteronuclear and homonuclear resonance conditions.

Contour plots of the decoupling efficiency shown in Fig. 2 ((a) , , (b) , , (c) , , and (d) , ) with the location of the theoretical minimum of the second-order cross terms plotted as a white dotted line. The resonance conditions are shown as white lines for purely homonuclear resonance conditions and as black line for heteronuclear and homonuclear resonance conditions.

Comparison of TPPM and CM decoupling for direct observation [(a) and (c)] and the spin-echo experiment [(b) and (d)]. The plots show the experimental peak height (peak height after Fourier transformation without an apodization function) of the group in -glycine ethylester under TPPM or CM decoupling as a function of the pulse length and the phase angle at a spinning frequency of and a rf-field amplitude of : (a) peak height under TPPM decoupling , (c) peak height under CM decoupling , (b) peak height after a spin-echo sequence with and TPPM decoupling , (d) peak height after a spin-echo sequence with and CM decoupling . The experiments were run on a Varian Infinity+ spectrometer with a proton Larmor frequency of using a double-resonance MAS probe. The phase resolution of the measurements was 0.25° and the time resolution was . The rf-field amplitudes were determined using a proton nutation experiment. The position of the highest intensity is marked by a white +.

Comparison of TPPM and CM decoupling for direct observation [(a) and (c)] and the spin-echo experiment [(b) and (d)]. The plots show the experimental peak height (peak height after Fourier transformation without an apodization function) of the group in -glycine ethylester under TPPM or CM decoupling as a function of the pulse length and the phase angle at a spinning frequency of and a rf-field amplitude of : (a) peak height under TPPM decoupling , (c) peak height under CM decoupling , (b) peak height after a spin-echo sequence with and TPPM decoupling , (d) peak height after a spin-echo sequence with and CM decoupling . The experiments were run on a Varian Infinity+ spectrometer with a proton Larmor frequency of using a double-resonance MAS probe. The phase resolution of the measurements was 0.25° and the time resolution was . The rf-field amplitudes were determined using a proton nutation experiment. The position of the highest intensity is marked by a white +.

## Tables

Experimental parameters of Fig. 2.

Experimental parameters of Fig. 2.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content