^{1,a)}and Stephen Wimperis

^{2}

### Abstract

Spin-locking of spin and nuclei in the presence of small resonance offset and second-order quadrupolar interactions has been investigated using both exact and approximate theoretical and experimental nuclear magnetic resonance(NMR) approaches. In the presence of second-order quadrupolar interactions, we show that the initial rapid dephasing that arises from the noncommutation of the state prepared by the first pulse and the spin-locking Hamiltonian gives rise to tensor components of the spin density matrix that are antisymmetric with respect to inversion, in addition to those symmetric with respect to inversion that are found when only a first-order quadrupolar interaction is considered. We also find that spin-locking of multiple-quantum coherence in a static solid is much more sensitive to resonance offset than that of single-quantum coherence and show that good spin-locking of multiple-quantum coherence can still be achieved if the resonance offset matches the second-order shift of the multiple-quantum coherence in the appropriate reference frame. Under magic angle spinning (MAS) conditions, and in the "adiabatic" limit, we demonstrate that rotor-driven interconversion of central-transition single- and three-quantum coherences for a spin nucleus can be best achieved by performing the spin-locking on resonance with the three-quantum coherence in the three-quantum frame. Finally, in the "sudden" MAS limit, we show that spin spin-locking behavior is generally similar to that found in static solids, except when the central-transition nutation rate matches a multiple of the MAS rate and a variety of rotary resonance phenomena are observed depending on the internal spin interactions present. This investigation should aid in the application of spin-locking techniques to multiple-quantum NMR of quadrupolar nuclei and of cross-polarization and homonuclear dipolar recoupling experiments to quadrupolar nuclei such as , , , , and .

I. INTRODUCTION

II. PULSE SEQUENCES FOR SPIN-LOCKING

III. THEORETICALMODEL

IV. EXPERIMENTAL DETAILS

V. RESULTS AND DISCUSSION

A. Static solids

B. Spin-locking under MAS

C. Rotary resonanceeffects in the sudden regime

VI. CONCLUSIONS

### Key Topics

- Coherence
- 34.0
- Dephasing
- 25.0
- Nuclear magnetic resonance
- 22.0
- Coherent effects
- 16.0
- Tensor methods
- 15.0

## Figures

Spin-locking pulse sequences allowing the observation of (a) and and (b) and coherence transfer. The duration, , and radiofrequency field strength, , of the spin-locking pulse are indicated. In each sequence, the duration and field strength of the preparation pulse are optimized for the excitation of the desired coherence order and in (b) the phase is cycled to select the conversion of the desired nQ coherence order back into observable 1Q coherence.

Spin-locking pulse sequences allowing the observation of (a) and and (b) and coherence transfer. The duration, , and radiofrequency field strength, , of the spin-locking pulse are indicated. In each sequence, the duration and field strength of the preparation pulse are optimized for the excitation of the desired coherence order and in (b) the phase is cycled to select the conversion of the desired nQ coherence order back into observable 1Q coherence.

Simulated NMR spectra showing the positions of the central-transition 1Q (solid), 3Q (dashed), and 5Q (dotted) spin and line shapes for (a) a single crystallite (with ), (b) a static powder, and (c) a powder under MAS. Simulations were performed with and 375 kHz , and .

Simulated NMR spectra showing the positions of the central-transition 1Q (solid), 3Q (dashed), and 5Q (dotted) spin and line shapes for (a) a single crystallite (with ), (b) a static powder, and (c) a powder under MAS. Simulations were performed with and 375 kHz , and .

Expectation values of spin spherical tensor operators, , created by rapid dephasing of an initial state (a, b) and (c, d) , as a function of , considering (a, c) the first-order and (b, d) the first- and second-order quadrupolar interactions. Tensors with coherence order (p) 0, 1, 2, and 3 are denoted by black, red, blue, and green lines, while tensors with rank (l) 1, 2, and 3 are denoted by solid, dotted, and dashed lines, respectively. Other parameters include , , , and .

Expectation values of spin spherical tensor operators, , created by rapid dephasing of an initial state (a, b) and (c, d) , as a function of , considering (a, c) the first-order and (b, d) the first- and second-order quadrupolar interactions. Tensors with coherence order (p) 0, 1, 2, and 3 are denoted by black, red, blue, and green lines, while tensors with rank (l) 1, 2, and 3 are denoted by solid, dotted, and dashed lines, respectively. Other parameters include , , , and .

The norm of the spin density operator created by rapid dephasing of an initial state (a) and (b) , as a function of , shown for either a first-order quadrupolar interaction (solid line) or, additionally, a second-order quadrupolar interaction (dashed line). Other parameters are as in Fig. 3.

The norm of the spin density operator created by rapid dephasing of an initial state (a) and (b) , as a function of , shown for either a first-order quadrupolar interaction (solid line) or, additionally, a second-order quadrupolar interaction (dashed line). Other parameters are as in Fig. 3.

Two-dimensional greyscale plot of the expectation values of (a, c) and (b, d) created by rapid dephasing of a spin initial state (a, b) and (c, d) , under a spin-locking Hamiltonian that includes the first- and second-order quadrupolar interactions, as a function of and . Positive signal intensity is shown in white while negative intensity in red. In each case, a cross section through the two-dimensional plot is shown for (dashed line). For comparison, similar cross sections considering the quadrupolar interaction to first order only (solid line) are also shown. Other parameters are as in Fig. 3.

Two-dimensional greyscale plot of the expectation values of (a, c) and (b, d) created by rapid dephasing of a spin initial state (a, b) and (c, d) , under a spin-locking Hamiltonian that includes the first- and second-order quadrupolar interactions, as a function of and . Positive signal intensity is shown in white while negative intensity in red. In each case, a cross section through the two-dimensional plot is shown for (dashed line). For comparison, similar cross sections considering the quadrupolar interaction to first order only (solid line) are also shown. Other parameters are as in Fig. 3.

(a, b) Expectation values of spin spherical tensor operators, , created by rapid dephasing of an initial state , under a spin-locking Hamiltonian that includes the first- and second-order quadrupolar interactions as a function of , with a resonance offset, , of (a) 7382 Hz and (b) −3164 Hz. Tensors with coherence order 0, 1, 2, and 3 are denoted by black, red, blue, and green lines, while tensors with rank (l) 1, 2, and 3 are denoted by solid, dotted, and dashed lines, respectively. (c) Norm, , of the spin density operator, created by rapid dephasing of an initial state , under a spin-locking Hamiltonian that includes either the first- (solid line) or, additionally, the second-order quadrupolar interaction as a function of with of 0 (dotted line), 7382 Hz (dashed line), and −3164 Hz (long-dashed line). Other parameters are as in Fig. 3.

(a, b) Expectation values of spin spherical tensor operators, , created by rapid dephasing of an initial state , under a spin-locking Hamiltonian that includes the first- and second-order quadrupolar interactions as a function of , with a resonance offset, , of (a) 7382 Hz and (b) −3164 Hz. Tensors with coherence order 0, 1, 2, and 3 are denoted by black, red, blue, and green lines, while tensors with rank (l) 1, 2, and 3 are denoted by solid, dotted, and dashed lines, respectively. (c) Norm, , of the spin density operator, created by rapid dephasing of an initial state , under a spin-locking Hamiltonian that includes either the first- (solid line) or, additionally, the second-order quadrupolar interaction as a function of with of 0 (dotted line), 7382 Hz (dashed line), and −3164 Hz (long-dashed line). Other parameters are as in Fig. 3.

Expectation values of spin single-element operators, , created by rapid dephasing of initial states (a) , (b) , and (c) as a function of , for Hamiltonians including first- (solid line) and, additionally, second-order (dashed line) quadrupolar interactions. Expectation values for , 3/2, and 5/2 are denoted by black, red, and blue lines, respectively. Other parameters include , , , and .

Expectation values of spin single-element operators, , created by rapid dephasing of initial states (a) , (b) , and (c) as a function of , for Hamiltonians including first- (solid line) and, additionally, second-order (dashed line) quadrupolar interactions. Expectation values for , 3/2, and 5/2 are denoted by black, red, and blue lines, respectively. Other parameters include , , , and .

Two-dimensional greyscale plot of the spin expectation values of single-element operators, , created by rapid dephasing of initial states , and , under a spin-locking Hamiltonian that includes the first- and second-order quadrupolar interactions, as a function of and . Other parameters are as in Fig. 7.

Two-dimensional greyscale plot of the spin expectation values of single-element operators, , created by rapid dephasing of initial states , and , under a spin-locking Hamiltonian that includes the first- and second-order quadrupolar interactions, as a function of and . Other parameters are as in Fig. 7.

Spin-locking of 3Q coherences with an initial state , simulated for a spin static powder using an exact density matrix approach. [(a)–(c)] 3Q amplitude as a function of the duration, , of a spin-locking pulse with (a) first- and (b, c) first- and second-order quadrupolar Hamiltonians included. In each case , , and . In (a, b), and are varied between 12.5 and 200 kHz. In (c), the offset was either 0 (black line) or chosen to match the 1Q (red line), 3Q (green line), or 3Q/3 (blue line) isotropic shift, with , 750, or 250 Hz, respectively, with a fixed of 25 kHz. (d) Average (over ) spin-locked 3Q amplitude as a function of , with , for either first-order (black line) or, additionally, second-order (red line) quadrupolar interactions.

Spin-locking of 3Q coherences with an initial state , simulated for a spin static powder using an exact density matrix approach. [(a)–(c)] 3Q amplitude as a function of the duration, , of a spin-locking pulse with (a) first- and (b, c) first- and second-order quadrupolar Hamiltonians included. In each case , , and . In (a, b), and are varied between 12.5 and 200 kHz. In (c), the offset was either 0 (black line) or chosen to match the 1Q (red line), 3Q (green line), or 3Q/3 (blue line) isotropic shift, with , 750, or 250 Hz, respectively, with a fixed of 25 kHz. (d) Average (over ) spin-locked 3Q amplitude as a function of , with , for either first-order (black line) or, additionally, second-order (red line) quadrupolar interactions.

Experimental (132.3 MHz) NMR of static powdered . (a) Central-transition and signal intensity as a function of the radiofrequency field strength of the spin-locking pulse with . (b) signal intensity as a function of the duration, , of a spin-locking pulse for a variety of field strengths. In (a), the relative intensity of the 3Q-filtered signal has been scaled by an estimated amount to take into account the efficiency of the reconversion pulse.

Experimental (132.3 MHz) NMR of static powdered . (a) Central-transition and signal intensity as a function of the radiofrequency field strength of the spin-locking pulse with . (b) signal intensity as a function of the duration, , of a spin-locking pulse for a variety of field strengths. In (a), the relative intensity of the 3Q-filtered signal has been scaled by an estimated amount to take into account the efficiency of the reconversion pulse.

Experimental (132.3 MHz) MAS NMR of powdered . Central-transition (solid line) and (dotted line) signal intensity as a function of the spin-locking duration . The relative intensity of 3Q-filtered signal has been scaled by an estimated amount to take into account the efficiency of the reconversion pulse. Spin-locking is performed with and , corresponding to (black line) and and , corresponding to (red line).

Experimental (132.3 MHz) MAS NMR of powdered . Central-transition (solid line) and (dotted line) signal intensity as a function of the spin-locking duration . The relative intensity of 3Q-filtered signal has been scaled by an estimated amount to take into account the efficiency of the reconversion pulse. Spin-locking is performed with and , corresponding to (black line) and and , corresponding to (red line).

Expectation values of spin spherical tensor operators, , created by rapid dephasing of an initial state, , under a spin-locking Hamiltonian that includes (a) the first-order and (b) first- and second-order quadrupolar interactions, as a function of the rotor phase . The adiabatic approximation has been assumed. Other parameters include , , , , and . Tensors with coherence order 0, 1, 2, and 3 are denoted by black, red, blue, and green lines, while tensors with rank (l) 1, 2, and 3 are denoted by solid, dotted, and dashed lines, respectively.

Expectation values of spin spherical tensor operators, , created by rapid dephasing of an initial state, , under a spin-locking Hamiltonian that includes (a) the first-order and (b) first- and second-order quadrupolar interactions, as a function of the rotor phase . The adiabatic approximation has been assumed. Other parameters include , , , , and . Tensors with coherence order 0, 1, 2, and 3 are denoted by black, red, blue, and green lines, while tensors with rank (l) 1, 2, and 3 are denoted by solid, dotted, and dashed lines, respectively.

Expectation values of spin single-transition operators (black line), (red line), and (blue line), created by rapid dephasing of an initial state , under a spin-locking Hamiltonian that includes (a) the first-order and (b-d) first and second-order quadrupolar interactions, as a function of the rotor phase , for resonance offsets of (a) 0, (b) 0, (c) 3058 Hz (1Q), and (d) −1163 Hz (3Q/3). The adiabatic approximation has been assumed. Other parameters are as in Fig. 12.

Expectation values of spin single-transition operators (black line), (red line), and (blue line), created by rapid dephasing of an initial state , under a spin-locking Hamiltonian that includes (a) the first-order and (b-d) first and second-order quadrupolar interactions, as a function of the rotor phase , for resonance offsets of (a) 0, (b) 0, (c) 3058 Hz (1Q), and (d) −1163 Hz (3Q/3). The adiabatic approximation has been assumed. Other parameters are as in Fig. 12.

Expectation values of spin single-transition operators (black line), (red line), and (blue line) operators, as a function of the duration of a spin-locking pulse under MAS conditions, simulated using an exact density matrix approach for a powder, including (a) first- and (b-d) first- and second-order quadrupolar interactions. The initial state was . Resonance offsets, , were (a) 0, (b) 0, (c) 1000 Hz (1Q), and (d) −1000 Hz (3Q/3). Other simulation parameters include , , , , and , corresponding to .

Expectation values of spin single-transition operators (black line), (red line), and (blue line) operators, as a function of the duration of a spin-locking pulse under MAS conditions, simulated using an exact density matrix approach for a powder, including (a) first- and (b-d) first- and second-order quadrupolar interactions. The initial state was . Resonance offsets, , were (a) 0, (b) 0, (c) 1000 Hz (1Q), and (d) −1000 Hz (3Q/3). Other simulation parameters include , , , , and , corresponding to .

Expectation value of spin single-transition operator simulated for a powder using an exact density matrix approach, plotted as a function of the duration of a spin-locking pulse with first- (black line) and first- and second-order (red, green, and blue lines) quadrupolar Hamiltonians included. The initial state was . Resonance offsets, , are 0 (black and red lines), −250 Hz (green line), or 250 Hz (blue line). Other simulation parameters include , , , , and , corresponding to .

Expectation value of spin single-transition operator simulated for a powder using an exact density matrix approach, plotted as a function of the duration of a spin-locking pulse with first- (black line) and first- and second-order (red, green, and blue lines) quadrupolar Hamiltonians included. The initial state was . Resonance offsets, , are 0 (black and red lines), −250 Hz (green line), or 250 Hz (blue line). Other simulation parameters include , , , , and , corresponding to .

Experimental (132.3 MHz) MAS NMR of powdered . (a) signal intensity as a function of (a) the duration, , of the spin-locking pulse and (b) the resonance offset . Other experimental conditions include and . In (a), resonance offsets from the chemical shift of −230 Hz (black line) and (red line) were included, while in (b) .

Experimental (132.3 MHz) MAS NMR of powdered . (a) signal intensity as a function of (a) the duration, , of the spin-locking pulse and (b) the resonance offset . Other experimental conditions include and . In (a), resonance offsets from the chemical shift of −230 Hz (black line) and (red line) were included, while in (b) .

Expectation values of single-transition operators (a, c) and (b) operators for (a, b) spin and (c) spin , as a function of the radiofrequency field strength of a spin-locking pulse under MAS conditions, simulated using an exact density matrix approach for a powder. The initial state was . Simulations use either a first-order quadrupolar Hamiltonian only (black line) or both first- and second-order Hamiltonians (red, green, and blue lines). Resonance offsets, , in (a, b) of −250 Hz (1Q) and (3Q/3) are shown by the green and blue lines, respectively. Other simulation parameters include , , , , and . Also shown in (a, c) is the corresponding central-transition nutation rate .

Expectation values of single-transition operators (a, c) and (b) operators for (a, b) spin and (c) spin , as a function of the radiofrequency field strength of a spin-locking pulse under MAS conditions, simulated using an exact density matrix approach for a powder. The initial state was . Simulations use either a first-order quadrupolar Hamiltonian only (black line) or both first- and second-order Hamiltonians (red, green, and blue lines). Resonance offsets, , in (a, b) of −250 Hz (1Q) and (3Q/3) are shown by the green and blue lines, respectively. Other simulation parameters include , , , , and . Also shown in (a, c) is the corresponding central-transition nutation rate .

(a) Experimental (132.3 MHz) MAS NMR of powdered . Central-transition (black lines) and (red lines) signal intensity as a function of the radiofrequency field strength of the spin-locking pulse. Resonance offsets, , of −230 Hz from the chemical shift (on resonance with the 1Q signal) and from the chemical shift (3Q/3) were used, denoted by solid lines and dashed lines, respectively. Other experimental conditions include and . The dip in spin-locking efficiency caused by rotary resonance recoupling of dipolar couplings is highlighted by . (b) Expectation values of spin central-transition (black lines) and (red lines) operators as a function of the radiofrequency field strength of a spin-locking pulse under MAS conditions, simulated using an exact density matrix approach for a powder. The initial state was . Both first- and second-order quadrupolar interactions were included, with resonance offsets of 230 Hz (1Q) and (3Q/3) denoted by solid lines and dashed lines, respectively. Other simulation parameters include , , , , and . Also shown is the corresponding central-transition nutation rate, .

(a) Experimental (132.3 MHz) MAS NMR of powdered . Central-transition (black lines) and (red lines) signal intensity as a function of the radiofrequency field strength of the spin-locking pulse. Resonance offsets, , of −230 Hz from the chemical shift (on resonance with the 1Q signal) and from the chemical shift (3Q/3) were used, denoted by solid lines and dashed lines, respectively. Other experimental conditions include and . The dip in spin-locking efficiency caused by rotary resonance recoupling of dipolar couplings is highlighted by . (b) Expectation values of spin central-transition (black lines) and (red lines) operators as a function of the radiofrequency field strength of a spin-locking pulse under MAS conditions, simulated using an exact density matrix approach for a powder. The initial state was . Both first- and second-order quadrupolar interactions were included, with resonance offsets of 230 Hz (1Q) and (3Q/3) denoted by solid lines and dashed lines, respectively. Other simulation parameters include , , , , and . Also shown is the corresponding central-transition nutation rate, .

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