^{1}and Malcolm H. Levitt

^{1,a)}

### Abstract

In a previous paper [M. Carravetta and M. H. Levitt, J. Chem. Phys.122, 214505 (2005)], we presented the theory of long-lived nuclear spin singlet states in low magnetic field. In this paper, we consider the spin locking of long-lived singlet states in high magnetic field by the application of resonant radio frequency irradiation. We present theoretical results for unmodulated irradiation, including approximate analytical expressions for the singlet decay rate constants. We show the results of numerical simulations, which indicate that modulated radio frequency fields may be used to achieve broadband spin locking of singlet states but only in the case of a small difference in Larmor frequencies between the members of the spin pair.

This research was supported by the EPSRC (UK). We would like to thank Salvatore Mamone and Marina Carravetta for useful discussions.

I. INTRODUCTION

II. SUPEROPERATORS

A. Notation

B. Irreducible spherical tensor superoperators

C. Operator basis

III. EQUATION OF MOTION

A. Master equation

B. Coherent interactions

C. Relaxation superoperator

IV. SINGLET SPIN LOCKING

A. Unmodulated rf field

1. Along the two diagonals

2. The vicinity of the main diagonal

B. Periodic rf fields

1. ALT of periodic systems

2. Numerical calculations

3. Analysis of modulation schemes

C. Inhomogeneous rf fields

V. CONCLUSIONS

### Key Topics

- Chemical shifts
- 44.0
- Eigenvalues
- 24.0
- Nuclear spin
- 23.0
- Nuclear magnetic resonance
- 15.0
- Spin relaxation
- 15.0

## Figures

Numerical simulations of rate constants for a system of two- in inequivalent chemical sites, irradiated by an unmodulated rf field. The behavior of the long-lived state is shown in bold. The simulation parameters are , , and . (a) The difference in chemical shift frequencies is held fixed at , while the sum of the resonance frequencies is varied. (b) The sum of the two chemical shift frequencies is held fixed at , while the difference in resonance frequencies is varied.

Numerical simulations of rate constants for a system of two- in inequivalent chemical sites, irradiated by an unmodulated rf field. The behavior of the long-lived state is shown in bold. The simulation parameters are , , and . (a) The difference in chemical shift frequencies is held fixed at , while the sum of the resonance frequencies is varied. (b) The sum of the two chemical shift frequencies is held fixed at , while the difference in resonance frequencies is varied.

Contour plots of the smallest nonzero value of , with , plotted as a function of the two resonance offset frequencies and . The simulation parameters are , and . (a) Unmodulated rf irradiation (cw), with a nutation frequency ; (b) WALTZ-16 irradiation (Ref. 40) with a nutation frequency ; (c) MLEV-16 irradiation (Ref. 37) with a nutation frequency ; (d) rf field with a cosine-modulated amplitude, with a peak nutation frequency and a modulation frequency .

Contour plots of the smallest nonzero value of , with , plotted as a function of the two resonance offset frequencies and . The simulation parameters are , and . (a) Unmodulated rf irradiation (cw), with a nutation frequency ; (b) WALTZ-16 irradiation (Ref. 40) with a nutation frequency ; (c) MLEV-16 irradiation (Ref. 37) with a nutation frequency ; (d) rf field with a cosine-modulated amplitude, with a peak nutation frequency and a modulation frequency .

Contour plot of the decay rate constant of the long-lived state as a function of the two resonance offset frequencies and , for unmodulated rf irradiation, at small values of the offset frequencies. The simulation parameters are ; , , and . (a) Numerical simulation. (b) Formula given in Eq. (48). (c) Formula given in Eq. (49).

Contour plot of the decay rate constant of the long-lived state as a function of the two resonance offset frequencies and , for unmodulated rf irradiation, at small values of the offset frequencies. The simulation parameters are ; , , and . (a) Numerical simulation. (b) Formula given in Eq. (48). (c) Formula given in Eq. (49).

Numerical simulations of rate constants for a system of two- in inequivalent chemical sites irradiated by a rf field with WALTZ-16 modulation. The behavior of the long-lived state is shown in bold. The simulation parameters are , , , and . (a) The difference in chemical shift frequencies is held fixed at , while the sum of the resonance frequencies is varied. (b) The sum of the two chemical shift frequencies is held fixed at , while the difference in resonance frequencies is varied.

Numerical simulations of rate constants for a system of two- in inequivalent chemical sites irradiated by a rf field with WALTZ-16 modulation. The behavior of the long-lived state is shown in bold. The simulation parameters are , , , and . (a) The difference in chemical shift frequencies is held fixed at , while the sum of the resonance frequencies is varied. (b) The sum of the two chemical shift frequencies is held fixed at , while the difference in resonance frequencies is varied.

Two expanded regions in Fig. 2(d) showing the rate constant for singlet decay plotted against the two resonance offset parameters for the case of cosine-modulated rf irradiation. (a) Region around the main diagonal. (b) Region around the antidiagonal.

Two expanded regions in Fig. 2(d) showing the rate constant for singlet decay plotted against the two resonance offset parameters for the case of cosine-modulated rf irradiation. (a) Region around the main diagonal. (b) Region around the antidiagonal.

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