The Wei–Norman Lie-algebraic technique applied to field modulation in nuclear magnetic resonance

Abstract

By implementing the Wei–Norman Lie algebra approach, this paper focuses on the development of a new method of solution to the quantum Liouville equation for NMR field modulation. First, the general method is reviewed and discussed for systems of any finite dimensional Lie algebra. The theory is then applied to arbitrary time-dependent Zeeman interactions [or SU(2) Hamiltonians] in which the complete arbitrary spin I≥1/2 density operator problem reduces to a Riccati equation. Novel exact analytical solutions of the complete spin density operator for a class of temporal field modulations are obtained. In particular, a spherical tensor operator basis is used to expand the density operator, and the solutions retain the physically appealing form of Wigner rotations with time-modulated rotation angles which are special functions with well known analytical properties. The exact solutions include the frequency swept hyperbolic secant pulse shapes, as well as any exponentially modulated amplitude pulse. In contrast to all other existing formulations, the present treatment provides the first known examples of exactly soluble nonrectangular pulse shapes for all resonance off-set, all field amplitudes, all time and valid for all spin I≥1/2. In addition, from the underlying Riccati equation, a new series solution for the complete spin density matrix is given which is valid for any well-behaved field modulations. Spin inversion profiles are also calculated for exponentially decaying pulses. These show applications to selective excitation.