Fictitious spin 1/2 operator formalism for multiple quantum NMR

Abstract

A formalism is presented that describes the time behavior of the spin density matrix of a nuclear spin system with arbitrary spin in terms of fictitious spin −(1/2) operators. This formalism is an extension of that used earlier for nuclei with spin I=1. For a spin system with n eigenstates we define for every pair of eigenstates ‖i〉 and ‖j〉 three operators I^i−j_p, with p=x, y, and z, according to the three 2×2 Pauli matrices σ_x, σ_y, and σ_z. These operators together constitute a complete set of n²−1 independent Hermitian operators, and we can write the n×n density matrix and the spin Hamiltonian of the system in terms of the I^i−j_p operators. The commutation relations among the operators make it possible in many cases to solve the equation of motion of the density matrix analytically. Three examples of the use of the I^i−j_p operators are presented. Firstly a system of noninteracting spins with I=1 is considered. The I^i−j_p operators for this case are compared with the I_q,k operator defined earlier. The cw signal intensities of the single and double quantum transitions for the I=1 spin system are derived. Secondly a spin system with I=3/2 is described as a simple extension of the use of the I^i−j_p operators. The excitation and the detection of the triple quantum coherence are discussed for noninteracting nuclei in solids. Finally the double quantum effects in a system of two interacting spins with I=1/2 are discussed.