The Basic Physics of Spin-1 Systems

In the following, we present a simple theory toolkit that is useful to compute the response of spin-1 systems to a variety of NMR experiments that are discussed in the two subsequent Chapters. The emphasis is on developing the relevant tools, instead of a formal treatment of the theory of spin-1 angular momentum. Following a general introduction to the problem at hand, we start with a quick and simple Summary of relevant aspects of spin-1 angular momentum theory, introducing a Hermitian, Cartesian operator basis, its matrix representation in two different eigenbases, and the Commutator algebra for spin-1. This is followed by a projection operator treatment of functions of Hermitian spin-1 operators, which is employed to build up an explicit matrix representation of rotations, effected for instance by hard pulses. The interactions relevant in spin-1 NMR are briefly discussed, followed by an introduction to the density matrix description of the state of spin-1 ensembles. This provides the background to evaluate evolution of spin-1 systems under various interactions, including quadrupolar coupling, scalar coupling to spin-1/2 and scalar coupling to spin-1. The rotation behavior of spin-1 wavefunctions is discussed both for selective two-level excitation (in the presence of quadrupolar coupling), as well as for non-selective three-level excitation, the latter in solution state. A simple coherence transfer function for weakly coupled systems of N inequivalent spins-1 is given next, based on the rotation matrix previously introduced. Finally, a brief treatment of quadrupolar relaxation in spin-1 systems is given, the emphasis once again being the derivation of relevant relaxation rates.