Analytical solution of Bloch NMR fluid flow space–time-dependent equation using laplace transform and complex inversion integral

Abstract

Nuclear magnetic resonance (NMR) is a phenomenon whereby magnetization is excited when static and time varying magnetic fields are applied simultaneously on a given medium such as human blood. The effect of the magnetization causes the protons of the medium to spinning. For several decades now, a set of three Bloch equations are used to describe the dynamics of the spinning protons. Exact solution of the Bloch equations has been the endeavors of many workers with partial success. In about a decade now, a milestone was the appearance of a single NMR fluid flow equation derived from the three set of Bloch equations. The single equation has been found insuperable up to now, defying all efforts to yield a closed form solution. Motivated by the exigency to achieve complete magnetization expression as a function of time and distance, for NMR signal calculations or experiments, we have actualized a closed form solution. We used the method of Laplace transforms and ultimately applied complex inversion theorem to obtain the inverse Laplace transforms. Our final expression is a labyrinth of several oscillatory systems that are characteristics of a nonlinear phenomenon. This is cognate with concepts of chaos.