Relaxation of spin 1/2 nuclei: two-state derivation

Introduction

When an NMR sample is placed in a static magnetic field and allowed to come to equilibrium it is found that a net magnetization of the sample along the direction of the applied field (traditionally the z-axis) is developed. Magnetization parallel to the applied field is termed longitudinal. This equilibrium magnetization arises from the unequal population of the two energy levels that correspond to the α and β spin states. In fact, the z-magnetization, M_z, is proportional to the population difference:

M_zz ∝ (n_α - n_β)

where n_α and n_β are the populations of the two corresponding energy levels. Ultimately, the constant of proportion just determines the absolute size of the signal we will observe. As we are generally interested in the relative size of magnetizations and signals we may just as well write:

M_zz = (n_α - n_β)

Mathematically this system can be treated similar to chemical kinetics.

Let the populations of the α and β states at time t be n_αt and n_βt , respectively. If these are not the equilibrium values, then for the system to reach equilibrium the population of one level must increase and that of the other must decrease. This implies that there must be transitions between the two levels i.e. something must happen which causes a spin to move from the α state to the β state or vice versa.