Conservation of energy, density of states, and spin lattice relaxation

Abstract

The starting point of all NMR experiments is a spin polarization which develops when we place the sample in static magnetic field B0. There are excess of spins aligned along B0 (spin up with lower energy) than spins aligned opposite (spin down with higher energy) to the field B0. A natural question is what is the source of this excess spin polarization because relaxation mechanisms can flip a up spin to a down spin and vice-versa. The answer lies in the density of states. When a molecule with spin down flips to spin up it loses energy. This energy goes into increasing the kinetic energy of the molecule in the gas/solution phase. At this increased kinetic energy, there are more rotational-translational states accessible to the molecule than at lower energy. This increases the probability the molecule will spend in spin up state (higher kinetic energy state). This is the source of excess polarization. In this article, we use an argument based on equipartition of energy to explicitly count the excess states that become accessible to the molecule when its spin is flipped from down to up. Using this counting, we derive the familiar Boltzmann distribution of the ratio of up vs down spins. Although prima facie, there is nothing new in this article, we find the mode counting argument for excess states interesting. Furthermore, the article stresses the fact that spin polarization arises from higher density of states at increased kinetic energy of molecules.