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 the population of each group of spin 1/2 atoms is divided into 2, with the number in each state determined by the Boltzmann Distribution:

N_h/N_l = e^-DE/kT

Let us call the two states α and β spin states. For a sample of ethanol (CH₃CH₂OH) there will be 3 sets of 2 states: the CH₃ pair, the CH₂ pair and the OH pair.

• N_βCH3/N_αCH3 = ratio for CH₃ at equilibrium
• N_βCH2/N_αCH2 = ratio for CH₂ at equilibrium
• N_βOH/N_αOH = ratio for OH at equilibrium

Once the sample is excited with a 90 degree Rf pulse the populations of the 3 sets of states will change to the maximum allowed by the different energies in the system:

• N_β90/N_α90 = ratio for CH₃ after 90 pulse
• N_β90/N_α90 = ratio for CH₂ after 90 pulse
• N_β90/N_α90 = ratio for OH after 90 pulse