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

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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:

Nh/Nl = 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. 
  • NβCH3/Nα3CH3 = ratio for CH3 at equilibrium
  • Nβ/Nα2 = ratio for CH2 at equilibrium
  • Nβ/Nα = 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:

  • Nβ90/Nα90 = ratio for CH3 after 90 pulse
  • Nβ90/Nα90 = ratio for CH2 after 90 pulse
  • Nβ90/Nα90 = ratio for OH after 90 pulse