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:
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/NαCH3 = ratio for CH3 at equilibrium
- NβCH2/NαCH2 = ratio for CH2 at equilibrium
- NβOH/NαOH = 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