Difference between revisions of "Relaxation of spin 1/2 nuclei: two-state derivation"
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N<sub>h</sub>/N<sub>l</sub> = e<sup>-<span style="font-family:symbol;">D</span>E/kT</sup> | N<sub>h</sub>/N<sub>l</sub> = e<sup>-<span style="font-family:symbol;">D</span>E/kT</sup> | ||
| − | Let us call the two states α and β spin states. For a sample of ethanol (CH<sub>3</sub>CH<sub>2</sub>OH) there will be 3 sets of 2 states: the CH<sub>3</sub> pair, the CH<sub>2</sub> pair and the OH pair. | + | Let us call the two states α and β spin states. For a sample of ethanol (CH<sub>3</sub>CH<sub>2</sub>OH) there will be 3 sets of 2 states: the CH<sub>3</sub> pair, the CH<sub>2</sub> pair and the OH pair. Each ratio will be different since the transition energy is different: |
*N<sub>βCH<sub>3</sub></sub>/N<sub>αCH<sub>3</sub></sub> = ratio for CH<sub>3</sub> at equilibrium | *N<sub>βCH<sub>3</sub></sub>/N<sub>αCH<sub>3</sub></sub> = ratio for CH<sub>3</sub> at equilibrium | ||
*N<sub>βCH<sub>2</sub></sub>/N<sub>αCH<sub>2</sub></sub> = ratio for CH<sub>2</sub> at equilibrium | *N<sub>βCH<sub>2</sub></sub>/N<sub>αCH<sub>2</sub></sub> = ratio for CH<sub>2</sub> at equilibrium | ||
Revision as of 03:42, 14 April 2020
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. Each ratio will be different since the transition energy is different:
- 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
