Difference between revisions of "Relaxation of spin 1/2 nuclei: two-state derivation"
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==Introduction== | ==Introduction== | ||
− | When an NMR sample is placed in a static magnetic field and allowed to come to equilibrium | + | 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<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. | |
+ | *N<sub>βCH<sub>3</sub></sub>/N<sub>α3CH<sub>3</sub></sub> = ratio for CH<sub>3</sub> at equilibrium | ||
+ | *N<sub>β</sub>/N<sub>α2</sub> = ratio for CH<sub>2</sub> at equilibrium | ||
+ | *N<sub>β</sub>/N<sub>α</sub> = 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<sub>β90</sub>/N<sub>α90</sub> = ratio for CH<sub>3</sub> after 90 pulse | |
− | + | *N<sub>β90</sub>/N<sub>α90</sub> = ratio for CH<sub>2</sub> after 90 pulse | |
− | + | *N<sub>β90</sub>/N<sub>α90</sub> = ratio for OH after 90 pulse | |
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Revision as of 03:39, 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.
- 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