Equilibrium population difference

If only single-quantum transitions (h, I2, S], and S ) were active as relaxation pathways, saturating S would not affect the intensity of I in other words, there will be no nOe at I due to S. This is fairly easy to understand with reference to Fig. 4.2. After saturation of S, the fMjpula-tion difference between levels 1 and 3 and that between levels 2 and 4 will be the same as at thermal equilibrium. At this point or relaxation processes act as the predominant relaxation pathways to restore somewhat the equilibrium population difference between levels 2 and 3 and between levels 1 and 4 leading to a negative or positive nOe respectively. 

For example, at 7.05 T magnetic field (a 300 MHz NMR instrument) and 25 °C, the population difference for protons is 0.00064% of the number of nuclei N. This equilibrium population difference is a constant throughout the NMR experiment and, as we perturb the equilibrium, the spins will always try to return to this equilibrium population distribution. Because the measureable signal from a nucleus in the ft state is exactly cancelled by the signal from a nucleus in the a state, it is this population difference that is the only material we have to work with and to detect in the NMR experiment. Because the difference is so small, the sensitivity of NMR is in many orders of magnitude lower than all other analytical techniques so low, in fact, that NMR is not considered a branch of analytical chemistry but rather a tool used by organic chemists and biologists. 

How big is this population difference at equilibrium The Boltzmann distribution defines the populations of the two states precisely, and it turns out that the equilibrium population difference APeq is proportional to the energy difference between the two states (a and ft)... 

What is the equilibrium population difference between ap and Pal After cross-relaxation, what is the percentage change in the z magnetization of Hb, and is it increased (enhanced) or decreased ... 

Perturbation of the equilibrium population difference for one nucleus (increased spin temperature) spreads over time to perturb the population difference (increase or decrease the spin temperature) of other nuclei that are nearby in space. For small molecules, increasing the spin temperature of one nucleus will decrease the spin temperature of nearby nuclei ( negative NOE ). This leads to an enhancement of peak intensities corresponding to the nearby nuclei. For large molecules, increasing the spin temperature of one nucleus will increase the spin temperature of nearby nuclei ( positive NOE ), leading to a reduction in peak intensity. 

Instead of acquiring an FID, we wait for a period of time rm (the mixing time) and allow relaxation to occur, dominated (for small molecules) by the DQ relaxation pathway pp -> aa. What is the equilibrium population difference between these two states From Figure 5.24, we see that AP = (N/2 + 28) — (N/2 — 28) = 48, or counting the circles... 

Note that the equilibrium population difference for the zero-quantum transition is zero because the two states have (essentially) the same energy. Now we can substitute the (indirectly) measurable quantities Ma and M for the population differences. For the disequilibrium of we have... 

Note that we do not start directly with 13 C magnetization because we want to take advantage of the larger (by a factor of 4) equilibrium population difference of lH compared to13 C, as well as the shorter T (faster relaxation) of 1H, which will permit shorter relaxation delays. We now know a lot of tricks, and the main one we need here is the heteronuclear INEPT transfer ... 

Here 7 and 70 are the transverse and longitudinal relaxation rates respectively Q is the Rabi frequency and wq is the equilibrium population difference. In the limit of strong phase diffusion, we may assume... 

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