Coherence Net Magnetization in the x-y Plane

The equilibrium state is characterized by a complete lack of coherence (random phase), a slight excess of population in the a state (M2 + 5), and a deficit in the p state (N/2 - 5). Anything that perturbs this equilibrium (e.g., an RF pulse) will be followed immediately by a process of relaxation back to the equilibrium state that can take as long as seconds to reestablish. Relaxation is extremely important in NMR because it not only determines how long we have to wait to repeat the data acquisition for signal averaging, but it also determines how quickly the FID decays and how narrow our NMR fines will be in the spectrum. Relaxation is also the basis of the nuclear Overhauser effect (NOE), which can be used to measure distances between nuclei one of the most important pieces of molecular information we can obtain from NMR. 

Immediately after a 90° pulse the net magnetization vector is in the x-y plane. This means that the z component of the net magnetization is zero and that there is no difference in population between the upper (P) and lower (a) energy states. The net magnetization vector will rotate (precess) in the x-y plane at the Larmor frequency, v0- The phase coherence 

The cosine and sine functions represent the rotation of the net magnetization vector it starts on the — y axis (—cos(0) = —1, sin (0) = 0) and moves toward the +x axis. After 1/4 counterclockwise rotation (v0t = 1/4) we have —cos(90°) = 0, sin(90°) = 1. The decaying exponential function represents the fanning out of individual vectors and loss of 

This is true regardless of the extent or the nature of the perturbation away from the Boltzmann distribution (90° pulse, 180° pulse, saturation, etc.). The z magnetization will always move toward M0 in this way, so that the distance to equilibrium (AMz) is decaying exponentially. For the specific case of a 90° pulse, we can describe the z magnetization by an exponential function that approaches M0 with time constant T  

At the end of a 180° pulse, the populations are inverted so that there is a slight excess (N/2 + 8) in the upper energy (P) state and a slight deficit (N/2 — 8) in the lower energy (a) state (Fig. 5.10, T =0.5 s). This is twice as far from equilibrium as the situation immediately after a 90° pulse (AP = —28 after a 180° pulse, 0 after a 90° pulse, and 28 at equilibrium). Spins drop down from the p state to the a state, reducing the population in the P state and increasing the population in the a state. After 8 spins have dropped down (time = ty2 = 0.693 Ti = 0.35 s), we have reached the point where populations are equal in the two states (Mz = 0). This is half of the way to equilibrium. Spins continue to drop down until in all 28 spins have dropped down from p to a, leaving a population of N/2 + 8 in the a state and N/2 - 8 in the p state. In mathematical terms, Mz = — M0 at the end of the 180° pulse, so 

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