Summary Of The Vector Model

The vector model is a way of visualizing the NMR phenomenon that includes some of the requirements of quantum mechanics while retaining a simple visual model. We will jump back and forth between a "classical" spinning top model and a "quantum" energy diagram with populations (filled and open circles) whenever it is convenient. The vector model explains many simple NMR experiments, but to understand more complex phenomena one must use the product operator (Chapter 7) or density matrix (Chapter 10) formalism. We will see how these more abstract and mathematical models grow naturally from a solid understanding of the vector model.

Consider a large population of identical spins—for example the protons in a liquid sample of 12CHCl3—in a strong, uniform magnetic field Bo oriented along the positive z axis.

1. Each *H nucleus can be in one of two states: aligned at a 45° angle to the +z axis ("up") or aligned at a 135° angle to the +z axis ("down").

2. Each 1H nucleus precesses about the z axis, with its spin axis tracing a conical path always at a 45° (or 135°) angle to the +z axis, at a rate equal to the Larmorfrequency vo:

vo = yBo/2n where y is a measure of the magnet strength of the nuclear magnet. The Larmor frequency is in the radio frequency range of the electromagnetic spectrum.

3. The "up" or a state, which is aligned with the laboratory magnetic field, is lower in energy than the "down" or j state, which is aligned against the field. This energy "gap" is proportional to the magnetic field strength and to the strength of the nuclear magnet:

4. At thermal equilibrium, slightly more than half of the population of spins is in the lower energy "up" state and slightly less than half is in the higher energy "down" state. The population difference is on the order of one spin in every 105 spins. At equilibrium each spin is randomly oriented around the cone at any moment in time, even though all spins precess at exactly the same frequency. Thus the "phase" of the precessing nuclei is random and spread equally around the two cones.

5. The net magnetization is defined as the vector sum of all of the nuclear magnets in the sample. If each vector's origin is moved to the origin of the coordinate system, the vectors can be added together for the whole population. The x and y components of the net magnetization are zero at equilibrium because the spins are spread equally around the two cones at any instant in time. The z components cancel for each pair of one "up" and one "down" spin, but because there is a slightly larger population of spins in the "up" state the net magnetization is a small vector pointing along the positive z axis. This macroscopic equilibrium net magnetization has a magnitude of Mo, and it is the starting point of all NMR experiments.

6. A radio frequency pulse has the effect of "organizing" the phase of the spins in their precession, so that at the end of a pulse all of the excess spins have the same phase. At any moment in time, all of these spins point in the same direction on the cone, so that they add together to make a net magnetization vector that rotates in the x-y plane at the Larmor frequency. We can say that the pulse created "phase coherence" or simply "coherence'' in the sample. The net magnetization is no longer stationary, and its rotation induces a voltage in the probe coil that oscillates at the Larmor frequency. After the pulse the spins begin to lose the organization imparted by the pulse, and the spins spread out on the cone until they are again randomly oriented on the cone at any moment in time. This loss of coherence causes the induced signal in the probe coil to decay exponentially to zero. The probe coil signal is called the free induction decay (FID). The information contained in the FID signal is the frequency of the oscillating voltage, which corresponds to the chemical shift of the nuclei in the sample, the amplitude of the voltage, which corresponds to the height of the peak, and the phase of the signal, which corresponds to the phase (absorptive positive, dispersive, etc.) of the NMR line.

7. The z component (Mz) of the net magnetization vector represents the difference in population between the two spin states ("up" and "down"). For example, immediately after a 90° pulse enough spins have been promoted from the lower energy ("up") state to the higher energy ("down") state to equalize the populations. At this moment the z components of the population of spins exactly cancel and the z component of the net magnetization vector is zero. The net magnetization vector is in the x-y plane, rotating at the Larmor frequency.

8. The radio frequency pulse is a very short (tens of microseconds), and a very high power (tens or hundreds of watts) pulse of radio frequency power applied to the probe coil at or very near the Larmor frequency. It has a rectangular envelope: the power turns on and instantly reaches full power, then at the end of its duration it goes instantly to zero. The pulse creates an oscillating magnetic field, which can be represented by a vector (the "B1 vector") that rotates in the x-y plane at the frequency of the pulse. The length of the B1 vector is equal to the amplitude of the radio frequency pulse.

9. After a 90° pulse, the phase coherence created by the pulse begins to be lost as the individual spins "fan out" around the cones due to slight local differences in magnetic field. The loss of coherence is exponential and goes to zero with time constant T2. At the end of the 90° pulse the populations of the two spin states are equal: half of the spins are in the "up" state and half are in the "down" state. Immediately spins begin to drop down from the higher energy ("down" or ft) state to the lower energy ("up" or a) state until the equilibrium population difference is reestablished. This process, which leads to an exponential growth of Mz with time constant Ti until it is equal to the full equilibrium value Mo, is called longitudinal relaxation. It is always slower than the loss of coherence, which is called transverse relaxation.

10. A 180° pulse rotates the equilibrium sample magnetization to the -z axis. Immediately after the pulse there is no phase coherence (no x or y component to the net magnetization) and no FID can be recorded. The population difference in now reversed: slightly more than half of the spins are in the higher energy ("down" or ft) state and slightly less than half are in the lower energy ("up" or a) state. This is the largest deviation from the equilibrium population distribution that can be achieved by an RF pulse. The reversal of populations actually occurs by moving every single spin that was in the "up" state to the "down" state, and every spin that was in the "down" state to the "up" state. After the pulse the slight excess of spins in the higher energy state begin to drop down and the net magnetization vector along the -z axis shrinks, passes through zero, and grows toward Mo along the +z axis in an exponential fashion.

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