Ensemble of identical spins

Net magnetization at equilibrium

Figure 7.1

complicated task. Instead, we add together all of the individual magnetic dipoles (vectors) to obtain the bulk magnetization of the sample, and we can work with this bulk feature only. At equilibrium, after the spins have been in the applied field for a while, there will be a slight excess of spins in the lower energy (aligned, a, "up") state. Because the energy difference is very small compared to the average thermal energy at room temperature, this population difference is only about one spin in 105. For this reason, for nearly every spin in the "up" state, there will be another in the "down" state pointing in exactly the opposite direction. These pairs can be erased from our picture as they do not contribute anything to the net magnetization. After erasing all these pairs, we are left with a much smaller number of spins in the upper cone and none in the lower cone. Of these, the individual spins are precessing at exactly the same frequency but at any moment if we took a snapshot of their orientations we would find that they are equally distributed around the cone; no one orientation is preferred. Thus, the components of magnetism in the x-y plane all cancel each other out and we are left only with the sum of the z components, which add together to give a macroscopic magnetic field oriented along the positive z axis. This equilibrium z magnetization simply means that we have weakly magnetized the sample by giving a slight preference to spins oriented with the Bo field.

This equilibrium magnetization is the stuff we have to work with in NMR. With pulses of radio frequency energy, we can make this vector move around, "dance," and tell us things of importance about molecular structure. NMR is like radar with a magnet: high-power pulses of radio frequency energy are sent to the sample, and an "echo" is received (the FID). Analysis of this echo provides information about the sample. In particular, we can learn about the relationships between different spins (atoms) within a molecule, both in the sense of the number of bonds separating them and the angles of those bonds, and in the sense of the direct through-space distance between the atoms.

In NMR, there is a very important distinction between the z-axis component of the net magnetization ("z magnetization") and the component of the net magnetization that lies in the x-y plane ("coherence"). z-Magnetization, which is the result of unequal populations in the two spin states ("up" and "down", a and ¡, aligned and disaligned, lower energy and higher energy), gradually relaxes to its equilibrium value Mo, defined by the Boltzmann distribution of spins between the two energy levels. It is not directly measurable because the net vector is stationary and does not rotate. Net magnetization in the x-y plane is called coherence because it results from the temporary organization (coherence) of the individual spins as they rotate around the cone. Coherence always relaxes to zero as the individual spins gradually get out of phase and lose their "memory" of the organizing pulse. Coherence is measurable because it rotates and creates the FID signal in the probe coil. The RF pulse converts z magnetization into coherence.

The effect of the RF pulse can be viewed more simply if we forget about the individual spins and think only about the net magnetization vector. The RF pulse is a magnetic field that rotates in the x-y plane at the frequency of the pulse. This signal is turned on for a very short time (about 10 |xs) and then abruptly turned off. During the pulse, we can view this rotating magnetic field as a vector, the B1 vector, which rotates in the x-y plane while the Bo field (about 20,000 times larger) is on the positive z axis, as is the equilibrium net magnetization of the sample. To make the analysis simpler, we rotate the x and y axes at the frequency of the pulse, so that the B1 vector stands still in this rotating frame of reference (Fig. 7.2(a)). In order to preserve the laws of physics in this artificial rotating frame, we have to remove the Bo field from our picture. Now we have only the stationary B1 vector in the x-y plane and the sample net magnetization on the z axis. During the pulse, the sample net magnetization vector, M, rotates around the B1 vector. This is analogous to the rotation of M around the Bo field during the FID. If we place the B1 vector on the x axis, for example (we can control this by setting the phase of the radio frequency), the M vector will rotate counterclockwise from the z axis to the —y axis, then to the — z axis, then to the +y axis

Bo After the pulse

(laboratory frame)

Bo After the pulse

(laboratory frame)

and then back to the +z axis. If we time the duration of the pulse correctly, we can rotate M by exactly 90° so that it ends up on the —y axis at the moment the pulse is turned off. This pulse is called a 90° pulse.

After a 90° pulse there is no z magnetization: all of it has been converted into x-y magnetization (coherence). This means there is no difference in population between the two spin states. We go back to the laboratory frame at this instant to look at the motion of the net magnetization M (Fig. 7.2(b)). It rotates in the x-y plane at the Larmor frequency, inducing the FID signal in the probe coil and gradually decaying (due to loss of coherence) to zero. At a bit slower rate, the population difference is reestablished between the two levels as a small percentage of spins fall down (relax) from the upper energy level to the lower energy level. This causes the z magnetization to grow and eventually return to the equilibrium magnitude, Mo, aligned along the positive z axis. When all coherence is gone and the z magnetization has returned to equilibrium (a few seconds), we can repeat the whole process of pulse, recording the FID and relaxation, adding the new FID to the previous one to obtain a better signal-to-noise ratio.

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