Figure 6.2

Figure 6.2

vector in a counterclockwise direction in the rotating frame at the frequency Av. If the NMR peak is in the upfield half (right half) of the spectrum, the Larmor frequency is less than the reference frequency (vo < vr) and we have a negative resonance offset (Av < 0). This corresponds to the motion of the net magnetization vector in a clockwise direction in the rotating frame at the frequency — Av. Remember that in the NMR spectrum the frequency scale runs from right to left, opposite to every other graphical scale known to man. In discussions of NMR theory, we will ignore the chemical shift scale (5 in ppm) and view the NMR spectrum as a frequency scale (Hz) with zero at the center, negative frequencies on the right-hand side, and positive frequencies on the left-hand side. The resonance offset is the key to understanding what happens to the net magnetization vector during a delay. We call this motion of the net magnetization vector in the X-y' plane of the rotating frame of reference "evolution." Sometimes to simplify the terms in equations, we will use the upper case Greek letter omega to represent the resonance offset in radians per second:

laboratory frame: &>o = 2nvo, rotating frame: ^ = 2nAv


The RF pulse is a short (~10 ^s) burst of a very high power (50-300 W) RF signal with a specific frequency, amplitude, and phase. The frequency is the same as the reference frequency, vr. The ideal pulse turns on and off instantly and has constant amplitude (Fig. 6.3), leading to a rectangular shape or envelope ("rectangular pulse"). The duration of the pulse is called the pulse width, usually measured in microseconds. The phase of the pulse is determined by its starting point in the sine function: starting at 0° the amplitude increases at first from zero to maximum; starting at 90° it decreases at first from maximum to zero; starting at 180° it decreases at first from zero to the negative peak; and starting at 270° it increases at first from the negative peak to zero. This can be precisely controlled by the hardware and

Figure 6.3

is programed into the pulse sequence. In the pulse sequence code that drives the hardware, the pulse phase is referred to as 0,1, 2, and 3 for 0o, 90o, 180°, and 270°, respectively. In the probe, the pulse is applied to a tuned circuit consisting of the probe coil (an inductance) and a variable capacitance that is used to tune the probe (Fig. 6.4). The actual circuit is much more complex, with at least two variable capacitors, but the simple tuned LC circuit is sufficient to understand how it works. The probe coil is saddle shaped, which is designed to produce a magnetic field oriented perpendicular to the NMR tube axis, that is, to the z-axis. When a pulse is applied to the coil, an oscillating magnetic field (B1) appears along an axis in the x-y plane. For example, if this is aligned on the x axis, we have B1 oscillating in time along the x axis: first zero, then growing to a maximum B1 field oriented on the positive x axis, then decreasing to zero, then growing to a maximum B1 field along the negative x axis, and then decreasing to zero again. This sequence repeats itself vr times per second, where vr is the frequency of the pulse (e.g., 300 MHz for 1H excitation on a 7.05-T instrument).

To get interaction with the individual magnetic vectors, which are precessing in a counterclockwise path around the cones at the Larmor frequency, we need a B1 field that is also rotating in a counterclockwise direction at the same frequency. We can divide the oscillating B1 field into two components: one that rotates clockwise in the x-y plane at frequency vr and the other that rotates counterclockwise in the x-y plane at the same frequency (Fig. 6.5). The vector sum of these two rotating vectors is the B1 vector, which oscillates in amplitude along the x axis alone. Of the two rotating components, only the counterclockwise one has any effect on the precessing spins; the other one is effectively 2vr away from the resonant frequency because it is rotating in the wrong direction. So we ignore this component and from now on we will describe the RF pulse as a magnetic vector of constant magnitude B1, which rotates counterclockwise in the x-y plane at the frequency vr. This explanation is

Figure 6.4

formally required to come up with a rotating B1 vector, but you can forget you ever heard of it if you like because we will always talk about the pulse as a rotating B\ vector from now on.

In the rotating frame of reference, the B\ vector always stands still because the reference frequency and the pulse frequency are the same. Changing the pulse phase changes the position of the B\ vector in the X-/ plane, so that a 0o phase corresponds to the X axis, a 90o phase to the y' axis, a 180o phase to the -X axis, and a 270o phase to the —y' axis. This ability to position the Bi vector wherever we want in the X-y' plane through the RF hardware allows us to control precisely the effect of the pulse. We can also change the amplitude of the pulse, which adjusts the length of the B1 vector and changes the strength of the magnetic field it represents.

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