The effect of the radiofrequency field
المؤلف:
Peter Atkins، Julio de Paula
المصدر:
ATKINS PHYSICAL CHEMISTRY
الجزء والصفحة:
ص534-535
2025-12-11
63
The effect of the radiofrequency field
We now consider the effect of a radiofrequency field circularly polarized in the xy plane, so that the magnetic component of the electromagnetic field (the only com ponent we need to consider) is rotating around the z-direction, the direction of the applied field B0, in the same sense as the Larmor precession. The strength of the rotating magnetic field is B1. Suppose we choose the frequency of this field to be equal to the Larmor frequency of the spins, νL = (γ/2π)B0; this choice is equivalent to selecting the resonance condition in the conventional experiment. The nuclei now experience a steady B1 field because the rotating magnetic field is in step with the precessing spins (Fig. 15.29a). Just as the spins precess about the strong static field B0 at a frequency γB0/2π, so under the influence of the field B1 they precess about B1 at a frequency γB1/2π. To interpret the effects of radiofrequency pulses on the magnetization, it is often useful to look at the spin system from a different perspective. If we were to imagine stepping on to a platform, a so-called rotating frame, that rotates around the direction of the applied field at the radiofrequency, then the nuclear magnetization appears stationary if the radiofrequency is the same as the Larmor frequency (Fig. 15.29b). If the B1 field is applied in a pulse of duration π/2γB1, the magnetization tips through 90° in the rotating frame and we say that we have applied a 90° pulse, or a ‘π/2 pulse’ (Fig. 15.30a). The duration of the pulse depends on the strength of the B1 field, but is typically of the order of microseconds. Now imagine stepping out of the rotating frame. To an external observer (the role played by a radiofrequency coil) in this stationary frame, the magnetization vector is now rotating at the Larmor frequency in the xy-plane (Fig. 15.30b). The rotating magnetization induces in the coil a signal that oscillates at the Larmor frequency and that
Fig. 15.29 (a) In a resonance experiment, a circularly polarized radiofrequency magnetic field B1 is applied in the xy-plane (the magnetization vector lies along the z-axis). (b) If we step into a frame rotating at the radiofrequency, B1 appears to be stationary, as does the magnetization M if the Larmor frequency is equal to the radiofrequency. When the two frequencies coincide, the magnetization vector of the sample rotates around the direction of the B1 field.
Fig. 15.30 (a) If the radiofrequency field is applied for a certain time, the magnetization vector is rotated into the xy-plane. (b) To an external stationary observer (the coil), the magnetization vector is rotating at the Larmor frequency, and can induce a signal in the coil.
can be amplified and processed. In practice, the processing takes place after subtraction of a constant high frequency component (the radiofrequency used for B1), so that all the signal manipulation takes place at frequencies of a few kilohertz. As time passes, the individual spins move out of step (partly because they are precessing at slightly different rates, as we shall explain later), so the magnetization vector shrinks exponentially with a time constant T2 and induces an ever-weaker signal in the detector coil. The form of the signal that we can expect is therefore the oscillating decaying free-induction decay (FID) shown in Fig. 15.31. The y-component of the magnetization varies as
My(t) = M0cos(2πνLt) e−t/T2
We have considered the effect of a pulse applied at exactly the Larmor frequency. However, virtually the same effect is obtained off resonance, provided that the pulse is applied close to νL. If the difference in frequency is small compared to the inverse of the duration of the 90° pulse, the magnetization will end up in the xy-plane. Note that we do not need to know the Larmor frequency beforehand: the short pulse is the ana logue of the hammer blow on the bell, exciting a range of frequencies. The detected signal shows that a particular resonant frequency is present.
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