< Classical spin precession in the rotating frame | Index | Spin precession from a quantum viewpoint >
We are now ready to introduce the key principle in magnetic resonance: the effect of a rotating magnetic field on the motion of a magnetic moment.
Let us first consider what happens if we apply a small rotating magnetic field, {$ b[\hat x \cos\omega t\, +\, \hat y \sin\omega t] $}, at right angles to a large static one, {$B\hat z$}, with angular velocity exactly equal to the Larmor frequency, {$\omega=\gamma B$}. Incidentally, we choose {$B$} large and {$b$} small because such is the condition with NMR, where the static field is provided by a big superconducting solenoid and the rotating field by radio frequency waves.
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It is convenient to describe the motion from the turntable again, rotating exacly at the same angular velocity, so that the rotating field itself looks stationary, along, say, the {$\hat x ^\prime$} axis of the turntable. In this case, recalling what we just concluded, the static effective field {$\mathbf{ B} - \Omega \hat z/\gamma$} vanishes and the motion in the rotating frame is determined entirely by the only non-vanishing torque, {$\mathbf{ m}\times\mathbf{ b}$}.
This motion is a precession (in the rotating frame) around the field {$\mathbf{ b}$} at an angular velocity {$\omega^\prime=\gamma b$}. If we assume for example that {$\mathbf{ m}$} is initially along the static field {$B\hat z$}, it will rotate in the {$ y ^\prime z$} plane. The same motion, seen from the laboratory frame, is generally called nutation and the trajectory of the arrow of the vector on a sphere will look like the trace in the sketch on the right: the composition of a very fast precession, at the Larmor frequency {$\omega$}, around {$B\hat z$}, with the much slower precession at frequency {$\omega^\prime$}.
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What happens now if we relax the condition that the frequency of the rotating field, {$\omega$}, is equal to the Larmor frequency of the static field, {$\gamma B$}? Let us look at it again in the rotating frame where {$b$} is stationary. The total effective field,
{$$ \mathbf{B}_e= \mathbf{ B}-\frac \omega \gamma + \mathbf{ b} $$}
will be generally large and directed almost along {$\hat z$}. Hence the precession will take place around that direction.
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| If, again, the moment lies initially along {$\hat z$}, the aperture of the precession cone will be very small.
The plot on the left shows the sine of the cone aperture angle {$\theta$} vs. frequency, after a simple geometrical calculation, which can be performed as an exercise. The ratio of the static to the rotating field was set at a rather typical value of {$B/b=500$}. This function describes a resonance which comes about because of the dynamics of precessions. We will see that all spin precession spectroscopies (including µSR, which does not make use of cavities) share this intrinsically resonant aspect.
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A consideration on the name: nutation is more generally known as the small amplitude, wobbling motion of the Earth axis, in its 25700 year long precession around the north ecliptical pole. Here, instead, it is principally an additional very large amplitude, low frequency motion of the magnetic moment. The motion is the same, the difference being that for the Earth the oscillatory perturbation (being due to the non collinearity of the angular momentum and the principal axes of the Earth moment of inertia) is non resonant with the main precession frequency. The different appearance of nutation close and far from resonance are also illustrated by the monochromats in the numerical simulation of the nutation, (Fig. 1 therein)
This simple example can be generalized in two ways:
- We shall see that nuclei experience a variety of different interactions; the same resonance criterion holds when any dominant stationary interaction, yielding a Larmor precession frequencies {$\omega_L$}, replaces the external static field {$\mathbf{ B}$}, and large nutation angles take place if {$\gamma b \ge \omega-\omega_L $};
- An oscillating magnetic field is much more practical to produce than a rotating field; fortunately it is almost entirely equivalent to it, being the superposition of two counter-rotating fields, only one of which may be in resonance conditions with the external field.
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We could summarize what we have seen as follows:
The effect of an oscillating magnetic field on a magnetic moment is (mostly) negligible: it produces precessions at its own frequency, which are of vanishing aperture, unless this frequency very nearly coincides with the Larmor frequency {$\gamma B$}
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