In the video shown previously (see Part One), Prof Callaghan introduced the phenomenon of resonance and demonstrated it with a spinning wheel. In MRI the resonance frequency is governed by a simple proportionality, as given in the Larmor equation. We will use this equation later on to establish different frequencies across an object, thereby encoding spatial information and yielding, ultimately, an image of that object. We will also see how the Larmor equation is used in the k-space formalism, so make sure you have a good understanding of this deceptively simple yet intuitively valuable equation.

An aside: Towards the end of the video, Prof Callaghan explains that a single coil is used to excite and detect the MR signals. This can be true, but for most modern MRI scanners the excitation is provided by a body coil (built-in behind the bore liner and invisible from the outside) and signal reception is performed by a "head coil." This has two beneficial effects: 1. the homogeneity of the transmission field is usually improved over the entire head by transmitting with a coil much larger than the head, and 2. the sensitivity to the signal is enhanced by receiving with a coil that is as close to head-sized as practicable. (There are some negative consequences of this scheme, but in general the benefits greatly outweigh the shortcomings.)

Before proceeding, let's pause for a moment and try a thought experiment. You have just seen the Larmor equation introduced. This simple relationship between resonance frequency and magnetic field strength is fundamental to all NMR and MRI experiments. Think about what this relationship might mean for the spinning wheel model of atomic spin that Prof Callaghan used earlier. If the precession frequency (i.e. wobbling rate) of the wheel in Earth's gravitational field acts like an atomic nucleus in a magnetic field, and we want to use a relationship like the Larmor equation to describe what we see, then what term in the Larmor equation corresponds to what physical property of the spinning wheel? It's quite simple. The proportionality constant, gamma, in the Larmor equation describes the atomic spin characteristic of the nucleus. Thus, in the wheel experiment, gamma would describe all the "spin" properties of the wheel, especially its mass and its radius. If Prof Callaghan swapped the wheel he used for one with twice the radius but with the same mass then we would expect the wobbling rate (the resonance frequency) to change. That's just like going from one type of nuclear spin (say the proton, or hydrogen nucleus) to another type of nuclear spin (say a carbon-13 nucleus).

You can probably guess what relationship we'll examine next. In the "wheel resonance" world, what is the equivalent of changing the magnetic field strength for an atomic spin? Clearly, it must be the strength of the gravitational field the wheel is experiencing! If you repeated the wheel-spinning experiment on the Moon (which has 1/6th the gravitational field of Earth) you'd expect the precession rate to differ from the rate it demonstrates here on Earth. Indeed, it would differ by a deterministic amount, too! So just like atomic nuclei in a magnetic field, spinning wheels in gravitational fields have their behavior governed by a simple, but intuitively very useful, relationship. Try to keep it in mind as we get into the depths of k-space.

Finally for this post, check out this cartoon of a bunch of spins initially at thermal equilibrium, to which a resonant excitation pulse is applied:

Notice how, before the radio waves (the radiofrequency (RF) excitation pulse) start, the direction of precession (the

*phase*) of the nuclei is random even though they are precessing at the same rate (frequency). It's subtle in the video, but if you look closely you'll see that the vector (arrow) indicating the direction of atomic spin is slightly different for each of the nuclei; they are precessing at the same

*frequency*but are out of

*phase*with one another. Then, once the RF field starts, the precession of the nuclei starts to become uniform until, at the point when the RF pulse is turned off, the nuclei are all precessing in-phase, i.e. their rotational motion is consistent, or coherent, across all of them. If you want to relate this phenomenon to what you've seen in previous videos, it's like having a bunch of spinning wheels and having lots of people use a finger to reorient each wheel's spin axis, as Prof Callaghan did in the video in Part One.

At the end of this cartoon, in the absence of the external RF field, the nuclei start to loose their phase coherence, gradually returning to the initial condition where the phase is essentially random across the nuclei. In a nutshell, this is how we get a signal out of a bunch of spins. We'll look in more detail at the nature of the signal in later posts.

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