Читать книгу Essential Concepts in MRI - Yang Xia - Страница 19

Now look at the situation when one places a single nucleus in an externally applied magnetic field B₀. Because of the dipole moment, the nucleus (which is represented by a magnetic moment µ) will interact with the external field B₀, from which the energy of the nucleus in the field will be given by a dot product

(2.2)

Since the nucleus also has an angular momentum, it experiences a torque, which can be expressed as a cross product, µ × B₀. Since this torque is equal to the time derivative of the angular momentum, by quoting Newton’s second law, we have

(2.3)

By multiplying the above equation on both sides with γ, we have

(2.4)

which is shown in Figure 2.3a. In classical mechanics, this equation describes a precessional motion [6] (Figure 2.3b) of µ around B₀ at an angular frequency ω0, known as the Larmor frequency,¹

Figure 2.3 (a) Precessional motion in classical mechanics. (b) The Larmor precession of a single nucleus with the aid of classical mechanics. (c) A spinning top can have a precessional motion.

(2.5)

where ω is expressed as an angular frequency in rad/s, which can be converted to the temporal/linear frequency f in Hz by noting ω = 2πf. Note that when γ is positive (which it is for proton [the nucleus of ¹H] and many other nuclei), the rotation will be clockwise, as shown in Figure 2.3b. γ can also be negative (e.g., ³He, ¹⁵N), where the rotation becomes counterclockwise.

Although the equation for this Larmor precession seems simple, it is the fundamental equation of the NMR phenomenon. The equation states that (a) the frequency of the nuclear precession is proportional to the externally applied magnetic field B₀, and (b) the proportionality constant is γ. Equation (2.4) and Eq. (2.5) are useful and accurate for describing the nuclear precession in the presence of an external field B₀.

On a macroscopic scale, one can find several analogs for the precessional motion of the nuclear spin. For example, a spinning top (Figure 2.3c) can have a precessional motion when the axis of rotation does not pass through the top’s center of gravity (i.e., the top is not standing up perfectly vertical), which yields a torque and an angular momentum for the top that induces it into a precessional motion. In a spinning top, the gravitational force mg that points vertically downward plays the same role of the magnetic field B0 in NMR. Since a top is a macroscopic object, the tipping angle of a spinning top can vary from 0˚ to a large angle continuously, while the tipping angle of a nuclear precessional motion is fixed and cannot be varied. A second common example is the precessional motion of our planet Earth in the solar system, which is tipped at a constant angle of 23.4˚ from an “axis” in space, with a period of the precessional motion of 26,000 years. The torque on Earth is exerted by the sun and the moon.