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2 Classical Description of Magnetic Resonance
2.1 FUNDAMENTAL ASSUMPTIONS
The states of atomic nuclei are quantum mechanical by nature. This means that the properties that we observe for a single nucleus belong to one in a discrete set of possibilities (i.e., quantum states). A deep understanding of NMR phenomenon therefore requires the assistance of quantum mechanics. In practical NMR and MRI experiments, however, we deal with an extremely large number of nuclei in any specimen (either a human or a tissue block or a drop of liquid). For example, if I give you a glass container that has 18.015 grams of liquid water, do you know how many water molecules are in the container? Well, we know precisely how many are in it: 6.022 × 1023 water molecules (Avogadro’s number, since one mole of water is 18.015 grams)! Consider taking just one gram of water from the container, which has a volume of one milliliter. If this one milliliter of water is further divided into 100 droplets, each tiny droplet still contains about 3.343 × 1020 water molecules, or about 6.686 × 1020 protons (i.e., hydrogen atoms) since each water molecule has two hydrogen atoms. It is an enormous number.
It is fortunate that these protons in a small water droplet act largely independently, so that at the macroscopic level, the collection of these protons appears continuous. (If these protons were to act completely independently, NMR would not have much practical use at all. If, on the other hand, these protons were to couple or interact tightly with each other, NMR also would not have much practical usage since we simply do not know how to solve the complex interactions among the enormous number of particles in any practical system.)
The simplest and most common nuclei used in NMR and MRI are hydrogen, or protons, a component of the water molecules in the liquid state. Since a proton is a spin-1/2 particle (another quantum mechanical concept) and often very mobile, we could ignore internuclear dipole interactions and scalar coupling between the protons. Hence all states of the nuclear ensemble may be characterized by a vector quantity that is referred to as the nuclear magnetization (M). The adoption of this vector quantity permits a classical description of magnetic resonance phenomena.
A quantum mechanical description is needed when nuclei experience mutual interactions or have a spin > 1/2 (even if they are independent, due to the presence of the quadrupole interaction). A quantum description is especially important in high-resolution NMR spectroscopy and some advance MRI techniques where the understanding of nuclear interactions is essential.
For the rest of this chapter, the physics of NMR will be described using classical mechanics. Since the classical mechanical description of NMR needs to use a few concepts in quantum mechanics, this type of classical mechanical approach can also be termed as a semi-classical description of NMR.
2.2 NUCLEAR MAGNETIC MOMENT
The theory of electricity and magnetism shows that any motion of a charged body has an associated magnetic field. For example, an electric current is due to the motion of electrons along a conductor on a macroscopic scale. If you bend this conducting wire into a loop, you have just made a coil (Figure 2.1). The coil with an electric current traveling in the wire has an associated magnetic moment, which is the product of the electric current and the area of the coil. When the current-carrying coil is placed in an external magnetic field, the coil will experience a mechanical torque.
Figure 2.1 (a) Moving charges at velocity v along a conducting wire form an electric current, which has an associated magnetic field B by the right-hand rule; moving charges carry momentum. (b) An electric current loop has an area and a current. The magnetic moment of the loop µ equals to the product of the area and current (which is valid for any shaped loop). n is the normal vector of the current loop; ϕ is the angle between n and B. The torque that causes the rotation of the loop in the magnetic field B is τ = µ × B = (area × current) B sinϕ. (c) The right-hand rule for the direction of the force on a current-carrying wire in the magnetic field.
This phenomenon can also be extended to the atomic scale: when electrons or nuclei possess angular momentum, there is an associated magnetic moment. Since on the atomic scale, angular momentum is quantized, that is, it can only take certain discrete values (one of the fundamental postulates in modern physics), the magnetic moment is also quantized. (Note that here the angular momentum is a vector quantity since we are using classical mechanics to describe the concept. Later in a quantum mechanical description [Chapter 3], the angular momentum keeps the same symbol I but becomes an operator.)
The angular momentum, labeled as I, is called the spin angular momentum or simply spin, which should be considered as a fundamental property of the nucleus. One could imagine the nucleus as a finite-sized ball spinning on its axis. (Such a spinning ball picture, however, remains valid only in classical mechanics and should be not taken too literally.) The spin I has the following properties:
I may have any half-integer or integer value such as 0, 1/2, 1, 3/2, etc. This value is known as the spin quantum number (I).
I will have a fixed value for a given nucleus (due to the even/odd mass and charge number of the nucleus). For example, I = 0 for 12C and 16O; I = 1/2 for 1H (proton), 13C, 19F, 31P; I = 1 for 2H (deuteron) and 14N; and I = 3/2
