LITMIR.BIZ

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.