Читать книгу Essential Concepts in MRI - Yang Xia - Страница 18

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 for 23Na.

If I = 0, then the nucleus has no spin and cannot be observed by NMR (e.g., there is no NMR for ¹²C, even though each ¹²C nucleus contains six protons). If I > 0, the nucleus will have an associated magnetic dipole moment µ, given by

(2.1)

where γ is called the gyromagnetic ratio, a characteristic constant for each nuclear species (γ = 2.675 × 10⁸ rad s^-1 T^-1 or 42.576 MHz T^-1 for protons), and ħ is the Planck’s constant (6.62607015 × 10⁻³⁴ J s) divided by 2π. (To convert γ between rad s^-1 T^-1 to MHz T^-1, consider rad/s as the angular velocity equal to 2π times the linear frequency.) The unit of µ is Joule per Tesla (J T^-1). Since γ, ħ, and I are all known constants, the magnitude of µ can be determined accurately.

Note that Eq. (2.1) could be also written as γ = µ/(ħI), which illustrates the fact that γ is the magnetic dipole moment divided by the angular momentum; hence, γ should be more properly named the magnetogyric ratio, not the gyromagnetic ratio that implies the inverse of the two quantities. The term magnetogyric ratio has indeed been used in some books and papers and also recommended by the 2001 International Union of Pure and Applied Chemistry (IUPAC) nomenclature [1]. In modern literature, however, γ is commonly known as the gyromagnetic ratio. Both magnetogyric ratio and gyromagnetic ratio refer to the same value.

The total number of the allowed spin states for a given nucleus is a discrete value, 2I + 1, which ranges from –I, –I + 1, –I + 2, …, to I. (It is another quantum mechanical concept that we need to cite in the classical description of nuclear magnetic resonance.) These values can be written as mI, the azimuthal quantum number. Hence, a nucleus with

 I = 1/2 has two spin states, mI = –1/2 and mI = 1/2. These two spin states are commonly illustrated in quantum mechanics by a two-level energy diagram as in Figure 2.2a, which we will discuss more in Chapter 3. Since there are only two spin states, we may use an arrow to describe the spin-1/2 particles. The mI = –1/2 state is called spin down (↓) while the mI = 1/2 state is called spin up (↑). The spin-down state has higher energy than the spin-up state, hence is the upper level in the energy level diagram (Figure 2.2a).Figure 2.2 The application of an external magnetic field B0 causes (a) a spin-1/2 system to have two discrete energy levels, and (b) a spin-1 system to have three energy levels. Each energy level can be labeled by the individual spin state.

 I > 1/2 has nuclear quadrupole moments that produce splitting of the resonant lines or line-broadening effects. For example, a deuteron (2H) has I = 1, which would have three spin states, corresponding to mI = –1, 0, 1. These three spin states can be used to label a three-level energy diagram as in Figure 2.2b. A quantum mechanical description must be used to understand the behavior of any spin with an angular momentum larger than 1/2.

Note that I is defined in this book as a dimensionless angular momentum; hence, a reduced Planck’s constant has been explicitly included in Eq. (2.1). This use of a dimensionless angular momentum can be found in many books, including those by Callaghan [2] and Hennel and Klinowski [3]. In contrast, I can also be defined as an angular momentum with in its definition [4]. With this definition, Eq. (2.1) would be written as µ = γI.

Table 2.1 lists some fundamental properties of several common nuclei (more lengthy tables can be found in many books and papers [1, 5]). On paper, the bigger γ yields better sensitivity (the column of relative sensitivity, which is defined for the equal numbers of nuclei at constant field.). In practice, one must consider the normal or natural concentration of a nucleus in the specimen (i.e., its availability). For example, the relative sensitivities of ¹H and ¹⁹F are similar (1 vs. 0.834); however, a human body is about 60% water (~75% in a newborn, 60–65% in men, and 55–60% in women), while the amount of ¹⁹F in human or other biological systems is many orders of magnitude smaller by comparison. This explains the challenges and difficulty in detecting the signals other than protons in biological experiments by NMR and MRI.

Table 2.1 Properties of common nuclei.

Isotope	Abundance (%)	Spin	γ (108 rad s-1 T-1)	Relative sensitivity	Resonance frequency f0 at 1T (MHz)
1H	99.9844	1/2	2.6752	1.00	42.577
2H	0.0156	1	0.4107	0.00964	6.536
13C	1.108	1/2	0.6726	0.0159	10.705
19F	100	1/2	2.5167	0.834	40.055
31P	100	1/2	1.0829	0.0664	17.235