Chapter 1
Production of net magnetization

Magnetic resonance (MR) is a measurement technique used to examine atoms and molecules. It is based upon the interaction between an applied magnetic field and a particle that possesses spin and charge. While electrons and other subatomic particles possess spin (or more precisely, spin angular momentum) and can be examined using MR techniques, this book focuses on nuclei and the use of MR techniques for their study, formally known as Nuclear Magnetic Resonance, or NMR. Nuclear spin, or more precisely nuclear spin angular momentum, is one of several intrinsic properties of an atom and its value depends on the precise atomic composition. Every element in the Periodic Table except argon and cerium has at least one naturally occurring isotope that possesses nuclear spin. Thus, in principle, nearly every element can be examined using MR, and the basic ideas of resonance absorption and relaxation are common for all of these elements. The precise details will vary from nucleus to nucleus and from system to system.

1.1 Magnetic fields

Magnetic fields are produced by and surround electric currents, whether these currents are macroscopic currents such as those running through wires or microscopic currents such as those around an atom of iron. The magnetic field can be represented as a vector, meaning that it has both a magnitude and a direction, and is usually denoted by the variable B.¹ For example, the B field at the center of a circular loop of current-carrying wire points in the direction of the axis of the loop (perpendicular to the plane of the loop and therefore perpendicular to the current flow) and it has a magnitude that is proportional to the current in the loop. The magnitude of the field is related to the strength of the magnetic force on wires or magnetic materials, and the direction of the field is perpendicular to the direction of the force.

Magnetic fields often vary over time and/or space, and will be coupled to the electric field, producing electromagnetic waves. Magnetic fields, particularly those in electromagnetic waves, are characterized by their frequency (the time between two consecutive “peaks” in the field). In MR, there are magnetic fields, which are constant in time, which vary at acoustic frequencies (a few kilohertz), and which vary at radio frequencies (RF) (several megahertz).

1.2 Nuclear spin

The structure of an atom is an essential component of the MR experiment. Atoms consist of three fundamental particles: protons, which possess a positive charge; neutrons, which have no charge; and electrons, which have a negative charge. The protons and neutrons are located in the nucleus or core of an atom; thus all nuclei are positively charged. The electrons are located in shells or orbitals surrounding the nucleus. The characteristic chemical reactions of elements depend upon the particular number of each of these particles. The properties most commonly used to categorize elements are the atomic number and the atomic weight. The atomic number is the number of protons in the nucleus and is the primary index used to differentiate atoms. All atoms of an element have the same atomic number and undergo the same chemical reactions. The atomic weight is the sum of the number of protons and the number of neutrons. Atoms with the same atomic number but different atomic weights are called isotopes. Isotopes of an element will undergo the same chemical reactions, but at different reaction rates.

A third property of the nucleus is spin or intrinsic spin angular momentum. Classically, nuclei with spin can be considered to be always rotating about an axis at a constant rate. This self-rotation axis is perpendicular to the direction of rotation (Figure 1.1). A limited number of values for the spin are found in nature; that is, the spin, I, is quantized to certain discrete values. These values depend on the atomic number and atomic weight of the particular nucleus. There are three groups of values for I: zero, integral, and half-integral values. A nucleus has no spin (I = 0) if it has an even atomic weight and an even atomic number; for example, ¹²C (6 protons and 6 neutrons) or ¹⁶O (8 protons and 8 neutrons). Such a nucleus does not interact with an external magnetic field and cannot be studied using MR. A nucleus has an integral value for I (e.g., 1, 2, 3) if it has an even atomic weight and an odd atomic number; for example, ²H (1 proton and 1 neutron) or ⁶Li (3 protons and 3 neutrons). A nucleus has a half-integral value for I (e.g., 1/2, 3/2, 5/2) if it has an odd atomic weight. Table 1.1 lists the spin and isotopic composition for several elements commonly found in biological systems. The ¹H nucleus, consisting of a single proton, is a natural choice for probing the body using MR techniques for several reasons. It has a spin of 1/2 and is the most abundant isotope for hydrogen. Its response to an applied magnetic field is one of the largest found in nature. Since the body is composed of tissues that contain primarily water and fat, both of which contain hydrogen, a significant MR signal can be produced naturally by normal tissues.

nfg001 — **Figure 1.1** A rotating nucleus (spin) with a positive charge produces a magnetic field known as the magnetic moment oriented parallel to the axis of rotation (a). This arrangement is analogous to a bar magnet in which the magnetic field is considered to be oriented from the south to the north pole (b).

c01-math-001 — **Table 1.1** Constants for Selected Nuclei of Biological Interest

c01-math-002 — **Table 1.1** Constants for Selected Nuclei of Biological Interest

Source: Adapted from Ian Mills (ed.), Quantities, Units, and Symbols in Physical Chemistry, IUPAC, Physical Chemistry Division, Blackwell, Oxford, UK, 1989.

While a rigorous mathematical description of a nucleus with spin and its interactions requires the use of quantum mechanical principles, most of MR can be described using the concepts of classical mechanics, particularly in describing the actions of a nucleus with spin. The subsequent discussions of MR phenomena in this book use a classical approach. In addition, while the concepts of resonance absorption and relaxation apply to all nuclei with spin, the descriptions in this book focus on ¹H (commonly referred to as a proton) since most imaging experiments visualize the ¹H nucleus.

1.3 Nuclear magnetic moments

Recall that the nucleus is the location of the positively charged protons. When this charge rotates due to the nuclear spin, a local magnetic field or magnetic moment is induced about the nucleus. This magnetic moment will be oriented parallel to the axis of rotation. Since the nuclear spin is constant in magnitude, its associated magnetic moment will also be constant in magnitude. This magnetic moment is fundamental to MR. A bar magnet provides a useful analogy. A bar magnet has a north and a south pole, or, more precisely, a magnitude and orientation to the magnetic field can be defined. The axis of rotation for a nucleus with spin can similarly be viewed as a vector with a definite orientation and magnitude (Figure 1.1). This orientation of the nuclear spin and the changes induced in it due to the experimental manipulations that the nucleus undergoes provide the basis for the MR signal.

In general, MR measurements are made on collections of spins rather than on an individual spin. It is convenient to consider such a collection both as individual spins acting independently (a “microscopic” picture) and as a single entity (a “macroscopic” picture). For many concepts, the two pictures provide equivalent results, even though the microscopic picture is more complete. Conversion between the two pictures requires the principles of statistical mechanics. While necessary for a complete understanding of MR phenomena, the nature of this conversion is beyond the scope of this book. However, the macroscopic picture is sufficient for an adequate description for most concepts presented in this book. When necessary, the microscopic picture will be used.

1.4 Larmor precession

Consider an arbitrary volume of tissue containing hydrogen atoms (protons) in the absence of an external magnetic field. Each proton has a spin vector (or magnetic moment) of equal magnitude. However, the spin vectors for the entire collection of protons within the tissue are randomly oriented in all directions; there is a continuous distribution of the spin orientations. Performing a vector addition (head-to-toe) of these spin vectors produces a zero sum; that is, no net magnetization is observed in the tissue (Figure 1.2).

nfg002 — **Figure 1.2** Microscopic and macroscopic pictures of a collection of spins in the absence of an external magnetic field. In the absence of a magnetic field, the spins have their axes oriented randomly (microscopic picture, left side of figure). The vector sum of these spin vectors is zero (macroscopic picture, right side).

If the tissue is placed inside a magnetic field B₀, the individual protons begin to rotate perpendicular to, or precess about, the magnetic field. The spin vectors for the protons are tilted slightly away from the axis of the magnetic field, but each axis of precession is parallel to B₀. This precession is at a constant rate and occurs because of the interaction of the magnetic field with the spinning positive charge of the nucleus. By convention, B₀ and the axis of precession are defined to be oriented in the z direction of a Cartesian coordinate system. (This convention is not universally followed, but it is the prevailing convention.) The motion of each proton can be described by a set of coordinates perpendicular (x and y) and parallel (z) to B₀. In the absence of other interactions the perpendicular, or transverse, coordinates are nonzero but vary cyclically with time as the proton precesses, while the parallel or longitudinal coordinate is constant with time (Figure 1.3). The rate or frequency of precession is proportional to the strength of the magnetic field and is expressed by the following equation, known as the Larmor equation:

1.1

where ω₀ is the Larmor frequency in megahertz (MHz),² B₀ is the magnetic field strength in tesla (T) that the proton experiences, and γ is a constant for each nucleus in MHz/T, known as the gyromagnetic ratio. Values for γ and ω at 1.5 T for several nuclei are tabulated in Table 1.1.

Figure 1.3 Inside a magnetic field, a proton precesses or revolves about the magnetic field. The precessional axis is parallel to the main magnetic field, B₀. The z component of the spin vector (projection of the spin onto the z axis) is the component of interest because it does not change in magnitude or direction as the proton precesses. The x and y components vary with time at a frequency ω₀ proportional to B₀ as expressed by equation (1.1).

An alternate picture known as a rotating frame of reference or rotating coordinate system is often used in MR. It is a convenient view when describing objects that undergo rotational motion. When viewed using in a rotating frame of reference, the coordinate system rotates about one axis while the other two axes vary with time. By choosing a suitable axis and rate of rotation for the coordinate system, the rotating object appears stationary.

For MR experiments, a convenient rotating frame uses the z axis, parallel to B₀, as the axis of rotation while the x and y axes rotate at the Larmor frequency, $c01-math-004$ . When viewed in this fashion, the precessing spin appears stationary in space with a fixed set of x, y, and z coordinates. Regardless of whether a stationary or rotating coordinate system is used, M₀ is of fixed amplitude and is parallel to the main magnetic field. For all subsequent discussions in this book, a rotating frame of reference with the rotation axis parallel to B₀ is used when describing the motion of the protons.

1.5 Net magnetization

If a vector addition is performed, as before, for the spin vectors inside the magnetic field, the results will be slightly different than for the sum outside the field. In the direction perpendicular to B₀, despite the precession of each spin, the spin orientations are still randomly distributed (covering a complete range of x and y values, both positive and negative) just as they were outside the magnetic field. There is therefore still no net magnetization perpendicular to B₀. However, in the direction parallel to B₀, there is a different result. Because there is an orientation to the precessional axis of the proton that is constant with time, there is a constant, nonzero interaction or coupling between the proton and B₀, known as the Zeeman interaction. Instead of a continuous range of values, this coupling causes the z component to be quantized to a limited or discrete number of values. For the ¹H nucleus, there are only two possible values for the z component: parallel or along B₀ and antiparallel or against B₀. This coupling also causes a difference in energy ΔE between these two orientations that is proportional to B₀ (Figure 1.4). This leads to a slight excess of positive z values compared to negative z values as described in detail below. If a vector sum is performed on this collection of protons, the x and y components sum to zero but a nonzero, positive z component will be left, the net magnetization M₀. In addition, since the z axis is the axis of rotation, M₀ does not vary with time.

Figure 1.4 Zeeman diagram. In the absence of a magnetic field (left side of figure), a collection of spins will have the configurations of z components equal in energy so that there is no preferential alignment between the spin-up and spin-down orientations. In the presence of a magnetic field (right side), the spin-up orientation (parallel to B₀) is of lower energy and its configuration contains more spins than does the higher-energy spin-down configuration. The difference in energy ΔE between the two levels is proportional to B₀.

The result of the Zeeman interaction is that spins in the two orientations, parallel (also known as spin up) and antiparallel (spin down), have different energies. Those spins oriented parallel to B₀ are of lower energy than those oriented antiparallel. For a collection of protons, more will be oriented parallel to B₀ than antiparallel; that is, there is residual polarization of the spins induced parallel to the magnetic field (Figure 1.5a). The exact number of protons in each energy level can be predicted by a distribution function known as the Boltzmann distribution:

1.2

where k is Boltzmann's constant, 1.381 × 10⁻²³ J K⁻¹ and N_UPPER and N_LOWER are the number of protons in the upper and lower energy levels, respectively. Since the separation between the energy levels ΔE depends on the field strength B₀, the exact number of spins in each level also depends on B₀ and the difference increases with increasing B₀. For a collection of protons at body temperature (310 K) at 1.5 T, there will typically be an excess of ∼1:10⁶ protons in the lower level out of the approximately 10²⁵ protons within the tissue. This unequal number of protons in each energy level means that the vector sum of spins will be nonzero and will point along the magnetic field. In other words, the tissue will become polarized or magnetized in the presence of B₀ with a value M₀, known as the net magnetization. The orientation of this net magnetization will be in the same direction as B₀ and, in the absence of other interactions, will be constant with respect to time (Figure 1.5b).

Figure 1.5 Microscopic (a) and macroscopic (b) pictures of a collection of spins in the presence of an external magnetic field. Each spin precesses about the magnetic field. If a rotating frame of reference is used with a rotation rate of ω₀, the collection of protons appears stationary. Whereas the z components are one of two values (one positive and one negative), the x and y components can be any value, positive or negative. The spins will appear to track along two cones, one with a positive z component and one with a negative z component. Because there are more spins in the upper cone, there will be a nonzero vector sum M₀, the net magnetization. It will be of constant magnitude and parallel to B₀.

1.6 Susceptibility and magnetic materials

The magnetic susceptibility describes the response of a substance to the applied magnetic field. While a complete analysis of the origin of this response is beyond the scope of this book, it is useful to describe the three levels of response that are encountered in MR. A diamagnetic response is found in all materials, arising from the electrons surrounding the nuclei. It is a very weak response ( $c01-math-008$ is very small and negative) except for superconductors. The diamagnetic response produces a slight repulsive force and is the only response for most materials, including tissue. A paramagnetic response is larger in magnitude than a diamagnetic response, but still relatively weak ( $c01-math-009$ is small but positive). It is found in molecules where there are so-called unpaired electrons, which align themselves in response to the external field and produce a mild attractive force. The final response level is known as ferromagnetic, typically found in certain metals in which $c01-math-010$ is very large and positive. The atoms in ferromagnetic materials align with each other to form relatively large (still microscopic) magnetic domains. The alignment of these domains produces a strong attractive force in response to an external field and also leaves the ferromagnetic material with a permanent residual magnetization after the external field is removed. While diamagnetic and paramagnetic materials are safe to be in or near MR scanners, ferromagnetic materials should remain outside the scan room.

Element	Nuclear composition		Nuclear spin I	Gyromagnetic ratio $c01-math-001$ (MHz T⁻¹)	% Natural abundance	$c01-math-002$ at 1.5 T (MHz)
	Protons	Neutrons
¹H, protium	1	0	1/2	42.5774	99.985	63.8646
²H, deuterium	1	1	1	6.53896	0.015	9.8036
³He	2	1	1/2	32.436	0.000138	48.6540
⁶Li	3	3	1	6.26613	7.5	9.39919
⁷Li	3	4	3/2	16.5483	92.5	24.8224
¹²C	6	6	0	0	98.90	0
¹³C	6	7	1/2	10.7084	1.10	16.0621
¹⁴N	7	7	1	3.07770	99.634	4.6164
¹⁵N	7	8	1/2	4.3173	0.366	6.4759
¹⁶O	8	8	0	0	99.762	0
¹⁷O	8	9	5/2	5.7743	0.038	8.6614
¹⁹F	9	10	1/2	40.0776	100	60.1164
²³Na	11	12	3/2	11.2686	100	16.9029
³¹P	15	16	1/2	17.2514	100	25.8771
¹²⁹Xe	54	75	1/2	11.8604	26.4	17.7906