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Chapter 2

The Basics

Magnetic Properties of Nuclei

The Boltzmann Distribution

The Larmor Equation



The Rotating Coordinate System

The Magnetic Resonance Signal

Frequency Analysis: Fourier Transform

02-02 Magnetic Properties of Nuclei

1H, 13C, 19F, 23Na, and 31P are among the most interesting nuclei for magnetic resonance imaging. All of these nuclei occur naturally in the body. The proton (¹H) is the most com­mon­ly used because the two major components of the hu­man body are water and fat, both of which contain hydrogen. They all have mag­ne­tic properties which dis­tin­gu­ish them from nonmagnetic isotopes.

Nuclei such as 12C and 16O which have even numbers of protons and neutrons do not produce magnetic resonance signals.

The hydrogen atom (¹H) consists of a single positively charged proton which spins around its axis. Spinning charged particles create an electromagnetic field, analogous to that from a bar magnet (Figure 02-02).

Figure 02-02:
A spinning charged particle possesses a characteristic magnetic moment μ and can be described as a magnetic dipole creating a magnetic field similar to a bar magnet (N = north, S = south).

When atomic nuclei with magnetic properties are placed in a magnetic field, they can absorb elec­tro­mag­ne­tic waves of characteristic frequencies. The exact frequency de­pends on the type of nucleus, the field strength, and the physical and chemical en­vi­ron­ment of the nucleus (Figure 02-03).

Figure 02-03:
The nuclei are able to absorb elec­tro­mag­ne­tic waves in both strong and weak mag­ne­tic fields. However, the absorption oc­curs at a field-strength-dependent fre­quen­cy, which is higher in the strong mag­ne­tic field than in the weak mag­ne­tic field.

The absorption and re-emission of such radiowaves is the basic phenomenon utilized in MR imaging and MR spectroscopy. To understand the magnetic re­so­nan­ce phe­no­me­non, two simple macroscopic parallels can be drawn:

First, let us consider a small magnetic needle placed in a magnetic field (Fi­gu­re 02- 04). If the needle is capable of rotating freely, it will orientate itself in the field in such a way that an equilibrium situation is attained. This equilibrium can be main­tai­ned in­de­fi­ni­te­ly if no external forces influence the system.

Figure 02-04:
The compass needle will seek the stable equilibrium state.
(a) When it is turned around with a finger, energy is brought in and it will be in an un­stab­le energy-rich position.
(b) As soon as the finger is taken away, the needle will return to its stable state.

A second example illustrates the influence of the external strain on the fre­quen­cy of the wave absorbed or re-emitted by the system. Imagine three iden­ti­cal guitar strings ex­po­sed to different tensions; the uppermost string of this in­stru­ment has no tension at all, the middle string weak tension, and the lower­most high tension. If we excite the strings, the resultant vibration is dependent on the tension of the strings (Figure 02-05).

Figure 02-05:
A string (the nucleus) cannot vibrate with­out being exposed to tension (the external magnetic field). The higher the tension, the higher will be the frequency of the vi­bra­tion.

In both examples, we have made comparisons between a macroscopic and the mi­cro­sco­pic nuclear system. In the first example, we compared the nuclei with small magnet needles and in the second, with strings.

Such parallels provide a mental picture of the phenomenon, but have their short­co­mings. One limitation of the models is that all physical phenomena on the molecular scale are quantified. For example, whereas an infinity of different orientations is pos­si­ble for the magnetic needle, no smooth continuous tran­si­tions between the equilibrium state and the unstable, energy-rich state exist for the magnetic nucleus; instead, quan­tum mechanics predicts that only jumps bet­ween these two states are pos­si­ble for nuc­lei with a spin of ½ such as protons (Figure 02-06).

Figure 02-06:
(a) Protons outside a magnetic field, and
(b) pro­tons in a magnetic field.
In the pre­sence of a magnetic field, nuclei populate two distinct energy levels. The se­pa­ra­tion between these levels increases li­ne­ar­ly with magnetic field strength, as does the population difference.

At equilibrium, we have a slightly larger population in the lower energy level, giving a net magnetization. To observe this population difference we have to pro­vide an amount of energy equal to ΔE (the energy difference between the two levels).

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