1.2.2.1 Atomic Nomenclature

Isotopes of a particular element are atoms that possess the same number of protons but a varying number of neutrons. For example, ¹H(protium), ²H(deuterium), and ³H(tritium) are isotopes of the element hydrogen and , , , , , , , and are isotopes of carbon. Isotones are atoms that possess the same number of neutrons but a different number of protons. For example, , , , , and are isotones, each contains three neutrons. Isobars are atoms with the same number of nucleons but a different number of protons and a different number of neutrons. For example, , , and are isobars, with six nucleons each. Finally, isomers represent different energy states for nuclei with the same number of neutrons and protons. Differences between isotopes, isotones, isobars, and isomers are illustrated in Table 1.2.

1.2.2.2 Electrons

Atoms in their normal state are neutral because the number of electrons outside the nucleus (i.e. the negative charge in the atom) equals the number of protons (i.e. the positive charge in the atom) of the nucleus. Electrons are positioned in energy levels, termed shells that surround the nucleus. The first or K-shell contains no more than two electrons, the second or L shell contains no more than eight electrons, and the third or M shell contains no more than 18 electrons (see Figure 1.2). The outermost electron shell of an atom, no matter which shell it is, never contains more than eight electrons. Electrons in this shell are called valence electrons and determine, to a large degree, the chemical properties of the atom. For example, atoms with an outer shell entirely filled with electrons seldom react chemically; these atoms constitute inert elements known as the noble gases (i.e. helium, neon, argon, krypton, xenon, and radon).

The energy levels for electrons are divided into sublevels separated from each other. To describe the position of an electron in the extranuclear structure of an atom, the electron is assigned four quantum numbers designated below as n, l, m_l, and m_s.

The principal quantum number, n, defines the main energy level or shell within which the electron resides (n = 1 for the K-shell, n = 2 for the L shell, etc.). The orbital angular momentum (also called azimuthal) quantum number, l, describes the electron’s angular momentum (l = 0, 1, 2, … n − 1). The orientation of the electron’s magnetic moment in a magnetic field is defined by the magnetic quantum number, m_l (m_l = −l, −l + 1, … l − 1, l). The direction of the electron’s spin upon its own axis is specified by the spin quantum number, m_s (m_s = +½ or −½). The Pauli exclusion principle states that no two electrons in the same atomic system may be assigned identical values for all four quantum numbers. Listed in Table 1.3 are quantum numbers for electrons in a few atoms with low atomic numbers.

The values of the orbital angular momentum quantum number, l = 0, 1, 2, 3, 4, 5, and 6, are also identified with the symbols, s, p, d, f, g, h, and i, respectively. This notation is known as “spectroscopic” because it is used to describe the separate emission lines observed when light emitted from a heated metallic vapor lamp is passed through a prism. From 1890s onward, observation of these spectra provided major clues about the binding energies of electrons in atoms of the metals under study. By the 1920s, it was known that the spectral lines above “s” (i.e. l = 0) could be split into multiple lines in the presence of a magnetic field. The lines were thought to correspond to “orbitals” or groupings of similar electrons within orbits. The modern view of this phenomenon is that while the “s” orbital is spherically symmetric (see Figure 1.3), the other orbitals are not. In the presence of a magnetic field, the “p” orbital can be in alignment along any one of the three axes of space: x, y, or z. Each of these three orientations has a slightly different energy corresponding to the three possible values of m_l (−1, 0, and 1). According to the Pauli exclusion principle, each orbital may contain two electrons, one with m_s = +½ and the other with m_s = –½.

The K-shell of an atom consists of one orbital, 1s, containing two electrons. The L shell consists of the 2s subshell that contains one orbital (two electrons) and the 2p subshell containing a maximum of three orbitals (i.e. six electrons). If the L shell of an atom is filled, its electrons will be noted in spectroscopic notation as 2s²2p⁶. This notation is summarized for three atoms: helium, carbon, and sodium, in Table 1.3.

Since the late 1920s, it has been understood that electrons in an atom do not behave exactly like tiny moons orbiting a planet-like nucleus. It is more accurate to define them as entities whose behavior is described by wave functions, rather than as point particles in orbits. Although a wave function itself is not directly observable, calculations performed with this function predict the location of the electron. In contrast to the calculations of classical mechanics, in which properties such as force, mass, and acceleration are used in equations to yield a definite answer for a quantity such as position in space, quantum mechanical calculations yield probabilities. At a particular location in space, for example, the square of the amplitude of a particle’s wave function yields the probability that the particle will appear at that location.

In Figure 1.3, this probability is depicted for several possible energy levels of a single electron surrounding a hydrogen nucleus (i.e. a single proton). In this illustration, a brighter shading implies a higher probability of finding the electron at that location. Locations at which the probability is maximized correspond roughly to the electron shell model discussed previously. However, it is important to emphasize that the probability of finding the electron at other locations, even in the middle of the nucleus, is not zero. This particular result explains a certain form of radioactive decay in which a nucleus captures an electron, a phenomenon that is better explainable by quantum mechanics.

1.2.2.3 Electron Energy Levels and Binding Energies

An electron in an inner shell of an atom is attracted to the nucleus by a force greater than that exerted by the nucleus on an electron farther away. An electron may be moved from one shell to a more distant shell only if energy is supplied by an external source. Binding energy is a negative value because it represents an amount of energy that must be supplied to remove an electron from an atom. The energy that must be imparted to an atom to move an electron from an inner shell to an outer shell is equal to the arithmetic difference in binding energy between the two shells. For example, the binding energy for an electron in the K-shell of hydrogen is −13.5 and −3.4 eV for an electron in the L shell. The energy required to move an electron from the K to the L shell in hydrogen is (−3.4 eV) − (−13.5 eV) = 10.1 eV. Electrons in inner shells of high-Z atoms are bound to the nucleus with a force much greater than that exerted upon the solitary electron in hydrogen (Table 1.4).

The electrons within a particular electron shell do not have exactly the same binding energy. Differences in binding energy among the electrons in a particular shell are determined by the orbital magnetic and spin quantum numbers, l, m_l, and m_s. The combinations of these quantum numbers provide three subshells for the L shell (L_I to L_III) and five subshells for the M shell (M_I–M_V), where the M subshells occur only if a magnetic field is present. Energy differences between the subshells are small when compared to differences between shells, but these differences are important in medical imaging as they explain certain properties of the emission spectra of X-ray tubes. Table 1.5 gives values for the binding energies of K, L, and M shell electrons for selected elements.

1.2.2.4 Electron Transitions, Characteristic, and Auger Emission

Alternatively, an outer shell electron may drop spontaneously to fill a vacancy in an inner shell. This spontaneous transition results in the release of energy. Spontaneous transitions of outer shell electrons falling to inner shells are depicted in Figure 1.4.

The energy released when an outer electron falls to an inner shell equals the difference in binding energy between the two shells involved in the transition, for example, an electron moving from the M to the K-shell of tungsten releases (−69 500 eV) − (−2810 eV) = −66 690 eV or −66.69 keV. The energy is released in one of the two forms. In its first form, the transition energy is released as a photon (see Figure 1.4b). As the binding energy of electron shells is a unique characteristic of each element, the emitted photon is called a characteristic photon or characteristic X-ray. The emitted photon may be described as a K, L, or M characteristic photon denoting the destination of the transition electron.

An electron transition creates a vacancy in the outer shell where the electron originated, and this vacancy may be filled by an electron transition from another shell, leaving yet another vacancy and so on. Thus, a vacancy in an inner electron shell produces a cascade of electron transitions that yield a range of characteristic photon energies. Electron shells farther from the nucleus are more closely spaced in terms of binding energy. Therefore, characteristic photons produced by transitions among outer shells have less energy than do those produced by transitions involving inner shells. For transitions to shells beyond the M shell, characteristic photons are no longer energetic enough to be considered X-rays.

An alternative to the release of the energy through characteristic X-ray production is for the atom to “eject” another electron to compensate for an electron vacancy (Figure 1.4c). In this process, the energy released during an electron transition is transferred to another electron. This energy is sufficient to eject the electron from its shell. The ejected electron is referred to as an Auger electron. The kinetic energy of the ejected electron will not equal the total energy released during the transition because some of the transition energy is used to free the electron from its shell. The Auger electron is usually ejected from the same shell that held the electron that made the transition to an inner shell, as shown in Figure 1.4c. In this case, the kinetic energy of the Auger electron is calculated by twice subtracting the binding energy of the outer shell electron from the binding energy of the inner shell electron. The first subtraction yields the transition energy and the second subtraction accounts for the liberation of the ejected electron.

Either characteristic photon emission or Auger electron emission may occur during an electron transition, and although it is impossible to predict which process will occur for a specific atom, the probability of characteristic emission is termed the fluorescence yield, ω, defined as

(1.6)

The fluorescence yield increases with atomic number as depicted in Figure 1.5. For a transition to the K-shell of calcium, for example, the probability is 0.19 that a K characteristic photon will be emitted and 0.81 that an Auger electron will be emitted. The fluorescence yield is a factor to consider in the selection of materials used to produce X-ray imaging system and also for radioactive sources for nuclear imaging where Auger electrons result in increased radiation dose to the patient because they do not travel far in tissue.

1.2.2.5 Electronic Conduction

Two electron energy bands of a solid are depicted in Figure 1.6. The lower energy band, called the valence band, consists of electrons that are tightly bound in the chemical structure of the material. The upper energy band, called the conduction band, contains electrons that are relatively loosely bound. Conduction band electrons can move in the material and may constitute an electrical current under the proper conditions. If no electrons populate the conduction band, then the material cannot support an electrical current under normal circumstances. However, if enough energy is imparted to an electron in the valence band to raise it to the conduction band, then the material can support an electrical current.

Solids can be separated into three categories based on the difference in energy between electrons in the valence and conduction bands. In conductors, there is little energy difference between the bands, so the electrons are continuously promoted from the valence to the conduction band by routine collisions between electrons. In insulators, the conduction and valence bands are separated by a wide band gap, also known as the “forbidden zone,” of 3 eV or more. Under this condition, application of voltage to the material usually will not provide enough energy to promote electrons from the valence band to the conduction band. Therefore, insulators do not support an electrical current under normal circumstances. Of course, there is always a “breakdown voltage” above which an insulator will support a current, although probably not without structural damage to the material.

If the band gap of the material is between 0 and 3 eV, the material exhibits electrical properties between those of an insulator and a conductor. Such a material, termed a semiconductor, will conduct electricity under certain conditions and act as an insulator under others. The conditions may be altered by the addition to the material of trace amounts of impurities that have allowable energy levels within the band gap of the solid, creating electron traps as depicted in Figure 1.6. Semiconductors have significant applications in radiation detection.

As previously stated, very little energy is required to promote electrons to the conduction band in a conductor. However, the electrons in the conduction band do not fully move freely. As electrons attempt to move through the material, they interact with other electrons and with imperfections such as impurities in the solid. At each encounter, some energy is transferred to atoms and molecules and ultimately transferred to heat. This transfer of energy is the basis of electrical resistance and is usually viewed as energy loss, unless explicitly intended for heat production as in an electric heating element. Otherwise, the loss or energy and mitigation for that loss can create a significant technological challenge as in an X-ray tube.

There is a class of material, called superconductors, in which there is very little resistance to the flow of electrons under certain conditions. First discovered in mercury [2], it has been described using the formalism of quantum mechanics [3,4] and is applicable to other materials at very low temperature. In a superconductor, the passage of an electron disturbs the structure of the material in such a way as to encourage the passage of another electron arriving after exactly the right interval of time. The passage of the first electron establishes an oscillation in the positive charge of the material that pulls the second electron through. This behavior has been analogized as “electronic waterskiing” where one electron is swept along in another electron’s “wake.” Thus, electrons tend to travel in what are known as Cooper pairs. Cooper pair electrons form a lower energy bound state where they can move as a unit with essentially no resistance to flow. Currents in superconducting loops of wire can continue for several years with no additional input of energy.

Many types of materials exhibit superconductive behavior when cooled to temperatures in the range of a few degrees Kelvin (room temperature is approximately 295 K); in fact, 26 elements and thousands of alloys and compounds exhibit this behavior. Lowering the temperature promotes superconductivity by decreasing molecular motion, thereby decreasing the kinetic energy of the material. However, maintenance of materials at very low temperatures often requires liquid helium as a cooling agent. Helium liquefies at 23 K, is relatively expensive, and is usually insulated from ambient conditions with another refrigerant such as liquid nitrogen, which adds to the cost. Superconducting technology is a foundation of modern magnetic resonance imaging (MRI) systems, where a string magnetic field is produced and maintained by superconducting coils. The science of superconducting has been leading to higher temperature possibilities in new material (e.g. barium–lanthanum–copper oxide and magnesium diboride) [5,6], with potential profound impact across electronic technology, which is sure to have a significant influence on medical imaging technology as well.

1.2.3.1 Nuclear Forces and Stability

1.2.3.2 Nuclear Fusion and Fission

The mass of an atom is less than the sum of the masses of its neutrons, protons, and electrons. This seeming paradox reflects Einstein’s famous equation E = mc² depicting the exchangeability of the binding energy of the nucleus and the masses of its constituent particles. When the nucleons are separate, they have their own individual masses, but when they are combined in a nucleus, some of their mass is converted into binding energy. The mass difference between the sum of the masses of the atomic constituents and the mass of the assembled atom is termed the mass defect. In Einstein’s equation, an energy E is equivalent to mass, m, multiplied by the speed of light in a vacuum, c, (i.e. 2.998 × 10⁸ m/s) squared. Because of the significant magnitude of the proportionality constant, c², in this equation, 1 kg of mass is equal to 9 × 10¹⁶ J, roughly equivalent to the energy released during detonation of 30 megatons of C₇H₅N₃O₆ (i.e. TNT). The energy equivalent of even 1 amu is significant:

Figure 1.7 depicts the average binding energy per nucleon as a function of the mass number A.

The potential energy bound in the nucleus can be released if a nucleus with a high mass number separates or fissions into two parts, each with an average binding energy per nucleon greater than that of the original nucleus. The energy release occurs because this split produces low-Z products with a higher average binding energy per nucleon than the original high-Z nucleus (see Figure 1.7). A transition from a state of lower binding energy per nucleon to a state of higher binding energy per nucleon results in the release of energy. This is reminiscent of the previous discussion of energy release that accompanies an L to K electron transition. However, the energy available from transition between nuclear energy levels is orders of magnitude greater than the energy released during electron transitions.

Certain high-A nuclei (e.g. ²³³U, ²³⁵U, and ²³⁹Pu) fission spontaneously after absorbing a slowly moving neutron. For ²³⁵U, a typical fission reaction is as follows:

The energy released is designated as Q and averages more than 200 MeV per fission. The energy is liberated primarily as ²³⁵U radiation, kinetic energy of radioactive fission products (e.g. and ), neutrons, and of course heat and light. Neutrons released during fission may interact with other nuclei and create the possibility of a chain reaction, provided that sufficient mass of fissionable material is contained within a small volume. The rate at which fission can occur in a material may be regulated by controlling the number of neutrons available to interact with fissionable nuclei; this is the method used to control nuclear reactors. Uncontrolled nuclear fission results in an “atomic explosion.”

The counterpart to fission is fusion, which is the combination of certain low-mass nuclei to produce a nucleus with an average binding energy per nucleon greater than that for either of the original nuclei. This process is accompanied by the release of large amounts of energy. A typical reaction is as follows:

In this particular reaction, Q = 18 MeV per fusion. To form products with higher average binding energy per nucleon, nuclei must be brought sufficiently near one another, so the strong nuclear force can initiate fusion. In the process, the electrostatic force of repulsion must be overcome as the two nuclei approach each other. Nuclei moving at very high velocities possess enough momentum to overcome this repulsive force. Adequate velocities may be attained by heating a sample containing low-Z nuclei to a temperature greater than 12 × 10⁶⁰ K, roughly equivalent to the temperature in the inner region of the Sun. Fusion is in fact the primary reaction fueling the energy production in the stars including our sun. Fusion has been used for military purposes where the high temperatures necessary in initiating a fusion (e.g. in a hydrogen bomb) are triggered with a fission igniter explosion. Although still out of reach, however, there is much hope and effort for harnessing fusion for safe energy production.

1.2.3.3 Nuclear Spin and Nuclear Magnetic Moments

Protons and neutrons behave like small magnets and have associated magnetic moments. The term moment has a strict meaning in physics. When a force is applied on a wrench to turn a bolt (see Figure 1.8a), for example, the mechanical moment is the product of force and length. The mechanical moment can be increased by increasing the length of the wrench, applying more force to the wrench, or a combination of the two. A magnetic moment (see Figure 1.8b) is likewise the product of the current in a circuit (i.e. a path followed by electrical charges) and the area encompassed by the circuit, so the magnetic moment is increased by increasing the current, the area, or a combination of the two. The magnetic moment is a vector, a quantity having both magnitude and direction. Like electrons, protons have a characteristic called spin, which can be explained by treating the proton as a small object spinning on its axis. In this model, the spinning charge of the proton produces a magnetic moment (see Figure 1.8c).

The spinning–charge model of the proton has two marked limitations. First, the mathematical prediction for the value of the magnetic moment is not quite what has been measured experimentally. From the model, a proton would have a fundamental magnetic moment known as the nuclear magneton, μ_n as

(1.7)

Here, e is the charge of the proton in coulombs, ℏ is Planck’s constant divided by 2π, and M_p is the mass of the proton. The magnetic moment magneton, μ_p, of the proton, however, has been measured as μ_p = 2.79μ_n. This classical observation has been explained using quantum mechanics and is used to describe the fundamental concepts of MRI (see Chapter 10). The unit of the nuclear magneton, energy (i.e. electron volt) per magnetic field strength (i.e. gauss), expresses the fact that a magnetic moment has a certain potential energy in a magnetic field. The second difficulty of the spinning proton model is that the neutron, an uncharged particle, exhibits a magnetic moment equal to 1.91μ_n. An explanation for these limitations is that neutrons and protons are composed of other more fundamental particles called quarks [7,8]. Quarks have fractional charges adding up to a unit charge. The nonuniform distribution of spinning charges of the quarks is the basis for the observed magnetic moments.

When magnetic moments exist near each other, as in the nucleus, they tend to form pairs with the vectors of the moments pointed in opposite directions. In nuclei with even numbers of neutrons and protons, this pairing cancels out the magnetic properties of the nucleus as a whole. Thus, an atom such as with six protons and six neutrons has no net magnetic moment because the neutrons and protons are paired up. An atom with an odd number of either neutrons or protons will have a net magnetic moment. For example, with six protons and seven neutrons has a net magnetic moment because it contains an unpaired neutron. Also, with seven protons and seven neutrons has a net magnetic moment because both proton and neutron numbers are odd. Table 1.7 lists several nuclides with net magnetic moments. The presence of a net magnetic moment for the nucleus is essential to MRI, as only nuclides with net magnetic moments can interact with the magnetic field of an MRI system to provide a signal to form an image.

1.3.1.1 Alpha Decay

Some heavy nuclei gain stability by the emission of an alpha (α) particle, which is two protons and two neutrons tightly bound as a nucleus of helium, [10]. The alpha particle is a relatively massive, poorly penetrating type of radiation that can be stopped by a sheet of paper or paint. An example of alpha decay is the decay of naturally occurring radium into the inert gas radon by the emission of an alpha particle:

A decay scheme for alpha decay is depicted in Figure 1.11.

1.3.1.2 Beta Decay

Nuclei with an N/Z ratio too high for stability (see Figure 1.10) decay as

where is an electron ejected from the nucleus and is an antineutrino, a massless neutral particle that is required for the conservation of energy and is the antimatter counterpart of the neutrino. An example of beta negative decay is that of ¹³⁷Cs:

A decay scheme for this transition is shown in Figure 1.12. A beta (β⁻) particle with a maximum energy (E_max) of 1.17 MeV is released in 5% of all decays; in the remaining 95%, a beta (β⁻) particle with an E_max of 0.51 MeV is accompanied by an isomeric transition of 0.66 MeV, where either a γ-ray is emitted or an electron is ejected by internal conversion. The transition energy is 1.17 MeV for the decay of ¹³⁷Cs.

Beta (β⁻) decay pathways are characterized by specific maximum energies; however, most beta (β⁻) particles are ejected with energies lower than these maxima. The average energy of beta (β⁻) particles is about 1/3 E_max along a specific pathway. The energy distribution of beta (β⁻) particles emitted during the decay of ³²P is shown in Figure 1.13. Spectra of similar shape but with different values of E_max and E_mean exist for the decay pathways of every beta particle emitting radioactive nuclide. In each particular decay, the difference in energy between E_max and the specific energy of the beta (β⁻) particle is carried away by the antineutrino. That is,

(1.8)

where E_max is the energy released during the beta (β⁻) decay process, E_k is the kinetic energy of the beta (β⁻) particle, and is the energy of the antineutrino.

1.3.1.3 Positron Decay and Electron Capture

Analogous to beta decay, positron decay or electron capture (EC) can be thought as a proton falling to a lower energy level reserved for neutrons, in which case the proton “becomes” a neutron, although conservation of mass dictates that a lower mass proton cannot truly change to a neutron. As in the case of beta rays, the progeny is most stable if it contains even numbers of protons and neutrons and least stable if it contains an odd number for either [10].

Positron decay results from the nuclear transition

where represents a positron ejected from the nucleus during decay and ν is a neutrino that accompanies the positron. The neutrino and antineutrino are similar, except that they have opposite spin and share the same energy balance with their corresponding positron or electron emission.

In positron decay, the N/Z ratio increases; hence, positron-emitting nuclides tend to be positioned below the N/Z stability curve shown in Figure 1.10. The decay of is representative of a positron decay:

The N/Z ratio of a nuclide may also be increased by electron capture, in which an electron is captured by the nucleus to yield the following transition:

Most electrons are captured from the K electron shell, although occasionally an electron may be captured from the L shell or a shell even farther from the nucleus. During electron capture, a hole is created in an electron shell within the atom. This vacancy is filled by an electron cascading from a higher shell, resulting in the release of characteristic radiation or one or more Auger electrons.

Many nuclei decay by both electron capture and positron emission, as illustrated below for ²²Na:

The electron capture branching ratio is the probability of electron capture per decay for a particular nuclide. For ²²Na, the branching ratio is 10% for electron capture and 90% of the nuclei decay by positron emission. Generally, positron decay prevails over electron capture when both decay modes occur.

A decay scheme for ²²Na is shown in Figure 1.14. The 2m_oc² listed alongside the vertical portion of the positron decay pathway represents the energy equivalent (i.e. 1.02 MeV or 2m_oc²) of the additional mass of the products of positron decay. This additional mass includes the greater mass of the neutron as compared with the proton together with the mass of the ejected positron. This amount of energy must be supplied by the transition energy during positron decay. Nuclei that cannot furnish at least 1.02 MeV for the transition do not decay by positron emission. These nuclei increase their N/Z ratios solely by electron capture.

There are a few nuclides that can decay by beta (β⁻) emission, positron emission, or electron capture. An example is ⁷⁴As (see Figure 1.15), which undergoes beta (β⁻) emission 32% of the time, positron emission 30%, and electron capture 38%. All modes of decay result in the transformation of the highly unstable “odd-odd” ⁷⁴As nucleus (Z = 33, N = 41) into a progeny nucleus with even Z and even N.

1.3.1.4 Isomeric Transition

Sometimes, one or more of the excited states of a progeny nuclide may exist for a finite lifetime. An excited state is termed a metastable state if its half-life exceeds 10⁻⁶ seconds (see Section 1.3.2.1). For example, the decay scheme for ⁹⁹Mo shown in Figure 1.16 exhibits a metastable energy state, ^99mTc, that has a half-life of six hours.

Just as electron transitions emit characteristic X-rays, metastatic nuclides emit γ-rays with characteristic energies. For example, ^99mTc emits photons of 142 and 140 keV and beta (β⁻) decay of ⁶⁰Co releases photons of 1.17 and 1.33 MeV (see Figure 1.17). In the latter case, the photons are released during a cascade of isomeric transitions of progeny ⁶⁰Ni nuclei from excited states to the ground energy state.

An isomeric transition can also occur by the interaction of the nucleus with an electron in one of the electron shells, i.e. internal conversion. When this happens, the electron is ejected with kinetic energy E_k equal to the energy E_γ released by the nucleus, reduced by the binding energy E_b of the electron:

(1.9)

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