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Atoms, Nuclei and Radioactivity

Elizabeth M. Parvin

CHAPTER OUTLINE Introduction Transverse and Longitudinal Waves Atomic Structure Electromagnetic Continuous Spectra and Characteristic Radiation The and the Nucleus The Stable and Unstable Levels Half-life Band Theory of Impurity Bands Decay Particles in Electric and Magnetic Fields Gamma Decay Electric Fields and Magnetic Fields Radioactive Decay Series The Lorentz Equation of Medical Interest Waves

INTRODUCTION fundamental particles; the and are composed of . Charges are, as is customary in , given as multiples of The aim of this first chapter is to lay some of the foundations of the the electronic charge, e, which is 1.602 10 19 C. The and pos- physics of radiotherapy. It starts, in the section titled Atomic Structure, itron have charges of +e and the electron has a charge of –e; all the other by looking at the main subatomic particles and the forces that hold particles listed are neutral. The fourth column gives the in kilo- them together in the atom. This on to an examination of the dif- grams, but in , it is common practice to express the ferent types of nuclei, with an emphasis on some of the important ones of a not in kilograms but in terms of its rest mass energy. This is used in medical physics. The behaviour of charged particles in electric based on Einstein’s famous equation, which gives the equivalence of and magnetic fields, central to much of the physics of radiotherapy, is mass and energy: covered in section titled Particles in Electric and Magnetic Fields. E ¼ mc2 Waves, including the , and the basics of 1.1 radioactive decay are introduced in the following sections. where m is the mass of the particle, c is the speed of in a vacuum For some readers, this chapter will be a reminder of previous knowl- (2.998 108 ms 1) and E is the energy. For an electron, the rest mass edge, for others it will be new territory. For the latter, the references energy associated with a mass of 9.109 10 31 kg is 8.187 10 14 should provide some more in-depth material that it has not been pos- joules (J). It is more convenient to express this very small magnitude sible to include here. For convenience, SI units are listed in the Physical of energy in units of the electron-volt (eV), where Units and Constants Section. 1eV¼ 1:602 1019 J 1.2

ATOMIC STRUCTURE The electron volt is the amount of energy acquired by an electron when Particles it is accelerated through a voltage of 1 volt (see the section titled Electric Most readers will be familiar with the idea that are composed Fields), hence the name electron volt. Using this conversion, we arrive at of chemically bonded together. Perhaps the most familiar exam- the values given in column 5 of Table 1.1. Note that the proton and neu- ple is the , which consists of two atoms tron (known collectively as ) are very much more massive than — bonded to one atom to give the well-known molecular the electron and , and that the has almost zero mass the exact value is still the subject of experiment. H2O. In radiotherapy, we are often more interested in the particles that make up the atom—these are known as subatomic particles. The positron is the of the electron, having the same β+ Table 1.1 lists the properties of the sub-atomic particles of most rel- mass but the opposite charge; it is emitted during decay (see the sec- evance to radiotherapy; the proton, , electron, positron, neutrino tion titled ) and is important in tomog- β and antineutrino. Strictly, only the electron, positron and are raphy (PET) (see Chapter 6). Neutrinos play a role in decay (see

2 CHAPTER 1 Atoms, Nuclei and Radioactivity 3 the section titled Beta Decay). The is the particle associated with respectively and are known as isotopes of . For many elements, electromagnetic radiation (see the section titled Waves). some of the isotopes are radioactive (see the section titled Radioactive Decay) and this fact can be very useful in clinical investigations because The Atom and the Nucleus the chemical behaviour of all the isotopes is the same. For example, 15 The atom is the smallest identifiable amount of an element. Each atom radioactive 8 O is taken up by the body in the same way as the stable 16 consists of a central nucleus, made up of protons and neutrons, which is (i.e. nonradioactive) , 8 O, and can be used in PET; the radioac- ‘ ’ 131 surrounded by a cloud of . The diameter of an atom and tive iodine isotope 58 I is taken up by the thyroid gland in the same way 10 14 127 nucleus are typically 10 m and 10 m, respectively. To put these as the stable isotope 58 I, so can be used to treat thyroid cancer. dimensions into a more accessible perspective, if the is represented by the point of a pencil (diameter approximately 0.5 mm) The Forces held in the centre of a medium-sized room (say 5 m 5 m), then the The next point to address is the question of what holds the atoms electron cloud surrounding the nucleus would extend to the walls of together. The protons in the nucleus are positively charged, and the the room. electrons surrounding the nucleus are negatively charged, so there is It is the number of protons in a nucleus that determines the type of an attractive between them. This electrostatic or force element. Because the protons in the nucleus are positively charged and depends on the product of the charges and is inversely proportional the electrons are negatively charged, a neutral atom must contain equal to the square of the between them. For one electron numbers of protons and electrons. It is the electrons, which surround (charge –e) and a nucleus (charge Z), the magnitude of the force the nucleus and are often described as orbiting it, that interact with (Fel) is given by the equation electrons from other atoms, thereby determining the chemical behav- Ze2 iour of the atom. Fel ¼ k 1.4 For example, a has one proton in the nucleus, r 2 has two, carbon has six and so on. This number is known as the atomic where k is a constant and r is the distance between the electron and the number, Z, of the element. The elements listed in order of increasing nucleus. This inverse-square relationship is analogous to the gravita- form the of the elements [1]. tional force between two masses and we could use the rules of classical As shown in Table 1.1, the neutrons in the nucleus carry no charge physics to calculate the orbits of the electrons around the nucleus (anal- but do have a similar mass to the protons. The electrons have a very ogous to the orbits of the planets around the Sun). However, there is small mass, so the mass of an atom is almost entirely due to from one big difference between the planetary orbits and the orbits of the the mass of the protons and neutrons. The sum of the number of neu- electrons around the nucleus; in the planetary case, it is possible to have trons (N) and protons in a nucleus is known as the num- any value of the radius (and therefore energy), whereas, in the atomic ¼ ber, A and A Z + N. Because both A and Z are needed to identify a case, theory only allows certain permitted orbits. This gives nucleus, the notation used is of the form rise to electron energy levels (or shells), which are the subject of the A section titled Electron Energy Levels. Z X 1.3 For like charges, the Coulomb force is repulsive, so, because the pro- The symbol shown here as X is the chemical symbol for the element—H tons in the nucleus are all positively charged, it might be expected that for hydrogen, He for helium, C for carbon and so on—and A and Z the Coulomb force would cause the nucleus to fly apart. However, there are the mass and atomic numbers. Because Z determines the is another force that acts on both protons and neutrons: the strong force. and therefore the element, strictly speaking, it is not necessary to have This force acts on protons and neutrons and other heavy particles called 12 the value of Z shown. For example, 6 C represents a carbon nucleus with ; it is independent of charge and is always attractive, but only at six protons and six neutrons, but it could be written simply as 12C, or even very short range. Fig. 1.1 shows the way in which the energy of a proton as carbon-12 because carbon always has six protons. However, to avoid varies depending on how far away from the nucleus it is. As a proton confusion, it is often easier to include both atomic and mass numbers. approaches the nucleus, it experiences a repulsive force but, if it has For any one element, the number of protons is always the same, but enough energy to overcome this ‘’ and gets within the number of neutrons, and hence A, can vary. For example, carbon 11 12 13 14 can exist as, 6 C, 6 C, 6 Cor6 C. These have 5, 6, 7 and 8 neutrons

−15 TABLE 1.1 Properties of the Subatomic ~10 m Coulomb Protons and neutrons Particles of Most Relevance to Radiotherapy potential within the nucleus Rest barrier Mass Energy Particle Symbol Charge, ea Mass (kg) (MeV) Proton Proton p +1 1.673 10 27 938 –r +r Neutron n 0 1.675 1027 940 Energy Electron e 1 9.109 1031 0.511 Positron e +1 9.109 1031 0.511 ν > > Neutrino e 0 0 0 Fig. 1.1 Schematic illustration of the energy of a proton as a of ν > > Antineutrino e 0 0 0 its distance, r, from the centre of the nucleus. As the proton approaches Photon γ 00 the nucleus, the repulsive Coulomb force increases, but close to the nucleus, this repulsion is overcome by the attractive strong force so ae is charge on an electron. the energy is reduced and the nucleons are held together in the nucleus. 4 Walter and Miller's Textbook of Radiotherapy the range at which the strong force works, then it has a much lower the simplest element, hydrogen. No energy is lost or gained while an elec- energy in the nucleus and stays there. An energy diagram like this is tron occupies a particular shell, and only discrete amounts of energy can known as a . be gained or lost by electrons when they move between shells. You can see from Fig. 1.2 that the lowest energy state for hydrogen has an energy Electron Energy Levels of –13.6 eV. Therefore the amount of energy that would be required to As mentioned in the section titled The Forces, planetary orbits around remove this electron from the atom is 13.6 eV. This is the ionisation the Sun and electron orbits around the nucleus differ in that, in the case energy of the hydrogen atom. The energy required to raise an electron of the electrons, quantum theory predicts that only certain orbits are from the lowest energy state to the second lowest energy state is equal allowed. This means that only certain orbit can occur—these to the difference between the energies of the two states, that is, (13.6 – different values of energy are referred to as energy levels or shells and 3.40) ¼ 10.2 eV, and so on for all other pairs of energy levels. were first hypothesised by in 1913. Large atoms have more complicated arrangements of electron A free electron, which is outside the nucleus, is said to have zero energy levels; the energy levels are numbered according to the principal energy; any electrons in levels closer to the nucleus have a lower, and , n (n ¼ 1, 2, 3 etc.) and are subdivided into other therefore negative energy. Fig. 1.2 shows the possible energy levels for energy levels, sometimes known as sub-orbitals, with more quantum numbers relating to angular and . These quantum numbers also dictate the number of electrons which can be in each E tot = 0 eV Continuum of positive . shell. Historically, the may also be repre- . energy states (E >0) E sented by the letters K (n ¼ 1), L (n ¼ 2), M (n ¼ 3) etc., so the shells tot 7 = –0.28 eV E = –0.38 eV are often referred to as the K, L, M shells etc. If all the electrons in an E6 5 = –0.54 eV atom are in the lowest possible energy states allowed by the rules, then E 4 = –0.85 eV the atom is said to be in the . For example, a hydrogen atom Excited E 3 = –1.51 eV states in the ground state has the electron in the state with energy 13.6 eV. Band Theory of Solids E 2 = –3.40 eV In individual atoms, outer electrons occupy specific energy levels. When atoms are brought together, as occurs in materials, inter- actions between atoms broaden these specific energy levels into ‘energy bands’. Electrons may occupy energy states only within these bands, between which are forbidden zones that normally do not have energy states for electrons to occupy, as illustrated in Fig. 1.3. The outermost energy bands within the solid material are termed the band and the conduction band. Electrons within the valence band are considered as linked to the chemical bonds between individual atoms and are therefore bound in place, although the term bound is used loosely because at normal such bonds may be con- tinually being broken and reformed. At an slightly above the valence band is the conduction band. Electrons within this band are surplus to any requirements for chemical bonding. At normal temper- atures, these electrons are not associated with specific atoms and chem- Ground n = 1–13.6 eV, etc. ical bonds, but migrate readily through the material. In some materials, state there are insufficient electrons to fill the available energy levels of the valence band, so the conduction band is empty. Where a large forbidden Fig.1.2 The possible energy levels of the electron in the hydrogen atom. zone exists, these materials are classed as nonconductors or insulators Note that when the electron is bound in the atom, the energy is negative (see Fig. 1.3A). Other materials may have more outer electrons than and can only take certain values. Outside the atom, the energy of the unbound electron is zero. The energy levels are numbered, n ¼ 1, 2 the valence band can accommodate, so that the lower levels of the con- and so on from the inside outwards. duction band are also occupied. In these materials, the conduction band

Conduction band Conduction band Conduction band

Conduction band Forbidden zone

Valence band Valence band Valence band Valence band

Energy A B C D

Fig.1.3 Simplified energy level diagram for solid materials: the shaded regions shows those levels that are nor- mally occupied by electrons for (A) an insulator; (B) a conductor; (C) a semiconductor (undoped); (D) material with impurity levels within the forbidden zone. CHAPTER 1 Atoms, Nuclei and Radioactivity 5 overlaps with the valence band and the forbidden zone disappears, as The Lorentz Equation shown in Fig. 1.3B. These materials will generally be good conductors The forces on a moving , which is subject to both an of . There are some materials in which the valence band is just electric and a magnetic , are complicated and are best dealt with filled, but the conduction band is effectively empty, and a small but sig- using the mathematics of vectors, which is beyond the scope of this nificant forbidden zone exists. These materials are classed as semiconduc- book. However, for the simple case where the is perpen- tors, illustrated in Fig. 1.3C. Any charges injected into a semiconductor dicular to the velocity of the charged particle, we can write will be free to travel through the material. It is to be stressed that this description is overly simplistic but serves as a basis for understanding F ¼ ðÞqE + qvB 1.7 the principles of solid-state dosimeters discussed in Chapter 3. The electric component of the force (qE) and the magnetic component Impurity Bands (qvB) are not necessarily in the same direction. This equation is known The introduction of impurities at low concentrations can alter the as the Lorentz equation after the Dutch Hendrik A. Lorentz, structure of the energy bands and may create energy bands that are and is an extremely important equation in many areas of physics. In located between the valence and conduction bands, within the forbid- radiotherapy, it is useful when considering the behaviour of electrons den zone as shown in Fig. 1.3D. The properties of the material so in a linear accelerator and of charged particles in a cyclotron or formed will depend upon whether these extra bands are normally occu- synchrotron. pied or empty of electrons, and their actual energy levels. The addition of impurities is critical to the formation of active semiconductor devices (see Chapter 3) and to the development and functioning of WAVES both and thermoluminescent materials (see Chapter 3). Transverse and Longitudinal Waves Energy, in the form of light, heat or sound, may be transmitted from PARTICLES IN ELECTRIC AND MAGNETIC FIELDS place to place by waves. The direction of propagation of a wave is the direction in which the energy is transported; however, the particles Electric Fields in the medium do not change their overall ; they simply vibrate As already explained, a single charge exerts either an attractive or a about an average position. We can distinguish two types of wave: repulsive force on another charge. A collection of charges will also exert longitudinal and transverse. a force on another charged particle. This force can be written as In a transverse wave (Fig. 1.5A), the oscillations are perpendicular to F ¼ qE 1.5 the direction of propagation of the wave. Water waves, or waves on a string, are good examples of transverse waves. where F is the force on the charge q and the quantity E is known as the In the case of longitudinal waves (Fig. 1.5B), the particles in the (caused by other charges in the vicinity). The direction of the medium move backwards and forwards in a direction parallel to the electric field is the direction in which a free positive charge would move; a direction of propagation, although their mean position stays the same. free negative charge would move in the opposite direction, so the force on A good example is a sound wave; the particles of the medium (e.g. air) a particle in an electric field is always parallel or antiparallel to the field. oscillate parallel to the direction of the sound wave and this gives rise to changes in along the wave. Magnetic Fields Fig. 1.5 shows the , λ, of the wave—the distance between The force on a charged particle in a magnetic field is more complicated. two adjacent peaks (or two adjacent troughs). Another important There is no force at all if the particle is not moving; if it is moving then parameter is the , f, which is the number of peaks that pass the force is, like the force in an electric field, dependent on the charge, q, a point per second. In all cases, the speed of the wave, c, is related to but it is perpendicular to the direction of both the magnetic field, B, and the wavelength and frequency by the equation the velocity, v, of the particle and depends on the angle between them. c ¼ f λ 1.8 Fig. 1.4 shows some examples. The largest force occurs when the velocity of the particle is perpen- This means that for a wave with constant speed, a larger wavelength dicular to the magnetic field. As the angle θ between B and v decreases, corresponds to a lower frequency and vice versa. Wavelength is mea- the force decreases; when the velocity and field are parallel θ ¼ 0 and sured in units of (m) and frequency in hertz (Hz). there is no force. In fact, the magnitude of the force is given by Longitudinal sound waves are important in ultrasound imaging (see F ¼ qvBsinðÞθ 1.6 Chapter 5), but for radiotherapy applications we are mostly concerned with the transverse waves of electromagnetic radiation and these are the ο So, when v and B are perpendicular, θ ¼ 90 and F ¼ qvB. subject of the next section.

F F

v v v B B B A B C

Fig.1.4 The force on a moving charge (positive in this case) in a magnetic field depends on the relative directions of the velocity (v) and the magnetic field (B). (a) v and B are perpendicular to each other. The force (F) is large and perpendicular to both v and B. (b) The angle between v and B is less than 90 degrees and the force is less but still perpendicular to both. (c) v and B are parallel: there is no force. 6 Walter and Miller's Textbook of Radiotherapy

Equilibrium Amplitude

A Transverse wave

Wavelength Equilibrium (of first segment) Longitudinal wave

Amplitude B

Fig.1.5 (A) Transverse and (B) longitudinal waves. Note the wavelength is the distance between two maxima.

Direction of oscillation of electric field vector e

Direction of wave propagation Direction of oscillation of Position magnetic field vector B Electric field λ

Magnetic field

Fig.1.6 The oscillations of electric and magnetic fields in an electromagnetic wave.

Electromagnetic Radiation waves and they could be explained in terms of wave physics. However, Electromagnetic radiation is so called because it can be described as the beginning of the saw the development of quantum waves in which the quantities that oscillate are electric and magnetic physics. Several key experiments, including the investigation of Comp- fields. Fig. 1.6 shows how the fields in these electromagnetic waves ton , an important process in radiotherapy, described in oscillate at right angles to each other. Chapter 2, showed that, when electromagnetic radiation interacts with Because electromagnetic waves depend only on electric and mag- , wave physics does not always predict the correct result. Instead netic fields, they can travel through any medium, including a vacuum. the radiation behaves as particles known as . A photon is a small In a vacuum, all electromagnetic waves travel at a speed of approxi- ‘packet’ or quantum of energy and each one has an energy given by mately 3 108 ms 1 (often known rather loosely as the in a vacuum) but the properties of the radiation vary greatly with wave- E ¼ hf 1.9 length and frequency. Fig. 1.7 shows the vast range of the electromag- netic spectrum; note that as the frequency increases, the wavelength where E is the energy, f is the frequency of the electromagnetic wave decreases, according to Equation 1.8. Radiotherapy physics is mostly and h is a constant, known as Planck’s constant and equal to 6.626 concerned with the high frequency/small wavelength end of the spec- 10 34 Js. trum, although radio waves are important in radiotherapy as they are As with the masses in the section titled Atomic Structure it is used to accelerate the electron beam in a linear accelerator. more usual to give these energy values in electron volts (eV) where By the end of the 19th century, were aware that all the 1eV¼ 1.602 10 19 J. This has the advantage of allowing an easy different types of radiation shown in Fig. 1.7 were electromagnetic comparison between the energy of a photon and the mass energy of CHAPTER 1 Atoms, Nuclei and Radioactivity 7 a particle and is especially useful when considering the transfer of electrons can be moved up into allowable, but normally empty, energy energy between a photon and particle or the conversion of a particle levels. This happens, for example, in an x-ray tube (see Chapters 5 and into radiation, radiation or radiation into particles, pair 8). This leaves the atom in an , that is, with a higher total production (see Chapter 2). Fig. 1.7 shows the energies in eV in addi- energy than in the ground state. After a of , the atom will tion to the and . return to the ground state as the excited electrons drop from the higher levels back to vacant lower energy states. When this happens, the excess Continuous Spectra and Characteristic Radiation electron energy is carried away as a photon of electromagnetic radia- In the section titled Electron Energy Levels we described the way in tion. Thus if we have two states with energies E1 and E2 then the energy which the electrons in atoms can only be in specific energy states, of the photon (Eγ) is the difference between E1 and E2. Using defined by their quantum numbers. The energies of these states are dif- Equation 1.9 to relate the energy of the photon to its frequency, we ferent for each element; for example, Fig. 1.8 shows the main energy arrive at: levels for tungsten, which is an element commonly used for the Target E E ¼ Eγ ¼ hf 1.10 in x-ray production, such as x-ray tubes (see Chapters 4 and 8), and 1 2 machines, such as linear accelerators (see Chapter 9). As previously described, to raise an electron from one energy level to The energy of photons produced is therefore dictated by the differences another, the energy required is equal to the difference in energy of the in energy between electron shells of the particular atom from which two states. If an atom is excited above its ground state by absorbing they are emitted. The spectrum of photons produced by an element energy from incoming particles (e.g. photons or electrons) then some is termed the characteristic radiation and will be different for each

Gamma-rays X-rays Infrared Microwaves Radio Wavelength (m) 10–12 10–6 1103 V Gamma-rays Ultraviolet i Microwave s i X-rays b Infrared Radio l e 1020 1018 1016 1014 1012 1010 108 106 Frequency, (Hz)

4 ´ 105 4 ´ 103 4 ´ 101 4 ´ 10–1 4 ´ 10–3 4 ´ 10–5 4 ´ 10–7 4 ´ 10–9 Energy (eV) 400 nm 700 nm Red Blue Violet Green Yellow Orange

7.5 ´ 1014 Hz 4.3 ´ 1014 Hz 3.1 eV 1.8 eV

Fig.1.7 The different types of radiation that form the electromagnetic spectrum. The visible spectrum covers a very small range of wavelength values and is expanded below the spectrum. 8 Walter and Miller's Textbook of Radiotherapy element. Fig. 1.8 shows the possible electron transitions leading to the N production of characteristic photons for tungsten. Distinction is made (number of neutrons) between electron transitions originating from different shells to the same destination shell by denoting the transition (and characteristic photon) with the final shell letter (K, L and so on) and adding a Greek letter suffix to indicate the originating shell, as shown in Fig. 1.8. If the electrons in tungsten are excited, as in an x-ray tube, then the emitted photons will be a mixture of the continuous background spec- trum (see Chapter 2) and the characteristic radiation from the tungsten atoms. This is shown in Fig. 1.9. The energy differences between the 126 levels in heavy elements tend to be much larger than for lighter ele- ments: note that for tungsten (see Fig. 1.8) the photon energies of the two main spectral lines are in the x-ray region at around 60 keV, which is in the x-ray range. They correspond to the L to K transition and the free electron to K transition. Contrast this with the energy of the largest possible transition in hydrogen, which is 13.6 eV, and in the ultraviolet region (see Fig. 1.7). 82 Type of (keV) Shell 0 n = ∞ b+ −0.1 n = 5 (O) 50 b– a −0.6 n = 4 (N) Fission −2.8 n = 3 (M) 28 Proton Neutron Stable − n 14 12.1 = 2 (L) Unknown 6

Lα, Lβ, Lγ 6 14 28 50 82 Z

−69.5 n = 1 (K) (number of protons) Fig.1.10 A graph of neutron number (A–Z) plotted against proton number (Z). The stable isotopes are shown in and the solid line represents Kα, Kβ, Kγ equal numbers of protons and neutrons. Nucleus Fig.1.8 Electron energy levels and transitions leading to characteristic photons for tungsten. The main transitions into each shell are marked. neutrons (A–Z) against the number of protons (Z) for stable and unsta- ble nuclei (Fig. 1.10). This figure shows the stable nuclei in black, which 1.2 is known as the stability line. It also shows the solid line corresponding 1 to equal numbers of protons and neutrons. We see that for low atomic number nuclei, an equal number of protons and neutrons is favoured, 0.8 whereas a greater proportion of neutrons to provide stability for large 0.6 nuclei. This may be explained by considering the increasing electro- 0.4 static force of repulsion between protons in the nucleus as the number 0.2 of protons is increased. For a more detailed, interactive, diagram which allows you to look up individual nuclei [2]. Relative photon fluence Relative 0 20 40 60 80 100 Evidence suggests that protons and neutrons within a nucleus adopt Photon energy (keV) a shell-like structure analogous to electron orbits and show particular stability when the number of protons or neutrons, or both, corresponds Fig.1.9 Characteristic spectral lines from tungsten superimposed on the . to a (2, 8, 20, 28, 50, 82, 126). The strength of the strong force that holds the nucleons together is associated with a nuclear bind- ing energy that must be overcome to break the nucleus apart. Essentially RADIOACTIVE DECAY the mass of a given nucleus is less than the sum of its constituent pro- tons and neutrons; this is known as the mass defect. Representing this in Stable and Unstable Isotopes terms of energy, using E ¼ mc2, gives the nuclear . In the section titled The Atom and the Nucleus we explained that Nuclei with an even number of protons or an even number of neutrons although the atomic number, Z, of a particular element is always the are more stable than those with an odd number of one or both. same, atomic , A, can vary so that each element can have A nucleus lying off the stability line shown in Fig. 1.10 is unstable and several different isotopes. Some of them will be stable: that is to say they decays by rearranging its numbers. This is achieved by releasing will not decay; others will be unstable and will undergo radioactive particles, changing a proton to a neutron or vice versa, or by absorbing decay. It is instructive to plot a graph which shows the number of nearby particles. The activity of an unstable, or radioactive, isotope is the CHAPTER 1 Atoms, Nuclei and Radioactivity 9 rate at which its nuclei decay, expressed in (Bq). One Becque- Half-lives can vary enormously—for example, the half-life of rel corresponds to one decay or disintegration per second. The -238 (not used for medical purposes!) is approximately the is a very small unit so the activity of sources used in medicine is generally same as the age of the , 4.5 billion years; the half-life of represented in MBq (1 106 disintegrations per second) or GBq (1 krypton-81, used in , is 13s, and others are even 109 disintegrations per second). (You may also occasionally come across shorter. The length of the half-life is an important consideration when the old unit of activity, the curie (Ci); 1 Ci ¼ 37 000 MBq.) Nuclei that choosing a for medical use; if the half-life is too long, then undergo radioactive decay are known as radionuclides. the patient may be radioactive for the rest of their life; if it is too short In the construction of practical radioactive sources, we are also then the activity will decay too fast for it to be useful. interested in the amount of material that is needed to manufacture a When radioactivity was first discovered at the end of the 19th cen- source with a required activity, determined by the specific activity, tury, three different types of emitted particle were identified and were the activity per unit mass (MBq kg 1). labelled alpha (α), beta (β) and gamma (γ) radiation. These names have remained, although there are now a few variants of them; the next four Half-life sections will cover these different types of decay. Radionuclides decay at very different rates soit isimportant to have some way of quantifying the rate of decay. If we have a collection of identical Alpha Decay nuclei, it is impossible to know exactly which one will decay next; one can Alpha (α) decay occurs most often from the unstable nuclei of heavy only predict the of decay. The number of nuclei, dN, which elements, such as uranium, or . The nucleus emits decay in a short time dt will depend on two factors: the decay constant, λ, an , which is actually a nucleus of helium, 4He. Having which is essentially the probability of decay, and N, the number of nuclei 2 an equal number of protons and neutrons (both of which are magic present at the start. This is expressed by the equation numbers) the helium nucleus is very stable. The parent nucleus (X) dN ¼λNdt 1.11 decreases its atomic number by 2 and its mass number by 4 so the gen- eral equation is: This equation can be rearranged and integrated to give λt A !A4 4 NtðÞ¼Noe 1.12 Z X Z2 Y+2He 1.14 ¼ where No is the number of nuclei present at t 0 and N(t) is the number 226 where Y is the daughter nucleus. For example, radium (88 Ra) decays to of nuclei present at time t. e is the . A graph of N(t) 222 plotted against time gives an curve similar to that radon (86 Rn) and an alpha particle. Energy must be conserved in the shown in Fig. 1.11. transformation, so any difference, Q betweenthenuclearbinding The interesting thing about an exponential decay is that the length energies of the parent and daughter nuclei is shared between the of time it takes for the number of undecayed nuclei to halve is always emitted alpha particle, in the form of kinetic energy and any photons the same. In Fig. 1.11, this time is shown as 2 (arbitrary) time units. This that are produced (see the section titled Gamma Decay). Alpha particles are typically emitted with kinetic energies of the order of time is known as the half-life, T½, and it is related to the decay constant, λ, by the equation several MeV. Being relatively heavy (approximately 4 the mass of a proton) and : ln2 0 693 highly charged (containing two protons), α particles are readily stopped T1=2 ¼ ¼ (1.13) λ λ in matter. For example, the 4.79 MeV α particle emitted from radium The term ln where the term ln represents the logarithm to base e;if has a range of less than 4 cm in air, or less than 0.04 mm in . This y ¼ ex then x ¼ ln y. means that alpha particles can be very useful for radiotherapy, but only

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2

1

0 012345678910 Time (arbitrary units) Fig.1.11 Exponential decay of radioactivity with time. Note that the half-life is the time taken to drop to one-half of the original value and is the same as the time taken to halve again to one-fourth and again to one-eighth. 10 Walter and Miller's Textbook of Radiotherapy if they can be released very close to the tumour tissue. Radium-226 was Depending on its energy, a positron will be stopped within a very used in some very early external radiotherapy treatments but has the short distance of the site of emission in tissue. The annihilation big disadvantage that the daughter product, radon, is a radioactive photons, on the other hand, at 0.511 MeV, each can pass relatively . More recently, Radium-223 is being trialled as an unsealed source easily through tissue. Detection of these coincident photons following of treatment for bone metastases. administration of a positron-emitting radionuclide, such as fluorine- 18 to a patient, therefore reveals where the annihilation event Beta Decay occurred and hence where the radionuclide was taken up within There are now known to be two types of beta decay. Beta minus (β ) the body. decay occurs frequently in naturally occurring radionuclides and involves the emission of an electron; beta plus (β+) decay occurs mainly Gamma Decay in artificially produced radionuclides and the particle emitted is a In the section titled Electron Energy Levels we showed that the elec- positron. trons in atoms could only occupy certain allowed energy levels. In A nucleus lying above the stability line in Fig. 1.10, is neutron-rich an analogous way, each nucleus can only have certain discrete energies and, by emitting an electron, can convert a neutron to a proton, thereby and transitions between two levels involve the emission or approaching the stability line. An example is iodine-131, which of photons of electromagnetic radiation. As with electron transitions, undergoes beta minus decay to become -131: the energy values of the levels are different for different elements; how- ever, in the nuclear case, the differences are much larger so the photons 131 !131 53 I 54 Xe 1.15 produced are generally of much higher energy. Following an α or β decay, the daughter nucleus is often left in an excited state. It will then Note that, because the total number of nucleons has not changed, the reach its ground state by emitting a photon with an energy correspond- mass number stays the same, but because a neutron has been converted ing to the difference between the two energy levels. These photons into a proton, the atomic number increases by one. The emitted β par- are known as gamma (γ) rays. Fig. 1.12 shows the decay scheme for ticle is a high-energy electron from the nucleus, not to be confused with -60, which is an isotope formerly widely used for external beam the orbiting electrons in the atom. therapy and now used in the Gamma Knife (see Chapter 8). By contrast, a nucleus lying below the stability line of Fig. 1.10, is In most cases, the emission of gamma rays occurs immediately proton-rich and therefore decays by converting a proton into a neu- after the alpha or beta decay, however occasionally, the nucleus tron. As with β decay the mass number does not change, but in this remains in an excited state and decays with a measurable half-life. case, since a proton is converted into a neutron, the atomic number + Such an excited state is known as a metastable energy state and is decreases by one. The emitted β particle is a positron, the denoted by the addition of an ‘m’ to the mass number. There is no equivalent of the electron. A good example, which is widely used in PET change in Z or A during the transition from the excited state of the imaging, is fluorine-18: metastable nucleus to the ground state, so this is known as an isomeric 18 !18 transition. One important example, widely used in nuclear medicine, 9F 8 O 1.16 is -99m (Fig. 1.13). This is a useful radionuclide because it Observation that the β particles produced display a spectrum of kinetic produces only gamma rays, with an energy of 140 keV, which can be energies, rather than the discrete energy difference between parent and used for imaging, and no short range α or β particles, which would daughter nuclei, indicates that a further particle must be involved. For only damage tissue. β+ ν β decay, this particle is the neutrino, e; for decay, it is its antipar- Note that both γ rays and x-rays are electromagnetic radiation at the ν ticle, e. The processes for iodine-131 and fluorine-18 are therefore: top end of the spectrum (see Fig. 1.7); the energy ranges overlap and indeed both are used in radiotherapy. They are only distinguished by 131 !131 0 β ν 53 I 54 Xe +1 + their origin: gamma rays coming from the nucleus and x-rays from the atomic electrons (see Chapter 2). and 18 !18 0 β ν 9F 8 O++1 + 1.17

The energy arising from the difference in masses of the initial and final 60 27 Co particles is carried away as kinetic energy of and the − neutrino. 5.272 years 0.31 MeV b 99.88% The range of a β particle is larger than that of an α particle as it is 0.12% much lighter. However, whereas the heavy alpha particles will travel 1.48 MeV b − 1.1732 MeV g along a straight path, the beta particles will follow a much more erratic path as they interact with atomic electrons. The range of a beta particle in tissue is only of the order of a few millimetres and will depend on the energy with which it was emitted. β+ An emitted positron ( ) travels through matter, rapidly losing 1.3325 MeV g kinetic energy through interactions with atomic electrons. When it col- lides with an electron, its antiparticle, both particles are annihilated and their energy converted into electromagnetic radiation. Using Einstein’s 2 60 mass-energy equivalence equation, E ¼ mc , the total energy of the radi- 28 Ni ation must be 2 0.511 MeV (see Table 1.1). Conservation of momen- Fig.1.12 Cobalt-60 decays via beta decay to give -60. The nickel tum demands that two photons, each with energy 0.511 MeV are nucleus is in an excited state and decays via the scheme shown, giving produced in opposite directions. This is the basis of PET (see Chapter 6). gamma rays with energies 1.1732 and 1.3325 MeV. CHAPTER 1 Atoms, Nuclei and Radioactivity 11

emitted from the nucleus carrying kinetic energy equal to the difference in between the parent and daughter nuclei. As with β+ decay, the mass number does not change but the atomic num- ber decreases by 1. Nuclei that decay by this method can be useful because there is no particulate emission. An example of a nuclide that decays by EC is iodine-125; it emits gamma rays of up to 35 keV, which can be used for (see Chapter 8). Another possible mode of decay is internal conversion (IC). An excited nucleus may de-excite by emitting a single photon, which inter- acts with the inner shell electron so that the electron is ejected from the atom. In contrast to beta decay, the emitted electron will have a single kinetic energy equal to the excitation energy of the nucleus minus the electron binding energy. The vacancy in the atomic shells left by the emitted electron will be filled by outer electrons, giving rise to char- acteristic radiation (see the section titled Continuous Spectra and Fig.1.13 Technetium-99m is produced from molybdenum-99 by beta Characteristic Radiation). decay and then decays to the ground state (Tc-99g) via an isomeric tran- sition with a half-life of 6 hours. The ground state technetium eventually Radioactive Decay Series decays to stable ruthenium-99 but with an extremely long half-life and therefore very low activity. There are many cases of radioactive nuclei that decay to give daughter nuclei, which are themselves radioactive and so on. This gives rise to a decay series. Fig. 1.14 shows the decay series of uranium-238, a nat- Electron Capture and Internal Conversion urally occurring radionuclide. At each stage, the α or β decay leads As an alternative to positron emission (β+ decay), the nucleus of a to a new nucleus which itself decays, the final product in this case proton-rich atom may capture one of its own inner shell electrons, being a stable isotope of . At each decay, the rate of growth of via electron capture (EC). The captured electron combines with a pro- the activity of the daughter nuclide depends on the relative values ton in the nucleus to produce a neutron and neutrino, the latter being of the decay constants (λ) of the parent and the daughter. Another

α

238 92U

109y β β γ 234 234 234 90T 91Pa 92U γ γ 24d 7h 105y α

230 T γ γ 90

226 104y α 88R

222 103y α α 86R

218 4d α 84P

3m β β γ

γ 214 214 214 82Pb 83Bi 84P

27m α 20m 10-4s α β β γ

210 210 210 210 81Tl 82Pb 83Bi 84P γ 1m 22y β 5d 140d α

206 82Pb

238 Fig.1.14 Decay series for 92 U. Half-lives are indicated in seconds (s), minutes (m), hours (h) and years (y). 12 Walter and Miller's Textbook of Radiotherapy example, of much more clinical importance, is the decay of TABLE 1.2 Characteristics of Some molybdenum-99 (see Fig. 1.13) to technetium-99m, which is widely Radionuclides Used in Radiotherapy as Either used for imaging. Unsealed Sources or Sealed Sources If we assume that there is no daughter present at time t¼ 0, and that all the disintegrations of the parent lead to the required daughter prod- Decay uct, then the activity of the daughter at time t, A2(t) is given by: Isotope Mechanism Half-Life Clinical Application ÀÁ Unsealed Sources λ2 λ t λ t A ðÞ¼t A ðÞ 1 2 11 + 2 1 0 e e (1.18) C β (2.0 MeV) 20 m PET imaging λ2 λ1 13N β+ (2.2 MeV) 10 m PET imaging ¼ 15 + where A1(0) is the initial activity of the parent at time t 0 and are the O β (2.8 MeV) 122 s PET imaging decay constants of parent and daughter, respectively. Equation 1.18 is 18F β+ (1.7 MeV) 109 m PET imaging of relevance to radionuclide generators as it allows calculation of the 32P β (695 keV) 14.3 d Polycythaemia vera optimum time between elutions of the daughter radionuclide. 89Sr β (500 keV) 50.5 d Bone metastases (palliation) If the decay of the parent is much slower than that of the daughter, 99mTc γ (143 keV) 6.0 h imaging λ >> λ 90 that is, if 2 1 then Equation 1.17 reduces to Y β (923 keV) 2.7 d Radiosynovectomy ÀÁ 131 β λ t λ t I (264 keV) 8.1 d Thyrotoxicosis and thyroid A ðÞ¼t A ðÞe 1 e 2 2 1 0 (1.19) γ (364 keV) cancer 223Ra α (5.7 MeV) 11.4 d Prostate cancer This is the situation for ionisation chamber consistency check devices containing a strontium-90 source. Strontium-90 undergoes beta decay Sealed Sources with a half-life of 28.7 years to -90, which itself decays via beta 60Co β, γ (1.17, 5.26 y External beam units and decay with a half-life of 64 hours. The activity of the long-lived stron- 1.33) MeV) gamma knife tium parent determines and maintains the activity of the short-lived 103Pd EC, γ (21 keV) 17 d Brachytherapy (seeds) yttrium daughter. 125I EC, γ (27– 60 d Brachytherapy (seeds) 36 keV) Radionuclides of Medical Interest 137Cs β, γ (662 keV) 30 y Brachytherapy (pellets) 192 Table 1.2 lists some common isotopes applied to radiotherapy and Ir β , γ (300–400) 74 d Brachytherapy (wire) nuclear medicine. The choice of isotope for a particular application keV) is based on type (γ, β (+ or )orα), product energy/ ies, half-life, specific activity (activity per unit mass) and availability. The α particles (heavy helium nuclei) have a very short range in tissue so will deposit energy close to the site at which a radionuclide is taken tissues in therapeutic applications. If greater penetration is required, up in the body; β particles (electrons) have a slightly longer, but still of the order of centimetre for brachytherapy, or if imaging of radioac- small, range. If the site of disease can be preferentially targeted by these tivity uptake through external detection of radiation is required, then emissions, this leads to significant sparing of surrounding normal photons (γ)orβ+ emissions will be the product of choice.

REFERENCES [1] Interactive Periodic Table Royal Society of Chemistry, http://www.rsc.org/ [2] Interactive Segre chart. http://people.physics.anu.edu.au/ecs103/chart/ periodic-table/. (Note that this plots Z against N—the opposite of Fig. 1.9. Both versions are commonly used).

FURTHER READING Dendy PP, Heaton B. Physics for diagnostic . 3rd ed. Baton Rouge: Grant IS, Phillips WR. The elements of physics. Oxford: Oxford University CRC Press; 2012. Press; 2001.