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Atoms, Nuclei and RadioactivityElizabeth M. Parvin

CHA P T E R OUT L I N E

IntroductionAtomic Structure

ParticlesThe Atom and the NucleusThe ForcesElectron Energy LevelsBand Theory of SolidsImpurity Bands

Particles in Electric and Magnetic FieldsElectric FieldsMagnetic FieldsThe Lorentz Equation

Waves

Transverse and Longitudinal WavesElectromagnetic RadiationContinuous Spectra and Characteristic Radiation

Radioactive DecayStable and Unstable IsotopesHalf-lifeAlpha DecayBeta DecayGamma DecayElectron Capture and Internal ConversionRadioactive Decay SeriesRadionuclides of Medical Interest

INTRODUCTIONThe aim of this first chapter is to lay some of the foundations of thephysics of radiotherapy. It starts, in the section titled Atomic Structure,by looking at the main subatomic particles and the forces that holdthem together in the atom. This leads on to an examination of the dif-ferent types of nuclei, with an emphasis on some of the important onesused in medical physics. The behaviour of charged particles in electricand magnetic fields, central to much of the physics of radiotherapy, iscovered in section titled Particles in Electric and Magnetic Fields.Waves, including the electromagnetic spectrum, and the basics ofradioactive decay are introduced in the following sections.

For some readers, this chapter will be a reminder of previous knowl-edge, for others it will be new territory. For the latter, the referencesshould provide some more in-depth material that it has not been pos-sible to include here. For convenience, SI units are listed in the PhysicalUnits and Constants Section.

ATOMIC STRUCTUREParticlesMost readers will be familiar with the idea that molecules are composedof atoms chemically bonded together. Perhaps the most familiar exam-ple is the water molecule, which consists of two hydrogen atomsbonded to one oxygen atom to give the well-known molecular formulaH2O. In radiotherapy, we are often more interested in the particles thatmake up the atom—these are known as subatomic particles.

Table 1.1 lists the properties of the sub-atomic particles of most rel-evance to radiotherapy; the proton, neutron, electron, positron, neutrinoand antineutrino. Strictly, only the electron, positron and neutrinos are

fundamental particles; the protons and neutrons are composed ofquarks. Charges are, as is customary in physics, given as multiples ofthe electronic charge, e, which is 1.602� 10�19 C. The proton and pos-itron have charges of +e and the electron has a charge of –e; all the otherparticles listed are neutral. The fourth column gives the masses in kilo-grams, but in nuclear physics, it is common practice to express the massof a particle not in kilograms but in terms of its rest mass energy. This isbased on Einstein’s famous equation, which gives the equivalence ofmass and energy:

E ¼mc2 1.1

where m is the mass of the particle, c is the speed of light in a vacuum(2.998 � 108 m s�1) and E is the energy. For an electron, the rest massenergy associated with a mass of 9.109 � 10�31 kg is 8.187 � 10�14

joules (J). It is more convenient to express this very small magnitudeof energy in units of the electron-volt (eV), where

1eV¼ 1:602�10�19 J 1.2

The electron volt is the amount of energy acquired by an electron whenit is accelerated through a voltage of 1 volt (see the section titled ElectricFields), hence the name electron volt. Using this conversion, we arrive atthe values given in column 5 of Table 1.1. Note that the proton and neu-tron (known collectively as nucleons) are very much more massive thanthe electron and positron, and that the neutrino has almost zero mass—the exact value is still the subject of experiment.

The positron is the antiparticle of the electron, having the samemass but the opposite charge; it is emitted during β+ decay (see the sec-tion titled Beta Decay) and is important in positron emission tomog-raphy (PET) (see Chapter 6). Neutrinos play a role in β decay (see

2

the section titled Beta Decay). The photon is the particle associated withelectromagnetic radiation (see the section titled Waves).

The Atom and the NucleusThe atom is the smallest identifiable amount of an element. Each atomconsists of a central nucleus, made up of protons and neutrons, which issurrounded by a ‘cloud’ of electrons. The diameter of an atom andnucleus are typically 10�10 m and 10�14 m, respectively. To put thesedimensions into a more accessible perspective, if the atomic nucleus isrepresented by the point of a pencil (diameter approximately 0.5 mm)held in the centre of a medium-sized room (say 5 m � 5 m), then theelectron cloud surrounding the nucleus would extend to the walls ofthe room.

It is the number of protons in a nucleus that determines the type ofelement. Because the protons in the nucleus are positively charged andthe electrons are negatively charged, a neutral atommust contain equalnumbers of protons and electrons. It is the electrons, which surroundthe nucleus and are often described as orbiting it, that interact withelectrons from other atoms, thereby determining the chemical behav-iour of the atom.

For example, a hydrogen atomhas one proton in the nucleus, heliumhas two, carbon has six and so on. This number is known as the atomicnumber, Z, of the element. The elements listed in order of increasingatomic number form the periodic table of the elements [1].

As shown in Table 1.1, the neutrons in the nucleus carry no chargebut do have a similar mass to the protons. The electrons have a verysmall mass, so the mass of an atom is almost entirely due to fromthe mass of the protons and neutrons. The sum of the number of neu-trons (N) and protons in a nucleus is known as the atomic mass num-ber, A and A ¼ Z + N. Because both A and Z are needed to identify anucleus, the notation used is of the form

AZX 1.3

The symbol shown here as X is the chemical symbol for the element—Hfor hydrogen, He for helium, C for carbon and so on—and A and Zare the mass and atomic numbers. Because Z determines the chemistryand therefore the element, strictly speaking, it is not necessary to havethe value of Z shown. For example, 126 C represents a carbon nucleus withsixprotons and sixneutrons, but it couldbewritten simply as 12C, or evenas carbon-12 because carbon always has six protons. However, to avoidconfusion, it is often easier to include both atomic and mass numbers.

For any one element, the number of protons is always the same, butthe number of neutrons, and hence A, can vary. For example, carboncan exist as, 116 C, 126 C, 136 C or 14

6 C. These have 5, 6, 7 and 8 neutrons

respectively and are known as isotopes of carbon. For many elements,some of the isotopes are radioactive (see the section titled RadioactiveDecay) and this fact can be very useful in clinical investigations becausethe chemical behaviour of all the isotopes is the same. For example,radioactive 15

8 O is taken up by the body in the same way as the stable(i.e. nonradioactive) isotope, 168 O, and can be used in PET; the radioac-tive iodine isotope 131

58 I is taken up by the thyroid gland in the same wayas the stable isotope 127

58 I, so can be used to treat thyroid cancer.

The ForcesThe next point to address is the question of what holds the atomstogether. The protons in the nucleus are positively charged, and theelectrons surrounding the nucleus are negatively charged, so there isan attractive force between them. This electrostatic or Coulomb forcedepends on the product of the charges and is inversely proportionalto the square of the distance between them. For one electron(charge –e) and a nucleus (charge Z), the magnitude of the force(Fel) is given by the equation

Fel ¼ kZe2

r21.4

where k is a constant and r is the distance between the electron and thenucleus. This inverse-square relationship is analogous to the gravita-tional force between two masses and we could use the rules of classicalphysics to calculate the orbits of the electrons around the nucleus (anal-ogous to the orbits of the planets around the Sun). However, there isone big difference between the planetary orbits and the orbits of theelectrons around the nucleus; in the planetary case, it is possible to haveany value of the radius (and therefore energy), whereas, in the atomiccase, quantum theory only allows certain permitted orbits. This givesrise to electron energy levels (or shells), which are the subject of thesection titled Electron Energy Levels.

For like charges, the Coulomb force is repulsive, so, because the pro-tons in the nucleus are all positively charged, it might be expected thatthe Coulomb force would cause the nucleus to fly apart. However, thereis another force that acts on both protons and neutrons: the strong force.This force acts on protons and neutrons and other heavy particles calledhadrons; it is independent of charge and is always attractive, but only atvery short range. Fig. 1.1 shows the way in which the energy of a protonvaries depending on how far away from the nucleus it is. As a protonapproaches the nucleus, it experiences a repulsive force but, if it hasenough energy to overcome this ‘Coulomb barrier’ and gets within

+r–rEnergy

~10−15 m

Proton

Coulombpotential

barrier

Protons and neutronswithin the nucleus

Fig. 1.1 Schematic illustration of the energy of a proton as a function ofits distance, r, from the centre of the nucleus. As the proton approachesthe nucleus, the repulsive Coulomb force increases, but close to thenucleus, this repulsion is overcome by the attractive strong force sothe energy is reduced and the nucleons are held together in the nucleus.

TABLE 1.1 Properties of the SubatomicParticles of Most Relevance to Radiotherapy

Particle Symbol Charge, ea Mass (kg)

Rest

Mass

Energy

(MeV)

Proton p +1 1.673� 10�27 938Neutron n 0 1.675� 10�27 940Electron e� �1 9.109� 10�31 0.511Positron e +1 9.109� 10�31 0.511Neutrino νe 0 >0 >0Antineutrino νe 0 >0 >0Photon γ 0 0

ae is charge on an electron.

3CHAPTER 1 Atoms, Nuclei and Radioactivity

the range at which the strong force works, then it has a much lowerenergy in the nucleus and stays there. An energy diagram like this isknown as a potential well.

Electron Energy LevelsAs mentioned in the section titled The Forces, planetary orbits aroundthe Sun and electron orbits around the nucleus differ in that, in the caseof the electrons, quantum theory predicts that only certain orbits areallowed. This means that only certain orbit energies can occur—thesedifferent values of energy are referred to as energy levels or shells andwere first hypothesised by Niels Bohr in 1913.

A free electron, which is outside the nucleus, is said to have zeroenergy; any electrons in levels closer to the nucleus have a lower, andtherefore negative energy. Fig. 1.2 shows the possible energy levels for

the simplest element, hydrogen.Noenergy is lost or gainedwhile an elec-tron occupies a particular shell, and only discrete amounts of energy canbe gained or lost by electrons when they move between shells. You cansee from Fig. 1.2 that the lowest energy state for hydrogen has an energyof –13.6 eV. Therefore the amount of energy that would be required toremove this electron from the atom is 13.6 eV. This is the ionisationenergy of the hydrogen atom. The energy required to raise an electronfrom the lowest energy state to the second lowest energy state is equalto the difference between the energies of the two states, that is, (13.6 –3.40) ¼ 10.2 eV, and so on for all other pairs of energy levels.

Large atoms have more complicated arrangements of electronenergy levels; the energy levels are numbered according to the principalquantum number, n (n ¼ 1, 2, 3 etc.) and are subdivided into otherenergy levels, sometimes known as sub-orbitals, with more quantumnumbers relating to angular momentum and spin. These quantumnumbers also dictate the number of electrons which can be in eachshell. Historically, the principal quantum number may also be repre-sented by the letters K (n ¼ 1), L (n ¼ 2), M (n ¼ 3) etc., so the shellsare often referred to as the K, L, M shells etc. If all the electrons in anatom are in the lowest possible energy states allowed by the rules, thenthe atom is said to be in the ground state. For example, a hydrogen atomin the ground state has the electron in the state with energy �13.6 eV.

Band Theory of SolidsIn individual atoms, outer electrons occupy specific energy levels.When atoms are brought together, as occurs in solid materials, inter-actions between atoms broaden these specific energy levels into ‘energybands’. Electrons may occupy energy states only within these bands,between which are forbidden zones that normally do not have energystates for electrons to occupy, as illustrated in Fig. 1.3.

The outermost energy bands within the solid material are termedthe valence band and the conduction band. Electrons within the valenceband are considered as linked to the chemical bonds between individualatoms and are therefore bound in place, although the term bound isused loosely because at normal temperatures such bonds may be con-tinually being broken and reformed. At an energy level slightly abovethe valence band is the conduction band. Electrons within this band aresurplus to any requirements for chemical bonding. At normal temper-atures, these electrons are not associated with specific atoms and chem-ical bonds, but migrate readily through the material. In some materials,there are insufficient electrons to fill the available energy levels of thevalence band, so the conduction band is empty. Where a large forbiddenzone exists, these materials are classed as nonconductors or insulators(see Fig. 1.3A). Other materials may have more outer electrons thanthe valence band can accommodate, so that the lower levels of the con-duction band are also occupied. In these materials, the conduction band

Excitedstates

Continuum of positiveenergy states (Etot> 0)

Groundstate

n = 1–13.6 eV, etc.

E2 = –3.40 eV

E3 = –1.51 eV

E4 = –0.85 eVE5 = –0.54 eVE6 = –0.38 eVE7 = –0.28 eV

...Etot = 0 eV

Fig.1.2 The possible energy levels of the electron in the hydrogen atom.Note that when the electron is bound in the atom, the energy is negativeand can only take certain values. Outside the atom, the energy of theunbound electron is zero. The energy levels are numbered, n ¼ 1, 2and so on from the inside outwards.

Forbidden zone

Valence band Valence band Valence band Valence band

Energy

Conduction band

Conduction band

Conduction band Conduction band

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 withimpurity levels within the forbidden zone.

4 Walter and Miller's Textbook of Radiotherapy

overlaps with the valence band and the forbidden zone disappears, asshown in Fig. 1.3B. These materials will generally be good conductorsof electricity. There are some materials in which the valence band is justfilled, but the conduction band is effectively empty, and a small but sig-nificant forbidden zone exists. These materials are classed as semiconduc-tors, illustrated in Fig. 1.3C. Any charges injected into a semiconductorwill be free to travel through the material. It is to be stressed that thisdescription is overly simplistic but serves as a basis for understandingthe principles of solid-state dosimeters discussed in Chapter 3.

Impurity BandsThe introduction of impurities at low concentrations can alter thestructure of the energy bands and may create energy bands that arelocated between the valence and conduction bands, within the forbid-den zone as shown in Fig. 1.3D. The properties of the material soformed will depend upon whether these extra bands are normally occu-pied or empty of electrons, and their actual energy levels. The additionof impurities is critical to the formation of active semiconductordevices (see Chapter 3) and to the development and functioning ofboth scintillator and thermoluminescent materials (see Chapter 3).

PARTICLES IN ELECTRIC AND MAGNETIC FIELDSElectric FieldsAs already explained, a single charge exerts either an attractive or arepulsive force on another charge. A collection of charges will also exerta force on another charged particle. This force can be written as

F ¼ qE 1.5

where F is the force on the charge q and the quantity E is known as theelectric field (caused by other charges in the vicinity). The direction of theelectric field is the direction inwhich a free positive charge wouldmove; afree negative chargewouldmove in the opposite direction, so the force ona particle in an electric field is always parallel or antiparallel to the field.

Magnetic FieldsThe force on a charged particle in a magnetic field is more complicated.There is no force at all if the particle is not moving; if it is moving thenthe force is, like the force in an electric field, dependent on the charge, q,but it is perpendicular to the direction of both the magnetic field, B, andthe velocity, v, of the particle and depends on the angle between them.Fig. 1.4 shows some examples.

The largest force occurs when the velocity of the particle is perpen-dicular to the magnetic field. As the angle θ between B and v decreases,the force decreases; when the velocity and field are parallel θ ¼ 0 andthere is no force. In fact, the magnitude of the force is given by

F ¼ q v B sin θð Þ 1.6

So, when v and B are perpendicular, θ ¼ 90ο and F ¼ q v B.

The Lorentz EquationThe forces on a moving charged particle, which is subject to both anelectric and a magnetic field, are complicated and are best dealt withusing the mathematics of vectors, which is beyond the scope of thisbook. However, for the simple case where the magnetic field is perpen-dicular to the velocity of the charged particle, we can write

F ¼ qE + qvBð Þ 1.7

The electric component of the force (qE) and the magnetic component(qvB) are not necessarily in the same direction. This equation is knownas the Lorentz equation after the Dutch physicist Hendrik A. Lorentz,and is an extremely important equation in many areas of physics. Inradiotherapy, it is useful when considering the behaviour of electronsin a linear accelerator and of charged particles in a cyclotron orsynchrotron.

WAVESTransverse and Longitudinal WavesEnergy, in the form of light, heat or sound, may be transmitted fromplace to place by waves. The direction of propagation of a wave isthe direction in which the energy is transported; however, the particlesin the medium do not change their overall position; they simply vibrateabout an average position. We can distinguish two types of wave:longitudinal and transverse.

In a transverse wave (Fig. 1.5A), the oscillations are perpendicular tothe direction of propagation of the wave. Water waves, or waves on astring, are good examples of transverse waves.

In the case of longitudinal waves (Fig. 1.5B), the particles in themedium move backwards and forwards in a direction parallel to thedirection of propagation, although their mean position stays the same.A good example is a sound wave; the particles of the medium (e.g. air)oscillate parallel to the direction of the sound wave and this gives rise tochanges in pressure along the wave.

Fig. 1.5 shows the wavelength, λ, of the wave—the distance betweentwo adjacent peaks (or two adjacent troughs). Another importantparameter is the frequency, f, which is the number of peaks that passa point per second. In all cases, the speed of the wave, c, is related tothe wavelength and frequency by the equation

c¼ fλ 1.8

This means that for a wave with constant speed, a larger wavelengthcorresponds to a lower frequency and vice versa. Wavelength is mea-sured in units of metres (m) and frequency in hertz (Hz).

Longitudinal sound waves are important in ultrasound imaging (seeChapter 5), but for radiotherapy applications we are mostly concernedwith the transverse waves of electromagnetic radiation and these are thesubject of the next section.

BB B

v v v

FF

A B C

Fig.1.4 The force on amoving charge (positive in this case) in amagnetic field depends on the relative directionsof the velocity (v) and themagnetic field (B). (a) v and B are perpendicular to each other. The force (F) is large andperpendicular to both v andB. (b) The angle between v andB is less than 90 degrees and the force is less but stillperpendicular to both. (c) v and B are parallel: there is no force.


Electromagnetic RadiationElectromagnetic radiation is so called because it can be described aswaves in which the quantities that oscillate are electric and magneticfields. Fig. 1.6 shows how the fields in these electromagnetic wavesoscillate at right angles to each other.

Because electromagnetic waves depend only on electric and mag-netic fields, they can travel through any medium, including a vacuum.In a vacuum, all electromagnetic waves travel at a speed of approxi-mately 3 � 108 m s�1 (often known rather loosely as the speed of lightin a vacuum) but the properties of the radiation vary greatly with wave-length and frequency. Fig. 1.7 shows the vast range of the electromag-netic spectrum; note that as the frequency increases, the wavelengthdecreases, according to Equation 1.8. Radiotherapy physics is mostlyconcerned with the high frequency/small wavelength end of the spec-trum, although radio waves are important in radiotherapy as they areused to accelerate the electron beam in a linear accelerator.

By the end of the 19th century, physicists were aware that all thedifferent types of radiation shown in Fig. 1.7 were electromagnetic

waves and they could be explained in terms of wave physics. However,the beginning of the 20th century saw the development of quantumphysics. Several key experiments, including the investigation of Comp-ton scattering, an important process in radiotherapy, described inChapter 2, showed that, when electromagnetic radiation interacts withmatter, wave physics does not always predict the correct result. Insteadthe radiation behaves as particles known as photons. A photon is a small‘packet’ or quantum of energy and each one has an energy given by

E ¼ hf 1.9

where E is the energy, f is the frequency of the electromagnetic waveand h is a constant, known as Planck’s constant and equal to 6.626 �10�34 J s.

As with the masses in the section titled Atomic Structure it ismore usual to give these energy values in electron volts (eV) where1 eV ¼ 1.602 � 10�19 J. This has the advantage of allowing an easycomparison between the energy of a photon and the mass energy of

Equilibrium (of first segment)

Equilibrium

Longitudinal wave

Transverse wave

Amplitude

Wavelength

Amplitude

A

B

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

Direction of oscillationof electric field vector e

Direction ofwave propagation

Ele

ctric

fiel

d

Position

λ

Magnetic field

Direction ofoscillation ofmagnetic fieldvector B

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


a particle and is especially useful when considering the transfer ofenergy between a photon and particle or the conversion of a particleinto radiation, annihilation radiation or radiation into particles, pairproduction (see Chapter 2). Fig. 1.7 shows the energies in eV in addi-tion to the wavelengths and frequencies.

Continuous Spectra and Characteristic RadiationIn the section titled Electron Energy Levels we described the way inwhich the electrons in atoms can only be in specific energy states,defined by their quantum numbers. The energies of these states are dif-ferent for each element; for example, Fig. 1.8 shows the main energylevels for tungsten, which is an element commonly used for the Targetin x-ray production, such as x-ray tubes (see Chapters 4 and 8), andmachines, such as linear accelerators (see Chapter 9).

As previously described, to raise an electron from one energy level toanother, the energy required is equal to the difference in energy of thetwo states. If an atom is excited above its ground state by absorbingenergy from incoming particles (e.g. photons or electrons) then some

electrons can be moved up into allowable, but normally empty, energylevels. This happens, for example, in an x-ray tube (see Chapters 5 and8). This leaves the atom in an excited state, that is, with a higher totalenergy than in the ground state. After a period of time, the atom willreturn to the ground state as the excited electrons drop from the higherlevels back to vacant lower energy states. When this happens, the excesselectron energy is carried away as a photon of electromagnetic radia-tion. Thus if we have two states with energies E1 and E2 then the energyof the photon (Eγ) is the difference between E1 and E2. UsingEquation 1.9 to relate the energy of the photon to its frequency, wearrive at:

E1�E2 ¼Eγ ¼ hf 1.10

The energy of photons produced is therefore dictated by the differencesin energy between electron shells of the particular atom from whichthey are emitted. The spectrum of photons produced by an elementis termed the characteristic radiation and will be different for each

Gamma-rays X-rays Ultraviolet Infrared Microwaves Radio

Gamma-rays

X-rays

Frequency, (Hz)

Energy (eV)4 ´ 105

7.5 ´ 1014 Hz

3.1 eV

4.3 ´ 1014 Hz

1.8 eV

4 ´ 103 4 ´ 101 4 ´ 10–1 4 ´ 10–3 4 ´ 10–5 4 ´ 10–7 4 ´ 10–9

400 nm

Vio

let

Blu

e

Gre

en

Yel

low

Ora

nge

Red

700 nm

Ultraviolet

Infrared

Microwave

Radio

10–12

1020 1018 1016 1014 1012 1010 108 106

10–6 1 103Wavelength (m)

Visible

Fig.1.7 The different types of radiation that form the electromagnetic spectrum. The visible spectrum covers avery small range of wavelength values and is expanded below the spectrum.


element. Fig. 1.8 shows the possible electron transitions leading to theproduction of characteristic photons for tungsten. Distinction is madebetween electron transitions originating from different shells to thesame destination shell by denoting the transition (and characteristicphoton) with the final shell letter (K, L and so on) and adding a Greekletter 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 theemitted photons will be a mixture of the continuous background spec-trum (see Chapter 2) and the characteristic radiation from the tungstenatoms. This is shown in Fig. 1.9. The energy differences between thelevels in heavy elements tend to be much larger than for lighter ele-ments: note that for tungsten (see Fig. 1.8) the photon energies ofthe 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 transitionand the free electron to K transition. Contrast this with the energyof the largest possible transition in hydrogen, which is 13.6 eV, andin the ultraviolet region (see Fig. 1.7).

RADIOACTIVE DECAYStable and Unstable IsotopesIn the section titled The Atom and the Nucleus we explained thatalthough the atomic number, Z, of a particular element is always thesame, atomic mass number, A, can vary so that each element can haveseveral different isotopes. Some of them will be stable: that is to say theywill not decay; others will be unstable and will undergo radioactivedecay. It is instructive to plot a graph which shows the number of

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, whichis known as the stability line. It also shows the solid line correspondingto equal numbers of protons and neutrons. We see that for low atomicnumber nuclei, an equal number of protons and neutrons is favoured,whereas a greater proportion of neutrons to provide stability for largenuclei. This may be explained by considering the increasing electro-static force of repulsion between protons in the nucleus as the numberof protons is increased. For a more detailed, interactive, diagram whichallows you to look up individual nuclei [2].

Evidence suggests that protons and neutrons within a nucleus adopta shell-like structure analogous to electron orbits and show particularstability when the number of protons or neutrons, or both, correspondsto amagic number (2, 8, 20, 28, 50, 82, 126). The strength of the strongforce that holds the nucleons together is associated with a nuclear bind-ing energy that must be overcome to break the nucleus apart. Essentiallythe mass of a given nucleus is less than the sum of its constituent pro-tons and neutrons; this is known as themass defect. Representing this interms of energy, using E ¼ mc2, gives the nuclear binding energy.Nuclei with an even number of protons or an even number of neutronsare more stable than those with an odd number of one or both.

A nucleus lying off the stability line shown in Fig. 1.10 is unstable anddecays by rearranging its nucleon numbers. This is achieved by releasingparticles, changing a proton to a neutron or vice versa, or by absorbingnearby particles. The activity of an unstable, or radioactive, isotope is the

n = ∞0

n = 5 (O)−0.1

n = 4 (N)−0.6

n = 3 (M)−2.8

n = 2 (L)

Lα, Lβ, Lγ

Kα, Kβ, Kγ

−12.1

n = 1 (K)

Nucleus

Energy (keV) Shell

−69.5

Fig.1.8 Electron energy levels and transitions leading to characteristicphotons for tungsten. The main transitions into each shell are marked.

0

0.2

0.4

0.6

0.8

1

1.2

20 40 60 80 100Photon energy (keV)

Rel

ativ

e ph

oton

flue

nce

Fig.1.9 Characteristic spectral lines from tungsten superimposed on thecontinuous spectrum.

N

(number of neutrons)

(number of protons)

825028146

6

14

28

50

82

126

Z

Type ofdecay

FissionProtonNeutronStable nuclideUnknown

b+

b–

a

Fig.1.10 A graph of neutron number (A–Z) plotted against proton number(Z). The stable isotopes are shown in black and the solid line representsequal numbers of protons and neutrons.


rate at which its nuclei decay, expressed in Becquerels (Bq). One Becque-rel corresponds to one decay or disintegration per second. The Becquerelis a very small unit so the activity of sources used in medicine is generallyrepresented in MBq (1 � 106 disintegrations per second) or GBq (1 �109 disintegrations per second). (You may also occasionally come acrossthe old unit of activity, the curie (Ci); 1 Ci ¼ 37 000 MBq.) Nuclei thatundergo radioactive decay are known as radionuclides.

In the construction of practical radioactive sources, we are alsointerested in the amount of material that is needed to manufacture asource with a required activity, determined by the specific activity,the activity per unit mass (MBq kg�1).

Half-lifeRadionuclidesdecayat verydifferent rates so it is important tohave someway of quantifying the rate of decay. If we have a collection of identicalnuclei, it is impossible to knowexactlywhichonewill decaynext; one canonly predict the probability of decay. The number of nuclei, dN, whichdecay in a short time dtwill depend on two factors: the decay constant, λ,which is essentially the probability of decay, andN, the number of nucleipresent at the start. This is expressed by the equation

dN ¼�λNdt 1.11

This equation can be rearranged and integrated to give

N tð Þ¼Noe�λt 1.12

whereNo is the number of nuclei present at t¼ 0 andN(t) is the numberof nuclei present at time t. e is the exponential function. A graph ofN(t)plotted against time gives an exponential decay curve similar to thatshown in Fig. 1.11.

The interesting thing about an exponential decay is that the lengthof time it takes for the number of undecayed nuclei to halve is alwaysthe same. In Fig. 1.11, this time is shown as 2 (arbitrary) time units. Thistime is known as the half-life, T½, and it is related to the decay constant,λ, by the equation

T1=2 ¼ ln2λ

¼ 0:693λ

(1.13)

The term ln where the term ln represents the logarithm to base e; ify ¼ ex then x ¼ ln y.

Half-lives can vary enormously—for example, the half-life ofuranium-238 (not used for medical purposes!) is approximately thesame as the age of the Earth, 4.5 billion years; the half-life ofkrypton-81, used in nuclear medicine, is 13s, and others are evenshorter. The length of the half-life is an important consideration whenchoosing a radionuclide for medical use; if the half-life is too long, thenthe patient may be radioactive for the rest of their life; if it is too shortthen the activity will decay too fast for it to be useful.

When radioactivity was first discovered at the end of the 19th cen-tury, three different types of emitted particle were identified and werelabelled alpha (α), beta (β) and gamma (γ) radiation. These names haveremained, although there are now a few variants of them; the next foursections will cover these different types of decay.

Alpha DecayAlpha (α) decay occurs most often from the unstable nuclei of heavyelements, such as uranium, radium or plutonium. The nucleus emitsan alpha particle, which is actually a nucleus of helium, 42He. Havingan equal number of protons and neutrons (both of which are magicnumbers) the helium nucleus is very stable. The parent nucleus (X)decreases its atomic number by 2 and its mass number by 4 so the gen-eral equation is:

AZX!A�4

Z�2 Y+42He 1.14

where Y is the daughter nucleus. For example, radium (22688 Ra) decays toradon (22286 Rn) and an alpha particle. Energy must be conserved in thetransformation, so any difference, Q between the nuclear bindingenergies of the parent and daughter nuclei is shared between theemitted alpha particle, in the form of kinetic energy and any photonsthat are produced (see the section titled Gamma Decay). Alphaparticles are typically emitted with kinetic energies of the order ofseveral MeV.

Being relatively heavy (approximately 4� the mass of a proton) andhighly charged (containing two protons), α particles are readily stoppedin matter. For example, the 4.79 MeV α particle emitted from radiumhas a range of less than 4 cm in air, or less than 0.04 mm in tissue. Thismeans that alpha particles can be very useful for radiotherapy, but only

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

Rel

ativ

e ac

tivity

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-halfof the original value and is the same as the time taken to halve again to one-fourth and again to one-eighth.


if they can be released very close to the tumour tissue. Radium-226 wasused in some very early external radiotherapy treatments but has thebig disadvantage that the daughter product, radon, is a radioactivegas. More recently, Radium-223 is being trialled as an unsealed sourceof treatment for bone metastases.

Beta DecayThere are now known to be two types of beta decay. Beta minus (β�)decay occurs frequently in naturally occurring radionuclides andinvolves the emission of an electron; beta plus (β+) decay occurs mainlyin artificially produced radionuclides and the particle emitted is apositron.

A nucleus lying above the stability line in Fig. 1.10, is neutron-richand, by emitting an electron, can convert a neutron to a proton, therebyapproaching the stability line. An example is iodine-131, whichundergoes beta minus decay to become xenon-131:

13153 I!131

54 Xe 1.15

Note that, because the total number of nucleons has not changed, themass number stays the same, but because a neutron has been convertedinto a proton, the atomic number increases by one. The emitted β� par-ticle is a high-energy electron from the nucleus, not to be confused withthe orbiting electrons in the atom.

By contrast, a nucleus lying below the stability line of Fig. 1.10, isproton-rich and therefore decays by converting a proton into a neu-tron. As with β� decay the mass number does not change, but in thiscase, since a proton is converted into a neutron, the atomic numberdecreases by one. The emitted β+ particle is a positron, the antimatterequivalent of the electron. A good example, which is widely used in PETimaging, is fluorine-18:

189F!18

8 O 1.16

Observation that the β particles produced display a spectrum of kineticenergies, rather than the discrete energy difference between parent anddaughter nuclei, indicates that a further particle must be involved. Forβ+ decay, this particle is the neutrino, νe; for β� decay, it is its antipar-ticle, νe. The processes for iodine-131 and fluorine-18 are therefore:

13153 I!131

54 Xe+0

�1β+ ν

and

189F!18

8 O+ 0+1β+ ν 1.17

The energy arising from the difference in masses of the initial and finalparticles is carried away as kinetic energy of beta particle and theneutrino.

The range of a β� particle is larger than that of an α particle as it ismuch lighter. However, whereas the heavy alpha particles will travelalong a straight path, the beta particles will follow a much more erraticpath as they interact with atomic electrons. The range of a beta particlein tissue is only of the order of a few millimetres and will depend on theenergy with which it was emitted.

An emitted positron (β+) travels through matter, rapidly losingkinetic energy through interactions with atomic electrons. When it col-lides with an electron, its antiparticle, both particles are annihilated andtheir energy converted into electromagnetic radiation. Using Einstein’smass-energy equivalence equation, E¼mc2, the total energy of the radi-ation must be 2 � 0.511 MeV (see Table 1.1). Conservation of momen-tum demands that two photons, each with energy 0.511 MeV areproduced in opposite directions. This is the basis of PET (see Chapter 6).

Depending on its energy, a positron will be stopped within a veryshort distance of the site of emission in tissue. The annihilationphotons, on the other hand, at 0.511 MeV, each can pass relativelyeasily through tissue. Detection of these coincident photons followingadministration of a positron-emitting radionuclide, such as fluorine-18 to a patient, therefore reveals where the annihilation eventoccurred and hence where the radionuclide was taken up withinthe body.

Gamma DecayIn the section titled Electron Energy Levels we showed that the elec-trons in atoms could only occupy certain allowed energy levels. Inan analogous way, each nucleus can only have certain discrete energiesand transitions between two levels involve the emission or absorptionof photons of electromagnetic radiation. As with electron transitions,the energy values of the levels are different for different elements; how-ever, in the nuclear case, the differences are much larger so the photonsproduced are generally of much higher energy. Following an α orβ decay, the daughter nucleus is often left in an excited state. It will thenreach its ground state by emitting a photon with an energy correspond-ing to the difference between the two energy levels. These photonsare known as gamma (γ) rays. Fig. 1.12 shows the decay scheme forcobalt-60, which is an isotope formerly widely used for external beamtherapy and now used in the Gamma Knife (see Chapter 8).

In most cases, the emission of gamma rays occurs immediatelyafter the alpha or beta decay, however occasionally, the nucleusremains in an excited state and decays with a measurable half-life.Such an excited state is known as a metastable energy state and isdenoted by the addition of an ‘m’ to the mass number. There is nochange in Z or A during the transition from the excited state of themetastable nucleus to the ground state, so this is known as an isomerictransition. One important example, widely used in nuclear medicine,is technetium-99m (Fig. 1.13). This is a useful radionuclide because itproduces only gamma rays, with an energy of 140 keV, which can beused for imaging, and no short range α or β particles, which wouldonly damage tissue.

Note that both γ rays and x-rays are electromagnetic radiation at thetop end of the spectrum (see Fig. 1.7); the energy ranges overlap andindeed both are used in radiotherapy. They are only distinguished bytheir origin: gamma rays coming from the nucleus and x-rays fromthe atomic electrons (see Chapter 2).

1.1732 MeV g

0.31 MeV b −5.272 years

0.12%

6027

99.88%

Co

6028Ni

1.48 MeV b −

1.3325 MeV g

Fig.1.12 Cobalt-60 decays via beta decay to give nickel-60. The nickelnucleus is in an excited state and decays via the scheme shown, givinggamma rays with energies 1.1732 and 1.3325 MeV.


Electron Capture and Internal ConversionAs an alternative to positron emission (β+ decay), the nucleus of aproton-rich atom may capture one of its own inner shell electrons,via electron capture (EC). The captured electron combines with a pro-ton in the nucleus to produce a neutron and neutrino, the latter being

emitted from the nucleus carrying kinetic energy equal to the differencein nuclear binding energy between the parent and daughter nuclei. Aswith β+ decay, the mass number does not change but the atomic num-ber decreases by 1. Nuclei that decay by this method can be usefulbecause there is no particulate emission. An example of a nuclide thatdecays by EC is iodine-125; it emits gamma rays of up to 35 keV, whichcan be used for brachytherapy (see Chapter 8).

Another possible mode of decay is internal conversion (IC). Anexcited nucleus may de-excite by emitting a single photon, which inter-acts with the inner shell electron so that the electron is ejected from theatom. In contrast to beta decay, the emitted electron will have a singlekinetic energy equal to the excitation energy of the nucleus minus theelectron binding energy. The vacancy in the atomic shells left bythe emitted electron will be filled by outer electrons, giving rise to char-acteristic radiation (see the section titled Continuous Spectra andCharacteristic Radiation).

Radioactive Decay SeriesThere are many cases of radioactive nuclei that decay to give daughternuclei, which are themselves radioactive and so on. This gives rise toa decay series. Fig. 1.14 shows the decay series of uranium-238, a nat-urally occurring radionuclide. At each stage, the α or β decay leadsto a new nucleus which itself decays, the final product in this casebeing a stable isotope of lead. At each decay, the rate of growth ofthe activity of the daughter nuclide depends on the relative valuesof the decay constants (λ) of the parent and the daughter. Another

Fig.1.13 Technetium-99m is produced from molybdenum-99 by betadecay 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 eventuallydecays to stable ruthenium-99 but with an extremely long half-life andtherefore very low activity.

γ

1m

γ

γ

γα

β

β

β

β

β

21081Tl

22y

21082Pb

γ7h

23491Pa

γ24d

23490T

20682Pb

5d

21083Bi

140d

21084P

γ

γ

γ

α

α

α

α

α

β

β

27m

21482Pb

20m

21483Bi

10-4s

21484P

3m

21884P 4d

22286R α103y

22688R α104y

23090T

α105y

23492U

109y

23892U

Fig.1.14 Decay series for 23892 U. Half-lives are indicated in seconds (s), minutes (m), hours (h) and years (y).


example, of much more clinical importance, is the decay ofmolybdenum-99 (see Fig. 1.13) to technetium-99m, which is widelyused for imaging.

If we assume that there is no daughter present at time t¼ 0, and thatall the disintegrations of the parent lead to the required daughter prod-uct, then the activity of the daughter at time t, A2(t) is given by:

A2 tð Þ¼ λ2λ2� λ1

A1 0ð Þ e�λ1t �e�λ2t� �

(1.18)

where A1(0) is the initial activity of the parent at time t¼0 and are thedecay constants of parent and daughter, respectively. Equation 1.18 isof relevance to radionuclide generators as it allows calculation of theoptimum time between elutions of the daughter radionuclide.

If the decay of the parent is much slower than that of the daughter,that is, if λ2>> λ1 then Equation 1.17 reduces to

A2 tð Þ¼A1 0ð Þ e�λ1t �e�λ2t� �

(1.19)

This is the situation for ionisation chamber consistency check devicescontaining a strontium-90 source. Strontium-90 undergoes beta decaywith a half-life of 28.7 years to yttrium-90, which itself decays via betadecay with a half-life of 64 hours. The activity of the long-lived stron-tium parent determines and maintains the activity of the short-livedyttrium daughter.

Radionuclides of Medical InterestTable 1.2 lists some common isotopes applied to radiotherapy andnuclear medicine. The choice of isotope for a particular applicationis based on decay product type (γ, β (+ or �) or α), 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 tissueso will deposit energy close to the site at which a radionuclide is takenup in the body; β� particles (electrons) have a slightly longer, but stillsmall, range. If the site of disease can be preferentially targeted by theseemissions, this leads to significant sparing of surrounding normal

tissues in therapeutic applications. If greater penetration is required,of the order of centimetre for brachytherapy, or if imaging of radioac-tivity uptake through external detection of radiation is required, thenphotons (γ) or β+ emissions will be the product of choice.

REFERENCES[1] Interactive Periodic Table Royal Society of Chemistry, http://www.rsc.org/

periodic-table/.[2] Interactive Segre chart. http://people.physics.anu.edu.au/�ecs103/chart/

(Note that this plots Z against N—the opposite of Fig. 1.9. Both versions arecommonly used).

FURTHER READINGDendy PP, Heaton B. Physics for diagnostic radiology. 3rd ed. Baton Rouge:

CRC Press; 2012.Grant IS, Phillips WR. The elements of physics. Oxford: Oxford University

Press; 2001.

TABLE 1.2 Characteristics of SomeRadionuclides Used in Radiotherapy as EitherUnsealed Sources or Sealed Sources

Isotope

Decay

Mechanism Half-Life Clinical Application

Unsealed Sources11C β+ (2.0 MeV) 20 m PET imaging13N β+ (2.2 MeV) 10 m PET imaging15O β+ (2.8 MeV) 122 s PET imaging18F β+ (1.7 MeV) 109 m PET imaging32P β� (695 keV) 14.3 d Polycythaemia vera89Sr β� (500 keV) 50.5 d Bone metastases (palliation)99mTc γ (143 keV) 6.0 h Gamma camera imaging90Y β� (923 keV) 2.7 d Radiosynovectomy131I β� (264 keV)

γ (364 keV)8.1 d Thyrotoxicosis and thyroid

cancer223Ra α (5.7 MeV) 11.4 d Prostate cancer

Sealed Sources60Co β�, γ (1.17,

1.33) MeV)5.26 y External beam units and

gamma knife103Pd EC, γ (21 keV) 17 d Brachytherapy (seeds)125I EC, γ (27–

36 keV)60 d Brachytherapy (seeds)

137Cs β�, γ (662 keV) 30 y Brachytherapy (pellets)192Ir β�, γ (300–400)

keV)74 d Brachytherapy (wire)


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