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3Atoms, Nuclei, Elementary Particles,and Radiations

3.1 Atoms

In ancient Greek philosophy the term atom (the indivisible) was used todescribe the small indivisible pieces of which matter consists. Father ofthe so-called Atomism theory was the philosopher Leucippus (450–370 B.C.)and his student Democritus (460–370 B.C.). According to them, matter isbuilt of identical, invisible, and indivisible particles, the atoms. Atoms arecontinuously moving in the infinite empty space. This infinite empty spaceexists without itself being made of atoms. Atoms show variations in theirform and size and they tend to be bound with other atoms. This behaviorof the atoms results in the building of the material world. According toDemocritus, the origin of the universe was the result of the incessantmovement of atoms in space. In this sense atoms were the elementaryparticles of nature.

It was in 1803, more than 2000 years later, when John Dalton (1766–1844)first provided evidence of the existence of atoms by applying chemicalmethods. In his theory, Dalton supported the concept that matter is built ofindivisible atoms of different weights. All atoms of a specific chemicalelement are identical in respect of their mass (weight) and their chemicalbehavior. Atoms are able to build compounds keeping their proportion toeach other in the form of simple integers. If these compounds decompose,the atoms involved emerge unchanged from this reaction. In 1896, DmitrijIwanowitsch Mendelejew (1834–1907) proposed the periodic law, accordingto which the properties of the elements are a periodic function of their atomicmasses. The definitive arrangement of the elements in the periodic tableaccording to their atomic number and not according to their atomic masseswas accomplished in 1914 by Henry Moseley (1887–1915).

In 1904, Sir Joseph John Thomson (1856–1940) proposed the first modelof the atom, according to which, the atom is a positively charged sphere ofradius about 10210 m with electrons interspersed over its volume.

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43

The Thomson model incorporates many of the known properties of theatom, i.e., size, mass, number of electrons, and electrical neutrality. In 1911,Ernest Rutherford (1871–1937) and his group disproved Thomson’s atommodel in a series of scatter experiments using alpha particles and thin metalfoils. These experiments revealed that besides scattering at small anglescorresponding to the Coulomb interaction of a-particles with a Thomson-atom, some a-particles were scattered at very large angles. To explain theseresults, Rutherford suggested that the positive atomic charge and almost allthe atomic mass is concentrated in a very small nucleus of about 10214 m,instead of being distributed over the whole volume of the atom, withelectrons distributed around it at comparatively large distances of about10210 m.

Niels Hendrick David Bohr (1885–1962) formulated his model for thehydrogen atom in 1913, during his stay in Rutherford’s laboratory. It wasbased on the results of Rutherford and the works of Max Karl Ernst LudwigPlanck (1858–1947) and Albert Einstein (1879–1955). According to Bohr’smodel, electrons move in orbits around the positive charged nucleus similarto the planetary movement around the sun. The role of the gravitationalforce in the solar system is undertaken by the Coulomb attractive force.In 1926, Werner Karl Heisenberg (1901–1976) and Erwin Schrodinger(1887–1961) founded a new approach, termed quantum mechanics, fordescribing microscopic phenomena where the deterministic approach ofclassical mechanics was replaced by probabilistic theory.

3.1.1 The Bohr Hydrogen Atom Model

Bohr’s model of the atom is reminiscent of the solar system. The stability ofthe solar system is ensured by equilibrium of the gravitational andcentrifugal forces. In the hydrogen atom a single electron (charge e2)circulates about a proton (charge eþ). The radius of the circular orbit is rand the electron of rest mass me moves with constant tangential speed v.The proton is assumed to be at rest. The attractive Coulomb force providesthe centripetal acceleration v 2/r, so:

F ¼e2

4p10

1

r2¼

mev2

rð3:1Þ

where 10 is the permittivity of free space (see Table A.2.2).Since an electron undergoes an accelerated movement in its orbit,

according to the classical electrodynamics theory it should continuouslyradiate energy in the form of electromagnetic waves. This, in turn, shouldresult in the collapse of the atom since by continuously losing energy theelectron should spiral on to the nucleus. Obviously, this atomic modelcontradicts two of the most significant experimental results, namely thestability of the atom and the discrete nature of atomic spectra. To overcomethese difficulties Bohr postulated the existence of stationary electron orbits in


The Physics of Modern Brachytherapy for Oncology44

the atom wherein the electron does not radiate energy. These orbits arecharacterized by definite values of angular momentum L, which is an integermultiplicate of Planck’s constant h:

L ¼ mevr ¼ nh

2p¼ n~ ð3:2Þ

where n, called the quantum number, is an integer of value n ¼ 1,2,…,1.Manipulating Equation 3.1 and Equation 3.2 it is found that electron orbits

lie at certain distances,

rn ¼ n2 4p10~2

mee2¼ n2 £ 0:529 £ 10210 m ð3:3Þ

and are characterized by definite energies:

En ¼ 21

n2

mee4

32p2120~

2¼ 2

1

n213:606 eV ð3:4Þ

These important results are very different from those we expect fromclassical physics. For example, a satellite may be placed into Earth’s orbit atany desired altitude when supplied with the proper speed. This is not truefor an electron’s orbit rn, for which, according to Equation 3.3 there are onlycertain, discrete allowed orbits. The same applies for the energy levels En,which appear in Equation 3.4; only certain values are allowed. They arequantized. The new theory proposed by Bohr is termed the quantum theoryof atomic processes.

Bohr postulated that, even though the electron does not radiate when it isin a certain energy state, it may make transitions from one state to a lowerenergy state. The energy difference between the two states is emitted as aquantum of radiation (photon) whose energy Ephoton is equal to this energydifference:

Ephoton ¼ Einitial 2 Efinal ð3:5Þ

For example, in a transition from initial state n ¼ 3 to final state n ¼ 2, theenergy Ephoton emitted is given by Equation 3.4 and Equation 3.5:

Ephoton ¼ E2 2 E3 ¼21

322

21

22

� �13:606 eV ¼ 2:520 eV

This photon has a wavelength l ¼ hc/Ephoton ¼ 656.1 nm, which is exactlythe measured wavelength of the visible, red line of the Balmer series.Likewise the wavelengths of all spectral lines were accurately predicted.

The absolute value of energy En in a state n is known as the binding energy,b, of the state (i.e., b ¼ lEnl since En is negative, indicating a bound electron–proton system). The state n ¼ 1, corresponds to the lowest energyE1 ¼ 213.606 eV. This is the ground state of the hydrogen atom. The atomcan remain unchanged over infinitely long periods of time in this groundstate. It is a stable state, from which the atom can only depart by energy


Atoms, Nuclei, Elementary Particles, and Radiations 45

absorption. An energy equal to binding energy b is just sufficient to break upthe hydrogen atom into a free electron and a proton, a process termedionization. If E . b is supplied to the hydrogen atom (e.g., photoelectricabsorption, Compton scattering, and other processes discussed in Chapter 4)the electron will leave the atom with a kinetic energy K ¼ E 2 b.

Higher energy states n $ 2 are excited states where the atom arrives uponabsorption of energy equal to En 2 E1. Excited states are usually short living(half-life ,1028 sec) and therefore unstable. The atom spontaneously decaysfrom an excited state to lower energy states, eventually returning to theground state, emitting upon the process quanta of radiation (photons)termed characteristic fluorescence radiation (see Section 3.1.4).

It is through these processes of excitation and ionization that radiationdeposits energy to matter.

The Bohr model provides a coherent picture of the hydrogen atom and itssize, explains the discrete nature of the atomic spectra, and accuratelypredicts the wavelengths of the emitted radiations (lacking, however, thepotential to make any prediction for their intensities). Bohr’s theory waslater modified and perfected; ellipsoidal orbits and motion of proton aboutthe common center of mass were considered and new quantum numberswere introduced to explain phenomena such as the fine structure (manyspectral lines are not single but actually composed of two closely spacedlines). However, the Bohr model remains a phenomenological andincomplete model applicable to atoms containing one electron and yetwith serious deficiencies (it violates the uncertainty principle and angularmomentum conservation, as discussed in Section 3.1.2). More generally, itdid not provide any insight as to why the concepts of classical mechanicsmust be renounced in order to describe the atomic processes. Thesedifficulties were overcome in 1826 when Heisenberg and Schrodingerproposed a quite new approach, termed quantum mechanics, for describingmicroscopic phenomena.

3.1.2 The Quantum Mechanical Atomic Model

The quantum theory was introduced by Planck in 1900 in order to describethe quantization of the energy emitted by a black body: the quantumhypothesis. In 1905, Einstein, based on Planck’s quantum hypothesis,succeeded in explaining the photoelectric effect (see Section 4.2.1) byassuming that the energy of light is bounded in light particles, photons,whose energy is quantized as described by Planck. The particle nature of theelectromagnetic radiation was further needed to explain the observationmade in 1922 by Arthur Holly Compton (1892–1962), known as theCompton effect or Compton scattering (see Section 4.2.3). In summary, it wasfounded that some effects, such as interference could be explained on thebasis of the wave nature of light whereas the explanation of others, such asthe photoelectric effect, required the assumption of the particle nature oflight. This was termed the wave-particle duality.



In 1924, Louis Victor Duc de Broglie (1892–1987) postulated in his Ph.D.thesis that all forms of matter exhibit wave as well as particle properties, asphotons do. According to this conception, an electron has a dual particle-wave nature. Accompanying the electron is a kind of wave that guidesthe electron through space. The corresponding wavelength, called the DeBroglie wavelength l of the particle, is given by

l ¼h

pð3:6Þ

where h is Planck’s constant and p is the momentum of the electron. ClintonJoseph Davisson (1881–1958) and Lester Halbert Germer (1896–1971), in1927, succeeded in providing experimental proof of de Broglie’s assumption.

Werner Karl Heisenberg (1901–1976), who had worked with Bohr in hislaboratory in Copenhagen during the period 1924–1927, began his workon the development of quantum mechanics in 1925, focusing on themathematical description of the frequencies and intensities (amplitudes) ofthe radiation emitted or absorbed by atoms. In 1927, Heisenberg formulatedthe uncertainty principle, which states that no experiment can ever beperformed yielding uncertainties below the limits expressed by thefollowing uncertainty relations:

DxDpx , ~ ð3:7Þ

DEDt , ~ ð3:8Þ

or any other relation combining two physical quantities whose productcan be expressed in ~ dimensions. According to the uncertainty principle,the simultaneous definition of neither the exact position and the exactmomentum of an electron in a given state, nor the exact energy of a state andits exact lifetime, are possible. For example, Bohr’s theory violates theuncertainty principle of Equation 3.7 since it allows for the simultaneousdefinition of radial distance r (thus Dr ¼ 0) and the momentum pr at theradial direction (it is actually zero and thus Dpr ¼ 0). Another example isgiven by Equation 3.8; it shows that the energy of stable ground state, thatcan remain unchanged for ever (thus Dt ¼ 1), can be determined with highaccuracy (DE ¼ 0). An excited state, however, usually presents a lifetimeof the order of 1028 sec (thus Dt < 1028 sec) and therefore DE < ~/Dt < 6.6 £ 1028 eV which corresponds to the natural width of spectrallines. Note that DE is the uncertainty in the energy of the excited state andalso the uncertainty in the energy of the photon emitted during de-excitation. In practice, the observed width of spectral lines is wider than thenatural width due to Doppler broadening caused by thermal motion.

Particle-waves are described by a complex-value wave function, usuallydenoted by the Greek letter c whose absolute square lcl2 yieldsthe probability of finding the particle at a given point at some instant.In 1926, Erwin Schrodinger (1887–1961) proposed a wave equation thatdescribes the manner in which particle-waves change in space and time.



The time-independent Schrodinger equation for the hydrogen atom reads:

2~

2

2mDcð~rÞ þ Vð~rÞcð~rÞ ¼ Ecð~rÞ ð3:9Þ

where

Vð~rÞ ¼e2

4p10

1

rð3:10Þ

is the potential and r is the radial distance of the electron from the proton.The solution of the Schrodinger equation for the hydrogen atom is a wavefunction cnlml

ð~rÞ where the three quantum numbers n, l, ml which emergefrom the theory are called:

† Principal quantum number n that takes the values n ¼ 1,2,3,…,1.

† Angular momentum quantum number l that, for given n, takes thevalues l ¼ 0,1,2,…,n 2 1.

† Magnetic quantum number ml that, for given l, takes the valuesml ¼ 2 l, 2 l þ 1, 2 l þ 2,…, þ l.

The quantized energy levels in the hydrogen atom are found to dependonly on the principal quantum number n and are given by Equation 3.4,i.e., they are exactly the same as predicted by Bohr. Note, however, that thisonly applies for the hydrogen atom due to the special form of the potential ofEquation 3.10 and its r 21 dependence (see also Section 3.1.3).

The angular momentum, however, is totally different from that postulatedby Bohr (Equation 3.2). According to quantum mechanics the orbital angularmomentum is a vector, ~L, for which we can only determine its magnitude lLland possible projections Lz along a given z-direction (e.g., a magnetic fieldapplied along the z-axis). The angular momentum quantum number lspecifies the magnitude L according to

lLl ¼ffiffiffiffiffiffiffiffiffilðlþ 1Þ

p~ ð3:11Þ

while the magnetic quantum number ml describes the orientation of theangular momentum in space; for a given l there are (2l þ 1) discrete values ofthe projection Lz of the angular momentum vector into a specific direction-zin space:

Lz ¼ ml~ ð3:12Þ

In a three-dimensional representation, the angular momentum vector ~Lmust lie on the surface of a cone which forms an angle u with the z-axis. Thevalues of u are also quantized and given by

cos u ¼mlffiffiffiffiffiffiffiffiffi

lðlþ 1Þp ð3:13Þ

Summarizing, the electron angular momentum is a vector ~L quantized inspace: for a given n there are n discrete values of the magnitude lLl givenby Equation 3.11. For each of these n discrete values of l, there are (2l þ 1)



(i.e., a total of n(2l þ 1)) discrete orientations in space given by Equation3.12 and/or Equation 3.13.

The next plausible step was the modification of Schrodinger’s equation forconsistency with the theory of special relativity. This was achieved in 1928 byDirac (1902–1984), who showed that the requirements imposed by relativityon the quantum theory have the following consequences: the electron hasintrinsic angular momentum and associated magnetic dipole moment andthere are fine structure corrections to the Bohr formula for the energy levelswhich do not depend only on the principal quantum number n. The intrinsicangular momentum or spin, ~S, is a quantum property of the electron. As forthe angular momentum, ~L, the spin vector ~S has a quantized magnitude lSlwhich in units of ~ is given by

lSl ¼ffiffiffiffiffiffiffiffiffiffisðsþ 1Þ

p~ ð3:14Þ

possible projections Sz along a given z-direction given by

Sz ¼ ms~ ð3:15Þ

and corresponding possible values of angle u with the z-axis, given by

cos u ¼msffiffiffiffiffiffiffiffiffi

lðlþ 1Þp ð3:16Þ

For electrons (as well as protons and neutrons) the spin quantum number shas a single value of s ¼ 1/2, that specifies the magnitude lSl. As for theorbital angular momentum, there is a spin magnetic quantum number ms

that takes the two quantized values ms ¼ ^1/2 that specify the orientation ofthe spin vector. Thus, Equation 3.14 through Equation 3.16 for s ¼ 1/2 yield:

lSl ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2

1

2þ 1

� �s~ ¼

ffiffi3p

2~; Sz ¼ ^

1

2~ and cos u ¼ ^

ffiffi3p

3

A vector model describing space quantization of an electron’s spin isshown in Figure 3.1. According to quantum mechanics there are two equallyprobable orientations of spin; either parallel or antiparallel to, for instance,an external magnetic field along the z-axis. These states are usually calledspin up and spin down. (Associated with the spin of electrons, protons aswell as neutrons, is a magnetic dipole moment and an interaction energy inthe presence of an external magnetic field and hence these states form twoseparate energy levels. Despite the equal quantum mechanical probability inspin orientations, the population of these two levels is not equal in ambienttemperature and this constitutes the physical basis for spectrometric orimaging techniques such as electron spin resonance (ESR) nuclear magneticresonance (NMR) and magnetic resonance imaging (MRI).)

A complete description of an electron state is achieved through the fourquantum numbers (n, l, ml, ms). For the ground state of hydrogen n ¼ 1,and therefore only l ¼ 0 and ml ¼ 0 are permitted. With the addition of



spin (ms ¼ ^1/2) the ground state is therefore either (1,0,0, þ 1/2) or(1,0,0, 2 1/2). These two states have the same energy, given by Equation 3.4for n ¼ 1. They are thus twofold degenerate (degeneracy, in general,disappears in the presence of an external magnetic field; e.g., Zeeman effect,MRI). For the first excited state, n ¼ 2 and thus the allowed values ofl are l ¼ 0 or l ¼ 1 (equally probable). For l ¼ 0, only ml ¼ 0 is permitted.For l ¼ 1, ml can take the values of 21, 0, and þ1. There are, therefore,2n 2 ¼ 2(22) eightfold degenerate states, namely (2,0,0, þ 1/2), (2,0,0, 2 1/2),(2,1,1, þ 1/2), (2,0,0, 2 1/2), (2,1,0, þ 1/2), (2,1,0, 2 1/2), (2,1, 2 1, þ 1/2),(2,1, 2 1, 2 1/2). For n ¼ 3 there are 2n 2 ¼ 2(32) ¼ 18 states.

For historical reasons, all states with the same principal quantum number,n, are said to form a shell. These shells are designated with the followingcapital letters:

Value of n 1 2 3 4 5 6…Shell symbol K L M N O P…

12

32 h

32

1 2 h–

33

cosq =

q

Z

.h

h

FIGURE 3.1A vector model describing space quantization of electron spin.



Likewise, there is a spectroscopic notation in which small-case letters areused to identify different l values:

Value of l 0 1 2 3 4 5 6Notation s p d f g h i

(the first four letters stand for sharp, principal, diffuse, and fundamental,which were terms originally used to describe atomic spectra before atomictheory was developed).

In spectroscopic notation the ground state of hydrogen is 1s where thevalue of n ¼ 1 is specified before the s. The first excited state is either 2s or 2p(the 2s state, as all s states, is single while the 2p state, as all p, d, f, andsubsequent states, is double due to spin–orbital momentum interaction inthe presence of the internal magnetic field, causing the effect of finestructure). For n ¼ 3 there are 3s, 3p, or 3d states. Not all transitions betweendifferent states are allowed. The transitions most likely to occur are thoseresulting to a change of l by one unit (i.e., those that conserve angularmomentum since the emitted photon has spin 1) and thus the selection rulefor allowed transitions is

Dl ¼ ^1 ð3:17Þ

Thus, a transition from n ¼ 3 to the n ¼ 1 state, that according to the Bohrmodel does not conserve angular momentum and therefore is not allowed,quantum mechanically could only be from 3p state to the 1s state. Transitionssuch as 3s ! 1s or 3d ! 1s are not allowed due to the selection rule ofEquation 3.17.

3.1.3 Multielectron Atoms and Pauli’s Exclusion Principle

An electronic state is completely specified by the four quantum numbers(n, l, ml, ms). In addition, we have seen that in the hydrogen atom the energyEn depends only on the principle quantum number n in Equation 3.4, andnot on the angular momentum quantum number l as a consequence of thespecial form of the potential (r 21 dependence). For atoms with more thanone electron (Z . 1) the energy of a quantum state depends on n as well ason the angular momentum quantum number l and, in general, the lower thevalue of l the lower the energy of the electron. For example a 4s state is oflower energy than a 3d state, as naturally provided quantum mechanically.States having the same values of n and l (for instance 4s or 3d) are known assubshells. The number of quantum states in a subshell is 2(2l þ 1). The(2l þ 1) factor comes from the number of different ml values for each l. Theextra factor of 2, comes from the two possible values of ms ¼ ^1/2. Thus,subshell 4s has two quantum states (as all s states) and 3d has ten quantumstates (as all d states). The way in which these different quantum states areoccupied by the electrons in a multielectron atom is based on the exclusionprinciple formulated in 1925 by Wolfgang Pauli (1900–1958): no two



electrons in a single atom can ever be in the same quantum state; that is, notwo electrons in the same atom can have the same set of the four quantumnumbers n, l, ml, and ms. In fact, particles that obey the exclusion principleare those of half-integer spin quantum numbers called fermions (such aselectrons, protons, and neutrons for which s ¼ 1/2) and not those of integerspin quantum numbers called bosons (such as photons of s ¼ 1).

One of the most important consequences of the exclusion principle is thatit explains the periodic table of the elements which constitutes the basis forstudying chemical behavior. The ordering of the energy levels in manyelectron atoms is well known. The two 1s states of the K-shell are the onesto be filled first, since these states correspond to the lowest energy. For theL-shell, the two 2s states are of lower energy than the six 2p states. Therefore,the electron configuration of the fluorine (Z ¼ 9) is: 1s 22s 22p 5, i.e., there aretwo electrons in the 1s states, two more in the 2s state and five electronsoccupy the 2p states. For neon (Z ¼ 10) the configuration is 1s 22s 22p 6 andfor sodium (Z ¼ 11) 1s 22s 22p 63s 1. Neon which has all its subshells filledis an inert gas (practically nonreactive) while its neighbors, fluorine andsodium, are among the most reactive elements.

3.1.4 Characteristic X-Rays. Fluorescence Radiation

Owing to historical reasons, x-rays immediately bring to mind theelectromagnetic radiation produced in an x-ray tube or linear acceleratorwhere electrons are rapidly decelerated in the anode (see Section 4.5). Thesex-rays present a continuous energy spectrum (bremsstrahlung) spreadingfrom zero up to a maximum energy, which depends on the appliedacceleration potential (e.g., for 100 kV potential the maximum energy ofthe spectrum is 100 keV). Superimposed on this continuous spectrum arediscrete x-ray line spectra, called characteristic x-rays, since they are emittedby the atoms of the anode (in general characteristic x-rays are those emittedby atoms while g-rays are those emitted by nuclei).

Since all the inner shells of an atom are filled, x-ray transitions do notnormally occur between these levels. However, once an inner electron isremoved by an atom (as in the case of photo-absorption which is of higherprobability for K-shell electrons, as discussed in Section 4.2.1) the vacancycreated is filled by outer electrons falling into it, and this process may beaccompanied by emission of fluorescent radiation or Auger electronemission. The fluorescent x-rays emitted in the process of filling a K-shellvacancy are known as K x-rays and may be of significant impact on thedosimetry of low-energy photon emitters such as 125I and 103Pd (see Section4.2.1, Chapter 5, and Chapter 9).

Consider the Ka x-rays, originating by the electron transition from the L tothe K-shell. An electron in the L-shell is screened by the two 1s electrons and soit faces an effective nuclear charge of Zeffective < Z 2 2. When one of these 1selectrons is removed and a K-shell vacancy is created, only the remaining1s electron screens the L-shell, and so Zeffective < Z 2 1. Bohr’s theory for the



hydrogen atom includes the nuclear electric charge only in Equation 3.1.Therefore the allowed energies (Equation 3.4) for Zeffective < Z 2 1 are given by

En ¼ 21

n2

mee4ðZ 2 1Þ2

32p2120~

2¼ 2

1

n2ðZ 2 1Þ213:606 eV ð3:18Þ

For n ¼ 1, this equation provides a crude yet useful approximation for theK-shell binding energy (see also Equation 4.15).

The energy E(Ka) of the Ka x-ray, i.e., transition from n ¼ 2 to n ¼ 1, isE2 2 E1:

EðKaÞ ¼3

4ðZ 2 1Þ213:606 eV ð3:19Þ

A plot of the square root of the Ka x-ray energy, i.e., {E(Ka)}1/2 as a functionof electric charge Z (and not Zeffective) is a straight line (the approximationZeffective < Z 2 1 is not crucial; it could easily be Zeffective < Z 2 k, where k isan unknown number to be evaluated from the intercept of the straight line).

This is the Moseley (1887–1915) law, formulated in 1913. It is a simple, yetpowerful way, to determine the atomic number Z of the atom, hence itsspecific place in the periodical table of the elements, which were previouslyordered according to increasing mass causing some abnormalities. While anelement is exclusively specified by its atomic number (number of protons),elemental atomic weight depends on the mass number (sum of protons andneutrons in its nucleus).

3.2 Atomic Nucleus

Rutherford suggested, in 1911, that the positive atomic charge and almost allthe atomic mass are concentrated in a very small central body, the nucleus.Continuing his experiments, in 1919, Rutherford discovered that there werehydrogen nuclei ejected from materials upon being bombarded by fastalpha-particles. He identified the ejected hydrogen nuclei, the protons, asnuclear constituents; the atomic nuclei contain protons.

Until 1932, physicists assumed that atomic nuclei are constructed ofprotons, alpha-particles, and electrons. In 1932, Sir James Chadwick(1891–1974) identified the neutron by interpreting correctly the results ofthe experiments carried out mainly by Jean Frederic (1897–1958) and IreneJoliot-Curie (1900–1956): neutrons, uncharged particles were ejected out ofberyllium nuclei after their bombardment with alpha-particles. Chadwickconsidered neutrons to be an electron–proton compound and added it to thenuclear mix.

In July 1932, Heisenberg published his neutron–proton nuclear model byassuming that neutrons and protons are constituents of the nucleus. Hismodel of the nucleus also contained electrons (nuclear electrons) eitherbound or unbound. The assumption of the existence of nuclear electrons was



definitively rejected in the late 1930s after the introduction of the neutrino byPauli in 1931 and the establishment of Enrico Fermi’s (1901–1954) theory ofbeta decay, published in 1933.

Contrary to the original idea of indivisible atom, it has been proved thatthe atomic nucleus has an internal structure. The nucleus contains two kindsof particles, the nucleons: the positive charged protons with a charge of þeand the uncharged neutrons.

3.2.1 Chart of the Nuclides

All nuclei are composed of protons and neutrons. The atomic number Zdenotes the number of protons and the neutron number N denotes thenumber of neutrons in a nucleus. The mass number A ¼ Z þ N indicates thetotal number of nucleons (protons and neutrons). A nucleus X specified byatomic number Z, neutron number N and mass number A is called a nuclideand it is symbolized by

ZAXN

Usually, however, the atomic number Z (implied by the chemical symbol)and the neutron number N (N ¼ A 2 Z) are omitted and a nuclide isidentified by its chemical symbol and its mass number A, e.g., 12C instead of

612C6, 192Ir instead of 77

192Ir115, etc.A convenient way of depicting nuclear information is offered by the chart

of the nuclides, first used by Segre, where each nuclide is represented by aunit square in a plot of atomic number Z vs. the neutron number N. A smallpart of such a chart is shown in Figure 3.2 (a complete chart can be found atreferences 1 and 2). In this chart isotopes (nuclides of the same atomicnumber Z) are arranged in horizontal lines; e.g., 1H, 2H, and 3H are isotopesof hydrogen. Isotopes have identical chemical properties, they arechemically undistinguishable but they have significantly different nuclearproperties. Isotones (nuclides of the same N) are arranged in vertical lines(e.g., 3H, 4He, 5Li…) and isobars (nuclides of the same A) fall alongdescending diagonals from left to right (e.g., 6Be, 6Li, 6He).

Figure 3.3 shows the general layout of a complete chart of nuclides. Thereare about 3000 nuclides known to exist (most of them artificial), divided intotwo broad classes: stable and radioactive. Stable nuclides remain unchangedover an infinitely long time, while radioactive nuclides are unstable,undergoing spontaneous transformations known as radioactive decays(see Section 3.3.1). Filled squares in Figure 3.2 and/or Figure 3.3 denotenatural nuclides, i.e., primordial nuclides (formed before the creation ofEarth) occurring in nature. These are either stable (there are only 274 stablenuclides in nature) or long-living radioactive nuclides surviving since thecreation of the Earth about five billion years ago (there are 14 including 40K,87Ru, 232Th, 234U, 235U, 238U). Natural nuclides form the stability curve orstability valley, discerned in Figure 3.3 (see also Section 3.4.2 and Figure 3.9in that section). For light nuclei, Z , 20, the greatest stability is achieved



FIGURE 3.2Part of the chart of nuclides. In this chart each nuclide is represented by a unit square in a plot of atomic number, Z, vs. neutron number, N. Shaded squarescorrespond to stable nuclides for which atomic mass in units of u and natural isotopic abundance are provided. For the remaining nuclides, which areunstable, atomic mass, half-life, decay mode, and decay energy in units of MeV are provided.

8

7

5

4

6

3

2

Z

N

1 Stable99.985%

Stable0.015%

12.33 yβ- 0.019

1H

1.0078250

2H

2.0141018

119.0 msβ- 10.653

8He

8.0339218

8.5 msβ- 20.610

11Li

11.04379624 E-22 sn 0.420

10Li

10.0354809178.3 msβ- 13.606

9Li

9.0267891138 ms

β- 16.004

8Li

8.0224867Stable92.5%

7Li

7.0160040Stable7.5%

6Li

6.01512233 E-22 sβ- 1.970

5Li

5.01253788 E-23 sp 3.100

4Li

4.0271823

3 E-22 s2p 2.200

7B

7.0299174

2 E-21 s2p 2.142

8C

8.037675126.5 ms

EC 16.498

9C

9.031040119.255 sEC 3.648

10C

10.016853120.39 monEC 1.982

11C

11.0114338Stable

98.93%

12C

12.0000000Stable1.07%

13C

12.00335485730 years

β- 0.156

14C

14.0032422.449 sβ- 9.772

15C

15.01059930.747 sβ- 8.012

16C

16.014701295 ms

β- 11.810

18C

18.026757149 ms

β- 16.970

19C

19.035248114 ms

β- 15.790

20C

20.0403224

18 msβ- 22.800

22N

22.034440387 ms

β- 17.170

21N

21.0270876142 ms

β- 17.970

20N

20.0233673290 ms

β- 12.527

19N

19.0170269624 ms

β- 13.899

18N

18.01408184.173 sβ- 8.680

17N

17.00844977.13 s

β- 10.419

16N

16.0061014Stable3.66%

Stable99.634%

15N

15.0001089

14N

14.0030740

13N

13.0057386

12N

12.0186132

11N

11.0267962

Unknown

21C

21.04934193 ms

β- 13.166

17C

17.0225837

5 E-21 s2p 1.372

6Be

6.0197258

53. 29dEC 0.862

7Be

7.01692923 E-16 s2α 0.092

8Be

8.0053051

Stable100%

9Be

9.01218211510000 y

β- 0.566

10Be

10.013533713.81 s

β- 11.506

11Be

11.021657723.6 ms

β- 11.708

12Be

12.0269206

3 E-21 sn 0.500

13Be

13.03613383 E-21 sβ- 16.220

14Be

14.0428155

5.08 msβ- 22.680

17B

17.0469314<190 psn 0.040

16B

16.039808810.5 ms

β- 19.094

15B

15.031097313.8 ms

β- 20.664

14B

14.025404117.36 msβ- 13.437

13B

13.017780320.20 msβ- 13.369

12B

12.0143521Stable80.1%

11B

11.0093055Stable19.9%

10B

10.01293708 E-19 s2α 0.277

9B

9.0133288770 ms

EC 17.979

8B

8.0246067

3 E-21 sn 0.440

7He

7.0280305806.7 msβ- 3.508

6He

6.01888818 E-22 sα0.890

5He

5.0122236Stable

99.99986%

4He

4.0026032

0.000137%

3He

3.0160293Stable

10.4 minβ- 0.782

0n

1.0086649

3H

3.0160493

0

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

9.965 minEC 2.220

11.000 msEC 17.338

13O13.0248104

14O14.0085953

8.58 msEC 17.765

70.606 sEC 5.143

15O15.0030654

122.24 sEC 2.754

16O15.9949146

Stable99.762%

17O16.9991315

Stable0.038%

18O17.9991604

Stable0.200%

26.91 sβ- 4.821

19O19.0035787

13.51 sβ- 3.814

20O20.0040761

3.42 sβ- 8.109

21O21.0086546

2.25 sβ- 6.490

22O22.0099672

82 msβ- 11.290

23O23.0156913

61 msβ- 11.400

24O24.0203699

3 E-22 sp 2.290


Atom

s,N

uclei,

Elem

entary

Particles,

and

Rad

iations

55

when the number of protons equals the number of neutrons, Z ¼ N. Forheavier nuclei, the instability caused by the Coulomb repulsion betweenthe protons is counterbalanced by a high-neutron excess, Z , N (seeSection 3.2.3). Neighboring the natural nuclides are radioactive nuclideswhich occupy the rest of the area bounded by the full line in Figure 3.3. Thesenuclides decay in such a way that they finally reach a stable nuclide lying onthe stability curve. The nuclides below the stability curve are b2-unstablewhile those above the stability curve are bþ-unstable (see Section 3.4.2).

From the three hydrogen isotopes 1H, 2H, and 3H, only 1H and 2H arenatural (actually stable nuclides, see Figure 3.2). 3H is a relatively short-living radioactive nuclide which has not survived since the creation of theEarth approximately 5 billion years ago. For carbon (Z ¼ 6) there are twonatural isotopes, 12C and 13C, with corresponding natural isotopicabundance of 98.89 and 1.11%. The short-living nuclide 14C is not considereda natural isotope although it is being constantly produced by cosmic rayneutron bombardment of 14N and this activity is continuously replenished(carbon dating is based on this isotope). Since 1961, the atomic weight hasbeen based on a 12C-scale; the atomic weight of this carbon isotope is exactly12. However, the atomic weight of carbon presented in the periodic table isnot exactly 12, but 0.9893 £ 12 þ 0.0107 £ 13.00033548 ¼ 12.0107 due tothe natural isotopic abundance of 13C. 238U is not stable but is long-living

120

100

80

60

40

20

00 20 40 60 80

Z=N

Neutron number, N

Ato

mic

num

ber,

Z

100 120 140 160

FIGURE 3.3An overview of the complete chart of nuclides. Shaded squares correspond to natural nuclides(forming the stability valley) while the remaining area bounded by the full line is occupied byradioactive nuclides.



(half-life T1/2 ¼ 4.51 £ 109 years) and is considered a natural nuclide withnatural isotope abundance of 99.27%. Through successive transformations,238U eventually arrives at the stable end product 206Pb. None of the membersof the corresponding radioactive series (14 overall) are considered as naturalnuclides although they are found in nature and sometimes are veryimportant, as was 226Ra in early brachytherapy practice. This is because theydo not have characteristic terrestrial compositions.

The chart of the nuclides provides some of the most important informationon each nuclide including its mass and binding energy, spin and magneticmoment, natural isotopic abundance, mode of decay, and decay constant if itis unstable, etc.

3.2.2 Atomic and Nuclear Masses and Binding Energies

Published tabulations present atomic rather than nuclear masses. Atomicmasses are known with high accuracy. This accuracy is achieved bymeasuring atomic masses relative to each other, and in particular, relative tothe neutral unexcited 12C atom which has arbitrarily been assigned 12.00000atomic mass units. The unit employed is the unified atomic mass unit, u,defined as the 1/12 of the mass of the neutral unexcited 12C atom. In SI units,the unified atomic mass unit u is therefore:

1 u ¼1

12£

12 £ 1023 kg

NA

¼ 1:66053873 £ 10227 kg ð3:20Þ

where NA is Avogadro’s number.The energy equivalence of the unified mass unit in MeV is, according to

Einstein’s mass–energy relation, E ¼ 1 u £ c 2 ¼ 931.494013 MeV, thus:

1 u ¼ 931:494013 MeV=c2 ð3:21Þ

The rest mass m in units of u and the rest energy mc 2 in units of MeV, forelectron, proton, and neutron are (see also Appendix 2):

Electron me ¼ 5.48597 £ 1024 u mec2 ¼ 0.511 MeV

Proton mp ¼ 1.008665 u mpc 2 ¼ 938.28 MeVNeutron mn ¼ 1.007277 u mnc2 ¼ 939.57 MeV

The rest energy of the hydrogen atom in its ground state is

mHydrogenc2 ¼ mec2 þmpc2 2 13:606 eV

that is, the binding energy b ¼ 13.606 eV is subtracted from the sum of therest energies of the electron and proton composing the hydrogen atom. Theelectron binding energy is only a , 1028 fraction of the atomic rest energywhich is 938.783 MeV. The rest mass of the hydrogen atom is 1.007825 u.



To a first approximation the atomic mass matom(Z,A) of an atom with Zprotons, A 2 Z neutrons and Z electrons in units of u or GeV/c 2 is

matomðZ;AÞ < A u < A GeV=c2 ð3:22Þ

Some times, instead of atomic mass the mass excess, D, defined as thedifference of the mass number, A, by the atomic mass, i.e. D ¼ matom(Z,A)2A,is used (see Appendix 1).

The nucleus contains almost all the mass of the atom. The nuclear massmnucleus(Z,A) of a certain nuclide of atomic number Z and mass number A,can be found from the atomic mass matom(Z,A) of the corresponding atom, bysubtracting the mass of the Z electrons with due consideration of the massreduction associated with the total binding energies be of all electrons in theatom. The equation for the rest energies is therefore:

mnucleusðZ;AÞc2 ¼ matomðZ;AÞc

2 2 Zmec2 2 be ð3:23Þ

The total binding energy be of all Z electrons expressed in eV, can beadequately approximated by the Thomas–Fermi empirical expression:

be ¼ 15:73 £ Z7=3eV ð3:24Þ

This is negligible when compared with the atomic masses and it is usuallyneglected in the calculations of the nuclear masses. Moreover, it usuallymakes no difference which of the atomic or nuclear mass is used, because thenumber of electrons, and hence electron masses, cancel out in most nuclearreactions (see for example Section 3.4.1).

The binding energy B of a nucleus ZAXN is defined as

B ¼ Zmp þNmn 2 mnucleusðZ;AÞn o

c2 ð3:25Þ

That is, as the difference in rest energies between the nucleus and itsconstituent Z protons and N neutrons. The binding energy is in that sense theamount of energy that has to be supplied in order to break up the nucleusinto free neutrons and protons. Equivalently, the nuclear binding energy canbe calculated from atomic mass ignoring the total binding energy be of theatomic electrons:

B ¼ Zðmp þmeÞ þ ðA 2 ZÞmn 2 matomðZ;AÞn o

c2 ð3:26Þ

Figure 3.4 presents the variation of binding energy per nucleon, B/A, withmass number, A, for the natural nuclides presented in Figure 3.3. The bindingenergy per nucleon starts at small values of 1.11 for 2H (B ¼ 2.22 MeV), rises toa maximum of 8.79 for 56Fe (B ¼ 492.25 MeV) and then falls to a value of 7.57for 238U (B ¼ 1801.69 MeV). A number of interesting conclusions can bedrawn by noting the features of the B/A vs. A plot:

1. B/A is approximately constant and equal to about 8 MeV/nucleon.This implies that the nuclear force acting between nucleons is dueto a short-range interaction. Indeed, if each nucleon interacted with



all the remaining (A 2 1) nucleons, the total binding energy Bwould be proportional to A(A 2 1) < A 2 and not to A.

2. B/A falls of at large A. This is due to Coulomb repulsion betweenprotons. Coulomb is a long-range interaction and therefore eachproton interacts with the remaining (Z 2 1) protons in the nucleus,resulting in a Coulomb repulsion energy which increases similarlyto Z(Z 2 1) < Z 2. Since Z 2 increases faster than A, heavy nuclei,as shown by Figure 3.3, have more neutrons than protons. Thiscompensation of the Coulomb repulsion between all protons by thestrong nuclear attraction between neighboring nucleons could notlast for ever; elements heavier than uranium do not occur in nature.

3. The B/A vs. A plot peaks at A < 60. This implies that bindingenergies can be increased by either splitting a heavy nucleus intotwo lighter nuclei or fusing two light nuclei together, highlightingthe importance of fission and fusion reactions in the production ofnuclear energy.

3.2.3 The Semiempirical Mass Formula

The general features of the B/A vs. A plot (Figure 3.4) indicate that nuclearbehavior resembles that of a drop of liquid. Using a semiempirical approach,

9

8

7

6

5

4

3

2

1

00 20 40 60 80 100

Mass number, A

B/A

(M

eV p

er n

ucle

on)

B/A

(M

eV p

er n

ucle

on)

120 140 160 180 200 220 240

200

2

4

6

8

10

12

14

16

40 60 80

Total

Surface

Volume

Coulomb

Assymetry

100Mass number, A

120 140 160 180 200 220

FIGURE 3.4The binding energy per nucleon, B/A, vs. mass number, A, for natural nuclides. The insetpresents corresponding data calculated using the semiempirical liquid drop model(Equation 3.27) as well as the relative magnitude of the terms in the model.



Weizsacher demonstrated, in 1935, that it is possible to achieve a quantitativeinterpretation of binding energy by regarding B as akin to a latent energy ofcondensation. According to this idea, the binding energy B(Z,A) of a nucleus

ZAX is given by a number of terms that are functions of A and Z:

BðZ;AÞ ¼ aVolumeA 2 aSurfaceA2=3 2 aCoulombZðZ 2 1Þ

A1=3

2 aAsymmetryðA 2 2ZÞ2

A2 aPairingdA23=4 ð3:27Þ

The first two terms in this formula indicate that there is an analog ofa nucleus with a drop of liquid. A liquid drop has volume and surface energydue to the fact that the attractive forces between molecules are short-rangedforces (intermolecular attractions—van der Waals forces). Each moleculeinteracts only with its neighbors, and the number of neighbors surroundinga molecule is independent of the overall size of the liquid drop. Therefore,the energy required to overcome these short-range forces and completelyevaporate the drop is, on average, the same for each molecule (volumeenergy). Molecules, however that lie on the surface of a drop are notsurrounded by neighbors on all sides, and consequently, they are not boundas tightly as the molecules in the interior of the drop. Hence, the contributionof these surface molecules to the total energy of the drop is, on average,diminished by a factor approximately equal to the surface of the drop (surfaceenergy). The attractive force between nucleons in a nucleus (regardless of thenucleon being a proton or a neutron), i.e., the strong nuclear force, is a short-range one. Nuclei are roughly spherical, with the nuclear radius R givenapproximately by

R ¼ R0A1=3 where R0 < 1:2 £ 10215 m ¼ 1:2 fm ð3:28Þ

Nuclear volume Vnucleus is

Vnucleus ¼4

3pR3

0A ! Vnucleus / A ð3:29Þ

and thus nuclear volume is directly proportional to the mass number A.Nuclear mass is also proportional to A (see Equation 3.22). In this way thenucleus can be seen to be analogous to a drop of incompressible liquid, whichhas a constant and very high-density (,1012 times greater than that ofordinary matter) independent of its size. The term (aVolumeA) in Equation 3.27represents a constant, bulk-binding energy per nucleon B/A (see inset ofFigure 3.4), similar to the cohesive energy of a simple liquid drop. The secondterm (aSurfaceA 2/3), representing the diminished contribution of the surfacenucleons to the total binding energy of the nucleus, is proportional to thesurface area of the nucleus ( / 4pA 2/3) and this surface term is subtracted bythe bulk-binding energy (see inset of Figure 3.4). As shown in Figure 3.4, forlight nuclei the fraction of nucleons on the surface is quite large and hencethere is a sharp fall in B/A. For heavy nuclei this term is less important.



The third term in Equation 3.27 represents the Coulomb repulsionbetween protons and has a simple explanation; it is the electrostatic energyof the nuclear charge distribution. Assuming that the nucleus is a uniformlycharged sphere of radius R0A 1/3 as in Equation 3.28, and total charge Ze, itsenergy would be

EC ¼3

5

ðZeÞ2

ð4p10ÞR0A1=3ð3:30Þ

Since each proton interacts with all other protons in the nucleus but itself,the Z 2 term in Equation 3.30 has been replaced by the term Z(Z 2 1) inEquation 3.27. The Coulomb term, similar to the surface term, has a negativesign and thus is subtracted from the total binding energy (see inset ofFigure 3.4). As shown in this figure, the Coulomb energy becomes verysignificant for heavy nuclei, since Z(Z 2 1) increases more rapidly than A(Coulomb is a long-range interaction, in contrast to the short-range nuclearinteraction). Note that it is the balance between the increase of Coulombenergy and the decrease of surface energy for heavier nuclei that producesthe maximum B/A at around A ¼ 60, shown in the plot of B/A vs. A.

The asymmetry term in Equation 3.27 originates from Pauli’s exclusionprinciple to which neutrons and protons, as fermions, obey. For a given A, itis energetically advantageous to maximize the number of neutron–protonpairs. The term (A 2 2Z)2/A is a simple empirical expression which, ifconsidered alone, indicates maximum stability is achieved when the neutronexcess N 2 Z ¼ A 2 2Z is minimum for a given A. It is sometimes called thesymmetry term since it tends to make nuclei symmetric with respect to thenumber of neutrons and protons. The Segre plot (Figure 3.3) shows that forlight nuclei the maximum stability is achieved when Z < N, while forheavier nuclei stability ensues only if there is a neutron excess, i.e., Z , N,due to the relatively higher importance of the Coulomb energy term. The(A 2 2Z) excess neutrons occupy higher energy quantum states andconsequently they are less tightly bound than the first 2Z nucleons whichoccupy the lower energy states.

The last term in Equation 3.27 is purely phenomenological in form and it’sA dependence. Nuclei display a systematic trend for pairing: those havingeven number of protons and neutrons (even–even) tend to be very stable(there are 165 even–even stable nuclides); those with even-Z and odd-N (55nuclides) or odd-Z and even-N (50 nuclides) are somewhat less stable; andthose with an odd number of Z and N (odd–odd) are mainly unstable (thereare only four stable odd–odd nuclides known: 2H, 6Li, 10B, and 14N). Also,whenever Z or N becomes equal to the so-called magic-numbers: 2, 4, 8, 20,50, 82, and 126, the corresponding nuclides have large binding energies. Toaccount for the pairing energy, the parameter d in Equation 3.27 takes thevalues of

d ¼ 21 for even– even nuclei



d ¼ 0 for even– odd nuclei

d ¼ þ1 for odd– odd nuclei

indicating that the largest stability (largest binding energy) is achieved foreven–even nuclei.

The five parameters in Equation 3.27 are evaluated by fitting the formulato known binding energies (hence the name semiempirical). Typical fittedvalues for these parameters are

aVolume ¼ 14 MeV

aSurface ¼ 13 MeV

aCoulomb ¼ 0:6 MeV

aAsymmetry ¼ 19 MeV

aPairing ¼ 33:5 MeV

For example, employing the above set of the five parameters yields abinding energy calculation of B ¼ 486.65 MeV for 56Fe (to be compared withthe actual value of 492.25 MeV) and B ¼ 1799.2 MeV for 238U (to be comparedwith the actual value of 1801.69 MeV).

Incorporating Equation 3.27 into Equation 3.25, the semiempirical massformula reads:

mnucleusðZ;AÞc2 ¼ Zmp þ ðA 2 2ZÞmn

n oc2

þ

(2 aVolumeAþ aSurfaceA2=3 þ aCoulomb

ZðZ 2 1Þ

A1=3

þaAsymmetryðA 2 2ZÞ2

Aþ aPairingdA23=4

)

(3.31)

3.3 Nuclear Transformation Processes

Nuclear transformation processes are those inducing transitions from onenuclear state to another. They can fall into two categories: those which occurspontaneously, referred to as decays and those which are initiated bybombardment with a particle from outside, called reactions. When a processoccurs spontaneously, conservation of energy requires that the final state beof lower energy than the initial state and the difference in these energies,called Q value, is liberated as kinetic energy of energetic particles beingemitted (see Section 3.3.3). These particles are rather easy to observe experi-mentally, and, in fact, it was their discovery in the 1890s that initiatedresearch in nuclear physics.



3.3.1 Radioactive Decay

As mentioned in Section 3.2.1 there are only 274 stable nuclides (forming thestability valley in Figure 3.3) and approximately 2800 unstable nuclides. Allunstable nuclei spontaneously transform into other nuclear species bymeans of different decay processes that change the Z and N numbers ofnucleus, until stability is reached. Such spontaneous nuclear processes arecalled radioactive decays. Among well-known radioactive decays are alphadecay, beta decay (including electron capture (EC)), and spontaneous fissionof heavy nuclei. Excited states of nuclei are also unstable (usually when anucleus decays by alpha- or beta-emission it is left in an excited state) andeventually decay to the ground state by emission of gamma radiation (nochange of Z or N occurs). All the above processes follow the laws ofradioactive decay. Radioactivity is thus a property of the nucleus or, to bemore precise, of a state of the nucleus, according to which the nucleus decaysto a more stable state.

The probability for a radioactive decay per unit time for a specific nuclideis constant and called the decay constant, l (see Section 2.6.1). Sinceradioactive decay is a stochastic, spontaneous process it is not possible toidentify which particular atoms out of an amount of a specific radionuclidewill undergo such decay at a specific time. It is only possible to predict themean number of disintegrated nuclei at a specific time, i.e., the activity A(t)defined as

AðtÞ ¼ 2dNðtÞ

dtð3:32Þ

where dN(t) is the number of decays observed during the time interval dt(the minus sign is included since dN(t)/dt is negative due to the decrease ofN(t) with time while activity, A(t), is a positive number). Experimentally, it isfound that the activity, A(t), at any instant of time t is directly proportional tothe number, N(t), of the radioactive parent nuclei present at that time:

AðtÞ ¼ lNðtÞ ð3:33Þ

where l is the decay constant. The SI unit of activity is the Becquerel (Bq,named after the discoverer of radioactivity): 1 Bq ¼ 1 disintegration persecond ¼ 1 sec21. Activity was traditionally measured in units of Curies (Ci)with one Ci originally defined as the activity of 1 g pure 226Ra(1 Ci ¼ 3.7 £ 1010 Bq).

Combining Equation 3.32 and Equation 3.33, gives

2dNðtÞ

dt¼ lNðtÞ ð3:34Þ

Integrating this differential equation results in

NðtÞ ¼ N0expð2ltÞ ð3:35Þ

where N0 is the (initial) number of radioactive nuclei at t ¼ 0, i.e., N0 ¼ N(0).Multiplying both sides of Equation 3.35 by the decay constant l and recalling



Equation 3.33, results in the following equation for the activity:

AðtÞ ¼ A0expð2ltÞ ð3:36Þ

where A0 is the (initial) activity at t ¼ 0, i.e., A0 ¼ A(0).Both Equation 3.35 and Equation 3.36 present the exponential law of

radioactive decay, which states that the number of nuclei that have notdecayed in the sample as well as the activity of the sample, both decreaseexponentially with time.

The time needed for half of the radionuclides to decay (or equivalently theactivity of a sample to be reduced to half its initial value) is called half-life,T1/2, and it can be calculated using Equation 3.35 for N(T1/2) ¼ N0/2 (orequivalent to Equation 3.36 for A(T1/2) ¼ A0/2):

N0

2¼ N0 expð2lT1=2Þ! T1=2 ¼

ln 2

lð3:37Þ

The mean lifetime t, i.e., the average lifetime of a given radioactivenucleus is the average value of t calculated as

t ¼ ktl ¼

ð1

0t dNðtÞ

ð1

0dNðtÞ

¼

N0

ð1

0lt expð2ltÞdt

N0

¼1

lð3:38Þ

The mean lifetime t, is the reciprocal of the decay constant l, and thisresult is natural since the decay constant has the physical meaning of thedisintegration probability, i.e., the fraction of decays taking place per unittime. Apparently, within time t the initial number of nuclei decreases by afactor of e.

Figure 3.5 presents the exponential decrease of unit activity with time forvarious radionuclides used in brachytherapy which are characterized byhalf-lives spanning from a couple of days to 30 years.

3.3.2 Radioactive Growth and Decay

Activity calculations in successive radioactive decays are more complicated.Suppose a chain decay of the form:

N1!l1 N2!

l2 N3 ð3:39Þ

In this chain, the parent radionuclide N1 with decay constant l1 decays todaughter nuclide N2 which is also radioactive and has a decay constant l2

and therefore decays to N3 (assumed for simplicity to be stable). It is clearthat there is a growth of N2 with time due to decay of N1, as well as a decay ofN2 since it is itself radioactive. This is also the case of growth and decay of theexcited states usually created when a parent nucleus decays by alpha- orbeta-emission.

A system of two differential equations can be written to describe the twosuccessive decays in Equation 3.39. For simplicity let the numbers N1, N2,



and N3 represent also the number of nuclei from each nuclide. If originallyonly the nuclide N1 is present, i.e., N1(0) – 0 is the number of parent nuclei attime t ¼ 0, then N2(0) ¼ N3(0) ¼ 0 and the differential equation for N1 is (seeEquation 3.34):

2dN1ðtÞ

dt¼ l1N1ðtÞ ð3:40Þ

which integrated results to

N1ðtÞ ¼ N1ð0Þ expð2l1tÞ ð3:41Þ

for the number of nuclei N1(t) present at time t. The parent activity A1(t) attime t is given by

A1ðtÞ ¼ A1ð0Þ expð2l1tÞ ð3:42Þ

The differential equation describing the growth and decay of the daughternuclei N2 is

dN2ðtÞ

dt¼ l1N1ðtÞ2 l2N2ðtÞ ð3:43Þ

137Cs60Co

192Ir125I

169Yb103Pd

198Au

1

0.9

0.8

0.7

0.6

0.5

0.4

Rel

ativ

e ac

tivity

, A(t

)

0.3

0.2

0.1

00 20 40 60 80 100

t (days)

120 140 160 180 200

FIGURE 3.5The exponential time decrease for a unit of initial activity of radionuclides used inbrachytherapy: 137Cs (T1/2 ¼ 30.20y), 60Co (T1/2 ¼ 5.27y), 192Ir (T1/2 ¼ 73.81d), 125I(T1/2 ¼ 59.49d), 169Yb (T1/2 ¼ 32.02d), 103Pd (T1/2 ¼ 16.99d), and 198Au (T1/2 ¼ 2.70d).



where the first term l1N1 on the right side of the equation corresponds to thegrowth of N2 due to the parent decay and the second term l2N2 correspondsto the daughter decay.

Substituting Equation 3.41 into Equation 3.43, results in

dN2ðtÞ

dtþ l2N2 ¼ l1N1ð0Þ expð2l1tÞ ð3:44Þ

which integrated gives

N2ðtÞ ¼l1

l2 2 l1

N1ð0Þ ðexpð2l1tÞ2 expð2l2tÞ� �

ð3:45Þ

for the number of daughter nuclei N2(t) present at time t. The daughteractivity A2(t) at time t is given by

A2ðtÞ ¼ A1ð0Þl2

l2 2 l1

ðexpð2l1tÞ2 expð2l2tÞ� �

ð3:46Þ

Daughter activity A2(t) reaches its maximum value when its timederivative vanishes, i.e., at time t for which dA2/dt ¼ 0 (or dN2/dt ¼ 0).This time is calculated as

tðA2 ¼ maxÞ ¼lnðl1=l2Þ

l1 2 l2

ð3:47Þ

While the parent activity A1(t) decreases exponentially with time t asin Equation 3.42, the daughter activity A2(t) in Equation 3.46 starts fromzero at t ¼ 0, increasing to its maximum value at time t(A2 ¼ max) as inEquation 3.47. At that time, and only at that time, the parent activity and theaccumulated daughter activity are equal. This can be easily deduced fromEquation 3.43, since dN2/dt ¼ 0.

There are many cases in nature for which l1 ,, l2, i.e., the daughter isshorter living than the parent. In such cases some approximations can beused. These are

l2 2 l1 ; l2 and expð2l2tÞ ; 0 when t . tðA2 ¼ maxÞ ð3:48Þ

Then it follows:

A2ðtÞ

A1ðtÞ¼

l2

l2 2 l1

expð2l1tÞ2 expð2l2tÞ

expð2l1tÞ

( )< 1

That is

A2ðtÞ ; A1ðtÞ ¼ A1ð0Þ expð2l1tÞ ð3:49Þ

This is a very interesting result since it allows for simple calculations ofdaughter activity when the daughter nuclear state is shorter living thanthe parent one. When the activities of parent and daughter are equal, thesituation is called ideal equilibrium. Figure 3.6 illustrates an example of thedecay of unitary activity of 226Ra and the corresponding growth and decay



of 222Rn versus time, in the alpha decay of 226Ra (T1/2 ¼ 1600 years) to 222Rn(T1/2 ¼ 91.8 hours) (see also Equation 3.68 and Figure 3.8 in Section 3.4.1).

Radioactive decay usually leaves daughter nuclides in an excited state.Excited states are usually short living. These states de-excite emittinggamma radiation. It is such gamma radiations that are mainly utilized inbrachytherapy. Chapter 5 presents brachytherapy-related radionuclides andtheir decay modes. For example, 137Cs is beta minus radioactive, decaying tothe stable nuclide 137Ba (see also Equation 3.76 and Figure 3.10 in Section3.4.2.1). However, this decay does not lead necessarily to the ground state of137Ba, but also to an excited state from which the 662 keV gamma radiationuseful to brachytherapy arises. In computing how gamma radiation outputof a 137Cs brachytherapy source changes with time, the decay constant of theparent nuclide 137Cs is used. In this example the half-time of 137Cs isapproximately 30 years. That means that in 30 years not only will the activityof a 137Cs source drop to half of its initial value, but also its gamma radiationoutput.

3.3.3 Nuclear Reactions

In a nuclear reaction, two nuclei or a nucleon and a nucleus come together atsuch close approach (of the order of 10215 m) that they interact through thestrong force. A nuclear reaction is accompanied by a redistribution of energyand momentum between both particles and this may lead to the formation

1

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0.8

0.7

0.6

Rel

ativ

e ac

tivity

, A(t

)

0.5

0.4

0.3

0.2

0.1

00 10 20 30 40

t (days)

222Rn

226Ra

50 60 70 80

FIGURE 3.6Relative activity plotted versus time for 226Ra and 222Rn.



of new particles. There are many different types of nuclear reactions.Depending on the particles responsible for these reactions they are usuallyclassified as neutron-induced reactions, reactions induced by chargedparticles or even by gamma radiation. The latter are associated withelectromagnetic interaction, but are usually referred to as nuclear since theinteraction takes place in the vicinity of the nucleus and results in itstransformation. Radioactive decays are also considered as nuclear reactions.

The most commonly encountered type of nuclear reaction involves a lightparticle a and a nucleus A, resulting in the formation of a light particle b anda nucleus B. This is called an (a,b) reaction and can be written in the followinggeneral form:

aþ A ! bþ B, Aða; bÞB ð3:50Þ

Suppose an incident particle a of rest mass ma and kinetic energy Ka is incollision with a target nucleus A of rest mass mA and kinetic energy KA ¼ 0.After the collision, the particle b has rest mass mb and kinetic energy Kb whilethe residual nucleus has rest mass mB and kinetic energy KB. Energyconservation for the reaction 3.50 implies that

mac2 þ Ka þmAc2 ¼ mbc2 þ Kb þmBc2 þ KB ð3:51Þ

An important aspect of a nuclear reaction is its energy balance called the Qvalue or reaction energy Q. The Q value for the reaction in Equation 3.50,taking into account Equation 3.51, is defined as

Q ¼ ðma þmAÞc2 2 ðmb 2 mBÞc

2 ¼ Kb þ KB 2 Ka ð3:52Þ

This equation shows that the rest energy balance equals the kinetic energybalance.

If Q . 0, the reaction is called exoergic and it is accompanied by aliberation of kinetic energy at the expense of the rest energy. Radioactivedecays are all exoergic reactions.

If Q , 0, the reaction is called endoergic and involves an increase in therest energy at the expense of the kinetic energy. Endoergic reactions can takeplace only when the incident particle has sufficient kinetic energy.

Finally, Q ¼ 0 corresponds to elastic scattering, denoted as A(a,a)A, inwhich there is no production of new particles and the kinetic energy isconserved.

Radionuclides used for the construction of brachytherapy sources (seeChapter 6) are either natural (e.g., radium 226Ra, see Section 3.1) or artificial.They can be fission products (e.g., 137Cs, 6.15 atoms formed per 100undergoing fission) or they can be obtained as a result of a neutron capturereaction of a stable nuclide, i.e., (n, g) reaction.

When a beam of particles is incident normally upon a thin sheet ofmaterial containing target nuclei, the probability of reaction is proportionalto the number of target nuclei per unit area of the sheet. The proportionalityconstant has the units of area and is called the cross-section s. A cross-section



may be visualized as the effective area a target nucleus presents to theincident particles, for undergoing the reaction; the probability of reaction isjust equal to the probability that the incident particle strikes within theeffective target area. Since the number of target nuclei per unit area is nix,where ni is the number of target per unit volume in the material and x isthe thickness of the sheet, the probability of the reaction occurring is equal tothe product nixs.

Assuming that the beam of particles has a fluence rate Fz

in particlescm22 sec21, the rate at which the reaction proceeds is

reaction rate _R ¼ nixF_s ð3:53Þ

Suppose that a certain radionuclide with decay constant l is produced in anuclear reactor with a constant rate, R

z. The differential equation describing

the production and decay of the radionuclide is

dNðtÞ

dt¼ _R 2 lNðtÞ ð3:54Þ

which solved for the number N gives

NðtÞ ¼_R

l1 2 expð2ltÞ� �

ð3:55Þ

The accumulated activity A(t) at a time t is

AðtÞ ¼ _R 1 2 expð2ltÞ� �

ð3:56Þ

The last equation shows that the accumulated activity A(t) depends onboth the reaction rate _R and the decay constant l through the term[1 2 exp(2lt)] (see also Section 6.1).

A more complicated situation is when the radionuclide of interest is adaughter product of a shorter-living nuclide as is the case of 125I which is thedaughter product of 125Xe. This decay chain and associated T1/2 values are

12554Xe !

16:9 h 12553I !

59:49 d 12552Te ð3:57Þ

Assuming that 125Xe is produced in a nuclear reactor at a constant rate _R,the accumulated activity AXe(t) is given from 3.56:

AXeðtÞ ¼ _R 1 2 expð2lXetÞ� �

ð3:58Þ

The differential equation for 125I is

dNIðtÞ

dt¼ lXeNXeðtÞ2 lINIðtÞ ¼ _R 1 2 expð2lXetÞ

� �2 lINIðtÞ ð3:59Þ

If NI(0) ¼ 0 then

NIðtÞ ¼ _R1 2 expð2lItÞ

lI2

expð2lXetÞ2 expð2lItÞ

lI 2 lXeð3:60Þ



and the accumulated activity AI(t) for a time t, becomes

AIðtÞ ¼ _R 1 2 expð2lItÞ2 lIexpð2lXetÞ2 expð2lItÞ

lI 2 lXe

� �ð3:61Þ

Figure 3.7 presents the increase of accumulated activity per unit reactionrate with time for 125Xe and 125I (see also Chapter 6).

3.4 Modes of Decay

The laws of radioactivity discussed so far, describe the rates at whichunstable nuclear states decay, and have nothing to do with the modes ofdecay themselves as well as the kind, energy, or intensity of emittedradiation. All these properties, including the decay constant, are character-istic for every nuclide and they univocally designate it.

The mode of decay is named according to the kind of radiation emitted, forwhich, historically, the Greek alphabet is used, i.e., a decay, b decay, and g

decay. The strong nuclear force is responsible for the alpha decay, the weaknuclear force for beta decay, and the electromagnetic for gamma decay. Allthese decay modes are spontaneous and the energy liberated, disintegrationenergy or Q-value (see Section 3.3.3), is distributed as kinetic energy among

1

0.9

0.8

0.7

0.6

0.5

Acc

umul

ated

act

ivity

per

uni

t rea

ctio

n ra

te

0.4

0.3

0.2

0.1

00 5 10 15 20 25

t (days)

30 35 40 45 50

125Xe

125I

FIGURE 3.7A plot of accumulated activity per unit reaction rate vs. time for 125Xe (T1/2 ¼ 16.9 h) and 125I(T1/2 ¼ 59.49d).



the decay products, i.e., the residual nucleus and the emitted particles.Conservation of energy and momentum specifies how this energy isdistributed, conservation of charge, nucleon, and lepton numbers specifythe particles emitted, and other conservation laws (parity, isospin) imposecertain restrictions on allowed transitions. These, however, are beyond thescope of this book, which focuses on the essential.

3.4.1 Alpha Decay

The peak of the mean binding energy per nucleon (B/A vs. A curve inFigure 3.4) at A , 60, indicates that it is advantageous for a heavy nucleus tosplit into two smaller nuclei, which together have a greater net bindingenergy. There are two processes of spontaneous heavy nuclei splitting,namely a decay and fission. In a decay, which is the principal modeof nuclear splitting, a heavy nucleus splits into a light 2

4He nucleus (thea-particle) and another heavy nucleus. 2

4He has the largest binding energyamong all light nuclides: B ¼ 28.29 MeV. In spontaneous fission, the nucleussplits into two more or less equal nuclei.

In a decay a parent nuclide, ZAX, decays spontaneously to the daughter

nuclide, Z22A24Y, and an a-particle is emitted, while energy Qa is liberated:

ZAX !Z22

A24Yþ aþQa ð3:62Þ

Conservation of energy demands:

mXc2 þ KX ¼ mYc2 þ KY þmac2 þ Ka ð3:63Þ

where rest masses are denoted by m and kinetic energies by K. Rewriting thelast equation and given zero kinetic energy of the parent nucleus (KX ¼ 0),yields:

Qa ; mXc2 2 mYc2 2 mac2 ¼ KY þ Ka ð3:64Þ

The last equation shows that the rest energy balance equals the kineticenergy balance. The disintegration energy Qa, as the rest energy balance(left side of Equation 3.64), can be accurately calculated from the knownmasses. Atomic masses can be readily used in these calculations instead ofcalculating the nuclear masses since the number of electrons, and henceelectron masses, cancels out as seen in Equation 3.62, and moreover, there isgood approximation to a balance of the electrons’ binding energies since themost tightly bound, K electrons practically remain unchanged.

The disintegration energy Qa, is distributed as kinetic energy of thea-particle and recoil energy of the daughter nucleus. Conservation ofmomentum in reaction 3.62 demands that the momentum pa of the a-particleis equal in magnitude to the momentum pY of the daughter nucleus, since theparent nucleus spontaneously decays at rest:

pa ¼ pY ð3:65Þ



For the energies considered herein, both the a-particle and daughternucleus can be treated as nonrelativistic ( p 2 ¼ 2 mK). Thus, Equation 3.65gives

Ka

KY

¼mY

ma

<A 2 4

4ð3:66Þ

where the mass ratio is approximated by mass number ratio as inEquation 3.22. From Equation 3.64 and Equation 3.66 the kinetic energy Ka

of the emitted a-particle and the recoil energy KY of the daughter nucleus areobtained:

Ka < Qa 1 24

A

� �and KY < Qa

4

Að3:67Þ

That is, the emitted a-particles have discrete (kinetic) energy whichconstitutes a considerable part of the disintegration energy Qa.

In the above calculations the daughter nucleus is assumed to be created inits ground state. This is not however mandatory; the daughter nucleus canbe left in an excited state. If this excited state has (excitation) energy E p abovethe ground state, the rest energy of the daughter nucleus is less than thisvalue, and consequently, the disintegration energy Qa in Equation 3.67 has tobe replaced by Qa 2 E p.

A typical example of an alpha decay is that of 226Ra (half-life T1/2 ¼ 1600years):

22688 Ra ! 222

86 Rnþ aþ 4:8706 MeV ð3:68Þ

The atomic masses are: mRa ¼ 226.025403 u, mRn ¼ 222.017571 u, andma ¼ 4.0026032 u yielding (according to Equation 3.64 and Equation 3.21)disintegration energy Qa ¼ 4.8706 MeV. This disintegration energy isdistributed as in Equation 3.67, resulting in Ka ¼ 4.7843 MeV andKRn ¼ 0.0863 MeV.

The above disintegration scheme occurs with 94.5% probability. In theremaining 5.5% of cases, the radon daughter nucleus is in an excited state( 86222Rnp) and decays to its ground state by emitting g-rays of 0.1862 MeV. In

this case, the kinetic energy of the emitted a-particle is (4.7843 2 0.1862)MeV ¼ 4.60 MeV. Figure 3.8 summarizes the simplified decay scheme of 226Ra.

3.4.2 Beta Decay

The semiempirical mass formula presented in Equation 3.31 reveals thatwhen nuclear masses m(Z,A) for a given mass number A are plotted vs. theatomic number Z, they form a parabola when A is odd or two parabolasdisplaced in mass by 2 £ aPairingdA 23/4 when A is even. The moststable nuclides lie at the base of the parabola. All nuclides belong toa corresponding A-value parabola. All these parabolas form a three-dimensional presentation of the two-dimensional Segre plot presented in



Figure 3.3 with nuclear mass as the third dimension and warranting thename stability valley.

For example, the odd-A parabolas for the isobaric families of A ¼ 125 andA ¼ 137 are presented in Figure 3.9. For A ¼ 125, only 125Te, which lies closeto the parabola’s minimum, is stable. All other members of the isobaric

124.95

124.94

124.93

125Ag

A = 125 A = 137

β–

β+

β–β+

125Cd

125In

125Sn

125Sb125Te

125I

125Xe

125Cs

125Ba

125La

125Ce

125Pr

137Gd

137Eu

137Sm

137Pm

137Nd

137Pr137Ce

137La137Ba

137Cs

137Xe

137I

137Te

137Sb

137Sn

124.92

Ato

mic

Mas

s (u

)

124.91

124.90

136.95

136.94

136.93

136.92

136.91

136.90 50 55

Atomic Number, Z60 50 55

Atomic Number, Z60 65

FIGURE 3.9A plot of the atomic mass of the isobaric families with A ¼ 125 and A ¼ 137 vs. atomic number Z

(i.e., the parabola on which these two odd-A isobaric families lie).

Qα=4.8706 Mev

α2 = 4.601 Mev (5.5%)

α1=4.7843 Mev (94.5%)

22688Ra

22286 Rn*

22286 Rn

T1/2 = 1602 years

γ = 0.1862 MeV

FIGURE 3.8The decay scheme of 226Ra.



family are radioactive, and by successive decays they finally reach the stablenuclide 125Te. For A ¼ 137, the stable nuclide is 137Ba. The mode of decaybetween isobars is named b decay from the Greek letter b (beta); weaknuclear force is responsible for b decay. There are three types of b decay:b2 (beta minus), bþ (beta plus), and electron capture. Members of theisobaric family that lie on the left arm of the parabola (Z , Zminimum), asthose presented in Figure 3.9, are b2 radioactive while members on the right-arm (Z . Zminimum) are bþ radioactive and/or decay by EC.

For even A, the two existing parabolas correspond to one with even Z andeven N (the lower of the two) and one with odd Z and odd N. In this case, thenumber of stable nuclides of an isobaric family could be one, two or eventhree in rare cases. An example of an isobaric family with two stable nuclidesis that with A ¼ 192 (the two stable nuclides are 192Pt and 192Os). 192Ir, whichis the most commonly used nuclide in current brachytherapy practice lies onthe upper parabola (since both Z and N are odd) and decays with either b2

to 192Pt (95%) or to 192Os by EC (5%) (see Chapter 5).

3.4.2.1 b2 Decay

In b2 decay a parent nuclide, ZAX, decays spontaneously to the daughter

nuclide, Zþ1AY, a b2-particle and an antineutrino, ne are emitted and energy

Qb2 is liberated:

AZX !Zþ1

AYþ b2 þ ne þQb2 ð3:69Þ

The b2-particle is an electron e2 whose presence in the reaction satisfiescharge conservation (since nucleons are conserved and there is a protonexcess in the right side of the reaction). The antineutrino ne has zero mass (orvery small), has no charge, and its presence in the reaction is to ensure leptonconservation.

Conservation of energy in reaction 3.69 demands:

mXc2 þ KX ¼ mYc2 þ KY þmec2 þ Kb2 þ Kn ð3:70Þ

where rest masses are denoted by m (excluding the massless ne) and kineticenergies by K (ne does have kinetic energy). Rewriting the last equation, andsince the kinetic energy of the parent nucleus is KX ¼ 0, gives

Qb2 ; mXc2 2 mYc2 2 mec2 ¼ KY þ Kb2 þ Kn ð3:71Þ

The masses mX and mY in Equation 3.71 are nuclear. In terms of atomicmasses MX and MY, if Equation 3.23 is recalled the right side of Equation 3.71gives

Qb2 ; MXc2 2 MYc2 ð3:72Þ

This is the energy condition for b2 decay; the atomic mass of parentnuclide must be greater than the atomic mass of daughter nuclide, sinceQb2 . 0 for a spontaneous decay.



Equation 3.71 shows that the disintegration energy, Qb2, is distributed askinetic energy among three particles: the daughter nucleus, the b2 particle,and the antineutrino ne. Since there is only one additional restriction due tomomentum conservation:

~pX ¼ 0 ¼ ~pY þ ~pb2 þ ~pn ð3:73Þ

there is no unique way of distributing the disintegration energy. Only aminute proportion of this energy can be carried away by the daughternucleus due to its huge mass relative to the electron-antineutrino pair.Therefore, almost all the energy goes to the lepton pair, the electron and theantineutrino, which present continuous and not discrete energy spectrum.Thus, the b2-particle can have any kinetic energy up to the maximumpossible value of Qb2, that is

Kb2;max ¼ Qb2 ; MXc22 MYc2 ð3:74Þ

It is customary to tabulate this maximum kinetic energy Kb2,max of theb2-particle (see for example Figure 3.2).

The simplest example of b2 decay is the decay of free neutron (seeFigure 3.2):

n ! pþ b2 þ �ne þ 0:782 MeV ð3:75Þ

The free neutron is not stable. It has a lifetime of T1/2 < 10.4 min!A typical example of b2 decay is that of 55

137Cs (half-life T1/2 ¼ 30.07 years)to 56

137Ba (see corresponding isobaric parabola in Figure 3.9):

13755 Cs ! 137

56 Baþ b2 þ ne þ 1:1756 MeV ð3:76Þ

Substituting the atomic masses of 137Cs (136.9070835 u) and 137Ba(136.9058214 u) into Equation 3.72 and accounting for the conversion factorof Equation 3.21, yields the disintegration energy of Qb2 ¼ 1.1756 MeV andhence Kb2,max ¼ 1.1756 MeV.

The above disintegration scheme occurs with 5.6% probability. In the other94.4% of cases the daughter nucleus is in an excited state (56

137Bap) and furtherdecays to its ground state by emitting g-rays of 0.6617 MeV. In this case, themaximum kinetic energy of the emitted b2-particle is (1.1756 2 0.6617)MeV ¼ 0.514 MeV. Figure 3.10 summarizes the decay scheme of 137Cs.

3.4.2.2 b1 Decay

In bþ decay a parent nuclide, ZAX, decays spontaneously to the daughter

nuclide Z21AY, a bþ-particle and a neutrino, ne are emitted, while energy Qbþ

is liberated:

AZX !Z21

AYþ bþ þ ne þQbþ ð3:77Þ

Conservation of energy in reaction 3.77 demands that

mXc2 þ KX ¼ mYc2 þ KY þmec2 þ Kbþ þ Kn ð3:78Þ



where rest masses are denoted by m (excluding the massless ne) and kineticenergies by K (ne does have kinetic energy). Rewriting the last equation,given that the kinetic energy of the parent nucleus is KX ¼ 0, gives

Qbþ ; mXc2 2 mYc2 2 mec2 ¼ KY þ Kbþ þ Kn ð3:79Þ

In terms of atomic masses MX and MY, if Equation 3.23 is recalled the rightside of Equation 3.79 gives

Qbþ ; MXc2 2 MYc2 2 2mec2 ð3:80Þ

This is the energy condition for bþ decay; the atomic mass of the parentnuclide must be greater by at least two times the electrons’ rest energy thanthe atomic mass of the daughter nuclide, since Qbþ . 0 for a spontaneousdecay. That is

MXc2 2 MYc2 . 2mec2 < 1:02 MeV ð3:81Þ

As for b2 decay, the disintegration energy Qbþ is practically distributedbetween the bþ-particle and the neutrino ne, both of which present acontinuous energy spectrum. Thus, the bþ-particle can have any kineticenergy up to a maximum value of Qbþ. That is

Kbþ;max ¼ Qbþ ; MXc2 2 MYc2 2 1:02 MeV ð3:82Þ

It is customary to tabulate this maximum kinetic energy Kbþ,max of thebþ-particle.

A typical example of bþ decay is that of 22Na (half-life of T1/2 ¼ 2.609years) to 22Ne (0.06% to the ground state of 22Ne and 89.84% to an excited

β1−, Eβ-,max = 0.514 MeV(94.4%)

β2−, Eβ-,max = 1.1756 MeV (5.6%)

Eγ = 0.6617 MeV

Qβ-=1.1756 MeV 137 Cs55

137 Ba *56

137 Ba56

T1/2=30.07 years

FIGURE 3.10The b2 decay scheme of 137Cs.



state of 22Nep with excitation energy E p ¼ 1.2746 MeV):

2211Na ! 22

10Neþ bþ þ ne þ 1:82027 MeV ð3:83Þ

Substituting the atomic masses of 22Na (21.9944368 u) and 22Ne(21.9913855) into Equation 3.82 and accounting for the conversion factorof Equation 3.21 yields the disintegration energy of Qbþ ¼ 1.82027 MeV andhence Kbþ,max ¼ 1.82027 MeV for decays to the ground state of 22Ne andQbþ ¼ 0.546 MeV and hence Kbþ,max ¼ 0.546 MeV for decays leading to thefirst excited state of 22Ne. Figure 3.11 summarizes the bþ decay schemeof 22Na (see also Figure 3.12).

3.4.2.3 Electron Capture

Electron capture (EC) is competitive to bþ decay. In EC, a parent nucleus, ZAX,

captures an electron from its own atomic electron shells and decays sponta-neously to the daughter nucleus Z21

AY, a neutrino, ne is emitted, while energyQEC is liberated:

AZX þ e2 ! A

Z21Yþ ne þQEC ð3:84Þ

Conservation of energy in reaction 3.84 demands that

mXc2 þ KX þmec2 þ Ke ¼ mYc2 þ KY þ Kn ð3:85Þ

where rest masses are denoted by m (excluding the massless ne) and kineticenergies by K (ne does have kinetic energy). Rewriting the last equation andignoring kinetic energies of both the parent nucleus and the atomic electron,

β2+, Eβ+,max = 1.820 MeV (0.06%)

22Na11

22Ne*10

22Ne10

β1+, Eβ+,max = 0.546 MeV (89.84%)

Qβ+=1.82027 MeV

T1/2 = 2.609 years

Eγ = 1.2746 MeV

FIGURE 3.11The bþ decay scheme of 22Na (see also Figure 3.12).



gives

QEC ; mXc2 þmec2 2 mYc2 ¼ KY þ Kn ð3:86Þ

The masses mX and mY appearing in 3.86 are nuclear. In terms of atomicmasses MX and MY, if Equation 3.23 is recalled the right side of Equation 3.86gives

QEC ; MXc2 2 MYc2 ð3:87Þ

This is the energy condition for electron capture; the atomic mass of theparent nuclide must be greater than the atomic mass of the daughter nuclide,since QEC . 0 for a spontaneous decay. That is

MXc2 2 MYc2 . 0 or MX . MY ð3:88Þ

The latter equation, in view of the energy condition for bþ decay ofEquation 3.81, implies that besides being competitive to bþ decay, ECprovides an alternative path of decay between two neighboring isobar nucleilying on the right arm of the isobar parabola and presenting a positive,yet less than 2mec 2 < 1.02 MeV atomic rest energy difference.

A characteristic example (see Figure 3.9) is that of 125I (atomicmass ¼ 124.9046242 u) which decays to 125Te (atomic mass = 124.9044247 u).In this example, the atomic rest energy difference is 0.186 MeV which fulfilscondition 3.88 for EC but not that for bþ decay in Equation 3.81. Therefore, asdiscussed in Chapter 5, 125I decays to 125Te exclusively by EC.

QEC =2.84227 MeV

EC (10.1%)

T1/2 = 2.609 years

Eγ = 1.2746 MeV

22Na11

22Ne*10

22Ne10

FIGURE 3.12The EC decay scheme of 22Na.



Equation 3.86 shows that the disintegration energy QEC, is distributed askinetic energy between two particles: the daughter nucleus A

Z21Y and theneutrino ne. Owing to momentum conservation:

~pY þ ~pn ¼ 0 ð3:89Þ

that is, the recoil momentum pY of the daughter nucleus is equal inmagnitude to the neutrinos’ momentum pn. Thus, the neutrino carries awayalmost the entire disintegration energy QEC in the form of kinetic energy. Ittherefore has a discrete and not a continuous energy spectrum:

Kn ¼ QEC ; MXc2 2 MYc2 ð3:90Þ

Given the high accuracy in QEC calculations from atomic masses,measurement of the daughter nucleus recoil energy was used by Davis(1952) to provide an estimate of the upper limit for the neutrino mass.

The captured electron is an inner-shell electron, usually from the K-shell(the electron capture process is alternatively named as K capture). Thevacancy created is filled by outer electrons falling into it and the process isaccompanied by emission of fluorescent radiation or Auger electronemission. This has significant impact on the dosimetry of low-energyemitters such as 125I and 103Pd (see Chapter 4 and Chapter 5).

The bþ decay of 22Na discussed in the previous section and presentedin Figure 3.11, reveals that there is an alternative 10.1% possibility for EC(bþ decay appears with a frequency of 89.9%). The rest energy difference is

QEC ; M22Nac22 M22Nec2 ¼ 2:842268 MeV ð3:91Þ

Figure 3.12 summarizes the EC decay for 22Na.

3.4.3 Gamma Decay and Internal Conversion

Alpha and beta decays discussed in previous sections (Section 3.4.1 andSection 3.4.2) reveal that the daughter nucleus in such decays is usuallycreated in an excited state. These states are usually short living and de-excitespontaneously emitting electromagnetic radiation named g-rays from theGreek letter g (gamma); the decay is called g decay. There are single transi-tions when a nucleus emits a single g-ray and at once falls to the ground state(see Figure 3.8 and Figure 3.10 through Figure 3.12) or cascade transitionswhen excitation is removed by a successive emission of several g-rays(see for example the decay scheme of 60Co and 192Ir in Chapter 5). Theenergy, Eg, of the g-rays is determined by the difference DE in the nuclearrest energy of the two levels involved:

DE ¼ Einitial 2 Efinal ¼ Eg þ Krecoil ð3:92Þ

where Krecoil is the kinetic energy of the recoil nucleus. Since the momentumof a g-ray (Eg/c) should be equal in magnitude to the recoil momentum due



to momentum conservation, the recoil energy Krecoil is

Krecoil ¼E2g

2mc2<ðDEÞ2

2mc2ð3:93Þ

Since DE is in the energy range of 10 keV to 5 Mev while the remainingenergy of the atom accounts for many GeV, the recoil energy is in the eVrange. Thus, the g-ray carries away an overwhelming part of the nuclearexcitation energy.

In addition to the g-ray emission, there is another mechanism by which theexcited daughter nucleus may de-excite; by transferring its energy excessdirectly to an inner orbital electron. The electron is then ejected by the atomwith a kinetic energy equal to the difference of the nucleus excess energy DEand the binding energy of the involved electron. This process is calledinternal conversion (IC) and the ejected electron is called an internalconversion electron. Although 137Cs decays to the excited state of 137Ba witha probability of 94.4% (see Figure 3.10) the 0.6627 MeV g-ray is emitted witha probability of 85.9% and not 94.4%. The remaining probability correspondsto de-excitation of the daughter nuclide by IC.

Both internal conversion electrons and g-rays present discrete line spectrawhich are characteristic for each nuclide. As in the case of the electroncapture process, IC is accompanied by atomic characteristic fluorescentradiation and Auger electrons. Obviously, neither gamma decay nor internalconversion change the atomic number Z, the mass number A, and theneutron number N of the nucleus.

3.5 Elementary Particles and the Standard Model

In the history of physics, during the last 85 years, we have witnessedtremendous progress in the area of atomic and subatomic physics which hashad a major impact on the frontier of technological advancement. In 1920,the laws of nature governing the behavior of the smallest matter entity, atthat time, the atom, started to be understood with the establishment ofquantum mechanics, followed by a breakthrough almost 50 years later, in1969, with paramount discoveries that established the Standard Model,proposed by Weinberg, Salam, and Glashow, as a true theory of nature.

With the advent of quantum mechanics we were able to interpret, in aprobabilistic way, the motion of small-sized objects, such as that of an atomfor instance, whose size is of the order of 10210 m or almost ten billion timessmaller than the height of a human being. At this microscopic scale the lawsof nature are not the same as those governing the motion of macroscopicobjects. We are not in a position to know the trajectory of such a small-sizedobject, of atomic or subatomic scale, merely because we cannot measure itsposition and its velocity simultaneously. This is actually the content of the



uncertainty principle (see Section 3.1.2), an inherent property of nature. Weare therefore unable to know the position and velocity of a microscopicparticle but we are in position to speak of the probability to find it here orthere. This probability is expressed through a function, known as wavefunction, which satisfies a certain equation, known as the Schroedingerequation, which is the fundamental law in quantum mechanics. The motionand interaction of atoms, the formation of molecules and chemical bonds areall explained with the rules of quantum mechanics. Knowing the laws ofnature at this microscopic level we have been able to understand themechanisms responsible for the behavior of matter at small scales, which hasbeen of paramount importance for technological developments.

The construction of new materials, for instance in contemporary timesthe silicon chip which is the heart of any computer machine, as well asother inventions used in modern medicine, such as MRI, positron emissiontomography, and other diagnostic tools, are based on the knowledge of ourmicrocosmos.

The present knowledge of the microcosm is based on research, conductedfrom 1930 to 1970, of a physics theory that completes quantum mechanics,in the sense that it can explain phenomena that quantum mechanics wasunable to interpret, and that occurs at a smaller scale, much smaller than thatof an atom or a nucleus. The Standard Model, which is the new landmark,was established in 1969 and it succeeds in explaining with unprecedentedaccuracy, all phenomena occurring at distances 10218 m, or equivalently onehundred million times smaller than the dimensions of an atom! But how canwe see at such small distances? The physics laws and the validity of theproposed theories can be tested with large accelerator machines, running atvery high energies, which are the eyes of the particle physicists, capable ofprobing such tiny distances.

In order to conceive the magnitude of such machines, the Large HadronCollider (LHC) accelerator being built at CERN (Switzerland), which willstart working in 2007, is a circular accelerator with a circumference of about27 km. In it, protons will be accelerated to total energies reaching 14 TeV(14 thousand billions eV!). With this accelerator we will be able to go deeperand probe at even smaller distances.

Perhaps one of the spectacular discoveries of past accelerators was the factthat protons and neutrons, the ingredients of the nuclei of the atoms, are notfundamental but are composed of other particles which are named quarks.The atom consists of electrons and a nucleus, the nucleus consists of protonsand neutrons, which are bound together by a sort of force known as thestrong force which must be stronger than the electromagnetic force toovercome the repulsion of protons within the nuclei due to electric forces.

The major discovery that protons, neutrons, as well as other particlesdiscovered since 1930 (which collectively we call hadrons), are notfundamental, had been theoretically founded and was experimentallyverified and it is at the heart of the Standard Model. According to it, thefundamental building blocks of matter are the leptons and the quarks. There



are six leptons and one of them is the well-known electron, one of theingredients of the atom. Quarks are the ingredients of protons, neutrons, andin general, of all hadrons. These fundamental particles carry spin equal to1/2 and they are grouped in three families which have exactly the samecharacteristics, such as electric charges, spins, and other quantum numbers,but different masses. These families are also called generations. The firstfamily, or generation, accommodates two leptons, the electron, e2, and itsneutrino, ne, and six quarks denoted by ua, da existing in three differentspecies, or colors, labeled by a ¼ 1, 2, 3. These quarks are called up anddown, respectively. Their names reflect the way they are grouped inmathematical entities used to describe the theory and will not concern ushere. The electric charge of the neutrino is zero and for this reason it does notinteract electromagnetically, that is it does not interact with other chargedparticles. Its mass is very small, almost vanishing, being therefore very fastalmost as fast as light. The only sort of interaction neutrinos can have is afeeble interaction, known as weak, which is responsible for the radioactivityof some particles or nuclei.

The neutron, for instance is unstable due to this kind of interaction and itdecays into a proton, an electron, and a neutrino (see Section 3.4.2.1). Thisprocess is known as b2 decay. Owing to the fact that neutrinos interact onlyweakly they can penetrate the Earth and travel galactic distances withoutbeing captured by matter, carrying information from remote parts of ourUniverse. Recently, great efforts have been made towards building neutrinotelescopes, which can see and study the properties of such cosmic neutrinosarriving at Earth, that could possibly reveal information concerning thecreation of our Universe.

Unlike neutrinos, the electric charge of the electron is not zero; it is actuallyjust opposite to that of the proton which conventionally is taken equal toþ1.Then, the charges of the ua, da quarks are fractional and equal to 2/3 and21/3 times the proton’s charge, respectively. The proton consists of threequarks, namely two up quarks u and one down quark d, whose total chargeis 2 £ 2/3 2 1/3 ¼ 1, the charge of the proton. Similarly, the neutron consistsof two d quarks and one u and their total charge is zero, which is actually thecharge of the neutron. The color a ¼ 1, 2, 3, labeling the quarks, is a quantumnumber analogous to the electric charge. In order for protons, neutrons, aswell as other hadrons to be composed of quarks, the latter have to be boundor glued together by some sort of force. This force feels not the electric chargebut the color instead and the part of the theory describing this kind ofinteraction is called quantum chromodynamics or QCD for short.

For every one of the three generations of the fundamental fermions therecorresponds a generation that includes their antiparticles. For instance, forthe first generation, which accommodates the electron e2, there is ageneration which includes its antiparticle eþ, the so-called positron havingopposite electric charge but exactly the same mass and spin, and also theantiparticles of the neutrino and up and down quarks. The leptons andquarks of the three generations as well as their antiparticles are shown



in Table 3.1 and Figure 3.13. Detailed information can be found in the reviewof particle properties.3 Anything we know in nature consists of thesefundamental building blocks accommodated in the three families of quarksand leptons. There is no experimental evidence, up to the time of the writing,for the existence of any other form of matter although at the theoretical levelthere are proposals advocating the existence of new forms of matterundiscovered in the laboratory as yet. New experiments scheduled to run inthe near future, starting with the LHC in 2007, will test some of thesetheoretical proposals and may discover new massive ingredients providingvital information to particle physics. It is worth noting that some of theseingredients are eagerly awaited in order to explain the missing mass of theUniverse, the so-called dark matter, which is one of the biggest mysteries ofmodern cosmology.

Except for the fundamental fermions accomodated in the three gener-ations the Standard Model also includes the carriers of the forces. With theexception of gravity, whose energy here plays little role, there are three

TABLE 3.1

Leptons, Quarks and Their Antiparticles

Leptons Quarks

Particle Name Symbol Antiparticle Particle Name Symbol Antiquark

Electron e 2 eþ Up u uNeutrino (e) ne ne Down d dMuon m 2 m þ Strange s sNeutrino (m) nm nm Charmed c cTau t 2 t þ Bottom b bNeutrino (t) nt nt Top t t

FIGURE 3.13

Quarks and Leptons (left side) and the Force Carriers (right side).



forces: the strong, the electromagnetic, and the weak. The strong force isresponsible for the binding of the protons and neutrons to form nuclei. Theelectromagnetic force is the one felt by charged matter, which at the classicallevel and for two charged particles of charges q1, q2, is given by the well-known Coulomb’s law F ¼ q1q2/r 2. The weak force, as well as the strong, isnot manifested as a classical force, and is responsible for a variety ofprocesses occurring in nuclear physics, among these being the b decaydiscussed earlier (see Section 3.4.2). Concerning the strength of these forces,if that of the strong force is taken equal to 1, the electromagnetic is roughly0.01, and that of the weak even smaller, 0.00001. The gravitational force is leftout of our discussion since it is imperceptibly small, 1040 times smaller thanthe strong force. Particles interact with each other because they exchange thecarriers, or mediators, of the corresponding force. For instance, twoelectrically charged particles interact with each other since they exchangea photon which is the carrier of the electromagnetic force. In order toconceive of this exchange mechanism, imagine two people ice-skating andone throwing a ball to the other when they approach at a close distance. Thetrajectory of the skater who throws the ball will change since the ball leavinghis hands carries momentum. The trajectory of the other skater who catchesthe ball will change as well for the same reason. This is how the interactiontakes place. The trajectories of the skaters change as a result of the exchangeof the ball which is the carrier, or mediator, of the interaction in this example.The carrier of the electromagnetic force is called a photon, it is usuallydenoted by the symbol g, and it has zero mass! The lack of mass is intimatelyconnected with the fact that the electromagnetic force is long range followingthe classical law F , 1/r 2. It should be noted that the light we see is a bunchof an enormous number of photons! In Figure 3.14, two electrons aredepicted interacting electromagnetically by exchanging a photon (wavyline). The weak force has three mediators named W þ, W 2, Z, which unlikethe photon are quite massive, each weighing roughly 100 times the mass ofthe proton. Owing to this, the weak force is of short range and cannotmanifest itself as a classical force. The carriers of the strong (QCD) forces areeight massless particles called gluons which will be collectively denoted bythe symbol g. Although they have zero mass, as the photon does, they do notinduce long-range forces, due to a special property inherent in the dynamics

e− e−

FIGURE 3.14Two electrons interact by exchanging a photon.



of the QCD, which prohibits the propagation of the force at large distances.All of these carriers have spin equal to 1, and accordingly in the terminologyof particle physics they are called gauge bosons. In Figure 3.13 all quarksand leptons are displayed along with the mediators of the three forces.

The Standard Model explains perfectly well all particle interactions to anunprecedented accuracy, as laboratory experiments have shown, forenergies less than about 200 GeV, that is two hundred billion eV, equivalentto distances 10218 m. However, there is still a missing link, escapingdetection as yet, which is a fundamental particle needed for the correctmathematical description of the theory. From the physics point of view, thisparticle, which bears the name Higgs boson, is spinless and electricallyneutral, and its presence is theoretically needed in order for the leptons andquarks, and also the carriers of the weak interactions W, Z, to acquire masses.Thus, its role is extremely important and searching for it will be one of theprimary tasks in the new accelerators that will run in the near future.The new accelerators will probe at distances smaller than 10218 m and willbe able to provide us with new information concerning the physics laws thatgovern the interactions of particles at smaller scales. Theoretical proposalspredict that when we probe more deeply, new species of particles will bediscovered which at present escape detection because of lack of energy. Withnew accelerators, of sufficiently high energy, these new degrees of freedomwill be produced in the laboratory, provided they exist, and their detectionwill open a new era in particle physics.

References

1. National Institute of Standards and Technology Physics Laboratory PhysicalReference Data Electronic Version, available online at http://physics.nist.gov/PhysRefData (Gaithersburg: National Institute of Standards and Technology),August 2004.

2. Nuclear Data Evaluation La, Table of Nuclides (http://atom.kaeri.re.kr). KoreaAtomic Energy Research Institute, 2000.

3. The Review of Particle Physics by Particle Data Group (PDG). Available online athttp://pdg.lbl.gov/



http://physics.nist.gov/

http://atom.kaeri.re.kr

http://pdg.lbl.gov/

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