Radiation Oncology Physics a handbook for teachers

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Radiation Oncology Physics: A Handbook for Teachers and Students E.B. Podgorsak Technical Editor Sponsored by the IAEA and endorsed by the COMP/CCPM, EFOMP, ESTRO, IOMP, PAHO and WHOChapter 1 BASIC RADIATION PHYSICS E.B. PODGORSAK Department of Medical Physics, McGill University Health Centre, Montreal, Quebec, Canada 1.1. INTRODUCTION 1.1.1. Fundamental physical constants (rounded off to four significant figures) 23 ● Avogadro’s number: N = 6.022 × 10 atoms/g-atom. A 23 ● Avogadro’s number: N = 6.022 × 10 molecules/g-mole. A 8 ● Speed of light in vacuum: c = 299 792 458 m/s (ª3 × 10 m/s). –19 ● Electron charge: e = 1.602 × 10 C. 2 ● Electron rest mass: m – = 0.5110 MeV/c . e 2 ● Positron rest mass: m + = 0.5110 MeV/c . e 2 ● Proton rest mass: m = 938.3 MeV/c . p 2 ● Neutron rest mass: m = 939.6 MeV/c . n 2 ● Atomic mass unit: u = 931.5 MeV/c . –34 ● Planck’s constant: h = 6.626 × 10 J·s. –12 ● Permittivity of vacuum: e = 8.854 × 10 C/(V·m). 0 –7 ● Permeability of vacuum: m = 4p × 10 (V·s)/(A·m). 0 –11 3 –1 –2 ● Newtonian gravitation constant: G = 6.672 × 10 m ·kg ·s . ● Proton mass/electron mass: m /m = 1836.0. p e 11 ● Specific charge of electron: e/m = 1.758 × 10 C/kg. e 1.1.2. Important derived physical constants and relationships ● Speed of light in a vacuum: 1 8 (1.1) c=ª31 ¥ 0 m/s em 00 1CHAPTER 1 ● Reduced Planck’s constant × speed of light in a vacuum: h = c== c 197.3 MeV◊ fmª 200 MeV◊ fm (1.2) 2p ● Fine structure constant: 2 e 11 a== (1.3) 4pe= c 137 0 ● Bohr radius: 2 == c 4pe () c 0 (1.4) a== = 0.Å 5292 0 22 2 amc e mc e e ● Rydberg energy: 2 2 2 ʈ 1 1 e mc 22 e (1.5) Em== c a= 13.61 eV ReÁ˜ 2 2 24pe ()= c ˯ 0 ● Rydberg constant: 2 22 2 2 ʈ E mc a 1 e mc R e e-1 1 (1.6) R== == 109 737cm •Á˜ 3 24 p== c pp c 44pe ˯ ()= c 0 ● Classical electron radius: 2 e (1.7) r== 2.818 fm e 2 4pe mc 0e ● Compton wavelength of the electron: h (1.8) l== 0.Å 0243 C mc e 2BASIC RADIATION PHYSICS 1.1.3. Physical quantities and units ●Physical quantities are characterized by their numerical value (magnitude) and associated unit. ● Symbols for physical quantities are set in italic type, while symbols for units are set in roman type (e.g. m = 21 kg; E = 15 MeV). ● The numerical value and the unit of a physical quantity must be separated by a space (e.g. 21 kg and not 21kg; 15 MeV and not 15MeV). ● The currently used metric system of units is known as the Système inter- national d’unités (International System of Units), with the international abbreviation SI. The system is founded on base units for seven basic physical quantities: Length l: metre (m). Mass m: kilogram (kg). Time t: second (s). Electric current I: ampere (A). Temperature T: kelvin (K). Amount of substance: mole (mol). Luminous intensity: candela (cd). All other quantities and units are derived from the seven base quantities and units (see Table 1.1). TABLE 1.1. THE BASIC AND SEVERAL DERIVED PHYSICAL QUANTITIES AND THEIR UNITS IN THE INTERNATIONAL SYSTEM OF UNITS AND IN RADIATION PHYSICS Physical Unit Units used in Symbol Conversion quantity in SI radiation physics 9 10 15 Length l m nm, Å, fm 1 m = 10 nm = 10 Å = 10 fm 2 2 –30 Mass m kg MeV/c 1 MeV/c = 1.78 × 10 kg 3 6 9 12 Time t s ms, ms, ns, ps 1 s = 10 ms = 10 ms = 10 ns = 10 ps 3 6 9 Current I A mA, mA, nA, pA 1 A = 10 mA = 10 mA = 10 nA –19 Charge Q C e 1 e = 1.602 × 10 C –2 Force F N 1 N = 1 kg·m·s –1 Momentum p N·s 1 N·s = 1 kg·m·s –19 –3 Energy E J eV, keV, MeV 1 eV = 1.602 × 10 J = 10 keV 3CHAPTER 1 1.1.4. Classification of forces in nature There are four distinct forces observed in the interaction between various types of particle (see Table 1.2). These forces, listed in decreasing order of strength, are the strong force, electromagnetic (EM) force, weak force and –6 –39 gravitational force, with relative strengths of 1, 1/137, 10 and 10 , respectively. 2 ●The ranges of the EM and gravitational forces are infinite (1/r dependence, where r is the separation between two interacting particles); ● The ranges of the strong and weak forces are extremely short (of the order of a few femtometres). Each force results from a particular intrinsic property of the particles, such as: — Strong charge for the strong force transmitted by massless particles called gluons; — Electric charge for the EM force transmitted by photons; 0 — Weak charge for the weak force transmitted by particles called W and Z ; —Energy for the gravitational force transmitted by hypothetical particles called gravitons. 1.1.5. Classification of fundamental particles Two classes of fundamental particle are known: quarks and leptons. ● Quarks are particles that exhibit strong interactions. They are constit- uents of hadrons (protons and neutrons) with a fractional electric charge (2/3 or –1/3) and are characterized by one of three types of strong charge called colour: red, blue and green. There are six known quarks: up, down, strange, charm, top and bottom. TABLE 1.2. THE FOUR FUNDAMENTAL FORCES IN NATURE Force Source Transmitted particle Relative strength Strong Strong charge Gluon 1 EM Electric charge Photon 1/137 0 –6 Weak Weak charge W and Z 10 –39 Gravitational Energy Graviton 10 4BASIC RADIATION PHYSICS ● Leptons are particles that do not interact strongly. Electrons (e), muons (m), taus (t) and their corresponding neutrinos (n , n , n ) are in this e m t category. 1.1.6. Classification of radiation As shown in Fig. 1.1, radiation is classified into two main categories, non- ionizing and ionizing, depending on its ability to ionize matter. The ionization potential of atoms (i.e. the minimum energy required to ionize an atom) ranges from a few electronvolts for alkali elements to 24.5 eV for helium (noble gas). ● Non-ionizing radiation (cannot ionize matter). ● Ionizing radiation (can ionize matter either directly or indirectly): —Directly ionizing radiation (charged particles): electrons, protons, a particles and heavy ions. —Indirectly ionizing radiation (neutral particles): photons (X rays and g rays), neutrons. Directly ionizing radiation deposits energy in the medium through direct Coulomb interactions between the directly ionizing charged particle and orbital electrons of atoms in the medium. Indirectly ionizing radiation (photons or neutrons) deposits energy in the medium through a two step process: ● In the first step a charged particle is released in the medium (photons release electrons or positrons, neutrons release protons or heavier ions); ● In the second step the released charged particles deposit energy to the medium through direct Coulomb interactions with orbital electrons of the atoms in the medium. Non-ionizing Radiation Directly ionizing (charged particles) electrons, protons, etc. Ionizing Indirectly ionizing (neutral particles) photons, neutrons FIG. 1.1. Classification of radiation. 5CHAPTER 1 Both directly and indirectly ionizing radiations are used in the treatment of disease, mainly but not exclusively for malignant disease. The branch of medicine that uses radiation in the treatment of disease is called radiotherapy, therapeutic radiology or radiation oncology. Diagnostic radiology and nuclear medicine are branches of medicine that use ionizing radiation in the diagnosis of disease. 1.1.7. Classification of ionizing photon radiation ● Characteristic X rays: resulting from electron transitions between atomic shells. ● Bremsstrahlung: resulting from electron–nucleus Coulomb interactions. ● g rays: resulting from nuclear transitions. ● Annihilation quanta: resulting from positron–electron annihilation. 1.1.8. Einstein’s relativistic mass, energy and momentum relationships m m 00 m() u===g m (1 .9) 0 22 1-b u ʈ 1- Á˜ ˯ c 2 E = m(u)c (1.10) 2 E = m c (1.11) 0 0 E = E – E = (g – 1)E (1.12) K 0 0 2 2 2 2 E = E + p c (1.13) 0 where u is the particle velocity; c is the speed of light in a vacuum; b is the normalized particle velocity (i.e. b = u/c); m(u) is the particle mass at velocity u; m is the particle rest mass (at velocity u = 0); 0 E is the total energy of the particle; E is the rest energy of the particle; 0 E is the kinetic energy of the particle; K p is the momentum of the particle. 6BASIC RADIATION PHYSICS ● For photons, E = hn and E = 0; thus using Eq. (1.13) we obtain p = hn/c = 0 h/l, where n and l are the photon frequency and wavelength, respec- tively. 1.1.9. Radiation quantities and units The most important radiation quantities and their units are listed in Table 1.3. Also listed are the definitions of the various quantities and the relationships between the old and the SI units for these quantities. 1.2. ATOMIC AND NUCLEAR STRUCTURE 1.2.1. Basic definitions for atomic structure The constituent particles forming an atom are protons, neutrons and electrons. Protons and neutrons are known as nucleons and form the nucleus of the atom. ● Atomic number Z: number of protons and number of electrons in an atom. ● Atomic mass number A: number of nucleons in an atom (i.e. number of protons Z plus number of neutrons N in an atom: A = Z + N). ● There is no basic relation between A and Z, but the empirical relationship A (1.14) Z= 23 / 1.. 98+ 0 0155A furnishes a good approximation for stable nuclei. ● Atomic mass M: expressed in atomic mass units u, where 1 u is equal to 12 2 1/12 of the mass of the C atom or 931.5 MeV/c . The atomic mass M is smaller than the sum of the individual masses of constituent particles because of the intrinsic energy associated with binding the particles (nucleons) within the nucleus. ● Atomic g-atom (gram-atom): number of grams that correspond to N A 23 atoms of an element, where N = 6.022 × 10 atoms/g-atom (Avogadro’s A number). The atomic mass numbers of all elements are defined such that A grams of every element contain exactly N atoms. For example: 1 A 60 60 60 g-atom of Co is 60 g of Co. In 60 g of Co (1 g-atom) there is 60 Avogadro’s number of Co atoms. 7CHAPTER 1 TABLE 1.3. RADIATION QUANTITIES, UNITS AND CONVERSION BETWEEN OLD AND SI UNITS Quantity Definition SI unit Old unit Conversion -4-4 10 C 10 C Exposure DQ 1 esu X= 2.58¥ R= 12 R=¥ .58 3 (X)Dm kg air cm air kg air air STP DE J erg ab Dose (D) 1 Gy = 100 rad 1 Gy= 1 1 rad= 100 D= kg g Dm Equivalent H = Dw 1 Sv 1 rem 1 Sv = 100 rem R dose (H) 1 Ci –1 10 –1 Activity (A) A = lN 1 Bq = 1 s 1 Ci = 3.7 × 10 s 1 Bq = 10 3.7¥ 10 DQ is the charge of either sign collected; Dm is the mass of air; air DE is the absorbed energy; ab Dm is the mass of medium; w is the radiation weighing factor; R l is the decay constant; N is the number of radioactive atoms; R stands for roentgen; Gy stands for gray; Sv stands for sievert; Bq stands for becquerel; Ci stands for curie; STP stands for standard temperature (273.2 K) and standard pressure (101.3 kPa). ● Number of atoms N per mass of an element: a N N a A = m A ● Number of electrons per volume of an element: N N N aa A Z ==rr Z Z V m A 8BASIC RADIATION PHYSICS ● Number of electrons per mass of an element: N Z a Z = N A m A Note that (Z/A) ª 0.5 for all elements, with the one notable exception of hydrogen, for which (Z/A) = 1. Actually, (Z/A) slowly decreases from 0.5 for low Z elements to 0.4 for high Z elements. A ● In nuclear physics the convention is to designate a nucleus X as X, where Z A is the atomic mass number and Z is the atomic number; for example, 60 60 226 226 the Co nucleus is identified as Co, the Ra nucleus as Ra. 27 88 ● In ion physics the convention is to designate ions with + or – superscripts. 4 + 4 4 2+ For example, He stands for a singly ionized He atom and He stands 2 2 4 for a doubly ionized He atom, which is the a particle. ● If we assume that the mass of a molecule is equal to the sum of the masses of the atoms that make up the molecule, then for any molecular compound there are N molecules per g-mole of the compound, where A the g-mole (gram-mole or mole) in grams is defined as the sum of the atomic mass numbers of the atoms making up the molecule; for example, a g-mole of water is 18 g of water and a g-mole of CO is 44 g of CO . Thus 2 2 18 g of water or 44 g of carbon dioxide contain exactly N molecules (or A 3N atoms, since each molecule of water and carbon dioxide contains A three atoms). 1.2.2. Rutherford’s model of the atom The model is based on the results of an experiment carried out by Geiger and Marsden in 1909 with a particles scattered on thin gold foils. The experiment tested the validity of the Thomson atomic model, which postulated that the positive charges and negative electrons were uniformly distributed over the spherical atomic volume, the radius of which was of the order of a few ångström. Theoretical calculations predict that the probability for an a particle to be scattered on such an atom with a scattering angle exceeding 90º is of the order of –3500 10 , while the Geiger–Marsden experiment showed that approximately 1 in 4 –4 10 a particles was scattered with a scattering angle q 90º (probability 10 ). From the findings of the Geiger–Marsden experiment, Rutherford in 1911 concluded that the positive charge and most of the mass of the atom are concentrated in the atomic nucleus (diameter a few femtometres) and negative electrons are smeared over on the periphery of the atom (diameter a few ångströms). In a particle scattering the positively charged a particle has a repulsive Coulomb interaction with the more massive and positively charged nucleus. 9CHAPTER 1 The interaction produces a hyperbolic trajectory of the a particle, and the scattering angle q is a function of the impact parameter b. The limiting case is a direct hit with b = 0 and q = p (backscattering) that, assuming conservation of energy, determines the distance of closest approach D in the backscattering a–N interaction: 22 zZ e zZ e a N a N E (a )=fi D= (1.15) K a-N 44 pe D pe E () a 0N a- 0K where z is the atomic number of the a particle; α Z is the atomic number of the scattering material; N E (a) is the initial kinetic energy of the a particle. K The repulsive Coulomb force between the a particle (charge +2e) and the 2 nucleus (charge +Ze) is governed by 1/r as follows: 2 2Ze F= (1.16) Coul 2 4pe r 0 resulting in the following b versus θ relationship: 1 q (1.17) bD= cot a-N 22 The differential Rutherford scattering cross-section is then expressed as follows: 2 dsÊ Dˆ 1 ʈ a-N = (1.18) Á˜ Á˜ 4 ˯ d4W˯ sin() q /2 R 1.2.3. Bohr’s model of the hydrogen atom Bohr expanded Rutherford’s atomic model in 1913 and based it on four postulates that combine classical, non-relativistic mechanics with the concept of angular momentum quantization. Bohr’s model successfully deals with one- electron entities such as the hydrogen atom, singly ionized helium atom, doubly ionized lithium atom, etc. 10BASIC RADIATION PHYSICS The four Bohr postulates are as follows: ● Postulate 1: Electrons revolve about the Rutherford nucleus in well defined, allowed orbits (shells). The Coulomb force of attraction F = Coul 2 2 Ze /(4pe r ) between the negative electrons and the positively charged 0 2 nucleus is balanced by the centrifugal force F = m u /r, where Z is the cent e number of protons in the nucleus (atomic number), r is the radius of the orbit, m is the electron mass and u is the velocity of the electron in the e orbit. ● Postulate 2: While in orbit, the electron does not lose any energy despite being constantly accelerated (this postulate is in contravention of the basic law of nature, which is that an accelerated charged particle will lose part of its energy in the form of radiation). ● Postulate 3: The angular momentum L = m ur of the electron in an e allowed orbit is quantized and given as L=n, where n is an integer referred to as the principal quantum number and  =h/(2p), where h is Planck’s constant. The simple quantization of angular momentum stipulates that the angular momentum can have only integral multiples of a basic value (). ● Postulate 4: An atom or ion emits radiation when an electron makes a transition from an initial orbit with quantum number n to a final orbit i with quantum number n for n n . f i f The radius r of a one-electron Bohr atom is given by: n 22 ʈʈ n n ra== 0.Å 529 n 0 Á˜Á˜ (1.19) Z Z ˯˯ where a is the Bohr radius (a = 0.529 Å). 0 0 The velocity u of the electron in a one-electron Bohr atom is: n Z cZ ʈʈ ua= c= (1.20) nÁ˜Á˜ ˯˯ n 137 n where a is the fine structure constant (a = 1/137). The energy levels for orbital electron shells in monoelectronic atoms (e.g. hydrogen, singly ionized helium and doubly ionized lithium) are given by: 22 Z Z ʈʈ (1.21) EE =- =-13.6 eV n RÁ˜Á˜ ˯˯ n n 11CHAPTER 1 where E is the Rydberg energy (13.61 eV); R n is the principal quantum number (n = 1, ground state; n 1, excited state); Z is the atomic number (Z = 1 for a hydrogen atom, Z = 2 for singly ionized helium, Z = 3 for doubly ionized lithium, etc.). The wave number k of the emitted photon is given by: ʈʈ 111 11 2-12 (1.22) kR ==Z -=- 109 737cm Z • Á22˜Á22˜ l nn nn ˯˯ fi fi where R is the Rydberg constant. � Bohr’s model results in the energy level diagram for the hydrogen atom shown in Fig. 1.2. 1.2.4. Multielectron atoms For multielectron atoms the fundamental concepts of the Bohr atomic theory provide qualitative data for orbital electron binding energies and electron transitions resulting in emission of photons. Electrons occupy allowed 2 shells, but the number of electrons per shell is limited to 2n , where n is the shell number (the principal quantum number). ● The K shell binding energies E (K) for atoms with Z 20 may be B estimated with the following relationship: 22 2 (1.23) EE (K)==Z E (Z-s)=E (Z- 2) B R eff R R where Z , the effective atomic number, is given by Z = Z – s, where s is eff eff the screening constant equal to 2 for K shell electrons. ● Excitation of an atom occurs when an electron is moved from a given shell to a higher n shell that is either empty or does not contain a full complement of electrons. ● Ionization of an atom occurs when an electron is removed from the atom (i.e. the electron is supplied with enough energy to overcome its binding energy in a shell). ● Excitation and ionization processes occur in an atom through various possible interactions in which orbital electrons are supplied with a given amount of energy. Some of these interactions are: (i) Coulomb 12BASIC RADIATION PHYSICS Continuum of electron kinetic energies 0 –0.9 eV n = 3 Excited –1.5 eV states n 1 n = 2 –3.4 eV Discrete energy levels Electron bound states Ground n = 1 state –13.6 eV n = 1 FIG. 1.2. Energy level diagram for a hydrogen atom (ground state: n = 1, excited states: n 1). interaction with a charged particle; (ii) the photoelectric effect; (iii) the Compton effect; (iv) triplet production; (v) internal conversion; (vi) electron capture; (vii) the Auger effect; and (viii) positron annihilation. ● An orbital electron from a higher n shell will fill an electron vacancy in a lower n atomic shell. The energy difference between the two shells will be either emitted in the form of a characteristic photon or it will be transferred to a higher n shell electron, which will be ejected from the atom as an Auger electron. ● Energy level diagrams of multielectron atoms resemble those of one- electron structures, except that inner shell electrons are bound with much larger energies, as shown for a lead atom in Fig. 1.3. ●The number of characteristic photons (sometimes called fluorescent photons) emitted per orbital electron shell vacancy is referred to as fluorescent yield w, while the number of Auger electrons emitted per orbital 13CHAPTER 1 electron vacancy is equal to (1 – w). The fluorescent yield depends on the atomic number Z of the atom and on the principal quantum number of a shell. For atoms with Z 10 the fluorescent yield w = 0; for Z ª 30 the K fluorescent yield w ª 0.5; and for high atomic number atoms w = 0.96, K K where w refers to the fluorescent yield for the K shell (see Fig. 1.9). K 1.2.5. Nuclear structure Most of the atomic mass is concentrated in the atomic nucleus consisting of Z protons and (A – Z) neutrons, where Z is the atomic number and A is the atomic mass number of a given nucleus. Continuum of electron kinetic energies 0 n = 3 M Eighteen electrons Excited –3 keV states n 1 n = 2 L Eight electrons –15 keV Discrete energy levels Electron bound states Ground n = 1 K Two electrons state –88 keV n = 1 FIG. 1.3. Energy level diagram for a multielectron atom (lead). The n = 1, 2, 3, 4… shells are referred to as the K, L, M, O… shells, respectively. Electronic transitions that end in low n shells are referred to as X ray transitions because the resulting photons are in the X ray energy range. Electronic transitions that end in high n shells are referred to as optical transitions because they result in ultraviolet, visible or infrared photons. 14BASIC RADIATION PHYSICS ● The radius r of the nucleus is estimated from: 3 rr= A (1.24) 0 where r is a constant (1.4 fm) assumed equal to ½ of r , the classical 0 e electron radius. ● Protons and neutrons are commonly referred to as nucleons and are bound in the nucleus with the strong force. In contrast to electrostatic and gravitational forces, which are inversely proportional to the square of the distance between two particles, the strong force between two nucleons is a very short range force, active only at distances of the order of a few femtometres. At these short distances the strong force is the predominant force, exceeding other forces by several orders of magnitude. ● The binding energy E per nucleon in a nucleus varies slowly with the B number of nucleons A, is of the order of 8 MeV/nucleon and exhibits a broad maximum of 8.7 MeV/nucleon at A ª 60. For a given nucleus it may be calculated from the energy equivalent of the mass deficit Dm as follows: E B 22 22 (1.25) ==Dmc/ A( Zmc+ A-Z)mc-Mc/A pn nucleon where 2 M is the nuclear mass in atomic mass units u (note that uc = 931.5 MeV); 2 m c is the proton rest energy; p 2 m c is the neutron rest energy. n 1.2.6. Nuclear reactions Much of the present knowledge of the structure of nuclei comes from experiments in which a particular nuclide A is bombarded with a projectile a. The projectile undergoes one of three possible interactions: (i) elastic scattering (no energy transfer occurs; however, the projectile changes trajectory); (ii) inelastic scattering (the projectile enters the nucleus and is re- emitted with less energy and in a different direction); or (iii) nuclear reaction (the projectile a enters the nucleus A, which is transformed into nucleus B and a different particle b is emitted). 15CHAPTER 1 ● Nuclear reactions are designated as follows: a + A Æ B + b or A(a, b)B (1.26) ● A number of physical quantities are rigorously conserved in all nuclear reactions. The most important of these quantities are charge, mass number, linear momentum and mass–energy. ● The threshold energy for a nuclear reaction is defined as the smallest value of a projectile’s kinetic energy at which a nuclear reaction can take thr place. The threshold kinetic energy E (a) of projectile a is derived from K relativistic conservation of energy and momentum as: 222 2 22 () mc+- m c(m c+m c) thr Bb A a (1.27) E () a= K 2 2mc A where m , m , m and m are the rest masses of the target A, projectile a A a B b and products B and b, respectively. 1.2.7. Radioactivity Radioactivity is characterized by a transformation of an unstable nucleus into a more stable entity that may be unstable and will decay further through a chain of decays until a stable nuclear configuration is reached. The exponential laws that govern the decay and growth of radioactive substances were first formulated by Rutherford and Soddy in 1902 and then refined by Bateman in 1910. ● The activity A(t) of a radioactive substance at time t is defined as the product of the decay constant l and the number of radioactive nuclei N(t): A(t) = lN(t) (1.28) ● The simplest radioactive decay is characterized by a radioactive parent nucleus P decaying with a decay constant l into a stable daughter P nucleus D: l P (1.29) P Æ D —The number of radioactive parent nuclei N (t) as a function of time t is P governed by the following relationship: 16BASIC RADIATION PHYSICS -l t P (1.30) Nt ()=N (0)e PP where N (0) is the initial number of parent nuclei at time t = 0. P —Similarly, the activity of parent nuclei A (t) at time t is given as: P -l t P (1.31) AA ()te= (0) PP where A (0) is the initial activity of parent nuclei at time t = 0. P ● The half-life t of a radioactive substance is the time during which the 1/2 number of radioactive nuclei decays to half of the initial value N (0) P present at time t = 0: -l t P 12 / (1.32) Nt() == t (12/)N(0)=N(0)e PP 12 / P ● The decay constant l and half-life (t ) for the parent are thus related as P 1/2 P follows: ln 2 l= (1.33) P t 12 / ● The specific activity a is defined as the parent’s activity per unit mass: A l N N N ln 2 PP A A (1.34) a== =l = P m m A At ( ) P PP 12 / where N is Avogadro’s number and A is the parent’s atomic mass A P number. ● The average (mean) life t of a radioactive substance represents the P average life expectancy of all parent radioactive atoms in the substance at time t = 0: • A () 0 -l t P P (1.35) AA ()00 t== ()et d PP P Ú l P 0 ● The decay constant l and average life t are thus related as follows: P P l = 1/t (1.36) P P resulting in the following relationship between (t ) and t : 1/2 P P 17CHAPTER 1 (t ) = t ln 2 (1.37) 1/2 P P ● A more complicated radioactive decay occurs when a radioactive parent nucleus P decays with a decay constant l into a daughter nucleus D P which in turn is radioactive and decays with a decay constant l into a D stable granddaughter G: ll PD (1.38) P ÆÆ D G —The activity of the daughter A (t) may then be expressed as: D l lltt D PD (1.39) AA ()te= (0)(-e ) D P ll- DP where A (0) is the initial activity of the parent nuclei present at time P t = 0 (i.e. A (0) = l N (0), where N (0) is the number of parent nuclei P P P P at t = 0). —The maximum activity of daughter nuclei occurs at time t given by: max ln(ll / ) DP (1.40) t= max ll- DP under the condition that N = 0 at time t = 0. D ● Special considerations in parent Æ daughter Æ granddaughter relation- ships: —For l l or (t ) (t ) we obtain the following general D P 1/2 D 1/2 P relationship: A l () ll t D D DP (1.41) = 1- e A ll- P DP —For l l or (t ) (t ) we obtain transient equilibrium with: D P 1/2 D 1/2 P A l D D = for t t (1.42) max A ll- P DP —For l l or (t ) (t ) we obtain secular equilibrium and D P 1/2 D 1/2 P A /A ª 1 (1.43) D P 18BASIC RADIATION PHYSICS 1.2.8. Activation of nuclides Activation of nuclides occurs when a stable parent isotope P is bombarded with neutrons in a nuclear reactor and transforms into a radioactive daughter D that decays into a granddaughter G: sf l D (1.44) P ÆÆ D G The probability for activation is determined by the cross-section s for the –24 2 nuclear reaction, usually expressed in barns per atom, where 1 barn = 10 cm . ● Activity of the daughter A (t) is expressed as: D sfl D-sft-l t D (1.45) A ()tN= (0)(e-e ) D P ls-f D where N (0) is the initial number of parent nuclei. P ● This result is similar to the P Æ D Æ G relationship above (Eq. (1.39)) in which an unstable parent P decays into an unstable daughter D that in turn decays into granddaughter G. However, the decay constant l in the P P Æ D Æ G decay relationship is replaced by sf, where s is the cross- 2 section for activation of the parent nuclei (cm /atom) and f is the fluence –2 –1 rate of neutrons in the reactor (cm ·s ). ● The time t at which the maximum activity A occurs in the activation max D process is then, similarly to Eq. (1.40), given by: l D ln sf t= (1.46) max ls-f D ● In situations where sf l , the daughter activity relationship of D Eq. (1.45) transforms into a simple exponential growth relationship: -l t D (1.47) A ()tN=- sf (01 )(e ) DP 60 ● An important example of nuclear activation is the production of the Co 59 isotope by bombarding Co with thermal neutrons in a nuclear reactor: 59 60 (1.48) Co + n Æ Co + g 27 27 19