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chapter 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 avogadro s number na 6.022 × 1023 atoms/g-atom 23 avogadro s number na 6.022 × 10 molecules/g-mole 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 me­ 0.5110 mev/c 2 positron rest mass me 0.5110 mev/c 2 proton rest mass mp 938.3 mev/c 2 neutron rest mass mn 939.6 mev/c 2 atomic mass unit u 931.5 mev/c ­34 planck s constant h 6.626 × 10 j·s ­12 permittivity of vacuum e0 8.854 × 10 c v·m ­7 permeability of vacuum m0 4p × 10 v·s a·m ­11 newtonian gravitation constant g 6.672 × 10 m3·kg­1·s­2 proton mass/electron mass mp/me 1836.0 11 specific charge of electron e/me 1.758 × 10 c/kg 1.1.2 important derived physical constants and relationships speed of light in a vacuum c 1 ª 3 ¥ 10 8 m/s e 0 m0 1.1 1

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chapter 1 reduced planck s constant × speed of light in a vacuum c h c 197.3 mev fm ª 200 mev fm 2p 1.2 fine structure constant a 1 e2 1 4pe 0 c 137 1.3 bohr radius a0 4pe c 2 c 20 0.5292 Å a mec 2 e mec 2 1.4 rydberg energy 1 1 Ê e 2 mec 2 e r mec 2a 2 Á 13.61 ev 2 2 Ë 4pe 0 c 2 ¯ 2 1.5 rydberg constant m c 2a 2 e 1 Ê e 2 mec 2 r· r e 109 737 cm -1 2p c 4p c 4p Á 4pe 0 c 3 Ë ¯ 2 1.6 classical electron radius re e2 2.818 fm 4pe 0 mec 2 1.7 compton wavelength of the electron lc h 0.0243 Å mec 1.8 2

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basic 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 international 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 quantity length mass time current charge force momentum energy symbol lmtiqfpe unit units used in in si radiation physics m kg sacnn·s j ev kev mev nm Å fm mev/c2 ms ms ns ps ma ma na pa e conversion 1 m 109 nm 1010 Å 1015 fm 1 mev/c2 1.78 × 10­30 kg 1 s 103 ms 106 ms 109 ns 1012 ps 1 a 103 ma 106 ma 109 na 1 e 1.602 × 10­19 c 1 n 1 kg·m·s­2 1 n·s 1 kg·m·s­1 1 ev 1.602 × 10­19 j 10­3 kev 3

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chapter 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 gravitational force with relative strengths of 1 1/137 10­6 and 10­39 respectively the ranges of the em and gravitational forces are infinite 1/r2 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 weak charge for the weak force transmitted by particles called w and z0 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 constituents 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 strong em weak gravitational source strong charge electric charge weak charge energy transmitted particle gluon photon w and z0 graviton relative strength 1 1/137 10­6 10­39 4

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basic radiation physics leptons are particles that do not interact strongly electrons e muons m taus t and their corresponding neutrinos ne nm nt are in this category classification of radiation 1.1.6 as shown in fig 1.1 radiation is classified into two main categories nonionizing 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 5

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chapter 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 electron­nucleus coulomb interactions g rays resulting from nuclear transitions annihilation quanta resulting from positron­electron annihilation 1.1.8 einstein s relativistic mass energy and momentum relationships mu m0 Êu 1 Á Ë c¯ 2 m0 1 b 2 g m0 1.9 e muc2 e0 m0c2 ek e ­ e0 g ­ 1e0 2 e2 e0 p2c2 1.10 1.11 1.12 1.13 where ucbmu m0 e e0 ek p is the particle velocity is the speed of light in a vacuum is the normalized particle velocity i.e b u/c is the particle mass at velocity u is the particle rest mass at velocity u 0 is the total energy of the particle is the rest energy of the particle is the kinetic energy of the particle is the momentum of the particle 6

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basic radiation physics for photons e hn and e0 0 thus using eq 1.13 we obtain p hn/c h/l where n and l are the photon frequency and wavelength respectively radiation quantities and units 1.1.9 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 z a 1.98 0.0155 a 2 3 1.14 furnishes a good approximation for stable nuclei atomic mass m expressed in atomic mass units u where 1 u is equal to 1/12 of the mass of the 12c atom or 931.5 mev/c2 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 na atoms of an element where na 6.022 × 1023 atoms/g-atom avogadro s number the atomic mass numbers of all elements are defined such that a grams of every element contain exactly na atoms for example 1 g-atom of 60co is 60 g of 60co in 60 g of 60co 1 g-atom there is avogadro s number of 60co atoms 7

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chapter 1 table 1.3 radiation quantities units and conversion between old and si units quantity exposure x dose d equivalent dose h activity a dq dmair deab dm wr l n r gy sv bq ci stp definition si unit old unit r 1 esu cm 3 airstp conversion x dq dmair 2.58 ¥ 10 -4 c kg air j kg 1 r 2.58 ¥ 10 -4 c kg air d de ab dm 1 gy 1 1 sv 1 rad 100 1 rem erg g 1 gy 100 rad 1 sv 100 rem h dwr a ln 1 bq 1 s­1 1 ci 3.7 × 1010 s­1 1 bq 1 ci 3.7 ¥ 10 10 is the charge of either sign collected is the mass of air is the absorbed energy is the mass of medium is the radiation weighing factor is the decay constant is the number of radioactive atoms stands for roentgen stands for gray stands for sievert stands for becquerel stands for curie stands for standard temperature 273.2 k and standard pressure 101.3 kpa number of atoms na per mass of an element na na m a number of electrons per volume of an element z na n n rz a rz avma 8

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basic radiation physics number of electrons per mass of an element z na z nmaa 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 zx where a is the atomic mass number and z is the atomic number for example the 60co nucleus is identified as 60 co the 226ra nucleus as 226 ra 27 88 in ion physics the convention is to designate ions with or ­ superscripts for example 4he stands for a singly ionized 4he atom and 4he2 stands 2 2 for a doubly ionized 4he 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 na molecules per g-mole of the compound where 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 co2 is 44 g of co2 thus 18 g of water or 44 g of carbon dioxide contain exactly na molecules or 3na atoms since each molecule of water and carbon dioxide contains 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 10­3500 while the geiger­marsden experiment showed that approximately 1 in 104 a particles was scattered with a scattering angle q 90º probability 10­4 from the findings of the geiger­marsden 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 9

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chapter 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 da­n in the backscattering interaction e k a where za z n e 2 4pe 0 da n fi da n za z n e 2 4pe 0 e k a 1.15 is the atomic number of the a particle z is the atomic number of the scattering material zn eka is the initial kinetic energy of the a particle the repulsive coulomb force between the a particle charge +2e and the nucleus charge +ze is governed by 1/r2 as follows fcoul 2ze 2 4pe 0 r 2 1.16 resulting in the following b versus relationship b 1 q d cot 2 a -n 2 1.17 the differential rutherford scattering cross-section is then expressed as follows 1 Ê da n Ê ds Á Ë dw r Á 4 sin 4 q /2 ¯ Ë ¯ 2 1.18 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 oneelectron entities such as the hydrogen atom singly ionized helium atom doubly ionized lithium atom etc 10

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basic 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 fcoul ze2 4pe0r2 between the negative electrons and the positively charged nucleus is balanced by the centrifugal force fcent meu2/r where z is the number of protons in the nucleus atomic number r is the radius of the orbit me is the electron mass and u is the velocity of the electron in the 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 meur of the electron in an 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 ni to a final orbit with quantum number nf for ni nf the radius rn of a one-electron bohr atom is given by Ê n2 Ê n2 rn a 0 Á 0.529 Å Á Ëz¯ Ëz¯ where a0 is the bohr radius a0 0.529 Å the velocity un of the electron in a one-electron bohr atom is 1.19 c Êz Êz un acÁ Ë n ¯ 137 Á n Ë ¯ 1.20 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 Êz Êz e n e r Á -13.6 ev Á Ë n¯ Ë n¯ 2 2 1.21 11

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chapter 1 where er n z is the rydberg energy 13.61 ev is the principal quantum number n 1 ground state n 1 excited state 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 k Ê 1 Ê 1 1 1 1 r· z 2 Á 2 2 109 737 cm -1z 2 Á 2 2 l Ë nf ni ¯ Ë nf ni ¯ 1.22 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 shells but the number of electrons per shell is limited to 2n2 where n is the shell number the principal quantum number the k shell binding energies ebk for atoms with z 20 may be estimated with the following relationship 2 e b k e r z eff e r z s 2 e r z 2 2 1.23 where zeff the effective atomic number is given by zeff z ­ s where s is 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 12

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basic radiation physics continuum of electron kinetic energies 0 ­0.9 ev excited states n>1 n=3 ­1.5 ev n=2 ­3.4 ev discrete energy levels electron bound states ground state n=1 n=1 ­13.6 ev 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 oneelectron 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 13

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chapter 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 wk 0 for z ª 30 the fluorescent yield wk ª 0.5 and for high atomic number atoms wk 0.96 where wk refers to the fluorescent yield for the k shell see fig 1.9 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 excited states n>1 n=3 m eighteen electrons ­3 kev n=2 l eight electrons ­15 kev discrete energy levels electron bound states ground n=1 state n=1 k two electrons ­88 kev 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 14

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basic radiation physics the radius r of the nucleus is estimated from r r0 3 a 1.24 where r0 is a constant 1.4 fm assumed equal to ½ of re the classical 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 eb per nucleon in a nucleus varies slowly with the 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 eb dmc 2 /a [zm p c 2 a z mn c 2 mc 2 a nucleon where 1.25 is the nuclear mass in atomic mass units u note that uc2 931.5 mev mpc2 is the proton rest energy mnc2 is the neutron rest energy m 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 reemitted 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 15

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