chapter-1-basics-radiation-physics.pdf

Chapter 1: Basic Radiation Physics Slide set of 195 slides based on the chapter authored by E.B. Podgorsak of the IAEA publication (ISBN 92-0-107304-6): Review of Radiation Oncology Physics: A Handbook for Teachers and Students Objective: To familiarize the student with basic princip...

Chapter 1: Basic Radiation Physics Slide set of 195 slides based on the chapter authored by E.B. Podgorsak of the IAEA publication (ISBN 92-0-107304-6): Review of Radiation Oncology Physics: A Handbook for Teachers and Students Objective: To familiarize the student with basic principles of radiation physics and modern physics used in radiotherapy. Slide set prepared in 2006 by E.B. Podgorsak (Montreal, McGill University) Comments to S. Vatnitsky: [email protected] Version 2012 IAEA International Atomic Energy Agency CHAPTER 1. TABLE OF CONTENTS 1.1. Introduction 1.2. Atomic and nuclear structure 1.3. Electron interactions 1.4. Photon interactions IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1. 1.1 INTRODUCTION 1.1.1 Fundamental physical constants  Avogadro’s number: NA  6.022  1023 atom/mol  Speed of light in vacuum: c  3  108 m/s  Electron charge: e  1.6  1019 As  Electron rest mass: me  0.511 MeV/c 2  Proton rest mass: mp  938.2 MeV/c 2  Neutron rest mass: mn  939.3 MeV/c 2  Atomic mass unit: u  931.5 MeV/c 2 IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.1.1 Slide 1 1.1 INTRODUCTION 1.1.2 Derived physical constants  Reduced Planck’s constant  speed of light in vacuum c = 197 MeV × fm » 200 MeV × fm  Fine structure constant e2 1 1   4 o c 137  Classical electron radius e2 1 re   2.818 MeV 4 o mec 2 IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.1.2 Slide 1 1.1 INTRODUCTION 1.1.2 Derived physical constants  Bohr radius: c 4pe 0 ( c)2 a0 = = 2 = 0.529 Å a mec 2 e mec 2  Rydberg energy: 2 1 1  e 2  m c 2 ER  mec 2 2    e  13.61 eV 2 2  4 o  ( c ) 2  Rydberg constant: ER mec 2a 2 R¥ = = = 109 737 cm-1 2p c 4p c IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.1.2 Slide 2 1.1 INTRODUCTION 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. For example: m  21 kg; E  15 MeV IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.1.3 Slide 1 1.1 INTRODUCTION 1.1.3 Physical quantities and units  Numerical value and the unit of a physical quantity must be separated by space. For example: 21 kg and NOT 21kg; 15 MeV and NOT 15MeV  The currently used metric system of units is known as the Systéme International d’Unités (International system of units) or the SI system. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.1.3 Slide 2 1.1 INTRODUCTION 1.1.3 Physical quantities and units The SI system of units is founded on base units for seven physical quantities: Quantity SI unit length meter (m) mass m kilogram (kg) time t second (s) electric current (I) ampère (A) temperature (T) kelvin (K) amount of substance mole (mol) luminous intensity candela (cd) IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.1.3 Slide 3 1.1 INTRODUCTION 1.1.4 Classification of forces in nature There are four distinct forces observed in interaction between various types of particles Force Source Transmitted particle Relative strength Strong Strong charge Gluon 1 EM Electric charge Photon 1/137 Weak Weak charge W+, W-, and Zo 10-6 Gravitational Energy Graviton 10-39 IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.1.4 Slide 1 1.1 INTRODUCTION 1.1.5 Classification of fundamental particles Two classes of fundamental particles are known:  Quarks are particles that exhibit strong interactions Quarks are constituents of hadrons with a fractional electric charge (2/3 or -1/3) and are characterized by one of three types of strong charge called color (red, blue, green).  Leptons are particles that do not interact strongly. Electron, muon, tau, and their corresponding neutrinos. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.1.5 Slide 1 1.1 INTRODUCTION 1.1.6 Classification of radiation Radiation is classified into two main categories:  Non-ionizing radiation (cannot ionise matter).  Ionizing radiation (can ionize matter). Directly ionizing radiation (charged particles) electron, proton, alpha particle, heavy ion Indirectly ionizing radiation (neutral particles) photon (x ray, gamma ray), neutron IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.1.6 Slide 1 1.1 INTRODUCTION 1.1.6 Classification of radiation Radiation is classified into two main categories: IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.1.6 Slide 2 1.1 INTRODUCTION 1.1.7 Classification of ionizing photon radiation Ionizing photon radiation is classified into four categories:  Characteristic x ray Results from electronic transitions between atomic shells.  Bremsstrahlung Results mainly from electron-nucleus Coulomb interactions.  Gamma ray Results from nuclear transitions.  Annihilation quantum (annihilation radiation) Results from positron-electron annihilation. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.1.7 Slide 1) 1.1 INTRODUCTION 1.1.8 Einstein’s relativistic mass, energy, and momentum m0 m0  Mass: m(u ) = = = g m0 æuö 2 1- b 2 1- ç ÷ è cø m(u ) 1 1  Normalized mass: = = =g m0 æuö 2 1- b 2 1- ç ÷ è cø  1 where  and   c 1-  2 IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.1.8 Slide 1 1.1 INTRODUCTION 1.1.8 Einstein’s relativistic mass, energy, and momentum m0 m0   m(u ) = = = g m0 c æuö 2 1- b 2 1- ç ÷ è cø m(u ) 1 1   1 = = =g 1-  2 m0 æuö 2 1- b 2 1- ç ÷ è cø IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.1.8 Slide 2 1.1 INTRODUCTION 1.1.8 Einstein’s relativistic mass, energy, and momentum  Total energy: E  m( )c 2  Rest energy: E0 = m0c 2  Kinetic energy: EK = E - E0 = (g - 1)E0 1  Momentum: p= E 2 - E02 c  1 with  and   c 1-  2 IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.1.8 Slide 3) 1.1 INTRODUCTION 1.1.9 Radiation quantities and units IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.1.9 Slide 1 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.1 Basic definitions for atomic structure  Constituent particles forming an atom are: Proton Neutron Electron Protons and neutrons are known as nucleons and form the nucleus.  Atomic number Z Number of protons and number of electrons in an atom. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.1 Slide 1 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.1 Basic definitions for atomic structure  Atomic mass number A Number of nucleons (Z + N) in an atom, where Z is the number of protons (atomic number) in an atom. N is the number of neutrons in an atom. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.1 Slide 2) 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.1 Basic definitions for atomic structure  There is no basic relation between the atomic mass number A and atomic number Z of a nucleus but the empirical relationship: A Z 1.98  0.0155A2/3 furnishes a good approximation for stable nuclei. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.1 Slide 3 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.1 Basic definitions for atomic structure  Atomic mole is defined as the number of grams of an atomic compound that contains exactly one Avogadro’s number of atoms, i.e., NA = 6.022 ´ 1023 atom/mol  Atomic mass number A of all elements is defined such that A grams of every element contain exactly NA atoms.  For example: 1 mole of cobalt-60 is 60 g of cobalt-60. 1 mole of radium-226 is 226 g of radium-226. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.1 Slide 4 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.1 Basic definitions for atomic structure  Molecular mole is defined as the number of grams of a molecular compound that contains exactly one Avogadro’s number of molecules, i.e., NA = 6.022 ´ 1023 molecule/mol  Mass of a molecule is the sum of masses of all atoms that make up the molecule.  For example: 1 mole of water (H2O) is 18 g of water. 1 mole of carbon dioxide (CO2 ) is 44 g of carbon dioxide. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.1 Slide 5 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.1 Basic definition for atomic structure  Atomic mass M is expressed in atomic mass units u: 1 u is equal to 1/12th of the mass of the carbon-12 atom or 931.5 MeV/c2. 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. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.1 Slide 6 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.1 Basic definition for atomic structure  Nuclear mass M is defined as the atomic mass with the mass of atomic orbital electrons subtracted, i.e., M  M - Zme , , where M is the atomic mass. Binding energy of orbital electrons to the nucleus is neglected. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.1 Slide 7 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.1 Basic definitions for atomic structure In nuclear physics the convention is to designate a nucleus X as AZ X , where A is the atomic mass number. Z is the atomic number. For example: Cobalt-60 nucleus with Z = 27 protons and A = 33 neutrons is 60 identified as 27 Co. Radium-226 nucleus with 88 protons and 138 neutrons is identified as 226 88 Ra. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.1 Slide 8 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.1 Basic definitions for atomic structure  Number of atoms Na per mass m of an element: Na NA  m A  Number of electrons Ne per mass m of an element: Ne Na NA Z Z m m A  Number of electrons Ne per volume V of an element: Ne Na NA  Z  Z V m A IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.1 Slide 9 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.1 Basic definitions for atomic structure  For all elements the ratio Z/A  0.5 with two notable exceptions: Hydrogen-1 for which Z/A  1.0. Helium-3 for which Z/A  0.67.  Actually, the ratio Z/A gradually decreases: From 0.5 for low atomic number Z elements. To ~0.4 for high atomic number Z elements.  For example: Z /A  0.50 for 42 He Z /A  0.45 for 60 27 Co Z /A  0.39 for 235 92 U IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.1 Slide 10 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.2 Rutherford’s model of the atom  Rutherford’s atomic model is based on results of the Geiger-Marsden experiment of 1909 with 5.5 MeV alpha particles scattered on thin gold foils with a thickness of the order of 10-6 m. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.2 Slide 1  1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.2 Rutherford’s model of the atom  At the time of the Geiger-Marsden experiment in 1909 Thomson atomic model was the prevailing atomic model.  Thomson model was based on an assumption that the positive and the negative (electron) charges of the atom were distributed uniformly over the atomic volume (“plum-pudding model of the atom”). IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.2 Slide 2 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.2 Rutherford’s model of the atom  Geiger and Marsden found that more than 99 % of the alpha particles incident on the gold foil were scattered at scattering angles less than 3o and that the distribution of scattered alpha particles followed a Gaussian shape.  Geiger and Marsden also found that roughly one in 104 alpha particles was scattered with a scattering angle exceeding 90o (probability 10-4)  This finding was in drastic disagreement with the theoretical prediction of one in 103500 resulting from the Thomson’s atomic model (probability 10-3500). IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.2 Slide 3 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.2 Rutherford’s model of the atom  Ernest Rutherford concluded that the peculiar results of the Geiger-Marsden experiment did not support the Thomson’s atomic model and proposed the currently accepted atomic model in which: Mass and positive charge of the atom are concentrated in the nucleus the size of which is of the order of 10-15 m. Negatively charged electrons revolve about the nucleus in a spherical cloud on the periphery of the Rutherford atom with a radius of the order of 10-10 m. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.2 Slide 4 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.2 Rutherford’s model of the atom  Based on his model and four additional assumptions, Rutherford derived the kinematics for the scattering of alpha particles on gold nuclei using basic principles of classical mechanics.  The four assumptions are related to: Mass of the gold nucleus. Scattering of alpha particles. Penetration of the nucleus. Kinetic energy of the alpha particles. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.2 Slide 5 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.2 Rutherford’s model of the atom  The four assumptions are: Mass of the gold nucleus M >> mass of the alpha particle ma. Scattering of alpha particles on atomic electrons is negligible. Alpha particle does not penetrate the nucleus, i.e., there are no nuclear reactions occurring. Alpha particles with kinetic energies of the order of a few MeV are non-relativistic and the simple classical relationship for the kinetic energy EK of the alpha particle is valid: m 2 EK  2 IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.2 Slide 6 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.2 Rutherford’s model of the atom As a result of the repulsive Coulomb interaction between the alpha particle (charge +2e) and the nucleus (charge +Ze) the alpha particle follows a hyperbolic trajectory. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.2 Slide 7 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.2 Rutherford’s model of the atom  Shape of the hyperbolic trajectory and the scattering angle  depend on the impact parameter b. The limiting case is a direct hit with b  0 and    (backscattering) that, assuming conservation of energy, determines the distance of closest approach D -N in a direct hit (backscattering) interaction. 2ZNe2 2ZNe2 EK   D -N  4 oD -N 4 oEK IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.2 Slide 8 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.2 Rutherford’s model of the atom  Shape of the hyperbolic trajectory and the scattering angle  are a function of the impact parameter b.  Repulsive Coulomb force between the alpha particle (charge ze, z = 2) and the nucleus (charge Ze) is governed by 1/ r 2 dependence: 2Ze 2 Fcoul  4 or 2 where r is the separation between the two charged particles. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.2 Slide 9 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.2 Rutherford’s model of the atom  Relationship between the impact parameter b and the scattering angle  follows from the conservation of energy and momentum considerations: 1  b  D -N cot 2 2  This expression is derived using: Classical relationship for the kinetic energy of the  particle: EK  m 2 / 2 Definition of D -N in a direct hit head-on collision for which the impact parameter b = 0 and the scattering angle   . IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.2 Slide 10 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.2 Rutherford’s model of the atom Differential Rutherford scattering cross section is: 2 ds é Da -N ù 1 =ê ú dW ë 4 û sin4 (q /2) IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.2 Slide 11 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.3 Bohr’s model of the hydrogen atom  Niels Bohr in 1913 combined Rutherford’s concept of nuclear atom with Planck’s idea of quantized nature of the radiation process and developed an atomic model that successfully deals with one-electron structures, such as hydrogen atom, singly ionized helium, etc. M nucleus with mass M me electron with mass me rn radius of electron orbit IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.3 Slide 1 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.3 Bohr’s model of the hydrogen atom  Bohr’s atomic model is based on four postulates: Postulate 1: Electrons revolve about the Rutherford nucleus in well-defined, allowed orbits (planetary-like motion). Postulate 2: While in orbit, the electron does not lose any energy despite being constantly accelerated (no energy loss while electron is in allowed orbit). Postulate 3: The angular momentum of the electron in an allowed orbit is quantized (quantization of angular momentum). Postulate 4: An atom emits radiation only when an electron makes a transition from one orbit to another (energy emission during orbital transitions). IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.3 Slide 2 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.3 Bohr’s model of the hydrogen atom Bohr’s atomic model is based on four postulates: Postulate 1: Planetary motion of electrons Electrons revolve about the Rutherford nucleus in well-defined, allowed orbits. Coulomb force of attraction between the electron and the positively charged nucleus is balanced by the centrifugal force 1 Ze 2 meue2 Fcoul = º Fcent = 4pe o re2 re IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.3 Slide 3 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.3 Bohr’s model of the hydrogen atom Bohr’s atomic model is based on four postulates: Postulate 2: No energy loss while electron is in orbit. While in orbit, the electron does not lose any energy despite being constantly accelerated. This is a direct contravention of the basic law of nature (Larmor’s law) which states that: “Any time a charged particle is accelerated or decelerated part of its energy is emitted in the form of photon (bremsstrahlung)”. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.3 Slide 4 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.3 Bohr’s model of the hydrogen atom Bohr’s atomic model is based on four postulates: Postulate 3: Quantization of angular momentum Angular momentum L  me r 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. Lowest possible angular momentum of electron in an allowed orbit is L =. All atomic orbital electron angular momenta are integer multiples of. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.3 Slide 5 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.3 Bohr’s model of the hydrogen atom Bohr’s atomic model is based on four postulates: Postulate 4: Emission of photon during atomic transition. Atom emits radiation only when an electron makes a transition from an initial allowed orbit with quantum number ni to a final orbit with quantum number nf. Energy of the emitted photon equals the difference in energy between the two atomic orbits. h  Ei - E f IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.3 Slide 6 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.3 Bohr’s model of the hydrogen atom  Radius rn of a one-electron Bohr atom is: é n2 ù é n2 ù rn = a0 ê ú = (0.53 Å) ´ ê ú ëZ û ëZ û  Velocity  n of the electron in a one-electron Bohr atom is: éZ ù c éZ ù -3 é Z ù un = a c ê ú = ê ú » 7 ´ 10 c ê ú ë n û 137 ë n û ënû IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.3 Slide 7 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.3 Bohr’s model of the hydrogen atom  Energy levels En of orbital electron shells in a one- electron Bohr atom are: 2 2 éZ ù éZ ù En = -ER ê ú = (-13.6 eV) ´ ê ú ënû ënû  Wave number k for transition from shell ni to shell nf :  1 1  k  R Z  2 - 2   109 737 cm-1 2  nf ni  IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.3 Slide 8 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.3 Bohr’s model of the hydrogen atom  Energy levels En of orbital electron shells in a one-electron Bohr atom are: 2 éZ ù En = -ER ê ú ënû 2 éZ ù = (-13.6 eV) ´ ê ú ënû  ER = Rydberg energy IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.3 Slide 9 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.3 Bohr’s model of the hydrogen atom Velocity of the orbital electron in the ground state n = 1 is less than 1 % of the speed of light for the hydrogen atom with Z = 1. un éZ ù 1 éZ ù éZ ù =aê ú= ê ú -3 » (7 ´ 10 ) ´ ê ú c ë n û 137 ë n û ënû Therefore, the use of classical mechanics in the derivation of the kinematics of the Bohr atom is justified. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.3 Slide 10 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.3 Bohr’s model of the hydrogen atom  Both Rutherford and Bohr used classical mechanics in their discoveries of the atomic structure and the kinematics of the electronic motion, respectively.  Rutherford introduced the idea of atomic nucleus that contains most of the atomic mass and is 5 orders of magnitude smaller than the atom.  Bohr introduced the idea of electronic angular momentum quantization. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.3 Slide 11 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.3 Bohr’s model of the hydrogen atom  Nature provided Rutherford with an atomic probe (naturally occurring alpha particles) having just the appropriate energy (few MeV) to probe the atom without having to deal with relativistic effects and nuclear penetration.  Nature provided Bohr with the hydrogen one-electron atom in which the electron can be treated with simple classical relationships. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.3 Slide 12 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.3 Bohr’s model of the hydrogen atom Energy level diagram for the hydrogen atom. n=1 ground state n>1 excited states Wave number of emitted photon 1  1 1  k   R Z  2 - 2  2   nf ni  R  109 737 cm-1 IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.3 Slide 13 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.4 Multi-electron atom  Bohr theory works well for one-electron structures but does not apply directly to multi-electron atoms because of the repulsive Coulomb interactions among the atomic electrons.  Electrons occupy allowed shells; however, the number of electrons per shell is limited to 2n2.  Energy level diagrams of multi-electron atoms resemble those of one-electron structures, except that inner shell electrons are bound with much larger energies than ER. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.4 Slide 1 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.4 Multi-electron atoms  Douglas Hartree proposed an approximation that predicts the energy levels and radii of multi-electron atoms reasonably well despite its inherent simplicity.  Hartree assumed that the potential seen by a given atomic electron is where Zeff is the effective atomic number Zeff e 1 2 V(r ) = - that accounts for the potential screening 4pe o r effects of orbital electrons (Zeff  Z ). Zeff for K-shell (n = 1) electrons is Z - 2. Zeff for outer shell electrons is approximately equal to n. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.4 Slide 2 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.4 Multi-electron atom Hartree’s expressions for atomic radii and energy level  Atomic radius In general For the K shell For the outer shell n2 n2 router shell » na0 rn = a0 r(K shell) = r1 = a0 Zeff Z-2  Binding energy In general For the K shell For outer shell 2 Zeff En  -ER 2 E(K shell)  E1  -ER (Z - 2)2 Eouter shell  -ER n IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.4 Slide 3 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.4 Multi-electron atom Energy level diagram for multi-electron atom (lead) Shell (orbit) designations: n=1 K shell (2 electrons) n=2 L shell (8 electrons) n=3 M shell (18 electrons) n=4 N shell (32 electrons) …… IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.4 Slide 4 1.2 ATOMIC AND NUCLEAR STRUCTURE 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 the atomic mass number (Rutherford-Bohr atomic model).  Protons and neutrons are commonly called nucleons and are bound to the nucleus with the strong force. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.5 Slide 1 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.5 Nuclear structure  In contrast to the electrostatic and gravitational forces that are inversely proportional to the square of the distance between two particles, the strong force between two particles is a very short range force, active only at distances of the order of a few femtometers.  Radius r of the nucleus is estimated from: r = r0 3 A , where r0 is the nuclear radius constant (1.25 fm). IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.5 Slide 2 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.5 Nuclear structure  Sum of masses of the individual components of a nucleus that contains Z protons and (A - Z) neutrons is larger than the mass of the nucleus M.  This difference in masses is called the mass defect (deficit) m and its energy equivalent mc 2 is called the total binding energy EB of the nucleus: EB  Zmpc 2  (A - Z)mnc 2 - Mc 2 IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.5 Slide 3 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.5 Nuclear structure Binding energy per nucleon (EB/A) in a nucleus varies with the number of nucleons A and is of the order of 8 MeV per nucleon. Nucleus EB/A (MeV) 2 1 H 1.1 3 2.8 1 H EB Zmp c  (A - Z)mnc - Mc 2 2 2 3 He 2.6  1 A A 4 He 7.1 1 60 27 Co 8.8 238 92 U 7.3 IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.5 Slide 4 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.6 Nuclear reactions  Nuclear reaction: Aa Bb or A(a,b)B Projectile a bombards target A which is transformed into reactants B and b.  The most important physical quantities that are conserved in a nuclear reaction are: Charge Mass number Linear momentum Mass-energy IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.6 Slide 1  1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.6 Nuclear reactions  Threshold kinetic energy (EK )athr for a nuclear reaction is calculated from the relativistic invariant and is the smallest value of projectile’s kinetic energy at which the reaction will take place: (m c 2 + m c 2 2 ) - (m c 2 + m c 2 2 ) (EK )thr = a B b A a 2mAc 2  Threshold total energy E thr a for a nuclear reaction to occur is: (mBc 2 + mbc 2 )2 - (mA2c 4 + ma2c 4 ) E thr a = 2mAc 2 mA , ma , mB, and mb are rest masses of A, a, B, and b, respectively. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.6 Slide 2 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.7 Radioactivity  Radioactivity is a process by which an unstable nucleus (parent) decays into a new nuclear configuration (daughter) that may be stable or unstable.  If the daughter is unstable, it will decay further through a chain of decays until a stable configuration is attained. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.7 Slide 1 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.7 Radioactivity  Henri Becquerel discovered natural radioactivity in 1896.  Other names used for radioactive decay are: Nuclear decay. Nuclear disintegration. Nuclear transformation. Nuclear transmutation. Radioactive decay. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.7 Slide 2 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.7 Radioactivity  Radioactive decay involves a transition from the quantum state of the parent P to a quantum state of the daughter D.  Energy difference between the two quantum states is called the decay energy Q.  Decay energy Q is emitted: In the form of electromagnetic radiation (gamma rays) or In the form of kinetic energy of the reaction products. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.7 Slide 3 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.7 Radioactivity  All radioactive processes are governed by the same formalism based on: Characteristic parameter called the decay constant  Activity A (t) defined as N(t) where N(t) is the number of radioactive nuclei at time t A (t)  N(t)  Specific activity a is the parent’s activity per unit mass: A (t) N(t) NA NA is Avogadro’s number. a   M M A A is atomic mass number. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.7 Slide 4 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.7 Radioactivity  Activity represents the total number of disintegrations (decays) of parent nuclei per unit time.  SI unit of activity is the becquerel (1 Bq = 1 s-1). Both becquerel and hertz correspond to s-1 yet hertz expresses frequency of periodic motion, while becquerel expresses activity.  The older unit of activity is the curie (1 Ci  3.7  1010 s-1) , originally defined as the activity of 1 g of radium-226. Currently, the activity of 1 g of radium-226 is 0.988 Ci. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.7 Slide 5 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.7 Radioactivity  Decay of radioactive parent P into stable daughter D:  P  P D  Rate of depletion of the number of radioactive parent nuclei NP (t) is equal to the activity A P (t) at time t: NP ( t ) t dNP (t) dNP (t ) dt  - AP (t)  - PNP (t)  NP (0) NP  -  P dt 0 where NP (0) is the initial number of parent nuclei at time t = 0. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.7 Slide 6 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.7 Radioactivity  Number of radioactive parent nuclei NP (t) as a function of time t is: - Pt NP (t)  NP (0)e  Activity of the radioactive parent AP (t) as a function of time t is: - Pt - Pt AP (t)  PNP (t)  PNP (0)e  AP (0)e where A P (0) is the initial activity at time t = 0. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.7 Slide 7 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.7 Radioactivity Parent activity A P (t) plotted against time t illustrating: Exponential decay of the activity. Concept of half life. Concept of mean life. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.7 Slide 8 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.7 Radioactivity  Half life (t1/2 )P of radioactive parent P is the time during which the number of radioactive parent nuclei decays from the initial value NP (0) at time t = 0 to half the initial value - lP (t1/2 )P NP (t = t1/2 ) = (1/ 2)NP (0) = NP (0)e  Decay constant P and the half life (t1/2 )P are related as follows ln2 lP = (t1/2 )P IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.7 Slide 9 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.7 Radioactivity  Decay of radioactive parent P into unstable daughter D which in turn decays into granddaughter G:   P  P D  D G  Rate of change dND /dt in the number of daughter nuclei D equals to supply of new daughter nuclei through decay of P given as PNP (t) and the loss of daughter nuclei D from the decay of D to G given as - DND (t) dND - t  PNP (t) - DND (t)  PNP (0) e P - DND (t) dt IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.7 Slide 10 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.7 Radioactivity  Number of daughter nuclei is: P ND (t )  NP (0) D - P e - t - e - t  P D  Activity of the daughter nuclei is: NP (0)P D - Pt D AD (t )  D - P  e - e - Dt = AP (0)  D - P e - Pt - e - Dt    D P     1 - Pt - Dt  AP ( 0) e -e  AP (t ) 1 - e -( D -P )t , 1- D - P D IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.7 Slide 11 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.7 Radioactivity At t  tmax parent and daughter activities are equal and the daughter activity reaches its maximum dAD =0 dt t=tmax and D ln P tmax  D - P   Parent and daughter activities against time for P  P D  D G. IAEA Review of Radiation Oncology Physics: A Handbook for Teachers and Students - 1.2.7 Slide 12 1.2 ATOMIC AND NUCLEAR STRUCTURE 1.2.7 Radioactivity   Special considerations for the P   P D  D G relationship: lD  For lD < lP or (t1/2 )D > (t1/2 )P AD = AP lD - lP { 1- e-( lD -lP )t } General relationship (no equilibrium)  For lD > lP or (t1/2 )D < (t1/2 )P AD D  Transient equilibrium for t  tmax AP D - P  For lD >> lP or (t1/2 )D