Atomic Structure and Radioactive Decay PDF

Atomic Structure and Radioactive Decay BME 229 Fall 2024 1 Electromagnetic (EM) Radiation Visible light, radio waves, x-rays No mass; unaffected by electrical or magnetic fields; constant speed in a given medium Travels in straight lines; trajectory...

Atomic Structure and Radioactive Decay BME 229 Fall 2024 1 Electromagnetic (EM) Radiation Visible light, radio waves, x-rays No mass; unaffected by electrical or magnetic fields; constant speed in a given medium Travels in straight lines; trajectory can be altered by interaction with matter Absorption – reduction of the radiation Scattering – change in trajectory 2  EM Wave Characteristics Waves characterized by amplitude, wavelength (), frequency (f ), and period (T) Speed (v or c), wavelength, and frequency related by  = vT OR v =  f EM wavelengths typically measured in nanometers (10-9 m); frequency expressed in hertz (Hz) (1 Hz = 1 cycle/sec = 1 sec-1) For EM waves, v is the speed of light. 3 Sinusoidal EM Wave 4  Sinusoidal Wave Function (From PCS 125) The sinusoidal wave function can be written in the form: y(x,t) = A sin(kx – t + ) 2 2 Wave number: k = Angular Frequency:  = 2 f =  T  is called the phase constant This is a 1D wave function. It could be expressed as a cosine function too. In the case of a 3D wave, x is replaced by position vector r ( x, y, z) Sinusoidal Wave Function - Generalized (kx t +  ) y( x, t ) = A sin / cos  (t kx +  ) 8 different forms for the sinusoidal wave function!  Sinusoidal EM Wave Electromagnetic radiation, in which energy is carried by oscillating electrical and magnetic fields traveling through space at the speed of light. 7 The Spectrum of the EM Waves 8  The Spectrum of the EM Waves Note the overlap between types of waves Visible light is a small portion of the spectrum Types are distinguished by frequency or wavelength 9  Notes on the EM Spectrum Radio Waves – Wavelengths of more than 104 m to about 0.1 m – Used in radio and television communication systems Microwaves – Wavelengths from about 0.3 m to 10-4 m – Well suited for radar systems – Microwave ovens are an application 10  Notes on the EM Spectrum, 2 Infrared waves – Wavelengths of about 10-3 m to 7×10-7 m – Incorrectly called “heat waves”! – Produced by hot objects and molecules – Readily absorbed by most materials Visible light – Part of the spectrum detected by the human eye – Most sensitive at about 5.5×10-7 m (yellow-green) 11  Visible Light Different wavelengths correspond to different colors The range is from red (λ ~ 7 × 10-7 m) to violet (λ ~ 4 × 10-7 m) 1 Visible Light, cont. 13  Notes on the EM Spectrum, 3 Ultraviolet light – Covers about 4×10-7 m to 6×10-10 m – Sun is an important source of UV light – Most UV light from the sun is absorbed in the stratosphere by ozone X-rays – Wavelengths of about 10-8 m to 10-12 m – Most common source is acceleration of high- energy electrons striking a metal target – Used as both diagnostic and therapeutic tools in medicine 14  Notes on the EM Spectrum, final Gamma rays – Wavelengths of about 10-10 m to 10-14 m – Emitted by radioactive nuclei – Highly penetrating and cause serious damage when absorbed by living tissue – Has both therapeutic and diagnostic applications in medicine Imaging an object with different portions of the spectrum can produce different information 15  Wavelengths and Information These are images of the Crab Nebula (a supernova remnant) They are (clockwise from upper left) taken with – x-rays – visible light – radio waves – infrared waves Supernova: The explosion of a star that has reached the end of its life. Supernovas radiate lots of energy into space and can briefly outshine the entire galaxies. 16  Prefixes 1  Photons Electromagnetic radiation can be described in terms of a stream of photons, which are massless particles each traveling in a wave-like pattern and moving at the speed of light. Each photon contains a certain amount (or bundle) of energy, and all electromagnetic radiation consists of these photons. The only difference between the various types of electromagnetic radiation is the amount of energy found in the photons. Radio waves have photons with low energies, microwaves have a little more energy than radio waves, infrared has still more, then visible, ultraviolet, x-rays, and the most energetic of all, gamma-rays. 18  EM Particle Characteristics When interacting with matter, EM radiation can exhibit particle-like behaviour Particle-like bundles of energy called photons; energy is given by E = h f = hv /  where h is the Planck’s constant (h = 6.6261×10-34 m2.kg/s) Energies of photons commonly expressed in electron volts (eV) 1 eV  1.6×10−19 J 19  EM Radiation in Imaging Gamma rays – originate within nuclei of radioactive atoms; used to image the distribution of radiopharmaceuticals X-rays – produced outside the nucleus; used in radiography and computed tomography Visible light – produced in detecting x- and gamma rays; used for observation and interpretation of images Radiofrequency EM in the FM radiowave region – used as the transmission and reception signal for MRI 20  Ionizing vs. Nonionizing Radiations EM radiation of higher frequency than near-ultraviolet region of spectrum carries enough energy per photon to remove bound electrons from atomic shells, producing ionized atoms and molecules Radiation in this region is called ionizing radiation Visible light, infrared, radio and TV broadcasts is called nonionizing radiation There are three main kinds of ionizing radiation in Biomedical Physics: alpha particles, which include two protons and two neutrons beta particles, which are essentially electrons (or positrons) gamma rays and x-rays, which are pure EM energy (photons) 21 http://www.epa.gov/rpdweb00/understand/ionize_nonionize.html 22  Particulate Radiation Protons – found in nuclei of all atoms; single positive charge Electrons – exist in atomic orbits; emitted by nuclei of some radioactive atoms (referred to as beta-minus particles (-), negatrons, or simply “beta particles”) Positrons – positively charged electrons (+); emitted from some nuclei during radioactive decay Neutrons – uncharged nuclear particle; released by nuclear fission and used for radionuclide production 23  Mass Energy Equivalence Einstein’s theory of relativity states that mass and energy are interchangeable E = mc 2 where E represents the energy equivalent to mass m at rest and c is the speed of light in vacuum 24  Fundamental Properties of Particulate Radiation Relative Approx. E Particle Symbol Charge (e) (MeV) Proton p +1 938 Electron e- -1 0.511 Positron e+ +1 0.511 Neutron n 0 940 25  Structure of the Atom Smallest division of an element in which the chemical identity of the element is maintained Composed of extremely dense positively charged nucleus containing protons and neutrons and an extra-nuclear cloud of light negatively charged electrons Electrically neutral in its nonionized state 26  Bohr Model of the Atom Electrons orbit around a dense positively charged nucleus at fixed distances Each electron occupies a discrete energy state in a given electron shell Shells assigned the letters K, L, M, N, …, with K denoting the innermost shell; also assigned quantum numbers 1, 2, 3, 4, …, with 1 designating the K shell Each shell can contain a maximum of (2n2) electrons, where n is the quantum number of the shell 27 Electron Shells Electron Energy Level 28  Binding Energy of Electron Energy required to remove an electron completely from an atom By convention, binding energies are negative with increasing magnitude for electrons in shells closer to the nucleus Binding energy of electrons in a particular orbit increases with the number of protons in the nucleus (i.e., atomic number, Z) 29 Binding Energy of Electron 30  Atomic Nucleus Composed of protons and neutrons (collectively, nucleons) Number of protons is atomic number (Z); total number of protons and neutrons (N) is the mass number (A) Notation specifying an atom with chemical symbol X is A Z X N 31  Nuclear Energy Levels Nucleus has energy levels that are analogous to orbital electron shells; often much higher in energy Lowest energy state is called the ground state Nuclei with energy in excess of the ground state are said to be in an excited state Excited states that exist longer than 10-12 seconds are called metastable or isomeric states; denoted by the letter m after the mass number of the atom (e.g., Tc-99m) 32  Nuclear Families Family Nuclides with Same Example Isotopes Atomic number (Z) I-131 and I-125 Isobars Mass number (A) Mo-99 and Tc-99 Isotones Number of neutrons (A – Z) 53I-131 and 54Xe- 132 Isomers Atomic and mass numbers Tc-99m and Tc-99; but different energy states E = 142 keV 33  Nuclear Stability Only certain combinations of neutrons and protons in the nucleus are stable A higher neutron-to-proton ratio is required in heavy elements to offset the repulsive electrostatic (coulombic) forces between protons by providing increased separation of protons Nuclei with odd number of neutrons and odd number of protons tend to be unstable 34 Nuclear Stability 35  Nuclear Structure The nucleus of an atom consists of nucleons such as: – Neutrons – Protons The number of protons in the nucleus is given by the atomic number Z. In a neutral atom, the number of protons equals the number of electrons in orbits around the nucleus The total number of protons and neutrons in the nucleus is given by the atomic mass number or nucleon number A The number of neutrons in nucleus is N Thus, the symbol for atom X is A Z X and A= Z +N 36  Nuclear Structure Nuclei that have the same number of protons Z, but different A’s are isotopes; Examples: 92 235 238 236 234 The radius of U 92 U 92 U 92 U potassium (A=39) is: r  4.1 10−15 m ◆ Protons and neutrons are clustered together in atom to form a spherical region, whose radius depends on the atomic mass number A by: −15 r  (1.2  10 1/ 3 m) A 37 Problem: For lead (Pb) find (a) the net electrical charge of nucleus, (b) the number of neutrons, (c) the number of nucleons, (d) the approximate radius of nucleus, and (e) the nuclear density. For lead, we have A = 208 and Z = 82. a. The net electrical charge of the nucleus is equal to the total number of protons multiplied by the charge on a single proton: qnet = 82(+1.6 10−19 C ) = +1.3110−17 C b. The number of neutrons is N = A – Z = 208 – 82 = 126 c. By inspection, the number of nucleons is A = 208 d. The approximate radius of the nucleus r = (1.2  10 –15 m) A1/ 3 = (1.2  10 –15 m)(208) 1/3 = 7.1  10 –15 m 38 Problem (con’d): For lead (Pb) find (a) the net electrical charge of nucleus, (b) the number of neutrons, (c) the number of nucleons, (d) the approximate radius of nucleus, and (e) the nuclear density. For lead, we have A = 208 and Z = 82. e. The nuclear density is the mass per unit volume of the nucleus. The total mass of the nucleus can be found by multiplying the mass of a single nucleon by the total number A of nucleons in the nucleus. Treating the nucleus as a sphere of radius r, the nuclear density is mtotal mnucleon A mnucleon A mnucleon  = = = = V 4 3 r 3 4  (1.2  10 –15 m) A 1/ 3 3 4 3  (1.2  10 –15 m) 3 3 1.67  10 –27 kg  = 4 = 2.3  10 17 kg / m 3 3  ( 1. 2  10 –15 3 m) 39  The Atomic Mass Unit (amu OR u) The atomic mass unit, u, is one-twelfth of the mass of 12C isotope of atom of carbon. The atomic mass unit, u, is about the mass of one proton or neutron. The atomic masses of the elements in u are average masses, taking into account the different isotopes that exist. The energy equivalent of one atomic mass unit: u = 1.4924×10-10 J = 931.5 MeV 40  The Mass Defect The standard is that one atom of carbon 12, the isotope of carbon with 6 protons and 6 neutrons, has a mass of exactly 12 u The mass of a proton is 1.00728 u; a neutron is 1.00866 u The mass of 6 protons and 6 neutrons is 12.0956 u, therefore the mass of a carbon nucleus is less than the sum of its particles The mass defect (m) is the difference in the mass of any nucleus and the sum of the separate masses of its protons and neutrons Einstein showed that mass and energy are really two different forms of the same thing; the "vanishing" mass of the protons and neutrons is simply converted to energy according to Einstein's equation E=m c2 The "binding energy" of a particular isotope is the amount of energy released at its creation. One can calculate it by finding the amount of mass that "disappears" by using Einstein's equation. The binding energy is also the amount of energy that needs to be added to a nucleus to break it apart into protons and neutrons again. 41  Binding energy of nucleus The binding energy reaches maximum at 8.7 MeV / nucleon (A = 60) Radioactive elements 42  27 Problem: Find the binding energy for aluminum, 13 Al (in MeV and J) (A = 26.981539 u). Aluminum contains Z = 13 protons and N = A – Z = 27 – 13 = 14 neutrons The mass of a proton is 1.007825 u. The mass of a neutron is 1.008665 u. The mass of an aluminum atom is given as 26.981539 u. Thus, the mass defect m is:  m = 13 (1.007825 u) + 14 (1.008665 u) – 26.981539 u m = 0.241496 u Since 1 u = 931.5 MeV, this mass defect corresponds to a binding energy of 931.5MeV 0.241496u[ ] = 225 MeV 1u The binding energy in Joules is: 225  106  1.6  10-19 = 3.6  10-11 J 43  The Nuclear Force n>p - The naturally occurring nuclei have a number N of neutrons that equals or exceeds the number Z of protons, with few exceptions Neutrons are electrically neutral; protons have n