Radioactive Decay and Radioactive Series

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1 Radioactive Decay and Radioactive Series by Michele Laino June 7, 2015 Abstract In this short paper I will explain some general aspects of radioactive decays, furthermore, some useful tables, concerning the three main radioactive series with type of nuclide, type of disintegration, half-life, disintegration constant and the energy of the involved particle in that type of radioactive decay, are presented. 1 Decay Probability. Single Substance Case The probability which a decay can occur from a specified atom, within the time interval dt is λdt, where λ is a constant quantity, whose dimensions are the reciprocal of a time, and whose name is decay constant. Such constant λ is related to a determined substance and to a type of decay, furthermore, it is independent from the age of that atom. The decay law, is a characteristic law of the random processes, and it can be applied to all phenomena of radioactive decay, for example: alpha decay, beta decay, K-capture, L-capture, spontaneous nuclear fission, and finally to the emission of electromagnetic radiation from atoms or excited nuclei. The simple application of the radioactive law, occur when we deal with a single radioactive substance, which, at the initial time, is composed by N (0) atoms, and such number is very large, so we can consider the same number of atoms at any time t, namely N (t), as a continuous variable. Under those hypothesis, we can write the decreasing of the number of atoms along the time t, as below: dn = λndt That equation, can be integrated with the initial number atoms N (0), and its solution is: N (t) = N (0) e λt 1

2 The quantity: τ = 1 λ is called half-life of that particular substance, whereas the time T within the initial number of atoms is halved, is called the half-time of that substance. So, by definition, we can write: e λt = e T/τ = 1 2, λt = T τ = ln 2 = Here is the simple computation which gives the half-life of a certain substance: τ = 1 N (0) 0 λtn (t) dt = 1 N (0) 0 N (0) e λt λtdt = 1 λ which computation, shows us, that τ, is really an average time. 2 Many Substances Case Frequently, one substance decays into another one, which in turn is radioactive too. In such case we say that between those two substances there is a genetic relationship: the first substance is called mother, whereas the second one is called daughter. That connection is not limited to the relationship motherdaughter: it can be extended, sometimes, for many generations. For some cases one radioactive substance can decay, through one or another process, for example alpha-emission or beta-emission, causing the production of two different daughter substances: in that case we have a dual decay or a branch. Examples of long chain of radioactive decays, are provided by the three natural radioactive families, namely: 1. The Actinium Series; 2. The Thorium Series; 3. The Uranium Series. Here are the corresponding tables, into which some useful data are collected. The half-life may be in days, hours, minutes, or seconds, abbreviated as y, d, h, m, s, respectively. 2

3 Table 2.1 The Actinium Series Radioactive Nuclide Type of Half-Life Disintegration Particle species disintegration constant energy ( sec 1 ) (MeV) Actinouranium (AcU) 92U 235 α y m Uranium Y (UY) 90Th 231 β 25.6 h Protoactinium (Pa) 91Pa 231 α y m Actinium (Ac) 89Ac 227 α, β 21.6 y α : 4.94 β : Radioactinium (RdAc) 90Th 227 α d m Actinium K (AcK) 87Fr 223 α, β 22 m α : 5.34 β : 1.2 Actinium X (AcX) 88Ra 223 α d Astatine At 219 α, β 0.9 m α : 6.27 Ac Emanation (An) 86Em 219 α 3.92 s m Bismuth Bi 215 α, β 8 m ? Actinium A (AcA) 84Po 215 α, β s α : 7.37 Actinium B (AcB) 82Pb 211 β 36.1 m Astatine At 215 α 10 4 s Actinium C (AcC) 83Bi 211 α, β 2.15 m α : m Actinium C (AcC ) 84Po 211 α 0.52 s m Actinium C (AcC ) 81Tl 207 β 4.79 m Actinium D (AcD) 82Pb 207 Stable 3

4 Table 2.2 The Thorium Series Radioactive Nuclide Type of Half-Life Disintegration Particle species disintegration constant energy ( sec 1 ) (MeV) Thorium (Th) 90Th 232 α y Mesothorium1 (MsTh1) 88Ra 228 β 6.7 y Mesothorium2 (MsTh2) 89Ac 228 β 6.13 h Radiothorium (RdTh) 90Th 228 α y m Thorium X (ThX) 88Ra 224 α 3.64 d m Th Emanation (Tn) 86Em 220 α 51.5 s Thorium A (ThA) 84Po 216 α, β 0.16 s Thorium B (ThB) 82Pb 212 β 10.6 h Astatine-216 ( At 216) 85At 216 α s Thorium C (ThC) 83Bi 212 α, β 60.5 m α : m β : 2.25 Thorium C (ThC ) 84Po 212 α s Thorium C (ThC ) 81Tl 208 β 3.10 m Thorium D (ThD) 82Pb 208 Stable 4

5 Table 2.3 The Uranium Series Radioactive Nuclide Type of Half-Life Disintegration Particle species disintegration constant energy ( sec 1 ) (MeV) Uranium I (UI) 92U 238 α y Uranium X 1 (UX 1 ) 90Th 234 β 24.1 d Uranium X 2 (UX 2 ) 91Pa 234 β 1.18 m Uranium Z (UZ) 91Pa 234 β 6.7 h Uranium II (UII) 92U 234 α y Ionium (Io) 90Th 230 α y m Radium (Ra) 88Ra 226 α 1620 y m Ra Emanation (Rn) 86Em 222 α 3.82 d Radium A (RaA) 84Po 218 α, β 3.05 m α : β :? Radium B (RaB) 82Pb 214 β 26.8 m Astatine-218 ( At 218) 85At 218 α s Radium C (RaC) 83Bi 214 α, β 19.7 m α : 5.51 m β : 3.17 Radium C (RaC ) 84Po 214 α s Radium C (RaC ) 81Tl 210 β 1.32 m Radium D (RaD) 82Pb 210 β 19.4 y Radium E (RaE) 83Bi 210 β 5.0 d Radium F (RaF) 84Po 210 α d Thallium-206 F ( Tl 206) 81Tl 206 β 4.2 m Radium G (RaG) 82Pb 206 Stable 3 Nuclear Branching Some radioactive substances decay through more than one way, for example β +, and β emissions, or, as in the previous tables α, and β emissions. Let s consider an α, and β branching and be λ α dt the probability of α emission of an atom within the time interval dt, and λ β dt the probability of β emission of the same atom within the time interval dt. Then the total probability of decay, of that atom, within the time interval dt for α or β emission, is (λ α + λ β ) dt. So we can write: dn dt = (λ α + λ β ) N 5

6 and the half-life of our substance, defined as the time after that the current quantity reduce itself from 1 to 1/e, is: 1 τ = λ α + λ β The ratio between the number of α particles and the number of β particles, is called branching ratio, and it is equal to λ α /λ β. Sometimes, the subsequent quantities: τ α = 1 λ α, τ β = 1 λ β which are called, improperly, the half-life for α decay, and β decay respectively, are introduced. Using those quantities, we can rewrite the half-life of our substance, as below: 1 τ = τ α τ β * * * 6

Radioactive Decay and Radiometric Dating

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