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Introduction to Nuclear Engineering, Fall 2020
Radioactive Decay
Fall, 2020
Kyoung-Jae Chung
Department of Nuclear Engineering
Seoul National University
2 Introduction to Nuclear Engineering, Fall 2020
Activity
Radiation: transportation of mass and energy through space Radioactivity: the process through which nuclei spontaneously emit subatomic
particles Radioactivity was discovered by Henri Becquerel in 1896. The terms radioactivity was suggested by Marie Curie about four years later.
Activity: the number of radioactive decays in a particular time SI unit: becquerel (Bq), 1 decay per second Old unit: curie (Ci), the activity of 1 g of radium-226 The becquerel is a measure only of quantity of radioactive material
1 Ci = 3.7x1010 Bq
Specific activity: the activity per unit mass
Unit conversion
3 Introduction to Nuclear Engineering, Fall 2020
Exponential decay law (Rutherford and Soddy, 1902)
Radioactive decay is a random process and has been observed to follow Poisson distribution. What this essentially means is that the rate of decay of radioactive nuclei in a large sample depends only on the number of decaying nuclei in the sample:
Activity
𝜆𝜆𝑑𝑑 : decay constant−𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
∝ 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
= −𝜆𝜆𝑑𝑑𝑑𝑑
𝐴𝐴 = −𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
= 𝜆𝜆𝑑𝑑𝑑𝑑 𝐴𝐴 = 𝐴𝐴0𝑒𝑒−𝜆𝜆𝑑𝑑𝑡𝑡 = 𝐴𝐴0𝑒𝑒−𝑡𝑡/𝜏𝜏
Half-life 𝑇𝑇1/2 =ln 2𝜆𝜆𝑑𝑑
= 0.693𝜏𝜏𝜏𝜏 : decay time
4 Introduction to Nuclear Engineering, Fall 2020
Mean lifetime
Given an assembly of elements, the number of which decreases ultimately to zero, the mean lifetime, 𝜏𝜏 (also called simply the lifetime) is the expected value of the amount of time before an object is removed from the assembly. The mean lifetime is the arithmetic mean of the individual lifetimes.
𝜏𝜏 = 𝑇𝑇 = �0
∞𝑑𝑑 exp −𝜆𝜆𝑑𝑑𝑑𝑑 𝜆𝜆𝑑𝑑𝑑𝑑𝑑𝑑 =
1𝜆𝜆𝑑𝑑
Expectation
Decay probability between 𝑑𝑑 and 𝑑𝑑 + 𝑑𝑑𝑑𝑑
Probability of surviving until time 𝑑𝑑
5 Introduction to Nuclear Engineering, Fall 2020
Exponential decay
6 Introduction to Nuclear Engineering, Fall 2020
Exponential decay
7 Introduction to Nuclear Engineering, Fall 2020
Specific activity
The specific activity of a sample is defined as its activity per unit mass, for example, Bq g–1 or Ci g–1.
Since the number of atoms per gram of the nuclide is 𝑑𝑑 = 6.02 × 1023/𝑀𝑀 (𝑀𝑀: atomic weight), the specific activity is given by
𝑆𝑆𝐴𝐴 =6.02 × 1023 𝜆𝜆𝑑𝑑
𝑀𝑀≈
4.17 × 1023
𝐴𝐴𝑇𝑇1/2
8 Introduction to Nuclear Engineering, Fall 2020
Serial radioactive decay
We can calculate the activity of a sample in which one radionuclide produces one or more radioactive offspring in a chain.
𝑑𝑑𝑑𝑑2𝑑𝑑𝑑𝑑 = 𝜆𝜆1𝑑𝑑1 − 𝜆𝜆2𝑑𝑑2 = 𝜆𝜆1𝑑𝑑10𝑒𝑒−𝜆𝜆1𝑡𝑡 − 𝜆𝜆2𝑑𝑑2 𝑑𝑑
𝑑𝑑𝑑𝑑1𝑑𝑑𝑑𝑑
= −𝜆𝜆1𝑑𝑑1 𝑑𝑑1(𝑑𝑑) = 𝑑𝑑10𝑒𝑒−𝜆𝜆1𝑡𝑡
𝑑𝑑1 ⟹ 𝑑𝑑2 ⟹ 𝑑𝑑3 ⟹ ⋯𝜆𝜆1 𝜆𝜆2 𝜆𝜆3
𝑑𝑑20 = 0𝑑𝑑2(𝑑𝑑) = 𝑑𝑑10
𝜆𝜆1𝜆𝜆2 − 𝜆𝜆1
𝑒𝑒−𝜆𝜆1𝑡𝑡 − 𝑒𝑒−𝜆𝜆2𝑡𝑡
The activity of the daughter nuclei
𝐴𝐴2 𝑑𝑑 = 𝜆𝜆2𝑑𝑑2 𝑑𝑑 = 𝜆𝜆2𝑑𝑑10𝜆𝜆1
𝜆𝜆2 − 𝜆𝜆1𝑒𝑒−𝜆𝜆1𝑡𝑡 − 𝑒𝑒−𝜆𝜆2𝑡𝑡
= 𝐴𝐴1(𝑑𝑑)𝜆𝜆2
𝜆𝜆2 − 𝜆𝜆11 − 𝑒𝑒−(𝜆𝜆2−𝜆𝜆1)𝑡𝑡
9 Introduction to Nuclear Engineering, Fall 2020
Serial radioactive decay
No equilibrium (𝑇𝑇1 < 𝑇𝑇2): When the daughter, initially absent (𝑑𝑑20 = 0), has a longer half-life than the parent, its activity builds up to a maximum and then declines. Because of its shorter half life, the parent eventually decays away and only the daughter is left. No equilibrium occurs.
10 Introduction to Nuclear Engineering, Fall 2020
Serial radioactive decay
Transient equilibrium (𝑇𝑇1 ≳ 𝑇𝑇2)
𝐴𝐴2 𝑑𝑑 = 𝜆𝜆2𝑑𝑑2 𝑑𝑑 = 𝐴𝐴1 𝑑𝑑𝜆𝜆2
𝜆𝜆2 − 𝜆𝜆11 − 𝑒𝑒− 𝜆𝜆2−𝜆𝜆1 𝑡𝑡 →
𝜆𝜆2𝜆𝜆2 − 𝜆𝜆1
𝐴𝐴1 𝑑𝑑
11 Introduction to Nuclear Engineering, Fall 2020
Serial radioactive decay
Secular equilibrium (𝑇𝑇1 ≫ 𝑇𝑇2 or 𝜆𝜆2 ≫ 𝜆𝜆1)
𝐴𝐴2 𝑑𝑑 = 𝜆𝜆2𝑑𝑑2 𝑑𝑑 = 𝐴𝐴1 𝑑𝑑𝜆𝜆2
𝜆𝜆2 − 𝜆𝜆11 − 𝑒𝑒− 𝜆𝜆2−𝜆𝜆1 𝑡𝑡 ≈ 𝐴𝐴1 𝑑𝑑 1 − 𝑒𝑒−𝜆𝜆2𝑡𝑡 → 𝐴𝐴1 𝑑𝑑
𝐴𝐴1 = 𝐴𝐴10𝑒𝑒−𝜆𝜆1𝑡𝑡
Total activity ~ 2𝐴𝐴1
A chain of 𝑛𝑛 short-lived radionuclides can all be in secular equilibrium with a long-lived parent. Then the activity of each member of the chain is equal to that of the parent and the total activity is 𝑛𝑛 + 1 times the activity of the original parent.
12 Introduction to Nuclear Engineering, Fall 2020
Bateman equation (1910)
General radioactive chain rule
Initial conditions
Solution
𝑑𝑑1(0) ≠ 0
13 Introduction to Nuclear Engineering, Fall 2020
Carbon dating
Formation of radiocarbon
14 Introduction to Nuclear Engineering, Fall 2020
Radioactive series
Most of the radionuclides found in nature are members of four radioactive series, with each series consisting of a succession of daughter products all ultimately derived from a single parent nuclide.
The reason that there are exactly four series follows from the fact that alpha decay reduces the mass number of a nucleus by 4.
15 Introduction to Nuclear Engineering, Fall 2020
Radioactive decay of Th-232 and U-238
16 Introduction to Nuclear Engineering, Fall 2020
Radon and radon daughters
The potential health hazard arises when radon in the air decays, producing nongaseous radioactive daughters. When inhaled, the airborne daughters can be trapped in the respiratory system, where they are likely to decay before being removed by normal lung-clearing mechanisms of the body.
Short-lived alpha emitters
17 Introduction to Nuclear Engineering, Fall 2020
100
101
102
103
104
105
106
0
10
20
30
40
50
60
70
80
90
100
Rn-222
Po-218
Pb-214
Bi-214
Po-214
Pb-210
Homework
J. Turner, Atoms, Radiation, and Radiation Detection, Wiley (2007), chapter 4Problems: 12, 13
Setup the decay equations and solve them (analytically or numerically). Plot the activities of each nuclide and the total activity versus time (up to 10 days). Initial activity of Rn-222 is 100 Bq and the others are zero.
Stable
10 sec
𝐴𝐴(𝑑𝑑)