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3/2003 Rev 1 I.2.8 – slide 1 of 31 Session I.2.8 Part I Review of Fundamentals Module 2Basic Physics and Mathematics Used in Radiation Protection Session 8Decay Chains and Equilibrium IAEA Post Graduate Educational Course Radiation Protection and Safety of Radiation Sources

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3/2003 Rev 1 I.2.8 – slide 2 of 31 Introduction Radioactive serial decay and equilibrium will be discussed Students will: learn the differences between secular and transient equilibrium identify when no equilibrium is possible understand how series decay works calculate ingrowth of a decay product from a radioactive parent

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3/2003 Rev 1 I.2.8 – slide 3 of 31 Content Secular equilibrium Transient equilibrium Case of no equilibrium Radioactive decay series Ingrowth of decay product from a parent radionuclide

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3/2003 Rev 1 I.2.8 – slide 4 of 31 Overview Radioactive decay chains (parent and single decay product) and equilibrium situations will be discussed

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3/2003 Rev 1 I.2.8 – slide 5 of 31 Types of Radioactive Equilibrium SecularHalf-life of parent much greater (> 100 times) than that of decay product

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3/2003 Rev 1 I.2.8 – slide 6 of 31 Types of Radioactive Equilibrium TransientHalf-life of parent only a little greater than that of decay product

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3/2003 Rev 1 I.2.8 – slide 7 of 31 90 Sr 90 Y 90 Zr Sample Radioactive Series Decay where 90 Sr is the parent (half-life = 28 years) and 90 Y is the decay product (half-life = 64 hours)

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3/2003 Rev 1 I.2.8 – slide 8 of 31 Differential Equation for Radioactive Series Decay = Sr N Sr - Y N Y dN Y dt Parent and Single Decay Product

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3/2003 Rev 1 I.2.8 – slide 9 of 31 Parent and Single Decay Product Differential Equation for Radioactive Series Decay N Y (t) = (e - t - e - t ) Sr Y Sr N Sr Y - Sr o Recall that Sr N o Sr = A o Sr which equals the initial activity of 90 Sr at time t = 0

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3/2003 Rev 1 I.2.8 – slide 10 of 31 General Equation for Radioactive Series Decay Y N Y (t) = (e - t - e - t ) Sr Y Y - Sr Y Sr N Sr o Activity of 90 Sr at time t = 0 Activity of 90 Y at time t or A Y (t)

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3/2003 Rev 1 I.2.8 – slide 11 of 31 Buildup of a Decay Product under Secular Equilibrium Conditions Secular Equilibrium A Y (t) = (1 - e - t ) Y A Sr

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3/2003 Rev 1 I.2.8 – slide 12 of 31 Secular Equilibrium Sr N Sr = Y N Y A Sr = A Y

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3/2003 Rev 1 I.2.8 – slide 13 of 31 Decay of 226 Ra to 222 Rn Secular Equilibrium A Rn (t) = A o (1 - e - t ) Rn Ra

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3/2003 Rev 1 I.2.8 – slide 14 of 31 226 Ra (half-life 1600 years) decays to 222 Rn (half-life 3.8 days). If initially there is 4000 kBq of 226 Ra in a sample and no 222 Rn, calculate how much 222 Rn is produced: a.after 7 half-lives of 222 Rn b.at equilibrium Sample Problem 1

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3/2003 Rev 1 I.2.8 – slide 15 of 31 The number of atoms of 222 Rn at time t is given by: Solution to Sample Problem = Ra N Ra - Rn N Rn dN Rn dt Solving: N Rn (t) = (1 - e - t ) Rn Ra N Ra Rn

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3/2003 Rev 1 I.2.8 – slide 16 of 31 Multiplying both sides of the equation by Rn : A Rn (t) = A Ra (1 - e - t ) Rn Solution to Sample Problem = 4000 x (0.992) = 3968 kBq of 222 Rn Let t = 7 T Rn Rn t = (0.693/T Rn ) x 7 T Rn = 0.693 x 7 = 4.85 Rn t = (0.693/T Rn ) x 7 T Rn = 0.693 x 7 = 4.85 e -4.85 = 0.00784 A Rn (7 half-lives) = 4000 kBq x (1 - 0.00784 )

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3/2003 Rev 1 I.2.8 – slide 17 of 31 Solution to Sample Problem 4000 kBq + 4000 kBq = 8000 kBq Rn N Rn = Ra N Ra or A Rn = A Ra = 4000 kBq Rn N Rn = Ra N Ra or A Rn = A Ra = 4000 kBq Note that the total activity in this sample is: Rn N Rn + Ra N Ra or A Rn + A Ra = Rn N Rn + Ra N Ra or A Rn + A Ra = Now, at secular equilibrium:

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3/2003 Rev 1 I.2.8 – slide 18 of 31 Transient Equilibrium D N D = D - P D P N P

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3/2003 Rev 1 I.2.8 – slide 19 of 31 Transient Equilibrium A D = D - P A P D

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3/2003 Rev 1 I.2.8 – slide 20 of 31 Time for Decay Product to Reach Maximum Activity Transient Equilibrium t mD = D - P ln D P

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3/2003 Rev 1 I.2.8 – slide 21 of 31 Example of Transient Equilibrium 132 Te Decays to 132 I Transient Equilibrium

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3/2003 Rev 1 I.2.8 – slide 22 of 31 The principle of transient equilibrium is illustrated by the Molybdenum-Technetium radioisotope generator used in nuclear medicine applications. Given that the generator initially contains 4000 MBq of 99 Mo (half-life 66 hours) and no 99m Tc (half-life 6 hours) calculate the: a. time required for 99m Tc to reach its maximum activity b. activity of 99 Mo at this time, and c. activity of 99m Tc at this time Sample Problem

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3/2003 Rev 1 I.2.8 – slide 23 of 31 Note that only 86% of the 99 Mo transformations produce 99m Tc. The remaining 14% bypass the isomeric state and directly produce 99 Tc Sample Problem

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3/2003 Rev 1 I.2.8 – slide 24 of 31 Tc = 0.693/(6 hr) = 0.12 hr -1 Tc = 0.693/(6 hr) = 0.12 hr -1 Mo = 0.693/(66 hr) = 0.011 hr -1 Mo = 0.693/(66 hr) = 0.011 hr -1 Solution to Sample Problem t mTc = Tc - Mo ln Tc Mo t mTc = 0.12 – 0.011 ln 0.12 0.011 = 21.9 hrs a)

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3/2003 Rev 1 I.2.8 – slide 25 of 31 (b) The activity of 99 Mo is given by A(t) = A o e - t = 4000 x e (-0.011/hr x 21.9 hr) = 4000 x (0.79) = 3160 MBq Solution to Sample Problem

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3/2003 Rev 1 I.2.8 – slide 26 of 31 c) The activity of 99m Tc at t = 21.9 hrs is given by: Solution to Sample Problem A Tc (t) = (e -(0.011)(21.9) - e -(0.12)(21.9) ) (0.12 – 0.011) (0.12)(4000 MBq)(0.86) = (3787) (0.785 - 0.071) = 2704 MBq of 99m Tc A Tc (t) = (e - t - e - t ) Mo Tc Tc - Mo Tc A Mo (see slide 10)

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3/2003 Rev 1 I.2.8 – slide 27 of 31 Solution to Sample Problem The maximum activity of 99m Tc is achieved at 21.9 hours which is nearly 1 day.

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3/2003 Rev 1 I.2.8 – slide 28 of 31 Types of Radioactive Equilibrium No EquilibriumHalf-life of parent less than that of decay product

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3/2003 Rev 1 I.2.8 – slide 29 of 31 No Equilibrium

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3/2003 Rev 1 I.2.8 – slide 30 of 31 Summary Secular equilibrium was defined Transient equilibrium was defined Case of no equilibrium was defined Series decay equations were developed Decay examples were discussed Problems in secular and transient equilibrium were solved

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3/2003 Rev 1 I.2.8 – slide 31 of 31 Where to Get More Information Cember, H., Johnson, T. E., Introduction to Health Physics, 4th Edition, McGraw-Hill, New York (2008) Martin, A., Harbison, S. A., Beach, K., Cole, P., An Introduction to Radiation Protection, 6 th Edition, Hodder Arnold, London (2012) Jelley, N. A., Fundamentals of Nuclear Physics, Cambridge University Press, Cambridge (1990) Firestone, R.B., Baglin, C.M., Frank-Chu, S.Y., Eds., Table of Isotopes (8 th Edition, 1999 update), Wiley, New York (1999)

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