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Unformatted text preview: Lesson 09 Radioactivity  02 MP200 Radiation Physics  2010 Duke Medical Physics Graduate Program 1 Exponential Decay and Decay Constant Consider a large number of radioactive atoms N at time t = 0. At any time t , number of atoms left is N . n = number of atoms disintegrated in time dt . n ∝ Ndt n = λNdt λ = decay constant Since all the kinds of decays are included here, the proportional con stant λ is called the total decay constant. Definition of λ λ = ( n N ) dt The rate of probability that an individual atom will decay is called the total radioactive decay constant ( λ ). Assume, the change in N , in time dt is dn . dn = ( N n ) N = n dn = λNdt When, t = 0→ t = t , N→ N Z N N dn N = λ Z t dt N = N e λt 2 Activity The number of disintegrations per second is called the Activity . A = dn dt ( negative sign is because of N decreasing) Units: SI units of activity: Becquerel ( Bq ) Bq is defined as the number of disintegration per second. 1 Bq = 1 dps Since, dn dt = λN The activity, A = λN The strength of a radioactive source is measured by its activity. The traditional unit of activity is curie ( Ci ). The Curie is defined as the number of disintegrations from 1 g of 226 88 Ra . 1 Ci = 3 . 7 × 10 10 Bq Since, N = N e λt λN = λN e λt A = A e λt 3 Total Activity If the nucleus has more than one type of disintegrations, (say, type A , type B etc. ) then, λ = λ A + λ B + λ C + ··· Total activity, Nλ = Nλ A + Nλ B + Nλ C + ··· The activity of the i th mode, λ i N = λ i N e λt Mean Life time( τ ) The time needed for an initial N nuclei to decay to 1 e of N , is called the mean life time ( τ ). N = N e λt N e = N e λt τ = 1 λ Units: s Half Life time ( T 1 2 ) The time required for onehalf of the initial number of nuclei N , to decay is called the halflife time. N = N e λt N 2 = N e λt 4 T 1 2 = ln 2 λ T 1 2 = . 693 λ Units: s Relation between Mean Life time( τ ) and Half Life time ( T 1 2 ) τ = 1 λ and T 1 2 = . 693 λ T 1 2 = 0 . 693 τ τ = 1 . 44 T 1 2 The following diagram illustrates the exponential decay and the con cepts of mean life time and half life time. 5 Specific Activity(SA) The specific activity (SA) of a radioactive sample,is defined as the ac tivity per unit mass. SA = dA dm ! where dA = Activity and, dm = mass Units: Bq.g 1 or Ci.g 1 From the definition of a mole, the number of atoms per gram is, N A M where N A = Avogadro’s number = 6 . 023 × 10 23 mol 1 and M = Atomic weight ( g.mol 1 ) Since, A = λN A M = λ N A M N A = 6 . 023 × 10 23 mol 1 Then, Specific activity, SA = A M = 6 . 023 × 10 23 λ M Also, λ = . 693 T 1 2 SA = 4 . 17 × 10 23 MT 1 2 6 If T is in seconds, here units of SA is, Bq.g 1 ....
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This note was uploaded on 10/30/2010 for the course MP 200 taught by Professor Guna during the Fall '10 term at Duke.
 Fall '10
 Guna

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