REVIEW OF ATOMIC AND NUCLEAR PHYSICS
3.8 THE RADIOACTIVE DECAY LAW
Radioactive decay is spontaneous change of a nucleus. The change may result in a new nuclide or simply change the energy of the nucleus. If there is a certain amount of a radioisotope at hand, there is no certainty that in the next second
"so many nuclei will decay" or "none will decay." One can talk of the probabil- ity that a nucleus will decay in a certain period of time.
The probability that a given nucleus will decay per unit time is called the decay constant and is indicated by the letter A. For a certain species, A is
1. The same for all the nuclei
2. Constant, independent of the number of nuclei present 3. Independent of the age of the nucleus
Consider a certain mass m of a certain radioisotope with decay constant A.
The number of atoms (or nuclei) in the mass m is equal to
where N, = 6.022 x loz3 = Avogadro's number A = atomic weight of the isotope
: v7 = 0.51 Figure 3.11 Electron-positron anni-
Ey = 0.51 1 MeV ,+ hilation.
104 MEASUREMENT AND DETECTION OF RADIATION
Figure 3.12 A complex decay scheme. For complete explanation of all the symbols and numbers see Ref. 4. Half-life is given for each element's ground state, and energy of each level is given at intermediate states. Q, is the neutron separation energy. Transition probabilities are indicated as percentages (from Ref. 4).
This number of atoms decreases with time, due to the decay according to Decrease per unit time = decay per unit time
or mathematically,
The solution of this equation is
where N(0) = number of atoms at t = 0.
The probability that a nucleus will not decay in time t-i.e., it will survive time t- is given by the ratio of
atoms not decaying in time t N(0)e-At
- - - - e - ~ t (3.70)
atoms at t = 0 N O )
The probability that the nucleus will decay between t and t + dt is
p ( t ) dt = (probability to survive to time t)(probablility to decay in dt) = ePA' A dt The average lifetime i of the nucleus is given by
One concept used extensively with radioisotopes in the half-life T, defined as the time it takes for half of a certain number of nuclei to decay. Thus, using Eq. 3.69,
N(T) 1 -A,
- - - -
N O ) 2 = e which then gives the relationship between A and T:
For a sample of N(t) nuclei at time t , each having decay constant A, the expected number of nuclei decaying per unit time is
where A(t) = activity of the sample at time t.
The units of activity are the Becquerel (Bq), equal to 1 decay/s, or the Curie (Ci) equal to 3.7 x 10'' Bq. The Becquerel is the SI unit defined in 1977.
The term specific activity (SA) is used frequently. It may have one of the two following meanings:
1. For solids,
activity
SA = -
mass (Bq/kg or Ci/g) 2. For gases or liquids,
activity
SA = - (Bq/m3 or Ci/cm3) volume
Example 3.8 What is the SA of 6 0 ~ o ? Answer The SA is
A AN In2 N,
S A = - = - = - m - = (In 2)(6.022 x loz3) m m Tm A (5.2 y)(3.16 X 10' s/y)(0.060 kg)
106 MEASUREMENT AND DETECIlON OF RADIATION
Example 3.9 What is the SA of a liquid sample of m3 containing
kg of 3 2 ~ ? I
Answer The SA is
A AN In2 NA
S A = - = - = - m - = (In 2)(1OP6 kgN6.022 x loz3)
V V VT A m3)(14.3 d)(864OO s/d)(O.OX kg)
There are isotopes that decay by more than one mode. Consider such an isotope decaying by the modes 1,2,3,. . . , i (e.g., alpha, beta, gamma, etc., decay), and let
A, = probability per unit time that the nucleus will decay by the ith mode The total probability of decay (total decay constant) is
A = A, + A 2 + -.. + A i + 0 . -
If the sample contains N(t) atoms at time t, the number of decays per unit time by the ith mode is
The term partial haZf-lz$e is sometimes used to indicate a different decay mode.
If T, is the partial half-life for the ith decay mode, using Eqs. 3.72 and 3.74, one obtains
It should be pointed out that it is the total decay constant that is used by Eqs.
3.69 and 3.73a, and not the partial decay constants.
Example 3.10 The isotope 2 5 2 ~ f decays by alpha decay and by spontaneous fission. The total half-life is 2.646 years and the half-life for alpha decay is 2.731 years. What is the number of spontaneous fissions per secondeper 10-' kg (1 g) of 252 ~ f ?
Answer The spontaneous fission activity is
The spontaneous fission half-life is, using Eq. 3.75,
Therefore,
Sometimes the daughter of a radioactive nucleus may also be radioactive and decay to a third radioactive nucleus. Thus, a radioactive chain
N, -, N2 -, N, -, etc.
is generated. An example of a well-known series is that of 2 3 5 ~ , which through combined a and P- decays ends up as an isotope of lead. The general equation giving the number of atoms of the ith isotope, at time t in terms of the decay constants of all the other isotopes in the chain was developed by at ern an.^ If
&(O) is the number of atoms of the ith isotope of the series at time t = 0 and
then the Bateman equation takes the form
Example 3.11 Apply the Bateman equation for the second and third isotope in a series.
Answer