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Total Decay Constants

- Consider a large number N of identical radioactive atoms
- We define as the total radioactive decay (or transformation) constant, which has the dimensions reciprocal time, usually expressed in inverse seconds (s-1)
- The product of by a time in consistent units (e.g., seconds), and that is <<1/ , is the probability that an individual atom will decay during that time interval

Total Decay Constants (cont.)

- The expectation value of the total number of atoms in the group that disintegrate per unit of time very short in comparison to 1/ is called the activity of the group, N
- This is also expressed in unit of reciprocal time, since N is a dimensionless number
- So long as the original group is not replenished by a source of more nuclei, the rate of change in N at any time t is equal to the activity:

Total Decay Constants (cont.)

- Separating variables and integrating from t = 0 (when N = N0) to time t, we have

whence

- So we can write for the ratio of activities at time t to that at t0 = 0

Partial Decay Constants

- If a nucleus has more than one possible mode of disintegration (i.e., to different daughter products), the total decay constant can be written as the sum of the partial decay constants i:

and the total activity is

Partial Decay Constants (cont.)

- The partial activity of the group of N nuclei with respect to the ith mode of disintegration can be written
- Note that each partial activity iN decays at the rate determined by the total decay constant , rather than i itself, since the stock of nuclei (N) available at time t for each type of disintegration is the same for all types, and its depletion is the result of their combined activity

Partial Decay Constants (cont.)

- Also note that the partial activities iN are always proportional to the total activity N, independent of time, since each i is constant
- That is, the iN/N are constant fractions, and their sum for all i modes of disintegration is unity

Units of Activity

- The old unit of activity was the curie (Ci), originally defined as the number of disintegrations per second occurring in a mass of 1 g of 226Ra
- Later the curie was divorced from the mass of radium, and was simply set equal to 3.7 1010 s-1
- Subsequent measurements of the activity of radium have determined that 1 g of 226Ra has an activity of 3.655 1010 s-1, or 0.988 Ci

Units of Activity (cont.)

- More recently it was decided by an international standards body to establish a new special unit for activity, the becquerel (Bq), equal to 1 s-1
- Thus

Units of Activity (cont.)

- In addition to the curie and becquerel a third option exists for expressing activity, but only for radium sources
- Such a source can be said to have an “activity” equal to the mass of 226Ra that it contains, typically in milligrams
- For historical reasons this usage is very common in spite of its irregularity and lack of consistency with the proper dimensions of activity (s-1)

Mean Life and Half Life

- The expectation value of the time needed for an initial population of N0 radioactive nuclei to decay to 1/e of their original number is called the mean life
- Thus

Mean Life and Half Life (cont.)

- The mean life has interesting and useful properties
- As its name implies, it represents the average lifetime of an individual nucleus from an arbitrary starting time t0 until it disintegrates at a later time t
- Here t - t0 may have any value from 0 to
- is also the time that would be needed for all the nuclei to disintegrate if the initial activity of the group, N0, were maintained constant instead of decreasing exponentially

Mean Life and Half Life (cont.)

- A second important characteristic time period associated with exponential decay is the half-life 1/2, which is the expectation value of the time required for one-half of the initial number of nuclei to disintegrate, and hence for the activity to decrease by half:

Radioactive Parent-Daughter Relationships

- Consider an initially pure large population (N1)0 of parent nuclei, which start disintegrating with total decay constant 1 at time t = 0
- The number of parent nuclei remaining at time t is N1 = (N1)0e-1t
- Let 1 be composed of partial decay constants 1A, 1B, and so on
- We focus our interest solely on the daughter product resulting from disintegrations of the A type, which occur with decay constant 1A

Radioactive Parent-Daughter Relationships (cont.)

- The rate of production of these daughter nuclei at time t is given by 1AN1 = 1A(N1)0e-1t
- Simultaneously they in turn will disintegrate with a total decay constant of 2A, where the 2 refers to the generation doing the decaying (i.e., daughter, or 2nd generation) and the A the type of parental disintegration that gave rise to the daughter in question
- Since we will not be concerned here with the fate of any other daughter products, we can simplify the terminology by dropping the A from the 2A

Radioactive Parent-Daughter Relationships (cont.)

- The rate of removal of the N2 daughter nuclei which exist at time t0 will be equal to the negative of their total activity, -2N2
- Thus the net rate of accumulation of the daughter nuclei at time t is

Radioactive Parent-Daughter Relationships (cont.)

- The activity of the daughter product at any time t, assuming N2 = 0 at t = 0, is
- The ratio of daughter to parent activities vs. time is

Radioactive Parent-Daughter Relationships (cont.)

- If the partial decay constant 1A of the parent were equal to its total decay constant 1 (i.e., only one daughter were produced by the parent), then
- We may ignore the influence of branching in the modes of parent disintegration until the final step when the activity of the daughter has been determined as a function of t on the basis of this equation, and then simply multiply by the ratio 1A/1 to decrease the daughter’s activity by the proper factor

Equilibria in Parent-Daughter Activities

- The activity of a daughter resulting from an initially pure population of parent nuclei will have the value zero both at t = 0 and
- Evidently 2N2 reaches a maximum at some intermediate time tm when

and therefore

and

Equilibria in Parent-Daughter Activities (cont.)

- This maximum occurs at the same time t = tm that the activities of the parent and daughter are equal if, and only if, 1A = 1(i.e., the parent has only one daughter)
- The specific relationship of the daughter’s activity to that of the parent depends upon the relative magnitudes of the total decay constants of parent (1) and daughter (2)

Daughter Longer-Lived than Parent, 2 < 1

- By changing signs we can obtain the following for the ratio of daughter to parent activities:

or, where only one daughter is produced,

Daughter Longer-Lived than Parent, 2 < 1 (cont.)

- This activity ratio is thus seen to increase continuously with t for all times
- Remembering that the parent activity at time t is

one can construct the activity curves vs. time for the representative case of metastable tellurium-131 decaying to its only daughter iodine-131; and thence to xenon-131:

Qualitative relationship of activity vs. time for Te-131m as parent and I-131 as daughter

Daughter Shorter-Lived than Parent, 2 > 1

- For t >> tm the value of the daughter/parent activity ratio becomes a constant, assuming as usual that N2 = 0 at t = 0:

or, where only a single daughter is produced,

Daughter Shorter-Lived than Parent, 2 > 1

- The existence of such a constant ratio of activities is called transient equilibrium, in which the daughter activity decreases at the same rate as that of the parent
- For 1A = 1, the daughter activity is always greater than that of the parent during transient equilibrium, and the two activities are equal at the time t = tm
- For 1A < 1, 2N2 still maximizes at tm, but the crossover of 1N1 occurs later, if it occurs at all

Daughter Shorter-Lived than Parent, 2 > 1

- For the special case where

the activity of the Ath daughter in transient equilibrium equals that of the parent

- Equality of daughter and parent during transient equilibrium is referred to as secular equilibrium, which will be discussed in the next section

Daughter Shorter-Lived than Parent, 2 > 1

- An interesting example of transient equilibrium, which also exhibits branching of the decay to more than one daughter, is provided by 99Mo (½ = 66.7 h)
- The total parent decay constant 1 = 0.0104 h-1
- In 86% of its - disintegrations, 99Mo decays to 99mTc, a metastable daughter having a 6.03-h half-life in decaying to its ground-state isomer 99Tc by -ray emission
- The other 14% decay by --emission to other excited states of 99Tc, which then promptly decay by -ray emission to the ground state

Daughter Shorter-Lived than Parent, 2 > 1

- The partial decay constant 1A for 99Mo disintegrating to 99mTc is 0.86 times the total decay constant for 99Mo, or 0.00894 h-1
- 99mTc itself decays to 99Tc, exhibiting a half-life of 6.03 h, so 2 = 0.115 h-1
- The time tm at which the activity of 99mTc reaches a maximum is given by:

Daughter Shorter-Lived than Parent, 2 > 1

- The ratio of daughter to parent activity at transient equilibrium in this case is
- If, hypothetically, 99mTc had been the only daughter of 99Mo, the ratio would have been

Only Daughter Much Shorter-Lived than Parent, 2 >> 1

- For long times (t >> 2) in this case
- That is, the activity of the daughter very closely approximates that of the parent
- Such a special case of transient equilibrium, where the daughter and parent activities are practically equal, is commonly called secular equilibrium, because it closely approximates that condition

Only Daughter Much Shorter-Lived than Parent, 2 >> 1

- The practical cases to which this terminology is applied usually include a very long-lived parent, hence the use of the word “secular” in its sense of “lasting through the ages”
- An example of this is the relationship of 226Ra as parent, decaying to 222Rn as daughter, thence to 218Po:

Only Daughter Much Shorter-Lived than Parent, 2 >> 1

- In this case

where both activities must be stated in the same units (e.g., Bq)

- Since 222Rn is the only daughter of 226Ra, its activity exactly equals that of its parent at tm = 66 days, and thereafter the equality is approximated within 7 parts per million

Only Daughter Much Shorter-Lived than Parent, 2 >> 1

- Thus 1 Ci of 226Ra sealed in a closed container at time t0 will, any time after 39 days later, be accompanied by 1 Ci (within 0.1%) of 222Rn, which is a noble gas
- The granddaughter product, 218Po, in turn decays to 214Pb, as shown in the following diagram, which gives the entire uranium series beginning with 238U
- It can be shown that in such a case all the progeny atoms will eventually be nearly in secular equilibrium with a relatively long-lived ancestor

Only Daughter Much Shorter-Lived than Parent, 2 >> 1

- Where 2 >> 1 with decay branching present, giving rise to more than one daughter, the ratio of the activity of the Ath daughter to that of its parent at long times can be gotten from:

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