Radioactive Decay I

1
Radioactive Decay I

• Decay Constants
• Mean Life and Half Life
• Parent-Daughter Relationships

2
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 ltlt1/ ?, is the
  probability that an individual atom will decay
  during that time interval

3
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

4
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

5
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

6
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

7
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

8
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

9
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

10
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)

11
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

12
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

13
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14
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

15
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

16
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

17
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

18
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

19
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 daughters activity
  by the proper factor

20
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

21
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 daughters
  activity to that of the parent depends upon the
  relative magnitudes of the total decay constants
  of parent (?1) and daughter (?2)

22
Daughter Longer-Lived than Parent, ?2 lt ?1

• By changing signs we can obtain the following for
  the ratio of daughter to parent activities
• or, where only one daughter is produced,

23
Daughter Longer-Lived than Parent, ?2 lt ?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

24
Qualitative relationship of activity vs. time for
Te-131m as parent and I-131 as daughter
25
Daughter Shorter-Lived than Parent, ?2 gt ?1

• For t gtgt 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,

26
Daughter Shorter-Lived than Parent, ?2 gt ?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 lt ?1, ?2N2 still maximizes at tm, but the
  crossover of ?1N1 occurs later, if it occurs at
  all

27
Daughter Shorter-Lived than Parent, ?2 gt ?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

28
Daughter Shorter-Lived than Parent, ?2 gt ?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

29
Daughter Shorter-Lived than Parent, ?2 gt ?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

30
Daughter Shorter-Lived than Parent, ?2 gt ?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

31
Example of transient equilibrium
32
Only Daughter Much Shorter-Lived than Parent, ?2
gtgt ?1

• For long times (t gtgt ?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

33
Only Daughter Much Shorter-Lived than Parent, ?2
gtgt ?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

34
Only Daughter Much Shorter-Lived than Parent, ?2
gtgt ?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

35
Only Daughter Much Shorter-Lived than Parent, ?2
gtgt ?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

36
Uranium-238 decay series
37
Only Daughter Much Shorter-Lived than Parent, ?2
gtgt ?1

• Where ?2 gtgt ?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