Radioactive Decay II

1
Radioactive Decay II

• Removal of Daughter Products
• Radioactivation
• Exposure-Rate Constant

2
Removal of Daughter Products

• In some cases, especially for diagnostic or
  therapeutic applications of short-lived
  radioisotopes, it is useful to remove the
  daughter product from its relatively long-lived
  parent, which continues producing more daughter
  atoms for later removal and use
• The greatest yield per milking will of course be
  gotten at time tm since the previous milking,
  assuming complete removal of the daughter product
  each time

3
Removal of Daughter Products (cont.)

• Waiting longer than tm is counterproductive, as
  the activity of the daughter present then begins
  to decline along with the parent
• Frequent (or continuous) milking would give a
  greater total yield of the daughter product,
  however

4
Removal of Daughter Products (cont.)

• Assuming that the initial parent activity is
  ?1(N1)0 and the initial Ath-daughter activity is
  zero at time t 0, the daughters activity at
  any later time t is obtained from

5
Removal of Daughter Products (cont.)

• This equation tells us how much Ath-daughter
  activity exists at time t as a result of the
  parent-source disintegrations, regardless of
  whether or how often the daughter has been
  separated from its source
• Thus the amount of daughter activity available to
  be removed from the source at time t is that
  given by this equation minus the daughter
  activity previously removed and still existing
  elsewhere at the same time t

6
Removal of Daughter Products (cont.)

• Alternatively, if we let ?1(N1)0 represent the
  initial activity of the parent source at time t
  0, and if the Ath daughter is completely removed
  at a later time t1 (not necessarily the first
  milking), then the additional Ath daughter
  activity that can be removed at a subsequent time
  t2 is given by

7
Removal of Daughter Products (cont.)

• If only a single daughter is produced (?1A ?1)
  and if we assume t1 0 and t2 t, then

8
Radioactivation by Nuclear Interactions

• Stable nuclei may be transformed into radioactive
  species by bombardment with suitable particles,
  or photons of sufficiently high energy
• Thermal neutrons are particularly effective for
  this purpose, as they are electrically neutral,
  hence not repelled from the nucleus by Coulomb
  forces, and are readily captured by many kinds of
  nuclei
• Tables of isotopes list typical reactions which
  give rise to specific radionuclides

9
Radioactivation by Nuclear Interactions (cont.)

• Let Nt be the number of target atoms present in
  the sample to be activated
• where NA Avogadros constant (atoms/mole)
• A gram-atomic weight (g/mole),
  and
• m mass (g) of target atoms only
  in the
• sample

10
Radioactivation by Nuclear Interactions (cont.)

• If ? is the particle flux density (s-1 cm-2) at
  the sample, assuming that the sample
  self-shielding is negligible, and ? is the
  interaction cross section (cm2/atom) for the
  activation process in question, then the initial
  rate of production (s-1) of activated atoms is
• assuming as usual that we are dealing with
  expectation values

11
Radioactivation by Nuclear Interactions (cont.)

• Correspondingly the initial rate of production of
  activity of the radioactive source being thus
  created is given by
• where ? is the total radioactive decay
  constant of the new species

12
Radioactivation by Nuclear Interactions (cont.)

• If we may assume that ? is constant and that Nt
  is not appreciably depleted as a result of the
  activation process, then the rates of production
  given by these equations are also constant
• As the population of active atoms increases, they
  decay at the rate ?Nact (s-1)
• Thus the net rate at which they accumulate can be
  expressed as

13
Radioactivation by Nuclear Interactions (cont.)

• After an irradiation time t gtgt ?, the rate of
  decay equals the rate of production, and the net
  rate of population increase becomes zero thus
  the equilibrium activity level is given directly
  by
• where the subscript e stands for equilibrium

14
Radioactivation by Nuclear Interactions (cont.)

• At any time t after the start of irradiation,
  assuming the initial activity to be zero (?Nact
  0 at t 0), the activity in becquerels can be
  shown to be related to its equilibrium activity
  by
• Or, assuming that no decay occurs during the
  irradiation period t (which will be approximately
  correct if t ltlt ?), the activity at time t may be
  approximated by

15
Growth of a radionuclide of decay constant ? due
to a constant rate of nuclear interaction
16
Radioactivation by Nuclear Interactions (cont.)

• Sometimes it is necessary to calculate the
  equilibrium activity level on the basis of the
  initial rate of growth of activity, without
  knowing the flux density or cross section for the
  interaction
• An example would be the prediction of the maximum
  activity level of a particular radionuclide that
  would be reached ultimately in a neutron shield,
  knowing only the activity resulting from a short
  initial irradiation period

17
Radioactivation by Nuclear Interactions (cont.)

• Combining the equations for initial rate of
  production of activity and for the equilibrium
  activity level, we have

18
Radioactivation by Nuclear Interactions (cont.)

• Therefore the equilibrium activity level is equal
  to the initial production rate of activity
  multiplied by the mean life ?
• This method of course required that the mean life
  (or the decay constant) be known for the
  radioactive product of interest

19
Exposure-Rate Constant

• The exposure-rate constant ?? of a radioactive
  nuclide emitting photons is the quotient of
  l2(dX/dt)? by A, where (dX/dt)? is the exposure
  rate due to photons of energy greater than ?, at
  a distance l from a point source of this nuclide
  having an activity A
• It is usually stated in units of R m2 Ci-1 h-1 or
  R cm2 mCi-1 h-1

20
Exposure-Rate Constant (cont.)

• This quantity was defined by the ICRU to replace
  the earlier specific gamma-ray constant ?, which
  only accounts for the exposure rate due to
  ?-rays, whereas ?? also included the exposure
  rate contributions (if any) of characteristic
  x-rays and internal bremsstrahlung, and
  establishes the arbitrary lower energy limit ?
  (keV) below which all photons are ignored

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22
Exposure-Rate Constant (cont.)

• It will be seen that ?? is greater than ? by by
  2 or less, except for Ra-226 (12) and I-125 (in
  which case ? is only about 3 of ?? because
  K-fluorescence x-rays following electron capture
  constitute most of the photons emitted)
• In extreme cases like this, where ? would be
  useless if defined literally (i.e., for ?-rays
  only), x-rays have been sometimes included in ?
  even though the definition did not call for it

23
Exposure-Rate Constant (cont.)

• In the following we will show how the specific
  ?-ray constant ? can be calculated for a given
  point source
• The exposure-rate constant ?? may be calculated
  in the same way by taking account of the
  additional x-ray photons (if any) emitted per
  disintegration

24
Exposure-Rate Constant (cont.)

• At a location l meters (in vacuo) from a ?-ray
  point source having an activity A Ci, the flux
  density of photons of the single energy Ei is
  given by
• where ki is the number of photons of energy
  Ei emitted per disintegration

25
Exposure-Rate Constant (cont.)

• This can be converted to energy flux density as
  follows
• in which Ei is to be expressed in MeV/photon
• It will be more convenient to express ?Ei in
  units of J/s m2, while still expressing Ei in
  MeV, in which case the above equation becomes

26
Exposure-Rate Constant (cont.)

• We can relate this energy flux density to the
  exposure rate by recalling

27
Exposure-Rate Constant (cont.)

• For photons of energy Ei the exposure rate is
  given by
• and the total exposure rate for all of the
  ?-ray energies Ei present is

28
Exposure-Rate Constant (cont.)

• Substituting the expression for the energy flux
  density, we obtain
• This can be converted into R/h, remembering that
  1 R 2.58 ? 10-4 C/kg and 3600 s 1 h

29
Exposure-Rate Constant (cont.)

• The specific ?-ray constant for this source is
  defined as the exposure rate from all ?-rays per
  curie of activity, normalized to a distance of 1
  m by means of an inverse-square-law correction
• where Ei is expressed in MeV and ?en/? in
  m2/kg

30
Exposure-Rate Constant (cont.)

• If (?en/?)Ei,air is given instead in units of
  cm2/g, the constant in this equation is reduced
  to 19.38
• ? may be obtained in units of R cm2/mCi h
  directly with this equation if (?en/?)Ei,air is
  expressed in cm2/g in place of m2/kg

31
Exposure-Rate Constant (cont.)

• For the special case of Ra-226 in equilibrium
  with its progeny, ? is usually expressed in R
  cm2/mg h, the activity of the Ra-226 being
  expressed in terms of its mass
• Also, the accepted value of 8.25 R cm2/mg h
  refers not to a bare point source, but rather
  to one in which the ?-rays are filtered through
  0.5 mm of Pt(10 Ir) in escaping

32
Exposure-Rate Constant (cont.)

• Applying this to an example, 60Co, we note first
  that each disintegration is accompanied by the
  emission of two photons, one at 1.17 MeV and the
  other at 1.33 MeV
• Thus the value of ki is unity at both energies
• The mass energy absorption coefficient values for
  air at these energies are

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Exposure-Rate Constant (cont.)

• Hence the equation for ? becomes
• which is close to the value given in the
  table, considering the difference in units

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Exposure-Rate Constant (cont.)

• The exposure rate (R/hr) at a distance l meters
  from a point source of A curies is given by
• where ? is given for the source in R m2/Ci h,
  and attenuation and scattering by the surrounding
  medium are assumed to be negligible

35
Exposure-Rate Constant (cont.)

• A quantity called the air kerma rate constant
  that is related to the exposure rate constant was
  also defined by the ICRU
• The defining equation is
• The units recommended are m2 J kg-1 or m2 Gy Bq-1
  s-1
• Unfortunately the ICRU chose the same symbol, ??,
  for this constant, which may cause confusion