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Radioactive Decay II Removal of Daughter Products Radioactivation Exposure-Rate Constant Removal of Daughter Products

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radioactive decay ii

Radioactive Decay II

Removal of Daughter Products

Radioactivation

Exposure-Rate Constant

removal of daughter products
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
removal of daughter products cont
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
removal of daughter products cont4
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 daughter’s activity at any later time t is obtained from
removal of daughter products cont5
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
removal of daughter products cont6
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
removal of daughter products cont7
Removal of Daughter Products (cont.)
  • If only a single daughter is produced (1A = 1) and if we assume t1 = 0 and t2 = t, then
radioactivation by nuclear interactions
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
radioactivation by nuclear interactions cont
Radioactivation by Nuclear Interactions (cont.)
  • Let Nt be the number of target atoms present in the sample to be activated:

where NA = Avogadro’s constant (atoms/mole)

A = gram-atomic weight (g/mole), and

m = mass (g) of target atoms only in the

sample

radioactivation by nuclear interactions cont10
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

radioactivation by nuclear interactions cont11
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

radioactivation by nuclear interactions cont12
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
radioactivation by nuclear interactions cont13
Radioactivation by Nuclear Interactions (cont.)
  • After an irradiation time t >> , 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

radioactivation by nuclear interactions cont14
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 << ), the activity at time t may be approximated by
slide15

Growth of a radionuclide of decay constant  due to a constant rate of nuclear interaction

radioactivation by nuclear interactions cont16
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
radioactivation by nuclear interactions cont17
Radioactivation by Nuclear Interactions (cont.)
  • Combining the equations for initial rate of production of activity and for the equilibrium activity level, we have
radioactivation by nuclear interactions cont18
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
exposure rate constant
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
exposure rate constant cont
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
exposure rate constant cont22
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
exposure rate constant cont23
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
exposure rate constant cont24
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

exposure rate constant cont25
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
exposure rate constant cont26
Exposure-Rate Constant (cont.)
  • We can relate this energy flux density to the exposure rate by recalling
exposure rate constant cont27
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

exposure rate constant cont28
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:
exposure rate constant cont29
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

exposure rate constant cont30
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
exposure rate constant cont31
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
exposure rate constant cont32
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:
exposure rate constant cont33
Exposure-Rate Constant (cont.)
  • Hence the equation for  becomes

which is close to the value given in the table, considering the difference in units

exposure rate constant cont34
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

exposure rate constant cont35
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