Transient equilibrium

In nuclear physics, transient equilibrium is a situation in which equilibrium is reached by a parent-daughter radioactive isotope pair where the half-life of the daughter is shorter than the half-life of the parent. Contrary to Secular equilibrium, the half-life of the daughter is not negligible compare to parent's half-life. An example of this is the Molybdenum 99 generator producing Technetium 99 for nuclear medicine diagnostic procedures. Such a generator is sometimes called cow because the daughter product, in this case Technetium 99 is milked at regular intervals.[1] Transient equilibrium occurs after four half-lives, on average.

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Activity in transient equilibrium

The activity of the daughter is given by the Bateman equation:

A_d = ([A_P(0)\frac{\lambda_d}{\lambda_d-\lambda_P} \times (e^{-\lambda_Pt}-e^{-\lambda_dt})] \times BR ) %2B A_d(0)e^{-\lambda_dt},

where A_P and A_d are the activity of the parent and daughter, respectively. T_P and T_d are the half-lives of the parent and daughter, respectively, and BR is the branching ratio.

In transient equilibrium, the Bateman equation cannot be simplified by assuming the daughter's half-life is negligible compared to the parent's half-life. The ratio of daughter-to-parent activity is given by:

\frac{A_d}{A_P} = \frac{T_P}{T_P-T_d} \times BR.

Time of maximum daughter activity

In transient equilibrium, the daughter activity increases and eventually reaches a maximum value that can exceed the parent activity. The time of maximum activity is given by:

t_{max} = \frac{1.44 \times T_P T_d}{T_P-T_d} \times ln(T_P/T_d),

where T_P and T_d are the half-lives of the parent and daughter, respectively. In the case of ^{99m}Tc-^{99}Mo generator, the time of maximum activity (t_{max}) is approximately 24 hours which makes it convenient for medical use. [2]

See also

References

  1. ^ transient equilibrium
  2. ^ S.R. Cherry, J.A. Sorenson, M.E. Phelps (2003). Physics in Nuclear Medicine. A Saunders Title; 3 edition. ISBN 072168341X.