Coulomb wave function

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 compared to parent's half-life. An example of this is a molybdenum-99 generator producing technetium-99 for nuclear medicine diagnostic procedures. Such a generator is sometimes called a cow because the daughter product, in this case technetium-99, is milked at regular intervals. Transient equilibrium occurs after four half-lives, on average.

Activity in transient equilibrium

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

$A_{d}=([A_{P}(0){\frac {\lambda _{d}}{\lambda _{d}-\lambda _{P}}}\times (e^{-\lambda _{P}t}-e^{-\lambda _{d}t})]\times BR)+A_{d}(0)e^{-\lambda _{d}t}$,

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.

See also

• Bateman Equation
• Secular equilibrium

References

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