Transient equilibrium

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

:$A\_d\; =\; A\_P(0)\backslash frac\{\backslash lambda\_d\}\{\backslash lambda\_d-\backslash lambda\_P\}\; \backslash times\; (e^\{-\backslash lambda\_Pt\}-e^\{-\backslash lambda\_dt\})\; \backslash times\; BR\; +\; A\_d(0)e^\{-\backslash lambda\_dt\},$

where $A\_P$ and $A\_d$ are the activity of the parent and daughter, respectively. $T\_P$ and $T\_d$ are the half-lives (inverses of reaction rates $\backslash lambda$ in the above equation modulo ln(2)) 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:

:$\backslash frac\{A\_d\}\{A\_P\}\; =\; \backslash frac\{T\_P\}\{T\_P-T\_d\}\; \backslash times\; BR.$

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:

:$t\_\{\backslash max\}\; =\; \backslash frac\{1.44\; \backslash times\; T\_P\; T\_d\}\{T\_P-T\_d\}\; \backslash times\; \backslash ln\backslash frac\{T\_P\}\{T\_d\},$

where $T\_P$ and $T\_d$ are the half-lives of the parent and daughter, respectively. In the case of ^{99\!m}Tc-^{99}Mo generator, the time of maximum activity ($t\_\{\backslash max\}$) is approximately 24 hours, which makes it convenient for medical use.{{cite book

|title=Physics in Nuclear Medicine

|author1=S.R. Cherry

|author2=J.A. Sorenson

|author3=M.E. Phelps

|publisher=A Saunders Title; 3 edition

|year=2003

|isbn=0-7216-8341-X

|url-access=registration

|url=https://archive.org/details/physicsinnuclear0000cher

}}

• Bateman equation
• Secular equilibrium

Category:Radioactivity

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