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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.<ref>[http://web.ead.anl.gov/marssim/acrogloss/dsp_wordpopup.cfm?word_id=504 transient equilibrium<!-- Bot generated title -->] {{dead link|date=June 2013}}</ref> Transient equilibrium occurs after four half-lives, on average.
 
==Activity in transient equilibrium==
The activity of the daughter is given by the Bateman equation:
 
:<math>A_d = ([A_P(0)\frac{\lambda_d}{\lambda_d-\lambda_P} \times (e^{-\lambda_Pt}-e^{-\lambda_dt})] \times BR ) + A_d(0)e^{-\lambda_dt}</math>,
 
where <math>A_P</math> and <math>A_d</math> are the activity of the parent and daughter, respectively. <math>T_P</math> and <math>T_d</math> 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:
 
:<math>\frac{A_d}{A_P} = \frac{T_P}{T_P-T_d} \times BR</math>.
 
==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:
 
:<math>t_{max} = \frac{1.44 \times T_P T_d}{T_P-T_d} \times ln(T_P/T_d)</math>,
 
where <math>T_P</math> and <math>T_d</math> are the half-lives of the parent and daughter, respectively. In the case of <math>^{99m}Tc</math>-<math>^{99}Mo</math> generator, the time of maximum activity (<math>t_{max}</math>) is approximately 24 hours which makes it convenient for medical use.
<ref name=brm>{{cite book
|title=Physics in Nuclear Medicine
|author=S.R. Cherry, J.A. Sorenson, M.E. Phelps
|publisher=A Saunders Title; 3 edition
|year=2003
|isbn=0-7216-8341-X
}}</ref>
 
==See also==
* [[Bateman Equation]]
* [[Secular equilibrium]]
 
==References==
<references/>
 
[[Category:Radioactivity]]
 
 
{{physics-stub}}

Revision as of 23:46, 6 September 2013

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.[1] Transient equilibrium occurs after four half-lives, on average.

Activity in transient equilibrium

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

Ad=([AP(0)λdλd−λP×(e−λPt−e−λdt)]×BR)+Ad(0)e−λdt,

where AP and Ad are the activity of the parent and daughter, respectively. TP and Td 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:

AdAP=TPTP−Td×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:

tmax=1.44×TPTdTP−Td×ln(TP/Td),

where TP and Td are the half-lives of the parent and daughter, respectively. In the case of 99mTc- 99Mo generator, the time of maximum activity (tmax) is approximately 24 hours which makes it convenient for medical use. [2]

See also

References

  1. ↑ transient equilibrium Template:Dead link
  2. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534


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