Radiative equilibrium

This is also known as radioactive equilibrium. When daughter nuclide B, which is produced by the decay of parent nuclide A, is further decaying, and the production and decay of the daughter nuclides are balanced, a steady state is said to be in radioactive equilibrium. Radioactive equilibrium can be divided into transient equilibrium and permanent equilibrium depending on the difference between the half-life τ _A of the parent nuclide and the half-life τ _B of the daughter nuclide. If the number of atoms of A is N _A , the number of atoms of B is N _B , and the decay constants are λ _A and λ _B , respectively, the following equation holds.

At t = 0, only A exists, and if we set its number of atoms as N _A ⁰ and solve the two differential equations,

N _A = N _A ⁰ e ^{- λ _A t} (3)

When τ _A ＞ τ _B , λ _A ＜ λ _B . When a sufficient amount of time has passed, the second term of equation (4) can be ignored, so equation (4) can be approximately written as

Therefore, the ratio of the number of atoms of A and B at this time is constant. In addition, the radioactivity ratio of A and B at this time is expressed by the following formula.

This state is called transient equilibrium. Also, when τ _A ≫ τ _B , λ _A ≪ λ _B , after sufficient time has passed, equation (4) is approximated as follows:

The radioactivity ratio at this time is

In other words, the radioactivity ratios of A and B are equal. This state is called permanent equilibrium. Incidentally, radioactive equilibrium does not exist when τ _A < τ _B.

Source: Morikita Publishing "Chemical Dictionary (2nd Edition)" Information about the Chemical Dictionary 2nd Edition

放射能平衡ともいう．親核種Aの崩壊で生じる娘核種Bがさらに崩壊している場合に，娘核種の生成と崩壊が釣り合って定常状態になることを放射平衡にあるという．放射平衡は，親核種の半減期 τ_A と娘核種の半減期 τ_B の大きさの違いにより，過渡平衡と永続平衡に分けられる．Aの原子数を N_A，Bの原子数を N_B，崩壊定数をそれぞれ λ_A，λ_B とすれば，次式が成立する．

t ＝ 0においてはAのみが存在し，その原子数を N_A⁰ として二つの微分方程式を解けば，

　N_A ＝ N_A⁰e^－λ_At　(3)　

となる．τ_A ＞ τ_B のとき λ_A ＜ λ_B，十分に時間が経過するときは式(4)の第2項は無視できるので，式(4)は近似的に

となる．したがって，このときのAとBの原子数の比は一定になる．また，そのときのAとBの放射能比は次式で表される．

このような状態を過渡平衡という．また，τ_A ≫ τ_B のとき λ_A ≪ λ_B，十分に時間が経過すると式(4)は次のように近似される．

このときの放射能比は，

となる．すなわち，AとBの放射能比は等しくなる．この状態を永続平衡という．ちなみに，τ_A ＜ τ_B のときには放射平衡は成立しない．

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