Find whether alpha decay or any of the beta decays are allowed for ${ }_{89}^{226} Ac$ .

Strategy For each of the possible four reactions, we first write the reaction, list the disintegration energy Q equation, look up the appropriate masses, and calculate the disintegration energy Q. If Q > 0, the decay is allowed.

Question

Find whether alpha decay or any of the beta decays are allowed for [latex]{ }_{89}^{226} Ac[/latex].

Strategy For each of the possible four reactions, we first write the reaction, list the disintegration energy Q equation, look up the appropriate masses, and calculate the disintegration energy Q. If Q > 0, the decay is allowed.

Accepted Answer

We outline the solution, but leave looking up the masses and doing the math to the student.

Alpha decay: [latex]{ }_{89}^{226} Ac ightarrow{ }_{87}^{222} Fr +\alpha[/latex]

[latex]\begin{aligned}Q &=\left[M\left({ }_{89}^{226} Ac ight)-M\left({ }^{222} Fr ight)-M\left({ }^{4} He ight) ight] c^{2} \&=5.54 MeV \quad ext { Alpha decay is allowed }\end{aligned}[/latex]

[latex]\beta ^{-}[/latex] decay: [latex]{ }_{89}^{226} Ac ightarrow{ }_{90}^{226} Th +\beta^{-}+\bar{ u}[/latex]

[latex]\begin{aligned}Q &=\left[M\left({ }_{89}^{226} Ac ight)-M\left({ }_{90}^{226} Th ight) ight] c^{2} \&=1.12 MeV \quad \beta^{-} ext {decay is allowed }\end{aligned}[/latex]

[latex]\beta ^{+}[/latex] decay: [latex]{ }_{89}^{226} Ac ightarrow{ }_{88}^{226} Ra +\beta^{+}+ u[/latex]

[latex]\begin{aligned}Q &=\left[M\left({ }_{89}^{226} Ac ight)-M\left({ }_{88}^{226} Ra ight)-2 m_{e} ight] c^{2} \&=-0.38 MeV \quad \beta^{+} ext {decay is not allowed }\end{aligned}[/latex]

Electron capture: [latex]{ }_{89}^{226} Ac +e^{-} ightarrow{ }_{88}^{226} Ra + u
[/latex]

[latex]\begin{aligned}Q &=\left[M\left({ }_{89}^{226} Ac ight)-M\left({ }_{88}^{226} Ra ight) ight] c^{2} \&=0.64 MeV \quad ext { Electron capture is allowed }\end{aligned}[/latex]

We find that α decay, [latex]\beta^{-}[/latex] decay, and electron capture are all possible from the same nucleus. Experiment shows that alpha decay occurs only 0.006% of the time for [latex]{ }^{226} Ac , \beta^{-}[/latex] decay 83%, and electron capture 17%.

Question 12.16: Find whether alpha decay or any of the beta decays are allow...

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