Show that the relations expressed for the disintegration energy Q in Equations (12.38), (12.41), and (12.44) are correct. Strategy We begin with the reaction for each of the beta decays (β^-, β^+, and EC) and change it to an energy equation. We neglect any neutrino mass and atomic binding energies

Question

Show that the relations expressed for the disintegration energy Q in Equations (12.38), (12.41), and (12.44) are correct.

[latex]Q=\left[M\left({ }_{Z}^{A} X ight)-M\left( {}_{z+1}^{\quad A}D ight) ight] c^{2} \quad \beta^{-} ext {decay }[/latex] (12.38)

[latex]Q=\left[M\left({ }_{Z}^{A} X ight)-M\left({ }_{z-1}^{\quad A} D ight)-2 m_{e} ight] c^{2} \quad \beta^{+} ext {decay }[/latex] (12.41)

[latex]Q=\left[M\left({ }_{Z}^{A} X ight)-M\left({ }_{z-1}^{\quad A} D ight) ight] c^{2} \quad ext { Electron capture }[/latex] (12.44)

Strategy We begin with the reaction for each of the beta decays ([latex]\beta^{-}, \beta^{+}[/latex], and EC) and change it to an energy equation. We neglect any neutrino mass and atomic binding energies and eventually use atomic masses.

Accepted Answer

[latex]\beta ^{-}[/latex] decay: We begin with [latex]\beta ^{-}[/latex] decay and write the mass-energy equation for the reaction in Equation (12.37).

[latex]{ }_{Z}^{A} X ightarrow{ }_{Z+1}^{\quad A} D+\beta^{-}+\bar{ u} \quad \beta^{-} ext {decay }[/latex] (12.37)

[latex]M_{ ext {nucl }}\left({ }_{Z}^{A} X ight)=M_{ ext {nucl }}\left({ }_{z+1}^{\quad A} D ight)+m_{e}+Q / c^{2}[/latex]

where we use [latex]M_{ ext {nucl }}[/latex] to indicate the nuclear mass. In order to change to atomic masses we add [latex]Zm _{e}[/latex] to each side above.

[latex]M_{ ext {nucl }}\left({ }_{Z}^{A} X ight)+Z m_{e}=M_{ ext {nucl }}\left({ }_{Z+1}^{\quad A} D ight)+(Z+1) m_{e}+Q / c^{2}[/latex]

Because we are neglecting the difference in atomic binding energies on the two sides of the equation, we now write this equation in terms of atomic masses.

[latex]M\left({ }_{Z}^{A} X ight)=M\left({ }_{z+1}^{\quad A} D ight)+Q / c^{2}[/latex]

We solve this equation for Q to determine Equation (12.38).

[latex]Q=\left[M\left({ }_{Z}^{A} X ight)-M\left({ }_{Z+1}^{\quad A} D ight) ight] c^{2}[/latex] [latex]\beta^{-} ext {decay }[/latex] (12.38)

[latex]\beta ^{+}[/latex] decay: We write the mass-energy equation for the reaction in Equation (12.40) and follow a procedure similar to that above.

[latex]{ }_{Z}^{A} X ightarrow{ }_{Z-1}^{\quad A} D+\beta^{+}+ u[/latex] (12.40)

[latex]\left.M_{ ext {nucl }}{ }_{Z}^{A} X ight)=M_{ ext {nucl }}\left({ }_{z-1}^{\quad A} D ight)+m_{e}+Q / c^{2}[/latex]

We again add [latex]Zm _{e}[/latex] to each side and have

[latex]M_{ ext {nucl }}\left({ }_{Z}^{A} X ight)+Z m_{e}=M_{ ext {nucl }}\left({ }_{z-1}^{\quad A} D ight)+(Z+1) m_{e}+Q / c^{2}[/latex]

In this case we only need [latex](Z-1) m_{e}[/latex] for the daughter atomic mass, which gives us a remaining mass of [latex]2 m_{e}[/latex].

[latex]M\left({ }_{Z}^{A} X ight)=M_{ ext {nucl }}\left({ }_{z-1}^{\quad A} D ight)+2 m_{e}+Q / c^{2}[/latex]

We solve this equation for Q to determine Equation (12.41).

[latex]Q=\left[M\left({ }_{Z}^{A} X ight)-M\left({ }_{z-1}^{\quad A} D ight)-2 m_{e} ight] c^{2} \quad \beta^{+} ext {decay }[/latex] (12.41)

Electron capture: The mass-energy equation for the electron capture reaction of Equation (12.43) is

[latex]{ }_{Z}^{A} X+e^{-} ightarrow{ }_{Z-1}^{\quad A} D+ u \quad ext { Electron capture }[/latex] (12.43)

[latex]M_{ ext {nucl }}\left({ }_{Z}^{A} X ight)+m_{e}=M_{ ext {nucl }}\left({ }_{z-1}^{\quad A} D ight)+Q / c^{2}[/latex]

We add [latex](Z-1) m_{e}[/latex] to each side above and obtain

[latex]M_{ ext {nucl }}\left({ }_{Z}^{A} X ight)+Z m_{e}=M_{ ext {nucl }}\left({ }_{Z-1}^{\quad A} D ight)+(Z-1) m_{e}+Q / c^{2}[/latex]

In this case we have just the right number of electron masses to change to atomic masses.

[latex]M\left({ }_{Z}^{A} X ight)=M\left({ }_{Z-1}^{\quad A} D ight)+Q / c^{2}[/latex]

We solve this equation for Q to find Equation (12.44).

[latex]Q=\left[M\left({ }_{Z}^{A} X ight)-M\left({ }_{Z-1}^{\quad A} D ight) ight] c^{2}[/latex] Electron capture (12.44)

Question 12.14: Show that the relations expressed for the disintegration ene...

This Question has been Answered.

Already have an account?

Login

Related Answered Questions