Question

Write the decay equations and expressions for the disintegration energy Q of the following decay: ${\beta }^{-}$ decay, ${\beta }^{+}$ decay, electron capture

Answer 1

In all the cases discussed here, we neglect any neutrino mass.
${\beta }^{-}decay:M_{nucl}{(}_Z^{A}X)=M_{nucl}{)}_{Z+1}^{A}D)+m_e+Q/c^2$
where, $M_{nucl}$ indicates the nuclear mass.

In order to convert it to atomic mass, we add $Z{m}_e$ on both sides:
$M_{nucl}{(}_Z^{A}X)+Z{m}_e=M_{nucl}{(}_{Z+1}^{A}D)+(Z+1)m_e+Q/c^2$

Since atomic binding energies are less than nuclear binding energies, they are neglected on the two sides of the equation.
$\left[M{(}_Z^{A}X\right)=M{(}_{Z+1}^{A}D\left)\right]c^2$

$Q$ for this equation $=\left[M{(}_Z^{A}X\right)=M{(}_{Z+1}^{A}D\left)\right]c^2$
${\beta }^{+}decay:M_{nucl}{(}_Z^{A}X)=M_{nucl}{(}_{Z-1}^{A}D)+m_e+Q/c^2$

In this case, only (Z-1)m is needed for the daughter atomic mass which gives us a remaining mass of $2m_e$.
$\left[M{(}_Z^{A}X\right)-M{(}_{Z-1}^{A}D)-2m_e]c^2$

$Q$ for this equation $=\left[M{(}_Z^{A}X\right)-M{(}_{Z-1}^{A}D)-2m_e]c^2$

We add $(Z-1)m_e$ to each side above and obtain:
$M_{nucl}{(}_Z^{A}X)+Z{m}_e=M_{nucl}{(}_{Z-1}^{A}D)+(Z-1)m_e+Q/c^2$

In this case, electron masses are balanced, i.e, cancelled out in equation:
$M{(}_Z^{A}X)-M{(}_{Z-1}^{A}D)+Q/c^2$

$Q$ for this equation $=\left[M{(}_Z^{A}X\right)-M{(}_{Z-1}^{A}D\left)\right]c^2$