# Energy conservation in nuclear reactions and radiactive decay

Reading "Fundamentals of Nuclear Physics" by Atam P. Arya, I understand that in a nuclear reaction, let say $x+X \to y+Y$ meaning that "when a particle $x$ strikes a target nucleus $X$, the outcome of the nuclear reaction is a recoil nucleus $Y$ and a particle $y$. In many cases more than one type of particle may be given out." Now, Arya applies the energy conservation to the nuclear reaction, writing $$E_i=E_f$$ where $$E_i=K_x+m_xc^2+K_X+M_Xc^2$$ and $$E_f=K_Y+M_Yc^2+K_y+m_yc^2$$ being $K_x, K_X, K_y, K_y$ the kinetic energies of the particle $x$, the target nucleus (parent), the recoil nucleus (daughter) and the particle $y$, respectively. And $M$ & $m$ its respective masses.

Rewriting energy conservation, he defines a quantity $Q$ (the disintegration energy): $$Q:=(K_Y+K_y)-(K_X+K_x)=(M_X+m_x)c^2-(M_Y+m_y)c^2 \qquad (1)$$ Now, my question is whether or not the equation (1) (and its interpretation) can be applied to alpha and beta decay (where there are no particle colliding with the nucleus).

Compare the data provided for $^{206}\mathrm{Pb}$ (stable) with the provided for $^{210}\mathrm{Pb}$ (unstable). Look in the section headed "Decay properties". Moreover, note that there is a separate $Q$ provided for each decay mode (but not for each channel).
In this instance, however you need to modify your understanding of what $Q$ means. It represents the energy excess of the parent over the total masses of all the products. In your notation:
$$Q_\mathrm{channel} = (M_X - M_Y - \sum_\mathrm{products} m_i)c^2 \quad .$$
• Your $Q_{channel}$ is my $Q$? If yes, then what is the difference between your equation and (1)? (Obviously there are no $m_x$). So you are saying that Q cannot be considered as the total change of kinetic energies? – Ana S. H. May 4 '13 at 23:56