The central quantity in nuclear-decay energetics is the Q-value. It measures how much energy is released
when a parent nucleus transforms into its daughter products. If the Q-value is positive, the decay is energetically allowed.
If it is negative, the chosen decay channel cannot happen spontaneously with the masses you entered.
General idea of the Q-value
In every decay process, the released energy comes from a difference in rest mass. If the total mass of the initial state
is larger than the total mass of the final state, that missing mass reappears as kinetic energy or photon energy. The
general relation is
Mass-energy definition of the Q-value.
\[
\begin{aligned}
Q &= \left(M_{\text{initial}} - M_{\text{final}}\right)c^2.
\end{aligned}
\]
In nuclear physics, masses are very often entered in atomic mass units, so it is convenient to use the standard conversion
\[
\begin{aligned}
1\ \mathrm{u}\,c^2 &\approx 931.494\ \mathrm{MeV}.
\end{aligned}
\]
This means that once the mass difference is known in atomic mass units, the Q-value in MeV follows immediately.
Atomic-mass formulas for α, β, and γ decay
This calculator uses atomic masses. That choice is practical because tables usually list atomic masses,
and electron contributions cancel in simple ways.
Alpha decay.
\[
\begin{aligned}
Q_{\alpha} &= \left(M_p - M_d - M_{\alpha}\right)c^2.
\end{aligned}
\]
Here \(M_{\alpha}\) is the atomic mass of the helium-4 atom. Using atomic masses works cleanly because the parent atom,
daughter atom, and helium atom contain the same total number of electrons on both sides.
Beta-minus decay.
\[
\begin{aligned}
Q_{\beta^-} &= \left(M_p - M_d\right)c^2.
\end{aligned}
\]
In β− decay, a neutron turns into a proton, an electron, and an antineutrino. When atomic masses are used,
the electron bookkeeping is already built into the parent-versus-daughter mass difference, so no extra electron term
appears in the formula.
Beta-plus decay.
\[
\begin{aligned}
Q_{\beta^+} &= \left(M_p - M_d - 2m_e\right)c^2.
\end{aligned}
\]
The \(2m_e\) subtraction appears because one electron mass is needed to create the emitted positron and another appears
through the atomic-mass bookkeeping when the daughter atom has one fewer orbital electron than the parent.
Gamma decay.
\[
\begin{aligned}
Q_{\gamma} &= \left(M_p^* - M_d\right)c^2.
\end{aligned}
\]
For γ decay, the parent and daughter are the same nuclide in different energy states. The excited state has a slightly
larger mass, and that excess becomes the emitted photon energy plus a tiny recoil correction.
Energy sharing
The Q-value does not always go into a single product. How it is shared depends on the decay type.
Alpha decay: two-body sharing.
\[
\begin{aligned}
K_{\alpha} &= Q \cdot \frac{M_d}{M_d + M_{\alpha}}, \\
K_d &= Q \cdot \frac{M_{\alpha}}{M_d + M_{\alpha}}.
\end{aligned}
\]
Since α decay is a two-body decay, momentum conservation fixes the kinetic-energy split. The α particle gets most of the
energy, while the much heavier daughter gets only a small recoil share.
Beta decay: continuous spectrum.
\[
\begin{aligned}
K_{\beta,\max} &\approx Q.
\end{aligned}
\]
In β decay, the emitted beta particle and the neutrino share the released energy continuously. That is why beta spectra
are not sharp lines. The calculator therefore reports the endpoint energy, which is the largest possible
kinetic energy of the emitted electron or positron.
Gamma decay: photon plus recoil.
\[
\begin{aligned}
K_d &\approx \frac{Q^2}{2 M_d c^2}, \\
E_{\gamma} &\approx Q - K_d.
\end{aligned}
\]
In γ decay the recoil is usually very small, so the photon energy is often almost equal to the Q-value.
Sample calculation: α decay of Po-210
A classic example is the α decay
Po-210 → Pb-206 + α
Using approximate atomic masses
\(M_p = 209.9828737\ \mathrm{u}\),
\(M_d = 205.9744653\ \mathrm{u}\),
and
\(M_{\alpha} = 4.0026033\ \mathrm{u}\),
the mass difference is
Step 1. Compute the mass defect.
\[
\begin{aligned}
\Delta m
&= M_p - M_d - M_{\alpha} \\
&= 209.9828737 - 205.9744653 - 4.0026033 \\
&\approx 0.005805\ \mathrm{u}.
\end{aligned}
\]
Step 2. Convert to the Q-value.
\[
\begin{aligned}
Q_{\alpha}
&= \Delta m \cdot 931.494 \\
&\approx 0.005805 \cdot 931.494 \\
&\approx 5.41\ \mathrm{MeV}.
\end{aligned}
\]
This is the total kinetic energy released to the α particle plus the daughter recoil. Because the daughter nucleus is
much heavier than the α particle, the α particle carries most of the energy.
Physical interpretation
Positive Q-values mean the decay products have lower total rest mass than the parent, so the decay can proceed without
external energy input. This is why α, β, and γ decays are spontaneous nuclear processes when the parent nucleus is unstable.
The released energy appears as measurable particle kinetic energy, photon energy, or recoil.
The form of the spectrum also reveals the decay type. α and γ decays are fundamentally two-body processes in their simplest
form, so they produce sharply defined energies. β decay includes a neutrino, so the beta particle can emerge with a range
of kinetic energies up to an endpoint set by the Q-value.
Advanced note
At a more advanced level, one studies fine recoil corrections, nuclear excitation branches, electron capture competing
with β+ decay, and selection rules that determine whether a decay is allowed or hindered. For β decay, the
neutrino makes the full kinematics richer than a simple two-body problem. Still, the Q-value remains the starting point
for all of those analyses because it sets the basic energy budget of the decay.
| Decay type |
Atomic-mass formula |
Main energy interpretation |
| α decay |
\(Q_{\alpha} = (M_p - M_d - M_{\alpha})c^2\) |
Shared between α kinetic energy and daughter recoil |
| β− decay |
\(Q_{\beta^-} = (M_p - M_d)c^2\) |
Endpoint scale for electron plus antineutrino energy sharing |
| β+ decay |
\(Q_{\beta^+} = (M_p - M_d - 2m_e)c^2\) |
Endpoint scale for positron plus neutrino energy sharing |
| γ decay |
\(Q_{\gamma} = (M_p^* - M_d)c^2\) |
Almost all goes into the photon, with tiny recoil |