Students will be able to use the unified atomic mass unit (u) as a unit of mass, convert between u, kg and MeV, and apply these concepts to calculations involving atoms, nuclei and radiation, in full accordance with Cambridge AS & A‑Level Physics 9702 syllabus requirements 11.1 & 11.2.
The unified atomic mass unit, symbol , is defined as one twelfth of the mass of a neutral carbon‑12 atom:
\[
1\ \text{u} = \frac{1}{12}\,m_{\text{C}^{12}}
\]
\[
1\ \text{u}=1.66053906660\times10^{-27}\ \text{kg}
\]
\[
1\ \text{u}=931.494\ \text{MeV}\;c^{-2}
\]
Thus a mass of \(m\) u corresponds to an energy \(E=m\times931.494\) MeV.
For most nuclides the atomic mass (in u) is close to the mass number:
\[
m \approx A\ \text{u}
\]
Deviations arise from the mass‑defect caused by nuclear binding energy.
By definition, \(\,^{12}_{6}\text{C}\) has a mass of exactly 12 u:
\[
12\ \text{u}=12\times1.66053906660\times10^{-27}\ \text{kg}=1.9926468799\times10^{-26}\ \text{kg}
\]
\[
m\ (\text{kg}) = m\ (\text{u})\times1.66053906660\times10^{-27}\ \text{kg u}^{-1}
\]
\[
E\ (\text{MeV}) = m\ (\text{u})\times931.494\ \text{MeV u}^{-1}
\]
| Radiation | Symbol | Charge | Mass (u) | Typical kinetic energy (MeV) | Decay equation |
|---|---|---|---|---|---|
| Alpha | \(\alpha\) | +2 e | 4.002603 u | 4–9 | \(\,^{A}{Z}\text{X}\rightarrow^{A-4}{Z-2}\text{Y}+^{4}_{2}\alpha\) |
| Beta‑minus | \(\beta^{-}\) | − e | 0.0005486 u (electron) | 0.01–10 | \(\,^{A}{Z}\text{X}\rightarrow^{A}{Z+1}\text{Y}+e^{-}+\bar{\nu}_{e}\) |
| Beta‑plus (positron) | \(\beta^{+}\) | + e | 0.0005486 u (positron) | 0.01–5 | \(\,^{A}{Z}\text{X}\rightarrow^{A}{Z-1}\text{Y}+e^{+}+\nu_{e}\) |
| Electron capture | \(EC\) | 0 | 0 u (the captured electron’s mass is included in the atomic mass of the parent) | – | \(\,^{A}{Z}\text{X}+e^{-}\rightarrow^{A}{Z-1}\text{Y}+\nu_{e}\) |
| Gamma | \(\gamma\) | neutral | 0 u (photon) | 0.01–10+ | \(\,^{A}{Z}\text{X}^{*}\rightarrow^{A}{Z}\text{X}+\gamma\) |
The Standard Model groups particles into two families:
Hadrons are colour‑neutral composites of quarks:
β‑decay can be described at the quark level as a change of flavour mediated by the weak interaction:
\[
d \;\rightarrow\; u + e^{-} + \bar{\nu}_{e}\qquad
u \;\rightarrow\; d + e^{+} + \nu_{e}
\]
The difference between the sum of the masses of the constituent nucleons (and electrons, when atomic masses are used) and the actual mass of the nuclide:
\[
\Delta m = \bigl(Z\,m{p}+N\,m{n}+Z\,m{e}\bigr)-M{\text{nuclide}}
\]
\[
E_{\text{b}} = \Delta m\,c^{2}= \Delta m \times 931.494\ \text{MeV u}^{-1}
\]
Binding energy per nucleon (\(E_{\text{b}}/A\)) is a useful indicator of nuclear stability.
\[
Q = \bigl(\text{mass of reactants} - \text{mass of products}\bigr)\,c^{2}
= \bigl(\Delta m\ \text{in u}\bigr)\times931.494\ \text{MeV}
\]
\[
^{238}{92}\text{U}\;\rightarrow\;^{234}{90}\text{Th}+^{4}_{2}\alpha+Q
\]
\[
\begin{aligned}
m(^{238}\text{U}) &= 238.050788\ \text{u}\\
m(^{234}\text{Th}) &= 234.043601\ \text{u}\\
m(^{4}\text{He}) &= 4.002603\ \text{u}
\end{aligned}
\]
\[
\Delta m = 238.050788-(234.043601+4.002603)=0.004584\ \text{u}
\]
\[
Q = 0.004584\ \text{u}\times931.494\ \text{MeV/u}\approx4.27\ \text{MeV}
\]
\[
^{14}{6}\text{C}\;\rightarrow\;^{14}{7}\text{N}+e^{-}+\bar{\nu}_{e}
\]
\[
\begin{aligned}
m(^{14}\text{C}) &= 14.003242\ \text{u}\\
m(^{14}\text{N}) &= 14.003074\ \text{u}\\
m(e^{-}) &= 0.0005486\ \text{u}
\end{aligned}
\]
\[
\Delta m = 14.003242-(14.003074+0.0005486)= -0.0003806\ \text{u}
\]
\[
Q = |\Delta m|\times931.494\ \text{MeV/u}\approx0.355\ \text{MeV}
\]
The negative sign simply indicates that the kinetic energy of the emitted electron and antineutrino equals the Q‑value.
\[
^{22}{11}\text{Na}\;\rightarrow\;^{22}{10}\text{Ne}+e^{+}+\nu_{e}
\]
\[
\begin{aligned}
m(^{22}\text{Na}) &= 21.994436\ \text{u}\\
m(^{22}\text{Ne}) &= 21.991385\ \text{u}\\
m(e^{+}) &= 0.0005486\ \text{u}
\end{aligned}
\]
\[
\Delta m = 21.994436-(21.991385+0.0005486)=0.002502\ \text{u}
\]
\[
Q = 0.002502\ \text{u}\times931.494\ \text{MeV/u}\approx2.33\ \text{MeV}
\]
| Nuclide | Symbol \(\,^{A}_{Z}\text{X}\,\) | Mass number (A) | Atomic mass (u) | Mass (kg) |
|---|---|---|---|---|
| Protium | \(^{1}_{1}\text{H}\) | 1 | 1.007825 | 1.674 × 10⁻²⁷ |
| Carbon‑12 | \(^{12}_{6}\text{C}\) | 12 | 12.000000 | 1.992 × 10⁻²⁶ |
| Nitrogen‑14 | \(^{14}_{7}\text{N}\) | 14 | 14.003074 | 2.324 × 10⁻²⁶ |
| Uranium‑235 | \(^{235}_{92}\text{U}\) | 235 | 235.043930 | 3.904 × 10⁻²⁵ |
| Uranium‑238 | \(^{238}_{92}\text{U}\) | 238 | 238.050788 | 3.953 × 10⁻²⁵ |
\[
^{22}{11}\text{Na}\;\rightarrow\;^{22}{10}\text{Ne}+e^{+}+\nu_{e}
\]
given:
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