23 – Nuclear Physics
23.1 Mass Defect and Nuclear Binding Energy
- Mass defect (Δm) – the difference between the total mass of the separate nucleons and the actual mass of the nucleus.
\[
\Delta m = Z\,m_{p}+N\,m_{n}-m_{\text{nucleus}}
\]
- \(Z\) – number of protons (atomic number)
- \(N\) – number of neutrons
- \(m_{p}=1.007276\;\text{u},\; m_{n}=1.008665\;\text{u}\)
- \(m_{\text{nucleus}}\) – measured nuclear mass (in atomic mass units, u)
- Binding energy (Ebinding) – the energy equivalent of the mass defect (Einstein’s \(E=mc^{2}\)).
\[
E_{\text{binding}} = \Delta m\,c^{2}
\qquad\text{or}\qquad
E_{\text{binding}}(\text{MeV}) = \Delta m(\text{u})\times 931.5\;\text{MeV u}^{-1}
\]
- Binding energy per nucleon – total binding energy divided by the mass number \(A=Z+N\). It is a convenient measure of nuclear stability.
Nucleus
(\(^{A}_{Z}\text{X}\)) |
Mass defect (u) |
Binding energy (MeV) |
Binding energy per nucleon (MeV) |
| \(^{4}_{2}\text{He}\) |
0.0304 |
28.3 |
7.1 |
| \(^{56}_{26}\text{Fe}\) |
0.5280 |
492.3 |
8.8 |
| \(^{238}_{92}\text{U}\) |
0.8620 |
1 786 |
7.5 |
Binding‑energy‑per‑nucleon curve
Insert a graph of binding energy per nucleon (y‑axis) versus mass number \(A\) (x‑axis). The curve rises sharply, peaks at about \(^{56}_{26}\text{Fe}\) (≈8.8 MeV per nucleon), then falls slowly for heavier nuclei.
- The peak explains why:
- Light nuclei (\(A<56\)) release energy by fusion (they move toward the peak).
- Heavy nuclei (\(A>56\)) release energy by fission (they also move toward the peak).
23.2 Radioactive Decay
| Decay type |
Particle emitted |
Charge |
Mass (u) |
Typical energy (MeV) |
Change in \(A\) and \(Z\) |
| α‑decay |
\(^4_{2}\text{He}\) (α‑particle) |
+2 |
4.002603 |
4–9 |
\(A\!-\!4,\;Z\!-\!2\) |
| β⁻‑decay |
electron (e⁻) |
–1 |
≈0 (≈5.5×10⁻⁴ u) |
0.1–3 |
\(A\) unchanged,\; \(Z\!+\!1\) |
| β⁺‑decay (positron emission) |
positron (e⁺) |
+1 |
≈0 |
0.1–3 |
\(A\) unchanged,\; \(Z\!-\!1\) |
| γ‑decay |
γ‑ray photon |
0 |
0 |
0.01–10 (often > 100 keV) |
\(A\) and \(Z\) unchanged |
- All decays obey conservation of nucleon number (\(A\)) and charge (\(Z\)).
- γ‑decay follows an α, β⁻ or β⁺ transition and removes excess nuclear excitation energy.
Activity, Decay Constant and Half‑Life
- Activity (A) – number of decays per unit time (Bq, where 1 Bq = 1 decay s⁻¹).
- Decay constant (λ) – probability per unit time that a single nucleus will decay (s⁻¹).
- Relationship: \[A = \lambda N\] where \(N\) is the number of undecayed nuclei.
- Half‑life (t½) – time required for half of the nuclei to decay.
\[
t_{½}= \frac{\ln 2}{\lambda}= \frac{0.693}{\lambda}
\]
Worked example – Activity of a 1 g sample of \(^{238}_{92}\text{U}\)
- Atomic mass of \(^{238}\text{U}\) ≈ 238.0508 u ⇒ \(N = \dfrac{(1\;\text{g})}{238.0508\;\text{g mol}^{-1}}\times N_{\!A}\)
\(N = \dfrac{1}{238.0508}\times 6.022\times10^{23}=2.53\times10^{21}\) nuclei.
- Half‑life of \(^{238}\text{U}\) = \(4.468\times10^{9}\) yr = \(1.41\times10^{17}\) s.
- Decay constant: \(\lambda = 0.693/t_{½}=4.9\times10^{-18}\;\text{s}^{-1}\).
- Activity: \(A = \lambda N = 4.9\times10^{-18}\times2.53\times10^{21}\approx1.2\times10^{4}\;\text{Bq}\).
- Thus 1 g of natural uranium produces about 12 kBq of α‑particles.
23.3 Nuclear Reactions
23.3.1 Nuclear Fusion
- Definition: Two light nuclei combine to form a single heavier nucleus. Because the product has a larger binding energy per nucleon, the excess energy is released.
Typical reaction (deuterium–tritium fusion)
\[
^{2}_{1}\text{H}+\,^{3}_{1}\text{H}\;\longrightarrow\;^{4}_{2}\text{He}+\,n+17.6\;\text{MeV}
\]
- All symbols are written in Cambridge form \(^{A}_{Z}\text{X}\).
- Key characteristics
- Requires kinetic energies ≈10 keV (≈\(10^{8}\) K) to overcome the Coulomb barrier.
- Energy appears as kinetic energy of the α‑particle and the neutron; in a reactor this kinetic energy is converted to heat.
- Occurs naturally in the cores of stars where gravitational pressure provides the required temperature and pressure.
- Sample calculation (mass‑defect method)
- Atomic masses: \(m_{^{2}\text{H}}=2.014102\;\text{u}\), \(m_{^{3}\text{H}}=3.016049\;\text{u}\), \(m_{^{4}\text{He}}=4.002603\;\text{u}\), \(m_{n}=1.008665\;\text{u}\).
- Mass of reactants = \(2.014102+3.016049 = 5.030151\;\text{u}\).
- Mass of products = \(4.002603+1.008665 = 5.011268\;\text{u}\).
- Mass defect \(\Delta m = 5.030151-5.011268 = 0.018883\;\text{u}\).
- Energy released \(=0.018883\times931.5 = 17.6\;\text{MeV}\) (matches the tabulated value).
Suggested diagram: Show D and T approaching, a short‑lived compound nucleus \(^{5}_{2}\text{He}^{*}\), and the outgoing \(^4_{2}\text{He}\) and neutron with arrows indicating kinetic energy.
23.3.2 Nuclear Fission
- Definition: A heavy nucleus absorbs a neutron and splits into two (or more) lighter fragments, releasing neutrons and a large amount of energy because the fragments have a higher binding energy per nucleon.
Typical reaction (thermal‑neutron‑induced fission of \(^{235}_{92}\text{U}\))
\[
^{235}_{92}\text{U}+\,n\;\longrightarrow\;^{141}_{56}\text{Ba}+\,^{92}_{36}\text{Kr}+3n+200\;\text{MeV}
\]
- Key characteristics
- The incident neutron makes the uranium nucleus unstable (U‑236*).
- The resulting fragments have a larger binding energy per nucleon; the difference appears as kinetic energy of the fragments and the emitted neutrons.
- Each fission typically emits 2–3 fast neutrons, which can induce further fissions → a self‑sustaining chain reaction.
- Control of the chain reaction (critical mass, moderators, control rods) is essential for a nuclear reactor.
- Sample calculation (mass‑defect method)
- Atomic masses (approximate):
\(m_{^{235}\text{U}}=235.043930\;\text{u}\)
\(m_{n}=1.008665\;\text{u}\)
\(m_{^{141}\text{Ba}}=140.914411\;\text{u}\)
\(m_{^{92}\text{Kr}}=91.926156\;\text{u}\)
- Initial mass = \(235.043930 + 1.008665 = 236.052595\;\text{u}\).
- Final mass = \(140.914411 + 91.926156 + 3\times1.008665 = 235.866562\;\text{u}\).
- Mass defect \(\Delta m = 236.052595 - 235.866562 = 0.186033\;\text{u}\).
- Energy released \(=0.186033\times931.5 \approx 173\;\text{MeV}\). The tabulated value of ≈200 MeV includes the kinetic energy of the fragments, the emitted neutrons and the γ‑rays that follow the fission.
Suggested diagram: Show an incoming neutron striking a \(^{235}_{92}\text{U}\) nucleus, the formation of an excited \(^{236}_{92}\text{U}^{*}\), and its split into two fragments (e.g., Ba and Kr) with three outgoing neutrons.
23.3.3 Comparison of Fusion and Fission
| Aspect |
Fusion |
Fission |
| Typical fuel nuclei |
Light isotopes of hydrogen (D, T, \(^{3}\text{He}\)) |
Heavy isotopes (e.g., \(^{235}\text{U}\), \(^{239}\text{Pu}\)) |
| Energy released per nucleon |
≈ 8 MeV · nucleon⁻¹ |
≈ 0.9 MeV · nucleon⁻¹ |
| Total energy per reaction |
~ 20 MeV (D–T) – 30 MeV (He–He) |
~ 200 MeV (U‑235 fission) |
| By‑products |
Helium nuclei and neutrons (low‑radioactivity) |
Varied radioactive fission fragments + neutrons |
| Technological challenges |
Achieving and maintaining \(10^{8}\) K and sufficient confinement (magnetic or inertial) |
Controlling the chain reaction, handling high‑level radioactive waste |
Key Points to Remember
- Mass defect \( \Delta m = \) (sum of nucleon masses) – (actual nuclear mass).
- Binding energy \(E_{\text{binding}} = \Delta m\,c^{2}\); use \(1\;\text{u}=931.5\;\text{MeV}\) for conversions.
- The binding‑energy‑per‑nucleon curve peaks at \(^{56}_{26}\text{Fe}\); nuclei move toward this maximum, giving rise to both fusion (light → heavier) and fission (heavy → lighter).
- Radioactive decay types (α, β⁻, β⁺, γ) conserve both nucleon number and charge; activity \(A\), decay constant \(λ\) and half‑life \(t_{½}\) are related by \(A=λN\) and \(t_{½}=0.693/λ\).
- Fusion: requires extreme temperature/pressure, produces mainly helium, offers a low‑radioactivity energy source.
- Fission: releases a large amount of energy per reaction, produces neutron‑driven chain reactions and radioactive waste; control is achieved with moderators, control rods and geometry.
- Both processes are fundamental to modern energy generation and to the life cycles of stars.