explain what is meant by nuclear fusion and nuclear fission

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 (E_binding) – 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.

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

• 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 \(^{238}_{92}\text{U}\)

1. 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.

2. Half‑life of \(^{238}\text{U}\) = \(4.468\times10^{9}\) yr = \(1.41\times10^{17}\) s.
3. Decay constant: \(\lambda = 0.693/t_{½}=4.9\times10^{-18}\;\text{s}^{-1}\).
4. Activity: \(A = \lambda N = 4.9\times10^{-18}\times2.53\times10^{21}\approx1.2\times10^{4}\;\text{Bq}\).
5. 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)

  1. Atomic masses: \(m{^{2}\text{H}}=2.014102\;\text{u}\), \(m{^{3}\text{H}}=3.016049\;\text{u}\), \(m{^{4}\text{He}}=4.002603\;\text{u}\), \(m{n}=1.008665\;\text{u}\).
  2. Mass of reactants = \(2.014102+3.016049 = 5.030151\;\text{u}\).
  3. Mass of products = \(4.002603+1.008665 = 5.011268\;\text{u}\).
  4. Mass defect \(\Delta m = 5.030151-5.011268 = 0.018883\;\text{u}\).
  5. 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)

  1. Atomic masses (approximate):
     

     \(m_{^{235}\text{U}}=235.043930\;\text{u}\)
     

     \(m_{n}=1.008665\;\text{u}\)
     

     \(m_{^{141}\text{Ba}}=140.914411\;\text{u}\)
     

     \(m_{^{92}\text{Kr}}=91.926156\;\text{u}\)

  2. Initial mass = \(235.043930 + 1.008665 = 236.052595\;\text{u}\).
  3. Final mass = \(140.914411 + 91.926156 + 3\times1.008665 = 235.866562\;\text{u}\).
  4. Mass defect \(\Delta m = 236.052595 - 235.866562 = 0.186033\;\text{u}\).
  5. 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

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.