Fission Knowledge-Based Study

Study notes

1. What is Nuclear Fission?

   Nuclear fission is a nuclear reaction in which a heavy, unstable atomic nucleus splits into two or more lighter daughter nuclei, accompanied by the release of several free neutrons and a significant amount of energy.

   The Liquid Drop Model and Nuclear Instability

   Heavy nuclei such as uranium-235 ($^{235}_{\ 92}\text{U}$) contain a large number of protons. The stability of a nucleus represents a delicate balance between:

   • The strong nuclear force: An attractive, short-range force acting between all nucleons (protons and neutrons).
   • The electrostatic (Coulomb) force: A repulsive, long-range force acting between positively charged protons.

   According to the liquid drop model, when a heavy nucleus absorbs a low-energy (thermal) neutron, it forms a highly excited compound nucleus ($^{236}_{\ 92}\text{U}^*$). The added excitation energy causes the compound nucleus to oscillate and deform into an elongated, dumbbell-like shape. Once the Coulomb repulsion between the two lobes overcomes the attractive strong force across the narrowing neck, the nucleus undergoes scission and splits apart.

   Energy Release in Fission

   The energy liberated during nuclear fission is directly related to the concept of binding energy per nucleon ($\text{BE}/A$). Heavy nuclei ($A \approx 235$) have a binding energy of approximately 7.6 MeV per nucleon, whereas medium-mass nuclei ($A \approx 90\text{–}140$) have a higher binding energy of approximately 8.5 MeV per nucleon.

   Because the daughter nuclei are more tightly bound than the original parent nucleus, their total rest mass is less than the initial mass of the reactants. This difference is known as the mass defect ($\Delta m$):

   $$\Delta m = \sum m_{\text{reactants}} - \sum m_{\text{products}}$$

   Mass-Energy Equivalence

   According to Einstein's mass-energy relation, the released energy $Q$ is:

   $$Q = \Delta m \cdot c^2$$

   For a typical single fission event of $^{235}_{\ 92}\text{U}$, approximately 200 MeV of energy is released. This energy is distributed among:

   • Kinetic energy of fission fragments: Accounting for roughly 80% to 85% ($\approx 168\text{ MeV}$) of the total energy.
   • Kinetic energy of prompt neutrons: About $5\text{ MeV}$.
   • Prompt gamma rays: About $7\text{ MeV}$.
   • Subsequent radioactive decays: Beta particles, delayed gamma rays, and antineutrinos account for the remaining $\approx 20\text{ MeV}$.

   Conservation Laws in Fission

   Every nuclear fission reaction strictly conserves:

   1. Total nucleon number (mass number, $A$): The sum of nucleons before and after the split remains constant.
   2. Total electric charge (atomic number, $Z$): The total proton count is conserved.
   3. Total relativistic energy and momentum.

   Representative Reaction

   Fission is an asymmetric process; a nucleus can split into a wide variety of daughter pairs. One well-known pathway involves the absorption of a thermal neutron by uranium-235:

   $$^{1}_{0}\text{n} + {}^{235}_{\ 92}\text{U} ightarrow {}^{236}_{\ 92}\text{U}^* ightarrow {}^{141}_{\ 56}\text{Ba} + {}^{92}_{\ 36}\text{Kr} + 3\ ^{1}_{0}\text{n} + Q$$

   In this specific event, barium-141 and krypton-92 are produced, alongside three free prompt neutrons. Notice that:

   • Mass numbers: $1 + 235 = 141 + 92 + 3(1) = 236$
   • Atomic numbers: $0 + 92 = 56 + 36 + 3(0) = 92$

   Both conservation requirements are fully satisfied.

   Self-Sustaining Nuclear Chain Reactions

   Because each fission event releases an average of 2 to 3 fast neutrons, these newly produced neutrons can induce subsequent fission events in neighboring fissile nuclei. This cascading sequence is known as a chain reaction.

   The Effective Neutron Multiplication Factor ($k$)

   The progression of a chain reaction is characterized by the effective multiplication factor, defined as:

   $$k = \frac{\text{Number of fissions in generation } n+1}{\text{Number of fissions in generation } n}$$

   • Subcritical ($k < 1$): The reaction rate decreases over time, and the chain reaction dies out because neutrons are lost to leakage or non-fission capture faster than they are produced.
   • Critical ($k = 1$): The reaction rate remains strictly constant and stable. Commercial nuclear power plants operate in this steady state.
   • Supercritical ($k > 1$): The reaction rate grows exponentially over successive generations. Uncontrolled prompt supercriticality is the operating regime of nuclear weapons.

   Critical Mass

   The minimum mass of fissile material required to sustain a chain reaction at $k \ge 1$ for a given geometry and purity is called the critical mass. A spherical geometry minimizes the surface-area-to-volume ratio, thereby reducing neutron leakage to the surrounding environment.

   Harnessing Fission in Thermal Reactors

   Nuclear power reactors utilize controlled fission to generate heat, which boils water into steam to turn electrical turbines. A standard thermal reactor relies on several key components:

   • Nuclear Fuel: Typically uranium dioxide ($\text{UO}_2$) enriched to 3%–5% $^{235}_{\ 92}\text{U}$, sealed inside fuel rods.
   • Moderator: Fission neutrons are emitted with high kinetic energy (fast neutrons, $\sim 2\text{ MeV}$). However, the probability (cross-section) of $^{235}\text{U}$ fission is several hundred times higher for thermal neutrons ($\sim 0.025\text{ eV}$). Moderators composed of light nuclei (such as light water, heavy water $\text{D}_2\text{O}$, or graphite) slow down neutrons via elastic collisions without absorbing them.
   • Control Rods: Materials with high neutron-absorption cross-sections, such as boron, cadmium, or hafnium. Inserting the rods decreases $k$, while withdrawing them increases $k$.
   • Coolant: A circulating fluid (commonly pressurized water, liquid sodium, or gas) that extracts thermal energy from the reactor core and prevents structural overheating.
   • Radiation Shielding: Dense concrete and lead walls designed to absorb escaping neutrons and gamma radiation, protecting personnel and the environment.