Nuclear fission is a radioactive decay process or a nuclear reaction in which a heavy nucleus splits into two or more smaller, lighter nuclei, releasing a tremendous amount of energy. Discovered by Otto Hahn and Fritz Strassmann in 1939, it underpins nuclear power reactors and atomic weapons.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition of nuclear fission, discovery by Hahn and Strassmann, fission of U-235 by thermal neutrons, release of secondary neutrons, chain reaction mechanism, and distinction between controlled (reactors) and uncontrolled (bombs) chain reactions.
PGT extension
Liquid drop model explanation of fission, Bohr-Wheeler theory of fission barrier, energy release calculation from mass defect, neutron reproduction factor (k-factor), and nuclear reactor kinetics including moderation and control.
Prerequisites
Binding energy per nucleon curve and mass defect
Neutron capture and compound nucleus formation
Coulomb electrostatic repulsion between protons
Mechanism of Nuclear Fission
When a thermal neutron (energy ~ 0.025 eV) is absorbed by a fissile nucleus like U-235, a compound nucleus U-236 is formed in an excited state.
The excitation energy exceeds the activation energy barrier for fission, causing the nucleus to deform into an elongated shape.
Electrostatic repulsion between the two halves overcomes the short-range nuclear attractive force, leading to scission into two primary fission fragments.
Energy Release in Fission
The binding energy per nucleon (B.E./A) for heavy nuclei like uranium is about 7.6 MeV, whereas for the medium-mass fission fragments (A ~ 90 to 140), it is about 8.5 MeV.
The difference in binding energy per nucleon (~0.9 MeV per nucleon) translates to a total energy release of approximately 200 MeV per fission event.
This energy appears primarily as kinetic energy of the fission fragments, accompanied by prompt neutrons, prompt gamma rays, and subsequent beta and gamma decay of the radioactive fragments.
Chain Reaction and Neutron Multiplication
Fission of U-235 releases on average 2.5 neutrons (typically 2 or 3), which can trigger further fissions in neighboring U-235 nuclei, establishing a chain reaction.
The multiplication factor k is defined as the ratio of the number of neutrons in any generation to the number in the immediately preceding generation.
For a sustained controlled chain reaction, k = 1 (critical state); for a subcritical state k < 1, and for an supercritical state k > 1.
Nuclear Reactor Components
Fuel: Enriched uranium (U-235) or plutonium (Pu-239) undergoes controlled fission.
Moderator: Light elements such as graphite, heavy water (D2O), or ordinary water (H2O) slow down fast neutrons to thermal energies via elastic collisions.
Control Rods: Materials with high neutron absorption cross-sections (cadmium, boron, or hafnium) are inserted into the core to regulate or halt the chain reaction.
Derivations
Energy Release Calculation from Mass Defect
Conditions: Consider the fission of Uranium-235 into Barium-141 and Krypton-92 with the release of 3 neutrons.
Calculate the total initial mass (m_i) of the reactants: mass of U-235 + mass of neutron.
Calculate the total final mass (m_f) of the products: mass of Ba-141 + mass of Kr-92 + 3 * mass of neutron.
Determine the mass defect using delta $m = m_i - m_f$.
Convert the mass defect into energy equivalent using Einstein's relation $Q = \delta m * c^2$, noting that 1 atomic mass unit (u) is equivalent to 931.5 MeV.
Result: Q approx 200\text{ MeV}
Bohr-Wheeler Fission Parameter based on Liquid Drop Model
Conditions: Assuming the nucleus behaves like an incompressible, charged liquid drop.
Express the surface energy of a spherical nucleus as $E_s = a_s A^{2/3}$ and the Coulomb electrostatic energy as $E_c = a_c Z^2 A^{-1/3}$.
Consider a small deformation of the nucleus into a spheroid of eccentricity epsilon.
The change in surface energy upon deformation is negative (opposes deformation) Delta $E_s = -2/5 a_s A^{2/3} \epsilon^2$, and the change in Coulomb energy is also negative (favors deformation) Delta $E_c = +1/5 a_c Z^2 A^{-1/3} \epsilon^2$.
The total change in energy is Delta $E = \delta E_s + \delta E_c$.
For fission to occur spontaneously without an activation barrier, Delta E <= 0, which yields the condition Z^2 / A >= 2 a_s / a_c.
P = Power, N = number of fissions, Q = energy per fission, m = mass consumed, M = molar mass, $N_A = Avogadro number$ Watt (W)
Steady-state reactor operation
Liquid Drop Coulomb Energy
$$E_c = a_c \frac{Z^2}{A^{1/3}}$$
$a_c = Coulomb coefficient$, Z = Atomic number, A = Mass number MeV
Spherical nucleus approximation
Liquid Drop Surface Energy
$$E_s = a_s A^{2/3}$$
$a_s = Surface tension coefficient$, A = Mass number MeV
Spherical nucleus approximation
Fission Barrier Condition
$$\frac{Z^2}{A} \ge 49$$
Z = atomic number, A = mass number Dimensionless
Threshold for spontaneous fission in liquid drop model
Graphs and experiments
Binding Energy per Nucleon versus Mass Number Curve
Draw: A plot of B.E./A on the y-axis against Mass Number A on the x-axis, peaking around $A = 50-60 (Iron)$ and sloping downwards for heavy nuclei (A > 200).
Interpret: Shows that splitting a heavy nucleus (A ~ 235) into two medium-mass nuclei results in a higher binding energy per nucleon, indicating an exothermic release of energy.
Exam use: Explains why nuclear fission is energetically favorable for heavy nuclei.
Fission Fragment Mass Distribution Curve
Draw: A double-peaked asymmetric curve plotting fission yield percentage on the y-axis against mass number of fragments on the x-axis, with peaks around A = 95 and A = 140.
Interpret: Indicates that symmetric fission is less probable than asymmetric fission due to shell closure effects in magic-number nuclei within the fragments.
Exam use: Tested in PGT multiple-choice questions regarding asymmetric mass yields.
Hahn and Strassmann Radiochemical Experiment Setup
Draw: Schematic diagram of neutron irradiation of Uranium target followed by chemical separation showing the presence of Barium isotopes.
Interpret: Proved chemically that neutron bombardment of uranium produces lighter barium nuclei, confirming the splitting of the uranium nucleus.
Exam use: Historical context of the discovery of nuclear fission.
Solved numericals
TGT core
Calculate the energy released in MeV when a single U-235 nucleus undergoes fission, given that the total mass defect in the reaction is 0.215 u.
Identify the mass defect: \Delta m = 0.215\text{ u}.
Use the conversion factor: 1\text{ u} \equiv 931.5\text{ MeV}.
A nuclear reactor generates a power of 100 MW. Assuming that 200 MeV of energy is released per fission of U-235 and that the reactor operates continuously, calculate the mass of U-235 consumed per day. (Avogadro number $N_A = 6.023 \times 10^{23}\text{ mol}^{-1}$, $M = 235\text{ g/mol})$
Calculate energy released per second (Power): $P = 100\text{ MW} = 10^8\text{ J/s}$.
Convert energy per fission to Joules: $E_{fiss} = 200\text{ MeV} = 200 \times 10^6 \times 1.6 \times 10^{-19}\text{ J} = 3.2 \times 10^{-11}\text{ J}$.
Calculate number of fissions per second: $n = \frac{P}{E_{fiss}} = \frac{10^8}{3.2 \times 10^{-11}} = 3.125 \times 10^{18}\text{ fissions/s}$.
Calculate number of fissions per day: $N_{day} = 3.125 \times 10^{18} \times 86400\text{ s} = 2.7 \times 10^{23}\text{ fissions/day}$.
Calculate mass consumed per day using $m = \frac{N_{day}}{N_A} \times M: m = \frac{2.7 \times 10^{23}}{6.023 \times 10^{23}} \times 235\text{ g} = 105.3\text{ g}$.
Answer: 105.3 g per day
PGT extension
Using the liquid drop model parameterization, evaluate whether a hypothetical nucleus with Z = 100 and A = 250 satisfies the threshold condition for spontaneous fission.
Calculate the fissility parameter value Z^2 / A for the given nucleus.
Substitute Z = 100 and $A = 250 into the expression: Z^2 / A = \frac{100^2}{250} = \frac{10000}{250} = 40$.
Compare with the critical threshold limit for spontaneous fission in the liquid drop model, which is approximately 49.
Since 40 < 49, the nucleus is stable against spontaneous fission in the framework of the basic liquid drop model (though quantum tunneling can still cause spontaneous fission).
Answer: $Z^2/A = 40 (less than critical limit 49; stable against classical spontaneous fission)$
Assumptions and limitations
The liquid drop model treats the nucleus as a continuous fluid drop, neglecting individual nucleon shell effects that are crucial for explaining asymmetric fission fragments and magic numbers.
Common misconceptions
Common error: Nuclear fission releases energy because mass is completely destroyed into pure energy.
Correct: Mass is not destroyed; rather, there is a mass defect where the total rest mass of the products is slightly less than the reactants, and this mass defect is converted into energy according to $E = \Delta m c^2$.
Mass-energy conservation holds; mass is converted into binding energy, not annihilated into non-mass energy.
Common error: Fast neutrons are more effective at causing fission in U-235 than slow thermal neutrons.
Correct: Thermal (slow) neutrons have a much higher fission cross-section for U-235 than fast neutrons.
The capture cross-section for low-energy neutrons matches the resonance and compound nucleus formation probability for U-235.
Quick revision
Nuclear fission is the splitting of a heavy nucleus into lighter fragments, releasing ~200 MeV per event.
Mass-Energy Equivalence: $Q = \Delta m \cdot c^2$
Energy Release per Fission (in MeV): $Q = (m_{reactant} - m_{product}) \times 931.5$
Must remember
Thermal neutrons have an energy of approximately $0.025\text{ eV}$ and are highly effective in causing fission of $\text{U-235}$.
The binding energy per nucleon increases from about $7.6\text{ MeV}$ for uranium to about $8.5\text{ MeV}$ for medium-mass fission fragments, releasing roughly $200\text{ MeV}$ per fission event.
The multiplication factor $k$ determines the state of a nuclear chain reactor: $k = 1$ (critical), $k < 1$ (subcritical), and $k > 1$ (supercritical).
Nuclear reactors utilize moderators like graphite, heavy water ($\text{D}_2\text{O}$), or ordinary water ($\text{H}_2\text{O}$) to slow down fast neutrons.
Control rods made of cadmium, boron, or hafnium have high neutron absorption cross-sections and are used to regulate or halt the chain reaction.
Glossary
Thermal NeutronA neutron in thermal equilibrium with its surroundings, having an average kinetic energy of about $0.025\text{ eV}$.
Multiplication Factor ($k$)The ratio of the number of neutrons in any generation to the number in the immediately preceding generation.
ScissionThe point during nuclear fission when the elongated compound nucleus splits into two primary fission fragments.
ModeratorA substance used in a nuclear reactor to slow down fast neutrons to thermal energies via elastic collisions without capturing them.
Mass Defect ($\Delta m$)The difference between the total rest mass of the separate nucleons and the actual mass of the nucleus.
State of Chain Reaction Based on Multiplication Factor
Condition
Multiplication Factor ($k$)
Reactor Behavior
Subcritical
$k < 1$
Neutron population decreases with time; reactor shuts down.
Critical
$k = 1$
Neutron population remains constant; steady-state power generation.
Supercritical
$k > 1$
Neutron population grows exponentially; power increases rapidly.
Absorbs neutrons to regulate or halt the reaction.
Practice Quiz
Choose an option to reveal the explanation.
1. What is the typical kinetic energy of a thermal neutron used to induce fission in $\text{U-235}$?
Explanation: Thermal neutrons possess very low kinetic energy, approximately $0.025\text{ eV}$, which matches thermal agitation energies and maximizes the fission cross-section for $\text{U-235}$.
2. When a thermal neutron is absorbed by $\text{U-235}$, the compound nucleus formed in an excited state is:
Explanation: Absorption of a neutron by $\text{U-235}$ adds one nucleon, resulting in the formation of the compound nucleus $\text{U-236}$ in an excited state.
3. Approximately how much energy is released in a single nuclear fission event of $\text{U-235}$?
Explanation: The difference in binding energy per nucleon between heavy nuclei ($~7.6\text{ MeV}$) and medium-mass fission fragments ($~8.5\text{ MeV}$) translates to roughly $200\text{ MeV}$ per fission.
4. For a nuclear reactor operating in a steady, controlled state, what is the value of the neutron multiplication factor $k$?
Explanation: A steady controlled chain reaction requires each fission event to trigger exactly one subsequent fission event, meaning $k = 1$ (critical state).
5. Which of the following materials is commonly used as a neutron moderator in nuclear reactors?
Explanation: Heavy water ($\text{D}_2\text{O}$), ordinary water, and graphite are light elements used as moderators to slow down fast neutrons via elastic collisions.
6. Control rods in a nuclear reactor are made of materials with:
Explanation: Materials like cadmium, boron, and hafnium have high neutron absorption cross-sections, allowing them to capture neutrons and control the reaction rate.
7. What is the primary form in which the $200\text{ MeV}$ energy released during fission appears?
Explanation: The majority of the energy released during fission appears as the kinetic energy of the two primary fission fragments, alongside neutrons and $\gamma$ rays.
8. The binding energy per nucleon for heavy nuclei like uranium is approximately:
Explanation: Heavy nuclei have a lower binding energy per nucleon of about $7.6\text{ MeV}$ compared to medium-mass nuclei ($~8.5\text{ MeV}$).s
9. In the liquid drop model, the Coulomb energy of a spherical nucleus is proportional to:
Explanation: According to the liquid drop model formula, the Coulomb energy is given by $E_c = a_c \frac{Z^2}{A^{1/3}}$.
10. What condition represents the threshold for spontaneous fission based on the liquid drop model parameter $\frac{Z^2}{A}$?
Explanation: The liquid drop model predicts spontaneous fission threshold when the fissility parameter $\frac{Z^2}{A} \ge 49$.
11. Which factor in the four-factor formula $k_{\infty} = \eta \cdot f \cdot p \cdot \epsilon$ represents the thermal utilization factor?
Explanation: In the four-factor formula, $f$ represents the thermal utilization factor, $\eta$ is the reproduction factor, $p$ is resonance escape probability, and $\epsilon$ is the fast fission factor.
12. Why are fast neutrons less effective than thermal neutrons in causing fission in $\text{U-235}$?
Explanation: The fission cross-section of $\text{U-235}$ is significantly higher for slow (thermal) neutrons than for fast neutrons.
13. What is the role of the fast fission factor ($\epsilon$) in a nuclear reactor?
Explanation: The fast fission factor $\epsilon$ accounts for the total number of fissions produced by fast neutrons relative to those produced by thermal neutrons alone.
14. On average, how many neutrons are released per fission event of $\text{U-235}$?
Explanation: Fission of $\text{U-235}$ typically releases on average 2.5 neutrons (usually 2 or 3), which sustain the chain reaction.
15. Which misconception about nuclear fission is scientifically incorrect?
Explanation: Mass is not annihilated into non-mass energy; rather, there is a mass defect where rest mass is converted into binding energy according to $E = \Delta m c^2$, adhering to mass-energy conservation.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. If the neutron multiplication factor $k$ of a nuclear reactor is set to $1.05$, the reactor is in which state?
Explanation: When $k > 1$ (in this case $1.05$), the reactor is in a supercritical state, meaning the neutron population and power level increase exponentially.
2. In the nuclear reaction $Q = \Delta m \cdot c^2$, if $\Delta m$ represents the mass defect in unified atomic mass units ($u$), the energy release in $\text{MeV}$ is approximately calculated by multiplying $\Delta m$ by:
Explanation: The energy release per atomic mass unit equivalent is approximately $931.5\text{ MeV}$, hence $Q = (m_{reactant} - m_{product}) \times 931.5\text{ MeV}$.
3. Which of the following pairs correctly identifies a fissile nucleus and a common moderator?
Explanation: \text{U-235} is a fissile fuel, and heavy water ($\text{D}_2\text{O}$) acts as an effective moderator. Cadmium and boron are used in control rods.
4. According to the liquid drop model, the nuclear surface energy term $E_s$ varies with mass number $A$ as:
Explanation: The surface energy in the liquid drop model is proportional to the surface area of the nucleus, which scales as $A^{2/3}$ ($E_s = a_s A^{2/3}$).
5. What is the primary function of control rods in a nuclear reactor?
Explanation: Control rods absorb neutrons due to their high absorption cross-section, allowing fine control over the chain reaction or shutting down the reactor.
6. Why does the binding energy per nucleon increase when a heavy nucleus undergoes fission into medium-mass fragments?
Explanation: Medium-mass nuclei ($A \sim 90-140$) have a higher binding energy per nucleon ($~8.5\text{ MeV}$) than heavy actinides ($~7.6\text{ MeV}$), resulting in energy release.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. If a nuclear power plant generates power using $\text{U-235}$ fission, what happens to the multiplication factor $k$ if control rods are inserted deeper into the core?
Explanation: Inserting control rods increases neutron absorption within the core, thereby reducing the number of neutrons available for subsequent fissions and lowering $k$.
2. Calculate the approximate energy released when $1\text{ u}$ of mass defect is converted into energy.
Explanation: By multiplying $1\text{ u}$ by $c^2$ in appropriate units, the mass-energy equivalence yields $931.5\text{ MeV}$.
3. Which of the following correctly lists the four factors in the four-factor formula for $k_{\infty}$?
Explanation: The four-factor formula is $k_{\infty} = \eta \cdot f \cdot p \cdot \epsilon$, corresponding to reproduction factor $\eta$, thermal utilization $f$, resonance escape $p$, and fast fission $\epsilon$.
4. In the context of nuclear fission, what is 'scission'?
Explanation: Scission refers to the exact moment when electrostatic repulsion overcomes nuclear forces and the deformed compound nucleus splits into two distinct fission fragments.
5. Why is ordinary water ($\text{H}_2\text{O}$) an effective moderator compared to heavier elements?
Explanation: In elastic collisions, maximum energy transfer occurs when colliding bodies have nearly equal masses. A proton (hydrogen nucleus) has almost the same mass as a neutron.
6. If $k = 0.95$ for a reactor configuration, the reactor is described as:
Explanation: When $k < 1$ (here $0.95$), the chain reaction cannot sustain itself and the reactor is subcritical.
7. What is the primary source of prompt $\gamma$ rays emitted during a nuclear fission event?
Explanation: Prompt $\gamma$ rays are emitted almost instantaneously as the newly formed primary fission fragments de-excite from high initial energy states.
8. The liquid drop model treats the nucleus analogously to a drop of liquid. Which nuclear force property supports the 'surface energy' term?
Explanation: Nucleons on the surface have fewer neighboring nucleons to interact with via the short-range attractive nuclear force, exactly analogous to surface tension in liquid drops.
9. Which of the following statements regarding the fission barrier is correct?
Explanation: Absorption of a thermal neutron forms an excited compound nucleus whose excitation energy exceeds the activation energy barrier, enabling nuclear deformation and fission.
10. What happens to the power output of a nuclear reactor if the multiplication factor increases from $k = 1$ to $k = 1.02$ and remains uncontrolled?
Explanation: When $k > 1$, the reactor is supercritical, causing the neutron population and consequently the power output to grow exponentially over successive generations.