Nuclear Physics PYQ Theory

Nuclear Physics & Radioactivity - Theory Answers

Prepared for Nakib | Based on KUET PYQ #uni/1-1/phy-1109 PHY-1109 Physics

1. Properties of the Nucleus

Q: Discuss the general properties of the nucleus.

The atomic nucleus is the central part of an atom, containing protons and neutrons (collectively called nucleons). Its key properties include:

1. Nuclear Size: The nucleus is extremely small compared to the atom. Its radius $R$ is related to the mass number $A$ by $R=R_0{A}^{1/3}$, where $R_0\approx 1.2\times 10^{-15}$ m (1.2 fermi).

2. Nuclear Mass: It makes up more than 99.9% of the atom’s mass. The mass is roughly $A\times m_n$ (mass of a nucleon).

3. Nuclear Density: Nuclear density is independent of the mass number $A$. It is extremely high, approximately $2.3\times 10^{17}{\text{ kg/m}}^3$, and is constant for all nuclei.

4. Nuclear Charge: The nucleus has a positive charge $+Ze$, where $Z$ is the atomic number and $e$ is the elementary charge.

5. Nuclear Spin: Nucleons have intrinsic spin angular momentum. The total nuclear spin depends on the number of protons and neutrons.

6. Magnetic Moment: Since nuclei have charge and spin, they possess a magnetic moment, though it is much smaller than the magnetic moment of an electron.

2. Nuclear Forces

Q: Discuss the properties of nuclear forces.

The strong nuclear force holds the nucleons together in the nucleus against the electrostatic repulsion of protons.

1. Short Range: It is effective only over very short distances (about 2-3 fermi). Beyond this range, it drops to zero rapidly.

2. Strongest Force: Within its range, it is the strongest force in nature (much stronger than gravitational or electromagnetic forces).

3. Charge Independent: The force between $p-p$, $n-n$, and $p-n$ is approximately the same (if they are in the same quantum state).

4. Spin Dependent: The force is stronger if the nucleons have parallel spins than if they have anti-parallel spins.

5. Saturation Property: A nucleon interacts only with its nearest neighbors, not with all other nucleons in the nucleus (explains why Binding Energy per nucleon is roughly constant).

6. Non-Central: It has a tensor component, meaning it depends on the orientation of the spin relative to the displacement vector.

3. Radioactivity & Decay

Q: What is radioactivity? State and explain radioactive decay law.

• Radioactivity: The spontaneous phenomenon where unstable atomic nuclei disintegrate by emitting radiation (alpha, beta, or gamma rays) to achieve a stable state.

• Radioactive Decay Law: The rate of disintegration of radioactive atoms at any instant is directly proportional to the number of radioactive atoms present at that instant.

  • Mathematically: $\frac{dN}{dt}=-\lambda N$

  • Where $N$ is the number of atoms, and $\lambda$ is the decay constant.

  • The negative sign indicates that the number of atoms decreases over time.

Q: Discuss radioactive dating.

• Radioactive Dating: A technique used to date materials such as rocks or carbon, usually based on a comparison between the observed abundance of a naturally occurring radioactive isotope and its decay products, using known decay rates.

• Concept: By measuring the ratio of parent nuclei (e.g., Uranium-238) to daughter nuclei (e.g., Lead-206) in a sample, and knowing the half-life, one can calculate the time elapsed since the sample was formed.

4. Types of Radiation (Alpha, Beta, Gamma)

Q: Write down the properties of alpha ($\alpha$), beta ($\beta$), and gamma ($\gamma$) rays.

Property	Alpha Particles (α)	Beta Particles (β)	Gamma Rays (γ)
Nature	Helium nuclei (${}_2^{4}H{e}^{2+}$)	Fast-moving electrons ($e^{-}$) or positrons	Electromagnetic waves (photons)
Charge	Positive ($+2e$)	Negative ($-e$) or Positive ($+e$)	Neutral (No charge)
Mass	Heavy ($4\text{ u}$)	Light ($1/1836\text{ u}$)	Rest mass is zero
Penetrating Power	Low (stopped by paper)	Moderate (stopped by Al foil)	High (stopped by thick lead)
Ionizing Power	Very High	Moderate ($1/100$ of $\alpha$)	Low ($1/100$ of $\beta$)
Velocity	$\approx 10^7\text{ m/s}$	$\approx 1-99\%$ of speed of light	Speed of light ($c$)
Field Deflection	Deflected by electric/magnetic fields	Strongly deflected	Not deflected

5. Radioactive Equilibrium

Q: Discuss secular equilibrium and transient equilibrium.

These concepts describe the relationship between a parent nucleus decaying into a radioactive daughter nucleus.

1. Secular Equilibrium:

   • Occurs when the parent has an extremely long half-life compared to the daughter (${\lambda }_{parent}\ll {\lambda }_{daughter}$).

   • The activity of the daughter becomes equal to the activity of the parent.

   • The amount of parent essentially remains constant over the observation period.

2. Transient Equilibrium:

   • Occurs when the parent’s half-life is longer than the daughter’s, but not infinitely so (${\lambda }_{parent}<{\lambda }_{daughter}$).

   • The ratio of the activities of the parent and daughter becomes constant, but both activities decrease with time according to the parent’s half-life.

6. Binding Energy

Q: What is nuclear binding energy? Discuss the significance of average binding energy per nucleon.

• Binding Energy (B.E.): The energy required to break a nucleus into its constituent protons and neutrons (or the energy released when they bind together). It arises from the mass defect ($\Delta m$).

• Significance of Average B.E. per Nucleon:

  • It is a measure of nuclear stability. Higher B.E./nucleon means the nucleus is more stable.

  • The curve of B.E./nucleon vs. Mass Number ($A$) peaks near Iron ($F{e}^{56}$), meaning iron is the most stable element.

  • Elements with very low or very high mass numbers have lower B.E./nucleon, making them susceptible to fusion (low mass) or fission (high mass) to move toward the stable center.

Q: What is Packing Fraction?

• Packing Fraction ($f$): It is the mass excess per nucleon.

  • Formula: $f=\frac{M-A}{A}$, where $M$ is the actual isotopic mass and $A$ is the mass number.

  • A negative packing fraction implies a stable nucleus (mass is converted to binding energy). A positive packing fraction implies instability.

Q: Explain the Binding Energy curve with the help of fission and fusion.

• The Curve: The graph of B.E./nucleon vs. Mass Number ($A$).

• Region of Stability: The curve peaks around $A\approx 56$ (Iron) with $\approx 8.8$ MeV/nucleon.

• Fusion: Light nuclei (low $A$, left side of curve) have low binding energy. If they fuse to form a heavier nucleus, the B.E./nucleon increases, releasing energy.

• Fission: Heavy nuclei (high $A$, right side of curve) also have lower binding energy (due to proton repulsion). If they split into two lighter nuclei, the resulting fragments have higher B.E./nucleon, releasing energy.

7. Nuclear Fission & Fusion

Q: Explain nuclear fission and fusion reactions with examples. What is the source of energy?

• Nuclear Fission: The process of splitting a heavy nucleus into two lighter nuclei of comparable mass, accompanied by the release of energy and neutrons.

  • Example: ${}_{92}^{235}U+{}_0^{1}n\to {}_{56}^{141}Ba+{}_{36}^{92}Kr+3{}_0^{1}n+\text{Energy}$

  • Source of Energy: The mass of the products is less than the mass of the reactants. This lost mass (mass defect) is converted into energy ($E=\Delta m{c}^2$).

• Nuclear Fusion: The process where two light nuclei combine to form a heavier nucleus, releasing energy.

  • Example: ${}_1^{2}H+{}_1^{3}H\to {}_2^{4}He+{}_0^{1}n+\text{Energy}$

  • Source of Energy: Similar to fission, the final mass is less than the initial mass, resulting in energy release.

Q: Discuss the C-N Cycle (Carbon-Nitrogen Cycle).

• This is a set of fusion reactions by which stars (like the Sun) convert hydrogen into helium, using Carbon, Nitrogen, and Oxygen as catalysts.

• It is the dominant energy source in stars heavier than the Sun.

• The net result is: $4({}_1^{1}H)\to {}_2^{4}He+2e^{+}+2\nu +\text{Energy}$.

8. Nuclear Reactors

Q: Explain the construction and working principle of a nuclear reactor.

A nuclear reactor initiates and controls a self-sustained nuclear chain reaction to produce energy.

Construction (Main Components):

1. Fuel: Usually enriched Uranium ($U^{235}$) or Plutonium ($P{u}^{239}$).

2. Moderator: Material (Heavy water $D_{2}O$, Graphite) used to slow down fast neutrons to thermal energies so they can cause further fission.

3. Control Rods: Materials (Cadmium, Boron) that absorb neutrons. Inserted or withdrawn to control the reaction rate.

4. Coolant: Liquid/Gas (Water, Liquid Sodium) that circulates through the core to extract heat.

5. Shielding: Thick concrete walls to prevent radiation from escaping.

Working Principle:

• Fission of fuel nuclei releases energy and fast neutrons.

• The moderator slows these neutrons down.

• Slow neutrons cause fission in other fuel nuclei, sustaining a chain reaction.

• The heat generated is removed by the coolant to produce steam, which drives a turbine to generate electricity.

Q: State and explain the criticality of a reactor.

• Criticality: Refers to the state of the chain reaction, defined by the multiplication factor $k$.

  • $k=\frac{\text{number of neutrons in one generation}}{\text{number of neutrons in the preceding generation}}$

Conditions for Chain Reaction:

1. Critical ($k=1$): The reaction is steady and self-sustaining (Normal operation).

2. Sub-critical ($k<1$): The reaction dies out (Shut down).

3. Super-critical ($k>1$): The reaction grows rapidly (Explosion/Bomb).

Controlled Chain Reaction: To maintain a controlled reaction, the reactor must be kept at Critical ($k=1$) state using control rods to absorb excess neutrons.