The nucleus

Q-value definition:

• Energy difference between initial and final constituents: $Q = U_i - U_f$
• Represents energy released (positive Q) or required (negative Q) in a nuclear process

Calculation for different decays:

• Alpha decay: $Q = [m(^A_Z\text{X}) - m(^{A-4}_{Z-2}\text{Y}) - m(^4_2\text{He})]c^2$
• Beta-minus decay: $Q = [m(^A_Z\text{X}) - m(^A_{Z+1}\text{Y})]c^2$
• Beta-plus decay: $Q = [m(^A_Z\text{X}) - m(^A_{Z-1}\text{Y}) - 2m_e]c^2$
• Electron capture: $Q = [m(^A_Z\text{X}) - m(^A_{Z-1}\text{Y})]c^2$

Significance:

• Determines if a reaction is energetically possible (Q > 0)
• Provides the kinetic energy available to the decay products
• Explains why isolated protons cannot beta-plus decay (Q would be negative)
• Critical for predicting possible decay modes and energy spectra

Electron Capture vs. Beta-Plus Decay:

Similarities:

• Both convert a proton to a neutron: p → n
• Both increase N by 1 and decrease Z by 1
• Both emit a neutrino
• Both occur in proton-rich nuclei moving toward stability

Differences:

• Electron Capture:

  • Captures orbital electron (usually K-shell)
  • Reaction: p + e⁻ → n + ν
  • Q-value: $Q = [m(^A_Z\text{X}) - m(^A_{Z-1}\text{Y})]c^2$
  • Produces X-rays as electrons fill atomic shell vacancy
  • No positron emitted

• Beta-Plus Decay:

  • Creates and emits positron
  • Reaction: p → n + e⁺ + ν
  • Q-value: $Q = [m(^A_Z\text{X}) - m(^A_{Z-1}\text{Y}) - 2m_e]c^2$
  • Requires at least 1.022 MeV (2m₉c²) more energy than EC

Preference factors:

• Beta-plus decay is energetically possible only if Q > 1.022 MeV
• Electron capture can occur at lower energy differences
• Probability of EC increases for higher Z elements (more orbital electrons)
• EC dominates when energy available is between 0 and 1.022 MeV
• When both are possible, there's competition between the processes

Deuterium's significantly lower binding energy per nucleon compared to helium-4 results from:

• Single nucleon pair interaction in deuterium versus multiple pairings in helium-4
• Pairing effects - helium-4 has two protons and two neutrons, creating a "doubly magic" configuration
• Volume-to-surface ratio - in deuterium, both nucleons are essentially "surface nucleons"
• Quantum mechanical effects - helium-4's closed shell configuration produces exceptional stability

This dramatic increase in binding energy explains:

1. Why fusion of deuterium into helium releases substantial energy (~24 MeV total)
2. The abundance of helium-4 in the early universe following Big Bang nucleosynthesis
3. Why alpha particles (helium-4 nuclei) are a common decay product from heavy nuclei

This represents a fundamental principle in nuclear physics: nuclei with fully paired nucleons arranged in complete shells exhibit enhanced stability.

The Lawson criterion:

• Fundamental requirement for energy-positive fusion
• Expressed as: $n\tau > 10^{14}$ s·cm$^{-3}$
• Where:
  • $n$ = particle density (particles per cm$^{3}$)
  • $\tau$ = confinement time (seconds)
  • Product $n\tau$ is called the Lawson number

For a viable fusion reactor:

• Must achieve Q > 1 (fusion energy output > energy input)
• Requires sufficient triple product of:
  • Density (10$^{20}$-10$^{22}$ particles/m$^3$)
  • Confinement time (1-10 seconds)
  • Temperature (10$^8$-10$^9$ K)
• D-T reaction is easiest, requiring lower temperatures (~10$^8$ K)
• D-D reactions require higher temperatures (~10$^9$ K)

This criterion explains why achieving practical fusion power remains challenging despite decades of research.

Proton-proton (PP) cycle:

Step 1: Two protons fuse to form deuterium with positron and neutrino emission $^1\text{H} + ^1\text{H} \rightarrow ^2\text{H} + e^+ + \nu_e$ (weak interaction)

Step 2: Deuterium captures another proton to form helium-3 $^2\text{H} + ^1\text{H} \rightarrow ^3\text{He} + \gamma$

Step 3: Two helium-3 nuclei combine to form helium-4 plus two protons $^3\text{He} + ^3\text{He} \rightarrow ^4\text{He} + 2 ^1\text{H}$

Net equation: $4 ^1\text{H} \rightarrow ^4\text{He} + 2e^+ + 2\nu_e + 2\gamma + 26.7 \text{ MeV}$

Viability in stars vs. laboratory:

1. Time scales:

   • Stars: Can sustain reactions over billions of years
   • Laboratory: Requires much higher reaction rates for practical energy

2. First step limitations:

   • Involves weak interaction (10^17 times slower than strong force reactions)
   • Average proton in Sun waits ~10^9 years before fusion
   • Dependent on quantum tunneling through Coulomb barrier

3. Density and confinement:

   • Stars: Gravitational confinement with immense mass
   • Laboratory: Cannot reproduce gravitational confinement

4. Temperature:

   • PP cycle optimized for ~15 million K
   • Laboratory fusion focuses on D-T or D-D reactions requiring much higher temperatures

This explains why fusion research focuses on deuterium and tritium rather than pure hydrogen fusion.

• Statistical uncertainty in radioactive decay:

  • Is inherent to the quantum nature of the process
  • Cannot be eliminated by improving measurement techniques
  • Follows precise mathematical laws (Poisson statistics)
  • Is a fundamental property of nature, not a limitation of equipment

• Whereas experimental measurement error:

  • Results from equipment limitations or environmental factors
  • Can potentially be reduced with better instruments
  • May follow different statistical distributions (often Gaussian)
  • Is a practical limitation, not a fundamental one

• Example: When monitoring a radioactive sample:

  • Even with perfect detection equipment, the number of decays per minute will naturally fluctuate
  • These fluctuations follow a predictable statistical pattern based on the sample size
  • The relative magnitude of fluctuations decreases with √N (where N is the number of atoms)

• Application: In radiation safety, statistical uncertainties must be accounted for when establishing safety margins

The asymmetric distribution occurs due to:

• Nuclear shell effects that favor the formation of nuclei with "magic numbers" of nucleons
• The lower potential energy configuration of asymmetric fragments compared to symmetric ones
• Quantum mechanical effects that make fragments with mass numbers around A≈95 and A≈140 more energetically favorable

In uranium-236 fission:

• Peak probability occurs at mass numbers A≈95 and A≈140
• The probability of equal-sized fragments (A≈118) is approximately 100 times lower
• This bimodal distribution is characteristic of thermal neutron-induced fission of uranium

Note: This asymmetry demonstrates that nuclear fission is not a simple splitting in half, but a complex quantum mechanical process governed by nuclear structure effects.

Control rods in a nuclear reactor serve to:

• Absorb neutrons to control the rate of fission reactions
• Regulate power output by adjusting their insertion depth into the core
• Provide emergency shutdown capability (SCRAM) when fully inserted
• Maintain criticality at exactly k=1 during normal operation
• Compensate for fuel depletion over time

Control rods are typically made of neutron-absorbing materials such as cadmium, boron, or hafnium.

Example: When operators need to reduce reactor power, control rods are inserted deeper into the moderator, capturing more neutrons that would otherwise cause additional fission events.

Inertial confinement fusion is a technique to achieve controlled nuclear fusion by:

• Using intense laser beams directed from multiple angles onto a D-T (deuterium-tritium) pellet • Rapidly compressing and heating the fuel pellet to fusion conditions • Creating pressure and temperature high enough to overcome Coulomb repulsion

Key characteristics: • Operates in pulses rather than continuous operation • Achieves ultra-high density (10³-10⁴ times initial density) for very brief periods • Self-heating occurs as fusion-produced alpha particles remain trapped in the compressed plasma

Unlike magnetic confinement (Tokamak), which uses magnetic fields to contain rarefied plasma for longer durations, inertial confinement relies on the fuel's inertia to maintain compression during the brief fusion reaction.

Classical vs. Quantum View of Nuclear Fission

Classical Physics Prediction:

• Fission impossible without sufficient external energy to overcome the entire barrier
• Energy conservation must be strictly maintained at all times
• Heavy nuclei should be permanently stable if they don't have enough energy to overcome the fission barrier

Quantum Mechanical Reality:

• Nuclei can "tunnel" through energy barriers they classically couldn't overcome
• Allows spontaneous fission without external energy input
• Temporary apparent violation of energy conservation permitted

Explanation via Heisenberg's Uncertainty Principle

• Energy-time uncertainty relation: $\Delta E \cdot \Delta t \geq \hbar/2$
• System can temporarily "borrow" energy $\Delta E = E_2 - E_1$ to penetrate barrier
• This borrowed energy must be "returned" within time $\Delta t$

Factors Affecting Penetration Probability

1. Barrier Height: Probability decreases exponentially with higher barriers ($\propto e^{-k\Delta E}$)
2. Barrier Width: Thinner barriers are more easily penetrated
3. Intermediate State Duration: Shorter durations increase penetration probability

This quantum tunneling mechanism explains why some heavy nuclei undergo spontaneous fission while others remain stable for extremely long periods, with half-lives determined by their specific barrier characteristics.