Nuclear Reactions, Fission & Thermonuclear Fusion

1. General Nuclear Reaction Formulation & Q-Value

A generic two-body nuclear reaction is denoted as $a + X \to Y + b$ or in compact Bethe notation $X(a, b)Y$, where projectile $a$ strikes stationary target nucleus $X$, producing ejectile $b$ and residual nucleus $Y$. The reaction $Q$-value is defined as the invariant rest-mass energy difference:

• Exothermic Reactions ($Q > 0$): Mass is converted into kinetic energy. The reaction can occur even at zero incident projectile energy ($T_a \to 0$).
• Endothermic Reactions ($Q < 0$): Kinetic energy is converted into mass. The reaction is forbidden unless the projectile delivers a kinetic energy exceeding a strict threshold energy $E_{\text{th}}$.

2. Laboratory vs Center-of-Mass (CM) Transformation

In the laboratory frame, the total momentum is $p_{\text{lab}} = p_a = \sqrt{2 M_a T_a}$. The velocity of the center-of-mass frame is:

The total kinetic energy available in the center-of-mass frame ($T_{\text{cm}}$) represents the fraction of laboratory energy available to induce nuclear transformations (the remainder $T_{\text{cm, motion}} = \frac{1}{2}(M_a + M_X)v_{\text{cm}}^2$ is locked up in rigid motion of the system):

where $\mu = \frac{M_a M_X}{M_a + M_X}$ is the reduced mass.

3. Derivation of Reaction Threshold Energy

For an endothermic reaction ($Q < 0$), the reaction can occur if and only if the center-of-mass kinetic energy is at least sufficient to supply the mass deficit: $T_{\text{cm}} \ge |Q|$. Substituting $T_{\text{cm}}$ gives the threshold laboratory kinetic energy:

If Coulomb forces are present (charged projectile and target), the projectile must also overcome or tunnel through the Coulomb barrier $V_C = \frac{1}{4\pi\varepsilon_0}\frac{Z_a Z_X e^2}{R_a + R_X}$, significantly elevating the effective practical threshold.

1. Reaction Cross-Section & Beam Attenuation

The reaction probability is quantified by the cross-section $\sigma$, possessing units of area (standard unit: $1\text{ barn (b)} \equiv 10^{-28}\text{ m}^2 = 100\text{ fm}^2$). For a uniform projectile beam of flux $\Phi = n_a v_a$ (particles per unit area per second) incident on a thin target containing $N_{\text{t}}$ target nuclei per unit area, the reaction rate $R$ is:

For a thick target of mass density $\rho$ and thickness $x$, the unscattered beam intensity decreases exponentially:

where $\Sigma_t$ is the macroscopic total cross-section ($\text{cm}^{-1}$).

2. Breit-Wigner Single-Level Resonance Formula

When the incident projectile energy matches a quasi-bound quantum level of the composite system ($E_0$), the cross-section displays a dramatic, sharp peak called a resonance. Gregory Breit and Eugene Wigner derived the cross-section for isolated s-wave ($\ell = 0$) resonance reactions using quantum scattering theory:

where:

• $\lambda\hspace{-0.45em}\bar{} = \frac{\lambda}{2\pi} = \frac{\hbar}{p_{\text{cm}}}$ is the reduced de Broglie wavelength of the incident channel.
• $\Gamma = \Gamma_a + \Gamma_b + \Gamma_\gamma + \dots$ is the total resonance energy width, related to the compound state lifetime by $\tau = \hbar / \Gamma$.
• $\Gamma_a, \Gamma_b$ are partial decay widths for the entrance and exit channels.
• $g = \frac{2 J + 1}{(2 s_a + 1)(2 I_X + 1)}$ is the spin statistical factor for compound nuclear spin $J$.

At the peak ($E = E_0$), the maximum cross-section is $\sigma_{\text{peak}} = 4\pi \lambda\hspace{-0.45em}\bar{}^2 g \frac{\Gamma_a \Gamma_b}{\Gamma^2}$. For slow neutrons where $\lambda \sim 10^{-10}\text{ m} \gg R$, thermal resonance cross-sections can reach tens of thousands of barns (e.g., $^{113}\text{Cd}$ has $\sigma \approx 20,000\text{ b}$, $^{135}\text{Xe}$ has $\sigma \approx 2.6 \times 10^6\text{ b}$).

3. The Optical Model Potential

To describe both elastic scattering and non-elastic absorption of nucleons by nuclei, Herman Feshbach introduced the optical model, representing the nucleus as a refractive, cloudy crystal ball with a complex potential:

The real depth $V(r) \approx 50\text{ MeV}$ governs nuclear refraction (elastic scattering), while the imaginary depth $W(r) \approx 5 - 15\text{ MeV}$ acts as an energy sink, absorbing flux from the incident channel to simulate all possible inelastic, transfer, and compound nuclear reactions.

1. Niels Bohr's Compound Nucleus Hypothesis (1936)

Niels Bohr proposed that low-energy nuclear reactions proceed through a distinct two-stage process:

1. Formation: The projectile enters the target and undergoes repeated collisions with multiple nucleons, dissipating its kinetic energy across the whole nucleus to form an excited compound nucleus $C^*$. The formation time is $\tau_{\text{form}} \sim 10^{-22}\text{ s}$ (nuclear transit time).
2. Independence Hypothesis: The compound nucleus survives for a long duration ($\tau_C \sim 10^{-16} - 10^{-18}\text{ s}$, thousands of times longer than transit time) during which all memory of the entrance channel ($a+X$) is completely erased, except for exact conserved quantum numbers (energy $E$, angular momentum $J$, and parity $\pi$).
3. Decay: De-excitation occurs through statistical evaporation when fluctuations concentrate sufficient energy on a single particle or photon channel:
   $$\sigma(a, b) = \sigma_{\text{form}}(C^*) \times P_{\text{decay}}(b)$$

Compound reactions produce isotropic or forward-backward symmetric angular distributions in the center-of-mass frame ($\frac{d\sigma}{d\Omega}(\theta) = \frac{d\sigma}{d\Omega}(\pi - \theta)$).

2. Direct Reactions (Stripping & Pickup)

At higher incident energies ($E \gtrsim 20 - 100\text{ MeV}$), the projectile traverses the nucleus in a single passage ($\tau \sim 10^{-22}\text{ s}$) and interacts with only one or two valence nucleons at the nuclear surface:

• Stripping Reactions: The projectile deposits one or more nucleons into a single-particle orbit of the target nucleus, while the remainder continues forward (e.g., $(d, p)$ stripping deposits a neutron).
• Pickup Reactions: The projectile scoops up a nucleon from the target (e.g., $(p, d)$ picks up a neutron).

Direct reactions display strong forward-peaked angular distributions whose diffraction peaks depend uniquely on the transferred orbital angular momentum $\ell$, providing a powerful experimental probe of single-particle shell structures.

1. Liquid Drop Model Explanation of Nuclear Fission

In 1939, Otto Frisch and Lise Meitner interpreted Otto Hahn and Fritz Strassmann's discovery of barium in neutron-irradiated uranium as the binary splitting—nuclear fission—of the heavy uranium nucleus. Niels Bohr and John Wheeler parameterized the stability of a spherical drop deformed into a prolate spheroid with eccentricity $\varepsilon$:

The deformed droplet is stable against spontaneous fission if $E_{\text{def}} > 0$. The threshold for spontaneous instability occurs when Coulomb repulsion exceeds twice surface tension:

The ratio $x = \frac{Z^2 / A}{(Z^2 / A)_{\text{crit}}} \approx \frac{Z^2 / A}{48}$ is the fissility parameter. For $^{238}_{92}\text{U}$, $Z^2/A \approx 35.6$, resulting in a finite fission barrier height $E_f \approx 5.8\text{ MeV}$.

2. Energy Release in Fission

For a typical heavy nucleus ($A \approx 236$), $B/A \approx 7.6\text{ MeV}$, while the medium-mass fission fragments have $B/A \approx 8.5\text{ MeV}$. The energy released per fission event is:

The $200\text{ MeV}$ is partitioned as follows:

• Fragment Kinetic Energy: $\approx 168\text{ MeV}$ ($84\%$, deposited immediately as heat within a few micrometers).
• Prompt Fission Neutrons: $\approx 5\text{ MeV}$ (average $2 - 3$ neutrons per fission, mean kinetic energy $\sim 2\text{ MeV}$).
• Prompt Gamma Rays: $\approx 7\text{ MeV}$.
• Delayed Beta Particles: $\approx 8\text{ MeV}$ from fragment radioactive decay chains.
• Delayed Antineutrinos: $\approx 12\text{ MeV}$ (escapes the reactor without depositing heat).

3. Mass Yield Asymmetry & Fission Neutrons

Low-energy thermal neutron fission of $^{235}\text{U}$ produces an asymmetric two-humped mass yield curve with peaks at light mass $A_L \approx 95$ and heavy mass $A_H \approx 140$. Symmetric fission ($A_1 = A_2 \approx 118$) is suppressed by a factor of 600 due to shell closures in the nascent fragments ($Z=50, N=82$). At high excitation energies ($E_n > 50\text{ MeV}$), shell effects wash out and symmetric fission dominates.

Fission releases an average of $\bar{\nu} \approx 2.43$ neutrons per event for $^{235}\text{U}$. Crucially, approximately $0.65\%$ of these neutrons are delayed neutrons emitted seconds to minutes later by beta-decay precursors (e.g., $^{87}\text{Br} \to {^{87}\text{Kr}} \to {^{86}\text{Kr}} + n$), providing the indispensable time delay required for mechanical control rods to safely stabilize nuclear power reactors.

1. The Neutron Multiplication Factor (k)

The continuity of a nuclear fission chain reaction is dictated by the effective multiplication factor $k_{\text{eff}}$, defined as the ratio of neutrons in generation $n+1$ to neutrons in generation $n$:

• $k_{\text{eff}} < 1$: Subcritical (neutron population and fission power die away exponentially).
• $k_{\text{eff}} = 1$: Critical (steady, self-sustaining stationary power generation).
• $k_{\text{eff}} > 1$: Supercritical (neutron flux grows exponentially).

2. Fermi's Four-Factor Formula ($k_\infty$)

In an infinitely extended homogeneous or heterogeneous reactor core (ignoring surface neutron leakage), the multiplication factor is governed by the Four-Factor Formula:

1. Fast Fission Factor ($\varepsilon \approx 1.03 - 1.08$): Ratio of total neutrons produced by both fast and thermal fissions to those produced solely by thermal fissions.
2. Resonance Escape Probability ($p \approx 0.85 - 0.92$): The probability that a fast fission neutron slows down through the dangerous intermediate resonance capture region of $^{238}\text{U}$ ($1 - 1000\text{ eV}$) without being absorbed. Heterogeneous fuel lump arrangements maximize $p$ through spatial self-shielding.
3. Thermal Utilization Factor ($f \approx 0.70 - 0.90$): The probability that a completely moderated thermal neutron is absorbed in the nuclear fuel rather than in the moderator, structural cladding, or control poisons:
   $$f = \frac{\Sigma_a^{\text{fuel}}}{\Sigma_a^{\text{fuel}} + \Sigma_a^{\text{other}}}$$
4. Neutron Reproduction Factor ($\eta \approx 1.3 - 2.1$): The average number of fission neutrons emitted per thermal neutron absorbed in the fuel:
   $$\eta = \nu \frac{\Sigma_f^{\text{fuel}}}{\Sigma_a^{\text{fuel}}}$$

3. Reactor Kinetics and Prompt Criticality

If all fission neutrons were prompt, the average neutron lifetime would be $\ell \sim 10^{-4}\text{ s}$ (in a thermal reactor). The reactor power would evolve as $P(t) = P_0 e^{(k - 1)t / \ell}$. Even a $0.1\%$ excess ($k = 1.001$) would cause power to escalate by $e^{10} \approx 22,000$ in one second, rendering reactor control physically impossible.

With delayed neutron fraction $\beta \approx 0.0065$, the effective mean lifetime is extended to $\bar{\ell} = (1 - \beta)\ell + \sum \beta_i \tau_i \approx 0.08 - 0.1\text{ seconds}$. Provided $k_{\text{eff}} < 1 + \beta$, the reactor is delayed critical and responds smoothly over human and mechanical timescales.

1. Stellar Nucleosynthesis: The Proton-Proton Chain

In main-sequence stars like our Sun ($T_{\text{core}} \approx 1.5 \times 10^7\text{ K}$, $k_B T \approx 1.3\text{ keV}$), stellar energy generation is powered by the fusion of four protons into helium-4 ($4 p \to {^4\text{He}} + 2 e^+ + 2 \nu_e + 26.73\text{ MeV}$). The dominant sequence is the $p$-$p$ chain:

1. $p + p \to {^2\text{H}} + e^+ + \nu_e$ ($Q = 1.442\text{ MeV}$): Weak interaction bottle-neck process ($p \to n + e^+ + \nu_e$) with an extraordinarily tiny cross-section ($\sigma \sim 10^{-47}\text{ cm}^2$). A proton in the solar core waits an average of $10^9\text{ years}$ to undergo this reaction, ensuring the Sun's multi-billion-year stability.
2. $^2\text{H} + p \to {^3\text{He}} + \gamma$ ($Q = 5.493\text{ MeV}$): Rapid electromagnetic capture ($\sim 1\text{ second}$).
3. $^3\text{He} + {^3\text{He}} \to {^4\text{He}} + 2 p$ ($Q = 12.86\text{ MeV}$, $pp\text{-I}$ branch): Completes the synthesis of $^4\text{He}$.

In heavier, hotter stars ($T > 2 \times 10^7\text{ K}$), the catalytic CNO cycle ($^{12}\text{C} \to {^{13}\text{N}} \to {^{13}\text{C}} \to {^{14}\text{N}} \to {^{15}\text{O}} \to {^{15}\text{N}} \to {^{12}\text{C}} + {^4\text{He}}$) dominates due to its steeper temperature dependence ($\epsilon_{\text{CNO}} \propto T^{17}$ vs $\epsilon_{pp} \propto T^4$).

2. Controlled Terrestrial Fusion: The D-T Reaction

For magnetic confinement fusion (tokamaks, stellarators), the most accessible reaction is Deuterium-Tritium fusion:

The D-T reaction possesses the lowest Coulomb barrier and highest cross-section ($\sigma_{\text{peak}} \approx 5.0\text{ b}$ at $E_{\text{cm}} \approx 64\text{ keV}$, accessible at thermal plasma temperatures $T \sim 15\text{ keV} \approx 1.7 \times 10^8\text{ K}$). The Gamow window represents the convolution of the Maxwell-Boltzmann tail $e^{-E / k_B T}$ with the quantum tunneling transmission $e^{-b / \sqrt{E}}$.

3. The Lawson Criterion & Triple Product

In 1957, J. D. Lawson formulated the ignition condition where thermonuclear self-heating by alpha particles ($E_\alpha = 3.5\text{ MeV}$) exceeds plasma Bremsstrahlung radiation and conduction losses without external heating:

where $n$ is fuel ion density and $\tau_E$ is the energy confinement time. For D-T fusion at the optimum temperature $T \approx 15\text{ keV}$, this requires the famous fusion triple product:

Magnetic confinement achieves this at low density and high confinement ($n \sim 10^{20}\text{ m}^{-3}, \tau_E \sim 3\text{ s}$), while inertial confinement uses extreme compression ($n \sim 10^{31}\text{ m}^{-3}, \tau_E \sim 10^{-10}\text{ s}$).