Unit 5: Nuclear Fission Mechanics & Reactor Physics

In December 1938, German radiochemists Otto Hahn and Fritz Strassmann at the Kaiser Wilhelm Institute in Berlin irradiated natural uranium with thermal neutrons. Expecting to identify transuranic elements ($Z > 92$), they performed fractional crystallization with barium carriers. Astonishingly, the radioactivity precipitated identically with barium ($Z = 56$), an element whose mass is barely half that of uranium.

The Meitner-Frisch Theoretical Interpretation (January 1939)

In exile in Sweden, Lise Meitner and her nephew Otto Frisch interpreted the result using George Gamow and Niels Bohr's Liquid Drop Model of the nucleus:

Frisch coined the term fission by analogy with binary fission in cellular biology.

Energetic Partition of Nuclear Fission

The total energy released per fission of uranium-235 is approximately $200\text{ MeV}$ ($\approx 3.204 \times 10^{-11}\text{ J}$). This immense thermodynamic yield arises from the drop in binding energy per nucleon: from $\approx 7.6\text{ MeV/nucleon}$ for $^{235}\text{U}$ to $\approx 8.5\text{ MeV/nucleon}$ for mid-mass fission fragments:

``` ENERGY COMPONENT TYPICAL VALUE FRACTION ───────────────────────────────────────────────────────────────────────────── Kinetic Energy of Fission Fragments (Prompt) ~ 168 MeV 84.0% Kinetic Energy of Prompt Neutrons (~2.4 per fission) ~ 5 MeV 2.5% Prompt Gamma-Ray Photons ~ 7 MeV 3.5% Beta- Particles from Radioactive Fission Fragments ~ 8 MeV 4.0% Antineutrinos (ν̄_e, escape reactor core completely) ~ 12 MeV 6.0% Delayed Gamma Photons from Fragment Decay Chains ~ 7 MeV 3.5% ───────────────────────────────────────────────────────────────────────────── TOTAL FISSION ENERGY RELEASE: ~ 207 MeV 100.0% RECOVERABLE THERMAL CORE ENERGY: ~ 195 MeV 94.0% ```

Over $80\%$ of the energy appears as kinetic energy of the two massive, highly charged fission fragments. They travel a mere $\sim 10\,\mu\text{m}$ in uranium metal before stopping, converting their kinetic energy into intense localized thermal heat via Coulomb ionization collisions.

In their 1939 paper, Niels Bohr and John Archibald Wheeler formulated the definitive classical theory of nuclear fission. They modeled the nucleus as an incompressible, charged liquid drop undergoing ellipsoidal quadrupole deformation.

Deformation Energy of a Liquid Drop

Consider a spherical nucleus of radius $R_0$ distorted into an axisymmetric prolate spheroid with semi-major axis $a = R_0(1 + \epsilon)$ and semi-minor axis $b = R_0(1 - \epsilon/2)$, where $\epsilon$ is the eccentricity deformation parameter:

• Volume: Conserved ($\frac{4}{3}\pi a b^2 = \frac{4}{3}\pi R_0^3$).
• Surface Area: Increases with deformation:

• Coulomb Self-Energy: Decreases with deformation (charge is pushed further apart):

The net change in nuclear potential energy $\Delta E$ relative to the spherical ground state is:

``` Potential Energy ΔE(ε) ▲ │ /\ Fission Barrier E_f │ / \ │ / \ │ / \____ Spontaneous Fission (Scission) │ ___ / 0 ┼─( O )/────────────────► Deformation ε Spherical Ground State ```

The Fissility Parameter ($x$)

The condition for spontaneous instability against infinitesimal deformation ($\Delta E < 0$) is:

We define the dimensionless Fissility Parameter $x$:

Substituting the SEMF expressions $E_S^0 = a_s A^{2/3}$ and $E_C^0 = a_c Z^2 / A^{1/3}$:

Using $a_c \approx 0.711\text{ MeV}$ and $a_s \approx 17.80\text{ MeV}$:

• If $x \ge 1.0$ ($Z^2/A \ge 50$): The spherical nucleus is unstable to immediate spontaneous fission ($\tau \sim 10^{-22}\text{ s}$).
• If $x < 1.0$: A finite Fission Barrier $E_f$ opposes deformation.

Fissile Versus Fertile Radionuclides

• Fissile Radionuclides: Capable of undergoing fission with zero-energy thermal neutrons ($0.025\text{ eV}$). Examples: $^{233}\text{U}, ^{235}\text{U}, ^{239}\text{Pu}, ^{241}\text{Pu}$.
• Fertile Radionuclides: Do not fission with thermal neutrons because neutron binding energy $S_n < E_f$. They require fast neutrons ($E_n > 1.0\text{ MeV}$) to fission, but can capture a thermal neutron to breed a fissile isotope. Examples: $^{238}\text{U} \to ^{239}\text{Pu}$, $^{232}\text{Th} \to ^{233}\text{U}$.

The difference is rooted in the pairing energy: When an odd-neutron nucleus ($^{235}_{92}\text{U}$) captures a neutron, it forms an even-even compound state ($[^{236}_{92}\text{U}^*]$), releasing an extra $+1.2\text{ MeV}$ of pairing binding energy ($S_n = 6.55\text{ MeV}$), which exceeds the $5.7\text{ MeV}$ fission barrier! Capturing a neutron on even-even $^{238}_{92}\text{U}$ forms an odd-neutron state ($[^{239}_{92}\text{U}^*]$) with zero pairing bonus ($S_n = 4.80\text{ MeV}$), falling short of the $6.2\text{ MeV}$ barrier.

When a heavy nucleus such as uranium-235 undergoes low-energy thermal fission, it does not split into two equal halves ($A_1 = A_2 \approx 117$). Instead, the fragment mass yield distribution exhibits a pronounced asymmetric double-humped morphology.

``` Fragment Yield (%) 10 ▲ Peak 1 (Light) Peak 2 (Heavy) │ A ≈ 95 A ≈ 138 8 │ (Kr, Sr, Zr) (Xe, Cs, Ba) │ __ __ 6 │ / \ / \ │ / \ / \ 4 │ / \ / \ │ / \ / \ 2 │ / \ Valley / \ │____/ \__(A ≈ 117)/ \____ 0 └────┴─────────────┴──────────┴─────────────┴────► Mass Number A 70 95 117 140 160 ```

The Double-Hump Peak Characteristics

For thermal neutron fission of $^{235}\text{U}$:

1. Light Fragment Group: Centered around $A_L \approx 95$ ($Z \approx 36 - 40$, elements $\text{Kr}, \text{Rb}, \text{Sr}, \text{Y}, \text{Zr}$).

2. Heavy Fragment Group: Centered around $A_H \approx 138$ ($Z \approx 53 - 57$, elements $\text{I}, \text{Xe}, \text{Cs}, \text{Ba}, \text{La}$).

3. Symmetric Valley: Symmetric fission ($A \approx 117$) occurs with a probability of less than $0.01\%$ ($\sim 600$ times less frequent than asymmetric fission).

Physical Origin: Quantum Shell Effects

The dominance of asymmetric fission is governed by nuclear shell closures: The heavy fragment peak is anchored by the proximity of the doubly magic spherical shell closure:

The extraordinary shell stabilization of nascent fragments near $Z = 50$ and $N = 82$ energetically favors asymmetric neck scission.

At high incident excitation energies ($E_n > 40\text{ MeV}$), individual single-particle shell structures wash out, and the fragment yield curve transitions into a single symmetric Gaussian peak centered at $A/2$.

In every fission event, an average of $\bar{\nu} \approx 2.4 - 3.0$ neutrons are released. Crucially for nuclear engineering, these neutrons are emitted via two distinct physical mechanisms:

1. Prompt Neutrons ($>99\%$)

Emitted directly from the scission neck and evaporating fragments within $\tau_{\text{prompt}} \sim 10^{-14}\text{ seconds}$ of scission. They exhibit a continuous Maxwellian fission spectrum (Watt spectrum):

2. Delayed Neutrons ($<1\%$)

Fission fragments are neutron-rich and undergo cascades of negative beta decays. In certain daughter nuclei (termed delayed neutron precursors), the beta decay Q-value exceeds the neutron separation energy ($Q_\beta > S_n$). The daughter is formed in an excited state that promptly ($<10^{-14}\text{ s}$) de-excites by boiling off a neutron:

``` FISSION FRAGMENT PRECURSOR (e.g., ⁸⁷₃₅Br, T₁/₂ = 55.6 s) │ ▼ (β⁻ decay, slow: governed by T₁/₂) EXCITED DAUGHTER [⁸⁷₃₆Kr] (Excitation E > S_n) │ ▼ (Prompt neutron emission, fast: <10⁻¹⁴ s) STABLE RESIDUAL ⁸⁶₃₆Kr + ¹₀n (DELAYED NEUTRON!) ```

The appearance of the delayed neutron is rate-limited not by nuclear forces, but by the half-life of the preceding beta decay!

Precursor Groups and the Delayed Neutron Fraction ($\beta$)

The fraction of all fission neutrons that are delayed is defined as $\beta$:

Delayed neutrons are categorized into six empirical precursor groups:

| Group $i$ | Representative Precursor | Half-Life $T_{1/2,i}$ | Decay Constant $\lambda_i$ ($\text{s}^{-1}$) | Yield Fraction $\beta_i$ ($^{235}\text{U}$) | | :--- | :--- | :--- | :--- | :--- | | 1 | $^{87}\text{Br}$ | $55.6\text{ s}$ | $0.0124$ | $0.00021$ | | 2 | $^{137}\text{I}$ | $24.5\text{ s}$ | $0.0283$ | $0.00142$ | | 3 | $^{138}\text{I}, ^{89}\text{Br}$ | $16.3\text{ s}$ | $0.0425$ | $0.00127$ | | 4 | $^{139}\text{I}, ^{93}\text{Kr}$ | $5.21\text{ s}$ | $0.1330$ | $0.00257$ | | 5 | $^{140}\text{I}, ^{91}\text{Br}$ | $2.37\text{ s}$ | $0.2920$ | $0.00075$ | | 6 | $^{97}\text{Rb}$ | $0.23\text{ s}$ | $3.0100$ | $0.00028$ | | Total | — | — | $\bar{\tau}_d \approx 12.7\text{ s}$ | $\beta = 0.00650$ |

Criticality and Reactor Control Physics

Without delayed neutrons, the average neutron generation lifetime in a thermal reactor would be $\Lambda \approx l_p \approx 10^{-4}\text{ seconds}$. If reactivity increased by just $\Delta k = +0.001$, reactor power would escalate as:

Power would multiply by 22,000 every single second, rendering mechanical control impossible.

With delayed neutrons, the effective neutron lifetime is dominated by precursor half-lives:

Power response slows by a factor of nearly $1,000$, enabling mechanical control rods to manage power levels safely.

• Prompt Criticality ($\rho \ge \beta$ or $k \ge 1 + \beta$): The reactor is critical on prompt neutrons alone; power explodes uncontrollably (Chernobyl scenario).
• Delayed Criticality ($1.000 \le k < 1 + \beta$): The normal, stable operational regime where criticality requires delayed neutrons.

Prompt fission neutrons are born with high kinetic energies averaging $E_0 \approx 2\text{ MeV}$. However, the fission cross-section of $^{235}\text{U}$ is hundreds of times higher for thermal neutrons ($E_{\text{th}} \approx 0.025\text{ eV}$). To sustain a thermal chain reaction, fast neutrons must be slowed down via elastic collisions with moderator nuclei.

Elastic Collision Kinematics in the LAB Frame

Consider a neutron of mass $m = 1$ colliding elastically with a stationary moderator nucleus of mass number $A$. By conservation of momentum and energy in the center-of-mass frame, the ratio of final neutron energy $E'$ to initial energy $E$ after scattering through CM angle $\theta$ is:

• Minimum Energy (Head-On Collision, $\theta = \pi$):

where $\alpha = \left(\frac{A - 1}{A + 1}\right)^2$ is the collision parameter.

• For hydrogen ($A = 1$): $\alpha = 0$. A neutron can transfer $100\%$ of its kinetic energy in a single collision!
• For carbon ($A = 12$): $\alpha = (11/13)^2 \approx 0.716$. The neutron retains at least $71.6\%$ of its energy.

The Average Logarithmic Energy Decrement ($\xi$)

Because neutron energy loss is multiplicative rather than additive, the slowing-down power is characterized by the average decrease in the natural logarithm of energy per collision:

For isotropic s-wave scattering in the CM frame, $P(E'/E) = \frac{1}{1 - \alpha}$. Evaluating the integral:

For $A > 10$, this is approximated by:

Number of Collisions to Thermalize ($N_{\text{coll}}$)

The average number of collisions required to slow a neutron from fission energy $E_0 = 2\text{ MeV}$ to thermal energy $E_{\text{th}} = 0.025\text{ eV}$ is:

| Moderator | Mass Number $A$ | $\alpha = \left(\frac{A-1}{A+1}\right)^2$ | Decrement $\xi$ | Collisions to Thermalize $N_{\text{coll}}$ | Moderating Ratio ($\xi \Sigma_s / \Sigma_a$) | | :--- | :--- | :--- | :--- | :--- | :--- | | Light Water ($\text{H}_2\text{O}$) | $1$ | $0.000$ | $1.000$ | $18$ | $71$ | | Heavy Water ($\text{D}_2\text{O}$) | $2$ | $0.111$ | $0.725$ | $25$ | $5,670$ | | Beryllium ($\text{Be}$) | $9$ | $0.640$ | $0.207$ | $86$ | $143$ | | Graphite ($\text{C}$) | $12$ | $0.716$ | $0.158$ | $115$ | $192$ |

Heavy water ($\text{D}_2\text{O}$) has the highest moderating ratio ($\xi \Sigma_s / \Sigma_a = 5,670$) because deuterium has an exceptionally small neutron capture cross-section ($0.5\text{ mb}$ vs $332\text{ mb}$ for $^1\text{H}$), enabling CANDU reactors to operate using natural, unenriched uranium.

The operational state of a nuclear reactor is governed by the effective neutron multiplication factor $k_{\text{eff}}$, defined as:

• Subcritical ($k_{\text{eff}} < 1$): Chain reaction dies out exponentially; power decreases.
• Critical ($k_{\text{eff}} = 1.0000$): Steady-state chain reaction; power is exactly constant.
• Supercritical ($k_{\text{eff}} > 1$): Chain reaction diverges; power increases exponentially.

The Four-Factor Formula for Infinite Media ($k_\infty$)

In an infinitely large reactor (where leakage is zero), the multiplication factor is governed by Fermi's Four-Factor Formula:

``` Fast Neutrons from Thermal Fission: η * f │ ▼ Fast Fission Bonus (* ε) Total Fast Neutrons: ε η f │ ▼ Resonance Escape (* p) Neutrons Reaching Thermal Energy: p ε η * f │ ▼ Thermal Utilization (* f) Thermal Neutrons Absorbed in Fuel: f p ε η f ──► Next Generation! ```

1. Reproduction Factor ($\eta$):

Average number of fission neutrons produced per thermal neutron absorbed in the fuel:

For natural uranium, $\eta \approx 1.34$; for $3.5\%$ enriched fuel, $\eta \approx 1.80$.

2. Fast Fission Factor ($\epsilon$):

Ratio of total fast neutrons (including those from fast fission of $^{238}\text{U}$) to neutrons from thermal fission alone:

3. Resonance Escape Probability ($p$):

Probability that a fast neutron slows down through the broad $^{238}\text{U}$ resonance absorption capture peaks ($10\text{ eV} - 1\text{ keV}$) without being absorbed:

Using heterogeneous fuel rods separated by moderator increases $p$ because neutrons slow down in the moderator away from $^{238}\text{U}$.

4. Thermal Utilization Factor ($f$):

Fraction of thermal neutrons absorbed in the nuclear fuel compared to total thermal absorptions across fuel, moderator, cladding, and structure:

The Six-Factor Formula for Finite Reactors ($k_{\text{eff}}$)

In a real finite reactor core, neutrons can leak out across the boundary:

where:

• $P_{NL,f} = \frac{1}{1 + M_f^2 B_g^2} \approx \frac{1}{1 + \tau_F B_g^2}$ is the fast non-leakage probability (Fermi age $\tau_F$).
• $P_{NL,\text{th}} = \frac{1}{1 + L_{\text{th}}^2 B_g^2}$ is the thermal non-leakage probability ($L_{\text{th}}$ = thermal diffusion length).
• $B_g^2$ is the geometric buckling of the core geometry (e.g., $B_g^2 = (\pi/H)^2 + (2.405/R)^2$ for a finite cylinder).

The reactivity $\rho$ of a reactor core is defined as:

Reactivity is measured in percent ($\%\Delta k/k$), parts per hundred thousand ($\text{pcm} = 10^{-5}$), or dollars ($\$ = \rho / \beta$).

Commercial nuclear power generation utilizes distinct reactor designs engineered around specific neutron moderation, cooling, and isotopic fuel cycles.

1. Pressurized Water Reactor (PWR)

The dominant global reactor technology ($>65\%$ of worldwide capacity).

• Coolant & Moderator: Light water ($\text{H}_2\text{O}$) under high pressure ($15.5\text{ MPa} \approx 155\text{ bar}$) to prevent bulk boiling at operating temperatures of $\sim 315^\circ\text{C}$.
• Steam Cycle: Two distinct circuits (primary and secondary). The hot radioactive primary water passes through U-tube steam generators, boiling secondary water to drive the steam turbine.
• Fuel: Low-enriched uranium ($\text{UO}_2$ pellets, $3.0 - 5.0\%$ $^{235}\text{U}$) in Zircaloy cladding.

2. Boiling Water Reactor (BWR)

• Coolant & Moderator: Light water at lower operating pressure ($7.0\text{ MPa} \approx 70\text{ bar}$).
• Steam Cycle: Direct single loop. Water boils directly in the reactor core ($12 - 15\%$ steam void fraction at core exit), and radioactive steam flows straight to the turbine.
• Control Rods: Inserted from the bottom of the pressure vessel (because steam voids at the top reduce moderation, shifting flux to the bottom).

3. CANDU (Canada Deuterium Uranium)

• Coolant & Moderator: Heavy water ($\text{D}_2\text{O}$) in separate circuits. The low-pressure moderator is contained in a horizontal calandria vessel traversed by hundreds of pressurized fuel channels.
• Fuel: Natural uranium ($0.72\% \, ^{235}\text{U}$). Enabled by the low neutron capture of deuterium.
• Refueling: On-line refueling while operating at full power.

4. Liquid Metal Fast Breeder Reactor (LMFBR)

• Moderator: None (operates on fast neutron spectrum, $E_n > 100\text{ keV}$).
• Coolant: Liquid sodium ($\text{Na}$) or lead-bismuth eutectic at atmospheric pressure, with thermal conductivity $\sim 100\times$ water.
• Breeding Cycle: Breeds more fissile $^{239}\text{Pu}$ from $^{238}\text{U}$ blankets than it consumes ($BR > 1.0$), multiplying available nuclear fuel reserves by a factor of 60.

``` REACTOR TYPE MODERATOR COOLANT FUEL ENRICHMENT THERMAL EFFICIENCY ────────────────────────────────────────────────────────────────────────────── PWR Light Water Water (15 MPa) 3.2 - 4.9% ²³⁵U ~ 33 - 34% BWR Light Water Steam/Water 3.0 - 4.5% ²³⁵U ~ 33 - 34% CANDU Heavy Water Heavy Water Natural (0.72%) ~ 30 - 31% LMFBR None (Fast) Liquid Sodium 15 - 20% ²³⁹Pu/U ~ 39 - 41% HTGR Graphite Helium Gas 8 - 15% ²³⁵U ~ 45 - 50% ```

Inherent Passive Safety Principles

Modern Generation III+ and IV reactors incorporate passive safety mechanisms that operate without electrical power or operator intervention:

• Negative Doppler Temperature Coefficient: As fuel temperature rises, thermal agitation broadens $^{238}\text{U}$ resonance capture peaks (Doppler broadening), absorbing more neutrons and automatically shutting down the chain reaction.
• Negative Moderator Void Coefficient: Boiling of water removes moderator, decreasing reactivity.

While binary fission splits a nucleus into two primary fragments, approximately 1 in every 500 thermal fissions of $^{235}\text{U}$ is a ternary fission event in which a third light charged particle (LCP) is emitted simultaneously from the scission neck.

Ternary Fission Mechanics

• Over $90\%$ of ternary particles are energetic alpha particles ($^{4}\text{He}^{2+}$ with kinetic energy $\bar{E} \approx 16\text{ MeV}$).
• Other light particles include tritons ($^3\text{H}$, $\sim 7\%$), deuterons ($^2\text{H}$), and trace lithium, beryllium, and carbon ions ($^8\text{Be}, ^{10}\text{Be}, ^{14}\text{C}$).
• The ternary particle is ejected nearly perpendicular to the fission axis ($90^\circ$) by the intense mutual Coulomb repulsion of the two receding heavy fragments. Ternary fission is the primary source of radioactive tritium ($^3\text{H}$) generated inside nuclear fuel rods.

Precursor Chemistry of Delayed Neutron Emitters

Delayed neutron precursors reside in specific chemical groups within the fission fragment distribution:

1. Halogen Precursors: Bromine ($^{87}\text{Br}, ^{88}\text{Br}, ^{89}\text{Br}$) and Iodine ($^{137}\text{I}, ^{138}\text{I}, ^{139}\text{I}$). They possess high beta-decay energies ($Q_\beta \sim 6 - 8\text{ MeV}$) feeding states above the neutron separation energy $S_n$ of the noble gas daughters ($\text{Kr}$ and $\text{Xe}$).

2. Alkali Precursors: Rubidium ($^{92}\text{Rb}, ^{93}\text{Rb}$) and Cesium ($^{141}\text{Cs}, ^{142}\text{Cs}$).

Minor Actinide Partitioning & Transmutation (Actinide Incineration)

In spent nuclear fuel, the dominant long-term radiotoxicity ($>1,000\text{ years}$) arises from minor actinides: neptunium ($^{237}\text{Np}$), americium ($^{241}\text{Am}, ^{243}\text{Am}$), and curium ($^{244}\text{Cm}$). To eliminate multi-millennial geological repository hazards, advanced fuel cycles develop Partitioning and Transmutation (P&T):

1. Pyrochemical Pyroprocessing: Spent oxide fuel is reduced to metal in molten lithium chloride-potassium chloride ($\text{LiCl-KCl}$) eutectic salt at $500^\circ\text{C}$ and electrorefined, separating actinides from fission products.

2. Fast Reactor & Accelerator-Driven System (ADS) Incineration:

In a fast neutron spectrum, the fission-to-capture cross-section ratio $\sigma_f / \sigma_c$ increases dramatically. Minor actinides are loaded into subcritical fast cores driven by spallation proton accelerators, where fast neutrons fission them into short-lived fission products, reducing waste storage lifespans from 300,000 years to under 300 years!

Comprehensive Comparison of Global Commercial Nuclear Reactor Architectures

| Reactor System | Full Name | Moderator | Coolant | Core Pressure | Outlet Temp | Fuel Type & Enrichment | Worldwide Fleet Share | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | PWR | Pressurized Water Reactor | $\text{H}_2\text{O}$ | $\text{H}_2\text{O}$ (liquid) | $15.5\text{ MPa}$ | $325^\circ\text{C}$ | $\text{UO}_2$ ($3.2 - 4.95\%$) | $68\%$ | | BWR | Boiling Water Reactor | $\text{H}_2\text{O}$ | Steam/Water | $7.2\text{ MPa}$ | $285^\circ\text{C}$ | $\text{UO}_2$ ($3.0 - 4.5\%$) | $15\%$ | | PHWR / CANDU | Pressurized Heavy Water | $\text{D}_2\text{O}$ | $\text{D}_2\text{O}$ | $10.0\text{ MPa}$ | $310^\circ\text{C}$ | Natural Uranium ($0.72\%$) | $11\%$ | | AGR | Advanced Gas-Cooled | Graphite | $\text{CO}_2$ gas | $4.2\text{ MPa}$ | $650^\circ\text{C}$ | $\text{UO}_2$ ($2.5 - 3.5\%$) in Stainless | $2\%$ | | SFR / LMFBR | Sodium Fast Reactor | None | Liquid Sodium | $0.1\text{ MPa}$ | $550^\circ\text{C}$ | $\text{MOX}$ ($15 - 20\% \, \text{Pu}$) | $<1\%$ | | VHTR | Very High Temperature Gas | Graphite | Helium gas | $7.0\text{ MPa}$ | $950 - 1000^\circ\text{C}$ | TRISO fuel particles ($10 - 15\%$) | Emerging Gen-IV |