Elementary Reactor Physics

Derivation of the Four-Factor Formula for Infinite Multiplication Factor

Conditions: Infinite thermal reactor containing natural or enriched uranium fuel and a moderator.

• Consider N_T thermal neutrons initiating fission in generation n. These neutrons produce N_T η fast neutrons, where η is the number of fast neutrons produced per thermal neutron absorbed in fuel.
• Due to fast fission of ²³⁸U by unmoderated neutrons, the total number of fast neutrons becomes N_T η ε, where ε is the fast fission factor (ε > 1).
• These fast neutrons slow down through the resonance absorption region of ²³⁸U. A fraction p, called the resonance escape probability, avoids capture and reaches thermal energies.
• The number of thermalized neutrons is therefore N_T η ε p.
• Out of these thermalized neutrons, a fraction f (thermal utilization factor) is absorbed specifically by the fuel rather than the moderator, structural materials, or fission products.
• The total number of thermal neutrons in generation (n+1) is N_T+1 = N_T ε p f η.
• By definition, the infinite multiplication factor k_inf is N_{T+1} / NT.}

Result: $k_inf = \epsilon \cdot p \cdot f \cdot \eta$

Condition for Criticality in a Finite Reactor

Conditions: Homogeneous thermal reactor considering neutron leakage.

• The effective multiplication factor k_eff accounts for both neutron production/absorption in an infinite medium and neutron leakage from the finite boundary.
• Let L_f be the probability that a fast neutron does not leak out during slowing down, and L_t be the probability that a thermal neutron does not leak out before absorption.
• Thus, k_eff = k_{inf} cdot L_f cdot L_t.
• Using Fermi age theory, the fast non-leakage probability is approximated as $L_f = e^{-B^2 \tau}$, where B² is the buckling of the reactor and τ is the Fermi age.
• Using one-group diffusion theory, the thermal non-leakage probability is approximated as $L_t = \frac{1}{1 + B^2 L_t^2}$, where L_t is the thermal diffusion length.
• Combining these yields the critical equation: $k_{inf} = (1 + B^2 \tau)(1 + B^2 L_t^2) e^{B^2 \tau} (or simplified$ as k_{inf} / (1 + B^2 $M^2) = 1$ where M is the migration length).

Result: $k_{eff} = \frac{k_{inf}}{(1 + B^2 L_t^2)(1 + B^2 L_f^2)} = 1$

Neutron Flux Distribution in a Finite Reactor

Draw: A bell-shaped cosine curve representing thermal neutron flux φ(r) plotted against spatial coordinate r from center to edge of a cylindrical or spherical reactor.

Interpret: Neutron flux is maximum at the center of the reactor core and drops smoothly to zero at (or slightly beyond, due to extrapolation distance) the extrapolated boundary.

Exam use: Tested conceptually in TGT/PGT to identify where fission rate and power density are maximal within a reactor core.

Fermi Age Moderation Spectrum

Draw: A Gaussian or lethargy distribution curve showing the slowing down density q(r, τ) of neutrons as a function of spatial displacement or age τ.

Interpret: Illustrates how fast fission neutrons spread out spatially as they lose energy through elastic collisions before reaching thermal energy.

Exam use: Used in PGT numericals and theoretical questions regarding slowing down length and Fermi age theory.

Reactor Startup and Reactivity Insertion Curve

Draw: Graph of neutron density (or power level on a log scale) versus time following a step increase in reactivity (ρ > 0).

Interpret: Shows an initial jump due to prompt neutrons followed by an exponential rise governed by the delayed neutron precursors (reactor period T).

Exam use: Evaluates understanding of reactor kinetics and the critical role of delayed neutrons in safe operation.

In a thermal nuclear reactor, if the reproduction factor η is 1.33, the thermal utilization factor f is 0.85, the resonance escape probability p is 0.90, and the fast fission factor ε is 1.03, calculate the infinite multiplication factor (k_inf).

• Write down the formula for the infinite multiplication factor: $k_{inf} = \epsilon \cdot p \cdot f \cdot \eta$
• Substitute the given values: $\epsilon = 1.03$, p = 0.90, f = 0.85, $\eta = 1.33$
• Calculate the product: $k_{inf} = 1.03 \times 0.90 \times 0.85 \times 1.33$
• Compute final numerical value: k_{inf} approx 1.045

Answer: 1.045

A large thermal reactor has an infinite multiplication factor $k_{inf} = 1.08$. The thermal diffusion length L_t is 5 cm and the Fermi age τ is 35 cm². Assuming critical condition ($k_{eff} = 1)$ and using one-group diffusion theory with L_f ≈ 1 - B²τ, find the geometric buckling B².

• Write the critical equation relation: $k_{eff} = \frac{k_{inf}}{(1 + B^2 L_t^2)(1 + B^2 \tau)} = 1$
• Rearrange to get: (1 + B^2 L_t^2)(1 + B^2 \tau) = k_{inf}
• Expand the binomial product, neglecting higher-order terms in B² if B² is small: 1 + B^2(L_t^2 + \tau) = k_{inf}
• Substitute M² = L_t^2 + $\tau = 5^2 + 35 = 25 + 35 = 60 cm^{2}$
• Substitute $k_{inf} = 1.08: 1 + B^2 (60) = 1.08$
• Solve for B²: 60 $B^2 = 0.08 => B^2 = \frac{0.08}{60} = 1.333 \times 10^{-3} cm^{-2}$

Answer: 1.333 \times 10^{-3} cm^{-2}

Calculate the reactivity ρ of a reactor whose effective multiplication factor k_{eff} is 1.05. Express your answer in dimensionless units and in percent.

• Recall the formula for reactivity: $\rho = \frac{k_{eff} - 1}{k_{eff}}$
• Substitute the given value: $k_{eff} = 1.05$
• Calculate the numerator: 1.05 - 1 = 0.05
• Divide by k_{eff}: \rho = \frac{0.05}{1.05} approx 0.0476
• Convert to percentage: 4.76%

Answer: 0.0476 (or 4.76%)