Reactor Kinetics, Reactivity Feedback, Control Rods & Poisoning

1. The Time-Dependent Diffusion Equation

When neutron populations vary with time, the net time rate of change of neutron density is governed by the time-dependent diffusion equation: $$\frac{1}{v} \frac{\partial \phi(\vec{r}, t)}{\partial t} = D \nabla^2 \phi(\vec{r}, t) - \Sigma_a \phi(\vec{r}, t) + S(\vec{r}, t)$$ In a multiplying medium, the source $S(\vec{r}, t)$ consists of two distinct components:

1. Prompt Neutrons: Fraction $1 - \beta$ of all fission neutrons, emitted promptly: $(1 - \beta) k_\infty \Sigma_a \phi$.
2. Delayed Neutrons: Produced via radioactive decay of six delayed precursor groups: $\sum_{i=1}^6 \lambda_i C_i(\vec{r}, t)$.

2. The Point Reactor Kinetics Equations (PRKE)

Assuming the spatial flux shape remains fixed during transients (the point reactor approximation $\phi(\vec{r}, t) \approx \psi(\vec{r}) P(t)$), integrating over core volume yields the classical Point Reactor Kinetics Equations for total neutron population $n(t)$ (or thermal power $P(t)$): $$\mathbf{\frac{dn(t)}{dt} = \frac{\rho(t) - \beta}{\Lambda} n(t) + \sum_{i=1}^6 \lambda_i C_i(t)}$$ $$\mathbf{\frac{dC_i(t)}{dt} = \frac{\beta_i}{\Lambda} n(t) - \lambda_i C_i(t) \quad (i = 1, 2, \dots, 6)}$$ where:

• $\rho(t) = \frac{k(t) - 1}{k(t)}$ is the net reactivity.
• $\beta = \sum_{i=1}^6 \beta_i$ is the total delayed neutron fraction ($\beta \approx 0.0065$ for $^{235}\text{U}$).
• $\Lambda \equiv \frac{l}{k} \approx l$ is the prompt neutron generation time ($\Lambda \sim 10^{-4}\text{ s}$ in PWRs, $\sim 10^{-7}\text{ s}$ in fast reactors).
• $C_i(t)$ is the precursor population of group $i$, decaying with decay constant $\lambda_i$.

1. The Purely Prompt Regime ($\rho \ge \beta$)

If a reactor receives a reactivity insertion equal to or exceeding the delayed neutron fraction: $$\rho \ge \beta \implies \rho - \beta \ge 0$$ The prompt production term $\frac{\rho - \beta}{\Lambda} n(t)$ becomes positive on its own! In this terrifying condition—known as Prompt Criticality ($\rho = 1\$$)—the chain reaction sustains itself entirely on prompt neutrons, completely bypassing the slow decay of delayed precursors! The neutron population explodes as: $$n(t) = n_0 \exp\left( \frac{\rho - \beta}{\Lambda} t \right)$$ For $\rho = 1.1 \beta$ ($\rho - \beta = 0.1 \beta \approx 6.5 \times 10^{-4}$) in a thermal reactor with $\Lambda = 10^{-4}\text{ s}$: $$\frac{\rho - \beta}{\Lambda} = \frac{6.5 \times 10^{-4}}{10^{-4}\text{ s}} = 6.5\text{ s}^{-1} \implies n(t) = n_0 e^{6.5 t}$$ In just one second, power multiplies by $e^{6.5} \approx 665$! In a fast reactor ($\Lambda = 10^{-7}\text{ s}$), $\frac{\rho - \beta}{\Lambda} = 6500\text{ s}^{-1}$, multiplying power by $10^{28}$ in one-hundredth of a second (a catastrophic explosive disassembly).

2. The Delayed Critical Buffer

For all normal reactor operations, reactivity is strictly confined to: $$\mathbf{0 \le \rho < \beta \quad (0 \le \rho < 1\$)}$$ In this delayed critical regime, the prompt term is negative ($\rho - \beta < 0$), forcing the system to wait for delayed precursors to decay before each new generation can be sustained.

1. Derivation of the Characteristic Equation

Assuming exponential solutions of the form $n(t) = n_0 e^{\omega t}$ and $C_i(t) = C_{i, 0} e^{\omega t}$ for constant reactivity $\rho$: Substitute into the precursor equation: $$\omega C_{i, 0} = \frac{\beta_i}{\Lambda} n_0 - \lambda_i C_{i, 0} \implies C_{i, 0} = \frac{\beta_i}{\Lambda (\omega + \lambda_i)} n_0$$ Substitute $C_{i, 0}$ into the neutron equation: $$\omega n_0 = \frac{\rho - \beta}{\Lambda} n_0 + \sum_{i=1}^6 \lambda_i \left[ \frac{\beta_i}{\Lambda (\omega + \lambda_i)} n_0 \right]$$ Dividing through by $n_0$ and multiplying by $\Lambda$: $$\omega \Lambda = \rho - \beta + \sum_{i=1}^6 \frac{\lambda_i \beta_i}{\omega + \lambda_i}$$ Since $\beta = \sum \beta_i$, we write $\rho = \omega \Lambda + \sum \beta_i - \sum \frac{\lambda_i \beta_i}{\omega + \lambda_i} = \omega \Lambda + \sum \beta_i \left( 1 - \frac{\lambda_i}{\omega + \lambda_i} \right) = \omega \Lambda + \sum \frac{\omega \beta_i}{\omega + \lambda_i}$. Setting reactor period $T \equiv 1/\omega$: $$\mathbf{\rho = \frac{\Lambda}{T} + \sum_{i=1}^6 \frac{\beta_i}{1 + \lambda_i T}}$$ This is the foundational Inhour Equation (named from historic units of inverse hours, $\text{hr}^{-1}$).

2. Roots and Asymptotic Stable Period

For any step reactivity $\rho$, the inhour equation possesses seven real roots: $\omega_0, \omega_1, \dots, \omega_6$. For $\rho > 0$, the largest root $\omega_0 > 0$ defines the stable reactor period $T = 1/\omega_0$. The remaining six roots $\omega_1, \dots, \omega_6$ are negative and die out within tens of seconds. After these transients decay, reactor power escalates cleanly as: $$\mathbf{P(t) = P_0 e^{t / T}}$$ The doubling time $T_d$ (time for power to double) is: $$\mathbf{T_d = T \ln 2 \approx 0.69315 \, T}$$

1. The Prompt Jump Phenomenon

When a positive step reactivity $\rho < \beta$ is inserted into a critical reactor at $t = 0$, the neutron population exhibits a two-stage response:

1. The Prompt Jump (0 to 10 ms): Because delayed precursors cannot change instantaneously ($C_i(0^+) = C_i(0^-) = \frac{\beta_i}{\Lambda} n_0$), the prompt neutron population jumps almost instantaneously until the prompt sink balances precursor emission: $$\frac{dn}{dt} \approx 0 \implies \frac{\rho - \beta}{\Lambda} n(0^+) + \sum \lambda_i C_i(0) = 0 \implies \frac{\beta - \rho}{\Lambda} n(0^+) = \frac{\beta}{\Lambda} n_0$$ $$\mathbf{n(0^+) = n_0 \frac{\beta}{\beta - \rho} = n_0 \frac{1}{1 - \rho/\beta} = n_0 \frac{1}{1 - \$}}$$
2. Precursor-Controlled Growth ($t > 1\text{ s}$): Following the prompt jump, power increases on the slow stable period $T \approx \frac{\beta - \rho}{\lambda_{\text{eff}} \rho}$: $$\mathbf{n(t) \approx n_0 \left[ \frac{\beta}{\beta - \rho} \exp\left( \frac{\lambda_{\text{eff}} \rho}{\beta - \rho} t \right) - \frac{\rho}{\beta - \rho} \exp\left( - \frac{\beta - \rho}{\Lambda} t \right) \right]}$$

1. Control Rod Worth via First-Order Perturbation Theory

Control rods contain strong neutron-absorbing materials (such as Boron Carbide $\text{B}_4\text{C}$, Cadmium, Silver-Indium-Cadmium Ag-In-Cd, or Hafnium). When a control rod is partially inserted to depth $z$ into a cylindrical reactor of active height $H$, first-order perturbation theory shows that the differential reactivity worth $d\rho/dz$ is proportional to the square of the unperturbed axial flux: $$\frac{d\rho}{dz} \propto \phi^2(z) = \phi_0^2 \sin^2\left( \frac{\pi z}{H} \right)$$ Integrating from $0$ to $z$ yields the cumulative Integral Control Rod Worth $\rho(z)$: $$\rho(z) = \rho_{\text{tot}} \frac{\int_0^z \sin^2(\pi z'/H) dz'}{\int_0^H \sin^2(\pi z'/H) dz'} = \mathbf{\rho_{\text{tot}} \left[ \frac{z}{H} - \frac{1}{2\pi} \sin\left( \frac{2\pi z}{H} \right) \right]}$$ This exhibits the characteristic S-curve:

• Near the top ($z \approx 0$) and bottom ($z \approx H$), flux is low, so rod movement produces tiny reactivity change.
• Near the core midplane ($z = H/2$), flux is maximal, so rod movement produces maximum differential worth: $\left.\frac{d\rho}{dz}\right|_{\max} = \frac{2 \rho_{\text{tot}}}{H}$.

2. Burnable Absorbers

To compensate for the large excess reactivity of freshly loaded fuel without requiring dozens of physical control rods, reactors incorporate burnable poisons (such as Gadolinium Oxide $\text{Gd}_2\text{O}_3$ or Boron in fuel cladding). These high cross-section isotopes burn away at the same rate as fuel fissile inventory decreases, keeping net core reactivity nearly flat throughout a 24-month operating cycle.

1. Reactivity Coefficients of Temperature

Any temperature increase in a reactor core alters material densities, thermal cross sections, and resonance absorption, producing net reactivity feedback: $$\frac{d\rho}{dt} = \alpha_F \frac{dT_{\text{fuel}}}{dt} + \alpha_M \frac{dT_{\text{mod}}}{dt} + \alpha_V \frac{d\alpha_{\text{void}}}{dt}$$

• Fuel Doppler Coefficient ($\alpha_F$): Promptly negative ($\alpha_F \sim -2\text{ to }-4\text{ pcm/}^\circ\text{C}$). Operates with zero thermal delay because heat is born directly inside the $\text{UO}_2$ crystals, instantly broadening ${}^{238}\text{U}$ resonance capture.
• Moderator Temperature Coefficient ($\alpha_M$): Governed by coolant thermal expansion. In an under-moderated PWR, heating decreases water density, reducing moderation and lowering $k_{\text{eff}}$ ($\alpha_M \sim -10\text{ to }-40\text{ pcm/}^\circ\text{C} < 0$).
• Void Coefficient of Reactivity ($\alpha_V$): In BWRs and PWRs, steam bubble formation displaces water. Because the core is under-moderated, voiding decreases reactivity ($\alpha_V < 0$). Conversely, in the Chernobyl RBMK reactor, the over-moderated graphite design caused a positive void coefficient ($\alpha_V > 0$), which contributed directly to the runaway power explosion in 1986.

Modern nuclear regulatory safety mandates an unconditionally negative power coefficient of reactivity under all operating states.

1. Xenon-135: The Super-Poison

Xenon-135 (${}^{135}\text{Xe}$) is the most potent neutron poison known to science, with a microscopic thermal absorption cross section of: $$\mathbf{\sigma_a({}^{135}\text{Xe}) \approx 2.65 \times 10^6\text{ barns} = 2.65 \times 10^{-18}\text{ cm}^2}$$ (Over $3{,}800$ times greater than ${}^{235}\text{U}$ fission!).

2. The Iodine-Xenon Dynamic System

${}^{135}\text{Xe}$ is formed through two pathways:

1. Direct fission yield: $\gamma_{\text{Xe}} \approx 0.003$ ($0.3\%$).
2. Radioactive decay of Iodine-135 (${}^{135}\text{I}$, yield $\gamma_I \approx 0.061 = 6.1\%$): $$\text{Fission} \longrightarrow {}^{135}_{53}\text{I} \xrightarrow[\beta^-, \, T_{1/2}=6.57\text{ h}]{} {}^{135}_{54}\text{Xe} \xrightarrow[\beta^-, \, T_{1/2}=9.14\text{ h}]{} {}^{135}_{55}\text{Cs}$$

The coupled non-linear differential equations are: $$\mathbf{\frac{d I(t)}{dt} = \gamma_I \Sigma_f \phi(t) - \lambda_I I(t)}$$ $$\mathbf{\frac{d X(t)}{dt} = \gamma_{\text{Xe}} \Sigma_f \phi(t) + \lambda_I I(t) - \lambda_{\text{Xe}} X(t) - \sigma_a^{\text{Xe}} X(t) \phi(t)}$$ where: $$\lambda_I = \frac{\ln 2}{6.57 \times 3600\text{ s}} \approx 2.93 \times 10^{-5}\text{ s}^{-1}, \qquad \lambda_{\text{Xe}} = \frac{\ln 2}{9.14 \times 3600\text{ s}} \approx 2.10 \times 10^{-5}\text{ s}^{-1}$$ At steady-state full power: $$I_0 = \frac{\gamma_I \Sigma_f \phi_0}{\lambda_I}, \qquad X_0 = \frac{(\gamma_I + \gamma_{\text{Xe}}) \Sigma_f \phi_0}{\lambda_{\text{Xe}} + \sigma_a^{\text{Xe}} \phi_0}$$

3. Post-Shutdown Xenon Peak ("The Iodine Pit")

When a reactor trips or is shut down from high power ($\phi \to 0$):

• Xenon destruction by neutron burnup ($\sigma_a^{\text{Xe}} X \phi$) abruptly drops to zero!
• Meanwhile, the large pre-existing inventory of Iodine-135 continues to decay into Xenon-135 at rate $\lambda_I I(t)$.
• Because $\lambda_I > \lambda_{\text{Xe}}$ ($T_{1/2}^I = 6.6\text{ h} < T_{1/2}^{\text{Xe}} = 9.1\text{ h}$), Xenon production initially outpaces its decay!

Xenon concentration swells to a colossal peak approximately $10\text{ to }11\text{ hours}$ after shutdown: $$t_{\text{peak}} = \frac{1}{\lambda_{\text{Xe}} - \lambda_I} \ln\left( \frac{\lambda_{\text{Xe}}}{\lambda_I} \left[ 1 + \frac{\lambda_{\text{Xe}} - \lambda_I}{\lambda_I} \frac{X_0}{I_0} \right] \right) \approx \mathbf{10.5\text{ hours}}$$ In high-flux reactors, this peak introduces a massive negative reactivity deficit ($\Delta\rho \sim -3000\text{ to }-5000\text{ pcm}$) that completely exceeds control rod withdrawal worth—locking the reactor in a temporary shutdown state known as Xenon Deadtime!