Neutron Slowing Down & Moderation Theory

1. Coordinate Systems for Two-Body Collisions

Neutrons born in fission possess an average kinetic energy of $\sim 2\text{ MeV}$, whereas thermal fission requires neutron energies of $\sim 0.0253\text{ eV}$—a reduction by eight orders of magnitude! This slowing down occurs via repeated billiard-ball elastic scattering collisions $(n, n)$ with stationary target nuclei of mass number $A$ ($m \approx A \, m_n$). To analyze the kinematics:

• Laboratory (LAB) Frame: Target nucleus is initially at rest ($V = 0$). Incident neutron has velocity $\vec{v}_1$ and kinetic energy $E_1 = \frac{1}{2} m_n v_1^2$. After scattering at angle $\theta$ relative to the incident direction, the neutron has velocity $\vec{v}_2$ and energy $E_2$.
• Center-of-Mass (CM) Frame: The center of mass moves with constant laboratory velocity: $$\vec{V}_{\text{CM}} = \frac{m_n \vec{v}_1 + M \vec{0}}{m_n + M} = \frac{1}{A + 1} \vec{v}_1$$ In the CM system, total linear momentum is identically zero both before and after collision.

2. CM Velocities and Elastic Conservation

Before collision in CM: $$u_1 = v_1 - V_{\text{CM}} = v_1 \left( 1 - \frac{1}{A+1} \right) = \frac{A}{A + 1} v_1$$ $$U_1 = 0 - V_{\text{CM}} = -\frac{1}{A + 1} v_1$$ Because elastic scattering conserves kinetic energy in the CM frame, the speeds of the particles are unchanged after collision; only their direction rotates through scattering angle $\theta_{\text{CM}}$: $$u_2 = u_1 = \frac{A}{A + 1} v_1, \qquad U_2 = |U_1| = \frac{1}{A + 1} v_1$$

1. Velocity Vector Addition and Energy Ratio

Transforming back to the Laboratory system, the final neutron velocity vector is the vector sum: $$\vec{v}_2 = \vec{u}_2 + \vec{V}_{\text{CM}}$$ Using the law of cosines: $$v_2^2 = u_2^2 + V_{\text{CM}}^2 + 2 u_2 V_{\text{CM}} \cos\theta_{\text{CM}}$$ Substituting $u_2 = \frac{A}{A+1} v_1$ and $V_{\text{CM}} = \frac{1}{A+1} v_1$: $$v_2^2 = v_1^2 \left[ \left(\frac{A}{A+1}\right)^2 + \left(\frac{1}{A+1}\right)^2 + \frac{2 A}{(A+1)^2} \cos\theta_{\text{CM}} \right]$$ Factoring out $(A+1)^2$: $$\frac{E_2}{E_1} = \frac{v_2^2}{v_1^2} = \frac{A^2 + 1 + 2 A \cos\theta_{\text{CM}}}{(A+1)^2}$$ We define the fundamental collision parameter $\alpha$: $$\alpha \equiv \left( \frac{A - 1}{A + 1} \right)^2$$ Rewriting $A^2 + 1 = \frac{1}{2} [(A+1)^2 + (A-1)^2]$: $$\mathbf{\frac{E_2}{E_1} = \frac{1 + \alpha}{2} + \frac{1 - \alpha}{2} \cos\theta_{\text{CM}}}$$

2. Extreme Collision Scenarios

• Glancing Collision ($\theta_{\text{CM}} = 0$, $\cos\theta_{\text{CM}} = 1$): $$\frac{E_2}{E_1} = \frac{1+\alpha}{2} + \frac{1-\alpha}{2} = 1 \implies E_2 = E_1 \quad (\text{No energy lost})$$
• Head-on Collision ($\theta_{\text{CM}} = \pi$, $\cos\theta_{\text{CM}} = -1$): $$\frac{E_2}{E_1} = \frac{1+\alpha}{2} - \frac{1-\alpha}{2} = \alpha \implies E_2 = \alpha E_1 \quad (\text{Maximum possible energy loss!})$$

The scattered neutron energy is bounded strictly in the interval: $$\alpha E_1 \le E_2 \le E_1$$ For hydrogen ($A = 1$): $\alpha = 0$, meaning a neutron can lose $100\%$ of its kinetic energy in a single head-on collision! For Carbon-12 ($A = 12$): $\alpha = (11/13)^2 = 0.716$, meaning at most $28.4\%$ can be lost per collision. For Uranium-238 ($A = 238$): $\alpha = (237/239)^2 = 0.983$, so at most $1.7\%$ can be lost.

3. Uniform Scattered Energy Probability Distribution

In the CM frame, for energies below $\sim 1\text{ MeV}$, elastic scattering is spherically symmetric (s-wave scattering): $$P(\theta_{\text{CM}}) \, d\Omega_{\text{CM}} = \frac{2\pi \sin\theta_{\text{CM}} \, d\theta_{\text{CM}}}{4\pi} = \frac{1}{2} d(\cos\theta_{\text{CM}})$$ Since $E_2$ depends linearly on $\cos\theta_{\text{CM}}$, $dE_2 = \frac{1-\alpha}{2} E_1 \, d(\cos\theta_{\text{CM}})$: $$P(E_1 \to E_2) \, dE_2 = \frac{dE_2}{(1 - \alpha) E_1} \quad \text{for } \alpha E_1 \le E_2 \le E_1$$ The probability distribution is perfectly flat (uniform) between $\alpha E_1$ and $E_1$, with zero probability outside.

1. Laboratory Scattering Angle Cosine $\mu_0$

Even though scattering is isotropic in the Center-of-Mass frame, the forward motion of the center of mass biases collisions forward in the Laboratory frame. From the velocity vector triangle: $$v_2 \cos\theta = u_2 \cos\theta_{\text{CM}} + V_{\text{CM}} = \frac{A v_1 \cos\theta_{\text{CM}} + v_1}{A + 1}$$ Dividing by $v_2 = v_1 \sqrt{\frac{1+\alpha}{2} + \frac{1-\alpha}{2}\cos\theta_{\text{CM}}}$: $$\mu_0 \equiv \cos\theta_{\text{LAB}} = \frac{1 + A \cos\theta_{\text{CM}}}{\sqrt{A^2 + 1 + 2 A \cos\theta_{\text{CM}}}}$$

2. The Average Cosine of the Scattering Angle $\bar{\mu}_0$

Averaging $\mu_0$ over all isotropic CM solid angles: $$\bar{\mu}_0 \equiv \langle \cos\theta_{\text{LAB}} \rangle = \int_{-1}^{1} \mu_0(\cos\theta_{\text{CM}}) \, \frac{d(\cos\theta_{\text{CM}})}{2}$$ Evaluating this integral yields the celebrated exact result: $$\mathbf{\bar{\mu}_0 = \frac{2}{3 A}}$$ Key values:

• For Hydrogen ($A = 1$): $\bar{\mu}_0 = 2/3 \approx 0.667$ (strongly forward-peaked).
• For Deuterium ($A = 2$): $\bar{\mu}_0 = 1/3 \approx 0.333$.
• For Carbon-12 ($A = 12$): $\bar{\mu}_0 = 2/36 \approx 0.056$.
• For Heavy Nuclei ($A \gg 1$): $\bar{\mu}_0 \to 0$ (nearly isotropic in LAB frame).

3. Transport Cross Section $\Sigma_{\text{tr}}$

Because forward-scattered neutrons preserve forward momentum, they diffuse farther than if scattering were isotropic. To account for this persistence of velocity, transport theory defines the transport cross section: $$\mathbf{\Sigma_{\text{tr}} = \Sigma_s (1 - \bar{\mu}_0) = \Sigma_s \left( 1 - \frac{2}{3A} \right)}$$ The corresponding transport mean free path $\lambda_{\text{tr}}$ is: $$\lambda_{\text{tr}} = \frac{1}{\Sigma_{\text{tr}}} = \frac{\lambda_s}{1 - \bar{\mu}_0}$$ In Hydrogen, $\lambda_{\text{tr}} = \frac{\lambda_s}{1 - 2/3} = 3 \, \lambda_s$: forward-peaked scattering triples the effective diffusion length per collision!

1. Definition of Logarithmic Energy Decrement $\xi$

Because the fractional energy remaining after a collision is independent of initial energy, the change in the natural logarithm of neutron energy is a constant property of the moderating nuclide: $$\xi \equiv \left\langle \ln\left(\frac{E_1}{E_2}\right) \right\rangle = \int_{\alpha E_1}^{E_1} \ln\left(\frac{E_1}{E_2}\right) P(E_1 \to E_2) \, dE_2$$ Substituting the uniform probability density $P(E_1 \to E_2) = \frac{1}{(1-\alpha)E_1}$ and letting $x = E_2 / E_1$: $$\xi = \frac{1}{1 - \alpha} \int_{\alpha}^{1} \ln\left(\frac{1}{x}\right) dx = -\frac{1}{1 - \alpha} [x \ln x - x]_\alpha^1$$ Evaluating at the integration limits: $$\mathbf{\xi = 1 + \frac{\alpha}{1 - \alpha} \ln \alpha}$$

2. Asymptotic Approximations

Using $\alpha = \left(\frac{A-1}{A+1}\right)^2$:

• For Hydrogen ($A = 1, \alpha = 0$): $$\xi_{\text{H}} = 1 + 0 = \mathbf{1.000}$$
• For Intermediate and Heavy Nuclei ($A > 10$): Taylor series expansion in powers of $1/A$ yields: $$\mathbf{\xi \approx \frac{2}{A + \frac{2}{3}} \approx \frac{2}{A}}$$ This approximation is accurate to within $1\%$ for all $A \ge 10$.

3. Average Number of Collisions to Thermalize

To slow a neutron from fission birth energy ($E_0 = 2.0\text{ MeV}$) to thermal energy ($E_{\text{th}} = 0.0253\text{ eV}$): $$\ln\left(\frac{E_0}{E_{\text{th}}}\right) = \ln\left(\frac{2.0 \times 10^6\text{ eV}}{0.0253\text{ eV}}\right) = \ln(7.905 \times 10^7) \approx 18.186$$ The average number of elastic collisions $N_{\text{coll}}$ required is: $$\mathbf{N_{\text{coll}} = \frac{\ln(E_0 / E_{\text{th}})}{\xi} = \frac{18.2}{\xi}}$$ Comparing candidate moderators:

• Hydrogen (${}^1\text{H}$): $\xi = 1.000 \implies N_{\text{coll}} \approx \mathbf{18\text{ collisions}}$
• Deuterium (${}^2\text{H}$): $\xi = 0.725 \implies N_{\text{coll}} \approx \mathbf{25\text{ collisions}}$
• Beryllium (${}^9\text{Be}$): $\xi = 0.207 \implies N_{\text{coll}} \approx \mathbf{88\text{ collisions}}$
• Carbon (${}^{12}\text{C}$): $\xi = 0.158 \implies N_{\text{coll}} \approx \mathbf{115\text{ collisions}}$
• Uranium (${}^{238}\text{U}$): $\xi = 0.0084 \implies N_{\text{coll}} \approx \mathbf{2170\text{ collisions}}$

1. The Lethargy Variable $u$

As a neutron slows down, its energy decreases over several orders of magnitude. In 1944, Enrico Fermi introduced the dimensionless lethargy variable $u$ to linearize the logarithmic energy scale: $$u \equiv \ln\left( \frac{E_0}{E} \right)$$ where $E_0$ is an arbitrary reference maximum energy (typically the top of the fission spectrum, $E_0 = 10\text{ MeV}$). Key properties:

• At high energy $E = E_0$: $u = 0$.
• As energy slows down ($E \to 0$): lethargy increases ($u \to \infty$).
• Differential relationship: $$du = - \frac{dE}{E} \implies \phi(u) \, du = \phi(E) \, dE \implies \phi(u) = E \, \phi(E)$$

2. Lethargy Gain Per Collision

The average increase in lethargy per elastic scattering collision is precisely equal to the logarithmic energy decrement: $$\langle \Delta u \rangle = \left\langle \ln\frac{E_1}{E_2} \right\rangle = \xi$$ In lethargy space, neutron slowing down appears as a steady drift toward increasing $u$ at an average rate of $\xi$ units per collision.

1. Slowing Down Power (SDP)

A superior moderator must not only reduce neutron energy rapidly per collision ($\xi$), but must also have a high collision probability per centimeter of travel ($\Sigma_s$). We define the Macroscopic Slowing Down Power: $$\mathbf{\text{SDP} \equiv \xi \, \Sigma_s\quad [\text{cm}^{-1}]}$$ This represents the average decrease in logarithmic energy per unit path length traveled by the neutron.

2. Moderating Ratio (MR)

A high slowing down power is useless if the material simultaneously swallows neutrons via radiative capture! The true figure of merit for a nuclear reactor moderator is the Moderating Ratio ($MR$): $$\mathbf{\text{MR} \equiv \frac{\xi \, \Sigma_s}{\Sigma_a} = \frac{\text{Slowing Down Power}}{\text{Thermal Absorption Cross Section}}}$$ Comparing the primary commercial moderators:

Moderator	$\xi$	$\Sigma_s\ (\text{cm}^{-1})$	$\Sigma_a\ (\text{cm}^{-1})$	SDP ($\xi\Sigma_s,\ \text{cm}^{-1}$)	Moderating Ratio ($\xi\Sigma_s/\Sigma_a$)
Light Water ($\text{H}_2\text{O}$)	$0.920$	$1.47$	$0.022$	$\mathbf{1.35}$ (Best SDP)	$\mathbf{61}$
Heavy Water ($\text{D}_2\text{O}$)	$0.509$	$0.35$	$3.3 \times 10^{-5}$	$0.18$	$\mathbf{5670}$ (Unmatched MR!)
Beryllium ($\text{Be}$)	$0.207$	$0.76$	$1.2 \times 10^{-3}$	$0.16$	$\mathbf{133}$
Graphite ($\text{C}$)	$0.158$	$0.38$	$3.2 \times 10^{-4}$	$0.06$	$\mathbf{192}$

Key Engineering Insights:

• $\text{H}_2\text{O}$ has the highest Slowing Down Power ($1.35\text{ cm}^{-1}$), allowing ultra-compact reactor cores (PWR, BWR). However, its moderate MR ($61$) requires enriched uranium fuel ($\ge 3\%\ {}^{235}\text{U}$) because natural uranium cannot achieve criticality in light water!
• $\text{D}_2\text{O}$ has an astonishing Moderating Ratio ($5670$) because deuterium captures virtually zero neutrons. This enables CANDU reactors to achieve criticality using cheap, un-enriched natural uranium ($0.72\%\ {}^{235}\text{U}$).

1. The Slowing Down Density $q(E)$

The slowing down density $q(E, \vec{r})$ is defined as the number of neutrons per unit volume per second that slow down past energy $E$. In a non-absorbing infinite medium with constant scattering cross section: $$q(E) = \xi \, \Sigma_s \, E \, \phi(E) = \text{constant} = S$$ where $S$ is the source rate of fast fission neutrons. Hence, the epithermal slowing down flux exhibits the famous universal $1/E$ dependence: $$\mathbf{\phi(E) = \frac{S}{\xi \, \Sigma_s \, E} \propto \frac{1}{E}}$$

2. Resonance Absorption and Escape Probability $p$

In real reactor fuels containing fertile ${}^{238}\text{U}$, the epithermal range ($1\text{ eV} \le E \le 1000\text{ eV}$) features hundreds of giant, razor-sharp $(n, \gamma)$ absorption resonances. As neutrons slow through an energy increment $dE$, the fraction absorbed is $\frac{\Sigma_a(E)}{\Sigma_t(E)} \frac{dE}{\xi E}$: $$\frac{dq}{q} = \frac{\Sigma_a(E)}{\xi \Sigma_s(E) + \Sigma_a(E)} \frac{dE}{E}$$ Integrating from fission energy $E_0$ down to thermal cut-off energy $E_{\text{th}}$ yields the Resonance Escape Probability ($p$): $$\mathbf{p = \exp\left( - \int_{E_{\text{th}}}^{E_0} \frac{\Sigma_a(E)}{\xi \Sigma_s + \Sigma_a(E)} \frac{dE}{E} \right)}$$ For a homogeneous fuel-moderator mixture where $\Sigma_a \ll \xi \Sigma_s$: $$\mathbf{p = \exp\left( - \frac{N_F}{\xi \Sigma_s} I_{\text{eff}} \right)}$$ where $I_{\text{eff}} \equiv \int_{E_{\text{th}}}^{E_0} \sigma_{a, \text{eff}}^F(E) \frac{dE}{E}$ is the Effective Resonance Integral (in barns). In heterogeneous lattices (fuel rods separated by moderator), resonance absorption is drastically reduced because neutrons moderate in the pure moderator without seeing fuel resonances, and self-shielding depresses the resonance flux inside the rod!