Neutron Interactions, Cross Sections & Nuclear Reaction Rates

1. The Energy Spectrum of Neutrons

Neutrons produced in nuclear fission are born with high kinetic energies averaging $\bar{E} \approx 2\text{ MeV}$, but subsequently moderate across nine orders of magnitude down to thermal equilibrium ($E \sim 0.025\text{ eV}$). Reactor physics categorizes neutrons into well-defined energy regimes:

2. Laboratory and Startup Neutron Sources

To start up a subcritical nuclear reactor safely, external neutron sources are installed to provide an initial detectable neutron flux for nuclear instrumentation:

• Alpha-Neutron $(\alpha, n)$ Sources: Utilize alpha decay from actinides mixed with beryllium-9: $${}^9_4\text{Be} + \alpha \longrightarrow {}^{12}_6\text{C} + n + 5.70\text{ MeV}$$ Standard commercial sources include $\text{Am-Be}$ ($T_{1/2} = 432\text{ yr}$) and $\text{Pu-Be}$.
• Photoneutron $(\gamma, n)$ Sources: Photons exceeding the low neutron binding energy of deuterium ($2.22\text{ MeV}$) or beryllium ($1.67\text{ MeV}$): $${}^9_4\text{Be} + \gamma (> 1.67\text{ MeV}) \longrightarrow {}^8_4\text{Be} + n$$ A common source is Antimony-Beryllium (${}^{124}\text{Sb}\text{-Be}$), where $^{124}\text{Sb}$ emits $1.69\text{ MeV}$ gammas.
• Spontaneous Fission Sources: Californium-252 (${}^{252}_{98}\text{Cf}$, $T_{1/2} = 2.645\text{ yr}$) decays $3.09\%$ by spontaneous fission, emitting an intense stream of $3.76$ neutrons per fission ($2.3 \times 10^{12}\text{ n/s per gram}$).

1. Definition of the Microscopic Cross Section $\sigma$

Consider a monodirectional, uniform beam of neutrons of intensity $I$ ($\text{neutrons/cm}^2\cdot\text{s}$) impinging perpendicularly upon a thin target containing a single isolated nucleus. The probability per unit time that a specific nuclear reaction occurs is directly proportional to the incident beam intensity $I$: $$\text{Interaction Rate} = \sigma \cdot I$$ The constant of proportionality $\sigma$ has dimensions of area ($\text{cm}^2/\text{nucleus}$) and represents the effective target area that the nucleus presents to the passing neutron.

Because nuclear radii are on the order of $R \sim 10^{-12}\text{ cm}$, typical geometric cross sections are: $$\sigma_{\text{geom}} \approx \pi R^2 \approx \pi (10^{-12}\text{ cm})^2 = 3.14 \times 10^{-24}\text{ cm}^2$$ To establish a convenient unit, nuclear physicists in the 1940s Manhattan Project coined the barn ($\text{b}$): $$1\text{ barn (b)} \equiv 10^{-24}\text{ cm}^2 = 10^{-28}\text{ m}^2$$ Sub-units include millibarns ($1\text{ mb} = 10^{-3}\text{ b} = 10^{-27}\text{ cm}^2$) and microbarns ($1\text{ }\mu\text{b} = 10^{-6}\text{ b}$).

2. Partial and Total Cross Sections

A neutron can interact with a nucleus through multiple mutually exclusive reaction channels: $$\sigma_t = \sigma_s + \sigma_a$$

1. Scattering Cross Section ($\sigma_s$): $$\sigma_s = \sigma_e + \sigma_i$$ where $\sigma_e$ is elastic potential and resonance scattering $(n, n)$ (kinetic energy conserved in CM frame), and $\sigma_i$ is inelastic scattering $(n, n')$ (leaving target nucleus in an excited state).
2. Absorption Cross Section ($\sigma_a$): $$\sigma_a = \sigma_\gamma + \sigma_f + \sigma_\alpha + \sigma_p$$ where $\sigma_\gamma$ is radiative capture $(n, \gamma)$, $\sigma_f$ is neutron-induced fission $(n, f)$, $\sigma_\alpha$ is alpha emission $(n, \alpha)$, and $\sigma_p$ is proton ejection $(n, p)$.

1. The Macroscopic Cross Section $\Sigma$

In a macroscopic medium containing target nuclei with atom density $N$ ($\text{nuclei/cm}^3$), the macroscopic cross section $\Sigma$ represents the total interaction probability per unit distance traveled by a neutron: $$\Sigma \equiv N \cdot \sigma \quad [\text{cm}^{-1}]$$ Physical significance of dimensions: $$\Sigma = \left( \frac{\text{nuclei}}{\text{cm}^3} \right) \times \left( \frac{\text{cm}^2}{\text{nucleus}} \right) = \mathbf{\text{cm}^{-1}}$$ For a homogeneous mixture of $K$ distinct isotopic species: $$\Sigma = \sum_{i=1}^K N_i \sigma_i = N_1 \sigma_1 + N_2 \sigma_2 + \dots + N_K \sigma_K$$ Total macroscopic cross section: $$\Sigma_t = \Sigma_s + \Sigma_a = \Sigma_e + \Sigma_i + \Sigma_\gamma + \Sigma_f$$

2. The Neutron Mean Free Path $\lambda$

The mean free path $\lambda$ is the average distance a neutron travels through a medium between two successive collisions. The probability that a neutron travels distance $x$ without interaction is $e^{-\Sigma_t x}$, and the probability of colliding in differential slice $dx$ is $\Sigma_t dx$. Thus, the collision probability density is: $$p(x) dx = \Sigma_t e^{-\Sigma_t x} dx$$ The expectation value of travel distance is: $$\lambda \equiv \langle x \rangle = \int_0^\infty x \, p(x) dx = \Sigma_t \int_0^\infty x e^{-\Sigma_t x} dx = \Sigma_t \left[ \frac{1}{\Sigma_t^2} \right] = \mathbf{\frac{1}{\Sigma_t}}$$ Partial mean free paths: $$\lambda_s = \frac{1}{\Sigma_s} \quad \text{(Mean free path for scattering)}, \qquad \lambda_a = \frac{1}{\Sigma_a} \quad \text{(Mean free path for absorption)}$$ Because $\Sigma_t = \Sigma_s + \Sigma_a$: $$\frac{1}{\lambda_t} = \frac{1}{\lambda_s} + \frac{1}{\lambda_a}$$ In light water, thermal neutrons have $\lambda_s \approx 0.3\text{ cm}$ and $\lambda_a \approx 45\text{ cm}$, demonstrating that a neutron scatters over a hundred times before undergoing absorption!

1. Derivation of the Exponential Attenuation Law

Consider a collimated, monoenergetic neutron beam of initial intensity $I_0$ ($\text{neutrons/cm}^2\cdot\text{s}$) entering a slab of material at normal incidence ($x = 0$). Let $I(x)$ denote the uncollided beam intensity at depth $x$. In traversing a differential layer of thickness $dx$: The number of collisions occurring in unit area per second is: $$-dI(x) = I(x) \cdot \Sigma_t \cdot dx$$ Rearranging into a differential equation: $$\frac{dI(x)}{dx} = -\Sigma_t I(x)$$ Integrating with boundary condition $I(0) = I_0$: $$\int_{I_0}^{I(x)} \frac{dI}{I} = -\Sigma_t \int_0^x dx \implies \ln\left( \frac{I(x)}{I_0} \right) = -\Sigma_t x$$ $$I(x) = I_0 \exp(-\Sigma_t x)$$ This is the fundamental exponential attenuation law for uncollided neutrons.

2. Transmission, Absorption, and Half-Value Thickness

For a shield slab of total physical thickness $t$:

• Transmission Probability ($T$): The fraction of the beam penetrating without undergoing any interaction: $$T \equiv \frac{I(t)}{I_0} = e^{-\Sigma_t t}$$
• Interaction Probability ($P_{\text{int}}$): The fraction undergoing at least one collision: $$P_{\text{int}} = 1 - T = 1 - e^{-\Sigma_t t}$$
• Half-Value Layer ($HVL$): The thickness of material required to reduce the uncollided beam intensity by exactly $50\%$ ($I/I_0 = 1/2$): $$e^{-\Sigma_t \cdot HVL} = \frac{1}{2} \implies HVL = \frac{\ln 2}{\Sigma_t} = \frac{0.69315}{\Sigma_t} \approx 0.693 \lambda_t$$
• Tenth-Value Layer ($TVL$): The thickness required to attenuate the uncollided beam by a factor of 10: $$TVL = \frac{\ln 10}{\Sigma_t} = \frac{2.3026}{\Sigma_t}$$

1. Scalar Neutron Flux $\phi(\vec{r}, E)$

In a reactor core, neutrons are not collimated into a single beam; they travel isotropically in all directions. If $n(\vec{r}, E) dE$ is the density of neutrons ($\text{neutrons/cm}^3$) at position $\vec{r}$ with energy in $dE$, and $v$ is their speed: The scalar neutron flux $\phi(\vec{r}, E)$ is defined as: $$\phi(\vec{r}, E) \equiv n(\vec{r}, E) \cdot v \quad [\text{neutrons/cm}^2\cdot\text{s}]$$ Physical interpretation: $\phi$ is the total track length swept out by all neutrons contained within a unit volume per second ($\text{cm of track} / \text{cm}^3\cdot\text{s} = \text{cm}^{-2}\text{s}^{-1}$).

2. Neutron Current Density Vector $\vec{J}(\vec{r}, E)$

While scalar flux $\phi$ is an orientation-independent scalar measure of neutron activity, the neutron current density $\vec{J}$ is a vector describing the net rate of flow of neutrons across a unit surface oriented perpendicular to $\vec{J}$: $$\vec{J}(\vec{r}, E) \equiv \int_{4\pi} \vec{\Omega} \, \psi(\vec{r}, \vec{\Omega}, E) \, d\Omega$$ where $\psi(\vec{r}, \vec{\Omega}, E) = v \cdot n(\vec{r}, \vec{\Omega}, E)$ is the angular flux. In an isotropic neutron field, the current vanishes ($\vec{J} = 0$), but the scalar flux remains non-zero ($\phi > 0$).

3. Volumetric Reaction Rate Density

The volumetric reaction rate density $R_x(\vec{r})$ (number of reactions of type $x$ occurring per unit volume per second) is given by: $$R_x(\vec{r}) = \int_0^\infty \Sigma_x(\vec{r}, E) \, \phi(\vec{r}, E) \, dE \quad [\text{reactions/cm}^3\cdot\text{s}]$$ For a monoenergetic or one-group thermal flux $\phi_{\text{th}}$: $$R_x = \Sigma_x \phi_{\text{th}} = N \sigma_x \phi_{\text{th}}$$ Integrating over the entire reactor core volume $V_{\text{core}}$ gives the total core reaction rate: $$\mathcal{R}_x = \int_{V_{\text{core}}} \Sigma_x(\vec{r}) \phi(\vec{r}) d^3r \quad [\text{reactions/s}]$$ For nuclear fission, the total thermal power $P$ released is: $$P = Q_f \cdot \mathcal{R}_f = Q_f \int_{V_{\text{core}}} \Sigma_f(\vec{r}) \phi(\vec{r}) d^3r \quad [\text{Watts}]$$

1. The $1/v$ Low-Energy Law

In the low-energy region below isolated resonances ($E \lesssim 0.1\text{ eV}$), quantum perturbation theory predicts that the probability of absorbing an $s$-wave ($l = 0$) neutron is proportional to the time the neutron spends in the vicinity of the nuclear potential well: $$\text{Interaction Time} \Delta t \propto \frac{R_{\text{nuc}}}{v} \propto \frac{1}{\sqrt{E}}$$ Consequently, for almost all non-threshold absorption reactions, the microscopic cross section follows the universal $1/v$ law: $$\sigma_a(v) = \sigma_a(v_0) \frac{v_0}{v} = \sigma_a(E_0) \sqrt{\frac{E_0}{E}}$$ where $v_0 = 2200\text{ m/s}$ and $E_0 = 0.0253\text{ eV}$. Plotted on a $\log\sigma_a$ versus $\log E$ graph, the $1/v$ cross section is a straight line with a characteristic slope of $-1/2$: $$\log\sigma_a = \text{const} - \frac{1}{2}\log E$$

2. Breit-Wigner Resonances and Doppler Broadening

In the intermediate energy regime ($1\text{ eV} \le E \le 10\text{ keV}$), incident neutron energies match discrete compound nuclear quasi-bound states. The cross section exhibits colossal resonance peaks accurately parameterized by the single-level Breit-Wigner dispersion formula: $$\sigma_\gamma(E) = \frac{\pi}{k^2} g_J \frac{\Gamma_n \Gamma_\gamma}{(E - E_0)^2 + (\Gamma/2)^2}$$ where $E_0$ is the resonance energy, $\Gamma = \Gamma_n + \Gamma_\gamma + \Gamma_f$ is the total resonance width, and $g_J$ is the statistical spin factor.

As the reactor fuel temperature $T$ increases, thermal vibrations of the actinide lattice nuclei broaden the relative velocity distribution between target nuclei and incident neutrons. This phenomenon—Doppler Broadening—flattens the peak height but broadens the resonance wings without altering the total area: $$\Psi(\theta, x) = \frac{\theta}{2\sqrt{\pi}} \int_{-\infty}^\infty \frac{\exp\left( -\frac{\theta^2}{4}(x - y)^2 \right)}{1 + y^2} dy, \quad \theta \equiv \frac{\Gamma}{\Delta_{\text{Doppler}}}$$ In thick fuel rods, Doppler broadening exposes more neutrons to resonance capture (reducing self-shielding), providing an instantaneous, inherently safe negative temperature reactivity feedback!

1. Maxwellian Average of $1/v$ Cross Sections

Because thermal neutrons are distributed across a continuous Maxwellian spectrum $n(v)$, the effective reaction rate in a thermal reactor is: $$R_a = \int_0^\infty N \sigma_a(v) v \, n(v) \, dv$$ If the absorber obeys the ideal $1/v$ law ($\sigma_a(v) = \sigma_0 v_0 / v$): $$R_a = N \sigma_0 v_0 \int_0^\infty n(v) dv = N \sigma_0 v_0 n_0$$ Defining the conventional 2200 m/s thermal flux $\phi_0 \equiv n_0 v_0$: $$R_a = N \sigma_0 \phi_0 = \Sigma_0 \phi_0$$ Remarkably, for a pure $1/v$ absorber, the reaction rate is independent of the moderator temperature $T$ when evaluated using the 2200 m/s cross section $\sigma_0$ and the standard flux $\phi_0 = n_0 v_0$!

2. The Westcott $g$-Factor for Non-$1/v$ Absorbers

For nuclides that possess low-lying resonances near thermal energies (such as $^{235}\text{U}$, $^{239}\text{Pu}$, $^{241}\text{Pu}$, and $^{113}\text{Cd}$), the absorption cross section deviates significantly from $1/v$. Carl Westcott introduced the dimensionless Westcott $g$-factor $g(T)$ to correct the 2200 m/s cross section: $$g(T) \equiv \frac{1}{\sigma_0 v_0} \frac{\int_0^\infty \sigma_a(v) v \, n(v) \, dv}{\int_0^\infty n(v) \, dv} = \frac{\bar{\sigma}_a(T)}{\sigma_0} \frac{\bar{v}}{v_0} \frac{\sqrt{\pi}}{2}$$ The true thermal reaction rate is then calculated as: $$R_a = g(T) \cdot \Sigma_0 \cdot \phi_0$$

• For an ideal $1/v$ absorber (such as Boron-10): $g(T) \equiv 1.000$ at all temperatures.
• For Uranium-235: $g_a(293.6\text{ K}) = 0.9780$, $g_f(293.6\text{ K}) = 0.9766$.
• For Plutonium-239: Due to a massive resonance at $0.296\text{ eV}$, $g_a(293.6\text{ K}) = 1.072$, rising above $1.4$ as coolant temperature rises to $600\text{ K}$!