Thermal Neutron Velocity

At 20°C (293 K), the thermal neutron energy corresponding to the most probable speed is kT = 0.0253 eV (where k is Boltzmann's constant). Using v = sqrt(2E/m) with E = 0.0253 eV (converted to joules: 4.05×10⁻²¹ J) and neutron mass m = 1.675×10⁻²⁷ kg, the most probable speed is about 2,200 m/s. This is the standard reference velocity used for defining thermal cross-sections (e.g., 2,200 m/s corresponds to the 1/v region).

Thermal neutrons are in thermal equilibrium with the moderator nuclei. As the moderator temperature increases, the kinetic energy of the neutrons increases, leading to higher average velocities. The most probable energy is E = kT (or 3/2 kT for the mean energy), so v ∝ sqrt(T). For example, at 293 K, v ≈ 2,200 m/s; at 573 K (300°C), the most probable speed increases to about 3,080 m/s. This temperature dependence affects the neutron cross-sections and is the basis for the Doppler effect.

In a Maxwell-Boltzmann distribution: the most probable speed is v_p = sqrt(2kT/m) (about 2,200 m/s at 20°C); the mean speed is = sqrt(8kT/(πm)) ≈ 1.128 v_p (≈2,480 m/s); and the RMS speed is v_rms = sqrt(3kT/m) ≈ 1.225 v_p (≈2,695 m/s). The RMS speed is often used in diffusion theory because it relates to the average kinetic energy, but the most probable speed is used for the standard 2,200 m/s reference.

For many nuclides, the absorption cross-section follows the 1/v law at thermal energies, meaning σ_a ∝ 1/v. At 2,200 m/s, the cross-section is tabulated as the 2,200 m/s value. If the neutron velocity is higher, the cross-section decreases proportionally. For example, if the neutron speed doubles, the absorption cross-section halves. This is crucial for reactor calculations because the fuel and absorber cross-sections must be corrected for the actual neutron spectrum (which is not monoenergetic).

The value 2,200 m/s corresponds to the most probable speed of a neutron in a Maxwell-Boltzmann distribution at 293.6 K (20.4°C), which is the standard reference temperature for nuclear data. It corresponds to a neutron energy of 0.0253 eV. Many nuclear cross-sections are tabulated at this velocity (e.g., the 2,200 m/s absorption cross-section for gold is 98.8 barns). Using this reference allows easy calculation of reaction rates when the neutron spectrum is known.

The de Broglie wavelength of a neutron is λ = h / (mv), where h is Planck's constant. As velocity increases, the wavelength decreases. For a thermal neutron at 2,200 m/s, λ ≈ 1.8 Å (angstroms), which is comparable to interatomic distances in solids. This wavelength is important for neutron diffraction and scattering experiments. The velocity formula v = sqrt(2E/m) directly gives the speed from energy, and the wavelength follows from the wave-particle duality.

Thermal neutron velocities are measured using time-of-flight (TOF) methods: a pulsed neutron beam travels a known distance, and the arrival time is measured. The velocity is derived from the flight time and distance. In a reactor, the velocity distribution is inferred from the neutron spectrum measured by choppers or by using multiple detectors with different absorption foils. The most common method is to use the Maxwellian distribution and derive the effective temperature from the neutron spectrum, then compute the most probable speed.

The neutron velocity distribution in a thermal reactor is approximately Maxwellian, with a temperature equal to the moderator temperature (sometimes slightly higher due to 'neutron temperature' effects). As the moderator temperature increases, the distribution shifts to higher velocities, which changes the average cross-sections (e.g., the fission cross-section of U-235 decreases with increasing neutron energy, reducing reactivity). This contributes to the moderator temperature coefficient of reactivity, which is typically negative in LWRs and helps stabilize the reactor.

In diffusion theory, the neutron flux is often expressed in terms of the average neutron speed times the neutron density. The average speed (or the RMS speed) is used because the diffusion coefficient D = / (3Σ_s) depends on the average velocity. The most probable speed is more relevant for 1/v absorption corrections, but for the bulk neutron transport, the mean speed is the appropriate quantity to compute the macroscopic cross-sections and the diffusion length.

The thermal neutron velocity distribution is determined by the moderator temperature, not by the type of moderator. Both light water and heavy water at the same temperature will have the same Maxwellian distribution of neutron velocities. However, the neutron spectrum in heavy water is slightly 'harder' (higher average energy) because the absorption is lower and the leakage is higher, but the thermal neutron velocity itself is the same function of temperature. The difference is in the slowing-down process and the effective temperature, which may be slightly higher in heavy water due to lower moderation efficiency, leading to a marginally higher average velocity.

The neutron temperature is the temperature that best fits the Maxwellian distribution of the neutron spectrum in a thermal reactor. It is often slightly higher than the actual moderator temperature because the neutrons are not fully thermalized and there is some 'heating' from the slowing-down process. The difference can be 20-100 K in some reactors. The thermal neutron velocity is calculated using the neutron temperature, not the moderator temperature, to account for this spectral shift. This is important for accurately predicting cross-sections.

Doppler broadening is caused by the thermal motion of the target nuclei (U-238), not by the neutron velocity itself. However, the neutron velocity determines the relative speed between the neutron and the nucleus. As the neutron velocity increases, the resonance absorption probability changes because the effective cross-section is a convolution of the resonance shape and the Maxwellian distribution of target velocities. At higher neutron velocities, the relative energy shift is larger, but the Doppler broadening effect is primarily a function of the target temperature, not the neutron velocity. The formula v = sqrt(2E/m) is used to relate the neutron energy to its speed, which is needed to compute the collision kinematics.

In a fast reactor, the neutron spectrum is dominated by high-energy neutrons (typically >100 keV) that are not in thermal equilibrium with the moderator (there may be no moderator at all). The neutrons do not follow a Maxwell-Boltzmann distribution; their energy spectrum is determined by fission and scattering without thermalization. Thus, the concept of a thermal neutron velocity is not relevant; instead, the neutron velocities are much higher (e.g., at 1 MeV, v ≈ 14,000 km/s), and cross-sections are treated as energy-dependent without a thermal reference.

Using E = (1/2) m v², we get v = sqrt(2E/m). With m = 1.675×10⁻²⁷ kg. For E in eV, first convert to joules: E_J = E_eV × 1.602×10⁻¹⁹ J/eV. Then v = sqrt(2 × E_eV × 1.602×10⁻¹⁹ / 1.675×10⁻²⁷). For example, for E = 0.0253 eV, v = sqrt(2 × 0.0253 × 1.602×10⁻¹⁹ / 1.675×10⁻²⁷) ≈ 2,200 m/s. A useful rule of thumb: v (m/s) ≈ 1.383 × 10⁴ × sqrt(E_eV).

The 1/v law arises because the neutron absorption probability is proportional to the time spent near a nucleus, which is inversely proportional to its speed. For many nuclides, at low neutron energies (thermal range), the absorption cross-section σ_a ∝ 1/v. Since v = sqrt(2E/m), this means σ_a ∝ 1/√E. This is used in reactor physics to correct cross-sections for deviations from the reference speed: σ(v) = σ_0 × (v_0 / v), where v_0 = 2200 m/s. The formula v = sqrt(2E/m) is essential for applying this correction.