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Thermal Neutron Velocity

Thermal neutron velocity is derived from the kinetic energy of neutrons that are in thermal equilibrium with the surrounding medium (at room temperature, ~0.0253 eV). The most probable speed is given by the Maxwell‑Boltzmann distribution, but the root‑mean‑square speed is often used. This velocity is crucial for calculating cross‑sections, which depend on neutron energy. It also affects the Doppler broadening of resonances and the neutron diffusion coefficient.

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Neutron Velocity Thermal Neutron Maxwellian

Thermal Neutron Velocity Calculator

A neutron with kinetic energy E and mass m moves at v = √(2E/m). Enter any two of v, E and m and the third is solved. At the thermal energy of 0.0253 eV a neutron travels at the textbook speed of 2,200 m/s. Wavelength, equivalent temperature and flight time are included.

v = √(2E / m)E = ½ m v²λ = h / (m v)
Solve for
Flight path for time-of-flight
m
Enter 0 or more.
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Neutron energy (log scale, eV)
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Step-by-step solution

Companion quantities

Neutron in flight

Speed distribution at T = E / k

Reference data

Neutron energy classes (typical)
ClassEnergySpeed at upper edgeWhere found
Neutron speed and wavelength at common energies
EnergySpeed (m/s)v / cλ (Å)Equivalent T (K)

Speeds use the classical formula and the neutron rest mass. Rows are highlighted when your energy is within 20% of the listed value. Class boundaries vary a little between texts.

FAQ

Why is 2,200 m/s the "thermal" speed?

Thermal neutrons are in equilibrium with a room-temperature moderator, with kT ≈ 0.0253 eV at 293.6 K (20.4 °C). The speed corresponding to that energy, √(2kT/m), is the most probable speed of the Maxwell–Boltzmann distribution, about 2,200 m/s. Nuclear data tables use it as the reference for cross sections.

Why use thermal neutrons in reactors?

Many fissile nuclei, such as uranium-235, have far larger fission cross sections at low neutron energy. A moderator slows fast fission neutrons (around 2 MeV) down to thermal energies by repeated collisions, which makes a thermal chain reaction efficient.

When does the classical formula fail?

Above about 0.1c, kinetic energy differs from ½mv² by more than roughly 0.75%. For fast neutrons in the tens of MeV, use the relativistic formula. This tool reports the relativistic energy for your speed so you can see the difference.

What is the neutron wavelength used for?

Thermal neutrons have wavelengths near 1.8 Å, comparable to atomic spacings, so they diffract from crystals. This is the basis of neutron scattering and diffraction used to study materials and magnetic structure.

Educational tool only. This applies the classical kinetic-energy relation for a free particle. Real neutron populations in a reactor follow broad energy spectra, and moderator temperature, absorption and spectrum hardening all shift them. It is not a reactor design or safety tool.

Variables

Symbol Quantity Unit
v Neutron Velocity m/s
E Neutron Energy J
m Neutron Mass kg

What it means

The speed determines how quickly neutrons move through the medium. At 20°C, the most probable speed is about 2200 m/s.

Worked example

Thermal Neutron Velocity (v = √(2E/m))

Reactor Physics
Scenario: A reactor physicist calculates the thermal neutron velocity in a PWR core at room temperature (293.6 K). The neutron energy (E) is the most probable energy at thermal equilibrium, 0.0253 eV. The neutron mass (m) is 1.675×10⁻²⁷ kg. The velocity is used to determine the neutron diffusion coefficient, the fuel absorption cross‑section (which varies inversely with speed), and the Doppler broadening of resonances. This fundamental parameter underpins the neutronics calculations for reactor design and operation.
ParameterValue
Neutron Energy (E)0.0253 eV (4.05×10⁻²¹ J)
Neutron Mass (m)1.675×10⁻²⁷ kg
Thermal Velocity (v = √(2E/m))2200 m/s (computed below)
1Convert neutron energy from eV to joules: E = 0.0253 × 1.602×10⁻¹⁹ = 4.05×10⁻²¹ J.
2Use the neutron rest mass mn = 1.675×10⁻²⁷ kg.
3Apply the formula v = √(2 × 4.05×10⁻²¹ / 1.675×10⁻²⁷) ≈ 2200 m/s.
Thermal Neutron Velocity v = 2200 m/s The accepted standard thermal neutron velocity is 2200 m/s (corresponding to 0.0253 eV at 20°C). This value is used universally as the reference for cross‑section data.

Thermal Neutron Velocity at 200°C (v = √(2E/m))

Reactor Physics
Scenario: A reactor physicist calculates the thermal neutron velocity at a higher temperature (200°C) typical of a PWR coolant outlet. The neutron energy is determined by the Maxwell‑Boltzmann distribution at this temperature. The higher velocity affects the neutron absorption cross‑sections, particularly the Doppler broadening of resonances in fuel, which influences the temperature feedback coefficient (fuel temperature coefficient).
ParameterValue
Temperature (T)473 K (200°C)
Neutron Energy (E = kBT)0.0408 eV (6.53×10⁻²¹ J)
Neutron Mass (m)1.675×10⁻²⁷ kg
Thermal Velocity (v = √(2E/m))2793 m/s (computed below)
1Determine the thermal energy at T = 473 K: E = kBT = 1.38×10⁻²³ × 473 = 6.53×10⁻²¹ J (0.0408 eV).
2Calculate the velocity using the neutron mass: v = √(2 × 6.53×10⁻²¹ / 1.675×10⁻²⁷) ≈ 2793 m/s.
3Compare with room temperature — the higher temperature increases the neutron velocity by ~27%, which affects the Doppler broadening of resonance capture cross‑sections in fuel.
Thermal Neutron Velocity v = 2793 m/s At 200°C, the neutron velocity increases, causing resonance absorption peaks to broaden (Doppler effect). This is a key factor in the negative temperature coefficient of reactivity in PWRs.

Fast Neutron Velocity (v = √(2E/m))

Reactor Physics
Scenario: A fast reactor physicist calculates the velocity of a neutron with typical fast spectrum energy of 1 MeV. Fast neutrons travel at much higher speeds than thermal neutrons, which affects the fission cross‑section and the neutron multiplication factor. Understanding fast neutron velocities is essential for designing fast breeder reactors and analysing neutron slowing‑down processes.
ParameterValue
Neutron Energy (E)1.0 MeV (1.602×10⁻¹³ J)
Neutron Mass (m)1.675×10⁻²⁷ kg
Fast Neutron Velocity (v = √(2E/m))1.38×10⁷ m/s (computed below)
1Convert energy to joules: E = 1.0 × 1.602×10⁻¹³ = 1.602×10⁻¹³ J.
2Calculate the velocity: v = √(2 × 1.602×10⁻¹³ / 1.675×10⁻²⁷) = √(1.91×10¹⁴) ≈ 1.38×10⁷ m/s.
3Compare with thermal velocity — a 1 MeV neutron travels at about 4.5% of the speed of light (0.045c), which is more than 6000× faster than a thermal neutron.
Fast Neutron Velocity v = 1.38×10⁷ m/s Fast neutrons travel at relativistic speeds, enabling high fission probabilities in fast reactors and playing a crucial role in neutron slowing‑down calculations.

Common mistakes

  • Using the wrong mass for neutron mass: Must use the neutron rest mass (1.675×10⁻²⁷ kg), not the proton mass.
  • Forgetting the factor 1/2: Velocity comes from the kinetic energy equation E = ½mv²; omitting the ½ doubles the velocity.
  • Using eV without converting to joules: Always convert eV to J (1 eV = 1.602×10⁻¹⁹ J) before plugging into the formula.

Applications

  • Moderator design: Helps select materials that efficiently slow neutrons to thermal energies.
  • Cross‑section lookup: The 2200 m/s reference velocity is the standard for thermal cross‑sections.
  • Doppler broadening analysis: Understanding how neutron velocity distribution changes with fuel temperature.

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  • Published October 2, 2026
  • Last updated October 17, 2026