Particle Physics Calculators
Free particle physics calculators: Lorentz factor, momentum, decay widths, cross-sections, cyclotron frequency, CKM mixing, Higgs decay, and QCD color factors.
Particle Physics Calculators - Special Relativity and High-Energy Physics
Particle physics studies matter at the smallest scales and highest energies, where speeds approach the speed of light and Einstein’s special relativity governs every calculation. These calculators cover the foundational relativistic quantities used across accelerator physics, cosmic ray studies, and high-energy experiments.
Special Relativity Fundamentals
Collider Kinematics and Scattering
Decay Physics
Neutrino and Electroweak Physics
Particle Properties and Detection
Detector Physics and Shower Development
Accelerator Physics
Quark Mixing and the Higgs
QCD and the Strong Force
What These Calculators Cover
Special relativity fundamentals. The Lorentz Factor Calculator computes γ, the multiplier that scales time dilation, length contraction, and every relativistic energy and momentum formula on this page. The Relativistic Momentum and Relativistic Kinetic Energy Calculators replace the Newtonian p = mv and KE = ½mv² with their exact relativistic forms, which diverge sharply from the classical result as a particle’s speed approaches c. The Relativistic Velocity Addition Calculator shows why two velocities near c never simply add - the result always stays below c. The Relativistic Doppler Effect Calculator extends the same physics to observed frequency and redshift for a relativistically moving source.
Collider kinematics and scattering. The Center of Mass Energy Calculator finds √s, the energy actually available to create new particles in a collision, verified against the LHC’s 13 TeV design energy. The Mandelstam Variables Calculator computes the Lorentz-invariant s, t, and u variables that parameterize any 2-to-2 scattering process independent of reference frame. The Cross-Section and Luminosity Calculator converts a theoretical cross-section into an expected event count for a real collider run, the calculation behind every “how many events do we need for 5-sigma discovery” estimate. The Threshold Energy and Pair Production Threshold Calculators find the minimum energy needed to create new particles in fixed-target and photon-photon collisions respectively.
Decay physics. The Branching Ratio Calculator and Particle Decay Width and Lifetime Calculator connect the two languages particle physicists use for instability: total decay width Γ (an energy) and mean lifetime τ, related by τ = ħ/Γ. The Breit-Wigner Resonance Calculator models the characteristic peaked lineshape of an unstable particle like the Z boson or Higgs boson as a function of collision energy. The Alpha and Beta Decay Q-Value Calculators compute the energy released in nuclear decay directly from atomic mass differences, and split that energy correctly between the emitted particle and the recoiling daughter nucleus.
Neutrino and electroweak physics. The Neutrino Oscillation Probability Calculator computes the two-flavor oscillation probability P = sin²(2θ)sin²(1.27Δm²L/E), the formula behind every neutrino oscillation experiment from Super-Kamiokande to long-baseline accelerator experiments. The PMNS Matrix Parameter Calculator extends this to the full three-flavor mixing matrix. The Electroweak Mixing Angle Calculator finds the Weinberg angle that sets the relative strength of the electromagnetic and weak forces after electroweak symmetry breaking.
Particle properties and detection. The Classical Electron Radius and Compton Wavelength Calculators compute two of the fundamental length scales associated with a charged particle. The Magnetic Rigidity Calculator gives Bρ = p/q, the quantity accelerator physicists use directly to size bending magnets for a beamline of known momentum. The Cherenkov Radiation Angle Calculator finds the characteristic light cone angle emitted by a charged particle exceeding the local speed of light in a medium, the effect that gives Cherenkov detectors (and the eerie blue glow of a nuclear reactor pool) their name.
Detector physics and shower development. The Bethe-Bloch Energy Loss Calculator computes the mean ionization energy loss (dE/dx) that lets tracking detectors register a charged particle’s passage and helps identify particle species by mass. The Radiation Length and Moliere Radius Calculators give the longitudinal and transverse length scales that govern how an electromagnetic shower develops inside a calorimeter, the two numbers detector designers use to size absorber depth and cell width. The Transition Radiation Threshold Calculator estimates the Lorentz factor needed for a relativistic particle to produce detectable transition radiation X-rays, the physics behind electron/pion separation in TRDs. The Synchrotron Radiation Critical Frequency Calculator finds the characteristic photon energy radiated by a relativistic electron bending through a dipole magnet, relevant both to synchrotron light sources and to radiative energy loss in circular colliders.
Accelerator physics. The Particle Accelerator Cyclotron Frequency Calculator finds the orbital frequency, period, and radius of a charged particle circling in a uniform magnetic field, and shows the relativistic frequency droop that forced early cyclotrons to evolve into synchrocyclotrons and synchrotrons. The Synchrotron Tune and Betatron Oscillation Calculator estimates the betatron tune and oscillation frequency of a storage ring using the smooth-focusing approximation, the same concept accelerator physicists use to avoid resonances that can lose the beam.
Quark mixing and the Higgs. The Cabibbo Angle and CKM Matrix Element Calculator finds theta_c from V_us and checks first-row CKM unitarity from V_ud, V_us, and V_ub, honestly reporting the small but real gap from 1 seen with current measured values. The Higgs Boson Decay Width Estimator computes the tree-level Standard Model partial decay width of the Higgs boson to a fermion pair, contrasting quark channels (which need higher-order QCD corrections this calculator omits) against lepton channels (which match the measured value closely at tree level).
QCD and the strong force. The Running Coupling Constant Estimator computes the one-loop QCD running coupling alpha_s at any energy scale, visualizing asymptotic freedom, the decrease of the strong coupling at high energy that earned Gross, Politzer, and Wilczek the 2004 Nobel Prize in Physics. The QCD Color Factor Calculator computes the underlying SU(N) Casimir color factors C_F and C_A, and the one-loop beta-function coefficient b0 that feeds directly into the running coupling calculation above.
Who Uses These Calculators
Undergraduate and graduate physics students use these calculators for special relativity and particle physics coursework, from the first relativistic momentum problem set to graduate-level neutrino oscillation and electroweak theory. Accelerator physicists and beamline designers use the magnetic rigidity, center-of-mass energy, cyclotron frequency, and synchrotron tune calculators for machine design. Experimental particle physicists use the cross-section/luminosity and branching ratio calculators for event-rate and discovery-significance estimates. Nuclear and particle astrophysics students use the alpha and beta decay Q-value calculators alongside the nuclear science section’s decay tools. Detector physicists use the Cherenkov radiation calculator for water Cherenkov and ring-imaging Cherenkov (RICH) detector design, and use the Bethe-Bloch, radiation length, Moliere radius, and transition radiation threshold calculators for tracking detector and calorimeter design. Synchrotron beamline scientists use the synchrotron radiation critical frequency calculator to estimate available photon energy ranges for a given machine. QCD phenomenology students use the running coupling constant estimator and color factor calculator together to see how group theory feeds directly into the asymptotic-freedom curve before moving on to multi-loop calculations. Precision electroweak physicists use the Cabibbo angle calculator to explore the CKM unitarity tension, and Higgs physics students use the decay width estimator to see the Yukawa mass-coupling relationship directly.
Constants Behind Particle Physics
The speed of light c = 2.998 x 10^8 m/s sets the ultimate speed limit for every calculation here, appearing in the Lorentz factor’s beta=v/c ratio and in every relativistic energy and momentum formula. The electron rest mass energy m_e c squared = 0.511 MeV is the natural energy scale for comparing particle energies against everyday relativistic effects.
Frequently Asked Questions
What is the Lorentz factor?
The Lorentz factor gamma is the dimensionless number that scales time dilation, length contraction, and relativistic momentum and energy for a moving object, equal to 1 at rest and diverging toward infinity as speed approaches the speed of light. The Lorentz Factor Calculator finds it from velocity or from beta directly.
Why does particle physics need special relativity?
Particles in accelerators and cosmic rays routinely travel at a significant fraction of the speed of light, where classical Newtonian mechanics breaks down and relativistic corrections (governed by the Lorentz factor) become essential to correctly predict energy, momentum, and time dilation.
What is the difference between decay width and lifetime?
Decay width Γ (measured in energy units, typically MeV or GeV) and mean lifetime τ (measured in seconds) describe the same instability from two angles, related by τ = ħ/Γ via the energy-time uncertainty relation. A short-lived particle like the Z boson (τ ≈ 3×10⁻²⁵ s) has a correspondingly large decay width (Γ_Z ≈ 2.5 GeV), while a long-lived particle like the neutron (τ ≈ 880 s) has an immeasurably tiny width. The Particle Decay Width and Lifetime Calculator converts between the two directly.
Why does the LHC quote its energy as 13 TeV when each proton beam is only 6.5 TeV?
For a symmetric collider where two beams of equal energy E collide head-on, the center-of-mass energy is √s = 2E - both beams' energy is fully available to create new particles, unlike a fixed-target experiment where much of the beam energy is "wasted" conserving momentum of the recoiling target. At the LHC, two 6.5 TeV proton beams collide to give √s = 13 TeV (later raised to 6.8 TeV per beam for √s = 13.6 TeV). The Center of Mass Energy Calculator computes this for both symmetric colliders and fixed-target setups, showing just how much more energy a collider makes available compared to a fixed target at the same beam energy.
What is Cherenkov radiation and why does it only happen in a medium?
Cherenkov radiation is emitted when a charged particle moves faster than the local phase speed of light in a medium (c/n, where n is the refractive index) - this never violates special relativity, since the particle is still slower than c in vacuum. The emitted light forms a cone at angle cosθ = 1/(nβ) around the particle's path, directly analogous to a sonic boom's Mach cone. Water Cherenkov detectors like Super-Kamiokande use this angle to reconstruct a neutrino-induced particle's speed and direction. The Cherenkov Radiation Angle Calculator computes the cone angle for any particle speed and medium refractive index.
What is neutrino oscillation and why does it prove neutrinos have mass?
Neutrino oscillation is the phenomenon where a neutrino created in one flavor (electron, muon, or tau) has a probability of being detected as a different flavor after traveling some distance, described by P = sin²(2θ)·sin²(1.27Δm²L/E) in the two-flavor approximation. This effect is only possible if the mass eigenstates that make up each flavor have different, non-zero masses (Δm² ≠ 0) - a discovery that directly contradicted the Standard Model's original assumption of massless neutrinos and earned the 2015 Nobel Prize in Physics. The Neutrino Oscillation Probability Calculator computes this probability for any mixing angle, mass-squared splitting, baseline distance, and neutrino energy.