Breit-Wigner Resonance Calculator
Compute the Breit-Wigner resonance lineshape and cross section at a given energy from a particle's mass and decay width.
📈 What is the Breit-Wigner Resonance Calculator?
This Breit-Wigner resonance calculator computes the normalized resonance lineshape amplitude f(E) = (Γ/2)² / [(E−M)² + (Γ/2)²] at any energy E, given a resonance's mass M and decay width Γ. Multiply by an optional peak cross section σ₀ to get the physical cross section σ(E) at that energy.
The Breit-Wigner distribution describes how the probability of interaction, or the measured cross section, varies with energy near an unstable particle's resonance. It appears throughout particle physics: the Z boson resonance measured at LEP, the J/ψ and other quarkonium states, and nuclear reaction resonances all follow this same characteristic peaked shape.
A common misconception is that the resonance peak has a sharp, well-defined energy. In reality, because the particle is unstable and decays after a finite mean lifetime, the energy-time uncertainty principle guarantees the resonance has a nonzero energy spread Γ, the wider the width, the shorter the particle's lifetime.
This calculator is useful for particle physics students studying resonances and cross sections, and for anyone who wants to visualize how a resonance's mass and width shape its characteristic peaked curve.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - Z boson at peak energy
Example 2 - Z boson off peak energy
Example 3 - J/ψ meson, a very narrow resonance
❓ Frequently Asked Questions
🔗 Related Calculators
What is the Breit-Wigner distribution?
The Breit-Wigner distribution describes the lineshape of an unstable particle or resonance as a function of energy, f(E) = (Γ/2)² / [(E−M)² + (Γ/2)²], where M is the resonance mass and Γ is its decay width. It peaks at E=M and falls off symmetrically on either side.
What is the formula for the Breit-Wigner resonance?
f(E) = (Γ/2)² / [(E−M)² + (Γ/2)²]. At E=M this equals 1 (the peak). To get a physical cross section, multiply by the peak cross section σ₀: σ(E) = σ₀ × f(E).
Why is Γ called the decay width?
Γ is exactly the full width at half maximum (FWHM) of the Breit-Wigner peak: the lineshape falls to exactly half its peak value when E is Γ/2 away from M on either side, so the total width between those two half-maximum points equals Γ.
What is a real example of a Breit-Wigner resonance?
The Z boson has mass M≈91.1876 GeV and width Γ≈2.4952 GeV, producing a broad resonance peak measured in electron-positron collision data at LEP. The J/ψ meson, by contrast, has mass M≈3.0969 GeV but a far narrower width of only Γ≈92.9 keV, producing an extremely sharp peak.
How does resonance width relate to particle lifetime?
A resonance's width Γ and its mean lifetime τ are related by τ=ħ/Γ, the energy-time uncertainty relation. Narrow resonances (small Γ) live much longer than wide resonances (large Γ).
What is the difference between the relativistic and non-relativistic Breit-Wigner forms?
This calculator uses the simplified non-relativistic form f(E)=(Γ/2)²/[(E−M)²+(Γ/2)²], which is an excellent approximation near the peak for narrow resonances. The full relativistic Breit-Wigner form replaces (E−M)² with (E²−M²)²/E², which matters mainly far from the peak or for very wide resonances.
What does the peak cross section σ₀ represent?
σ₀ is the maximum cross section at the resonance peak (E=M), a measured quantity with units of area (commonly barns or nanobarns in particle physics). Multiplying the dimensionless lineshape f(E) by σ₀ converts it into a physical cross section at any energy.
Can this calculator be used for atomic or nuclear resonances too?
Yes, the Breit-Wigner lineshape describes resonances across physics, from nuclear reaction cross sections to atomic spectral lines to particle physics resonances, as long as you supply the resonance's central energy M and its width Γ in consistent units.
Why does the resonance curve look symmetric around the peak?
The simplified Breit-Wigner formula depends only on (E−M)², which is identical for equal positive and negative offsets from M, making the curve exactly symmetric around the peak. The full relativistic form introduces a small asymmetry, more noticeable far from the peak.
What happens to f(E) far from the resonance peak?
As E moves far from M, (E−M)² grows much larger than (Γ/2)², so f(E) falls toward 0, meaning the resonance contributes negligibly to the cross section far from its central mass.