Schwinger rate density estimated by this calculator
This Schwinger pair-production calculator examines a distinctive prediction of quantum electrodynamics: an electric field can create charged particle pairs when it is sufficiently strong. In the familiar electron case, the particles are electron-positron pairs. The page evaluates the leading Schwinger rate per unit volume and per unit time for an ideal constant, spatially uniform electric field. It answers a narrow question: for particle mass m, charge q, and field strength E, how large is the leading tunnelling rate?
The Schwinger output is a local rate density in cubic metres and seconds, not a predicted yield for a complete experiment. Estimating a total number of pairs would additionally require an effective field volume and duration, plus a decision about whether a constant-field description is suitable. This calculator performs the first physics step only: it turns mass, charge, and field inputs into a rate density, a critical field, and the dimensionless ratio E/Ec.
The prefilled mass and charge are electron values. The field is deliberately left for you to choose because the calculated rate is extraordinarily field-sensitive. At low field the exponential suppression makes the result negligible; close to the critical field the leading estimate rises very quickly.
Choosing mass, charge, and electric-field inputs for the Schwinger estimate
For this Schwinger calculation, enter Particle Mass m (kg) as the particle rest mass in SI kilograms. The standard electron value in the form is 9.10938356 × 10−31 kg. When comparing particle species, remember that the exponent contains m2: increasing the mass raises the critical field substantially and makes tunnelling more difficult.
Enter Charge q (C) in coulombs. The script uses the magnitude of charge, so either a signed value or a positive magnitude gives the same rate. The electron charge magnitude is 1.602176634 × 10−19 C. A larger charge magnitude lowers the critical field because the applied electric field couples more strongly to the particle.
The Electric Field E (V/m) input must be positive for this model. Field strength appears as E2 in the prefactor, but its more consequential role is in the exponential. If E is far below Ec, the exponent is very negative. As the field approaches Ec, that suppression becomes much weaker.
Before calculating an electron result, it is useful to think in powers of ten. The electron critical field is roughly 1.3 × 1018 V/m. A value near 1014 V/m should be overwhelmingly suppressed, whereas a field on the scale of 1018 V/m is near the regime where the leading estimate changes dramatically. This check can reveal unit mistakes such as entering kV/m as V/m.
Leading Schwinger formula used for the rate density
The Schwinger rate calculation uses the leading constant-field term in SI units:
The same Schwinger inputs set the particle-specific critical field:
These expressions account for the behaviour in the results panel. The prefactor grows polynomially with field, while the exponential depends on Ec/E. Consequently, a sweep through field strengths can look nearly abrupt: pair production is heavily suppressed far below criticality and grows rapidly near it.
The full constant-field result can be written as a series over integer terms. This page intentionally retains only the leading n = 1 contribution, which is the term implemented by the JavaScript. It is a useful first-pass estimate, not a substitute for a calculation that includes pulse shape, inhomogeneity, magnetic fields, or backreaction.
Electron-field example for the Schwinger leading term
For an electron mass and charge with an electric field of 1.0 × 1018 V/m, the critical field is about 1.3 × 1018 V/m. The resulting ratio E/Ec is therefore below, but not far below, one. That immediately indicates that the tunnelling exponent is no longer catastrophically large in magnitude.
For those electron inputs, the exponent is approximately −πEc/E ≈ −4.1, making the exponential factor about 0.016. The SI prefactor is nevertheless very large, so the leading estimate is on the order of 1054 m−3s−1. This is a local ideal-model rate density, not a direct count of pairs an apparatus must observe.
At 1.0 × 1016 V/m, the electron field ratio is only roughly 0.0076 and the exponent is about −414. The result is fantastically small, although it can still be represented by ordinary floating-point arithmetic. At still lower fields the exponential can underflow numerically to zero; in that case the displayed zero means the rate is below the calculation’s numerical range, not that the mathematical expression has become exactly zero.
Interpreting the Schwinger results panel
The Schwinger results panel returns three quantities. Γ is the leading pair-production rate density in m−3s−1. The critical field Ec is in V/m for the particle you entered. The field ratio E/Ec is dimensionless and usually provides the fastest physical interpretation: a tiny ratio corresponds to strong suppression, while a ratio approaching one means the exponential is weakening rapidly.
Very small and very large displayed rates are expected for the Schwinger expression, not necessarily signs of a form error. A rate close to zero places the chosen field deeply in the tunnelling-suppressed regime. An enormous rate signals vacuum instability within the constant-field model, where effects omitted here—such as field depletion, pulse structure, and environmental interactions—may matter.
Electron-positron Schwinger rate comparison by field scale
This electron-focused comparison shows why the Schwinger calculation is dominated by electric-field scale. It is an orientation guide for the leading constant-field term, rather than a replacement for entering your own particle properties.
Order-of-magnitude guide for electron fields
| Electric field |
Approximate ratio E/Ec |
Interpretation |
| 1014 V/m |
about 10−4 |
Overwhelming exponential suppression; the rate is effectively zero for this model. |
| 1017 V/m |
about 0.08 |
Still strongly suppressed, though far less hopeless than the previous case. |
| 1018 V/m |
about 0.76 |
Suppression weakens dramatically and the leading-term estimate becomes extremely large. |
For a non-electron scenario, use the reported critical field rather than the table’s electron values. Comparing the applied field with that critical scale is the most direct route from the raw SI inputs to an interpretation of the result.
Physical assumptions behind this Schwinger rate estimate
This Schwinger calculator assumes a constant, spatially uniform electric field and uses only the leading term. Actual strong-field environments may be pulsed, inhomogeneous, combined with magnetic fields, or changed by plasma effects and backreaction. The output treats the vacuum response locally, so it does not itself predict a detector count, spectrum, or beam profile.
Charge sign does not affect the rate because the formula uses |q|. Mass and charge are treated as fixed properties of one particle species. The script uses standard floating-point arithmetic, so an extremely negative exponent can produce a numerical zero. That result should be read as “too small to resolve in this computation,” rather than a literal claim that the exact rate is zero.
The calculator also omits the higher terms of the infinite Schwinger series. Keeping the leading term makes the model transparent for scale estimates. When a near-critical field produces a huge rate, however, the simplest constant-field picture may no longer capture all relevant physics. Precision theory or a direct experimental comparison requires a more complete treatment.
A practical Schwinger workflow is to choose sensible SI inputs, calculate the rate, compare E with the displayed Ec, and then decide whether the constant-field approximation is credible for the field geometry and duration. If the answer depends critically on pulse duration, volume, or depletion, the calculator has still supplied an important first conclusion: further modelling is needed.
Why the electric-field ratio controls Schwinger tunnelling
The key organizing quantity in this Schwinger calculation is the ratio E/Ec. Mass and charge alter the critical field, but once that scale is known, the ratio reveals whether the applied field sits in a deeply suppressed regime or near the regime where the leading rate grows quickly.
When comparing possible Schwinger scenarios, change one input at a time. Increasing mass raises Ec; increasing charge magnitude lowers it; increasing the applied field raises E/Ec. The field has the most visually dramatic effect in the output because it also appears in the tunnelling exponent. Double-check the field unit and its power of ten before drawing a conclusion from an apparently extreme result.