Schwinger Pair Production Rate Calculator

Assumptions & limitations (validity of this calculator)

• Constant, uniform electric field: Uses the classic Schwinger result for a spatially uniform, time-independent field. Rapidly varying/inhomogeneous laser fields can change the rate substantially.
• Vacuum QED (no medium): Assumes pair creation in vacuum, not in matter/plasma/solids. (Condensed-matter “Schwinger-like” analogs require different parameters.)
• Leading-term approximation: Implements only the first term of the Schwinger series. When E is not far below the critical field, higher-order terms can contribute.
• No backreaction: Ignores depletion/screening of the field by the created pairs and ignores subsequent cascade dynamics.
• Magnitude of charge: The standard formula depends on |q|. If you enter a negative charge, the magnitude is what matters for the rate.
• Physical domain: The rate is meaningful for E > 0. At extremely large fields, additional physics (higher terms, strong-field effects, non-ideal geometry) may be required.

Result definition & units

The calculator returns the pair production rate density (often written \(\Gamma\)): number of pairs produced per unit time per unit volume. In SI units this is typically reported as m⁻³·s⁻¹.

Key formulas used (leading term)

In its simplest (leading-term) form for spin-1/2 particles in a constant electric field, the Schwinger rate density is:

A convenient reference scale is the critical field: \(E_c = \dfrac{m^2 c^3}{|q|\hbar}\). For electrons, \(E_c\) is about \(1.3\times 10^{18}\,\text{V/m}\), which is why the rate is usually extremely small at laboratory-scale static fields.

Inputs (what to enter)

• Particle mass \(m\) (kg): Electron default is \(9.10938356\times 10^{-31}\,\text{kg}\).
• Charge \(q\) (C): Use the magnitude of the particle charge. Electron magnitude is \(1.602176634\times 10^{-19}\,\text{C}\).
• Electric field \(E\) (V/m): Must be positive. The rate becomes appreciable only as \(E\) approaches \(E_c\).

How to interpret the result

The prefactor scales like \(E^2\), but the dominant behavior is the exponential suppression \(\exp(-\pi E_c/E)\). That means:

• If \(E \ll E_c\), the exponent is very large and negative, so the rate is effectively zero.
• If \(E\) is within an order of magnitude of \(E_c\), the suppression weakens quickly and the rate can become enormous (subject to the limitations above).

Worked example (order-of-magnitude)

Suppose you keep the electron defaults for \(m\) and \(|q|\), and try \(E = 1\times 10^{17}\,\text{V/m}\). Then \(E/E_c \approx 0.077\), so the exponential factor is roughly \(\exp[-\pi(E_c/E)] \approx \exp(-\pi/0.077)\), which is extraordinarily small. Even though the \(E^2\) prefactor is large, the total rate density remains negligible. This illustrates why reaching near-critical fields is essential for observable Schwinger production.

Quick comparison: how the rate changes with field strength

References

• J. Schwinger, “On Gauge Invariance and Vacuum Polarization,” Physical Review 82, 664 (1951).
• Standard QED/strong-field texts and reviews discussing the Schwinger effect and constant-field pair production.