In the classical problem of electron - ion Bremsstrahlung, an electron passes near an ion, accelerates due to electric forces and emits radiation. In this entry we discuss a similar process, where two electrons pass near each other. The main difference between the two processes is that electron ion involves dipole radiation, while electron electron involves quadrupole radiation. We recall that quadrupole radiation is given by

$ P \approx \frac{\left(d^3 Q / dt^3\right)^2}{c^5} $

where $ Q $ is the quadrupole moment and $ c $ is the speed of light. The second derivative of the dipole moment is given by $ \ddot{d} \approx q^3 / m b^2 $ where $ q $ is the elementary charge, $ m $ is the electron mass and $ b $ is the impact parameter, so the third time derivative of the quadrupole moment is given by

$ d^3 Q /dt^3 \approx \ddot{d} v $

where $ v $ is the relative velocity between the electrons. The peak power from a single fly by is therefore

$ P \approx \frac{q^6 v^2}{m^2 b^4 c^5} $

The energy emitted during the collision is $ \Delta U \approx P \cdot b/v $. The collision rate is given by $ 1/\tau \approx n b^2 v $. The emission is dominated by collisions with the smallest possible impact parameter, which is given by the Heisenberg uncertainty principle $ b \approx \hbar / m v $. The emissivity is given by

$ \varepsilon_{ee} \approx \frac{q^6 v^3 n^2}{m c \hbar} \approx \frac{v^2}{c^2} \varepsilon_{ei} $

where $ \varepsilon_{ei} $ is the electron ion Bremsstrahlung emissivity. For a plasma in thermal equilibrium the velocity can be replaced by the thermal velocity $ v \approx \sqrt{k T/m} $.