Bremsstrahlung - Dipole Approximation

Dipole Approximation

Suppose that a particle of charge experiences an acceleration which is collinear with its velocity (this is the relevant case for linear accelerators). Then, the relativistic expression for the angular distribution of the bremsstrahlung (considering only the dominant dipole radiation contribution), is

where and is the angle between and the point of observation.

Integrating over all angles then gives the total power emitted as

where is the Lorentz factor.

The general expression for the total radiated power is

where signifies a time derivative of . Note, this general expression for total radiated power simplifies to the above expression for the specific case of acceleration parallel to velocity, by noting that and . For the case of acceleration perpendicular to the velocity (a case that arises in circular particle accelerators known as synchrotrons), the total power radiated reduces to

The total power radiated in the two limiting cases is proportional to or . Since, we see that the total radiated power goes as or, which accounts for why electrons lose energy to bremsstrahlung radiation much more rapidly than heavier charged particles (e.g., muons, protons, alpha particles). This is the reason a TeV energy electron-positron collider (such as the proposed International Linear Collider) cannot use a circular tunnel (requiring constant acceleration), while a proton-proton collider (such as the Large Hadron Collider) can utilize a circular tunnel. The electrons lose energy due to bremsstrahlung at a rate times higher than protons do.