Engineering & Science
Synchrotron Radiation: Critical X-Ray Photon Energy & Energy Loss per Turn
Calculate bending-magnet synchrotron critical photon energy E_crit = 0.665 * E_GeV^2 * B_Tesla (in keV) and relativistic electron radiative energy loss per storage-ring revolution U0 = 88.46 * E_GeV^4 / rho_m (in keV/turn).
See why circular electron colliders hit an E^4 synchrotron wall—while X-ray light sources thrive on it!
Bend an ultra-relativistic electron beam of energy E (in GeV) through dipole magnets of field B (Tesla) and bending radius rho (meters), and relativistic Lorentz beaming compresses its dipole radiation into a brilliant forward searchlight of hard X-rays! Two engineering formulas govern every synchrotron light source (APS, ESRF, NSLS-II, Diamond) and circular electron-positron ring (LEP, FCC-ee):
First, the Critical Photon Energy E_crit(keV) = 0.665 * (E_GeV ^ 2) * B_T divides the radiated power spectrum into two equal halves (50% of X-ray power above E_crit, 50% below)! Second, Julian Schwinger's relativistic Larmor integral gives the radiative energy lost by each electron on every single turn around the ring: U0(keV/turn) = 88.46 * (E_GeV ^ 4) / rho_m! In a 3.0 GeV X-ray light source (B = 1.40 T, rho = 7.14 m), E_crit = 8.38 keV (ideal for protein crystallography!) and U0 = 1,003.5 keV/turn (~1 MeV RF replenishment per lap); doubling beam energy to 6.0 GeV at double the radius (14.28 m) quadruples E_crit to 33.52 keV and jumps U0 eight-fold to 8,028.3 keV/turn!
| Storage Ring Design | Beam E (GeV), Field B, Radius rho | Critical X-Ray E_crit (keV) | RF Loss per Turn U0 (keV/turn) |
|---|---|---|---|
| 3rd/4th-Gen Medium X-Ray Ring | 3.0 GeV, 1.40 T, rho = 7.14 m | 8.38 keV (Cu K-alpha class) | 1,003.5 keV/turn |
| High-Energy Hard X-Ray Ring | 6.0 GeV, 1.40 T, rho = 14.28 m | 33.52 keV (4x harder X-rays!) | 8,028.3 keV/turn (2^4 / 2 = 8x!) |
| Invalid Ring Radius rho <= 0 | Division by Zero Guard | Guarded Input Error | "Invalid Storage Ring Inputs" |
Bending-Magnet Critical X-Ray Photon Energy E_crit (keV)
Evaluates 0.665 * (E_GeV ^ 2) * B_Tesla in kiloelectron-volts.
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Storage Ring Inputs", ROUND(0.665 * (B2 ^ 2) * B3, 2))Electron Radiative Energy Loss per Ring Revolution U0 (keV/turn)
Evaluates 88.46 * (E_GeV ^ 4) / rho_meters across isomagnetic bending dipoles.
=IF(OR(B2 <= 0, B4 <= 0), "Invalid Storage Ring Inputs", ROUND((88.46 * (B2 ^ 4)) / B4, 1))Try the example
Use the original values or change them to test your understanding.