Synchrotron Radiation Critical Frequency Calculator
Find the critical photon energy of synchrotron radiation from a relativistic electron bending through a dipole magnet, from beam energy and bending radius or field.
🌈 What is the Synchrotron Radiation Critical Frequency Calculator?
This synchrotron radiation critical frequency calculator finds the critical photon energy Ec and critical angular frequency omega_c of the synchrotron radiation spectrum emitted by a relativistic electron bending through a dipole magnet, a standard result from classical accelerator physics (Jackson's Classical Electrodynamics).
Synchrotron light source facilities, from compact university machines to national labs like the APS and NSLS-II, are built entirely around this physics: bending high-energy electrons through dipole magnets produces intense, broadband X-ray beams used for materials science, structural biology, and chemistry research. Accelerator physicists use the critical energy to characterize where a beamline's usable X-ray spectrum is centered.
This calculator offers two equivalent ways to specify the machine geometry. Bending Radius mode lets you enter beam energy and bending radius directly, the two quantities that appear explicitly in the critical frequency formula. Magnetic Field mode instead lets you enter beam energy and dipole field strength, deriving the bending radius automatically, useful when you know the magnet strength rather than the ring geometry.
This calculator is useful for accelerator physics students, synchrotron beamline scientists estimating available photon energy ranges, and particle physics students studying radiative energy loss in circular accelerators.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - Compact light source, bending radius mode
Example 2 - Higher-energy ring, bending radius mode
Example 3 - Same beam, magnetic field mode
❓ Frequently Asked Questions
🔗 Related Calculators
What is synchrotron radiation critical frequency?
The critical frequency omega_c divides a relativistic electron's synchrotron radiation spectrum so that exactly half the total radiated power is emitted above it and half below it. It sets the characteristic photon energy scale of the whole emitted spectrum, from a classical accelerator-physics result found in Jackson's Classical Electrodynamics.
What is the formula for the critical frequency and critical energy?
omega_c = 1.5 gamma cubed c / rho, where gamma is the electron's Lorentz factor and rho is the bending radius in metres. The critical photon energy is Ec = hbar omega_c, converted to electron volts.
Why does critical energy grow so fast with beam energy?
Because omega_c scales as gamma cubed, and gamma is directly proportional to beam energy for an ultra-relativistic electron, doubling the beam energy multiplies the critical frequency (and critical energy) by a factor of about eight. This cubic growth is why higher-energy storage rings reach much harder X-ray spectra.
What is the difference between the two input modes?
Bending Radius mode lets you enter the beam energy and bending radius directly, the two quantities that appear explicitly in the critical frequency formula. Magnetic Field mode instead lets you enter beam energy and dipole field strength, and derives the bending radius automatically using pc[GeV] approximately equal to 0.299792458 x B[T] x rho[m].
How is bending radius derived from magnetic field?
For an ultra-relativistic electron, momentum in GeV is approximately equal to beam energy in GeV (the 0.000511 GeV rest mass is negligible), so rho[m] = E[GeV] / (0.299792458 x B[T]). A stronger field or lower beam energy gives a tighter bending radius.
What is a typical critical energy for a real synchrotron light source?
Third-generation light sources with beam energies of roughly 3 to 8 GeV and bending radii of a few to tens of metres typically reach critical energies from a few keV up to tens of keV, in the hard X-ray range useful for materials science and structural biology beamlines.
Why does this calculator assume electrons rather than protons or other particles?
Synchrotron light sources overwhelmingly use electron (or positron) beams because radiated power scales as 1/mass to the fourth power, an electron radiates vastly more synchrotron power than a proton at the same energy and bending radius, which is exactly why electron storage rings make useful light sources and proton accelerators do not.
What does it mean that half the power is radiated above the critical frequency?
The synchrotron radiation spectrum is broad and continuous, not a single sharp line. The critical frequency is defined precisely so that the area under the power spectrum curve above omega_c exactly equals the area below it, making it the natural single number to characterize where the spectrum is centered.
What beam energy and bending radius does this calculator default to?
The default is a 3 GeV beam energy with a 5 m bending radius, representative of a compact third-generation synchrotron light source, giving a critical energy of about 11,978.75 eV (about 11.98 keV).
Is this the same critical energy used to describe synchrotron radiation damage in particle accelerators?
Yes, the same critical energy formula describes both deliberate synchrotron light sources and unwanted synchrotron radiation losses in circular particle colliders like LEP or the proposed FCC-ee, where it directly determines radiofrequency power requirements and radiation shielding needs.