Synchrotron Radiation Energy Loss Calculator

Introduction

In a synchrotron or electron storage ring, bending magnets force the beam to follow a curved trajectory. Because the electrons are continuously accelerated sideways, they emit electromagnetic radiation (synchrotron radiation). This radiation is a feature for dedicated light sources, but it is also an unavoidable energy drain that must be replenished by the RF system every turn.

This calculator estimates (1) the energy loss per turn for ultra-relativistic electrons and (2) the corresponding total radiated power for a stored beam current. It is intended for quick, order-of-magnitude planning and for comparing scenarios (for example, changing energy, ring size, or current).

Why synchrotron radiation matters

The key scaling is steep: for electrons, the energy loss per turn grows with the fourth power of beam energy. That means modest energy upgrades can dramatically increase RF power requirements and heat loads on the vacuum chamber. Increasing the bending radius reduces losses linearly, which is one reason high-energy electron rings tend to be physically large.

In practical terms, synchrotron radiation shows up in several places at once. The RF system must provide enough voltage to replace the lost energy each turn, and the RF amplifiers must deliver enough power to sustain the beam current. At the same time, the vacuum chamber and photon absorbers must handle the heat load from the emitted radiation fan. Even if the beam is stable, insufficient cooling or shielding can limit operating current.

Synchrotron radiation is also the reason modern x-ray light sources exist. The same emission that complicates accelerator engineering produces extremely bright, highly collimated photon beams used for crystallography, spectroscopy, imaging, and time-resolved studies. For a storage ring facility, the radiated power is not just a number on a datasheet; it is tied to component lifetime, vacuum performance, and the achievable brightness delivered to beamlines.

How to use the calculator

1. Beam energy (GeV): Enter the electron beam energy in giga-electronvolts.
2. Bending radius (m): Enter the average bending radius of the arcs (not the full ring circumference). Use meters.
3. Beam current (A): Enter the stored beam current in amperes. Use 0 if you only want the per-turn loss.
4. Select Compute radiation to see the energy loss per turn and total radiated power.

Tip: If you are comparing two designs, keep units consistent and change only one parameter at a time to see the scaling clearly. If you are unsure about the bending radius, start with a rough estimate based on the dipole field and beam energy, then refine it using lattice documentation.

Formulas and units

For ultra-relativistic electrons in a ring with an effective bending radius , a widely used practical approximation for the energy loss per turn is:

keV per turn

• is the beam energy in GeV.
• is the bending radius in meters.
• is returned in keV per turn.

The calculator also converts keV to joules using 1 keV = 1.602176634×10⁻¹⁶ J. This conversion is helpful when you want to compare the per-turn loss to other energy scales, or when you are building a more detailed power budget.

To estimate total radiated power, the page uses the convenient relationship (with in keV/turn and current in A):

watts

This works because beam current already represents charge flow per second; multiplying by the per-turn energy loss (expressed as an equivalent voltage) yields power. In accelerator terms, if the beam loses keV each turn, the RF system must provide at least that much energy gain per turn to maintain the beam energy (plus margin for stability and operational overhead).

Worked example

Suppose an electron storage ring operates at 3.0 GeV with an average bending radius of 100 m and a stored current of 0.50 A.

• Energy loss per turn: U = 88.5 × 3.0⁴ / 100 ≈ 71.7 keV/turn
• Total radiated power: P = 71.7 × 1000 × 0.50 ≈ 35,850 W ≈ 35.9 kW

If you keep the same radius and current but double the energy to 6 GeV, the loss increases by a factor of 16, which quickly pushes power into the hundreds of kilowatts. This is why energy upgrades often require RF and cooling upgrades as well.

Typical parameter comparisons

The following representative values illustrate the strong energy dependence and the linear benefit of larger bending radius. These are approximate and intended for intuition.

Use these comparisons as a quick check: if your inputs produce results that differ by orders of magnitude from these trends, verify that you entered GeV (not MeV), meters (not millimeters), and amperes (not milliamps). Also confirm that the radius you are using is the bending radius of the dipoles rather than the overall ring radius.

Limitations and assumptions

• Electron beams only: The constant and scaling are for electrons. Heavier particles (protons/ions) radiate far less at the same energy because radiation scales strongly with particle mass.
• Ultra-relativistic regime: The approximation assumes electrons are highly relativistic (typical for GeV-scale rings).
• Average bending radius: Real lattices have multiple dipoles and straight sections; using an effective/average is an approximation.
• No lattice-dependent corrections: Effects such as insertion devices (wigglers/undulators), edge radiation, and detailed magnet distribution can increase total radiated power beyond the simple arc-only estimate.
• No beam dynamics: Quantum excitation, energy spread, emittance, and RF bucket constraints are not modeled.

For conceptual design and teaching, these formulas are often sufficient. For engineering design, consult a full accelerator model and facility-specific parameters.

How to interpret the results

The calculator reports two primary outputs: energy loss per turn and total radiated power. The per-turn loss is useful for estimating the minimum RF voltage needed to maintain energy (in practice, the RF voltage must exceed the loss to provide stable longitudinal focusing). The total radiated power is useful for thermal and electrical planning: it indicates how much power is being deposited as photons into the ring environment and how much RF power must ultimately be supplied (before accounting for RF efficiency and overhead).

Keep in mind that the power computed here is the power radiated by the beam due to bending. The wall-plug power required to run the RF system can be significantly higher because of amplifier efficiency, cryogenic loads (for superconducting RF), and operational margins. Conversely, if the beam current is low, the radiated power may be modest even at higher energies, but the per-turn loss can still be large and can set constraints on RF voltage and momentum acceptance.

Practical notes for choosing inputs

Beam energy: Use the nominal operating energy of the stored beam. If you are evaluating an upgrade, run the calculator at both the current and proposed energies to see the E⁴ scaling directly.

Bending radius: Many rings have multiple dipole families and non-uniform curvature. If you have a lattice file or design report, look for an “effective bending radius” or compute it from the dipole field and beam rigidity. If you only know the approximate ring size, do not substitute the ring circumference divided by 2π unless the ring is close to a perfect circle and the arcs dominate.

Beam current: Use the total stored current (sum over all bunches). If you are planning a current increase, note that power scales linearly with current, so doubling current doubles radiated power and doubles the RF power needed to replenish losses.

Common mistakes and quick checks

The most common input mistakes are unit mix-ups. A quick sanity check is to remember that a few-GeV ring with a bending radius on the order of tens to hundreds of meters typically loses tens of keV per turn, not eV and not MeV. If you see micro-watts or gigawatts for a typical light source current, re-check the current units (A vs mA) and the radius.

Another common confusion is between energy loss per turn and energy loss per second. The per-turn loss is a property of the orbit and energy; the per-second loss depends on how much charge is circulating, which is why current appears in the power calculation. The calculator’s power output is the quantity most directly tied to heat load and RF power.

Background: where the constant comes from (high level)

The numerical factor 88.5 in the practical formula is a compact way to package fundamental constants and unit conversions for electrons. Starting from classical electrodynamics, an accelerating charge radiates power; in the relativistic regime the radiation depends strongly on the Lorentz factor and the curvature of the trajectory. Integrating the radiated power around one full revolution and expressing the result in keV per turn, with energy in GeV and radius in meters, yields the convenient coefficient used here.

In more detailed accelerator physics treatments, you may see the same relationship written using the radiation constant and the integral of around the ring. Those forms are useful when the bending radius varies along the lattice. This page intentionally uses the single-radius approximation because it is easy to apply and is often accurate enough for early-stage comparisons.

Additional questions (plain-language)

Is this only for storage rings? The per-turn formula is naturally phrased for circular machines, including storage rings and synchrotrons. For a ramping synchrotron, the energy changes with time, so the loss changes during the ramp; you would evaluate multiple energies or integrate over the cycle.

What about protons? Proton synchrotron radiation is usually negligible at comparable energies because the radiated power drops dramatically with increasing particle mass. That is why very high-energy hadron colliders can be circular while very high-energy electron machines tend to be linear.

Does the result include beamline extraction? No. The calculator estimates the radiation emitted by the circulating beam due to bending. How much of that radiation is delivered to beamlines depends on lattice design, insertion devices, apertures, and beamline optics.