Synchrotron Tune and Betatron Oscillation Calculator

Betatron tune Q is the number of transverse betatron oscillations a particle completes in one full turn around a storage ring. In the ring-averaged smooth-focusing approximation used here, Q = R / average beta function, where R is the average ring radius and the beta function describes how strongly the lattice focuses the beam at each point.

These are two unrelated quantities that happen to share the Greek letter beta. The beta function (written beta(s) or average beta here) is a lattice optics property measured in metres, describing how the beam envelope's width varies around the ring. The velocity fraction beta = v/c is dimensionless and describes how close the particle's speed is to the speed of light. This calculator labels them average beta function and velocity fraction beta_v to keep them clearly separate.

A betatron oscillation is the transverse wobble a particle makes around the ideal closed orbit as it circles a storage ring, caused by the focusing and defocusing quadrupole magnets that keep the beam from spreading out. Every particle in the beam oscillates transversely at the betatron frequency as it travels around the ring, in both the horizontal and vertical planes.

If the tune Q lands on or near a simple fraction (an integer, half-integer, or other low-order rational number), small imperfections in the lattice reinforce on every turn instead of averaging out, driving a resonance that can grow the beam size until particles are lost. Real machines are carefully tuned to sit away from these resonance lines in what is called a tune diagram.

Revolution frequency f_rev is how often the particle completes one lap of the ring, f_rev = beta_v * c / C. Betatron oscillation frequency f_beta is how often the particle completes one transverse wobble, f_beta = Q * f_rev. Since Q is rarely a whole number, the transverse wobble and the physical lap around the ring are not synchronized, which is exactly what avoiding a resonance requires.

This tool uses the smooth-focusing (ring-averaged) approximation from introductory accelerator physics textbooks, replacing the real lattice's varying beta function with a single average value. A real ring's beta function varies significantly from point to point around a FODO cell, so an exact tune calculation needs a full lattice simulation, not just circumference and an average beta.

Average ring radius is R = C / (2 * pi), the radius of a circle with the same circumference as the actual ring. Real rings are rarely perfect circles, they alternate straight sections and bending arcs, so R is a convenient average used throughout smooth-focusing calculations.

Betatron wavelength is lambda_beta = 2 * pi * average beta function, the physical distance along the ring over which one betatron oscillation cycle completes, in the smooth-focusing approximation. It equals the ring circumference divided by the tune, C / Q.

Synchrotrons are almost always operated with particles that are already ultra-relativistic, so beta_v = v/c is extremely close to 1 (often 0.999999 or higher for electron machines). This calculator defaults to 0.999999 to match typical synchrotron operating conditions, though you can lower it for a non-relativistic or partially relativistic ring.

Circumference is entered in metres, average beta function in metres, and velocity fraction as a dimensionless number between 0 and 1. Revolution frequency is reported in kilohertz, betatron oscillation frequency in megahertz, and tune as a dimensionless number.

Not necessarily, tune Q = R / average beta depends on the ratio of ring radius to average beta function, not radius alone. A larger ring with proportionally stronger focusing (smaller average beta function) can have the same or higher tune than a smaller ring with weaker focusing.