Magnetic Rigidity Calculator

Magnetic rigidity, Bρ, is the product of a bending magnet's field strength B and the radius of curvature ρ of a charged particle's circular path through it. It equals the particle's momentum divided by its charge (Bρ = p/q), making it the single quantity accelerator engineers use to design bending magnet systems.

Bρ = p/q, where p is the particle's momentum and q is its electric charge. In practical accelerator-physics units, this is often written Bρ[T·m] = p[GeV/c] / (0.299792458 × Z), where Z is the charge state in units of the elementary charge.

Particle momenta at accelerator energies are enormous in SI units but conveniently sized when expressed in GeV/c (giga-electronvolts per speed of light), since particle energies are already routinely quoted in GeV or TeV. This calculator performs the conversion to SI internally.

Given a beam's magnetic rigidity and a chosen dipole magnet field strength B, the required bending radius follows directly as ρ = Bρ/B. This lets engineers size ring circumferences, choose magnet field strengths, and design beam transport lines for a target particle momentum.

The LHC's 7 TeV/c proton beams have an enormous magnetic rigidity, requiring either extremely strong bending magnets or a very large bending radius (or, in practice, both) to keep the beam circulating. Its 27 km circumference and 8.3 T superconducting dipole magnets are a direct consequence of the magnetic rigidity needed at its design energy.

Magnetic rigidity is inversely proportional to charge state Z, so a heavy ion carrying multiple units of charge (like a fully stripped lead nucleus with Z=82) has much lower magnetic rigidity than a proton at the same momentum, allowing the same magnet ring to handle far higher-momentum heavy ion beams than protons.

No, though closely related, momentum is a property of the particle's motion alone, while magnetic rigidity specifically combines momentum with charge to describe how that particle behaves in a magnetic field, the quantity actually relevant to accelerator magnet design.

Tesla-metres (T·m), reflecting its definition as the product of magnetic field strength (tesla) and bending radius (metres). This calculator reports the result in T·m.

It is most directly relevant to circular accelerators and any beamline section using bending (dipole) magnets, since it describes circular motion in a magnetic field. Linear accelerators (linacs) do not bend the beam, so magnetic rigidity is not directly applicable to their main accelerating sections, though it still matters for any beam transport or steering magnets they include.

It can be inferred by measuring a particle's actual bending radius in a known magnetic field (ρ measured directly from the beam trajectory, B measured from the magnet), then computing Bρ, providing an experimental cross-check against the value calculated from the beam's known momentum and charge.