In ion optics, there is a concept called Magnetic Rigidity. When an charged ion passing through a magnetic field, it will bend and rotate. The magnetic rigidity is the B\rho, where B is the magnetic field strength in Tesla, and \rho is the rotation radius.

using basics physics, we have the centripetal force equals to the Lorentz force,

\displaystyle \frac{mv^2}{\rho} = Q \vec{v}\times \vec{B}

Simplify,

B\rho = mv/Q


The above is non-relativistic, and I always assume the mv can be replaced by relativistic momentum p = m \gamma \beta . Now I give a prove.

The relativistic centripetal force is

\displaystyle \vec {F} = m \gamma \vec{a_\perp} = - m \gamma\frac{ v^2}{\rho} \hat{\rho}

Thus,

\displaystyle m \gamma \frac{v^2}{\rho} = QvB

\displaystyle m \gamma \beta = p = cQB \rho

where c is speed of light. In the last equation, the unit of p is MeV/c, c = 299.792458 mm/ns, and \rho in meter.

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