What affects the value of cyclotron frequency for protons?

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SUMMARY

The cyclotron frequency for protons is influenced by relativistic effects and the vertical magnetic field configuration. As proton energies increase, the mass in the denominator of the cyclotron frequency equation becomes relativistic, resulting in a decrease in resonance frequency. Additionally, for stable vertical orbits, the vertical magnetic field Bz(r) must decrease as a function of the radius of the orbit. These factors are critical for accurate calculations of cyclotron resonance in high-energy physics applications.

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Can anyone tell me the value of cyclotron frequency with full mathematical proof and explanation?
 
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Two things the above formula for proton cyclotron resonance misses:

1) At higher proton energies, the mass in the denominator of the equation becomes relativistic, causing the resonance frequency to drop.

2) For vertical orbit stability, the vertical B field Bz(r) must decrease as a function of r (radius) of orbit.
 

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