What affects the value of cyclotron frequency for protons?

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The cyclotron frequency for protons is influenced by relativistic effects at higher energies, which cause the resonance frequency to decrease due to the increased mass in the denominator of the equation. Additionally, for stable vertical orbits, the vertical magnetic field must decrease as a function of the radius of the orbit. The Wikipedia article provides a satisfactory overview of cyclotron resonance, but it does not address these critical factors. Understanding these influences is essential for accurate calculations of cyclotron frequency. Comprehensive mathematical proof and explanation are necessary for a complete understanding of these concepts.
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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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