# The theory (2 of 2) | Cyclotron resonance

Electron cyclotron resonance - Motion of an electron in a magnetic field

Let us consider that a charged particle, namely an electron enters the magnetic field B of the Earth with a velocity v that is perpendicular to the magnetic field lines as shown in Figure 1. The magnetic field will exert a Lorenz force which will act as a centripetal. Under its influence the electron will run a cycle around the magnetic field lines, and it will actually continue running circles. In other words, it will gyrate or it will perform a cyclotronic motion.

Figure 1: Electron enters a magnetic field B with a velocity v that is perpendicular to B. It will perform a circular motion of radius R.

The natural frequency of rotation in such a magnetic field is called cyclotron frequency or gyrofrequency. It is given by the equation:

ω(ce)= e*B/m(e)

A useful numerical formula is f(ce) = 28 GHz/Tesla   (Chen 1984 (p.85) [1])

Calculations based on the above (cf. https://en.wikipedia.org/wiki/Electron_cyclotron_resonance) show that the electron gyration/cyclotron frequency in the Earth's magnetic field (of approximately 50μΤ) is 1.4 MHz.

In the previous section we referred to the electron spin resonance frequency or electron Larmor frequency. This is the frequency of precession of the electron spin around the magnetic field lines. (The precession is a special rotation where the axis writes a cone).
The useful numerical formula which corresponds to the magnetogyric ratio γ is 28.025 GHz/T.
The electron spin resonance frequency for 50μΤ is 1.4 MHz.

We obtain the same result.
Figure 2 below illustrates the difference between the magnetic resonance frequency and the cyclotron resonance frequency.

Figure 2: Illustration of the difference between the magnetic resonance frequency and the cyclotron resonance frequency.

Proton cyclotron resonance frequency and proton magnetic resonance frequency

What is the cyclotron frequency of the proton for 50μΤ?

It is 762,1 Hz. [ref.]

What is the proton magnetic resonance frequency for 50μΤ?

It is 2129 Hz or 2.1 KHz. This would be the frequency for an MRI at 50μΤ.

https://en.wikipedia.org/wiki/Gyromagnetic_ratio#For_a_nucleus

http://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/nmr.html#c1

Ion cyclotron frequencies and magnetic resonance frequencies

What are the cyclotron frequencies for different ions?

Ca2+=38,3 Hz, Cl2+=21,5 Hz, K+=19,7 Hz, Mg2+= 63,2 Hz, Na+=33,5 Hz. [ref.], (Zhadin et al 1999)

What are the magnetic resonance frequencies for different ions?

Cl=209 Hz, K = 99 Hz, P = 863 Hz, Na= 563 Hz  [ref.]