## The Cyclotron - Cyclotron frequency f(c) or gyrofrequency - Cyclotron resonance condition

The cyclotron is an apparatus for accelerating charged particles by causing them to revolve in orbits of increasing diameter*. It consists of two concave semi-cylindrical segments in “D” shape (“Dees”) that are connected to the poles of an alternative current power source of very large frequency. Between the “Dees” there is a gap and at its center A there is the source of the particles that we wish to accelerate. The entire setting is found in a magnetic field of large intensity with its magnetic field lines being perpendicular. The electric field is confined in the gap. To avoid collisions with air molecules that would slow down the particles, the apparatus is placed in very low pressure or vacuum.

Figure 1: Cyclotron

Let us consider a positive particle, e.g. a proton at point A. If the electric field between the “Dees” has the direction shown on the left, then the proton will be accelerated towards D2. When it is inside D2, the magnetic field will exert a force (Lorentz force) which will as a centripetal force.

Due to this it will perform uniform circular motion with a radius R1 that is given by the following relationships:

F(Lorentz) = F(centripetal) ⇔ qvB = mv^2/R

R1=mv1/qB

When it will exit D2 and enter the gap, it will be accelerated by the electric field to a velocity v2 (v2>v1) and will then enter D1.

When in D1 it will perform uniform circular motion with a radius R2 which will be equal to R2=mv2/qB. We are expecting that R2>R1.

The particle will exit D1 and will be accelerated again in the gap. This will be repeated several times resulting in successive accelerations of the particle until it exits the apparatus.

In order for the particle to be accelerated every time it is found in the gap, the direction of the electric field must be the same as its velocity.

The particle runs a semi-circle when inside the “Dees” and therefore it exits them every half a period or T/2. The period is T=2π/ω with the angular velocity ω being equal to v/R.

By substituting R from above we find that the period is T=2πm/qB.

Note that the period does not depend on the radius R of the circular movement of the particle.

We could say informally that the direction of the electric field should change every T/2, but this is the definition of the alternative electric field as we know it from our everyday experience. The AC voltage changes from (+) to (-) every T/2.

Therefore, the period of the circular motion of the particle should equal the period of the alternative electric field.

T(alt)=T(circ)=2πm/qB

The associated frequency corresponded to the revolution (circle, gyre, gyration) of the particle  is called cyclotron frequency f(c) or gyrofrequency.

f(c)=qB/2πm

In order to accelerate the particle, we have to apply an alternating voltage V(alt) of the same frequency f(alt) as the cyclotron frequency f(c).

The requirement of  f(alt)=f(c) is called the “resonance condition”. Upon matching this requirement we will obtain acceleration as a response.

The above description and Figure 1 is based on a secondary education textbook

Video reference: https://youtu.be/cNnNM2ZqIsc?t=156

*Reference: etymonline.com

## Electron Cyclotron Resonance

Electron cyclotron resonance is a phenomenon where electrons are accelerated by being led to revolve in orbits of increasing diameter upon the effect of an alternating electric field such as that of microwaves of the same frequency f(alt) as the electron cyclotron frequency f(c).

In accordance with what was mentioned for the cyclotron, the electron cyclotron frequency is:

f(c)=eB/2πm(e)

where e is the electron charge and m is its mass.

If we want to apply the commonly used microwave frequency f(alt) of 2.45 GHz, then for the electron charge and mass, the resonance condition is met when B=0.0875 T as mentioned at https://en.wikipedia.org/wiki/Electron_cyclotron_resonance.

It is also noted that "the circular motion may be superimposed with a uniform axial motion, resulting in a helix" etc.

Applications of electron cyclotron resonance

Energized/accelerated electrons - heating or use of heating power: (a) ionization of gas to generate plasma, (b) driving a current

“The inverse process of electron cyclotron emission can be used as a diagnostic of the radial electron temperature profile.”

Under the influence of a magnetic field, electrons revolve in orbits (gyrate) in a volume with low pressure gas.

As mentioned at https://en.wikipedia.org/wiki/Electron_cyclotron_resonance#ECR_ion sources: "the alternating electric field of the microwaves is set to be synchronous with the gyration period of the free electrons of the gas, and increases their perpendicular kinetic energy. Subsequently, when the energized free electrons collide with the gas in the volume they can cause ionization if their kinetic energy is larger than the ionization energy of the atoms or molecules."

This link provides an animation demonstrating cyclotron resonance between a charged particle and an alternating electric field in the presence of a magnetic field that is perpendicular to the page (directed out towards the observer). A screen capture is shown in Figure 2.

Figure 2: Electron cyclotron resonance - Screen capture from animation by Lynnbwilsoniii - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=40080147

## Ion cyclotron resonance - Mass spectrometer principle (Dempster)

Ion cyclotron resonance "is used for accelerating ions in a cyclotron, and for measuring the masses of an ionized analyte in mass spectrometry."

A description of the principle of the mass spectrometer (Dempster) is provided.

Let us consider the apparatus below. A positive ion exits at ion source S and is accelerated by an electric field of voltage V found between E and A. At A it enters a uniform magnetic field B with the direction shown (perpendicular, towards the observer).

Due to the Lorentz force which acts as a centripetal force, the ion will run a semicircle of radius R1 with a velocity v1. In general:

F(Lorentz) = F(centripetal) ⇔ qvB = mv^2/R

v=qBR/m (1)

During the movement of the ion in the electric field from S to E, energy equal to qV is transferred to the ion. This is equal to its kinetic energy:

qV=1/2mv^2 (2)

From (1),(2):

m/q = B^2R^2/2V

The above formula allows us to calculate the mass to charge ratio m/q of the ion if we measure V, B and R.

The green shape of the figure represents a photographic plate where the ions fall.

If two ions with the same charge enter the magnetic field, then we can calculate their mass. The figure depicts at I2 an ion that is heavier than the one at I1.

Also from (1) given that v=ωR=2π f R :

m/q=B/2πf

An electric excitation signal having a frequency f will resonate with ions having a mass-to-charge ratio m/q.

The above description and Figure 3 is based on a secondary education textbook

Figure 3: Mass spectrometer principle (Dempster)