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



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.




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



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

Additional reference

Video reference:




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:


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

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 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,





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 :




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)




Vacuum Electron Devices (VEDs) are microwave and millimeter wave power sources

Their invention lead to the genesis of the engineering branch of electronics.
Their applications include RADAR and defense in general, communications, plasma research and industrial heating.
"Vacuum electron devices (VEDs) are the most powerful and efficient sources of coherent radiation throughout the microwave and millimeter wave bands."


Indicative classification based on


1. Crossed field tubes 
a. Magnetron: "Magnetrons are crossed field devices, with E l B, where E is the electric field and B the magnetic field. Electrons are emitted from the cathode and in-travelling to the anode excite the resonant cavities and give up energy to the RF field. 


2. Linear field tubes
"Linear beam cubes have E || B, and consist of an electron gun, an interaction region where the electron beam is confined magnetically and interacts with a slow wave structure, and a collector."

a. Reflex Klystrons: "In this tube, the electron beam passes once through the resonant cavity and is then reflected back through the interaction gap before being collected on the finely machined RF structure. 
b. Travelling Wave Tubes (TWTs)


3. Fast Wave Tubes
a. Gyrotron: "The gyrotron is based upon the electron cyclotron effect, where an electron in a DC magretic field performs orbits with an angular frequency ω=eB/nγm where e/m is the electron charge/mass, γ the relativistic mass factor, B is the axial magnetic flux density and n is the harmonic number. In the beam, most of the electron energy is required to be transverse to the tube axis, and it is this energy which is converted with high efficiency into radiation. In the gyrotron, angular bunching of the electron beam occurs because of the relativistic mass effect and when correctly tuned, the electrons give up their transverse energy to the TE01 RF field (n=1). 
A major field of application for gyrotrons has been for electron cyclotron resonance heating of plasmas in fusion research machines." 



Vacuum Electronic Device Research Proposal Solicitation by DARPA in 2015


"Those microwaves that heat the food in your microwave oven come from a magnetron, the vacuum tube that made radar possible in the first half of the 20th century. Traveling wave tubes (TWTs), not solid-state amplifiers, generate the strong electromagnetic signals in communication satellites because of their exceptional on-orbit reliability and high power efficiency. And it’s the unique ability of vacuum tube electronic devices to generate high-frequency signals at chip-melting operating powers that makes possible modern aviation radar systems for navigation and collision avoidance. What’s more, there are more than 200,000 vacuum electronic devices (VEDs) now in service in the Department of Defense, powering critical communications and radar systems that cover the land, sea, air, and space.

With its new Innovative Vacuum Electronic Science and Technology (INVEST) program, DARPA aims to develop the science and technology base for new generations of more capable VEDs."


From DARPA*: “While most VEDs in common use today (traveling wave tubes (TWTs), klystrons, crossed-field amplifiers, magnetrons, gyrotrons and others) were invented in the first half of the 20th century, ongoing, intense development efforts have produced dramatic advances in their performance and reliability. Space-qualified TWTs are used for nearly all satellite communications (...)"





Traveling Wave Tube (TWT)


A Traveling Wave Tube (TWT) "is a specialized vacuum tube that is used in electronics to amplify radio frequency (RF) signals in the microwave range." "The radio wave is amplified by absorbing power from a beam of electrons as it passes down the tube."


The RF signal is applied on a helix copper wire inside the tube. A cathode emits electrons and these are focused on a narrow beam at the center of the tube. Due to the structural features of the helix wire, the electric field will not move with the speed of light but with a slower velocity. This electric field will interact with the beam. The positive half cycle will push electrodes forward and therefore accelerate them, while the negative half will push electrodes backward and thereby decelerate them. This will result in accelerated electrons in one segment catching up wih decelerated electrons in the adjoining segment. Due to this, there will be regions of high and low electron concentration or, in other words, electrons will tend to aggregate or bunch, and will travel in bunches. This is termed "bunching" and constitutes the result of velocity modulation linked to current density modulation. The electron beam is structured spatially by the RF input. The resulting pattern of electron density in the beam is an analog of the original RF signal. The continuous beam will be transformed in a beam with bunches which could correspond to a pulsed beam (cf LINACs Fig.32 


The electron bunches give their energy to the helix wire as they repel its the electrons and therefore increase the amplitude of the wave on the helix. The slightly amplified wave causes a denser electron bunch which then amplifies the signal further. The amplification is continuous as the RF wave and the electron beam travel down the length of the tube.


Additional reference:

Relevant video:


FB post




The Magnetron


Transcript from video at link slightly modified - added notes.


Consider a magnetron oscillator composed of a continuous cathode and an anode of three segments. The first and the third segment are connected to the positive (+) pole of an inductor and the second segment is connected to the negative (-) pole of an inductor. We consider that we have a capacitor-inductor setting (LC circuit).


Between the cathode and the anode there will exist a steady DC electric field. Also there will exist an RF field of the LC circuit. Consider an instant when the alternate anode segments have positive and negative values and the field of the RC setting (cf. circuit) is the one shown in Figure 1. 



Figure 1.


Τhe resultant of the combined fields will have different directions in region one and two, as shown in Figure 2.



Figure 2.


"Since electrons tend to move in cycloids of right angles to the direction of the electric field, an electron leaving the cathode in region one would move" as shown in Figure 3. "It would strike an RF electric field in its proper phase relationship and give off energy to the RF field. However, an electron leaving the cathode and entering region two will not be in the proper phase relationships to give up energy to the RF field and will quickly be returned to the cathode. The net result is that more energy is given to the RF field by the electron in region 1 than is taken away from the field by the electron in region 2. In this way two blockers are overcome and oscillation can be maintained." 



Figure 3.


"The energy given up by the electrons to the RF field will change the polarity of those fields so that at the end of one half cycle the RF field will be reversed. Thus the electron which has completed one half cycle in region one will now strike region two in the proper phase relationship to give energy to that RF field (Figure 4). 



Figure 4.


This process will continue until the electron eventually reaches the anode. All electrons which move from the cathode to the anode will follow this course" (Figure 5).



Figure 5.


"Now let's look at a typical magnetron (Figure 6). Imagine the six anode segments of the plane magnetron arranged in a circle around the cathode. The distributed inductance and the inter electrode capacitance form the necessary tank circuit for oscillation in the microwave region. 


With a magnetron oscillating at this frequency, electrons leaving the cathode are governed by the same forces as those in the plane magnetron. The electrons leaving the cathode will move in the usual cycloid, passing two segments per cycle of the oscillator, until they finally reach the anode. By using a segmented anode the electrons are made to work against an alternating electric field crosswise to the steady field. In this mode, the RF field gives the electron a finite transit time which is two anode segments per cycle." 



Figure 6.


"In actual practice, large numbers of electrons will leave the cathode at the same instant. Those entering the RF field out of phase of course will be returned immediately to the cathode. But those entering in phase will follow the usual orbit until they strike the anode. Thus, there will be bunches or clouds of electrons in the phase region and very few in the outer phase regions (Figure 7). And the cloud will revolve in the same direction as the individual electrons always approaching a RF negative segment of the anode. In this way, the kinetic energy which the electrons obtain from the DC potential is given up to the RF field and oscillations are sustained."



Figure 7.



Please refer to section starting at for introductory knowledge or background.


(Note: Klystron




"How a Microwave Oven Works" - The Magnetron "The real engineering in the microwave oven lies in creating the magnetron that generates high-powered radio waves. It's truly an amazing and revolutionary device. The vacuum tube is inside here. These are cooling fins, thin pieces of metal that dissipate the heat as the magnetron operates. The key parts are these two magnets and the vacuum tube." "You apply a large voltage across both the inner filament and the circular copper outside. This voltage boils electrons off the center filament and they fly toward the circular copper section. (...) The magnets bend these electrons so that they swing back towards the center filament. We adjust the magnetic strength so that the now orbiting electrons just brush past the opening of these cavities, like blowing over a half-filled pop bottle to make it whistle. This creates an oscillating wave, the microwave radiation that heats food."




"How Microwaves Work" (Maglab) - The Magnetron

"The copper rod is a cathode or negatively charged electrode. The copper ring is an anode, a positively charged electrode. An electric field exists between these two electrodes. When you turn the microwave on, a current runs through the filament. Negative charged electrons travel through it, generating a lot of heat. Eventually the filament gets so hot that electrons boil off the surface, so to speak, because there is a vacuum between the cathode copper rod and the anode copper ring.
Those boiled off electrons are easily propelled by the electric field attracted to the positive anode. This is where the magnet come in. The north end of one of the magnets faces the south end of the second so the copper ring and rod are inside the magnetic field created between those two magnets. This magnetic field bends the path of the electrons moving from the filament to the copper ring. Called the Lorentz force, this force causes those electrons to spiral as they leave the filament forming a sort of pinwheel in the vacuum.
The tips of this pinwheel, and a negative charge concentrated there, pass over grooves cut into the inner surface of the copper ring at regular intervals. As the negative charge comes and goes across the mouth of each groove, it creates electromagnetic waves inside that groove of a very particular frequency, a microwave. An antenna directs those microwaves into a waveguide which then delivers them to the chamber where the food is."

Cavity Resonators

The Feynman Lectures Vol.II Ch.23

A capacitor with circular plates at AC voltage of high frequency develops a changing magnetic field with a strength that is dependent on the radius from the center axis. This magnetic field creates a changing electric field. These dependencies necessitate a series of corrections for the magnetic and electric field which result in a solution which includes an infinite series and a function known as the Bessel function.


Based on these calculations for a capacitor with circular plates we can calculate the electric and magnetic field inside a closed can which can be considered to constitute a resonant cavity.


As expected there is a dependence of the electric and magnetic fields on the radius from the center.

Other notions (FB post):
A resistor has some inductance and capacitance.
A coil as an inductance in series with some resistance and there is also capacitance between the turns.

Resonant Cavities and Waveguides 




Microwave oven: an electromagnetic cavity resonator or a radiofrequency (RF) cavity where standing waves are generated "A cavity resonator is a hollow closed conductor such as a metal box or a cavity within a metal block, containing electromagnetic waves (radio waves) reflecting back and forth between the cavity's walls. When a source of radio waves at one of the cavity's resonant frequencies is applied, the oppositely-moving waves form standing waves, and the cavity stores electromagnetic energy.
"Since the cavity's lowest resonant frequency, the fundamental frequency, is that at which the width of the cavity is equal to a half-wavelength (λ/2), cavity resonators are only used at microwave frequencies and above, where wavelengths are short enough that the cavity is conveniently small in size."
Determination of the resonant frequency and the standing wave pattern in a microwave oven
"The ovens metal walls only reflect waves of a length that fits inside the oven."
The reflections result in the generation of a standing wave.
"This standing wave causes hot and cold spots inside the oven. The three-dimensional pattern of waves it's difficult to predict but the principle can be seen by looking at the waves in a single dimension. The peaks and valleys in the wave represent the greatest energy of the wave, while the nodes here correspond to the cold spots inside the chamber. If I measure the distance between melted cheese spots, I find about two and a half inches. That would be half the wavelength, the distance between nodes and is pretty close to the actual wavelength of microwave radiation used. Using that wavelength I can estimate the microwave radiation frequency."

Determination of the electromagnetic resonance frequency of a cavity using Finite-Difference Time-Domain (FDTD) modelling

Example 1 - 2D parallel plate resonator
Example 2 - 3D air filled cavity resonator
Note: Sampling Biorthogonal Time-Domain (SBTD) modelling is also used