Acceleration with alternating electric and magnetic fields




Electromagnetic waves

Electromagnetic(EM) waves consist of oscillating electric and magnetic fields. The electric and magnetic field components of EM waves are mutually perpendicular, and also perpendicular to the direction of propagation. So, EM waves are transverse. Electromagentic waves are generated by accelerating charges, and are most important to us. EM waves are prduced by the stars(sun), and can also be man-made. Light(what we see) is electromagnetic radiation, as are radio waves(for example). EM radiation spans a huge frequency range, and it is frequency which determines the type of wave. For example, the visible region of EM radiation(light) ranges in wavelength from about 430 nanometers to 690 nanometers. This corresponds to frequencies on the order of 1014 - 1015 cycles per second(Hz).




Broadcast Frequency Bands

Mankind generates and detects EM waves for many purposes. We transmit and receive radio, TV, and cell phone signals, for example. We separate these signals by frequency, and have defined frquency bands as shown in the table below.


 Frequency Range

 Broadcast Band

Human Hearing:    20 Hz - 20 KHz VLF (Very Low Frequency) 
3 Hz - 30 KHz
  LF (Low Frequency)
30 - 300 KHz
AM Radio:    5 - 1.5 MHz MF (Medium Frequency)
300 KHz - 3 MHz
FM Radio:    88 - 108 MHz VHF (Very High Frequency)
30 - 300 MHz
TV:    55 - 890 MHz UHF (Ultra High Frequency)
300 MHz - 3000 MHz

Cell Phones:    800 - 900 MHz 
                     1800 - 1900MHz

 
Blue tooth, Microwave ovens, Phones:  2.4 GHz  
Satellite TV:     10.7 - 12.5 GHz SHF (Super High Frequency)
3 - 30 GHz
Police Radar :   2 GHz EHF ( Extra High Frequency)
30 - 300 GHz


Particle Acceleration

Particle accelerators typically use alternating electric fields rather than static electric fields to accomplish the acceleration. While DC acceleration(static field) is still used for some applications, limitations in the techonology often cause alternating fields to be employed. Acceleration using time-varying electromagnetic fields is called RF(Radio Frequency) acceleration, and the technology used to do this acceleration is loosely referred to as RF. The term RF is not meant to imply that particle acceleration is is accomplished with a radio, but rather that the operating frequency of acceleration components often lies within the radio frequency range of the electromagnetic spectrum. Machines accelerating electrons rather than protons often have operation frequencies in the radar range. However, accelerating systems are now always referred to as RF systems regardless of their actual operating frequency. The operating frequencies of accelerating systems in the Fermilab machines are given in the table below. In some machines, the frequency changes as the particle energy increases. In these cases the low frequency end of the range corresponds to the operating frequency at the initial particle energy, and the high frequency end corresponds to the operating frequency at the final particle energy.

 

              Accelerator    Frequency Range ( MHz)
          Low Energy Linac    201.24 MHz    
          High Energy Linac    805 MHz
          Booster    37.8 - 52.8 MHz
          Main Injector    52.8 - 53.1 MHz
          Tevatron    53.103 - 53.104 MHz
 

Direction

A beam is accelerated in the direction of beam propagation. The direction of beam propagation is called the longitudinal direction. The two directions contained in a cross-sectional plane perpendicular to the direction of propagation are called the transverse directions. A cartoon of the directional definitions for motion along a straight line, as in a linear accelerator, is shown below:

When the motion of the beam has curvature, the direction of beam propagation at any point along the curve is tangent to the line of curvature at that point. The tangential direction is the longitudinal direction, while the transverse directions lie in a plane perpendicular to this tangent line, as shown below.

 

Why not use magnets (instead of electric fields) to accelerate the beam? The force that the electric charge (beam) feels due to a magnetic field is always perpendicular to path the charge follows. Since the magnetic force always acts at right angles to the motion of a charge, it can only turn the charge, it cannot do work on the charge. So, electric fields, which can be oriented to act parallel to the motion of the particles, are used to accelerate particles. Although a particle accelerator complex often has many magnets, these are used not to increase the beam energy, but to control the direction of motion of the particles, pointing or focusing the beam.

Explaining this with a simple analogy, imagine the chair in which you are sitting. You want to roll across the room. The chair is pushing you with an upward force, while you are naturally pushing against it with a downward force. This upward force is perpendicular to the direction you want to go. In order to go across the room, you need another force that will push you parallel to the floor, i.e. perpendicular to the normal force of the chair. You still want the have the force of the chair as well, to keep you off the floor/carpet itself (because dragging on the carpet really burns). Now replace yourself with a small particle, replace the chair with a magnetic field, and replace the person nice enough to push you across the room with an electric field. Now you have a simplified acceleration situation, except in the acceleration process, the beam is propelled to travel because of an electric potential difference, or voltage difference, due to an electric field.
Direction

A beam is accelerated in the direction of beam propagation. The direction of beam propagation is called the longitudinal direction. The two directions contained in a cross-sectional plane perpendicular to the direction of propagation are called the transverse directions. A cartoon of the directional definitions for motion along a straight line, as in a linear accelerator, is shown below:

When the motion of the beam has curvature, the direction of beam propagation at any point along the curve is tangent to the line of curvature at that point. The tangential direction is the longitudinal direction, while the transverse directions lie in a plane perpendicular to this tangent line, as shown below.

 

Why not use magnets (instead of electric fields) to accelerate the beam? The force that the electric charge (beam) feels due to a magnetic field is always perpendicular to path the charge follows. Since the magnetic force always acts at right angles to the motion of a charge, it can only turn the charge, it cannot do work on the charge. So, electric fields, which can be oriented to act parallel to the motion of the particles, are used to accelerate particles. Although a particle accelerator complex often has many magnets, these are used not to increase the beam energy, but to control the direction of motion of the particles, pointing or focusing the beam.

Explaining this with a simple analogy, imagine the chair in which you are sitting. You want to roll across the room. The chair is pushing you with an upward force, while you are naturally pushing against it with a downward force. This upward force is perpendicular to the direction you want to go. In order to go across the room, you need another force that will push you parallel to the floor, i.e. perpendicular to the normal force of the chair. You still want the have the force of the chair as well, to keep you off the floor/carpet itself (because dragging on the carpet really burns). Now replace yourself with a small particle, replace the chair with a magnetic field, and replace the person nice enough to push you across the room with an electric field. Now you have a simplified acceleration situation, except in the acceleration process, the beam is propelled to travel because of an electric potential difference, or voltage difference, due to an electric field.


Standing-wave resonant accelerating structures

At present, particles may be accelerated in either standing-wave or traveling-wave accelerating structures. Resonant cavities, of which the pill-box cavities (described below) are simple models, are used to set up standing electromagnetic waves suitable for particle acceleration. In order to work as an accelerator, the electric field must be along the direction of beam propagation. Typically, the electric field is driven by a sinusoidal voltage, and so the polarity of the electric field will be accelerating only half the time, the rest of the time the field is oriented in such a way as to decelerate the beam.




Thus, when alternating fields are used to accelerate particles, the beam cannot be a continuous stream of particles, for then half the particles would be decelerated instead of accelerated. The beam must consist of bursts of particles with space in between the bursts. A burst of particles timed so as to see the accelerating portion of the field is called a 'bunch' (yep, that's the technical term).



If more than one bunch is being accelerated, the frequency of bunch arrival must match the frequency of the high voltage waveform (often called the RF waveform). When using standing-wave resonant cavities to accelerate, the transit time of a particle through a cavity might be several RF wavelengths long. (RF wavelengths = accelerating voltage wavelengths) In this case, the bunches must be shielded from the decelerating portion of the voltage cycle, a job that can be accomplished by shielding tubes (called drift tubes). Particles within the shielding tubes do not 'see' the fields outside the tubes, the inside of a tube a field-free region.




If the particle velocity increases with every kick from the electric field, then it will cover more distance in the same amount of time as it accelerates, and so the spacing between drift tubes and the length of the tubes must increase as the particle travels through the accelerator.



Courtesy Fermilab Operations Concepts Rookie Book


One way to generate an alternating electric field (not how it is done for an accelerator) would be to drive a large parallel plate capacitor, say with circular plates, with an alternating voltage source. If it were used to accelerate particles, there would have to be a little hole in the plates along the axis so the particles could move through the capacitor always along the direction of the field. Since the electric field is changing along the axis, a magnetic field would be induced azimuthally.


Let's ignore edge effects and examine how the E and B fields change within the capacitor as a function of time. The picture below shows the amplitude variation of E and B with time, and seven specific times are labeled on this graph. Cartoons of E and B within the capacitor plates are shown below the first graph for these seven times.




Notice that the induced B field is 90o out of phase with the applied E field. The stored energy is being exchanged between the electric and magnetic fields. Now to make a pillbox style cavity, make the cylindrical capacitor into a can by connecting the plates with a conducting cylinder. Drive the cavity with a coupling loop, which couples the power from a transmission line into the cavity. The coupling loop most likely will come in through the cylindrical side, while the beam will pass in and out of the cavity through the circular end plates.



Courtesy Fermilab Operations Concepts Rookie Book

     The pillbox cavity behaves like an LRC resonant circuit. Energy is exchanged between the magnetic field and the electric field. The resonant frequency of the cavity depends on its geometry, in particular, the radius of the circular end plates. Although the electric filed of the accelerating mode is uniform along the axis of the cavity, it is not uniform radially, but decreases as the radial position increases. This decrease in the field with radial position occurs also for the cylindrical capacitor at high frequency. There is a great derivation of the form of the field in the 'Feynman Letures on Physics' by Feynman, Leighton and Sands, ISBN 0-201-02117-X. A similar discussion follows. Begin by driving the capacitor or cavity with a sinusoidal field, Eapplied = Eoexp(jwt), pointing along the axis of the cylinder.
Note that  exp(jwt) = cos(jwt) + jsin(jwt), where j º Ö-1, is a convenient representation for a sinusoidal excitation. It is understood that the actual excitation cannot be complex, and that for any actual signal it is the real part of the expression that represents the signal.

     The changing electric field(it is varying sinusoidally) induces a B field, call it B1. The induced B1 field is also changing with time, and so it induces and E field, call it E1 This goes on an on:
Eapplied ® B1
B1 ® E1
E1 ® B2
B2 ® E2
E2 ® B3
B3 ® E3
E3 ® B4
.
.
.


     The total E field at any given moment must be a sum of all E fields, applied an induced (and similarly for B).

              Etotal = Eapplied + E1 + E2 + E3 + ...
Btotal = B1 + B2 + B3 + ...

     It is possible to find expressions for each of the induced fields in terms of the applied field. Use the Ampere-Maxwell law to get the induced magnetic fields, and Faradays's law to get the induced electric fields. The first few terms are calculated this way below.
     Symmetry and the right-hand-rule indicate that B must be azimuthal and constant at a given radius, r.


      Pick an Amperian loop of constant radius with respect to the axis of symmetry and then find B using the Ampere-Maxwell law. Since B is the same anywhere on the loop, it may be taken outside of the loop integral. Similarly, E is constant through the area of the loop, and may be taken outside of the flux integral. Also, moeo = 1/c2, where c is the speed of light. (Plug the numbers in, and you'll see.)



So, the magnetic field induced directly from the applied field is B1 = (jwr / 2c2) X (Eapplied). Notice that B1 a w, so higher frequencies allow more energy to exchange between E and B. If w = 0, we recover the DC case, there is no induced magnetic field.
Next on the agenda, use Faraday's law to find E1 induced from B1. The loop integral for Faraday's law is for the electric field, while the area integral is for the magnetic field. It is wise to choose an area for the flux integral such that B is perpendicular to the surface (parallel to the unit vector specifying the direction of the area). Meanwhile, it is also convenient if E along any given side of the loop is either constant, or perpendicular to that side. A loop such as that shown in the figure below satisfies these conditions. Notice that since E is parallel to the cylinder axis, it is perpendicular to two sides of the loop, and parallel to the other two sides. The magnetic field is perpendicular to the area everywhere.





Longitudinal particle motion with an accelerating field

Each burst of particles that is injected into an RF system has a finite (but not zero) time duration, and a finite (but not zero) spread of energies around some central, ideal energy. Each such grouping of particles is called a 'bunch'. Consider a standing wave RF structure. An ideal particle with just the right energy would travel from accelerating gap to accelerating gap remaining in perfect synchronism with the driving electromagnetic wave. Such a particle is called a 'synchronous particle'. Since the driving wave (RF) varies with time (usually sinusoidally), particles arriving at the RF gap at the correct time will experience a different voltage than particles arriving at a slightly different time. For the cases of particles not yet close to light speed, or else traveling in a circular machine, the dependence of the value of the accelerating voltage on particle arrival time can be used to advantage, providing a restoring force for particles that are not synchronous.



The case of no acceleration

Sometimes it is desirable to maintain the particles of the beam in a bunched structure, but not accelerate. This might be the case, for example, when beam is injected into a machine before the acceleration starts. The synchronous particle should see no voltage at the accelerating gap in order to maintain the same energy. So, the synchronous particle should arrive at the center of the gap at the zero crossing of the RF voltage waveform. The figure below shows qualitatively the different voltages experienced at an accelerating gap by three particles of different energy. The figure is a sketch of the voltage level at an accelerating gap versus time. Time zero (t0=0) corresponds to the correct arrival time for the synchronous particle. The positive time axis corresponds to times later than t0, while the negative time axis has times earlier than t0.


Types of accelerating structures

  Superconducting linac accel


     Courtesy Fermilab Visual Media Services