DC acceleration, Electrostatics

Energy gain of charged particles in electrostatic fields, eletron-volts

An unbound charged particle in a uniform electric field will be accelerated by the electric field and so gain energy. This is analogous to the energy gain of a freely falling mass in a uniform gravitational field. The energy gain of the particle is due to work done by the field on the particle, this work being done by exertion of electrostatic force on the particle. An electrostatic field is conservative (as is a gravitational field), and so as a particle 'falls' through a potential difference, it gains kinetic energy. The energy units of choice at an accelerator facility are electron volts (eV) and multiples of electron volts.

An electron volt (eV) is the amount of work done on a single electron when it moves through a potential difference of one volt. This is just like the more familiar joule (J) except that one uses electron volt when dealing on small scales, like the atomic and subatomic scale where joules would have to be in incredibly tiny amounts. One electron volt is equal to 1.602*10^-19 Joules. You can see how hard it would be to work in joules when discussing subatomic particles.

KeV1,000eV103 eV
MeV 1,000,000 eV 106 eV
GeV 1,000,000,000 eV 109 eV
TeV 1,000,000,000,000 eV 1012 eV

The units eV make explicit the fact that a charge (e is a symbol for the magnitude of charge of an electron) moving through a potential difference (volts) will experience an energy change; Energy [eV] = (charge)(potential difference). The total energy of a free electric charge is given by two contributions, the kinetic energy and the rest energy. All mass, even if not in motion, has an energy just due to the mass itself, given by Einstein's famous relation E=mc2 (m is the mass and c is the speed of light). The rest energy (as it is called) of an electron is .511 MeV and the rest energy of a proton is 938 MeV. The rest energy doesn't change. The kinetic energy is the energy of motion, and increases as a charge is accelerated by an electric field. So, the total energy of a particle in an accelerator is given by Etotal=Ekinetic+Erest where the kinetic energy is increased by application of an electric field directed along the line of particle motion.

  Cockcroft Walton

The first accelerators utilized electrostatic fields. The first stage of acceleration at Fermilab is done with an electrostatic field generated by the Cockroft-Walton accelerator (shown below). There H- particles travel through a steady electric field in an accelerating column with a potential difference of 750 KV from end to end. (An arrow is drawn in the picture along the accelerating column for reference.) The upstream end of the column (where acceleration starts) has a potential of -750 KV, and the downstream end of the column is at ground potential. Charge is transported to the -750 KV terminal electrically, via a voltage multiplying circuit. The high voltage is distributed uniformly along the acceleration tube; each electrode is an equipotential surface, and is perpendicular to the orientation of the uniform electric field. The work done on a particle moving through a uniform electric field is given by W= -q(Vf - Vi). The total charge of an H-, which is one proton and two electrons, is the same as the charge of an electron. So, the work done on the H- ions by the field is then W = -(-e)[ 0 volts - (-750 kilovolts) ] = 750 KeV (see discussion of electron-volts above). The H- particles emerge from the Cockroft-Walton with a kinetic energy gain of 750 KeV.

Cockroft-Walton generators are limited to about 1 MeV before they break down.
        Courtesy Fermilab Visual Media Services

A technical limitation of accelerators having an accelerating field that does not vary with time is that at voltages of a few MeV/meter, air breaks down, allowing the formation of conducting paths to ground. At low electric field values, air is a good insulator; electrons attach to atoms faster than they are knocked loose by collisions. However, when the electric field strength becomes high enough, electrons are knocked loose by collisions faster than they recombine with atoms. When such breakdown occurs, there can be arcing through the air, and the electrodes of the accelerator will discharge. The mechanism for lightning is similar; when enough charge collects in localized regions of clouds, the resulting high fields cause creation of conducting paths to ground via ionization of the atoms in the atmosphere.

       Courtesy: NOAA Photo Library, NOAA Central Library; OAR/ERL/National Severe Storms Laboratory (NSSL)
       Photographer: C.Clark

Notice that the conducting surfaces of the Cockroft-Walton are smooth, avoiding sharp points in favor of larger radiuses of curvature. Charge collects at points, making the electric field near the surface of those places larger than elsewhere. Given the same electric potential, the electrodes with smoother shapes are less susceptible to breakdown.

There is a model demonstrating this explicitly in The Feynman Lectures (ISBN 0-201-02117-X Addison-Wesley Publishing Company), the gist of which is repeated here. Suppose there are two conducting spheres connected by a conducting wire, as shown in the figure below.

The Pelletron

To be complete, there are two large electrostatic accelerators used in the Fermilab accelerating complex.The Cockroft-Walton is the first stage of acceleration for the entire complex of accelerators, taking H- ions up in energy to 750 KeV before they are injected into the Linear accelerator.The Pellatron is a 4.3 MV electrostatic accelerator used to produce a high current beam of electrons (.5 amps in 2005). This electron beam is used for the specific purpose of cooling antiprotons in the Recycler Ring, and is used only in one section of that storage ring. The electron beam is transported from the Pelletron into a section of the Recycler Ring,where it then co-propagates with the antiprotons for about 20 meters, so as to cool the antiprotons. The electron beam then returns to the Pelletron to complete the circulation path.

Electrostatic accelerators provide steady electric fields for acceleration via charged electrodes.Somehow the electrodes must be charged up. Van de Graaf accelerators, one of the earliest types of electrostatic accelerators, charged up the electrode via a moving belt; the belt carried charges and deposited them on the conductor. A Pelletron is an electrostatic accelerator with an improved belt design; it has a moving belt made of metal pellets connected with nylon links to carry the charge.

  Pelletron, one deck of the accelerator's column
        Courtesy Fermilab Visual Media Services
  Pelletron, external view
        Courtesy Fermilab Visual Media Services