The next picture simulates a part of a circular beam of electrons.  Photons are emitted along a tangent to the beam.  Electron synchrotrons can be powerful sources of ultra-violet and X-rays.  The radiation is strongly polarized.  The effect is rather like the sparks from a catherine wheel.

How does this work?  It is actually an originally unwanted side effect in electron synchrotrons.

Because the process electron > electron + photon cannot conserve both momentum and energy, the magnetic field must provide the required components to make the balance.  The electron beam is rather like the current in an electric motor, except that it is not in a wire.

In another type of electron accelerator, the betatron, the magnetic field is changed rapidly.  The electron beam then behaves like the current the secondary winding of a transformer and is accelerated.

The electron synchrotron is a conceptually simple device which was designed to give large amounts of energy to electrons.  It is very hard to give large amounts of energy to a particle in a controlled way, so it has to be done in small steps.  As the electrons very quickly approach the speed of light, the equipment is likely to become very large.  Indeed, an electron  accelerator in California is two miles long.  To keep it small, the accelerator can be bent around into a circle.  One advantage is that the parts are re-used thousands of times as the electrons go round and round.  By kicking the electrons forward at just the right instants, they are given more energy.  This is made easier by the fact that the electrons are going at a virtually constant speed, the speed of light.  To bend the path of the electrons they go through very strong magnets.  Other magnets keep the beam tightly focussed.  

There is a problem.  There always is.  This one is very subtle.  Particles like electrons have some properties which seem very peculiar from our large scale perspective.  One of these is that particles can undergo processes which are rather like chemical reactions - they can split into pieces, or they can collide with others and make new combinations.  In a chemical reaction the materials before and after the reaction contain exactly the same number of atoms of each type . All that happens in a reaction is that they swap partners.  An extreme example occurs in sexual reproduction when strings of DNA join forces to combine of the four bases A, T, C and G in different ways.

But with particles there is a difference - the basic particles present before and after can be quite different, as long as certain rules are obeyed.  Conservation of energy and conservation of momentum are two of these.  Some particles can turn into combinations of others with no external influence.  They are unstable, rather like radioactive substances. Some reactions that could happen in theory are impossible because of energy and momentum conservation.  But they can happen in an evanescent way, rather like the evanescent waves described in Interference.  

The particle behaves in a sense as if it were a part of the time a composite system.  An electron is accompanied by virtual photons.  If a source of energy and momentum can be supplied, some virtual photons can be given reality.  And this is what prevents a circular electron accelerator surpassing a certain energy without unacceptable loss of energy.  as its path curves in the magnetic field of the accelerator, it is as if some of the virtual photons continue straight on, like the drops of water coming off a spun umbrella.  The electron loses energy for each photon shaken off, while the magnetic field takes up or supplies the small energy and momentum to maintain conservation.  This is the reason why electron synchrotrons are often used as light sources and not accelerators.  There is even a bonus - the photons are strongly polarised, a very desirable feature for use in analyzing materials bombarded by the photon beam.

Long before the synchrotron was understood, a closely related phenomenon was discovered by Röntgen in the form of X-rays.  His beam of cathode rays - electrons - struck the glass wall of his vacuum vessel, and were sharply decelerated.  Rays were emitted, which Röntgen soon found to be capable of penetrating substances which were opaque to visible light.  They had, of course, already penetrated the glass of the cathode ray tube.   Like the steam engine and many other inventions, these X-rays came into use long before they were understood.  Although people soon discovered that X-rays were probably like light, but with shorter wavelengths, a full explanation requires quantum electrodynamics, which was worked out many years later.  A peculiar thing about these thrown off particles is that the electron remains exactly the same.  It is as if a magician could remove thousands of rabbits from the same hat.  Because of the sudden deceleration of the electrons, the radiation was called bremsstrahlung - braking radiation.  The diagram below gives some idea of the mechanism.

This is called a Feynman diagram.  We can imagine time running from left to right.  The lines do not represent the paths of particles - only their existence.  By drawing all the diagrams that are physically possible (the easy bit), and calculating the dynamics of each one (the hard bit), people can calculate very precisely what happens.  What is going on here?  The net result is that a photon exists that was not there before.  The vertex where the photon is created cannot conserve energy and momentum  using real particles, but the other two vertices at the left perform a cunning trick.  The vertical photon carries some energy or momentum to or from the nucleus - just enough, in fact, to allow the electron and the final photon to balance energy and momentum.  Actually these other two vertices cannot balance either, with real particles.  

What appears to happen is that between the first and last vertices the masses of the objects are not the physical ones.  As long as this is unobservable, it does not matter.  They are called virtual particles.  How can they not be observed?  Because Heisenberg's principle states that values of certain pairs of observables must not simultaneously be more precise than a certain quantity.

Electrons are not alone in being able to throw off other objects, photons can do it too.  If a sufficiently energetic photon passes close enough to an atomic nucleus, it can be converted into a pair of electrons, one positive, and one negative.  This process cannot occur in free space, because it cannot simultaneously conserve both energy and momentum.  That is where the spectator nucleus comes in - its recoil does the trick and takes up enough energy and momentum to balance.

Elementary particles can produce the same effect.  If a beam of pions or kaons are directed on to a target, some of them are scattered elastically, that is, no other particles are produced.  At least a part of this process behaves like a diffraction pattern, because the particles have wave properties.  But as in the case of the electrons, there is a splitting effect also.  Both types of particles can also diffract in a way which produces a pair of particles, namely rho meson and a pion from an incoming pion, and K* meson and a kaon from an incoming kaon.  Although a great many different types of particles are known, very few different pairs are produced by this mechanism.  This is because the initial and final states must have all quantum numbers the same.

Even more amazing - near a black hole, the gravitational field near a black hole can generate particles by the same mechanism.  In a sense, a vacuum behaves as if it were partially composed of evanescent particle-anti-particle pairs.  Near a black hole, the gradient of the gravitational field is so strong that if a pair of electrons is created, the energy released by one falling into the black hole is enough to pay for the release of the other.  And so black holes are not so black after all - Stephen Hawking discovered that they not only emit radiation - they do so exactly in accordance with the laws of black body radiation - even their entropy is correct.  Black Holes  This is very satisfying to physicists, who believe in the unity of nature, and in the possibility of universal descriptions.  Michael Faraday in particular was a deep believer in the unity of all things, and contributed to the discovery that electricity and magnetism are closely related.