# Why Fat Particles Radiate Less

I am just reading Knocking on Heaven’s Door by Lisa Randall which has a chapter on the impressive machinery of Large Hadron Collider. The LHC has been built to smash proton beams against each other: Protons, not electrons. Why protons? I stumbled upon the following statement:

“But accelerated particles radiate, and the lighter they are, the more they do so”.

Electrons would cause higher radiation losses and less energy would be available for the creation of new particles in collisions.

But why is this so? In order to prove this, you would go through a calculation of the electromagnetic field generated by the moving particles based on Maxwell’s equations (which are relativistic per default).

I think you can understand it qualitatively from this chain of reasoning:

If a particle is forced to move on a curved path, it is accelerated – such as planets are accelerated all the time by the gravitational force exerted by the sun.

Consider a curved part of the LHC’s trajectory – the radius is given. The acceleration of particles moving in circles is equal to v2/R with R being the radius of curvature and v the speed of the particle. So acceleration increases with increasing speed.

Charged particles lose energy via electromagnetic radiation when they are accelerated. This can be understood from conservation of energy: If a particle would be slowed down in free space (friction due to collisions with particles in the atmosphere being not an option), the energy has to go somewhere. If a particle is accelerated, some force does work on it (which is also true for an orbiting particle).
This argument has been used to prove the classical model of the atom as a miniature solar system wrong: If an electron would orbit round the core it would lose energy and finally ‘fall down’ into the core. So we need a quantum mechanics to explain the stability of atoms.

Particles are accelerated by electrical fields: the energy transferred to particles of the same electrical charge would be the same for a proton or an electron (except the sign). For particles with velocities close to the speed of light relativistic effects cannot be neglected so the energy of a particle of rest mass m and velocity v is (c = speed of light) $E = \frac{m c^2}{\sqrt{1-\frac{v^2}{c^2}}}$

For smaller velocities this reduces to the sum of the ‘rest mass energy’ mc2 and the kinetic energy mv2/2.

If the energy is given a particle with higher rest mass would exhibit a smaller velocity. Thus its acceleration in a toroidal tube (~v2) would be smaller.

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Further reading and notes

– Note that I prefer to call the ‘rest mass’ just ‘mass’ – I would not introduce a so-called relativistic mass.
– As usual, I am recommending Feynman’s Physics Lectures. Even in Volume 1 he gives a concise introduction on the radiation of charges. Actually he states that he developed an expression for the electrical field caused by a single point charge for the purpose of this lecture only that had not been published elsewhere before.  Volume 2 comprises electrodynamics in depth.

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Reading this again after some years I honestly can’t say anymore if this is too simplified an argument. Now it sounds borderline crackpot to me. Even if the result is right, it may be a coincidence. Maybe, someday I will get back to that and compare each step to the relativistically correct calculation.

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