acceleration - How many particles can a particle accelerator accelerate at once?

Does a particle accelerator lose its accelerating effectiveness as the number of particles being accelerated increases? According to Wikipedia, the mean acceleration of a proton in the Large Hadron Collider is 190,000,000 g's. Could the LHC accelerate one gram worth of protons at the rate while using the same amount of power? I suspect not, but I don't know why.

Answer

In general the total voltage $V$ seen by a particle (for a 1 TeV proton must be 1 trillion volts), multiplied by the beam current $I$, gives you the beam power: $ P = VI $. So in principle you may have some trade off between the two of them, but often there are other limitations both to $V$ and to $I$. The designs of most of the existing machines have stretched these limits as much as possible (either because of cost or physical/technological problems) so it will not be immediately possible trade $V$ for $I$ or vice versa.

For instance at the LHC the energy is limited by the maximum bending field that the magnets are capable to produce, so even accelerating a single proton, you wont be able to push it beyond the nominal energy (currently 6.5 TeV). On the other hand you cannot arbitrary increase the current as collective behaviours of the beams may lead to instabilities and beam losses. Moreover, due to field quality issues, the LHC cannot even accept particles at an energy lower than 450 GeV.

So basically the number of particle that an accelerator can accept is the design one. Most of the time, at an operating machine, it cannot be easily increased even giving up something else (a counter example is the fully-loaded linac of CTF3). During the design phase you can optimize it, but this may lead to completely different design! Have a look at the (experimental) fusion reactors: those are basically very-low energy and quality accelerators (they just need to overcome the Coulomb barrier and particles scatter everywhere) which however store a huge amount of particles. Their designs do not share much with the high energy accelerators.