The main role of the RF (Radiofrecuency) cavities is to keep the 2808 proton bunches tightly bunched to ensure high luminosity at the collision points and hence maximize the number of collisions (luminosity). They also deliver radiofrequency (RF) power to the beam during acceleration to the top energy. Superconducting cavities with small energy losses (synchrotron radiation) and large stored energy are the best solution.

The LHC use eight cavities per beam, each delivering 2 MV (an accelerating field of 5 MV/m) at 400 MHz. The cavities operate at 4.5 K — the LHC magnets use superfluid helium at 1.9 K.

 

 

 

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For the LHC, cavities are grouped in fours cryomodules, with two cryomodules per beam, and installed in a long straight section of the machine where the interbeam distance will be increased from the normal 195 mm to 420 mm.

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Every proton passing thru the RF Cavities is affected for:

2 x 8 MV = 16 MV

So, it receives an extra energy:   16 MeV

Since every proton goes around 11245 laps per second the total energy received per second is:

(16 MeV/lap) x (11245 laps/s)= 1.8·10⁵ MeV/s  or  0.18 TeV/s    (1)

From SPS every proton enters LHC with 0.45 TeV , so the amount of energy that cavities has to provide is:

7 - 0.45 = 6.55 TeV

From (1) we can calculate the length of time required to accelerate the beam to full energy:

6.55 / 0,18 = 36,4 s

The right result is about 20 minutes. The reason for this difference, at least part,  is that protons are not fully affected by the total voltage of the Cavity. So, more time is needed. More important is to keep the bunches compact to increase of the chances of a collision.

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We define the Harmonic Number as a ratio of the particle revolution period to the period of the accelerating RF voltage changes

Harmonic Number = T_particle/T_RF   or   Harmonic Number = T_particle·f_RF

T_particle=LHC_length/c    ⇒   T_particle = 26.659/300000 = 8.9·10^-5 s

 Harmonic Number = (8.9·10^-5)·(400·10⁶)

Harmonic Number ≈ 35600

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For a more complete discussion go here...