Oleg Zabluda's blog
Tuesday, July 10, 2012
All (?) cyclic particle colliders, including Tevatron, were colliding protons with antiprotons.
All (?) cyclic particle colliders, including Tevatron, were colliding protons with antiprotons. [1] It provides more available energy, after the law of charge conservation is satisfied (important for lower-energy colliders), and allows using same magnets for both beams.

But LHC collides protons with protons, which necessitates using 2 sets of magnets (~40% of the accelerator cost [2]), only cryo can be partially shared.

The reason is to increase luminosity (лучше хуже да больше), until we learn how to cheaply produce such a shitload of collimated antiprotons (2808 bunches of 115 billion protons each = 0.54 nanogram).

[1] or electrons with positrons, like LEP/LEP2. Ion colliders are different.

[2] 1232 dipole magnets at €1M each = €1.232G. The cost of the whole LHC is €3.1G.


Random LHC calculations:

bunches are 27km/2808=9.6 m apart.
2*2808*115e9*mass of proton=1.1 nanogram
7 Tev / 938 MeV = 7,463 = Lorentz Factor
1.1 nanogram * 7,463*c^2 / 4.2e9 joules=0.18 tons of TNT -stored energy
27 km / 7,463 = 3.6 m (how long LHC is in proton's frame)
27km/2/pi = 4.3 km - radius in Lab or Proton's frame

The ring consists of circular sections with 8.1 Tesla dipole magnets bending and straight sections accelerating. Bending sections radius is:

7,463*(proton mass)*c/(proton charge)/8.1 Tesla =  2.9 km.

7,463^2*c^2/2.9 km=1.76e20 g (proper centripetal acceleration in proton frame)
c^2/2.9 km = 3.1e12 g (centripetal acceleration of proton in lab frame)
For comparison, it's exactly 15x of the neutron star surface - 2e11 g.

3.1e12 g*7,463*(proton mass) =7 Tev/2.9 km=3.9e-10 N (centripetal force on 1 proton in Lab frame)
3.9e-10*115e9 = 4.6kg (force on the bunch)

7,463-1=7,462(Thomas precession is 7462 complete revolutions per orbit or 7462*11245 Hz = 8.4e7 Hz)

(1 mm/16 microns)^2=3,906 (ratio of areas before/after focus at collision)
1/(2808*11245Hz) = 31.6 ns on average. See [3] why the min is 25 ns.

For comparison, for ground-state electron in a hydrogen atom
speed = 1/137 c
Loretz factor = 1.00002664
(c/137)^2/(bohr radius)=9e21 g (centripetal acceleration of electron in lab frame) ~= (proper centripetal acceleration in electron frame)
(electron mass)*9e21 g=8.2 e-9 kg (force on electron)

freq = 1/137^2*(electron mass)/h*c^2 = 6.6e15 Hz
1.00002664-1=2.664e-5 (Thomas precession, revolutions per orbit)
6.6e15*(1.00002664-1)=1.8e11 (Thomas precession per second, freq)

[3] http://lhc-machine-outreach.web.cern.ch/lhc-machine-outreach/collisions.htm
The bunch spacing in the LHC is 25 ns., however, there are bigger gaps (e.g. to allow dump kickers the time to get up etc.). 

A 25 ns. beam  gives us a peak crossing rate of 40 MHz. Because of the gaps we get  an average crossing rate = number of bunches * revolution frequency = 2808 * 11245 = 31.6 MHz.

Proton-proton inelastic collision at 7 TeV is 60 mbrns: 60*e-3*10^-24m^2.
naive calculation of collision rate is

(60e-24*1e-3)/16e-6^2*115e9^2=3,099,609 collision per bunch crossing. The actual number is 19, due to geometry (115e9^2 is wrong). Apparently, only sqrt(19/((60e-24*1e-3)/16e-6^2))=285M protons out of 115G (0.3%) are actually colliding at the crossover point.

Quoting url above:

Inelastic event rate at nominal luminosity therefore 10e34*60*e-24 = 600 million/second per high luminosity experiment - around 19 inelastic events per crossing. I don't really know where 10^34 comes from.


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