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.