The socalled “Beam Lifetime” (ζ) is the time interval after which the intensity of the beam has reached 1/e of its initial value (e is Euler’s number).
We are going to estimate this value by considering any of the more important situations wich can cause decreasing of the number of protons
First we consider the collisions among protons in the interaction points. The total protonproton cross section at 7 TeV is approximately 110 mbarns (milibarns)
(with 1 barn = 10^{24} cm^{2}= 10^{28} m^{2}).
The total collision rate at nominal luminosity is:
N_{event/s} = L·σ _{event}
10^{34} x [(110 x10^{3}) x 10^{24}] ~ 10^{9 }collisions/second
With 2808 bunches:
10^{9}/2808 ⇒ 3.6·10^{5} collisions per bunch and second.
We can consider these collisions as a proton “decay” process with a probability, λ :
Since initially the total number of protons per bunch is N_{p0 }~ 1.15·10^{11} protons:
λ = 3.6·0^{5}/1.15·10^{11} ⇒ λ = 3·10^{6} s^{1}
This value represents the probability that one proton collide with an opposing proton per second.
So, the variation of the number of protons is given by:
dN_{p}/dt = – λ × N_{p}
So,
N_{p(t)} = N_{p0} × e^{}^{λ× t}
where N_{p}(t) is the number of protons per bunch as a function of time and N_{p0} the initial number of protons per bunch.
If we now solve the last equation with N_{p}_{(t)} /N_{p0} = 1/e and t = ζ we get:
1/e = e^{}^{λ×} ^{ζ}
and finally,
ζ_{ }=1/λ
With λ = 3·10^{6} s^{1} , we obtain:
ζ = 3·10^{5} s (~ 80 h)
So, by considering only protons collisions in the interactions point the Beam Lifetime would be over 80 hours.
A second important process which decreases the number of protons is the the inelastic scattering produced by protongas collisions (see Ideal Gases Equation section). Elastic interactions are not considered because in this process the kinetic energy of the proton is conserved, and magnetic multipoles act to correct these deviations by focussing the bunch.
The main gases are H_{2}, CH_{4}, CO, CO_{2}, H_{2}O and nobles gases, but we will consider that all molecules are H_{2} (In fact, for calculation the rest of molecules "are converted into" H_{2} equivalent, by introducing corrector parameters).
For protonhydrogen nucleus collision (pH_{1}) at 7 TeV, the inelastic scattering crosssection is ~ 40 mb; therefore, for protonhydrogen molecule (pH_{2}) this value is σ ~ 80 mb (crosssection, σ, for protongas collisions represents a hypothetical area which describes the probability of a proton being scattered by a hydrogen molecule. See more ...).
σ ~ 80 mb ⇒ σ ~ 80·10^{3} x 10^{28} ⇒ σ ~ 8·10^{30} m^{2}
For this calculation, we consider the 2.45kmlong LHC arcs, where the gas density (ρ_{m}) has significant importance ( see Ideal Gases Equation section).
In this case ρ_{m} _{ }~ 1.4·10^{15 }molecules/m^{3}
The distance of the eight 2.45kmlong is d = 8 × 2450 ⇒ d ~ 2·10^{4} m
Then, for a single bunch (N_{p} ~ 10^{11} protons) circulating, the number (N_{lap}) of bunchgas interactions per lap over a distance d is:
N_{lap} ~ σ× ρ_{m }× N_{p }× d
Then,
N_{lap} ~ 8·10^{30} × 1.4·10^{15 }_{ }× 1.15·10^{11}_{ }× 2·10^{4}
N_{lap} ~ 26 collisions/lap per bunch
Since bunches do 11245 laps per second (f ), the rate of bunchgas interactions (R_{int}) is:
R_{int} = N_{lap }× f
R_{int} ~ 26 × 11245 ~ 2.9·10^{5} (290 kHz)
So we have around 3·10^{5} lost protons per bunch in a second when beams begin to circulate around the LHC.
We can consider again this situation as a proton “decay” process with a probability, λ :
λ = (2.9·10^{5})/(1.15·10^{11})
λ = 2.5·10^{6} s^{1}
This value represents the probability that any proton collide against any gas molecule per second.
Using again,
ζ_{ }=1/λ
we finally obtain:
ζ_{ }= 4·10^{5} s (~ 110 h)
So, taking only into account protonsgas collisions the Beam Lifetime would be over 110 hours.
By considering the both two process together, the Beam Life time will be:
1/ζ = 1/ζ_{1} + 1/ζ_{2}
1/ζ ~ 1/80 + 1/110
ζ ~ 45 h
Other mechanisms such as limited efficiency of the correction and focalization systems, Coulomb scattering of protons travelling together (Touschek effect included), or travelling in differents bunches when crossing in interaction points, or operating errors, also contribute in decreasing the Beam Lifetime.
Due to this, finally the Beam Lifetime for the LHC is about 10 hours.
