The so-called “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). ag

We are going to estimate this value by considering any of the more important situations wich can cause decreasing of the number of protons

Keep in mind that we will consider the parameters corresponding to the initial start-up of the LHC, so they should be updated, if necessary, with the values corresponding to the specific moment we wish to evaluate.

First we consider the collisions among protons in the interaction points. The total proton-proton cross section at 7 TeV is approximately 110 mbarns (milibarns)

(with 1 barn = 10^-24 cm²= 10^-28 m²).

The total collision rate at nominal luminosity is:

N_event/s = L·σ _event

10³⁴ x [(110 x10^-3) x 10^-24] ~ 10⁹ collisions/second

With 2808 bunches:

10⁹/2808  ⇒  3.6·10⁵ 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 ~ 1.15·10¹¹ protons:

λ  = 3.6·0⁵/1.15·10¹¹   ⇒  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 = 3·10⁵ 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 proton-gas 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 H₂, CH₄, CO, CO₂, H₂O and nobles gases, but we will consider that all molecules are H₂ (In fact, for calculation the rest of molecules "are converted into" H₂equivalent, by introducing corrector parameters).

For proton-hydrogen nucleus collision (p-H₁) at 7 TeV, the inelastic scattering cross-section  is ~ 40 mb; therefore, for proton-hydrogen molecule (p-H₂) this value is σ ~ 80 mb (cross-section,  σ, for proton-gas 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 m²

For this calculation, we consider the 2.45-km-long LHC arcs, where the gas density (ρ_m) has significant importance ( see Ideal Gases Equation section).

In this case  ρ_m  ~ 1.4·10¹⁵ molecules/m³     

The distance of the eight 2.45-km-long is  d = 8 × 2450  ⇒  d ~ 2·10⁴ m

Then, for a single bunch (N_p ~ 10¹¹ protons) circulating, the number (N_lap) of bunch-gas interactions per lap over a distance d  is:

N_lap ~  σ× ρ_m × N_p × d

Then,

N_lap ~ 8·10^-30 × 1.4·10¹⁵  × 1.15·10¹¹ × 2·10⁴

       N_lap ~ 26 collisions/lap  per bunch

Since bunches do 11245 laps per second (f ), the rate of bunch-gas interactions (R_int) is:

R_int = N_lap × f

R_int ~ 26 × 11245 ~ 2.9·10⁵  (290 kHz)

So we have around 3·10⁵ 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 = (2.9·10⁵)/(1.15·10¹¹)

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 = 4·10⁵ s (~ 110 h)

So, taking only into account protons-gas 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/ζ₁ + 1/ζ₂

1/ζ ~ 1/80 + 1/110

ζ ~ 45 h

 A third group of processes is related to 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.

The number of beam protons lost per second due to these processes is approximatente 2,5·10⁶ protons per second and per bunch when beams begin to circulate around the LHC..

We can consider again this situation as a proton “decay” process with a probability, λ :

λ = (2.5·10⁶)/(1.15·10¹¹)

λ = 2.2·10^-5 s^-1

This value represents the probability of losing one proton per second per bunch.

Using again,

ζ =1/λ

we finally obtain:

ζ = 4,5·10⁴ s (~ 13 h)

Taking this third type of processes the Beam Lifetime would be over 13 hours.

If we consider all the processes we have just studied at the same time, the Beam Lifetime will be:

1/ζ = + 1/ζ₁ + 1/ζ₂ + 1/ζ₃

1/ζ ~ 1/80 + 1/110 + 1/13

ζ ~ 10 h

Taking all phenomena into account, the Beam Lifetime for the LHC is finally around 10 hours.

Again, we recall that these calculations have been performed with the nominal starting parameters of the LHC.