Luminosity

Taking a closer look at LHC

Luminosity (L) is one of the most important parameters of an accelerator.

It´s a measurement of the number of collisions that can be produced in a detector per cm2 and per second. The bigger is the value of L, the bigger is the number of collisions. To calculate the number of collisions we need also to consider the cross section.
 

L can be obtained semiqualitatively from:

     N2 : number of protons, because each particle in a bunch might collide with anyone from the bunch approaching head on.

     t  :  time between bunches.

     - Seff :  section effective of collision that depends on the cross section of the bunch(“effective” because the beam profile doesn’t have a sharp edge); the formula for this is given by : Seff =4·π·σ2 with σ=16 microns or 16·10-4 cm (transversal size of the bunch at Interaction Point).

Other parameter to be considered is F, the geometric luminosity reduction factor (≤ 1), due to the crossing angle at the interaction point (IP). But in 2011 F ~ 0.95 , so it can be taken as 1.

So we get:

   L ~  N2/(t·Seff )  

 Now, with    N2 = (1,15·1011)2

t = 25·10-9 s  ,  Seff =4·π(16·10-4)2 cm2

~   1034 cm-2·s-1


If we use the bunches crossing frecuency (fin this case 40·106) and Seff = 4·π·σ2,  we can express the Luminosity in a more well-known way:

 

   ~ f·N2 /(4·π·σ2)  

 And considering different number of protons per crossing bunches, and x and y components for σ separately: 

  L =  f· N1N2 /(4πσx σy)   

 We can also express the Luminosity in terms of ε (emittance) and βeta (amplitude function)as: 

  L = f·N2/ (4·ε·β*)   (see here)

 

This value, 1034 cm-2· s-1 , means that in the LHC detectors might produce 1034 collisions per second and per cm2.

Since in the LHC the value of L is 100 times greater than that of LEP or Tevatron makes CERN a leader in this field.


After the LHC will have operated for some years at nominal parameters, it will be necessary to upgrade it for significantly higher luminosity. The most direct way of increasing luminosity isto focus the beam more tightly at the collision point (reduce Seff , or more especifically the so-called β* parameter) which calls for a redesign of the machine optics in the Interaction Regions (IR) and a replacement of the final-focusing quadrupole magnets. The time scale forreplacing IR magnets is in part determined by the lifetime of the present magnets under high radiation doses. It can be estimated as being around 2015. The need for restructuring the injector chain will be assessed at the time of commissioning and can be envisioned as on the 2020 horizon.

Other options can be considered to raise the LHC luminosity, such as increasing the number of bunches or increasing the number of protons per bunch. However, there are limitations on how far these parameters can be pushed, such as the beam-beam limit and the long-range beam-beam interactions, the electron cloud effects, the implication on collimations and machine protection, pile up of events in the experiments and so on.

To see the relation between L and transverse emittanceε , and the amplitude functionβ, go to Beta and Emittance Section.


The integral of the delivered luminosity over time is called integrated luminosity. It is a measurement of the collected data size, and it is an important value to characterize the performance of an accelerator.

Usually, it is expressed in inverse of cross section (i.e. 1/nb or nb-1 - nanobarn-1 ;  1/pb or pb-1- picobarn-1 ; 1/fb or  1fb-1  - femtobarn-1).

 

The next image shows the Integrated luminosity (nb-1) delivered to the LHC experiments  (3.5 TeV proton energy) through July 14, 2010. (Image copyright CERN).


The integrated luminosity recorded by the ATLAS and CMS experiments in 2010 was around 45 inverse picobarns, which translates to over 3000 billion collisions recorded.

And only in the first month of operation in 2011, the LHC had already accumulated an integrated luminosity of 28 pb-1, which corresponds to over 50% of the total delivered to the experiments in 2010.

In the first seven months of 2012, the LHC had delivered more than twice as many collisions to the ATLAS and CMS experiments this year as it did in all of 2011. On 4 August, the integrated luminosity recorded by each of the experiments passed the 10 fb–1 mark. Last year, they each recorded data corresponding to around 5.6 fb–1. On 22 August this year, the more specialized LHCb experiment passed 1.11 fb–1, the same as its entire data sample for 2011.

The LHC is well on its way towards its goal of delivering in the order of 15 fb–1 in 2012. Indeed, at the beginning of September, CMS and ATLAS had already recorded more than 13 fb–1.


In the next graphic below we can see the relation between luminosity and integrated luminosity, considering design parameters and an LHC upgrade.

Taken from Ruggiero F. (2005). “LHC Upgrade (accelerator)”. CERN 8th ICFA Seminar, Daegu, Korea 29/09/2005.

For more information about LHC Luminosity see here.

AUTHORS


Xabier Cid Vidal, PhD in experimental Particle Physics for Santiago University (USC). Research Fellow in experimental Particle Physics at CERN from January 2013 to Decembre 2015. Currently, he is in USC Particle Physics Department (Spanish Postdoctoral Junior Grants Programme).

Ramon Cid Manzano, secondary school Physics Teacher at IES de SAR (Santiago - Spain), and part-time Lecturer (Profesor Asociado) in Faculty of Education at the University of Santiago (Spain). He has a Degree in Physics and in Chemistry, and is PhD for Santiago University (USC).

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