Luminosity Calibration

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Bunch population N= nuclei
Colliding bunch pairs nbb=
Number of bunches per beam kb=
Revolution frequency frev= Hz
Beam energy E= GeV
Particle mass m= GeV
Beta* β*= m
Normalised transverse emittance    εN= m
Full crossing angle (xz) α= μrad
Transverse separation (y) d= μm
Bunch length (RMS) σz= m
Process cross section σproc= mb   (10-27 cm2)
Planned run time T= days
Overall run factor H=

Lorentz factor γ =          γ = E / m
Transverse emittance ε = m          ε = εN / γ
Transverve beam size at IP   σx = σy = μm          σxσy = ε β*
Effective area x-size at IP   Σx = μm          Σx2 = 2σx2cos2(α/2) + 2σz2sin2(α/2)
Effective area y-sizeat IP   Σy = μm          Σy2 = 2σy2
Geometric factor S =         S = 21/2 σx / Σx
Separation factor F =         F = exp(-d2 / 2Σy2 )
Luminosity per bunch pair Lbb = Hz/cm2          Lbb = frev N2 cos2(α/2) F / (2πΣxΣy) = frev N2 cos2(α/2) S F / (4πσxσy)
Average number of processes per crossing   μ =          μ = σproc Lbb / frev
Luminosity (all bunches) L = Hz/cm2          L = nbb Lbb
Integrated luminosity Lint = pb-1          Lint = L H T
Rate of processes R = Hz          R = L σproc
Lumi half life (burn-off 1 IP) τ = h          τ = (1 h / 3600 s) (21/2-1) N / (Lbb σproc)
Stored beam energy Estored = MJ          Estored = E N kb

For non-equal beta star and emittance values, the luminosity formula (for zero crossing angle) is Lbb = frev N2 / {2π (εx1β*x1x2β*x2)1/2y1β*y1y2β*y2)1/2}  , where the indices 1,2 are for the beam and x,y for the plane   (   σij = (εijβ*ij)1/2  ).

For the burn-off half-life one must give the value of the total cross section in σproc