Luminosity Calculator |
Bunch population, number of bunches and colliding bunches, emittance, beta* and the corresponding crossing angles in the experiments are taken from the talk of Run 3 Configuration WG report. The process cross section of 80mb is the agreed value established by the LPC to allow for comparision of pileup values.
The separations in ALICE and LHCb have been chosen such that the interaction rate in ALICE is close to 500 kHz and that the luminosity in LHCb is close to 2E34.
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Lorentz factor | γ | = | γ = E / m | |
Transverse emittance | ε | = | m | ε = ε_{N} / γ |
Transverve beam size at IP | σ_{x} = σ_{y} | = | μm | σ_{x} = σ_{y} = (ε β*)^{1/2} |
Effective area x-size at IP | Σ_{x} | = | μm | Σ_{x}^{2} = 2σ_{x}^{2}cos^{2}(α/2) + 2σ_{z}^{2}sin^{2}(α/2) |
Effective area y-size at IP | Σ_{y} | = | μm | Σ_{y}^{2} = 2σ_{y}^{2} |
Geometric factor | S | = | S = 2^{1/2} σ_{x} / Σ_{x} | |
Separation factor | F | = | F = exp(-d^{2} / 2Σ_{y}^{2} ) | |
Luminosity per bunch pair | L_{bb} | = | Hz/cm^{2} | L_{bb} = f_{rev} N^{2} cos^{2}(α/2) F / (2πΣ_{x}Σ_{y}) = f_{rev} N^{2} cos^{2}(α/2) S F / (4πσ_{x}σ_{y}) |
Avg interactions per crossing | μ | = | μ = σ_{proc} L_{bb} / f_{rev} | |
Luminosity (all bunches) | L | = | Hz/cm^{2} | L = n_{bb} L_{bb} |
Integrated luminosity | L_{int} | = | pb^{-1} | L_{int} = L H T |
Rate of processes | R | = | Hz | R = L σ_{proc} |
Lumi half life (burn-off 1 IP) | τ | = | h | τ = (1 h / 3600 s) (2^{1/2}-1) N / (L_{bb} σ_{proc}) |
Stored beam energy | E_{stored} | = | MJ | E_{stored} = E N k_{b} |
Transverse beam spot | σ_{xy}^{trans} | = | μm | σ_{xy}^{trans} = σ_{x} / 2^{1/2} (correct for head-on colliding bunches) |
Longitudinal beam spot | σ_{z}^{long} | = | cm | σ_{z}^{long} = σ_{z} / 2^{1/2} * S (correct for head-on colliding bunches) |
For non-equal beta star and emittance values, the luminosity formula (for zero crossing angle) is L_{bb} = f_{rev} N^{2} / {2π (ε_{x1}β*_{x1}+ε_{x2}β*_{x2})^{1/2}(ε_{y1}β*_{y1}+ε_{y2}β*_{y2})^{1/2}} , where the indices 1,2 are for the beam and x,y for the plane ( σ_{ij} = (ε_{ij}β*_{ij})^{1/2} ).
For the burn-off half-life one must give the value of the total cross section in σ_{proc}