SIMPLE LHC LUMINOSITY CALCULATOR
INPUT:
Bunch population
N
=
nuclei
Colliding bunch pairs
n
bb
=
Number of bunches per beam
k
b
=
Revolution frequency
f
rev
=
Hz
Beam energy
E
=
GeV
Particle mass
m
=
GeV
Beta*
β*
=
m
Normalised transverse emittance
ε
N
=
m
Full crossing angle (xz)
α
=
μrad
Transverse separation (y)
d
=
μm
Bunch length (RMS)
σ
z
=
m
Process cross section
σ
proc
=
mb (10
-27
cm
2
)
Planned run time
T
=
days
Overall run factor
H
=
OUTPUT:
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*
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-sizeat 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
)
Average number of processes 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
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