CERN Program Library Long Writeup W5013 - CERNLIB ...

The functions F i (Z,X,Y) (i = σ, l) have the form:
F i (Z, X, Y )=F i0 (X, Y )+ZF i1 (X, Y ) (7)
where F ij (X, Y ) denotes a function constructed from two polynomials
⎧
⎪⎨
F ij (X, Y )=
⎪⎩
P neg
ij (X, Y ) if Y ≤ 0
P pos
ij (X, Y ) if Y > 0
(8)
and P neg
ij , P pos
ij fulfil the conditions:
P neg
ij (X, Y ) Y =0 = P pos
∂P neg
ij
ij (X, Y ) Y =0 = ∂Ppos
ij
∂Y ∣ Y =0
∂Y ∣ Y =0
The detailed form of the P ij polynomials is:
with
P neg
i0 (X, Y ) = (C 1 + C 2 X + ···+ C 6 X 5 )+(C 7 + C 8 X + ···+ C 12 X 5 )Y
+ (C 13 + C 14 X + ···+ C 18 X 5 )Y 2 + ···+(C 31 + ···+ C 36 X 5 )Y 5
P pos
i0 (X, Y ) = (C 1 + C 2 X + ···+ C 6 X 5 )+(C 7 + C 8 X + ···+ C 12 X 5 )Y
+ (C 37 + C 38 X + ···+ C 42 X 5 )Y 2 + ···+(C 55 + ···+ C 60 X 5 )Y 5 (9)
P neg
i1 (X, Y ) = (C 61 + C 62 X + ···+ C 65 X 4 )+(C 66 + C 67 X + ···+ C 70 X 4 )Y
+ (C 71 + C 72 X + ···+ C 75 X 4 )Y 2 + ···+(C 81 + ···+ C 85 X 4 )Y 4
P pos
i1 (X, Y ) = (C 61 + C 62 X + ···+ C 65 X 4 )+(C 66 + C 67 X + ···+ C 70 X 4 )Y
+ (C 86 + C 87 X + ···+ C 90 X 4 )Y 2 + ···+(C 96 + ···+ C 100 X 4 )Y 4
X =ln E m µ
Y =ln E c
v j E
j = σ, l
where v σ,l are additional parameters of the fit. By a numerical (twofold) integration of the formulae (5)
and (6) above, we have calculated 1800 data points in the range Z =1, 6, 13, 26, 50, 82, 92 and the energy
ranges 1GeV ≤ T ≤ 10 TeV and 4m ≤ E c ≤ T and performed a least-squares fit to determine the
parameters. The fitted values of the parameters are in the DATA statements in the functions GPRSGM and
GPRELM, which compute the formulae (5) and (6) respectively. The accuracy of the fit is:
∆σ
σ
∆E pair
loss
E pair
loss
{
10% if T ≤ 5 GeV
≤
5% if T>5 GeV
{
10% if T ≤ 5 GeV
≤
4% if T>5 GeV
The function GPRELM contains a second formula to calculate the total energy lost by the muon due to direct
e + e − -production used when E c ≥ Emax = E − 0.75 √ em µ Z 1/3 . This formula describes the total energy
loss with an error less than 1%:
E pair
loss
(Z, T ) = Epair
loss (Z, T, E c = E max ) (10)
= Z(Z +1)E max [d 1 +(d 2 X + d 3 Y )+(d 4 X 2 + d 5 XY + d 6 Y 2 )+
+ ···+(d 22 X 6 + d 23 X 5 Y + ···+ d 28 Y 6 )]
where X =ln(E/m µ ) and Y = Z 1/3 . The fitted parameters d i can be found in the DATA statement in the
function GPRELM.
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