Parameterisation of energy loss and total cross-section

The cross-sections from [] have been used to calculate data points for () and (). These have been parameterised as:

$\sigma (Z,T,k$c)	$=$	$Z[Z+\; \xi$σ(1+ γlnZ)][ln{kmaxkc}]αFσ(Z,X,Y)	$in\; barn$
$E$lossbrem(Z,T,kc)	$=$	$Z[Z+\xi$l(1+δlnZ)][ln{kcE}]βFl(Z,X,Y)	$in\; GeV\; barn$

where $k$_max is the maximum possible value of the photon energy. The functions $F$_i(Z,X,Y) ($i=\sigma ,l$ ) are polynomials:

$F$i(Z,X,Y)	$=$	$(C$1+C2X+...+C6X5)+(C7+C8X+...+C12X5)Y
	$+$	$...+\; (C$31+C32X+...+C36X5)Y5
	$+$	$Z[(C$37+C38X+...+C40X3)+(C41+C42X+ ...+C44X3)Y
	$+$	$...+\; (C$48+C49X+...+C52X3)Y3)]

A least-squares fit has been performed on more than 2000 points for Z = 1, 6, 13, 26, 50, 82, 92 and 1 GeV $\le T\; \le$ 10 TeV and 10 keV $\le k$_c≤T . The resulting values of $\xi$_σ,l , $\gamma$ , $\alpha$ ,$C$_i , $\xi$_l , $\delta$ and $\beta$

can be found in the DATA statements within the functions GBRSGM and GBRELM which compute formulae () and () respectively.

The accuracy of the fit can be estimated as:

The contribution of the bremsstrahlung to the total energy loss of the muons is less than 1% for $T\; \le$ 5 GeV.

When $k$_c≥k_max , a parameterisation different from () can be used for the total muon energy loss due to bremsstrahlung:

$E$lossbrem(Z,T)	$=$	$E$lossbrem(Z,T,k=kmax)
	$=$	$Z(Z+1)\; k$max[d1+(d2X+d3Y) + (d4X2+d5XY+d6Y2)
	$+$	$...+(d$22X6+d23X5Y+...+d28Y6)]

where $Y=Z1/3$ . The accuracy of the formula () for $1\; \le Z\; \le 100$

is rather good:

$\{\Delta E$_loss^bremE_loss^brem}=

$\le 1.5$	$if$	$T\; >\; 1\; \hspace{1em}GeV$
$\le 1$	$if$	$T\; >\; 5\; \hspace{1em}GeV$

$.$

PHYS441