Method

When a muon of energy E moves in the field of an atom of charge Z, it can radiate a -pair ($4th$ order QED process) with differential cross-section []: $\{2\sigma \nu \hspace{1em}\rho \}=\alpha 4\{23\; \pi \}(Z\; \lambda )2\{1-vv\}[\; \phi$_e+ ({m_em_μ})²φ_μ]

where

and

$\alpha$

1/137, fine structure constant;

$\lambda$

$3.8616\; x10-11$ cm, electron Compton wavelength;

v

$k/E$ fraction of energy transferred to the pair;

T

E-M kinetic energy of the muon.

$E\pm$

the energy of the $e\pm$ .

The explicit form of the terms $\phi$_e and $\phi$_μ

can be found in []. The kinematic ranges of $\nu$ and $\rho$ are:

$\{4m$eE}=νmin	$\le$	$\nu$	$\le$	$\nu$max	$=$	$1\; -0.75e\{m$μE}Z1/3
$0=\rho$min	$\le$	$|\rho (\nu )\; |$	$\le$	$\rho$max(ν)	$=$	$[1-\{6m$μ2E2(1-ν)}]1- {4meνE}

where $e\; =\; 2.718...$ .

$E$_c ( PPCUTM in the program) is the energy cut-off; below this energy -pair are treated as continuous energy loss, and above they are explicitly generated and $v$_c= E_c/E . The mean value of the energy lost by the incident muon due to -pair with energy below $E$_c is: $E$_loss^pair(Z,T,E_c) = 2E ∫_νmin^ν_c dν ν∫₀^ρ_max(ν) dρ {²σν ρ}&sp;GeV barn/atom

whereas the total cross-section for the emission of a hard -pair is: $\sigma (Z,T,E$_c) = 2 ∫_νc^ν_maxd ν ∫₀^ρ_max(ν)d ρ {²σ ν ρ}&sp;barn/atom

• Parameterisation of energy loss and total cross-section