Probability of Interaction with a Shell

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Probability of Interaction with a Shell

To calculate the probability of the interaction with a particular shell we use the jump ratios defined as: $J$ _shell= {σ(E_shell+δE)σ(E_shell-δE)}

where $δE →0$ . In addition we assume that the jump ratio is also valid away from the edges.
From () it follows that the probability $p$ _shell

to interact with a shell is: $p$ _shell= 1-{1J_shell}

We use the following parametrisation of the jump ratios for K and $L$ _I shells[]: $J$ _K= {125Z}+ 3.5

$J$ _{L_I}= 1.2

For the $L$ _II and $L$ _III shells we adopt approximation of the formulae calculated by Gavrila []:

$σ$ _{L_II} $=$ $γβ{m$ _eE_γ}γ³-5γ²+24γ-16 -(γ²+3γ-8){log(γ(1+β))γβ}

and

$σ$ _{L_III} $=$ $γβ{m$ _eE_γ}4γ³-6γ²+5γ+3 -(γ²-3γ+4){log(γ(1+β))γβ}

where

$γ, β$ are the emitted photoelectron Lorentz gamma and beta factors;
$E$ _γ is the incident radiation energy;
$m$ _e is the electron mass.

Below an example of the shell interaction probability calculations for $E$ _γ> E_K is given.
If

$Σ$ _II,III $=$ $σ$ _{L_II}+σ_{L_III}

$r$ _{L_II} $=$ ${σ$ _{L_II}Σ_II,III}

$r$ _{L_III} $=$ ${σ$ _{L_III}Σ_II,III}

then from () one can find that

$p$ _K $=$ $1-{1 J$ _K}

$p$ _L₁ $=$ $(1-p$ _K)(1 - {1J_L₁})

$p$ _{L_II} $=$ $(1-p$ _K-p_{L_I})r_{L_II}

$p$ _{L_III} $=$ $(1-p$ _K-p_{L_I})r_{L_III}

After simple calculations one obtains the probability distribution function which is used to select the shell.

Janne Saarela
Mon Apr 3 12:46:29 METDST 1995

$γ, β$	are the emitted photoelectron Lorentz gamma and beta factors;
$E$ _γ	is the incident radiation energy;
$m$ _e	is the electron mass.

$Σ$ _II,III	$=$	$σ$ _{L_II}+σ_{L_III}
$r$ _{L_II}	$=$	${σ$ _{L_II}Σ_II,III}
$r$ _{L_III}	$=$	${σ$ _{L_III}Σ_II,III}

$p$ _K	$=$	$1-{1 J$ _K}
$p$ _L₁	$=$	$(1-p$ _K)(1 - {1J_L₁})
$p$ _{L_II}	$=$	$(1-p$ _K-p_{L_I})r_{L_II}
$p$ _{L_III}	$=$	$(1-p$ _K-p_{L_I})r_{L_III}