Angular distributions of photoelectrons

The angular distributions of photoelectrons should be calculated using the partial wave analysis for the outgoing electron. Since this method is very time consuming we use approximations of the angular distributions calculated by F. von Sauter [] [] (K shell) and Gavrila [] [] (K and L shells). We use the cross-section which is correct only to zero order in $\alpha Z$ . This approximation foresees zero scattering amplitude in the direction of incident photon while experimentally the cross-section at 0 angle is non-vanishing []. If

$X$

$=$

$1-\beta cos\Theta$

then for K and $L$_I shells we use:

$\{d\sigma$K,LId(cosΘ)}

$\sim$

$\{sin2\Theta X4\}1+\; \{12\}\gamma (\gamma -1)(\gamma -2)$

and for $L$_II and $L$_III shells we have:

$\{d\sigma$LIId(cosΘ)}	$\sim$	$\{(\gamma 2-1)\{12\}\gamma 5(\gamma -1)5\}\{\gamma (3\gamma +1)2\; X4\}-\{\gamma 2(9\gamma 2+30\gamma -7)8\; X3\}.$
$$	$$	$+\{\gamma 3(\gamma 3+6\gamma 2+11\gamma -2)4\; X2\}-\{\gamma 4(\gamma -1)(\gamma +7)8\; X\}$
$$	$$	$+sin2\Theta .\; (\; \{(\gamma +1)X5\}-\{\gamma (\gamma +1)X4\}-\{\gamma 2(3\gamma +1)(\gamma 2-1)X3\})$

$\{d\sigma$LIIId(cosΘ)}	$\sim$	$\{(\gamma 2-1)\{12\}\gamma 5(\gamma -1)5\}-\{\gamma (3\gamma -1)2\; X4\}+\{\gamma 2(3\gamma 2-1)X3\}.$
$$	$$	$\{\gamma 3(\gamma 3-3\gamma 2+2\gamma +1)X3\}-\{\gamma 4(\gamma -2)(\gamma -1)2\; X\}$
$$	$$	$+sin2\Theta .\; (\; \{(\gamma +1)X5\}-\{\gamma (\gamma +1)(3\gamma -1)X4\}+\{\gamma 2(\gamma 2-1)X3\})$

The azimuthal angle distribution is uniform.