Analytical calculation: ANB

The following approximations were used to derive an analytical function from the complicated intensity expression (eq. 4): (i) All two dimensional transversal distributions are assumed to be Gaussian and approximated by azimuthal symmetrical Gaussian. (ii) A mean MS variance is used, averaged over the crystal. (iii) A combined total electron divergence (ED) distribution $w_{\text{\text{\sl ED}}}$ from a folding of the MS and BD distributions is used instead. (iv) The variation of g_t, being in second order of $\delta\Omega$ and therefore much smaller than the variation in g_l, is neglected in the intensity. With that the uncollimated coherent intensity (eq. 4) in terms of the variation lof g_l due to ED is expressed as follows:

  

The sole dependence of the intensity on l allows the conversion of the two dimensional integral to an integral over l in consideration of the kinematical constraint of the pancake. Eq. 5a has to be calculated numerically but for small BD ( $\sigma_e \lessapprox 10^{-2}$) the integral may be solved analytically and will be presented in a forthcoming paper. A collimation function is derived in the following, which includes experimental deficiencies. In the electron coordinate system the circular collimator is characterised by a lateral displacement distribution $w_c(\delta r_c=\delta \rho z_c/E_0,\varphi_c)$ of its centre. Due to the azimuthal asymmetry, only the integral over the displacement $\delta \rho$ has to be calculated. The collimator angle is now a function of the displacement and azimuthal photon angle $U(\rho,\phi)$, which results in a convolution of the intensity[4]:

 $$\begin{align} I^{\text{coh}}_c &= \int \rho d\rho \, d\phi \, w_c(\rho) \int_0... ...\phi)} du \, I^{\text{coh}}_{\text{\text{\sl ED}}}(x) \delta(u-u(x)) \end{align}$$

Owing to the delta function the $\phi$ integration is trivial and the expression separates into the coherent intensity in terms of the $\Psi_i$ functions and a collimation function $I^{\text{coh}}_c = \sum_{\vec g} I^{\text{coh}}_{\text{\text{\sl ED}}}(x,\vec g) C(U(x))$. The treatment of collimation in the incoherent case works analogous, but the different angular dependence leads to a remaining integral (note: v=1/(1+U_c²):

 $$\begin{align} I^{\text{inc}}_c &= \int dv \; c(v) I^{\text{inc}}(v) & \text{with} \quad c(v) &= -\frac{1}{2v\sqrt{v-v^2}}\frac{dC(U)}{dU} \end{align}$$

Therefore, a single collimation function accounts for experimental deficiencies in both cases of coherent and incoherent bremsstrahlung production. After these derivations, C(U) and c(v) have to be calculated numerically only once and the remaining evaluation of the intensities is a closed analytical calculation providing very fast results.