Treatment of experimental deficiencies

Up to now an ideal electron beam was assumed, but in experiments a deficient electron beam (tab. 1) affects the photon spectra, especially the collimated ones. A finite beam spot size, characterised by the distribution $w_{\text{\sl BS}}$ of the impact positions $\underline{s}=(s_x,s_y)$ of the electrons on the radiator, has the same effect like a collimator with a fuzzy edge and smears out the collimator cutoff in the photon spectra at x_c (eq. 3b). The primary divergence of the electron beam, described by the distribution $w_{\text{\sl BD}}$, has a similar effect on x_c but causes in addition a variation of the crystal angles with respect to their nominal values $\Omega_0$ changing the intensity due to the dependence of the momentum transfer on these angles. The deflection of the electron is not given by the beam divergence alone but is enhanced because the electron undergoes many small angle scattering processes mainly due to Coulomb interaction with atoms while traversing the radiator (thickness z_R). This distribution is well represented by Molières theory[11], which uses a Gaussian approximation for small angles defined by the variance $\sigma^m_{\text{plane}}(z)$being a function of medium properties and pathlength z, the particle has travelled.

 

Table: experimental deficiencies and their influence on the photon spectra

source	distr.	effect	influence
diamond temperature	-	Debye Waller factor	$I^{\text{coh}}/I^{\text{inc}}$
BS: beam spot size	$w_{\text{\text{\sl BS}}}(\underline{s})$	"fuzzy" collimator	xc
BD: beam divergence	$w_{\text{\text{\sl BD}}}(\underline{p})$	+ variation of $\Omega_0$	xd
MS: multiple scattering	$w_{\text{\text{\sl MS}}}(\underline{m})$	increases BD	xd
ES: beam energy spread	$w_{\text{\text{\sl ES}}}(p)$	smears out peaks	$I^{\text{coh}}$

The experimental photon intensity is a sum over all these effects weighted with the appropriate distributions. Due to the collimation condition (a collimator with radius r_c is situated at distance z_c) the boundary of the integration volume in eq. 4 is topological non-trivial.

 $$\begin{multline} I^{\text{exp}}_c = \frac{1}{z_R} \int_{\text{\sl R}} dz \int_{... ...r_c>\left\vert\underline{r}_\gamma(z_c,\underline{s})\right\vert} \end{multline}$$

Underlined vectors denote the transversal component of the respective unit vectors. This situation is sketched in fig. 2 in order to clarify the relations used to calculate the experimental spectra via eq. 4.

  

Figure: (Left) Sketch of the momenta and vectors used in describing BS, BD and MS (see tab. 1) and the resulting electron divergence ( ED) vector $\underline{e}$. (Right) Angles and vectors, i.e. the reciprocal basis vectors $\underline{b}_i$, in reference to the lab system ($\hat e_i$). See also eqs. A5c and A2.

Two approaches will be presented: (ii) An accurate Monte Carlo method: MCB which permits the study of collimation effects on the photon beam and its polarisation in full dependence of all electron beam deficiencies[12]. (i) An analytical one: ANB which is approximative but very fast for quick first results.