Monte Carlo simulation: MCB

Measured electron beam parameters and their standard deviation as well as radiator and collimator properties are the basic input for calculations based on the Monte Carlo technique. Starting from a given number of electrons N_e, depending on the desired statistical accuracy, a certain set of physical values are chosen randomly in parameter space. First the direction of an incident electron $\underline{p}$ with energy E₀ impinging at $\underline{s}$ on the radiator is chosen from the beam energy $w_{\text{\text{\sl ES}}}$ and divergence $w_{\text{\text{\sl BD}}}$ distributions, which are assumed to be of Gaussian shape with known parameters $\sigma_{E_0}$, $\sigma_p^{x,y}$ and $\sigma_s^{x,y}$ respectively. The mean polar angle deviation $\underline{m}(\sigma^m_{\text{plane}}(z))$ from the incident direction depend via Molières theory[11] on the depth z of the bremsstrahl process in the radiator, which is chosen randomly from a homogenous distribution within the radiator thickness z_R. To calculate the coherent bremsstrahlung for this particular electron the lattice has to be rotated into its coordinate system, involving a transformation of the crystal angles $\Omega_0$. The total transversal electron deflection $\underline{e}$ due to multiple scattering and beam divergence and the transformation of the crystal $\hat b_1(\Omega_0)$ axis in the electron system $\hat b_e$ is calculated (eq. A5c and fig. 2). Then a lattice vector is chosen uniformly in reciprocal space $V_{\vec g}$ with the Miller indices h,k,l, the intensity $I^{\text{coh}}(\vec \Sigma)$ is calculated with these parameters $\vec \Sigma = (h,k,l,E_0,z,\underline{s},\underline{m},\underline{p}, \underline{k},x)$ and the photon momentum $\underline{k}'$ is transformed back in the lab system. The resulting cross section is differential in photon energy k and angle, which is the azimuthal ($\psi_k$) in coherent bremsstrahlung and is the polar angle ( $\vartheta_k$) in the incoherent case. As an example the polarisation for a rectangular collimator compared to a circular one, both producing the same tagging efficiency, is shown in fig. 3.

  

Figure 3: Enhancement of polarisation at low energies (right) by use of a rectangular collimator instead of a circular one (left).