Kinematics and cross sections

Using energy and momentum conservation of the bremsstrahlung process a decomposition of the involved momenta with respect to the incident electron momentum in longitudinal q_l and transversal q_t components permits the formulation of kinematical limits for the momentum transfer $\vec{q}\,$:

 $$\begin{align} \frac{\delta_x}{x} \geq & \; q_l \geq \frac{q^2_t}{2E_0} + \delta_x & \text{with} \quad \delta_x = \frac{x}{2E_0(1-x)} \end{align}$$

This momentum transfer range, depending on the electron energy E₀ and relative photon energy x=k/E₀, is referred to as the `pancake' due to its large lateral extension ( $1 \gtrsim q_t \geq 0$). Calculating the bremsstrahlung cross section[3,4] but retaining the photon polarisation $\vec \epsilon$, the following asymptotic term is obtained: $\frac{d\sigma}{d\Omega} \propto \frac{1}{k} \cos^2\phi$. This shows that the cross section drops in first order with 1/k and that the maximal linear polarisation, with $\phi$ being the azimuthal angle of $\vec \epsilon$ with respect to the scattering plane, is found in that plane. When employing an amorphous radiator, the electron scatters off a single atom which means that the momentum transfer $\vec q$ may lie at any point in the pancake leading to an isotropic distribution of $\vec \epsilon$, hence to an unpolarised beam. In contrast to this incoherent contribution an additional process can be observed in crystal radiators due to their regular structure: Whenever the momentum transfer coincides with a reciprocal lattice vector $\vec q = \vec g$the recoil is absorbed by the whole lattice enhancing the yield by coherently adding the contributions of each atom. Also the momentum transfer is fixed in space which therefore leads to a polarised photon beam. Raising the photon energy x increases the coherent contribution and polarisation monotonically as well as q_lwhich depends like q_t on the orientation of the crystal $\Omega=(\theta,\alpha)$(fig. 2 and eq. A2) and leads to a discontinuity at $x_d = \left(1 + 1/(2E_0g_l-g^2_t)\right)^{-1}$ , where a given lattice vector does not satisfy the pancake condition anymore and thus no longer contributes to the coherent bremsstrahlung spectrum. The total cross section of a crystal[5,6] is a sum of coherent and incoherent (including electron-electron) bremsstrahlung where the Debye-Waller factor $f_{\text{Deb}}(q^2) \in [0,1]$, which is a function of temperature and crystal properties, governs the fractioning into coherent and incoherent contributions to the total cross section. After introducing the intensity I per atom in units of $\bar \sigma = \alpha^2 Z^2 = 0.57947$ Z²mb the cross section and polarisation P are expressed as a sum over lattice vectors $\vec g$ of the functions $\Psi_{1,2,3}$ (eq. A1) and read in common notation:

  

In the coherent contribution due to kinematical constraints for a given lattice vector the photon polar angle $U=E_0\vartheta_\gamma$ is a function of photon energy (eq. 3a), apart from a negligible dependence on the azimuthal angle[4]. Collimating the photon beam enhances the ratio of coherent to incoherent bremsstrahlung, thus increasing the degree of polarisation. Whereas the incoherent cross section is reduced approximately by f_c = u_c/(1+u_c) using a collimation angle of U_c (note: u=U²), the coherent one stays unaffected in the energy range $x(\vec{g}\,) \in \left[x_c,x_d\right]$ but vanishes elsewhere:

  

For the incoherent contribution instead of using the Bethe-Heitler cross section (eq. 3BSb in ref.[7]) the Hubbell[8] cross section derived from the Schiff cross section (eq. 3BSe), which has a more accurate dependence on photon energy, radiator charge Z and collimation angle u_cwas employed (see eq. A3 and fig. 1).

  

Figure: Comparison of Z, photon energy and collimator angle U_c dependence of Bethe Heitler and Hubbell intensity. In the right panel the intensity was normalised to unity whereas in the insert the intensity was set to one at the origin (I(0)=1) for the sake of comparison.

For a calculation of the incoherent contribution from a crystal radiator compared to an amorphous one, the Debye Waller factor $f_{\text{Deb}}$ has to be taken into account, which leads to a modified form-factor: $(1-F_{\text{Deb}})^2=(1-f_{\text{Deb}})(1-F_r)^2$ denoted by $F_{\text{Deb}}$. Here F_r means a realistic carbon form-factor from a Hartree Fock calculation[9]. Due to the fact that an analytical integration of the Schiff cross section as performed by Hubbell for the dipole form-factor with screening constant C (eq. A3) seems no longer feasible, two approximate treatments were investigated[4]. The use of an effective screening constant seemed superior because in addition the temperature dependence of $f^{\text{Deb}}(T)$ is easier implemented there. The screening constant $C_{\text{eff}}$ for the Hubbell cross section was determined via a fitting procedure for the total intensity $\int d\Gamma \sigma_{\text{BH}}(1-F_{\text{Deb}})^2 = \int d\Gamma \sigma_{\text{Hub}}(C_{\text{eff}})$with $d\Gamma=dx d\vartheta_\gamma d\vartheta_e d\phi_{e\gamma}$, resulting in $C_{\text{eff}}=91$ for the amorphous and 33 for the incoherent intensity.

Also an improved description[10] of the electron-electron bremsstrahlung I^ewas used, which takes into account the binding energy of the radiator electrons and which has a non trivial Z and photon energy dependence, see (eq. A4). For this contribution the asymptotic Bethe-Heitler angular distribution f(U) = U/(1+U²)², which leads to the collimation reduction factor f_c (s. above) was adopted.