Polarization of atomic bremsstrahlung in coincidence studies

We present a theoretical study of bremsstrahlung produced by high-energy electrons scattered by heavy atomic targets. Considering coincident observation of the emitted photons and the scattered electrons, we pay special attention to the polarization degree and direction of the outgoing light. To investigate these properties of atomic bremsstrahlung, we apply the density matrix approach and solutions of the Dirac equation. Detailed calculations are performed for initial electron energies ranging from 100 to 500 keV and different fixed electron scattering angles. The results of these calculations are compared with predictions obtained under the assumption that the scattered electrons remain unobserved. This comparison reveals that both the degree and the direction of linear polarization of bremsstrahlung are very sensitive to the direction of the scattered electron.

Figure 1

Geometry of the radiative scattering of an electron by an atom. The $z$ axis is chosen to be along the asymptotic momentum of the incident electron ${\mathit{p}}_i$. Together with the photon momentum $\mathit{k}$, it defines the $xz$ plane. The angle ${\theta }_k$ is the emission angle of the photon and the angles ${\theta }_p$ and ${\phi }_p$ define the direction of the scattered electron.

Figure 2

Scaled triple-differential bremsstrahlung cross section (7) as a function of the photon emission angle ${\theta }_k$. The calculations have been performed for the scattering of initially unpolarized electrons on neutral silver atoms, for the initial and final electron energies ${\varepsilon }_i=180\mathrm{keV}$ and ${\varepsilon }_f=100\mathrm{keV}$ and the fixed electron scattering angles ${\phi }_p=0^{\circ }$ and ${\theta }_p={30}^{\circ }$. Our predictions (solid line) are compared to the calculations of Tseng [16] (dashed line) and Shaffer et al. [15] (dotted line). The experimental results are from Aehlig and Scheer [10] (solid squares).

Figure 3

Polarization correlation $C_{200}$ as a function of the photon emission angle ${\theta }_k$ for the scattering of 300 keV electrons on a gold target. The outgoing electrons with an energy of ${\varepsilon }_f=200$ keV were detected under the angles ${\phi }_p=0^{\circ }$ and ${\theta }_p={45}^{\circ }$. Our predictions (solid line) are compared with the results by Tseng [16] (dashed line), Haug [27] (dotted line), and the experiment by Mergl et al. [12] (solid squares).

Figure 4

Degree of linear polarization $P_L$ (upper row) and polarization angle $\chi$ (lower row) of bremsstrahlung radiation as a function of the photon emission angle ${\theta }_k$ for three different azimuthal scattering angles ${\phi }_p=0^{\circ }$ (left column), ${\phi }_p={45}^{\circ }$ (center column), and ${\phi }_p={90}^{\circ }$ (right column) of the electron. The calculations were performed for initially unpolarized electrons, $Z=79$, an initial electron energy ${\varepsilon }_i=100\mathrm{keV}$, a final electron energy ${\varepsilon }_f=1\mathrm{keV}$, and polar electron scattering angles of ${\theta }_p={180}^{\circ }$ (solid line), ${\theta }_p={135}^{\circ }$ (dashed line), and ${\theta }_p={90}^{\circ }$ (dash-dotted line). For comparison, we also present the results obtained for the scenarios where the scattered electron is not observed (dotted line).

Figure 5

Degree of linear polarization $P_L$ (upper row) and polarization angle $\chi$ (lower row) of bremsstrahlung radiation as a function of the photon emission angle ${\theta }_k$ for three different azimuthal scattering angles ${\phi }_p=0^{\circ }$ (left column), ${\phi }_p={45}^{\circ }$ (center column), and ${\phi }_p={90}^{\circ }$ (right column) of the electron. The calculations were performed for electrons polarized perpendicularly within the emission plane [${\mathit{P}}_e=(1,0,0)$], $Z=79$, an initial electron energy ${\varepsilon }_i=100\mathrm{keV}$, a final electron energy ${\varepsilon }_f=1\mathrm{keV}$, and polar electron scattering angles of ${\theta }_p={180}^{\circ }$ (solid line), ${\theta }_p={135}^{\circ }$ (dashed line), and ${\theta }_p={90}^{\circ }$ (dash-dotted line). For comparison, we also present the results obtained for the scenarios where the scattered electron is not observed (dotted line).

Figure 6

Polarization angle $\chi$ for different photon energies. The calculations were performed for ${\mathit{P}}_e=(1,0,0),Z=79$, and initial electron energy ${\varepsilon }_i=100\mathrm{keV}$, photon scattering angle ${\theta }_k={130}^{\circ }$, and electron scattering angles of ${\theta }_p={30}^{\circ }$ (solid circles, solid line), ${\theta }_p={60}^{\circ }$ (solid squares, dashed line), and ${\theta }_p={90}^{\circ }$ (solid triangles, dash-dotted line) with ${\phi }_p=0^{\circ }$ always. For comparison, the results $-\chi$ after integration over ${\Omega }_p$ are shown (open circles, dotted line).