6 Polarization matrix elements of the reemitted photons

The polarization matrix elements of the reemitted photons can be obtained from the previous equation describing the evolution of the atomic subsystem in interaction with the incident photons and the colliding particles in the presence of a magnetic field. We denote by $\phi({\textbf{\textit{v}}}_{\scriptscriptstyle{\rm {A}}},{\textbf{\textit B}})$ this matrix in the magnetic field frame $\left(PX^{\prime}Y^{\prime}Z^{\prime}\right)$ where the quantization axis $\left(PZ^{\prime}\right)$ is parallel to the magnetic field vector. The polarization matrix elements $\phi^k_q({\textbf{\textit{v}}}_{\scriptscriptstyle{\rm {A}}}, {\textbf{\textit B}})$ of the reemitted photons are given (as number of re-emitted photons per unit time, per unit volume and per unit velocity distribution) by

$$\begin{array}{l} {\displaystyle \phi_{q}^{k} \left({\textbf{... ...riptscriptstyle{\rm {A}}}\right) } \; c(k) \Bigg], \end{array}$$ (13)

where $N_{\scriptscriptstyle{\rm {A}}}$ is the number of scattering atoms per unit volume, $\gamma=\omega\,\tau$ ($\tau$ being the lifetime of the upper level of the atomic transition), and $B_{\rm lu}$ is Einstein's coefficient for absorption. The coefficient c(k) is defined by

$$\begin{array}{l} c(k)=\left\{ \begin{array}{r} 1\;\;\;\mbox{... ... c_{ul}\;\;\;\mbox{if}\;\;\;k=2 \end{array}\right. \end{array}$$ (14)

the coefficient c_ul being defined in Sahal-Bréchot et al. (1998) (see also Landi Degl'Innocenti 1984; Landi Degl'Innocenti & Landi Degl'Innocenti 1988).

Then, the polarization matrix elements of the reemitted photons written in the frame of the magnetic field are given by

$$\begin{array}{l} {\displaystyle \phi^0_0({\textbf{\textit{v}}}_{\... ...\prime}^2_q({\textbf{\textit{v}}}_{\scriptscriptstyle{\rm {A}}}) }. \end{array}$$ (15)

The other elements $\phi^2_{-1}({\textbf{\textit{v}}}_{\scriptscriptstyle{\rm {A}}}, {\textbf{\textit B}})$ and $\phi^2_{-2}({\textbf{\textit{v}}}_{\scriptscriptstyle{\rm {A}}}, {\textbf{\textit B}})$ can be obtained using the conjugation property (Bommier 1977)

$$\phi^k_{-q}=(-1)^q\;{\phi^k_{q}}^* .$$ (16)

To get the Stokes parameters of the scattered photons, we should calculate the density matrix elements of the reemitted radiation written in the laboratory frame where the quantization axis is along the line of sight. We denote this frame by (PXYZ), where (PZ) is directed along the line of sight. (PXYZ) is obtained from the solar frame (Pxyz) by an Euler rotation with angles $(-\frac{\pi}{2},\theta,\frac{\pi}{2})$. $\theta$ (angle between (Pz) and (PZ)) is the scattering angle.

It is more useful to rewrite these polarization matrix elements of the reemitted radiation first in the solar frame (Pxyz) and then in the frame of the line of sight (PXYZ). This is to reduce the number of rotation angles in the final expressions. The atomic frame (Pxyz) is obtained from the magnetic frame $\left(PX^{\prime}Y^{\prime}Z^{\prime}\right)$ by a rotation with angles $(0,-\eta,-\psi)$. We note with ${\phi^\prime}^k_q({\textbf{\textit{v}}}_{\scriptscriptstyle{\rm {A}}}, {\textbf{\textit B}})$ and ${\Phi}^k_q({\textbf{\textit{v}}}_{\scriptscriptstyle{\rm {A}}}, {\textbf{\textit B}})$, respectively, the polarization matrix elements of the scattered photons (developed on the irreducible tensor basis) in the solar frame (Pxyz) and in the frame of the line of sight (PXYZ).

The polarization matrix elements of the reemitted photons ${\phi^\prime}^k_q({\textbf{\textit{v}}}_{\scriptscriptstyle{\rm {A}}}, {\textbf{\textit B}})$ written in (Pxyz) are given by

$$\begin{array}{l} {\phi^\prime}^0_0({\textbf{\textit{v}}}_{\script... ...Re{\rm e}(\phi^2_2) +i\,\cos\eta\; \Im{\rm m}(\phi^2_2) \Big\}\cdot \end{array}$$ (17)

In (PXYZ), the polarization matrix elements of the scattered radiation are given by

$$\begin{array}{l} {\Phi}^0_0({\textbf{\textit{v}}}_{\scriptscripts... ...ft. - i\,\cos\theta\, \Im{\rm m}({\phi^{\prime}}^2_2) \right\}\cdot \end{array}$$ (18)