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Up: Polarization for pair annihilation


2 Transition rates

A single photon of energy $\omega>2m/{\sin\theta}$, with $\theta$ the photon emission angle with respect to the magnetic field, may be produced by electron - positron annihilation in a strong magnetic field. This first order process is permitted in the presence of an external magnetic field, which can absorb the transverse momentum difference between the initial and final particles.

The S-matrix element for this process may be written in the form:

\begin{displaymath}S_{fi} = (-ie) \int_{-\infty}^{\infty} {\rm d}t \int {\rm d}^{3}y
\bar\Psi_f(y) \gamma_\mu A^\mu(y)\Psi_i(y)
\end{displaymath} (1)

with: $A^\mu = {{\epsilon^\mu}/(2\omega V)^{1/2}}{\rm e}^{iky}$. $\Psi_f(y)$ and $\Psi_i(y)$ are the wave functions for positron and electron, respectively. We used the wave functions for ${\rm e}^-$ and ${\rm e}^+$ defined by Sokolov & Ternov (1983).

The photon has four possible polarizations to consider. The two linear polarizations are defined by the unit vectors:

    $\displaystyle \hat{\varepsilon}^{(1)} = -\cos\theta\cos\phi\cdot\hat{i}-\cos\theta
\sin\phi\cdot\hat
{j}+\sin\theta\cdot\hat{k}$  
    $\displaystyle \hat{\varepsilon}^{(2)}
= \sin\phi\cdot\hat{i}-\cos\phi\cdot\hat{j}$ (2)

where the photon momentum vector is given by:

\begin{displaymath}\vec{k} = \omega (\sin\theta\cos\phi\cdot\hat{i}+\sin\theta\sin\phi\cdot\hat{j}+
\cos\theta\cdot\hat{k}).
\end{displaymath} (3)

The unit vectors $\hat{\varepsilon}^{(1)}$, $\hat{\varepsilon}^{(2)}$ and $\hat{k}$ form a right handed triad:

\begin{displaymath}\hat{\varepsilon}^{(1)} \times \hat{\varepsilon}^{(2)} = \hat{k}.
\end{displaymath} (4)

The two circular polarizations (right-handed and left-handed respectively) are given by:

\begin{displaymath}\hat\varepsilon_{\pm}=\mp\frac {1}{\sqrt{2}}(\hat\varepsilon^{(1)}\pm
i\hat\varepsilon^{(2)}).
\end{displaymath} (5)

The energies for ${\rm e}^-$ (E) and ${\rm e}^+$ ($E^\prime$) are:
$\displaystyle E {=} [ m^2{+}{p_z}^2 {+} 4N\gamma)]^{1/2};~
E^\prime{=} [ m^2{+}{p^\prime_z}^2 {+} 4N^\prime \gamma)]^{1/2}.$     (6)

Here pz is the longitudinal momentum for ${\rm e}^-$, $p^\prime_z$ is the longitudinal momentum for ${\rm e}^+$, N=0, 1, 2, ... is the Landau level of ${\rm e}^-$, $N^\prime $ is the Landau level of ${\rm e}^+$, and $\gamma =\frac {eB} {2c\hbar}$. In natural units ($c=\hbar=1$), which we adopt hereafter, $\gamma = \frac {eB} {2}$.

The probability for the annihilation process is:

    $\displaystyle \vert S_{fi}\vert^2 = \frac {\alpha (2\pi)^4 T
{I_{s,s^\prime}}^2...
...E^\prime {E_{\rm o}}^\prime E E_{\rm o} }
\delta(p_z + {p_z}^\prime - \omega_z)$  
    $\displaystyle \delta(p_y + {p_y}^\prime - \omega_y)
\delta(E+ E^\prime - \omega) \vert N_{\rm ann}\vert^2$ (7)

with:
    $\displaystyle \vert N_{\rm ann}\vert^2 = [{\varepsilon^{(\lambda)}}_z {{\vareps...
..._z}^*
(E^\prime E + {p_z}^\prime p_z - r r^\prime {E_{\rm o}}^\prime E_{\rm o})$  
    $\displaystyle [ ({E_{\rm o}}^\prime - r^\prime m) (E_{\rm o} + r m)
{I_{N-1,N^\prime-1}}^2 ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle + ({E_{\rm o}}^\prime + r^\prime m) (E_{\rm o} - r m)
{I_{N,N^\prime}}^2 ({\omega _\perp}^2 / 2eB )]$  
    $\displaystyle + r r^\prime \sqrt{2eBN} (p_z {E_{\rm o}}^\prime + r r^\prime {p_z}^\prime E_{\rm o})$  
    $\displaystyle [({E_{\rm o}}^\prime - r^\prime m) I_{N-1,N^\prime-1} ({\omega _\perp}^2 / 2eB)
I_{N,N^\prime-1} ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle ({\varepsilon^{(\lambda)}}_z {{\varepsilon^{(\lambda)}}_-}^* {\rm...
... +
{\varepsilon^{(\lambda)}}_- {{\varepsilon^{(\lambda)}}_z}^* {\rm e}^{i\phi})$  
    $\displaystyle + ({E_{\rm o}}^\prime + r^\prime m) I_{N-1,N^\prime} ({\omega _\perp}^2 / 2eB)
I_{N,N^\prime} ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle ({\varepsilon^{(\lambda)}}_+ {{\varepsilon^{(\lambda)}}_z}^* {\rm...
...
{\varepsilon^{(\lambda)}}_z {{\varepsilon^{(\lambda)}}_+}^* {\rm e}^{i\phi}) ]$  
    $\displaystyle - \sqrt{2eBN^\prime} (- p_z {E_{\rm o}}^\prime - r r^\prime {p_z}^\prime E_{\rm o})$  
    $\displaystyle [(E_{\rm o} + r m) I_{N-1,N^\prime-1} ({\omega _\perp}^2 / 2eB)
I_{N-1,N^\prime} ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle ({\varepsilon^{(\lambda)}}_z {{\varepsilon^{(\lambda)}}_+}^* {\rm...
...+
{\varepsilon^{(\lambda)}}_+ {{\varepsilon^{(\lambda)}}_z}^* {\rm e}^{-i\phi})$  
    $\displaystyle + (E_{\rm o} - r m) I_{N,N^\prime-1} ({\omega _\perp}^2 / 2eB)
I_{N,N^\prime} ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle ({\varepsilon^{(\lambda)}}_- {{\varepsilon^{(\lambda)}}_z}^* {\rm...
...{\varepsilon^{(\lambda)}}_z {{\varepsilon^{(\lambda)}}_-}^* {\rm e}^{-i\phi}) ]$  
    $\displaystyle - 4eB\sqrt{NN^\prime} {\varepsilon^{(\lambda)}}_z {{\varepsilon^{...
...\prime E^\prime E - r r^\prime {p_z}^\prime p_z + {E_{\rm o}}^\prime E_{\rm o})$  
    $\displaystyle I_{N-1,N^\prime-1} ({\omega _\perp}^2 / 2eB)
I_{N,N^\prime} ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle + (E^\prime E - {p_z}^\prime p_z + r r^\prime {E_{\rm o}}^\prime ...
...(\lambda)}}_- {{\varepsilon^{(\lambda)}}_-}^*
({E_{\rm o}}^\prime - r^\prime m)$  
    $\displaystyle (E_{\rm o} - r m)
{I_{N,N^\prime-1}}^2 ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle +{\varepsilon^{(\lambda)}}_+ {{\varepsilon^{(\lambda)}}_+}^*
({E_{\rm o}}^\prime + r^\prime m) (E_{\rm o} + r m)
{I_{N-1,N^\prime}}^2$  
    $\displaystyle ({\omega _\perp}^2 / 2eB)]$  
    $\displaystyle + 2eB\sqrt{NN^\prime}
( r r^\prime E^\prime E - r r^\prime {p_z}^\prime p_z + {E_{\rm o}}^\prime E_{\rm o})$  
    $\displaystyle I_{N,N^\prime-1} ({\omega _\perp}^2 / 2eB)
I_{N-1,N^\prime} ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle ({\varepsilon^{(\lambda)}}_- {{\varepsilon^{(\lambda)}}_+}^* {\rm...
...\varepsilon^{(\lambda)}}_+ {{\varepsilon^{(\lambda)}}_-}^* {\rm e}^{-i2\phi}) ]$ (8)

with $E_{\rm o}=(m^2+2NeB)^{1/2}$, ${E_{\rm o}}^\prime=(m^2+2N^\prime eB)^{1/2}$, and ${\varepsilon^{(\lambda)}}_\pm = {\varepsilon^{(\lambda)}}_x \pm i
{\varepsilon^{(\lambda)}}_y$. Formulae for ${\varepsilon ^{(\lambda )}}_\pm $ are given explicity in Table 1. The polarization of the electron spin is $r=\pm1$, except for N=0 which has r=-1 only. The polarization of the positron spin is $r^\prime=\pm1$, except for $N^\prime =0$ which has $r^\prime =+1$ only.


 

 
Table 1: ${\varepsilon ^{(\lambda )}}_\pm $ for different photon polarizations
$\lambda=1$ $\lambda=2$ $\lambda=+$ $\lambda=-$
${\varepsilon^{(\lambda)^{^{^{^{}}}}}}_+$ $-\cos \theta {\rm e}^{i \phi}$ $-i {\rm e}^{i \phi}$ $-(1-\cos \theta) {\rm e}^{i \phi}/\sqrt{2}$ $-(1+\cos \theta) {\rm e}^{i \phi}/\sqrt{2}$
${\varepsilon^{(\lambda)}}_-$ $-\cos \theta {\rm e}^{-i \phi}$ $i {\rm e}^{-i \phi}$ $(1+\cos \theta) {\rm e}^{-i \phi}/\sqrt{2}$ $ (1-\cos \theta) {\rm e}^{-i \phi}/\sqrt{2}$
       


The Laguerre function is:

\begin{displaymath}I_{N,R} (x) = \sqrt{R!/{N!}} {\rm e}^{-x/2} x^{(N-R)/2} {L_R}^{N-R}(x).
\end{displaymath} (9)

If we sum over radial quantum number $s^\prime$:

\begin{displaymath}\sum\limits_{s^\prime=0}^\infty {I_{s,s^\prime}}^2 (x) = 1.
\end{displaymath} (10)

However, the sum over s is limited to some maximum value $s_{\rm max}$ (see Sokolov & Ternov 1983), so

\begin{displaymath}\sum\limits_{s=0}^{s_{\rm max}} \sum\limits_{s^\prime=0}^\inf...
...\omega _\perp}^2 / 2eB) =
\frac{m^2 B}{2 \pi B_{\rm cr}} L^2.
\end{displaymath} (11)

The evaluation of the probability for the 1 photon annihilation process using Sokolov's spinors is different with respect to the evaluation using Herold's spinors (Herold 1979) as follows. We obtain one more term ${I_{s,s^\prime}}^2 ({\omega _\perp}^2 / 2eB)$and only two laws of conservation:
$\displaystyle p_z + {p_z}^\prime = \omega_z$   $\displaystyle E + E^\prime = \omega.$ (12)

In comparison Wunner et al. (1986) has one more conservation law $p_y + {p_y}^\prime = \omega_y$.

The transition rate is given by:

    $\displaystyle W= \lim_{T \rightarrow \infty} \frac {1}{T}
\sum_{\rm final~states~~} \sum_{\rm initial~states} \vert S_{fi}\vert^2$  
    $\displaystyle = \lim_{T \rightarrow \infty} \frac {1}{T}
\int \int \frac {\alph...
..._y L_z m^2 16 E^\prime {E_{\rm o}}^\prime E E_{\rm o}} \vert N_{\rm ann}\vert^2$ (13)

with $p_y + {p_y}^\prime = \omega_y$ and $E + E^\prime = \omega$. $\alpha$ denotes the fine structure constant ( $\alpha= {\rm e}^2/{\hbar c}$ or ${\rm e}^2$ in natural units). The differential transition rate is:

\begin{displaymath}{{\rm d}W_{r.r^\prime}}^{N,N^\prime} (\omega, \lambda)\!\! =\...
...rime {E_{\rm o}}^\prime E E_{\rm o}} \vert N_{\rm ann}\vert^2.
\end{displaymath} (14)

Restricting to the case of zero longitudinal momentum for the electron (pz=0), we obtain:
    $\displaystyle {{\rm d}W_{r.r^\prime}}^{N,N^\prime}(\omega, \lambda) =
\alpha (\...
...B_{\rm cr}}{m^3 B}
\frac {{\rm d}\phi}{2 E^\prime {E_{\rm o}}^\prime E_{\rm o}}$  
    $\displaystyle [ (E_{\rm o} - r m)({E_{\rm o}}^\prime + r^\prime m) (E^\prime - r r^\prime {E_{\rm o}}^\prime)
{I_{N,N^\prime}}^2$  
    $\displaystyle ({\omega _\perp}^2 / 2eB)
{\varepsilon^{(\lambda)}}_z {{\varepsilon^{(\lambda)}}_z}^*$  
    $\displaystyle + (E_{\rm o} + r m)({E_{\rm o}}^\prime + r^\prime m) (E^\prime + r r^\prime {E_{\rm o}}^\prime)
{I_{N-1,N^\prime}}^2$  
    $\displaystyle ({\omega _\perp}^2 / 2eB)
{\varepsilon^{(\lambda)}}_+ {{\varepsilon^{(\lambda)}}_+}^*$  
    $\displaystyle + (E_{\rm o} + r m)({E_{\rm o}}^\prime - r^\prime m) (E^\prime - r r^\prime {E_{\rm o}}^\prime)
{I_{N-1,N^\prime-1}}^2$  
    $\displaystyle ({\omega _\perp}^2 / 2eB)
{\varepsilon^{(\lambda)}}_z {{\varepsilon^{(\lambda)}}_z}^*$  
    $\displaystyle + (E_{\rm o} - r m)({E_{\rm o}}^\prime - r^\prime m) (E^\prime + r r^\prime {E_{\rm o}}^\prime)
{I_{N,N^\prime-1}}^2$  
    $\displaystyle ({\omega _\perp}^2 / 2eB)
{\varepsilon^{(\lambda)}}_- {{\varepsilon^{(\lambda)}}_-}^*$  
    $\displaystyle + \sqrt{2eBN} {p_{z\rm o}}^\prime [({E_{\rm o}}^\prime + r^\prime m)
I_{N,N^\prime} ({\omega _\perp}^2 / 2eB)
I_{N-1,N^\prime}$  
    $\displaystyle ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle ({\varepsilon^{(\lambda)}}_+ {{\varepsilon^{(\lambda)}}_z}^* {\rm...
...}
+{\varepsilon^{(\lambda)}}_z {{\varepsilon^{(\lambda)}}_+}^* {\rm e}^{i\phi})$  
    $\displaystyle + ({E_{\rm o}}^\prime - r\prime m) I_{N-1,N^\prime-1}$  
    $\displaystyle ({\omega _\perp}^2 / 2eB)
I_{N,N^\prime-1} ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle ({\varepsilon^{(\lambda)}}_z {{\varepsilon^{(\lambda)}}_-}^* {\rm...
...
+{\varepsilon^{(\lambda)}}_- {{\varepsilon^{(\lambda)}}_z}^* {\rm e}^{i\phi})]$  
    $\displaystyle + r r^\prime \sqrt{2eBN^\prime} {p_{z\rm o}}^\prime [(E_{\rm o} + r m)
I_{N-1,N^\prime-1}$  
    $\displaystyle ({\omega _\perp}^2 / 2eB)
I_{N-1,N^\prime} ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle ({\varepsilon^{(\lambda)}}_+ {{\varepsilon^{(\lambda)}}_z}^* {\rm...
...}
+{\varepsilon^{(\lambda)}}_z {{\varepsilon^{(\lambda)}}_+}^* {\rm e}^{i\phi})$  
    $\displaystyle +(E_{\rm o} - r m)
I_{N,N^\prime} ({\omega _\perp}^2 / 2eB)
I_{N,N^\prime-1} ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle ({\varepsilon^{(\lambda)}}_z {{\varepsilon^{(\lambda)}}_-}^* {\rm...
...
+{\varepsilon^{(\lambda)}}_- {{\varepsilon^{(\lambda)}}_z}^* {\rm e}^{i\phi})]$  
    $\displaystyle + 2eB\sqrt{NN^\prime}[({E_{\rm o}}^\prime + r r^\prime E^\prime)
I_{N-1,N^\prime} ({\omega _\perp}^2 / 2eB)
I_{N,N^\prime-1}$  
    $\displaystyle ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle ({\varepsilon^{(\lambda)}}_- {{\varepsilon^{(\lambda)}}_+}^* {\rm...
...\varepsilon^{(\lambda)}}_+ {{\varepsilon^{(\lambda)}}_-}^* {\rm e}^{-i2\phi}) ]$  
    $\displaystyle + 2 r r^\prime ( E^\prime - r r^\prime {E_{\rm o}}^\prime)
I_{N,N^\prime} ({\omega _\perp}^2 / 2eB)
I_{N-1,N^\prime-1}$  
    $\displaystyle ({\omega _\perp}^2 / 2eB)
{\varepsilon^{(\lambda)}}_z {{\varepsilon^{(\lambda)}}_z}^*]].$ (15)

Here ${p_{z\rm o}}^\prime$ is the longitudinal momentum of the positron when the electron longitudinal momentum is zero.

The energy of the annihilation radiation is given by:

$\displaystyle \omega$ = $\displaystyle \frac {(E{-}p_z \cos \theta) {+} \sqrt{(E{-}p_z \cos \theta)^2
{-} 4 \sin^2 \theta \gamma (N{-} N^\prime)}} {\sin^2 \theta}$  
  = $\displaystyle \frac {(E{-}p_z \cos \theta)} {\sin^2 \theta} \!\!\left[\!1\!{+}\...
...\prime) m^2 (B/B_{\rm cr}) \sin^2\theta }{(E{-}p_z
\cos \theta)^2}}\right]\cdot$ (16)

This follows from the two conservation laws $p_z + {p_z}^\prime = \omega \cos \theta$ and $E + E^\prime = \omega$.

The coefficients above related to the polarization of the photon are evaluated and listed in Table 2. By averaging over polarization of the positron, the differential transition rate for a polarized electron ($r=\pm1$) and unpolarized positron (if $p_z \ne 0$) is:

    $\displaystyle {{\rm d}W_r}^{N,N^\prime} (\omega, \lambda) =
\frac { \alpha \pi ...
...ac {{\rm d}\phi}{2 E^\prime E E_{\rm o}}
\frac {(1 + \delta _{N^\prime,0})} {2}$  
    $\displaystyle [{\varepsilon^{(\lambda)}}_z {{\varepsilon^{(\lambda)}}_z}^*
[(E_...
...o} + r m)(E^\prime E + {p_z}^\prime p_z + r E_{\rm o} m)
{I_{N-1,N^\prime-1}}^2$  
    $\displaystyle ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle +(E_{\rm o} - r m)(E^\prime E + {p_z}^\prime p_z - r E_{\rm o} m)
{I_{N,N^\prime}}^2 ({\omega _\perp}^2 / 2eB)]$  
    $\displaystyle + \sqrt{2eBN} [({p_z}^\prime E_{\rm o} - r p_z m)
I_{N-1,N^\prime-1} ({\omega _\perp}^2 / 2eB)
I_{N,N^\prime-1}$  
    $\displaystyle ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle ({\varepsilon^{(\lambda)}}_z {{\varepsilon^{(\lambda)}}_-}^* {\rm...
... +
{\varepsilon^{(\lambda)}}_- {{\varepsilon^{(\lambda)}}_z}^* {\rm e}^{i\phi})$  
    $\displaystyle +({p_z}^\prime E_{\rm o} + r p_z m)
I_{N-1,N^\prime} ({\omega _\perp}^2 / 2eB)
I_{N,N^\prime} ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle ({\varepsilon^{(\lambda)}}_+ {{\varepsilon^{(\lambda)}}_z}^* {\rm...
...
{\varepsilon^{(\lambda)}}_z {{\varepsilon^{(\lambda)}}_+}^* {\rm e}^{i\phi}) ]$  
    $\displaystyle + \sqrt{2eBN^\prime} p_z [ (E_{\rm o} + r m)
I_{N-1,N^\prime-1} ({\omega _\perp}^2 / 2eB)
I_{N-1,N^\prime}$  
    $\displaystyle ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle ({\varepsilon^{(\lambda)}}_z {{\varepsilon^{(\lambda)}}_+}^* {\rm...
...+
{\varepsilon^{(\lambda)}}_+ {{\varepsilon^{(\lambda)}}_z}^* {\rm e}^{-i\phi})$  
    $\displaystyle + (E_{\rm o} - r m) I_{N,N^\prime-1} ({\omega _\perp}^2 / 2eB)
I_{N,N^\prime} ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle ({\varepsilon^{(\lambda)}}_- {{\varepsilon^{(\lambda)}}_z}^* {\rm...
...{\varepsilon^{(\lambda)}}_z {{\varepsilon^{(\lambda)}}_-}^* {\rm e}^{-i\phi}) ]$  
    $\displaystyle - 2eB\sqrt{NN^\prime} E_{\rm o}
[2{\varepsilon^{(\lambda)}}_z {{\...
...n^{(\lambda)}}_z}^*
I_{N-1,N^\prime-1} ({\omega _\perp}^2 / 2eB)
I_{N,N^\prime}$  
    $\displaystyle ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle - I_{N,N^\prime-1} ({\omega _\perp}^2 / 2eB)
I_{N-1,N^\prime} ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle ({\varepsilon^{(\lambda)}}_- {{\varepsilon^{(\lambda)}}_+}^* {\rm...
...\varepsilon^{(\lambda)}}_+ {{\varepsilon^{(\lambda)}}_-}^* {\rm e}^{-i2\phi}) ]$  
    $\displaystyle + [(E_{\rm o} - r m)(E^\prime E - {p_z}^\prime p_z - r E_{\rm o} ...
...\varepsilon^{(\lambda)}}_- {{\varepsilon^{(\lambda)}}_-}^*
{I_{N,N^\prime-1}}^2$  
    $\displaystyle ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle + (E_{\rm o} + r m)(E^\prime E - {p_z}^\prime p_z + r E_{\rm o} m)
{\varepsilon^{(\lambda)}}_+ {{\varepsilon^{(\lambda)}}_+}^*
{I_{N-1,N^\prime}}^2$  
    $\displaystyle ({\omega _\perp}^2 / 2eB)]].$ (17)


 

 
Table 2: Coefficients related to the polarization of the photon
Coefficient $\sum\limits_{\lambda}$ $\lambda=1$ $\lambda=2$ $\lambda=\pm$
${\varepsilon^{(\lambda)}}_- {{\varepsilon^{(\lambda)}}_-}^*$ $1+{\rm cos}^2 \theta$ ${\rm cos}^2 \theta$ 1 $(1 \pm \cos \theta)^2/2$
${\varepsilon^{(\lambda)}}_- {{\varepsilon^{(\lambda)}}_{z}}^*$ $-\sin(2\theta) {\rm e}^{-i\phi}/2$ $-\sin(2\theta) {\rm e}^{-i\phi}/2$ 0 $ \mp (1 \pm \cos \theta) \sin \theta {\rm e}^{-i\phi}/2$
${\varepsilon^{(\lambda)}}_- {{\varepsilon^{(\lambda)}}_+}^*$ $-{\rm sin}^2 \theta {\rm e}^{-i2\phi}$ $ {\rm cos}^2 \theta {\rm e}^{-i2\phi}$ $-{\rm e}^{-i2\phi}$ $ - {\rm sin}^2 \theta {\rm e}^{-i2\phi}/2 $
${\varepsilon^{(\lambda)}}_{z} {{\varepsilon^{(\lambda)}}_-}^*$ $-\sin(2\theta) {\rm e}^{i\phi}/2$ $-\sin(2\theta) {\rm e}^{i\phi}/2$ 0 $ \mp (1 \pm \cos \theta) \sin \theta {\rm e}^{i\phi}/2$
${\varepsilon^{(\lambda)}}_{z} {{\varepsilon^{(\lambda)}}_{z}}^*$ ${\rm sin}^2 \theta$ ${\rm sin}^2 \theta$ 0 $ ({\rm sin}^2 \theta) /2$
${\varepsilon^{(\lambda)}}_{z} {{\varepsilon^{(\lambda)}}_+}^*$ $-\sin(2\theta) {\rm e}^{-i\phi}/2$ $-\sin(2\theta) {\rm e}^{-i\phi}/2$ 0 $ \pm (1 \mp \cos \theta) \sin \theta {\rm e}^{-i\phi}/2$
${\varepsilon^{(\lambda)}}_+ {{\varepsilon^{(\lambda)}}_-}^*$ $-{\rm sin}^2 \theta {\rm e}^{i2\phi}$ $ {\rm cos}^2 \theta {\rm e}^{i2\phi}$ $-{\rm e}^{i2\phi}$ $ - {\rm sin}^2 \theta {\rm e}^{i2\phi}/2 $
${\varepsilon^{(\lambda)}}_+ {{\varepsilon^{(\lambda)}}_{z}}^*$ $-\sin(2\theta) {\rm e}^{i\phi}/2$ $-\sin(2\theta) {\rm e}^{i\phi}/2$ 0 $ \pm (1 \mp \cos \theta) \sin \theta {\rm e}^{i\phi}/2$
${\varepsilon^{(\lambda)}}_+ {{\varepsilon^{(\lambda)}}_+}^*$ $1+{\rm cos}^2 \theta$ ${\rm cos}^2 \theta$ 1 $(1 \mp \cos \theta)^2/2$
       


For the case of zero electron longitudinal momentum (pz = 0) this reduces to:

    $\displaystyle {{\rm d}W_\mp}^{N,N^\prime} (\omega, \lambda) =
\frac {\alpha \pi...
...\frac {{\rm d}\phi}{2 E^\prime E_{\rm o}} \frac {(1 + \delta _{N^\prime,0})}{2}$  
    $\displaystyle [{\varepsilon^{(\lambda)}}_z {{\varepsilon^{(\lambda)}}_z}^*
[(E_{\rm o} \mp m)(E^\prime \mp m)
{I_{N-1,N^\prime-1}}^2 ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle +(E_{\rm o} \pm m)(E^\prime \pm m)
{I_{N,N^\prime}}^2 ({\omega _\perp}^2 / 2eB)]$  
    $\displaystyle + {p_{z\rm o}}^\prime \sqrt{2eBN} [
I_{N-1,N^\prime-1} ({\omega _\perp}^2 / 2eB)
I_{N,N^\prime-1} ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle ({\varepsilon^{(\lambda)}}_z {{\varepsilon^{(\lambda)}}_-}^* {\rm...
... +
{\varepsilon^{(\lambda)}}_- {{\varepsilon^{(\lambda)}}_z}^* {\rm e}^{i\phi})$  
    $\displaystyle + I_{N-1,N^\prime} ({\omega _\perp}^2 / 2eB)
I_{N,N^\prime} ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle ({\varepsilon^{(\lambda)}}_+ {{\varepsilon^{(\lambda)}}_z}^* {\rm...
...
{\varepsilon^{(\lambda)}}_z {{\varepsilon^{(\lambda)}}_+}^* {\rm e}^{i\phi}) ]$  
    $\displaystyle - 2eB\sqrt{NN^\prime}
[2{\varepsilon^{(\lambda)}}_z {{\varepsilon^{(\lambda)}}_z}^*
I_{N-1,N^\prime-1} ({\omega _\perp}^2 / 2eB)
I_{N,N^\prime}$  
    $\displaystyle ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle - I_{N,N^\prime-1} ({\omega _\perp}^2 / 2eB)
I_{N-1,N^\prime} ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle ({\varepsilon^{(\lambda)}}_- {{\varepsilon^{(\lambda)}}_+}^* {\rm...
...\varepsilon^{(\lambda)}}_+ {{\varepsilon^{(\lambda)}}_-}^* {\rm e}^{-i2\phi}) ]$  
    $\displaystyle + [(E_{\rm o} \pm m)(E^\prime \pm m)
{\varepsilon^{(\lambda)}}_- {{\varepsilon^{(\lambda)}}_-}^*
{I_{N,N^\prime-1}}^2 ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle + (E_{\rm o} \mp m)(E^\prime \mp m)
{\varepsilon^{(\lambda)}}_+ {{\varepsilon^{(\lambda)}}_+}^*
{I_{N-1,N^\prime}}^2 ({\omega _\perp}^2 / 2eB)]].$  
      (18)

Alternatively, if we average ${{\rm d}W_{r.r^\prime}}^{N,N^\prime} (\omega, \lambda)$over the polarization of the electron, the differential probability for a polarized positron ( $r^\prime=\pm1$) and an unpolarized electron is obtained (for $p_z \ne 0$):


    $\displaystyle {{\rm d}W_{r^\prime=\mp 1}}^{N,N^\prime} (\omega, \lambda) =
\fra...
...c {{\rm d}\phi} {2 E^\prime E {E_{\rm o}}^\prime} \frac {(1+\delta _{N,0})} {2}$  
    $\displaystyle [{\varepsilon^{(\lambda)}}_z {{\varepsilon^{(\lambda)}}_z}^*
[({E...
...(E^\prime E + {p_z}^\prime p_z \pm {E_{\rm o}}^\prime m)
{I_{N-1,N^\prime-1}}^2$  
    $\displaystyle ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle +({E_{\rm o}}^\prime \mp m)(E^\prime E + {p_z}^\prime p_z \mp {E_{\rm o}}^\prime m)
{I_{N,N^\prime}}^2$  
    $\displaystyle ({\omega _\perp}^2 / 2eB)]$  
    $\displaystyle + {p_z}^\prime \sqrt{2eBN} [({E_{\rm o}}^\prime \pm m)
I_{N-1,N^\prime-1} ({\omega _\perp}^2 / 2eB)
I_{N,N^\prime-1}$  
    $\displaystyle ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle ({\varepsilon^{(\lambda)}}_z {{\varepsilon^{(\lambda)}}_-}^* {\rm...
... +
{\varepsilon^{(\lambda)}}_- {{\varepsilon^{(\lambda)}}_z}^* {\rm e}^{i\phi})$  
    $\displaystyle +({E_{\rm o}}^\prime \mp m)
I_{N-1,N^\prime} ({\omega _\perp}^2 / 2eB)
I_{N,N^\prime} ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle ({\varepsilon^{(\lambda)}}_+ {{\varepsilon^{(\lambda)}}_z}^* {\rm...
...
{\varepsilon^{(\lambda)}}_z {{\varepsilon^{(\lambda)}}_+}^* {\rm e}^{i\phi}) ]$  
    $\displaystyle - \sqrt{2eBN^\prime} (\pm {p_z}^\prime m - p_z {E_{\rm o}}^\prime)
I_{N-1,N^\prime-1} ({\omega _\perp}^2 / 2eB)
I_{N-1,N^\prime}$  
    $\displaystyle ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle ({\varepsilon^{(\lambda)}}_z {{\varepsilon^{(\lambda)}}_+}^* {\rm...
...+
{\varepsilon^{(\lambda)}}_+ {{\varepsilon^{(\lambda)}}_z}^* {\rm e}^{-i\phi})$  
    $\displaystyle + \sqrt{2eBN^\prime} (\pm {p_z}^\prime m + p_z {E_{\rm o}}^\prime)
I_{N,N^\prime-1} ({\omega _\perp}^2 / 2eB)
I_{N,N^\prime}$  
    $\displaystyle ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle ({\varepsilon^{(\lambda)}}_- {{\varepsilon^{(\lambda)}}_z}^*{\rm ...
...
+{\varepsilon^{(\lambda)}}_z {{\varepsilon^{(\lambda)}}_-}^* {\rm e}^{-i\phi})$  
    $\displaystyle - 4eB\sqrt{NN^\prime} {E_{\rm o}}^\prime
{\varepsilon^{(\lambda)}...
...n^{(\lambda)}}_z}^*
I_{N-1,N^\prime-1} ({\omega _\perp}^2 / 2eB)
I_{N,N^\prime}$  
    $\displaystyle ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle + ({E_{\rm o}}^\prime \pm m)(E^\prime E - {p_z}^\prime p_z \pm {E...
...\varepsilon^{(\lambda)}}_- {{\varepsilon^{(\lambda)}}_-}^*
{I_{N,N^\prime-1}}^2$  
    $\displaystyle ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle + ({E_{\rm o}}^\prime \mp m)(E^\prime E - {p_z}^\prime p_z \mp {E...
...\varepsilon^{(\lambda)}}_+ {{\varepsilon^{(\lambda)}}_+}^*
{I_{N-1,N^\prime}}^2$  
    $\displaystyle ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle + 2eB \sqrt{NN^\prime} {E_{\rm o}}^\prime
I_{N,N^\prime-1} ({\omega _\perp}^2 / 2eB)
I_{N-1,N^\prime} ({\omega _\perp}^2 / 2eB)$  
    $\displaystyle ({\varepsilon^{(\lambda)}}_- {{\varepsilon^{(\lambda)}}_+}^* {\rm...
...\varepsilon^{(\lambda)}}_+ {{\varepsilon^{(\lambda)}}_-}^* {\rm e}^{-i2\phi})].$ (19)


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