4 Results and discussion

The decay rate per unit volume is calculated from the cross-section by (e.g. Wunner et al. 1986):

$${R_{r,r^\prime}}^{N,N^\prime}(p_{z},{p_{z}}^\prime) = {\sigma _{r... ...}^{N,N^\prime}(p_{z},{p_{z}}^\prime) ~n_{{\rm e}^-}(N, p_{z}, r) ~n_{{\rm e}^+}$$

$$(N^\prime, {p_{z}}^\prime, r^\prime) ~\vert p_{z}/E - {p_{z}}^\prime /E^\prime\vert.$$ (28)

Numerical integration over the distributions for longitudinal momenta of ${\rm e}^-$ and ${\rm e}^+$ yields the total decay rate per unit volume. We use a Gaussian distribution for the longitudinal momentum, p, of the electron:

$$(\pi ^{1/2} \Delta p)^{-1} \exp[-(p-p_{\rm o})^2/ (\Delta p)^2]$$ n(p) = (29)

with parameters $\Delta p$, and $p_{\rm o}$. This is normalized to unity. For the distribution of positron longitudinal momentum, q, we also use the Gaussian distribution, but with parameters $\Delta q$, and $q_{\rm o}$. The resulting decay rate is then per unit volume, per unit density of positrons, per unit density of electrons.

In order to carry out the integration over the longitudinal momentum distributions, we use the relation giving p_z for the electron from the conservation laws, which is:

$$p_{z} = \frac {1}{2\omega {\rm sin}^2 \theta} [\cos \theta [\omega ^2 {\rm sin}^2 \theta + 4 \gamma (N- N^\prime)] +$$

$$[{\rm cos}^2 \theta [\omega ^2 {\rm sin}^2 \theta + 4 \gamma (N- N^\prime)]^2 +{\rm sin}^2 \theta [16 \gamma^2 (N- N^\prime)^2$$

$$-8 \gamma \omega^2 (N- N^\prime) (1+{\rm cos}^2 \theta) + \omega ^4 (1-{\rm cos}^2 \theta)^2 - 4 \omega ^2$$

$$(m^2 +4 \gamma N^\prime) ]^{1/2}].$$ (30)

If $N=N^\prime =0$, this simplifies to:

$$\frac {\omega \sin(2\theta)/2 + \sqrt{\omega ^2 {\rm sin}^2 \theta - 4 m^2}} {2 \sin \theta}$$ p_z = (31)

with the positron longitudinal momentum given by ${p_{z}}^\prime= \omega \cos \theta - p_{z}$, and $\omega$ satisfies the condition $\omega>2m/{\sin\theta}$.

Figure 1: The $1-\gamma$ annihilation rate as a function of photon emission angle for $B=1 \ 10^{13}$ Gauss

Figure 2: The $1-\gamma$ annihilation rate as a function of photon emission angle for $B=B_{\rm crit}=4.414 \ 10^{13}$ Gauss

Figure 3: The annihilation cross-section normalized to the free-space 2-photon nonrelativistic cross-section, $\sigma _{\rm o}$, for ${\rm e}^-$ with N=0 and r=-1 (spin down), and ${\rm e}^+$ with $N^\prime =0$ and $r^\prime =+1$ (spin up). For all cases pz=0. $\sigma 1_0$ is for photon polarization $\hat{\varepsilon}^{(1)}$ and $p_{z}^\prime \simeq 0$and is equal to the total unpolarized cross-section since the $\hat{\varepsilon}^{(2)}$ cross-section is zero, $\sigma p_0$( $=\sigma 1_0/2$) is for $\hat{\varepsilon}^{(+)}$ and $p_{z}^\prime \simeq 0$(and is equal to that for the $\hat{\varepsilon}^{(-)}$ case). $\sigma 1_{2~mc}$ is for $\hat{\varepsilon}^{(1)}$ and $p_{z}^\prime=2~mc$ and is equal to the total unpolarized cross-section since the $\hat{\varepsilon}^{(2)}$ cross-section is zero, $\sigma p_{2~mc}$( $=\sigma 1_{2~mc}/2$) is for $\hat{\varepsilon}^{(+)}$ and $p_{z}^\prime=2~mc$, (and is equal to that for the $\hat{\varepsilon}^{(-)}$ case)

Figure 4: The annihilation cross-section normalized to $\sigma _{\rm o}$ for ${\rm e}^-$ with N=1 and r=-1 (spin down), and ${\rm e}^+$ with $N^\prime =0$ and $r^\prime =+1$ (spin up). For all cases pz=0. $\sigma 1_0$ is for photon polarization $\hat{\varepsilon}^{(1)}$ and $p_{z}^\prime \simeq 0$and is equal to the total unpolarized cross-section since the $\hat{\varepsilon}^{(2)}$ cross-section is zero, $\sigma p_0$( $=\sigma 1_0/2$) is for $\hat{\varepsilon}^{(+)}$ and $p_{z}^\prime \simeq 0$(and is equal to that for the $\hat{\varepsilon}^{(-)}$ case). $\sigma u_{2~mc}$ is the total unpolarized cross-section for $p_{z}^\prime=2~mc$, $\sigma 1_{2~mc}$ is for $\hat{\varepsilon}^{(1)}$ and $p_{z}^\prime=2~mc$, $\sigma p_{2~mc}$ is for $\hat{\varepsilon}^{(+)}$ and $p_{z}^\prime=2~mc$, $\sigma m_{2~mc}$ is for $\hat{\varepsilon}^{(-)}$ and $p_{z}^\prime=2~mc$, and $\sigma2_{2~mc}$ is for $\hat{\varepsilon}^{(2)}$ and $p_{z}^\prime=2~mc$

Figure 5: The annihilation cross-section normalized to $\sigma _{\rm o}$ for ${\rm e}^-$ with N=1, and ${\rm e}^+$ with $N^\prime =0$. For all cases pz=0 and $p_{z}^\prime \simeq 0$. $\sigma 1_{du}$ is for photon polarization $\hat{\varepsilon}^{(1)}$ and r=-1 (spin down) and $r^\prime =+1$ (spin up) (and is equal to the total unpolarized cross-section since the $\hat{\varepsilon}^{(2)}$ cross-section is zero). $\sigma p_{du}$( $=\sigma 1_{du}/2$) is for $\hat{\varepsilon}^{(+)}$and r=-1 (spin down) and $r^\prime =+1$ (spin up), (and is equal to that for the $\hat{\varepsilon}^{(-)}$ case). $\sigma 2_{uu}$ is for $\hat{\varepsilon}^{(2)}$ and r=+1 (spin up) and $r^\prime =+1$ (spin up) (and is equal to the total unpolarized cross-section since the $\hat{\varepsilon}^{(1)}$ cross-section is zero). $\sigma p_{uu}$( $=\sigma 2_{uu}/2$) is for $\hat{\varepsilon}^{(+)}$ and r=+1 (spin up) and $r^\prime =+1$ (spin up) (and is equal to that for the $\hat{\varepsilon}^{(-)}$ case)

Figure 6: The annihilation cross-section normalized to $\sigma _{\rm o}$ for ${\rm e}^-$ with N=2, and ${\rm e}^+$ with $N^\prime =1$. For all cases pz=0 and $p_{z}^\prime \simeq 0$ and the cross-section for unpolarized radiation is plotted. $\sigma _{du}$ is for r=-1 (spin down) and $r^\prime =+1$ (spin up). $\sigma _{uu}$ is for r=+1 (spin up) and $r^\prime =+1$ (spin up). $\sigma _{dd}$ is for r=-1 (spin down) and $r^\prime =-1$ (spin down). $\sigma _{ud}$ is for r=+1 (spin up) and $r^\prime =-1$ (spin down)

Figure 7: The annihilation cross-section normalized to $\sigma _{\rm o}$ as a function of N and $N^\prime$. For all cases $p_{z}=p_{z}^\prime=2~mc$, ${\rm e}^-$ has r=-1 (spin down) and ${\rm e}^+$ has $r^\prime =+1$ (spin up), and the cross-section for unpolarized radiation is plotted. $\sigma _{18}$ is for $N=N^\prime =18$. $\sigma _{15}$ is for $N=N^\prime =15$. $\sigma _{10}$ is for $N=N^\prime =10$. $\sigma _{5}$ is for $N=N^\prime =5$. $\sigma _{3}$ is for $N=N^\prime =3$

Figure 8: The annihilation cross-section normalized to $\sigma _{\rm o}$ as a function of N and $N^\prime$. For all cases the total energy of the electron and positron are each 2 mc, the cross-section is for average over electron and positron spins, and the cross-section is for unpolarized radiation. $\sigma _{15}$ is for $N=N^\prime =15$. $\sigma _{10}$ is for $N=N^\prime =10$. $\sigma _{5}$ is for $N=N^\prime =5$. $\sigma _{3}$ is for $N=N^\prime =3$. $\sigma _{1}$ is for $N=N^\prime =1$. $\sigma _{0}$ is for $N=N^\prime =0$. For each curve (except $N=N^\prime =0$) there is a maximum B, for which the parallel energy drops to zero

For the specific cases presented here for the annihilation rate, we take the electron and positron distributions to be the same. We take the case $p_{\rm o}=q_{\rm o}=0$ and $\Delta p=\Delta q= 50$ keV, $N=N^\prime =0$, and unpolarized radiation. Figure 1 shows the annihilation rate as a function of photon emission angle for $B=1 \ 10^{13}$ Gauss. Figure 2 shows the same but for $B=B_{\rm cr}$. Figure 1 is consistent with the result of Daugherty & Bussard (1980) (however we give, in addition, the value of the decay rate): the radiation is confined to a narrow beam centered at $\theta = \pi/2$ for low magnetic field and small $\Delta p$. Figure 2 shows that the annihilation rate increases rapidly with increasing B (a factor of $\sim200$ for an increase in B by a factor of 4.4) but that the radiation remains well beamed. We find that the beam is narrow for small $\Delta p (<< mc)$ and wide for large $\Delta p (\geq m)$, for all values of B.

Next we show some results of numerical evaluations of the cross-sections. First we present cross-sections as a function of magnetic field for the different photon polarizations. The non-relativistic free-space $2\gamma$ annihilation cross-section for ${\rm e}^-~{\rm e}^+$ pairs with opposite spins and for the electron at rest (e.g. Wunner et al. 1986) is: $\sigma_{\rm o}=2\pi \alpha^2 {/\lambda_{\rm c}}^2/v^\prime$, with $v^\prime$ the velocity of the positron. We present our cross-sections normalized by the non-relativistic free-space $2\gamma$ annihilation cross-section. The expression for $v^\prime$ valid for also for a relativistic positron and in the positron in any Landau state is: $v^\prime=p_{z}^\prime/ \sqrt((m^2)+({p_{z}^\prime}^2)+2N^\prime eB)$. We note that our calculations of the normalized cross-sections for the cases given in Wunner et al. (1986) agree with their calculations.

For the case of $p_{z}=p_{z}^\prime$ and electron with spin down and positron with spin up, the following relations can be shown to hold: $\sigma_{1\gamma,\hat{\varepsilon}^{(1)}} = 2 \sigma_{1\gamma,\hat{\varepsilon}^{(\pm)}}$ and $\sigma_{1\gamma,\hat{\varepsilon}^{(2)}}=0$. For $N=N^\prime =0$, the above is valid for any values of p_z and $p_{z}^\prime$.

Figure 3 gives the annihilation cross-section, normalized to $\sigma _{\rm o}$, for ${\rm e}^-$ with N=0 and r=-1 (spin down), and ${\rm e}^+$ with $N^\prime =0$ and $r^\prime =+1$ (spin up). For all cases the electron has p_z=0. The positron has $p_{z}^\prime \simeq 0$ (in practise we use a small value, like 10^-20 MeV/c to numerically evaluate the cross-section) in one case (with subscript 0), and $p_{z}^\prime=2~mc$ for the second case (with subscript 2 mc). Since $\sigma_{1\gamma,\hat{\varepsilon}^{(1)}} = 2 \sigma_{1\gamma,\hat{\varepsilon}^{(\pm)}}$ and $\sigma_{1\gamma,\hat{\varepsilon}^{(2)}}=0$, only the cross-sections for photon polarizations $\hat{\varepsilon}^{(1)}$ (labelled $\sigma 1$) and $\hat{\varepsilon}^{(+)}$ (labelled $\sigma p$) are plotted. The cross-sections for unpolarized radiation (sum over polarization for outgoing states) are not plotted, since they are equal to the polarization $\hat{\varepsilon}^{(1)}$ cross-sections in this case.

We find that the cross-sections rise as a function of B rapidly to a peak at $B=\sim2-3B_{\rm cr}$ and decrease slowly for larger B. The peak values are larger than the non-relativistic free-space annihilation cross-section by factors of $\sim20$ to $\sim160$. For nonrelativistic parallel momentum, $p_{z}^\prime \simeq 0$, one has: for unpolarized radiation or for polarization $\hat{\varepsilon}^{(1)}$, $\sigma_{1\gamma}>\sigma_{\rm o}$ for $B>0.245B_{\rm cr}$; and for radiation polarization $\hat{\varepsilon}^{(+)}$ or $\hat{\varepsilon}^{(-)}$, $\sigma_{1\gamma}>\sigma_{\rm o}$ for $B>0.245B_{\rm cr}$. For large parallel momentum (>mc), the annihilation rate is suppressed compared to the small parallel momentum case. For a relativistic parallel momentum of $p_{z}^\prime=2~mc$, one has: for unpolarized radiation or for polarization $\hat{\varepsilon}^{(1)}$, $\sigma_{1\gamma}>\sigma_{\rm o}$ for $B>0.484B_{\rm cr}$; and for radiation polarization $\hat{\varepsilon}^{(+)}$ or $\hat{\varepsilon}^{(-)}$, $\sigma_{1\gamma}>\sigma_{\rm o}$ for $B>0.552B_{\rm cr}$. For $p_{z}^\prime=4~mc$, one has: for unpolarized radiation or for polarization $\hat{\varepsilon}^{(1)}$, $\sigma_{1\gamma}>\sigma_{\rm o}$ for $B>0.950B_{\rm cr}$; and for radiation polarization $\hat{\varepsilon}^{(+)}$ or $\hat{\varepsilon}^{(-)}$, $\sigma_{1\gamma}>\sigma_{\rm o}$ for $B>1.132B_{\rm cr}$.

Figure 4 gives the cross-section normalized to $\sigma _{\rm o}$ for the same cases as for Fig. 3, except that the electron is now in an excited Landau level (${\rm e}^-$ with N=1). The positron has $p_{z}^\prime \simeq 0$ in one case (with subscript 0), and $p_{z}^\prime=2~mc$ for the second case (with subscript 2 mc). For $p_{z}^\prime \simeq 0$, one has $\sigma_{1\gamma,\hat{\varepsilon}^{(1)}} = 2 \sigma_{1\gamma,\hat{\varepsilon}^{(\pm)}}$ and $\sigma_{1\gamma,\hat{\varepsilon}^{(2)}}=0$, so only the cross-sections for photon polarizations $\hat{\varepsilon}^{(1)}$ (labelled $\sigma 1$) and $\hat{\varepsilon}^{(+)}$ (labelled $\sigma p$) are plotted. The cross-section for unpolarized radiation (sum over polarization for outgoing states) for $p_{z}^\prime \simeq 0$ is equal to the polarization $\hat{\varepsilon}^{(1)}$ cross-section. However for nonzero positron momentum (here $p_{z}^\prime=2~mc$), the cross-sections for the different polarizations are all different. The cross-section for unpolarized radiation (labelled $\sigma u$) is the largest of the 5 curves that peak below a value of 13, and in decreasing order follow $\sigma 1$, $\sigma p$, $\sigma m$(for photon polarization $\hat{\varepsilon}^{(-)}$), and $\sigma2$(for polarization $\hat{\varepsilon}^{(2)}$). The $N=1, N^\prime=0$ case has lower (unpolarized radiation) cross-sections by a factor of 2.5-3 compared to the N=0 case. However it looks very different in $\hat{\varepsilon}^{(2)}$ radiation, which is not allowed in the N=0, $N^\prime =0$ case. $\sigma_{1\gamma}/\sigma_{\rm o}$ reaches a maximum as a function of B. The maximum value and the B of maximum depend on N and $N^\prime$. E.g. for N=1 and $N^\prime =0$ (Fig. 4), the maximum value is 64.7, at $B=1.33B_{\rm cr}$.

The cross-section decreases with increasing longitudinal momentum. E.g. for $p_{z}^\prime=2~mc$ and N=1, $N^\prime =0$ (Fig. 4), the cross-section at $B=0.1B_{\rm cr}$ is $1.05\ 10^6$ times smaller than for $p_{z}^\prime \simeq 0$, and the cross-section at $B=B_{\rm cr}$ is 11.4 times smaller. From Figs. 3 and 4, it can be seen that the reduction in cross-section due to larger $p_{z}^\prime$ is smaller for larger B. A second effect of larger $p_{z}^\prime$ is that the cross-section reaches maximum at larger B, and, for non-zero $p_{z}^\prime$, the B of maximum depends on polarization. E.g., for $p_{z}^\prime=2~mc$ N=1, $N^\prime =0$ (Fig. 4): the maximum of $\sigma_{1\gamma}/\sigma_{\rm o}$ has value at the magnetic field of maximum as follows: 12.5 at $B=3.07B_{\rm cr}$ for sum over polarization; 11.6 at $B=2.89B_{\rm cr}$ for polarization $\hat{\varepsilon}^{(1)}$; 1.17 at $B=6.63B_{\rm cr}$ for polarization $\hat{\varepsilon}^{(2)}$; 9.60 at $B=3.48B_{\rm cr}$ for polarization $\hat{\varepsilon}^{(+)}$; and 3.18 at $B=2.23B_{\rm cr}$ for polarization $\hat{\varepsilon}^{(-)}$. For all values of B and r=-1, $r^\prime =+1$ one has: $\sigma_{1\gamma,\hat{\varepsilon}^{(1)}} > \sigma_{1\gamma,\hat{\varepsilon}^{... ...ma_{1\gamma,\hat{\varepsilon}^{(-)}} > \sigma_{1\gamma,\hat{\varepsilon}^{(2)}}$.

We next compare the cross-sections for different spin states of the electron. In Fig. 5, the normalized cross-section is presented for two different spin states: ${\rm e}^-$ with N=1, p_z=0 and $r=\pm1$, and ${\rm e}^+$ with $N^\prime =0$, $p_{z}^\prime \simeq 0$ and $r^\prime =+1$. The upper two curves ($\sigma 1$ for photon polarization $\hat{\varepsilon}^{(1)}$ and $\sigma p$for polarization $\hat{\varepsilon}^{(+)}$) are for r=-1, $r^\prime =+1$ (denoted by subscript du in the figure) and are the same as for Fig. 4. The lower two curves ($\sigma2$ for photon polarization $\hat{\varepsilon}^{(2)}$ and $\sigma p$for polarization $\hat{\varepsilon}^{(+)}$) are for r=+1, $r^\prime =+1$ (denoted by subscript uu). Here and in what follows the first arrow indicates whether the electron spin is up or down and the second arrow indicates whether the positron spin is up or down (for the figures we use u and d to indicate spin up or down). For the case of $\downarrow\uparrow$ we have $\sigma_{1\gamma,\hat{\varepsilon}^{(1)}} = 2 \sigma_{1\gamma,\hat{\varepsilon}^{(\pm)}}$ and $\sigma_{1\gamma,\hat{\varepsilon}^{(2)}}=0$. However for the case of electron and positron both spin-up ( $\uparrow\uparrow$), we have $\sigma_{1\gamma,\hat{\varepsilon}^{(2)}} = 2 \sigma_{1\gamma,\hat{\varepsilon}^{(\pm)}}$ and $\sigma_{1\gamma,\hat{\varepsilon}^{(1)}}=0$. The cross-section for spins $\downarrow\uparrow$ is larger than that for spins $\uparrow\uparrow$ (e.g. by factor 22.0 at $B=0.1B_{\rm cr}$), the difference decreasing with increasing B(e.g. by factor 1.56 at $B=9.9B_{\rm cr}$).

Generally for N>0 and $N^\prime>0$ (so both spins are allowed for positron and electron) and if $p_{z}=p_{z}^\prime$, one has the following relation for the cross-sections for different photon polarizations: For the cases with antiparallel spins ( $\downarrow\uparrow$ or $\uparrow\downarrow$) we have $\sigma_{1\gamma,\hat{\varepsilon}^{(1)}} = 2 \sigma_{1\gamma,\hat{\varepsilon}^{(\pm)}}$ and $\sigma_{1\gamma,\hat{\varepsilon}^{(2)}}=0$. For the cases with parallel spins ( $\downarrow\downarrow$ or $\uparrow\uparrow$) we have $\sigma_{1\gamma,\hat{\varepsilon}^{(2)}} = 2 \sigma_{1\gamma,\hat{\varepsilon}^{(\pm)}}$ and $\sigma_{1\gamma,\hat{\varepsilon}^{(1)}}=0$. In the above two cases we have the unpolarized cross-section is the same as the $\hat{\varepsilon}^{(\pm)}$ cross-sections. For summing over polarization we have for all B: $\sigma_{1\gamma,\downarrow\uparrow} > \sigma_{1\gamma,\uparrow\uparrow} > \sigma_{1\gamma,\downarrow\downarrow} > \sigma_{1\gamma,\uparrow\downarrow}$. Figure 6 shows the normalized cross-section for unpolarized radiation and for four different spin states: ${\rm e}^-$ with N=2, p_z=0 and $r=\pm1$, and ${\rm e}^+$ with $N^\prime =1$, $p_{z}^\prime \simeq 0$ and $r^\prime=\pm1$. The cross-section for r=-1, $r^\prime=1$ is denoted $\sigma _{du}$, for r=1, $r^\prime=1$ is denoted $\sigma _{uu}$, for r=-1, $r^\prime =-1$ is denoted $\sigma _{dd}$, and for r=1, $r^\prime =-1$ is denoted $\sigma _{ud}$.

Next we consider the dependence of the cross-section on N and $N^\prime$. Figure 7 shows the normalized cross-section for ${\rm e}^-,{\rm e}^+$ spins $\downarrow\uparrow$, summed over photon polarization, for five sets of values of (N,$N^\prime$): (3, 3), (5, 5), (10, 10), (15, 15) and (18, 18). Here we have taken the case of longitudinal momentum: $p_{z}=p_{z}^\prime=2~mc$. (The case of zero longitudinal momentum and sum over polarization has been considered by Wunner et al. 1986.) Since $p_{z}=p_{z}^\prime$ the relations in the last paragraph above give the cross-sections for the individual polarizations. I.e., $\sigma_{1\gamma,\hat{\varepsilon}^{(1)}} = 2 \sigma_{1\gamma,\hat{\varepsilon}^{(\pm)}}$ and $\sigma_{1\gamma,\hat{\varepsilon}^{(2)}}=0$. In Fig. 7, the number label on $\sigma$ represents the values of N and $N^\prime$. For small B ( $<B_{\rm cr}$), the cross-sections at first increase with N and $N^\prime$ then reach a maximum and decrease for larger N and $N^\prime$. E.g. for $p_{z}=p_{z}^\prime=2~mc$, and $B=0.12B_{\rm cr}~(=5.30\ 10^{12}$ Gauss), the maximum cross-section is for $N^\prime= N = 12$, and $\sigma_{1\gamma,\sum_{\rm spin},N^\prime= N = 12}/ \sigma_{1\gamma,\sum_{\rm spin},N^\prime= N = 0}=8.81\ 10^{5}$. For large B ( $>B_{\rm cr}$), one finds: $\sigma_{1\gamma}$ is largest for for $N^\prime= N=0$; and $\sigma_{1\gamma}$ for $N^\prime= N$ decreases monotonically with increasing N. Generally we have the following relations for the relative size of the cross-sections for different electron spins at fixed N and $N^\prime$. (a) For $N^\prime= N$, $\sigma_{1\gamma,\downarrow \uparrow} > \sigma_{1\gamma,\uparrow \uparrow} = \sigma_{1\gamma,\downarrow \downarrow} >\sigma_{1\gamma,\uparrow \downarrow}$ (except for $N^\prime= N=0$, where we only have the possibility for spins of $\downarrow\uparrow$). (b) For $N^\prime < N$, $\sigma_{1\gamma,\downarrow\uparrow} > \sigma_{1\gamma,\uparrow\uparrow} > \sigma_{1\gamma,\downarrow\downarrow} > \sigma_{1\gamma,\uparrow\downarrow}$ (except for $N^\prime =0$, where we only have the possibilities for spins of $\uparrow\uparrow$ and $\downarrow\uparrow$). (c) For $N^\prime > N$, $\sigma_{1\gamma,\downarrow \uparrow} > \sigma_{1\gamma,\downarrow \downarrow} > \sigma_{1\gamma,\uparrow \uparrow} >\sigma_{1\gamma,\uparrow \downarrow}$ (except for N= 0, where we only have the possibilities for spins of $\downarrow\uparrow$ and $\downarrow\downarrow$).

Finally, we show a set of cross-sections for different N and $N^\prime$(with $N^\prime= N$), but rather than at fixed p_z and $p_{z}^\prime$, they are at fixed $E_{\rm e-}$ and $E_{\rm e+}$. In this case, because the ${\rm e}^-$ (or ${\rm e}^+$) energy increases with B for N>0 (or $N^\prime>0$), there is a maximum B at constant $E_{\rm e-}$ for N>0 (or at constant $E_{\rm e+}$ for $N^\prime>0$). Since we take $E_{\rm e-}=E_{\rm e+}$, the maximum B is the same for both ${\rm e}^-$ and ${\rm e}^-$. Figure 8 shows the resulting normalized cross-sections as a function of $B/B_{\rm cr}$for six sets of values of (N,$N^\prime$): (0, 0), (1, 1), (3, 3), (5, 5), (10, 10), and (15, 15) and for ${\rm e}^-$ with spin down and ${\rm e}^+$with spin down and unpolarized radiation. In all cases we have $p_{z}=p_{z}^\prime$, so the relation for the different photon polarization holds: $\sigma_{1\gamma,\hat{\varepsilon}^{(1)}} = 2 \sigma_{1\gamma,\hat{\varepsilon}^{(\pm)}}$ and $\sigma_{1\gamma,\hat{\varepsilon}^{(2)}}=0$.

In summary, we have presented formulae for the one photon pair annihilation cross-section for different photon polarizations which are valid for any magnetic field and any Landau and spin states of the electron and positron. These results are generalizations of those of Wunner et al. (1986). This is an important generalization, since the new results presented here allow calculation of the polarization of the emitted radiation, which is significantly different from zero, and strongly dependent on the momenta, spin and Landau states of the electron and positron.

Acknowledgements

DAL thanks the Natural Sciences and Engineering Research Council of Canada for support.