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

(28) |

Numerical integration over the distributions for longitudinal momenta of and yields the total decay rate per unit volume. We use a Gaussian distribution for the longitudinal momentum,

n(p) |
= | (29) |

with parameters , and . This is normalized to unity. For the distribution of positron longitudinal momentum,

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:

(30) |

If , this simplifies to:

p_{z} |
= | (31) |

with the positron longitudinal momentum given by , and satisfies the condition .

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
and
keV,
,
and unpolarized radiation.
Figure 1 shows the annihilation rate as a function of photon emission angle
for
Gauss.
Figure 2 shows the same but for
.
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
for low magnetic field and small .
Figure 2 shows that the annihilation rate increases rapidly with increasing
*B* (a factor of
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
and wide for large
,
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 annihilation cross-section for pairs with opposite spins and for the electron at rest (e.g. Wunner et al. 1986) is: , with the velocity of the positron. We present our cross-sections normalized by the non-relativistic free-space annihilation cross-section. The expression for valid for also for a relativistic positron and in the positron in any Landau state is: . 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
and electron with spin down
and positron with spin up,
the following relations can be shown to hold:
and
.
For
,
the above is valid for any values of *p*_{z} and
.

Figure 3 gives the annihilation cross-section, normalized to
,
for
with *N*=0 and *r*=-1 (spin down), and
with
and
(spin up). For all cases the electron
has *p*_{z}=0. The positron has
(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
for the second case (with subscript 2 *mc*).
Since
and
,
only the cross-sections
for photon polarizations
(labelled )
and
(labelled )
are plotted.
The cross-sections for unpolarized radiation (sum over polarization for
outgoing states) are not plotted, since they are equal to the
polarization
cross-sections in this case.

We find that the cross-sections rise as a function of *B* rapidly to a
peak at
and decrease slowly for larger *B*.
The peak values
are larger than the non-relativistic free-space annihilation cross-section
by factors of
to .
For nonrelativistic parallel momentum,
,
one has:
for unpolarized radiation or for polarization
,
for
;
and for radiation polarization
or
,
for
.
For large parallel momentum (>*mc*), the annihilation rate is suppressed
compared to the small parallel momentum case.
For a relativistic parallel momentum of
,
one has:
for unpolarized radiation or for polarization
,
for
;
and for radiation polarization
or
,
for
.
For
,
one has:
for unpolarized radiation or for polarization
,
for
;
and for radiation polarization
or
,
for
.

Figure 4 gives the cross-section normalized to
for the same cases
as for Fig. 3, except that the electron is now in an excited Landau level
(
with *N*=1).
The positron has
in one case (with subscript 0),
and
for the second case (with subscript 2 *mc*).
For
,
one has
and
,
so only the cross-sections
for photon polarizations
(labelled )
and
(labelled )
are plotted.
The cross-section for unpolarized radiation (sum over polarization for
outgoing states) for
is equal to the
polarization
cross-section.
However for nonzero positron momentum (here
),
the cross-sections for the different polarizations are all different.
The cross-section for unpolarized radiation (labelled )
is
the largest of the 5 curves that peak below a value of 13, and in
decreasing order follow ,
,
(for photon polarization
), and (for polarization
).
The
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
radiation,
which is not allowed in the *N*=0,
case.
reaches a maximum as a function of *B*.
The maximum value and the *B* of maximum depend on *N* and .
E.g. for *N*=1 and
(Fig. 4),
the maximum value is 64.7, at
.

The cross-section decreases with increasing longitudinal momentum. E.g.
for
and *N*=1,
(Fig. 4), the cross-section at
is
times smaller than for
,
and the cross-section at
is 11.4 times smaller.
From Figs. 3 and 4, it can be seen that the
reduction in cross-section due to larger
is smaller for
larger *B*.
A second effect of larger
is that the cross-section reaches
maximum at larger *B*, and, for non-zero
,
the *B* of
maximum depends on polarization.
E.g., for
*N*=1,
(Fig. 4): the maximum of
has value
at the magnetic field of maximum as follows:
12.5 at
for sum over polarization;
11.6 at
for polarization
;
1.17 at
for polarization
;
9.60 at
for polarization
;
and
3.18 at
for polarization
.
For all values of *B* and *r*=-1,
one has:
.

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:
with *N*=1, *p*_{z}=0 and ,
and
with
,
and
.
The upper two curves (
for photon polarization
and for polarization
)
are for *r*=-1,
(denoted by subscript du in the figure) and are the same as for Fig. 4.
The lower two curves (
for photon polarization
and for polarization
)
are for *r*=+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
we have
and
.
However for the case of electron and positron both spin-up
(
), we have
and
.
The cross-section for spins
is larger than that for
spins
(e.g. by factor 22.0 at
), the
difference decreasing with increasing *B*(e.g. by factor 1.56 at
).

Generally for *N*>0 and
(so both spins are allowed for positron
and electron) and if
,
one has the following relation for the cross-sections for different photon
polarizations:
For the cases with antiparallel spins (
or
)
we have
and
.
For the cases with parallel spins (
or
)
we have
and
.
In the above two cases we have the unpolarized cross-section is the same
as the
cross-sections.
For summing over polarization we have for all *B*:
.
Figure 6 shows the normalized cross-section for unpolarized radiation and for
four different spin states:
with *N*=2, *p*_{z}=0 and ,
and
with
,
and
.
The cross-section for *r*=-1,
is denoted
,
for *r*=1,
is denoted
,
for *r*=-1,
is denoted
,
and for *r*=1,
is denoted
.

Next we consider the dependence of the cross-section on *N* and .
Figure 7 shows the normalized cross-section for
spins
,
summed over photon polarization,
for five sets of values of (*N*,):
(3, 3), (5, 5), (10, 10), (15, 15) and (18, 18).
Here we have taken the case of longitudinal momentum:
.
(The case of zero longitudinal momentum and sum over polarization has been
considered by Wunner et al. 1986.)
Since
the relations in the last paragraph above give the
cross-sections for the individual polarizations.
I.e.,
and
.
In Fig. 7, the number label on
represents the values of
*N* and .
For small *B* (
), the cross-sections at first
increase with *N* and
then reach a maximum and
decrease for larger *N* and .
E.g. for
,
and
Gauss), the maximum cross-section
is for
,
and
.
For large *B* (
), one finds:
is largest for for
;
and
for
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 .
(a) For
,
(except for
,
where we only have the possibility for spins of
).
(b) For
,
(except for
,
where we only have the possibilities for spins of
and
).
(c) For
,
(except for *N*= 0,
where we only have the possibilities for spins of
and
).

Finally, we show a set of cross-sections for different *N* and (with
), but rather than at fixed *p*_{z} and
,
they are
at fixed
and
.
In this case, because the
(or )
energy increases with *B* for *N*>0 (or
), there is a maximum
*B* at constant
for *N*>0 (or at constant
for
).
Since we take
,
the maximum *B* is the same for both
and .
Figure 8 shows the resulting normalized cross-sections as a function of
for six sets of values of (*N*,): (0, 0), (1, 1),
(3, 3), (5, 5), (10, 10), and (15, 15) and for
with spin down and with spin down and unpolarized radiation.
In all cases we have
,
so the relation for the different photon
polarization holds:
and
.

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.

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

Copyright The European Southern Observatory (ESO)