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Using energy and momentum conservation of the bremsstrahlung process a decomposition of the involved momenta
with respect to the incident electron momentum in longitudinal q_{l} and transversal q_{t} components
permits the formulation of kinematical limits for the momentum transfer :
This momentum transfer range, depending on the electron energy E_{0} and relative photon energy x=k/E_{0},
is referred to as the `pancake' due to its large lateral extension (
).
Calculating the bremsstrahlung cross section[3,4] but
retaining the photon polarisation
,
the following asymptotic term is obtained:
.
This shows that the cross section drops in first order with 1/k and that the maximal linear polarisation,
with
being the azimuthal angle of
with respect to the scattering plane, is found in that plane.
When employing an amorphous radiator, the electron scatters off a single atom which means that
the momentum transfer
may lie at any point in the pancake leading to an isotropic distribution
of
,
hence to an unpolarised beam.
In contrast to this incoherent contribution an additional process can be observed
in crystal radiators due to their regular structure:
Whenever the momentum transfer coincides with a reciprocal lattice vector
the recoil is absorbed by the whole lattice enhancing the yield by coherently adding the contributions
of each atom.
Also the momentum transfer is fixed in space which therefore leads to a polarised photon beam.
Raising the photon energy x increases the coherent contribution and polarisation monotonically as well as q_{l}which depends like q_{t} on the orientation of the crystal
(fig. 2 and eq. A2) and leads to a discontinuity at
,
where a given lattice vector does not satisfy the pancake condition anymore and thus no longer contributes
to the coherent bremsstrahlung spectrum.
The total cross section of a crystal[5,6] is a sum of coherent and incoherent (including electronelectron)
bremsstrahlung where the DebyeWaller factor
,
which is a function of temperature
and crystal properties, governs the fractioning into coherent and incoherent contributions to the total cross section.
After introducing the intensity I per atom in units of
Z^{2}mb
the cross section and polarisation P are expressed as a sum over lattice vectors
of the functions
(eq. A1) and read in common notation:
In the coherent contribution due to kinematical constraints for a given lattice vector the photon polar angle
is a function of photon energy (eq. 3a),
apart from a negligible dependence on the azimuthal angle[4].
Collimating the photon beam enhances the ratio of coherent to incoherent bremsstrahlung,
thus increasing the degree of polarisation. Whereas the incoherent cross section is reduced approximately by
f_{c} = u_{c}/(1+u_{c}) using a collimation angle of U_{c} (note: u=U^{2}),
the coherent one stays unaffected in the energy range
but
vanishes elsewhere:
For the incoherent contribution instead of using the BetheHeitler cross section (eq. 3BSb in ref.[7])
the Hubbell[8] cross section derived from the Schiff cross section (eq. 3BSe),
which has a more accurate dependence on photon energy, radiator charge Z and collimation angle u_{c}was employed (see eq. A3 and fig. 1).
Figure:
Comparison of Z, photon energy and collimator angle U_{c} dependence of Bethe Heitler and Hubbell intensity.
In the right panel the intensity was normalised to unity whereas in the insert the intensity was set to one
at the origin (I(0)=1) for the sake of comparison.

For a calculation of the incoherent contribution from a crystal radiator compared to an
amorphous one, the Debye Waller factor
has to be taken into account, which leads to a modified
formfactor:
denoted by
.
Here F_{r} means a realistic carbon formfactor from a Hartree Fock calculation[9].
Due to the fact that an analytical integration of the Schiff cross section as performed by Hubbell
for the dipole formfactor with screening constant C (eq. A3) seems no longer feasible,
two approximate treatments were investigated[4].
The use of an effective screening constant seemed superior because in addition the temperature dependence of
is easier implemented there.
The screening constant
for the Hubbell cross section was determined via a fitting procedure
for the total intensity
with
,
resulting in
for
the amorphous and 33 for the incoherent intensity.
Also an improved description[10] of the electronelectron bremsstrahlung I^{e}was used, which takes into account the binding energy of the radiator electrons
and which has a non trivial Z and photon energy dependence, see (eq. A4).
For this contribution the asymptotic BetheHeitler angular distribution
f(U) = U/(1+U^{2})^{2}, which leads to the collimation reduction factor f_{c} (s. above) was adopted.
Next: Treatment of experimental deficiencies
Up: PRODUCTION OF LINEARLY POLARISED
Previous: Introduction
Frank Natter
19990716