FUNDAMENTALS OF X-RAY FLUORESCENCE - Total-Reflection X-Ray Fluorescence Analysis and Related Methods - page 37

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At the angles θ .

According to Equation 1.29, the corresponding angles in S meet the condition
of sin Δ θ =± 1/ γ and cos Δ θ = β . Consequently, Δ θ is about ± 1/ γ . The radiation
is beamed in the direction of the particles motion within

=± π /2, the intensity of the emitted radiation is zero in S

1/ γ <Δ θ <+ 1/ γ .It
leads to the forward-pointing narrow cone of Figure 1.17 where the observation
line and the velocity vector of the accelerated particles coincide within a
horizontal emission angle of about 2/ γ . This is also valid for the vertical
emission angle. The higher the speed of the electrons, the narrower the
momentary emission cone of the photons become. Within a certain time
interval the cone describes a fan that may be 10 times wider.
The emitted radiation of a single relativistic electron is a flux with a large
number of photons. They all do not have the same energy but carry very

Figure1.17. The emitted radiation of relativistic electrons deflected by a homogeneous magnetic
field B vertical to the orbit plane. The rest frame S

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of the electron (a) shows an isotropic dipole
pattern while the relativistic frame S of the observer (b) gives a narrow cone in forward direction.
Three-dimensional view (left). Cross-section of the orbit plane (right). θ and Δ θ are emission angles
between velocity υ and Lorentz force F or acceleration a . Its sum is 90 ° or π /2. Figure from Ref. [53],
reproduced with permission from K. Wille.