11 Application of Accelerators and Storage Rings
675
much weaker, or if protons are considered (e.g. at LHC/CERN), the resulting time
profile of the radiation field is shorter than shown in Fig. 11.5 and assumed for
the calculation of the synchrotron radiation spectrum. As the time profile gets
shorter, its Fourier transform extends to higher frequencies, thus the spectrum
gets “harder”. In contrast to increasing B or γ , this hardening is, however, not
accompanied by increased flux as the instantaneous properties of emission depend
only on the local magnetic field strength which does not change by shortening the
magnet.
A spectrum hardening effect similar to short-magnet radiation takes place if the
magnetic field rises at the entry face of the magnet (or drops at the exit, respectively)
over a distance comparable to or smaller than given by Eq. (11.32). In such case,
the time profile of the radiation field exhibits a rising (or falling, respectively) edge
steeper than that seen in Fig. 11.5. As a consequence, the Fourier transform extends
to rather high frequencies. At high-energy proton synchrotrons this is being used to
extend the spectrum towards wavelengths which are easy to observe.
11.1.1.4 Wigglers and Undulators
Definitions
Undulators and wigglers provide a periodic magnetic field over a part of the
synchrotron’s circumference. In most cases, the magnetic field perpendicular to the
electron beam’s design orbit can be described by a pure sinusoidal—at least within
the small spatial area where the electron beam is present. If the field acting on the
electron beam has only one non-zero Cartesian component, the device is called a
planar undulator (or wiggler, respectively). The field close to the axis can then be
described by
B y (z) = B 0 sin (k u z) .
(11.33)
Here, B 0 is the field amplitude and k u = 2π/λ u is the undulator wave number,
with λ u the undulator’s period. The z-axis is along the electron beam’s initial design
momentum, and we have chosen arbitrarily the magnetic field to be in the vertical
y-direction.
Typically, the length L u of these devices is a few meters, and the field integral
I 1 =
L u
0
B y (s)ds
(11.34)
is made zero such that there is no over-all deflection of the electron beam’s
trajectory.
675
much weaker, or if protons are considered (e.g. at LHC/CERN), the resulting time
profile of the radiation field is shorter than shown in Fig. 11.5 and assumed for
the calculation of the synchrotron radiation spectrum. As the time profile gets
shorter, its Fourier transform extends to higher frequencies, thus the spectrum
gets “harder”. In contrast to increasing B or γ , this hardening is, however, not
accompanied by increased flux as the instantaneous properties of emission depend
only on the local magnetic field strength which does not change by shortening the
magnet.
A spectrum hardening effect similar to short-magnet radiation takes place if the
magnetic field rises at the entry face of the magnet (or drops at the exit, respectively)
over a distance comparable to or smaller than given by Eq. (11.32). In such case,
the time profile of the radiation field exhibits a rising (or falling, respectively) edge
steeper than that seen in Fig. 11.5. As a consequence, the Fourier transform extends
to rather high frequencies. At high-energy proton synchrotrons this is being used to
extend the spectrum towards wavelengths which are easy to observe.
11.1.1.4 Wigglers and Undulators
Definitions
Undulators and wigglers provide a periodic magnetic field over a part of the
synchrotron’s circumference. In most cases, the magnetic field perpendicular to the
electron beam’s design orbit can be described by a pure sinusoidal—at least within
the small spatial area where the electron beam is present. If the field acting on the
electron beam has only one non-zero Cartesian component, the device is called a
planar undulator (or wiggler, respectively). The field close to the axis can then be
described by
B y (z) = B 0 sin (k u z) .
(11.33)
Here, B 0 is the field amplitude and k u = 2π/λ u is the undulator wave number,
with λ u the undulator’s period. The z-axis is along the electron beam’s initial design
momentum, and we have chosen arbitrarily the magnetic field to be in the vertical
y-direction.
Typically, the length L u of these devices is a few meters, and the field integral
I 1 =
L u
0
B y (s)ds
(11.34)
is made zero such that there is no over-all deflection of the electron beam’s
trajectory.