direction—they are the so-called transverse waves. But when we discussed the
Lorentz forces, we argued that to add energy to the particle, we need an electric
field along the direction of motion; hence we need a longitudinal component to the
electric field.
As illustrated in Fig. 2.4, the first trick that machine physicists employ in
designing accelerator cavities is to use special, usually cylindrical, shapes so that
electric field components are allowed only along the direction of propagation. The
example shown is for a cylindrical “pillbox” cavity, for which the z-component
(along the beam axis) E z of the electric field of an appropriate travelling wave is
given (in cylindrical coordinates) by:
E z ¼ E 0 J 0 k c r
ð ÞÁ exp i ωt À kz
ð
Þ
½
ð 2:4Þ
where ω is the angular frequency, k is the wave number, k c is the cutoff wave
number, r is the radial coordinate, and J 0 is a Bessel function (Appendix D).
The second issue has to do with timing. The particles are moving at close to the
speed of light, but the phase velocity of the traveling electromagnetic wave is
actually greater than the speed of light. To slow that velocity down so that the
particles can successfully “surf” the cavity, machine physicists insert metal irises
into the cavity at periodic intervals. The mathematical description of such a “discloaded” waveguide is complicated [11], but the bottom line is that the particle bunch
arrival in each section can be timed to result in acceleration by the longitudinal
electric field.
As the particle bunch propagates down the cavity, some early particles are ahead
of the voltage maximum, while some slightly later particles experience greater
accelerating voltages. As shown in Fig. 2.4, the net results are that (a) particles
become more tightly bunched and (b) the bunches gain energy from the electromagnetic wave as they are propelled down the linac. In a typical modern instrument, they
gain about ~10–20 MeV per meter. For example, each of the pair of 3.5 m final
sections of the linac for the ALBA source provides an energy gain of 52 MeV or
~15 MeV/m [12].
Fig. 2.4 Left: schematic of the electromagnetic fields inside a typical circular waveguide. Right:
illustration of a disc-loaded waveguide sequence with electrons arriving at proper time for
acceleration
2.2 The Linear Accelerator
15
Lorentz forces, we argued that to add energy to the particle, we need an electric
field along the direction of motion; hence we need a longitudinal component to the
electric field.
As illustrated in Fig. 2.4, the first trick that machine physicists employ in
designing accelerator cavities is to use special, usually cylindrical, shapes so that
electric field components are allowed only along the direction of propagation. The
example shown is for a cylindrical “pillbox” cavity, for which the z-component
(along the beam axis) E z of the electric field of an appropriate travelling wave is
given (in cylindrical coordinates) by:
E z ¼ E 0 J 0 k c r
ð ÞÁ exp i ωt À kz
ð
Þ
½
ð 2:4Þ
where ω is the angular frequency, k is the wave number, k c is the cutoff wave
number, r is the radial coordinate, and J 0 is a Bessel function (Appendix D).
The second issue has to do with timing. The particles are moving at close to the
speed of light, but the phase velocity of the traveling electromagnetic wave is
actually greater than the speed of light. To slow that velocity down so that the
particles can successfully “surf” the cavity, machine physicists insert metal irises
into the cavity at periodic intervals. The mathematical description of such a “discloaded” waveguide is complicated [11], but the bottom line is that the particle bunch
arrival in each section can be timed to result in acceleration by the longitudinal
electric field.
As the particle bunch propagates down the cavity, some early particles are ahead
of the voltage maximum, while some slightly later particles experience greater
accelerating voltages. As shown in Fig. 2.4, the net results are that (a) particles
become more tightly bunched and (b) the bunches gain energy from the electromagnetic wave as they are propelled down the linac. In a typical modern instrument, they
gain about ~10–20 MeV per meter. For example, each of the pair of 3.5 m final
sections of the linac for the ALBA source provides an energy gain of 52 MeV or
~15 MeV/m [12].
Fig. 2.4 Left: schematic of the electromagnetic fields inside a typical circular waveguide. Right:
illustration of a disc-loaded waveguide sequence with electrons arriving at proper time for
acceleration
2.2 The Linear Accelerator
15