Figure III.3 is a schematic representation of the Alvarez Linac. As compared with the Wideroe Linac, it is characterized by the polarization of the drift tubes, due to the TM010 mode. The voltage drop is concentrated along the gaps between the isopotential drift tubes. On average, the voltage gradient is constant along the axial direction. Since the length of the drift tubes, as

in the Wideroe case, has to increase as the particle is being accelerated, in order to keep the synchronism, it follows that the voltage drop between two adjacent drift tubes increases with the particle energy, in contrast to the case of the Wideroe accelerator where this voltage drop is independent of the particle energy.

In the case shown on figure III.3, the particle travels one drift tube length during one period of the RF. This means that

I = j3X. (III.39)

As compared to the Wideroe I = (f3/2)X relation, this implies a length of the accelerator twice as long. For this reason, side-coupled cavity Linacs are used at the highest energies, where it is very important to reduce the length of the Linac as much as possible. In the side-coupled Linac, cavities work on the I = (f3/2)X mode but particles cross only one cavity out of two, the odd one for example. This way the particles still see cavities with the same polarization but in the I = (f3/2)X mode, which allows division by two of the length of the accelerator.

III.1.2 Phase stability

Up to now we have assumed that particles were exactly in phase with the RF and that they were accelerated at the maximum field. However, in this case, any small deviation of the particle energy from the resonance energy would lead to a loss of synchronism. For example, a particle with less than the resonance energy would take more time than the synchronous particle to reach the following gap and thus be accelerated by a reduced electric field and, therefore, become even more distant in energy from the synchronous particle. These particles would eventually be lost. Due to this mechanism, beam intensities would be extremely reduced. It has been realized that, if the particles cross the inter-drift tube gap at a time when the electric field is rising, particles with energies larger than that of the resonant one cross the next gap at an earlier phase and, thus, for a smaller field value and, there­fore, their energy comes closer to that of the resonant particles. On the contrary, particles with lower energy than that of the resonant one will cross the next gap at a later phase, and, thus, for a larger field value; therefore their energy again comes closer to that of the resonant particles. This principle, which allows capture in the accelerated beam of particles with non­resonant energies, is called the phase stability principle. Although it makes a low loss acceleration of intense beams possible, it has some drawbacks:

• It decreases the effective accelerating field, and thus requires longer accelerators.

• It leads to a weaker focusing, or even a defocusing of the beam. This can be seen in figure III.4. Indeed, the figure shows that, in static fields, particles

Figure III.4. Partial representation of the electric field line in an Alvarez Linac. The tank and the drift tubes are in black. Electric field lines are shown with dashed lines. A non-axial particle trajectory is shown with a dotted line. Note the modification of the field lines due to the presence of the drift tubes. While the drift tubes are equipotential the full potential difference appears along the gaps.

exiting from the drift tube at an angle to the beam are first deflected towards the beam axis. In the second half of the gap they are deflected off the beam axis, but less strongly than they are towards it in the first half, since the deflecting field becomes smaller the closer it is to the beam axis. The net result of these deflections is a focusing effect. However, if the electric field is increasing while the particle crosses the gap, the deflection off the beam axis in the second half of the gap may exceed that in the first half. The net result may be a defocusing of the beam, or at least a weaker focusing. It is usually necessary to correct this effect by the interposition of magnetic focusing devices such as quadrupoles along the beam axis. Note that focusing and defocusing by the electric field are only significant at low energies where the increment of the particle energy within the gap is noticeable with respect to the particle energy itself.