Why the drift tubes in an RF linac get longer

A metal tube provides a place to wait

A positive ion speeds up in a forward electric field. An RF field, however, repeatedly reverses direction. The push that helps now can pull backward a little later. If the ion remains exposed for equal portions of both directions, energy gained during one part can be taken back during the other.

A drift tube shelters the ion during an unhelpful part of the cycle. In the ideal shielding approximation, there is no accelerating axial field inside the metal tube. The ion coasts through at its existing speed, then emerges into an exposed gap where the field can do work. Tube length controls the waiting time; gaps supply the energy.

Phase describes where an arrival falls within the repeating field cycle. Shift the arrival by half a cycle and a forward push becomes a backward pull. Two ions can therefore encounter the same peak voltage and still leave with very different energies, simply because they arrived at different times.

Sources: [1][2]

Higher speed needs a longer sheltered journey

In an Alvarez drift-tube arrangement, neighbouring gaps share the same field phase, called zero mode. A chosen design particle is timed to travel between neighbouring gap centres in one full RF period. It has moved to another gap, but the field has returned to the same part of its cycle.

That reference particle becomes faster after each useful crossing. It covers more distance during the next period, so the later tubes grow longer. Their extra length does not provide extra accelerating work. It keeps the faster particle inside the shield until the next suitable moment to emerge.

A different drift-tube arrangement can put neighbouring gaps in opposite phases and use a half-period timing rule. Those rules cannot be interchanged. Multiplying an entrance speed by a period starts a spacing estimate. The finished design follows the changing speed through each finite gap.

Sources: [1][2]

Early and late particles need the right feedback

There is one design particle but a source supplies an ensemble. Some ions leave earlier; others start with more energy. Even a well-timed reference does not ensure that nearby particles stay close to it. Their small timing errors can shrink or grow as energy gain changes their subsequent speed.

Consider a forward field that is becoming stronger. An early particle receives a smaller push and then travels a little more slowly. A late one receives a larger push and travels a little faster. This feedback can tighten their arrival times. Longitudinal bunching means bringing particles closer together along their direction of travel.

On the falling side of the field, the mean gain can be similar while the feedback reverses: early particles become faster and late ones slower. Arrival times spread apart. Equal average exit energy therefore does not establish equal bunch quality. The range of initial errors and the remaining flight distance also limit how far the restoring argument applies.

Sources: [1]

Even a narrow gap takes time to cross

The field keeps changing while an ion crosses a gap. It may weaken, or even reverse, before the crossing finishes. Actual work accumulates along the moving trajectory. Peak voltage alone cannot say how much energy every ion receives from each gap.

For a uniform gap with nearly constant crossing speed, a transit-time factor describes the difference between this accumulated action and an ideal instantaneous encounter. Slow ions, wide gaps and higher frequencies let the field rotate further during a crossing. When acceleration changes the speed appreciably, that change must be solved along with the flight.

Measurements can follow concrete events: two timing pickups within one shielded tube provide a flight time over a known distance. With a separately known ion species, that speed gives kinetic energy. Comparing the energies in neighbouring tubes tests the work done in the intervening gap. Distances, clocks and pickup responses each need calibration.

The terminal arrival distribution shows how tightly the detected ensemble reaches the exit. A narrow pulse does not by itself demonstrate cooling: a larger energy spread can accompany a smaller timing spread. Both coordinates belong in longitudinal phase space. Acceleration and bunching can be explained without claiming lower temperature or successful transmission of every incoming particle.

Sources: [3][1]