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In this pulsed mode of operation much higher peak power pulses can feed the structure, increasing the accelerating field. As a consequence only pulsed beams can be accelerated leading to small duty cycles. Standing-wave structures can also be used for ultrarelativistic particles. In that case the π mode of operation is efficient, where the field has opposite phase in two adjacent cells. This type of structure as shown on Fig. 8, often called "nose cone structure", is very similar to the drift tube one in which the length of the tubes has been made very small.

So finally in practice it is necessary to use external magnetic fields, such as those given by solenoids or quadrupoles, to ensure a stable transverse motion within the aperture of the linac. 4 . 5 it is better to use high-β linac structures. 04 successful activity in designing low-β structures is very recent (in the past, the solution consisted of applying a very high voltage on the gun to extract particles at energies that could match a drift-tube structure). The biggest success has been the invention of the radio frequency quadrupole (RFQ) which combines three functions: electric focusing, bunching and acceleration.

The azimuthal position z is generally taken as the independant variable instead of t. 1 Non-relativistic case – Adiabatic damping The rate of energy gain for the synchronous particle is given by: d d 1 dWs =  mvs2  = ( mvs ) = eEo cos φ s   dt dz 2 dz where Eo would take into account the transit time in the case of standing-wave drift-tube linacs. For other particles the energy gain can be expressed in reduced variables: [ dw = eEo cos(φ s + ϕ ) − cos φ s dz ] In addition one has: 1 1  dt dt dϕ = ω − s  = ω −   dz dz  dz  v vs  which turns out to be: dϕ ω =− 3w dz mvs having assumed that: w = W − Ws = ( ) 1 m v 2 − vs2 ≅ mvs (v − vs ) 2 Considering small deviations from the synchronous particle one gets the following set of first-order linear equations: ω dϕ =− 3w dz mvs dw = −eEo sin φ s ϕ dz which respresent a phase harmonic oscillation: d 2ϕ 2 2 + Ωs ϕ = 0 dz with angular frequency relative to the independant variable z: Ω 2s = − eEoω sin φ s mvs3 108 showing that sin φs has to be negative for stable solutions.

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CAS-CERN Accelerator School - Basic (general) Accelerator Physics

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