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6 10-2, will travel a distance of roughly 1 meter in half a cycle. Clearly the length of the drift tubes will soon become prohibitive at higher energies unless the input RF frequency is increased. Higher-frequency power generators only became available after the second world war, as a consequence of radar developments. However at higher frequencies the system, which is almost capacitive, will radiate a large amount of energy; as a matter of fact if one considers the end faces of the drift tubes as the plates of a capacitor, the displacement current flowing through it is given by I = ω CV where C is the capacitance between the drift tubes, V the accelerating voltage and ω the angular frequency in use.

The disks act like capacitive loads and reduce the speed of propagation as in loaded transmission lines. It is usual to draw the Brillouin diagram for the type of propagating wave under consideration. This diagram relates the frequency to the propagation factor (Fig. 16). Fig. 16 Brillouin diagram The straight line vp = c separates the two domains corresponding respectively to slow and fast waves. For the latter, as obtained in a normal guide, the relation 97 ω 2 ω 2 ω c2 = 2 − 2 v 2p c c gives a hyperbola for a given ωc.

For a slow wave it will exit an operating point P in the diagram and the corresponding phase velocity is given by tgα = vp/c. If ω varies, P moves on a certain curve; the slope of this curve at point P is: tgθ = d (ω / c) 1 dω 1 = = vg c dβ c d ω / vp ( ) where vg = (dβ/dω)−1 is called the group velocity and happens to be equal to the velocity of the energy flow in the waveguide: vg = ve Exercise: Calculation of the energy flow velocity The average power which flows through a transverse cross-section of a waveguide is given by the integral of the Poynting vector: P= 1 Re ( ET × HT ) dS 2 ∫S where only the transverse components of the field have to be considered.

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

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