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Power loss is another important consideration in cavity design, for reasons of both feasibility and economics. Cost obviously rises with power consumption and heat dissipation is power that

cannot be reclaimed. Also, the heating of accelerator components can seriously degrade their performance (e.g. by increasing material resistivities, and thus compounding the losses problem).

RF power (usually generated by a klystron) enters an accelerator cavity and begins filling the TM010 mode with energy. Because the walls of the cavity are lossy, they begin to heat and drain

energy from the TM010 mode. When a particle bunch passes through the cavity, it gains energy

from the TM010 mode, but also loses energy by radiating wakefields. In this section, we define the

accelerator cavity figures of merit (namely, the quality factor and the shunt impedance) relating to the power lost to the cavity materials. Discussions on the calculation of power losses for different materials will also be presented along with examples using the pillbox geometry.

2.3.1 Quality factor

The quality factor, Q, is a measure of the lifetime of an electromagnetic mode in a lossy resonant cavity. It is defined for the nth resonant mode of a cavity by

Qn=

ωnUn

Pn

(2.20)

where ωn is the resonant frequency of the nth mode, Un is the stored energy, and Pn is the power

loss. Since the power loss Pn =−dUn/dt, then Un(t) = Un,0exp(−ωnt/Qn), so that the quantity,

Qn/2π, is the number of oscillation periods before the energy of that mode in the cavity falls to 1/e

of its original value. A higher Q reduces the input power required to sustain a given electromagnetic energy in the cavity (for the accelerating mode). The Q-factor is also an important indicator of troublesome non-TM010 resonant modes (called higher-order modes or HOMs) that contribute to

the wakefields; HOMs with high Qs are likely to cause problems as long-range wakefields (wakefields that affect later bunches in a bunch train) because of their longevity.

2.3.2 Shunt impedance

The shunt impedance, Rshunt, for the accelerating mode is a measure of the power required

to produce a given accelerating voltage. It is defined by

Rshunt =

V_acc2

P (2.21)

where P is the power lost to the cavity materials and Vacc, the accelerating voltage, is

Vacc≡

Z L

0

Ez(ρ = 0, z, t = z/c) dz = EaccL. (2.22)

Obviously, a larger shunt impedance is desirable. The much higher losses in copper cavities (as opposed to superconducting) demand shape optimizations to maximize the shunt impedance. For example, increasing the beam tube radius generally reduces the shunt impedance, which explains why it is usually smaller in normal conducting cavities than in superconducting cavities (also, a larger beam tube can help suppress wakefields—superconducting cavities can afford the associated decrease in shunt impedance).

2.3.3 Electromagnetic power loss in conducting walls

For conducting cavities, the Q-factor of the accelerating mode is determined by the ohmic losses in the walls, which depend on the surface magnetic fields. The differential loss per area at a conducting surface (for “good” conductors such as copper) can be estimated by

dP da = 1 2σδ|Hs| 2 ₌ Rs 2 |Hs| 2 _(2.23)

where σ is the conductivity, δ is the skin depth (the depth over which the fields inside the conductor are appreciable), Hs is the magnetic field at the surface, and Rs = 1/δσ is called the surface

resistance [37, 64]. For good conductors, where the fields do not differ much from the perfectly conducting case, Hsis usually taken to be the surface field for the perfect conductor. This technique

is used extensively in computer codes, since perfectly conducting boundary conditions are much easier to simulate. The total ohmic losses for a cavity are then given by the integral of Eq. 2.23

over the entire cavity surface. By Eq. 2.20, the Q-factor is then Qn= ωnUn Pn = µ0ωn R V |H|2dv RsR_A|Hs|2da (2.24)

where A is the cavity surface and V is the cavity volume. Since the ratio of the integrals scales as 1/ωn, Eq. 2.24 is often written as

Qn=

Gn

Rs

(2.25)

where Gnis the “geometry” factor for mode n (the form of which can be inferred from Eqs. 2.24 and

2.25. As its name indicates, the quantity Gnis independent of the cavity size (that is, a proportional

scaling of the entire cavity). The geometry factor for the TM010 mode in the pillbox is 257 Ω.

Thus, for a surface resistance of 0.028 Ω (copper at 12 GHz), Q010 = 9300. A superconducting

pillbox cavity with a surface resistance of 20 nΩ would have a Q010 of 1.3× 1010 [64]. Similarly,

integrating Eq. 2.23 using the analytic TM010 surface fields, the shunt impedances for the copper

and superconducting pillboxes are R010 = 1.8× 106 Ω (copper at 12 GHz) and R010 = 2.5× 1012

Ω (Rs= 20 nΩ), respectively.

2.3.4 Electromagnetic power loss in dielectrics

Dielectrics also incur losses, quantified by the loss tangent, or imaginary part of the permit- tivity. If we define an isotropic complex permittivity as ε(x) = εr(x) + iεi(x), then the power lost

to the heating of dielectrics for mode n is calculated from

Pdiel= 1 2ωn Z V εi(x)|En(x)|2d3x (2.26)

where V is the cavity volume and εi(x) = 0 outside the dielectric [37]. Using Eq. 2.20, Eq. 2.26,

and the low-loss approximation to the stored energy, Eq. 2.10, the Q-factor due purely to dielectric heating is Qdiel≈ R V εr(x)|En(x)|2d3x R V εi(x)|En(x)|2d3x (2.27) ≈ _{tan δ}1 _UUn n,diel (2.28)

tan(δ)_|| 4.8_{× 10}−6 _{2.7× 10}−8 _{5.0× 10}−9

tan(δ)⊥ 9.1× 10−6 5.9× 10−8 5.0× 10−9

Table 2.1: Loss tangents of single-crystal sapphire for different temperatures near 10 GHz resonant frequency. Subscripts (_{||) and (⊥) refer to responses in the direction of the c-axis of the crystal and} transverse to the c-axis, respectively [48].

where Un,diel is the energy in the dielectric volume and tan δ≡ εi/εr is called the loss tangent, the

inverse of which is seen to be a lower limit on the Qdiel of the cavity (if the dielectric completely

fills the cavity or if the electric fields for mode n are confined to the dielectric volume, then Qn → 1/ tan(δ)). An example of a very low-loss dielectric is sapphire with loss tangents near

10−5 _{for GHz frequencies at room temperature—decreasing with temperature (see Table 2.1). In}

principle, if a TM010 mode can be confined with sapphire as efficiently as it can be confined with

copper, Q-factors and shunt impedances 100 times larger are feasible.

2.3.5 Section summary

Wall losses in superconducting cavities are miniscule; thus, the acceleration efficiencies (shunt impedances) are enormous. Unfortunately, the critical magnetic field enforces a strict limit on the maximum accelerating gradient, and thus the length of a superconducting accelerator. Low-loss dielectrics like sapphire may be the next best bet, since their losses can be orders of magnitude lower than those of conductors and their Eacc-limit could be larger (especially under the right

conditions—high frequency, short electromagnetic pulses). Confining radiation with conductors, however, is much easier than confining radition solely with dielectrics. Photonic crystals may alleviate this problem and are discussed in great detail in the next chapter.