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In the following, all quantities are assumed to apply to the accelerating mode in the steady state. For cell number j, define an incoming power Pj,in (power from cell j− 1) and an outgoing

power Pj,out (power to cell j + 1). The difference is the power lost in that cell due to heating of the

cavity walls and the acceleration of a particle beam, i.e.

Pj,in− Pj,out= Pj,wall+ Pj,beam (2.80)

= ωUj Qj + IVj (2.81) = ωUj Qj + 2IpkjUj (2.82)

where Uj is the energy, Vj is the accelerating voltage, Qj is the quality factor, and kj is the loss

factor in cell j (Eq. 2.41 was used to obtain the final expression). Since beams come in bunches, the beam current looks like: I = νq where ν is the bunch frequency (i.e. number of bunches passing

a given point per second) and q is the charge of a single bunch.

Dividing Eq. 2.82 by the cell length L and taking the limit N ≫ 1, we get the following differential equation for the power flow along the cavity as a function of cell number (or simply position z):

dP (z) dz =−

ωu(z)

Q(z) − 2Ipk(z)u(z) (2.83) where we have defined a spatially-dependent energy per unit length u(z) _{≈ U}j/L, loss factor per

unit length k(z)_{≈ k}j/L, and quality factor Q(z)≈ Qj. The linear energy density can be expressed

in terms of the power flow via

u(z) = P (z) vg(z)

(2.84)

where vg(z) is the group velocity. Equation 2.83 becomes

dP (z) dz =− ω vg(z)Q(z) P (z)− 2I s k(z) vg(z) pP (z). (2.85)

The cavity parameters k(z), vg(z), and Q(z) are determined by periodic single-cell simulations at

frequency ω and the target phase advance for cell geometries equal to those at position z in the multicell cavity.

For multicell cavities of identical cells (usually termed constant impedance cavities) it is well-known that (for small beam currents) the power decays exponentially along the structure (as is evident from the form of Eq. 2.85 for small enough I). In this case, the accelerating gradient E(z) = 2pkP (z)/vg also decays exponentially, which means that the constant impedance structure

is a poor use of linac real estate. Ideally, the accelerating gradient would be constant along the cavity, at a level slightly below the breakdown threshold of each cell; indeed, this is the type of cavity used in high-energy accelerators today (called a constant gradient cavity).

A constant gradient is achieved by decreasing the group velocity along the length of the cavity, so that electromagnetic energy has more time to build up in later cells (where the power flow is weaker) before “leaking” into subsequent cells. Assuming a constant Q-factor and loss factor per unit length, (approximately true in most cases), the following is an extremely simple,

constant-gradient solution to Eq. 2.85: P (z) = Pin− Pin− Pout N L z (2.86) vg(z) = vg,in− vg,in− vg,out N L z (2.87)

where Pin is the power put into the first cell of the cavity, Pout is the power removed from the

last cell, and vg,in and vg,out are the group velocities of the first and last cells, respectively. The

constant electric field is given by

E = 2 q

kP (z)/vg(z) (2.88)

which gives the constraint

Pin vg,in = Pout vg,out = P (z) vg(z) . (2.89)

Plugging the solution into Eq. 2.85 gives

IV = ηcgPin (2.90)

where V = N LE (ultimately a function of Pin and vg,in) and ηcg is the constant-gradient steady-

state efficiency: ηcg= ∆vg vg,in − φN Q c vg,in (2.91)

where ∆vg = vg,in− vg,out and φ is the phase advance per cell. The first term describes a baseline

efficiency due to the power exiting the cavity at its final cell; obviously, this is wasted power (it is sent to an absorbing load). However, it is necessary, since reducing the group velocity anywhere along the structure increases the fill time of the cavity (that is, the time between the input of RF energy and the injection of the beam)—the fill time is unaccounted for in the above steady-state analysis. The second term is the hit in efficiency given by the cavity material losses.

The above constant gradient solution gives only a rough idea of the behavior of a more realistic cavity. Two major complications arise. First, the constant gradient structure described above assumed a steady-state with beam loading; however, during the filling of the cavity, a beam is not present. Thus, the fields in the structure will be higher than predicted by the above, and the

cavity will be more likely to break down. Knowledge of both the loaded (with beam) and unloaded (without beam) gradient as a function of cell number for a given structure is required. Second, the loss factor per unit length and the Q-factor change with cell geometry. These complications necessitate the direct numerical solution of Eq. 2.85. Regardless, it is clear that a multicell cavity should be designed with a tapering group velocity.

2.5.7 Wakefields

Wakefield simulations require the calculation of frequencies up to the bunch cutoff frequency, which can be several times that of the fundamental. For these HOMs, weak coupling may no longer apply. Even so, an uncoupled calculation can be performed to give a rough idea of the wake potential for a chain of slowly varying cells. It was shown in [3] that for the lowest dipole modes of a multicell pillbox cavity, the uncoupled calculation can be accurate for many accelerating mode periods behind an exciting bunch. Examining the highest group velocity in the wakefield frequency range of interest can give an estimate of a trailing distance smax below which the wake potential

should be accurate in an uncoupled calculation. For a HOM excited in one cell, the time it takes for the next cell to feel its effects is on the order of L/vg where vg is the group velocity of the HOM;

thus smax ∼ Lc/vg,max where vg,max is the highest group velocity within the HOMs of interest (in

fact, to the advantage of the method, Ref. [3] showed this to be a large underestimate).

In the uncoupled wakefield calculation, the net wake potential is determined by an average over the wake potentials in each cell, where each cell is treated as periodic. The net wake potential per unit length is

Wz(s, r, r′) = 1 N N X i W_z(i)(s, r, r′) (2.92)

where Wz(i)(s, r, r′) is the wake potential per unit length for a periodic single cell with the geometry

of cell i. The above method is commonly used to characterize new multicell cavity designs since the periodic single cell calculations need be performed only once (also, results can often be interpolated between very similar geometries).

To calculate the wake potential for a periodic single cell Wz(i)(s, r, r′) to be used in Eq. 2.92,

either the Condon method or time-domain method can be used. In the Condon method, the loss factors of all synchronous modes need to be calculated up to the bunch cutoff frequency. Then, Wz(i)(s, r, r′) for cell i is reconstructed from the impulse wake potentials. In the time-domain

method, with sufficient computational resources, a multicell cavity of identical cells (with the geometry of cell i) can be simulated to approximate the wake potential in the periodic case. As the number of cells increases, the effect of the end cells diminishes [31]. Figure 2.12 shows the asymptotic effect on the wake potential of increasing the number of simulated cells in the case of the pillbox. In all simulations, particles are emitted from and absorbed by conducting plates in the beam tubes of the end cells.