Worked example 7.8.3: A copper pillbox cell of an S-band linac

About the worked examples of Section 7.8

Three short calculations tied to the applications, with resonant frequencies, $Q$ , bandwidth, stored energy and wall loss. They use only the results of this section. A good way to use them: read the given quantities, produce the formula from the corresponding application yourself, and only then put the numbers in. Constants: $c = 3 \times 1 0^{8} m/s$ , $ε_{0} = 1 0^{- 9} /36 π F/m$ , $η_{0} = 120 π Ω$ , copper $σ_{c} = 5.8 \times 1 0^{7} S/m$ , $c /2 = 15 GHz cm$ , $c /2 π = 4.775 GHz cm$ .

Worked example 7.8.3

A copper pillbox cell of an S-band linac

cf. Application 7.8.3

Given. A copper TM $_{010}$ cell at $f = 2.856 GHz$ (the frequency of many electron linacs), length $d = λ /2$ , on-axis peak field $E_{0} = 20 MV/m$ . The electrons travel at $v \approx c$ .

Find. (a) $λ$ , $a$ , $d$ , (b) $Q$ , the bandwidth and the field time constant $2 Q / ω$ , (c) the stored energy and the wall loss, (d) the energy gain per cell, $V = E_{0} d T$ with $T = \frac{s i n ( π d / λ )}{π d / λ}$ , and the shunt impedance $R = V^{2} / P$ .

Try it yourself first.