Vyacheslav Yakovlev PHY862 “Accelerator Systems”

1. Vyacheslav Yakovlev PHY862 “Accelerator Systems” Homework, Lectures 6-9

2. 1a. Why it is impossible to accelerate particles in the straight waveguide? Show that in a waveguide vph> c. Use dispersion relation for the waveguide, Lecture 6, slide 18: vph=ω/kz; 1b. Show that in a waveguide the product vph∙vgr=c2. Use dispersion relation for the waveguide, see task 1a. Group velocity is vgr =dω/dkz; 1c. Estimate the ratio of acceleration gain on the cavity axis over the gain at the cavity aperture for a/λ =0.25 and β=0.5 (Lecture 6, slide 19); note that k=2π/λ; γ = 1/(1- β2)1/2) V. Yakovlev | RF Accelerating Structures, 25-27 September, 2018

3. 2. Estimate resonant frequency f0 and R/Q for the axisymmetric cavity shown below for β=1: Note: use a simple equivalent circuit and estimate L and C suggesting that B does not depend on z, 0<z<L, and electric field E =const in the gap. Calculate f0 and R/Q for: b= 20 cm; a=5 cm; L=10 cm; h=0.1 cm L 2b 2a h ω0=(LC)-1/2 R/Q = 2(L/C) 1/2T2 T = sin(k0h/2)/(k0h/2), k0 = ω0 /c V. Yakovlev | RF Accelerating Structures, 25-27 September, 2018

4. 3. The cavity considered in previous Task is used for acceleration of the beam having the average current Ib. The energy gain is V. • Neglecting the Ohmic losses in the cavity wall estimate (Lecture 6, slide 53): • Required loaded QL; • Power Pb required for acceleration. • Calculate Pb and QL for • V=100 kV; • Ib=100 mA. L 2b 2a h V. Yakovlev | RF Accelerating Structures, 25-27 September, 2018

5. 4. The cavity has the frequency of 500 MHz, R/Q =100 Ohms and G=250 Ohms. The voltage V is 1 MV. Estimate Q0 the wall losses Ps for copper walls (Lecture 6, Slides 41 and 47) V. Yakovlev | RF Accelerating Structures, 25-27 September, 2018

6. 5. HB cavity for the PIP II project has the following parameters: • Frequency: 650 MHz; • R/Q: 630 Ohm; • G: 255 Ohm; • Voltage V: 20 MV; • Surface resistance Rs: 8.5 nOhm • Estimate: • Wall loss; • Stored energy. • (Lecture 6, Slides 41 and 47) V. Yakovlev | RF Accelerating Structures, 25-27 September, 2018

7. 6. HB cavity for the PIP II project has the following parameters: • Frequency f : 650 MHz; • R/Q: 630 Ohm; • G: 255 Ohm; • Voltage V: 20 MV; • Surface resistance: 8.5 nOhm • R.m.s. microphonics δf: 5 Hz • Beam current Ib: 2 mA • Synchronous phase φis zero • Estimate required power from the RF source (Lecture 8, slides 36-38). V. Yakovlev | RF Accelerating Structures, 25-27 September, 2018