Longitudinal beta is given by1: τη τη 1 βL = = − ;γ L = 2 2 2 π ES β ν s eV0ω rf ES β cos ϕ s βL The bucket equation is given by2
%π ( 1− ' − ϕ s * tan ϕ s &2 ) ΔEb = 2 ω rf β L 3 For coasting beam, this becomes
2 4eV0 Es β 2 2eV0 Es β 2 ΔEb = = = ω rf β L τ η ω rf τ η frf π
The required voltage will then be given by 2
! 1 $ ( ΔEb ) τ η frf π V0 = # & "e% 2Es β 2 2
! 1 $ ( ΔEb ) C η frf π =# & "e% 2cEs β 3
2
! 1 $ ( ΔEb ) η hπ =# & " e % 2Es β 2
1 From my USPAS Longitudinal Motion Lecture, http://home.fnal.gov/~prebys/misc/uspas_2014/08-‐ Longitudinal%20Motion%201.pptx, p. 8 2 ibid. p. 12 3 You can also get this from Chao and Tigner, “Accelerator Physics”, 2.1.2, eq. 12, by plugging in φ=φs=0 and rearranging some terms.