From Syphers and Edwards, eq. 2.15, the contours of particle motion are given by
2 β 2 Es eV β 2 Es eV 2 (cos φ + φ sin φ s ) = ΔE + (cos φ + φ sin φ s ) = constant ΔE + ηω RFτ ηπ h 2
For coasting beam below transition, η < 0 , φ s = 0 , and the endpoints of stable motion are φ = ±π / 2 . To find the constant for the separatrix, we plug in φ = π / 2,ΔE = 0 and get
β 2 Es eV constant = ηπ h
We can then find the bucket height by pluggin in φ = 0 to obtain
(ΔEb )
2
2 β 2 Es eV = ηπ h 2 1 η π h ( ΔEb ) →V = e 2Es β 2
