Longitudinal calculations cheat sheet

Relativistic relationships

Mass energy
$E_{0} = m_{0} c^{2}$ $E_0=m_0 c^2$
Total energy
$E = E_{k i n} + E_{0}$ $E=E_{\mathrm{kin}}+E_0$
Momentum
$E = \sqrt{(p c)^{2} + E_{0}^{2}}$ $E=\sqrt{(p c)^2+E_0^2}$
$p = γ m_{0} β c$ $p=\gamma m_0 \beta c$
Relativistic velocity
$β = \frac{v}{c} = \frac{p c}{E} = \sqrt{1 - \frac{1}{γ^{2}}}$ $\beta = \frac{v}{c} = \frac{p c}{E} = \sqrt{1-\frac{1}{\gamma^2}}$
Lorentz factor
$γ = \frac{E}{E_{0}} = \frac{1}{\sqrt{1 - β^{2}}}$ $\gamma = \frac{E}{E_0} = \frac{1}{\sqrt{1-\beta^2}}$
Differential relationships
$\frac{d p}{p} = \frac{1}{β^{2}} \frac{d E}{E} = γ^{2} \frac{d β}{β}$ $\frac{dp}{p}=\frac{1}{\beta^2}\frac{dE}{E}=\gamma^2\frac{d\beta}{\beta}$

Magnetic rigidity
$B ρ = \frac{p}{q}$ $B\rho = \frac{p}{q}$
Revolution period/frequency
$T = \frac{2 π R}{v} = \frac{1}{f} = \frac{2 π}{ω}$ $T=\frac{2\pi R}{v}=\frac{1}{f}=\frac{2\pi}{\omega}$
RF period/frequency
$f_{r f} = h f = \frac{ω_{r f}}{2 π} = \frac{1}{T_{r f}}$ $f_{\mathrm{rf}}=h f=\frac{\omega_{\mathrm{rf}}}{2\pi}=\frac{1}{T_{\mathrm{rf}}}$
Linear momentum compaction factor (and transition factor)
$α_{c} = \frac{Δ R / R}{Δ p / p} = \frac{1}{γ_{t}^{2}}$ $\alpha_c=\frac{\Delta R/R}{\Delta p/p}=\frac{1}{\gamma^2_{t}}$
Linear phase slippage factor
$η = \frac{Δ T / T}{Δ p / p} = - \frac{Δ f / f}{Δ p / p} = α_{c} - \frac{1}{γ^{2}}$ $\eta=\frac{\Delta T/T}{\Delta p/p}=-\frac{\Delta f/f}{\Delta p/p}=\alpha_c-\frac{1}{\gamma^2}$
Other useful relationship
$p R = \frac{β^{2} E}{ω}$ $pR=\frac{\beta^2 E}{\omega}$

Equation of motion 1 - Phase slippage (drift) along the ring (continuous)
$\frac{d ϕ}{d t} = \frac{h η ω}{p R} (\frac{Δ E}{ω}) = \frac{h η ω^{2}}{β^{2} E} (\frac{Δ E}{ω})$ $\frac{d\phi}{dt}= \frac{h \eta \omega}{p R} \left(\frac{\Delta E}{\omega}\right)= \frac{h \eta \omega^2}{\beta^2 E} \left(\frac{\Delta E}{\omega}\right)$
Equation of motion 2 - Energy kick with single RF cavity (continuous)
$\frac{d}{d t} (\frac{E}{ω}) = \frac{q V}{2 π} (\sin ϕ - \sin ϕ_{s})$ $\frac{d}{dt}\left(\frac{E}{\omega}\right)=\frac{qV}{2\pi}\left(\sin\phi-\sin\phi_s\right)$
Equation of motion 1 - Phase slippage (drift) along the ring (discretized over $T$ )
$ϕ_{n + 1} = ϕ_{n} + 2 π h η \frac{Δ E_{n}}{β^{2} E}$ $\phi_{n+1} = \phi_n + 2 \pi h \eta \frac{\Delta E_n}{\beta^2 E}$
Equation of motion 2 - Energy kick in cavity (discretized over $T$ , $\dot\omega$ neglected)
$Δ E_{n + 1} = Δ E_{n} + q V \sin (ϕ_{n + 1}) - U_{0}$ $\Delta E_{n+1} = \Delta E_n + q V \sin (\phi_{n+1}) - U_0$
Beam energy gain per turn
$U_{0} = q V \sin ϕ_{s}$ $U_0=qV\sin\phi_s$

Bucket height
$Δ E_{m a x} = β \sqrt{\frac{2 q V E}{π h | η |}} Y (ϕ_{s})$ $\Delta E_{\mathrm{max}}=\beta\sqrt{\frac{2qVE}{\pi h |\eta|}}Y\left(\phi_s\right)$
Bucket height reduction factor
$Y (ϕ_{s}) = {| \cos ϕ_{s} - \frac{π - 2 ϕ_{s}}{2} \sin ϕ_{s} |}^{1 / 2}$ $Y\left(\phi_s\right)=\left|\cos\phi_s-\frac{\pi-2\phi_s}{2}\sin\phi_s\right|^{1/2}$
Bucket area
$A_{b k} = 16 \frac{β}{ω_{r f}} \sqrt{\frac{q V E}{2 π h | η |}} α (ϕ_{s})$ $A_{\mathrm{bk}}=16\frac{\beta}{\omega_{\mathrm{rf}}}\sqrt{\frac{qVE}{2\pi h |\eta|}}\alpha\left(\phi_s\right)$
Bucket area reduction factor
$α (ϕ_{s}) \approx \frac{1 - \sin ϕ_{s}}{1 + \sin ϕ_{s}}$ $\alpha\left(\phi_s\right)\approx\frac{1-\sin\phi_s}{1+\sin\phi_s}$
Angular synchrotron frequency (small amplitudes)
$Ω_{s}^{2} = 2 π f_{s} = - ω^{2} \frac{h η q V \cos ϕ_{s}}{2 π β^{2} E}$ $\Omega_s^2=2\pi f_s=-\omega^2\frac{h\eta qV\cos\phi_s}{2\pi\beta^2E}$
Non-linear synchrotron frequency for a maximum amplitude in phase $\phi_u$
$\frac{Ω (ϕ_{u})}{Ω_{s}} \approx 1 - \frac{ϕ_{u}^{2}}{16}$ $\frac{\Omega\left(\phi_u\right)}{\Omega_s}\approx1-\frac{\phi_u^2}{16}$
Synchrotron tune
$Q_{s} = \frac{Ω_{s}}{ω}$ $Q_s = \frac{\Omega_s}{\omega}$

Variables	Equations
$B, p, R$	$\frac{d p}{p} = γ_{t}^{2} \frac{d R}{R} + \frac{d B}{B}$ $\frac{dp}{p}=\gamma_t^2\frac{dR}{R}+\frac{dB}{B}$
$f, p, R$	$\frac{d p}{p} = γ^{2} \frac{d f}{f} + γ^{2} \frac{d R}{R}$ $\frac{dp}{p}=\gamma^2\frac{df}{f}+\gamma^2\frac{dR}{R}$
$B, f, p$	$\frac{d B}{B} = γ_{t}^{2} \frac{d f}{f} + \frac{γ^{2} - γ_{t}^{2}}{γ^{2}} \frac{d p}{p}$ $\frac{dB}{B}=\gamma_t^2\frac{df}{f}+\frac{\gamma^2-\gamma^2_t}{\gamma^2}\frac{dp}{p}$
$B, f, R$	$\frac{d B}{B} = γ^{2} \frac{d f}{f} + (γ^{2} - γ_{t}^{2}) \frac{d R}{R}$ $\frac{dB}{B}=\gamma^2\frac{df}{f}+\left(\gamma^2-\gamma_t^2\right)\frac{dR}{R}$