Longitudinal calculations cheat sheet

Relativistic relationships

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

Machine parameters

• Magnetic rigidity
  $$B\rho = \frac{p}{q}$$
• Revolution period/frequency
  $$T=\frac{2\pi R}{v}=\frac{1}{f}=\frac{2\pi}{\omega}$$
• RF period/frequency
  $$f_{\mathrm{rf}}=h f=\frac{\omega_{\mathrm{rf}}}{2\pi}=\frac{1}{T_{\mathrm{rf}}}$$
• Linear momentum compaction factor (and transition factor)
  $$\alpha_c=\frac{\Delta R/R}{\Delta p/p}=\frac{1}{\gamma^2_{t}}$$
• Linear phase slippage factor
  $$\eta=\frac{\Delta T/T}{\Delta p/p}=-\frac{\Delta f/f}{\Delta p/p}=\alpha_c-\frac{1}{\gamma^2}$$
• Other useful relationship
  $$p=\frac{\beta^2 E}{\omega}$$

Longitudinal equations of motion

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

Bucket parameters and synchrotron motion

• Bucket height
  $$\Delta E_{\mathrm{max}}=\beta\sqrt{\frac{2qVE}{\pi h |\eta|}}Y\left(\phi_s\right)$$
• Bucket height reduction factor
  $$Y\left(\phi_s\right)=\left|\cos\phi_s-\frac{\pi-2\phi_s}{2}\sin\phi_s\right|^{1/2}$$
• Bucket area
  $$A_{\mathrm{bk}}=16\frac{\beta}{\omega_{\mathrm{rf}}}\sqrt{\frac{qVE}{2\pi h |\eta|}}\alpha\left(\phi_s\right)$$
• Bucket area reduction factor
  $$\alpha\left(\phi_s\right)\approx\frac{1-\sin\phi_s}{1+\sin\phi_s}$$
• Angular synchrotron frequency (small amplitudes)
  $$\Omega_s^2=2\pi f_s=-\omega^2\frac{h\eta qV\cos\phi_s}{2\pi\beta^2E}$$
• Non-linear synchrotron frequency for a maximum amplitude in phase $\phi_u$
  $$\frac{\Omega\left(\phi_u\right)}{\Omega_s}\approx1-\frac{\phi_u^2}{16}$$
• Synchrotron tune
  $$Q_s = \frac{\Omega_s}{\omega}$$

Differential relationships

C. Bovet et al., A selection of formulae and data useful for the design of A.G. synchrotrons, CERN-MPS-SI-Int-DL-70-4

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