ImpactX Longitudinal Spatial Coordinate

[Binabek]

Hello, I have a question on the sign of the ImpactX longitudinal "t" coordinate.
My understanding is $t_{impactX} = c \Delta t$, where $\Delta t = t - t_o$ or $\Delta t = t_o - t$. Which is the correct direction?
Here $t = \frac{s}{\beta c}$ and $t_o = \frac{s}{\beta_o c}$, where $s$ is an arbitrary point along the beam line and $\beta$ and $\beta_o$ are the relativistic velocities of the test and reference particle respectively.

[cemitch99]

@Binabek ImpactX uses the lattice position $s$ as the independent variable. At a fixed lattice position $s$, different particles will arrive at different times $t$. If $t_0$ denotes the arrival time of the reference particle, then for each particle $t_{impactX}=c\Delta t$, where $\Delta t=t-t_0$. The relationships that you give between $s$ and $t$ are correct only if there is no acceleration and all particles start the simulation with an initial value of $t=0$. (In general, this is not the case. For example, even at the source, different particles are emitted at different times.)

[Binabek]

To make sure I understand, In a drift space the conversion of the particle offset from the reference particle would then be $z = -\beta_o t_{impactX}$.

[cemitch99]

What you wrote is exactly correct for a particle moving at the reference velocity (with no transverse momentum or momentum deviation). For a general particle, it is correct to linear order in the phase space variables, and this is what I usually use if asked to plot the distribution in $z$. For most purposes, this is fine.

Strictly speaking, there are nonlinear corrections that could be important at low energy. To second order: $z=-\beta_0t_{impactX}+t_{impactX}p_t/\gamma_0^2+\ldots$. It is helpful to think of two pairs of canonical variables $(t,p_t)$ and $(z,\delta)$ related by a mixed-variable generating function $F(z,P_t)=z\left(\sqrt{1+P_t^2-\frac{2P_t}{\beta_0}}-1\right)$.

[Binabek]

That is very helpful, thanks a bunch Chad!