Results to compare for CSR benchmarking
from CSR Workshop 2002

This is an updated version of parameters we want to compare among the various codes. In the following $E_o$ is the beam initial total energy (it is assumed that the beam momentum $p_o$ is $p_o c \simeq E_o$), $\Lambda(s)$ stands for the charge distribution along the bunch with $\int_{-\infty}^{+\infty} ds \Lambda(s) = 1$. Given the longitudinal csr-induced field $W_{\parallel}(s,z)$ along the bunch at a given position $z$ of the beamline, we define the accumulated energy loss (from the beamline entrance to $z$) as: $\Delta E(s) =e \int_0^{z} d\tilde{z} W_{\parallel}(s,\tilde{z})$.

1. plot of $\Delta E/E_o$ vs $s(=ct)$ along the bunch and accumulated over the whole chicane (i.e. downstream of the 2 m drift). From this accumulated wake, compute the average and rms energy loss defined as:
   
   
   $$\langle \Delta E/E_o \rangle = \int_{-\infty}^{+\infty} ds (\Delta E/E_o)(s) \Lambda (s)$$
   
   
   and
   
   $$\sigma_{\Delta E/E_o } = \left[ \int_{-\infty}^{+\infty} ds ... ...ngle \Delta E/E_o \rangle \right)^2 \Lambda (s) \right]^{1/2}$$
   
   

   conventions: particles with $s$-coordinate $>0$ belong to the tail. The units are: $s$ in $\mu$m, and $\Delta E/E_o$ in %.

2. when available: plot of transverse kick (in the bending plane) purely due to the transverse CSR forces vs $s$ accumulated over the whole chicane. From this accumulated transverse kick, compute the average and rms transverse kick:
   
   
   $$\langle \Delta \zeta' \rangle = \frac{1}{E_o} \int_{-\infty}^{+\infty} ds (e\Delta W_{\perp}/E_o) (s) \Lambda (s)$$
   
   
   and
   
   $$\sigma_{\zeta ' } = \left[ \int_{-\infty}^{+\infty} ds \lef... ...lta W_{\perp}/E_o \rangle \right)^2 \Lambda (s) \right]^{1/2}$$
   
   
   where $\Delta W_{\perp}$ stands for the transverse component of the CSR-induced field.
   

   conventions: The unit for $\Delta \zeta'$ is $\mu$rad.

3. Plot of $\Delta x$ and $\Delta x'$ vs $s$. The quantities $\Delta x$ and $\Delta x'$ are the changes in position and angle, due to CSR, accumulated over the whole chicane (i.e. downstream of the 2 m drift). Similarly to (1) and (2) compute the mean and rms for $\Delta x$ and $\Delta x'$.
   conventions: The units are: $\mu$rad for $\Delta x'$ and $\mu$m for $\Delta x$ .

4. Plot temporal distribution at the end of the chicane (units are kA vs $\mu$m (again $s>0$ corresponds the bunch tail). Comment as to whether the $T_{566}$ contribution is included.
   

5. Plot of the normalized slice emittance $\varepsilon_x$ versus $s$ computed at the end of the beamline (suggestion use 1 $\mu$m bin).
   conventions: Units are mm-mrad for emittance and $\mu$m for $s$.

ASCII format request
In order to generate plots that superimpose the results of the various codes you should provide the ASCII files:

• items (1), (2) and (4) can be gathered in a 3-column file whose nomenclature is:
  column 1: $s$ in $\mu$m ($s>0$ corresponds to tail),
  column 2: $I$ in kA,
  column 3: $\Delta E/E_o$ accumulated over the whole chicane in %.
• item (3) should be provided in a separated 3-column file:
  column 1: $x$ in $\mu$m,
  column 2: $\Delta x$ in $\mu$m,
  column 3: $\Delta x'$ in $\mu$rad.
• item (5) should be provided in a separate 2-column file:
  column 1: $s$ in $\mu$m, (center of the bin within which emittance is computed, $s>0$ corresponds to tail),
  column 2: $\varepsilon_x$ in mm-mrad.