Simulation

We have used Tracy2 [4] to evaluate the chromaticity during the ramping. The model used is the 4th order symplectic integrator. This model allows us to do a good simulation of the integrated sextupole component of the dipoles. The sextupole component is treated in the same way than the dipolar and quadrupolar component. Table 2 shows the parameters for the booster vacuum chamber and magnets.

Table 2: Parameters for the combined-function magnets.

Vacuum chamber wall thickness	D	0.7	mm
Steel conductivity	$\kappa$	$1.3\times 10^6$	$\Omega\cdot$ m
Gap in D-magnets		23.3	mm
Gap in F-magnets		26.2	mm
Minimum Energy	E_injection	0.1	GeV
Maximum Energy	E_extraction	2.4	GeV
	$\alpha_E$	1.0869
Maximum B_D dipolar field	B_extractionD	0.714	T
bending radius at maximum field	$\rho_D$	11.22	m
Maximum B_F dipolar field	B_extractionF	0.158	T
bending radius at maximum field	$\rho_F$	50.72	m
Sextupole S_F,D half-aperture		18	mm

Figure 3 shows the energy ramping in function of time for the SLS booster. We assume that the injection takes place at the start of the ramping and extraction at the end. Figure 4 shows the required ramp for the dipolar field of the bending magnets.

**Figure 3:** Energy ramping.
$\begin{figure} \begin{center} \mbox{\epsfig{figure=energy.eps, width=.9\textwidth} } \end{center}\end{figure}$

**Figure 4:** Dipolar fields (B_F,d) during ramping.
$\begin{figure} \begin{center} \mbox{\epsfig{figure=bfield.eps, width=.9\textwidth} } \end{center}\end{figure}$

The sextupolar component produced by eddy currents, during the ramping is shown in figure 5. As expected, the contribution is more important at low energies, with the maximum at 0.2 GeV (0.025 s after injection). Also, the contribution of the B_Dmagnets is much more important that the one from B_F magnets, due to the smaller bending radius in the B_D magnets. The chromaticity during the ramping cycle is shown in figure 6.

**Figure 5:** Sextupolar component generated by eddy currents in the vacuum chamber of the bending magnets.
$\begin{figure} \begin{center} \mbox{\epsfig{figure=vsext.eps, width=.9\textwidth} } \end{center}\end{figure}$

The horizontal chromaticity remains positive during the process, but we need to compensate the vertical one, that goes very soon to negative values at the start of the ramping, with a maximum of $\approx$ -5 at 0.2 GeV, and remain negative up to $\approx$ 2 GeV. However, due to the integrated sextupole component in the bending magnets, the value of the vertical chromaticity remains relatively small and the compensation with the two sextupoles families S_F and S_D will be easy.

**Figure 6:** Chromaticity during the ramping, with pole shims and eddy current contributions.
$\begin{figure} \begin{center} \mbox{\epsfig{figure=chrom1.eps, width=.9\textwidth} } \end{center}\end{figure}$