Higher-order formulas of amplitude-dependent tune shift caused by a sextupole magnetic field distribution

Nowadays, designs for ring-based light sources use multibend lattices for achieving a very small emittance of around 100 pmrad. In this type of storage ring, the chromaticity correcting sextupoles generally have greater strengths than those used in typical third-generation light sources. Therefore, controlling lattice nonlinearity such as amplitude-dependent tune shift (ADTS) is important for enabling stable operations and smooth beam commissioning. As the strength of the sextupoles increases, their higher-order terms contribute significantly to ADTS, rendering well-known lowest-order formulas inadequate for describing tune variations at large horizontal amplitudes. In response, we have derived explicit expressions of ADTS up to the fourth order in sextupole strength based on the canonical perturbation theory, assuming that the amplitude of a vertical betatron oscillation is smaller compared with the horizontal one. The new formulas express the horizontal and vertical betatron tune variations as functions of the action variables: $J_x$ and $J_y$ up to $O(J_x^2)$ and $O(J_y)$. The derived formulas were applied to a five-bend achromat lattice designed for the SPring-8 upgrade. By comparing the calculated results with the tracking simulations, we found that (1) the formulas accurately express ADTS around a horizontal amplitude of $\sim 10\mathrm{mm}$ and (2) the nonlinear terms of the fourth order in sextupole strength govern the behaviors of circulating electrons at large horizontal amplitudes. In this paper, we present explicit expressions of fourth-order formulas of ADTS and provide some examples to illustrate their effectiveness.

Figure 1

Examples of ADTS for three different sets of sextupole strengths for the 5BA lattice for the SPring-8 upgrade. Betatron tunes calculated by the second-order (dashed curves) and fourth-order (solid curves) perturbation formulae are shown together with those by tracking simulations (markers).

Figure 2

The relationship between the action variable $J_x$ and the normalized coordinate $x/\sqrt{{\beta }_x}$ for the case (a) of Fig. 1.

Figure 3

The horizontal phase-space (Poincare map) for the case (a) of Fig. 1. The position and angle of electrons are normalized by the Twiss parameters ${\beta }_x$ and ${\gamma }_x$ at an observation point. Red dots represent tracking simulation results and solid curves represent perturbation calculations up to the $n$th-order in sextupole strength, with $n=0$, 1, 2, and 3.