Intrinsic nonlinear effects of dipole magnets in small rings

We find that dynamic aperture depends significantly on the bending radii of dipole magnets when designing a small storage ring for Tsinghua Thomson scattering X-ray source (TTX) mainly because of the nonlinearity of the dipole field. In this paper, we present systematic studies on the intrinsic-geometric nonlinearity of dipole magnets. The Hamiltonian approach is used to determine the expressions of the geometric nonlinear potential and the corresponding third-order resonance strengths. Simulations are conducted to study these resonances. Our analysis results agree well with the tracking results at the third-order resonances $3{\nu }_x=\ell$ and ${\nu }_x\pm 2{\nu }_z=\ell$, where $\ell$’s are the integer multiple of the number of superperiods.

Figure 1

Schematic of the example ring. The lattice consists of 4 dipoles (indices from B1 to B4) and 2 quadrupoles (Q1 and Q2). The Frenet-Serret coordinate system is represented by $(\mathbf{x},\mathbf{z},\mathbf{s})$.

Figure 2

Layout and the corresponding betatron amplitude and dispersion functions of a 4.8-m ring, which serves as an example to analyze the $3{\nu }_x=\ell$ like third-order resonances.

Figure 3

Poincaré map showing the tracking data points of the particles in the horizontal normalized phase space $(X,P_x)$. Each color denotes the phase-space trajectory of one particle.

Figure 4

Tracking data points and fitted Hamiltonian tori in the horizontal action-angle space.

Figure 5

Resonance strength in cases with different bending radii $\rho$.

Figure 6

Betatron amplitude and dispersion functions of a 4.8 m ring, which is used to study the third-order coupling resonance at ${\nu }_x-2{\nu }_z=\ell$.

Figure 7

The upper two plots are the horizontal and vertical normalized phase space, respectively. The lower left plot shows the actions $J_x$, $J_z$, and $J_2$ vs turns. The lower right plot shows the tracking data of the test particle, the corresponding linear fit equation, and the R-squared of the fitting.

Figure 8

Tracking data and the corresponding fitted curve.