Emittance sharing and exchange driven by linear betatron coupling in circular accelerators

The influence of linear betatron coupling due to constant-in-time skew quadrupolar fields on the transverse emittances is discussed using both a simplified model of a smooth circular accelerator and a more realistic strong-focusing lattice with localized sources of coupling (thin lens). New formulas for the coupled transverse emittances are derived that include the initial emittances, the coupling strengths, and the tune distance from the resonance. By using the more powerful Lie algebra and the resonance driving terms formalism, equivalent formulas are derived that provide a better understanding of some counterintuitive effects, otherwise not understandable in the smooth approximation. The new formulas have been tested both numerically and experimentally by using data of the CERN Proton Synchrotron showing a remarkable agreement.

Figure 1

RMS emittances against $\Delta =Q_x-Q_y-l$, $l\in \mathbf{N}$ from multiparticle simulations compared with Eqs. (3, 4) for $|C|=0.01$ (top) and $|C|=0.002$ (bottom).

Figure 2

Vertical (top) and horizontal (bottom) turn-by-turn RMS emittance ($|C|=0.0128$). Simulations are repeated for three working points $\Delta =0.028,0.01,0.0$. Horizontal lines correspond to the average RMS emittance. Stars are derived from Eqs. (54, 55).

Figure 3

$|C_0|$ against $\Delta$ for $|C|=0.01$ (left) and $|C|=0.002$ (right).

Figure 4

The same RMS emittances of Fig. 1 compared with Eqs. (3, 4) and the new formulas, Eqs. (56, 57), for $|C|=0.01$ (top) and $|C|=0.002$ (bottom).

Figure 5

Multiparticle simulations of the turn-by-turn RMS emittance oscillations for two different amounts of coupling and three working points. The three lines correspond to different locations where the emittances were computed.

Figure 6

Single-particle invariants, $2I_{x,y}$ (left) and phase of $f_{1001}$ $q$ (right) against $\Delta =Q_x-Q_y-l$, during the resonance crossing (numerical simulation).

Figure 7

Dynamical crossing from multiparticle simulations: the RMS emittances are plotted together with Eqs. (77, 78) and Eqs. (30, 31) for several starting points. The coupling strength is $|C|=0.01$.

Figure 8

$|{\overline{f}}_{1001}|$ from Eq. (40) (blue line) compared with $|f_{1001}|$ from Eq. (88) against $\Delta$. Note how on the resonance, $\Delta =0$, $|{\overline{f}}_{1001}|=1/4$ whereas $|f_{1001}|=\pi /8$. The coupling strength is $|C|=0.02$

Figure 9

Time dependence of the tunes $Q_{x,y}$ as set in the control room to cross the resonance (upper plot). Measured eigentunes $Q_{u,v}$ for the case with the skew quadrupole current of $-0.5\mathrm{A}$ (bottom plot): the resulting coupling strength $|C|=0.055$ is inferred from the closest-tune approach.

Figure 10

Transverse physical emittances at $2\sigma$ measured in the PS while crossing the resonance with a coupling strength $|C|=0.055$. The dashed lines correspond to the curves predicted by Eqs. (30, 31).

Figure 11

Emittance-exchange curves measured in the PS for two extreme cases with large coupling, $|C|=0.120$ (upper plot), and small coupling, $|C|=0.016$ (bottom plot). The dashed lines correspond to the curves predicted by Eqs. (30, 31). The curves denote the physical emittances at $2\sigma$.