Notes from an informal meeting on the Muon Collider Lattice Design

Date: 29^th November 2022

Present: C. Carli, K. Oide, R. Tomas, K. Skoufaris

Aim: Presentation of an updated version of the Final Focusing and local Chromatic Compensation (CC) sections for a 10 TeV com muon collider and to get  feedback from experts in the field.

Main points of the presentation and discussion:

• Changes compared to the version presented on 18th November:
  • There are no sextupole pairs with almost vanishing phase advance any more. The reason is that exactly zero phase advance is not possible with finite betatron functions and distances. Now, the phase advance between sextupole pairs is π in both transverse planes.
  • The FF triplet has been replaced by four quadrupoles allowing to have smaller betatron functions (significantly smaller than inside the FF region) out side the FF region. This results in some not dramatic increase of the Montague W functions inside the FF region.
  • There are two sextupole pairs for local chromaticity correction (no 3^rd pair anymore for 2^nd order chromaticity).
  • Only the FF and CC sections, but not yet the whole lattice, have been re-designed.
• Discussion on symmetry - asymmetry of Twiss parameters in the CC section:
  • The dispersion is now identical at the two positions of the first sextupoles pair acting (mainly) on the horizontal W function. Nevertheless, the dispersion is quite different at the two sextupoles of the 2^nd sextupole pair acting mainly on the vertical W function. This should result in some 2^nd order chromaticity. This is a consequence of the present scheme based on a FODO structure with π/2 phase advance per cell (the Twiss parameters are not the periodic ones of this FODO cell). Other optics solutions, in particular between the two parts of the CC section, will be studied.
  • Some questioning on the asymmetry of transverse betatron functions (maximum horizontal ones smaller than the maximum vertical ones) in the CC section. The behavior depends on the optics before the CC section (Twiss parameters anyhow not matched to FODO structure).
• The significant 2^nd order chromaticity may have an impact on the dynamic aperture.
• Plots of betatron functions and the dispersion for particles with momentum offset may allow to better understand the optics.
• Some discussion (and confusion) about the betatron phase advance between the FF quadrupoles and the sextupoles of the CC section:
  • The phase advances are close to multiples of π in all cases. Between locations belonging to the same sextupole pairs, the phase advance is exactly π.
  • In principle the Montague W functions could be corrected as well with an additional phase advance of about π/2. Nevertheless, this probably does not allow an effective compensation for large relative momentum offsets δ and W functions (maximum of δ*W >> 1 in our case).
• Detuning with amplitude could possibly be studied even without having so far a closed lattice by tracking "large" amplitude particle along the FF and CC sections.