Review of dynamic aperture studies

Yiton YanSLAC

Dynamic aperture (DA) studies• tracking

• Default method: symplectic element by element tracking – however, attention will be more on nonlinear maps.

• Taylor-map tracking – accurate, but not exactly symplectic

• Generating-type mixed-variable map tracking – implicit procedure, accurate, symplectic, but with singularities.

• Irwin factorization

• Integrable polynomial

• Resonance basis map and nPB tracking

• Optimization of dynamic aperture – parameterized Lie map for Resonance basis map vs. direct DA optimization (which uses GA).

SSC collider ring

• Largest ring for DA study.

• Highly nonlinear • small DA • small interest

region • fast convergence• Accurate one-turn

Taylor map

Direct 12th-order and 11th-order Taylor map tracking

• Compare the 11th-order with the 12th –order Taylor map tracking

• Not exactly symplectic, seems OK with this much turn?

Directly 12th-order Taylor map tracking

• Compared with element-by-element tracking

• 5 cm dipole

• SSC 4-cm dipole case

• 1 million turn• 11th-order

Taylor map• Seems OK.

• But 10th-order Taylor map tracking shows biased artificial diffusion due to not syplectic.

• But is it accurate?

Lie transformation

Dragt-Finn factorization

Can be trasnferred back to Taylor map with one-order (or more orders) higher

• The 10th-order Taylor map is re-expanded to the 11th-order Taylor

map and works!• Degree of accuracy symplecticity

Mixed variable map and Generating function

• It is a mixed variable Vps obtained through, implicitly, generating function, but no need to get the generating function.

• It retains the same accuracy compared to the same-order Taylor

• It is symplectic at any truncated order• Tracking time is about twice of the same-order Taylor map

tracking, but of course much faster compared to several orders higher Taylor map.

• But with singularities - OK.

Conversion

Implicit mixed-variable power-series tracking

Direct iteration will do.

The same SSC 4-cm diameter dipole injection lattice with mixed-variable map tracking

Irwin (Kick) Factorization

• A map is completely reconstructed into a minimum number of kick maps such with an order by order accuracy while push the errors to higher order.

Very elegant idea. Very fast. However, Spurious terms present unknown accuracy concern

Explicit integrable polynomial

where

Symplectic integrator is then used for separating the integrable polynomials.

Lower homogeneous order terms uses higher order symplectic integrator while higher homogenous order terms uses lower order symplectic integrator or even just separate them.

Computation time is slower than minimized-term Irwin factorization. However, it is more naturally close to the original Lie map and therefore generally less concerns about accuracy.

Use of symplectic integrator for separation of homgeneous integrable polynomial

Resonance basis map and nPB tracking for PEP-II HER and LER dynamics aperture stuides

• Taylor map can be trasnformed into a Deprit-type Lie map and then further transformed into the following resonance-basis map

This is a goal-moving soccer game

nPB tracking

nPB converge very fast. Not exactly symplectic. OK for PEP-II HER and LER tracking 1000 turns are needed for PEP-II due to damping

PEP II Swamp plot

• See resonances• Easier to

choose an working tune

Side issue – tune shift with amplitude

• Normalized tune shift and resonance driving terms

• They are analytical

• Can this be used for indirect dynamic aperture optimization

otherwise the direct dynamic aperture optimization is not analytical and so must use GA which usually take longer time.

All done. Thanks.