10.79 QUAD—A quadrupole implemented as a matrix, up to 3rd order. Use KQUAD for symplectic tracking.

A quadrupole implemented as a matrix, up to 3rd order. Use KQUAD for symplectic tracking.
Parallel capable? : yes
GPU capable? : yes
Back-tracking capable? : yes

A quadrupole implemented as a matrix, up to 3rd order. Use KQUAD for symplectic tracking.

This element simulates a quadrupole using a matrix of first, second, or third order.

Length specification As of version 29.2, this element incorporates the ability to have different values for the insertion and effective lengths. This is invoked when LEFFECTIVE is positive. In this case, the L parameter is understood to be the physical insertion length. Using LEFFECTIVE is a convenient way to incorporate the fact that the effective length may differ from the physical length and even vary with excitation, without having to modify the drift spaces on either side of the quadrupole element.

Fringe effects By default, the element has hard edges and constant field within the defined length, L. However, this element supports two different methods of implementing fringe fields. Which method is used is determined by the FRINGE_TYPE parameter.

Edge integral method The most recent and preferred implementation of fringe field effects is based on edge integrals and is invoked by setting FRINGE_TYPE to “integrals”. This method is compatible with the use of LEFFECTIVE. However, it provides a first-order matrix only.

The model is based on publications of D. Zhuo et al. [34] and J. Irwin et al. [35], as well as unpublished work of C. X. Wang (ANL). The fringe field is characterized by 10 integrals given in equations 19, 20, and 21 of [34]. However, the values input into elegant should be normalized by K₁ or K₁², as appropriate.

For the exit-side fringe field, let s₁ be the center of the magnet, s₀ be the location of the nominal end of the magnet (for a hard-edge model), and let s₂ be a point well outside the magnet. Using K_1,he(s) to represent the hard edge model and K₁(s) the actual field profile, we define the normalized difference as (s) = (K₁(s) - K_1,he(s))∕K₁(s₁). (Thus, (s) = (s)∕K₀, using the notation of Zhou et al.)

The integrals to be input to elegant are defined as

Normally, the effects are dominated by i₁^- and i₁⁺.

Trapazoidal models This method is based on a third-order matrix formalism and the assumption that the fringe fields depend linearly on z. Although the third-order matrix is computed, it is important to note that the assumed fields do not satisfy Maxwell’s equations.

To invoke this method, one specifies “inset” or “fixed-strength” for the FRINGE_TYPE parameter and then provides a non-zero value for FFRINGE. If FFRINGE is zero (the default), then the magnet is hard-edged regardless of the setting of FRINGE_TYPE. If FFRINGE is positive, then the magnet has linear fringe fields of length FFRINGE*L/2 at each end. That is, the total length of fringe field from both ends combined is FFRINGE*L.

Depending on the value of FRINGE_TYPE, the fringe fields are modeled as contained within the length L (“inset” type) or extending symmetrically outside the length L (“fixed-strength” type).

For “inset” type fringe fields, the length of the “hard core” part of the quadrupole is L*(1-FFRINGE). For “fixed-strength” type fringe fields, the length of the hard core is L*(1-FFRINGE/2). In the latter case, the fringe gradient reaches 50% of the hard core value at the nominal boundaries of the magnet. This means that the integrated strength of the magnet does not change as the FFRINGE parameter is varied. This is not the case with “inset” type fringe fields.

Misalignments

There are three modes for implementing alignment errors. Which is used is controlled by the value of the MALIGN_METHOD parameter:

• MALIGN_METHOD=0 — This selects the original method, which was the only one available before version 2021.1. The misalignment is referenced to the entrance face. The YAW and PITCH parameters are ignored.
• MALIGN_METHOD=1 — This selects a linearized method based on M. Venturini’s work [58], with misalignment referenced to the entrance face. The YAW and PITCH parameters are implemented.
• MALIGN_METHOD=2 — This selects a linearized method based on M. Venturini’s work [58], with misalignment referenced to the magnet center. The YAW and PITCH parameters are implemented.
• MALIGN_METHOD=3 — This selects an exact method based on M. Venturini’s work [58], with misalignment referenced to the entrance face. The YAW and PITCH parameters are implemented.
• MALIGN_METHOD=4 — This selects an exact method based on M. Venturini’s work [58], with misalignment referenced to the magnet center. The YAW and PITCH parameters are implemented.

For elements with non-zero TILT, error displacements and rotations are performed in the lab frame.