QUAD

A quadrupole implemented as a matrix, up to 3rd order.
Parallel capable? : yes

Parameter Name	Units	Type	Default	Description
L	$M$	double	0.0	length
K1	$1/M^{2}$	double	0.0	geometric strength
TILT	$RAD$	double	0.0	rotation about longitudinal axis
DX	$M$	double	0.0	misalignment
DY	$M$	double	0.0	misalignment
DZ	$M$	double	0.0	misalignment
FSE		double	0.0	fractional strength error
HKICK	$RAD$	double	0.0	horizontal correction kick
VKICK	$RAD$	double	0.0	vertical correction kick
HCALIBRATION		double	1	calibration factor for horizontal correction kick
VCALIBRATION		double	1	calibration factor for vertical correction kick
HSTEERING		long	0	use for horizontal steering?
VSTEERING		long	0	use for vertical steering?
ORDER		long	0	matrix order
EDGE1_EFFECTS		long	1	include entrance edge effects?
EDGE2_EFFECTS		long	1	include exit edge effects?
FRINGE_TYPE		STRING	fixed-strength	type of fringe: "inset", "fixed-strength", or "integrals"
FFRINGE		double	0.0	For non-integrals mode, fraction of length occupied by linear fringe region
I0P	$M$	double	0.0	i0+ fringe integral
I1P	$M^{2}$	double	0.0	i1+ fringe integral
I2P	$M^{3}$	double	0.0	i2+ fringe integral
I3P	$M^{4}$	double	0.0	i3+ fringe integral
LAMBDA2P	$M^{3}$	double	0.0	lambda2+ fringe integral
I0M	$M$	double	0.0	i0- fringe integral
I1M	$M^{2}$	double	0.0	i1- fringe integral
I2M	$M^{3}$	double	0.0	i2- fringe integral
I3M	$M^{4}$	double	0.0	i3- fringe integral

A quadrupole implemented as a matrix, up to 3rd order.

Parameter Name	Units	Type	Default	Description
LAMBDA2M	$M^{3}$	double	0.0	lambda2- fringe integral
RADIAL		long	0	If non-zero, converts the quadrupole into a radially-focusing lens
GROUP		string	NULL	Optionally used to assign an element to a group, with a user-defined name. Group names will appear in the parameter output file in the column ElementGroup

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

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.

The first method is based on a third-order matrix formalism and the assumption of linear fringe fields. 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.

A more recent implementation of fringe field effects is based on integrals and is invoked by setting FRINGE_TYPE to ``integrals''. However, this method provides a first-order matrix only. This 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 $K_1$ or $K_1^2$, as appropriate.

For the exit-side fringe field, let $s_1$ be the center of the magnet, $s_0$ be the location of the nominal end of the magnet (for a hard-edge model), and let $s_2$ be a point well outside the magnet. Using $K_{1,he}(s)$ to represent the hard edge model and $K_1(s)$ the actual field profile, we define the normalized difference as $\tilde{k}(s) = (K_1(s) - K_{1,he}(s))/K_1(s_1)$. (Thus, $\tilde{k}(s) = \tilde{K}(s)/K_0$, using the notation of Zhou et al.)

The integrals to be input to elegant are defined as

$$i_0^- = \int_{s_1}^{s_0} \tilde{k}(s) ds i_0^+ = \int_{s_0}^{s_2} \tilde{k}(s) ds$$ (59)

$$i_1^- = \int_{s_1}^{s_0} \tilde{k}(s) (s-s_0) ds i_1^+ = \int_{s_0}^{s_2} \tilde{k}(s) (s-s_0) ds$$ (60)

$$i_2^- = \int_{s_1}^{s_0} \tilde{k}(s) (s-s_0)^2 ds i_2^+ = \int_{s_0}^{s_2} \tilde{k}(s) (s-s_0)^2 ds$$ (61)

$$i_3^- = \int_{s_1}^{s_0} \tilde{k}(s) (s-s_0)^3 ds i_3^+ = \int_{s_0}^{s_2} \tilde{k}(s) (s-s_0)^3 ds$$ (62)

$$\lambda_2^- = \int_{s_1}^{s_0} ds \int_s^{s_0} ds\prime \tilde{k}(s) \tilde{k}(s\prime) (s\prime-s) \lambda_2^+ = \int_{s_0}^{s_2} ds \int_s^{s_2} ds\prime \tilde{k}(s) \tilde{k}(s\prime) (s\prime-s)$$ (63)

Normally, the effects are dominated by $i_1^-$ and $i_1^+$.