NIBEND

A numerically-integrated dipole magnet with various extended-fringe-field models.
Parallel capable? : yes

Parameter Name	Units	Type	Default	Description
L	$M$	double	0.0	arc length
ANGLE	$RAD$	double	0.0	bending angle
E1	$RAD$	double	0.0	entrance edge angle
E2	$RAD$	double	0.0	exit edge angle
TILT		double	0.0	rotation about incoming longitudinal axis
DX	$M$	double	0.0	misalignment
DY	$M$	double	0.0	misalignment
DZ	$M$	double	0.0	misalignment
FINT		double	0.5	edge-field integral
HGAP	$M$	double	0.0	half-gap between poles
FP1	$M$	double	10	fringe parameter (tanh model)
FP2	$M$	double	0.0	not used
FP3	$M$	double	0.0	not used
FP4	$M$	double	0.0	not used
FSE		double	0.0	fractional strength error
ETILT	$RAD$	double	0.0	error rotation about incoming longitudinal axis
ACCURACY		double	0.0001	integration accuracy (for nonadaptive integration, used as the step-size)
MODEL		STRING	linear	fringe model (hard-edge, linear, cubic-spline, tanh, quintic, enge1, enge3, enge5)
METHOD		STRING	runge-kutta	integration method (runge-kutta, bulirsch-stoer, modified-midpoint, two-pass modified-midpoint, leap-frog, non-adaptive runge-kutta)
SYNCH_RAD		long	0	include classical synchrotron radiation?
ADJUST_BOUNDARY		long	1	adjust fringe boundary position to make symmetric trajectory? (Not done if ADJUST_FIELD is nonzero.)

A numerically-integrated dipole magnet with various extended-fringe-field models.

Parameter Name	Units	Type	Default	Description
ADJUST_FIELD		long	0	adjust central field strength to make symmetric trajectory?
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

For the NIBEND element, there are various fringe field models available. In the following descriptions, $l_f$ is the extend of the fringe field, which starts at $z=0$ for convenience in the expressions. Also, $K = \frac{1}{g}\int_-\infty^\infty F_y(z) (1-F_y(z)) dz$ is K. Brown's fringe field integral (commonly called FINT), where $g$ is the full magnet gap and $\vec{F} = \vec{B}/B_0$, $B_0$ being the value of the magnetic field well inside the magnet.

• Linear fringe field:
  

  $$F_y = z F_a$$ (23)

  $$F_z = y F_a$$ (24)

  $$F_a = 1/(6 K g)$$ (25)

  
  
  For this model, the user specifies FINT and HGAP only.

• Cubic-spline fringe field:
  

  $$F_y = F_a z^2 + F_b z^3 + y^2 (-F_a - 3 F_b z)$$ (26)

  $$F_z = (2 F_a z + 3 F_b z^2) y$$ (27)

  $$F_a = 3/l_f^2$$ (28)

  $$F_b = -2/l_f^3$$ (29)

  $$l_f = 70 K g/9$$ (30)

  
  
  For this model, the user specifies FINT and HGAP only.

• Tanh-like fringe field:
  

  $$F_y = \frac{1}{2} (1 + \tanh F_a z) + \frac{1}{2} (y F_a {\rm sech} F_a z )^2 \tanh F_a z +$$ (31)

  $$\frac{1}{24} (y F_a {\rm sech} F_a z)^4 {\rm sech} F_a z (11 \sinh F_a z - \sinh 3 F_a z)$$ (32)

  $$F_z = \frac{1}{2} y F_a {\rm sech}^2 F_a z + \frac{1}{6} (y F_a {\rm sech} F_a z)^3 {\rm sech} F_a z (2 - \cosh 2 F_a z)) +$$ (33)

  $$\frac{1}{120} (y F_a {\rm sech} F_a z)^5 {\rm sech} F_a z (33 - 26 \cosh 2 F_a z + cosh 4 F_z z)$$ (34)

  $$F_a = 1/(2 K g)$$ (35)

  $$l_f = P_1/F_a$$ (36)

  
  
  For this model, the user specifies FINT and HGAP, along with the parameter FP1, which is the quantity $P_1$ in the last equation. It determines the length of the fringe field that is integrated.

• Quintic-spline fringe field, to third order in y:
  

  $$F_y = (F_a z^3 + F_b z^4 + F_c z^5) + y^2 z (3 F_a + 6 F_b z + 10 F_c z^2)$$ (37)

  $$F_z = y (3 F_a z^2 + 4 F_b z^3 + 5 F_c z^4) + y^3 (-F_a - 4 F_b z - 10 F_c z^2)$$ (38)

  $$F_a = 10/l_f^3$$ (39)

  $$F_b = -15/l_f^4$$ (40)

  $$F_c = 6/l_f^5$$ (41)

  $$l_f = 231 K g /25$$ (42)

  
  

  For this model, the user specifies FINT and HGAP only.

• Enge model with 3 coefficients:
  

  $$F_0 = \frac{1}{1 + e^{a_1 + a_2 z/D + a_3 (z/D)^2}}$$ (43)

  $$F_y = F_0 - \frac{1}{2} y^2 F_0^{(2)} + \frac{1}{24} y^4 F_0^{(4)}$$ (44)

  $$F_z = y F_0^{(1)} - \frac{1}{6} y^3 F_0^{(3)} + \frac{1}{120} y^5 F_0^{(5)}$$ (45)

  
  
  where $F_0^{(n)} = \frac{\partial^n F_0}{\partial z^n}$.

  The user may choose ``enge1'', ``enge3'', or ``enge5'', where the number indicates the order of the expansion of $F_z$ with respect to $y$.

  The need only specify FINT and HGAP. The Enge parameters are then automatically determined to give the correct linear focusing.

  However, if user gives non-zero value for FP2, then FINT and HGAP are ignored. FP2, FP3, and FP4 and taken as the Enge coefficients $a_1$, $a_2$, and $a_3$, respectively.