QUAD †
A quadrupole magnet.
- DISFRIN
- If nonzero, the nonlinear Maxwellian fringe is suppressed.
- DISKIN
- If nonzero, the nonlinear term of the kinematic term is suppressed.
- DISRAD
- If nonzero, the synchrotron radiation in the particle-tracking is inhibited.
- DX
- Horizontal displacement of magnet. This applied before the rotation by
ROTATE.
- DY
- Vertical displacement of magnet. This applied before the rotation by
ROTATE.
- F1, F2
- F1 and F2 are parameters to characterize the slope of the field at the edges
defined as:
\[ F_1 = SIGN(\sqrt{a},a) \]
\[ a = 24(I_0^2/2 - I_1) \]
\[ F_2 = I_2 - I_0^3/3 \]
with
\[ I_n = \int _{-\infty} ^{\infty}(s-s_0)^n \frac{K_1}{K_{10}}ds \]
where 
is the location of the edge where the effective length is defined,
and 
is the nominal value of 
, given by the keyword .
The effects only in the first ordef of K1 is taken into account.
- FRINGE
- The effects of the linear fringe (characterized by F1 and F2), and the
nonlinear Mexwellian fringe are controled as:
DISFRIN=0 DISFRIN<>0
Nonlinear Linear Nonlinear Linear
FRINGE=0 entr & exit none none none
FRINGE=1 entr entr none entr
FRINGE=2 exit exit none exit
FRINGE=3 entr & exit entr & exit none entr & exit
- K1
- The normal 4-pole magnetic field component (times the length L).
where L is the length of the component. Positive sign means horizontal
focusing.
- L
- Effective length.
- ROTATE
- Rotation in x-y plane. After displacing the magnet by DX and DY,
rotate the magnet around the local s-axis by -(amount given by ROTATE),
then place the component. At the exit rotate back the magnet around
the local s-axis at the exit, then take out displacement.
- transformation
- The transformation in a QUAD is a sequence of:
- (nonlinear fringe at entrance)
canonical transformation by a generating function
where
\begin{array}{rcl}
H_0 &=& p_{x2} d_{x1} + p_{y2} d_{y1} \\
d _{x1} &=& x_{1} (\frac{a}{3} + b) \\
d _{y1} &=& -y _{1} (a + \frac{b}{3}) \\
a &=& \frac{K_1 x_1^2}{4p_1} \\
b &=& \frac{K_1 y_1^2}{4p_1}
\end{array}
}}
.
- (linear fringe at entrance)
\begin{array}{rcl}
p_{x2} &=& \exp(-a) p_{x1} \\
p_{y2} &=& \exp(a) p_{y1} \\
x_2 &=& \exp(a) x_1 + b p_{x1} \\
y_2 &=& \exp(-a) y_1 - b p_{y1} \\
z_2 &=& z_1 - (a x_1 + b (1 + a/2) p_{x2}) p_{x1} + (a y_1 + b (1 - a/2) p_{y2}) p_{y1}
\end{array}
}}
where 
, 
.
- (body of quad)
The body is subdivided in n = 1 + floor(10 abs(K1 L)/EPS) (EPS = 1
is used when EPS = 0), then a transversely linear transformation
exp(:H:) is done in each slice with
\[ H = ( -p + \frac{p_x^2 + p_y^2}{2p} +E/v_0 ) L +\frac{K_1 (x^2 - y^2)}{2n} \]
Between slices applied is the correction exp(:dH:) for the kinematical
term with
\[ dH=(-\sqrt{p^2-p_x^2-p_y^2}+p-\frac{p_x^2 + p_y^2}{2 p})\frac{L}{n} \]
- In a solenoid, the forms of H and dH are modified.
- (linear fringe at exit)
\begin{array}{rcl}
p_{x2} &=& \exp( a) p_{x1} \\
p_{y2} &=& \exp(-a) p_{y1} \\
x_2 &=& \exp(-a) x_1 + b p_{x1} \\
y_2 &=& \exp( a) y_1 - b p_{y1} \\
z_2 &=& z_1 + (a x_1 - b (1 - a/2) p_{x2}) p_{x1}
- (a y_1 - b (1 + a/2) p_{y2}) p_{y1} \\
\end{array}
}}
where , .
- (nonlinear fringe at exit)
canonical transformation by a generating function
\[ G(x_1,p_{x2},y_{1},p_{y2},p_1) = H_0(x_1,p_{x2},y_1,p_{y2},p_1)+ \frac{D[H_0, x_1] D[H_0, p_{x2}]+ D[H_0, y_1] D[H_0, p_{y2}]}{2} \]
where
\begin{array}{rcl}
H_0 &=& p_{x2} dx_1 + p_{y2} dy_1 \\
dx_1 &=& x_1 (\frac{a}{3} + b) \\
dy_1 &=& -y_1 (a + \frac{b}{3}) \\
a &=& - \frac{K_1 x_1^2}{4 p_1} \\
b &=& -\frac{K_1 y_1^2}{4 p_1}
\end{array}
}}
.
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