elements.asciidoc

  F(z) = \frac{1}{1 + e^{\sum\limits_{n=0}^{N_{order}} c_{n} (z/D)^{n}}}
\begin{aligned}
B_0 &= \text{Field amplitude (T)} \\
R &= \text{Bend radius (m)} \\
n &= -\frac{R}{B_{y}}\frac{\partial B_y}{\partial x} \text{ (Field index, set using the parameter } \mathrm{K1} \text{)} \\
F(z) &= \left\{
\begin{array}{lll}
    & F_{entrance}(z_{entrance}) \\
    & F_{center}(z_{center}) = 1 \\
    & F_{exit}(z_{exit})
\end{array}
\right.\end{aligned}
\begin{aligned}
y &\equiv \text{Vertical distance from magnet mid-plane} \\
\Delta_x &\equiv \text{Perpendicular distance to reference trajectory (see Figures)} \\
\Delta_z &\equiv \left\{
\begin{array}{lll}
    & \text{Distance from entrance Enge function origin perpendicular to magnet entrance face.} \\
    & \text{Not defined, Enge function is always 1 in this region.} \\
    & \text{Distance from exit Enge function origin perpendicular to magnet exit face.}
\end{array}
\right.\end{aligned}
\begin{aligned}
\nabla \cdot \vec{B} &= 0 \\
\nabla \times \vec{B} &= 0
\end{aligned}
\begin{aligned}
F'(z) &\equiv   \frac{\mathrm{d} F(z)}{\mathrm{d} z} \\
F''(z) &\equiv  \frac{\mathrm{d^{2}} F(z)}{\mathrm{d} z^{2}} \\
F'''(z) &\equiv \frac{\mathrm{d^{3}} F(z)}{\mathrm{d} z^{3}}
\end{aligned}
\begin{aligned}
B_x(\Delta_x, y, \Delta_z) &= -\frac{B_0 \frac{n}{R}}{\sqrt{\frac{n^2}{R^2} +  \frac{F''(\Delta_z)}{F(\Delta_z}}} e^{-\frac{n}{R} \Delta_x} \sin \left[ \left( \sqrt{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z)}} \right) y \right] F(\Delta_z) \\ \\
B_y(\Delta_x, y, \Delta_z) &= B_0 e^{-\frac{n}{R} \Delta_x} \cos \left[ \left( \sqrt{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z)}} \right) y \right] F(\Delta_z) \\ \\
B_z(\Delta_x, y, \Delta_z) &= B_0 e^{-\frac{n}{R} \Delta_x} \left\{\frac{F'(\Delta_z)}{\sqrt{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z)}}} \sin \left[ \left( \sqrt{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z)}} \right) y \right] \right. \\ \\
&- \frac{1}{2 \sqrt{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z)}}} \left(F'''(\Delta_z) - \frac{F'(\Delta_z) F''(\Delta_z)}{F(\Delta_z)} \right) \left[ \frac{\sin \left[ \left( \sqrt{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z)}} \right) y \right]}{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z)}} \right. \\ \\
&- \left. \left. y \frac{\cos \left[ \left( \sqrt{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z)}} \right) y \right]}{\sqrt{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z)}}} \right] \right\}\end{aligned}
\begin{aligned}
B_x(\Delta_x, y, \Delta_z) &\approx -B_0 \frac{n}{R} e^{-\frac{n}{R} \Delta_x} y \\
B_y(\Delta_x, y, \Delta_z) &\approx B_0 e^{-\frac{n}{R} \Delta_x} \left[ F(\Delta_z) - \left( \frac{n^2}{R^2} F(\Delta_z) + F''(\Delta_z) \right) \frac{y^2}{2} \right] \\
B_z(\Delta_x, y, \Delta_z) &\approx B_0 e^{-\frac{n}{R} \Delta_x} y F'(\Delta_z)
\end{aligned}
\begin{aligned}
B_x(\Delta_x, y, \Delta_z) &\approx -B_0 \frac{n}{R} e^{-\frac{n}{R} \Delta_x} y \\
B_y(\Delta_x, y, \Delta_z) &\approx B_0 e^{-\frac{n}{R} \Delta_x} \left[ 1 -  \frac{n^2}{R^2} \frac{y^2}{2} \right] \\
B_z(\Delta_x, y, \Delta_z) &\approx 0
\end{aligned}
\begin{aligned}
  c_{0} &= 0.478959 \\
  c_{1} &= 1.911289 \\
  c_{2} &= -1.185953 \\
  c_{3} &= 1.630554 \\
  c_{4} &= -1.082657 \\
  c_{5} &= 0.318111\end{aligned}
 V = z f_0(x,s) + \frac{z^3}{3!} f_1(x,s) + \frac{z^5}{5!} f_2(x,s) + \ldots
B_z = f_0 = T(x) \cdot S(s)
 S(s) = \frac{1}{2} \left[ \tanh \left( \frac{s + s_0}{\lambda_{left}} \right) -
 \tanh \left( \frac{s - s_0}{\lambda_{right}} \right) \right]
B_\phi = \sum_{n=0} f_{2n+1}(\psi) h(r) \left(\frac{z}{r}\right)^{2n+1}

B_r = \sum_{n=0}  \left[ \frac{k-2n}{2n+1} f_{2n} - \tan(\delta) f_{2n+1} \right] h(r) \left(\frac{z}{r}\right)^{2n+1}

B_z = \sum_{n=0} f_{2n}(\psi) h(r) \left(\frac{z}{r}\right)^{2n}
B_x = \sum_n B_0 \exp(mz) \frac{1}{m} \partial_x f_n y^n

B_y = \sum_n B_0 \exp(mz) \frac{n+1}{m} f_{n+1} y^n

B_z = \sum_n B_0 \exp(mz) f_n y^n
\mathbf{E} = \big(0, 0, E_0(t)\sin[2\pi f(t) t+\phi(t)]\big)
g(t) = p_0 + p_1 t + p_2 t^2 + p_3 t^3.
\begin{aligned}
    \mathbf{E_{entrance}}(\mathbf{r}, t) &= \mathbf{E_{from-map}}(\mathbf{r}) \cdot \mathrm{VOLT} \cdot \cos \left( 2\pi \cdot \mathrm{FREQ} \cdot t
        + \phi_{entrance} \right) \\
    \mathbf{E_{exit}}(\mathbf{r}, t) &= \mathbf{E_{from-map}}(\mathbf{r}) \cdot \mathrm{VOLT} \cdot \cos \left( 2\pi \cdot \mathrm{FREQ} \cdot t
        + \phi_{exit} \right)
  \end{aligned}
\begin{aligned}
    \mathbf{E} ( \mathbf{r},t ) &= \frac{\mathrm{VOLT}}{\sin (2 \pi \cdot \mathrm{MODE})} \\
    & \times \Biggl\{ \mathbf{E_{from-map}} (x,y,z) \cdot \cos \biggl( 2 \pi \cdot \mathrm{FREQ} \cdot t + \mathrm{LAG}+ \frac{\pi}{2} \cdot \mathrm{MODE} \Bigr) + \\
    & \mathbf{E_{from-map}}(x,y,z+d) \cdot \cos \biggl( 2 \pi \cdot \mathrm{FREQ} \cdot t + \mathrm{LAG} + \frac{3 \pi}{2} \cdot \mathrm{MODE} \Bigr) \Biggr\}
  \end{aligned}
\begin{aligned}
    &L = N\lambda + 2\,\text{fringe},\\
    &\text{fringe} = 2\lambda,
\end{aligned}
\begin{aligned}
    &B_x = B_0\cosh(kr)\sin(kz)\cos(\alpha),\\
    &B_y = B_0\cosh(kr)\sin(kz)\sin(\alpha),\\
    &B_z = B_0\sinh(kr)\cos(kz),
\end{aligned}
\begin{aligned}
    &B_x = B_0\cosh(kr)kze^{-(kz)^2/2}\cos(\alpha),\\
    &B_y = B_0\cosh(kr)kze^{-(kz)^2/2}\sin(\alpha),\\
    &B_z = B_0\sinh(kr)e^{-(kz)^2/2}.
\end{aligned}