Reference System for Bending Magnets

Both rectangular and sector bending magnets have a curved reference orbit. For both types of magnets

$$R=\left(\begin{array}{c} \rho(\cos\alpha-1) \\ 0 \\ \rh... ... & 1 & 0 \\ \sin\alpha & 0 & \cos\alpha \end{array}\right),$$

where $\alpha$ is the bend angle. A positive bend angle represents a bend to the right, i.e. towards negative $x$ values. For sector bending magnets, the bend radius is given by $\rho$, and for rectangular bending magnets it has the value

$$\rho = L / 2 \sin(\alpha/2).$$

If the magnet is rotated about the $s$-axis by an angle psi, $R$ and $S$ are transformed by

$$R^{*} = T R, \qquad S^{*} = T S T^{-1}.$$

where $T$ is the orthogonal rotation matrix

$$T=\left(\begin{array}{ccc} \cos\psi & -\sin\psi & 0 \\ \sin\psi & \cos\psi & 0 \\ 0 & 0 & 1 \\ \end{array}\right).$$

The special value $\psi = \pi/2$ represents a bend down.

Figure 1.4: Reference System for a Rectangular Bending Magnet; the signs of pole-face rotations are positive as shown.

Figure 1.5: Reference System for a Sector Bending Magnet; the signs of pole-face rotations are positive as shown.