| Parameter Description | Argument Order | Argument Name | Units | Default Value |
|---|---|---|---|---|
| Length | 1 | L | cm | 0.0 |
| Axial Field | 2 | B | kG | 0.0 |
| Effective Aperture | 3 | a | degrees | 0.0 |
The letters "C" and "c" denote a solenoid. The first parameter ($L$) is the effective length of the magnet in cm. It is defined as follows:
$$ L = \frac{1}{B_0} \int_{-\infty}^{\infty} B_s(s) ds $$The second parameter is the magnetic field in kG. The third parameter, $a$ is the effective aperture radius of the solenoid. When $a=0$ - or equivalently, is simply omitted- $L$ is the same as the physical length and the transfer matrix is that of a hard edge solenoid i.e.
$$ M = \left[ \begin{array}{cccc} \frac{1+c}{2} & \frac{s}{k} & \frac{s}{2} & \frac{1-c}{k} \\ -\frac{ks}{4} & \frac{1+c}{2} & -k\frac{1-c}{4} & \frac{s}{2} \\ -\frac{s}{2} & -\frac{1-c}{k} & \frac{1+c}{2} & \frac{s}{k} \\ k\frac{1-c}{4} & -\frac{s}{2} & -\frac{ks}{4} & \frac{1+c}{2} \end{array} \right] \;\; \begin{eqnarray} s & = & \sin{kL} \\ c & = & \cos{kL} \\ k & = & \frac{eB_0}{2p_0} \end{eqnarray} $$with $B_0$ representing the magnetic field amplitude at the center of the solenoid. When $a$ is non-zero, a correction related the extent of the solenoid edge field region is introduced. This correction results in a decrease of the total focusing compared to the ideal hard-edge case. OptiM assumes the following analytical form to model the on-axis magnetic field:
$$ \begin{eqnarray} B_z(s) & = & \frac{1}{2}B_0 [ 1 + \tanh{( (s+ L/2)/a )} ] \; s < 0 \\ & = & \frac{1}{2}B_0 [ 1 - \tanh{( (s- L/2)/a )} ] \; s > 0 \end{eqnarray} $$Using this form, the resulting strength of the focusing correction kicks at each edge is: $$ \Phi_{\hbox{edge}} = -\frac{1}{2} \left[\frac{e}{p}\right]^2 \left[ \int_{-\infty}^{\infty} B_s^2(s) ds - B_0^2L \right] = -\frac{k^2 a}{2} $$ where $k$ is the cyclotron wavenumber. Since $$ k^2 = \left[\frac{e}{2p}\right]^2 B_0^2 $$ one has $$ a = \frac{1}{B_0^2} \int B_s^2(s) ds - L $$ In practice, given a measured on-axis field profile, one would use the later to determine $a$ from the above. Note that the position of the edge focusing kick is determined by the effective length. This means that the edge kick corrections are applied at positions that are slightly shifted with respect with those of the physical edges. In tracking simulations, the program takes into account the first nonlinear focusing term $(\Delta\Phi r^3)$.
Examples: C1 L[cm]=10 B[kG]=20 c1 L[cm]=1100 B[kG]=1