Quadrupoles in Electron Lens Design

1.2: The short quadrupole: The bell-shaped model

The best analytical model is now Glaser's bell-shaped model

$k\left(z\right)=\frac{1}{{\left\{1+{\left(z/d\right)}^2\right\}}^2}$

  (26)

or in the asymmetric case, two half-bells:

$\begin{array}{ll}k\left(z\right)=\frac{1}{{\left\{1+{\left(z/p\right)}^2\right\}}^2}& z⩽0\\ k\left(z\right)=\frac{1}{{\left\{1+{\left(z/q\right)}^2\right\}}^2}& z⩾0\end{array}$

  (27)

The asymmetric case is considered separately in §1.3. The excitation parameter β² is now given by

${\beta }^2=\frac{V}{a^2{\mathrm{\Phi }}_0\left(1+\varepsilon {\mathrm{\Phi }}_0\right)}\left(1-\frac{a^2}{3d^2}\right)$

  (28a)

or

${\beta }^2={\mu }_0\eta \frac{\mathit{NI}}{a^2}\frac{1}{{\left\{{\mathrm{\Phi }}_0\left(1+\varepsilon {\mathrm{\Phi }}_0\right)\right\}}^{1/2}}\left(1-\frac{a^2}{3d^2}\right)$

  (28b)

where as usual, a denotes the bore-radius (Hardy, 1967) and for a short ...