Quadrupoles in Electron Lens Design

1. c. Quadrupoles and octopoles only. The formulae of §b expressed in terms of the quantities β², k(z) and n defined in Eqs. (11a–11f). (n = −1 corresponds to purely electrostatic quadrupoles, n = 0 to magnetic quadrupoles and, as we shall see later, n = σ² or nonrelativistically, n = 1 corresponds to achromatic quadrupoles.)

   $\begin{array}{ll}F& =\frac{1}{6}{\beta }^4{k}^2\left(\frac{3n^2}{{\sigma }^2}-2n+3\right)+\frac{1}{24}{\beta }^2{k}^{\prime \prime }-\mathrm{\Xi }\\ \overline{F}& =\frac{1}{6}{\beta }^4{k}^2\left(\frac{3n^2}{{\sigma }^2}-2n+3\right)-\frac{1}{24}{\beta }^2{k}^{\prime \prime }-\mathrm{\Xi }\\ K& =-\frac{1}{2}{\beta }^4{k}^2\left(\frac{n^2}{{\sigma }^2}-2n+1\right)+3\mathrm{\Xi }\\ L& =-\frac{3}{8}{\beta }^2{k}^{\prime }.\end{array}$

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2. d. Quadrupoles and octopoles: asymptotic aperture aberrations. (Formulae in which the potential functions appear undifferentiated.)

   $\begin{array}{l}\begin{array}{ll}{\alpha }^3:\left(0030\right)=& {\int }_{-\infty }^{\infty }\left\{\frac{1}{96}\left(\frac{7+16\mathit{\varepsilon V}}{V^2}{D}^2-24\mathit{\eta \sigma }\frac{\mathit{DQ}}{V^{3/2}}+32{\eta }^2\frac{Q^2}{V}\right)H_x^4\right.\\ \left.-\frac{1}{8}\left(\frac{\mathit{\sigma D}}{V}-4\eta \frac{Q}{V^{1/2}}\right)H_x^2{H}_x^{\prime 2}\right\}\mathit{dz}-{\int }_{-\infty }^{\infty }\mathrm{\Xi }{H}_x^4\mathit{dz}\\ {\gamma }^3:\left(0003\right)=& {\int }_{}^{}\end{array}\end{array}$