A simple accurate expression of the potential in electrostatic lenses. Part I : Two cylinder lenses.

Abstract

A simple accurate method is presented of calculating the potential of two-cylinder lenses separated by a finite gap under the form of an expansion of Bessel functions. The first order properties and the spherical aberration constant are calculated over a large range of potential ratios and gap widths. The results are compared with those obtained by various other methods

Expressions diverses du potentiel électrostatique et de ses dérivées pour des systèmes à symétrie de révolution composés de cylindres de même rayon et à charge d'espace nulle

“…Enfin tout dernièrement Bonjour [15,16] donne les caractéristiques optiques et les coefficients d'aberration pour des lentilles à deux ou trois cylindres en Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/rphysap:0198000150106100 déterminant le champ à partir d'une méthode décrite par Bertram [17] utilisant l'expression de l'énergie potentielle avec une approximation du troisième ordre pour le potentiel aux limites.…”

Atomic Physics with Positrons

“…Cook and Heddle (1976) applied a variational principle to derive TABLE 1 Axial potential in a two cylinder lens the approximate expressions, but were constrain~d by the need to approximate the potential in the gap of the lens by a linear variation. Bonjour (1979) showed that a cubic gave slightly better results. I have since found that it is easy to allow for the non-linear variation of potential in the gap by increasing the value used in the calculation by a small amount.…”

A simple accurate expression of the potential in electrostatic lenses. Part II : triple cylinder lenses

“…Calculation of the parameters a and b. -When the potential within the lens is expressed by (6), the electrostatic energy stored inside it is written as The terms Wi, come from the functions Vn(r, zx + gl2) representing the potential created by the lefthand gap. The terms Xij come from the functions Vn(r, zx ± dl2) related to the right-hand gap.…”

“…The values of the parameters a and b are obtained from the condition that 03B5 is an extremum with respect to variations of a and b. By writing the following values of a and b are obtained The total energy 9 is obtained by integrating the square of the electric field modulus over the system inner volume, and this gives the following expression vit VII,..., Vv express the potential in the five regions of figure 1, obtained by substituting the appropriate expression for Bertram' functions Vn (r, z) in (6).…”

“…For instance, figure 2 shows that, for the lens g = 1, a differs by more than 10 % from its limiting value when x/g is less than 0.72. The values of the axial potential are modified through expression (6); for instance, in the gap centres, the slope of the potential curve, which is proportional to a + 1, is increased of about 4 %. In a general way, the potential varies more rapidly over the gaps than it does in the case of a 1 double-cylinder lens gap, and for the same value of the axial potential, the curve slope is greater.…”