Geometric Aspects of Optimizing the Acceptance of a Static Mass Analyzer

Abstract

The axial aberration of the ion-optical system of a static mass analyzer is one of the main factors limiting the transmission and sensitivity of the instrument. We described an algorithm for selecting the optimal collimating system for a mass analyzer, which makes it possible to decrease this aberration effectively, with minimal loss in the intensity of the mass spectral lines. The algorithm is based on the representation of the considered aberration in the phase space as an ellipse and the optimal approximation of this aberration ellipse by a polygon with a given number of vertices. In the absence of constraints, the solution of the corresponding optimization problems is in phase polygons generated by regular polygons inscribed in a unit circle or circumscribed around a unit circle with the same number of vertices. After that, the unit circle along with the polygons is “stretched” to the size of the specified aberration ellipse along its main axes. Additional freedom of the rotation of the original polygon around the center of the unit circle can be used to position auxiliary slits of the collimating system more conveniently.

APPENDIX

Statement 1. Of all N-gons inscribed in the unit circle, a regular N-gon has the largest area.

Evidence. An N-gon inscribed in a unit circle, up to rotation around the center of the circle, is uniquely characterized by a set of central angles φ₁, φ₂, …, φ_N formed by radii connecting the center of the circle and the vertices of the N-gon (Fig. 4a). If not all angles φ₁, φ₂, …, φ_N are the same, then there are two adjacent angles, one of which is larger than the other. Let, for example, φ₁ < φ₂. If we change these two angles according to the rule φ₂ = φ₂ – δ, ${{\hat {\varphi }}_{1}}$ = φ₁ + δ, where δ = (φ₂ – φ₁)/2 > 0, leaving the remaining angles unchanged, then the condition ${{\hat {\varphi }}_{1}}$ = ${{\hat {\varphi }}_{2}}$ = (φ₁ + φ₂)/2 is fulfilled, and the area of the inscribed N-gon increases (Figs. 4b and 4c). Indeed, the sum of the areas of triangles A₁OA₂ and A₂OA₃ is

$$\frac{1}{2}{{R}^{2}}\sin {{\varphi }_{1}} + \frac{1}{2}{{R}^{2}}\sin {{\varphi }_{2}} = \frac{1}{2}{{R}^{2}}\sin \Phi \cos \varphi ,$$

Fig. 4.

Polygons inscribed in a circle: (a) a one-to-one relationship with a set of central angles, (b) a local area between two unequal central angles, and (c) a locally optimized local area.

where R is the radius, Φ = (φ₁ + φ₂)/2, and φ = (φ₂ – φ₁)/2 and reaches a maximum at φ = 0. Therefore, an N-gon that differs from a regular N-gon, in which all central angles are the same, is not optimal.

Statement 2. Of all N-gons circumscribed around the unit circle, a regular N-gon has the smallest area.

Evidence. An N-gon circumscribed around the unit circle, up to rotation around the center, is uniquely characterized by a set of central angles φ₁, φ₂, …, φ_N formed by the radii connecting the center of the circle and the points of tangency of the sides of the N-gon with the circle, since as each tangent line and the points of intersection of the tangents (the vertices of the N-gon) are uniquely restored from the points of tangency (Fig. 5a). If not all angles φ₁, φ₂, …, φ_N are the same, then there are two adjacent angles, one of which is larger than the other. Let, for example, φ₁ < φ₂. If we change these two angles according to the rule ${{\hat {\varphi }}_{2}} = {{\varphi }_{2}}$ – δ, ${{\hat {\varphi }}_{1}} = {{\varphi }_{1}}$ + δ, where δ = (φ₂ – φ₁)/2 > 0, leaving the remaining angles unchanged, then the condition ${{\hat {\varphi }}_{1}} = {{\hat {\varphi }}_{2}}$ = (φ₁ + φ₂)/2 is fulfilled, and the area of the circumscribed N-gon decreases (Figs. 5b and 5c). Indeed, the sum of the areas of triangles B₁OA₁, B₁OA₂, B₂OA₂, and B₂OA₃ is

Fig. 5.

Polygons circumscribed around a circle: (a) a one-to-one relationship with a set of central angles, (b) a local area between two unequal central angles, and (c) a locally optimized local area.

$$\begin{gathered} \frac{1}{2}{{R}^{2}}\tan \left( {\frac{{{{\varphi }_{1}}}}{2}} \right) + \frac{1}{2}{{R}^{2}}\tan \left( {\frac{{{{\varphi }_{1}}}}{2}} \right) + \frac{1}{2}{{R}^{2}}\tan \left( {\frac{{{{\varphi }_{2}}}}{2}} \right) \hfill \\ + \,\,\frac{1}{2}{{R}^{2}}\tan \left( {\frac{{{{\varphi }_{2}}}}{2}} \right) = \frac{{2{{R}^{2}}\sin \Phi }}{{\cos \Phi + \cos \varphi }}, \hfill \\ \end{gathered} $$

where R is the radius, Φ = (φ₁ + φ₂)/2, and φ = (φ₂ – φ₁)/2 and reaches a minimum at φ = 0. Therefore, an N-gon that differs from a regular N-gon, in which all central angles are the same, is not optimal.

Statement 3. Of all N-gons circumscribed around the unit circle, a regular N-gon has the least maximum distance from the center of the circle to its vertices.

Evidence. An N-gon circumscribed around a unit circle, up to rotation around the center of the circle, is uniquely characterized by a set of central angles φ₁, φ₂, …, φ_N formed by radii connecting the center of the circle and the points of tangency of the sides of the N-gon with the circle (Fig. 5a). The maximum distance from the center of the circle to the vertex of the N-gon is achieved for the maximum central angle φ_k and is R/cos(φ_k/2). If not all angles φ₁, φ₂, …, φ_N are the same, then there are two adjacent angles, one of which has the maximum value, and the neighboring one is smaller. For example, let these be the angles φ₁ and φ₂, with φ₁ < φ₂. If we change these two angles according to the rule ${{\hat {\varphi }}_{2}} = {{\varphi }_{2}}$ – δ, ${{\hat {\varphi }}_{1}} = {{\varphi }_{1}}$ + δ, where δ = (φ₂ – φ₁)/2 >0, leaving the remaining angles unchanged, then the condition ${{\hat {\varphi }}_{1}}$ = ${{\hat {\varphi }}_{2}}$ = (φ₁ + φ₂)/2 is fulfilled. Then the distance OB₂ decreases, and the adjacent distance OB₁ does not increase so much as to exceed the new value for OB₂ (Figs. 5b and 5c). Repeating this operation for all central angles that have the maximum value at the moment, we get an N-gon with a smaller maximum distance from the center to the vertices. Therefore, an N-gon that differs from a regular N-gon, in which all central angles are the same, is not optimal.

Note. To avoid an infinite chain of iterations, it is necessary to select adjacent angles φ₁ and φ₂ at each step so that the larger of them is greater than 2π/N, and the smaller of them is less than 2π/N (which is always achievable, since in the sequence φ₁, φ₂, …, φ_N, the angles can be swapped to produce equivalent N-gons with the same values for any of the considered criteria). New angles ${{\hat {\varphi }}_{2}} = {{\varphi }_{2}}$ – δ, ${{\hat {\varphi }}_{1}} = {{\varphi }_{1}}$ + δ, where δ > 0, must be chosen so that at least one of them becomes 2π/N. As a result, for a finite number of steps, a regular N-gon is obtained (the only one suitable up to rotation around the center), while the criterion being optimized is invariably improved at each step. Now the statement is obvious that the only regular N-gon up to rotation ensures the largest (smallest) value for the criterion under consideration, and that the optimum is attainable rather is a limiting point, to which one can approach arbitrarily close but never reach it. The use of trigonometric expressions in the above calculations was needed only for a more concise presentation of the data, since the same results can be obtained from purely geometric considerations, and more elegantly.