Hemispherical Deflection Analyzer (HDA)¶

Idealized HDA showing electrons with energies 1000 eV (red), 900 eV (green), and 1100 eV (blue).

2D cross-section of previous figure.

Potential energy (PE) view of previous figure.

The Hemispherical Deflection Analyzer (HDA) consists of two concentric hemispherical electrodes held at different potentials. Charged particles of only a certain kinetic energy (KE), called the pass energy, can pass through the electrodes at constant radius, and particles at energies sufficiently lower or higher will hit the walls, so the system acts as a narrow band KE filter.

A picture of a SIMION simulation of an HDA is shown in Figures 1-3. This HDA allows through electrons with energies between roughly 900 and 1100 eV. The HDA also has a focusing effect: 1000 eV electrons (red) focused at the top entrance regain focus at the bottom entrance.

A real HDA has fringe fields at the entrance and exit, but for easier understanding we may approximate it without fringe fields. This idealized HDA assumes the same electric field as the so-called spherical capacitor, consisting of two concentric spherical electrodes held at different potentials. The spherical capacitor has simple analytic formulas for potential $V$ and electric field $\vec{E}$ at radius $r$ :

(1) $V(r) = -(k/r) + c$

(2) $\vec{E}(r) = - \bnabla V = -(k/r^2) \hat{r}$

where

(3) $\begin{align*} c & = (R_2 V(R_2) - R_1 V(R_1)) / (R_2 - R_1) k \\ & = (V(R_2) - V(R_1)) \cdot R_1 \cdot R_2 / (R_2 - R_1) \\ \hat{r} & = \textrm{unit vector in radial direction} \end{align*}$

The pass energy $w$ at $R_0$ is $w = K(R_0) = q k / (2 R_0)$ , for particle charge $q$ .

Modeling¶

Typical GEM files for sperical capacitors are given below. Here, $k$ = -200000, $c$ = -2000, $R0$ = 100, $K(R0)$ = 1000 eV, and $V(R0)$ = 0. The inner and outer radii are $R1$ = 80 mm and $R_2$ = 120 mm.

Here is a typical 2D GEM file:

; sc2d.gem
pa_define(130,130,1,cylindrical,xy)
; outer electrode, R2
e(-333.3333) { fill { notin  { circle(0,0,119) } } }
; inner electrode, R1
e( 500.0000) { fill { within { circle(0,0,80) } } }

Here is a typical 3D GEM file:

; sc3d.gem
pa_define(130,130,130,planar,xyz)
; outer electrode, R2
e(-333.3333) { fill { notin  { sphere(0,0,0,119) } } }
; inner electrode, R1
e( 500.0000) { fill { within { sphere(0,0,0,80) } } }

Notice that the radius in the notin command is one grid unit less than R2 for improved accuracy. If creating a PA# file, you may replace the two voltages with 1 and 2 to permit fast adjusting.

The 3D GEM file consumes a lot more RAM than the 2D version, but if you need to add any feature to this geometry that breaks the 2D symmetry (e.g. entrance/exit slits), you likely will need the 3D form.

If you omit the “x” mirroring in the pa_define statement, then an idealized HDA will be represented (i.e. just one half of the sphere). The boundary condition for potentials on the x=0 plane is assumed by SIMION to be approximately a Neumann condition (derivative of potential along normal to the x=0 plane equal to zero), which is the same condition that exists in the spherical capacitor. In theory, an idealized HDA could be constructed by covering the X=0 plane with an ideal (100% transmission) grid that has a variable potential over its surface exactly matching Equation (1). In practice, such a grid can only be approximated (e.g. with a series of thin wires), and some fringing fields will always exist near this X=0 plane.

To model a real HDA with fringe fields, you would include in your potential array the space and electrodes surrounding the entrace/exit (like in the SIMION “hda” example – Figure 4).

SIMION “hda” example (Zouros), run at higher mesh resolution and displayed with cross section cut-out.

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Hemispherical Deflection Analyzer (HDA)¶

Modeling¶

See also¶

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