Sensitive High-Resolution Ion Microprobe

A spectrometer designed to measure the energy, energy spread, and mass-to-charge ratio of a particle beam, can be simulated by solving the equations of motion for ions in an electric and magnetic field. Such a device can be most easily understood by looking at the ideal design, where the electric and magnetic fields are at their theoretically optimum values, which are easily derived.

Model Definition

Using some basic mathematics, the magnitude and direction of the electric and magnetic fields (or to be precise, the magnetic flux density) can be computed to ensure 100 % transmission for an ion of a given initial energy and charge to mass ratio.

The geometry of a basic Sensitive High-Resolution Ion Microprobe (SHRIMP) is shown below. The basic operating principle involves isolating particles of a specified energy and mass-to-charge ratio by subjecting the beam to a radial electric field followed by a uniform magnetic field.

Figure 1: Basic SHRIMP geometry. Particles enter through the emitter, and are subjected first to an electric field directed radially inward, and then to a uniform out-of-plane magnetic flux density.

force balance

The toroidal electrostatic analyzer filters particles according to their kinetic energy. For a particle to propagate through the analyzer along a circular arc of radius r (SI unit: m), the centrifugal force acting on the particle must balance the electric force, that is:

(1)

where

•

m (SI unit: kg) is the particle mass,

•

v (SI unit: m/s) is the particle velocity magnitude,

•

q (SI unit: C) is the absolute value of the particle charge, and

•

Er (SI unit: V/m) is the radial component of the electric field.

Using the classical definition of kinetic energy, ε = mv2/2, the radial component of the electric field should be:

(2)

This can be conveniently entered into COMSOL using an expression.

The magnetic sector filters particles according to their mass-to-charge ratio. For particles to propagate through a toroidal region along a circular arc of radius r while subjected to a uniform magnetic flux density B (SI unit: T), the centrifugal force must once again balance the magnetic force:

(3)

where v (SI unit: m/s) is the particle velocity. Since the force should only be produced in the xy-plane, the magnetic flux density should only have a z-component:

(4)

For a sulfur hexafluoride ion with an initial energy of 50 eV, electrostatic analyzer with mean radius 1.272 m, and magnetic sector of mean radius 1 m, this results in an electric field magnitude of 78.616 V/m and a magnetic flux density magnitude of 0.012346 T.

The goal of the model is to compute the nominal trajectory of the particle beam and the fraction of ions that reach various points along the beam path (transmission probability).

release mechanism

A realistic velocity distribution of the ions entering the modeling domain is necessary. The feature provides may different ways to characterize the initial beam position and velocity. A common way of characterizing the beam is to use Twiss parameters and emittance, which give a measure of the transverse velocity distribution of the particles. It is also possible to specify the phase space ellipse dimensions directly, as indicated in Figure 2.

Figure 2: Beam specification using phase space ellipse dimensions.

In this case, the maximum transverse displacement is equal to the size of the inlet aperture and the maximum transverse velocity is 0.001 times the velocity norm, corresponding to a beam which diverges very slowly. Despite this, a significant fraction of the beam current will be lost as the beam travels through 4 additional apertures en route to the detector.

Results and Discussion

The trajectories of the particle beam are indicated in Figure 3 below. The beam starts at (0,0,0) and travels along the y direction. Part of the beam is lost at the first aperture. After traveling 2 m in the y direction, the beam encounters the radial electrostatic force, whose magnitude is selected to ensure maximum transmittance. If there were ions present with a different charge number or initial energy, they would not reach the third aperture, at the other end of the electrostatic separator.

At the boundaries where the electric and magnetic forces change discontinuously, a node with a condition is used. This improves the accuracy of the calculation of the particle trajectories during time steps in which they enter or leave one of these regions.

Figure 3: Plot of the particle trajectories through the system.

Once the ions cross the third aperture, they travel in the negative x direction until they enter the domain containing the magnetic filter. The uniform out-of-plane magnetic flux density causes the beam to follow a circular arc in this region. The magnetic flux density is chosen so that only the sulfur hexafluoride ions are transmitted through the forth aperture toward the detector. Any ions with a different initial energy or charge to mass ratio would be separated during this stage.

The nominal trajectory of the beam is shown in Figure 4. This is the average position of the beam, where the averaging only occurs for particles which reach the detector. Each feature in the model creates variables to filter particles based on the domain or boundary they occupy, allowing convenient visualization of the nominal trajectory of a beam which connects a source and a detector.

Figure 4: Plot of the nominal trajectory of the particles transmitted from the inlet to the detector.

The transmission probability of the beam in this example is about 0.3.

Reference

1. https://en.wikipedia.org/wiki/Sensitive_high-resolution_ion_microprobe

Particle_Tracing_Module/Charged_Particle_Tracing/sensitive_high_resolution_ion_microprobe

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Geometry 1

In the toolbar, click and choose .

Browse to the model’s Application Libraries folder and double-click the file sensitive_high_resolution_ion_microprobe_geom_sequence.mph.

In the dialog box, click .

In the toolbar, click

Load the model’s variables, which define the electric and magnetic fields.

Definitions

Variables 1

In the window, under right-click and choose .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file sensitive_high_resolution_ion_microprobe_variables.txt.

Global Definitions

Load the model’s physics parameters.

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file sensitive_high_resolution_ion_microprobe_phys_parameters.txt.

Define selections.

Definitions

Electrostatic Analyzer

In the toolbar, click

In the window for , type Electrostatic Analyzer in the text field.

Select Domain 6 only.

Magnetic Sector

In the toolbar, click

In the window for , type Magnetic Sector in the text field.

Select Domain 3 only.

Sample Chamber

In the toolbar, click

In the window for , type Sample Chamber in the text field.

Select Domain 8 only.

Detector

In the toolbar, click

In the window for , type Detector in the text field.

Select Domain 1 only.

Filters

In the toolbar, click

In the window for , type Filters in the text field.

Locate the section. From the list, choose .

Select Boundaries 5–8, 14–20, 39–47, 51, 52, 63–69, 72, 75–80, 83–86, and 91–94 only.

Apertures

In the toolbar, click

In the window for , type Apertures in the text field.

Locate the section. From the list, choose .

Select Boundaries 23, 28, 52, and 80 only.

Click the

button in the toolbar.

Specify the particle mass and charge. Then define the forces that will act on the particles.

Charged Particle Tracing (cpt)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Locate the section. Select the check box.

Particle Properties 1

In the window, under click .

In the window for , locate the section.

In the mp text field, type Mr*mp_const.

Locate the section. In the Z text field, type 1.

Electric Force 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. Specify the E vector as


Ex	x
Ey	y
Ez	z

Magnetic Force 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. Specify the B vector as


Bx	x
By	y
Bz	z

Particle Beam 1

In the toolbar, click

and choose .

Select Boundary 74 only.

In the window for , locate the section.

In the N text field, type 5000.

Locate the section. From the list, choose .

In the xm text field, type 1[cm].

In the xm′ text field, type 0.001.

Locate the section. In the E text field, type Ei0.

Add features at the detector as well as at the lens apertures. The particle counters will be used to visualize the nominal trajectory of the beam ions that reach the detector.

Particle Counter 1

In the toolbar, click

and choose .

Select Boundary 11 only.

In the window for , locate the section.

From the list, choose .

Particle Counter 2

In the toolbar, click

and choose .

Select Boundary 65 only.

In the window for , locate the section.

From the list, choose .

Particle Counter 3

In the toolbar, click

and choose .

Select Boundary 68 only.

In the window for , locate the section.

From the list, choose .

Particle Counter 4

In the toolbar, click

and choose .

Select Boundary 51 only.

In the window for , locate the section.

From the list, choose .

Particle Counter 5

In the toolbar, click

and choose .

Select Boundary 8 only.

In the window for , locate the section.

From the list, choose .

Wall 2

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Specifying a condition at boundaries where the force changes discontinuously will increase the accuracy of the solution.

Mesh 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Free Triangular 1

In the toolbar, click

and choose .

Select Boundaries 3, 11, 61, and 74 only.

Size 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Click

Free Triangular 2

In the window, under right-click and choose .

In the window for , locate the section.

From the list, choose .

Size 1

In the window, expand the node, then click .

In the window for , locate the section.

From the list, choose .

Click

Click the

button in the toolbar.

Free Tetrahedral 1

In the toolbar, click

In the window for , click

Because was selected from the list, the default solver uses explicit time stepping. This can significantly improve simulation time in some classes of nonstiff problems.

Study 1

Step 1: Time Dependent