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Introduction An Einzel lens is an electrostatic device used for focusing charged particle beams. The beam focusing depends on the initial particle energy, the voltage on the Einzel lens and the initial beam collimation (initial radius and transverse velocity of the charged particles). The problem is almost axisymmetric: the electrostatic field is axisymmetric but the particle velocity transverse to the beam axis is random, which spoils the symmetry. Thus, the model is solved in 3D. The model uses COMSOL Multiphysics with the Particle Tracing Module.

Geometry 3 identical cylinders arranged on same axis
Outer cylinders are grounded Middle cylinder has non-zero voltage This particular model uses cylinders, but other geometries are possible. 𝑉=0 𝑉= 𝑉 0 ≠0 𝑉=0

Focusing By Fringe Fields
Since the focusing is accomplished via the fringe fields, then it is important to model this region accurately. This includes an accurate model of the cylinder edges (filleted) and a dense mesh since the spatial variation of the field is highest near the edges.

Notes on COMSOL Model The dimensions are listed in the Parameters table. The geometry is constructed in a plane, which is then rotated 180 o and mirrored. Domains and boundaries are grouped in Selections (under Definitions) to make the rest of the model clear. The mesh is also constructed in the plane and then swept 180 o in both directions. The electrostatics problem is solved first. The particle trajectories are computed in a separate study.

Initial Set Up The first part is a 3D electrostatics calculation.

Parameters Add a table of parameters under Global Definitions.

Geometry – Work Plane Draw a 2D cross-section of the geometry in a Work Plane. Later, the plane will be revolved and mirrored to get the 3D geometry. The Work Plane is 𝑦=0.

Geometry – Rectangle With Filleted Corners
Draw a Rectangle in the Work Plane, and then Fillet all 4 corners.

Geometry – Array Make copies of the filleted rectangle using a 1×3 array.

Geometry – Vacuum Chamber Rectangle
Add the large rectangle representing the vacuum chamber, including an inner box using “Layers”.

Geometry – Work Plane Add a point that will eventually trace out a circle for the beam inlet. The finished 2D geometry.

Geometry – Revolve Revolve all objects in the Work Plane by 180 o about its axis.

Geometry – Mirror Mirror all of the revolved objects about the xz-plane, whose normal is 0,1,0 . Keep the original set of objects as well!

Selections Add Selections to clearly group domains and boundaries for later use. Right click on Definitions and choose “Selections > Explicit”.

Selections – Cylinder 1 Click on “Wireframe Rendering” in the Graphics window. Click on “Go to ZX View” in the Graphics window. Click on “Select Box” in the Graphics window and drag the box around the cylinder on the left. Right-click to add the domain numbers to the selection list. Rename “Explicit 1” to “Cylinder 1”. Step 1 Step 2 Step 3

Selections – Cylinder 2 Right-click on Definitions and choose “Selections > Explicit”. Click on “Select Box” in the Graphics window and drag the box around the cylinder in the middle. Right-click to add the domain numbers to the selection list. Rename “Explicit 2” to “Cylinder 2”. Step 2

Selections – Cylinder 3 Right-click on Definitions and choose “Selections > Explicit”. Click on “Select Box” in the Graphics window and drag the box around the cylinder on the right. Right-click to add the domain numbers to the selection list. Rename “Explicit 3” to “Cylinder 3”. Step 2

Selections – Vacuum Domains
Right-click on Definitions and choose “Selections > Complement”. Click on “Add” (the “+” sign) and choose all 3 cylinders. Rename “Complement 1” to “Vacuum domain”.

Selections – Boundaries
Define Selections to group the boundaries of the cylinders individually and collectively. Right-click on “Cylinder 1” and choose “Duplicate”. Change “Output Entities” from “Selected domains” to “Adjacent boundaries”. Rename “Cylinder 1.2” to “Cylinder 1 – boundaries. Repeat for the other 2 cylinders.

Selections – All cylinder boundaries
Right-click on Definitions and choose “Selections > Union”. Under “Geometric Entity Level”, change Level from Domain to Boundary. Press the Add button and select all of the cylinder boundaries defined previously. Rename “Union 1” to “All cylinder boundaries”.

Electrostatics Right-click on Electrostatics and choose “Sort by Space Dimension”. Under “Domain Selection”, change “All domains” to “Vacuum domains”. (The potential inside each cylinder is equal to the value on the surface, so it does not need to be computed.)

Electrostatics – Vacuum Properties
We could define a material called “Vacuum”, but only the permittivity is needed so just set the value in “Charge Conservation 1”. Click on “Charge Conservation 1” and under “Electric Field”, change “Relative permittivity” from “From material” to “User defined”. Leave the value as 1.

Electrostatics – Boundary Conditions
The default boundary condition is “Zero Charge”, which means 𝑛∙𝐷=𝑛∙ −𝜖𝛻𝑉 =0, that is, the surface charge density is zero. The “Zero Charge” boundary condition is fine for the ends of the vacuum chamber. For the other surfaces, the potential will be fixed. Right-click on “Boundaries” and choose Ground. Select all of the surfaces on the sides of the vacuum chamber. (2, 3, 6, 8, 11, 13, 59, 61, 63, 111, 112, 113) Rename “Ground 1” to “Ground – vacuum chamber walls”.

Electrostatics – Outer Cylinders
Right-click on “Boundaries” and choose “Ground”. Under “Boundary Selection”, change “Manual” to “Cylinder 1 - boundaries”. Rename “Ground 2” to “Ground – Cylinder 1”. Repeat for Cylinder 3.

Electrostatics – Middle Cylinder
Right-click on “Boundaries” and choose “Electric Potential”. Under “Boundary Selection”, change “Manual” to “Cylinder 2 - boundaries”. Change “Electric potential” to “V0”, defined in the Parameters table above. Rename “Electric Potential 1” to “Electric Potential - Cylinder 2”.

Mesh – Edge The particle trajectories will be near the cylindrical axis and so construct a dense mesh there. Right-click on “Mesh 1” and choose “More Operations > Edge”. Add the edges on the axis to the Selection list. (122, 125, 128) Right-click on “Edge 1” and choose “Distribution”. Change “Number of elements” from 5 to 120. Press “Build Selected”.

Mesh – Free Triangular Right-click on “Mesh 1” and choose “More Operations > Free Triangular”. Add boundaries 85, 87 and 89 to the Selection list. Right-click on “Free Triangular 1” and choose “Size”. In “Size 1”, clear boundaries 85 and 89 from the Selection list, so that only boundary 87 remains. Under “Element Size”, change “Predefined” from “Normal” to “Extremely fine”. Select “Custom” and change “Maximum element size” to 0.01. Press “Build Selected”.

Mesh – Swept Right-click on “Mesh 1” and choose “Swept”.
Add the domains, source faces and destination faces as shown on the left. Right-click on “Swept 1” and choose “Distribution”. Change “Number of elements” from 5 to 10. Press “Build Selected”.

Mesh – Swept Right-click on “Swept 1” and choose “Duplicate”.
In the newly created “Swept 2”, change the domains to 1,3,5, which are the other half of the vacuum chamber. Press “Build All”. Right click on “Study 1” and choose Compute!

Data Sets – Cut Plane: y=0
Create 3 new data sets to plot a cut-away view of the equipotential surfaces near the Einzel lens Right-click on “Data Sets” and choose “Cut Plane” Under “Plane Data”, change Plane to “xz-planes” Rename “Cut Plane 1” to “Cut Plane: y=0”

Data Sets – Solution 1 – behind lenses only
Under “Data Sets”, right-click on “Solution 1” and choose Duplicate Rename “Solution 2” to “Solution 1 – behind lenses only” Right-click on “Solution 1 – behind lenses only” and choose Selections Change “Geometric entity level” to “Domain” and add domain 3, which is one of the half cylinder sections near the Einzel lens

Data Sets – Solution 3 – cylinder surfaces
Right-click on “Data Sets” and choose “Solution” Rename “Solution 3” to “Solution 3 – cylinder surfaces” Right-click on “Solution 3 – cylinder surfaces” and choose Selection Change “Geometric entity level” to Boundary Change Selection to “All cylinder boundaries”

Adding Views Add a “View” for the upcoming plot so that the perspective and zoom level can remain fixed, even after more model steps are added Right-click on “Show” and check “Advanced Results Options”, which has the effect of adding a new node called “Views” under Results Right-click on “Results > Views” and choose “View 3D”. In “View 3D 3”, uncheck “Show grid” and “Show axis orientation”.

Plots – Equipotential Surfaces
Rename “Electric Potential” to “Equipotential surfaces near Einzel lens” Under “Plot Settings”, change the View to “View 3D 3” and uncheck the box “Plot data set edges” Right-click on “Equipotential surfaces near Einzel lens” and choose Contour. Also add Isosurface and Surface. The plot settings for each are shown on the next slide.

Plot Settings – Equipotential Surfaces
Contour Isosurface Surface

Equipotential Surfaces
Rotate and zoom-in to obtain a plot similar to the one below

2D Plot of Fringe Field Plot the fringe field in the cut plane y=0.
Right-click on Results and choose “2D Plot Group”. Rename “2D Plot Group 2” to “Fringe Field”. Change “Data set” to “Cut Plane: y=0”. Right-click on “Fringe Field” and choose “Surface”. Also add “Contour” and “Arrow Surface”. The plot settings are shown on the next slide.

Plot Settings for Fringe Field
Surface Contour Arrow Surface

Fringe Field Zoom-in to obtain a plot of the fringe field similar to the one below.

Right-click on “Model 1” and choose “Add Physics”. Choose “AC/DC > Charged Particle Tracing”. Choose “Time Dependent” and uncheck “Electrostatics” below.

Particle Tracing – Vacuum Domains
Once “Charged Particle Tracing” has been added to the model, right-click on it and choose “Sort By Space Dimension”. Change “Domain Selection” to “Vacuum domains”.

Particle Tracing – Electric Force
By default, there are no forces on the particles. Right-click on “Charged Particle Tracing > Domains” and choose “Electric Force”. The electric force only needs to be computed near the Einzel lens. (It is negligible elsewhere.) Add only domains 3 and 4 to the selection list. The electric field is computed in “Electrostatics”, so change “Electric field” from “User defined” to “Electric field (es/ccn1)”. The gradient of the electrostatic potential is not continuous across mesh element boundaries, so then neither is the electric field. Check the box beside “Use piecewise polynomial recovery on field” to apply smoothing.

Particle Tracing – Coulomb Repulsion
The particle density in the beam is very low so the Coulomb repulsion between particles is negligible. If the particle density were high enough so that the Coulomb repulsion between particles is significant, then this effect could be added by right-clicking on Domains and choosing “Particle-Particle Interaction”. “Particle-Particle Interaction” is not added to the model.

Particle Tracing – Random Function
Define a random function in the model under “Global Definitions”. This will be used in the initial particle velocity components transverse to the beam axis. Change the “Distribution” of random numbers to Normal. The number of arguments is 1, which is the seed for the random number generator.

Initial Conditions of the Particles
The initial conditions are the position and the velocity. The initial position is on the small circle at the end of the vacuum chamber. Right-click on Boundaries and choose Inlet. Select boundaries 57, 58, 84 and 86. Add the initial velocity. The x- and y-components are small and random using seeds “x/initial_beam_radius” and “y/initial_beam_radius”, respectively. With “Initial position = Mesh based” and “Refinement factor = 1”, there will be 1 particle released on each mesh element on the boundary.

Electrons Click on “Global > Particle Properties 1” and verify that the particles are electrons. The charge number is -1 for the electron, that is, the charge on the particle is (charge number) x (charge on proton). The mass is set to “me_const”, where “_const” signifies a built-in physical constant. To see a list of all built-in physical constants, click on “Help > Documentation”, search with “_const” and click on “Physical Constants”.

Particle Tracing – Time Dependent Study
Click on “Study 2 > Step 1: Time Dependent” and specify the times for the particle tracing calculation. Note that “T” is the time it would take a particle to travel from one end of the vacuum chamber to the other in the absence of any external force. The end time of the study is 5% higher in case the particles slow down near the Einzel lens. (“T” is defined in “Global Definitions > Parameters”.) Verify that only “Charged Particle Tracing” will be computed, not “Electrostatics”. To use the electrostatics solution from the previous study, check the box beside “Values of variables not solved for”, change “Method” to “Solution” and change “Study” to “Study 1, Stationary”. Right-click on “Study 2” and choose Compute!

Create a New Plot View To create a new plot view, right-click on “Results > Views” and choose “View 3D”. Clear all of the boxes. The angle and zoom level for the following plots can be set to this particular view, “View 3D 5”. If we want to change the view while doing something else in the Model above, the view for the plots using “View 3D 5” will not be changed.

Plot – Particle Trajectories
Change the plot settings for “Particle Trajectories (cpt)” to those below. The default is to plot the particle position using a point at the final time. Change this to a line to trace out the trajectory. Change the color so that it is the ratio of the particle kinetic energy, “cpt.Ep”, to the initial kinetic energy, “E0”.

Particle Trajectories and Energies
Modify the perspective to obtain a plot similar to the one below. Focusing depends on the initial position and velocity as well as the charge-to-mass ratio and the voltage on the Einzel lens.