Pierce Electron Gun

An electron gun must be able to draw a sufficient current and accelerate the electrons to the desired speed. First the electrons are drawn from an electron source, such as a metal cathode or plasma. Then the electrons are accelerated using electric and magnetic forces.

The first part of an electron gun geometry, where electrons are first extracted, presents unique design challenges because the emitted electron speeds are usually lowest there, and therefore the space charge density usually reaches a maximum in this region. The high space charge density can cause an electron beam to spread out.

The Pierce electron gun design uses electrodes with a particular shape to counteract the Coulomb repulsion between electrons in the beam. As a result, the electrons in the beam propagate in straight lines. The emitted electrons at the cathode are assumed to be space charge limited; the initial thermal distribution of electron velocities is neglected.

Model Definition

The Pierce electron gun design uses electrodes of a specific shape, so that the outward component of the electric field at the edge of the beam is zero. This requires the electrodes to perfectly counterbalance the Coulomb repulsion between the beam electrons.

Inside the beam, both the cathode and anode are flat edges. Here the anode is understood to be a metal grid so that the electrons may pass through. At higher current it may be necessary to shoot the electron beam through a gap in the anode.

Outside the beam, the bottom electrode makes an angle of 67.5 degrees with the beam propagation direction. The anode is shaped so that

where r (SI unit: m) is the distance from the point where the flat part of the cathode meets the incline, d (SI unit: m) is the gap thickness between the cathode and anode at the beam center, and θ (SI unit: rad) is the angle that a line from this point makes with the beam propagation direction.

The determination of these electrode shapes is sometimes called the Pierce method of gun design. The full derivation is given in Appendix: Derivation of the Cathode Shape.

The gun geometry is shown in Figure 1. The interior edge separating the two domains is only kept for meshing purposes, since the space charge density distribution within the beam can be calculated more accurately using a finer mesh as shown in Figure 2.

Figure 1: Geometry of the Pierce electron gun.

Figure 2: A fine mapped mesh is used in the beam path. The mesh coarsens as the distance from the beam is increased.

Note that the lower end of the rectangular domain and the inclined electrode do not meet at a common vertex; the rectangle is a bit higher. This is because the electrons are released using the feature. This feature emits model particles from a virtual cathode a short distance away from the real cathode. This is to avoid the infinite space charge density that occurs at the real cathode when the thermal distribution of released electron velocities is not taken into account.

Results and Discussion

The electric potential and some electric field streamlines are shown in Figure 3. Ideally the electric field streamlines within the beam path and the electron trajectories should be perfectly vertical.

Figure 3: Surface plot of the electric potential. Electric field streamlines are shown in black.

The electron trajectories are shown in Figure 4. The beam does become about 3% thinner by the time it reaches the anode, but a small deviation can be expected because this finite-size geometry is actually a truncated version of the ideal Pierce design. For more details, see Appendix: Derivation of the Cathode Shape. To better visualize the shape of the electric potential distribution in the beam, a contour plot with electric field streamlines is shown in Figure 5. A node was used to hide part of the geometry.

Figure 4: Particle trajectories are shown as lines. The color expression is the particle speed.

Figure 5: Contour plot of the electric potential, Electric field streamlines are shown in black.

Reference

1. S. Humphries, Charged Particle Beams, Dover Publications, New York, 2013.

2. J. R. Pierce, “Rectilinear electron flow in beams”, Journal of Applied Physics, vol. 11, no. 8, pp. 548-554, 1940.

Particle_Tracing_Module/Charged_Particle_Tracing/pierce_electron_gun

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

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In the tree, select .

Click .

Click

In the tree, select .

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Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
d	5[cm]	0.05 m	Gap thickness
db	1[mm]	0.001 m	Virtual cathode height
w1	1[cm]	0.01 m	Cathode width
w2	20[cm]	0.2 m	Focusing electrode width
alpha	22.5[deg]	0.3927 rad	Focusing electrode angle
V0	1[kV]	1000 V	Anode potential

Geometry 1

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type w1.

In the text field, type d-db.

Locate the section. In the text field, type db.

The bottom edge of the rectangle is the flat virtual cathode. The top edge is a flat anode grid that the released electrons can pass through. At high current it may be necessary to replace the grid with a gap in the anode so that the beam does not melt the grid.

Next add the curved part of the anode.

Parametric Curve 1 (pc1)

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type theta.

In the text field, type 62*pi/180.

This maximum parameter value is somewhat arbitrary but must be less than 67.5 degrees.

Locate the section. In the text field, type w1+d*sec(4*theta/3)^0.75*sin(theta).

In the text field, type d*sec(4*theta/3)^0.75*cos(theta).

Add a to define the focusing electrode. An ideal Pierce gun extends infinitely, but here the polygon is cut off at a sufficient distance from the beam.

Polygon 1 (pol1)

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Locate the section. In the table, enter the following settings:


x (m)	y (m)
w1	d
w1	0
w2	w2*tan(alpha)
w2-d*tan(alpha)	w2*tan(alpha)+d

Convert to Solid 1 (csol1)

In the toolbar, click

and choose .

Select the objects and only.

In the window for , click

Click the

button in the toolbar. The plot should look like Figure 1.

Definitions

Integration 1 (intop1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Select Point 4 only.

This coupling will be used to evaluate the electric potential at the end of the virtual cathode, in order to avoid discontinuities in the potential.

Materials

Material 1 (mat1)

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	1	1	Basic

Charged Particle Tracing (cpt)

Particle Properties 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Electric Force 1

In the window, click .

In the window for , locate the section.

From the E list, choose .

Multiphysics

Space Charge Limited Emission 1 (scle1)

In the toolbar, click

and choose .

Select Boundary 2 only.

In the window for , locate the section.

In the os text field, type db.

The node has two purposes: to release electrons and to determine the electric potential at the virtual cathode.

Electric Particle Field Interaction 1 (epfi1)

In the window, click .

In the window for , locate the section.

Select the check box.

In the β text field, type 10.

When is selected, the space charge density of the electrons is gradually increased over the first few iterations of the study. This prevents the space charge density from being overestimated, which could possibly cause the initial particle velocity to point in the wrong direction.

Set up the boundary conditions for the Electrostatics interface. Specify the potentials on the anode and the focusing electrode. The left boundary is a symmetry plane so the default condition may be used.

Electrostatics (es)

In the window, under click .

Anode

In the toolbar, click

and choose .

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

Select Boundaries 3 and 8 only.

Locate the section. In the V0 text field, type V0.

Focusing Electrode

In the toolbar, click

and choose .

In the window for , type Focusing Electrode in the text field.

Select Boundary 5 only.

Adjacent to Virtual Cathode

In the toolbar, click

and choose .

In the window for , type Adjacent to Virtual Cathode in the text field.

Select Boundary 4 only.

Locate the section. In the V0 text field, type intop1(scle1.V0)*(y/db)^(4/3).

This expression uses the coupling defined earlier. It ensures continuity between the zero potential at the focusing electrode and the nonzero potential on the edge of the virtual cathode.

Mesh 1

Define a very fine mapped mesh where the electron beam will propagate, to accurately accumulate the space charge density. The mesh is allowed to become coarser in the charge-free region away from the beam.

Mapped 1

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Select Domain 1 only.

Distribution 1

Right-click and choose .

Select Boundaries 2 and 3 only.

In the window for , locate the section.

In the text field, type 20.

Distribution 2

In the window, right-click and choose .

Select Boundaries 1 and 6 only.

In the window for , locate the section.

In the text field, type 100.

Free Triangular 1

In the toolbar, click

In the window for , click

. The plot should look like Figure 2.

Study 1

The study step alternates between stationary calculations of the electric potential and transient calculations of the particle trajectories, allowing the particles and the electric potential to influence each other.

Step 1: Bidirectionally Coupled Particle Tracing

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the text field, type range(0,1,10).

Locate the section. In the text field, type 20.

In the toolbar, click

Results

Streamline 1

The default plot shows the electric potential in the domain. First add some electric field lines. In order to show that the focusing electrode offsets the space charge of the beam, the electric field lines in the beam should be vertical.

Right-click and choose .

Select Boundaries 3 and 8 only.

In the window for , locate the section.

In the text field, type 60.

In the toolbar, click

. The plot should look like Figure 3.

Particle Trajectories 1

In the window, expand the node, then click .

In the window for , locate the section.

Find the subsection. From the list, choose .

In the toolbar, click

. The plot should look like Figure 4.

To better visualize the electric potential, add a dataset to fill in the left half of the geometry.

Mirror 2D 1

In the toolbar, click

and choose .

Using this new dataset, create a contour plot of the electric potential.

Contours with Field Lines

In the toolbar, click

In the window for , type Contours with Field Lines in the text field.

Locate the section. From the list, choose .

Locate the section. Clear the check box.

Contour 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Select the check box.

Filter 1

Right-click and choose .

In the window for , locate the section.

In the text field, type x<5[cm].

Streamline 1

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

In the text field, type range(-0.05,0.002,0.05).

In the text field, type 0.045.

Filter 1

Right-click and choose .

In the window for , locate the section.

In the text field, type x<5[cm].

In the toolbar, click

Click the

button in the toolbar. The plot should look like Figure 5.

Appendix: Derivation of the Cathode Shape

In a charge-free region of space, the electric potential V (SI unit: V) potential must satisfy Poisson’s equation,

The Pierce method for determining the shape of the electrodes takes advantage of the observation that any function of the complex variable y + ix will satisfy Poisson’s equation.

Define u = y + ix and consider an arbitrary function f(y + ix). It can be shown by repeated applications of the chain rule,

The function f must be defined so that it matches the space charge limited potential in the beam,

So the electric potential is

Then the cathode is the equipotential surface where V = 0, and the anode is the equipotential surface where V = Va. It is somewhat easier to determine the shape of these equipotential surfaces by expressing u in cylindrical polar coordinates,

Thus

Since r is real,

Invoking Euler’s theorem yields

The solution for V = 0 is a straight line with

(1)

That is, θ = 3π/8 or 67.5 degrees. Recalling that the beam propagation direction is the real axis, this means the equipotential surface is a straight line that makes a 67.5 degree angle with the beam propagation direction.

The solution for V = Va is

which can be rearranged and solved for r,

(2)

In fact, the line given by Equation 1 is the asymptote of Equation 2 as the radial coordinate r becomes very large. The ideal Pierce electron gun therefore has a thin sliver-like gap between the electrodes that is infinitely long, but the electrodes will never touch. In this example, the electrodes are simply truncated at a sufficient distance from the beam. The line cutting off this gap is drawn perpendicular to the lower boundary because this is approximately the direction in which the electric field will point there, so the disruption from imposing the default boundary condition along such an edge will be minimized.