Electron Beam Divergence Due to Self Potential

When modeling the propagation of a charged particle beam at a high current, the electric field due to the space charge of the beam significantly affects the trajectories of the charged particles. Perturbations to these trajectories, in turn, affect the space charge distribution. In order to accurately predict the properties of the beam, the particle trajectories and fields must be computed in a self-consistent manner. The Charged Particle Tracing interface can use an iterative procedure to efficiently compute the strongly coupled particle trajectories and electric field for systems operating under steady-state conditions. Such a procedure reduces the required number of model particles by several orders of magnitude, compared to methods based on explicit modeling of Coulomb interactions between the beam particles. A mesh refinement study confirms that the solution agrees with the analytical expression for the shape of a nonrelativistic, paraxial beam envelope.

This application requires the Particle Tracing Module.

Model Definition

This model computes the shape of an electron beam propagating through free space. When the magnitude of the beam current is large enough that Coulomb interactions are significant, the shape of the beam may be determined by solving a set of strongly coupled equations for the beam potential and the electron trajectories,

The beam electrons are assumed to be nonrelativistic so that magnetic forces can be neglected. Modeling the beam electrons and the resulting electric potential using a time-dependent study would require a very large number of model particles to be released at a large number of time intervals. Instead, this model computes the shape of the electron beam by coupling a time-dependent study step for computing the particle trajectories to a stationary step for computing the electric potential. This algorithm is suitable for modeling beams which operate at steady-state conditions. It consists of the following steps:

Compute the particle trajectories, assuming no space charge effects are present, using a time-dependent solver. From these trajectories, compute the space charge density using the node.

Compute the electric potential due to the space charge density of the beam, using a stationary solver. The model uses an region to apply appropriate boundary conditions for a beam propagating in free space.

Use the electric potential calculated in step 2 to compute the perturbed particle trajectories. Recalculate the space charge density using these perturbed trajectories.

Repeat steps 2 and 3 until a specified number of iterations has been reached.

After several iterations, the particle trajectories and the corresponding space charge density and electric field reach a stable, self-consistent solution. For a nonrelativistic, paraxial beam of electrons, the shape of the beam envelope is given by Ref. 1 as

(1)

where z is the distance from the beam waist, R0 is the waist radius, K is the generalized beam perveance,

χ is the ratio of the beam radius to the beam waist radius, and

(2)

This analytical expression for the relationship between axial position and beam envelope radius is used to determine the accuracy of the solution. A mesh refinement study confirms that the agreement between the solutions improves as the mesh element size is reduced.

In this example, is selected from the list in the physics interface section. As a result, each model particle represents a continuous stream of particles, released at regular time intervals, rather than the instantaneous position of a charge. For the purpose of modeling particle-field interactions, each model particle leaves behind a trail of space charge in its wake. The contribution of each model particle to the total space charge density of the beam is found by evaluating the sum

where e is the elementary charge, Z is the charge number of the particles, δ is the Dirac delta function, and frel is the effective frequency of release of the particle. The frequency of release is the number of particles per model particle per second. To avoid the infinite potential associated with an infinitesimally small point charge, the space charge density is distributed uniformly over each mesh element before the electrostatics problem is solved.

Results and Discussion

After several iterations, the model reaches a self-consistent solution for the trajectories of the electrons and the beam potential. The electron trajectories are plotted in Figure 1. The expression r-at(0,r) is used to define a color expression for the trajectories. The at operator is used to evaluate an expression at the initial time, rather than the current time. Thus the color expression gives the radial displacement of each particle due to space charge effects.

Figure 1: A beam of electrons with a waist located at z = 0 diverges due to transverse beam forces. The color represents the radial displacement of each electron from its initial position.

The electric potential distribution in the beam is shown in Figure 2. Since the beam propagates from left to right, and the beam electrons initially move in the positive z direction, the left end of the plot corresponds to the beam waist. This is also the location where the beam radius is smallest in magnitude.

Figure 2: Plot of the electric potential of the electron beam. The potential is greatest in magnitude close to the beam waist.

A mesh refinement study confirms that the shape of the beam envelope agrees more closely with the analytical expression of Equation 1 as the maximum mesh element size is reduced. The distance from the beam waist as a function of beam radius is compared to the result of this expression, and the relative error is plotted in Figure 3. For all values of the maximum element size, the error shown is computed after three iterations of the solver loop.

These results show that a self-consistent solution for the particle trajectories and the fields due to their space charge density can be obtained using an iterative solver sequence. This requires much less time and memory than a fully coupled time-dependent study of the individual beam particles and their fields. The accuracy of the solution clearly improves as the mesh element size is reduced, enabling more accurate computation of the electron trajectories and beam potential.

Figure 3: The results of a mesh refinement study indicate that the observed relationship between beam radius and distance from the beam waist converges to the expected value as the mesh is refined. The comparison is made at a location 0.2 meters from the waist.

Reference

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

ACDC_Module/Particle_Tracing/electron_beam_divergence

Modeling Instructions

From the menu, choose .

New

In the window, click .

Model Wizard

In the window, click .

In the tree, select .

Click .

In the tree, select .

Click .

Global Definitions

Parameters

On the toolbar, click .

To save time, the parameters can be loaded from a file.

In the window for , locate the section.

Click .

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

Geometry 1

Cylinder 1 (cyl1)

On the toolbar, click .

In the window for , locate the section.

In the text field, type r2.

In the text field, type L.

Click to expand the section. In the table, enter the following settings:


Layer name	Thickness (m)
Layer 1	tlayer

Right-click and choose .

Work Plane 1 (wp1)

On the toolbar, click .

In the window for , locate the section.

From the list, choose .

Find the subsection. Select the toggle button.

On the object , select Boundary 10 only.

Click .

Circle 1 (c1)

On the toolbar, click and choose .

In the window for , locate the section.

In the text field, type r0beam.

Right-click and choose .

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	Name	Value	Unit	Property group
Relative permittivity	epsilonr	1	1	Basic

Definitions

Variables 1

On the toolbar, click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
qr	sqrt(qx^2+qy^2)	m	Radial distance from beam axis
qrmax	cpt.cptmaxop1(qr)	m	Beam radius
z_avg	cpt.cptaveop1(qz)	m	Average z-coordinate
chi	qrmax/at(0,qrmax)		Ratio of beam radius to waist radius

To model a drifting electron beam propagating in a large, open area, surround the modeling domain with an . This results in appropriate boundary conditions for the electric potential.

Infinite Element Domain 1 (ie1)

On the toolbar, click .

In the window for , locate the section.

From the list, choose .

Select Domains 1, 2, 4, and 5 only.

Charged Particle Tracing (cpt)

Inlet 1

In the window, under right-click and choose .

Select Boundary 12 only.

In the window for , locate the section.

In the I text field, type Ibeam.

Locate the section. From the list, choose .

In the N text field, type 1000.

Locate the section. Specify the v0 vector as


0	x
0	y
v0beam	z

Electric Force 1

In the window, under click .

Select Domain 3 only.

In the window for , locate the section.

From the E list, choose .

Locate the section. Select the check box.

Electrostatics (es)

On the toolbar, click and choose .

Ground 1

In the window, under right-click and choose .

Select Boundaries 2, 3, 14, and 22 only.

Mesh 1

Free Tetrahedral 1

In the window, under right-click and choose .

In the window for , locate the section.

From the list, choose .

Select Domain 3 only.

Size 1

Right-click and choose .

In the window for , locate the section.

Click the button.

Locate the section. Select the check box.

In the associated text field, type hmax.

Distribution 1

In the window, right-click and choose .

Right-click and choose .

In the window for , click .

Study 1

Step 1: Bidirectionally Coupled Particle Tracing

In the window for , locate the section.

Click .

In the dialog box, type 1e-8 in the text field.

In the text field, type 21e-8.

Click .

In the window for , locate the section.

From the list, choose .

In the text field, type qrmax.

In the text field, type 1E-5.

In the text field, type 0.015.

In the text field, type 8.

Use a to confirm that a finer mesh results in closer agreement with the expected beam envelope size.

Parametric Sweep

On the toolbar, click .

In the window for , locate the section.

Click .

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
hmax

Click .

In the dialog box, type 0.01 in the text field.

In the text field, type -0.001.

In the text field, type 0.003.

Click .

The smallest mesh size requires about 5 GB RAM.

On the toolbar, click .

Results

Plot the trajectories of the electrons, using a to observe their radial displacement over time.

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 .

Color Expression 1

In the window, expand the node, then click .

In the window for , locate the section.

In the text field, type qr-at(0,qr).

On the toolbar, click .

Click the button on the toolbar. This plot should look like Figure 1.

Data Sets

The electric potential can be visualized more clearly by excluding the selection of the from the data set.

Study 1/Parametric Solutions 1 (sol2)

In the window, expand the node, then click .

Selection

On the toolbar, click .

In the window for , locate the section.

From the list, choose .

Select Domain 3 only.

Electric Potential (es)

In the window, under click .

In the window for , click to expand the section.

Locate the section. From the list, choose .

Click the button on the toolbar. This plot should look like Figure 2.

For each mesh size, compare the beam envelope shape to the analytical solution for a paraxial, nonrelativistic beam.

1D Plot Group 3

On the toolbar, click .

In the window for , locate the section.

From the list, choose .

Global 1

Right-click and choose .

In the window for , type Relative Error in Longitudinal Beam Displacement in the text field.

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


Expression	Unit	Description
abs((r0beam/sqrt(2K)integrate(1/sqrt(log(s)),s,1+eps,chi)-z_avg)/z_avg)		Relative error in z(r)

This expression is Equation 1, where the first argument of the integrate operator is the integrand of Equation 2. The lower limit of integration is increased by the floating point relative accuracy eps (machine epsilon, 2−52 or about 2.2204×10−16) to avoid division by zero during numerical integration.

Locate the section. From the list, choose .

On the toolbar, click . This plot should look like Figure 3.