COMSOL 6.4 - Multipactor Saturation

The multipactor effect is a sometimes deleterious phenomenon in which the number of electrons in a microwave cavity can grow exponentially. Free electrons within the cavity are accelerated by an RF signal, causing them to strike the cavity walls. If the electrons hit the walls with sufficient energy, the walls may then release secondary electrons. Under the right conditions, a single electron impact can cause the release of multiple secondary electrons, thus causing the total number of electrons in the cavity to increase.

If a certain relationship exists between the size of the cavity and the frequency and amplitude of the RF signal, then the electron emission occurs in resonance with the driving signal; the emitted secondary electrons are accelerated to the other end of the cavity, where they cause an even greater number of secondary electrons to be released, and so on.

When the number of electrons in the cavity becomes sufficiently large, then the mutual electrostatic repulsion between the electrons can disrupt their resonance with the driving RF signal, so that the total population of electrons does not grow without bound. Instead, the population of electrons eventually reaches a dynamic equilibrium in which the effects of space charge and the RF signal counterbalance each other. This verification model is a quasi-1D, two-way coupled particle field interaction model that shows how growth of space charge in a microwave cavity leads to multipactor saturation.

Model Definition

This example uses the Electrostatics interface to solve for the electric potential in a cavity and the Charged Particle Tracing interface to track electrons within the cavity. The RF voltage is applied by using the boundary condition at one end of the geometry and the boundary condition at the other end, where the specified potential is a sinusoidal function of time,

where V0 (SI unit: V) is the amplitude of the signal and f (SI unit: 1/s) is the frequency.

The direction from the boundary condition to the boundary condition is taken to be the positive x direction. Let D (SI unit: m) represent the gap thickness, the perpendicular distance between these two surfaces. The dimensions of the cavity in the y and z directions are assumed to be much larger than in the x direction. In addition, a constant, uniform magnetic field is applied within the cavity,

Multipaction can occur for specific combinations of the values of D, V0, f, and B0. This can include situations in which B0 = 0. A stability analysis of such a one-dimensional cavity geometry is demonstrated in Ref. 1. The model parameters used in this example are also based on Ref. 1: V0 = 1078 V, B0 = 360 G, f = 2.5 GHz, and D = 0.16 cm.

When an electron hits the wall at either end of the microwave cavity, the incident electron is removed from the simulation using a node with the wall condition. Depending on the kinetic energy of the incident particle, it may release one or more secondary particles into the domain using the subnode. The relationship between the kinetic energy of the incident particle and the number of emitted secondary electrons to release is called the secondary electron yield or SEY, an energy-dependent expression also from Ref. 1 and plotted in Figure 1,

Figure 1: Secondary electron yield (SEY) as a function of incident electron kinetic energy.

Although the model geometry in this example is three-dimensional, the model is essentially 1D-3V, meaning that only one component of particle position and all three components of the particle velocity are relevant. The spatial density of particles and the electric potential are assumed to be completely uniform in the y and z directions. To enforce this uniformity without introducing spurious y- and z-components of the electric field, a highly structured mesh was used with only a single element in these directions and a large number of very thin elements in the x direction, as shown in Figure 2.

Figure 2: Structured mesh to simulate the multipactor as a 1D-3V model.

To account for space-charge-induced debunching, in which the mutual electrostatic repulsion of electrons causes some of the model particles to slow down before hitting the wall (and thus release fewer secondary electrons), the particles were bidirectionally coupled to the electric potential in the cavity via the multiphysics coupling. At every time step, each model particle contributes to a volumetric space charge density term within the domain mesh element that it occupies, and then the electric potential is updated based on this space charge density term. Because the mesh only has a single element in the y and z directions, and the space charge density of particles is considered uniform within each mesh element, this method of coupling the particles and field does not introduce any y- or z-components of the electric field.

The actual number of electrons in the multipactor can be very large, so to reduce the computational cost a macroparticle-based approach was implemented. Each model particle represents n electrons for the purpose of computing the volumetric space charge density, where in this example n = 50,000. This value is entered into the text field in the settings for the multiphysics coupling node. To check that the model is statistically converged, consider recomputing with a lower value of n, such as 10,000. This might require an increase in the (see below), but the total charge of particles hitting the cavity walls should be unaffected if the model is statistically converged (see Figure 3).

The secondary electron yield is a continuous function of energy and is usually not integer-valued. For instance, consider a value of 2.4. In this scenario the boundary condition has been configured to always release two secondary electrons, and to release a third electron with a 40% probability. For this reason, each node has two attributes, one for a whole number of guaranteed secondary electrons, and one for an additional secondary electron with an emission probability between 0 and 1.

It is important to allocate a sufficiently large number of secondary particles so that all of the secondary electron emission in the model can be simulated. The can be controlled in the physics interface section. Because electrons are constantly added to and removed from the simulation domain, it is highly beneficial to recycle the degrees of freedom associated with the removed model particles. In the physics interface section, the option was selected from the . The default behavior is not to recycle degrees of freedom so that every emitted secondary particle has a unique index, but at the cost of significantly more DOFs overall.

The constant, spatially uniform magnetic field was applied using the node. Because the particles in this example are nonrelativistic, the contribution of the moving electrons to the background magnetic field was deemed negligible.

The length of the simulation domain in the y and z directions should not affect the solution, so any length can be selected that allows the solution to be viewed easily in the Graphics window. When a particle hits one of the surfaces parallel to the xy- or zx-planes, the particle is immediately moved to the opposite face and then carries on with its precollision velocity. Particles can be mapped from a source boundary to a destination boundary using the feature. This example uses a pair of nodes, one for the two surfaces parallel to the xy-plane and another for the two surfaces parallel to the zx-plane.

Results and Discussion

The total charge of impacting particles on the cavity walls per half period is plotted over 20 RF cycles in Figure 3. Compare this figure to Figure 6a in Ref. 1.

After saturation has been reached, two distinct populations of electron can be observed by carefully inspecting every frame of the default plot. For many electrons, after being released from one wall, they are accelerated enough so that they hit the opposite wall about half of one RF cycle later. For some other particles, the trajectory is sufficiently curved by the magnetic field that they do not reach the opposite wall before the RF signal reverses direction, so they are instead accelerated back toward the wall that released them.

Another way to observe the behavior after saturation is to plot and animate the distribution of space charge density in a plot through the model geometry. Figure 4 shows one such plot at a time after saturation has been reached. The dark band is the main band containing most of the newly released secondary electrons. The lighter band includes some recently released secondary particles that lead or trail the main band due to space charge effects, as well as electrons that did not quite reach the opposite wall during the last RF cycle. The space charge density uses constant shape functions within each mesh element, so the sharp changes in color just indicate where one element ends and the next element begins. To make plot look smoother, consider refining the mesh in the x direction (which might also require a larger number of model particles for statistical convergence) or smoothing the solution during results processing.

You can also try animating either the plot or the plot shown in Figure 4 to visualize how the band of maximum charge density moves in phase with the applied RF voltage.

Figure 3: Total charge of impacting electrons per unit area, per RF half-cycle.

Figure 4: Space charge density in a cross section of the model geometry after saturation.

Reference

1. S. Riyopoulos, “Multipactor saturation due to space-charge-induced debunching,” Phys. Plasmas, vol. 4, no. 5, pp. 1448–1462, 1997.

Particle_Tracing_Module/Charged_Particle_Tracing/multipactor_saturation

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

Global Definitions

Parameters 1

Load the model parameters from a file.

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 multipactor_saturation_parameters.txt.

Variables 1

In the toolbar, click

and choose .

Load the global variables from a file.

In the window for , locate the section.

Click

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

These global variables will be used to suppress the release of secondary electrons when the RF field exerts a force in the outward normal direction.

Piecewise 1 (pw1)

In the toolbar, click

and choose > .

In the window for , locate the section.

Find the subsection. In the table, enter the following settings:


Start	End	Function
0	300	2.8((x/300)exp(1-x/300))^0.6
300	5000	2.8((x/300)exp(1-x/300))^0.2

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

In the text field, type 1.

Click

. Compare the resulting plot to Figure 1. This function will be used to define the secondary electron yield (SEY) as a function of the kinetic energy of the incident particle.

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Block 1 (blk1)

In the toolbar, click

In the window for , locate the section.

In the text field, type D.

In the text field, type L.

Click

Any value of L should work, since the geometry extends infinitely far in the y and z directions, so a value was chosen that allows the geometry to be viewed easily in the window.

Definitions

Variables 2

In the toolbar, click

In the window for , locate the section.

Click

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

Some of these variables will be used to define the number of released secondary electrons at the boundaries. Others are postprocessing variables that will be used to plot the number of electron impacts at the cavity walls over time. Note how the at operator is used to evaluate an expression at a previous solution time.

Electrostatics (es)

Ground 1

In the toolbar, click

and choose .

Select Boundary 1 only.

Electric Potential 1

In the toolbar, click

and choose .

Select Boundary 6 only.

In the window for , locate the section.

In the V0 text field, type V0*sin(2*pi*f0*t).

Next, allow to be shown if they are not already shown. This is required to access the settings that govern recycling of degrees of freedom for the Charged Particle Tracing interface.

Root

Click the

button in the toolbar.

In the dialog, in the tree, select the checkbox for the node > .

Click .

Charged Particle Tracing (cpt)

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the text field, type 1000.

Click to expand the section. From the list, choose .

Electric Force 1

In the window, under > click .

In the window for , locate the section.

From the E list, choose .

Magnetic Force 1

In the toolbar, click

and choose .

Select Domain 1 only.

In the window for , locate the section.

Specify the B vector as


0	x
0	y
B0	z

Release from Grid 1

In the toolbar, click

and choose .

In the window for , locate the section.

In the qx,0 text field, type range(0.01,0.02,0.99)*D.

In the qy,0 text field, type L/2.

In the qz,0 text field, type L/2.

The initial particle positions are distributed across the width of the cavity so that at least some of these seed charges will hit the walls at sufficiently high energy to start the electron avalanche.

Next, two nodes will be added to nodes on opposite sides of the cavity. The first releases a guaranteed number of secondary electrons equal to the largest integer less than the secondary electron yield. The second releases one more electron with a probability based on the remainder of the secondary electron yield. Two nodes have been used because each has an additional requirement that the force from the RF signal should point into the simulation domain, or else no particles can be released.

Wall 2

In the toolbar, click

and choose .

Select Boundary 1 only.

In the window for , locate the section.

From the list, choose .

Secondary Emission 1

In the toolbar, click

and choose .

In the window for , locate the section.

In the Ns text field, type SEY_int_left.

From the list, choose .

In the T text field, type T0.

Secondary Emission 2

Right-click and choose .

In the window for , locate the section.

From the list, choose .

In the γ text field, type SEY_frac_left.

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

Wall 2

In the window, click .

Accumulator 1

In the toolbar, click

and choose .

In the window for , locate the section.

Click

In the dialog, type surfacecharge in the text field.

In the tree, select > .

Click .

In the window for , locate the section.

In the R text field, type e_const*n.

These accumulated variables will be used in results processing to track the total number of electron impacts on each surface over time.

Now duplicate this boundary condition and apply the copy to the opposite wall, with the appropriate modifications.

Wall 3

Right-click and choose .

Select Boundary 6 only.

Secondary Emission 1

In the window, expand the node, then click .

In the window for , locate the section.

In the Ns text field, type SEY_int_right.

Secondary Emission 2

In the window, click .

In the window for , locate the section.

In the γ text field, type SEY_frac_right.

Periodic boundary conditions will be used on the remaining surfaces to represent the infinite extent of the cavity in the y and z directions. Each node is applied to one pair of boundaries on opposite sides.

Periodic Condition 1

In the toolbar, click

and choose .

Select Boundaries 3 and 4 only.

Periodic Condition 2

In the toolbar, click

and choose .

Select Boundaries 2 and 5 only.

Multiphysics

Electric Particle–Field Interaction 1 (epfi1)

In the window, under > click .

In the window for , click to expand the section.

In the n text field, type n.

This multiplication factor is the number of real electrons that are represented by each model particle for the purpose of computing the space charge density. This prevents an inordinate number of particle degrees of freedom from being required.

Mesh 1

A structured mesh will be designed with a single element in the y and z directions, but very fine resolution in the x direction in which the space charge density may vary greatly.

Mapped 1

In the toolbar, click

and choose .

Select Boundary 1 only.

Distribution 1

In the toolbar, click

Select Edges 1, 2, 4, and 6 only.

In the window for , locate the section.

In the text field, type 1.

Swept 1

In the toolbar, click

Distribution 1

In the toolbar, click

In the window for , locate the section.

In the text field, type 100.

Click

. Compare the resulting plot to Figure 2.

Study 1

Now adjust the study settings to trace the particles for up to 20 RF cycles. A rather small time step is used in order to create higher-quality animations in postprocessing, but a larger time step could also be used to reduce file size.

Step 1: Time Dependent

In the window, under click .

In the window for , locate the section.

In the text field, type range(0,1/(100*f0),20/f0).

Solution 1 (sol1)

In the toolbar, click

In the window, expand the node.

In the window, expand the > > > node.

Right-click > > > and choose .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

If the in the settings for the Charged Particle Tracing interface is too low, a warning might appear, saying that some secondary particles were not released. For the parameter values in these instructions, 1000 secondary particles is expected to suffice and this warning should not appear. If you later reduce the value of the model parameter n, the charge multiplication factor, then an increase in the preallocated secondary particles may be required.

Results

Some default plots, as well as a default dataset, are created. These can be viewed at any time. To see the quantitative information that indicates whether saturation has been reached, create a to track the number of electron impacts over time.

1D Plot Group 5

In the toolbar, click