Ion Cyclotron Motion

If a charged particle is placed in a uniform magnetic field, it moves in a helical pattern about a fixed gyro radius. The gyro radius, which is also known as the Larmor or cyclotron radius, is given by the simple equation:

•

rL (SI unit: m) is the Larmor radius,

•

(SI unit: m/s) is the velocity component orthogonal to the magnetic field,

•

Z (dimensionless) is the particle charge number,

•

e = 1.602176634 × 10−19 C is the elementary charge,

•

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

•

B (SI unit: T) is the magnitude of the magnetic flux density.

This model computes the trajectory of an ion in a uniform magnetic field using the Newtonian, Lagrangian, and Hamiltonian formulations available in the Mathematical Particle Tracing interface.

Model Definition

The equations of motion for a charge in a magnetic field can be determined from the Lagrange equations:

(1)

where v is the particle velocity, q is the particle position, and L (SI unit: J) is the Lagrangian, which is defined as:

This form of the Lagrangian is valid for nonrelativistic particles; that is, the particle velocity is much less than the speed of light. The contribution due to the electric potential is neglected. The Hamiltonian is related to the Lagrangian via:

Introducing the generalized momentum of the particle, P (SI unit: kg·m/s), the Hamiltonian becomes:

In order to derive the equations of motion for the Newtonian formulation, start with the right-hand side of Equation 1:

(2)

The left-hand side of Equation 1 becomes

(3)

Equating Equation 2 and Equation 3 and canceling like terms yields

(4)

for a stationary magnetic field. Here, the magnetic flux density has been introduced as

(5)

When the particle velocity is small compared to the speed of light Equation 4 yields the classical equation of motion for a charged particle in a stationary, uniform magnetic field

Results and Discussion

The model is solved in COMSOL using the Lagrangian, Hamiltonian, and Newtonian formulations. The Larmor radius is compared to the analytic solution and given in Table 1. All formulations agree with the analytic expression to within 0.05 %. The two coupled first-order differential equations give a slightly different result from a single second-order differential equation for each coordinate.

Table 1: table comparing the Larmor radius for the different formulations.
	Analytic	Lagrangian	Hamiltonian	Newtonian, 2nd order	Newtonian, 1st order
Larmor radius (μm)	414.57	414.55	414.55	414.55	414.57

The particle trajectories for the three different formulations are plotted below.

Figure 1: Plot of the ion trajectory for the Lagrangian formulation.

Figure 2: Plot of the ion trajectory for the Hamiltonian formulation.

Figure 3: Plot of the particle trajectory for the Newtonian formulation.

Figure 4: Plot of the particle trajectory for the first-order Newtonian formulation.

Reference

1. L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, 4th ed., Elsevier, 2005.

Particle_Tracing_Module/Charged_Particle_Tracing/ion_cyclotron_motion

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

Cylinder 1 (cyl1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 2e-3.

In the toolbar, click

Global Definitions

Define parameters for the particle mass, magnetic flux density, initial particle velocity, and Larmor radius. The Larmor radius is only used during results processing.

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
mp	0.04[kg/mol]/N_A_const	6.6422E-26 kg	Ion mass
B	2[T]	2 T	Magnetic flux density
v0	2E3[m/s]	2000 m/s	Particle velocity, perpendicular to the magnetic field
rL	mpv0/(e_constB)	4.1457E-4 m	Larmor radius

Definitions

Define an analytic expression for the magnetic vector potential, which results in a uniform magnetic field in the z direction.

Variables 1

In the window, expand the node.

Right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
Ax	1[Wb/m]*y[1/m]	Wb/m	Magnetic vector potential, x-component
Ay	-1[Wb/m]*x[1/m]	Wb/m	Magnetic vector potential, y-component
Az	0[Wb/m]	Wb/m	Magnetic vector potential, z-component
Bx	d(Az,y)-d(Ay,z)	T	Magnetic flux density, x-component
By	d(Ax,z)-d(Az,x)	T	Magnetic flux density, y-component
Bz	d(Ay,x)-d(Ax,y)	T	Magnetic flux density, z-component

Mathematical Particle Tracing (pt)

Release a single particle at the origin with an initial velocity in the x direction so that the Lorentz force is nonzero. Also add a small initial velocity in the z direction so that you can clearly see the particle trajectory after solving.

Release from Grid 1

In the window, under right-click and choose .

In the window for , locate the section.

Specify the v0 vector as


v0	x
0	y
1e2	z

The first formulation you will use is . The Lagrangian for a particle in a magnetic field is the sum of the particle kinetic energy, which is here defined as pt.Ep, and the dot product of the particle velocity and the magnetic potential, multiplied by the particle charge.

In the window, click .

In the window for , locate the section.

From the list, choose .

Particle Properties 1

In the window, click .

In the window for , locate the section.

In the mp text field, type mp.

Locate the section. In the L text field, type pt.Ep+e_const*(pt.vx*Ax+pt.vy*Ay+pt.vz*Az).

Mesh 1

Use a coarse mesh. The field is entered using an analytic expression, so the accuracy of the solution is independent of the mesh element size.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Click

Study 1

Step 1: Time Dependent

In the window, under click .

In the window for , locate the section.

In the text field, type range(0,5.0e-8,2.0e-5).

For all of the studies in this example, a tight user-defined tolerance will be used to ensure particle kinetic energy is conserved.

From the list, choose .

In the text field, type 1.0E-6.

In the window, click .

In the window for , type Lagrangian Study in the text field.

In the toolbar, click

Results

Lagrangian Results

In order to be able to see the radius of the particle orbit, plot the y-component of the particle location as a color expression.

In the window for , type Lagrangian Results in the text field.

Locate the section. Clear the check box.

Particle Trajectories 1

Render the particle trajectory as a ribbon. The default ribbon orientation is in the direction of the unit binormal, or the direction out of the plane tangent to the curved trajectory.

In the window, expand the node, then click .

In the window for , locate the section.

Find the subsection. From the list, choose .

Select the check box. In the associated text field, type 4E-5.

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 qy/2.

Locate the section. Click

In the dialog box, select in the tree.

Click .

In the toolbar, click

Click the

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

Particle Evaluation 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

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

Click the arrow next to the button and click .

Particle Evaluation 2

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the text field, type timemax(0,2E-5,qy)/2.

Click the arrow next to the button and select .

Now switch formulation from to . When you do this, the particle momentum components are added as additional degrees of freedom. The momentum has three components: px, py, and pz. This results in a doubling of the number of degrees of freedom in the model.

Mathematical Particle Tracing (pt)