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Trajectories |

Our basic explanation is split into several steps

Usually electron trajectory calculations use the magnetic field strength and the Lorenz force. Here we

calculate the vector potential field directly,and use force equations derived from the change in momentum.

calculate the vector potential field directly,and use force equations derived from the change in momentum.

In the VPPEM the momentum from the vector potential is converted

into the off-axis momentum of an angular image by suddenly

terminating the field.

To understand this action in detail we have created a computer

program to simulate an electron trajectory where the Lorenz force has

been rewritten using the forces due to changes in the vector potential.

into the off-axis momentum of an angular image by suddenly

terminating the field.

To understand this action in detail we have created a computer

program to simulate an electron trajectory where the Lorenz force has

been rewritten using the forces due to changes in the vector potential.

The vector potential is given by equation 1:

(1)

Where A(r) is the vector potential at position r, J(r ́ ) is the current

density at position r ́, and Ω is the volume of the current density.

density at position r ́, and Ω is the volume of the current density.

Integrating Equation 1 around a circular loop of current gives a vector

potential field that is zero on the axis, and is a maximum at the radius.

The direction of the momentum is around the axis in the direction of

the current. The curl (∇×) of this field distribution is a constant. From

Maxwell’s equations the magnetic field is:

potential field that is zero on the axis, and is a maximum at the radius.

The direction of the momentum is around the axis in the direction of

the current. The curl (∇×) of this field distribution is a constant. From

Maxwell’s equations the magnetic field is:

(2)

The canonical momentum of a charged particle in a magnetic field is:

(3)

Where **p** is the momentum, m is the electron mass, and q is the

electron charge.

Using standard analytical mechanics we write the Lagrangian for an

electron in a vector potential field:

electron charge.

Using standard analytical mechanics we write the Lagrangian for an

electron in a vector potential field:

(4)

After using Lagrange’s equations:

(5)

We obtain the resolved force equations:

(6)

(7)

(8)

The force equations are used to track trajectories in a simple model system.

Figure 1 shows a model system with two current loops.

Figure 1 shows a model system with two current loops.

Figure 1. Two current loop model for electron trajectory simulations using the

vector potential. The current in loop b is opposite to the current in loop a, and

reduces the total field at the center of loop b to zero.

vector potential. The current in loop b is opposite to the current in loop a, and

reduces the total field at the center of loop b to zero.

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We have arranged the respective currents in the current loops to produce a

zero field at the center of the second loop. After passing the second loop the

field is numerically set to zero to simulate the electron passing through a

ferromagnetic shield.

zero field at the center of the second loop. After passing the second loop the

field is numerically set to zero to simulate the electron passing through a

ferromagnetic shield.

All terms in Az are zero in equations 6-8 because there is no resultant

vector potential out of the plane of the current loops.

For any position in the electron trajectory Ax and Ay, and their

differentials are numerically calculated by integrating equation 1 around

the current loops.

The initial field at the center of loop a along the z axis, Bz, is calculated

from the curl of the vector potential, or simply:

vector potential out of the plane of the current loops.

For any position in the electron trajectory Ax and Ay, and their

differentials are numerically calculated by integrating equation 1 around

the current loops.

The initial field at the center of loop a along the z axis, Bz, is calculated

from the curl of the vector potential, or simply:

(9)

From the dimensions and current shown in Figure 1 the initial

Bz = 0.06283 Tesla, in agreement with the standard calculation.

The shape of the axial B field with z is shown in Figure 2.

Bz = 0.06283 Tesla, in agreement with the standard calculation.

The shape of the axial B field with z is shown in Figure 2.

Figure 2 Axial magnetic field Bz of the model in Figure 1. Inset is a detail that

shows the ‘knee’ of the magnetic field.

shows the ‘knee’ of the magnetic field.

Starting at z=0, x=0, and y=10 microns the simulated electron trajectory

moves smoothly along a field line with its distance from the axis in y given

approximately by:

moves smoothly along a field line with its distance from the axis in y given

approximately by:

(10)

With higher energy electrons there is a significant deviation from this

relationship due to non-adiabatic behavior. This is seen in the additional

deflection in the x direction.

Figure 3 shows the results of a simulation of a 2.0 eV electron leaving the

central field parallel to the z axis with an initial y offset of 10 microns.

Figure 3 shows that the final angle of the trajectory in the y direction is

largely determined by the curve in the trajectory following the field line.

In contrast, the angle of the trajectory in the x direction is directly due to the

force created by the termination of the field.

The combination of the two deflections causes a rotation of the trajectory

out of the y plane.

relationship due to non-adiabatic behavior. This is seen in the additional

deflection in the x direction.

Figure 3 shows the results of a simulation of a 2.0 eV electron leaving the

central field parallel to the z axis with an initial y offset of 10 microns.

Figure 3 shows that the final angle of the trajectory in the y direction is

largely determined by the curve in the trajectory following the field line.

In contrast, the angle of the trajectory in the x direction is directly due to the

force created by the termination of the field.

The combination of the two deflections causes a rotation of the trajectory

out of the y plane.

Figure 3 A simulated 2.0 eV electron trajectory resolved on the x and y axis for

the model shown in Figure 1.

the model shown in Figure 1.

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The angle of rotation is a constant of the model system, and does not

depend on the starting position in x and y.

The resultant angular images have two angular properties, the first in the

direction of the magnetic vector potential (φ the transverse rotational

direction, the x direction), and the second in the response to the magnetic

field lines diverging from the axis at the field termination (θ the off-axis

inclination, the y direction)

depend on the starting position in x and y.

The resultant angular images have two angular properties, the first in the

direction of the magnetic vector potential (φ the transverse rotational

direction, the x direction), and the second in the response to the magnetic

field lines diverging from the axis at the field termination (θ the off-axis

inclination, the y direction)

It can also be seen that the apparent position of the source of the angled

trajectory is at a significantly different position along the z axis for the x and

y directions. This shift in position makes a true focus of a bundle of

Figure 4 shows the forces in Fx and Fy in the area of the knee in the

magnetic field. As the electron trajectory approaches the ‘knee’ in the

magnetic field due to the opposing small loop field, it sees a rapid

change in the magnetic field going to zero at the center of the second

loop, and the trajectory is no longer adiabatic.

The force due to the change in the vector potential in the x direction

increases more than the force in the y direction, and the electron is

deflected away from the axis.

trajectory is at a significantly different position along the z axis for the x and

y directions. This shift in position makes a true focus of a bundle of

Figure 4 shows the forces in Fx and Fy in the area of the knee in the

magnetic field. As the electron trajectory approaches the ‘knee’ in the

magnetic field due to the opposing small loop field, it sees a rapid

change in the magnetic field going to zero at the center of the second

loop, and the trajectory is no longer adiabatic.

The force due to the change in the vector potential in the x direction

increases more than the force in the y direction, and the electron is

deflected away from the axis.

Figure 4 Resolved forces on an electron at the termination of the magnetic

field: x (blue line), y (red line). The forces rapidly increase as the electron is

driven away from the magnetic field line, and then decay as the electron leaves

the field. The actual position of the magnetic field knee is shown

superimposed as the black line.

field: x (blue line), y (red line). The forces rapidly increase as the electron is

driven away from the magnetic field line, and then decay as the electron leaves

the field. The actual position of the magnetic field knee is shown

superimposed as the black line.

More information on how it works

How it works |