Theory

In the following section we will provide a short introduction into the physics of charged particles. Here, we will first provide an abreviated description of the continuous system and then outline the basic numerical simulation techniques.

Continuous Description

Let us first provide an overview on the principle interaction behaviour of charged particles. To this end let us describe the interplay between particles and electromagnetic fields; as well as many particle phenomena.

Charged Particles

We start with the concept of a point-particle, sometimes called Dirac particle. A point-particle is a rather theoretical concept, describing a particle that occupies exactly on point in phase-space (i.e. it is located at an exact location \(\mathrm{x}\) with exact velocity \(\mathrm{v}\)) having a Mass \(\mathrm{m}\). Therefore, the point-particle concept is a principle concept o classical mechanics. Its motion is described by Newtons second law \[\begin{aligned} \mathbf{F} &= \mathrm{m} \, \mathbf{a}, \end{aligned}\] where \(\mathbf{F}\) is a principal newtonian force. If the above particle also comprises an electric charge \(\mathrm{q}\), thus making it an charged point-particle, we can specify the acting force to be the Lorentz force \[\begin{aligned} \mathbf{F}_L&= \mathrm{q} \left( \mathbf{E} + \mathbf{v} \times \mathbf{B}\right), \end{aligned}\] where \(\mathbf{E}\) and \(\mathbf{B}\) are the electric field and the magnetic flux density, respectively. Indeed, basic physic tells us that a charged particle is accelerated by an electric field and deflected by an magnetic field (this is exactly the meaning of the Lorentz Force). With this knowledge at hand we can now derive the velocities and the positions by simple integration \[\begin{aligned} \mathbf{a} (t) &= \tfrac{\mathrm{q}}{\mathrm{m}} \left( \mathbf{E} + \mathbf{v} \times \mathbf{B}\right),\\ \mathbf{v} (t) &= \tfrac{\mathrm{q}}{\mathrm{m}} \left( \mathbf{E} + \mathbf{v} \times \mathbf{B}\right) t + \mathbf{v}_0,\\ \mathbf{x} (t) &= \tfrac{1}{2} \tfrac{\mathrm{q}}{\mathrm{m}} \left( \mathbf{E} + \mathbf{v} \times \mathbf{B}\right) t^2 + \mathbf{v}_0 t + \mathbf{x}_0. \end{aligned}\] Observe the immanent charge-mass relation factor \(\tfrac{\mathrm{q}}{\mathrm{m}}\) that determines the coupling strength.

Maxwells Equations with Charged Particles

With the principal particle description at hand, we may now continue with the field description. We know that electromagnetic fields are described by Maxwell’s equations \[\begin{aligned} \nabla \cdot \mathbf{D} &= \rho,\\ \nabla \times \mathbf{H} &= \tfrac{\mathrm{d} \mathbf{D}}{ \mathrm{d} t} + \mathbf{J},\\ \nabla \cdot \mathbf{B} &= 0,\\ \nabla \times \mathbf{E} &= - \tfrac{\mathrm{d} \mathbf{B}}{ \mathrm{d} t}. \end{aligned}\] We can split this set of coupled equations into two parts: source equations (i.e. Gauss’ Law and Ampere’s law) and source-free equations (i.e. Faradays Law and magnetic Gauss law). From the previous discussion we know that a particle \(i\) carries a charge \(\mathrm{q}_i\). Moreover, moving charges also create a current (microscopically speaking this is the acctual origin of currents: the macroscopic current in a wire is created by multiple moving electrons) \[\begin{aligned} \mathbf{J}_i &= \mathrm{q}_i \, \mathbf{v}_i. \end{aligned}\] With this in mind we can directly write down macroscopic charges and currents \[\begin{aligned} \rho = \sum_i \mathrm{q}_i \quad \mathrm{and} \quad \mathbf{J}_{\mathrm{part}} = \sum_i \mathrm{q}_i \, \mathbf{v}_i. \end{aligned}\] Here, \(\rho\) describes an ensemble of point-particle charges and directly provides a source for Gauss’ law. If one combines the particle current with the possible conductor current, i.e. \(\mathbf{J} = \sigma \mathbf{E} + \mathbf{J}_{\mathrm{part}}\), the resulting total current \(\mathbf{J}\) may be used as a (current) source for Ampere’s law.

Vlasov Equation and Plasma Physics

Discrete Description

Particle In Cell Methods

The PIC method for electric and magnetic field calculations consists of the following procedure. Discrete charge density and current density arrays are defined with values in each cell. The charge density in a cell at time \(t\) equals the product of the number of computational particles in the cell times the charge per computational particle divided by the cell volume. The current density is the average vector velocity of computational particles multiplied by their charge divided by the cell area.
The space-charge and current density functions are combined with a Finite Element solution of the Maxwell equations to generate electric and magnetic field values. The PIC procedure involves one pass through the array of computational particles assigning them to cells, followed by a field computation. The number of mathematical operations is linearly proportional to the number of particles and the number of cells. In a simulation involving thousands of particles, the PIC process takes much less time than a direct evaluation of inter-particle forces.
An electrostatic computer simulation using the PIC method for field calculations consists of the following operations at each time step:

  1. At time \(t\), the electric field at the location of each computational particle is interpolated from the field result. The field is used to advance the vectors \(\mathbf{x(t)}\) and \(\mathbf{v(t)}\) for the particles to time \(t+\Delta t\) using an accurate difference scheme.

  2. The charge function is evaluated at the mesh points by assigning the charge of the computational particles according to their position.

  3. The electrostatic equation is solved to find electric fields at \(t+\Delta t\), and the process is repeated.

The procedure continues until the beam advances to the desired final state.

Particle Stepper

Particle Smearing

The Testparticle Method

Fixed Source Emission Model (Type 0)

The fixed source emission model allows defining fixed velocities and particle numbers to emission geometry.

Given Velocity Field Emission Model (Type 3)

The ’Given Velocity Field’ emission model allows defining particle number and initial velocities through a tabular field. This can be for instance the velocities resulting from CFD simulations.

Thermal Emission Models

Thermionic emission is the thermally induced flow of charged particles from a surface. This occurs because the thermal energy given to the particles overcomes the escape energy of the material. The charged particles can be for instance electrons or ions. The classical example of thermionic emission is that of electrons from a hot cathode into a vacuum (also known as thermal electron emission or the Edison effect) in a vacuum tube. The hot cathode can be a metal filament, a coated metal filament, or a separate structure of metal. The below table lists typical escape energies and work temperatures of some materials. Following are thermionic source types explained that are available in MAGNETICS.

Material Escape Energy typical work
Wa [eV] temperature [K]
Metal cathodes Cäsium 1.94
Molybdän 4.29
Nickel 4.91
Platin 5.30
Tantal 4.13 2400
Thorium 3.35
Wolfram 4.50 2600
Metal film cathodes Barium film on Wolfram 1.5-2.1 1200
Cäsium film on Wolfram 1.4
Thorium film on Wolfram 2.8 2000
Barium film on Wolframoxide 1.3
Oxide cathodes Barium oxide 1.0-1.5
Barium oxide with 0.9-1.3 1100
Strontium oxide


Table: Experimentally determined work functions for the thermal emission of electrons from hot cathodes (Krieger data)

Ideal Gas with mean velocities (Type 1 and 5)

The number of particles is given by the user. For the computation of initial velocities \(v\) there are two options possible:

  1. Type 5: Ideal Gas with mean velocities, local temperatures: Velocities are computed by the mean of Maxwell Speed Distribution for ideal gas which is given as \[v=\sqrt{\frac{3 k T}{m}}\] with \(k\) the Boltzmann’s constant, \(m\) the mass of an particle and \(T\) the locally varying temperatures which can be applied as field or precomputed by thermal simulation.

  2. Type 1: Random Velocity at a fixed (global) Temperature: Velocities are computed through the same mean of Maxwell Speed Distribution for ideal gas as for type 5, but directions are randomly defined. Temperatures are not local but fixed by the user (global).

Richardson-Dushman (Type 6 and 7)

Richardson’s equation describes the current density \(J\) of electrons emitted from a metal at high temperatures. it is \[J=A T^{2} e^{-W / k T}\] where \[A=\frac{4 \pi e m k^{2}}{h^{3}}=1.2 \times 10^{6} \mathrm{A} / \mathrm{m}^{2} \mathrm{K}^{2}\] Here, \(T\) is the temperature, \(W\) the work functions (escape energy) for electrons, \(k\) the Boltzmann constant and \(A\) the Richardson constant. This source type assumes that the electron current emitted by an electrode is independent of the applied voltage, and that it depends only on the temperature, work function and emission constant of the material. This is also called thermal saturation limit. The number of particles and the initial velocities are both derived automatically if this type is chosen. Temperatures can be locally varying, applied as initial constraint, field or precomputed by another simulation.
Initial velocities \(v\) are derived from the escape energy by \[v=\sqrt{\frac{2 W}{m}}\] with \(W\) the escape energy in electron volt and \(m\) the particle mass. There are two options available that control the velocity directions:

  1. Type 6: Directions Orthogonal

  2. Type 7: Random directions

The user should chose the input parameters (temperature and escape energy) carefully because in many cases there will be either none or a huge number of particles created.

Child’s Law (Type 8 and 9)

Child’s law (or Child, Langmuir) gives the maximum current density that can be carried in a beam of charged particles across a one dimensional accelerating gap. In order to apply this equation within the program, an accelerating gap width \(d\) and a voltage difference \(V_{a}\) must be applied. The equation only applies to infinite planar emitters. For electrons, the current density \(J [A/m^{2}]\) is written: \[J=\frac{I_{a}}{S}=\frac{4 \epsilon_{0}}{9} \sqrt{2 e / m} \frac{V_{a}^{3 / 2}}{d^{2}}\] where \(I_{a}\) is the anode current and \(S\) the anode surface inner area; \(e\) is the magnitude of the charge of the electron and \(m\) is its mass.

Particle velocities \(v\) are calculated after time step 1 with the assumption of a constant electric field \(E\) and energy conservation through Newton’s law \(F=m a\) and Lorentz force \(F=q E\) what results as \[v = \frac{q}{m} E \Delta t\] Using those velocities \(v\) at time step 1 and \(J\) from Child’s law the number of particles \(n\) can be computed with \(J=n q v\).

Finally the initial velocities and particle positions must be found. There are two options available for this:

  1. Type 8: Zero Initial Velocity: Particles are initialized with zero velocity at time step 0 at the source elements.

  2. Type 9: Non Zero Initial Velocity: With the knowledge of velocity and position at time step 1 particles are propagated to time step 0. The resulting initial velocities are non zero and initial positions are near to their emission surfaces.

References