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.
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.
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.
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.
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:
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.
The charge function is evaluated at the mesh points by assigning the charge of the computational particles according to their position.
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.
The fixed source emission model allows defining fixed velocities and particle numbers to emission geometry.
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.
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)
The number of particles is given by the user. For the computation of initial velocities \(v\) there are two options possible:
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.
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’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:
Type 6: Directions Orthogonal
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 (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:
Type 8: Zero Initial Velocity: Particles are initialized with zero velocity at time step 0 at the source elements.
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.
Roger W. Hockney, James W. Eastwood: Computer Simulation Using Particles. CRC Press, 1988
Jones and Langmuir GE Review, The Characteristics of Tungsten Filaments as Functions of Temperature 30 (1927) Part I pp. 310–19, Part II pp. 354–61, Part III pp. 408–12
Melissinos Experiments in Modern Physics, 1966, pp. 65–80
Preston and Dietz The Art of Experimental Physics, 1991, pp. 141–47, 152–61
Blakemore Solid State Physics, 1974, pp. 188–95
Koller The Physics of Electron Tubes, 1937, Ch. I, VI, VIII
Script "Thermionic Emission" Chapter 5.
Found at http://www.physics.csbsju.edu/370/thermionic.pdf
Hanno Krieger: "Strahlungsquellen für Technik und Medizin"