PartIII.Bookmark 1: Acute repulsion between electrons in a classical collision.
Part III.Bookmark 4 : Paired electrons: the repulsive force from the centers of charge approximates the centers of mass, allowing a stable configuration.
Part III.Bookmark 4 : Paired electrons: the repulsive force from the centers of charge approximates the centers of mass, allowing a stable configuration.
2. Summary
2. Summary
3. Electron Collision Simulator
3. Electron Collision Simulator
3.1 Description of the Simulator
3.1 Description of the Simulator
The purpose of this simulator is to visualize and provide an intuition on the dynamics of two electrons when they are shot one against the other, taking into account their spin orientation, their initial position and velocity and the presence of external electromagnetic fields in a relativistic framework. The formalism is relativistic and therefore there is no restriction on the value of the maximum energy allowed for the analysis. This is not a hidden variables model. Here all classical variables have a very well defined interpretation like positions, velocities and orientations. The dynamical equations are non-linear and a very small modification of the angular variables that define the orientation of the spins or the initial internal phase of the position of the CC’s produces a completely different scattering process, which closely resembles what we find in experimental results. Experimental physicists interested in this topic can numerically calculate average values of scattering processes by integrating the dynamical equations leaving some of the parameters free and averaging them, for example keeping constant initial spin orientations and averaging over the internal phases ψ1 and ψ2 of the two particles. If the electrons of the beam are depolarized the average should be done on the θ and ϕ variables that control the spin orientation of the two particles.As we will see in the equations derived in later sections, there are terms with , and the Poincaré invariant interaction becomes singular when = , even if the two particles are far away. The numerical integration breaks down when this happens and sometimes, in the Poincaré numerical integration, if the numerical value of the expression · > 1, the squared root , is a complex number and the numerical integration fails. A small change in the boundary conditions can help to repair this fault, related to the numerical integration routine and not to the physical system. In any case a small ϵ is added solving most of the divergence problems: In the next subsections we have the complete code for the two electrons collision simulator ending with a user configurable visual model of the interactions (Manipulate 3D plot).In later sections we will provide the theoretical derivation of the dynamical equations that are the basis for this model and the analysis of some electron collision cases.
1
1−·
u1
u2
u1
u2
u1
u2
1−·
u1
u2
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1−·+ϵ
u1
u2
3.2 User Interface
3.2 User Interface
The user interface of the Electron Collision Simulator depicts particle 1 in blue and particle 2 in red. Full lines are used for plotting the evolution of the CC’s while dotted lines represent the motion of the CM’s.
It depicts the initial CM velocity of each particle and also at different CM positions the CM spin vector of each particle (the user can select the number of spin vectors, and scale for better visualization and even a trace of their evolution over the spins ). The final CM spin is shown at the final CM position of each particle, blue for particle 1 and red for particle 2, to see how the spin orientation has changed during the interaction.
The boundary conditions panel is shown on the left hand side. It allows you to manually customize the exact value desired for each boundary variable and press ENTER to update the plot with the new integration. To increase the responsiveness, the new integration is only performed when there is a change in parameters that affect the calculation.
- TMax is the maximum value of time that is calculated. The integration goes from 0 to TMax.
- TimeRange provides two cursors to define the time interval that is visualized, a subrange of what has been calculated with TMax.
- Visualization can also be improving by “zooming” with the values of the plot margins padding, that can be positive or negative.
- TimeCursor allows to move the time position of little points that mark the time instant in the CC & CM trajectories of both particles. Rblue and Rred measure the instantaneous distance between CC and CM, the current “radius”, which at slow speeds (v) should be around 0.5, and always in the range [0,1] if there has not been a numerical calculation error.
- Particles initial location (dx,dy,dz), speed (v,β,λ) and spin orientation and phase (θ,ϕ, ψ) can be customized. Also the presence of electric (ex,ey,ez) and magnetic (bx,by,bz) fields with their frequencies.
It is also possible to obtain an animation of consecutive integrals by activating the AutoExec with a customizable number of animation steps and length of the trace.
A sample of many possible interactions are included as referred bookmarks in the notebook that can be accessed through the + bookmark menu in the upper right corner.
In the figure subtitle we show the latest bookmark selected and the initial values of the numerical experiments.
Clicking the button on the bottom will save the plotted image (or video .gif if autoexec is running) in the directory where the Mathematica Notebook is located.
It depicts the initial CM velocity of each particle and also at different CM positions the CM spin vector of each particle (the user can select the number of spin vectors, and scale for better visualization and even a trace of their evolution over the spins ). The final CM spin is shown at the final CM position of each particle, blue for particle 1 and red for particle 2, to see how the spin orientation has changed during the interaction.
The boundary conditions panel is shown on the left hand side. It allows you to manually customize the exact value desired for each boundary variable and press ENTER to update the plot with the new integration. To increase the responsiveness, the new integration is only performed when there is a change in parameters that affect the calculation.
- TMax is the maximum value of time that is calculated. The integration goes from 0 to TMax.
- TimeRange provides two cursors to define the time interval that is visualized, a subrange of what has been calculated with TMax.
- Visualization can also be improving by “zooming” with the values of the plot margins padding, that can be positive or negative.
- TimeCursor allows to move the time position of little points that mark the time instant in the CC & CM trajectories of both particles. Rblue and Rred measure the instantaneous distance between CC and CM, the current “radius”, which at slow speeds (v) should be around 0.5, and always in the range [0,1] if there has not been a numerical calculation error.
- Particles initial location (dx,dy,dz), speed (v,β,λ) and spin orientation and phase (θ,ϕ, ψ) can be customized. Also the presence of electric (ex,ey,ez) and magnetic (bx,by,bz) fields with their frequencies.
It is also possible to obtain an animation of consecutive integrals by activating the AutoExec with a customizable number of animation steps and length of the trace.
A sample of many possible interactions are included as referred bookmarks in the notebook that can be accessed through the + bookmark menu in the upper right corner.
In the figure subtitle we show the latest bookmark selected and the initial values of the numerical experiments.
Clicking the button on the bottom will save the plotted image (or video .gif if autoexec is running) in the directory where the Mathematica Notebook is located.
3.3 Code
3.3 Code
3.3.0 Clean previous outputs
3.3.0 Clean previous outputs
3.3.1 Vector definition of different observables
3.3.1 Vector definition of different observables
3.3.2 Poincare Initial Conditions
3.3.2 Poincare Initial Conditions
3.3.3 Dynamical equations
3.3.3 Dynamical equations
3.3.4 Solve the dynamical equations
3.3.4 Solve the dynamical equations
3.3.5 Bookmarks
3.3.5 Bookmarks
3.3.6 Manipulate definition
3.3.6 Manipulate definition
3.3.7 Output
3.3.7 Output
In[]:=
manipulatePlot
Out[]=
3.3.8 GIF Images Generator
3.3.8 GIF Images Generator
4. Spinning Particle Model
4. Spinning Particle Model
4.0 Basic Equations
4.0 Basic Equations
This model represents the circular motion, at the speed of light, of the center of charge (CC) of the electron in the center of mass (CM) frame, as described by the dynamical equation (=1/2 contained in the plane orthogonal to the Spin. The center of mass (CM) is always a different point than the center of charge (CC). The spin has the opposite direction to the angular velocity (due to the minus sign in Part I.(30) =-m). The radius of this motion is = , in this frame. The angular velocity is ω = .
6
). In the free motion the CC describes a circle of radius R
0
S
ω
S
r
u
R
0
ℏ
2mc
2
2
mc
ℏ