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https://github.com/shahabahreini/magnetic-nozzle-particle-dynamics

M.Sc. Thesis in University of Saskatchewan - Charged Particle in EM Fields
https://github.com/shahabahreini/magnetic-nozzle-particle-dynamics

adiabatic-invariants charged-particles conservation-laws electric-field energy-conversion feagin14 julia magnetic-fields momentum-conservation numerical-analysis plotly python runge-kutta thesis usask

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M.Sc. Thesis in University of Saskatchewan - Charged Particle in EM Fields

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README 

        
# Particle Motion Simulation in Electromagnetic Fields



[![DOI](https://zenodo.org/badge/841295299.svg)](https://doi.org/10.5281/zenodo.14451939)



This repository provides a modern and professional framework for simulating particle motion in electromagnetic fields. It leverages **Julia** for high-performance numerical simulations and **Python** for advanced visualization and data analysis. The project offers a streamlined workflow, from running simulations via a user-friendly terminal interface to analyzing results with detailed plots and visualizations.



## Table of Contents



1. [Key Features](#key-features)

2. [Installation](#installation)

    1. [Prerequisites](#prerequisites)

    2. [Setup](#setup)

3. [Usage](#usage)

    1. [Running Simulations](#running-simulations)

    2. [Visualization and Post-Processing](#visualization-and-post-processing)

4. [Configuration](#configuration)

5. [Physics and Equations of Motion](#physics-and-equations-of-motion)

6. [Appendix](#appendix)

7. [Contributors](#contributors)



## Key Features



- **High-Performance Simulations**: Utilizes Julia for efficient numerical computations.

- **Interactive Visualization**: Python-based tools for generating 2D and 3D plots, energy and momentum conservation analysis, and peak detection.

- **User-Friendly Interface**: A menu-driven terminal interface for running simulations and configuring parameters.

- **Comprehensive Simulation Modes**: Supports 1D, 2D, and 3D simulations in cylindrical coordinates.

- **Adiabatic Invariants Analysis**: Tools for detecting extrema and analyzing invariants in particle motion.

- **Customizable Initial Conditions**: Easily modify initial parameters without changing source code.



## Installation



### Prerequisites



Ensure the following software is installed on your system:



- **Julia**: [Download Julia](https://julialang.org/downloads/)

- **Python**: [Download Python](https://www.python.org/downloads/)

- **Pip**: [Install Pip](https://pip.pypa.io/en/stable/installation/)



### Setup



#### Method 1: Automated Setup



Use the provided Julia and Python scripts for automated package management by selecting the **Package Management**:



```bash

julia Start.jl

```



and when you run the `visualization.py` it automatically check and install all requirements:



```bash

python3 visualization.py

```



#### Method 2: Manual Setup



Alternatively, install dependencies manually:



1. **Julia Packages**:



    ```bash

    julia install_packages.jl

    ```



2. **Python Packages**:



    ```bash

    python3 -m pip install -r requirements.txt

    ```



## Usage



### Running Simulations



1. **Launch the Simulator**:



    ```bash

    julia Start.jl

    ```



2. **Select Simulation Mode**:

    - [1] 1D Simulation

    - [2] 2D Simulation

    - [3] 3D Simulation



3. **Modify Initial Conditions**:

    Update the `initial_conditions.txt` file to set parameters such as:



    ```plaintext

    theta_0 = π/60

    alpha_0 = π/2

    beta_0 = 0

    phi_0 = 0

    epsilon = 0.002

    eps_phi = 0.0

    kappa = 0.01

    delta_star = 0.01

    time_end = 300.0

    ```



4. **Run Simulations**:

    Monitor progress and receive completion feedback in the terminal.



**Screen Recording**:



[![Video Thumbnail](https://via.placeholder.com/640x360.png?text=Click+to+Watch+Video)](https://github.com/user-attachments/assets/b58eab73-5906-48b0-a356-a0eec0d89a57)



### Visualization and Post-Processing



After running simulations, use the `visualization.py` script for all visualization tasks. The available options include:



- **Unified Terminal Interface**: Run 1D, 2D, and 3D simulations directly from a Julia-based menu-driven UI.

- **Interactive Parameter Configuration**: Modify initial conditions and simulation settings without changing source code.

- **Visualization Tools**: Python scripts for generating 2D and 3D plots, analyzing energy and momentum conservation, and detecting peaks in particle trajectories.

- **Comprehensive Simulation Modes**:

  - **1D Simulation**: Particle motion along a single axis.

  - **2D Simulation**: Particle motion in a plane with radial and vertical dynamics.

  - **3D Simulation**: Full spatial dynamics in cylindrical coordinates.

- **Peak Detection**: Identify extrema and analyze adiabatic invariants in particle motion.

- **Energy and Momentum Conservation**: Visualize energy and momentum trends over time.



Example usage:



```bash

python3 visualization.py

```



Follow the interactive prompts to select the desired analysis or plotting mode.



## Configuration



### Initial Conditions



Set initial conditions in the `initial_conditions.txt` file. Key parameters include:



- **theta_0**: Initial trajectory angle in radians.

- **alpha_0**: Initial pitch angle in radians.

- **beta_0**: Initial azimuthal angle in radians.

- **phi_0**: Initial azimuthal position in radians.

- **epsilon**: Dimensionless parameter representing the initial velocity magnitude.

- **eps_phi**: Electric field perturbation parameter.

- **kappa**: Magnetic field gradient strength.

- **delta_star**: Drift correction factor.

- **time_end**: Total simulation duration.



## Physics and Equations of Motion



### 2D Simulation



The 2D simulation models particle motion in **cylindrical coordinates** (`ρ`, `z`, `φ`). The key equations solved include:



- **Equation of Motion for `ρ` and `z`**:



```math

\frac{d^2 \rho}{dt^2}, \frac{d^2 z}{dt^2}

```



These describe the particle's motion in radial and vertical directions under the influence of electromagnetic fields.



### 3D Simulation



The 3D simulation solves for **radial distance** (`ρ`), **vertical position** (`z`), and **azimuthal angle** (`φ`). The key equations of motion involve:



```math

\frac{d{\widetilde{R}}^2}{{d\tau}^2}=\frac{ \widetilde{R}.\widetilde{z}}{{(\widetilde{R}^2+\widetilde{z}^2)}^{3/2}} \frac{d\phi}{d\tau}+ \widetilde{R} ({\frac{d\phi}{d\tau}})^2-\epsilon_\phi\frac{\partial \widetilde{\Phi}}{\partial \widetilde{R}}

```



```math

\frac{d{\widetilde{\phi}}^2}{{d\tau}^2}=\space(\frac{1}{{(\widetilde{R}^2+\widetilde{z}^2)}^{3/2}} (\widetilde{R}\frac{d\widetilde{z}}{d\tau}-\widetilde{z}\frac{d\widetilde{R}}{d\tau})- 2\frac{d\widetilde{R}}{d\tau}\frac{d\phi}{d\tau})/\widetilde{R}

```



```math

\frac{d{\widetilde{z}}^2}{{d\tau}^2}=- \frac{\widetilde{R}^2}{{(\widetilde{R}^2+\widetilde{z}^2)}^{3/2}} \frac{d\phi}{d\tau}-\epsilon_\phi\frac{\partial \widetilde{\Phi}}{\partial \widetilde{z}}

```



This simulates the full trajectory in a 3D cylindrical coordinate system under magnetic and electric fields. The guiding center approximation can also be employed for specific scenarios.



- **Electric Potential Function**:



   ```math

    \Phi = \kappa  \Phi _0 \left(1-\delta_*^2 \left(1-\frac{z^2}{R^2+z^2}\right)\right) \ln \left(\frac{1}{R^2+z^2}\right)+\frac{1}{2} \phi _0 \left(1-\frac{z^2}{R^2+z^2}\right)

  ```



- **Expanded Derivatives**:



   Radial derivative:



    ```math

  \frac{\partial \Phi}{\partial R}=\frac{R \Phi _0 \left(-2 \kappa  R^2+2 \delta_*^2 \kappa  \left(R^2-z^2 \log

   \left(\frac{1}{R^2+z^2}\right)\right)+(1-2 \kappa )

   z^2\right)}{\left(R^2+z^2\right)^2}

    ```



   Vertical derivative:



    ```math

  \frac{\partial \Phi}{\partial z}=-\frac{z \Phi _0 \left((2 \kappa +1) R^2-2 \delta_*^2 \kappa  R^2 \left(\log

   \left(\frac{1}{R^2+z^2}\right)+1\right)+2 \kappa 

   z^2\right)}{\left(R^2+z^2\right)^2}

    ```



These equations describe the magnetic flux function and its derivatives in the radial and vertical directions, which are crucial for understanding the magnetic field structure in the simulation.



## Appendix



For additional information, refer to the documentation and examples provided in the repository.



## Contributors



- Shahab Bahreini Jangjoo