Nuclear Fusion

This simulation has been developed as a part of a project in one of my courses. It attempts to recreate the situations under which self sustaining nuclear fusion reactions can occur. It involves a plethora of idealisations and ad-hoc assumptions, therefore the viewer is advised to interpret the observations made here with a pinch of salt.




Enter number of nuclei (and click on restart):




Components

  1. System

    The system is a rectangular area that contains positive nuclei and allows for diffusion of heat. It is enclosed by a rigid wall. The world begins with only hydrogen nuclei (blue). As fusion takes place, some of them fuse to form helium nuclei (yellow). Further fusion leads to the formation of heavier nuclei (light gray).

  2. Dynamics

    The motion of nuclei obtained using the following equations: \[ a = F / m \] \[ v(t + \Delta t) = v(t) + a \Delta t \] \[ x(t + \Delta t) = x(t) + v(t + \Delta t) \Delta t \] Where \( \Delta t = 10^{-3} \ units \). Further, when a nucleus moves through a heated cell, it gets agitated, resulting in an increase in it's velocity. Lighter nuclei are more influenced by heat than heavier nuclei.

  3. Coulombic Force of Repulsion

    Every nuclei is positively charged. Hence it feels a repulsive force due to the presence of all other nuclei. The magnitude of this force between two charges is directly proportional to the product of their charges and inversely proportional to the square of the distance of separation between them: \[ \vec{F_{12}} = \frac{k q_1 q_2}{r^2} \vec{r_{12}} \]

  4. Nuclear Fusion

    If two nuclei have sufficient kinetic energy to overcome the repulsive force, then they undergo the process of nuclear fusion. In this simulation, the condition for nuclear fusion to take place is: \[ r_{12} < min(r_1, r_2) \] Nuclear fusion results in emission of heat. Certain fusion reactions generate more heat than others.

  5. Diffusion of Heat

    The world is divided into a rectangular array of cells. Whenever two nuclei fuse, they release heat into their corresponding cell, thereby increasing it's temperature. Relatively hotter cells are rendered with a brighter shade of red. Heat spreads across the world according to the diffusion equation: \[ \frac{\partial T}{\partial t} = \alpha \nabla^2 T \ \ where \ \ \nabla^2 \equiv \begin{bmatrix} 0.05 & 0.2 & 0.05 \\ 0.2 & -1 & 0.2 \\ 0.05 & 0.2 & 0.05 \end{bmatrix} \] Heat also gets dissipated at a constant rate.

  6. Introduction of Energy

    When number density of nuclei is low, or if nuclei have less kinetic energy, then fusion events are rare. In such cases, energy can be introduced in the form of heat. One can drag the mouse pointer across the screen to introduce heat in the corresponding cells. One can also instantaneously heat the entire container to a given temperature. Inertial confinement involves heating as well as compression.


Analysis

In the default scenario, we have 300 hydorgen nuclei at the start. The number density is high enough for self ignition to take place. The heat generated in initial reactions is enough to sustain the process for a few seconds. With time, more and more heavier nuclei are formed. Heavier nuclei get less agitated by heat and are less likely to fuse with each other. Hence, the reaction eventually stops. At this stage, heating up the container or carrying out inertial confinement will be of little help. It is also noticed that if we begin the simulation with more nuclei, the process will be of higher intensity but will terminate faster. This is similar to how massive stars have lower lifespans since they burn out their fuel at a faster rate.

With 50 - 70 hydrogen nuclei, nuclear fusion is a sporadic occurence. However, either heating the container to 100 units of temperature or carrying out inertial confinement will trigger some fusion reactions. The process can be sustained for upto a minute by adding 10 hydrogen nuclei whenever the number of nuclei drops below 50.

Inertial confinement involves heating the outer layer of fuel-laden capsules and compressing it's contents. We have attempted to recreate such an process here. When inertial confinement is turned on, the edges of the system get heated. The nuclei are directed towards the center of the container in a batchwise manner. However, inertial confienemnt is relatively useless when the system involves higher proportions of heavier nuclei.

The observation that high number density and high temperature are two approaches to initiate and maintain fusion reactions in this system suggest that we have appreciably recreated Lawson's criteria within the confines of this simulation.



Note:
  1. This simulation should preferably be viewed on desktops. Weird behaviour is observed on mobiles.
  2. Avoid introducing more than 1000 nuclei. This simulation involves calculation of forces among all pairs of nuclei, application of laws of dynamics on all nuclei, collision detection between nuclei as well as calculation of Laplacians for all cells. All of this happens in a single frame. This simulation is computationally heavy and may lag on lower-end devices.
  3. In order to lower total potential energy, lot of nuclei (especially heavier ones) get stuck to the boundaries. This sometimes creates conditions of electric confinement wherein all the nuclei stuck to the wall repel any nuclei coming close to the wall.
  4. When number density of nuclei are low, fusion reactions are more likely to happen at the corners, since nuclei have less freedom to move away from approaching nuclei.
  5. The units of temperature used in this simulation are completely arbitrary.

Developed by ChanRT | Fork me at GitHub