Abstract
This research project focuses on the simulation of a Hydrogen ion (H + ) in a cyclotron by using Python. The plot of the motion of the particle in 2D and the plot of the velocity of the proton will be attained with both classical and relativistic perspectives. The paper will be concluded by emphasising the differences between classical mechanics and special relativity. The importance of the calculations for classical and relativistic physics will be understood with the simulation of a cyclotron.
1. Introduction
A cyclotron is a particle accelerator operating with a magnetic and electric field. It consists of two parallel plates in the shape of a "D", which has a constant magnetic field inside. This magnetic field creates a circular motion of the particle with a constant radius. Between the "D" shaped plates, there is an electric field () that changes direction constantly to provide particle acceleration in the gap between the two “D” shaped magnetic plates. This voltage difference, and thus the electric field, make particles travel in a circle with a greater radius. In the TRIUMF cyclotron based in Canada, a foil about 0.025 mm thick is used to separate electrons from protons when high-speed ions are rotating in the outer circles with a greater radius. By this method, researchers can make Hydrogen ions (H+), protons, leave the cyclotron. This process gives the cyclotron many other various purposes, including proton therapy in the medical field.
Lawrence, E. O. (1934, February 20). Image of the principles of a cyclotron. [Image]. File:CyclotronPatent.Png.
According to the TRIUMF accelerator, it would take a potential difference of over 500 million volts to get Hydrogen ions (protons) up to about 3/4 of the speed of light. (Freeman & Pavan, 2009) When a particle reaches 75% of the speed of light, relativistic effects start to be observed. Therefore, it is important to consider Einstein’s special relativity and implement the Lorentz factor to avoid the error of only using Newtonian mechanics while working with cyclotrons. Einstein’s special relativity was developed to work at high speeds like the speed of light. Briefly, the theory explains that the speed of light is constant in any circumstance. It proposes the famous equation which relates energy with mass. Lastly, it explains how time can dilute with high velocities and be experienced differently depending on the speed (Stein, 2021). The Lorentz factor is used in special relativity to calculate the degree of time dilation, length contraction, and the relativistic mass of an object moving relative to an observer (Definition of Lorentz Factor in Physics., n.d. 2 ). This paper will focus on the difference between classical and relativistic mechanics by implementing a cyclotron simulation using Python. Further components that differ between classical and relativistic mechanics, such as motion and velocity, will be discussed and simulated. With the cyclotron simulation using both classical and relativistic perspectives, the difference between classical mechanics and special relativity will be further understood. Differentiating these two physics aspects will provide a better understanding for future research dealing with higher speeds close to the speed of light. Furthermore, understanding this case study of a cyclotron could give insights into predicting scientific results and making inferences.
2. Methodology
Python programming will be used to simulate the motion of a proton in the cyclotron. First, the motion of a Hydrogen ion (H+) will be plotted on a XY-plane graph in two dimensions. This plot will be attained twice using classical mechanics and special relativity calculations. After the motion plot is obtained, the velocity of the particle will be plotted twice using Newtonian mechanics and Einstein’s special relativity to compare the effects of relativity observed in the particle accelerator. The calculations for special relativity will be using the Lorentz factor in the relativistic equations. The Lorentz factor provides the change in the mass of the particle as its speed changes. In addition, the simulation will ignore the effects of gravity and friction. In real life, gravity and friction also exert a force on the particle. However, in the case of the simulation, since the effects of the forces will be extremely small compared to the magnetic and electric forces, they will be ignored. After the calculations from the simulation are obtained, both the position and absolute velocity graphs will be compared with each other in both calculations. This comparison will provide a better understanding of the difference and importance of classical and relativistic mechanics. The implementation of relativistic motion calculations will be better understood in a cyclotron.
3. The Simulation of a Cyclotron with Python
The plots of motion and speed (absolute velocity) for an accelerating Hydrogen ion (H + ), a proton, are obtained with Python two times using slightly different codes. The differences in these codes are only the calculations of the electric force with updated mass and the proton’s speed limit. The data and the results of the cyclotron simulation will be discussed and interpreted to highlight some key concepts in classical and relativistic physics.
3.1. Hydrogen Ion (H + )’s Motion in a Cyclotron with Classical Mechanics
A cyclotron design is created on a plot according to real-life parameters such as the radius of the cyclotron
and the separation between the two “D” shaped plates
Notably, the shape does not look exactly like a cyclotron. This is because the gap between the plates is very narrow (90 micrometers) compared to the overall size of the cyclotron. Therefore, the gap is seen as two overlapping lines.
Before plotting the motion of the particle, arrays and lists for the x and y position of the particle, the velocity, and time have been created. This ensures that all the different values attained at different times will be stored in the list so that every different position the particle moves through can be seen on the plot.
Arrays and lists created for the positions:
The array and list created for the velocity:
The list created for time:
The initiator for the motion of the proton is the electric and magnetic force. Both forces can be expressed in one equation with the Lorentz Force Law:
The code below provides the function (particle_motion) for the spiral motion of the Hydrogen ion (the proton) in the cyclotron:
In order to ensure that lists are empty before starting to store values, they are cleared. Then the values of force, velocity, positions, and time are defined with the previous function:
Lastly, the motion of the Hydrogen ion (H + ) in the cyclotron can be plotted.
This image explains that the proton is spiraling with the magnetic force and making circles with a greater radius every time it passes through the gap in the middle. The electric force in the gap between two “D” shaped plates initiates this acceleration and motion. The plot ensures that the motion of the Hydrogen ion (H + ) is following the path that it would in an actual real-life cyclotron. This means that the particle is not exceeding the borders of the cyclotron outside its radius and follows a spiraling path due to the electric field between the “D” shaped magnetic plates. The plot demonstrates the ideal situation with a proton accelerating inside a cyclotron.
3.2. Absolute Velocity of the Hydrogen Ion (H + ) with Classical Mechanics
Each time the proton goes through the gap between two “D”s, the electric field affects the particle with the electric force. The electric force accelerates and consequently increases the proton’s absolute velocity. The starting velocity of the particle was negative
and changed in every time frame. Thus, it is important to have the absolute value of the velocity to compare the magnitude of the velocity. Therefore, the speed of the proton is obtained and collected in a list to create the following plot of “time vs. absolute velocity”.
It can be concluded that the velocity increases exponentially in a cyclotron for a proton. The notable point in this plot is that the velocity increases to m/s, which is more than the speed of light ( m/s). Special relativity suggests that it is impossible to reach the speed of light or even exceed it. Only particles with no mass, such as photons, which make up light, have the ability to move at that speed. No mass can be accelerated to the speed of light because it would require an infinite amount of energy (Cosmic Speed Limit, n.d.). As the proton exceeds the speed of light in the Python simulation, it shows that the calculations did not consider relativity. Below is a graph that illustrates the change in kinetic energy as speed increases according to both special relativity and classical mechanics.
The plot clearly shows the difference between Newtonian (classical) mechanics and special relativity. The main difference is the relationship between the kinetic energy and the speed of the object. Classical mechanics shown with the pink line easily exceeds the speed of light and the kinetic energy does not increase that much. However, relativistic seed, as shown by the red line, cannot reach the speed of light; it can only come very close. As the speed increases, the kinetic energy skyrockets. This demonstrates the proposal in Einstein’s special relativity that an object with a mass needs an infinite amount of energy to reach the speed of light. Hence, it’s impossible for an object to reach the speed of light if it has a mass. In other words, the speed of light is an asymptote for any object with mass. This means that it can never reach the speed of light.
3.3. Relativistic Motion of a Hydrogen Ion (H + ) in a Cyclotron
The principle of plotting the motion is nearly the same as it is with classical mechanics. The only difference is in the calculations of the electric force. As stated in the previous section, it requires an infinite amount of energy to accelerate a particle with a mass. In addition, in the continuation of this rule, the Hydrogen ion’s (H + ) mass would increase with the higher velocities it gains through the acceleration in the cyclotron. The change in mass can be expressed as:
is the initial mass of the proton when it is at rest and is the Lorentz factor, which can be expressed as:
Moving on with the expression of force in the effects of relativistic motion, although is the popular format of Newton’s 2 nd Law of Motion, using is more logical because of the change in the mass of the proton with the effect of high velocities.
With the multiplication rule of derivatives, the expression takes the shape of the following expression:
In this equation, the expression with the Lorentz factor can be represented in terms of velocity, acceleration, and speed of light. Therefore, the equation can be updated explicitly as shown below:
When the equation is solved for acceleration (), the following equation is obtained:
Since velocity and force are perpendicular to each other in a magnetic field, . Therefore, for the magnetic field, the acceleration becomes:
Meanwhile, for the electric field, the equation becomes more complicated than the magnetic field because the velocity and the force vectors are in the same direction. As a result, both vectors have effects on each other. Hence, the equation cannot be shortened as it is in the magnetic field. The acceleration in the electric can be expressed as the following by only specifying the second previous equation for the electric field:
In Python code, these equations are implemented as
and updated as
With the final implementation of the calculations in the code, the relativistic motion graph of the Hydrogen ion (the proton) is obtained.
As can be seen in the plot, the proton’s motion is very similar and nearly the same as the previous plot of the Hydrogen ion’s motion with classical calculations. It follows the same spiraling motion. However, when the plot is examined closely, it can be noticed that it is not the same path as the previous one. It is noticeable that the last line the Hydrogen ion (the proton) follows is just a little outside the borders of the cyclotron’s radius. In addition, while the proton in the previous plot obtained with classical calculations leaves the cyclotron on the right side (+x direction), the proton in this plot obtained with relativistic calculations leaves the cyclotron on the left side (-x direction). These small differences indicate that there is a significant difference between the calculations of classical and relativistic mechanics. This plot explains that there will be a difference in the results between classical and relativistic calculations in a scenario like a cyclotron where an object gets close to the speed of light with acceleration. Therefore, a scenario in physics should be considered with its conditions first before deciding to implement the calculations with either method. For example, a particle accelerator and a traveling train would require different calculations to use.
3.4. Relativistic Speed of the Hydrogen Ion (H + )
The particle's velocity is determined by updating the velocity as a function of force. This can be calculated from the following equations:
The only difference in the coding will be the calculations of the electric force, mentioned earlier, because of special relativity. In addition, since special relativity restricts the speed limit to the speed of light due to the need for infinite energy, the code includes a statement as the following. This code only proceeds if the statement is satisfied.
This ensures one of the basic principles of special relativity that the speed of light should have the same constant value in any such frame. The following plot of the speed of the Hydrogen ion (H+), the proton, is obtained by making some adjustments in the code for the calculation of the electric force and the speed limit for the proton.
As can be seen in the plot, the speed of the proton increases exponentially nearly until . This value is the time limit that the proton’s speed exceeds the speed of light with classical mechanics calculations. Furthermore, when the plot is continued until time frame, it can be observed that the speed remains constant from . The speed of light is m/s as mentioned previously. In the plot, it is explicit that the speed limit remains just under the value of the speed of light, and the absolute velocity of the Hydrogen ion (the proton) does not exceed . This plot clearly demonstrates one of the main rules of Einstein’s special relativity that a massive object cannot exceed the limit of light’s speed. In addition, the slope of the line decreases because as time passes and the proton gains more speed, it becomes harder for it to accelerate. This is because it requires more energy every time the proton increases its speed.
4. Importance of Classical and Relativistic Calculations
In the implementation of the cyclotron simulation, both classical mechanics and special relativity calculations can be observed. It can be implicitly seen that one of the basic differences between the two calculations was the speed shown in the plot. Newton’s classical mechanics laws explain the surroundings that people come across in everyday life. Mathematical calculations enable people to predict nearly every motion and an object’s properties, such as its mass. Meanwhile, in the theory of special relativity, Albert Einstein keeps the postulate that the equations of motion do not depend on the reference frame but assumes that the speed of light () is invariant. As a result, position and time in two reference frames are attained with the Lorentz transformation instead of the Galilean transformation (27.3: Relativistic Quantities, 2020).
A Galilean transformation is used in physics to modify the coordinates of two reference frames that vary only by constant relative motion within Newtonian physics frameworks (What Does Galilean Transformation Mean?, n.d.). In contrast, the Lorentz transformation describes the interaction between two separate coordinate frames that move at a constant velocity relative to each other (What Is Lorentz Transformation?, 2021).
Although classical physics in general and classical relativity have limitations, they are excellent approximations for massive, slow-moving objects. It would not be possible to launch satellites or build bridges without working with classical physics. Relativistic mechanics becomes classical mechanics in the classical limit for items bigger than microscopic level and traveling slower than 1% of the speed of light (Libretexts, 2021). However, classical mechanics rules do not apply to objects moving close to the speed of light because the object’s mass changes depending on the speed, as seen in the Lorentz factor. Therefore, it is highly important to classify the case according to the object’s speed in physics.
5. Conclusion
This paper mainly focused on the differences between classical mechanics and special relativity, their calculation’s implementation on a cyclotron, and the utilisation of these calculations in real-life situations. First, the cyclotron’s working principle was explained as two "D" shaped magnetic plates, which have a constant magnetic field inside and an everchanging electric field in the very narrow gap between these two magnetic plates.
The motion of a Hydrogen ion (H + ) was first plotted with classical mechanics calculations, which do not change the proton’s mass or energy relative to its speed. Therefore, a perfect spiraling plot was attained as a result. Then, the same process was repeated with special relativistic calculations which update the Hydrogen ion’s (H + ) mass according to its velocity and relative to the speed of light. This change in the calculation only influenced the electric force, which was present only between the two "D" shaped magnetic plates in the cyclotron. The magnetic force was not influenced by this change since the magnetic force and velocity are perpendicular vectors. The plot of the proton’s motion in the cyclotron obtained with special relativistic calculations was very similar to the previous plot with classical calculations.
However, some details indicated the difference between the force, velocity, and position. These small details demonstrate the difference between classical mechanics and special relativity in circumstances in which a massive object gets close to the speed of light as it is in a cyclotron.
After examining the motion in two dimensions, absolute velocity plots of the Hydrogen ion (H+) were obtained with both classical mechanics and special relativistic calculations and rules. In the plot with classical calculations, it was seen that the speed of the proton increased exponentially as time passed. In addition, the last value that it reached was m/s, which is more than the speed of light. However, Einstein’s special relativity suggests that an object with a mass cannot reach the speed of light because this would require an infinite amount of energy, which is impossible to gain. Later, the plot of the absolute velocity of the Hydrogen ion (H+) with relativistic calculations showed that the speed cannot reach the speed of light, although it gets extremely close to light’s speed. This graph also increased exponentially, similar to the plot of the absolute velocity in classical mechanics. However, the graph stopped at some point near the speed of light where the acceleration decreased significantly.
The illustration of both the motion and absolute velocity plots of a Hydrogen ion in a cyclotron portrayed the difference between what classical mechanics and special relativity distinguish. It is crucial to analyze the situation before applying the calculations because, depending on the circumstances and some values such as the velocity or speed, the need to use different physics calculations may arise. While classical mechanics is useful for larger objects with low speeds, special relativity should be used for cases dealing with smaller objects that get close to the speed of light, even if it’s only 5% of light’s speed.
This research paper serves as a case study for understanding the difference between classical and relativistic physics through a cyclotron simulation with Python. In further research, the cyclotron application could be enhanced further. Advancements can be made by taking the differences and consequences of using classical mechanics and special relativity in calculations into account. These advancements could be in any field. For example, proton therapy mentioned in the introduction could be simulated beforehand to anticipate the results of an actual real-life application. This anticipation could lead to a significant improvement in different branches of physics and science. Physicists and scientists would have the opportunity to predict the results, avoid any unnoticed errors, and receive support with any unpredicted results. Furthermore, especially in theoretical physics, computationally visualizing the theoretical concepts could help reaching the results of complex problems and pave the way for further research and exploration.
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