Mathematical Modelling of Electromagnetic Radiation Emitted by a Bremsstrahlung Electron

Abstract

Terrestrial Gamma-ray Flashes exhibit slopes of ionizing radiation associated with bremsstrahlung. Bremsstrahlung has a continuous spectrum of radiation from radio waves to ionizing radiation. The Poynting vector of the emitted radiation, i.e., the radiation pattern around a single particle under the external lightning electric field during interaction with other particles or atoms, is not quite well known. The overall radiation pattern arises from the combination of radiation of parallel and perpendicular motions of a particle caused by the acceleration from the lightning electric field and the bremsstrahlung. The calculations and displays of radiation patterns are generally limited to a low-frequency approximation for radio waves and separate parallel and perpendicular motions. Here, in chapter 1 of the thesis is about the particle acceleration and bremsstrahlung radiation patterns and presents the main outcomes of this project. In addition, chapter 1 is the published chapter of the thesis. Chapter 1 reports the radiation patterns of combined parallel and perpendicular motions from accelerated relativistic particles at low and high frequencies of the bremsstrahlung process with an external lightning electric field. The primary outcome is that radiation patterns have four relative maxima with two forward peaking and two backward peaking lobes. The asymmetry of the radiation pattern, i.e., the different intensities of forward and backward peaking lobes, are caused by the Doppler effect. A novel outcome is that bremsstrahlung has an asymmetry of the four maxima around the velocity vector caused by the curvature of the particle's trajectory as it emits radiation. This mathematical modeling helps to better understand the physical processes of a single particle's radiation pattern, which might assist the interpretation of observations with networks of radio receivers and arrays of gamma-ray detectors.

Moreover, chapter 2 is about converting a single particle system to a multi-particle system using the Lagrangian method and tries to provide an idea and formulation to predict internal force and interaction between particles through magnetic vector potential field, A. Following chapter 2, chapter 3 tries to explain how particle in periodic motion emits spiral radiation pattern. Chapter 3 demonstrates how the forward-backward peaking radiation pattern found in chapter 1 can transform into a spiral radiation pattern. The periodic motion of the particle could be due to negatively charged particles being trapped between two positive leader tip electric fields. The forward peaking radiation pattern of a single particle with an increasing particle velocity is well-established knowledge. Further details of a single particle radiation pattern suggest that a particle also has a backward peaking radiation pattern and two associated asymmetries coming from the Doppler and spiral trajectory bremsstrahlung effects. Relativistic particle under periodic motion emits spiral radiation pattern which is measured as short pulses by the sensors. However, the transition from peaking to spiral radiation pattern as particle transits from discrete to continuous periodic motion is not clear. This paper reports a possible physical asymmetric effect caused by the bremsstrahlung spiral trajectory that could be responsible for the spiral radiation pattern emitted by the periodic particle motion. The bremsstrahlung asymmetry changes within each period that change the radiation intensity symmetry continuously, within a specific minimum and maximum range. This maximum and minimum range is defined by the limits of the bremsstrahlung asymmetry, R. This change in radiation intensity, independent of particle periodicity, emits circular waves of the varying radius that forms a spiral radiation pattern. Hence, the spiral of a periodic motion comes from a parameter, R, that is independent of periodicity. Hence, parameter R can change during a period by preserving the periodicity of the incoming particle motion. Overall, a continuous periodic motion is important in predicting the experimental observations as incoming particle interacts with multiple target particles meaning the same bremsstrahlung process repeats periodically. Chapter 4, introduces a 3 particle system and looks at the overall radiation pattern through wave interference. Finally, chapter 5 Introduces a new technique of using the Golden ratio spiral to locate Schumann resonant frequencies on the bremsstrahlung radiation patterns. These Schumann resonant frequency points are demonstrated to be related to each other in odd, even, and whole number Schumann triads. Novel Octave technique is introduced and demonstrated to predict Schumann frequency points that are on the same radiation lobe. The Schumann resonances are low-frequency waves and they can also exist on high-frequency bremsstrahlung radiation patterns at the regions closer to the source particle. Summary of the details of the chapter 5 are as follows;

Although lightning discharge is not the only source or only physical phenomenon that affects the Schumann resonances, they have the highest contribution to the Schumann resonances oscillating between the ground the ionosphere. Schumann resonances are predicted through several different numerical models such as the transmission-line matrix model or partially uniform knee model. Here we report a different prediction method for Schumann resonances derived from the first principle fundamental physics combining both particle radiation patterns and the mathematical concept of the Golden ratio. This prediction allows the physical understanding of where Schumann resonances originate from radiation emitted by a particle that involves many frequencies that are not related to Schumann resonances. In addition, this method allows to predict the wave propagation direction of each frequency value in the Schumann frequency spectrum. Particles accelerated by lightning leader tip electric fields are capable of contributing most of the Schumann resonances. The radiation pattern of a single particle consists of many frequencies that there are only specific ones within this pattern that contribute to the Schumann radiation. Vast majority of Schumann resonances distribute quite nicely obeying the Golden ratio interval. This property used in conjunction with the full single-particle radiation patterns also revealed that high-frequency forward-backward peaking radiation patterns as well as low-frequency radiation patterns can contribute to Schumann resonances. Moreover, this also allows to locate them on the full radiation pattern. Furthermore, theoretical analysis using Golden ratio spiral predicts that there are more Schumann resonances in high frequency forward-backward peaking radiation pattern of relativistic particle than low frequency dipole radiation pattern.

There are different numerical models, such as the transmission-line matrix model or partially uniform knee model used to predict Schumann radiation. This report introduces a new method build on the previously stated idea of locating Schumann resonances on a single particle’s radiation pattern using a Golden ratio. In addition, this different prediction method for Schumann resonances derived from the first principle fundamental physics combining both particle radiation patterns and the mathematical concept of the Golden ratio spiral that expands at the rate of Golden ratio. Moreover, extending the idea of ratios to a specific ratio called octaves used in standing waves that identify the identical sounding notes with different frequencies. Knowing the value of initial Schumann resonant frequency, this method allows us to predict the magnitude of other Schumann resonances on the radiation pattern of a single accelerated charged particle conveniently. In addition, it also allows us to find and match Schumann resonances that are on the same radiation lobe, which is named electromagnetic Schumann octaves. Furthermore, it is important to find Schumann octaves as they propagate in the same direction and have a higher likelihood of wave interference.

As the golden ratio seems to be part of the Schumann resonances, it is helpful in understanding to know why this is the case. The main method used in the reasoning of the existence of golden ratio in Schumann resonances is the eigenfrequency modes, square root of {n(n+1)} in the spherical harmonic model. It has been found that eigenfrequency modes have two a start off points, n_0 = 0 or n_0 = (square root{5}-1)/ 2 where the non-zero one is exactly the golden ratio. This allows to extend the existing eigenfrequency modes to square root of {(n_0+n)^2+(n_o+n)} in order to explain why golden ratio exist within Schumann resonances.

Date of Award	23 Mar 2022
Original language	English
Awarding Institution	University of Bath
Sponsors	Engineering and Physical Sciences Research Council & The Met Office
Supervisor	Martin Fullekrug (Supervisor) & Biagio Forte (Supervisor)