Analogy between a moving line source illuminating a metallic wire and Compton scattering experiment

This paper presents an electromagnetic analysis of a moving infinitely long line source illuminating an infinitely long metallic wire at rest. The study uses the full‐wave numerical finite difference time domain (FDTD) method, which is based on the spatial and temporal discretization of Maxwell's equations. Movement is computed within the FDTD technique by varying the position of the line source at each time loop. An analogy is proposed between the wavelength spectrum of the simulated electromagnetic field and the Compton scattering experiment. A good agreement is obtained between theoretical analysis, numerical results, and Compton experimental data.


| INTRODUCTION
Applied electromagnetism relies mainly on the utilization of Maxwell's equations.The finite difference time domain (FDTD) method is based on a spatial and temporal discretization of Maxwell's equations.Its algorithm was proposed in 1966, using Yee cell and finite differences. 14][5] Today, the FDTD technique is an important tool for designing components, circuits and systems, from DC to optics.
For the analysis of electromagnetic problems with moving structures, a simple numerical approach consists in varying the positions of objects at each time loop, in the FDTD algorithm.This technique has been applied successfully for a moving half-space dielectric with oblique electromagnetic plane wave incidence. 6e hundred years ago, Compton 7 published experimental and theoretical results showing the scattering of a graphite sample illuminated by monochromatic electromagnetic X-rays.Compton described the results theoretically by means of the scattering of high-frequency photons after interaction with electrons.
In this work, an analysis of the electromagnetic field for a moving line source illuminating a metallic wire is proposed.A full-wave FDTD technique, where motion is implemented in the FDTD time loop, 6 is used.By studying the wavelength spectrum at different angles, an analogy exists between the simulated results and the measurements in the Compton scattering experiment.The remainder of the paper is organized as follows.Section 2 presents the numerical setup for the FDTD simulations.In Section 3, the simulated results are presented and analyzed.Section 4 describes the analogy between the obtained data and the Compton scattering experiment.The importance and contribution of this work are discussed in Section 5. Concluding remarks are given in Section 6.

| NUMERICAL SETUP
Figure 1 shows the FDTD setup: an infinitely long line source moves with speed v toward an infinitely long metallic wire.The line source is made of z-polarized current sources (TM z problem).The metallic wire is modeled by annulling the vertical component of the electric field at the position of the wire (E = 0 z ).Perfectmatched layers are used in boundaries parallel to yoz and zox planes.Perfect electric conductors are used in boundaries parallel to xoy plane.
The space mesh, time step, and dimensions of the problem are chosen to operate in the same frequency range of the Compton experiment.The space mesh and the time step are sufficiently small to mitigate numerical dispersion and to avoid instability.The space mesh is δ = 2.4e − 12m x , the time step is δ = 4.6e − 21s t , and the diameter of the wire is2.2e − 10m.For the motion of the line source, we use ∕ v c = 0.032, where c is the speed of light in vacuum.
For the excitation, a Gaussian pulse modulated by a Sine function is used: where the following values of the parameters are considered: σ = 7.7e − 18s 0 , μ = 2.3e − 17s 0 , and ∕ ω = 2.6e + 19rad s 0 .The excitation signal is plotted in Figure 2.

| RESULTS AND ANALYSIS
The electric field component in z-axis is computed at different angle θ as illustrated in Figure 1.The result is plotted in Figure 3, in wavelength domain, for θ = 90°, as an example.In Figure 4, the electric field distribution is shown at a time instant.
The results in the wavelength spectrum show one wavelength peak at θ = 0°, and two wavelength peaks at F I G U R 1 FDTD setup: a moving line source illuminates a metallic wire at rest.The total field is analyzed at different angle θ.The line source is chosen as the origin of the angle θ, at a time instant, for simplicity.In reality, the observer is sufficiently far from the source and wire, such that any point around them could be chosen as origin.FDTD, finite difference time domain; PEC, perfect electric conductor; PML, perfect matched layer.other angles.The difference between these two peaks increases with θ.
This phenomenon can be analyzed by using the Doppler effect.As validated with the FDTD approach, the Doppler effect for a moving source can be expressed , where λ i is the wavelength of the source at rest, v is the speed of the moving source, c is the speed of light and θ is the angle between the direction of propagation of the wave and the direction of motion of the source.For θ = 0°, we have λ . From this, we can write: The two wavelengths λ 0 and λ θ correspond to the two wavelength peaks observed in the FDTD simulations.Equation (2) agrees with the simulated results as shown in Figure 5.

| ANALOGY WITH COMPTON EXPERIMENT
Compton analyzed in his paper 7 the scattering of X-rays from electrons in a carbon target.A detector placed behind the target could measure the intensity of radiation scattered in any direction with respect to the direction of the incident electromagnetic X-ray beam.The scattering angle θ is the angle between the direction of the scattered beam and the direction of the incident beam.For all scattering angles (except for θ = 0°), two intensity peaks were measured.One peak is located at the wavelength λ 0 .The other peak is located at some other wavelength λ θ .The two peaks are separated by λ Δ , which depends on the scattering angle of the outgoing beam (in the direction of observation).The separation, which is called the Compton shift, increases with the scattering angle according to the Compton formula: where h is the Planck constant and m 0 is the resting mass of an electron.Compton explained and modeled the data by assuming a particle (photon) nature for light and applying conservation of energy and conservation of momentum to the collision between the photon and the electron.The scattered photon has lower energy and, therefore a longer wavelength.By comparing Equations ( 2) and ( 3), the two equations can be made equivalent if the following formula is verified: The Formula (4) expresses the conservation of momentum for a moving electron with speed v, as derived by Louis de Broglie. 8This demonstrates the correctness of the proposed model.
Using an analogy with the Compton experiment, the moving line source, in the FDTD simulations, behaves like a radiating free electron moving with speed v, whereas the metallic wire behaves like a scattering bound electron or a scattering atom.We conclude that the X-rays wave incident on the target, in the proposed analog model of the Compton experiment, must make the free electrons move toward the inside of the target in the direction of the wave incidence.
Table 1 presents a comparison between the wavelengths measured with the FDTD method, and those measured in a Compton scattering experiment (experiment 9 ), at different directions.A good agreement is obtained between FDTD data and experimental results.
The Compton formula is routinely verified experimentally by using different X-ray or gamma-ray sources. 10The proposed model can be used for these different experiments by modifying the frequency of excitation and using the velocity of motion of the line source that satisfy (4).
F I G U R E 5 Difference of the two simulated peak wavelengths versus θ (Figure 3) and analytical formula.FDTD, finite difference time domain.The Compton effect is frequently cited as a pillar of the quantum theory.The Compton equation ( 3) is usually developed using the photon model.It is also possible to do a derivation based on a wave model of light.2][13][14][15][16][17] This work has proposed a theoretical and numerical model of the Compton effect based on Maxwell's equations and the FDTD method.Maxwell's theory of light is applied and Planck's constant is implemented to control the motion of particles to deal with the quantum domain.The employment of a full-wave numerical tool can provide physical insight into problems where particles are in motion.
In the quantum approach for the Compton experiment, X-rays whose wavelengths have been changed in the scattering process (shifted peaks) have scattered from free electrons in the target material.For the unshifted peaks, the corresponding X-rays have scattered from a lightly bound electron, an electron bound tightly to its carbon atom, or conceivably even the nucleus of a carbon atom.The line source of the proposed FDTD model acts like a free electron in the quantum model and the metallic wire acts like a bound electron, an atom or a nucleus.Thus, there is a correspondence between the proposed model based on Maxwell's electromagnetic theory and the quantum model.
Further studies are required to test the applicability of the proposed method for other particle scattering experiments.

| CONCLUSION
This paper has presented an analysis of a moving line source illuminating a metallic wire, by using the fullwave FDTD method.An analogy between the Compton scattering experiment and the proposed electromagnetic study has been proposed.The proposed model agrees with de Broglie quantum theory.

F
I G U R E 2 Excitation signal in time domain.F I G U R E 3 Observed FDTD signal in wavelength domain at the angle of 90°around the line source.FDTD, finite difference time domain.F I G U R E 4 FDTD electric field distribution around the source at t = 0.02 fs.FDTD, finite difference time domain; PML, perfect matched layer.

T A B L E 1
Abbreviation: FDTD, finite difference time domain.