Recent advancements in Transmission Electron Microscopy is built upon the remarkable achievements of the transmission electron microscope, especially, with the aberration corrected object lens, which itself is the incoherent integration of the particle electron optics and modern wave imaging technology.This involves the particle-wave duality of electrons.This book answers questions by applying the de Broglie Hypothesis and Einstein’s Theory of Relativity on the relationship between particles and electromagnetic waves to shed some light onto the electron microscopy. The first chapter explains what an electron is, which includes: (a) using the transmission electron microscope to observe the wave-particle duality of electrons, (b) the internal structure of the electron, (c) the electron as a confined electromagnetic vortices field in a corpuscle space. The following chapters, then, decipher the enigmatic relationship between the de Broglie wave of the electron and the internal electromagnetic flux circulatory motion, and analyze the spatiotemporal modification of the traveling electron corpuscle as it passes through the electron gun and magnetic lens. Based on the de Broglie wave of the traveling electron corpuscle at a certain velocity, the author defines the electron microscopy as the technology steering the velocity of the electron corpuscle assemble which encodes the information of periodic spatial grating constructed by the atoms and electromagnetic potential field, which surround the flying electrons. Then the author uses the space-time Talbot effects of the electrons to interpret the high resolution images, which was first developed by Ijima-Cowley, and expounds the contrast of the high resolution electron microscopy images as the intensity distribution of the assemble of corpuscle electron torus pulses train or beam at near field and at its far-field, which is a diffraction pattern. The final chapter of the book elaborates on how to understand the quantum electron microscopy. This book offers a comprehensive understanding what the quantum electron microscopy is, that may bring the microscopy field beyond the atom’s spatial periodicity in materials.
Recent advancements in Transmission Electron Microscopy is built upon the remarkable achievements of the transmission electron microscope, especially, with the aberration corrected object lens, which itself is the incoherent integration of the particle electron optics and modern wave imaging technology.This involves the particle-wave duality of electrons.This book answers questions by applying the de Broglie Hypothesis and Einstein’s Theory of Relativity on the relationship between particles and electromagnetic waves to shed some light onto the electron microscopy. The first chapter explains what an electron is, which includes: (a) using the transmission electron microscope to observe the wave-particle duality of electrons, (b) the internal structure of the electron, (c) the electron as a confined electromagnetic vortices field in a corpuscle space. The following chapters, then, decipher the enigmatic relationship between the de Broglie wave of the electron and the internal electromagnetic flux circulatory motion, and analyze the spatiotemporal modification of the traveling electron corpuscle as it passes through the electron gun and magnetic lens. Based on the de Broglie wave of the traveling electron corpuscle at a certain velocity, the author defines the electron microscopy as the technology steering the velocity of the electron corpuscle assemble which encodes the information of periodic spatial grating constructed by the atoms and electromagnetic potential field, which surround the flying electrons. Then the author uses the space-time Talbot effects of the electrons to interpret the high resolution images, which was first developed by Ijima-Cowley, and expounds the contrast of the high resolution electron microscopy images as the intensity distribution of the assemble of corpuscle electron torus pulses train or beam at near field and at its far-field, which is a diffraction pattern. The final chapter of the book elaborates on how to understand the quantum electron microscopy. This book offers a comprehensive understanding what the quantum electron microscopy is, that may bring the microscopy field beyond the atom’s spatial periodicity in materials.
The author has worked with high-resolution transmission electron microscope for many years, which provides the most direct evidence of wave-particle duality. This relates to Crookes and J. J. Thomson's cathode rays tubes to determine that electron beams are of particle nature. This also relates to Röntgen's discovery that when an electron beam in the cathode rays tube strikes the glass wall, it produces X-rays with extremely high penetrating power. This behavior is consistent with Maxwell-Lorentz’s electrodynamics that electromagnetic waves must be produced whenever electricity carriers alter their velocity. The wave-particle duality proposed by de Broglie refers to accelerated electron beam, rather than single electron. De Broglie emphasized that the crystallographic diffraction pattern produced by his electron beam was completely consistent with the crystallographic diffraction pattern of von Laue by X-rays, therefore, he proposed that electrons possess wave-particle duality. This is actually the selected area electron diffraction pattern in transmission microscope. It is X-rays, not the waves of the electrons themselves to produce diffraction pattern. He extended this to the wave-particle duality of all matter and light. Schrödinger directly introduced de Broglie's electron beam behavior into electron wave in atoms. It was just at this stage that illogical reasoning occurred, making the quantum theory increasingly difficult to understand.
Atomic resolution transmission electron microscopy (TEM) is an indispensable tool for interdisciplinary investigations of radiation soft or hard materials that evolves rapidly because emerging technologies enable remarkable new experiments [1, 2]. Nowadays, many of them embrace the time domain [2-4]. Beyond materials research, electron microscopy is uniquely suited to study fundamental quantum-mechanical aspects. In fact, it remains concerning that aspects addressing the wave-particle duality, time-dependent measurements, or the entanglement of wave functions and their collapse at a detector still require detailed investigations to guide progress. Certainly, this is no academic matter because such aspects become the bottleneck for the deployment of microscopes that enable studying dynamic processes at high spatiotemporal resolution. It is established practice to describe the scattering of electron ensembles by either a particle picture (Figure 1a) or a wave picture (Figure 1b). Sometimes both pictures are mixed. In spite of calls for solving Schrödinger’s time dependent equation [5], the static Schrödinger equation of wave functions is almost exclusively used to reliably simulate the image formation process using multislice calculations for coherent-elastically scattered electron ensembles [6, 7]. Moreover, the Copenhagen convention of quantum mechanics [8] is adopted, which emphasizes the importance of an observation for the collapse of wave functions (Figure 1b). In order to foster further insight into fundamental quantum-mechanical aspects we pursue a series of experiments with low electron dose rates that guarantee the presence of single electrons in the microscope column at any given time [9]. Consequently, we address the scattering of single electrons (Figure 1c) and build up intensities from electron ensembles over time. In such conditions energy filtered TEM recently revealed the presence of image intensity at sample/vacuum interfaces that cannot be simulated using the established (static) calculation tools even if frozen phonon excitations are considered [10]. In contrast, these intensities can be explained by the time-dependent Schrödinger equation assuming that an energy exchange ΔE occurs during any interaction of the electron with the sample in the Heisenberg limit ΔE = ℏ/2 Δt−1 (coherent-inelastic scattering). In this process a wave package is formed by the superposition of partial waves and a decoherence phase of 0.5-1 radian [10]. The time-dependent Schrödinger equation yields two parameters that characterize the static and the dynamic behavior of any wave function ψ(r, t) = ψ(r) ϕ(t), namely xstatic = ℏ/(2mΔE)1/2 and xdynamic = ℏc/ΔE [11]. The former is the de Broglie wavelength if ΔE is a kinetic energy and the latter becomes an expression for self-coherence if the scattering process is coherent-inelastic in the Heisenberg limit. In these circumstances, the product of Planck’s constant and the speed of light hc is given by the product of the expression for temporal coherence λ2/Δλ and the energy loss ΔE (hc = (λ2/Δλ) ΔE). The validity of the relation is shown in Figure 2 [11]. Thus, the temporal part of Schrödinger’s equation adds contributions of a finite self-coherence length (or time if the electron velocity is considered) to any scattering process, which is neglected if only the static expression is exploited. Further, the formation of wave packages rationalizes the wave-particle duality. The width of wave packages is described by a self-coherence length ls that can vary by over 40 orders of magnitude depending on the strength of interactions [11]. For Coulomb interactions, ls explains the vanishing phase contrast from EF-HRTEM images when the energy loss exceeds values of ΔE > 200 eV [11]. A procedure to experimentally determine the self-coherence length was established earlier by measuring the energy dependent localization of scattered matter waves in surface proximity [10]. Therefore, electron microscopy can be used as a general tool to measure the self-coherence length in real space images on a scale of nanometers and relate this these values to a self-coherence time that reaches attoseconds [12]. Particle and wave descriptions of electron scattering. a) Scattering of electron ensembles in a particle picture. The process is often described by Feynman diagrams, which involves the emission of a virtual photon of lifetime given by Heisenbergs Uncertainty Principle. b) Scattering of electron ensembles in a wave picture that underlies the multi-slice calculations. Infinite wave functions entangle source, sample and detector. The Copenhagen convention requires a wave function collapse upon observation. c) Proposed scattering process of a single electron wave function. The energy exchange occurs in the Heisenberg limit ΔEloss = E2-E1= ℏ/2 (t2-t1)−1= ℏ/2 Δt−1 when a wave package is formed during Δt because of existing energy (wavelength) differences ΔEloss of partial waves [11]. Measurements of self-coherence length and the related self-coherence time at the calculated Heisenberg limit. The colored background marks the quantum realm. The solid lines represent the Heisenberg limit in space and time, which includes the constant ℏc [11].
Wave-particle duality is widely regarded as one of the conceptual pillars of modern quantum theory, providing the framework through which the behavior of microscopic matter can be consistently interpreted. According to this principle, microscopic entities such as electrons, photons, atoms and even complex molecules exhibit both wave-like and particle-like properties depending on the experimental configuration used to observe them. This duality is quantitatively formalized through the De Broglie relation, which associates a wavelength with any particle possessing momentum. The present work provides a comprehensive theoretical analysis of the physical and mathematical foundations of wave-particle duality, focusing particularly on the De Broglie wavelength and the kinetic energy of the electron in both non-relativistic and relativistic regimes. The connection between the De Broglie hypothesis and Schrödinger’s equation is discussed in detail, highlighting the role of the kinetic energy operator and the interpretation of matter waves in terms of quantum wave packets. Experimental evidence supporting the wave nature of matter, including electron diffraction experiments and interferometry techniques, is also reviewed. Beyond their conceptual importance, these principles underpin several modern technologies, including electron microscopy, nanoscale semiconductor devices and quantum metrology. The analysis demonstrates that the De Broglie relation provides a unifying framework connecting momentum, wavelength and kinetic energy, thereby forming a fundamental bridge between classical mechanics, quantum theory and modern technological applications.
A theory employing the vortex shape of the electron was presented to resolve the enigma of the wave-particle duality. Conventions such as “particle” and “wave” were used to describe the behavior of quantum objects such as electrons. A superfluid vacuum formed the base to describe the basic vortex structure and properties of the electron, whereas various formulations derived from hydrodynamic laws described the electron vortex circumference, radius, angular velocity and angular frequency, angular momentum (spin) and magnetic momentum. A vortex electron fully explained the associations between momentum and wave, and hydrodynamic laws were essential in deriving the energy and angular frequency of the electron. In general, an electron traveling in space possesses internal and external motions. To derive the angular frequency of its internal motion, the Compton wavelength was used to represent the length of one cycle of the internal motion that is equal to the circumference of the electron vortex. The angular frequency of the electron vortex was calculated to obtain the same value according to Planck’s theory. A traveling vortex electron has internal and external motions that create a three-dimensional helix trajectory. The magnitude of the instantaneous velocity of the electron is the resultant of its internal and external velocities, being equal to the internal velocity reduced by the Lorentz factor (whose essence is presented in a detailed formulation). The wavelength of the helix trajectory represents the distance traveled by a particle along its axis during one period of revolution around the axis, resulting in the same de Broglie wavelength that corresponds to the helix pitch of the helix. Mathematical formulations were presented to demonstrate the relation between the energy of the vortex and its angular frequency and de Broglie’s wavelength; furthermore, Compton’s and de Broglie’s wavelengths were also differentiated.
As for DE Broglie's hypothesis of matter waves, many experimental physicists have experimentally verified it by using cathode ray tube and proved that matter waves do exist, so there is a mysterious hypothesis of wave-particle duality.In this paper, according to the electric field force of electrons in the cathode ray tube and the continuity equation of electrons, Under assumptions such as inelastic approximation, it is proved that the wave-particle duality is actually the macroscopic motion of real particles in the form of waves.In this paper, it is proved theoretically that the component of electron velocity in the tube satisfies the wave equation, and the wave velocity varies with the change of electric potential in the tube.From the discussion in this paper, it can be seen that there are still some problems to be studied about the motion of electrons in the cathode ray tube, so that the special motion law of electrons in the cathode ray tube can be further explored from both theoretical and experimental aspects.At the same time, it is expected that the theory in this paper can provide theoretical guidance for the manufacture and debugging of electron microscopes and the design of large-scale integrated circuits, and thus influence other related fields...This will involve a vast area of science and technology!This paper is devoted to the in-depth analysis of the objective truth of the wave-particle duality presented by microscopic particles.
One of the astounding consequences of quantum mechanics is that it allows the detection of a target using an incident probe, with only a low probability of interaction of the probe and the target. This 'quantum weirdness' could be applied in the field of electron microscopy to generate images of beam-sensitive specimens with substantially reduced damage to the specimen. A reduction of beam-induced damage to specimens is especially of great importance if it can enable imaging of biological specimens with atomic resolution. Following a recent suggestion that interaction-free measurements are possible with electrons, we now analyze the difficulties of actually building an atomic resolution interaction-free electron microscope, or "quantum electron microscope". A quantum electron microscope would require a number of unique components not found in conventional transmission electron microscopes. These components include a coherent electron beam-splitter or two-state-coupler, and a resonator structure to allow each electron to interrogate the specimen multiple times, thus supporting high success probabilities for interaction-free detection of the specimen. Different system designs are presented here, which are based on four different choices of two-state-couplers: a thin crystal, a grating mirror, a standing light wave and an electro-dynamical pseudopotential. Challenges for the detailed electron optical design are identified as future directions for development. While it is concluded that it should be possible to build an atomic resolution quantum electron microscope, we have also identified a number of hurdles to the development of such a microscope and further theoretical investigations that will be required to enable a complete interpretation of the images produced by such a microscope.