The new results reported here mainly include: 1) recognition that phonon is carrier of electromagnetic interaction between its lattice wave mode and electrons; 2) recognition that binding energy of electron pairs of high-temperature superconductivity is due to escape of optical threshold phonons, of electron pairs at or near Fermi level, from crystal by direct radiation; 3) recognition that binding energy of electron pairs of low-temperature superconductivity is possibly due to escape of non-optical threshold phonons by anharmonic crystal interactions; and, 4) recognition of a possible mechanism explaining why some crystals never have a superconducting phase. While electron pairing is phonon-mediated in general, HTS should be associated with electron pairing mediated by optical phonon at or near Fermi level (EF), so the rarity of HTS corresponds to the rarity of such pairing match.
A model of electron pairing, with depletion of mediating phonons at fermi sur...Qiang LI
We present a model of electron pairing based on nonstationary interpretation of electron-lattice interaction. Electron-lattice system has an intrinsic time dependent characteristic as featured by Golden Rule, by which electrons on matched pairing states are tuned to lattice wave modes, with pairing competition happening among multiple pairings associated with one electron state. The threshold phonon of an electron pair having a good quality factor can become redundant and be released from the pair to produce a binding energy. Lattice modes falling in a common linewidth compete with one another, like modes competing in a lasing system. In cuprates, due to near-parallel band splitting at and near Fermi Surface (EF), a great number of electron pairs are tuned to a relatively small number of lattice wave modes, leading to strong mode competition, transfer of real pairing-mediating phonons from EF towards the “kink”, and depletion of these phonons at and near EF.
(v3) Phonon as carrier of electromagnetic interaction between vibrating latti...Qiang LI
With emphasis on time-dependency of electron-lattice system, we suggest the fallacy of presumed quantization in the context of electron-lattice system and propose the definition of phonons as carriers of electromagnetic interaction between electrons and vibrating lattice. We have investigated behaviors of electron-lattice system relating to “measured” energy, identified non-stationary steady state of electrons engaging in “electron pairing by virtual stimulated transitions”, recognized some origins of binding energy of electron pairs in crystals, and explained the state of electrons under pairing. Moreover, we have recognized the behavior and role of threshold phonon, which exists in electron pairing and is released by the electron from excited state, and have recognized the redundancy of the threshold phonon when the electrons under pairing have entered non-stationary steady state. We have also studied the effect of the stability of lattice wave on the evolution of the function of transition probability and on the stability of phonon-mediated electron pairs, the competition among multiple pairings associated with one same ground state, and determination of presence/absence of superconductivity by such competition.
candidate mechanisms of electron pairing near EF at T=0, binding energy of the electron pairs thus formed, and superconductivity in mono-atom crystals have been proposed.
Once EM wave modes are established in the ranges of their associated lattice chains of a crystal concerned, which ranges can be long or even macroscopic, electron-pairs are produced in the crystal’s electron system over these ranges. As EM wave modes with frequencies below certain value (corresponding to an energy value Δ) may have little contribution to stimulated transitions of electrons and electron-pairing, at T=0 each of the electrons at and near EF pairs with one of the electrons at energy levels of EF-hωM/(2π)≤E≤EF-Δ (where ωM is the maximum frequency of lattice wave modes of the system, which is often associated with a specific crystal orientation), resulting in a binding energy of at least Δ for each of these pairs at T=0.
electron pairing and mechanism of superconductivity in ionic crystals Qiang LI
The behaviors of valence electrons and ions, particularly ion chains, in ionic crystals are important to understanding of the mechanism of superconductivity. The author has made efforts to establish a candidate mechanism of electron-pairing and superconductivity in ionic crystals.
Analyses are first made to a one-dimensional long ion lattice chain model (EDP model), with the presence of lattice wave modes having frequency ω. A mechanism of electron pairing is established.
Analyses are then extended to scenarios of 3D ionic crystals, particularly those with a donor/acceptor system, with emphasis being given to the interpretation and understanding of binding energy of electron pairs formed between electrons at the top/bottom of donor/acceptor band and the bottom/top of conducting/full band.
It is established that once the lattice/EM wave modes are established in its range, which can be long or even macroscopic, electron pairs are produced in the crystal’s electron system over the same range by stimulated transitions induced by the EM wave mode. The lattice wave mode having the maximum frequency ωM is of special significance with respect to superconductivity, for electron pairs produced by it can be stabilized in the context of a combination of some special factors (including energy level structure featured by donor/acceptor band and ωM) with a binding energy typically no smaller than hωM/(2π). A candidate mechanism of electron pairing in ion crystals and therefore of superconductivity is provided.
candidate mechanisms of electron pairing near EF at T=0, binding energy of the electron pairs thus formed, and superconductivity in mono-atom crystals have been proposed.
Once EM wave modes are established in the ranges of their associated lattice chains of a crystal concerned, which ranges can be long or even macroscopic, electron-pairs are produced in the crystal’s electron system over these ranges. As EM wave modes with frequencies below certain value (corresponding to an energy value Δ) may have little contribution to stimulated transitions of electrons and electron-pairing, at T=0 each of the electrons at and near EF pairs with one of the electrons at energy levels of EF-hωM/(2π)≤E≤EF-Δ (where ωM is the maximum frequency of lattice wave modes of the system, which is often associated with a specific crystal orientation), resulting in a binding energy of at least Δ for each of these pairs at T=0.
The document discusses the internal mechanism of electric potential in superconductors carrying a superconducting current. It reveals that the potential difference is due to the difference in electron attraction energies between the normal and superconducting states of the material. This difference is caused by resonant electron-phonon interactions. The potential is analyzed at extreme temperatures of 0K and the critical temperature, using concepts from quantum mechanics, relativity, and BCS theory. The potential difference provides an explanation for electric fields in superconductors without resistance.
Polarization bremsstrahlung on atoms, plasmas, nanostructures and solidsSpringer
This document discusses the quantum electrodynamics approach to describing bremsstrahlung, or braking radiation, of a fast charged particle colliding with an atom. It derives expressions for the amplitude of bremsstrahlung on a one-electron atom within the first Born approximation. The amplitude has static and polarization terms. The static term corresponds to radiation from the incident particle in the nuclear field, reproducing previous results. The polarization term accounts for radiation from the atomic electron and contains resonant denominators corresponding to intermediate atomic states. The full treatment allows various limits to be taken, such as removing the nucleus or atomic electron, reproducing known results from quantum electrodynamics.
This document reviews experimental approaches to analyze spin wave dynamics in ferromagnetic nanoscale structures. It describes recent developments in frequency- and field-swept spectroscopy to determine the resonant response of nanoscale ferromagnets. It also describes time-resolved measurements in the GHz frequency and picosecond time domains to analyze the relaxation of magnetization after microwave excitation. Examples are presented for analyzing and manipulating different mechanisms for the relaxation of magnetization into its ground state.
A model of electron pairing, with depletion of mediating phonons at fermi sur...Qiang LI
We present a model of electron pairing based on nonstationary interpretation of electron-lattice interaction. Electron-lattice system has an intrinsic time dependent characteristic as featured by Golden Rule, by which electrons on matched pairing states are tuned to lattice wave modes, with pairing competition happening among multiple pairings associated with one electron state. The threshold phonon of an electron pair having a good quality factor can become redundant and be released from the pair to produce a binding energy. Lattice modes falling in a common linewidth compete with one another, like modes competing in a lasing system. In cuprates, due to near-parallel band splitting at and near Fermi Surface (EF), a great number of electron pairs are tuned to a relatively small number of lattice wave modes, leading to strong mode competition, transfer of real pairing-mediating phonons from EF towards the “kink”, and depletion of these phonons at and near EF.
(v3) Phonon as carrier of electromagnetic interaction between vibrating latti...Qiang LI
With emphasis on time-dependency of electron-lattice system, we suggest the fallacy of presumed quantization in the context of electron-lattice system and propose the definition of phonons as carriers of electromagnetic interaction between electrons and vibrating lattice. We have investigated behaviors of electron-lattice system relating to “measured” energy, identified non-stationary steady state of electrons engaging in “electron pairing by virtual stimulated transitions”, recognized some origins of binding energy of electron pairs in crystals, and explained the state of electrons under pairing. Moreover, we have recognized the behavior and role of threshold phonon, which exists in electron pairing and is released by the electron from excited state, and have recognized the redundancy of the threshold phonon when the electrons under pairing have entered non-stationary steady state. We have also studied the effect of the stability of lattice wave on the evolution of the function of transition probability and on the stability of phonon-mediated electron pairs, the competition among multiple pairings associated with one same ground state, and determination of presence/absence of superconductivity by such competition.
candidate mechanisms of electron pairing near EF at T=0, binding energy of the electron pairs thus formed, and superconductivity in mono-atom crystals have been proposed.
Once EM wave modes are established in the ranges of their associated lattice chains of a crystal concerned, which ranges can be long or even macroscopic, electron-pairs are produced in the crystal’s electron system over these ranges. As EM wave modes with frequencies below certain value (corresponding to an energy value Δ) may have little contribution to stimulated transitions of electrons and electron-pairing, at T=0 each of the electrons at and near EF pairs with one of the electrons at energy levels of EF-hωM/(2π)≤E≤EF-Δ (where ωM is the maximum frequency of lattice wave modes of the system, which is often associated with a specific crystal orientation), resulting in a binding energy of at least Δ for each of these pairs at T=0.
electron pairing and mechanism of superconductivity in ionic crystals Qiang LI
The behaviors of valence electrons and ions, particularly ion chains, in ionic crystals are important to understanding of the mechanism of superconductivity. The author has made efforts to establish a candidate mechanism of electron-pairing and superconductivity in ionic crystals.
Analyses are first made to a one-dimensional long ion lattice chain model (EDP model), with the presence of lattice wave modes having frequency ω. A mechanism of electron pairing is established.
Analyses are then extended to scenarios of 3D ionic crystals, particularly those with a donor/acceptor system, with emphasis being given to the interpretation and understanding of binding energy of electron pairs formed between electrons at the top/bottom of donor/acceptor band and the bottom/top of conducting/full band.
It is established that once the lattice/EM wave modes are established in its range, which can be long or even macroscopic, electron pairs are produced in the crystal’s electron system over the same range by stimulated transitions induced by the EM wave mode. The lattice wave mode having the maximum frequency ωM is of special significance with respect to superconductivity, for electron pairs produced by it can be stabilized in the context of a combination of some special factors (including energy level structure featured by donor/acceptor band and ωM) with a binding energy typically no smaller than hωM/(2π). A candidate mechanism of electron pairing in ion crystals and therefore of superconductivity is provided.
candidate mechanisms of electron pairing near EF at T=0, binding energy of the electron pairs thus formed, and superconductivity in mono-atom crystals have been proposed.
Once EM wave modes are established in the ranges of their associated lattice chains of a crystal concerned, which ranges can be long or even macroscopic, electron-pairs are produced in the crystal’s electron system over these ranges. As EM wave modes with frequencies below certain value (corresponding to an energy value Δ) may have little contribution to stimulated transitions of electrons and electron-pairing, at T=0 each of the electrons at and near EF pairs with one of the electrons at energy levels of EF-hωM/(2π)≤E≤EF-Δ (where ωM is the maximum frequency of lattice wave modes of the system, which is often associated with a specific crystal orientation), resulting in a binding energy of at least Δ for each of these pairs at T=0.
The document discusses the internal mechanism of electric potential in superconductors carrying a superconducting current. It reveals that the potential difference is due to the difference in electron attraction energies between the normal and superconducting states of the material. This difference is caused by resonant electron-phonon interactions. The potential is analyzed at extreme temperatures of 0K and the critical temperature, using concepts from quantum mechanics, relativity, and BCS theory. The potential difference provides an explanation for electric fields in superconductors without resistance.
Polarization bremsstrahlung on atoms, plasmas, nanostructures and solidsSpringer
This document discusses the quantum electrodynamics approach to describing bremsstrahlung, or braking radiation, of a fast charged particle colliding with an atom. It derives expressions for the amplitude of bremsstrahlung on a one-electron atom within the first Born approximation. The amplitude has static and polarization terms. The static term corresponds to radiation from the incident particle in the nuclear field, reproducing previous results. The polarization term accounts for radiation from the atomic electron and contains resonant denominators corresponding to intermediate atomic states. The full treatment allows various limits to be taken, such as removing the nucleus or atomic electron, reproducing known results from quantum electrodynamics.
This document reviews experimental approaches to analyze spin wave dynamics in ferromagnetic nanoscale structures. It describes recent developments in frequency- and field-swept spectroscopy to determine the resonant response of nanoscale ferromagnets. It also describes time-resolved measurements in the GHz frequency and picosecond time domains to analyze the relaxation of magnetization after microwave excitation. Examples are presented for analyzing and manipulating different mechanisms for the relaxation of magnetization into its ground state.
1. The document discusses the derivation of de Broglie's equation relating the wavelength of matter waves to the momentum of particles. It then derives different forms of de Broglie's wavelength equation using kinetic energy and potential energy.
2. It lists properties of matter waves including that lighter particles have greater wavelengths. It derives the Schrodinger time-independent and time-dependent wave equations.
3. It applies the time-independent equation to a particle in an infinite square well, finding the wavefunctions and energy levels based on boundary conditions and normalization.
Smell in real noses: how the environment changes vibrationsVorname Nachname
This document discusses Benjamin Yiwen Färber's research into how the vibrational frequency of odor molecules is affected by the environment inside the nose. It begins by summarizing the biological process of smell and Turin's theory that odor discrimination is based on the vibrational frequencies of molecules. It then models odor molecules as diatomic harmonic oscillators and examines how the oscillator's frequency is altered by environmental factors like external charges and binding to olfactory receptors. The document relates the harmonic oscillator model to the quantum mechanical Morse potential model and tight binding model of diatomic molecules.
I do not have enough context to answer these questions. The document provided is a lecture on interactions of radiation with matter and does not contain questions.
In computational physics and Quantum chemistry, the Hartree–Fock (HF) method also known as self consistent method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system or many electron system in a stationary state
1. Hartree-Fock theory describes molecules using a linear combination of atomic orbitals to approximate molecular orbitals. It treats electrons as independent particles moving in the average field of other electrons.
2. The Hartree-Fock method involves iteratively solving the Fock equations until self-consistency is reached between the input and output orbitals. This approximates electron correlation by including an average electron-electron repulsion term.
3. The Hartree-Fock method satisfies the Pauli exclusion principle through the use of Slater determinants, which are antisymmetric wavefunctions that go to zero when the spatial or spin coordinates of any two electrons are identical.
1) The Born-Oppenheimer approximation separates the molecular Schrodinger equation into electronic and nuclear parts based on the large mass difference between electrons and nuclei.
2) It assumes that over short time periods, electrons adjust instantaneously to nuclear motions. This allows treating electronic motions separately for fixed nuclear positions.
3) Solving the electronic Schrodinger equation for different nuclear configurations provides the potential energy surface for nuclear vibrations and rotations.
1. The Stern-Gerlach experiment discovered that silver atoms split into two beams, indicating the presence of an intrinsic "spin" angular momentum of 1/2 beyond orbital angular momentum.
2. Elementary particles are classified as fermions, with half-integer spin, and bosons, with integer spin. The spin of the electron is represented by a two-component spinor.
3. In a magnetic field, the spin precesses around the field direction at the Larmor frequency, independent of initial spin orientation. This principle underlies paramagnetic resonance and nuclear magnetic resonance spectroscopy.
This document discusses voltammetry, an electroanalytical technique used in qualitative and quantitative analytical chemistry. It introduces the basic concepts and principles of voltammetry, including instrumentation, excitation signals, types of voltammetry, and features of voltammograms. Specifically, it discusses the fundamentals of voltammetric cells, electrodes, hydrodynamic voltammetry, and common shapes of voltammograms including linear scan and peak voltammograms. The overall purpose is to explain the fundamental concepts and applications of voltammetry as an analytical technique.
Effect of isotopic subsitution on the transition frequenciesApurvaSachdeva
This document discusses isotopic substitution, which is the replacement of atoms in a molecule with isotopes of different mass. Isotopic substitution is useful for vibrational spectroscopy because it changes the reduced mass and normal modes of vibration, leading to different vibrational frequencies. Specifically, substituting heavier isotopes lowers vibrational frequencies due to an increase in reduced mass. Examples given are substituting deuterium for hydrogen in HCl, which lowers frequencies by a factor of 1.35-1.41, and substituting 13C for 12C in CO, which also lowers vibrational energy levels.
This document provides an introduction to quantum mechanics concepts including:
1. It describes Schrodinger's wave equation and its applications, including quantized energy levels and tunneling effects.
2. Wave-particle duality is discussed through experiments demonstrating the wave-like and particle-like properties of electrons.
3. The uncertainty principle and solutions to Schrodinger's wave equation for simple potential wells are presented, showing energy levels are quantized.
1) Maxwell's equations describe electromagnetic waves propagating through space and time. For time-varying fields, the full set of Maxwell's equations must be used.
2) By assuming time-harmonic fields with a sinusoidal time variation, Maxwell's equations can be simplified to phasor forms containing only spatial derivatives.
3) The phasor forms of Maxwell's equations can be reduced to Helmholtz wave equations for the electric and magnetic fields. Plane wave solutions representing uniform electromagnetic waves propagating in a given direction can be derived from these equations.
The document discusses the state of thermal excitation, or "THE THEXI STATE". It defines ground and excited states, with the ground state being the lowest energy state and excited states having higher energy. Thermal excitation occurs when electrons absorb energy, such as from light or collisions, promoting them to an excited state. The properties of thermally excited states, or "Thexi states", are described, including that they are unstable and have short lifetimes. Methods to prepare Thexi states include irradiation or energy transfer. Thexi states emit radiation as they return to the ground state. Spectroscopy techniques can provide information about Thexi state energies and structures, though the actual structures are difficult to determine.
The document summarizes NMR spectroscopy techniques. It discusses how nuclei with spin produce magnetic moments that can be aligned in an external magnetic field. It describes how this spin property gives rise to quantized energy levels for nuclei and how transitions between these levels appear as signals in NMR spectra. It explains concepts like chemical shift, spin-spin coupling, and how NMR spectra provide information about molecular structure. Examples are given to illustrate NMR phenomena for nuclei like 1H, 13C, 11B, 19F, and how spectra change with conditions like temperature.
This document summarizes key concepts in condensed matter physics related to interacting electron systems.
It introduces the Hartree and Hartree-Fock approximations for modeling interacting electrons, which improve upon treating electrons independently but still do not fully capture electron correlation. The Hartree approximation models the average electrostatic potential felt by each electron from other electrons. Hartree-Fock further includes an "exchange" term to account for the Pauli exclusion principle.
It then discusses limitations of these approximations in capturing electron correlation, where the motion of each electron is correlated with all others due to both Coulomb repulsion and the Pauli principle. Capturing electron correlation is important for obtaining more accurate descriptions of materials' properties.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
This document discusses magnetic materials and their properties. It begins by explaining the origins of magnetism from electron orbits and spins. It then classifies magnetic materials as diamagnetic, paramagnetic, ferromagnetic, antiferromagnetic, ferrimagnetic, and discusses their characteristics. The document also covers Weiss molecular field theory of ferromagnetism, hysteresis curves, hard and soft magnetic materials, ferrites and their applications. Finally, it discusses superconductivity including the BCS theory and applications of superconductors such as SQUIDs.
This document discusses the quantum theory of light dispersion using time-dependent perturbation theory. It describes how bound electrons in materials contribute to the permittivity and optical properties when subjected to an external electric field. The perturbation leads to polarization of electron orbitals and possible transitions to excited states. Absorption of light occurs when the photon energy matches the energy difference between bound states. The permittivity and refractive index are derived in terms of oscillator strengths, and dispersion is explained through resonant absorption at certain photon frequencies.
Microscopic Mechanisms of Superconducting Flux Quantum and Superconducting an...Qiang LI
We have provided microscopic explanations to superconducting flux quantum and (superconducting and normal) persistent current. Flux quantum is generated by current carried by "deep electrons" at surface states. And values of the flux quantum differs according to the electronic states and coupling of the carrier electrons. Generation of persistent carrier electrons does not dissipate energy; instead there would be emission of real phonons and release of corresponding energy into the environment; but the normal carrier electrons involved still dissipate energy. Even for or persistent carriers,there should be a build-up of energy of the middle state and a build-up of the probability of virtual transition of electrons to the middle state, and the corresponding relaxation should exist accordingly.
Energy scale in ARPES data suggesting Bogoliubov quasiparticles as excitation...Qiang LI
A dip is identified in existing ARPES spectra of Bi2223. The energy separation between the dip and Bogoliubov (BQP) peak is well-defined at a value of about 68 meV. More remarkably, it is shown that, with a lattice-mode-specific modification detailed below, the strength of the dip is in well qualitative agreement with that of the BQP peak. These results strongly suggest an origin of BQPs as excitations by phonons of a very small number of lattice modes, which could be a direct clue to understanding the interactions leading to nodal and antinodal energy gap features and even high-temperature superconductivity itself.
Energy gap as a measure of pairing instability, Bogoliubov quasiparticles as ...Qiang LI
The magnitude of an apparent energy gap is recognized as a measure of relative instability of electron pairing at the gap location, for it indicates that stabilized pairing can only be realized at a greater binding energy. At a low temperature, the chemical potential of a system like Bi2212 is determined by the most stable pairing, and will drop by about the energy of the mediating mode when pairing is stable. Bogoliubov quasiparticles are explained as excitations by lattice modes that win a mode competition among mediating modes and therefore have a large number of phonons depleted from losing modes. Thus, the energy of BQP peak is that of upper states of pairs mediated by the winning modes (~70 meV), while the energy of the nodal kink is that of the base states of these pairs, which shifts upward due to band topology on leaving the node. The “superconducting gap” corresponds to a kinked band section, a kink at the lower edge of which varnishes as nodal gap transform into the antinodal one.
D:\Edit\Super\For Submission 20100306\12622 0 Merged 1267687011Qiang LI
This document discusses a proposed mechanism for electron pairing and superconductivity in ionic crystals. It analyzes a one-dimensional ion lattice chain model and establishes a mechanism for electron pairing driven by lattice vibration modes. The analysis is extended to 3D ionic crystals, focusing on donor-acceptor systems. Electron pairing occurs between energy levels matched to the maximum vibration frequency ωM. Introducing an acceptor band can stabilize electron pairs across the acceptor and full bands, with a binding energy estimated to be at least hωM/(4π).
Mechanism of electron pairing in crystals, with binding energy no smaller tha...Qiang LI
(A prepring publication)
Establishment of mechanism of electron pairing with a lower limit of binding energy is necessary for understating of superconductivity. Due to conservation of wavevector, photon absorption/emission by an electron in crystal can only be allowed across at least on band gap, which is also true for virtual photon absorption/emission inducing electron pairing in crystal. Therefore, it is clearly explained that electron pairs, formed by virtual stipulated transition, can only exist between electrons across a band gap, with a binding energy no smaller than the width of the band gap.
1. The document discusses the derivation of de Broglie's equation relating the wavelength of matter waves to the momentum of particles. It then derives different forms of de Broglie's wavelength equation using kinetic energy and potential energy.
2. It lists properties of matter waves including that lighter particles have greater wavelengths. It derives the Schrodinger time-independent and time-dependent wave equations.
3. It applies the time-independent equation to a particle in an infinite square well, finding the wavefunctions and energy levels based on boundary conditions and normalization.
Smell in real noses: how the environment changes vibrationsVorname Nachname
This document discusses Benjamin Yiwen Färber's research into how the vibrational frequency of odor molecules is affected by the environment inside the nose. It begins by summarizing the biological process of smell and Turin's theory that odor discrimination is based on the vibrational frequencies of molecules. It then models odor molecules as diatomic harmonic oscillators and examines how the oscillator's frequency is altered by environmental factors like external charges and binding to olfactory receptors. The document relates the harmonic oscillator model to the quantum mechanical Morse potential model and tight binding model of diatomic molecules.
I do not have enough context to answer these questions. The document provided is a lecture on interactions of radiation with matter and does not contain questions.
In computational physics and Quantum chemistry, the Hartree–Fock (HF) method also known as self consistent method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system or many electron system in a stationary state
1. Hartree-Fock theory describes molecules using a linear combination of atomic orbitals to approximate molecular orbitals. It treats electrons as independent particles moving in the average field of other electrons.
2. The Hartree-Fock method involves iteratively solving the Fock equations until self-consistency is reached between the input and output orbitals. This approximates electron correlation by including an average electron-electron repulsion term.
3. The Hartree-Fock method satisfies the Pauli exclusion principle through the use of Slater determinants, which are antisymmetric wavefunctions that go to zero when the spatial or spin coordinates of any two electrons are identical.
1) The Born-Oppenheimer approximation separates the molecular Schrodinger equation into electronic and nuclear parts based on the large mass difference between electrons and nuclei.
2) It assumes that over short time periods, electrons adjust instantaneously to nuclear motions. This allows treating electronic motions separately for fixed nuclear positions.
3) Solving the electronic Schrodinger equation for different nuclear configurations provides the potential energy surface for nuclear vibrations and rotations.
1. The Stern-Gerlach experiment discovered that silver atoms split into two beams, indicating the presence of an intrinsic "spin" angular momentum of 1/2 beyond orbital angular momentum.
2. Elementary particles are classified as fermions, with half-integer spin, and bosons, with integer spin. The spin of the electron is represented by a two-component spinor.
3. In a magnetic field, the spin precesses around the field direction at the Larmor frequency, independent of initial spin orientation. This principle underlies paramagnetic resonance and nuclear magnetic resonance spectroscopy.
This document discusses voltammetry, an electroanalytical technique used in qualitative and quantitative analytical chemistry. It introduces the basic concepts and principles of voltammetry, including instrumentation, excitation signals, types of voltammetry, and features of voltammograms. Specifically, it discusses the fundamentals of voltammetric cells, electrodes, hydrodynamic voltammetry, and common shapes of voltammograms including linear scan and peak voltammograms. The overall purpose is to explain the fundamental concepts and applications of voltammetry as an analytical technique.
Effect of isotopic subsitution on the transition frequenciesApurvaSachdeva
This document discusses isotopic substitution, which is the replacement of atoms in a molecule with isotopes of different mass. Isotopic substitution is useful for vibrational spectroscopy because it changes the reduced mass and normal modes of vibration, leading to different vibrational frequencies. Specifically, substituting heavier isotopes lowers vibrational frequencies due to an increase in reduced mass. Examples given are substituting deuterium for hydrogen in HCl, which lowers frequencies by a factor of 1.35-1.41, and substituting 13C for 12C in CO, which also lowers vibrational energy levels.
This document provides an introduction to quantum mechanics concepts including:
1. It describes Schrodinger's wave equation and its applications, including quantized energy levels and tunneling effects.
2. Wave-particle duality is discussed through experiments demonstrating the wave-like and particle-like properties of electrons.
3. The uncertainty principle and solutions to Schrodinger's wave equation for simple potential wells are presented, showing energy levels are quantized.
1) Maxwell's equations describe electromagnetic waves propagating through space and time. For time-varying fields, the full set of Maxwell's equations must be used.
2) By assuming time-harmonic fields with a sinusoidal time variation, Maxwell's equations can be simplified to phasor forms containing only spatial derivatives.
3) The phasor forms of Maxwell's equations can be reduced to Helmholtz wave equations for the electric and magnetic fields. Plane wave solutions representing uniform electromagnetic waves propagating in a given direction can be derived from these equations.
The document discusses the state of thermal excitation, or "THE THEXI STATE". It defines ground and excited states, with the ground state being the lowest energy state and excited states having higher energy. Thermal excitation occurs when electrons absorb energy, such as from light or collisions, promoting them to an excited state. The properties of thermally excited states, or "Thexi states", are described, including that they are unstable and have short lifetimes. Methods to prepare Thexi states include irradiation or energy transfer. Thexi states emit radiation as they return to the ground state. Spectroscopy techniques can provide information about Thexi state energies and structures, though the actual structures are difficult to determine.
The document summarizes NMR spectroscopy techniques. It discusses how nuclei with spin produce magnetic moments that can be aligned in an external magnetic field. It describes how this spin property gives rise to quantized energy levels for nuclei and how transitions between these levels appear as signals in NMR spectra. It explains concepts like chemical shift, spin-spin coupling, and how NMR spectra provide information about molecular structure. Examples are given to illustrate NMR phenomena for nuclei like 1H, 13C, 11B, 19F, and how spectra change with conditions like temperature.
This document summarizes key concepts in condensed matter physics related to interacting electron systems.
It introduces the Hartree and Hartree-Fock approximations for modeling interacting electrons, which improve upon treating electrons independently but still do not fully capture electron correlation. The Hartree approximation models the average electrostatic potential felt by each electron from other electrons. Hartree-Fock further includes an "exchange" term to account for the Pauli exclusion principle.
It then discusses limitations of these approximations in capturing electron correlation, where the motion of each electron is correlated with all others due to both Coulomb repulsion and the Pauli principle. Capturing electron correlation is important for obtaining more accurate descriptions of materials' properties.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
This document discusses magnetic materials and their properties. It begins by explaining the origins of magnetism from electron orbits and spins. It then classifies magnetic materials as diamagnetic, paramagnetic, ferromagnetic, antiferromagnetic, ferrimagnetic, and discusses their characteristics. The document also covers Weiss molecular field theory of ferromagnetism, hysteresis curves, hard and soft magnetic materials, ferrites and their applications. Finally, it discusses superconductivity including the BCS theory and applications of superconductors such as SQUIDs.
This document discusses the quantum theory of light dispersion using time-dependent perturbation theory. It describes how bound electrons in materials contribute to the permittivity and optical properties when subjected to an external electric field. The perturbation leads to polarization of electron orbitals and possible transitions to excited states. Absorption of light occurs when the photon energy matches the energy difference between bound states. The permittivity and refractive index are derived in terms of oscillator strengths, and dispersion is explained through resonant absorption at certain photon frequencies.
Microscopic Mechanisms of Superconducting Flux Quantum and Superconducting an...Qiang LI
We have provided microscopic explanations to superconducting flux quantum and (superconducting and normal) persistent current. Flux quantum is generated by current carried by "deep electrons" at surface states. And values of the flux quantum differs according to the electronic states and coupling of the carrier electrons. Generation of persistent carrier electrons does not dissipate energy; instead there would be emission of real phonons and release of corresponding energy into the environment; but the normal carrier electrons involved still dissipate energy. Even for or persistent carriers,there should be a build-up of energy of the middle state and a build-up of the probability of virtual transition of electrons to the middle state, and the corresponding relaxation should exist accordingly.
Energy scale in ARPES data suggesting Bogoliubov quasiparticles as excitation...Qiang LI
A dip is identified in existing ARPES spectra of Bi2223. The energy separation between the dip and Bogoliubov (BQP) peak is well-defined at a value of about 68 meV. More remarkably, it is shown that, with a lattice-mode-specific modification detailed below, the strength of the dip is in well qualitative agreement with that of the BQP peak. These results strongly suggest an origin of BQPs as excitations by phonons of a very small number of lattice modes, which could be a direct clue to understanding the interactions leading to nodal and antinodal energy gap features and even high-temperature superconductivity itself.
Energy gap as a measure of pairing instability, Bogoliubov quasiparticles as ...Qiang LI
The magnitude of an apparent energy gap is recognized as a measure of relative instability of electron pairing at the gap location, for it indicates that stabilized pairing can only be realized at a greater binding energy. At a low temperature, the chemical potential of a system like Bi2212 is determined by the most stable pairing, and will drop by about the energy of the mediating mode when pairing is stable. Bogoliubov quasiparticles are explained as excitations by lattice modes that win a mode competition among mediating modes and therefore have a large number of phonons depleted from losing modes. Thus, the energy of BQP peak is that of upper states of pairs mediated by the winning modes (~70 meV), while the energy of the nodal kink is that of the base states of these pairs, which shifts upward due to band topology on leaving the node. The “superconducting gap” corresponds to a kinked band section, a kink at the lower edge of which varnishes as nodal gap transform into the antinodal one.
D:\Edit\Super\For Submission 20100306\12622 0 Merged 1267687011Qiang LI
This document discusses a proposed mechanism for electron pairing and superconductivity in ionic crystals. It analyzes a one-dimensional ion lattice chain model and establishes a mechanism for electron pairing driven by lattice vibration modes. The analysis is extended to 3D ionic crystals, focusing on donor-acceptor systems. Electron pairing occurs between energy levels matched to the maximum vibration frequency ωM. Introducing an acceptor band can stabilize electron pairs across the acceptor and full bands, with a binding energy estimated to be at least hωM/(4π).
Mechanism of electron pairing in crystals, with binding energy no smaller tha...Qiang LI
(A prepring publication)
Establishment of mechanism of electron pairing with a lower limit of binding energy is necessary for understating of superconductivity. Due to conservation of wavevector, photon absorption/emission by an electron in crystal can only be allowed across at least on band gap, which is also true for virtual photon absorption/emission inducing electron pairing in crystal. Therefore, it is clearly explained that electron pairs, formed by virtual stipulated transition, can only exist between electrons across a band gap, with a binding energy no smaller than the width of the band gap.
phonon as carrier of electromagnetic interaction between lattice wave modes a...Qiang LI
The new results reported here mainly include: 1) recognition that phonon is carrier of electromagnetic interaction between its lattice wave mode and electrons; 2) recognition that binding energy of electron pairs of high-temperature superconductivity is due to escape of optical threshold phonons, of electron pairs at or near Fermi level, from crystal by direct radiation; 3) recognition that binding energy of electron pairs of low-temperature superconductivity is possibly due to escape of non-optical threshold phonons by anharmonic crystal interactions; and, 4) recognition of a possible mechanism explaining why some crystals never have a superconducting phase. While electron pairing is phonon-mediated in general, HTS should be associated with electron pairing mediated by optical phonon at or near Fermi level (EF), so the rarity of HTS corresponds to the rarity of such pairing match.
Explaining cuprates antinodal psuedogap features lq111203Qiang LI
We provide an explanation of anti-nodal pseudogap features of cuprates on the basis of a proposed electron pairing model, and recognize the bosonic modes responsible for electron pairings leading to the anti-nodal pseudogap features, as having and energy range estimated at about 25-30 meV.
The document proposes a mechanism for electron pairing and superconductivity in metals at low temperatures. It suggests that constant virtual stimulated transitions of valence electrons, driven by electromagnetic wave modes coupled to lattice vibrations, can lead to electron pairing. At zero temperature, each electron near the Fermi energy pairs with another electron at a lower energy level, giving the pairs a minimum binding energy equal to an energy value Δ related to lattice vibration frequencies. This electron pairing produces superconductivity in metals below a critical temperature that depends on the strength of electromagnetic wave modes at low frequencies associated with long-range lattice vibrations.
Mechanism of electron pairing in crystals, with binding energy no smaller tha...Qiang LI
The document discusses the mechanism of electron pairing in crystals with a binding energy greater than or equal to one band gap. It explains that due to conservation of wavevector, virtual photon absorption/emission inducing electron pairing can only occur across at least one band gap. Therefore, electron pairs formed by virtual stimulated transitions must have a binding energy greater than or equal to the width of the band gap. It also reviews how electron pairing is established and explains why virtual stimulated transitions in low-frequency ranges are negligible.
The document discusses the internal mechanism of electric potential in superconductors carrying a superconducting current. It reveals that the potential difference is due to the difference in electron attraction energies between the normal and superconducting states of the material. This difference is caused by resonant electron-phonon interactions. The potential is analyzed at extreme temperatures of 0K and the critical temperature, using concepts from quantum mechanics, relativity, and BCS theory. The potential difference provides an explanation for electric fields in superconductors without resistance.
The document summarizes the author's theory on the internal mechanism that causes electric potential differences in superconductors carrying superconducting currents. The author argues that at zero Kelvin, the energy of electron attraction in Cooper pairs differs between the normal and current-carrying states of the superconductor. This difference in energy corresponds to the potential difference of the superconducting current. Near the critical temperature, resonant electron-phonon interactions are responsible for equalizing the energies of electrons and phonons, resulting in a loss of superconductivity. The author presents the theory from an engineering perspective using quantum mechanical and relativistic concepts.
1) According to the free electron model, conduction electrons exist in metals that are not bound to individual atoms but are free to move throughout the crystal lattice.
2) The electrons occupy discrete quantum states that can be modeled as plane waves. The lowest energy state is filled first according to the Pauli exclusion principle.
3) Key properties of the free electron gas model include the Fermi energy (EF), Fermi temperature (TF), and Fermi momentum (kF) and sphere, which describe the highest occupied electron state at 0K.
A Mechanism Of Electron Pairing Relating To SupperconductivityQiang LI
Mechanism of superconductivity is based on a mechanism of electron pairing near EF with definite binding energy. A candidate mechanism of such electron pairing is described in this paper. Electron pairs are induced by EM wave modes generated by corresponding lattice wave modes. The pairs are formed between E(k) faces from different Brillouin zones, with definite binding energies. The binding energy of an electron pair is characterized by the frequency of the EM mode that induces the pairing.
Energy bands and electrical properties of metals newPraveen Vaidya
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A free electron model is the simplest way to represent the
electronic structure and properties of metals.
According to this model, the valence electrons of the constituent
atoms of the crystal become conduction electrons and travel
freely throughout the crystal.
The classical theory fails to explain the heat capacity and the
magnetic susceptibility of the conduction electrons. (These are
not failures of the free electron model, but failures of the classical
Maxwell distribution function.)
Condensed matter is so transparent to conduction electrons
This document outlines topics related to semiconductor physics and optoelectronics physics, including:
1. Free electron theory of metals, Bloch's theorem, energy band diagrams, direct and indirect bandgaps, density of states, and the types of electronic materials including metals, semiconductors and insulators.
2. Lasers, which use stimulated emission of radiation to produce an intense, coherent beam of light. Key concepts covered include spontaneous emission, stimulated absorption, population inversion, and semiconductor lasers.
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3) The Franck-Condon principle which states that electronic transitions occur rapidly without changes in internuclear distance, leading to vertical transitions between vibrational levels.
4) Dissociation of electronically excited molecules and the relationship between dissociation energies and excitation energies.
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Elementary plasma-chemical reactions can be described by micro-kinetic characteristics like cross-sections and reaction probabilities. There are several types of ionization processes including direct electron impact ionization, stepwise ionization, and ionization by photons or heavy particles. Collision parameters include the cross-section, probability, mean free path, and reaction rates. Direct ionization follows the Thomson formula at high energies. Molecular ionization is affected by the Frank-Condon principle where atomic positions remain fixed. Stepwise ionization occurs through excitation to an energy above the ionization potential. High-energy electrons like in beams follow the Bethe-Bloch formula for energy loss per unit length.
This document summarizes key concepts in plasma chemistry from Chapter 2 of the reference book, including:
1) Elementary plasma reactions are determined by micro-kinetic characteristics like cross-sections and reaction probabilities, as well as kinetic distribution functions.
2) Collisions can be elastic, inelastic, or superelastic depending on whether the total kinetic energy and internal energies change during the collision.
3) Ionization processes include direct electron impact ionization, stepwise ionization, ion-molecule collisions, photoionization, and surface ionization.
4) The Thomson formula describes direct electron impact ionization cross-sections at high energies, while the Frank-Condon principle applies to ionization
lecture classical and Quantum Free electron theory (FERMI GAS) (23-24).pdfLobnaSharaf
The electron theory of solids explains properties through electronic structure. It applies to metals and nonmetals. The theory developed in three stages:
1. Classical free electron theory treated electrons as free gas particles. It could not explain many properties.
2. Quantum free electron theory incorporated quantum mechanics. Electrons occupy discrete energy levels according to Fermi-Dirac statistics.
3. Band theory views electrons moving in periodic potentials of atom arrays. It explains conductivity, effective mass, and the origin of band gaps.
The document discusses approximations used to model the helium atom with two electrons. It begins by introducing the independent particle model, which ignores interactions between electrons and treats each as moving independently. This approximation yields the wrong energy for helium's ground state. The screening approximation is then introduced to account for each electron partially screening the nucleus charge from the other. Finally, the need for a symmetric wavefunction accounting for electron indistinguishability is discussed.
The Propagation and Power Deposition of Electron Cyclotron Waves in Non-Circu...IJERA Editor
The document summarizes a numerical study of the propagation and power deposition of electron cyclotron waves in non-circular HL-2A tokamak plasmas. The ray trajectories and power deposition were simulated by solving the plasma equilibrium equation, ray equations, and quasi-linear Fokker-Planck equation. The results show that shaping effects and temperature profiles have little influence on ECRH, while plasma density significantly affects propagation and power deposition. When ordinary mode EC waves are launched from the mid-plane and low-field side, ray trajectories bend as the parallel refractive index increases and can even recurve to the low-field side when the index reaches a certain value. Single absorption decreases with increasing both poloidal and toroidal
The document discusses the Fourier transform and Laplace transform. Some key points:
- The Fourier transform represents periodic and non-periodic signals as a sum of complex exponentials, allowing representation of aperiodic signals. It converges if the signal has finite energy.
- The Laplace transform generalizes the Fourier transform to a broader class of signals by using a complex variable 's' instead of just 'jw'. It can represent signals that are not absolutely integrable.
- The Fourier transform is a special case of the Laplace transform when s = jw, along the imaginary axis where σ = 0.
- Both transforms allow mapping between a signal and its frequency spectrum, with applications in analyzing signals and systems.
Similar to phonon as carrier of electromagnetic interaction between lattice wave modes and electrons and its role in superconductivity (20)
In the rapidly evolving landscape of technologies, XML continues to play a vital role in structuring, storing, and transporting data across diverse systems. The recent advancements in artificial intelligence (AI) present new methodologies for enhancing XML development workflows, introducing efficiency, automation, and intelligent capabilities. This presentation will outline the scope and perspective of utilizing AI in XML development. The potential benefits and the possible pitfalls will be highlighted, providing a balanced view of the subject.
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The presentation aims to deliver a comprehensive overview of AI usage in XML development, providing attendees with the necessary knowledge to make informed decisions. Whether you’re at the early stages of adopting AI or considering integrating it in advanced XML development, this presentation will cover all levels of expertise.
By highlighting the potential advantages and challenges of integrating AI with XML development tools and languages, the presentation seeks to inspire thoughtful conversation around the future of XML development. We’ll not only delve into the technical aspects of AI-powered XML development but also discuss practical implications and possible future directions.
Dive into the realm of operating systems (OS) with Pravash Chandra Das, a seasoned Digital Forensic Analyst, as your guide. 🚀 This comprehensive presentation illuminates the core concepts, types, and evolution of OS, essential for understanding modern computing landscapes.
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Moving to the realm of mobile devices, Das unravels the dominance of Android and iOS. Android's open-source ethos fosters a vibrant ecosystem of customization and innovation, while iOS boasts a seamless user experience and robust security infrastructure. Meanwhile, discontinued platforms like Symbian and Palm OS evoke nostalgia for their pioneering roles in the smartphone revolution.
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This presentation provides valuable insights into effective cost-saving techniques on AWS. Learn how to optimize your AWS resources by rightsizing, increasing elasticity, picking the right storage class, and choosing the best pricing model. Additionally, discover essential governance mechanisms to ensure continuous cost efficiency. Whether you are new to AWS or an experienced user, this presentation provides clear and practical tips to help you reduce your cloud costs and get the most out of your budget.
Digital Marketing Trends in 2024 | Guide for Staying AheadWask
https://www.wask.co/ebooks/digital-marketing-trends-in-2024
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Taking AI to the Next Level in Manufacturing.pdfssuserfac0301
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Ready to take your DeFi project to the next level? Partner with Intelisync for expert DeFi development services today!
5th LF Energy Power Grid Model Meet-up SlidesDanBrown980551
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Power Grid Model
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phonon as carrier of electromagnetic interaction between lattice wave modes and electrons and its role in superconductivity
1. Phonon as carrier of electromagnetic interaction between lattice
wave modes and electrons and its role in superconductivity
Qiang LI
Jinheng Law Firm
1004, Quantum Plaza, 23, Zhichun Road, Beijing 100191, China
lq@jinheng-ip.com
Abstract
The new results reported here mainly include: 1) recognition that phonon is
carrier of electromagnetic interaction between its lattice wave mode and electrons; 2)
recognition that binding energy of electron pairs of high-temperature
superconductivity is due to escape of optical threshold phonons, of electron pairs at or
near Fermi level, from crystal by direct radiation; 3) recognition that binding energy
of electron pairs of low-temperature superconductivity is possibly due to escape of
non-optical threshold phonons by anharmonic crystal interactions; and, 4) recognition
of a possible mechanism explaining why some crystals never have a superconducting
phase. While electron pairing is phonon-mediated in general, HTS should be
associated with electron pairing mediated by optical phonon at or near Fermi level
(EF), so the rarity of HTS corresponds to the rarity of such pairing match.
Keywords: the essence of phonon; origin of binding energy of electron pair;
superconductivity; Heisenberg Uncertainty Principle; threshold phonon; stability of
electron pair; non-stationary stable state; anharmonic crystal interactions; direction of
electron pairing; multiple-pairing; graph theory
PACS numbers: 74.20.Mn 74.25.F-
The central roles of electromagnetic (EM) wave modes generated by lattice
wave modes and the mechanism of “electron pairing by virtual stimulated transitions”
were proposed with strong suggestions that such electron pairing would result in a
threshold (binding) photon released by the electron at the excited state [1], but the
origin of binding energy of electron pairs was not clearly identified in spite of
attempts to attribute it to Heisenberg Uncertainty Principle and the threshold photon.
While phonon is widely believed to be the mediator of electron pairing relating
to superconductivity, it has a somewhat awkward status in physics; although it is
typically defined as “a quasiparticle characterized by the quantization of the modes of
lattice vibrations of periodic, elastic crystal structures of solids” and/or “a quantum
mechanical description of a special type of vibrational motion” [2], what kind of
interaction (electromagnetic, gravitational, or etc.) phonon is associated with seems
not having been well-addressed so far? If phonon is the carrier of electromagnetic
interaction, then phonon should be essentially the same as photon. In this paper it is
explained that phonons are indeed the carriers of electromagnetic interaction between
electrons and lattice wave modes, and that photon can be regarded as a special kind of
phonon.
With the recognition of identity of phonon to photon, some detailed
1
2. interpretations of the origin of binding energy of electron pairs in crystals are given
hereinbelow. The electron pairs are generally phonon-mediated, but some special ones
are “optical phonon-mediated”, and it seems that this difference constitutes a water
parting between high-temperature superconductivity (HTS) and lower temperatures
superconductivity (LTS).
For discussing some details of non-stationary behavior of electron, let us begin
with examining the exemplary dual energy system with one electron as shown in Fig.
1, which is subject to the interaction by an electromagnetic (EM) wave mode hν=E2-
E1.
According to well-known time-dependent perturbation approach, the matrix
element to the first order is
a nk=δnk+a nk1 (1)
k
a n 1 =2π/(ih) ∫Vnk(t1) exp (i(En-Ek)t1/h)dt1 (2)
where the integral is carried over [t0,t], and
Vnk(t1)=<φn|Vnk(r,t1) |φk> (3)
The potential of the EM mode (hν) the system shown in Fig. 1 is
V(r,t)=V0(r,t)cos(2πνt) (4)
When the EM mode (hν) is stable, its magnitude remains time-independent as
V0(r,t)=V(r) (5)
So Vnk(t1)=Vnkcos(2πνt1) (6)
with
Vnk=<φn|Vnk(r) |φk> (7)
Then, Equation (2) becomes
a nk1 ~(Vnk/h){exp[2πi(Enk+hν)t/h]-1}/ (Ep+hν)
-{exp[2πit(Enk-hν)t/h]-1}/ (Enk-hν) (8)
According to Equation (8), at Enk=±hν there is a nk1 ∝t, so all a nk components
except those at Enk=±hν will be normalized to zero with increasing time t, resulting in
a nk1=1 for Enk=±hν. Remarkably, the matrix a nk becomes time t-independent because t
can be taken away from the matrix at t→∞; thus, the system becomes stable but is not
at any eigenstate of energy.
It is to be further noted that this result of “stable state” should not be limited by
the approximation of perturbation approach in that all the higher terms of perturbation
should either be normalized off or become time-independent.
While the establishment above is a well-known problem in quantum mechanics,
I would request special attention to the precondition “the EM mode (hν) is stable so
its magnitude is constant”, as this is a critically important factor to stability of electron
pairs in crystals, as will be explained below.
For an EM wave mode of (m + 1/2)hν (with m=0,1,2…), if the number m of
photon fluctuates, the EM wave mode is no longer stable and Equation (5) no longer
valid; conversely, V0(r,t) in Equations (4) and (5) becomes a step function of time t,
the integral of Equation (2) becomes segmented, and the result of Equation (8) is no
longer a single term uniform over the entire range of integration (0,t) but becomes a
summation like
a nk1 ~ΣCjVnkj/h{exp[2πi(Enk+hν)(tj-tj-1)/h]-1}/ (Ep+hν)
-{exp[2πit(Enk-hν)(tj-tj-1)/h]-1}/ (Enk-hν) (9)
where the summation is over index j; in each time segment (tj-1,tj), the number m of
photon of the EM wave mode remains unchanged, but the number m assumes
2
3. different values in different time segments (tj-1,tj), and Cj will denote a random
complex number. Then, Equation (9) becomes a summation of a series of random
complex numbers, so the matrix element a nk1 including such a summation not only
cannot normalize off other matrix elements (a nk) but also goes to zero statistically. In
conclusion, when the photon number m of the EM mode fluctuates, transition of the
electron in the system no longer converges to Enk=±hν.
Let us now consider the energy relation of the system as shown in Fig. 1. Some
interpretation says that at the limit of t→∞ Equation (8) indicates that the electron is
exchanging phonon (photon) with the EM wave mode or the outside. But such an
interpretation is problematic. First, Equation (8) does not indicate the requirement of
real phonon emission/absorption. Second, at finite time t transitions other than
Enk=±hν at sufficiently low temperature (T→0) is allowable according to Equation
(8), but the EM mode and the outside environment will surely not provide such
phonon or photon, indicating that the transitions as indicated by Equation (8) do not
require real emission/absorption of phonon/photon. (Hereinbelow transition not
including real emission/absorption of quantum would be referred to as “virtual
transition”, and that including real emission/absorption of quantum as “real
transition”.)
As I understand it, virtual transition could be explained on two bases: 1) the
significance of “measurement”, and 2) the Heisenberg uncertainty principle.
“Measurement” is usually interpreted as an intervention to the system to be measured,
which makes the system “collapse” to an eigenstate of which the eigenvalue is the
result of the measurement. Apparently, virtual transitions seem not in conformity with
the requirement of energy conservation. But all energy terms we observe are
“observable”, some “non-observable” energy terms can get involved in a virtual
transition, and the relationship of energy conservation shall cover both “observable”
and “non-observable” terms of energy involved in the virtual transition concerned.
For example, the EM wave mode at its ground state (with m=0) still might “lend” a
threshold photon to the electron for transition from E1 to E2, and the electron then
could return the threshold photon to the EM wave mode in the subsequent transition
from E2 to E1. As such a “lend/return” process is transient, the phonon/energy
exchanges could be “non-observable”, because the system cannot collapse to a state in
which the EM mode has a negative number of photon like m= -1.
According to Heisenberg Uncertainty Principle, the energy of the electron in the
system of Fig. 1 has a spread ΔE, which is related to a lifetime Δt, by which the
system stays in a (stationary) state associated with an energy level, as
ΔEΔt≥h/(4π) (10)
It is important to note here that the energy spread ΔE is associated with the energy of
system (electron) concerned, not with an energy level a stationary state, and it is likely
that the system concerned is not “at” any stationary state at all.
Therefore, in the system as shown in Fig. 1, as long as the electron transits
between energy states of E1 and E2 at a transition frequency of νE with
νE≥4πν (11)
there will be
ΔE≥2(E2-E1) (11.1)
Which means the energy spread ΔE of an electron, which is “originally” at
stationary state of E1, is broad enough to cover the excited energy level E2, so the
electron may transit to E2 without actually absorbing any photon/phonon. In fact, since
the notion “transit” relates to “stationary state”, it becomes somewhat meaningless in
3
4. such a situation where no stationary state exists at all. Hereinbelow, the stabilized
non-stationary state, at which an electron has an energy spread ΔE covering at least
two stationary levels (as E1 and E2 shown in Fig. 1), is referred to as “non-stationary
stable (NSS) state”.
Jan Hilgevoord et al, at discussing ΔEΔt≥h/(4π), make a good comment:
“Evidently a point particle and a point of space are very different things.
Nevertheless they are not always clearly distinguished.” [3] We could make a
similar comment as: an energy eigenvalue and the energy of an electron are very
different things even if the electron is associated with the eigenstate of the energy
eigenvalue, so they are to be clearly distinguished. At a stationary state, an electron
may “steadily” stay at an energy eigenstate with a lifetime of Δt→∞, so its energy
uncertainty spread becomes ΔE→0; then, “the energy of the electron” equates to “the
energy eigenvalue” of the eigenstate. Under a non-stationary state, however, an
electron does stays at any single eigenstate and can be associated with a plurality of
energy eigenstates.
The electron system as shown in Fig. 1 differs from one in real crystals in that
the effect of wavevector selection rule is not taken into account. The wavevector
selection rule relates to the physical significance of phonon as the carrier of
electromagnetic interaction in crystals, just like photon is the carrier of
electromagnetic interaction. This interpretation can be made on the basis of the well-
discussed mathematical operation concerning transition in electron-phonon
interaction, as made in lattice representation by Huang [4], which is summarized in
English in Italic below due to its high relevance:
The first order approximation of potential variation δVn of the atom at the nth
lattice point Rn caused by a lattice wave mode is
δVn≈-μn•▽V(r-Rn) (7-86)
where V(r) is the potential of one atom, and
μn=Aecos2π(q•Rn-νt) (7-87)
represents displacement of the atom by the lattice wave mode, with e being the unit
vector in the wave direction, A being the magnitude of the lattice wave mode, ν being
the frequency of the lattice wave mode, and q being the wavevector of the lattice wave
mode under elastic wave approximation. Then, the potential variation of the entire
lattice is
ΔH=ΣδVn=-(A/2)exp(-2πiνt)Σexp(2πiq•Rn)e•▽V(r-Rn)
-(A/2)exp(2πiνt)Σexp(-2πiq•Rn)e•▽V(r-Rn) (7-88)
where the summation is over all the lattice points (n).
ΔH can be treated as a perturbation. With Equation (7-88), transition from k1 to
k2 has the energy relation of
E2(k2)=E1(k1)±hν (7-90)
The normalized wave function can be written as
Ψk(r)=1/(N)1/2exp(-2πikr)μk(r)
where N is the number of primitive cells in the limited crystal concerned. The matrix
element can be written as
(A/2)(e•RIkk’)[(1/N)Σexp{2πi(k1-k2±q)•Rn}] (7-91)
where the summation is over all the lattice points (n), and
Ikk’ =∫exp{-2πi(k2-k1)•ξ}μ*k’(ξ)μk(ξ)▽V(ξ)dξ (7-92)
Of special importance is the summation in the matrix element:
4
5. (1/N)Σexp{2πi(k1-k2±q)•Rn}
which yields
(1/N)Σexp{2πi(k1-k2±q)•Rn}=1 for
k1-k2±q=-Kn (7-93)
and zero otherwise.
The above presentation is given without introducing the concept of “phonon”;
the two key parameters-the wavevector q and frequency ν of the lattice wave mode-
are taken directly from the deduction of lattice wave modes based on lattice dynamics
[5] [6], where “phonon” is also not introduced.
The dispersion as arising in Equation (7-87) is the attribute of the lattice; it can
be seen from Equations (7-86) to (7-93) that each lattice wave mode endows its
attribute of dispersion, its wavevector q, to its carriers of electromagnetic interaction
(phonons), as particularly indicated by Equations (7-88) and (7-90) to (7-93).
Equations (7-86) to (7-90) indicate that phonon is the (energy) carrier of
electromagnetic interaction between electron and the atoms/ions of oscillating lattice.
Specifically, Equations (7-88), (7-91) and (7-93) also indicate that phonons of some
special lattice wave modes, the optical lattice wave modes, show no difference from
photons. So a photon can be understood as being a special form of phonon.
Moreover, we shall distinguish a phonon from a quantum of oscillation of the
lattice wave mode associating with the phonon. Here, again in analogy to the above
comment by Jan Hilgevoord et al, we should say that phonons are not their lattice
wave modes, although their lattice wave modes are often characterized by
them. As phonons necessarily relate to the time-dependent electric field of their
lattice wave modes, they are always associated with non-stationary process in
crystals.
An optical wave mode can interact with incident electromagnetic wave of the
same wavevector and frequency [7], so an optical phonon can escape its crystal as a
photon, while a non-optical phonon cannot. This important difference is identified as
a candidate water parting between HTS and LTS, as will be explained below.
Hereinbelow “optical phonon” refers to the phonon of an optical lattice wave mode,
“non-optical phonon” refers to the phonon of a non-optical lattice wave mode, and
“phonon” is used to cover both “optical phonon” and “non-optical phonon”.
Back to “measurement” of electrons, it is to be understood as involving not only
intervention by interaction between incident photons and electrons (as in AREPS or
the like) but also electron-phonon interactions in all “real transitions”, particularly the
transitions in the process of electric resistance; therefore, the “real transitions” are
also referred to as “measurement” hereinbelow. If an energy process cannot be
realized by human-performed “measurement”, it also cannot be realized by the
electron-phonon process in electric resistance mechanism. For the system as shown in
Fig. 1, assuming that the system is at ground state E1 at t=0, that the system is
isolated, and that the EM wave mode (hν) is in its ground state, then after time t1, the
energy of the electron can only be “measured” as E 1, as in conformity with the
requirement of energy conservation. But this does not mean that the electron keeps
staying at the eigenstate of E1 all the time, rather it just indicates that “measurement”
can only “collapse” the electron to the eigenstate of E1. Conversely, according to
quantum mechanics, the electron shall transit between the eigenstates of E 1 and E2
during the time period [0, t1], or it may be said that the electron is in an NSS state that
incorporating both energy eigenstates of E1 and E2.
5
6. The condition “the EM wave mode is stable” means “its magnitude (number of
phonons) remains unchanged”. But “number of phonons remains unchanged” can
hardly be ensured unless the lattice wave mode is at its ground state and the system
concerned is at sufficiently low temperature; only at the ground state of a lattice wave
mode can its number of phonons be reliably kept constant (zero). The higher the
frequency of the lattice wave mode is and/or the lower the temperature is, the more
likely and reliable that its phonon number is kept at constant zero.
Let us now consider the two electron system as shown in Fig. 2, which differs
from the system shown in Fig. 1 in that the excited state E2 originally has an electron
too. As I proposed in a previous paper [1], in the system of Fig. 2 “electron pairing by
virtual stimulated transitions” will happen as long as wavevectors k1 and k2 satisfy
wavevector selection rule indicated in Equation (7-93). As has been explained above,
the electron pairing in a system as shown in Fig. 2 is phonon-mediated in general. It
should be easier for the two electron system as shown in Fig. 2 to enter into an NSS
state than the one electron system as shown in Fig. 1, for the electron at the excited
state E2 may emit a threshold phonon, which will balance off the energy deficit as
apparent in the system of Fig. 1. However, once the two electrons in Fig. 2 are in NSS
state, the threshold phonon becomes redundant as far as the NSS state is maintained.
The fate of this threshold phonon is critical in deciding the binding energy of the
electron pairs in crystals and the superconducting attributes of the crystal concerned,
as will be explained shortly later.
For each state (E, k), its pairing candidate could be determined as the
intersections of laminated plot of hν-q dispersion curves [8] of lattice wave modes and
the plot of E-k bands, with the origin of the hν-q plot being placed at the (E, k) point;
for determining pairing candidates for the excited state in the pairing, the hν-q plot
should be placed upside-down. Obviously, each electron usually has more than one
matches of phonon-mediated electron pairing. The collection of all these matches
covers all possible (one phonon)-electron interactions of the subject electron. If all
these phonon-mediated electron pairs can “normally” become superconducting
carriers, HTS would be omnipresent, which is definitely not in conformity with the
rarity of HTS in reality.
Some exemplary scenarios of the phonon-mediated electron pairing by
stimulated transition are shown in Figs. 3 and 4, where exemplary pairings are
indicated by dotted or dashed lines with double arrows. As shown in Figs. 3 and 4, an
electron pairing typically occurs slantingly due to the dispersion of the mediating
phonon. But a few of the pairs are between nearly vertically separated electrons; these
are “optical phonon-mediated” (OPM) pairs, which are indicated by thick dashed lines
in Figs. 3 and 4. The optical threshold phonon of OPM pair of the longitudinal optical
(LO) branch should have the maximum frequency (νM) of all phonons in the crystal.
We now discuss the fate of the threshold phonon and its effect on binding
energy of its electron pair. As mentioned above, the electron originally at excited state
(E2) may emit a threshold phonon, which can be absorbed by the electron originally at
the ground state (E1), so the system of Fig. 2 can easily enter NSS state. Once the
system is in NSS state, the threshold phonon becomes redundant and can be absorbed
by the lattice wave mode (here the significance of distinguishing a phonon from its
lattice wave mode is seen.) But the phonon cannot be emitted to the outside of the
crystal, unless it is an optical phonon. After the phonon is absorbed by the lattice
wave mode, due to the stimulation of the EM wave mode associated with the lattice
wave mode, the phonon is easily taken back by one of the two electrons for that
6
7. electron to perform real transition, and the cycle restarts as the system begins to re-
establish NSS state. As explained above with reference to Equation (9), such frequent
exchanges of the threshold phonon between the lattice wave mode and the pair of
electrons destroy the dominance of matrix elements a 12 and a 21over other matrix
element components, which make the NSS state to collapse into the stationary energy
eigenstates, at which the threshold phonon has to be retrieved by the electron
collapsing to the excited state. As such, a non-optical phonon-mediated electron pair
would not be stable and could not become superconducting carriers.
But an optical phonon-mediated (OPM) pair is different in that the redundant
optical threshold phonon has a definite and substantial probability of escaping the
crystal by radiation, although it can also be absorbed with certain probability by the
lattice wave mode. When the optical lattice mode is at ground state and the
temperature is sufficiently low, once the threshold phonon escapes, the lattice wave
mode will surely be stable and, most notably, each of the two electrons can only
collapse to the ground state (E1) -this is the origin of binding energy of OPM pairs. As
the optical threshold phonon can escape without affecting the stability of the lattice
wave mode, the NSS state will not be affected by the escape of the threshold phonon
and will be maintained until the lattice wave mode received a new optical threshold
phonon, whence the new threshold phonon will allow one of two electrons to collapse
to the excited state. Therefore, the escape of the optical threshold phonon is self-
consistent.
Let us now further consider the fate of a redundant non-optical threshold
phonon. Lattice wave modes may couple with one another by anharmonic crystal
interactions [9] [10], by which a redundant threshold phonon may be taken away from
its lattice wave mode so that each of the two electrons in the pair can only collapse to
the ground state. This might be the origin of binding energy corresponding to low
temperature superconductivity (LTS). However, the probability with which the
threshold phonon escapes by anharmonic crystal interactions must have to compete
with the probability of occurrence of thermal noise phonon of the lattice wave mode.
Thus, even if escape of the threshold phonon by anharmonic crystal interactions can
win over occurrence of thermal noise phonon, non-optical phonon-mediated (NOPM)
electron pairs should be stabilized only below temperatures much lower than those for
OPM pairs. These may indicate that binding energy along may not decide
superconducting temperature (Tc), as the stability of electron pairs is subject to the
effects of a plurality of factors, including the strength of anharmonic crystal
interactions, the presence/absence of optical phonon-mediated pairing matches, and so
on. Of course, if escape of the threshold phonon by anharmonic crystal interactions
cannot win over occurrence of thermal noise phonon, the crystal will not have a
superconducting phase.
While “multiple pairing” is explained above as being common, additional
pairing between E1 and an energy level E3>E1 do not affect stability of pairing
between E1 and <E2 (E3 is not shown in the drawings). We are now explaining this.
Assuming that a third energy level E3 is present in the system shown in Fig. 2, with
E1<E3<E2, and E3 has unstable NOPM pairing with E1 while E2 has OPM pairing with
E1. Then, the matrix elements A13 and A31 will oscillate between 0 and a value having
a modulus less than one, and A12 and A21 will also oscillate, but this will not affect the
NSS states of the electrons on levels E1 and E2. This can be clearly seen from the
composition of the system, which has three electrons, a optical threshold phonon, and
a non-optical threshold phonon; the optical threshold phonon will escape soon or later,
7
8. then at sufficiently low temperature no new optical threshold phonon will enter the
system, so the two electrons originally at E1 and E2 can only stay at NSS states and
collapse to ground state E1 upon “being measured”, no matter what state the third
electron is in.
Now let us consider the fact that both electrons in a stabilized pair can only
collapse to the ground state (E1) upon “being measured”. Obviously, this means that
both electrons condensate to the ground state; the condensation here is in a non-
stationary static (NSS) state, which is a “measured” state, so condensation represents
a “measurement effect”; it does not indicate that the electrons are co-staying on the
stationary ground state (E1); conversely, the electrons are “staying” on a plurality of
stationary states including the original excited state (E2). Moreover, insofar that the
ground state may be the common lower state of multiple pairings as discussed above,
all electrons in these pairings will condensate to the common ground state (E 1) when
their pairs get stabilized.
The effect of an additional pairing between E1 and an energy level E4<E1 varies
(E4 is not shown in the drawings). First, we do not know the exact amount of energy
spread ΔE of any electron. However, if we assume ΔE approximately corresponds to
the energy of the threshold phonon, then the answer to the above question would be
NO in view of the limitation of Pauli Exclusion Principle. This can be explained that,
in the situation shown in Fig. 2, for example, if an electron at NSS state also pairs up
with an electron at E4<E1, its energy spread ΔE might no longer cover the level of E 2
so it could not keep its association with the eigenstate of E2. Thus, the two eigenstates
of E4 and E1 would be co-occupied by the two electrons originally at the states of E4
and E1, but the eigenstate of E1 must also be co-occupied by the electron originally at
E2 if the latter electron is to keep in NSS state, resulting in that the “degree of
occupancy” of eigenstate of E1 would exceed one, which is not in conformity with the
requirement of Pauli Exclusion Principle.
If this explanation is valid, candidate pairings having the same ground state (E1)
have to compete with all its “lower neighbors” (the candidate pairings with eigenstate
E1 being their excited state) to realized themselves. As each candidate pairing can be
characterized by its threshold phonon, whether the electron at a level (such as E1) is
“pairing upward” or “pairing downward” can be said to depend on the competition
between its “upper threshold phonon(s)” and “lower threshold phonon(s)”, with the
definite rule that if one of the “upper threshold phonons” wins then all the “upper
threshold phonons” win (and vise versa).
Obviously, the “upper threshold phonons win” outcome is pro-
superconductivity. It seems that the threshold phonon with greater energy (binding
energy) would have an edge, but magnitude of a matrix element depends on, among
other things, degree of coupling between the two states concerned, and anharmonic
crystal interactions may play a very important role. The question may be that whether
anyone of the upper threshold phonon can eventually dissolve itself into the lower
threshold phonon and something else (as T→0). If it can, the outcome will surely be
“upper threshold phonons win”; but it cannot, the situation could be more
complicated, and superconducting phase (if one exists) could possibly be unstable
and/or uncertain. So no general answer to this question can be given at this time,
except that an optical threshold phonon of LO mode, which corresponds to HTS and
escapes by simple and direct radiation with definite and substantial probability at a
very early stage of the temperature-decreasing process, will definitely win. On the
other hand, it may be likely that all electrons at or near EF cannot get any win in each
8
9. of their candidate pairings. If this happens, the crystal will never have a
superconducting phase.
As explained above, the collection of all candidate threshold phonons of an
electron corresponds to all possible phonon-electron interactions of the electron,
which is also all the candidate pairing options of the electron (except for levels above
EF). Thus, an electron system in lattice can be treated as a network, wherein each
eigenstate E=E(k) is a node with each of its candidate threshold phonons representing
one of its connecting edges, so the electron system can be treated with tools of graph
theory. All electrons in the system might not be in one single graph. And the
characteristic or attribute of the edges (threshold phonons or pairing state) will vary
with temperature, such as that some of them may become “directional” (i.e. it can
only pairing upward or downward, as discussed above) below certain temperature.
As a few more comments on virtual transitions, such as those discussed above
with reference to Fig. 1, a reasonable understanding seems that virtual transition is the
“normal” or “general” form of transition (at least for a system like that shown in Fig.
1) while real transitions are “abnormal” or “special”. In virtual transitions, a process
of virtual lending-returning of boson (phonon) happens in a continuous way while the
system is in a non-stationary stable state; when a real transition occurs, the “normal”
process of lending/returning of boson is interrupted and the system “temporarily”
collapses to an eigenstate; then virtual transitions take place again and the system
begins to re-establish its non-stationary stable state. So an eigenstate corresponds to a
transient process triggered by a real transition (at least in the time-dependent system
as shown in Fig. 1). In other words, like in a time-dependent system at low
temperature, real transitions and associated collapses to eigenstates are occasional
events happening on a continuous background of virtual transitions and non-stationary
stable state, like scattered small islands on a background of vast ocean, while at
sufficiently high temperatures such islands of real transition become so frequent that
they merge into a continent of “common” process of real transitions.
By far, I have
1) identified phonons as carriers of electromagnetic interaction between
electrons and lattice wave modes; the lattice wave modes act on electrons by the EM
wave modes they generate;
2) recognized a promising origin of binding energy of electron pairs in crystals
for high-temperature superconductivity, as relating to electron pairs mediated by
optical phonons, which may escape by direct radiation;
3) identified a possible origin of binding energy of electron pairs in crystals for
low-temperature superconductivity, as relating to electron pairs mediated by non-
optical phonons, which may escape by anharmonic crystal interactions;
4) established a picture of electron condensation, where all electrons in one or
more pairings condensate to their common ground state when their pair(s) gets
stabilized; the condensation is a non-stationary static (NSS) state, which is a
“measured” state, and represents a “measurement effect”;
5) identified a possible mechanism explaining why some crystals never have a
superconducting phase, as that all electrons at or near EF cannot get any win in pairing
competitions for each of their candidate pairings;
6) investigated some virtual behaviors (particularly relating to “measured”
energy) of one-electron system and two-electron system on the basis of Heisenberg
Uncertainty Principle, and identified non-stationary stable state of electrons engaging
9
10. in “electron pairing by virtual stimulated transitions” and its importance in
establishing superconductivity;
7) recognized the behavior and role of threshold phonon, released in non-
stationary stable state by electron from excited state; recognized the redundancy of the
threshold phonon in non-stationary stable state;
8) identified a mechanism by which the stability of lattice wave mode and/or
temperature affect the stability of electron pairs mediated by the phonon
corresponding to the lattice wave mode;
9) identified and discussed the situations and effects relating to “multiple-
pairing”; recognized “pairing competition” and “direction of pairing” and some
details of the mechanism in which the “direction of pairing” is determined; and
10) proposed that electron pairing, and the electron system in crystal generally,
may be treated with tools of graph theory, with each eigenstate E=E(k) in the system
being a node and each of its candidate threshold phonons representing one of its
connecting edges.
In summary, a promising origin of binding energy of electron pairs in crystals
has been recognized, and the identity of phonon as carrier of electromagnetic
interaction between electrons and lattice wave modes has been recognized. While
electron pairing is phonon-mediated in general, HTS should be associated with
electron pairing mediated by optical phonon at or near EF, so the rarity of HTS
corresponds to the rarity of such pairing match. The origin of binding energy of
electron pairs, and of superconductivity in general, may relate to some attributes of
physics as manifesting in virtual transitions and non-stationary stable process of an
electron system subjected to an EM wave mode.
References:
[1] “A mechanism of electron pairing relating to superconductivity”, Qiang LI,
http://www.paper.edu.cn/index.php/default/releasepaper/content/42033
[2] “Phonon”, http://en.wikipedia.org/wiki/Phonon
[3] “Time in Quantum Mechanics”, Jan Hilgevoord et al,
http://atkinson.fmns.rug.nl/public_html/Time_in_QM.pdf.
[4] “Solid State Physics”, by Prof. HUANG Kun, published (in Chinese) by People’s
Education Publication House, with a Unified Book Number of 13012.0220, a
publication date of June 1966, and a date of first print of January 1979, page
201-205.
[5] Pages 102-112 of [4].
[6] Kittel Charles Introduction To Solid State Physics 8Th Edition, pages 95-99 and
Figs, 7, 8(a) and 8(b) of Chapter 4.
[7] Fig. 5-9 and page 108 of [4].
[8] Figs, 7, 8(a), 8(b), and 11 of Chapter 4 of [6]
[9] Pages 119-120 of [6].
[10] Page 120-121 of [4].
10
11. E2(k2)
Stimulated transition of
electron
EM mode hν=E2- E1
E1(k1)
electron
Figure 1 A dual energy system of one electron subject to an electromagnetic (EM)
wave mode hν=E2-E1.
11
12. :
E2(k2)
Electron 2
Electron pairing
EM mode hν=E2- E1
E1(k1)
Electron 1
Figure 2 A dual energy system of two electrons subject to an electromagnetic (EM)
wave mode hν=E2-E1.
12
13. E= E(k) Band E2(k2)
EF
Pairing with
E2(k2)-E1(k1)~hνM Energy Gap
Pairing with
E2(k2)-E1(k1)<hνM Band gap
Peak
Band E1(k1)
k
Figure 3 An exemplary scenario of the phonon-mediated electron
pairing by stimulated transition
13
14. E= E(k) E1(k1) E2(k2)
EF
wi
Gap
Peak
Pairing
k
Figure 4 Another exemplary scenario of the phonon-mediated electron pairing
by stimulated transition
14