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.
To find the susceptibility arising due to water in the solution of MnCl2 , ionic molecular susceptibility ,magnetic moment of the Mn++ using quinche's Method
Branislav K. Nikoli
ć
Department of Physics and Astronomy, University of Delaware, U.S.A.
PHYS 624: Introduction to Solid State Physics
http://www.physics.udel.edu/~bnikolic/teaching/phys624/phys624.html
The integral & fractional quantum hall effectSUDIPTO DAS
Introductory idea of integral & fractional quantum hall effect and by imposing the idea of composite fermions showing the existence of fractional charge.
Optical band gap measurement by diffuse reflectance spectroscopy (drs)Sajjad Ullah
Introduction to Optical band gap measurement
by electronic spectroscopy and diffuse reflectance spectroscopy (DRS) with comparison of the results obtained suing different equation and measurement techniques.
The role of scattering in extinction of light as it passes through media is briefly discussed.
A presentation on White Light Upconversion Emissions from Tm3+ + Ho3+ + Yb 3+ Codoped Tellurite and Germanate Glasses on Excitation with 798 nm Radiation made by Deepak Rajput. It was presented as a course requirement at the University of Tennessee Space Institute.
An introduction to the fundamental physics of transparent conducting oxides including a review of the electrical and optical properties of common materials.
This document introduces nano-materials and discusses their properties and applications. It defines nano-materials as low-dimensional semiconductor structures between a few nanometers to tens of nanometers in size, including quantum wells, wires, and dots. Electron behavior changes from plane waves in bulk semiconductors to quantized energy levels in nano-structures. Nano-materials are of interest because they combine advantages of semiconductors and atomic systems by allowing controllable electron confinement. Common fabrication methods include top-down patterning and bottom-up self-assembly. Nano-materials exhibit properties like ballistic transport, tunneling, and discrete optical transitions useful for applications in lasers, detectors, and other optoelectronic devices
To find the susceptibility arising due to water in the solution of MnCl2 , ionic molecular susceptibility ,magnetic moment of the Mn++ using quinche's Method
Branislav K. Nikoli
ć
Department of Physics and Astronomy, University of Delaware, U.S.A.
PHYS 624: Introduction to Solid State Physics
http://www.physics.udel.edu/~bnikolic/teaching/phys624/phys624.html
The integral & fractional quantum hall effectSUDIPTO DAS
Introductory idea of integral & fractional quantum hall effect and by imposing the idea of composite fermions showing the existence of fractional charge.
Optical band gap measurement by diffuse reflectance spectroscopy (drs)Sajjad Ullah
Introduction to Optical band gap measurement
by electronic spectroscopy and diffuse reflectance spectroscopy (DRS) with comparison of the results obtained suing different equation and measurement techniques.
The role of scattering in extinction of light as it passes through media is briefly discussed.
A presentation on White Light Upconversion Emissions from Tm3+ + Ho3+ + Yb 3+ Codoped Tellurite and Germanate Glasses on Excitation with 798 nm Radiation made by Deepak Rajput. It was presented as a course requirement at the University of Tennessee Space Institute.
An introduction to the fundamental physics of transparent conducting oxides including a review of the electrical and optical properties of common materials.
This document introduces nano-materials and discusses their properties and applications. It defines nano-materials as low-dimensional semiconductor structures between a few nanometers to tens of nanometers in size, including quantum wells, wires, and dots. Electron behavior changes from plane waves in bulk semiconductors to quantized energy levels in nano-structures. Nano-materials are of interest because they combine advantages of semiconductors and atomic systems by allowing controllable electron confinement. Common fabrication methods include top-down patterning and bottom-up self-assembly. Nano-materials exhibit properties like ballistic transport, tunneling, and discrete optical transitions useful for applications in lasers, detectors, and other optoelectronic devices
Photonic crystals are periodic dielectric structures that have a band gap that forbids propagation of a certain frequency range of light. This property enables one to control light with amazing facility and produce effects that are impossible with conventional optics.Photonic crystals can be fabricated for one, two, or three dimensions. One-dimensional photonic crystals can be made of layers deposited or stuck together. Two-dimensional ones can be made by photolithography, or by drilling holes in a suitable substrate. Fabrication methods for three-dimensional ones include drilling under different angles, stacking multiple 2-D layers on top of each other, direct laser writing, or, for example, instigating self-assembly of spheres in a matrix and dissolving the spheres
The document summarizes the synthesis of tungsten disulfide (WS2) nanosheets using chemical vapor deposition (CVD) with a gas-phase sulfur reactant. It is observed that the WS2 grows laterally and that domains coalesce over time. The number of layers can be controlled by adjusting the reaction time. Additionally, a graphene/WS2 heterostructure is shown to have properties suitable for photo detection.
This document discusses key characteristics of optical fibers that affect their performance as a transmission medium. It describes how wavelength, frequency, reflection, refraction, polarization, and attenuation properties influence fiber optic communication. Specific bands used in optical fibers, including O, C, E, S and L bands, are defined. The document also examines intrinsic and extrinsic factors contributing to fiber attenuation, as well as dispersion which limits bandwidth by spreading out light pulses over time as they travel through the fiber.
This document provides an overview of nonlinear optics and second harmonic generation. It begins with an introduction to lasers and their components. It then discusses symmetry operations in crystals and how centrosymmetric and noncentrosymmetric materials affect nonlinear polarization. Maxwell's equations are presented for linear media. The document introduces nonlinear optics and lists various nonlinear optical effects such as second harmonic generation. It derives the wave equation for nonlinear media and shows how second harmonic generation leads to frequency doubling. Examples of nonlinear crystals used for second harmonic generation are also provided.
The document discusses Mott physics and the metal-insulator transition. It introduces concepts such as Fermi liquids in metals, Mott insulators arising from electron-electron interactions, and the competition between kinetic energy and interaction energy leading to a Mott transition from metal to insulator with increasing interaction strength. It also distinguishes between Slater insulators driven by antiferromagnetism and Mott insulators where insulating behavior does not require magnetic ordering.
Heterostructures, HBTs and Thyristors : Exploring the "different"Shuvan Prashant
The document discusses heterostructures, heterojunction bipolar transistors (HBTs), and thyristors. It begins by explaining homojunctions and heterojuctions, how they differ in material composition and resulting energy band structures. It then describes HBTs, noting they can achieve higher speeds than bipolar junction transistors (BJTs) due to reduced injection of minority carriers into the emitter. Finally, it discusses thyristors, four-layer pnpn semiconductor devices that can operate in either conducting or blocking states, and diacs, bidirectional thyristor variants used in alternating current switching applications.
The document summarizes research on the magnetic properties and magnetocaloric effect of two materials: La2NiMnO6 nanocrystals and a single crystal of La1.2Sr1.8Mn2O7. For both materials, the document examines structural properties, magnetic phase transitions, critical behavior near the Curie temperature, and magnetocaloric effects. Key results include determining the materials undergo second-order phase transitions and exhibit short-range ferromagnetic order. The magnetocaloric effect is also investigated through measurements of magnetic entropy change and development of universal curves for both materials.
This presentation include :
1. Multiferroic
2. Possible cross- couplings
3. History and Applications
4. Ferromagnetic nature
5. Ferroelectric nature
6. Piezoelectric and Subgroups
7. Perovskite structure and Perovskite based multiferroics
8. My future work on particular type of multiferroic material .............
This document provides an overview of quantum electrodynamics (QED). It begins by discussing cross sections and the scattering matrix, defining cross section as the effective size of target particles. It then derives an expression for cross section in terms of the transition rate and flux of incident particles. Next, it summarizes the derivation of the differential cross section and decay rate formulas in QED using relativistic quantum field theory and Feynman diagrams. It concludes by briefly reviewing the historical development of QED and the equivalence of the propagator approach and other formulations.
The document discusses various methods and examples of intercalating guest species such as polymers into host lattices like layered clay materials. It describes how intercalation can lead to intercalated or exfoliated nano-composite structures depending on factors like the degree of interlayer spacing expansion. Examples discussed include polyurethane intercalated into montmorillonite clay, acrylamide polymer exfoliated into montmorillonite layers, and polyaniline intercalated into layered double hydroxides. Applications mentioned are in potentiometric sensors, energy storage, sensors, actuators and transistors.
This document discusses solitons in optical fiber communication. It begins with an introduction to solitons as pulses that maintain their shape despite dispersion and nonlinearities. The history of discovering solitons in fiber optics is described, including key experiments in the 1980s and 1990s that demonstrated their use for long-distance, high-capacity data transmission. The document outlines how solitons form in fibers due to a balance between dispersion and the Kerr effect. It describes the properties and equations that characterize fundamental and higher-order soliton pulses. Parameters like dispersion length and peak power are also defined. Finally, the document discusses optimizing soliton width and spacing for high bit rates.
1) Schrödinger's thought experiment involves placing a cat in a box with a device that may release poison, triggered by the radioactive decay of an atom over 10 minutes.
2) According to quantum mechanics, the cat is in a superposition of being both alive and dead until the box is opened and the wave function collapses.
3) The paradox illustrates how quantum mechanics describes reality in terms of a wave function that collapses into a definite outcome only when observed, raising questions about the role of consciousness.
Graphene is a one-atom-thick planar sheet of sp2-bonded carbon atoms that are densely packed in a honeycomb crystal lattice
The name ‘graphene’ comes from graphite + -ene = graphene
This document discusses nonlinear optics and the dynamical Berry phase. It introduces nonlinear optics and summarizes early experiments. It then discusses how the Berry phase is related to nonlinear optical effects like second harmonic generation (SHG). Computational methods are presented for calculating SHG and other nonlinear optical properties from first principles using time-dependent density functional theory and the dynamical Berry phase. Examples of applying these methods to study SHG in semiconductors are provided.
Perovskite solar cells are a promising photovoltaic technology that has seen rapid increases in efficiency from 3.8% in 2009 to 19.3% in 2014. Perovskites have a unique crystal structure and can be prepared through various methods like spin coating and inkjet printing. They offer benefits such as high absorption, tunable bandgaps, and flexibility. However, challenges remain around stability issues from oxygen, moisture, UV light and heat that can be addressed through material engineering and encapsulation. With further research into replacing lead and improving stability, perovskite solar cells have the potential to become a leading solar technology of the future.
1. Drude's classical theory of electrical conduction models a metal as composed of stationary ions and free-moving valence electrons. Electrons move randomly between collisions with ions or other electrons.
2. The drift velocity of electrons in an electric field is proportional to the field strength, resulting in a net current of electrons. This explains metals' conductivity.
3. However, the classical model fails to fully explain experimental observations such as the temperature dependence of resistivity and heat capacity. Quantum mechanics provides a more accurate description of electron behavior in metals.
Metals are opaque and highly reflective. They reflect most visible light due to their continuous empty electron states that allow electrons to absorb and re-emit light of the same wavelength. Silver reflects almost all light wavelengths, while gold's color is due to some light photons not being reemitted visibly. Nonmetals may absorb, reflect, refract, or transmit light depending on their electron band structure and properties like index of refraction.
This document summarizes Ankit Master's final presentation on microwave components. It describes several types of couplers - branchline, Wilkinson, modified Wilkinson, and ratrace couplers. It also discusses the design and measurement results of a gain block, low noise amplifier, and oscillator. Measurements of the S-parameters and other specifications are provided to analyze the performance of each circuit.
NANO106 is UCSD Department of NanoEngineering's core course on crystallography of materials taught by Prof Shyue Ping Ong. For more information, visit the course wiki at http://nano106.wikispaces.com.
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.
Photonic crystals are periodic dielectric structures that have a band gap that forbids propagation of a certain frequency range of light. This property enables one to control light with amazing facility and produce effects that are impossible with conventional optics.Photonic crystals can be fabricated for one, two, or three dimensions. One-dimensional photonic crystals can be made of layers deposited or stuck together. Two-dimensional ones can be made by photolithography, or by drilling holes in a suitable substrate. Fabrication methods for three-dimensional ones include drilling under different angles, stacking multiple 2-D layers on top of each other, direct laser writing, or, for example, instigating self-assembly of spheres in a matrix and dissolving the spheres
The document summarizes the synthesis of tungsten disulfide (WS2) nanosheets using chemical vapor deposition (CVD) with a gas-phase sulfur reactant. It is observed that the WS2 grows laterally and that domains coalesce over time. The number of layers can be controlled by adjusting the reaction time. Additionally, a graphene/WS2 heterostructure is shown to have properties suitable for photo detection.
This document discusses key characteristics of optical fibers that affect their performance as a transmission medium. It describes how wavelength, frequency, reflection, refraction, polarization, and attenuation properties influence fiber optic communication. Specific bands used in optical fibers, including O, C, E, S and L bands, are defined. The document also examines intrinsic and extrinsic factors contributing to fiber attenuation, as well as dispersion which limits bandwidth by spreading out light pulses over time as they travel through the fiber.
This document provides an overview of nonlinear optics and second harmonic generation. It begins with an introduction to lasers and their components. It then discusses symmetry operations in crystals and how centrosymmetric and noncentrosymmetric materials affect nonlinear polarization. Maxwell's equations are presented for linear media. The document introduces nonlinear optics and lists various nonlinear optical effects such as second harmonic generation. It derives the wave equation for nonlinear media and shows how second harmonic generation leads to frequency doubling. Examples of nonlinear crystals used for second harmonic generation are also provided.
The document discusses Mott physics and the metal-insulator transition. It introduces concepts such as Fermi liquids in metals, Mott insulators arising from electron-electron interactions, and the competition between kinetic energy and interaction energy leading to a Mott transition from metal to insulator with increasing interaction strength. It also distinguishes between Slater insulators driven by antiferromagnetism and Mott insulators where insulating behavior does not require magnetic ordering.
Heterostructures, HBTs and Thyristors : Exploring the "different"Shuvan Prashant
The document discusses heterostructures, heterojunction bipolar transistors (HBTs), and thyristors. It begins by explaining homojunctions and heterojuctions, how they differ in material composition and resulting energy band structures. It then describes HBTs, noting they can achieve higher speeds than bipolar junction transistors (BJTs) due to reduced injection of minority carriers into the emitter. Finally, it discusses thyristors, four-layer pnpn semiconductor devices that can operate in either conducting or blocking states, and diacs, bidirectional thyristor variants used in alternating current switching applications.
The document summarizes research on the magnetic properties and magnetocaloric effect of two materials: La2NiMnO6 nanocrystals and a single crystal of La1.2Sr1.8Mn2O7. For both materials, the document examines structural properties, magnetic phase transitions, critical behavior near the Curie temperature, and magnetocaloric effects. Key results include determining the materials undergo second-order phase transitions and exhibit short-range ferromagnetic order. The magnetocaloric effect is also investigated through measurements of magnetic entropy change and development of universal curves for both materials.
This presentation include :
1. Multiferroic
2. Possible cross- couplings
3. History and Applications
4. Ferromagnetic nature
5. Ferroelectric nature
6. Piezoelectric and Subgroups
7. Perovskite structure and Perovskite based multiferroics
8. My future work on particular type of multiferroic material .............
This document provides an overview of quantum electrodynamics (QED). It begins by discussing cross sections and the scattering matrix, defining cross section as the effective size of target particles. It then derives an expression for cross section in terms of the transition rate and flux of incident particles. Next, it summarizes the derivation of the differential cross section and decay rate formulas in QED using relativistic quantum field theory and Feynman diagrams. It concludes by briefly reviewing the historical development of QED and the equivalence of the propagator approach and other formulations.
The document discusses various methods and examples of intercalating guest species such as polymers into host lattices like layered clay materials. It describes how intercalation can lead to intercalated or exfoliated nano-composite structures depending on factors like the degree of interlayer spacing expansion. Examples discussed include polyurethane intercalated into montmorillonite clay, acrylamide polymer exfoliated into montmorillonite layers, and polyaniline intercalated into layered double hydroxides. Applications mentioned are in potentiometric sensors, energy storage, sensors, actuators and transistors.
This document discusses solitons in optical fiber communication. It begins with an introduction to solitons as pulses that maintain their shape despite dispersion and nonlinearities. The history of discovering solitons in fiber optics is described, including key experiments in the 1980s and 1990s that demonstrated their use for long-distance, high-capacity data transmission. The document outlines how solitons form in fibers due to a balance between dispersion and the Kerr effect. It describes the properties and equations that characterize fundamental and higher-order soliton pulses. Parameters like dispersion length and peak power are also defined. Finally, the document discusses optimizing soliton width and spacing for high bit rates.
1) Schrödinger's thought experiment involves placing a cat in a box with a device that may release poison, triggered by the radioactive decay of an atom over 10 minutes.
2) According to quantum mechanics, the cat is in a superposition of being both alive and dead until the box is opened and the wave function collapses.
3) The paradox illustrates how quantum mechanics describes reality in terms of a wave function that collapses into a definite outcome only when observed, raising questions about the role of consciousness.
Graphene is a one-atom-thick planar sheet of sp2-bonded carbon atoms that are densely packed in a honeycomb crystal lattice
The name ‘graphene’ comes from graphite + -ene = graphene
This document discusses nonlinear optics and the dynamical Berry phase. It introduces nonlinear optics and summarizes early experiments. It then discusses how the Berry phase is related to nonlinear optical effects like second harmonic generation (SHG). Computational methods are presented for calculating SHG and other nonlinear optical properties from first principles using time-dependent density functional theory and the dynamical Berry phase. Examples of applying these methods to study SHG in semiconductors are provided.
Perovskite solar cells are a promising photovoltaic technology that has seen rapid increases in efficiency from 3.8% in 2009 to 19.3% in 2014. Perovskites have a unique crystal structure and can be prepared through various methods like spin coating and inkjet printing. They offer benefits such as high absorption, tunable bandgaps, and flexibility. However, challenges remain around stability issues from oxygen, moisture, UV light and heat that can be addressed through material engineering and encapsulation. With further research into replacing lead and improving stability, perovskite solar cells have the potential to become a leading solar technology of the future.
1. Drude's classical theory of electrical conduction models a metal as composed of stationary ions and free-moving valence electrons. Electrons move randomly between collisions with ions or other electrons.
2. The drift velocity of electrons in an electric field is proportional to the field strength, resulting in a net current of electrons. This explains metals' conductivity.
3. However, the classical model fails to fully explain experimental observations such as the temperature dependence of resistivity and heat capacity. Quantum mechanics provides a more accurate description of electron behavior in metals.
Metals are opaque and highly reflective. They reflect most visible light due to their continuous empty electron states that allow electrons to absorb and re-emit light of the same wavelength. Silver reflects almost all light wavelengths, while gold's color is due to some light photons not being reemitted visibly. Nonmetals may absorb, reflect, refract, or transmit light depending on their electron band structure and properties like index of refraction.
This document summarizes Ankit Master's final presentation on microwave components. It describes several types of couplers - branchline, Wilkinson, modified Wilkinson, and ratrace couplers. It also discusses the design and measurement results of a gain block, low noise amplifier, and oscillator. Measurements of the S-parameters and other specifications are provided to analyze the performance of each circuit.
NANO106 is UCSD Department of NanoEngineering's core course on crystallography of materials taught by Prof Shyue Ping Ong. For more information, visit the course wiki at http://nano106.wikispaces.com.
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.
The document summarizes the theoretical framework for studying quantum resonances in diatomic molecules using the Born-Oppenheimer approximation. It considers the Hamiltonian for a diatomic molecule, with coordinates for the two nuclei and one electron. By fixing angular momentum and applying a rotation, the problem can be reduced to studying an effective one-dimensional Hamiltonian as a function of the internuclear distance R. Under certain assumptions about the electronic eigenvalues and effective potentials, the Hamiltonian takes the form of particle in overlapping potential wells, setting up the problem of studying resonances between the wells.
Quantum chemistry is the application of quantum mechanics to solve problems in chemistry. It has been widely used in different branches of chemistry including physical chemistry, organic chemistry, analytical chemistry, and inorganic chemistry. The time-independent Schrödinger equation is central to quantum chemistry and can be used to model chemical systems like the particle in a box, harmonic oscillator, and hydrogen atom. Molecular orbital theory is also important in quantum chemistry for describing chemical bonding in molecules.
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 approach 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. 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.
This document provides an overview of density functional theory and methods for modeling strongly correlated materials. It discusses the limitations of standard DFT approaches like LDA for strongly correlated systems and introduces model Hamiltonians and correction methods like LDA+U, LDA+DMFT, self-interaction correction, and generalized transition state to better account for electron correlation effects. The document outlines the basic theory and approximations of DFT, including Kohn-Sham equations and the local density approximation, and discusses basis set approaches like plane waves, augmented plane waves, and pseudopotentials.
Ultracold atoms in superlattices as quantum simulators for a spin ordering mo...Alexander Decker
This document discusses using ultracold fermionic atoms in optical lattices to simulate spin ordering models. It begins by describing how atoms can be trapped in optical lattices using laser light. It then proposes how a spin ordering Hamiltonian could be used to achieve superexchange interaction in a double well system. Finally, it suggests going beyond double wells to study resonating valence bond states in a kagome lattice, which could provide insights into phenomena like high-temperature superconductivity.
This document discusses atomic structure, beginning with the hydrogen atom and one-electron atoms. It then discusses the Hamiltonian and solutions of the Schrodinger equation for these systems. It introduces quantum numbers and describes the orbitals and energy levels. For polyelectronic atoms, it discusses separating the Schrodinger equation and introduces Hartree-Fock self-consistent field approximations. It describes Slater determinants which satisfy the Pauli exclusion principle for many-electron wavefunctions.
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.
Modern theory of magnetism in metals and alloysSpringer
This document provides an introduction to magnetism in solids. It discusses how magnetic moments originate from electron spin and orbital angular momentum at the atomic level. In solids, electron localization determines whether magnetic properties are described by localized atomic moments or collective behavior of delocalized electrons. The key concepts of metals and insulators are introduced. The document then presents the basic Hamiltonian used to describe magnetism in solids, including terms for kinetic energy, electron-electron interactions, spin-orbit coupling, and the Zeeman effect. It also discusses how atomic orbitals can be used as a basis set to represent the Hamiltonian and describes the symmetry properties of s, p, and d orbitals in cubic crystals.
02 - Ab initio Methods converted into wwWalidHarb2
This document summarizes different computational chemistry methods, including ab initio methods that are derived directly from theoretical principles without experimental data. It describes the Hartree-Fock approximation and how it breaks the many-electron Schrodinger equation into simpler one-electron equations. It also discusses the limitations of Hartree-Fock in accounting for electron correlation and summarizes post-Hartree-Fock methods like Moller-Plesset perturbation theory and configuration interaction that include electron correlation.
This document presents a framework for determining the rotational-vibrational spectra and stability of diatomic molecules using quantum mechanics. It summarizes:
1) The framework models the diatomic molecule using a Hamiltonian that accounts for the kinetic energy of the nuclei and electrons, as well as the potential energies between nuclei, electrons, and nuclei-electron interactions.
2) It approximates the molecule as rigid to simplify the problem, treating nuclear motion as a perturbation. This allows separating the wavefunction into electronic and nuclear components.
3) It then solves for rotational states using a rigid rotor model that describes nuclear rotation, yielding energy levels dependent on the angular momentum quantum number. This provides insights into diatomic molecular stability and
The document discusses ab initio molecular dynamics simulation methods. It begins by introducing molecular dynamics and Monte Carlo simulations using empirical potentials. It then describes limitations of empirical potentials and the need for ab initio molecular dynamics which calculates the potential from quantum mechanics. The document outlines several ab initio molecular dynamics methods including Ehrenfest molecular dynamics, Born-Oppenheimer molecular dynamics, and Car-Parrinello molecular dynamics. It provides details on how these methods treat the quantum mechanical potential and classical nuclear motion.
(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.
This presentation is the introduction to Density Functional Theory, an essential computational approach used by Physicist and Quantum Chemist to study Solid State matter.
Sergey seriy thomas fermi-dirac theorySergey Seriy
This document presents modern ab-initio calculations based on Thomas-Fermi-Dirac theory with quantum, correlation, and multishell corrections. It summarizes extensions made to the statistical model by including additional energy terms to account for quantum corrections, exchange energy, and correlation energy. This leads to a quantum- and correlation-corrected Thomas-Fermi-Dirac equation involving a density term, kinetic energy density term, and modified potential function. Solving this quartic equation in the electron density provides a way to determine the electron density distribution as a function of distance from the nucleus.
Microbial interaction
Microorganisms interacts with each other and can be physically associated with another organisms in a variety of ways.
One organism can be located on the surface of another organism as an ectobiont or located within another organism as endobiont.
Microbial interaction may be positive such as mutualism, proto-cooperation, commensalism or may be negative such as parasitism, predation or competition
Types of microbial interaction
Positive interaction: mutualism, proto-cooperation, commensalism
Negative interaction: Ammensalism (antagonism), parasitism, predation, competition
I. Mutualism:
It is defined as the relationship in which each organism in interaction gets benefits from association. It is an obligatory relationship in which mutualist and host are metabolically dependent on each other.
Mutualistic relationship is very specific where one member of association cannot be replaced by another species.
Mutualism require close physical contact between interacting organisms.
Relationship of mutualism allows organisms to exist in habitat that could not occupied by either species alone.
Mutualistic relationship between organisms allows them to act as a single organism.
Examples of mutualism:
i. Lichens:
Lichens are excellent example of mutualism.
They are the association of specific fungi and certain genus of algae. In lichen, fungal partner is called mycobiont and algal partner is called
II. Syntrophism:
It is an association in which the growth of one organism either depends on or improved by the substrate provided by another organism.
In syntrophism both organism in association gets benefits.
Compound A
Utilized by population 1
Compound B
Utilized by population 2
Compound C
utilized by both Population 1+2
Products
In this theoretical example of syntrophism, population 1 is able to utilize and metabolize compound A, forming compound B but cannot metabolize beyond compound B without co-operation of population 2. Population 2is unable to utilize compound A but it can metabolize compound B forming compound C. Then both population 1 and 2 are able to carry out metabolic reaction which leads to formation of end product that neither population could produce alone.
Examples of syntrophism:
i. Methanogenic ecosystem in sludge digester
Methane produced by methanogenic bacteria depends upon interspecies hydrogen transfer by other fermentative bacteria.
Anaerobic fermentative bacteria generate CO2 and H2 utilizing carbohydrates which is then utilized by methanogenic bacteria (Methanobacter) to produce methane.
ii. Lactobacillus arobinosus and Enterococcus faecalis:
In the minimal media, Lactobacillus arobinosus and Enterococcus faecalis are able to grow together but not alone.
The synergistic relationship between E. faecalis and L. arobinosus occurs in which E. faecalis require folic acid
Mechanisms and Applications of Antiviral Neutralizing Antibodies - Creative B...Creative-Biolabs
Neutralizing antibodies, pivotal in immune defense, specifically bind and inhibit viral pathogens, thereby playing a crucial role in protecting against and mitigating infectious diseases. In this slide, we will introduce what antibodies and neutralizing antibodies are, the production and regulation of neutralizing antibodies, their mechanisms of action, classification and applications, as well as the challenges they face.
BIRDS DIVERSITY OF SOOTEA BISWANATH ASSAM.ppt.pptxgoluk9330
Ahota Beel, nestled in Sootea Biswanath Assam , is celebrated for its extraordinary diversity of bird species. This wetland sanctuary supports a myriad of avian residents and migrants alike. Visitors can admire the elegant flights of migratory species such as the Northern Pintail and Eurasian Wigeon, alongside resident birds including the Asian Openbill and Pheasant-tailed Jacana. With its tranquil scenery and varied habitats, Ahota Beel offers a perfect haven for birdwatchers to appreciate and study the vibrant birdlife that thrives in this natural refuge.
Signatures of wave erosion in Titan’s coastsSérgio Sacani
The shorelines of Titan’s hydrocarbon seas trace flooded erosional landforms such as river valleys; however, it isunclear whether coastal erosion has subsequently altered these shorelines. Spacecraft observations and theo-retical models suggest that wind may cause waves to form on Titan’s seas, potentially driving coastal erosion,but the observational evidence of waves is indirect, and the processes affecting shoreline evolution on Titanremain unknown. No widely accepted framework exists for using shoreline morphology to quantitatively dis-cern coastal erosion mechanisms, even on Earth, where the dominant mechanisms are known. We combinelandscape evolution models with measurements of shoreline shape on Earth to characterize how differentcoastal erosion mechanisms affect shoreline morphology. Applying this framework to Titan, we find that theshorelines of Titan’s seas are most consistent with flooded landscapes that subsequently have been eroded bywaves, rather than a uniform erosional process or no coastal erosion, particularly if wave growth saturates atfetch lengths of tens of kilometers.
ESA/ACT Science Coffee: Diego Blas - Gravitational wave detection with orbita...Advanced-Concepts-Team
Presentation in the Science Coffee of the Advanced Concepts Team of the European Space Agency on the 07.06.2024.
Speaker: Diego Blas (IFAE/ICREA)
Title: Gravitational wave detection with orbital motion of Moon and artificial
Abstract:
In this talk I will describe some recent ideas to find gravitational waves from supermassive black holes or of primordial origin by studying their secular effect on the orbital motion of the Moon or satellites that are laser ranged.
PPT on Sustainable Land Management presented at the three-day 'Training and Validation Workshop on Modules of Climate Smart Agriculture (CSA) Technologies in South Asia' workshop on April 22, 2024.
Authoring a personal GPT for your research and practice: How we created the Q...Leonel Morgado
Thematic analysis in qualitative research is a time-consuming and systematic task, typically done using teams. Team members must ground their activities on common understandings of the major concepts underlying the thematic analysis, and define criteria for its development. However, conceptual misunderstandings, equivocations, and lack of adherence to criteria are challenges to the quality and speed of this process. Given the distributed and uncertain nature of this process, we wondered if the tasks in thematic analysis could be supported by readily available artificial intelligence chatbots. Our early efforts point to potential benefits: not just saving time in the coding process but better adherence to criteria and grounding, by increasing triangulation between humans and artificial intelligence. This tutorial will provide a description and demonstration of the process we followed, as two academic researchers, to develop a custom ChatGPT to assist with qualitative coding in the thematic data analysis process of immersive learning accounts in a survey of the academic literature: QUAL-E Immersive Learning Thematic Analysis Helper. In the hands-on time, participants will try out QUAL-E and develop their ideas for their own qualitative coding ChatGPT. Participants that have the paid ChatGPT Plus subscription can create a draft of their assistants. The organizers will provide course materials and slide deck that participants will be able to utilize to continue development of their custom GPT. The paid subscription to ChatGPT Plus is not required to participate in this workshop, just for trying out personal GPTs during it.
PPT on Alternate Wetting and Drying presented at the three-day 'Training and Validation Workshop on Modules of Climate Smart Agriculture (CSA) Technologies in South Asia' workshop on April 22, 2024.
Evidence of Jet Activity from the Secondary Black Hole in the OJ 287 Binary S...Sérgio Sacani
Wereport the study of a huge optical intraday flare on 2021 November 12 at 2 a.m. UT in the blazar OJ287. In the binary black hole model, it is associated with an impact of the secondary black hole on the accretion disk of the primary. Our multifrequency observing campaign was set up to search for such a signature of the impact based on a prediction made 8 yr earlier. The first I-band results of the flare have already been reported by Kishore et al. (2024). Here we combine these data with our monitoring in the R-band. There is a big change in the R–I spectral index by 1.0 ±0.1 between the normal background and the flare, suggesting a new component of radiation. The polarization variation during the rise of the flare suggests the same. The limits on the source size place it most reasonably in the jet of the secondary BH. We then ask why we have not seen this phenomenon before. We show that OJ287 was never before observed with sufficient sensitivity on the night when the flare should have happened according to the binary model. We also study the probability that this flare is just an oversized example of intraday variability using the Krakow data set of intense monitoring between 2015 and 2023. We find that the occurrence of a flare of this size and rapidity is unlikely. In machine-readable Tables 1 and 2, we give the full orbit-linked historical light curve of OJ287 as well as the dense monitoring sample of Krakow.
Evidence of Jet Activity from the Secondary Black Hole in the OJ 287 Binary S...
M sc sem iv u iii
1. 4PHY-3(ii) : CONDENSED MATTER PHYSICS-II
Unit-III
Interacting electron gas, Hartee & Hartee-Fock approximation, Correlation energy, Screening,
dielectric function, Thomas-Fermi and Lindhard Theory, Frequency dependent Lindhard
screening, Screening of Hartee-Fock approximation. Introduction of Fermi Liquid Theory.
Book Referred :
Introduction to Solid State Theory
By Otfried Madelung
Solid State Physics
By N W Ashcroft
&
Introduction to Solid State Physics
By C. Kittel
2. 3.1 Many-Particle Hamiltonians
The Schrodinger equation for a system with more than one particle is
..............1i H
t
where H and ψ are the many-particle Hamiltonian and wave function, respectively.
For a set of
non interacting particles, the Hamiltonian of the system is
2
2
( ) .........2
2
i i ii
H H V r
m
where the independent-particle solution ψn for each Hi satisfies
2
2
( ) .............3
2
i i n n nV r E
m
Now once we consider interaction between electron and electron, lattice ions and
electrons, with equal positive and negative charge. The motion of any particle is now
correlated with all other particles, and such an interacting many-particle system
may be described by the following Hamiltonian
..................4ekin ikin e e e i i iH H H H H H
Or,
2 2 2
2 2 1 1
( , ) ( , )..........5
2 2 2 2
i i ii i i j i
i j
e
H V r R V R R
m M r r
where ri, m and -e are used to represent the coordinates, mass and charge of
electrons, while Rα, Rβ and M are the corresponding quantities of the lattice ion.
However, it is impossible to solve the Schrodinger equation for the Hamiltonian (5)
in order to have extract solution. Since it is difficult to get results directly, various
approximations and models were introduced with for simplifying the problem.
Under the adiabatic approximation, the coupling between electrons and ions in (5)
is ignored, and we only need to investigate the independent subsystems: one is the
interacting ion system and the other is the interacting electron system . Thus
Hamiltonion for interacting electron system can be written from (5) as,
2 2
2
2 2
2
1
( , )
2 2
simply
1
( ) ..........6
2 2
i ii i j i
i j
i ii i i j
i j
e
H V r R
m r r
or
e
H V r
m r r
2
1
, ( )
2
i i j
i j
e
where V r
r r
Thus Schrodinger equation can be written as,
2 2
2
1 1
1
( ) ( ,.... ) ( ,.... )..........7
2 2
i i N Ni i i j
i j
e
V r r r E r r
m r r
3.2 The Hartree Approximation
According to hartree the effect of electron-electron interaction on a certain electron ( sat at
r) should be given by electrostatic Coulomb potential generated by all other electrons on
average at position r’. The equation (7) is for interacting electron system.
Thus the total wave function for a system with N electrons could be written as the product
of one-electron wave functions
3. 1
1
( ,.... ) ( )..........8
N
N i i
i
r r r
for which Hartree suggested a variational calculation of E by
| |
.............9
|
H
E
If ψ were the exact ground state wave functions of the system, then E would be the
ground state energy. Thus it leads to a set of Hartree equations
2
2 2
*( ') ( ')
( ) ' ( ) ( )..........10
2 '
j j
i i ij
r r
V r e dr r E r
m r r
This is the Hartree approximation equation. Where,
2 3
*( ') ( ')
( ) ' the effective potential.
'
j j
eff j
r r
V V r e d r is
r r
where second term represent Hartree potential, which is Coulomb potential between an electron and
average charge density of rest of electron.
Thus in terms of average charge density the potencial can
2 3
be written as
( ')
( ) '
'
Hartree
n r
V r e d r
r r
Where n(r’) is the electron density.
Initially the electron density is not known. For this purpose self consistent field method is
used to calculate n(r’).
And energy eigen values from (9) will be
2
*( ') ( ') *( ) ( )| | 1
'
| 2 '
j j i i
ii i j
r r r rH
E E e drdr
r r
The second term is the coulomb interaction energy. If this interaction is dropped from
Hartree equation then the energy of the Hartree electron is purely kinetic as in case of
free electron gas.
The total energy per electron calculated from Hartree energy equation is small
enough to explain metallic cohesion (binding). Thus Hartree approximation needs
modification.
3.3 The Hartree Fock Approximation
The Hartree approximation can be applied to real solids by replacing the original
electron-ion potential by a new one-electron potential . However, to obtain more
realistic interaction, we must go beyond it to introduce the Hartree–Fock
approximation. Because electrons are fermions, the Pauli principle must be
considered. Therefore the total wavefunction for a system with N electrons could be
written as Slater determinant as,
1 1 1
1
( ) ( )
1
( )
( ) ( )
N
i
N N N
r r
r
N
r r
……………………11
The Hartree-Fock approximation can be expressed in terms of the Rayleigh Ritz variational
principle, in which the many-particle wave function is written as a single Slater
determinant.
The Hamiltonian operator is expressed as
4. 𝐻̂ = [∑ −
ħ2
2𝑚
∇2
+ Vions(ri)
𝑖
] +
1
2
∑
𝑒2
| 𝑟𝑖 − 𝑟𝑗|
𝑖≠𝑗
This is Hartree–Fock (HF) approximation. Now from variational calculation by putting
(11) into (9 ) leads to a set of Hartree–Fock equations:
2
2 2 2
*( ') ( ') *( ') ( ')
( ) ' ( ) ' ( ) ( )..........12
2 ' '
j j j i
i j i ij j
r r r r
V r e dr r e dr r E r
m r r r r
We may obtain a value for the total energy in the Hartree–Fock approximation and this
will
again contain a extra term is the exchange interaction. Thus energy eigen values in
Hartree-Fock approximation from (9) will be
2
2
*( ') ( ') *( ) ( )| | 1
'
| 2 '
*( ') ( ') *( ) ( )1
- '
2 '
j j i i
ii i j
j i i j
i j
r r r rH
E E e drdr
r r
r r r r
e drdr
r r
In the application of HF equations, it is usually assumed that the spatial part of the
wavefunction
is the same for spin-up and spin-down electrons, i.e., every orbital is doubly occupied,
and the
wavefunctions of the Slater determinant are spin singlets. This is so-called restricted
Hartree–Fock (HF) method, and can be reasonably used in many problems not
involving magnetism. In magnetic problems the HF equations are necessarily different.
Average energy per particle calculated from Hartree-Fock equation is better than
that from Hartree equation, but still binding is too weak. Major difficulty with Hartree-
Fock approximation is that the density of states at the Fermi level goes to zero.
The particle density of the other electrons felt by Hartree-Fock particle, looks like that
shown in Fig.
The concentration of the
electrons of like spin is
lowered in the
neighborhood of the
investigated electron. The
difference between the
Hartree and the Hartree-
Fock approximation is
that density of particles
for Hartree electron only
depends on position of
other electrons it is the same for
each position of the observed particle. But density of other electrons for Hartree-Fock
electrons depends on the position of the observed particle, i.e., on the position of the
particle for which we are actually solving the Hartree-Fock equation.
If the investigated electron i is at position r, then all other electrons of like spin are
displaced from position r. Due to the Pauli principle the electrons of like spin do not
move independently of each other, but their motion is correlated, because in its
neighborhood an electron displaces the other electrons. Another correlation due to the
Coulomb repulsion for all electrons, is included in an averaged way in Hartree as well as
in Hartree-Fock theory, so that the correlation resulting from the Coulomb repulsion is
missing in both theories. Hartree-Fock therefore contains a part of the correlation, the so-
called Pauli correlation.
3.4 Hartree Fock Theory of Free Electrons
In case of free electron gas the wave function is a plane wave of type,
5. 𝜓𝑖( 𝑟) =
𝑒 𝑖𝑘 𝑖.𝑟
√ 𝑉
𝑋 𝑠𝑝𝑖𝑛 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
Because of free electrons the electron charge density is uniform all over the specimen.
Hence coulomb potential due to the electron is constant. Similarly ions are fixed at their
positions and having same charge density as that of electrons. Therefore potentials are
cancelled together and only the exchange term survives.
The Fourier transform of the term
𝑒2
|𝑟−𝑟′|
is given by,
𝑒2
|𝑟−𝑟′|
= 4𝜋𝑒2 1
𝑉
∑
1
𝑞2 𝑒−𝑖𝑞.(𝑟−𝑟′)
→ 4𝜋𝑒2
∫
𝑑𝑞
(2𝜋)3
1
𝑞2 𝑒−𝑖𝑞.(𝑟−𝑟′)
Using this value in exchange term in Hartree Fock equation and after simplifying we get,
𝐸( 𝑘) =
ħ2 𝑘2
2𝑚
−
2𝑒2
𝜋
𝑘 𝐹 𝐹(𝑥), where, 𝑥 =
𝑘
𝑘 𝐹
and 𝐹( 𝑥) =
1
2
+
1−𝑥2
4𝑥
𝑙𝑛 |
1+𝑥
1−𝑥
|
To compute the contribution of all electrons , say N, then total energy is obtained by
multiplying first term by 2 ( because of double spin), and dividing second term by 2 (
because when we are considering interaction electron with other electrons we are
counting each electron pair twice). That is
𝐸 = 2 ∑
ħ2 𝑘2
2𝑚
−
𝑒2 𝑘 𝐹
𝜋
∑ 𝐹(𝑥),
𝐸 = 2 ∑
ħ2 𝑘2
2𝑚
−
𝑒2 𝑘 𝐹
𝜋
∑
1
2
+
𝑘 𝐹
2
−𝑘2
2𝑘 𝐹 𝑘
𝑙𝑛 |
𝑘 𝐹+𝑘
𝑘 𝐹−𝑘
| since x = k/kF
By transforming summation to integration and simplifying we get,
𝐸 = 𝑁 [
3
5
𝐸 𝐹 −
3
4
𝑒2
𝑘 𝐹
𝜋
]
3.5 Correlation
The Hartree method is a good starting point for the discussion of electron-electron
interactions, but there are some shortcoming that we have assumed that at any particular
instant an electron does not care positions of others. The technical term for this is the
neglect of correlation. In reality electron motions are correlated for two reasons:
(a) Coulomb Correlation
Since electrons repel each other they will keep as far apart from each other as possible. If
we take the example of the hydrogen molecule from we can easily accept that at any
instant it would be highly unlikely for both electrons to be “on” the same atom. If we
know where electron one is then we can predict with good certainty where electron two
is, just on the basis of electrostatics. In the Hartree approximation we assume that any
particular electron does not know where other electron is at any moment, but only their
time-averaged positions. As a result the Hartree approximation allows electrons to
occasionally come very close to each other, Thus the Hartree approximation slightly
overestimates electron-electron repulsions.
(b) Exchange
Since electrons are not distinguishable and have half-integer spin, the wavefunction of an
N electron system must change sign on interchange of any two of its particles. Writing
each one electron wavefunction as the product of a “space function” and a “spin
function”, it can be shown that this fundamental requirement introduces a special form of
electron correlation that is electrons with parallel spins tend to avoid each other. Each
electron is said to carry around an exchange hole, a region in which other electrons with
the same spin are excluded.
The Exchange Energy
Because electrons are Fermions and obey Pauli’s principle, the total wave function of the
system must be antisymmetric (i.e. the wave function changes sign when two electrons
are interchanged). That prevent two electrons of the same spin to come close to each
other. This has nothing to do with Coulomb repulsion, it is a purely quantum mechanical
effect. To illustrate this, consider just a two electron wave function (where the electrons
6. have the same spin). Then by definition, antisymmetry of the wave function says that Ψ
(r1,r2)= -Ψ (r2,r1) where r1 and r2 are the positions of electrons 1 and 2, respectively. But
that means the wave function is identically zero when r1 = r2 (i.e., the electrons are on top
of each other). Since the probability is proportional to |Ψ|2
, and since the wave function
is a smooth function (i.e., it will be small even when r1 and r2 are similar), this is
equivalent to “keeping the electrons apart”. But because this quantum effect keeps the
electrons apart, we have overestimated the repulsive electron-electron interaction term in
the Coulomb energy above. The correction is called the exchange energy and it has the
value
𝐸𝑒𝑥𝑐ℎ𝑎𝑛𝑔𝑒 = −
0.916
𝑟
There is one other term, even when the electron gas is uniform. This arises from the fact
that even electrons of opposite spin avoid each other, because of the Coulomb force. This
is called a correlation energy. This is also a negative energy. It goes to a finite value as
the electron spacing parameter goes to zero (infinite density limit). And then there is an
energy that arises from the non-uniformity of the electron gas.
the correlation energy is usually defined as the difference between the exact and self-
consistent Hartree-Fock energies.
3.6 Thomas Fermi screening and Dielectric function
If the external potential V(r ) is applied to the electron gas then average electron
density no longer remain constant because electrons will get attracted towards maximum
of V(r ).
If we write spatially varying electron density as,
( ) ( )........1n r n n r
Where n is the uniform density when V=0, then we can define induce charge density as,
( ).............2en r
This induce charge density creates induced electrostatic potential δV, given by Poisson’s
equation as,
2
4 ..................3V
Then total potential can be written as,
total total
............................4
consider this in Fourier space then V(k), V ( ), ( ), ( ) are the Fourier transform of V(r), V ( ),
( ), ( ) respectively. Then equation [4] become
totalV V V
k V k k r
V r r
V
2
( ) ( ) ( )............................5
Thus dielectric function is defined as,
( )
( ) ...............................6
( )
Since, k=i , equation [3] can be written as,
( ) 4 ( )
or,
total
total
k V k V k
V k
k
V k
k V k k
2
2
2
4
( )
k
Therefore, equation [5] become,
4
( ) ( )
k
( ) 4
, ( ) 1 ...............[7]
( ) k ( )
total
total total
V k
V k V k
V k
or k
V k V k
7. We now need to find that is induced in presence of ( ).
To compute the value of we assume the slowly varying V(r)
so that the system is remain in local equilibrium at every position of r.
T
totalV k
k total B
k B
(E -eV (k)- )/k
0 total
0 (E - )/k
hen in such case the probability of finding an electron with wave
vector k at postion r is given by Fermi function,
1
f(k,r)=
e 1
or, f(k,r)=f (k,μ+eV (k))
1
so, f (k,μ)=
e
T
3
0 total 03
3
0
3
3
2
03
2
, which is equilibrium distribution when V=0.
1
d
Thus (r)=-e f (k,μ+eV (k))-f (k,μ)
4
dfd k
or, δρ(r)=-e ( )
4π
d d k
or,δρ(r)=-e ( ) f ( )
4π
dn( )
or,δρ(r)=-e ( ) , where
T
total
total
total
k
eV r
d
V r k
d
V r
d
3
03
2
2
2
2
d k
n( )= f ( ) is the equilibrium density.
4π
dn( )
So also, δρ(k)=-e ( )
δρ(k) dn( )
or, =-e
( )
so, equation [6] become,
4 dn( )
( ) 1 e .........................[8]
k
This is Thomas F
total
total
k
V k
d
V k d
k
d
2
2 20
02
ermi equation for dilectric function and it can also be
written in the form as,
dn( )
( ) 1 , where 4 e
k
k
k k
d
To illustrate the significance of k0 , let us consider a point charge Q is placed in the metal
at a point r. Then external potential V is given by,
𝑉( 𝑟) =
𝑄
𝑟
𝑜𝑟 𝑉(𝑘) =
4𝜋𝑄
𝑘2
The potential I the metal will then be,
𝑉𝑡𝑜𝑡𝑎𝑙( 𝑘) =
𝑉( 𝑞)
𝜖( 𝑘)
=
4𝜋𝑄
𝑘2
2
0
2
1 ,
k
k
=
4𝜋𝑄
𝑘2 + 𝑘0
2
By inverting through Fourier transform we get,
𝑉𝑡𝑜𝑡𝑎𝑙( 𝑟) = ∫
𝑑3
𝑘
(2𝜋)3
𝑒 𝑖𝑘.𝑟
4𝜋𝑄
𝑘2 + 𝑘0
2 =
𝑄
𝑟
𝑒−𝑘0 𝑟
Thus the total potential is of Coulomb form times the exponential damping factor. Thus
potential reduces to negligible size at a distance greater than 1/k0. This form of potential
is known as screened Coulomb potential or Yukawa potential.
The Thomas Fermi method has the advantages that it is applicable even when a linear
relation between induced charge density and the potential does not hold. It has a
disadvantage that it is only for slowly varying external potentials.
8. 3.7 Lindhard Dielectric function
Consider a potential V(r) applied to the electron gas . To calculate the change in
electron density δn(r) we could calculate effect of V(r) on electron eigen sates.
Using Rayleigh-Schrodinger stationary perturbation theory for lowest order in V(r),
the eigen states become,
|𝜓 𝑘〉 = |𝑘〉 + ∑
|𝑘′〉⟨ 𝑘′| 𝑣(𝑟)| 𝑘⟩
𝐸𝑘 − 𝐸𝑘′
k'
Where |𝑘〉 is the unperturbed plane wave eigen state with energy E 𝑘=
ħ2 𝑘2
2𝑚
and
|𝜓 𝑘〉 is the new eigen function results from v(r).
The electron density as a function of position for wave function |𝜓 𝑘〉 is |〈 𝑟|𝜓 𝑘〉|2
SO chane in electron density due to perturbation is
|〈r|ψk〉|2
-|〈r|k〉|2
=〈ψk|r〉〈r|ψk〉 - 〈k|r〉〈r|k〉
=[〈r|k〉 + ∑
〈r|k'〉〈k'|v|k〉
𝐸 𝑘−𝐸 𝑘′
𝑘′ ] [〈k|r〉 + ∑
〈k'|r〉〈k|v|k'〉
𝐸 𝑘−𝐸 𝑘′
𝑘′ ] − 〈k|r〉〈r|k〉
To linear order in V above equation become
=∑ {
〈r|k〉〈k'|r〉〈k|v|k'〉
𝐸 𝑘−𝐸 𝑘′
+
〈k|r〉〈r|k'〉〈k'|v|k〉
𝐸 𝑘−𝐸 𝑘′
}𝑘′
Now we have , 〈r|k〉 =
𝑒 𝑖𝒌⃗⃗ .𝒓⃗
√ 𝑉
where V is the volume,
〈k|r〉 =
𝑒−𝑖𝒌⃗⃗ .𝒓⃗
√ 𝑉
and
〈k'|v|k〉 = ∫
1
𝑉
𝑒−𝑖𝒌′⃗⃗⃗⃗ .𝒓⃗
𝑣( 𝑟) 𝑒 𝑖𝒌⃗⃗ .𝒓⃗
𝑑3
𝑟
〈k'|v|k〉 =
1
𝑉
∫ 𝑒
−𝑖(𝒌′⃗⃗⃗⃗ −𝒌⃗⃗ ).𝒓⃗
𝑣(𝑟)𝑑3
𝑟
〈k'|v|k〉 =
1
𝑉
𝑣 𝒌′⃗⃗⃗⃗ −𝒌⃗⃗ is the Fourier transform of v(r).
So above equation become
〈k'|v|k〉 =
1
𝑉2 ∑ {
𝑒−𝑖(𝒌′⃗⃗⃗⃗ −𝒌⃗⃗ ).𝒓⃗
𝑣 𝒌⃗⃗ −𝒌′⃗⃗⃗⃗
𝐸 𝑘 − 𝐸 𝑘′
+
𝑒−𝑖(𝒌⃗⃗ −𝒌′⃗⃗⃗⃗ ).𝒓⃗
𝑣 𝒌′⃗⃗⃗⃗ −𝒌⃗⃗
𝐸 𝑘 − 𝐸 𝑘′
}
𝑘′
So total induced electron density will be obtained by summing over all
occupied states
δn(r) = 2 ∑ fk
k
1
𝑉2
∑
{
𝑒
−𝑖( 𝒌′⃗⃗⃗⃗
−𝒌⃗⃗ ).𝒓⃗⃗
𝑣 𝒌⃗⃗ −𝒌′⃗⃗⃗⃗
𝐸 𝑘 − 𝐸 𝑘′
+
𝑒−𝑖( 𝒌⃗⃗ −𝒌′⃗⃗⃗⃗ ).𝒓⃗⃗
𝑣 𝒌′⃗⃗⃗⃗ −𝒌⃗⃗
𝐸 𝑘 − 𝐸 𝑘′
}
𝑘′
where factor ‘2’ appears due to spin degeneracy and fk is the Fermi occupation
function fk =
1
𝑒
(𝐸 𝑘−𝜇) 𝑘 𝛽 𝑇⁄
+1
, where 𝑘 𝛽 is the Boltzmann constant and μ is chemical
potential.
9. Fourier transform to get δn(q)
δn(q) = ∫ 𝑒−𝑖𝒒⃗⃗ .𝒓⃗⃗
𝛿𝑛(𝑟)𝑑
3
𝑟
δn(q) =
1
𝑉2
2 ∑ 𝑓 𝑘 {
𝑉𝛿 𝑞,𝒌⃗⃗ −𝒌′⃗⃗⃗⃗ 𝑣 𝒌⃗⃗ −𝒌′⃗⃗⃗⃗
𝐸 𝑘 − 𝐸 𝑘′
+
𝑉𝛿 𝑞,𝒌′⃗⃗⃗⃗ −𝒌⃗⃗ 𝑣 𝒌′⃗⃗⃗⃗ −𝒌⃗⃗
𝐸 𝑘 − 𝐸 𝑘′
}
𝑘,𝑘′
now use 𝛿′
𝑠𝑡𝑜 𝑑𝑜 𝑠𝑢𝑚 𝑜𝑣𝑒𝑟 𝑘′ we get
δn(q) =
2
𝑉
∑ 𝑓 𝑘 {
𝑣 𝑞
𝐸 𝑘 − 𝐸 𝑘−𝑞
+
𝑣 𝑞
𝐸 𝑘 − 𝐸 𝑘+𝑞
}
𝑘
so,
δn(q)
𝑣 𝑞
=
2
𝑉
∑ 𝑓 𝑘 {
1
𝐸 𝑘−𝐸 𝑘−𝑞
+ 1
𝐸 𝑘−𝐸 𝑘+𝑞
}𝑘 now put 𝑘′
= 𝑘 − 𝑞
δn(q)
𝑣 𝑞
=
2
𝑉
∑
𝑓 𝑘+𝑞 − 𝑓 𝑘
𝐸 𝑘+𝑞 − 𝐸 𝑘
𝑘
δn(q)
𝑣 𝑞
= ∫
𝑓 𝑘+𝑞 − 𝑓 𝑘
𝐸 𝑘+𝑞 − 𝐸 𝑘
𝑑
3
𝑟
4𝜋3
For electrostatic potential 𝑣 𝑞 = −𝑒𝑉𝑞
𝑡𝑜𝑡𝑎𝑙
and 𝛿𝜌 = −𝑒𝛿𝑛, so
𝛿𝜌(𝑞)
𝑉𝑞
𝑡𝑜𝑡𝑎𝑙 =
−𝑒𝛿𝑛(𝑞)
𝑣(𝑞)
(−𝑒)⁄
𝛿𝜌(𝑞)
𝑉𝑞
𝑡𝑜𝑡𝑎𝑙 =
𝑒2
𝛿𝑛(𝑞)
𝑣(𝑞)
𝛿𝜌(𝑞)
𝑉𝑞
𝑡𝑜𝑡𝑎𝑙 = 𝑒2
∫
𝑓 𝑘+𝑞−𝑓 𝑘
𝐸 𝑘+𝑞−𝐸 𝑘
𝑑3 𝑟
4𝜋3
………………..[1]
For small q 𝑓𝑘+𝑞 − 𝑓𝑘 ≈
𝜕𝑓
𝜕𝑞
. 𝑞
And 𝐸 𝑘+𝑞 − 𝐸 𝑘 ≈
𝜕𝐸
𝜕𝑞
. 𝑞
Therefore [1] become
𝛿𝜌(𝑞)
𝑉𝑞
𝑡𝑜𝑡𝑎𝑙
= 𝑒2
∫
𝜕𝑓
𝜕𝐸
𝑑3
𝑟
4𝜋3
= 𝑒2
∫
𝜕𝑓
𝜕𝐸
𝑑𝐸 𝑔(𝐸)
As 𝑇 → 0,
𝜕𝑓
𝜕𝐸
→ −𝛿(𝐸 − 𝐸 𝐹)
So we get,
𝛿𝜌(𝑞)
𝑉𝑞
𝑡𝑜𝑡𝑎𝑙 = −𝑒2
𝑔(𝐸 𝐹)
As dielectric function , 𝜖( 𝑞) = 1 −
4𝜋
𝑞2
𝛿𝜌(𝑞)
𝑉𝑞
𝑡𝑜𝑡𝑎𝑙 , we get
𝜖( 𝑞) = 1 +
4𝜋
𝑞2
𝑒2
𝑔(𝐸 𝐹) ……………………………..2
Thus it is seen that Lindhard dielectric function is same as Thomas –Fermi function.
10. Friedel Oscillations (Linhard dielectric function at higher ‘q’)
We have fron equation [2],
𝜖( 𝑞) = 1 +
4𝜋
𝑞2
𝑒2
𝑔(𝐸 𝐹)
𝜖( 𝑞) = 1 +
4𝜋
𝑞2
𝑒2 ∑
𝑓 𝑘−𝑓 𝑘+𝑞
𝐸 𝑘+𝑞−𝐸 𝑘
𝑘 ……………….3
Since
𝐸 𝑘 =
ħ2 𝑘2
2𝑚
we get,
𝐸 𝑘+𝑞 − 𝐸 𝑘 =
ħ2( 𝑘 + 𝑞)2
2𝑚
−
ħ2
𝑘2
2𝑚
=
ħ2
𝑞2
2𝑚
+
2ħ2
𝐤. 𝐪
2𝑚
=
( 𝑞2
+ 2𝐤. 𝐪)ħ2
2𝑚
Therefore equation [3] become
𝜖( 𝑞) = 1 +
4𝜋
𝑞2
𝑒2 2𝑚
ħ2 ∫
𝑑3 𝑘
8𝜋3𝑅
1
(𝑞2+2𝐤.𝐪)
× 2 × 2
Here R represent the region of k-
space where k if full and k+q is
empty First factor ‘2’ appears in
above equation because of spin up
and spin down similarly second ‘2’
appears when Fermi sphere k+q if
full and k is empty.
If q increases then region R is also increases. We can show it graphically as follow,
The integral
𝜖( 𝑞) = 1 +
4𝜋
𝑞2
𝑒2
2𝑚
ħ2
∫
𝑑3
𝑘
8𝜋3
𝑅
1
( 𝑞2 + 2𝐤. 𝐪)
× 2 × 2
By solving the integral explicitly we get,
𝜖( 𝑞) = 1 +
4𝜋𝑒2
𝑞2
𝑞( 𝐸 𝐹) [
1
2
+
1 − 𝑥2
4𝑥
𝑙𝑛 |
1 + 𝑥
1 − 𝑥
|]
Where 𝑥 =
𝑞
2𝑘 𝐹
As 𝑥 → 0, 𝑡ℎ𝑒 𝑏𝑟𝑎𝑐𝑘𝑒𝑡 𝑡𝑒𝑟𝑚 =
1 𝑎𝑛𝑑 𝑤𝑒 𝑔𝑒𝑡 𝑏𝑎𝑐𝑘 𝑇ℎ𝑜𝑚𝑎𝑠 𝐹𝑒𝑟𝑚𝑖 𝑑𝑖𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
𝑎𝑛𝑑 𝑎𝑡 𝑥 = 1 𝑡ℎ𝑎𝑡 𝑖𝑠 𝑓𝑜𝑟 𝑞 = 2𝑘 𝐹, 𝑡ℎ𝑒𝑛 𝜖( 𝑞) ~
1
𝑟3
cos(2 𝑘 𝐹 𝑟)
So
𝜖( 𝑞) 𝑜𝑟 𝑖𝑛𝑑𝑢𝑐𝑒𝑑 𝑐ℎ𝑒𝑟𝑔𝑒 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑑𝑒𝑐𝑎𝑦𝑠 𝑚𝑜𝑟𝑒 𝑠𝑙𝑜𝑤𝑙𝑦 𝑎𝑛𝑑 𝑜𝑠𝑐𝑖𝑙𝑙𝑎𝑡𝑒𝑠 𝑎𝑠𝑎 𝑐𝑜𝑠𝑖𝑔𝑛 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛.
This is known as Friedel oscillations.