One corresponding to a plane wave, or a free electron, and another corresponding to a periodic correction factor. Topics covered include elementary quantum theory, quantum statistics and manyparticle systems, and energy bands in crystals. However, the perturbation theory cannot be applied when the potential cannot be. In the thomasfermi model, one makes two assumptions. That is, the electron eigenstates are initially assumed to. Mppt,althoughin the beginning largely ignored, had a strong impact on the development of quantum chemical ab initio methods in the past 40 years. By means of the nearly free electron model explained in this lecture.
This is true for any k,but a periodic perturbation has nonzero matrix elements only between particular states. The nearlyfree electron model key reference for weeks 1012. Electrical conductivity of metallic hydrogen in the nearly. The idea is to start with a simple system for which a mathematical solution is known, and add an additional perturbing hamiltonian representing a weak disturbance to the system. We first find the band structure using the nearly free electron model.
Summary of free electron model free electron model simplest way to describe electronic properties of metals. Atomic selectrons are delocalised due to overlap of outer orbits. Sidebottom, fundamentals of condensed matter and crystalline physics, chapters 12 so far we have seen that the free electron gas model can explain a surprising number of the properties of metals, including their heat capacity and bulk modulus. Assumption a allows one to use the perturbation theory whereas assumption b casts this theory into a quasiclassical form. Free electron model a free electron model is the simplest way to represent the electronic structure of metals. Tsymbal 4 lattice vector, if the bloch form 14 holds for k, it will also hold for k. Energy bands nearlyfree electron model electron diffraction and. Although the free electron model is a great oversimplification of the reality, surprisingly in many cases it works pretty well, so that it is able. So in the absence of the perturbation we have the free electron case. Quantum mechanics for applied physics and engineering. An example is given below for a nearly free electron model. The free electron model gives us considerable information about several electronic properties of the socalled simple metals.
At the microscopic level, these two states are characterized by. Besides, in the free electron model electrons can travel long. Perturbation theory the goal here is to treat the periodic potential as a perturbation to the free electron hamiltonian. We begin our treatment of free energy differences by examining the problem of transforming a system from one thermodynamic state to another. First and second order perturbation theory in quantum mechanics. The nearly free electron model is a modification of the freeelectron gas model which includes a weak periodic perturbation meant to model the interaction between the conduction electrons and the ions in a crystalline solid. According to the perturbation theory the energy is given by. The free electron approximation remains a good approximation well away from the edges of the brillouin zone only wavevectors close to a multiple of. A nearly free electron model for crystalline mgh2 request pdf. Unit 9 blochs theorem for wavefunction of a particle in a.
Nfe model is good for na, k, al etc, in which the lattice potential is only a small perturbation to the electron sea. The nearly free electron model is a modification of the free electron gas model which includes a weak periodic perturbation meant to model the interaction between the conduction electrons and the ions in a crystalline solid. On the 200zone boundary collective mode in al within. Solid state physics nearly free electron model nanopdf. Quantum mechanics for applied physics and engineering is devoted to the use of quantum mechanics in applied physics and engineering. In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. To this end, we consider the effect of the potential on the free electron wavefunctions given by and on the interval. The energy e of free electrons which is plotted versus k in fig.
Timeindependent nondegenerate perturbation theory timeindependent degenerate perturbation theory. By means of the tight binding model explained in lecture 7. Now, lets see how the wave functions are modified by weak crystal potential. In solidstate physics, the nearly free electron model or nfe model is a quantum mechanical. Within the nearly free electron model we start from the dispersion relation of free. According to the stationary perturbation theory it can be written. Twodimensional harmonic oscilator 3 timedependent perturbation theory 4 literature igor luka cevi c perturbation theory.
Phys 446 solid state physics summary last lecture lecture 8. Since the exact solution of the problem should ful. A simple, free electron like model for the saline hydrides is discussed. This model, like the freeelectron model, does not take into account electronelectron interactions. For free particles, we know that any positive energy is an allowed eigenenergy i. If 0, we assume the electron to be traveling in a weak potential, and can thus expand the wave function into two parts. This is a model that is suitable for simple metals like li, na, k, etc.
Fermi energy the energy of the highest occupied electronic level at t 0 k. It is shown that a pair of 200 bragg planes gives rise to another pole in the energy loss function im1. Although the free electron model is a great oversimplification of the reality, surprisingly in many cases it works pretty well, so that it is able to describe many important properties of metals. The alkali metals or nobel metals can be regarded as simple metals. Nov 22, 2010 drawing a dispersion relation i the reduced zone scheme for free electron and nearly free electrons case things we need to know. The free electron model cannot explain why some elements are metals and others insulators. Because of the complexity of many physical problems, very few can be solved exactly unless they involve only small hilbert spaces. A natural extension of the free electron model is the nearly free electron nfe model, which applies to crystals, because it is assumed that the electrons are subjected to a periodic potential inside them.
Handout 6 electrons in a periodic potential, blochs theorem and bloch functions, electron bragg scattering, nearly free electron approach and perturbation theory, finite basis expansion, physical origin of bandgaps, location of bandgaps, zone folding, introduction to energy bands. In solidstate physics, the free electron model is a simple model for the behaviour of. Nearlyfree electron model starts from the free electron perspective v 0 and adds a very weak periodic potential to represent the ions. Perturbation theory also fails to describe states that are not generated adiabatically from the free model, including bound states and various collective phenomena such as solitons.
Energy bands nearlyfree electron model electron diffraction and energy gap bloch theorem the central equation emptylattice approximation tightbinding model see chap 9 nfe model is good for na, k, al etc, in which the lattice potential is only a small perturbation to the electron sea. Nearly free electron in the tight bonding model, the electron eigenstates are initially assumed to be localized around the atoms, then a hopping term is introduced to allow them to delocalize. The electrical resistivity of metallic hydrogen has been calculated. What should reassure you is that the properties that emerge are consistent with the nearlyfree electron model. The nearlyfree electron model university of oxford. The free electron model gives us a good insight into many properties of. To zeroth order the valence electrons are assumed to form a free gas. Phys 446 solid state physics summary last lecture lecture. To do this, the perturbation theory in terms of electronproton interaction for the reciprocal relaxation time characterizing the electrical conductivity has been employed. Drawing a dispersion relation i the reduced zone scheme for free electron and nearly free electrons case things we need to know. In this exercise, we will find the band structure of this crystal in two ways.
This model, like the free electron model, does not take into account electron electron interactions. Since the lattice is periodic we may expand the perturbation into a fourier series where g are a set of vectors and v g are fourier coe. Free electron model simplest way to describe electronic properties of metals. March 23, 2018 in general, the wave function of conduction electrons can be expressed by a plane wave state k with the wave function eikr in a periodic potential. The aschroft and sturm twobandmodel on the optical properties of al is extended for finite k values and the dielectric function. Other states with nonzero matrix elements to these states are relatively much further away in energy, and can be treated using ordinary perturbation theory. The terms of interest arise in second order perturbation theory and are e2k x0 k0 jhk0jvrjkij2 e0k. In such cases, time dependence of wavefunction developed through. Generalized perturbational molecular orbital pmo theory. Within the single active electron model of an atom, for an initial s or p state we define all dynamical parameters in.
Electron behaves essentially as a free particle, except the wave vectors close to the boundaries of the zone. If the wavelength of the electron is related to the lattice parameter a, then the electron cannot. To do this, the perturbation theory in terms of electron proton interaction for the reciprocal relaxation time characterizing the electrical conductivity has been employed. Energy spectrum of a nearlyfree electron model in 1d the eigenstates of free problem in 1d.
A natural extension of the freeelectron model is the nearly free electron nfe model, which applies to crystals, because it is assumed that the electrons are subjected to a periodic potential inside them. Perturbation theory analysis of attosecond photoionization. The secondand thirdorder terms as well as an approximate expression for the series sum have been calculated in detail. The opposite extreme to the nearlyfree electron model assumes the electrons in the. In the nearly free electron model, we take the opposite approach. Rubin december 2, 2010 lecture 3241 november 10 december 3, 2010 1 stationary state perturbation theory 1. March 23, 2018 in general, the wave function of conduction electrons can be expressed by a plane wave state k with the wave function eikr.
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