Within the nearly free electron model we start from the dispersion relation of free. In solidstate physics, the free electron model is a simple model for the behaviour of. Because of the complexity of many physical problems, very few can be solved exactly unless they involve only small hilbert spaces. One corresponding to a plane wave, or a free electron, and another corresponding to a periodic correction factor. The free electron model cannot explain why some elements are metals and others insulators. In the thomasfermi model, one makes two assumptions. In such cases, time dependence of wavefunction developed through. 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. Within the single active electron model of an atom, for an initial s or p state we define all dynamical parameters in. The nearlyfree electron model university of oxford. 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.
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. Since the exact solution of the problem should ful. The free electron model gives us considerable information about several electronic properties of the socalled simple metals. Summary of free electron model free electron model simplest way to describe electronic properties of metals. If 0, we assume the electron to be traveling in a weak potential, and can thus expand the wave function into two parts. Assumption a allows one to use the perturbation theory whereas assumption b casts this theory into a quasiclassical form.
First and second order perturbation theory in quantum mechanics. So in the absence of the perturbation we have the free electron case. The terms of interest arise in second order perturbation theory and are e2k x0 k0 jhk0jvrjkij2 e0k. Nfe model is good for na, k, al etc, in which the lattice potential is only a small perturbation to the electron sea. An example is given below for a nearly free electron model. 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. Phys 446 solid state physics summary last lecture lecture. By means of the tight binding model explained in lecture 7. Drawing a dispersion relation i the reduced zone scheme for free electron and nearly free electrons case things we need to know. For free particles, we know that any positive energy is an allowed eigenenergy i. What should reassure you is that the properties that emerge are consistent with the nearlyfree electron model. Tsymbal 4 lattice vector, if the bloch form 14 holds for k, it will also hold for k.
We first find the band structure using the nearly free electron model. The nearlyfree electron model key reference for weeks 1012. 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. Phys 446 solid state physics summary last lecture lecture 8. A simple, free electron like model for the saline hydrides is discussed. Solid state physics nearly free electron model nanopdf. 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. It is shown that a pair of 200 bragg planes gives rise to another pole in the energy loss function im1. 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. Unit 9 blochs theorem for wavefunction of a particle in a. The opposite extreme to the nearlyfree electron model assumes the electrons in the.
This is true for any k,but a periodic perturbation has nonzero matrix elements only between particular states. A nearly free electron model for crystalline mgh2 request pdf. The secondand thirdorder terms as well as an approximate expression for the series sum have been calculated in detail. Energy bands nearlyfree electron model electron diffraction and. 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. The energy e of free electrons which is plotted versus k in fig. 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. Quantum mechanics for applied physics and engineering is devoted to the use of quantum mechanics in applied physics and engineering. That is, the electron eigenstates are initially assumed to. In the nearly free electron model, we take the opposite approach. 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.
Timeindependent nondegenerate perturbation theory timeindependent degenerate perturbation theory. Topics covered include elementary quantum theory, quantum statistics and manyparticle systems, and energy bands in crystals. Rubin december 2, 2010 lecture 3241 november 10 december 3, 2010 1 stationary state perturbation theory 1. To this end, we consider the effect of the potential on the free electron wavefunctions given by and on the interval. According to the stationary perturbation theory it can be written. The alkali metals or nobel metals can be regarded as simple metals. Electrical conductivity of metallic hydrogen in the nearly. 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. Quantum mechanics for applied physics and engineering. At the microscopic level, these two states are characterized by. This model, like the freeelectron model, does not take into account electronelectron interactions. Perturbation theory the goal here is to treat the periodic potential as a perturbation to the free electron hamiltonian.
According to the perturbation theory the energy is given by. To do this, the perturbation theory in terms of electronproton interaction for the reciprocal relaxation time characterizing the electrical conductivity has been employed. In this exercise, we will find the band structure of this crystal in two ways. This is a model that is suitable for simple metals like li, na, k, etc. The free electron model gives us a good insight into many properties of. We begin our treatment of free energy differences by examining the problem of transforming a system from one thermodynamic state to another. 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. If the wavelength of the electron is related to the lattice parameter a, then the electron cannot. By means of the nearly free electron model explained in this lecture. Fermi energy the energy of the highest occupied electronic level at t 0 k. The aschroft and sturm twobandmodel on the optical properties of al is extended for finite k values and the dielectric function. 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. 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. Nearlyfree electron model starts from the free electron perspective v 0 and adds a very weak periodic potential to represent the ions.
Twodimensional harmonic oscilator 3 timedependent perturbation theory 4 literature igor luka cevi c perturbation theory. Generalized perturbational molecular orbital pmo theory. Perturbation theory analysis of attosecond photoionization. Electron behaves essentially as a free particle, except the wave vectors close to the boundaries of the zone. On the 200zone boundary collective mode in al within. 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 free electron approximation remains a good approximation well away from the edges of the brillouin zone only wavevectors close to a multiple of. Besides, in the free electron model electrons can travel long.
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. Atomic selectrons are delocalised due to overlap of outer orbits. To do this, the perturbation theory in terms of electron proton interaction for the reciprocal relaxation time characterizing the electrical conductivity has been employed. To zeroth order the valence electrons are assumed to form a free gas. The electrical resistivity of metallic hydrogen has been calculated. Energy spectrum of a nearlyfree electron model in 1d the eigenstates of free problem in 1d. However, the perturbation theory cannot be applied when the potential cannot be.
This model, like the free electron model, does not take into account electron electron interactions. 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. 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. Other states with nonzero matrix elements to these states are relatively much further away in energy, and can be treated using ordinary perturbation theory.
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