To understand how electrons move in a solid, we need to consider two things:
- The potential energy- The wave function of the electrons.
The potential energy is the energy that an electron has due to its position in the material. The wave function is a mathematical function that describes the probability of finding an electron in a certain position and with a certain momentum. The potential energy and the wave function are related by a mathematical equation called the Schrödinger equation, which is the basic equation of quantum mechanics.
The potential energy of an electron in a solid is not constant but varies periodically according to the arrangement of the atoms in the material. The atoms form a regular pattern called a crystal lattice, and the potential energy of an electron depends on how close or far it is from the nuclei of the atoms. The potential energy is lower when the electron is closer to the nuclei, and higher when it is farther away. The potential energy also repeats itself after a certain distance, called the lattice constant, which is the length of the smallest unit of the crystal lattice.
The wave function of an electron in a solid is also not simple, but a complex function that has both amplitude and phase. The amplitude is the height of the wave function, and the phase is the angle of the wave function. The amplitude and the phase of the wave function depend on the position and the momentum of the electron. The wave function also has a property called periodicity, which means that it repeats itself after a certain distance, called the wavelength. The wavelength is related to the momentum of the electron by a mathematical formula called the de Broglie relation.
The problem of finding the wave function and the potential energy of an electron in a solid is very difficult to solve exactly because the potential energy is very complicated and the wave function is very large. Therefore, physicists use different models and approximations to simplify the problem and get some useful results. The web page that you are viewing introduces four models that are commonly used to study the electrons in a solid: the free electron model, the nearly free electron model, the Kronig-Penney model, and the effective mass model. I will briefly explain each of these models and their main results.
The free electron model is the simplest model, which assumes that the potential energy of an electron in a solid is zero everywhere. This means that the electron is free to move in any direction and at any speed, without any interaction with the atoms in the material. The wave function of a free electron is a plane wave, which is a wave that has a constant amplitude and a constant phase. The plane wave has a wave number, which is the inverse of the wavelength, and a wave vector, which is a vector that points in the direction of the wave propagation. The wave number and the wave vector are related to the momentum and the energy of the electron by the de Broglie relation and the kinetic energy formula.
The free electron model can explain some basic properties of metals, such as the electrical conductivity, the heat capacity, and the Fermi energy. The Fermi energy is the highest energy that an electron can have in a metal at absolute zero temperature. The free electron model cannot explain some other properties of metals, such as the band structure, the band gap, and the effective mass. The band structure is the relationship between the energy and the wave number of the electrons in a solid. The band gap is the region of energy where no electrons can exist in a solid. The effective mass is the apparent mass of an electron in a solid, which may be different from its actual mass.
The nearly free electron model is a more realistic model, which assumes that the potential energy of an electron in a solid is zero everywhere, except at some points where it is very high. These points are called the Bragg planes, and they correspond to the planes of atoms in the crystal lattice. The Bragg planes act as barriers that reflect some of the electrons and transmit some of the electrons. The wave function of a nearly free electron is a superposition of two plane waves, one that is moving forward and one that is moving backward. The superposition of two plane waves creates a standing wave, which is a wave that has a variable amplitude and a constant phase. The standing wave has a wave number, which is the inverse of the wavelength, and a wave vector, which is a vector that points in the direction of the average wave propagation. The wave number and the wave vector are related to the momentum and the energy of the electron by the de Broglie relation and the kinetic energy formula.
The nearly free electron model can explain some properties of metals that the free electron model cannot, such as the band structure, the band gap, and the effective mass. The nearly free electron model shows that the energy of the electrons in a solid is not continuous, but discrete and that there are regions of energy where no electrons can exist. These regions are called band gaps, and they occur when the wave number of the electrons is equal to the wave number of the Bragg planes. The band gaps separate the regions of energy where the electrons can exist, which are called the energy bands. The energy bands have different widths and shapes, depending on the strength and the periodicity of the potential energy. The effective mass of the electrons in a solid is not constant but varies according to the curvature of the energy bands. The effective mass of the electrons affects their acceleration and mobility in a solid.
The Kronig-Penney model is a simplified model, which assumes that the potential energy of an electron in a solid is periodic and consists of a series of square wells separated by barriers. The square well is a region of space where the potential energy is constant and low, and the barrier is a region of space where the potential energy is constant and high. The width of the square well is equal to the width of the barrier, and the period of the potential energy is equal to the lattice constant. The wave function of an electron in a Kronig-Penney model is a piecewise function, which has different forms in the square well and in the barrier. The wave function in the square well is a sinusoidal function, which has a variable amplitude and a variable phase. The wave function in the barrier is an exponential function, which has a decreasing amplitude and a constant phase. The wave function in the Kronig-Penney model has to satisfy some boundary conditions, which are the rules that the wave function has to follow at the edges of the square well and the barrier. The boundary conditions ensure that the wave function is continuous and smooth everywhere.
The Kronig-Penney model can explain the same properties of metals as the nearly free electron model, such as the band structure, the band gap, and the effective mass. The Kronig-Penney model can also derive a mathematical equation that relates the wave number and the energy of the electrons in a solid, which is called the characteristic equation. The characteristic equation is a transcendental equation, which means that it cannot be solved algebraically, but only numerically or graphically. The characteristic equation has a real solution for the wave number only when the cosine of the wave number times the lattice constant is between -1 and +1. This condition ensures that the electron has a permissible energy in the solid and that there is a band gap between the energy bands. The condition also determines the width of the energy bands and the size of the band gaps, which depend on the ratio of the mass of the electron, the height of the barrier, and the wave number of the electron in the square well.
The effective mass model is a general model, which assumes that the potential energy of an electron in a solid is periodic and arbitrary. The effective mass model does not specify the exact form of the potential energy or the wave function of the electron, but only uses some properties of the energy bands and the wave number of the electron. The effective mass model can approximate the behavior of the electron in a solid by using a simple equation, which is called the effective mass equation. The effective mass equation is similar to the kinetic energy formula, but it replaces the actual mass of the electron with the effective mass of the electron, and it replaces the actual momentum of the electron with the reduced Planck constant times the wave vector of the electron. The effective mass equation can be used to calculate the energy of the electron in a solid, given the effective mass and the wave vector of the electron.
The effective mass model can explain some properties of solids that the other models cannot, such as the transport properties, the optical properties, and the magnetic properties.
The transport properties are the properties that describe how the electrons move in a solid when they are subjected to an external force, such as an electric field or a magnetic field. The transport properties depend on the effective mass of the electrons, which affects their acceleration and mobility in a solid. For example, the electrical conductivity of a solid is proportional to the inverse of the effective mass of the electrons, which means that the lower the effective mass, the higher the conductivity. The effective mass of the electrons can vary depending on the direction and the magnitude of the applied force, which can lead to phenomena such as anisotropy and nonlinearity.
The optical properties are the properties that describe how the electrons interact with light in a solid, such as the absorption, the reflection, and the emission of light. The optical properties depend on the energy gap between the valence band and the conduction band, which are the highest occupied and the lowest unoccupied energy bands of the electrons in a solid. The energy gap determines the minimum amount of energy that a photon of light needs to have to excite an electron from the valence band to the conduction band, or vice versa. The energy gap also determines the wavelength of the light that is absorbed or emitted by the solid, which can be used to identify the type and the state of the solid. For example, the color of a solid is related to the energy gap of the electrons, which means that the different colors of the same material can indicate different levels of doping or impurities.
The magnetic properties are the properties that describe how the electrons generate and respond to magnetic fields in a solid, such as the magnetization, the susceptibility, and the Hall effect. The magnetic properties depend on the spin of the electrons, which is a quantum mechanical property that gives the electrons a magnetic moment. The spin of the electrons can align or oppose the external magnetic field, depending on the energy and the temperature of the solid. The spin of the electrons can also interact with each other, creating different types of magnetic order, such as ferromagnetism, antiferromagnetism, or paramagnetism. The magnetic properties of a solid can be used to store, manipulate, and transmit information, which is the basis of spintronics.
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