Menu Outline Exercise Questions Appendices Lectures Student Projects Books Sections Introduction Atoms Molecules Crystal Structure Crystal Diffraction Crystal Binding Photons Phonons Electrons Band Model Crystal Physics Semiconductors

PHY.K02UF Molecular and Solid State Physics

## Introduction

Molecular and solid state physics is a vast subject. There are many interesting and useful molecules in food, medicine, fuels, pollutants, cleaning solutions, and living cells. These molecules have many properties such as color, density, viscosity, chemical reactivity, toxicity, combustibility, melting point, boiling point, and magnetic moment. Typically all these properties change as the pressure and temperature change. Solids also have many properties such as piezoelectricity, refractive index, density, electric susceptibility, magnetic susceptibility, thermal expansion coefficient, pyroelectricity, tensile strength, electrical conductivity, thermal conductivity, Hall effect, bulk modulus, and specific heat. The properties of solids are important for a wide range of applications including microelectronics, cars, ships, planes, bridges, buildings, satellites, and motors.

Our task in this course is to understand the how to describe the many properties of a class of billions of molecules and solids. Fortunately, every property of every material can be calculated with great precision using quantum mechanics and statistical physics. It is not terribly difficult to write down the equations that must be solved. However, these equations are incredibly difficult to solve. Even though we can write a program to solve these equations, the most powerful computers that we have cannot solve them. The equations are mind-bogglingly difficult to solve.

In order to make any progress with this difficult problem, we need to make some approximations. In this course we discuss the simplest approximations. We begin with a microscopic description of molecules and solids where we will specify the positions of all atoms in molecule or solid. We will then show how the knowledge of the microscopic quantum states can be used to calculate the macroscopic properties.

The course starts with a review of atomic physics. If an atom has just one electron like hydrogen, it is relatively simple to determine the electron wave functions that solve the Schrödinger equation. However, if there is more than one electron, it is becomes much more difficult. There is a standard approximation that is used where the multi-electron wave functions are constructed from products of hydrogen wave functions. We will show how this approximation can be used to calculate the properties of many-electron atoms.

After atoms, we consider molecules. Since the many electron problem is so difficult to solve, we take away all of the electrons of a molecule except one and calculate the wave function of just one electron moving in the potential given by the positive nuclei of all the atoms of the molecule. This wave function is called a molecular orbital. The many-electron wave functions of the molecule can be constructed from the molecular orbitals much in the same way that the multi-electron wave functions for atoms can be constructed from the hydrogen wave functions. Once an approximate wave function is known, we can calculate the energy associated with that wave function. This turns out to be very powerful. To calculate a bond length: guess the distance between the atoms, calculate the molecular orbitals, construct the multi-electron wave function, and find the energy of the multi-electron wave function. Then guess a new distance between the atoms and continue the process until you find the distance that minimizes the energy. This distance will be the bond length. Approximately the same procedure can be used to determine the bond angle or the whole shape of a molecule. If you want to know if a molecule is linear or forms a ring, construct the multi-electron wave functions for both cases and see which one has a lower energy. Once you have constructed the multi-electron wave functions for a molecule, every property of the molecule can be calculated.

Solids are just large molecules so the concepts developed for molecules can be used to describe solids. Most of the solids we will consider crystals. In crystals, the atoms are arranged in a periodic pattern. In the crystal structure section, the possible periodic patterns are considered. Crystals are classified according to their symmetries.