Statistical Aspects in Material Science

This lecture is designed for students who have basic knowledge in material science. It starts with some

0. Introduction

1. Some Basics of classical Thermodynamics

1.1. 0th Axiom of Thermodynamics

1.2. First Axiom of Thermodynamics

1.3. Second Axiom of Thermodynamics

"Irreversible Processes exist"

The inverse temperature as an integrating factor

1.4. Motivation for subsequent sections

1.5. Thermodynamic Contacts

The isolated system

A system in thermal equilibrium

Temperature and pressure are defined by the surrounding area

Examples for other contacts

1.6. Systematic description of a thermodynamic system

1.7. The free energy as an example for a thermodynamic potential

1.8 The transformation of thermodynamic potentials

Why not just replace a coordinate by its adjacent force?

The Legendre-Transformation in 1D

From the free energy to the inner energy

Calculation of the free energy of an ideal gas

Without this basics the concept of statistical mechanics is hard to understand.

2. Statistical Mechanics

2.1. The aim of statistical mechanics

Principle: Maximize "Degree of Uncertainty"

Justification of this principle

The differential operator

2.2 The microcanonic, canonic and the grandcanonic ensemble

Calculation of the canonic ensemble

Calculation of the microcanonic ensemble

Calculation of the grand canonical ensemble

First results from the calculation of the state sum

2.3 Small summary

What is temperature?

When do we used which Potential?

Isn't that easy?!?

2.4 The classical and quantum mechanical phase volume

2.5 The classical ideal gas

Calculation of the micro canonic state sum (phase volume)

Summing up all approximations

The grand canonical potential and variations of the number of particles

Isn't that easy?!?

3. Specific Heat Capacity

3.1. Calculation of the inner energy

3.2. The equipartition law of classical thermodynamics

3.3. Specific heat capacitance of phonons (Bosons)

One-dimensional lattice vibrations

Simple approximations for lattice vibrations

Quantum mechanical description of lattice vibrations

The Debye Model

The Einstein Model

3.4. Specific heat capacity of the free electron gas (Fermions)

4. Description of non equilibrium

4.1. Einstein's interpretation of the Bose statistics

Einstein's interpretatio

The spectral density of radiation

Planck's radiation law as a balance between absorption and emission

4.2 Essentials for the amplification of electromagnetic radiation

From amplifier to oscillator

4.3 The semiconductor LASER

Description of non equilibrium in a semiconductor

The injection LASERFrom amplifier to oscillator

4.4. Guided tour through synergy

The most simple non differential equation of a LASER

LASER and LIFE

5. The Boltzmann equation

5.1 Derivation of the Boltzmann equation

5.2 The relaxation time approximation in the Boltzmann equation

5.3. Particle and energy current

Current flow through a non degenerated semiconductor

Electrical current flow

Diffusion

5.4. The pn-Junction

The Concept

The Equations

The topics of the lecture are free to change, if the autitorium makes adequate suggestions