Mathematical methods for materials science

Dear students,

Welcome to MSE-487 !

For this first class on Monday, we will introduce the structure of the class and how it will be organized.

We will then start with some more theoretical aspects of algebra and number theory. We will see that despite the seemingly abstract nature of some of the concepts introduced, we can already find interesting practical use in engineering, and materials science. We will revisit in particular symmetries and point groups in crystallography, explaining what a group is as a mathematical object. 

On Tuesday, we will continue the class and Prof. Carter will give us a first computational tutorial on the visualization of symmetries computationally. There will be no exercise this week, we will do the exercise session the following week (on Tuesday).

In Week 2, we will continue introducing some basic notions of numbers including prime and co-prime numbers. We will apply these concepts to review the important notions of Miller indices, as well as crystal planes and directions. 

This week, we will start by reviewing reciprocal spaces. We will then discuss basic properties of rational, real and complex numbers. We wil in particular see how one can construct the set of complex numbers, their different representations and operations, and how it enables to simplify the handling of many trigonometric operations. We will see their use in engineering particularly regarding their importance in revealing phase and dissipation phenomena in wave propagation, and discuss X-ray diffraction as a way to define in a different way reciprocal spaces.

On Tuesday, there will be 15 mn (9:15-9:30) or so to finish the class if needed, and then exercises until 11h. 

Note that this week I put two set of exercises: a regular document of the exercises of the week, and additional ones. The regular exercises one is for you to apply directly the important concepts seen in class, and are more applied to concrete problems. The additional exercises are a bit more theoretical, or add other exercises to train on, that you can do now or when you review the material for the exam.

This week, we will start linear algebra that we will review over the next three weeks. Here, I put all the slides on this part for weeks 4, 5 and 6. For this particular week, we will start with an example of matrix manipulation last week and review matrix formalism and manipulation. We will then discuss the postulates of quantum mechanics and see the linear algebra formalism required to understand these postulates. We will then dive a bit deeper into the notions of basis and characteristic polynomials.

The following weeks, we will introduce notions such as unitary or self-adjoint operators, Hilbert spaces and spectral theorem, that are essential in many aspects of engineering and particularly in quantum technologies.

We will do two hours of class on Monday, and 2 hours of exercises on Tuesday, when we will finish exercise-related examples introduced in class.

This week we will revisit the linear algebra notions introduced in weeks 4 and 5, and give concrete examples with the translation operators and the Bloch theorem, as well as the Brillouin zone and the splitting of the energy levels. These are key notions in the fonctional properties of materials to understand.

We will have classes on Monday 3:15 pm to 4 pm, and then Prof. Carter will make a tutorial to explain and visualize  computationally these notions. This will be via Zoom, at 4:15 pm until 5 pm, at this address: mit.zoom.us/my/wcraigcarter

It is easier if you log on and follow the tutorial on your own computer but we can also project on the screen in MXF1 for those of you there without a computer.

This week, we will start to review concepts regarding functions: limits, continuity, differentiability and Taylor expansions. We will see many examples of their appearance in Engineering problems, particularly in binary phase diagrams and the Lennard-Jones potential.

Note that the recording of the tutorial from Prof. Carter last week on Bloch Waves and Brillouin zones is up on the video channel of the course.

This week we will continue visiting concepts of analysis regarding Taylor expansion, integration, parametric functions and multi-variable functions.

We will see an example of taylor expansion relating the the mechanical properties of materials. We will also discuss the concept of exact and inexact differentials for multi-variable functions often seen in the thermodynamics of materials.

This week we will continue visiting concepts of analysis regarding  multi-variable functions, and in particular we will derive the diffusion equation and introduce important results regarding manipulations of limits, derivatives and integration.

On Monday at 4:15 pm, Prof. Carter will give a tutorial on computational visualization of multi-variable functions concepts regarding extremum and saddle point, and the matrix approach to partial derivations.

You can join the tutorial via Zoom here: mit.zoom.us/my/wcraigcarter

On Tuesday during the second hour, to solve the diffusion equation, we will then introduce the Fourier transform. This week and next, we will revise the main properties of Fourier transform, and give examples of its use in solving differential equations, in reciprocal space and X-ray analysis.

This week we will continue our study of Fourier transform via its use in reciprocal space and x-ray analysis.

We will then introduce examples of ordinary differential equations via the Beer Lambert law and the Lorentz-Drude models regarding optical properties of materials. We will discuss general rules to solve these equations.

We will then introduce the Lapalce transform as an extension of the Fourier transform, and show how it can be used to solve differential equations. Visco-elastic materials will be discussed as part of the exercise session.

This week, we will continue looking at the application of probability and statistics concepts in statistical physics and solid state physics. We will in particular discuss the important result of having the same probability for each microstate for a micro-canonical ensemble. We will remind the concepts of entropy from a statistical point of view, and use it in the study of canonical and grand canonical ensembles. We will use the implications of these results on a fermion gas, and on the physics of semiconductors.

We will have classes on Monday and Tuesday this week. The exercise session will be on Monday next week. On Tuesday next week, we will conclude the term by summarizing the concepts reviewed.

This week on Monday will be two hours of exercise session

Tuesday class will start at 10:15 am (no class between 9:15 and 10 am). For this last class, we will summarize the important notions seen during the semester.

It will be via Zoom, you can join here: https://epfl.zoom.us/j/4258417268