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 an example of how mathematics shape materials discovery, via the understanding of black body radiation from materials. 

We will then start to introduce 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, there will be no exercise this week, we will do the exercise session Tuesday next week.  

In Week 2 on Monday, during the first hour we will continue introducing some basic notions of numbers including divisibility, prime and co-prime numbers, and apply these concepts to Miller indices, as well as crystal planes and directions, discussed in Crystallography.  

During the second hour at 4:15 pm, Prof. Carter (MIT, USA) will give a 45 mn tutorial on the symmetry and crystal planes concepts discussed in class, that can be visualized via computational methods. 

The zoom link is the following: mit.zoom.us/my/wcraigcarter. You can bring your computer to follow the tutorial (easier). If needed, we can also project it in class. 

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. 

Remember: Monday 22nd is a day off, no class. 

The class will be on Tuesday. On Monday of the following week, we will finish the class and do the exercise session. 

This week, on Monday we will finish the class on complex numbers and see another application of this formalism to the understanding of reciprocal spaces via X-ray diffraction. In particular, we will derive the condition of Laue and the Bragg law. This will be between 15h15 and 15h35-40, followed by an exercise session on the exercises of week 3 until 5 pm. 

On Tuesday, 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 and review matrix formalism and manipulation. We will review operators and vector spaces of finite and infinite dimensions, and the concept of basis. 

The following weeks, 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. We will continue and 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.

This week we will discuss the spectral theorem and revisit the linear algebra notions introduced in weeks 4 and 5 to 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 the class on these notions on Tuesday. 

On Monday from 3:15 pm to 4 pm, Prof. Carter will teach a tutorial to introduce and visualize computationally these notions. This will be via Zoom at this address: mit.zoom.us/my/wcraigcarter

A regular exercises session will follow between 4 and 5 pm on Monday. 

This week on Monday, we will finish a last example on the opening of bandgap and the Brillouin zone edge that reviews the notions seen in weeks 04 to 06. We will then continue with an exercise session. 

On Tuesday, 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.

This week we will continue visiting concepts of analysis regarding  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 start to discuss the concept of exact and inexact differentials for multi-variable functions often seen in the thermodynamics of materials. 

We will have the class on these notions on Tuesday. 

On Monday from 3:15 pm to 4 pm, Prof. Carter will teach a tutorial to introduce and visualize computationally these notions. This will be via Zoom at this address: mit.zoom.us/my/wcraigcarter

A regular exercises session will follow between 4 and 5 pm on Monday. 

This week we will continue visiting concepts of analysis regarding  multi-variable functions, and look into extremums and saddle point on Monday, before doing the exercise session. 

On Tuesday, we will derive the diffusion equation and introduce important results regarding manipulations of limits, derivatives and integration. We will then introduce the Fourier transform, revise its main properties, and give examples of its use in solving differential equations, in reciprocal space and X-ray analysis (this week and next).

This week on Monday, we will continue our study of Fourier transform via its use in reciprocal space and x-ray analysis. We will then do the exercises session. 

On Tuesday, we will start by introducing 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 to illustrate the use of Laplace transforms in Materials Science. 

This week, on Monday we will finish the treatment of diffusion via a random walk approach by deriving the variance, and making the link with the exercises of the week. Then it will be a regular exercise session. 

On Tuesday, and to be continued next week, we will look 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.

This week on Monday we will continue the notions seen in class to solve exercise 1, and leave time for an exercise session and questions.  

Tuesday we will finish the class on the notions to solve exercise 2, and summarize the important notions seen during the semester.