Content:

  • Class 1: Discussion about classification of materials, meaning of structure, how symmetry is indispensable for classifying structure, a broader view, importance and implication of Symmetry - a short proof of Noether’s theorem in the case of translational symmetry resulting in conservation of linear momentum.
  • Class 2: a discussion on rotational symmetry, defining the rotational matrix for an arbitrary angle, solving the particular case of C4 (90degree rotations) and it’s multiples by taking into consideration their respective rotation matrices, how and why the properties of a group apply to all elements of the set C4, Multiplication table for cyclic groups C3 and C4.
  • Class 3: discussion on other point symmetry elements - Inversion, Reflection and improper rotation (Roto-reflection). Special emphasis and analysis of Roto-reflection and the sub groups of roto-reflections of even order.
  • Class 4: Product of two Symmetry operations, tracking all possible Symmetry elements of an eclipsed Ethane molecule, mapping out the complete multiplication table for eclipsed Ethane by performing consecutive symmetry operations on a ball and stick model - which turns out to be the Multiplication table (not character table) for the group D3h.
  • Class 5: Deriving all possible point groups and their orders. Groups discussed in this lecture - Cn, Cs, Ci, Cnh, Cnvband Dn.
  • Class 6: Discussion on groups Dnh and Dnd. Looking at the already worked out multiplication table of D3h and figuring out all subgroups within it.
  • Class 7: Going beyond Cn rotation axes and perpendicular C2 axes. Simplest example - structures having C3 axis with other perpendicular/non perpendicular C3 axes (Tetrahedron). C3 with C4 and C4 with C4 (Octahedron) and like wise all possible platonic solids. Working out all symmetry elements of a Tetrahedron forming the group Td.
  • Class 8: Figuring the pure rotation subgroup of a Tetrahedron, point group - T. And the group Th. Working out all symmetry operations in an Octahedron - the point group Oh. It’s pure rotation sub group O. For all symmetry operations of the next platonic solid - Cube. We find that Cube is a ‘DUAL’ of the Octahedron, and hence shares the same symmetry operations and same point group - Oh. Dodecahedron and Icosahedron are duals of each other. Figuring out Symmetry operations of each. Both belong to point group Ih. This marks the end of discussion on all possible point groups.

Notes

Quiz

  • Quiz 1 Marking Scheme
  • Quiz 1 Discussion
  • Marking Scheme details and common errors:
  • Computing the conjugacy classes correctly or showing that any cyclic group is Abelian and then proving that for Abelian groups the conjugacy classes are each element (Singleton sets).
  • Find Multiplicative Identity: many students missed this step. Note that if the operation on Integers is addition, Identity element is 0 whereas for multiplication it is 1. Identity element depends on the operation and not the set. Show that an inverse element may not exist for at least one element. Many students also confused associativity: (ab)c=a(bc) with commutativity: ab=ba. Groups don’t have to have commutativity- take all invertible n*n matrices they form a group but matrix multiplication is not commutative.
  • Showing all symmetry operations:{E, C2x,C2y,C2z,v,v’,h,S2} -2, Inversion i=S2 and hence not listed separately-1, point group-1.

Final Exam

X-Ray Diffraction (XRD)

Crystallography (Bragg’s Law)

  • tutorial questions

  • EVENT Notice: I would encourage all of you to attend the following outreach event organised by TIFR, Bombay on Crystallography. Chai and Why? is a series of talks that run as a part of TIFR’s Science Popularisation and Public Outreach Programme. This particular edition on crystal growth titled “Make it crystal clear!” has caught our attention and we can’t contain our excitement. The rumours are that they are going to grow some real crystals during the talk! With help from the kind folks in the technical council who are just as excited about this, we are organizing transport to and from the talk. Please register with this link https://forms.gle/yAuYUMbdGrPwwTsE8. ‘Make it crystal clear!’ by Ruta Kulkarni (TIFR). What does the word “crystal” bring to mind? Sugar crystals in your chai? Precious crystals in jewellery? Solar cells and mobile phones maybe? Or eggshells and chocolate? Crystalline materials are all around us, and especially single crystals are key to several of our 21st-century devices. Let’s take a look at what crystals are, how they form in nature, how they can be grown in the lab (or the kitchen!) and how they impact the world today. Of course, you can try your hand at growing some crystals during the session as well! About the speaker: Ruta Kulkarni works on the growth of single crystals of various materials at the department of condensed matter physics and materials science at TIFR.

Symmetry in Particle Physics

  • Particles and their interactions are governed by symmetries and breaking of symmetries ie. by symmetries with imperfections. The defects in the symmetries lead to some particles being heavy and some light, some interactions being strong and others weak, and to much else besides.

References:

  • MIT 3.60 Symmetry, Structure, Tensor Properties of Materials by Prof. Bernhardt Wuensch youtube videos, MIT-OCW page
  • Frank Hoffman and Michael Sartor (Universitat Hamburg) have made a brilliant youtube video series on symmetry in crystals, slides and pdfs on this site
  • Cotton Chemical applications of group theory
  • Elements of X-Ray Diffraction- Cullity
  • M. Artin Algebra Chapter 6 Symmetry
  • Used this doc maintained by me as a common platform to discuss doubts, problems and communicate.
  • Crystallography Minerals Web

Extra Questions

  • Show that if the order of a group is p where p is a prime number then the group must be cyclic.
  • Find out the relationship between Quaternion Group and rotational isometries of the cube. Use this to find the Automorphism group of Q8.

Extra Reading (beyond this course) for those interested in the subject

  • Notes Section on Group Theory and Symmetry