Crystals Symmetry and Tensors

By Prof. Rajesh Prasad   |   IIT Delhi
Learners enrolled: 72
Crystallography is fundamental to physics, chemistry, metallurgy, materials science, mechanical engineering, chemical engineering, and any other disciplines dealing with solid state. Despite this importance, due to constraints in design of curriculua a full course is rarely available on this topic. The present course intends to fill this gap. It will begin with basic fundamentals and take the students to advanced concepts of point and space groups symmetry and vector and tensor applications in crystallography.

INTENDED AUDIENCE: Students interested in solid state science and engineering will be benefited. UG and PG students of Physics, Chemistry, Metallurgy, Materials Science, Mechanical Engineering, Chemical Engineering will find the course useful.
Course Status : Upcoming
Course Type : Elective
Duration : 12 weeks
Start Date : 23 Jan 2023
End Date : 14 Apr 2023
Exam Date : 30 Apr 2023 IST
Enrollment Ends : 30 Jan 2023
Category :
  • Metallurgy and Material science & Mining Engineering
Credit Points : 3
Level : Undergraduate/Postgraduate

Page Visits

Course layout

Week 1: Geometrical crystallography: crystals, lattice and motif, lattice points and translation vectors, unit cells: primitive and non-primitive. Miller indices of planes and directions.
Week 2: Reciprocal lattice, use of reciprocal lattice in crystallographic calculations, angle between planes, angle between a plane and a direction, d-spacing of planes, volume of a crystal. Structure and metric matrices.
Week 3: Concept of Symmetry, Symmetry operations. Proper and Improper operations. Point and Space operations. Rotation axes: Roto-reflection and Roto-inversion axes. Mirror and glide planes. Inversion centre. Screw axes.
Week 4: Definition and properties of mathematical groups. Subgroups. Cosets. Lagrange’s theorem.
Week 5: Stereographic projection and matrix representation of symmetry operations.
Week 6: Symmetry based Classification of lattices: 7 crystal systems and 14 Bravais lattices. 17 plane groups.
Week 7: Symmetry based classification of crystals: 32 Crystal classes and 230 space groups. Possible proper rotation axes for crystals. Possible combination of pure rotation axes: Euler’s construction.
Week 8: Development of 32 point groups.
Week 9: Space groups and International Tables.
Week 10: Cartesian tensors: definition, rank, representation quadric, magnitude of a property in a given direction.
Week 11: Second-rank tensor properties: electrical and thermal conductivity, thermal expansion coefficient.
Week 12: Elasticity and fourth rank tensors. Elastic stiffness and compliance. Calculation of useful elastic properties such as modulus, Poisson ratio etc using tensor methods.

Books and references

1. Aroyo, M.I., International Tables for Crystallography, 6th. Edn., Wiley, 2016.
2. Hammond, C., The Basics of Crystallography and Diffraction, 4th. Edn., Oxford, 2015.
3. Ladd, M.F.C., The Essence of Crystallography, World Scientific, 2019.
4. Nye, J.F., Physical Properties of Crystals: Their Representation by Tensors and Matrices, Oxford University Press, 1985 .

Instructor bio

Prof. Rajesh Prasad

IIT Delhi
Professor Rajesh Prasad (B.Tech., IIT-BHU, Varanasi; M.E., IISc, Bangalore; Ph.D., Cambridge) began teaching Materials Science at University of Cambridge as a graduate teaching assistant for the undergraduate course Crystalline Materials. He now has about three decades of experience in teaching materials science courses at both undergraduate and graduate levels at the Indian Institutes of Technology, at Varanasi, Kanpur and Delhi. He received A.A. Krishnan Gold Medal for his M.E. thesis at the Indian Institute of Science, Bangalore. He has been awarded a Teaching Excellence Award in 2012 by the Indian Institute of Technology Delhi. In 2013, he received the Distinguished Alumnus Award of the Department of Metallurgical Engineering, IIT-BHU, Varanasi.

Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams: 30 April 2023 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
Registration url: Announcements will be made when the registration form is open for registrations.
The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.


Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Madras. It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team

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