Time reversal symmetry and

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Presentation transcript:

Time reversal symmetry and PHY 745 Group Theory 11-11:50 AM MWF Olin 102 Plan for Lecture 28: Time reversal symmetry and Magnetic groups Chap. 16 in DDJ Effects of time reversal symmetry on point groups and space groups Effects of magnetic configurations on point groups and space groups Note: These slides contain materials from an electronic version of the Dresselhaus textbook. 3/31/2017 PHY 745 Spring 2017 -- Lecture 28

3/31/2017 PHY 745 Spring 2017 -- Lecture 28

Time reversal symmetry in physics: -t  t 3/31/2017 PHY 745 Spring 2017 -- Lecture 28

Properties of the time-reversal symmetry 3/31/2017 PHY 745 Spring 2017 -- Lecture 28

For a wavefunction not involving electron spin More details For a wavefunction not involving electron spin Krammer’s theorem: All energy levels of a system containing an odd number of electrons must be at least doubly degenerate (unless magnetic fields remove the degeneracy. 3/31/2017 PHY 745 Spring 2017 -- Lecture 28

Effects of time reversal symmetry on point groups Wigner’s possibilities: D and D* are equivalent to the same real irreducible representation. D and D* are inequivalent. D and D* are equivalent but cannot be made real. 3/31/2017 PHY 745 Spring 2017 -- Lecture 28

Wigner’s possibilities: D and D* are equivalent to the same real irreducible representation. D and D* are inequivalent. D and D* are equivalent but cannot be made real. Example: 3/31/2017 PHY 745 Spring 2017 -- Lecture 28

Some details: 3/31/2017 PHY 745 Spring 2017 -- Lecture 28

Time reversal symmetry effects on the group of the wavevector Possibilities according to C. Herring: 3/31/2017 PHY 745 Spring 2017 -- Lecture 28

Effects of time reversal symmetry on a Bloch state 3/31/2017 PHY 745 Spring 2017 -- Lecture 28

3/31/2017 PHY 745 Spring 2017 -- Lecture 28

Examples: 3/31/2017 PHY 745 Spring 2017 -- Lecture 28

Examples: 3/31/2017 PHY 745 Spring 2017 -- Lecture 28

3/31/2017 PHY 745 Spring 2017 -- Lecture 28

Some details 3/31/2017 PHY 745 Spring 2017 -- Lecture 28

3/31/2017 PHY 745 Spring 2017 -- Lecture 28

Magnetic space groups 3/31/2017 PHY 745 Spring 2017 -- Lecture 28

Example 3/31/2017 PHY 745 Spring 2017 -- Lecture 28

3/31/2017 PHY 745 Spring 2017 -- Lecture 28

Types of magnetic point groups total=122 3/31/2017 PHY 745 Spring 2017 -- Lecture 28