1. Symmetries in Superconductors By: Hugh Heldenbrand Yoon-Mi Kim January 15, 2001 Computational Chemistry Seminar

2. Problem Statement How can the different symmetries of a crystal be used to describe it?

3. Introduction Superconductors. State University of New York at Binghamton: Preparation, Structure and Properties of a High- temperature Superconductor (YBa 2 Cu 3 O 7 ). “Model the system using CAChe software...” (Observations and Questions). MacSpartan vs. CAChe (demo version)

4. Computational Method Our program was CAChe, “the happy union on computational chemistry and highly sophisticated graphics.” (CAChe 4.1 A Chemist’s Guide) When we went to enter the space group and fractional coordinates for YBa 2 Cu 3 O 7 we got an error message that we thought was related to the fact that we were using demo version software.

7. Computational Method This is the point where the computer would say “path not found for space groups.”

9. Point Group Symmetries There are four main types of symmetry operations: Rotation (C n )--the object appears identical if rotated about an axis by  = 360/n = 2  /n degrees. Improper Rotation (S n )—a combination of a rotation and a mirror plane reflection perpendicular to the axis of rotation.

10. Point Group Symmetries Inversion (center of symmetry, i)--each point in the object is converted to an identical point by projecting through a common center and extending an equal distance beyond this center. Reflection (mirror plane,  )--each point in the object is converted to an identical point by projecting through a mirror plane and extending an equal distance beyond this plane.

13. Space-Group Symmetries 230 space groups The International Union of Crystallography publishes them in Volume A of International Tables for Crystallography.

14. The Herman-Mauguin System The space group of YBa 2 Cu 3 O 7 is: P mmm, a= 3.820, b=3.886, c=11.683

15. Herman-Mauguin System The first letter identifies the centering of the lattice P = Primitive I = Body centered F = Face centered C = C-centered B = B-centered A = A-centered Our crystal is P, so it has a primitive Bravais lattice (there are no atoms outside the eight that make up the corners).

16. Herman-Mauguin System The “mmm” in the space group for YBa 2 Cu 3 O 7 means that it has three “mirror plane” transitions. A mirror plane is simply a plane through which the crystal can be reflected identically on both sides.

17. Herman-Mauguin System The a= 3.820, b=3.886, c=11.683 in the space group gives the dimensions of the crystal. Since a, b, and c are all different numbers, the crystal is orthorhombic.

18. Database of Superconductors Here is a website with 3-D models of superconductors, which could be produced with a program like CAChe. http://barns.ill.fr/dif/xtal-3d.super.html

19. Conclusion Why does a crystal’s symmetry matter?

20. Bibliography http://www.chem.ox.ac.uk/icl/heyes/structure_of_ solids/Lecture4/Lec4.html http://imr.chem.binghamton.edu/labs/super/superc. html http://barns.ill.fr/dif/xtal-3d.super.html Huheey, James et al. Inorganic Chemistry: Principle of Structure and Reactivity. New York: Harper Collins, 1993.