Selected Topics in Condensed

Matter Physics: Quasicrystals

0321.4076

Dan Shechtman's discovery of quasicrystals, over 15 years ago, has spawned an exciting new field of research which today is being actively pursued by hundreds of scientists worldwide. This new phase of ordered matter—a quasiperiodic crystal—has completely changed our understanding of the most basic concepts of condensed matter physics, such as crystallinity, long range order, and symmetry. With promising new industrial applications, and with many unexplained experimental observations, this field of research is sure to stay at the forefront of science for years to come.

Ho-Mg-Zn single crystal.
I.R. Fisher et al., PRB 59 (1999) 308-321.

The course is intended as an introduction to this young field for students of varied backgrounds. Basic knowledge of condensed matter physics is recommended, but all important concepts will be reviewed. By the end of the course students should be able to understand current literature and begin their own research in the field. Students from other schools (math, chemistry, engineering, etc.) are encouraged to take the course.

Lecturer

Dr. Ron Lifshitz
Shenkar Physics building, Room 310
Phone: 03-6405145
E-mail: ronlif@post.tau.ac.il

Lectures

Tuesdays, 13:00-16:00
Shenkar Physics building, Room 204
Excluding: Purim (21/3/2000), Pesach (18,25/4/2000), and Memorial Day (9/5/2000).

Syllabus

Introduction: a bit of history; forbidden symmetries (the crystallographic restriction); the new definition of 'crystal'; phenomenology of incommensurate crystals, quasicrystals, and approximants.
Long range order: order parameters; Landau mean field theory; symmetry braking; Goldstone's "theorem"; the connection with diffraction theory.
Quasiperiodic order: point sets; 1-dimensional tilings (the "Fibonacci sequence"); 2-dimensional tilings (the "Penrose tiling"); the use of substitution rules, matching rules, cut and projection, and multigrids for generating quasiperiodic tilings; calculating the Fourier transform.
Symmetry: review of relevant concepts from group theory; symmetry of periodic and quasiperiodic crystals, high-dimensional descriptions (the "superspace approach"), the notion of indistinguishability, and the "Fourier space approach"; color and magnetic symmetry.
Physical models of quasiperiodicity: the Frenkel-Kontorova model; Landau mean field theory; ANNNI spin model; Peierls instability and charge density waves; Faraday waves.
Models of quasicrystals: atomic decorations of: perfect tilings, random tilings, and overlapping tilings; and the question of growth and stability of quasicrystals.
Structural analysis: crystallography: Patterson analysis; the "phase problem"; microscopy.
Physical properties: the Schroedinger equation with quasiperiodic potentials: algebraic decay of the wave functions; elementary excitations: electrons, phonons, and phasons; transport properties; magnetism; elasticity; defects.

Bibliography

Main textbooks

M. Senechal, Quasicrystals and geometry, (Cambridge University Press, 1996).
C. Janot, Quasicrystals: A primer, Second edition, (Oxford University Press, 1997).
Z.M. Stadnik (Editor), Physical properties of quasicrystals, (Springer Verlag, 1999).

Additional books on quasicrystals

P.J. Steinhardt and S. Ostlund (Editors), The physics of quasicrystals, (World Scientific, 1987).
M.V. Jaric (Editor), Introduction to quasicrystals, (Academic Press, 1988).
M.V. Jaric (Editor), Introduction to the mathematics of quasicrystals, (Academic Press, 1989).
P.J. Steinhardt and D.P. DiVincenzo (Editors), Quasicrystals: The state of the art, (World Scientific, 1991).
F. Hippert and D. Gratias (Editors), Lectures on quasicrystals, (Editions De Physique, 1994).
F. Axel and D. Gratias (Editors), Beyond quasicrystals, (Springer, 1995).

General background in condensed matter physics

N.W. Ashcroft and N.D. Mermin, Solid state physics, (Saunders College, 1976).
P.M. Chaikin and T.C. Lubensky, Principles of condensed matter physics, (Cambridge University Press, 1995).
R.H. Silsbee and J. Drager, Simulations for solid state physics: An interactive resource for students and teachers, (Cambridge University Press, 1997).

General background in symmetry and crystallography

H. Weyl, Symmetry, (Princeton University Press, 1952).
M. Senechal, Crystalline symmetries: An informal mathematical introduction, (Adam Hilger, 1990).

Course requirements

Students are NOT required to hand in their homework assignments, but they may do so if they wish.
Students are required to write one paper on some aspect of quasicrystals.
No exam will be given.

Homework assignments (in postscript): 1 (22 February, 2000), 2 (29 February, 2000), 3 (7 March 2000), 4 (14 March 2000), 5 (28 March 2000), 6 (11 April 2000).

For more information

Visit the Home page of Dr. Lifshitz, or
E-mail him at ronlif@post.tau.ac.il

Last updated: April 13, 2000, by Ron Lifshitz.