Extrema of P-invariant functions on the Brillouin zone
ZONES DE BRILLOUIN
THEORIE DE MORSE
MAXIMUMS ET MINIMUMS
CRISTALLOGRAPHIE
MATHEMATIQUES
SYMETRIE
PHYSIQUE
Exapnded version of a lecture fiven at Naples, on October 25, 1991 at a Colloquium in memory of Léon Vanhove
Abstract : This paper studies the number of extrema (and their positions) of a countinuous Morse function on the Brillouin zone, when it is invariant by the point group symmetry of the crystal. Forty years ago, Vanhove had shown the importance of this problem in physics, but he could use only the crystal translational symmetry. In that case Morse theory predicts at least eight extrema. With the added use of general symmetry arguments we show that this number is larger for six of the 14 classes of Bravais lattices ; moreover it is possible to give the position of the extrema (and their nature) for 30 of the 73 arithmetic classes.This paper is written for a larger audience than that of solid state physicists ; it also defines carefully the necessary crystallographic concepts which are generally poorly understood in the solid state literature.
MICHEL
P/92/16
IHES
04/1992
A4
13 f.
EN
TEXTE
PREPUBLICATION
P_92_16.pdf
1992
Physical implications of crystal symmetry and time reversal
RESEAUX CRISTALLINS
SYMETRIE
ESPACE ET TEMPS
Lectures given at the international school on Symmetry and Structural Properties of Condensed Matter. August 28 - September 5, 1996, Zajaczkowo (Posnan), Poland.
MICHEL
P/96/80
IHES
12/1996
A4
13 f.
EN
TEXTE
PREPUBLICATION
P_96_80.pdf
1996
Symmetry in condensed matter physics
SYMETRIE
MATIERE CONDENSEE
MICHEL
P/81/58
IHES
11/1981
A4
8 f.
EN
TEXTE
PREPUBLICATION
P_81_58.pdf
1981
What is a Crystal ?
CRISTAUX
MATHEMATIQUES
PAVAGE
FONCTIONS
Abstract : An historical survey of our knowledge of crystal structure and of the mathematical problem of paving space with one or a few types of stones show that a crystal has not necessarily a translation lattice. The recently discovered crystals of alloys Al with 14% Mn have an ecosahedral symetry (incompatible qith translation lattice). Their electron micrographs show a structure similar to a Penrose tiling. It seems that long range order characterizing crystals can be obtained from projection of a thin slab of genuine crystal in higher dimension so that crystal physical properties can be described by quasi-periodic functions (as generalizing periodic ones) or may be almost periodic functions.
MICHEL
P/85/68
IHES
12/1985
A4
8 f.
EN
TEXTE
PREPUBLICATION
P_85_68.pdf
1985