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.]]>

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.

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

IHES

IHES

MICHEL

https://repo-archives.ihes.fr/FONDS_IHES/I_Prepublications/MICHEL/1990-1999/P_92_16/P_92_16.pdf

Oui

Bures-sur-Yvette

BRILLOUIN

MORSE

VANHOVE

VAN HOVE

BRAVAIS

MONTROLL

SMOLLETT

HIGGS

RYDBERG

HESSE

BETTI

POINCARE

EULER

RIEMANN

EUCLIDE

GRAMM

JORDAN

HERMANN

MAUGUIN

FEDOROV

SCHONFLIES

ABEL