Structure and classification of band representations
CRISTALLOGRAPHIE
SOLIDES
ASTRONOMIE
ORBITE
ETOILES
MODELISATION
ENERGIE
ATOMES
Abstract : Band representaitons in solids are investigated in the general framework of induced representations by using the concepts of orbits (stars) and strata (Wyckoff positions) in their construction and classification. The connection between band representations and irreducible representations of space groups is established by reducing the former in the basis of quasi-Bloch functions wich are eigenfunctions of translations but ar not, in general, eigenfunctions of the Hamiltonian. While irreducible representations of space group ar fnite-dimensional and are induced from infinite-order little groups Gk for vectors K in the Brillouin zone, band representations are infinite-dimensional adn are induced from finite-order little groups Gr for vectors r in the Wigner-Seitz cell. This connection between irreductible representations and band representations of space groups shedd new light on the duality properties of the Brillouin zone and the Wigner-Seitz cell. As an introduction to band representations the induced representations of point groups wich id applied to the investigation of th equivalency of band representations. Based on this connection and on the properties of the crystallographic point groups a necessary condition id established for the inequivalency of band representations induced from maximal isotropy groups. For using this condition there is need fro the induced representaitons of oint groups and a full list of them is given in the paper. One is especially interested in irreducible-band representations which form the elementary building bricks for band representations. From the point of view of the physics, irreducible-band representations correspond to energy bands with minimal numbers of branches. A method id developped for finding all the inequivalent irreducible-band representations of space groups by using the induction from maximal isotropy groups. As a rule the latter leads to inequivalent irreducible-band representations. There are, however, few exceptions to this rule. A full list of such exceptions is tabulated in the paper. With this list at hand one can construct all the different irreducible-band representations of 2-dimensional space groups. For them we list the continuity chords of all their irreducible-band representations.
MICHEL
BACRY
ZAK
P/86/35
IHES
06/1986
A4
46 f.
EN
TEXTE
PREPUBLICATION
P_86_35.pdf
1986
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
Complete description of the Voronoï cell of the Lie Algebras An weight lattice. On the Bounds for the number of d-faces of the n-dimensional Voronoï cells
ALGEBRES DE LIE
POLYGONES DE VORONOI
Version étendue d'une conférence faite le 9 janvier 1997 au Centre de Recherches Mathématiques de l'Université de Montréal au cours du colloque pour le 60e anniversaire de Jiri Patera et Pavel Winternitz.
Abstract :Denoting these bounds by Nd(n), 0 ? d ? n we prove that Nd(n)/(n + 1)! is a polynomial Pd(n) of degree d with rational coefficients. We give the polynomials for d ? 5 explicitly. The proof uses the fact that these bounds Nd(n) are also the number of d-faces of the Voronoï cell of the weight lattice of the Lie algebra An (it is also the Cayley diagram of the symmetric group Sn+1, which is isomorphic to the Weyl group of An). Each d-face of this cell is a zonotope that can be defined by a symmetry group ~ G d (?), (d-dimensional reflection subgroup of the A n Weyl group. We show that for a given d and n large enough, all such subgroups of A n are represented, and we compute explicitly N(G d (?),n) the number of d-faces of type G d (?) in the Voronoï cell of L = A w n. The final result is obtained by summing over a. That also yields the simple expression Nd(n)=(n+1?d)!S(n+1?d)n+1Nd(n)=(n+1?d)!Sn+1(n+1?d) where the last symbol is the Stirling number of second kind.
MICHEL
P/97/53
IHES
07/1997
A4
11 f.
EN
TEXTE
PREPUBLICATION
P_97_53.pdf
1997
Bravais classes, Voronoï cells, Delone symbols
CRISTALLOGRAPHIE
MATHEMATIQUES
GROUPES SPATIAUX
POLYGONES DE VORONOI
CLASSES DE BRAVAIS
SYMBOLES DE DELONE
This text correspond to a set of lectures given at Third international school at Zajaceskowo (Poznan) Polane on Symmetry and structural Properties of condensed matter
Abstract : We give the most refined intrinsic classification of three dimensional Euclidean lattices by combining the 14 Bravais classes, the 5 combinatorial types of Voronoï cells and the 24 Delone symbols. After recalling the fondamental concepts of group actions, we define Bravais classes and Voronoï celles in arbitrary dimension. We are quite explicit for the application to two and three dimensions.
MICHEL
P/94/63
IHES
12/1994
A4
21 f.
EN
TEXTE
PREPUBLICATION
P_94_63.pdf
1994