ICSD (Inorganic Crystal Structure Database) http://www.fiz- karlsruhe.de/icsd.html/; NAVY http://cst-www.nrl.navy.mil/lattice/ • Crystallography Open Database http://www.crystallography.net/ How to visualize them? • VESTA (Visualization for Electronic and Structural Analysis) http://jp-minerals.org/vesta/en/ • XCrysDen (X-window CRYstalline Structures an DENsities) http://www.xcrysden.org/ Some useful crystallographic webs
programs and utilities free of charge? • Bilbao Crystallographic Server http://www.cryst.ehu.es/ • ISOTROPY http://stokes.byu.edu/iso/isotropy.htm/ Bilbao Crystallographic Server Database from International Tables for Crystallography (Volumes A, A1,E)+ database on incommensurate structures, and a k-vector database with Brillouin-zone figures and classification tables of the wave vectors. Some useful crystallographic webs
application field related to group theory, crystallography and solid-state physics applications. Interface tabulated data. Standard and default settings. Input: space group number. GENPOS: generators and general positions in different settings. Bilbao Crystallographic Server
letters, multiplicities and symbols of the site-symmetry groups. HKLCOND: gives access to general and special reflection conditions. MAXSUB: gives access to the maximal subgroups of a given SG, characterized by the index and transformation matrix- column pair (P, p) that relates the standard bases of group and subgroup. KVEC: wave-vector database and Brillouin zone figures. (CDML: primitive reciprocal vectors, ITA: conventional reciprocal vectors). Different lattice parameters relations can lead to different Brillouin-zone figures. Time for exercises 1, 2 and 3. Bilbao Crystallographic Server
all, read the basics on subgroup-group relationships... SUBGROUPGRAPH: retrieves all possible group-subgroup chains of intermediate maximal subgroups that relate the group G and the subgroup H and a graphical representation. Example: group-subgroup relations between G=P622 No. 177, and H=C2, No. 5. a) without index: table with the possible intermediate space groups and the indices. b) with index: the matrix transformations are also given. WYCKSPLIT: lists the splittings of the Wyckoff positions from G to H. You need to know the matrix transformation. Bilbao Crystallographic Server
compatible with a specific multiple of the unit cell of G (ik index). COMMONSUBS: searches for common subgroups H between two space groups G1 and G2 without group- subgroup relation. Input: G1,G2,Z1,Z2 and index (the number of formula units in H is the same for both structures). Go to exercise 4 APPLICATION: STUDY OF PHASE TRANSITIONS Based on symmetry aspects: 1) Group-subgroup related: structures related by small atomic displacements (displacive in Buerger’s notation). Bilbao Crystallographic Server
of the low symmetry to the high-symmetry (η=0) structure. The low-symmetry phase approaches the transition to higher symmetry continuously. T-driven transition: usually the symmetry of the l.t. phase is a subgroup of that of the h.t. phase. p-driven transition: not known. Bilbao Crystallographic Server
quite abrupt (reconstructive) (no order parameter). One phase transits in a continuous way to the other following a transition path (atomic displacements + lattice strains). Transition involves an intermediate structure whose space group is subgroup of the two end phases. Order parameter or reaction coordinate: a degree of freedom of the intermediate structure. Symmetry restrictions: 1) The number of formula units of the intermediate structure can not change along the path. 2) Atoms must remain in the same types of Wyckoff positions along the path. Bilbao Crystallographic Server
and lattice strains are in general favoured (read the notes on structural analysis of a transition). Energetics: A symmetry analysis cannot predict the energetically most favourable transition path. It requires to explore the energy landscape and, in particular, the energy barrier that separates both phases. 1) A crude approximation: interpolation of the structural parameters (defined in the intermediate structure) of the limit phases and calculation of the Gibbs free energy along the path. Bilbao Crystallographic Server
(atomic coordinate, lattice parameters ratio...). Example: B3/B1 phase transition in Zn0, ZnS and SiC under p. B3: zinclende (F4-3m) Z=4; M ¼ ¼ ¼ (4c) X 0,0,0 (4a) aI B1: rocksalt (Fm-3m) Z=4; M ½ ½ ½ (4b) X 0,0,0 (4a) aII Intermediate state: R3m Z=1 M x,x,x (3a) X 0,0,0 (3a) Order parameter x(M) (¼ -- ½) B3: aR =aI /2 αR = 60º B1: aR =aII /2 αR = 60º Go to exercises 5, 6 and 7. Bilbao Crystallographic Server
space groups G1 and G2 without group-subgroup relation. (COMMONSUBS). Every path is checked for compatibility of the Wyckoff position splittings (WYCKSPLIT). The WP’s occupied by a given atom in G1 and G2 must give rise to the same WP por that atom in H. The lattice strain (STRAIN) and atomic shifts are compared to threshold values. INPUT: Z1,Z2, description of the G1 and G2 structures (space group, lattice parameters, number, type and coordinates of atomic positions, maximum strain and atomic shifts allowed and maximum cell multiplication. Go to exercise 8 and 9. Bilbao Crystallographic Server
irreducible representations of the Γ-point vibrational modes. In a crystal, 3n phonons (n= number of atoms in the primitive cell), 3 acoustic ones. site species for atomic displacements (Tx,Ty,Tz) in site group Bilbao Crystallographic Server site species for lattice vibrations in the crystal Γcrys= Γeq. set 1 + Γeq. set 2 + ….. Γvib = Γcrys - Γacous INPUT: space group (factor group) and occupied WPs. CORRELATION TABLES
structural analysis. Developers: Koichi Momma & Fujio Izumi. Runs on Windows, Mac OS X and Linux. Input files: 42 formats: CIF, ICSD, PDB...Output in *.VESTA format. Exports graphic-data files. To run just type VESTA in the folder containing the executable. It opens the MAIN WINDOW consisting of: Menu bar: “File”, “Edit”, “View”, “Objects”, “Utilitites”, “Help”. Horizontal bar: tools for viewing along axis, rot+trans Vertical bar: tools for selecting atoms, distances, angles... Graphical area: the structures are displayed. Text area: information on the structures. VESTA
(to create a new structure) or Open (to edit a given structure). A dialog box appears with several tabs. Unit cell tab: lattice parameters & symmetry. Clicking the Option button a transformation matrix can be selected. Structure parameters tab: symbols, labels, charges, x,y,z, occupancies of atoms. To search for bonds: Edit Bonds. To add a new bond specification, click the New botton, select atoms to bond and minimun and maximum lenghts. Several searches modes Lattice planes: Edit Lattice planes. VESTA
bonded to A1” or “Search atoms bonded to A1” should be selected. A1: central atom. Objects tab: Structural models: ball-stick, space filling polyhedra..... Properties: General (unit cell, axis...), atoms (resolution, atom style, radius, color), bonds(resolution, style), polyhedra (types), isosurfaces (isosurface level). “File” menu and: Save (.VESTA) format, export images (raster, vector). Go to exercises 9 and 10. VESTA
Lewis picture: Carbon core, K-shell Double bond, B(C=O). Multiplicity determined by the number of basins, not the charge Oxygen core, K(O) Oxygen lone pairs, LP(O). O-C=O↔O=C=O 53% 47% - + Implementation in solids
define Ionic properties (superbasins) →similar to QTAIM Absence of bond basins Only closed-shell basins, spherical and complete charge transfer : K(Na) (≈2 ē),L(Na) (≈8 ē) K(F) (≈2 ē),L(F) (≈8 ē) Implementation in solids