1 / 17

MATERIALS SCIENCE & ENGINEERING

MATERIALS SCIENCE & ENGINEERING. Part of. A Learner’s Guide. AN INTRODUCTORY E-BOOK. Anandh Subramaniam & Kantesh Balani Materials Science and Engineering (MSE) Indian Institute of Technology, Kanpur- 208016 Email: anandh@iitk.ac.in, URL: home.iitk.ac.in/~anandh.

valentine-mcgee
Télécharger la présentation

MATERIALS SCIENCE & ENGINEERING

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. MATERIALS SCIENCE & ENGINEERING Part of A Learner’s Guide AN INTRODUCTORY E-BOOK Anandh Subramaniam & Kantesh Balani Materials Science and Engineering (MSE) Indian Institute of Technology, Kanpur- 208016 Email:anandh@iitk.ac.in, URL:home.iitk.ac.in/~anandh http://home.iitk.ac.in/~anandh/E-book.htm

  2. Symmetry of Solids • We consider the symmetry of some basic geometric solids (convex polyhedra). • Important amongst these are the 5Platonic solids(the only possible regular solids* in 3D):  Tetrahedron Cube  Octahedron(identical symmetry) Dodecahedron Icosahedron(identical symmetry)The symbol  implies the “dual of”. • Only simple rotational symmetries are considered (roto-inversion axes are not shown). • These symmetries are best understood by taking actual models in hand and looking at these symmetries. • Certain semi-regular solids are also frequently encountered in the structure of materials (e.g. rhombic dodecahedron). Some of these can be obtained by the truncation (cutting the edges in a systematic manner) of the regular solids (e.g. Tetrakaidecahedron, cuboctahedron) * Regular solids are those with one type of vertex, one type of edge and one type of face (i.e. ever vertex is identical to every other vertex, every edge is identical to every other edge and every face is identical to every other face)

  3. Symmetry of the Cube {4,3}* 4- fold axes pass through the opposite set of face centres3 numbers Pink2-fold Blue3-fold Yellow4-fold * The schläfli symbol for the cube is {4,3}  4-sided squares are put together in 3 numbers at each vertex

  4. The body diagonals are 3-fold axes (actually a 3 axis) 4 numbers

  5. 2-fold axes pass through the centres of opposite edges 6 numbers

  6. 3 mirrors Centre of inversion at the body centre of the cube

  7. Important Note: • These are the symmetries of the cube (which are identical to those present in the cubic lattice) • A crystal based on the cubic unit cell could have lower symmetry as well • A crystal would be called a cubic crystal if the 3-folds are NOT destroyed

  8. Symmetry of the Octahedron {3,4} • Octahedron has symmetry identical to that of the cube • Octahedron is the dual of the cube (made by joining the faces of the cube as below) 3-fold is along centre of opposite faces 2-fold is along centre of opposite edges 4-fold is along centre of opposite vertices Centre of inversion at the body centre

  9. Yellow4-fold Pink2-fold Blue3-fold

  10. Symmetry of the Tetrahedron {3,3} 3-fold connects vertex to opposite face 2-fold connects opposite edge centres No centre of inversion No 4-fold axis

  11. Yellow3-fold Pink2-fold Blue3-fold

  12. Symmetry of the Dodecahedron {5,3} Yellow5-fold Blue3-fold Pink2-fold

  13. Symmetry of the Icosahedron {3,5} Yellow5-fold Blue3-fold Pink2-fold

  14. Truncated solids • Certain semi-regular solids can be obtained by the truncation of the regular solids. • Usually truncation implies cutting of all vertices in a systematic manner (identically) • E.g. Tetrakaidecahedron, cuboctahedron can be obtained by the truncation of the cube. In these polyhedra the rotational symmetry axes are identical to that in the cube or octahedron. • Tetrakaidecahedron {4,6,6}:  Two types of faces: square and hexagonal faces Two types of edges: between square and hexagon & between hexagon and hexagon • Cuboctahedron {3,4,3,4}={3,4}2: Two types of faces: square and triangular faces Cuboctahedron Tetrakaidecahedron Cuboctahedron formed by truncating a CCP crystal

  15. Cuboctahedron Pink2-fold Yellow4-fold Blue3-fold

  16. Space filling solids • Space filling solids are those which can ‘monohedrally’ tile 3D space (i.e. can be put together to fill 3D space such that there is no overlaps or no gaps). • In 2D the regular shapes which can monohedrally tile the plane are: triangle {3}, Square {4} and the hexagon {5}. • The non regular pentagon can tile the 2D plane monohedrally in many ways. • The cube is an obvious space filling solid. None of the other platonic solids are space filling. • The Tetrakaidecahedron and the Rhombic Dodecahedron are examples of semi-regular space filling solids. Tetrahedral configuration formed out the space filling units Video: Space filling in 2D Video: Space filling in 3D

More Related