unit_cell - Part of MATERIALS SCIENCE AALearners...

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MATERIALS SCIENCE & ENGINEERING ENGINEERING Anandh Subramaniam & Kantesh Balani Materials Science and Engineering (MSE) Indian Institute of Technology, Kanpur- 208016 Email: [email protected], URL: home.iitk.ac.in/~anandh AN INTRODUCTORY E-BOOK AN INTRODUCTORY E-BOOK Part of http://home.iitk.ac.in/~anandh/E-book.htm http://home.iitk.ac.in/~anandh/E-book.htm A Learner’s Guide A Learner’s Guide A Learner’s Guide A Learner’s Guide
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UNIT CELLS (UC) UNIT CELLS (UC) A unit cell (also sometimes causally referred to as a cell) is a representative unit of the structure. which when translationally repeated (by the basis vector(s) ) gives the whole structure. The term unit should not be confused with ‘having one’ lattice point or motif (The term primitive or sometimes simple is reserved for that). If the structure is a lattice, the unit cell will be unit of that (hence will have points* only). If the structure under considerations is a crystal , then the unit cell will also contain atoms (or ions or molecules etc.). Note: Instead of full atoms (or other units) only a part of the entity may be present in the unit cell (a single unit cell) The dimension of the unit cell will match the dimension of the structure**: If the lattice is 1D the unit cell will be 1D, if the crystal is 3D then the unit cell will be 3D, if the lattice is nD the unit cell will be nD. Unit cell Crystal Lattice of a Will contain lattice points only Will contain entities which decorate the lattice * Strangely in crystallography often we even ‘split a point’ (and say that 1/8 th belongs to the UC). ** One can envisage other possibilities– e.g. a 2D motif may be repeated only along one direction (i.e. the crystal is 3D but the repeat direction is along 1D)
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ADDITIONAL POINTS A cell is a finite representation of the infinite lattice/crystal A cell is a line segment (1D) or a parallelogram (2D) or a parallelopiped (3D) with lattice points at their corners This is the convention If the lattice points are only at the corners, the cell is primitive. Hence, a primitive unit cell is one, wherein the lattice points are only at the corners of the unit cell (or the ends of a line segment unit cell in 1D) If there are lattice points in the cell other than the corners, the cell is non-primitive . Why Unit Cells? Instead of drawing the whole structure I can draw a representative part and specify the repetition pattern. Consider an infinite pattern made of squares This can be thought of as a single square repeated in x and y directions This way the infinite information content of a crystal can be reduced to the information required to specify the contents of a unit cell (along with the lattice translation vectors).
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In general the following types of unit cells can be defined: Primitive unit cell Non-primitive unit cells Voronoi cells Wigner-Seitz cells
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1D
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