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THE “MOST IMPORTANT” CRYSTAL STRUCTURES

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NOTE!! Much of the discussion & many figures in what follows was again constructed from lectures posted on the web by Prof. Beşire GÖNÜL in Turkey. She has done an excellent job of covering many details of crystallography & she illustrates her topics with many very nice pictures of lattice structures. Her lectures on this are posted Here: Her homepage is Here: THE “MOST IMPORTANT” CRYSTAL STRUCTURES

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3 THE “MOST IMPORTANT” CRYSTAL STRUCTURES Sodium Chloride Structure Na + Cl - Cesium Chloride Structure Cs + Cl - Hexagonal Closed-Packed Structure Diamond Structure Zinc Blende

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1 – Sodium Chloride Structure Sodium chloride also crystallizes in a cubic lattice, but with a different unit cell. The sodium chloride structure consists of equal numbers of sodium & chlorine ions placed at alternate points of a simple cubic lattice. Each ion has six of the other kind of ions as its nearest neighbors.

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NaCl Structure

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This structure can also be considered as a face-centered-cubic Bravais lattice with a basis consisting of a sodium ion at 0 and a chlorine ion at the center of the conventional cell, at position LiF, NaBr, KCl, LiI, have this structure. The lattice constants are of the order of 4-7 Angstroms.

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Take the NaCl unit cell & remove all “red” Cl ions, leaving only the “blue” Na. Comparing this with the FCC unit cell, it is found to be that they are identical. So, the Na ions are on a FCC sublattice. NaCl Structure

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NaCl Type Crystals

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2 - CsCl Structure

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Cesium Chloride, CsCl, crystallizes in a cubic lattice. The unit cell may be depicted as shown. (Cs+ is teal, Cl- is gold) Cesium Chloride consists of equal numbers of Cs and Cl ions, placed at the points of a body-centered cubic lattice so that each ion has eight of the other kind as its nearest neighbors. 2 - CsCl Structure

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The translational symmetry of this structure is that of the simple cubic Bravais lattice, and is described as a simple cubic lattice with a basis consisting of a Cs ion at the origin 0 and a Cl ion at the cube center CsBr & CsI crystallize in this structure.The lattice constants are of the order of 4 angstroms. CsCl Structure

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8 cells CsCl Structure

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CsCl Crystals

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The Ancient “Periodic Table”

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4 - Diamond Structure The Diamond Lattice consists of 2 interpenetrating FCC Lattices. There are 8 atoms in the unit cell. Each atom bonds covalently to 4 others equally spaced about a given atom. The Coordination Number = 4. The diamond lattice is not a Bravais lattice. C, Si, Ge & Sn crystallize in the Diamond structure.

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Diamond Lattice The Cubic Unit Cell Diamond Lattice

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The Zincblende Structure has equal numbers of zinc and sulfur ions distributed on a diamond lattice, so that Each has 4 of the opposite kind as nearest-neighbors. This structure is an example of a lattice with a basis, both because of the geometrical position of the atoms & because two types of atoms occur. Some compounds with this structure are: AgI, GaAs, GaSb, InAs, – Zinc Blende or ZnS Structure

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Zincblende (ZnS) Lattice Zincblende Lattice The Cubic Unit Cell

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Diamond & Zincblende Structures A brief discussion of both of these structures & a comparison. These two are technologically important structures because many common semiconductors have Diamond or Zincblende Crystal Structures They obviously share the same geometry. In both structures, the atoms are all tetrahedrally coordinated. That is, atom has 4 nearest-neighbors. In both structures, the basis set consists of 2 atoms. In both structures, the primitive lattice Face Centered Cubic (FCC). In both the Diamond & the Zincblende lattice there are 2 atoms per fcc lattice point. In Diamond: The 2 atoms are the same. In Zincblende: The 2 atoms are different.

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Diamond & Zincblende Lattices Diamond Lattice The Cubic Unit Cell Zincblende Lattice The Cubic Unit Cell Other views of the cubic unit cell

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A view of the tetrahedral coordination & the 2 atom basis Zincblende & Diamond Lattices Face Centered Cubic (FCC) lattices with a 2 atom basis

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The Wurtzite Structure A structure related to the Zincblende Structure is the Wurtzite Structure Many semiconductors also have this lattice structure. In this structure there is also Tetrahedral Coordination Each atom has 4 nearest-neighbors. The Basis set is 2 atoms. Primitive lattice hexagonal close packed (hcp). 2 atoms per hcp lattice point. A Unit Cell looks like

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The Wurtzite Lattice Wurtzite Lattice Hexagonal Close Packed (HCP) Lattice + 2 atom basis View of tetrahedral coordination & the 2 atom basis.

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Diamond & Zincblende crystals The primitive lattice is FCC. The FCC primitive lattice is generated by r = n 1 a 1 + n 2 a 2 + n 3 a 3. The FCC primitive lattice vectors are: a 1 = (½)a(0,1,0), a 2 = (½)a(1,0,1), a 3 = (½)a(1,1,0) NOTE: The a i ’s are NOT mutually orthogonal! Diamond: 2 identical atoms per FCC point Zincblende: 2 different atoms per FCC point Primitive FCC Lattice cubic unit cell

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Wurtzite Crystals The primitive lattice is HCP. The HCP primitive lattice is generated by r = n 1 a 1 + n 2 a 2 + n 3 a 3. The hcp primitive lattice vectors are: a 1 = c(0,0,1) a 2 = (½)a[(1,0,0) + (3) ½ (0,1,0)] a 3 = (½)a[(-1,0,0) + (3) ½ (0,1,0)] NOTE! These are NOT mutually orthogonal! Wurtzite Crystals 2 atoms per HCP point Primitive HCP Lattice: Hexagonal Unit Cell Primitive Lattice Points

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Each of the unit cells of the 14 Bravais lattices has one or more types of symmetry properties, such as inversion, reflection or rotation,etc. SYMMETRY INVERSIONREFLECTIONROTATION ELEMENTS OF SYMMETRY

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Typical symmetry properties of a lattice. Some types of operations that can leave a lattice invariant. OperationElement InversionPoint ReflectionPlane RotationAxis RotoinversionAxes

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Inversion A center of inversion: A point at the center of the molecule. (x,y,z) --> (-x,-y,-z) A center of inversion can only occur in a molecule. It is not necessary to have an atom in the center (benzene, ethane). Tetrahedral, triangles, pentagons don't have centers of inversion symmetry. All Bravais lattices are inversion symmetric. Mo(CO) 6

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A plane in a cell such that, when a mirror reflection in this plane is performed, the cell remains invariant. Rotational Invariance & Invariance on Reflection Through a Plane Rotational Invariance about more than one axis Invariance on Reflection through a plane

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Examples A triclinic lattice has no reflection plane. A monoclinic lattice has one plane midway between and parallel to the bases, and so forth.

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There are always a finite number of rotational symmetries for a lattice. A single molecule can have any degree of rotational symmetry, but an infinite periodic lattice – can not. Rotational Symmetry

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This is an axis such that, if the cell is rotated around it through some angles, the cell remains invariant. The axis is called n-fold if the angle of rotation is 2π/n. 90 º 120° 180° Rotational Symmetries

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Axes of Rotation

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This type of symmetry is not allowed because it can not be combined with translational periodicity! 5-Fold Symmetry

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90° Examples A Triclinic Lattice has no axis of rotation. A Monoclinic Lattice has a 2-fold axis (θ = [2π/2] = π) normal to the base.

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Examples

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