Presentation on theme: "Crystal Structure Continued!"— Presentation transcript:
1Crystal Structure Continued! NOTE!!Much of the discussion & many figures in what follows was\constructed from lectures posted on the web by Prof. BeşireGÖNÜL in Turkey. She has done an excellent job of coveringmany details of crystallography & she illustrates her topics withmany very nice pictures of lattice structures. Her lectures on thisare posted Here:Her homepage is Here:
3Lattice Translation Vectors In GeneralMathematically, a lattice is defined by 3 vectors calledPrimitive Lattice Vectorsa1, a2, a3 are 3d vectors which depend on the geometry.Once a1, a2, a3 are specified, thePrimitive Lattice Structureis known.The infinite lattice is generated by translating through aDirect Lattice Vector: T = n1a1 + n2a2 + n3a3n1,n2,n3 are integers. T generates the lattice points. Each lattice point corresponds to a set of integers (n1,n2,n3).
4Rn n1a + n2b 2 Dimensional Lattice Translation Vectors Consider a 2-dimensional lattice (figure). Define the2 Dimensional Translation VectorRn n1a + n2b(Sorry for the notation change!!)a & b are 2 d Primitive Lattice Vectors, n1, n2 are integers.Once a & b are specified by the lattice geometry & an origin is chosen, all symmetrically equivalent points in the lattice are determined by the translation vector Rn. That is, the lattice has translational symmetry. Note that the choice of Primitive Lattice vectors is not unique! So, one could equally well take vectors a & b' as primitive lattice vectors.Point D(n1, n2) = (0,2)Point F(n1, n2) = (0,-1)
5The Basis Crystal Structure ≡ Primitive Lattice + Basis DIRECT LATTICE (or basis set) The set of atoms which, when placed at eachlattice point, generates the Crystal Structure.Crystal Structure≡ Primitive Lattice + BasisTranslate the basis through all possible lattice vectorsT = n1a1 + n2a2 + n3a3to get the Crystal Structure or theDIRECT LATTICE
6Primitive r' = r + T (1) T = n1a1 + n2a2 + n3a3 The periodic lattice symmetry is such that the atomic arrangement looks the same from an arbitrary vector position r as when viewed from the pointr' = r + T (1)where T is the translation vector for the lattice:T = n1a1 + n2a2 + n3a3Mathematically, the lattice & the vectors a1,a2,a3 arePrimitiveif any 2 points r & r' always satisfy (1) with a suitable choice of integers n1,n2,n3.
7Don’t think of a1,a2,a3 as a mutually orthogonal set! In 3 dimensions, no 2 of the 3 primitive lattice vectors a1,a2,a3 can be along the same line. But,Don’t think of a1,a2,a3 as a mutually orthogonal set!Usually, they are neither mutually perpendicular nor all the same length!For examples, see Fig. 3a (2 dimensions):
8they are neither mutually perpendicular nor all the same length! The Primitive Lattice Vectors a1,a2,a3 aren’t necessarily a mutually orthogonal set!Usuallythey are neither mutually perpendicular nor all the same length!For examples, see Fig. 3b (3 dimensions):
9Crystal Lattice Types Bravais Lattice An infinite array of discrete points with anarrangement & orientation that appears exactly thesame, from whichever of the points the array is viewed.A Bravais Lattice is invariant under a translationT = n1a1 + n2a2 + n3a3Nb film
10Non-Bravais Lattices In a Bravais Lattice, not only the atomic arrangement but also the orientations must appearexactly the same from every lattice point.2 Dimensional Honeycomb LatticeThe red dots each have a neighbor to the immediate left. The blue dot has a neighbor to its right. The red (& blue) sides are equivalent & have the same appearance. But, the red & blue dots are not equivalent. If the blue side is rotated through 180º the lattice is invariant. The Honeycomb Lattice is NOT a Bravais Lattice!!Honeycomb Lattice
11Five (5) & ONLY Five Bravais Lattices! It can be shown that, in 2 Dimensions, there areFive (5) & ONLY Five Bravais Lattices!
122-Dimensional Unit Cells Unit Cell The Smallest Componentof the crystal (group of atoms, ions or molecules),which, when stacked together with pure translational repetition, reproduces the whole crystal.2D-CrystalSabSSUnit CellSSSSSSSSSSSS
13Unit Cell The Smallest Component of the crystal (group of atoms, ions or molecules),which, when stacked together with puretranslational repetition, reproduces the whole crystal.Note that the choice of unit cell is not unique!2D-CrystalSSS
142-Dimensional Unit Cells Artificial Example: “NaCl”Lattice points are points withidentical environments.
152-Dimensional Unit Cells: “NaCl” Note that the choice of origin is arbitrary!the lattice points need not be atoms, butThe unit cell size must always be the same.
162-Dimensional Unit Cells: “NaCl” These are also unit cells!It doesn’t matter if the origin is at Na or Cl!
172-Dimensional Unit Cells: “NaCl” These are also unit cells.The origin does not have to be on an atom!
182-Dimensional Unit Cells: “NaCl” Empty space is not allowed! These are NOT unit cells!Empty space is not allowed!
192-Dimensional Unit Cells: “NaCl” In 2 dimensions, these are unit cells.In 3 dimensions, they would not be.
202-Dimensional Unit Cells Why can't the blue triangle be a unit cell?
21Can you find the “Unit Cell” in this painting? Example: 2 Dimensional, Periodic Art! A Painting by Dutch Artist Maurits Cornelis Escher ( )Escher was famous for his so called “impossible structures”, such as Ascending & Descending, Relativity,..Can you find the “Unit Cell” in this painting?
26Conventional Unit Cells A Conventional Unit Cell just fills space when translated through a subset of Bravais lattice vectors.The conventional unit cell is larger than the primitive cell, but with the full symmetry of the Bravais lattice.The size of the conventional cell is given by the lattice constant a.FCC Bravais LatticeThe full cube is theConventional UnitCell for the FCCLattice
27NOT Mutually Orthogonal! Conventional & Primitive Unit CellsFace Centered Cubic LatticePrimitive LatticeVectorsa1 = (½)a(1,1,0)a2 = (½)a(0,1,1)a3 = (½)a(1,0,1)Note that the ai’s areNOT MutuallyOrthogonal!Primitive Unit Cell(Shaded)LatticeConstantConventional Unit Cell (Full Cube)
28Elements That Form Solids with the FCC Structure
30NOT mutually orthogonal! Body Centered Cubic Lattice Primitive Lattice Conventional & Primitive Unit CellsBody Centered Cubic LatticePrimitive LatticeVectorsa1 = (½)a(1,1,-1)a2 = (½)a(-1,1,1)a3 = (½)a(1,-1,1)Note that the ai’s areNOT mutuallyorthogonal!Primitive Unit CellLatticeConstantConventional Unit Cell (Full Cube)
31Elements That Form Solids with the BCC StructureNote: This was the end of lecture 1
32Cubic Lattices Conventional & Primitive Unit Cells Simple Cubic (SC)Primitive Cell = Conventional CellFractional coordinates of lattice points:000, 100, 010, 001, 110,101, 011, 111Body Centered Cubic (BCC)Primitive Cell Conventional CellFractional coordinates of the lattice points in the conventional cell:000,100, 010, 001, 110,101, 011, 111, ½ ½ ½Primitive Cell = Rombohedron
33Conventional & Primitive Unit Cells Cubic Lattices Face Centered Cubic (FCC)Primitive Cell Conventional CellThe fractional coordinates of lattice points in the conventional cell are:000,100, 010, 001,110,101, 011, 111,½ ½ 0, ½ 0 ½, 0 ½ ½,½ 1 ½, 1 ½ ½ , ½ ½ 1
35Conventional & Primitive Unit Cells Points of the Primitive Cell120oHexagonal Bravais LatticePrimitive Cell = Conventional CellFractional coordinates of lattice points in conventional cell:100, 010, 110, 101,011, 111, 000, 001
36Hexagonal Close Packed (HCP) Lattice: A Simple Hexagonal Bravais Lattice with a 2 Atom Basis The HCP lattice is not a Bravais lattice, because the orientation of the environment of a point varies from layer to layer along the c-axis.
37General Unit Cell Discussion For any lattice, the unit cell &, thus, the entire lattice, is UNIQUELY determined by 6 constants (figure):a, b, c, α, β and γwhich depend on lattice geometry.As we’ll see, we sometimes want to calculate the number of atoms in a unit cell. To do this, imagine stacking hard spheres centered at each lattice point & just touching each neighboring sphere. Then, for the cubic lattices, only 1/8 of each lattice point in a unit cell assigned to that cell. In the cubic lattice in the figure,Each unit cell is associated with (8) (1/8) = 1 lattice point.
38Primitive Unit Cells & Primitive Lattice Vectors In general, a Primitive Unit Cell is determined by the parallelepiped formed by the Primitive Vectors a1 ,a2, & a3 such that there is no cell of smaller volume that can be used as a building block for the crystal structure.As we’ve discussed, a Primitive Unit Cell can be repeated to fill space by periodic repetition of it through the translation vectorsT = n1a1 + n2a2 + n3a3.The Primitive Unit Cell volume can be found by vector manipulation:V = a1(a2 a3)For the cubic unit cell in the figure, V = a3
39Primitive Unit Cells A 2 Dimensional Example! P = Primitive Unit Cell Note that, by definition, the Primitive Unit Cell must contain ONLY ONE lattice point.There can be different choices for the Primitive Lattice Vectors, but the Primitive Cell volume must be independent of that choice.A 2 Dimensional Example!P = Primitive Unit CellNP = Non-PrimitiveUnit Cell