Presentation on theme: "Solid State Physics (1) Phys3710"— Presentation transcript:
1 Solid State Physics (1) Phys3710 Crystal structure 2Lecture 2Department of PhysicsDr Mazen AlshaaerSecond semester 2013/2014Ref.: Prof. Charles W. Myles, Department of Physics, Texas Tech University
2 Crystal structuresCrystal: atoms are arranged so that their positions are periodic in allthree dimensionsAtoms are bound to one another →well defined equilibrium separations;many identical atoms →minimum energy requires everyidentical atom to be inidentical environment → 3D periodicityIdeal crystal: perfect periodicityReal crystals are never perfect:•surface•impurities and defects•thermal motion of atoms (lattice vibrations)
5 Basic definitionsThe periodic array of points is called crystal lattice.For every lattice point there is a group of atoms (orsingle atom) called basis of the latticeDon't confuse with a1, a2, a3 - basis vectorsparallelogram formed by the basis vectors –unit cellif a unit cell contains only one lattice point, it iscalled a primitive cell (minimum volume)Bravais lattices – all lattice pointsare equivalent
7 Bravais Lattice (1)A fundamental concept in the description of crystalline solids is that of a“Bravais lattice”. A Bravais lattice is an infinite arrangement of points (oratoms) in space that has the following property:The lattice looks exactly the same when viewed from any lattice pointA 1D Bravais lattice:bA 2D Bravais lattice:cb
9 Bravais Lattice (3) A Bravais lattice has the following property: The position vector of all points (or atoms) in the lattice can be written asfollows (Translational Lattice Vectors) :Example (1D):Example (2D):
12 Non-Bravais LatticeNot only the arrangement but also the orientation must appear exactly the same from every point in a bravais lattice.HoneycombThe red side has a neighbour to its immediate left, the blue one instead has a neighbour to its right.Red (and blue) sides are equivalent and have the same appearanceRed and blue sides are not equivalent. Same appearance can be obtained rotating blue side 180º.
13 Five Bravais Lattices in 2D General or ObliqueRectangularCentered rectangularSquareHexagonal
14 aren’t necessarily a mutually orthogonal set! The Primitive Lattice Vectors a1,a2,a3aren’t necessarily a mutually orthogonal set!UsuallyUsually, they are neither mutually perpendicular nor all the same length!For examples, see Fig. 3b (3 dimensions):
15 2-Dimensional Unit Cells Unit Cell The smallest component of the crystal (group of atoms, ions or molecules), which, when stacked together with pure translational repetition, reproduces the whole crystal.S2D-CrystalabSSUnit CellSSSSSSSSSSSS
16 The choice of unit cell is not unique! Unit Cell The smallest component of the crystal (group of atoms, ions or molecules), which, when stacked together with pure translational repetition, reproduces the whole crystal.The choice of unit cell is not unique!2D-CrystalSSS
17 2-Dimensional Unit Cells Artificial Example: “NaCl”Lattice points are points with identical environments.
18 2-Dimensional Unit Cells: “NaCl” The choice of origin is arbitrary - lattice points need not be atoms - but the unit cell size must always be the same.
19 2-Dimensional Unit Cells: “NaCl” These are also unit cells it doesn’t matter if the origin is at Na or Cl !
20 2-Dimensional Unit Cells: “NaCl” These are also unit cells the origin does not have to be on an atom!
21 2-Dimensional Unit Cells: “NaCl” These are NOT unit cells - empty space is not allowed!
22 2-Dimensional Unit Cells: “NaCl” In 2 dimensions, these are unit cells – in 3 dimensions, they would not be.
23 2-Dimensional Unit Cells Why can't the blue triangle be a unit cell?
27 Bravais Lattices in 3DThere are 14 different Bravais lattices in 3D that are classified into 7 different crystal systems (only the unit cells are shown below)
28 Simple Cubic (SC) Structure Simple Cubic Lattice:Unit Cell:It is very cumbersome to draw entire lattices in 3D so some small portion of the lattice, having full symmetry of the lattice, is usually drawn. This small portion when repeated can generate the whole lattice and is called the “unit cell” and it could be larger than the primitive cell
33 Conventional & Primitive Unit Cells Body Centered Cubic Lattice Primitive Lattice Vectorsa1 = (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)
34 Elements That Form Solids with the BCC Structure Note: This was the end of lecture 1
35 Conventional & Primitive Unit Cells Cubic Lattices 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 lattice points in conventional cell: ,100, 010, 001, 110,101, , ½ ½ ½
36 Conventional & Primitive Unit Cells Cubic Lattices Face Centered Cubic (FCC)Primitive Cell Conventional CellFractional coordinates of lattice points in conventional cell: ,100, 010, 001, 110,101, 011,111, ½ ½ 0, ½ 0 ½, 0 ½ ½ ½1 ½ , 1 ½ ½ , ½ ½ 1
38 Conventional & Primitive Unit Cells Hexagonal Bravais Lattice Points of Primitive Cell120oHexagonal Bravais LatticePrimitive Cell = Conventional CellFractional coordinates of lattice points in conventional cell: , 010, 110, 101, , 000, 001
39 Hexagonal Close Packed (HCP) Structure: (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.
40 General 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.
41 Primitive 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
42 Primitive Unit Cells A 2 Dimensional Example! 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-Primitive Unit Cell