Solid State Physics: Crystal Structure Chapter 1 PDF

Damanhour university Faculty of science Physics Department Phy303 Solid State Physics ‫فيزياء الجوامــد‬ 1  Chapter (1) Crystal Structure 2  What is solid state physi...

Damanhour university Faculty of science Physics Department Phy303 Solid State Physics ‫فيزياء الجوامــد‬ 1  Chapter (1) Crystal Structure 2  What is solid state physics? Solid state physics Explains the properties of solid materials. Explains the properties of a collection of atomic nuclei and electrons interacting with electrostatic forces. Formulates fundamental laws that govern the behavior of solids. 3 Crystalline Solids Crystalline materials are solids with an atomic structure based on a regular repeated pattern. The majority of all solids are crystalline. More progress has been made in understanding the behavior of crystalline solids than that of non-crystalline materials since the calculation are easier in crystalline materials. Understanding the electrical properties of solids is right at the heart of modern society and technology. 4 Interatomic forces Solids are stable structures, and therefore there exist interactions holding atoms in a crystal together. For example a crystal of sodium chloride is more stable than a collection of free Na and Cl atoms. This implies that the Na and Cl atoms attract each other, i.e. there exist an attractive interatomic force, which holds the atoms together. This also implies that the energy of the crystal is lower than the energy of the free atoms. The amount of energy which is required to pull the crystal apart into a set of free atoms is called the cohesive energy of the crystal. Cohesive energy = energy of free atoms – crystal energy 5  Magnitude of the cohesive energy varies for different solids from 1 to 10 eV/atom, except inert gases in which the cohesive energy is of the order of 0.1eV/atom.The cohesive energy controls the melting temperature. 6  The corresponding energy U0 is the cohesive energy. The interatomic force is determined by the gradient of t If we apply this to the curve in Fig.1, we see that F(R)R0. This means that for large separations the force is attractive, tending to pull the atoms together. On the other, hand F(R)>0 for R 0 K. The crystal is always distorted to a lesser or greater degree, depending on T. “main source of electrical resistivity in metal” 3. Because of the existing of some foreign atoms, even with the best crystal-growing techniques. ✓ So, we deal with the Real Crystals “are not perfect or ideal”. ✓ A real crystal always has a large number of imperfections in the lattice. ✓ Since real crystals are of finite size, they have a surface to their boundary. ✓ At the boundary, atomic bonds terminate and hence the surface itself is an imperfection. ✓ One can reduce crystal defects considerably, but can never eliminate them entirely.  In 3-dimension ✓ There are different common crystal structure in 3-D: Simple lattice (S) Body center lattice (BC) Face center lattice (FC) Has points only at Has one additional point at Has six additional points, the corners the center of the cell one on each face  Crystal Properties 1. Isotropic: Properties of a material are identical in all directions 2. Anisotropic: Properties of a material depend on the direction Properties Types 1. Intensive - Properties that do not depend on the amount of the matter present. Density, viscosity, electrical resistivity, heat capacity, hardness, temperature, pressure, velocity and acceleration. 2. Extensive - Properties that do depend on the amount of matter present. Energy, mass, weight, particles number, volume , momentum and electrical charge.  BASIC DEFINITIONS 1-The crystal lattice Crystallography focuses on the geometric properties of crystals. So, one imagine each atom replaced by a geometrical point at the equilibrium position of that atom. The result is a pattern of points having the same geometrical properties as the crystal, devoid of any physical contents. This geometrical pattern is the crystal lattice, or the lattice, all the atomic sites have been replaced by lattice sites. A Lattice is Defined as an Infinite Array of Points in Space In which each point has identical surroundings to all others. The points are arranged exactly in a periodic manner. The simplest structural unit for a given solid is called BASIS Or is the building block consist of atoms Crystal Structure ≡ Lattice + Basis Crystal Structure Lattice   Basis   Basis  Crystal Structure  Lattice ⎯→ A Two-Dimensional Bravais Lattice with Different Choices for the Basis  Crystal Structure = Lattice + Basis Basis   Basis vector ≡ Lattice Translation Vectors Mathematically (in 3D), a lattice is defined by three vectors called: Primitive Lattice Translation Vectors form a set of Basis Vectors for the lattice, in terms of which the positions of all lattice points can be expressed by the following equation: R = n1a + n2b + n3c a, b, c are three arbitrary independent vectors n1, n2, n3 are integers (-ve, zero or +ve ) Each lattice point corresponds to a set of integers (n1,n2,n3) The infinite lattice is generated by translating through a Direct Lattice Vector “R”  2 Dimensional Lattice Translation Vectors 2 Dimensional Translation Vector be R = n1a + n2b. 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 R. 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)  Unit cell The area of the parallelogram whose sides are the basis vectors a and b is called the unit cell of the lattice. The unit cell is usually the smallest area that the lattice may be viewed as composed of a large number of equivalent unit cells placed side by side. The choice of a unit cell for one and the same lattice is not unique, according to the choice of basis vectors. The shape of unit cells is not the same, depending on the choice of the basis vectors. Simple Cubic (sc) Body Centered Cubic (bcc) Cell = Primitive cell Cell ≠ Primitive cell ▪ The vectors a1, a2, can be chosen as a basis set, in which case the unit cell is the parallelogram S1. “Primitive unit cell” ▪ The vectors a and b form the unit cell of the area S2. ”Non- primitive unit cell”. ▪ The area of the non-primitive cell is an integral multiple of the primitive cell. “The multiplication factor here is two”  Fundamental Types of Lattices Only rotations Cn with n = 2, 3, 4 and 6 are compatible with the translational symmetry. There’re 32 crystallographic point groups (classes). Lattices with the same maximal point group are said to belong to the same crystal systems. There’re only 7 crystal systems in 3-D. Besides the primitive lattice (denoted by P or R ), some crystal systems may allow other centered lattices (denoted by C, A, F, or I ). → There’re 14 Bravais lattices (lattice types) in 3-D and 5 in 2-D.  In 2 Dimensions, ONLY Five Bravais Lattices!  B2. Bravais lattices in 2D – 5 types – general case : oblique lattice |a1|≠|a2| , (a1,a2)=φ – special cases : square lattice: |a1|=|a2| , φ= 90° hexagonal lattice: |a1|=|a2| , φ= 120° rectangular lattice: |a1|≠|a2| , φ= 90° centered rectangular lattice: |a1|≠|a2| , φ= 90°  In 3 Dimensions The number of Bravais lattices is 14, grouped into seven crystal systems, depending on their shape and symmetry of the unit cell.  Crystal Plane & Miller Indices The planes passing through lattice points are called lattice planes. Surface or plane through the crystal can be described by: 1. Taking the intercepts of the plane along the , b and c axes used for the lattice. 2. Reciprocal these intercepts. 3. If it fraction, multiply them to the smallest integers to form the plane set (hkl) that is known as the Miller indices. Miller indices is defined as the reciprocals of the intercepts made by the plane on the three axes with no common factor.  Index System for Crystal Planes Miller indices of a crystal plane: 1. Express the intercepts of the plane with the crystal axes in units of lattice constants a1 , a2 , a3. 2. Take the reciprocal of these numbers. 3. Reduce them to integers of the same ratio: (h,k,l). Intercepts at 3a1, 2a2, and 2a3. Reciprocals are (1/3, 1/2, 1/2). Miller indices = (233).  Example-1 Axis X Y Z Intercept points 1 ∞ ∞ Reciprocals 1/1 1/ ∞ 1/ ∞ Smallest Ratio 1 0 0 Miller İndices (100) (1,0,0) Crystal Structure 47 Example-2 Axis X Y Z Intercept points 1 1 ∞ Reciprocals 1/1 1/ 1 1/ ∞ Smallest Ratio 1 1 0 (0,1,0) Miller İndices (110) (1,0,0) Crystal Structure 48 Example-3 Axis X Y Z (0,0,1) Intercept points 1 1 1 Reciprocals 1/1 1/ 1 1/ 1 Smallest (0,1,0) Ratio 1 1 1 (1,0,0) Miller İndices (111) Crystal Structure 49  (100) (110) (111) (200) (100 ) Three common Unit Cell in 3D Crystal Structure 51 Unit Cell The unit cell and, consequently, the entire lattice, is uniquely determined by the six lattice constants: a, b, c, α, β and γ. Only 1/8 of each lattice point in a unit cell can actually be assigned to that cell. Each unit cell in the figure can be associated with 8 x 1/8 = 1 lattice point. 52