Department of Applied Science and Humanities SAITM, Gurgaon M.D.U Syllabus Section A: Crystal Structure Quantum Physics Section B: Nano-Science Free Electron Theory Section C: Band Theory of solids Photoconductivity and Photovoltaics Section D: Magnetic properties of Solids Dr. sangeeta Negi solids Non crystalline or Amorphous solids Strongly bonded molecules but no geometrical regularity Do not have directional properties, so they are called as isotropic substance. Wide range of melting point Ex: Glass, plastics, rubber Crystalline solids Perfect periodicity of atomic structure Has directional properties and therefore called as anisotropic substance. Sharp melting point Ex:metallic crystal of Cu, Ag, Al, Mg Non metallic crystal: carbon, silicon Figures 6 Materials and Packing atoms pack in periodic, 3D arrays Crystalline materials... -metals -many ceramics -some polymers atoms have no periodic packing Noncrystalline materials... crystalline SiO 2 noncrystalline SiO 2 "Amorphous" = Noncrystalline Adapted from Fig. 3.23(b), Callister & Rethwisch 8e. Adapted from Fig. 3.23(a), Callister & Rethwisch 8e. SiOxygen typical of: Space lattice A lattice is a regular and periodic arrangement of points in three dimensions. An infinite array of points in three dimensions in which every point has surroundings identical to that of every other points in the array. Figure: Two dimensional arrays Crystal= Lattice + Basis A crystal structure is formed by associating every lattice point with unit assembly of atoms or molecules or ions identical in composition, arrangement and orientation. This unit assembly is called the basis. Figure: Unit cell A unit cell is defined as a fundamental building block of a crystal structure, which can generate the complete crystal by repeating its own dimension in various direction Figure: Crystallographic Axes: Unit cell consisting of three mutually perpendicular edges OA, OB and OC. Denoted by x, y, z Primitives: OA, OB OC be the intercepts made by the unit cell along the crystallographic axes. These intercepts are known as primitives. Denoted by a, b, c Interaxial angle: The angle between axes. Denoted by ,, . Unit Cell Primitive Cell: Smallest unit cell in volume constructed by Primitives. Atoms at corners of the unit cell Ex: Simple cubic unit cell Non primitive cell: Atom at corner and other points Ex: Bcc and fcc unit cell Symbols Primitive lattice P: lattice points only at corner of unit cell Body centered lattice BCC: I- corner + body centre of unit cell Face centered lattice FCC: F- point at corner + Face centre Base centered lattice : C- Corner+ top + bottom base symmetry Centre of symmetry Plane of symmetry Axis of symmetry Bravais lattices ( 1948) 14 types of unit cells under these 7 crystal systems are possible, they commonly called Bravais Lattices. Cubic system:- i. Simple ii. Body centered iii. Face centered Tetragonal system:- i. Simple ii. Body centered Monoclinic system i. Simple ii. End face centered Orthorhombic system i. Simple ii. Body centered iii. End face centered iv. Face centered Triclinic Rhombohedral hexagonal Seven crystal system and bravais lattices Figures of crystal systems Important parameters Number of atoms molecules or ions per unit cell (n). Coordination number (CN): It is the number of nearest neighboring atom molecules or ions to a particular atom. Atomic Radius: It is radius of an atom or half the distance between two nearest neighboring atoms in a crystal. Atomic packing factor or density of packing: ratio of the volume occupy by the molecules atom and ion in a unit cell to the total volume of the unit cell. APF=v/V Simple cubic structure A simple cubic unit cell consists of eight corner atoms. The total number of atoms present in a unit cell 1/8x8=1 Coordination number: There are four nearest neighbors in its own plane. There is another nearest neighbor in another plane, which lie just below this atom. Therefore the total number of nearest neighbor is six. Hence the CN =6 21 Simple Cubic Structure (SC) Rare due to low packing density (only Po has this structure) Close-packed directions are cube edges. Coordination # = 6 (# nearest neighbors) Cont.. Atomic Radius: r=a/2 Atomic packing factor APF=v/V v=1x4/3r 3 V=a 3 APF=/6 substituting r=a/2 Figure: 23 Atomic Packing Factor (APF):SC APF for a simple cubic structure = 0.52 APF = a (0.5a) 3 1 atoms unit cell atom volume unit cell volume APF = Volume of atoms in unit cell* Volume of unit cell *assume hard spheres Adapted from Fig. 3.24, Callister & Rethwisch 8e. close-packed directions a R=0.5a contains 8 x 1/8 = 1atom/unit cell Body centered cubic structure A body centered cubic structure has eight corner atoms and one body centered atom. In bcc unit cell, each and every corner atom is shared by eight adjacent unit cells Total number of atoms contributed by the corner atom is 1/8x8=1 Total number of atoms present in bcc unit cell 1+1=2 CN=8 Cont. Atomic radius r=3/4 a APF=v/V, v=2x4/3r 3, V=a 3 APF=0.68 Figure: 26 Body Centered Cubic Structure (BCC) Coordination # = 8 Adapted from Fig. 3.2, Callister & Rethwisch 8e. Atoms touch each other along cube diagonals. --Note: All atoms are identical; the center atom is shaded differently only for ease of viewing. ex: Cr, W, Fe ( ), Tantalum, Molybdenum 2 atoms/unit cell: 1 center + 8 corners x 1/8 Click once on image to start animation (Courtesy P.M. Anderson) 27 Atomic Packing Factor: BCC a APF = 4 3 (3a/4) 3 2 atoms unit cell atom volume a 3 unit cell volume length = 4R = Close-packed directions: 3 a APF for a body-centered cubic structure = 0.68 a R Adapted from Fig. 3.2(a), Callister & Rethwisch 8e. a 2 a 3 Coordination # = 12 Adapted from Fig. 3.1(a), Callister 6e. (Courtesy P.M. Anderson) Close packed directions are face diagonals. --Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing. FACE CENTERED CUBIC STRUCTURE (FCC) 29 Atomic Packing Factor: FCC APF for a face-centered cubic structure = 0.74 maximum achievable APF APF = 4 3 (2a/4) 3 4 atoms unit cell atom volume a 3 unit cell volume Close-packed directions: length = 4R = 2 a Unit cell contains: 6 x 1/2 + 8 x 1/8 =4 atoms/unit cell a 2 a Adapted from Fig. 3.1(a), Callister & Rethwisch 8e. 30 A sites B B B B B BB C sites C C C A B B ABCABC... Stacking Sequence 2D Projection FCC Unit Cell FCC Stacking Sequence B B B B B BB B sites C C C A C C C A HEXAGONAL CLOSE-PACKED STRUCTURE (HCP) Ideally, c/a = for close packing However, in most metals, c/a ratio deviates from this value 32 Hexagonal Close-Packed Structure (HCP) Coordination # = 12 ABAB... Stacking Sequence APF = D Projection 2D Projection Adapted from Fig. 3.3(a), Callister & Rethwisch 8e. 6 atoms/unit cell ex: Cd, Mg, Ti, Zn c/a = c a A sites B sites A sites Bottom layer Middle layer Top layer COMPARISON OF CRYSTAL STRUCTURES Crystal structurecoordination #packing factorclose packed directions Simple Cubic (SC)60.52cube edges Body Centered Cubic (BCC)80.68body diagonal Face Centered Cubic (FCC)120.74face diagonal Hexagonal Close Pack (HCP)120.74hexagonal side Close packed crystals A plane B plane C plane A plane ABCABCABC packing [Face Centered Cubic (FCC)] ABABAB packing [Hexagonal Close Packing (HCP)] LATTICE CONSTANT mass in each unit=a 3 If M is molecular weight and N the Avogadros number, then Mass of each molecule= M/N Mass in each unit cell=nM/N A lattice constant a=(nM/N ) 1/3 THEORETICAL DENSITY, Density = mass/volume mass = number of atoms per unit cell * mass of each atom mass of each atom = atomic weight/avogadros number Structure of NaCl STRUCTURE OF OTHER SYSTEMS (Courtesy P.M. Anderson) Structure of Carbon Graphite Diamond Sodium chloride structure Diamond structure Miller Indices Miller indices is defined as the reciprocals of the intercepts made by the plane on the three axes. The orientation of planes or faces in a crystal can be described in terms of their intercepts on the three axes. Miller introduced a system to designate a plane in a crystal. The set of three number is known as miller indices of concerned plane. Procedure for findings miller indices Determined the intercepts of the plane along the axes x, y, z in terms of lattice constant a, b, c Determine the reciprocals of these numbers Find the least common denominators (LCD) and multiply each by this LCD. The result is written in paranthesis This is called Miller indices of the plane in the form (hkl) Important features of miller indices A plane which is parallel to any one of the coordinate axes has an intercept of infinity. Therefore the miller index for that axis is zero. A plane passing through origin is defined in terms of a parallel plane have non-zero intercept. All equally spaced parallel planes have same miller indices, the miller indices do not only defence a particular plane but also a set of parallel planes. Miller planes Figures: CRYSTALLOGRAPHIC PLANES Crystallographic planes specified by 3 Miller indices as (hkl) Procedure for determining h,k and l: If plane passes through origin, translate plane or choose new origin Determine intercepts of planes on each of the axes in terms of unit cell edge lengths (lattice parameters). Note: if plane has no intercept to an axis (i.e., it is parallel to that axis), intercept is infinity ( ) Determine reciprocal of the three intercepts (2 4 2) If necessary, multiply these three numbers by a common factor which converts all the reciprocals to small integers (1 2 1) The three indices are not separated by commas and are enclosed in curved brackets: (hkl) (121) If any of the indices is negative, a bar is placed in top of that index THREE IMPORTANT CRYSTAL PLANES Parallel planes are equivalent EXAMPLE: CRYSTAL PLANES Construct a (0,-1,1) plane Interplanar spacing The relation between the interplanar distance and interatomic distance is given by d=a/(h 2 +K 2 +l 2 ) Angle between planes given by braggs law: X-rays are electromagnetic waves with wavelengths thousand times smaller than the visible light. To measure the wavelength, a grating of corresponding dimensions is required and hence simple grating can not be used. Bragg explains the simple explanation of the observed angle of the diffracted beams from a crystal. Consider Incoming X-rays diffract from crystal planes, following Braggs law: n = 2dsin( ) Measurement of: Critical angles, c, for X-rays provide atomic spacing, d. Adapted from Fig. 3.2W, Callister 6e. X-RAYS TO CONFIRM CRYSTAL STRUCTURE Experimental crystal structure determination Three methods of X-rays crystallography allows for this in the following ways Laue Technique: A stationary single crystal is irradiated by a range of x-rays wavelengths. Rotating crystal method: A single crystal specimen is rotated in a beam of monochromatic x-rays Powder techniques: A polycrystalline powder is kept stationary in a beam of monochromatic radiation. Laue method A single crystal is mounted on a goniometer, which enables the crystal to be rotate through known angles in two perpendicular planes and maintain a stationary in a beam of x-rays ranging in wavlength from about A0. Crystal select out and diffract those values of wavelength for which plane exist of spacing d and angle . A flat photographic film is placed to receive either transmitted diffracted beam or the reflected diffracted beam. figure Rotating crystal method A small crystal is mounted on a goniometer which is fixed to spindle so the crystal can rotate abour a fixed axis The specimen is usually oriented with one of the crystallographic axes parallel to the axis of the rotation. The resulting variation in brings difference lattice planes into position for reflection and diffracted images are recorded. Powder diffraction method A monochromatic x-ray beam is allowed to fall on a small specimen and contained in a thin walled glass capillary tube. Since the orientation of the minute crystal is completely random a certain of them will be with any given set of lattice planes making exactly the correct angle with the incident beam for reflection. After taking n=1 in braggs equation, there are still a number of combinations of d and that would satisfy the braggs law. figure For each combination of d and , one cone of reflection results, which is coaxially with the axis of the incident beam and with a semi apex angleof twice of braggs angle 2. There are many cones of reflection emitted by the powder specimen. Imperfection (defect) in crystals No crystal is perfectly regular. Any deviation from this perfect atomic periodicity is an imperfection called lattice defect. Structure insensitive properties- density, dielectric capacitivity, specific heat,elastic properties etc. Structure sensitive properties- electricalresistance, diffusion crystal growth etc. Defects Crystal imperfection Thermal vibrations Point defectsLine defectsSurface defects Electronic imperfection Conduction Electron/holes Transient imperfection Photon/ beam of charged particle Point defect Interstitial atom: This is an atom inserted into the voids between the regularly occupied sites. Vacancies: lattice sited from which the atoms are missing. Impurity atom: This is a defect in which a foreign atom occupies a regular lattice site. Schottky Defect There are irregularities of the atomic arrays in which atoms are missing at some lattice point, such a point is called a vacancy (also called Schottky defect). Derivation as per discussed in class Frenkel defect When an interstitial is caused by transferring an atom from a lattice site to an intrstitial position, a vacancy is created. The associated vacancy and interstitital atom is called Frenkel defect. Derivation as discussed in class. Bonding in solids The atoms, ions or molecules in a crystal hold together by electrostatic forces and the ability to hold is called bonding. Crystals are classified into following categories; Ionic crystals: bonding by strong electrostatic attraction between ions of opposite sign. Ex; KBr, NaCl, CsCl Covalent crystals: bonding by sharing of electrons between atoms, ex; C, Si, Ge Metallic crystals: bonding by electrostatic attraction between lattice of ion core and the free electron gas. Ex; Cu, Na, Al Inert gas crystals: Bonding by vander waals forces ex; solid CH 4, Solid He Hydrogen bonded crystals: bonding by hydrogen bonds ex; HF, H 2 0 Ionic crystals Electrons are transferred from one kind of atoms to the other kind, so that the atoms become positive and negative ions. These ions arrange themselves in such a configuration that Coulomb attraction between ion of opposite signs is stronger than the Coulomb repulsion between ions of the same sign. This electrostatic interaction of oppositely charged ions results ionic bond Cohesive energy Energy of an ionic crystal is defined as the energy that would be liberated in the form ation of the crystals from the individual neutral atoms B= e2/ 4o r o n-1 Born Lande equation Figure: Properties of ionic solids Hard cubic crystals Brittle Transparent to visible radiation Poor conductors of electricity Easily soluable in polar liquids like water Ionic bonds are non directional Covalent crystal In covalent crystals, one or more electron are detached from two adjacent atoms and are shared equally by both atoms. Figure; properties Strong bond Cohesive energies of covalent crystal 6 to 12 eV/ atom No sharp distinction between ionic and covalent crystal Covalent crystal with large bond energies are very hard High melting point and transparent to visible light Conductivity of covalent crystals varies over a wide range Solids are hard, brittle and posses crystalline structure Soluable in non-polar solvent such as benzene