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## Unit VII Crystal structure

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**Unit VIICrystal structure**• Dr. Ravindra H.J & Dr. P.S. Aithal**Objectives**• Definition of Space lattice, Bravais lattice, Unit cell, Primitive cell and Lattice parameters • Seven crystal systems and 14 Bravais lattices • Direction and planes in a crystal, Miller indices and interplanar spacing • Co-ordination number of cubic structure, Number of atoms per unit cell, Relation between Atomic radius and Lattice constant, and Atomic packing factor • Bragg’s Law and Bragg’s x-ray spectrometer • Crystal structures of NaCl and diamond.**Introduction TO CRYSTALS**• Gas • Liquids • Solids - crystals Amorphous Crystal: The solids which possess the long range, three dimensional ordering are known as crystals. Definition: a solid which possesses long range, three dimensional atomic or molecular arrangements**Motif**Lattice points Space lattice • A crystal structure is built of infinite repetition of identical atoms or group of atoms in three dimensions. • A group of atoms is called the basis or motif. • The set of mathematical points to which the basis is attached is called a lattice.**Space lattice**• A crystal lattice or space latticeis an infinite pattern of lattice points in three dimensions, each of which must have the same surroundings in the same orientation. OR Space lattice is a well defined repeating unit (motif or basis) and the each motif is associated to a lattice point so that the overall lattice is a three dimensional array of lattice points which are geometrically identical. Space lattice = Lattice * Motif (Basis) here the symbol * refers to mean “associated with” In simple metallic crystals the number of atoms allocated to each lattice point is just one and in the case of a protein crystal the number of atoms allocated to each lattice point may be thousands.**Motif**Bravais lattice and non-Bravais lattice • The array of equivalent lattice points in two or in three dimensions is referred as Bravais lattice. • The array of non-equivalent lattice points is referred as non-Bravais lattice.**D**C B A Basis vector and translational vector • A vector is a mathematical quantity having both magnitude and direction. • A vector having unit magnitude is generally referred as a unit vector. • The vectors which gives the magnitude and direction of separation between two nearest lattice points is called basis vector.**D**C B A Translational vector or lattice vector • If we chose three basis vectors, , and to define a right-handed, orthogonal, three dimensional Cartesian co-ordinate systems then any vector T which gives the position and direction of the any identical lattice point in three dimensional lattices can be represented as • Where, is the translational vector or lattice vector and u, v, and w are integers. The coordinates of the lattice point in three dimension is represented by (u,v,w). E**Unit cell**• The parallelepiped-shaped volume which, when reproduced by close packing in three dimensions, gives the whole crystal structure or space lattice itself is called the unit cell. • Definition: Unit cell is the smallest geometrical volume in a space lattice, defined by three basis vectors, which on translation along three basis vector directions yields the complete crystal structure or space lattice itself.**Primitive cell and non-primitive cell**• All fundamental unit cells associated with a single lattice point in it are called primitive cells or P-cells. • A primitive cell is a one in which lattice points are present only at the edges (or corners) of the unit cell. • The unit cells associated with more than one lattice point are referred as non-primitive or multi unit cells. • A non-primitive cell is a one in which there are lattice points present in the unit cell in addition to the lattice points at the edges of the unit cell.**Lattice parameters**• Unit cell parameters is defined as a set of six parameters, lengths of the three non-collinear basis vectors and three interfacial angles, that together defines a unit cell. • Basically the parallelepiped formed by the three basis vectors , and , defines the unit cell of the lattice, with edges of length a, b and c, is selected as unit cell.**Crystal systems & Bravais lattices**• The systematic work of describing the space lattices was done initially by Frankenheim, who, in 1835, proposed that there were fifteen space lattices. • Bravais In 1848 proposed there are only 14 space lattices • These 14 Bravais lattices fall into seven crystal systems • Cubic • Tetragonal • Orthorhombic • Monoclinic • Triclinic • Rhombohedra (or Trigonal) and • Hexagonal.**Polymorphism**• Ability of any substance to crystallize in several solid phases that possess different lattice structures. • Example : Fe – BCC at room temperature • At 9100C BCC to FCC • >14000C FCC to BCC • Type of the lattice also affects the properties of the Iron.**Directions and planes in a crystal lattice**• Any lattice point T can be represented by the equation • The direction in space corresponding to the vectors is written in square brackets, without using commas to separate the digits, as [uvw]. • For example [213] and it should be read as ‘two one three’.**Planes in a crystal and Miller indices**• A set of three integers which are used to represent the plane are referred as Miller integers. • A set of planes is defined by three integers h,k, and l , known as plane indices or Miller indices. Procedure to find Miller indices • Find the intercept of the planes along three basis vectors. • Either reduce the fractional intercepts to set of integers p, q • and r or express the intercepts as fractional multiples of • basis vector lengths • Take the reciprocals • Find the least common multiple of the denominator and • multiply this LCM to above ratios yields set of three integers • h, k and l referred as Miller indices.**Coordination number**• The coordination number is defined as the number of nearest neighbors surrounding the central or a given atom within the space lattice. • Simple cubic- coordination number is 6 • Body centered cubic – coordination number is 8 • Face centered cubic – coordination number is 12 • Number of atoms per unit cell • Simple cubic - (1/8 X 8) = 1 • Body centered cubic - (1/8 X 8) +1 =2 • Face centered cubic - (1/8 X 8 ) +(1/2 X 6) =4**Relation between Atomic radius and Lattice constant**• Simple cubic: a = 2R • Body centered cubic • Face centered cubic**Atomic packing factor (APF)**• Atomic packing factor is the ratio of the total volume of the atoms present in one unit cell to that of total volume of the unit cell. • APF = volume of the number of atoms present in a unit cell/ volume of the unit cell • SC • BCC FCC**Expression for a space lattice constant “a” for a cubic**lattice The density of the unit cell is the ratio of total mass of the atoms or molecules belonging to unit cell to that of volume of unit cell**Crystal structure of NaCl**• Type of the lattice is determined by considering only one type of atom • NaCl is an ionic compound • Coordination number is 6 • Distance between two same type ion is 5.62 Å • Number of NaCl molecules per unit cell is 4 • Number of atoms per unit cell is 8**Calculation of lattice constant of NaCl**• Atomic weight of Na is 23 • Atomic weight of Cl is 35.45 • Density of NaCl is 2180 kg/m3 • Number of NaCl molecules in a Unit cell is 4**Crystal structure of diamond**• The diamond structure can be described as two interpenetrating fcc structures that are displaced relative to one another along the main diagonal. • The four carbon atoms together forms tetrahedron structure • The nearest neighbor distance in the • diamond structure is • there are 8 atoms in a unit cell • packing factor Is or 0.34. • The elements which crystallize similar to the structure of diamond are Germanium (Ge) and Silicon (Si).**Problems**• Draw the following planes in a cubic cell [312] • A monochromatic X-ray beam of wavelength 1.5Å undergoes second order Bragg reflection from the plane (2 1 1) of cubic crystal at a glancing angle of 54.38o. Calculate the lattice constant. • Calculate the glancing angle for incidence of X-rays of wavelength 0.58 Å on the plane (1 3 2) of NaCl which results in second order diffraction maxima taking the lattice as 3.81 Å. • The minimum order of Bragg’s reflection occurs at an angle of 20o in the plane [212]. Find the wavelength of X-ray if lattice constant is 3.615 Å. • In a cubic unit cell the distance between the two atoms present at the coordinates (100) and (011) is 5.8 Å. A monochromatic electron beam of unknown wavelength undergoes second order Bragg reflection from the plane (025) at a glancing angle of 45.95o. Calculate the wavelength of the electron beam.