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CONDENSED MATTER PHYSICS PHYSICS PAPER A BSc. (III) (NM and CSc.) Harvinder Kaur Associate Professor in Physics PG.Govt College for Girls Sector -11, Chandigarh

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Prof. Harvinder Kaur PG.Govt College for Girls Sector -11, Chandigarh

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OUTLINE Crystal Structure Unit Cell Symmetry Operations Bravais Lattice Characteristics of Unit Cell of cubic system Closed packed structure Miller Indices

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CRYSTAL STRUCTURE Crystal structure is a unique arrangement of atoms, molecules or ions constructed by the infinite repetition of identical structural units(called unit cell) in space.The structure of all crystals can be described in terms of lattice & basis. lattice : regular periodic arrangements of identical points in space lattice : regular periodic arrangements of identical points in space Basis : A group of atoms or ions Basis : A group of atoms or ions

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UNIT CELL Primitive Unit cell has one lattice point A primitive unit cell of a particular crystal structure is the smallest possible volume one can construct with the arrangement of atoms in the crystal such that, when stacked, completely fills the space. This primitive unit cell will not always display all the symmetries inherent in the crystal. Unit cell : A building block that can be periodically duplicated to result in the crystal structure, is known as the unit cell. Unit Cell is of two types Primitive Unit Cell Non-Primitive Unit Cell Non Primitive Unit cell has more than one lattice point

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A Physicist Wigner Seitz gave a geometrical way to design a primitive unit cell known as Wigner Seitz cell Steps for the construction of Wigner Seitz Cell Draw lines to connect a given lattice point to all nearby lattice points At the midpoint and normal to these lines draw new lines or planes The smallest volume enclosed in this way is Wigner- Seitz primitive cell WIGNER SEITZ PRIMITIVE CELL

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SYMMETRY OPERATIONS A symmetry operation is the one that leaves the crystal and its environment invariant. Symmetry operations performed about a point are called point group symmetry operations like Rotation, Reflection and Inversion Types of Symmetry operations Translation Symmetry Rotation Reflection Inversion

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TRANSLATION SYMMETRY The translation symmetry is the manifestation of the order of crystalline solids. r’= r + T= r + n 1 a +n 2 b +n 3 c b a c Translational operator, T is defined in terms of three fundamental vectors, a,b and c T = n 1 a+n 2 b+n 3 c Translational symmetry means that when the operator T is applied on any point r in the crystal, the resulting point r’ is exactly identical in all respects to the original point r

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ROTATION A lattice is said to possess the rotational symmetry about an axis if the rotation of the lattice by some angle leaves it invariant. Since the lattice remains invariant by rotation of 2 , so must be equal to 2 /n with n an integer. The integer n is called the multiplicity of the rotation axis.

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REFLECTION A lattice is said to possess reflection symmetry about a plane (or a line in two dimensions) if it is left unchanged after being reflected in a plane. In other words the plane divides the lattice into two identical halves which are mirror images of each other.

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INVERSION A crystal structure possesses an inversion symmetry if for each point located at r relative to a lattice point there exists an identical point at –r. Inversion is applicable in three dimensional lattices only.

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BRAVAIS LATTICE Bravais lattices :The space lattices which are invariant under one or more point of the symmetry operation are known as Bravais lattices. There are five Bravais lattice in two dimensions and 14 unique Bravais lattices in three dimensions In two dimensions, there are five Bravais lattices. These are 1. Oblique 2. Rectangular 3.Centered Rectangular 4. Hexagonal 5.Square

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CRYSTAL SYSTEM In three dimensions the 14 Bravais lattices are grouped into 7 crystal systems according to the seven types of conventional cells. They are : Triclinic - 1 Bravais Lattice, least symmetric Monoclinic – 2 Bravais Lattices Orthorhombic – 4 Bravais Lattices Rhombohedral/Trigonal -1 Bravais Lattice Tetragonal – 2 Bravais Lattices Hexagonal – 1 Bravais Lattices Cubic - 3 Bravais Lattices, most symmetric

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CRYSTAL SYSTEM CONTINUED TriclinicMonoclinic SimpleBase-Centered Orthorhombic Simple Base-Centered Base-Centered Face-Centered

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CRYSTAL SYSTEM CONTINUED SimpleBody- Centered Hexagon al SimpleBody- Centered Face- Centered Cubic Rhombohedral Tetragonal

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TRICLINIC a b c

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MONOCLINIC Simple Base Centered a b c

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ORTHORHOMBIC Simple Base- Centered Body- Centered Face- Centered = β= = 90

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RHOMBOHEDRAL or TRIGONAL

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TETRAGONAL SimpleBody-Centered = β= = 90

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HEXAGONAL = β=90, = 120

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CUBIC SimpleBody-CenteredFace-Centered

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CHARACTERISTICS OF THE UNIT CELL OF THE CUBIC SYSTEM Volume : The volume of unit cell is a 3 Atoms per unit cell : Simple Cubic - 1 Body Centered Cubic – 2 Face Centered Cubic - 4 Simple Body- Centered Face-Centered

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Cooridination Number : It is equal to the number of nearest neighbour that surrounds each atom. Simple Cubic - 6 Body Centered Cubic – 8 Face Centered Cubic - 12

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FCC Atomic Radius (r) : Simple Cubic - r= a/2 Body Centered Cubic – r = ( 3/4)a Face Centered Cubic - r = ( 2/4)a

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Atomic packing factor = Volume of atoms in a unit cell ----------------------------------------- Volume of the unit cell For Simple cubic P.F = (1x(4 /3)r 3 )/a 3 = /6 = 0.524 For Body centered cubic P.F = (2x(4 /3)r 3 )/a 3 = 3 /8 = 0.680 For Face centered cubic P.F = (4x(4 /3)r 3 )/a 3 = ( 2)/6 = 0.740

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CLOSE PACKED STRUCTURE

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ABAB STACKING GIVE RISE TO HEXAGONAL CLOSED PACKED STRUCTURE

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CLOSE PACKED STRUCTURE ABCABC.. STACKING GIVE RISE TO FACE CENTERED CUBIC STRUCTURE

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NaCl Crystal Structure The NaCl lattice is face –centered cubic; the basis consists of one Na atom and in Cl atom separated by one-half the body diagonal of unit cube. There are four units of NaCl in each unit of cube, with atoms in the positions Cl: 000 ½½0 ½0½ 0½½ Na : ½½½ 00½ 0½0 ½00 The NaCl structure has ionic bonding with each atom having 6 nearest neighbour and 12 next nearest neighbour. It has primitive unit cell which is simple cubic Atomic Packing fraction = 52.4%

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Diamond Crystal Structure The Diamond lattice is face –centered cubic; the basis consists of two identical C atoms separated by one-fourth the body diagonal of unit cube. C: 000 ½½0 ½0½ 0½½ C : ¼¼¼ ¾¾¼ ¾¼¾ ¼¾¾ The Diamond structure has tetrahedral bonding with each atom having 4 nearest neighbour and 12 next nearest neighbour. Atomic packing fraction = 34%

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Miller indices are a notation system in crystallography for planes in crystal (Bravais) lattices. MILLER INDICES Steps for calculating Miller Index Take any lattice point as origin in the crystal lattice and erect coordinate axis from this point in the direction of three basis vectors, a,b and c Identify the intercepts on these axis made by a plane of the set of a parallel planes of interest in terms of lattice constant Take the reciprocals of these intercepts and reduce these into smallest set of integers h,k,l The miller Indices of a set of parallel planes – (h k l)

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MILLER INDICES Planes with different Miller indices in cubic crystals

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EXAMPLES

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Examples

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LATTICE DIRECTION Generally the square brackets are used to indicate the direction i.e., [h,k,l]

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