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Crystal Structure, BCC ,FCC,HCP

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The document discusses different types of solids and their crystal structures. It defines crystalline and amorphous solids. Crystalline solids have an orderly arrangement of atoms and distinct crystal structures, while amorphous solids lack any orderly pattern. Common crystalline structures include face-centered cubic, body-centered cubic, and hexagonal close-packed. These structures differ in how atoms are arranged within the unit cell. The document also discusses properties like atomic radius, packing efficiency, and symmetry elements of different crystal structures.

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MD MOYNUL ISLAM ID:20151107004
Solid: A solid is one of the three states of matter, in which atoms are closely packed as compared to the other two states, e.g., gas and liquid. The atoms in the solid are not stationary but are vibrating around fix points, giving rise to the orderly arrangement of crystal structures. Solid may be classified as ο‚· Crystalline solid, also called true solid, ο‚· Amorphous solid and ο‚· A combination of two. Crystalline solids: In which solids, the atoms, molecules or ions are arranged in a systematic and characteristic geometrical shape or pattern are called crystalline solid. In a crystalline solid, the constituents are arranged in a regular, repeating three-dimensional pattern called the crystal lattice. Crystalline solid is two types. They are οƒ˜ Metallic crystals: In the formation of meal crystal the ions are connected only indirectly, through the free electrons surrounding them. Structures of some of the more important metallic crystals are: o Face centered cubic (FCC): 𝐢𝑒, 𝐴𝑔, 𝐴𝑙, 𝑁𝑖, 𝑃𝑏 etc. o Body centered cubic (BCC): π‘Š, π‘€π‘œ, πΆπ‘Ÿ etc. o Hexagonal close-packed (HCP): 𝑀𝑔, 𝑍𝑛, 𝑇𝑖, 𝐢𝑑, π‘π‘Ÿ etc. οƒ˜ Non-metallic crystals: Non-metallic crystals may have covalent or ionic bonding or any of a great variety of mixed bonds. Atoms of two or more kinds are frequently involved. Simple examples of non-metallic crystals are crystalline carbon and crystallized polymers or plastics.
Amorphous solids: Amorphous solids (Gr amorphos = no form) have atoms, molecules and ions arranged at random and in an irregular fashion or pattern and lacks the ordered crystalline lattice. In amorphous materials, the elementary particles are mixed together in a disorderly manner, their positions bearing no fixed relationship to those of their neighbor. The examples are glass, silica, plastics, rubbers, polymers etc. The difference between crystalline solids and amorphous solids are given below: Properties Crystalline solid Amorphous solid Geometrical shape The crystals of every crystalline solid have a defined geometrical shape due to definite and orderly arrangement of particles in three- dimensional shape. An amorphous solid does not have any orderly pattern of arrangement of particles and therefore, does not have any definite geometrical shape. Melting point The crystalline solids which directly change into liquid state have definite melting point. Amorphous solids like glass do not have definite melting point. Cleavage planes Crystalline solids have definite cleavage planes which are inclined to one another at a particular angle for a given crystalline solid. Amorphous solids do not have such well-defined cleavage planes. Anisotropic and isotropic properties As the orderly arranged particles in crystalline solids are different in different directions, these solids show anisotropic properties. As in amorphous solids, as those of liquids and gases, the arrangement of particles is random, these solids show isotropic properties. Symmetry Crystalline solids have crystal symmetry. Amorphous solids do not have symmetry. Tensile strength The tensile strength of crystalline solid is very high. The tensile strength of amorphous solid is low. Fracturing The crystalline solids fracture in a ductile manner. The amorphous solids fracture in a brittle manner. Examples Metals, π‘π‘ŽπΆπ‘™, 𝑍𝑛𝑂, 𝐾𝐢𝑙, 𝐢𝑠𝐢𝑙, carbon etc. Glass, plastics, polymers, rubbers, silica etc.
Crystal: A crystal is a solid whose constituent atoms or molecules are arranged in a systematic geometrical pattern. The term β€˜crystal’ of a material may be defined as a small body having a regular polyhedral form bounded by smooth surfaces which are acquired under the action of its intermolecular forces. In other words, the three dimensional periodic arrangement of atoms in a solid is known as crystal. An example for crystal is π‘π‘ŽπΆπ‘™. Unit Cell: The unit cell is the simplest repeating unit in the crystal. Crystal Lattice The crystal lattice is used to describe the lattice of a real crystal. A lattice point in a crystal lattice represents the position of a sodium ion or a chloride ion. The atoms arrange themselves in a distinct pattern in space called a space lattice. In other words, the three dimensional network of imaginary lines connecting the atoms in a crystalline solid is called the space lattice or crystal lattice. Bravais Lattice Bravais lattices are more mathematical and abstract than crystal lattices. They are pretty much the same as crystal lattices. Unlike the crystal lattice, however, lattice points in the Bravais lattice no longer represent a position of a particular atom. Instead a lattice point represents a position in which an atom can be placed. What matters in a Bravais lattice are not the points themselves, but how they are arranged (i.e. symmetry). Ex-Na,Li
Symmetry Crystal symmetry refers to the balanced pattern of the atomic structure which is reflected in the external (crystal) shape. Different species vary in the symmetrical arrangement of faces. These arrangements have certain β€˜planes’ and β€˜axes’ of symmetry. These form part of the definition of the crystal system to which specific gemstones belong. Plane of Symmetry An imaginary plane dividing a body into two parts such that each is the reflected image of the other. Crystals may have more than one plane of symmetry. i.e. a cube has nine planes of symmetry. Axis of Symmetry An imaginary axis is placed through a perfect crystal so that during a single rotation about this axis the outline of the crystal form appears identically more than once; 2, 3, 4 or 6 times. Centre of Symmetry: (Centro-Symmetry) Often present, it exists when every face of a perfect crystal is exactly opposite a similar face on the other side of the crystal.
Thelatticeparameterrelationshipforthesevencrystalsystemisgivenbelow: Crystalsystem Relationbetweenprimitives Interfaceangles Cubic π‘Ž = 𝑏 = 𝑐 𝛼 = 𝛽 = 𝛾 = 90Β° Monoclinic π‘Ž β‰  𝑏 β‰  𝑐 𝛼 = 𝛾 = 90Β°; 𝛽 β‰  90Β° Triclinic π‘Ž β‰  𝑏 β‰  𝑐 𝛼 β‰  𝛽 β‰  𝛾 β‰  90Β° Tetragonal π‘Ž = 𝑏 = 𝑐 𝛼 = 𝛽 = 𝛾 = 90Β° Orthorhombicorrhombic π‘Ž β‰  𝑏 β‰  𝑐 𝛼 = 𝛽 = 𝛾 = 90Β° Rhombohedralortrigonal π‘Ž = 𝑏 = 𝑐 𝛼 = 𝛽 = 𝛾 β‰  90Β° Hexagonal π‘Ž = 𝑏 β‰  𝑐 𝛼 = 𝛽 = 90Β°; 𝛾 = 120Β°
Crystal structure: The structure of a compound in which the atoms are arranged in a regular pattern or crystal form is known as crystal structure. Crystals are two types according to the arrangement of atoms or molecules: ο‚· Single crystal structure and ο‚· Polycrystalline structure. Single crystal structure: A single crystal is solid throughout which the atoms or molecules are arranged in a regular repetitive pattern. Examples are π‘π‘ŽπΆπ‘™, diamond etc. In the simplest form, the atomic pattern is continuous throughout the whole mass of the metal. When this occurs the metal is said to be a single crystal. Such a state is uncommon. Polycrystalline structure: A crystalline solid in which arrangements of many crystals are separated by well-defined boundaries is called polycrystalline structure. Usually the crystal pattern is discontinuous and changes its inclination from place to place. Thus polycrystalline structure is formed.
Face centered cubic (FCC) structure: A FCC unit cell has an atom at each corner of the cube and in addition, one atom at the intersection of the diagonals of each of the six faces of the cube but none at the center of the cube. It is also known as cubic packed structure. It is more common among metals than the BCC structure. 𝐴𝑙, 𝐢𝑒, 𝑃𝑏, 𝑁𝑖, πΆπ‘œ, 𝐴𝑔, 𝛾 𝐹𝑒 etc. posses FCC arrangement. It is also formed in ceramic crystal. We know total number of atoms in a cubic unit cell is 𝑁 = 𝑁𝑐 8 + 𝑁𝑓 2 + 𝑁𝑖 1 Here, 𝑁𝑐 = Total number of corner atom in unit cell 𝑁𝑓 = Total number of face atom in unit cell 𝑁𝑖 = Number of interior or center atom in unit cell For FCC, 𝑁𝑐 = 8, 𝑁𝑓 = 6 and 𝑁𝑖 = 0 Therefore, 𝑁 = 8 8 + 6 2 + 0 1 = 1 + 3 + 0 = 4 Hexagonal close packed (HCP) structure: A HCP unit cell has one atom at each corner of the hexagon, one atom at each center of the two hexagonal face (basal planes) and one atom at each center of the line connecting the perpendiculars in case of three rhombuses, namely 𝐷𝐸𝐹𝐺, 𝐡𝐷𝐺𝐻 and 𝐡𝐾𝐸𝐷, which combine and form the HCP structure. We know total number of atoms in a hexagonal unit cell is 𝑁 = 𝑁𝑐 6 + 𝑁𝑓 2 + 𝑁𝑖 1 Here, 𝑁𝑐 = Total number of corner atom in unit cell 𝑁𝑓 = Total number of face atom in unit cell 𝑁𝑖 = Number of interior or center atom in unit cell For HCP, 𝑁𝑐 = 12, 𝑁𝑓 = 2 and 𝑁𝑖 = 3 Therefore, 𝑁 = 12 6 + 2 2 + 3 1 = 2 + 1 + 3 = 6
Atomic radius: Atomic radius may be defined as half the distance between the centers of two neighboring atoms. It is possible to calculate the atomic radius by assuming that atoms are spherical in shape and are in contact in a crystal. Atomic radius of SC structure: We know that each simple cubic structure has one atom at each of the corner of the cube. If β€²π‘Žβ€² is the lattice parameter (length of the cube edge) and β€²π‘Ÿβ€² is the atomic radius, then π‘Ž = 2π‘Ÿ β‡’ π‘Ÿ = π‘Ž 2 Atomic radius of BCC structure: we know a BCC unit cell has one atom in the center of the cube and one atom each at all the corners. Let β€²π‘Žβ€² is the lattice parameter and β€²π‘Ÿβ€² is the atomic radius. From figure we see, 𝐴𝐺 = π‘Ÿ + 2π‘Ÿ + π‘Ÿ = 4π‘Ÿ β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― (1) Again, 𝐸𝐺2 = π‘Ž2 + π‘Ž2 = 2π‘Ž2 β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― (2) Then, 𝐴𝐺2 = 𝐸𝐺2 + 𝐴𝐸2 = 2π‘Ž2 + π‘Ž2 = 3π‘Ž2 β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― (3) From equation (1) and (3) above, 𝐴𝐺2 = 4π‘Ÿ 2 = 16π‘Ÿ2 = 3π‘Ž2 β‡’ π‘Ÿ2 = 3π‘Ž2 16 β‡’ π‘Ÿ = 3π‘Ž 4
Atomic radius of FCC structure: We know that a FCC unit cell has an atom at each corner of the cube and in addition, one atom at the intersection of the diagonals of each of the six faces of the cube but none at the center of the cube. Let β€²π‘Žβ€² is the lattice parameter and β€²π‘Ÿβ€² is the atomic radius. From figure we see, 𝐴𝐺 = π‘Ÿ + 2π‘Ÿ + π‘Ÿ = 4π‘Ÿ β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― (1) Then considering βˆ†π΄π΅πΊ, 𝐴𝐺2 = 𝐴𝐡2 + 𝐡𝐺2 β‡’ 4π‘Ÿ 2 = π‘Ž2 + π‘Ž2 β‡’ 16π‘Ÿ2 = 2π‘Ž2 β‡’ π‘Ÿ2 = 2π‘Ž2 16 β‡’ π‘Ÿ = π‘Ž2 8 β‡’ π‘Ÿ = π‘Ž 2 2 β‡’ π‘Ÿ = 2π‘Ž 4 Atomic packing factor or density of packing or packing efficiency: The packing of atoms in a unit cell of the crystal structure of a material is known as atomic packing. It has been defined as the ratio of the volume of the atoms per unit cell to the total volume of the unit cell. Then, APF = actual volume of atoms per unit cell total volume of the unit cell = 𝑣 𝑉 APF for SC structure: Let β€²π‘Žβ€² is the lattice parameter and β€²π‘Ÿβ€² is the atomic radius. Volume of unit cell = π‘Ž Γ— π‘Ž Γ— π‘Ž = π‘Ž3 . Atoms per unit cell = 1. Volume of one atom = 4 3 πœ‹π‘Ÿ3 . For SC structure, we know, π‘Ž = 2π‘Ÿ. Therefore, 𝑣 = 4 3 πœ‹ π‘Ž 2 3 = 4πœ‹π‘Ž3 3 Γ— 8 = πœ‹π‘Ž3 6 Hence, 𝐴𝑃𝐹 = πœ‹π‘Ž3 6 π‘Ž3 = πœ‹ 6 = 0.52
APF for BCC structure: Let β€²π‘Žβ€² is the lattice parameter and β€²π‘Ÿβ€² is the atomic radius. Volume of unit cell = π‘Ž Γ— π‘Ž Γ— π‘Ž = π‘Ž3 . Atoms per unit cell = 2. Volume of one atom = 2 Γ— 4 3 πœ‹π‘Ÿ3 . For BCC structure, we know, π‘Ÿ = 3π‘Ž 4. Therefore, 𝑣 = 8 3 πœ‹ 3π‘Ž 4 3 = 24 3πœ‹π‘Ž3 3 Γ— 64 = 3πœ‹π‘Ž3 8 Hence, 𝐴𝑃𝐹 = 3πœ‹π‘Ž3 8 π‘Ž3 = 3πœ‹ 8 = 0.68 APF for FCC structure: Let β€²π‘Žβ€² is the lattice parameter and β€²π‘Ÿβ€² is the atomic radius. Volume of unit cell = π‘Ž Γ— π‘Ž Γ— π‘Ž = π‘Ž3 . Atoms per unit cell = 4. Volume of one atom = 4 Γ— 4 3 πœ‹π‘Ÿ3 . For FCC structure, we know, π‘Ÿ = 2π‘Ž 4. Therefore, 𝑣 = 16 3 πœ‹ 2π‘Ž 4 3 = 32 2πœ‹π‘Ž3 3 Γ— 64 = 2πœ‹π‘Ž3 6 Hence, 𝐴𝑃𝐹 = 2πœ‹π‘Ž3 6 π‘Ž3 = 2πœ‹ 6 = 0.74
AFP for HCP structure: Let β€²π‘Žβ€² is the lattice parameter, ′𝑐′ is the distance between two basal planes and β€²π‘Ÿβ€² is the atomic radius. Then from βˆ†π»πΊπ·, area, 𝐴 = 1 2 Γ— π‘π‘Žπ‘ π‘’ Γ— β„Žπ‘’π‘–π‘”β„Žπ‘‘ β‡’ 𝐴 = 1 2 Γ— π‘Ž Γ— π‘Ž sin 60Β° β‡’ 𝐴 = 1 2 π‘Ž2 sin 60Β° Therefore total area, 𝐻𝐺𝐹𝐸𝐾𝐡 = 6 Γ— 𝐴 β‡’ 𝐻𝐺𝐹𝐸𝐾𝐡 = 6 Γ— 1 2 π‘Ž2 sin 60Β° β‡’ 𝐻𝐺𝐹𝐸𝐾𝐡 = 3π‘Ž2 sin 60Β° Hence the volume of a HCP unit cell, 𝑉 = 3π‘Ž2 sin 60Β° Γ— 𝑐 β‡’ 𝑉 = 3π‘Ž2 𝑐 Γ— 3 2 Now, atoms per unit cell = 6. Volume of six atoms = 6 Γ— 4 3 πœ‹π‘Ÿ3 = 8πœ‹π‘Ÿ3 . For HCP structure, π‘Ÿ = π‘Ž 2. So, volume of six atoms, 𝑣 = 6 Γ— 4 3 πœ‹ π‘Ž 2 3 = πœ‹π‘Ž3 . Hence, 𝐴𝑃𝐹 = πœ‹π‘Ž3 3π‘Ž2 𝑐 Γ— 3 2 = 2πœ‹π‘Ž 3 3𝑐 = 2πœ‹ 3 3 𝑐 π‘Ž = 2πœ‹ 3 3 Γ— 1.633 = 0.74 𝑐 π‘Ž = 1.633
Derive an equation to calculate the density of crystal materials. Let, 𝑛 = Number of atoms or molecules per unit cell 𝑀 = Molecular weight of the crystalline materials 𝑁 = Avogadro’s number = 6.023 Γ— 1023 Now, Weight of 𝑁 molecules or atoms = 𝑀 𝑔 Weight of 𝑛 molecules or atoms = 𝑀 Γ— 𝑛 𝑁 𝑔 = weight of the unit cell We know that weight of unit cell is πœŒπ‘Ž3 . Where, 𝜌 = Density of crystal materials π‘Ž = Lattice parameter Therefore, πœŒπ‘Ž3 = 𝑀 Γ— 𝑛 𝑁 β‡’ 𝜌 = 𝑀 Γ— 𝑛 π‘Ž3 Γ— 𝑁
1. Aluminium has FCC structure and a density of 𝟐. πŸ”πŸ—πŸ— Γ— 𝟏𝟎 πŸ‘ π’Œπ’ˆ/π’Ž πŸ‘ . Calculate its atomic radius. Atomic weight of aluminium is πŸπŸ”. πŸ—πŸ– π’ˆ/π’Žπ’π’π’† and 𝑡 = πŸ”. πŸŽπŸπŸ‘ Γ— 𝟏𝟎 πŸπŸ‘ π’‚π’•π’π’Žπ’”/ π’Žπ’π’π’†. We know, 𝜌 = 𝑛𝑀 π‘Ž3 𝑁 β‡’ 𝜌 = 𝑛𝑀 4π‘Ÿ 2 3 𝑁 β‡’ 𝜌 = 𝑛𝑀 Γ— 2 2 16π‘Ÿ3 𝑁 β‡’ π‘Ÿ3 = 𝑛𝑀 Γ— 2 2 16πœŒπ‘ β‡’ π‘Ÿ3 = 4 Γ— 26.98 Γ— 2 2 16 Γ— 2.7 Γ— 6.023 Γ— 1023 β‡’ π‘Ÿ3 = 2.93 Γ— 10βˆ’24 β‡’ π‘Ÿ = 1.43 Γ— 10βˆ’8 ∴ π‘Ÿ = 1.43 Γ— 10βˆ’8 π‘π‘š Given, 𝑛 = 4 for FCC 𝑀 = 26.98 𝑔/π‘šπ‘œπ‘™π‘’ π‘Ž = 4π‘Ÿ 2 𝜌 = 2.699 Γ— 103 π‘˜π‘”/π‘š3 = 2.7𝑔/π‘π‘š3 𝑁 = 6.023 Γ— 1023 π‘Žπ‘‘π‘œπ‘šπ‘ /π‘šπ‘œπ‘™π‘’ π‘Ÿ =?
Bragg’s equation: In 1913 the father-and-son, William Lawrence Bragg and William Henry Bragg worked out a mathematical relation to determine interatomic distances from X-ray diffraction patterns. This relation is called the Bragg’s equation. They showed that: i. The X-ray diffracted from atoms in crystal planes obeys the laws of reflection. ii.The two rays reflected by successive planes will be in phase if the extra distance travelled by the second ray is an integral number of wavelengths. The equation is π‘›πœ† = 2𝑑 sin πœƒ Where, πœƒ = diffraction angle πœ† = wave length of the X-ray 𝑑 = distance between the planes Derivation: Let a beam of X-rays is falling on the crystal surface. Two successive atomic planes of the crystal are shown separated by a distance 𝑑. Let the X-rays of wavelength πœ† strikes the planes at an angle πœƒ. For the first plane, 𝐴𝐡 is incident ray and 𝐡𝐢 is reflected ray. For the second plane, 𝐷𝐸 is incident ray and 𝐸𝐹 is reflected ray. Now we draw perpendicular lines 𝐡𝐺 and 𝐡𝐻 on 𝐷𝐸 and 𝐸𝐹 respectively. After falling on first plane some of the rays will be reflected at the same angle. Some of the rays will penetrate and get reflected from the second plane. These rays will reinforce those reflected from the first plane if the extra distance travelled by them (𝐺𝐸 + 𝐸𝐻) is equal to integral number, 𝑛, of wavelengths. That is, π‘›πœ† = 𝐺𝐸 + 𝐸𝐻 β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― (1) Geometry shows that 𝐺𝐸 = 𝐸𝐻 = 𝐡𝐸 sin πœƒ From equation (1), we get, π‘›πœ† = 𝐡𝐸 sin πœƒ + 𝐡𝐸 sin πœƒ β‡’ π‘›πœ† = 𝑑 sin πœƒ + 𝑑 sin πœƒ β‡’ π‘›πœ† = 2𝑑 sin πœƒ This is Bragg’s equation. [Derived]
Types Of Bonds The properties of a solid can usually be predicted from the valence and bonding preferences of its constituent atoms. Four main bonding types are discussed here: ionic, covalent, metallic, and molecular. Hydrogen-bonded solids, such as ice, make up another category that is important in a few crystals. There are many examples of solids that have a single bonding type, while other solids have a mixture of types, such as covalent and metallic or covalent and ionic. ο‚·
Lattice energy of crystal system The stability of an ionic solid is measured in terms of its lattice energy (U0). It is defined as the energy released when on mole of an ionic crystal is formed from one mole of gaseous positive and one mole of gaseous negative ions, when these are separated from each other y infinite distance. Greater the value of lattice energy, more stable, more stable is the ionic solid. The lattice energy of sodium chloride is - 786 kj/mole. The negative sign indicates that the process is exothermic and that the energy of the system is lowered as the solid is formed. The high value suggests the stability of the solid. Factors Affecting Lattice Energy ο‚· as the charge of the ions increases, the lattice energy increases ο‚· as the size of the ions increases, the lattice energy decreases Lattice energies are also important in predicting the solubility of ionic solids in H2O. Ionic compounds with smaller lattice energies tend to be more soluble in H2O. Cohesive energy: The amount of energy evolved when a crystalline solid is formed from infinitely separated atoms or the amount of energy required to separate atoms in a crystalline solid to infinite distance is known as cohesive or binding energy of the crystal. The interatomic forces which hold the atoms together in a solid are responsible for crystal formation. So, the energy of the crystal is lower than that of the free atoms by an amount equal to the energy required to pull the atoms to an infinite distance. This energy is called the binding energy of the crystal.
What are defects and imperfections in crystals? Describe them with neat sketches. Defects and imperfections in crystals: A crystal is a solid composed of atoms, ions or molecules arranged in a pattern which is repetitive in three dimensions. In an ideal crystal, the atomic arrangement is perfectly regular and continuous throughout. But, real crystals are never perfect; lattice distortion and various imperfections, irregularities or defects are generally present in them. If the species in crystal are arranged in irregular or random pattern, it is said to be defect or imperfection in crystal. All defects and imperfections in crystals can be conveniently classed under four main divisions, namely, a) Point defects or imperfections or zero dimensional imperfections i. Vacancies ii. Interstitialcies iii. Impurities iv. Substitutional v. Schottky defect b) Line defects or imperfections i. Edge dislocation ii. Screw dislocation c) Planar, surface, interfacial or grain boundaries defects or imperfections i. Grain boundaries ii. Tilt boundaries iii. Twin boundaries d) Volume defects or imperfections
Point defects: The imperfections which take place due to imperfect packing of atoms during crystallization are known as point imperfections. The point imperfections also take place due to vibrations of atoms at high temperature. i. Vacancies: Whenever one or more atoms are missing from a normally occupied position within a crystal lattice is known as vacancy. ii. Interstitialcies: An interstitial defect arises when an atom occupies a definite position in the lattice that is not normally occupied in the perfect crystal. iii. Impurities: Impurities may be small particles (such as slag inclusions in metals) embedded in the structure or foreign (metal) atoms in the lattice. Impurities give rise to compositional defects. iv. Electronic defects or Schottky defects: Electronic defects are the result of errors in charge distribution in solids.
Line defects: The most important two-dimensional or line defect is the dislocation. A dislocation may be defined as a disturbed region between two substantially perfect parts of a crystal. i. Edge dislocation: An edge dislocation lies perpendicular to its Burger’s vector. It moves (in its slip plane) in the direction of the Burger’s vector (slip direction) ii. Screw dislocation: An screw dislocation lies parallel to its Burger’s vector. It moves (in the slip plane) in a direction perpendicular to the Burger’s vector (slip direction). Figure 1: Edge dislocation
Surface or planar defects: The defects which are take place on the surface of a material are known as surface defects. i. Grain boundaries: Grain boundaries are those planar imperfects ions in polycrystalline materials that separate crystals (grains) of different orientation. ii. Tilt boundaries: Till boundary may be regarded as an array of edge dislocations (βŠ₯). Actually tilt boundary is a class of low angle boundaries.
i. Twin boundaries: A twin boundary separates two parts of crystal having the same orientation and they look like mirror image of each other. Volume defects: Volume defects such as cracks or stacking faults may arise where there is only small dissimilarity (electrostatically) between the stacking sequence of closed packed planes in FCC and HCP metals. It is possible for one atom layer to be out of sequence relative to the atoms of the layers above and below, giving a mistake or defect.
1. Whatisunderstoodbythetermβ€˜solidsolution’? Solid solution: A homogeneous mixture of two or more crystalline metals in varying proportion is known as solid solution. The metal which is present in a small quantity in a solid solution is called solute while the metalwhich ispresentinlargequantityiscalled solute.Solid solutionhasonlyonephase. Solid solution forms more rapidly when the solvent and solute atoms have similar size and comparable electron structure. For example brass is a solid solution of 𝐢𝑒 and 𝑍𝑛; a typical composition of 𝐢𝑒 64% and 𝑍𝑛 36%,where 𝐢𝑒 atomsare solvent and 𝑍𝑛 atomsare solute. Therearetwotypesofsolid solution:  Interstitialsolidsolutionand  Substitutionalsolid solution. 1. What are the conditions for the formation of solid solution? Three conditions must be fulfilled to form a solid solution. Such as – οƒ˜ Two or more metals that are involved to form a solid solution should have similar crystal structure. οƒ˜ Metals that are involved to form a solid solution should be of same group in the periodic table and should have same oxidation number. οƒ˜ Metals that are involved to form a solid solution should have nearly same atomic structure or atomic size.
1. Explain with neat sketches the substitutional solid solution and interstitial solid solution. Substitutional solid solution: A solid solution in which solute atoms directly replace some solvent atoms so that they lie at normal atoms site of the crystal structure then this type of solid solution is called substitutional solid solution. Substitutional solid solution forms when the solute and solvent atoms posses equal or approximately equal diameter. For example atomic diameter of 𝐢𝑒 is 2.522β„« and 𝑁𝑖 is 2.487β„« and these two form substitutional solid solution. Substitutional solid solutions are two types: ο‚« Random or disordered substitutional solid solution and ο‚« Ordered substitutional solid solution. Random or disordered substitutional solid solution: A solid solution in which the solute atoms replace the solvent atoms randomly is called random substitutional solid solution and this type of substitution is called random substitution. Following is the structure of random substitutional solid solution. Ordered substitutional solid solution: A substitutional solid solution in which the solute atoms replace by the solvent atoms orderly is called ordered substitutional solid solution and this type of substitution is known as ordered substitution. 𝐢𝑒 βˆ’ 𝑍𝑛, 𝐴𝑒 βˆ’ 𝐢𝑒, 𝐢𝑒2 𝑀𝑛𝐴𝑙 are some examples of ordered structure. Following is the structure of ordered substitutional solid solution. Interstitial solid solution: A solid solution in which solute atoms are located in the interstices of the solvent lattice is called interstitial solid solution. Interstitial solid solution can form only when the solute are small enough to fit into the space between the solvent atoms. They are unable to substitute solvent atoms and can only to fit interstices or spaces in the crystal lattice of solvent atoms. For example FCC 𝐹𝑒 is the solvent for 𝐢.
Miller indices: Miller indices are a system of notation that denotes the orientation of the faces of a crystal and the planes and directions of atoms within that crystal. Some General Principles: ο‚· If a Miller index is zero, the plane is parallel to that axis. ο‚· The smaller a Miller index, the more nearly parallel the plane is to the axis. ο‚· The larger a Miller index, the more nearly perpendicular a plane is to that axis. ο‚· Multiplying or dividing a Miller index by a constant has no effect on the orientation of the plane ο‚· Miller indices are almost always small. Determination of Miller indices: Following are the rules for determination of Miller indices: a) At first we should have to determine the intercepts of the plane on the axes π‘₯, 𝑦, 𝑧 in terms of lattice constant. b) Now we have to take the reciprocal terms of the intercepts. c) If the results are fractions, then they are changed to the small integers having the same ratio i.e., multiplied each reciprocal or fraction by the same number such as the common denominator. d) At last the result is enclosed in parenthesis (β„Ž π‘˜ 𝑙). Example: π‘₯ 𝑦 𝑧 Intercept 1 ∞ 1 Reciprocal 1 1 = 1 1 ∞ = 0 1 1 = 1 Miller indices β„Ž π‘˜ 𝑙 = (1 0 1) π‘₯ 𝑦 𝑧 Intercept 1 3 2 Reciprocal 1 1 = 1 1 Γ— 6 = 6 1 3 = 1 3 Γ— 6 = 2 1 2 = 1 2 Γ— 6 = 3 Miller indices β„Ž π‘˜ 𝑙 = (6 2 3)

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Crystal Structure, BCC ,FCC,HCP

  • 2. Solid: A solid is one of the three states of matter, in which atoms are closely packed as compared to the other two states, e.g., gas and liquid. The atoms in the solid are not stationary but are vibrating around fix points, giving rise to the orderly arrangement of crystal structures. Solid may be classified as ο‚· Crystalline solid, also called true solid, ο‚· Amorphous solid and ο‚· A combination of two. Crystalline solids: In which solids, the atoms, molecules or ions are arranged in a systematic and characteristic geometrical shape or pattern are called crystalline solid. In a crystalline solid, the constituents are arranged in a regular, repeating three-dimensional pattern called the crystal lattice. Crystalline solid is two types. They are οƒ˜ Metallic crystals: In the formation of meal crystal the ions are connected only indirectly, through the free electrons surrounding them. Structures of some of the more important metallic crystals are: o Face centered cubic (FCC): 𝐢𝑒, 𝐴𝑔, 𝐴𝑙, 𝑁𝑖, 𝑃𝑏 etc. o Body centered cubic (BCC): π‘Š, π‘€π‘œ, πΆπ‘Ÿ etc. o Hexagonal close-packed (HCP): 𝑀𝑔, 𝑍𝑛, 𝑇𝑖, 𝐢𝑑, π‘π‘Ÿ etc. οƒ˜ Non-metallic crystals: Non-metallic crystals may have covalent or ionic bonding or any of a great variety of mixed bonds. Atoms of two or more kinds are frequently involved. Simple examples of non-metallic crystals are crystalline carbon and crystallized polymers or plastics.
  • 3. Amorphous solids: Amorphous solids (Gr amorphos = no form) have atoms, molecules and ions arranged at random and in an irregular fashion or pattern and lacks the ordered crystalline lattice. In amorphous materials, the elementary particles are mixed together in a disorderly manner, their positions bearing no fixed relationship to those of their neighbor. The examples are glass, silica, plastics, rubbers, polymers etc. The difference between crystalline solids and amorphous solids are given below: Properties Crystalline solid Amorphous solid Geometrical shape The crystals of every crystalline solid have a defined geometrical shape due to definite and orderly arrangement of particles in three- dimensional shape. An amorphous solid does not have any orderly pattern of arrangement of particles and therefore, does not have any definite geometrical shape. Melting point The crystalline solids which directly change into liquid state have definite melting point. Amorphous solids like glass do not have definite melting point. Cleavage planes Crystalline solids have definite cleavage planes which are inclined to one another at a particular angle for a given crystalline solid. Amorphous solids do not have such well-defined cleavage planes. Anisotropic and isotropic properties As the orderly arranged particles in crystalline solids are different in different directions, these solids show anisotropic properties. As in amorphous solids, as those of liquids and gases, the arrangement of particles is random, these solids show isotropic properties. Symmetry Crystalline solids have crystal symmetry. Amorphous solids do not have symmetry. Tensile strength The tensile strength of crystalline solid is very high. The tensile strength of amorphous solid is low. Fracturing The crystalline solids fracture in a ductile manner. The amorphous solids fracture in a brittle manner. Examples Metals, π‘π‘ŽπΆπ‘™, 𝑍𝑛𝑂, 𝐾𝐢𝑙, 𝐢𝑠𝐢𝑙, carbon etc. Glass, plastics, polymers, rubbers, silica etc.
  • 4. Crystal: A crystal is a solid whose constituent atoms or molecules are arranged in a systematic geometrical pattern. The term β€˜crystal’ of a material may be defined as a small body having a regular polyhedral form bounded by smooth surfaces which are acquired under the action of its intermolecular forces. In other words, the three dimensional periodic arrangement of atoms in a solid is known as crystal. An example for crystal is π‘π‘ŽπΆπ‘™. Unit Cell: The unit cell is the simplest repeating unit in the crystal. Crystal Lattice The crystal lattice is used to describe the lattice of a real crystal. A lattice point in a crystal lattice represents the position of a sodium ion or a chloride ion. The atoms arrange themselves in a distinct pattern in space called a space lattice. In other words, the three dimensional network of imaginary lines connecting the atoms in a crystalline solid is called the space lattice or crystal lattice. Bravais Lattice Bravais lattices are more mathematical and abstract than crystal lattices. They are pretty much the same as crystal lattices. Unlike the crystal lattice, however, lattice points in the Bravais lattice no longer represent a position of a particular atom. Instead a lattice point represents a position in which an atom can be placed. What matters in a Bravais lattice are not the points themselves, but how they are arranged (i.e. symmetry). Ex-Na,Li
  • 5. Symmetry Crystal symmetry refers to the balanced pattern of the atomic structure which is reflected in the external (crystal) shape. Different species vary in the symmetrical arrangement of faces. These arrangements have certain β€˜planes’ and β€˜axes’ of symmetry. These form part of the definition of the crystal system to which specific gemstones belong. Plane of Symmetry An imaginary plane dividing a body into two parts such that each is the reflected image of the other. Crystals may have more than one plane of symmetry. i.e. a cube has nine planes of symmetry. Axis of Symmetry An imaginary axis is placed through a perfect crystal so that during a single rotation about this axis the outline of the crystal form appears identically more than once; 2, 3, 4 or 6 times. Centre of Symmetry: (Centro-Symmetry) Often present, it exists when every face of a perfect crystal is exactly opposite a similar face on the other side of the crystal.
  • 6. Thelatticeparameterrelationshipforthesevencrystalsystemisgivenbelow: Crystalsystem Relationbetweenprimitives Interfaceangles Cubic π‘Ž = 𝑏 = 𝑐 𝛼 = 𝛽 = 𝛾 = 90Β° Monoclinic π‘Ž β‰  𝑏 β‰  𝑐 𝛼 = 𝛾 = 90Β°; 𝛽 β‰  90Β° Triclinic π‘Ž β‰  𝑏 β‰  𝑐 𝛼 β‰  𝛽 β‰  𝛾 β‰  90Β° Tetragonal π‘Ž = 𝑏 = 𝑐 𝛼 = 𝛽 = 𝛾 = 90Β° Orthorhombicorrhombic π‘Ž β‰  𝑏 β‰  𝑐 𝛼 = 𝛽 = 𝛾 = 90Β° Rhombohedralortrigonal π‘Ž = 𝑏 = 𝑐 𝛼 = 𝛽 = 𝛾 β‰  90Β° Hexagonal π‘Ž = 𝑏 β‰  𝑐 𝛼 = 𝛽 = 90Β°; 𝛾 = 120Β°
  • 7. Crystal structure: The structure of a compound in which the atoms are arranged in a regular pattern or crystal form is known as crystal structure. Crystals are two types according to the arrangement of atoms or molecules: ο‚· Single crystal structure and ο‚· Polycrystalline structure. Single crystal structure: A single crystal is solid throughout which the atoms or molecules are arranged in a regular repetitive pattern. Examples are π‘π‘ŽπΆπ‘™, diamond etc. In the simplest form, the atomic pattern is continuous throughout the whole mass of the metal. When this occurs the metal is said to be a single crystal. Such a state is uncommon. Polycrystalline structure: A crystalline solid in which arrangements of many crystals are separated by well-defined boundaries is called polycrystalline structure. Usually the crystal pattern is discontinuous and changes its inclination from place to place. Thus polycrystalline structure is formed.
  • 8. Face centered cubic (FCC) structure: A FCC unit cell has an atom at each corner of the cube and in addition, one atom at the intersection of the diagonals of each of the six faces of the cube but none at the center of the cube. It is also known as cubic packed structure. It is more common among metals than the BCC structure. 𝐴𝑙, 𝐢𝑒, 𝑃𝑏, 𝑁𝑖, πΆπ‘œ, 𝐴𝑔, 𝛾 𝐹𝑒 etc. posses FCC arrangement. It is also formed in ceramic crystal. We know total number of atoms in a cubic unit cell is 𝑁 = 𝑁𝑐 8 + 𝑁𝑓 2 + 𝑁𝑖 1 Here, 𝑁𝑐 = Total number of corner atom in unit cell 𝑁𝑓 = Total number of face atom in unit cell 𝑁𝑖 = Number of interior or center atom in unit cell For FCC, 𝑁𝑐 = 8, 𝑁𝑓 = 6 and 𝑁𝑖 = 0 Therefore, 𝑁 = 8 8 + 6 2 + 0 1 = 1 + 3 + 0 = 4 Hexagonal close packed (HCP) structure: A HCP unit cell has one atom at each corner of the hexagon, one atom at each center of the two hexagonal face (basal planes) and one atom at each center of the line connecting the perpendiculars in case of three rhombuses, namely 𝐷𝐸𝐹𝐺, 𝐡𝐷𝐺𝐻 and 𝐡𝐾𝐸𝐷, which combine and form the HCP structure. We know total number of atoms in a hexagonal unit cell is 𝑁 = 𝑁𝑐 6 + 𝑁𝑓 2 + 𝑁𝑖 1 Here, 𝑁𝑐 = Total number of corner atom in unit cell 𝑁𝑓 = Total number of face atom in unit cell 𝑁𝑖 = Number of interior or center atom in unit cell For HCP, 𝑁𝑐 = 12, 𝑁𝑓 = 2 and 𝑁𝑖 = 3 Therefore, 𝑁 = 12 6 + 2 2 + 3 1 = 2 + 1 + 3 = 6
  • 9. Atomic radius: Atomic radius may be defined as half the distance between the centers of two neighboring atoms. It is possible to calculate the atomic radius by assuming that atoms are spherical in shape and are in contact in a crystal. Atomic radius of SC structure: We know that each simple cubic structure has one atom at each of the corner of the cube. If β€²π‘Žβ€² is the lattice parameter (length of the cube edge) and β€²π‘Ÿβ€² is the atomic radius, then π‘Ž = 2π‘Ÿ β‡’ π‘Ÿ = π‘Ž 2 Atomic radius of BCC structure: we know a BCC unit cell has one atom in the center of the cube and one atom each at all the corners. Let β€²π‘Žβ€² is the lattice parameter and β€²π‘Ÿβ€² is the atomic radius. From figure we see, 𝐴𝐺 = π‘Ÿ + 2π‘Ÿ + π‘Ÿ = 4π‘Ÿ β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― (1) Again, 𝐸𝐺2 = π‘Ž2 + π‘Ž2 = 2π‘Ž2 β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― (2) Then, 𝐴𝐺2 = 𝐸𝐺2 + 𝐴𝐸2 = 2π‘Ž2 + π‘Ž2 = 3π‘Ž2 β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― (3) From equation (1) and (3) above, 𝐴𝐺2 = 4π‘Ÿ 2 = 16π‘Ÿ2 = 3π‘Ž2 β‡’ π‘Ÿ2 = 3π‘Ž2 16 β‡’ π‘Ÿ = 3π‘Ž 4
  • 10. Atomic radius of FCC structure: We know that a FCC unit cell has an atom at each corner of the cube and in addition, one atom at the intersection of the diagonals of each of the six faces of the cube but none at the center of the cube. Let β€²π‘Žβ€² is the lattice parameter and β€²π‘Ÿβ€² is the atomic radius. From figure we see, 𝐴𝐺 = π‘Ÿ + 2π‘Ÿ + π‘Ÿ = 4π‘Ÿ β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― (1) Then considering βˆ†π΄π΅πΊ, 𝐴𝐺2 = 𝐴𝐡2 + 𝐡𝐺2 β‡’ 4π‘Ÿ 2 = π‘Ž2 + π‘Ž2 β‡’ 16π‘Ÿ2 = 2π‘Ž2 β‡’ π‘Ÿ2 = 2π‘Ž2 16 β‡’ π‘Ÿ = π‘Ž2 8 β‡’ π‘Ÿ = π‘Ž 2 2 β‡’ π‘Ÿ = 2π‘Ž 4 Atomic packing factor or density of packing or packing efficiency: The packing of atoms in a unit cell of the crystal structure of a material is known as atomic packing. It has been defined as the ratio of the volume of the atoms per unit cell to the total volume of the unit cell. Then, APF = actual volume of atoms per unit cell total volume of the unit cell = 𝑣 𝑉 APF for SC structure: Let β€²π‘Žβ€² is the lattice parameter and β€²π‘Ÿβ€² is the atomic radius. Volume of unit cell = π‘Ž Γ— π‘Ž Γ— π‘Ž = π‘Ž3 . Atoms per unit cell = 1. Volume of one atom = 4 3 πœ‹π‘Ÿ3 . For SC structure, we know, π‘Ž = 2π‘Ÿ. Therefore, 𝑣 = 4 3 πœ‹ π‘Ž 2 3 = 4πœ‹π‘Ž3 3 Γ— 8 = πœ‹π‘Ž3 6 Hence, 𝐴𝑃𝐹 = πœ‹π‘Ž3 6 π‘Ž3 = πœ‹ 6 = 0.52
  • 11. APF for BCC structure: Let β€²π‘Žβ€² is the lattice parameter and β€²π‘Ÿβ€² is the atomic radius. Volume of unit cell = π‘Ž Γ— π‘Ž Γ— π‘Ž = π‘Ž3 . Atoms per unit cell = 2. Volume of one atom = 2 Γ— 4 3 πœ‹π‘Ÿ3 . For BCC structure, we know, π‘Ÿ = 3π‘Ž 4. Therefore, 𝑣 = 8 3 πœ‹ 3π‘Ž 4 3 = 24 3πœ‹π‘Ž3 3 Γ— 64 = 3πœ‹π‘Ž3 8 Hence, 𝐴𝑃𝐹 = 3πœ‹π‘Ž3 8 π‘Ž3 = 3πœ‹ 8 = 0.68 APF for FCC structure: Let β€²π‘Žβ€² is the lattice parameter and β€²π‘Ÿβ€² is the atomic radius. Volume of unit cell = π‘Ž Γ— π‘Ž Γ— π‘Ž = π‘Ž3 . Atoms per unit cell = 4. Volume of one atom = 4 Γ— 4 3 πœ‹π‘Ÿ3 . For FCC structure, we know, π‘Ÿ = 2π‘Ž 4. Therefore, 𝑣 = 16 3 πœ‹ 2π‘Ž 4 3 = 32 2πœ‹π‘Ž3 3 Γ— 64 = 2πœ‹π‘Ž3 6 Hence, 𝐴𝑃𝐹 = 2πœ‹π‘Ž3 6 π‘Ž3 = 2πœ‹ 6 = 0.74
  • 12. AFP for HCP structure: Let β€²π‘Žβ€² is the lattice parameter, ′𝑐′ is the distance between two basal planes and β€²π‘Ÿβ€² is the atomic radius. Then from βˆ†π»πΊπ·, area, 𝐴 = 1 2 Γ— π‘π‘Žπ‘ π‘’ Γ— β„Žπ‘’π‘–π‘”β„Žπ‘‘ β‡’ 𝐴 = 1 2 Γ— π‘Ž Γ— π‘Ž sin 60Β° β‡’ 𝐴 = 1 2 π‘Ž2 sin 60Β° Therefore total area, 𝐻𝐺𝐹𝐸𝐾𝐡 = 6 Γ— 𝐴 β‡’ 𝐻𝐺𝐹𝐸𝐾𝐡 = 6 Γ— 1 2 π‘Ž2 sin 60Β° β‡’ 𝐻𝐺𝐹𝐸𝐾𝐡 = 3π‘Ž2 sin 60Β° Hence the volume of a HCP unit cell, 𝑉 = 3π‘Ž2 sin 60Β° Γ— 𝑐 β‡’ 𝑉 = 3π‘Ž2 𝑐 Γ— 3 2 Now, atoms per unit cell = 6. Volume of six atoms = 6 Γ— 4 3 πœ‹π‘Ÿ3 = 8πœ‹π‘Ÿ3 . For HCP structure, π‘Ÿ = π‘Ž 2. So, volume of six atoms, 𝑣 = 6 Γ— 4 3 πœ‹ π‘Ž 2 3 = πœ‹π‘Ž3 . Hence, 𝐴𝑃𝐹 = πœ‹π‘Ž3 3π‘Ž2 𝑐 Γ— 3 2 = 2πœ‹π‘Ž 3 3𝑐 = 2πœ‹ 3 3 𝑐 π‘Ž = 2πœ‹ 3 3 Γ— 1.633 = 0.74 𝑐 π‘Ž = 1.633
  • 13. Derive an equation to calculate the density of crystal materials. Let, 𝑛 = Number of atoms or molecules per unit cell 𝑀 = Molecular weight of the crystalline materials 𝑁 = Avogadro’s number = 6.023 Γ— 1023 Now, Weight of 𝑁 molecules or atoms = 𝑀 𝑔 Weight of 𝑛 molecules or atoms = 𝑀 Γ— 𝑛 𝑁 𝑔 = weight of the unit cell We know that weight of unit cell is πœŒπ‘Ž3 . Where, 𝜌 = Density of crystal materials π‘Ž = Lattice parameter Therefore, πœŒπ‘Ž3 = 𝑀 Γ— 𝑛 𝑁 β‡’ 𝜌 = 𝑀 Γ— 𝑛 π‘Ž3 Γ— 𝑁
  • 14. 1. Aluminium has FCC structure and a density of 𝟐. πŸ”πŸ—πŸ— Γ— 𝟏𝟎 πŸ‘ π’Œπ’ˆ/π’Ž πŸ‘ . Calculate its atomic radius. Atomic weight of aluminium is πŸπŸ”. πŸ—πŸ– π’ˆ/π’Žπ’π’π’† and 𝑡 = πŸ”. πŸŽπŸπŸ‘ Γ— 𝟏𝟎 πŸπŸ‘ π’‚π’•π’π’Žπ’”/ π’Žπ’π’π’†. We know, 𝜌 = 𝑛𝑀 π‘Ž3 𝑁 β‡’ 𝜌 = 𝑛𝑀 4π‘Ÿ 2 3 𝑁 β‡’ 𝜌 = 𝑛𝑀 Γ— 2 2 16π‘Ÿ3 𝑁 β‡’ π‘Ÿ3 = 𝑛𝑀 Γ— 2 2 16πœŒπ‘ β‡’ π‘Ÿ3 = 4 Γ— 26.98 Γ— 2 2 16 Γ— 2.7 Γ— 6.023 Γ— 1023 β‡’ π‘Ÿ3 = 2.93 Γ— 10βˆ’24 β‡’ π‘Ÿ = 1.43 Γ— 10βˆ’8 ∴ π‘Ÿ = 1.43 Γ— 10βˆ’8 π‘π‘š Given, 𝑛 = 4 for FCC 𝑀 = 26.98 𝑔/π‘šπ‘œπ‘™π‘’ π‘Ž = 4π‘Ÿ 2 𝜌 = 2.699 Γ— 103 π‘˜π‘”/π‘š3 = 2.7𝑔/π‘π‘š3 𝑁 = 6.023 Γ— 1023 π‘Žπ‘‘π‘œπ‘šπ‘ /π‘šπ‘œπ‘™π‘’ π‘Ÿ =?
  • 15. Bragg’s equation: In 1913 the father-and-son, William Lawrence Bragg and William Henry Bragg worked out a mathematical relation to determine interatomic distances from X-ray diffraction patterns. This relation is called the Bragg’s equation. They showed that: i. The X-ray diffracted from atoms in crystal planes obeys the laws of reflection. ii.The two rays reflected by successive planes will be in phase if the extra distance travelled by the second ray is an integral number of wavelengths. The equation is π‘›πœ† = 2𝑑 sin πœƒ Where, πœƒ = diffraction angle πœ† = wave length of the X-ray 𝑑 = distance between the planes Derivation: Let a beam of X-rays is falling on the crystal surface. Two successive atomic planes of the crystal are shown separated by a distance 𝑑. Let the X-rays of wavelength πœ† strikes the planes at an angle πœƒ. For the first plane, 𝐴𝐡 is incident ray and 𝐡𝐢 is reflected ray. For the second plane, 𝐷𝐸 is incident ray and 𝐸𝐹 is reflected ray. Now we draw perpendicular lines 𝐡𝐺 and 𝐡𝐻 on 𝐷𝐸 and 𝐸𝐹 respectively. After falling on first plane some of the rays will be reflected at the same angle. Some of the rays will penetrate and get reflected from the second plane. These rays will reinforce those reflected from the first plane if the extra distance travelled by them (𝐺𝐸 + 𝐸𝐻) is equal to integral number, 𝑛, of wavelengths. That is, π‘›πœ† = 𝐺𝐸 + 𝐸𝐻 β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― β‹― (1) Geometry shows that 𝐺𝐸 = 𝐸𝐻 = 𝐡𝐸 sin πœƒ From equation (1), we get, π‘›πœ† = 𝐡𝐸 sin πœƒ + 𝐡𝐸 sin πœƒ β‡’ π‘›πœ† = 𝑑 sin πœƒ + 𝑑 sin πœƒ β‡’ π‘›πœ† = 2𝑑 sin πœƒ This is Bragg’s equation. [Derived]
  • 16. Types Of Bonds The properties of a solid can usually be predicted from the valence and bonding preferences of its constituent atoms. Four main bonding types are discussed here: ionic, covalent, metallic, and molecular. Hydrogen-bonded solids, such as ice, make up another category that is important in a few crystals. There are many examples of solids that have a single bonding type, while other solids have a mixture of types, such as covalent and metallic or covalent and ionic. ο‚·
  • 17. Lattice energy of crystal system The stability of an ionic solid is measured in terms of its lattice energy (U0). It is defined as the energy released when on mole of an ionic crystal is formed from one mole of gaseous positive and one mole of gaseous negative ions, when these are separated from each other y infinite distance. Greater the value of lattice energy, more stable, more stable is the ionic solid. The lattice energy of sodium chloride is - 786 kj/mole. The negative sign indicates that the process is exothermic and that the energy of the system is lowered as the solid is formed. The high value suggests the stability of the solid. Factors Affecting Lattice Energy ο‚· as the charge of the ions increases, the lattice energy increases ο‚· as the size of the ions increases, the lattice energy decreases Lattice energies are also important in predicting the solubility of ionic solids in H2O. Ionic compounds with smaller lattice energies tend to be more soluble in H2O. Cohesive energy: The amount of energy evolved when a crystalline solid is formed from infinitely separated atoms or the amount of energy required to separate atoms in a crystalline solid to infinite distance is known as cohesive or binding energy of the crystal. The interatomic forces which hold the atoms together in a solid are responsible for crystal formation. So, the energy of the crystal is lower than that of the free atoms by an amount equal to the energy required to pull the atoms to an infinite distance. This energy is called the binding energy of the crystal.
  • 18. What are defects and imperfections in crystals? Describe them with neat sketches. Defects and imperfections in crystals: A crystal is a solid composed of atoms, ions or molecules arranged in a pattern which is repetitive in three dimensions. In an ideal crystal, the atomic arrangement is perfectly regular and continuous throughout. But, real crystals are never perfect; lattice distortion and various imperfections, irregularities or defects are generally present in them. If the species in crystal are arranged in irregular or random pattern, it is said to be defect or imperfection in crystal. All defects and imperfections in crystals can be conveniently classed under four main divisions, namely, a) Point defects or imperfections or zero dimensional imperfections i. Vacancies ii. Interstitialcies iii. Impurities iv. Substitutional v. Schottky defect b) Line defects or imperfections i. Edge dislocation ii. Screw dislocation c) Planar, surface, interfacial or grain boundaries defects or imperfections i. Grain boundaries ii. Tilt boundaries iii. Twin boundaries d) Volume defects or imperfections
  • 19. Point defects: The imperfections which take place due to imperfect packing of atoms during crystallization are known as point imperfections. The point imperfections also take place due to vibrations of atoms at high temperature. i. Vacancies: Whenever one or more atoms are missing from a normally occupied position within a crystal lattice is known as vacancy. ii. Interstitialcies: An interstitial defect arises when an atom occupies a definite position in the lattice that is not normally occupied in the perfect crystal. iii. Impurities: Impurities may be small particles (such as slag inclusions in metals) embedded in the structure or foreign (metal) atoms in the lattice. Impurities give rise to compositional defects. iv. Electronic defects or Schottky defects: Electronic defects are the result of errors in charge distribution in solids.
  • 20. Line defects: The most important two-dimensional or line defect is the dislocation. A dislocation may be defined as a disturbed region between two substantially perfect parts of a crystal. i. Edge dislocation: An edge dislocation lies perpendicular to its Burger’s vector. It moves (in its slip plane) in the direction of the Burger’s vector (slip direction) ii. Screw dislocation: An screw dislocation lies parallel to its Burger’s vector. It moves (in the slip plane) in a direction perpendicular to the Burger’s vector (slip direction). Figure 1: Edge dislocation
  • 21. Surface or planar defects: The defects which are take place on the surface of a material are known as surface defects. i. Grain boundaries: Grain boundaries are those planar imperfects ions in polycrystalline materials that separate crystals (grains) of different orientation. ii. Tilt boundaries: Till boundary may be regarded as an array of edge dislocations (βŠ₯). Actually tilt boundary is a class of low angle boundaries.
  • 22. i. Twin boundaries: A twin boundary separates two parts of crystal having the same orientation and they look like mirror image of each other. Volume defects: Volume defects such as cracks or stacking faults may arise where there is only small dissimilarity (electrostatically) between the stacking sequence of closed packed planes in FCC and HCP metals. It is possible for one atom layer to be out of sequence relative to the atoms of the layers above and below, giving a mistake or defect.
  • 23. 1. Whatisunderstoodbythetermβ€˜solidsolution’? Solid solution: A homogeneous mixture of two or more crystalline metals in varying proportion is known as solid solution. The metal which is present in a small quantity in a solid solution is called solute while the metalwhich ispresentinlargequantityiscalled solute.Solid solutionhasonlyonephase. Solid solution forms more rapidly when the solvent and solute atoms have similar size and comparable electron structure. For example brass is a solid solution of 𝐢𝑒 and 𝑍𝑛; a typical composition of 𝐢𝑒 64% and 𝑍𝑛 36%,where 𝐢𝑒 atomsare solvent and 𝑍𝑛 atomsare solute. Therearetwotypesofsolid solution:  Interstitialsolidsolutionand  Substitutionalsolid solution. 1. What are the conditions for the formation of solid solution? Three conditions must be fulfilled to form a solid solution. Such as – οƒ˜ Two or more metals that are involved to form a solid solution should have similar crystal structure. οƒ˜ Metals that are involved to form a solid solution should be of same group in the periodic table and should have same oxidation number. οƒ˜ Metals that are involved to form a solid solution should have nearly same atomic structure or atomic size.
  • 24. 1. Explain with neat sketches the substitutional solid solution and interstitial solid solution. Substitutional solid solution: A solid solution in which solute atoms directly replace some solvent atoms so that they lie at normal atoms site of the crystal structure then this type of solid solution is called substitutional solid solution. Substitutional solid solution forms when the solute and solvent atoms posses equal or approximately equal diameter. For example atomic diameter of 𝐢𝑒 is 2.522β„« and 𝑁𝑖 is 2.487β„« and these two form substitutional solid solution. Substitutional solid solutions are two types: ο‚« Random or disordered substitutional solid solution and ο‚« Ordered substitutional solid solution. Random or disordered substitutional solid solution: A solid solution in which the solute atoms replace the solvent atoms randomly is called random substitutional solid solution and this type of substitution is called random substitution. Following is the structure of random substitutional solid solution. Ordered substitutional solid solution: A substitutional solid solution in which the solute atoms replace by the solvent atoms orderly is called ordered substitutional solid solution and this type of substitution is known as ordered substitution. 𝐢𝑒 βˆ’ 𝑍𝑛, 𝐴𝑒 βˆ’ 𝐢𝑒, 𝐢𝑒2 𝑀𝑛𝐴𝑙 are some examples of ordered structure. Following is the structure of ordered substitutional solid solution. Interstitial solid solution: A solid solution in which solute atoms are located in the interstices of the solvent lattice is called interstitial solid solution. Interstitial solid solution can form only when the solute are small enough to fit into the space between the solvent atoms. They are unable to substitute solvent atoms and can only to fit interstices or spaces in the crystal lattice of solvent atoms. For example FCC 𝐹𝑒 is the solvent for 𝐢.
  • 25. Miller indices: Miller indices are a system of notation that denotes the orientation of the faces of a crystal and the planes and directions of atoms within that crystal. Some General Principles: ο‚· If a Miller index is zero, the plane is parallel to that axis. ο‚· The smaller a Miller index, the more nearly parallel the plane is to the axis. ο‚· The larger a Miller index, the more nearly perpendicular a plane is to that axis. ο‚· Multiplying or dividing a Miller index by a constant has no effect on the orientation of the plane ο‚· Miller indices are almost always small. Determination of Miller indices: Following are the rules for determination of Miller indices: a) At first we should have to determine the intercepts of the plane on the axes π‘₯, 𝑦, 𝑧 in terms of lattice constant. b) Now we have to take the reciprocal terms of the intercepts. c) If the results are fractions, then they are changed to the small integers having the same ratio i.e., multiplied each reciprocal or fraction by the same number such as the common denominator. d) At last the result is enclosed in parenthesis (β„Ž π‘˜ 𝑙). Example: π‘₯ 𝑦 𝑧 Intercept 1 ∞ 1 Reciprocal 1 1 = 1 1 ∞ = 0 1 1 = 1 Miller indices β„Ž π‘˜ 𝑙 = (1 0 1) π‘₯ 𝑦 𝑧 Intercept 1 3 2 Reciprocal 1 1 = 1 1 Γ— 6 = 6 1 3 = 1 3 Γ— 6 = 2 1 2 = 1 2 Γ— 6 = 3 Miller indices β„Ž π‘˜ 𝑙 = (6 2 3)