classification of cubic lattice structure
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brief description of simple cubic lattice, miller indices, interplanar spacing
classification of cubic lattice structure
- 1. Crystal Structure Arpan Deyasi Dept. of ECE, RCCIIT, Kolkata Arpan Deyasi Electron Device
- 2. Crystal A crystal is a homogeneous, anisotropic solid body having long-range 3D ordering (order may be Translational/orientational/combination of both), i.e., constructed by infinite repetition of identical structural units. Features: • Natural shape of polyhedron • Chemical compound • Under the action of interatomic forces • End phase generally becomes solid when passing from phase transformation under suitable conditions Arpan Deyasi Electron Device
- 3. Lattice Space-lattice is a mathematical concept, defined as infinite number of regular periodic arrays of points in 3D configuration having long-range ordering (both in translational & orientational), with the property that arrangement of points about any particular point is isotropically identical. a2 a 3 a1 u2a2 u 3 a 3 1 1 2 2 3 3 r u a u a u a = + + where are translational vectors 1 2 3 , , a a a Arpan Deyasi Electron Device
- 4. Basis It is the group of atoms, identical in composition, arrangement and orientation, when repeated in space periodically and isotropically, forms crystal structure. Basis Lattice Crystal Arpan Deyasi Electron Device
- 5. Crystalline Solid Amorphous Solid Atomic Arrangement Atoms are arranged in a regular, repeating pattern Atoms are arranged randomly, lacking a long-range order Melting Point Have a sharp, distinct melting point Melt over a range of temperatures, not a specific point Hardness When broken, crystalline solids tend to cleave along specific planes due to their ordered structure Break with irregular or curved surfaces, not along defined planes X-ray Diffraction Produce sharp, well-defined X- ray diffraction patterns due to their ordered structure Produce broad, diffuse patterns or no diffraction pattern at all, indicating the lack of long-range order Anisotropy/ Isotropy Can be anisotropic (physical properties vary depending on direction) Generally isotropic, with similar properties in all directions Example Diamond, quartz, salt Glass, rubber, plastic Arpan Deyasi Electron Device
- 6. Unit Cell Smallest geometric arrangement whose repetition in 3D space gives actual crystal structure. This elementary pattern of minimum number of points carry all the characteristics of the crystal. Unit cell Crystal formation Arpan Deyasi Electron Device
- 7. Primitive Cell It is the unit cell which contains lattice points at corners only. It is the minimum volume unit cell, and will fill all the space by repetition of suitable crystal translational operations. Each primitive cell is associated with one lattice point. No. of atoms in a primitive cell is always the same for a given crystal structure. 1 2 3 . p V a a a = Crystal formation Arpan Deyasi Electron Device
- 8. Primitive Cell Unit Cell Repeating pattern simplest repeating unit in a crystal structure larger repeating unit that can be primitive or non-primitive No of points Contains only one lattice point, which is typically located at the corners of the cell Non-primitive unit cells (also called centered unit cells) contain additional lattice points within the cell, along with corner point Example A cubic primitive unit cell has lattice points only at the corners BCC unit cell has lattice points at the corners and one in the center. FCC unit cell has lattice points at the corners and in the center of each face Arpan Deyasi Electron Device
- 9. Bravis Lattice Network of part where position vectors can be expressed in the form 1 1 2 2 3 3 r u a u a u a = + + where the choice of basis vectors is not unique. A primitive unit cell is called Bravis lattice. Arpan Deyasi Electron Device
- 10. Classification of Cubic Lattice Cubic Lattice Simple Cubic Body-Centered Cubic (BCC) Face-Centered Cubic (FCC) Arpan Deyasi Electron Device
- 11. A few properties Coordination number: It is defined as number of equidistant neighbor that an atom has in the given structure Packing Fraction: It is defined as the maximum proportion of available volume in a unit cell that can be occupied by the closed pack spheres Arpan Deyasi Electron Device
- 12. Simple Cubic Lattice Unit cell contains 8 atoms, 1 in each corner Each corner atom is shared among 8 cells, so Total no of atoms per unit cell = 8⨯(1/8)=1 No. of atoms per unit cell = 1 Coordination no. = 6 Distance between any two corner atoms = ‘a’ So, radius of closed-packed sphere r = 0.5a Atomic radius = 0.5a Packing fraction = 3 3 4 3 r a 3 3 4 3.8 a a = 0.523 6 = = P.F = 0.523 Arpan Deyasi Electron Device
- 13. Body-Centered Cubic Lattice Unit cell contains 9 atoms, 1 in each corner and 1 in center Each corner atom is shared among 8 cells, so Total no of atoms per unit cell = 8⨯(1/8) + 1 = 2 No. of atoms per unit cell = 2 Coordination no. = 8 Lattice constant = ‘a’ So, length of body diagonal = √3. a = 4r Atomic radius = (√3. a)/4 This length contains 3 atoms, where no of corner atoms = 2 Packing fraction = 3 3 4 3 2 r a 3 3 4 .3 3. 2 3.8.8 a a = 3 0.68 8 = = P.F = 0.68 Arpan Deyasi Electron Device
- 14. Face-Centered Cubic Lattice Unit cell contains 14 atoms, 1 in each corner and 6 in faces Each corner atom is shared among 8 cells, and each face atom is shared among 2 cells, so total no of atoms per unit cell = 8⨯(1/8) + 6⨯(1/2) = 4 No. of atoms per unit cell = 4 Coordination no. = 12 Lattice constant = ‘a’ So, length of face diagonal = √2. a = 4r Atomic radius = 2√2.a This length contains 3 atoms, where no of corner atoms = 2 Packing fraction = 3 3 4 3 4 r a 3 3 4 4 3.16. 2 a a = 0.74 3 2 = = P.F = 0.74 Arpan Deyasi Electron Device
- 15. Comparative Study between SC, BCC, FCC Feature Simple Cubic (SC) Body-Centered Cubic (BCC) Face-Centered Cubic (FCC) Atom Locations Atoms only at the 8 corners of the cube Atoms at the 8 corners and 1 atom at the body center Atoms at the 8 corners and at the center of each of the 6 faces Atoms per Unit Cell 1 2 4 Coordination Number 6 8 12 ( Relationship between Edge Length (a) and Atomic Radius (r) a = 2r a = 4r/√3 a = 2√2 r Packing Efficiency (APF) 52.4% 68% 74% Arpan Deyasi Electron Device
- 16. Miller Indices Position and orientation of a crystal plane is determined by any three non-collinear points in the plane. The notation by which the plane is described, is called Miller Indices. a2 a 3 a1 u2a2 u 3 a 3 1 1 2 2 3 3 r u a u a u a = + + where are translational vectors 1 2 3 , , a a a Arpan Deyasi Electron Device
- 17. How to calculate Miller Indices Determine intercept of each plane along each of 3 crystallographic directions. Calculate reciprocals of intercepts. If fraction appears, multiply each by the denominator of the smallest fraction. X Y Z X Y Z X Y Z Arpan Deyasi Electron Device
- 18. Interplanar Spacing Perpendicular distance between two adjacent parallel planes in a crystal lattice is called interplanar spacing X Y Z P γ β α O Plane (hkl) intercepts the coordinate system at (a/h), (b/k), (c/l) respectively Interplanar spacing dhkl = OP cos( ) / hkl d a h = cos( ) / hkl d b k = cos( ) / hkl d c l = Arpan Deyasi Electron Device
- 19. Interplanar Spacing X Y Z P γ β α O 2 2 2 cos ( ) cos ( ) cos ( ) 1 + + = 2 2 2 1 / / / hkl hkl hkl d d d a h b k c l + + = 2 2 2 2 2 2 1 hkl d h k l a b c = + + For cubic cell ( ) 2 2 2 hkl a d h k l = + + Arpan Deyasi Electron Device