crystal lattice

1. Arpan Deyasi Crystal Structure Electron Device Arpan Deyasi Dept. of ECE, RCCIIT, Kolkata

2. Crystal Arpan Deyasi 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. Electron Device 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 • • • •

3. Lattice Arpan Deyasi 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. Electron Device u3a3 a3 = + + r 1 1 u a u a 3 3 u a a2 u2a2 2 2 a1 where are translational vectors 1 2 3 , , a a a

4. Basis It is the group of atoms, identical in composition, arrangement and orientation, when repeated in space periodically and isotropically, forms crystal structure. Arpan Deyasi Electron Device Lattice Crystal Basis

5. Arpan Deyasi Crystalline Solid Amorphous Solid Atomic Arrangement Melting Point Atoms are arranged in a regular, repeating pattern Have a sharp, distinct melting point When broken, crystalline solids tend to cleave along specific planes due to their ordered structure Produce sharp, well-defined X- ray diffraction patterns due to their ordered structure Atoms are arranged randomly, lacking a long-range order Melt over a range of temperatures, not a specific point Break with irregular or curved surfaces, not along defined planes Electron Device Hardness X-ray Diffraction Produce broad, diffuse patterns or no diffraction pattern at all, indicating the lack of long-range order Generally isotropic, with similar properties in all directions Anisotropy/ Isotropy Can be anisotropic (physical properties vary depending on direction) Diamond, quartz, salt Example Glass, rubber, plastic

6. Unit Cell Arpan Deyasi 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. Electron Device Unit cell Crystal formation

7. Primitive Cell Arpan Deyasi 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. Electron Device 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. Crystal formation =  V a a . a p 1 2 3

8. Arpan Deyasi Primitive Cell Unit Cell Repeating pattern No of points simplest repeating unit in a crystal structure Contains only one lattice point, which is typically located at the corners of the cell Electron Device larger repeating unit that can be primitive or non-primitive Non-primitive unit cells (also called centered unit cells) contain additional lattice points within the cell, along with corner point BCC unit cell has lattice points at the corners and one in the center. Example A cubic primitive unit cell has lattice points only at the corners FCC unit cell has lattice points at the corners and in the center of each face

9. Bravis Lattice Arpan Deyasi Network of part where position vectors can be expressed in the form r u a u a = + Electron Device + 3 3 u a 1 1 2 2 where the choice of basis vectors is not unique. A primitive unit cell is called Bravis lattice.

10. Classification of Cubic Lattice Arpan Deyasi Electron Device Cubic Lattice Face-Centered Cubic (FCC) Body-Centered Cubic (BCC) Simple Cubic

11. A few properties Coordination number: It is defined as number of equidistant neighbor that an atom has in the given structure Arpan Deyasi Electron Device 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

12. Simple Cubic Lattice Arpan Deyasi Unit cell contains 8 atoms, 1 in each corner Electron Device 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 43  3 r Coordination no. = 6 Packing fraction = 3 a  Distance between any two corner atoms = ‘a’  3 4 3.8 a a So, radius of closed-packed sphere r = 0.5a = = = 0.523 6 3 Atomic radius = 0.5a P.F = 0.523

13. Body-Centered Cubic Lattice Arpan Deyasi 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 Electron Device No. of atoms per unit cell = 2 43  3 r  2 Coordination no. = 8 Packing fraction = 3 a Lattice constant = ‘a’   3 3 8 4 .3 3. 3.8.8 a So, length of body diagonal = √3. a = 4r = = =  0.68 2 This length contains 3 atoms, where no of corner atoms = 2 3 a Atomic radius = (√3. a)/4 P.F = 0.68

14. Face-Centered Cubic Lattice Arpan Deyasi Unit cell contains 14 atoms, 1 in each corner and 6 in faces Electron Device 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 43 No. of atoms per unit cell = 4  3 r  4 Packing fraction = 3 Coordination no. = 12 a Lattice constant = ‘a’ So, length of face diagonal = √2. a = 4r   3 4 a = = 0.74 =  4 3 2 3 3.16. 2 a This length contains 3 atoms, where no of corner atoms = 2 P.F = 0.74 Atomic radius = 2√2.a

15. Comparative Study between SC, BCC, FCC Arpan Deyasi Body-Centered Cubic (BCC) Atoms at the 8 corners and 1 atom at the body center Face-Centered Cubic (FCC) Atoms at the 8 corners and at the center of each of the 6 faces Feature Simple Cubic (SC) Electron Device Atoms only at the 8 corners of the cube Atom Locations 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%

16. Miller Indices Arpan Deyasi 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. Electron Device u3a3 a3 = + + r 1 1 u a u a 3 3 u a a2 u2a2 2 2 a1 where are translational vectors 1 2 3 , , a a a

17. How to calculate Miller Indices Z Determine intercept of each plane along each of 3 crystallographic directions. Arpan Deyasi Calculate reciprocals of intercepts. Electron Device Y If fraction appears, multiply each by the denominator of the smallest fraction. Z Z X Y Y X X

18. Interplanar Spacing Z Perpendicular distance between two adjacent parallel planes in a crystal lattice is called interplanar spacing Arpan Deyasi Plane (hkl) intercepts the coordinate system at (a/h), (b/k), (c/l) respectively Electron Device P γ Y β Interplanar spacing dhkl = OP O d a h α  = hkl cos( ) / d b k d c l  = X  = hkl cos( ) hkl cos( ) / /

19. Interplanar Spacing Z cos ( ) cos ( ) cos ( ) 1   + Arpan Deyasi +  = 2 2 2 2 2 2                   d b k Electron Device d a h d c l + + = hkl hkl hkl 1 / / / P γ Y β O 1 = d α hkl       2 2 2 h a k b l c + + 2 2 2 X a = d For cubic cell ( ) hkl + + 2 2 2 h k l