Crystal Structures
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This document discusses crystal structures, including the definitions of a lattice, basis, and unit cell. It then discusses molecules and bonding principles, showing diagrams of force vs. interatomic separation and energy vs. interatomic separation. It also provides examples of the crystal structures of several common materials like copper, iron, zinc, silicon, and sodium chloride. It defines directions and planes in crystal structures, including Miller indices. It provides examples of how to determine crystal directions and plane intercepts. It discusses properties like planer concentration and its significance. Finally, it poses numerical problems involving calculating planer concentration for different planes in an FCC copper structure.
![Crystal Direction
13© Md Imran Momtaz, Dept. of EEE, BUET
1. Take the x, y and z axis intercepts. a/2, b, c/2
Result
2. Normalize them w.r.t. lattice constants. 1/2, 1, 1/2
3. Put them within [] without comma. [½ 1 ½]
4. Multiply with suitable number, if needed. [1 2 1]
a/3, b/2, c
Another example
1/3, 1/2, 1
[1/3 ½ 1]
[2 3 6]](https://image.slidesharecdn.com/eee307l2crystalstructure-140521221905-phpapp01/85/Crystal-Structures-13-320.jpg)
![Crystal Direction
14© Md Imran Momtaz, Dept. of EEE, BUET
Note: If the intercept is negative, place a
bar over the number.
-a, b, c
Another example
-1, 1, 1
[-1 1 1]
[111]
Note: If the direction does not start from
origin, shift that direction to some unit
cell so that its initial point starts from
origin.
Family of planes:
It is represented by <> .
<100> means
[100],[010],[001],[100],[010] [001]and
<110> means
[110],[011],[101],[110],[110] .etc](https://image.slidesharecdn.com/eee307l2crystalstructure-140521221905-phpapp01/85/Crystal-Structures-14-320.jpg)
![Important Point
18© Md Imran Momtaz, Dept. of EEE, BUET
[hkl] direction is perpendicular to (hkl)
plane. This is applicable for cubic crystal
system ONLY.
[hkl] direction is perpendicular to (hkl)
plane. This is applicable for cubic crystal
system ONLY.](https://image.slidesharecdn.com/eee307l2crystalstructure-140521221905-phpapp01/85/Crystal-Structures-18-320.jpg)
Crystal Structures
- 1. Crystal structures 1© Md Imran Momtaz, Dept. of EEE, BUET
- 2. The Crystalline State • Lattice • Basis • Unit Cell 2© Md Imran Momtaz, Dept. of EEE, BUET Crystal = Lattice + Basis at each lattice pointCrystal = Lattice + Basis at each lattice point
- 3. Molecules and general bonding principles 3© Md Imran Momtaz, Dept. of EEE, BUET (a) Force vs. interatomic separation (b) Energy vs. interatomic separation
- 4. Molecules and general bonding principles 4© Md Imran Momtaz, Dept. of EEE, BUET • In previous slide, only electron scenario has been shown. Many electron case can be analyzed in the same manner.
- 5. Crystal structure of some well known material 5© Md Imran Momtaz, Dept. of EEE, BUET • Copper, Cu: FCC unit cell Crystal structure of Cu An FCC unit cell Reduced sphere representation 4 atoms per unit cell4 atoms per unit cell 2 2 R a = 3 2 APF π =
- 6. Crystal structure of some well known material 6© Md Imran Momtaz, Dept. of EEE, BUET • Iron, Fe: BCC structure of Fe Reduced sphere representation 2 atoms per unit cell2 atoms per unit cell 3 4 R a = 3 8 APF π =
- 7. Crystal structure of some well known material 7© Md Imran Momtaz, Dept. of EEE, BUET • Zinc, Zn: HCP Crystal structure of Zn Unit cell Smallest unit cell (Hexagonal unit cell) 74%APF =
- 8. Crystal structure of some well known material 8© Md Imran Momtaz, Dept. of EEE, BUET • Silicon, Si & Germanium, Ge: Diamond Cubic Crystal Structure of Si (Same for Ge as well) 8 atoms per unit cell8 atoms per unit cell Note: A diamond cubic crystal structure can be formed by superimposing two FCC structures by (a/4, a/4, a/4) distance
- 9. Crystal structure of some well known material 9© Md Imran Momtaz, Dept. of EEE, BUET • ZnS & GaAs (and some other well known III-V semiconductor as well): Zinc Blend Cubic Crystal Structure of Si (Same for Ge as well) 8 atoms per unit cell8 atoms per unit cell Note: A Zinc Blend cubic crystal structure can be formed by superimposing two FCC structures by (a/4, a/4, a/4) distance
- 10. Crystal structure of some well known material 10© Md Imran Momtaz, Dept. of EEE, BUET • NaCl & CsCl: FCC Crystal of NaCl BCC Crystal of CsCl
- 11. Numerical Problem 11© Md Imran Momtaz, Dept. of EEE, BUET • Consider FCC structure of Cu crystal. a) How many atoms are there per unit cell? b) Prove that, c) Determine APF. d) Determine atomic concentration and density of crystal. Given, atomic mass of Cu is 63.55 gm mol-1 and Rcu = 0.128 nm. 2 2R a=
- 12. Crystal Direction & Plane 12© Md Imran Momtaz, Dept. of EEE, BUET a, b, c, α, β and γ are known as lattice parameters. For Cubic Crystal structure, a = b = c and α = β = γ = 90° For Cubic Crystal structure, a = b = c and α = β = γ = 90° For Hexagonal Close structure, a = b ≠ c and α = β = 90°, γ = 120° For Hexagonal Close structure, a = b ≠ c and α = β = 90°, γ = 120°
- 13. Crystal Direction 13© Md Imran Momtaz, Dept. of EEE, BUET 1. Take the x, y and z axis intercepts. a/2, b, c/2 Result 2. Normalize them w.r.t. lattice constants. 1/2, 1, 1/2 3. Put them within [] without comma. [½ 1 ½] 4. Multiply with suitable number, if needed. [1 2 1] a/3, b/2, c Another example 1/3, 1/2, 1 [1/3 ½ 1] [2 3 6]
- 14. Crystal Direction 14© Md Imran Momtaz, Dept. of EEE, BUET Note: If the intercept is negative, place a bar over the number. -a, b, c Another example -1, 1, 1 [-1 1 1] [111] Note: If the direction does not start from origin, shift that direction to some unit cell so that its initial point starts from origin. Family of planes: It is represented by <> . <100> means [100],[010],[001],[100],[010] [001]and <110> means [110],[011],[101],[110],[110] .etc
- 15. Crystal Planes: Miller Indices 15© Md Imran Momtaz, Dept. of EEE, BUET 1. Take the x, y and z axis intercepts. a/2, b, ∞ 2. Normalize them w.r.t. lattice constants. 1/2, 1, ∞ 3. Invert them, put them within () without comma. (2 1 0) 4. Multiply with suitable number, if needed. (2 1 0) Result This is Miller Indices.
- 16. Crystal Planes: Miller Indices 16© Md Imran Momtaz, Dept. of EEE, BUET
- 17. Crystal Plane 17© Md Imran Momtaz, Dept. of EEE, BUET Note: If the intercept is negative, place a bar over the number. -a, b, c Another example -1, 1, 1 (-1 1 1) (111) Note: If the direction starts from origin, shift that direction to a unit cell. Family of planes: It is represented by {} . {100} means (100),(010),(001),(100),(010) (001)and {110} means (110),(011),(101),(110),(110) .etc
- 18. Important Point 18© Md Imran Momtaz, Dept. of EEE, BUET [hkl] direction is perpendicular to (hkl) plane. This is applicable for cubic crystal system ONLY. [hkl] direction is perpendicular to (hkl) plane. This is applicable for cubic crystal system ONLY.
- 19. Planer Concentration 19© Md Imran Momtaz, Dept. of EEE, BUET • Significance: It can justify different properties of the materials. e.g. It can justify the oxide growth probability. It also can justify the required amount of force required to penetrate the layer. •Planer Concentration atomsinacertain plane areaof that plane =
- 20. Numerical Problem 20© Md Imran Momtaz, Dept. of EEE, BUET • Consider FCC structure of Cu crystal. a) Determine planer concentration at (100) plane b) Determine planer concentration at (110) plane