![Crystallographic Directions: Moving Between Atomic
Positions
1. Vector repositioned (if necessary) to pass
through origin.
2. Read off projections in terms of
unit cell dimensions a, b, and c
3. Adjust to smallest integer values
z
x
Algorithm
y
1/19/2017 28
Example
1, 0, ½ => 2,0,1 => [201]
-1,1,1 => [111] , the overbar represents negative index.
Notation for crystallographic direction
Enclose in square brackets, no commas
[uvw]](https://image.slidesharecdn.com/module2-170119185414/85/Module2-28-320.jpg)
![29
Example 2: The vector is obtained by subtracting the coordinates
of the tip of the vector from the coordinates of tail
-4, 1, 2
families of directions <uvw>
z
x
where the overbar represents a
negative index
[412]=>
y
Step 1:
pt. 1 x1 = a, y1 = b/2, z1 = 0
pt. 2 x2 = -a, y2 = b, z2 = c
=> -2, 1/2, 1
pt. 2
head
pt. 1:
tail
Step 2: Multiplying by 2 to eliminate the fraction
1/19/2017](https://image.slidesharecdn.com/module2-170119185414/85/Module2-29-320.jpg)
![1/19/2017 30
Example 3: In a cubic unit cell find the vector with indices [123]
Step 1: Divide all indices with the largest integer.
X component = 1/3
Y component = 2/3
Z component = 3/3
Step 2: Note the coordinates (1/3, 2/3, 1)
Step 3: Draw the vector](https://image.slidesharecdn.com/module2-170119185414/85/Module2-30-320.jpg)
![Linear Density
Linear Density of Atoms: LD =
a
[110]
Unit length of direction vector
Number of atoms
1/19/2017 36](https://image.slidesharecdn.com/module2-170119185414/85/Module2-36-320.jpg)
![Examples of Linear Density
a
[110]
1/19/2017 38
Linear density of Al in [110] direction
a = 0.405 nm
# atoms
length
1
3.5 nm
a2
2
LD -
==](https://image.slidesharecdn.com/module2-170119185414/85/Module2-38-320.jpg)
Module2
- 1. Structure of Metals and their Physical Properties 1
- 2. 2 Atomic arrangement of solids Source: The Principles of Engineering Materials [Craig R. Barrett, Alan S. Tetelman, William D. Nix Solids Crystalline solids Amorphous solids Long range order Repetitive pattern Periodic arrangement of atoms Bricks on a wall Random arrangement Short range order -metals -ceramics -some polymers
- 3. 3 The basic principles of many materials characterization techniques such as X-ray diffraction (XRD), are based on crystallography. Crystalline cristoballite Silica glass
- 4. Energy of Crystalline and Amorphous Structures • Non-dense, random packing • Dense, ordered packing Dense, ordered packed structures tend to have lower energies. Energy r typical neighbor bond length typical neighbor bond energy Energy r typical neighbor bond length typical neighbor bond energy 4
- 5. Energy r typical neighbor bond length typical neighbor bond energy 5 • Atomic radii of metallic atoms do not differ dramatically • Metallic bonding is non-directional • Nearest neighbor distances tend to be small • Electron cloud shields cores from each other Smallest repeating structure as unit cell Dense ordered packing in metallic structures.
- 6. 6 Space Lattice It is a three-dimensional array of points. Also defined as periodic configuration of points in space, defines 3D shapes that fill space Bravais Lattice 14 different ways of arranging lattice points so that each lattice point has identical surrounding. 14 different lattices are called ‘Bravais lattices’
- 7. 7 14 different unit cells which are grouped to form 7 crystal systems. 7 systems of Crystal symmetry--- Crystal Systems Crystal Systems Bravais Lattices Cubic 3 Bravais Lattice Tetragonal 2 Bravais Lattice Orthorhombic 4 Bravais Lattice Monoclinic 2 Bravais Lattice Triclinic 1 Bravais Lattice Hexagonal 1 Bravais Lattice Rhomdohedral 1 Bravais Lattice
- 8. 7 systems of crystal symmetry Source: R.E. Smallman, Ch1, F1.7 8
- 9. Seven Crystal Systems Source: R.E. Smallman, Ch1, F1.7 9
- 10. 10 Cubic Crystal Systems – 3 Bravais Lattice arrangement Source: Solid-State Physics: Introduction to the Theory ; By James Patterson, Bernard Baile Simple Cubic Lattice Body Centered Cubic Lattice Face Centered Cubic Lattice
- 11. Enforcing Terminology Unit Cell of Crystal Structure: Describes the smallest repetitive pattern and explains the complete lattice pattern of the crystal. • a, b, and c are the lattice parameters • α, β, γ are the angles between axes 11
- 12. Using Crystal Structure to Predict Properties For each of the cubic based crystal structures, we will now identify the: Closed packed directions (where the hard spheres can touch) Identify the number of nearest neighbors to each atom Calculate the atomic packing factor Atomic coordination number Calculate the theoretical density APF = Volume of atoms in unit cell* Volume of unit cell 12
- 13. Unit Cells and Percentage of Atoms 13
- 14. • Rare due to low packing density (only Po has this structure) • Close-packed directions are cube edges. (Courtesy P.M. Anderson) Simple Cubic Structure (SC) 14 Atomic coordination number = ?
- 15. Atomic Packing Factor (APF) for SC close-packed directions a R=0.5a 15 Step 1: Volume of Atoms in the unit cell : Given Atomic radii is ‘R’ Information required -> Number of atoms in the unit cell Each corner contains 1/8th of the atom So, total number of atoms in the unit cell = 8 x 1/8 = 1 atom per unit cell Volume of atoms =1 atom x ( 4/3 𝝅𝑹 𝟑) Step 2: Identify the close-packed direction in SC unit cell i.e, ‘a’ = 2R Volume of unit cell = a3=(2R)3 = 8R3 Atomic Packing Factor = 4 3 𝜋𝑅3 8𝑅3 = 0.52
- 16. Body Centered Cubic Source: R.E. Smallman, Ch1, F1.15 16 • Atoms touch each other along cube diagonals. --Note: All atoms are identical; the center atom is shaded differently only for ease of viewing. ex: Cr, W, Tantalum, Molybdenum Atomic coordination number = 8
- 17. Atomic Packing Factor (APF) for BCC a aR Adapted from Fig. 3.2(a), Callister 7e. a2 a3 17 Step 1: Number of atoms per unit cell = 1 atom at center + 1/8 th of 8 corner atoms = 2 atoms Step 2: Identify close packed direction 3 𝑎 = 4𝑅 Step 3: Volume of unit cell =𝑎3=( 4𝑅 3 )3 Atomic Packing Factor = 𝑽𝒐𝒍𝒖𝒎𝒆 𝒐𝒇 𝒂𝒕𝒐𝒎𝒔 𝑽𝒐𝒍𝒖𝒎𝒆 𝒐𝒇 𝒖𝒏𝒊𝒕 𝒄𝒆𝒍𝒍 = 𝟐𝒙 𝟒 𝟑 𝝅𝑹 𝟑 𝒂 𝟑 = 0.68
- 18. • Coordination # = 12 Adapted from Fig. 3.1, Callister 7e. (Courtesy P.M. Anderson) • Atoms touch each other along face diagonals. --Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing. Face Centered Cubic Structure (FCC) ex: Al, Cu, Au, Pb, Ni, Pt, Ag 4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8 18
- 19. Atomic Packing Factor (APF) for FCC 19 Close packed direction length, 4R= a. 2 => a= 4𝑅 √2 Atomic Packing Factor= 𝑽𝒐𝒍𝒖𝒎𝒆 𝒐𝒇 𝒂𝒕𝒐𝒎𝒔 𝑽𝒐𝒍𝒖𝒎𝒆 𝒐𝒇 𝒖𝒏𝒊𝒕 𝒄𝒆𝒍𝒍 = 𝟒𝒙 𝟒 𝟑 𝝅𝑹 𝟑 𝒂 𝟑 = 0.74 Step 1: Number of atoms per unit cell = 6 x ½ + 8 x 1 8 = 4 atoms per unit cell Step 2: Close packed direction in FCC are along the face diagonals.
- 20. Hexagonal Close Packed For ideally packed hcp, c/a=1.633 c/a(Zn)=1.856, c/a(Ti)=1.587 Source: R.E. Smallman, Ch1, F1.15 20 Coordination number =12 APF=0.74 with 6 atoms/unit cell ABAB… stacking sequence
- 21. 21 • The face-centered atom and the three mid-layer atoms form a tetrahedron MNOP which has sides equal to a (as atoms at vertices touch each other) and height of c/2. • Using this tetrahedron it can be shown that for an ideal hexagonal crystal c/a ratio = 1.633 Hexagonal Lattice
- 22. Stacking: Difference between HCP (0.74) and FCC (0.74) FCC: ABCABCABC HCP: ABABAB Source: R.E. Smallman, Ch1, F1.16 22
- 23. FCC vs. HCP 23
- 24. Two or more distinct crystal structures for the same material (allotropy/polymorphism) Each structure is stable under different conditions such as temperature or pressure the crystal structure with the lowest free energy will be stable BCC FCC BCC 1538ºC 1394ºC 912ºC Fe Gamma -Fe alpha-Fe liquid iron system 24 Carbon Diamond (high pressure) Graphite (stable at normal conditions)
- 25. 25 Where, n= number of atoms/unit cell A=atomic weight (g/mol) Vc= Volume of unit cell = a3 for cubic NA= Avogadro’s number= 6.023 x 1023atoms/mol Density = = VC NA n A = CellUnitofVolumeTotal CellUnitinAtomsofMass Theoretical Density Example: Chromium - with BCC crystal structure. A=52 g/mol and atomic radius R=0.125 nm a=4R/ 3 𝜌 = 2×52.0 6.023×1023 ×𝑎3
- 26. Examples of Metallic Crystal Structure Unit Cells: SC, BCC, FCC, HCP z x y 1/19/2017 26
- 27. Point Coordinates: Determining where atoms sit. Point coordinates for unit cell center are (½ , ½, ½) Point coordinates for unit cell corner are (1, 1, 1) Translation: integer multiple of lattice constants identical position in another unit cell Notation for point coordinates -> (a,b,c) z x y a b c 000 111 y z · 2c · · · b b 1/19/2017 27
- 28. Crystallographic Directions: Moving Between Atomic Positions 1. Vector repositioned (if necessary) to pass through origin. 2. Read off projections in terms of unit cell dimensions a, b, and c 3. Adjust to smallest integer values z x Algorithm y 1/19/2017 28 Example 1, 0, ½ => 2,0,1 => [201] -1,1,1 => [111] , the overbar represents negative index. Notation for crystallographic direction Enclose in square brackets, no commas [uvw]
- 29. 29 Example 2: The vector is obtained by subtracting the coordinates of the tip of the vector from the coordinates of tail -4, 1, 2 families of directions <uvw> z x where the overbar represents a negative index [412]=> y Step 1: pt. 1 x1 = a, y1 = b/2, z1 = 0 pt. 2 x2 = -a, y2 = b, z2 = c => -2, 1/2, 1 pt. 2 head pt. 1: tail Step 2: Multiplying by 2 to eliminate the fraction 1/19/2017
- 30. 1/19/2017 30 Example 3: In a cubic unit cell find the vector with indices [123] Step 1: Divide all indices with the largest integer. X component = 1/3 Y component = 2/3 Z component = 3/3 Step 2: Note the coordinates (1/3, 2/3, 1) Step 3: Draw the vector
- 31. Crystallographic Planes Adapted from Fig. 3.9, Callister 7e. 1/19/2017 31
- 32. 32 Crystallographic Planes Miller Indices: (hkl) is a specific crystallographic plane. {hkl} is family of planes with each having same atomic arrangement. 1/19/2017
- 33. 33 Example 1 z x y a b c Read off intercepts of plane with X, Y and Z axes in terms of a, b, c Intercepts a b c 1 1 1 Take reciprocals of intercepts Reciprocals 1/1 1/1 1/ꝏ Reduce to smallest integer values Reduction 1 1 0 Enclose in parentheses, no commas (hkl) Miller indices (110) 1/19/2017
- 34. 34 Identify the Miller indices of given plane z x y a b c · · · 4. Miller Indices (634) example 1. Intercepts 1/2 1 3/4 a b c 2. Reciprocals 1/½ 1/1 1/¾ 2 1 4/3 3. Reduction 6 3 4 1/19/2017
- 35. a b c z x y a b c 4. Miller Indices (100) 1. Intercepts 1/2 2. Reciprocals 1/½ 1/ 1/ 2 0 0 3. Reduction 2 0 0 Example 2 1/19/2017 35
- 36. Linear Density Linear Density of Atoms: LD = a [110] Unit length of direction vector Number of atoms 1/19/2017 36
- 37. Properties vary with direction: anisotropic. -Example: the modulus of elasticity (E) in BCC iron: Importance of Linear/ Planar Densities E (diagonal) = 273 GPa E (edge) = 125 GPa 1/19/2017 37
- 38. Examples of Linear Density a [110] 1/19/2017 38 Linear density of Al in [110] direction a = 0.405 nm # atoms length 1 3.5 nm a2 2 LD - ==
- 39. Planar Density of (100) Iron At T < 912 C iron has the BCC structure. (100) R 3 34 a = Adapted from Fig. 3.2(c), Callister 7e. 2D repeat unit 1/19/2017 39 =Planar Density = a2 1 atoms 2D repeat unit = nm2 atoms 12.1 m2 atoms = 1.2 x 1019 1 2 R 3 34 area 2D repeat unit
- 40. 40 Planar Density of (111) Iron-BCC Solution (cont): (111) plane 1 atom in plane/ unit surface cell atoms in plane atoms above plane atoms below plane ah 2 3 = a2 1/2 = = nm2 atoms 7.0 m2 atoms 0.70 x 1019 3 2 R 8 Planar Density = atoms 2D repeat unit area 2D repeat unit 1/19/2017
- 41. Spacing between equivalent planes dhkl = a h2 + k2 + l2 What is the spacing between (111) planes in FCC Nickel? The atomic radius is 0.124 nm. a) 0.20 nm b) 0.35 nm c) 0.30 nm 1/19/2017 41
- 42. 42 X-Rays to Determine Crystal Structure X-ray intensity (from detector) θ θc d = nλ 2 sinθc Measurement of critical angle, c, allows computation of planar spacing, d. • Incoming X-rays diffract from crystal planes. Adapted from Fig. 3.22, Callister & Rethwisch 9e. reflections must be in phase for a detectable signal spacing between planes d θ λ θ extra distance travelled by wave “2” 1/19/2017
- 43. 43 X-Ray Diffraction Pattern Adapted from Fig. 3.22, Callister 8e. (110) (200) (211) z x y a b c Diffraction angle 2θ Diffraction pattern for polycrystalline α-iron (BCC) Intensity(relative) z x y a b c z x y a b c dhkl-spacing increases 1/19/2017