UCSD NANO106 - 02 - 3D Bravis Lattices and Lattice Computations
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The document discusses 3D Bravais lattices, detailing the derivation and characteristics of different crystal systems such as triclinic, monoclinic, and cubic. It explains the relationship between lattice parameters, symmetry, and the derivation of additional lattice structures through node placements. Additionally, the document covers lattice visualization tools, file formats, and methods for calculating distances using a metric tensor in crystallographic contexts.
![3D crystal systems
¡ The lattices you have seen thus far are all primitive and corresponds
to the seven 3D crystal systems.
¡ The 3D crystal systems can be ranked according to their symmetry.
NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2
Distort along [001]
direction.
Distort along [111]
direction to make
all angles ≠ 90.
Make all lengths
unequal butone lattice
vector perpendicular to
other two Make all angles
unequal
Distort along
[100] or [010].
Decreasing
symmetry
Make one angle ≠
90.
Make lengths unequal
and all angles = 90.](https://image.slidesharecdn.com/02-3dbravislatticesandlatticecomputations-160328031653/85/UCSD-NANO106-02-3D-Bravis-Lattices-and-Lattice-Computations-16-320.jpg)
![Adding new lattice nodes
¡ Remember how we derived the additional centered
rectangular net by adding a node to the primitive
rectangular net?
¡ In 3D, there are five possible locations where an
additional node can be placed:
¡ In the center of the cell at (½, ½, ½) [Body centering, denoted by
the symbol I]
¡ At the center of all faces of the cell at (½, ½, 0), (½, 0, ½), (0, ½, ½)
[Face centering, denoted by the symbol F]
¡ At the center of one of the faces at (½, ½, 0) or (½, 0, ½) or (0, ½,
½) [Base centering, denoted by the symbols A, B or C,
depending on whether the additional node is on the b-c, a-c or
a-b base planes respectively.]
NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2](https://image.slidesharecdn.com/02-3dbravislatticesandlatticecomputations-160328031653/85/UCSD-NANO106-02-3D-Bravis-Lattices-and-Lattice-Computations-18-320.jpg)
UCSD NANO106 - 02 - 3D Bravis Lattices and Lattice Computations
- 1. 3D Bravais Lattices, Lattice Planes and the Reciprocal Lattice Shyue Ping Ong Department of NanoEngineering University of California, San Diego
- 2. Readings ¡Chapter 3 of Structure of Materials NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2
- 3. Brief Recap ¡Concepts of symmetry – translational and rotational ¡Lattice vectors and parameters ¡Derivation of the five 2D Bravais nets ¡Definition of unit cells NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2
- 4. Derivation of 3D Bravais lattices NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 ¡ Following the same general principle as 2D Bravais nets, we consider possible values for the six lattice parameters ¡ All 3D crystal systems can be derived by adding an additional vector to existing an 2D crystal system a = Length of a b = Length of b c = Length of c α = Angle between b and c β = Angle between a and c γ = Angle between a and b a,b,c,α,β,γ{ }
- 5. Conventional for Crystallographic Axes ¡ Always right-handed ¡ c lattice vector is usually the direction of highest rotational symmetry (if unique) NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 a.(b×c) > 0 a b c
- 6. Triclinic lattice ¡ No special conditions ¡ Lowest symmetry (no rotational symmetry at all) NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 a ≠ b ≠ c α ≠ β ≠γ Example: K2Cr2O7 {7.574Å, 7.632Å, 13.661Å, 95.374°, 97.483°, 90.955°}
- 7. Monoclinic lattice ¡ What symmetry does the monoclinic lattice have? (hint: think of the relationship with a particular 2D net) NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 α =γ = 90° β ≠ 90° Example: Monoclinic sulfur {10.668Å 10.690Å 10.816Å, 90°, 95.637°, 90°}
- 8. Hexagonal lattice NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 a = b α = β = 90° γ =120° Example: ZnO {3.25Å, 3.25Å, 5.21Å, 90°, 90°, 120°}
- 9. Rhombohedral/Trigonal lattice NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 a = b = c α = β =γ ≠ 90° Example: CaCO3 {6.36Å, 6.36Å, 6.36Å, 46.1°, 46.1°, 46.1°} Note: All rhombohedral cells can be represented in a hexagonal setting
- 10. Orthorhombic lattice NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 α = β =γ = 90° Example: BaSO4 {8.87Å, 5.54Å, 7.15Å, 90°, 90°, 90°}
- 11. Tetragonal lattice NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 a = b ≠ c α = β =γ = 90° Example: SnO2 {3.784Å, 3.784Å 9.514Å, 90°, 90°, 90°}
- 12. Cubic lattice NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 a = b = c α = β =γ = 90° Example: CsCl {4.115Å, 4.115Å, 4.115Å, 90°, 90°, 90°}
- 13. File formats and Viewing Software for Crystals ¡ VESTA (Visualization for Electronic and STructural Analysis) ¡ Open-source 3D visualization program for structural models, volumetric data such as electron/nuclear densities, and crystal morphologies. ¡ Create lattice planes, calculate atomic distances, etc. ¡ Download at http://jp-minerals.org/vesta/en/. ¡ Main supported file formats used in NANO106 ¡ Crystallographic information file (CIF) for crystals ¡ XYZ files for molecules (mainly for demonstration of point group symmetries) NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2
- 14. Where to get Crystal Structures ¡Inorganic Crystal Database (ICSD) ¡ http://icsd.fiz-karlsruhe.de ¡The Materials Project ¡ https://www.materialsproject.org/ NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2
- 15. Demo NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 2
- 16. 3D crystal systems ¡ The lattices you have seen thus far are all primitive and corresponds to the seven 3D crystal systems. ¡ The 3D crystal systems can be ranked according to their symmetry. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 Distort along [001] direction. Distort along [111] direction to make all angles ≠ 90. Make all lengths unequal butone lattice vector perpendicular to other two Make all angles unequal Distort along [100] or [010]. Decreasing symmetry Make one angle ≠ 90. Make lengths unequal and all angles = 90.
- 17. The 7 3D Crystal Systems and Primitive Lattices NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2
- 18. Adding new lattice nodes ¡ Remember how we derived the additional centered rectangular net by adding a node to the primitive rectangular net? ¡ In 3D, there are five possible locations where an additional node can be placed: ¡ In the center of the cell at (½, ½, ½) [Body centering, denoted by the symbol I] ¡ At the center of all faces of the cell at (½, ½, 0), (½, 0, ½), (0, ½, ½) [Face centering, denoted by the symbol F] ¡ At the center of one of the faces at (½, ½, 0) or (½, 0, ½) or (0, ½, ½) [Base centering, denoted by the symbols A, B or C, depending on whether the additional node is on the b-c, a-c or a-b base planes respectively.] NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2
- 19. Generating additional 3D lattices ¡ We now apply all five node placements (I, F, A, B, and C) to all seven primitive 3D lattices (35 possibilities). ¡ Not all node placements are compatible with all primitive 3D lattices, and some do not result in a new lattice. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 Example: Tetragonal lattice • Analogous to the 2D square net, C- centering does not result in a new lattice. • A or B centering can be redefined into monoclinic cell. • The I node placement result in a new tetragonal lattice. • F-centering does not result in a new lattice (problem set exercise)
- 20. The 14 3D Bravais Lattices NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 P: primitive C: C-centered I: body-centered F: face-centered (upper case for 3D) a: triclinic (anorthic) m: monoclinic o: orthorhombic t: tetragonal h: hexagonal c: cubic
- 21. 3D unit cells ¡ Infinite number of unit cells for all 3D lattices ¡ Always possible to define primitive unit cells for non-primitive lattices, though the full symmetry may not be retained. ¡ The Wigner-Seitz cell is similarly defined for 3D. NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 Conventional cF cell Primitive unit cell Wigner-Seitz cell
- 22. Crystallographic Calculations NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 2
- 23. Readings ¡Chapter 4 of Structure of Materials NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2
- 24. Positions and Directions in Lattices ¡Recall that the lattice vectors form a basis for 3D space, i.e., any point in 3D space can be represented as a linear combination of the lattice vectors ¡We then denote the coordinates as (u, v, w) NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 ua + vb+ wc
- 25. Calculating Distances in Lattices ¡Consider the (1, 1, 1) points in two lattices, one cubic and another triclinic. What is the distance from the origin (0, 0, 0) to (1, 1, 1)? NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2
- 26. Calculating Distances in Lattices a) Effectively a Cartesian frame. From Pythagoras’ Theorem, we can calculate the distance as b) We can convert this to a Cartesian frame by multiplying by the basis vectors, and then calculating the distance Though this is a perfectly valid approach, we will introduce an alternative, general method of calculating distances using what is known as a metric tensor (for now, just think of this as a 3x3 matrix, we will get into a more formal definition of a tensor later) NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 d = a2 + a2 + a2 = a 3 d = 1a +1b+1c
- 27. Derivation of the Metric Tensor NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 Blackboard
- 28. Using the metric tensor in dot products ¡By an extension of the previous derivation, we can see that a dot product in crystal coordinates can be computed as: NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 x.y = pT gq
- 29. Form of the metric tensor NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 ¡How can we write the metric tensor in terms of the lattice parameters? ¡What is the special form of the metric tensor for a hexagonal lattice? What about the orthorhombic and tetragonal lattices? Blackboard
- 30. Metric Tensors for the 7 Crystal Systems NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 gtriclinic = a2 abcosγ accosβ abcosγ b2 bccosα accosβ bccosα c2 ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ,gmonoclinic = a2 0 accosβ 0 b2 0 accosβ 0 c2 ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ gorthorhombic = a2 0 0 0 b2 0 0 0 c2 ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ,gtetragonal = a2 0 0 0 a2 0 0 0 c2 ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ,gcubic = a2 0 0 0 a2 0 0 0 a2 ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ghexagonal = a2 − a2 2 0 − a2 2 a2 0 0 0 c2 ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ,grhombohedral = a2 a2 cosα a2 cosα a2 cosα a2 a2 cosα a2 cosα a2 cosα a2 ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟
- 31. Example ¡ You have a orthorhombic lattice with lattice parameters {4, 6, 8, 90, 90, 90}. ¡ What is the length of a vector from the origin to (0.5, 0.5, 0.5)? ¡ You have two atoms P and Q at locations (0.2, 0.2, 0.2) and (0.4, 0.5, 0.6) respectively. What is the distance between them? What is the angle between them and the origin, i.e., the angle POQ? NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2 Blackboard example