Crystal Structure Continued!

1. Crystal Structure Continued! • NOTE!! • Much of the discussion & many figures in what follows was\ • constructed from lectures posted on the web by Prof. Beşire • GÖNÜLin Turkey. She has done an excellent job of covering • manydetails of crystallography & she illustrates her topics with • many very nice pictures of lattice structures. Her lectures on this • are posted Here: • http://www1.gantep.edu.tr/~bgonul/dersnotlari/. • Her homepage is Here:http://www1.gantep.edu.tr/~bgonul/.

2. 2 d examples

3. Lattice Translation Vectors In General • Mathematically, a lattice is defined by 3 vectors called Primitive Lattice Vectors a1, a2, a3are 3d vectors which depend on the geometry. • Once a1, a2, a3are specified, the Primitive Lattice Structure is known. • The infinite lattice is generated by translating through a Direct Lattice Vector:T = n1a1 + n2a2 + n3a3 n1,n2,n3 are integers. T generates the lattice points. Each lattice point corresponds to a set of integers (n1,n2,n3).

4. 2 Dimensional Lattice Translation Vectors Consider a 2-dimensional lattice (figure). Define the 2 Dimensional Translation Vector Rn n1a+ n2b (Sorry for the notation change!!) a &b are 2 d Primitive Lattice Vectors, n1, n2 are integers. • Once a & b are specified by the lattice geometry & an origin is chosen, all symmetrically equivalent points in the lattice are determined by the translation vector Rn. That is, the lattice has translational symmetry. Note that the choice of Primitive Lattice vectors is not unique! So, one could equally well take vectors a & b' as primitive lattice vectors. Point D(n1, n2) = (0,2) Point F(n1, n2) = (0,-1)

5. The Basis (or basis set)  The set of atoms which, when placed at each lattice point, generates the Crystal Structure. Crystal Structure ≡Primitive Lattice + Basis Translate the basis through all possible lattice vectors T = n1a1 + n2a2 + n3a3 to get the Crystal Structure or the DIRECT LATTICE

6. The periodic lattice symmetry is such that the atomic arrangement looks the same from an arbitrary vector position r as when viewed from the point r' = r + T (1) where T is the translation vector for the lattice: T = n1a1 + n2a2 + n3a3 • Mathematically, the lattice & the vectors a1,a2,a3 are Primitive if any 2 points r & r' always satisfy (1) with a suitable choice of integers n1,n2,n3.

7. In 3 dimensions, no 2 of the 3 primitive lattice vectors a1,a2,a3 can be along the same line. But, Don’t think ofa1,a2,a3as a mutually orthogonal set! Usually, they are neither mutually perpendicular nor all the same length! • For examples, see Fig. 3a (2 dimensions):

8. The Primitive Lattice Vectors a1,a2,a3aren’t necessarily a mutually orthogonal set! Usually they are neither mutually perpendicular nor all the same length! • For examples, see Fig. 3b (3 dimensions):

9. Nb film Crystal Lattice Types Bravais Lattice An infinite array of discrete points with an arrangement & orientation that appears exactly the same, from whichever of the points the array is viewed. A Bravais Lattice is invariant under a translation T = n1a1 + n2a2 + n3a3

10. Honeycomb Lattice Non-Bravais Lattices In a Bravais Lattice, not only the atomic arrangement but also the orientations must appear exactly the same from every lattice point. 2 Dimensional Honeycomb Lattice The red dots each have a neighbor to the immediate left. The blue dot has a neighbor to its right. The red (& blue) sides are equivalent & have the same appearance. But,the red & blue dotsare not equivalent.If the blue side is rotated through 180º the lattice is invariant.  The Honeycomb Lattice is NOT a Bravais Lattice!!

11. It can be shown that, in 2 Dimensions, there are Five (5) & ONLY FiveBravais Lattices!

12. 2-Dimensional Unit Cells S S S S S S S S S S S S S b a Unit CellThe Smallest Component of the crystal (group of atoms, ions or molecules), which, when stacked together with pure translational repetition, reproduces the whole crystal. 2D-Crystal S Unit Cell S

13. S S S Unit CellThe Smallest Component of the crystal (group of atoms, ions or molecules), which, when stacked together with pure translational repetition, reproduces the whole crystal. Note that the choice of unit cell is not unique! 2D-Crystal

14. 2-Dimensional Unit Cells Artificial Example: “NaCl” Lattice pointsare points with identical environments.

15. 2-Dimensional Unit Cells: “NaCl” Note that the choice of origin is arbitrary! the lattice points need not be atoms, but The unit cell size must always be the same.

16. 2-Dimensional Unit Cells: “NaCl” These are also unit cells! It doesn’t matter if the origin is atNa orCl!

17. 2-Dimensional Unit Cells: “NaCl” These are also unit cells. The origin does not have to be on an atom!

18. 2-Dimensional Unit Cells: “NaCl” These areNOTunit cells! Empty space is not allowed!

19. 2-Dimensional Unit Cells: “NaCl” In 2 dimensions, these areunit cells. In 3 dimensions, they would not be.

20. 2-Dimensional Unit Cells Why can't the blue triangle be a unit cell?

21. Example: 2 Dimensional, Periodic Art!A Painting byDutch Artist Maurits Cornelis Escher (1898-1972) Escher was famous for his so called “impossible structures”, such as Ascending & Descending, Relativity,.. Can you find the “Unit Cell” in this painting?

22. 3-Dimensional Unit Cells

23. 3-Dimensional Unit Cells 3 Common Unit Cells with Cubic Symmetry Simple Cubic Body Centered Face Centered (SC) Cubic (BCC) Cubic (FCC)

24. Conventional & Primitive Unit Cells Simple Cubic(SC) Conventional Cell=Primitive cell Body Centered Cubic (BCC) Conventional Cell≠Primitive cell

25. Face Centered Cubic (FCC) Structure

26. Conventional Unit Cells • A Conventional Unit Cell just fills space when translated through a subset of Bravais lattice vectors. • The conventional unit cell is larger than the primitive cell, but with the full symmetry of the Bravais lattice. • The size of the conventional cell is given by the lattice constanta. FCC Bravais Lattice The full cubeis the Conventional Unit Cellfor the FCC Lattice

27. Conventional & Primitive Unit Cells Face Centered Cubic Lattice Primitive Lattice Vectors a1 = (½)a(1,1,0) a2 = (½)a(0,1,1) a3 = (½)a(1,0,1) Note that theai’s are NOT Mutually Orthogonal! Primitive Unit Cell (Shaded) Lattice Constant Conventional Unit Cell (Full Cube)

28. Elements That Form Solidswith the FCC Structure

29. Body Centered Cubic (BCC) Structure

30. Conventional & Primitive Unit Cells Body Centered Cubic Lattice Primitive Lattice Vectors a1 = (½)a(1,1,-1) a2 = (½)a(-1,1,1) a3 = (½)a(1,-1,1) Note that theai’s are NOT mutually orthogonal! Primitive Unit Cell Lattice Constant Conventional Unit Cell (Full Cube)

31. Elements That Form Solids with the BCC Structure

32. Conventional & Primitive Unit Cells Cubic Lattices Simple Cubic (SC) Primitive Cell =Conventional Cell Fractional coordinates of latticepoints: 000, 100, 010, 001, 110,101, 011, 111 Body Centered Cubic (BCC) Primitive Cell Conventional Cell Fractional coordinates of the latticepoints in the conventional cell: 000,100, 010, 001, 110,101, 011, 111, ½ ½ ½ Primitive Cell =Rombohedron

33. Conventional & Primitive Unit Cells Cubic Lattices Face Centered Cubic (FCC) Primitive Cell Conventional Cell The fractional coordinates of latticepoints in the conventional cell are: 000,100, 010, 001, 110,101, 011, 111, ½ ½ 0, ½ 0 ½, 0 ½ ½, ½ 1 ½, 1 ½ ½ , ½ ½ 1

34. Simple Hexagonal Bravais Lattice

35. Points of the Primitive Cell 120o Conventional & Primitive Unit Cells Hexagonal Bravais Lattice Primitive Cell =Conventional Cell Fractional coordinates of latticepoints in conventional cell: 100, 010, 110, 101, 011, 111, 000, 001

36. Hexagonal Close Packed (HCP) Lattice:A Simple Hexagonal Bravais Latticewith a 2 Atom Basis The HCP latticeis not a Bravais lattice,because the orientation of the environment of a point varies from layer to layer along the c-axis.

37. General Unit Cell Discussion • For any lattice, the unit cell &, thus, the entire lattice, is UNIQUELYdetermined by 6 constants (figure): • a, b, c, α, β and γ • which depend on lattice geometry. • As we’ll see, we sometimes want to calculate the number of atoms in a unit cell. To do this, imagine stacking hard spheres centered at each lattice point & just touching each neighboring sphere. Then, for the cubic lattices, only1/8 of each lattice point in a unit cell assigned to that cell. In the cubic lattice in the figure, • Each unit cell is associated with (8)  (1/8) = 1 lattice point.

38. Primitive Unit Cells & Primitive Lattice Vectors • In general, a Primitive Unit Cellis determined by the parallelepiped formed by the Primitive Vectorsa1 ,a2, & a3 such that there is no cell of smaller volumethat can be used as a building block for the crystal structure. • As we’ve discussed, a Primitive Unit Cell can be repeated to fill space by periodic repetition of it through the translation vectors • T = n1a1 + n2a2 + n3a3. • The Primitive Unit Cellvolume can be found by vector manipulation: • V = a1(a2 a3) • For the cubic unit cell in the figure, V = a3

39. Primitive Unit Cells • Note that, by definition,the Primitive Unit Cell must contain ONLY ONE lattice point. • There can be different choices forthe Primitive Lattice Vectors, but the Primitive Cell volume must be independent of that choice. A 2 Dimensional Example! P = Primitive Unit Cell NP = Non-Primitive Unit Cell